Deterministic and stochastic braking distance
prediction of disc-braked rail vehicles
Deterministische und stochastische Bremswegermittlung von
scheibengebremsten Schienenfahrzeugen
vorgelegt von
M.Sc.
Marc Ehret
ORCID: 0000-0001-5027-3865
an der Fakultät V - Verkehrs- und Maschinensysteme
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
-Dr.-Ing.-
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr.-Ing. Jörg Krüger
Gutachter: Prof. Dr.-Ing. Markus Hecht
Gutachter: Prof. Dr.-Ing. Tjark Siefkes
Tag der wissenschaftlichen Aussprache:
15. April 2024
Berlin 2024
Vorwort und Danksagung
Diese Arbeit entstand im Rahmen des gemeinsamen Forschungsprojektes "Virtual Train"
des Instituts für Systemdynamik und Regelungstechnik des Deutschen Zentrums für Luft-
und Raumfahrt (DLR) und der Knorr-Bremse Systeme für Schienenfahrzeuge GmbH
(KB).
Ich bedanke mich ausdrücklich bei allen Kolleginnen und Kollegen des DLR und der
KB, die das Forschungsvorhaben ermöglicht und mich während des Projektes und der
wissenschaftlichen Ausarbeitung unterstützt haben. Insbesondere danke ich der KB für
die Bereitstellung der Messdaten, ohne die diese Arbeit nicht möglich gewesen wäre.
Mein besonderer Dank gilt Dr.-Ing. Andreas Heckmann für die hervorragende wis-
senschaftliche Betreuung und das sehr gute Mentoring während meiner Tätigkeit als
wissenschaftlicher Mitarbeiter am DLR. Weiterhin möchte ich mich ausdrücklich bei Ernst
Hohmann für die Unterstützung und den sehr wertvollen Austausch zur Interpretation
der Forschungsergebnisse seitens KB bedanken.
Prof. Dr.-Ing. Markus Hecht und Prof. Dr.-Ing. Tjark Siefkes danke ich für die Betreuung
und Begutachtung meiner Dissertation an der TU Berlin.
Eine Promotion ist ein Marathon bei dem immer wieder Täler durchschritten werden
müssen, um voranzukommen. Mit Deiner Unterstützung habe ich alle Täler hinter mir
gelassen. Danke Lisa.
iii
Abstract
A rapid modal shift to rail is inevitable to reduce transportation-related greenhouse gas
emissions while meeting the ever-increasing demand for mobility. Due to the nature of
rail-bound traffic, the prediction of the braking performance plays a key role in improving
the infrastructure capacity without compromising safety. Against this background, this
works addresses the complexity of frictional brake forces and contributes to an improved
brake performance prediction of disc-braked rail vehicles.
By analyzing almost 2000 experimental brake applications conducted on a test rig, this
work provides fundamental insights into the deterministic and stochastic behavior of
a typical brake pad material applied in rail vehicles. Based on the available data and
a thorough literature review, a new friction model is developed and identified for the
investigated material. The model is capable of predicting the time-variant and non-linear
behavior of the friction forces prevailing during the braking process and outperforms
state-of-the-art friction models. In combination with a temperature model, additionally
developed and identified in this work, the braking distance resulting from a single brake
unit is predicted with an accuracy of 2%. The models are validated using data from more
than 80 vehicle brake applications conducted with a multiple unit. The comparison of
these measurements with the simulations reveals a very good agreement of instantaneous
deceleration, friction forces and thermal loads occurring in the brake discs. For braking
scenarios with an initial velocity of 120
km
h
, the braking distance of the train is predicted
with an accuracy of 5%. No significant deviations are observed for other load cases.
Moreover, this work presents a novel probabilistic approach that allows to consider
the stochastic nature of frictional brake forces when predicting the brake performance.
Based on this approach, it is found that the friction-related scatter prevailing in the
brake units of rail vehicles depends on the initial velocity and is a superposition of
global and individual phenomena. These are fundamental findings with respect to the
meaningfulness of probabilistic analysis of brake applications, whose results are closely
related to the capacity utilization during an operation with the European Train Control
System (ETCS). In fact, an exemplary probabilistic analysis conducted in this work
reveals that an improved consideration of the friction characteristics offers the potential to
reduce the safety margins for ETCS braking curves by up to 14% without compromising
safety.
v
Kurzfassung
Die Reduzierung der Treibhausgasemissionen bei gleichzeitig steigendem Mobilitätsbedarf
erfordert eine rasche Verlagerung des Verkehrs auf die Schiene. Aufgrund des schienenge-
bundenen Verkehrs spielt die Vorhersage der Bremsleistung eine Schlüsselrolle, um die
Kapazität der Infrastruktur ohne Beeinträchtigung der Sicherheit zu erhöhen. Vor diesem
Hintergrund widmet sich diese Arbeit der Komplexität von Reibungsbremskräften und
leistet einen Beitrag zur Verbesserung der Vorhersage der Bremsleistung von scheibenge-
bremsten Schienenfahrzeugen.
Durch die Analyse von nahezu 2000 auf einem Prüfstand durchgeführten Bremsvorgängen
liefert diese Arbeit grundlegende Erkenntnisse über das deterministische und stochastische
Verhalten eines Schienenfahrzeugbremsbelages. Auf Basis der Daten und einer umfang-
reichen Literaturstudie wird ein neues Reibmodell entwickelt und für den untersuchten
Belag identifiziert. Dieses Modell ermöglicht die Vorhersage des zeitvarianten und nicht-
linearen Verhaltens der Reibungskräfte während des Bremsvorgangs und übertrifft die in
der Literatur vorherrschenden Reibmodelle. In Verbindung mit einem in dieser Arbeit
entwickelten Temperaturmodell wird der Bremsweg, der aus einem Bremsvorgang mit
nur einer einzigen Bremsscheibe resultiert, mit einer Genauigkeit von 2% vorhergesagt.
Die Modelle werden anhand von Daten aus über 80 Zugbremsversuchen validiert. Der
Vergleich dieser Messungen mit den Simulationen zeigt eine sehr gute Übereinstimmung
der momentanen Verzögerung, der Reibungskräfte und der in den Bremsscheiben auftre-
tenden thermischen Belastungen. Für Bremsszenarien mit einer Ausgangsgeschwindigkeit
von 120
km
h
wird der Bremsweg des Zuges mit einer Genauigkeit von 5% vorhergesagt.
Für andere Lastfälle werden keine signifikanten Abweichungen beobachtet.
Darüber hinaus wird ein neuer probabilistischer Ansatz vorgestellt, der es ermöglicht, die
stochastische Natur der Reibungskräfte bei der Prognose zu berücksichtigen. Auf der
Grundlage dieses Ansatzes zeigt sich, dass die in Schienenfahrzeugen vorherrschende rei-
bungsbedingte Streuung von der Ausgangsgeschwindigkeit abhängt und eine Überlagerung
von globalen und individuellen Effekten darstellt. Dies sind grundlegende Erkenntnisse
im Hinblick auf die Aussagekraft probabilistischer Analysen von Bremsvorgängen, deren
Ergebnisse eng mit der Streckenauslastung während des Betriebs mit dem European
Train Control System (ETCS) verknüpft sind. In der Tat offenbart eine in dieser Arbeit
durchgeführte probabilistische Analyse, dass eine verbesserte Berücksichtigung der Rei-
bungscharakteristik das Potenzial bietet, die Sicherheitsmargen für ETCS Bremskurven
um bis zu 14% zu reduzieren ohne die Sicherheit zu beeinträchtigen.
vii
Contents
1 Introduction 1
1.1 Problemcontext............................... 1
1.2 The complexity of friction forces . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Scientific need for action . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Research objectives and outline . . . . . . . . . . . . . . . . . . . . . . 6
2 State-of-the-art in railway brake technology 9
2.1 Introduction to railway brake systems . . . . . . . . . . . . . . . . . . . 9
2.1.1 Classification ............................ 9
2.1.2 The compressed air brake . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Disc brake units of railway vehicles . . . . . . . . . . . . . . . . 14
2.2 Brakemechanics .............................. 18
2.2.1 Brake mechanics of a wheel set . . . . . . . . . . . . . . . . . . 18
2.2.2 Brake kinetics of a train considered as single mass . . . . . . . . 22
2.3 Brake performance assessment . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Brake assessment with regard to operation with ETCS . . . . . 25
2.3.2 Deterministic brake calculations according to EN 14531 . . . . . 29
2.3.3 Stochastic analysis of vehicle brake applications . . . . . . . . . 31
2.4 Friction in disc brake units . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1 Fundamental laws of friction . . . . . . . . . . . . . . . . . . . . 34
2.4.2 The origin of friction in disc brake units . . . . . . . . . . . . . 34
2.4.3 Friction phenomena observed in disc brake units . . . . . . . . . 36
2.5 Modeling the coefficient of friction . . . . . . . . . . . . . . . . . . . . . 39
2.5.1 Staticmodels ............................ 40
2.5.2 Dynamicmodels .......................... 40
2.5.3 Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . 41
3 Analysis of experimental data 43
3.1 Description of experiments . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Dynamometer test rig . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.2 Friction measurement . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.3 Additional measurements . . . . . . . . . . . . . . . . . . . . . . 45
3.1.4 Brake disc and pad specimen . . . . . . . . . . . . . . . . . . . 46
3.1.5 Testing procedure and braking scenario . . . . . . . . . . . . . . 47
ix
3.2 Instantaneous coefficient of friction . . . . . . . . . . . . . . . . . . . . 48
3.2.1 Dataprocessing........................... 48
3.2.2 Instantaneous mean values . . . . . . . . . . . . . . . . . . . . . 50
3.2.3 Instantaneous standard deviation . . . . . . . . . . . . . . . . . 52
3.2.4 Instantaneous distribution . . . . . . . . . . . . . . . . . . . . . 54
3.3 Friction coefficient averaged over braking distance . . . . . . . . . . . . 56
3.3.1 Meanvalues............................. 57
3.3.2 Standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.3 Distribution............................. 59
3.4 Surface temperature of the disc . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1 Mean values and variance . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 Correlation of surface temperature and friction . . . . . . . . . . 61
3.5 Key findings of the data analysis . . . . . . . . . . . . . . . . . . . . . 62
4 Development of a brake model for disc brake units 63
4.1 Deterministic friction model . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Selection of model approach . . . . . . . . . . . . . . . . . . . . 63
4.1.2 Modelsynthesis........................... 65
4.1.3 Model identification . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.4 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Deterministic temperature model . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Modelsynthesis........................... 75
4.2.2 Model identification . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.3 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Deterministic brake model . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Dynamometer model . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.2
Deterministic analysis of stopping-brake scenarios with a single
discbrakeunit ........................... 83
4.4 Stochastic brake model . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.1 Stochastic modeling approach . . . . . . . . . . . . . . . . . . . 86
4.4.2 Stochastic identification . . . . . . . . . . . . . . . . . . . . . . 87
4.4.3
Stochastic analysis of stopping-brake scenarios with a single disc
brakeunit.............................. 90
4.5 Key findings of the model development . . . . . . . . . . . . . . . . . . 96
5 Analysis of vehicle brake applications 97
5.1 Experimental field tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.1.1 Vehicle configuration and braking scenarios . . . . . . . . . . . . 98
5.1.2 Post-processing ........................... 100
5.2 Deterministic analysis of field tests . . . . . . . . . . . . . . . . . . . . 101
5.2.1 Synthesis of the train model . . . . . . . . . . . . . . . . . . . . 101
5.2.2 Parameters and simulation scenarios . . . . . . . . . . . . . . . 101
5.2.3 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.4 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Stochastic analysis of field tests . . . . . . . . . . . . . . . . . . . . . . 114
5.3.1 Monte-Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . 114
5.3.2 Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.3 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . 119
5.4 Key findings of vehicle simulations . . . . . . . . . . . . . . . . . . . . 122
6 Suggestions for brake system design and operation 123
6.1 Modelidentification............................. 124
6.2 Systemdesign................................ 125
6.3 Virtualtesting................................ 126
6.4 Probabilistic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 ETCSbrakingcurves............................ 130
6.5.1 Dynamic braking curves . . . . . . . . . . . . . . . . . . . . . . 130
6.5.2 Passive safety in braking curves . . . . . . . . . . . . . . . . . . 131
6.6 Controlled emergency brake applications . . . . . . . . . . . . . . . . . 132
7 Conclusion 135
7.1 Answers to research questions . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Outlook ................................... 139
A Appendix 141
A.1 Appendix to state-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . 141
A.1.1
Test program on dynamometer test rig for disc brake units accord-
ingtoEN15328........................... 141
A.1.2 Brake kinetics based on mean values according to EN 14531-1 . 142
A.1.3 Equivalent brake force . . . . . . . . . . . . . . . . . . . . . . . 143
A.1.4 COF averaged over the braking distance . . . . . . . . . . . . . 144
A.1.5 Emergency Brake Confidence Level (EBCL) . . . . . . . . . . . 146
A.2 Appendix to data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.2.1 Testing procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.2.2 Fault propagation of friction measurement at dynamometer . . . 148
A.2.3 COF with respect to instantaneous velocity (Pad Material A) . 149
A.2.4
Instantaneous mean values and standard deviations of COF gained
from different specimen (Pad Material A) . . . . . . . . . . . . . 150
A.2.5 Instantaneous distribution of COF (Pad Material A) . . . . . . 154
A.2.6 Estimation of confidence intervals . . . . . . . . . . . . . . . . . 156
A.2.7
Polynomial regression of mean values and standard deviations of
COF averaged over braking distance (Pad Material A). . . . . . 157
A.2.8
Distribution of COF averaged over braking distance (Pad Material
A) .................................. 158
A.2.9
Temperature evolution measured by single sensors in brake disc
surface................................ 159
A.3 Appendix to model development . . . . . . . . . . . . . . . . . . . . . . 161
A.3.1 Deterministic model identification . . . . . . . . . . . . . . . . . 161
A.3.2 Deterministic brake model . . . . . . . . . . . . . . . . . . . . . 163
A.3.3 Deterministic temperature model . . . . . . . . . . . . . . . . . 163
A.3.4 Stochastic brake model . . . . . . . . . . . . . . . . . . . . . . . 164
A.4 Appendix to analysis of vehicle brake tests . . . . . . . . . . . . . . . . 165
Own Publications
This thesis is based in parts on the following own publications:
[E1]
Ehret, M.: Identification of a dynamic friction model for railway disc brakes. In:
Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail
and Rapid Transit 235. no. 10 (2021), pp. 1214–1224.
[E2]
Ehret, M. ; Hohmann, E. ; Heckmann, A.: On the modelling of the fric-
tion characteristics of railway vehicle brakes. In: International Journal of Rail
Transportation 11. no. 1 (2022), pp. 50–68.
xiii
List of Figures
1.1 Relation between track occupancy, stopping distance and safety margin. 2
1.2
Deceleration of a disc-braked rail vehicle recorded during brake applications
with different initial velocities [E1][E2]. . . . . . . . . . . . . . . . . . . 3
1.3
Structure of this work and relation of chapters to formulated research
questions. .................................. 7
2.1
Classification of brake systems for rail vehicles according to [51], [52] and
[53]. ..................................... 10
2.2 Commonly applied friction brakes for rail vehicles. . . . . . . . . . . . . 11
2.3
Sketch of an indirect compressed air brake for the actuation of disc brake
units (the pressure values are relative to an ambient pressure of 1 bar). 12
2.4
Sketch to compare the built-up times, brake pipe pressure and signals of
the different brake modes during an emergency brake applications initiated
at t0. .................................... 14
2.5
Full-scaled dynamometer test rig of the Knorr-Bremse SfS GmbH applied
with AMD brake and caliper unit (Picture provided by Knorr-Bremse SfS
GmbH). ................................... 15
2.6
Derivation of tolerance intervals according to EN 15328 based on friction
measurements (data was artificially generated). . . . . . . . . . . . . . 16
2.7
Example of organic and sintered brake pads with dimensions (Pictures
provided by Knorr-Bremse SfS GmbH. Pictures are edited for the illustra-
tion)...................................... 17
2.8
Sketch to visualize the acting forces during a brake application with a
single WMD disc brake unit (Forces and torques: red, dedree of freedom:
blue)...................................... 19
2.9
Instantaneous deceleration
|a|
(
t
), velocity
v
(
t
), stopping distance
s
(
t
)and
clamping forces
FC,i
(
t
)of a train measured during brake applications with
discbrakeunits. .............................. 24
2.10
Methods for the brake performance assessment of rail vehicles and deriva-
tion of ETCS braking curves for γ-trains.................. 25
2.11
Probability density of brake deceleration caused by deviations related to
brake system and degraded adhesion conditions according to [72]. . . . 27
2.12
Methods proposed by EN 14531 to calculate the brake performance [16][17].
........................................ 29
xv
List of Figures
2.13
Derivation of correction factor
Kdry
based on probabilistic analysis and
EBCL..................................... 31
2.14 Simplified sketch of pad surface topography based on [58] and [81]. . . . 35
2.15
Simplified sketch of friction phenomena observed for brake units according
to [85],[30],[87],[86] and [91]. . . . . . . . . . . . . . . . . . . . . . . . . 36
2.16 Influences, phenomena and model approaches for the COF. . . . . . . . 39
3.1 Simplified sketch of a full-scale dynamometer test rig FSD. . . . . . . . 44
3.2 Position of temperature sensors [E2]. . . . . . . . . . . . . . . . . . . . 45
3.3
Instantaneous progression of COF
µ
, clamping force
FC
, velocity
v
, local
disc temperatures
T1−6
and mean temperature
Tfric
during a stopping
brakeapplication............................... 47
3.4
Comparison of
µ
(
t
)of all 144 brake applications (left) measured during
load case V0140_Case2 and temporary mean value and range of variation
(right)..................................... 48
3.5
Comparison of instantaneous COF of all brake applications measured for
all load cases including temporary mean values and ranges of variation
[E2]. ..................................... 50
3.6 Comparison of mean values ¯µ(t)of all load cases. . . . . . . . . . . . . 51
3.7 Comparison of standard deviation σµ(t)and σµ(v)of all load cases. . . 52
3.8 Potential causes for the dispersion of the COF observed on the test rig. 53
3.9
Comparison of
µ
(
t
)of all 144 brake applications (left) measured during
load case V0140_Case2 and temporary histogram of µ(tj= 25s)(right). 55
3.10 Comparison of µmsamples by box plots for each load case. . . . . . . . 56
3.11
Comparison of empirical mean values
¯µm
and standard deviations
σµm
from samples of µmvalues of each load case. . . . . . . . . . . . . . . . 57
3.12
Comparison of averaged surface temperature
Tfric
(
t
)of all brake applica-
tions including mean values and range of variation [E2]. . . . . . . . . . 60
3.13 Comparison of ¯µ(¯
TFric)for each load case. . . . . . . . . . . . . . . . . 61
4.1 Comparison of optimization criteria cf,i of all 12 load cases. . . . . . . . 69
4.2
Comparison of measured and simulated instantaneous COF over time
based on identified model parameters including relative error [E2]. . . . 71
4.3
Comparison of measured and simulated COF based on identified model
parameters for each load case with respect to velocity and distance. . . 72
4.4
Comparison of measured and simulated derivative of the COF with respect
to time as well as progression of single terms based on identified parameters.
73
4.5 Comparison of optimization criteria cf,i of different model approaches. . 74
4.6
Comparison of measured and simulated
µ
(
t
)based on different model
approaches. ................................. 74
4.7 Two capacity model for one ring of WMD including heat flows [E2]. . . 76
xvi
List of Figures
4.8
Comparison of criteria
cf,i
of all 12 load cases. Due to the strong reduction
of criteria with respect to the initial values, the y-axis is logarithmically
scaled..................................... 80
4.9
Comparison measured average temperatures at the disc surface with
simulated temperature based on identified parameter values [E2]. . . . 81
4.10
Structure and interaction of submodels within dynamometer brake model
to conduct virtual stopping brake scenarios with a single disc brake unit
[E2]. ..................................... 82
4.11
Comparison of COF averaged over braking distance gained from deter-
ministic simulation
µm,sim
with mean value from measurements
¯µm,meas
including confidence intervals with a confidence level of 95% (see Appendix
A.2.6)[E2]. ................................. 84
4.12
Comparison of 200 simulated brake applications during MCS of load case
V0140_Case2 (left) and instantaneous mean values and range of variation
(right)..................................... 91
4.13
Comparison of measured (black) and simulated (red) distributions of
friction curves at t= 25sduring load case V0140_Case2. . . . . . . . . 91
4.14
Comparison of instantaneous mean values and ranges of variations from
measurements and MCS [E2]. . . . . . . . . . . . . . . . . . . . . . . . 92
4.15
Comparison of 1
σµm
resulting from measurements and MCS including
confidence intervals based on a confidence level of 95% (see Appendix
A.2.6)[E2]. ................................. 93
4.16
Comparison of stopping distance samples resulting from measurements
and MCS by box plots (Explanation of box plot is given in Section 3.3). 94
4.17
Comparison of 1
σsst
resulting from measurements and MCS including
confidence intervals based on a confidence level of 95% (see Appendix
A.2.6)..................................... 95
5.1 Configuration of friction brakes applied in investigated multiple unit. . 98
5.2
Structure of train model to simulate stopping brake applications of multiple
unit. ..................................... 102
5.3
Comparison of measured and simulated deceleration for emergency brake
applications conducted with PN and empty load condition. . . 103
5.4
Comparison of measured and simulated disc temperatures. Note that
measurements represent a single position at the disc, while simulations
represent the estimated average temperature of the surface Tfric. . 106
5.5
Comparison of
µm,eff
(
t
)resulting from measurements and deterministic
simulations with respect to the stopping distance. . . . . . . . . . . . . 107
5.6
Comparison of empirical mean values and standard deviations of
µm,eff
-
samples resulting from measurements and deterministic simulations. Con-
fidence intervals correspond to a confidence level of 95%. . . . . . . . . 108
5.7 Comparison of measured and simulated stopping distances. . . . . . . . 110
xvii
List of Figures
5.8
Comparison of empirical mean values and standard deviations of stopping
distances resulting from measurements and deterministic simulations for
brake applications with
v0
= 120
km
h
. Confidence intervals correspond to a
confidencelevelof95%............................ 111
5.9
Classification of potential sources of deviations between test rig experi-
ments and vehicle brake tests. . . . . . . . . . . . . . . . . . . . . . . . 112
5.10 Illustrative comparison of correlated and uncorrelated COF. . . . . . . 115
5.11
Comparison of empirical mean values and standard deviations of
µm,eff
samples resulting from measurements, deterministic simulation as well
as MCS100 and MCS75/25 [E2]. Confidence intervals correspond to a
confidencelevelof95%............................ 117
5.12
Comparison of standard deviations of stopping distances resulting from
measurements, deterministic and stochastic simulations for brake ap-
plications with
v0
= 120
km
h
[E2]. Confidence intervals correspond to a
confidencelevelof95%............................ 119
6.1
Identification and application of a friction model during brake system
designandoperation............................. 123
6.2
Relative lengths of confidence intervals related to estimation of empirical
mean value and standard deviation with respect to sample size. Intervals
correspond to a confidence level of 95%. . . . . . . . . . . . . . . . . . 124
6.3
Cumulative probability
P
(
|a|
|a|n
)resulting from MCS with different assump-
tions regarding the scatter of the pad friction. Shaded areas represent
confidence intervals with respect to a confidence level of 95%. . . . . . 129
6.4 Comparison of dynamic and static braking curves. . . . . . . . . . . . . 131
6.5
Comparison of static and dynamic braking curves with and without passive
safety included, while the permissible final speed at the supervised location
is 18km
h. ................................... 132
6.6 Concept of adaptive DCC with dynamic target value. . . . . . . . . . . 133
6.7 Deceleration with uncontrolled brake system, DCC and adaptive DCC. 133
7.1 Instantaneous mean values ¯µ(t)and dispersion ¯µ(t)±3σµ(t)of the COF
gained from test rig experiments and MCS based on the stochastic brake
model..................................... 136
7.2
Comparison of measured vehicle deceleration during stopping brake ap-
plications with simulation results based on the brake model developed in
thiswork. .................................. 137
7.3
Braking potential during emergency brake applications with different
initial velocities based on static and dynamic braking curves. . . . . . . 138
A.1
Extract of test program for brake pads of classes B1 and C1 according to
EN15328,Page2[55]............................ 141
xviii
List of Figures
A.2
Comparison of instantaneous empirical mean values
¯µ
(
v
)and standard
deviation 1
σµ
(
v
)of all load cases of Pad Material A analyzed in Chapter
3........................................ 149
A.3
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single speci-
menandallspecimen. ........................... 150
A.4
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single speci-
menandallspecimen. ........................... 151
A.5
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single speci-
menandallspecimen. ........................... 152
A.6
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single speci-
menandallspecimen. ........................... 153
A.7
Comparison of
µ
(
t
)of all 160 brake applications (left) measured during
load case V0160_Case2 and temporary histogram of µ(t= 25s)(right). 154
A.8
Comparison of
µ
(
v
)of all 160 brake applications (left) measured during
load case V0160_Case2 and temporary histogram of
µ
(
v
= 90
km/h
)
(right).
154
A.9
Polynomial regression of mean values and standard deviations of
µm
with
respect to the initial velocity. Data points represent values of
¯µm
and
σµm
listed Table 3.4 (The unit of v0in the fitted polynomials is [km
h]). . . . 157
A.10
Comparison of sample distributions of
µm
of each load case and all speci-
mens...................................... 158
A.11
Instantaneous mean values of temperatures measured by each sensor during
all brake applications of each load case with respect to time. . . . . . . 159
A.12
Instantaneous standard deviation of temperatures measured by each sensor
during all brake applications of each load case with respect to time. . . 160
A.13
Comparison of measured
¯µmeas
(
t
)and simulated values
µsim
(
t
)for two
load cases based on manually tuned parameter values. . . . . . . . . . . 161
A.14
Comparison of measured and simulated deceleration based on deterministic
model..................................... 165
A.15
Comparison of measured and simulated deceleration based on deterministic
model..................................... 166
A.16
Comparison of measured and simulated deceleration based on deterministic
model..................................... 167
A.17
Comparison of measured and simulated deceleration based on deterministic
model..................................... 168
A.18
Comparison of measured and simulated deceleration based on deterministic
model..................................... 169
A.19
Comparison of measured and simulated deceleration based on deterministic
model..................................... 170
A.20
Comparison of measured and simulated deceleration based on deterministic
model..................................... 171
xix
List of Figures
A.21
Comparison of measured and simulated deceleration based on deterministic
model..................................... 172
A.22
Comparison of measured and simulated deceleration based on deterministic
model..................................... 173
A.23
Comparison of average measured and simulated driving resistance. Under-
lying data corresponds to all brake applications conducted for each initial
velocity. ................................... 174
A.24
Time series of ambient temperature and relative humidity recorded during
all 88 vehicle brake tests. . . . . . . . . . . . . . . . . . . . . . . . . . . 175
xx
List of Tables
2.1
Extract of the classification of brake pads for different train types and disc
brakes according to EN 15328 [55]. . . . . . . . . . . . . . . . . . . . . 15
2.2
Extract of deterministic friction models to predict the instantaneous COF
of friction brakes applied in rail and automotive vehicles. The limits of
applicability of the models are given in the references. Column Dyno.
specifies whether the given parameters were identified using measurements
from POD or FSD test rigs. - means that there was no specification in the
reference. Note that the symbols in the formulas of the table correspond
to the ones applied in the references, which may deviate from the glossary
ofthiswork.................................. 42
3.1 Pad, disc and caliper parameters. . . . . . . . . . . . . . . . . . . . . . 46
3.2
Number of brake applications performed for each load case and specimen.
46
3.3
P-values resulting from
χ2
-tests conducted for samples of
µi
(
t
)at
tj
=
5
,
15
,
25 and 35
s
gained from all load cases. Grey cells indicate that
Pχ2<0.05, i.e. the null hypothesis is rejected with respect to α= 5%. . 55
3.4
Comparison of statistical quantities calculated for
µm
samples for all load
cases. Gray cells in the last column indicate that
Pχ2<
0
.
05, i.e. the
sample is not normally distributed with respect to α= 5%........ 58
4.1 Reference values of the friction model. . . . . . . . . . . . . . . . . . . 67
4.2
Comparison of start and identified parameter values resulting from nu-
merical optimization of the friction model valid for Pad Material A. . . 69
4.3
Comparison of start and identified parameter values resulting from numer-
ical optimization of the friction model valid for the investigated friction
pairing. ................................... 80
4.4
Comparison of stopping distances from deterministic simulation
sst,sim
with mean values of measurements
¯sst,meas
including length of confidence
intervals ∆
L
related to the estimation of
¯sst,meas
based on a confidence
level of 95% (see Appendix A.2.6). . . . . . . . . . . . . . . . . . . . . 85
xxi
List of Tables
4.5
Results of sensitivity analysis based on 8 optimization problems. During
each optimization only one coefficient is individually tuned, whereas the
others remain at their start values. ∆
p
represents the change of the fitted
value with respect to the start value.
|¯
∆p|
is the mean of the absolute
changes of each coefficient for both fits.
csum
is the final value of the
objective function according to Equation 4.8. . . . . . . . . . . . . . . . 89
4.6
Comparison of mean values and standard deviation of stopping distance
resulting from MCS and measurements. . . . . . . . . . . . . . . . . . . 95
5.1 Parameter values of multiple unit and applied brake units. . . . . . . . 98
5.2
Number of brake applications conducted during field tests with respect to
braking scenario and initial velocity
v0
[E2]. Act. type defines if brakes
are actuated by EP or PN. Values of
mdyn
and
FC
represent mean values
ofall16brakeunits. ............................ 99
5.3
Nominal stopping distances resulting from det. simulations
¯sst,sim
and
measurements
¯sst,meas
including length of confidence intervals ∆
L
related
to the estimation of ¯sst,meas with respect to a confidence level of 95%. . 111
5.4
Comparison of empirical standard deviation of effective COF averaged
over the braking distance 1
σµm,eff
resulting from samples of experimental
vehicle brake tests, deterministic simulation, MCS100 and MCS75/25
(error=1σµm,meas −1σµm,sim). ....................... 118
5.5
Empirical standard deviations of stopping distances resulting from samples
of experimental vehicle brake tests, deterministic simulation, MCS100 and
MCS75/25 (error=1σsst,meas −1σsst,sim). ................. 119
6.1
Characterization of distributed parameters applied during MCS (Nominal
values of pCand mdyn correspond to mean values of all axles). . . . . . 128
6.2
Nominal deceleration, correction factor and save deceleration derived for
safety target of 10−4, i.e. EBCL = 4.................... 129
A.1
Emergency Brake Confidence Levels (EBCL) and related probabilities
P(EBCL) according to the definition in the ERTMS/ETCS system specifi-
cations [74]. Column Countries indicates what countries use which values
accordingto[75]. .............................. 146
A.2
Exemplary testing procedure comprising 40 subsequent Case1 brake ap-
plications................................... 147
A.3 Fault propagation for µfor dynamometer test-rig. . . . . . . . . . . . . 148
A.4
p-values resulting from
χ2
-tests conducted for samples of
µi
(
v
)at
vj
=
10
,
50
,
90 and 130
km
h
gained from all load cases. Gray cells indicate that
Pχ2<0.05, i.e. the null hypothesis is rejected with respect to α= 5%. . 155
A.5
Identified parameters of friction models to predict the mean instantaneous
COF of Pad Material A for all investigated load cases. . . . . . . . . . 162
xxii
List of Tables
A.6
Comparison of
µm,sim
resulting from deterministic simulation with mean
value from measurements
¯µm,meas
including length of confidence intervals
∆
L
related to the estimation of
¯µm,meas
with respect to a confidence level
of 95% (see Appendix A.2.6). . . . . . . . . . . . . . . . . . . . . . . . 163
A.7
Dimension, material properties and heat transfer coefficients of gray cast
iron disc according to geometry data and literature. The intervals corre-
spond to the range of data found in the references. . . . . . . . . . . . 163
A.8
Comparison of mean values and standard deviations of
µm
resulting from
MCSandmeasurements........................... 164
A.9
Comparison of empirical mean values of effective COF averaged of the
braking distance
¯µm,eff
resulting from samples of experimental vehicle
brake tests and deterministic simulation. Length of confidence intervals
∆
L
are related to the estimation of empirical mean value of measurements
with respect to a confidence level of 95% (see Appendix A.2.6). . . . . . 175
xxiii
List of Abbreviations
AMD Axle-Mounted Disc Brake
ANN Artificial Neural Networks
ATO Automatic Train Operation
AVADH Available Adhesion Weighting Factor
BP Brake Pipe
COF Coefficient of Friction
DCC Deceleration Control
EBCL Emergency Brake Confidence Level
EBD Emergency Brake Deceleration Curve
EP Direct Electro-Pneumatic Brake
ETCS European Train Control System
EU European Union
FSD Full-Scale Dynamometer
MCS Monte-Carlo Simulation
NNTR Notified National Technical Regulations
PN Indirect Pneumatic Brake
POD Pin-On-Disc Dynamometer
RQ Research Questions
TSI Technical Specification for Interoperability
UIC International Union of Railways
WMD Wheel-Mounted Disc Brake
WSP Wheel-Slide Protection
xxv
List of Symbols
Symbol Description Unit
Greek symbols
αLevel of significance −
χ2
1−α
2
Quantiles of a Chi
2
-distribution with respect to a sample
size of n−1and a probability P= 1 −α
2
−
∆LLength of confidence interval −
∆pDifference of parameters −
ϵRelative error %
ηCEfficiency of cylinder −
λThermal conductivity W
mK
µCoefficient of friction −
µnNominal friction coefficient according to UIC B126 −
µ0Initial coefficient of friction −
µeff
Effective coefficient of friction all brake units applied in
rail vehicle −
µlLocal coefficient of friction at infinitesimal area dA −
µmed Mean friction coefficient according to UIC B126 −
µmCoefficient of friction averaged over braking distance −
ψ
Amount of wear particles within the frictional boundary
layer −
σyStandard deviation of stochastic quantity ˜y−
¨φAngular acceleration kg
m2
ξDynamic state of tribo-system −
ζ1−α
2
Quantile of a t-Distribution with respect to a sample size
of n−1and a probability P= 1 −α
2
−
xxvii
List of Symbols
Latin symbols
aAcceleration m
s2
|a|brake_safe
Corrected deceleration that shall guarantee a safe operation
m
s2
|a|EEquivalent deceleration m
s2
|a|nNominal deceleration m
s2
ACCylinder surface area m2
Asurf Friction surface of disc brake m2
Avent Ventilation surface of disc brake m2
Cdisc Heat capacity of bulk disc J
K
cf,i Optimization criterion −
Cfric Heat capacity of friction zone J
K
ciCoefficients of friction model 1
s
cpSpecific heat capacity J
kgK
csum Objective function −
eiExponents of friction model −
ex Exponent of heat transfer equation −
FBE Equivalent brake force N
FBBrake force N
FCClamping force N
fdFunction describing destruction processes of patches −
Ffric Friction force −
fgFunction describing growth processes of patches −
FLAxle load N
FMSupporting force acting on load cell of test rig N
FNNormal force N
Fref Reference force N
FRE Equivalent resistance force N
FRDriving resistance force N
FSSpring force N
FTTangential wheel force N
xxviii
List of Symbols
fx
Coefficient of adhesion between wheel and rail in longitu-
dinal direction −
GThermal conductance W
K
Gref Thermal conductance at reference temperature W
K
hSpecific heat transfer coefficient W
m2K
h0
Specific heat transfer coefficient for free convection (veloc-
ity=0)
W
m2K
href
Specific heat transfer coefficient (forced convection at ref-
erence velocity)
W
m2K
iCTransmission ration of cylinder −
iTTotal transmission ration of brake unit −
JRotational interia kg
m2
Kdry
Correction factor within ETCS for good adhesion condi-
tions −
Kwet
Correction factor for within ETCS for degraded adhesion
conditions −
lThickness m
L+
1−α
Upper side of confidence interval with respect to a confi-
dence level of 1−α−
L−
1−α
Lower side of confidence interval with respect to a confi-
dence level of 1−α−
lFLever arm m
mStatic mass kg
MBBrake torque Nm
mdyn
Dynamic mass of wheel set taking account for static and
rotating inertia
kg
mtrain
Dynamic mass of entire train set taking account for all
static and rotating inertia
kg
nNumber −
nbNumber of brake units applied in train −
nrot Number of rotating wheel sets of train −
PProbability −
Pχ2
P-value of Chi
2
-test. If
Pχ2
is larger
α
, null hypothesis is
rejected −
xxix
List of Symbols
Pref Reference power W
pParameter vector −
p+
Upper boundary of parameter tuning interval applied dur-
ing optimization −
p−
Lower boundary of parameter tuning interval applied dur-
ing optimization −
pident
Parameter vector containing identified parameter values of
optimization −
TP
Patch temperature. Corresponds to the temperature acting
in the true contacts of the frictional boundary layer
K
pstart
Parameter vector containing initial parameter values for
optimization −
pBP Brake pipe pressure N
m2
pCCylinder pressure N
m2
plLocal pressure at infinitesimal area dA N
m2
˙
Qcond Heat conduction W
˙
Qfric Frictional heat input W
˙
Qsurf Convectional heat flow at friction surface area W
˙
Qvent Convectional heat flow at ventilation area W
R1Rolling resistance N
R2Velocity dependent resistance N
m
s
R3Aerodynamic resistance N
m2
s2
rfric Friction radius m
rwheel Wheel radius m
s0Initial position m
s1
Response distance: Distance traveled until friction brakes
are fully actuated
m
s2
Braking distance: Distance from the point of time when
friction brakes are fully actuated to standstill
m
sst Entire stopping distance during braking (s1+s2)m
sEEquivalent stopping distance m
Tdisc Temperature of the bulk disc K
Tfric Average temperature at the surface of the brake disc K
xxx
List of Symbols
T∞Ambient temperature K
Tref Reference temperature ◦C
t95
Point of time when 95% of the demanded clamping force
is reached
s
tEEquivalent response time s
tjTime instant of discrete time series s
tst Point of time when standstill is reached s
uiTime series of input variables −
vVelocity m
s
v0Initial velocity m
s
¨xTranslational acceleration m
s2
¯yMean value of stochastic quantity ˜y−
˜yStochastic quantity −
xxxi
1 Introduction
1.1 Problem context
Due to its efficiency and low environmental pollution, rail transport plays a key role in
future mobility. The specific greenhouse gas emissions related to passenger transport
are about three to four times higher for passenger cars and domestic aviation than the
emissions caused by rail transport [1]
1
. However, the modal split between rail and other
modes of transport is clearly expandable. Although the overall demand for mobility
of goods and passengers has increased strongly over the last two decades, the share of
rail-related passenger transport in the European Union (EU) remains below 10% [2].
For freight transport, it even declined from 16% to 13% [2]. It is clear that a modal
shift to rail is only possible, if rail transport becomes a more competitive alternative to
other modes of transport, i.e. it needs to be reliable, available, affordable, flexible and
of course safe [3]. Hence, the performance and attractiveness of rail transport must be
greatly improved. However, building new infrastructure to enhance rail transport is a
very complex and time-consuming task, as many stakeholders are involved or affected.
As the northern approach to the Brenner Base Tunnel shows, potential noise nuisance
and impacts on the environment as well as high costs for planning and construction
hinder the public acceptance and realization of new infrastructure [4][5]. A rapid modal
shift is therefore not only possible by building new tracks, but also requires an optimized
use of the existing infrastructure, i.e. by pushing its capacity limits.
Due to the track-guided motion, the capacity and safety of rail transport are closely related
to the performance and reliability of the braking process, in particular during emergency
brake applications. Failures to comply with speed limits, as well as overruns of danger
points due to a defect brake system or a misjudgment of the braking performance, may
result in severe implications for passengers and vehicles, as analyzed in [6]. Consequently,
the brake system design and rail operation need to guarantee that these fatal events are
highly unlikely. With the introduction of the European Train Control System (ETCS),
the train protection and the derivation of safety margins are based on a clear separation of
responsibilities between vehicle operators and infrastructure management [7]. Accordingly,
vehicle operators need to provide an estimate of both the nominal brake deceleration as
well as possible deviations from the nominal performance expected during an emergency
1Passenger cars: 164, Domestic aviation: 214, Rail transport 27-51 gram per passenger kilometer [1].
1
1 Introduction
brake application. Based on these parameters, a so-called braking curve is derived with
respect to a safety target prescribed by the infrastructure management, as indicated by
Figure 1.1. By monitoring the vehicle speed in relation to this curve, it shall be ensured
that the probability of an overrun of the end of authority during an emergency brake
application is not higher than the safety target. Hence, the braking curve marks the
guaranteed braking capability with respect to a certain probability.
However, studies show that conservative braking curves applied within different migration
levels of ETCS hinder the exploitation of the full potential expected from the introduction
of the new and harmonized signalling system compared to conventional national signalling
systems based on track-side or operational safety margins [8][9][10][11][12]. This reveals
that the estimation of the vehicle brake performance based on the probabilistic approach
indicated by Figure 1.1 is not yet precises enough to justify smaller safety margins without
compromising safety. In fact, Fehlauer argues that there is still a lack of experience
regarding the input parameters needed for the probabilistic derivation of safety margins
[8]. Hence, it must be specified in more detail, which parameters of the braking system
contribute to the deviations of the braking potential in operation and to what extent.
Moreover, it is obvious that an accurate estimation of the braking capability and its
variations is also essential for the realization of advanced modes of rail operation aimed
at increasing the infrastructure capacity, such as driving in absolute braking distance
(Moving block) or even in relative braking distance (Virtual Coupling). In addition to
these operational aspects, the global market is forcing brake suppliers to frequently push
the limits of the system design in order to remain competitive. The ability to precisely
predict the system performance from the very beginning of the development is therefore
becoming increasingly important to provide competitive offers [13]. In conclusion, the
overarching goal of harmonized, more efficient and competitive rail transportation results
in an urgent need to better understand and predict of the braking behavior of rail vehicles.
This applies to both brake system design and rail operation.
Safety
margin
Braking distance
(Nominal deceleration)
Track occupancy
Supervised location
(End of authority)
Estimated variance of
braking performance
Braking distance
(Guaranteed deceleration)
Speed Probability density
Safety
target
Figure 1.1: Relation between track occupancy, stopping distance and safety margin.
2
1.2 The complexity of friction forces
1.2 The complexity of friction forces
Predicting the brake performance of a disc-braked rail vehicle is a challenging task,
as Figure 1.2 exemplifies. The curves represent the deceleration of a multiple unit
measured during brake applications with different initial velocities. Although the brake
applications are conducted with identical loads and brake demands under good adhesion
conditions, i.e. dry and clean rails, the single curves differ significantly from each other.
In addition, the brake forces are subject to significant changes during the braking process.
These observations are mainly related to the characteristic behavior of friction forces
generated by the brake pads. During the braking process the frictional boundary layers
between pads and discs are exposed to large thermal and mechanical loads, which affect
the transmissible friction forces. Moreover, friction forces are subject to an immanent
stochastic behavior caused by continuous adaptions between pad and disc. Finally, wear
and tear as well as environmental impacts affect the frictional properties. As a result,
frictional brake forces are subject to systematic and probabilistic influences, which evoke
a time-variant, non-linear and stochastic brake performance.
Figure 1.2:
Deceleration of a disc-braked rail vehicle recorded during brake applications with
different initial velocities [E1][E2].
As a thorough characterization of the friction properties is often lacking, highly simplified
assumptions are made when calculating the braking performance. This can lead to
significant deviations in the estimated braking performance and increases the risk of a
system design that does not meet the requirements. Hence, lengthy experimental tests
and adjustments are required to validate the brake design. In addition, rough estimates
of the friction characteristics to be expected during operation require the application of
large safety margins and hinder the full utilization of infrastructure capacity. The ability
to predict friction phenomena in a reproducible manner is thus a key challenge to improve
the estimation of the braking performance for system design and rail operations.
3
1 Introduction
1.3 Scientific need for action
The computational verification of the braking capability of rail vehicles is prescribed
by the Technical Specification for Interoperability (TSI) for locomotives and passenger
passenger rolling stock [14][15]. The TSI refers to the European Standard EN 14531,
which proposes standardized procedures for the computational assessment of the brake
performance of rail vehicles considered as a lumped mass [14][16][17]. Moreover, tools,
such as TrainBraC, TTBS01 or LABRADOR, have been developed especially for the
brake assessment and provide generic libraries for estimating brake dynamics with respect
to the vehicle composition and brake system configuration based on numerical simulation
[18][19][20][21]. TrainBraC and TTBS01 also comply with EN 14531 [18][20][21]. However,
the results of these calculations are highly dependent on the assumptions regarding the
parameters of the brake systems, in particular the Coefficient of Friction (COF) between
pad and disc. Alturbeh et al. demonstrate that their results are in better agreement
with reference measurements when the pad material is less vulnerable to fading during
high energy brake applications [19]. Analogously, Pugi et al. show that estimation of
the braking distance is significantly improved when they include the dependency of pad
friction on the vehicle speed in their performance calculations [21].
Although these results are common observations when comparing brake performance
calculations with experimental brake tests, there is still a lack of research on the friction
phenomena of disc brakes applied in rail vehicles. Many friction models used for rail
applications are based on the formulation of Karwatzki from 1955, which was originally
developed for gray cast iron blocks [22][23][24][25][26]. However, the introduction of
composite materials for pads and blocks for rail vehicles to cope with ever-increasing
operating speeds has greatly complicated the mathematical description of friction forces.
First, these materials are subject to complex nonlinear characteristics that additionally
depend on the actual load case, as exemplified in [27],[21],[28],[25],[29] and [30]. Thus,
conventional models developed for tread brakes lose their validity. Secondly, there is no
limit to the diversity of compositions of synthetic friction materials. This is especially
true for materials applied in disc-braked rail vehicles, which is reflected in the large
number of certified pad materials listed in leaflet 541-3 of the International Union of
Railways (UIC) [31]. Due to the individual behavior of the various composite materials,
it is not possible to make a general formulation for brake pads of disc brake units. In
contrast, the diversity of block materials applied in tread brakes of freight trains is limited
in order to ensure the safe operation of a train composed of wagons with different braking
equipment [32].
The aforementioned reasons explain why international standards, such as EN 14531,
require experimental testing to determine the COF applied for the calculation of the
brake performance [16]. This means that data is usually generated during the design
process, which can be exploited to retrieve an enhanced description of friction forces.
4
1.3 Scientific need for action
The identification of a friction model for railway disc brakes with composite pads based
on test rig data is carried out by Hendrichs, Lee and Kang as well as Yuan et al.
[29][33][34]. However, the models of Hendrichs and Yuan are fitted to single load cases
and are therefore not able to capture the frictional behavior of other brake scenarios.
In contrast, Lee and Kang propose a model that takes into account the influence of
the temperature and friction velocity, which are calculated simultaneously during the
simulation of the braking process. However, their model is only applied in a laboratory
test environment and is not validated by experimental vehicle brake tests. Validation in
terms of predicting the vehicle brake performance is also lacking in Yuan’s investigations.
Hence, a thorough method for the identification, verification and validation based on
both test rig experiments and vehicle brake tests is still missing for friction models of
disc brake units used in rail vehicles.
In addition, many friction models are purely empirical approaches that are fitted to
test rig observations. Examples are artificial neural networks or models using polyno-
mial regression, as found in [35],[29],[24],[36] and [37]. These models lack of a solid
phenomenological background, making their prediction accuracy highly dependent on the
amount and quality of available input/output data. These shortcomings limit the ability
to predict braking scenarios not investigated during the test campaign. In contrast,
the phenomenological friction models of Ostermeyer et al., Wan et al. and Ricciardi et
al. were derived from investigations of actual pad topographies [38][39][40][41]. These
sophisticated approaches attempt to simulate processes taking place within the frictional
boundary to predict the behavior of the COF during braking. However, the models were
developed for automotive disc brakes, which are subject to large differences regarding
geometry and power consumption compared to rail applications. Nevertheless, the results
motivate to consider the state-of-the-art friction theory in the model description and to
transfer the findings to rail applications.
Finally, most research on friction focuses on deterministic phenomena and neglects its
stochastic behavior. Some research dealing with the uncertainty of friction results from the
stability analysis of friction oscillators, as shown in [42],[43] and [44]. Thorough research
on the stochastic behavior of disc brakes applied in rail vehicles is still lacking. The need
for knowledge in this research area is additionally amplified by the introduction of ETCS.
As described above, the derivation of safety margins is based on a probabilistic approach.
This method requires information on the scatter of the COF in service. Although the
UIC working group B126 worked out a method based on Monte-Carlo Simulation (MCS)
in 2003-2006 [45][46][47], there is hardly data available on the stochastic behavior of
friction brakes applied in rail vehicles. Moreover, the few publications on this topic use
different approaches to consider friction-related scatter within their probabilistic analysis,
as shown in [46],[48],[49] and [50]. This reveals that there is still a lack of understanding
regarding the extent of friction-related scatter in operation, the main drivers of this
scatter, and how the uncertainty of the individual brake units is propagated during brake
applications with an entire rail vehicle.
5
1 Introduction
1.4 Research objectives and outline
The previous sections show that there is an urgent need for a better understanding of
the friction forces of disc brake units in rail vehicles in order to achieve optimized and
safe rail operations as well as a competitive and successful system design. To address the
identified gaps in a systematic and methodical manner, the following Research Questions
(RQ) are derived for this work:
1.
How does the braking scenario affect the friction forces of disc brakes?
This question includes to
•
characterize the deterministic and stochastic behavior of a typical pad material
applied in rail vehicles and to
•identify the significant factors that drive this behavior.
2.
Is it possible to predict the instantaneous friction forces generated by a
single disc brake unit? This question addresses both the estimation of
•systematic effects associated with the braking scenario and
•stochastic phenomena.
3.
Is it possible to predict the deterministic and stochastic behavior of a
disc-braked rail vehicle during stopping brake applications? This question
includes to
•
quantify the prediction accuracy with respect to the nominal brake performance
and to
•
investigate how the uncertainty of friction forces is propagated in a train with
multiple brake units.
4.
How does improved friction prediction enrich rail operation? This question
deals with suggestions for an improved
•brake system design
•and rail operation.
With respect to the formulated RQ, the remainder of this work is structured according
to Figure 1.3. Chapter 2 gives a brief introduction to railway brake systems and the
brake performance assessment by calculation. These sections provide the theoretical
background necessary for the comprehension of the following chapters. In addition, a
thorough literature review on friction theory and modeling is given, which paves the way
for the contributions of this work.
6
1.4 Research objectives and outline
To answer RQ 1, this work presents a comprehensive analysis of data obtained from a
full-scale test rig. Measurements of nearly 2000 stopping brake applications performed
with a single disc brake unit are discussed and exploited to characterize the deterministic
and the stochastic behavior of a typical pad material applied in rail vehicles. Due to the
large amount of data, the results of this chapter are of particular interest in addressing
the lack of knowledge about the stochastic behavior of friction forces.
The objective of Chapter 4 is to develop and identify a brake model based on the results
of Chapters 2 and 3, which is suitable for the simulation of stopping brake scenarios with
a single disc brake unit. The method proposed in this chapter is the main contribution
of this work and represents a new approach to predict both deterministic and stochastic
friction forces. In addition, the results are verified based on the large amount of available
measurement data. Accordingly, Chapter 4 deals with RQ 2.
To answer RQ 3, the findings of Chapter 3 and Chapter 4 are transferred to an entire rail
vehicle in Chapter 5 by simulating stopping-brake scenarios conducted with a multiple
unit. Measurements from 88 experimental vehicle brake tests are available to evaluate
the suitability of the brake model for predicting the brake performance. In addition, the
large amount of experimental data as well as stochastic simulations based on the model
approach developed in Chapter 4 are used to draw conclusions regarding the propagation
of uncertainty during the braking process.
In Chapter 6, suggestions arising from this work to transfer the findings to rail operation
and brake system design are presented, as addressed by RQ 4. Finally, the contributions
of this work are concluded and an outlook for future research is given.
Chapter 2
State-of-the art in railway brake technology
Chapter 3
Analysis of measurement data
Chapter 4
Development of a brake model
Chapter 5
Analysis of vehicle brake applications
Chapter 7
Conclusion and outlook
RQ 1
Deterministic and stochastic behavior?
RQ 2
Simulate brake applications with single brake unit?
RQ 3
Transfer to vehicle?
Chapter 1
Introduction and derivation of research questions of this work
Chapter 6
Suggestions for design and operation
RQ 4
Transfer to design and rail operation?
Figure 1.3:
Structure of this work and relation of chapters to formulated research questions.
7
2 State-of-the-art in railway brake
technology
In the following, a brief introduction to rail brake systems is given, which presents the
basic working principles, components and friction requirements of disc brake units applied
in rail vehicles. Subsequently, the brake mechanics of disc-braked rail vehicle considered
as lumped mass are derived. Based on these results, the brake performance assessment of
rail vehicles is discussed to identify the main parameters determining the brake potential
of a disc-braked rail vehicle with regard to the operation with ETCS. Finally, this chapter
gives a comprehensive literature review of theories dealing with the complex processes
occurring in friction brakes and concludes with an overview of modeling approaches to
predict common friction phenomena related to disc brake units.
2.1 Introduction to railway brake systems
2.1.1 Classification
The main tasks of brake systems applied in rail vehicles are to decrease the vehicle
velocity partly or to full stop by converting the kinetic energy, to hold the velocity during
a falling gradient of the track and to prevent rolling of stationary vehicles due to wind or
gradients [51]. The TSI for locomotives and passenger rolling stock assigns these basic
tasks to a main brake function and a parking brake function [14]. The main brake function
is applied to decelerate the vehicle during operation for service and emergency cases and
shall be continuous and automatic. This means that the brake signal is controlled by
a central command and provided by a continuous line throughout the entire train. A
disruption of this control line shall lead to an automatic activation of a brake application
of all vehicles. Furthermore, it is required that the main brake systems is inexhaustible,
i.e. that is capable to perform successive applications and releases. In general, the TSI
differs between three basic brake operations [14]:
•
Emergency braking applications to stop the rail vehicle with a predefined level
of brake performance (main brake function).
9
2 State-of-the-art in railway brake technology
•
Service braking applications to control the vehicle velocity partly or to full stop
as well as to temporarily hold a stationary vehicle (main brake function).
•
Park braking applications to permanently hold a stationary vehicle without
providing energy on board (park brake function).
Figure 2.1 shows a classification of common brake systems to perform the main brake
functions [51][52][53]. In general, there are two ways to transfer retardation forces on the
vehicle. Adhesion dependent brakes, such as friction and dynamic brakes, apply braking
torques on the rotating axles of the vehicle. The braking torques generate reaction forces
in the contacts between wheels and rails, which decelerated the vehicle. The braking
effect is therefore dependent on the magnitude of the torque, the radius of the wheel
and the limits of adhesion between wheel and rail. In comparison, adhesion independent
brake systems do not make use of the contact between wheel and rail to decelerate the
vehicle. Thus, the application of these brakes allows to meet brake demands which exceed
the limits of adhesion.
Adhesion dependent brakes
Friction brakes
•Tread brake
•Disc brake
−Axle-mounted
−Wheel-mounted
Brake systems for rail vehicles
Adhesion independent brakes
Dynamic brakes
•Electro-dynamic brake
−Regenerative brake
−Rheostatic brake
•Rotating eddy current brake
•Hydro-dynamic brake
Track brakes
•Magnetic track brake
•Linear eddy current brake
Other
•Aerodynamic brake
Additional brake systems
•Support, relieve and substitute main brake system
•Enlarge permissible deceleration
•Recuperate kinetic energy
Main brake system
•Continuous
•Automatic
•Inexhaustible
Figure 2.1: Classification of brake systems for rail vehicles according to [51], [52] and [53].
Due to their high reliability and high level of safety, friction brakes are applied in nearly
every type of rail vehicle as main brake system [53]. Tread brakes have a long tradition
in railway brake applications and are still widely used today. In particular, freight trains
are applied with tread brakes, since they usually do not operate faster than 120
km
h
. As
shown in Figure 2.2, tread brakes generate brake torques by pressing brake blocks against
the wheels in order to convert the kinetic energy of the vehicle into thermal energy of
10
2.1 Introduction to railway brake systems
the frictional brake components. To cope with increasing brake demands due to ever
higher operational velocities and masses of rail vehicles, disc brake units were introduced.
They dissipate the kinetic energy of the train by pressing brake pads against rotating
discs. In case of an Axle-Mounted Disc Brake (AMD), the disc is directly mounted on
the axle, as depicted in Figure 2.2. In comparison, a Wheel-Mounted Disc Brake (WMD)
are directly attached to the wheel and applied, if constructional reasons leave no space
to equip the shaft of the axle with a brake disc.
Tread brake Axle-mounted disc brake
(AMD)
Wheel-mounted disc brake
(WMD)
Figure 2.2: Commonly applied friction brakes for rail vehicles.
The main brake system may be complemented by additional brake systems to support,
relieve or substitute the friction brakes [14]. Dynamic brakes enable to perform controlled
brake applications free from wear. Their application reduces the abrasion of friction
components and allows to recuperate braking energy, in particular for service braking.
Track brakes are fixed at the bogie of the rail vehicle and directly generate retardation
forces between bogie and rail. Magnetic track brakes are widely used to support friction
brakes during emergency cases and realize brake demands, which exceed the limits of
adhesion. Since they always apply the maximum brake force and strongly strain rails and
magnets, they are not used for service braking. In comparison, linear eddy current brakes
do not touch the rail, but induce a magnetic field in the rail to decelerate the vehicle.
Since the resulting eddy currents heat up the rails and the electromagnetic compatibility
to the rail side equipment is not guaranteed everywhere, the areas of application are
however limited. In the following, disc brakes and their actuation are briefly presented.
For details to other brake systems the interested reader is referred to [51] and [52].
11
2 State-of-the-art in railway brake technology
2.1.2 The compressed air brake
2.1.2.1 The indirect brake (PN)
Friction brakes applied for mainline rail vehicles are in general pneumatically actuated
1
.
This requires the availability of compressed air within the entire rail vehicle. The principle
of the indirect compressed air brake invented by George Westinghouse in 19th century
still represents the backbone of rail brake systems, due to the high level of safety provided
by the continuous and automatic working principle [51].
The design of an Indirect Pneumatic Brake (PN) is visualized by Figure 2.3. The Brake
Pipe (BP) running through the entire train is pressurized by an air compressor up to
5 bar and linked to so-called distributor valves, which are mounted in each wagon or
bogie of the train. As soon as the distributor valve detects a certain decrease of brake
pipe pressure
pBP
, either initiated via the drivers brake valve during a regular braking
process or in case of an accidental separation of the single wagons, a nozzle opens, which
pressurize the brake cylinders with compressed air from a local reservoir in the wagon.
The caliper unit transforms the cylinder pressure
pC
into a clamping force, which presses
the brake pads against the rotating discs to retard the vehicle.
A disadvantage of the PN is that the brake units along the train are not actuated at
the same time, since the propagation and extent of the pressure decrease in the BP is
limited. In particular, for long freight trains, which may exceed 700m in length, this
leads to strong longitudinal coupler forces acting between front and rear wagons [54].
Distributer valve
Brake pipe: 𝑝𝐵𝑃=0-5 bar
Air reservoir: 𝑝𝑅=0-5 bar
Cylinder pipe:
𝑝𝐶=0-3.8 bar
Pneumatic coupler
Drivers brake valve
Main air reservoir:
𝑝𝑀𝑅 =8.5-10 bar
Air compressor
Rail
Brake disc
Caliper unit
Brake cylinder
Bogie
Wheel
Figure 2.3:
Sketch of an indirect compressed air brake for the actuation of disc brake units
(the pressure values are relative to an ambient pressure of 1 bar).
1In contrast, friction brakes in trams usually base on hydraulic actuation [53].
12
2.1 Introduction to railway brake systems
2.1.2.2 The direct electro-pneumatic brake (EP)
In order to minimize the braking distance and longitudinal oscillations, passenger trains
are commonly equipped with an additional Direct Electro-Pneumatic Brake (EP). The EP
allows to actuate all brake cylinders at the same time by an electric signal without a time
delay. Therefore, the vehicle is equipped with an additional electronic line throughout
the entire train. The actuation is conducted by electro-magnetic valves in each wagon or
bogie to transform the electronic brake signal into a pneumatic precontrol pressure. As
soon as the precontrol pressure increases, a relay valve vents the brake cylinders with
compressed air from the reservoir. Distributor valve and electro-magnetic valves are
redundant components to actuate the brake cylinders. Hence, the indirect brake is still
available as fall-back level of the EP in order to guarantee a consistent level of safety.
2.1.2.3 Load dependency
Compressed air brakes are commonly equipped with a gradual or continuous load
dependency to assign the height of the cylinder pressure
pC
with respect to the axle load.
This is important to avoid an exceed of the load dependent adhesion limits between
wheel and rail by scaling the clamping forces and thus brake torques to the weight of the
vehicle. In particular, rail vehicles with varying load conditions during operation, such as
mass transit or freight trains, rely on this load dependency. In addition, the deceleration
of the vehicles is becoming less dependent on the load. This reduces longitudinal forces
between unequally loaded coaches or wagons of a train.
2.1.2.4 Brake modes and positions
The build-up time characterizes the time interval for the pressurization of the brake
cylinders by the indirect brake. In general, there are fast-acting modes and (brake
positions P and R) and a slow-acting mode (brake position G). As indicated by Figure
2.4, the pressurization for fast acting modes takes between 3-5s, whereas for the slow
acting position G it takes 18-30s. The latter is applied to master longitudinal forces
acting between single wagons of long freight trains. The so-called rapid brake position R
yields higher cylinder pressure to meet larger brake demands for rail vehicles operating
with velocities larger than 120km
h.
It needs to be considered that the build-up time describes the delay between the brake
signal and the cylinder pressurization. Due to the limited propagation of the pneumatic
brake signal within the BP of long trains, the pressurization of brake cylinders in the
rear wagons does not start simultaneously to the initiation of the indirect braking process.
This time lag is described by the so-called delay time. The sum of delay and built-up time
yields the actual response time, which characterizes the time interval between initiation
13
2 State-of-the-art in railway brake technology
and fully actuated brakes. During brake applications with the EP the brake units of the
train are actuated at almost the same point of time. This actuation strongly reduces
delay times and allows to realize faster built-up times, as indicated by Figure 2.4.
Relative pressure (bar) / Electronic signal (boolean)
Time (s)
Indirect Brake: emergency brake
Electronic brake signal
𝑝𝐶(R)
Direct EP: emergency brake
𝑝𝐵𝑃
𝑝𝐶(R)
𝑝𝐶(P)
𝑝𝐶(G)
3-5 s
18-30 s
1
0
5
𝑡0
Figure 2.4:
Sketch to compare the built-up times, brake pipe pressure and signals of the
different brake modes during an emergency brake applications initiated at t0.
2.1.3 Disc brake units of railway vehicles
2.1.3.1 Friction requirements
The amount of energy which needs to be dissipated by friction brakes during an emergency
stop depends on the braked mass mdyn and the quadratic initial velocity v2
0:
Ekin =1
2·mdyn ·v2
0(2.1)
As shown Table 2.1, the european standard EN 15328 classifies brake pads applied in disc
brake units of rail vehicles according to energy and power consumption of a single disc
during a stopping brake application. The maximum velocity ranges between 120
km
h
and
400
km
h
. The braked mass takes values between 5
tons
to 12
tons
. This yields a maximum
amount of energy to be transformed by a disc brake unit between 5
MJ
and 40
MJ
. In
comparison, one of the four disc brakes applied in a passenger car of 2
tons
needs to
transfer less than 1
MJ
during a stop braking application with 160
km
h
. This reveals
the enormous amount of energy disc brake units of rail vehicles are exposed to during
operation.
14
2.1 Introduction to railway brake systems
Table 2.1:
Extract of the classification of brake pads for different train types and disc brakes
according to EN 15328 [55].
Class
Train Type Disc
Type
Maximum
Velocity
[km
h]
Maximum
Energy
[MJ]
Maximum
Power
[kW ]
Braked
Mass
[ton]
Mean
decel.
[m
s2]
A1 Freight Wagons
AMD
120 6.3 338 11.25 0.9
B1 Multiple Unit
WMD
160 7.9 427 8 1.2
C1 Multiple Unit
WMD
200 12.3 533 8 1.2
D1 Multiple Unit
AMD
250 15.7 361 6.5 0.8
E1 High Speed Train
WMD
300 27.8 533 8 0.8
F1 High Speed Train
AMD
350 23.6 389 5 0.8
G1 High Speed Train
AMD
400 37 533 6 0.8
The standards EN 15328 and EN 14535-3 specify the requirements for disc brakes
and brake pads related to power consumption, durability and frictional properties
[55][56]. Furthermore, testing programs are specified to verify the requirements based
on experimental brake applications conducted with a single brake unit on a Full-Scale
Dynamometer (FSD) test rig, as shown in Figure 2.5
2
. Appendix A.1.1 shows an extract
of the testing program for brake pads of class B1 and C1. The program includes about
170 brake applications on the test rig, which are performed in a predefined sequence.
Each program contains dry and wet brake applications with and without cooling phases
conducted for different braked masses and clamping forces.
Figure 2.5:
Full-scaled dynamometer test rig of the Knorr-Bremse SfS GmbH applied with
AMD brake and caliper unit (Picture provided by Knorr-Bremse SfS GmbH).
To verify the frictional behavior of the applied brake pads, EN 15328 determines permis-
sible tolerances for the COF averaged over the braking distance
µm
. As exemplified by
Figure 2.6, a regression line is retrieved from data gained during the test rig experiments,
which relates
µm
to the initial velocity. In general, the line shall lie between 0.28 and
2A detailed description of the test rig will be given in Chapter 3.
15
2 State-of-the-art in railway brake technology
0.47. Furthermore, the
µm
values of Priority 1 brake applications, i.e dry brake appli-
cations with the highest nominal clamping force for each mass according to EN 15328,
should lie within a tolerance interval of
±10%
with respect to the regression line, while
12% of all Priority 1 brake applications may lie out of this interval. For lower priority
brake applications, tolerances of up to
±30%
are permitted, which represents significant
variations and corresponds to a very wide acceptance range.
0,3
0,35
0,4
0,45
0,5
050 100 150 200 250
µm[ ]
Initial Velocity [km/h]
Data
Linear Regression
Tolerance Interval +10%
Tolerance Interval - 10%
Figure 2.6:
Derivation of tolerance intervals according to EN 15328 based on friction mea-
surements (data was artificially generated).
2.1.3.2 Friction pairing
Due to the large spectrum of loads revealed by Table 2.1, materials and designs of disc
and pad, also referred to as friction pairing, need to be selected with regard to the
individual application and expected operating conditions. Apart from functional and
durability requirements, the selection is also affected by economic demands.
Cast iron is usually applied as material for the disc, if the loads are not too high. For
medium loads, spheroidal graphite cast iron and for high loads cast steel is used [51].
In rare cases where a light weight construction is desired, disc brakes made of particle-
strengthened aluminum or ceramic matrix composite are deployed. The outer diameter
of an AMD varies between 450
mm
and 700
mm
[51]. A WMD has an outer diameter of
600mm to 1100mm and an inner diameter of 300mm to 800mm [51]. In order to avoid
thermal overloads, AMD and WMD are mostly equipped with cooling fins to enable an
enhanced heat exchange with the ambient air. However, these fins might cause ventilation
losses during high rotational speed of the disc [57].
The pads are usually made of composites materials based on organic polymers or sintered
metals. Two examples as well as their dimensions are shown in Figure 2.7. The envelope
geometry is specified by EN 15328 and yields a nominal friction area of 400
cm2
[55].
The initial thickness of the pads is about 3
.
5
cm
[51]. As shown, the pads are segmented
16
2.1 Introduction to railway brake systems
to counteract the deformation caused by thermo-mechanical loads during the braking
process and to guarantee a uniform pressure distribution, force transmission and wear.
Organic Pad Sintered Pad
40 cm
14 cm
400 cm2
Figure 2.7:
Example of organic and sintered brake pads with dimensions (Pictures provided
by Knorr-Bremse SfS GmbH. Pictures are edited for the illustration).
So-called organic brake pads are applied for brake applications with medium thermal
loads. Organic pads are made of composite materials, i.e a mixture of multiple materials
with different properties. The components have different functions and are classified
according to the following categories [58][59]:
•
Abinder, often made of phenolic resin, to cohere all materials within a thermally
stable matrix.
•
Structural reinforcement components, such as metallic or carbon fibers, to provide
mechanical strength.
•
Frictional additives to guarantee the stability of frictional behavior and to control
wear. These are abrasives practicals, such as alumina or silica and solid lubricants,
such as metal sulfides or graphite.
•
Fillers are usually added to reduce the costs of the composite material without
affecting the performance.
If the temperature of the pads exceed a certain value (300
−
500
◦C
), the organic binder is
disintegrated yielding a strong decay of the friction properties [30][51][60]. This so-called
fading effect limits the application of organic pads. To cope with higher loads and
resulting temperatures, sintered pads are applied for high speed rail applications. The
bonding between the components in this pad type is provided by a metal matrix. During
the sinter process, pressure and heat is applied to a mixture of metal powders and other
components yielding the metal bonded pad material. The resulting sintered composite
material withstands higher temperatures without losing its frictional properties.
17
2 State-of-the-art in railway brake technology
2.2 Brake mechanics
In the following, the mechanical laws are discussed, which rule the longitudinal kinetics of
rail vehicles considered as a lumped mass. Initially, the equation of motions are derived
for a wheel-set equipped with a single disc brake. Subsequently, the governing equations
are transferred to the entire vehicle.
2.2.1 Brake mechanics of a wheel set
Figure 2.8 shows a sketch of a wheel set equipped with a WMD. Beneath the sketch,
forces and torques are illustrated, which act during a brake application. The actuation
of the brake unit is initiated by an increasing cylinder pressure
pC
acting on the surface
area
AC
. As soon as the resulting pressure force (
pC·AC
) exceeds the counteracting
force of the return spring
FS
, both pads are pressed against the disc via the rigging of
the caliper, as shown by sketch (I) in Figure 2.8. The acting normal forces are thus given
by
FN=
0for pC·AC≤FS,
(pC·AC−FS)·ηC·iCfor pC·AC> FS,(2.2)
while
ηC
represents the efficiency and
iC
the transmission ratio of the caliper. The forced
contact evokes a friction force
Ffric
at each side of the disc, which acts opposed to the
sliding direction. The friction force is the product of the normal force and the Coefficient
of Friction (COF), denoted by µ,
Ffric =µ·FN.(2.3)
The point of attack of the friction force is defined as the friction radius
rfric
. Assuming
that rfric and µare identical for both contacts, yields the resulting brake torque MB
MB= 2 ·Ffric ·rfric = 2 ·FN·µ·rfric.(2.4)
Introducing the total clamping force
FC
as the sum of all normal contact forces acting in
the disc brake unit arrangement
FC= 2 ·FN(2.5)
and integrating the doubling into the total transmission ratio
iT
= 2
·iC
finally, gives:
18
2.2 Brake mechanics
MB=FC·µ·rfric = (pC·AC−FS)·ηC·iT·µ·rfric.(2.6)
Equation 2.6 reveals the parameters, which influence the magnitude of the brake torque
of a disc brake unit.
Wheel
Compact Caliper Unit
Bogie
Cylinder pressure pipe
Rail
Brake disc
Axle bearing
Primary suspension
Driving direction
Caliper unit (I) Brake pad (II) Axle and rail (III)
𝐹𝑇
𝐹𝑇
𝐹𝐿
𝑥
𝜑
𝐽, 𝑚
𝑟𝑤ℎ𝑒𝑒𝑙
𝐹𝐿
𝐹𝐿
𝑀𝐵
Cylinder
Rigging Pad
𝐹𝑁
𝐹𝑁
𝐹
𝑠
𝑝𝐶𝐴𝐶
𝑝𝐶
𝑟
𝑓𝑟𝑖𝑐
𝐴𝐶
𝑟
𝑓𝑟𝑖𝑐
𝑑𝐴 𝜇𝑙, 𝑝𝑙
𝑟𝑙
𝑟
𝑓𝑟𝑖𝑐
𝐹
𝑓𝑟𝑖𝑐
Figure 2.8:
Sketch to visualize the acting forces during a brake application with a single WMD
disc brake unit (Forces and torques: red, dedree of freedom: blue).
Strictly speaking,
MB
results from the pressure distribution
pl
and local friction properties
µl
integrated over the pad area, as indicated by Sketch (II) in Figure 2.8.
µ
and
rfric
are
thus equivalent values, which yield the same brake torque as the integral of the differential
friction torques
FN·µ·rfric =Zpl·µl·rldA. (2.7)
Presuming constant friction properties along the surface,
rfric
may be estimated from
Equation 2.7 with respect to a certain pressure distribution and pad geometry [23][30][55].
19
2 State-of-the-art in railway brake technology
Analogously,
µ
corresponds to the integral of the differential friction torques related to
normal force and the equivalent value of the friction radius
µ=1
FN·rfric ·Zpl·µl·rldA. (2.8)
As shown in [61] and [30], a detailed analysis of local friction properties requires spacial
discretisation of the pad area and the application of elaborate numerical methods, such
as finite element methods, which are related to a high computational effort. For the
investigation of the longitudinal vehicle dynamics, the consideration of equivalent values
is however sufficient.
As indicated by sketch (III) in Figure 2.8, the brake torque generates a counteracting
tangential force
FT
in the wheel/rail contact, which decelerates the wheel-set. Neglecting
lateral and vertical movements, such a pitch and yaw, as well as rolling, the equations
for the rotational and longitudinal motion of the wheel set are thus given by
J·¨φ= 2 ·FT·rwheel −MB(2.9)
and
m·¨x=−2·FT.(2.10)
¨φ
and
¨x
are the angular and translational acceleration.
J
represents the rotational inertia
of the wheel set and mthe static mass, respectively.
The transferable tangential forces and thus the brake force are limited by the coefficient
of adhesion prevailing in longitudinal direction
fx
between wheel and rail and the axle
load FL:
FT≤FT,max =FL·fx.(2.11)
fx
dependents on the condition of the rail (dry, wet, sanded, contaminated) and the
longitudinal slip between wheel and rail, which is commonly defined as the ratio between
the circumferential velocity of the wheel and the driving velocity of the vehicle. Mea-
surements show that
fx
takes values between 0.1 and 0.4 [62][25]. The TSI demands
that the design of the brake systems shall guarantee that the required value of
fx
does
generally not exceed 0.15 during an emergency brake application to avoid macroscopic
slip and a blockage of the wheel [14]. In case of degraded adhesion conditions due to wet
or contaminated rail heads, a Wheel-Slide Protection (WSP) system, if installed, detects
20
2.2 Brake mechanics
and intervenes the undesirable blockage and sliding of the wheels by rapidly dearating
the cylinder pressure.
In this work, only brake applications are investigated, which do not exceed
the limits of adhesion. They represent the basis for the assessment of the nominal
brake performance of railway vehicles according to EN 14531 and EN 16834 [16][63]. In
this case, only microscopic slip occurs between wheel and rail during braking and the
assumption of pure rolling is valid, i.e.:
¨φ≈¨x
rwheel
.(2.12)
By inserting this kinematic relation and Equation 2.10 in Equation 2.9, it follows
¨x·(J
rw2+m) = ¨x·mdyn =−MB
rwheel
,(2.13)
where the dynamic mass
mdyn
is given by the the sum of the static mass
m
and the
equivalent rotating mass J
rw2related to a defined wheel radius.
The ratio of brake torque and wheel radius yields the brake force FB
FB=MB
rwheel
=FC·µ·rfric
rwheel
.(2.14)
The resulting translational acceleration
a
of a wheel set braked by a single brake unit is
thus given by
a= ¨x=−FB
mdyn
=−FC·µ·rfric
mdyn ·rwheel
=−(pC·AC−FS)·ηC·iC·µ·rfric
mdyn ·rwheel
,(2.15)
if pure rolling is assumed. As indicated by the negative sign, the vehicle is decelerated.
The deceleration
|a|
corresponds to the absolute value of the acceleration. Accordingly,
the brake performance is directly proportional to the COF, the clamping force as well as
the ratio of friction and wheel radius.
21
2 State-of-the-art in railway brake technology
2.2.2 Brake kinetics of a train considered as single mass
An entire train may be considered as a single lumped mass, if the longitudinal oscilla-
tions occurring between the wagons during braking are small compared to the overall
deceleration caused by the brake system. This generally applies for brake applications
of multiple-units, as demonstrated by the measurements in Figure 1.2
3
. The lumped
dynamic mass of a train
mtrain
is composed of the total static train mass
m
and the sum
of the equivalent rotating masses of all nrot wheel sets
mtrain =
nrot
X
i=1
Ji
rw,i
+m. (2.16)
For a train with
nb
brake units, the total brake force applied by the brake system is thus
the sum of all single brake forces
FB=
nb
X
i=1
FB,i =
nb
X
i=1
rfric,i
rwheel,i ·FC,i ·µi.(2.17)
The total deceleration acting on a disc-braked train results from all brake and resistance
forces
a= ¨x=−
nb
P
i=1
FB,i +FR
mtrain
=−
nb
P
i=1
rfric,i
rwheel,i ·FC,i ·µi
mtrain −FR
mtrain
,(2.18)
where
FR
represents additional retardation forces acting on the train, which result from
the driving resistance, curvature or the slope of the track.
The driving resistance on a straight and even track is primarily influenced by the
vehicle velocity
v
. Various formulas exists to describe this dependency for different
vehicle types, as shown in [25] and [54]. In general, they are all based on the polynomial
approach made by Davis in 1926 [66]
FR(v) = R1+R2·v+R3·v2.(2.19)
This equation is still valid and commonly applied to identify the driving resistance of
rail vehicles by determining the coefficients
R1
,
R2
and
R3
based on coasting or drag
experiments [67]. The first two terms are related to rolling and impulse resistances. The
3
In comparison, long freight trains may be subject to strong longitudinal oscillations due to delayed
built-up times throughout the train, as described in Section 2.1.2.1. In this case the train needs to
be considered as system of coupled masses, as shown in [64],[65] and [54].
22
2.2 Brake mechanics
third term expresses the aerodynamic resistance, which increases by the square of the
velocity. Thus, the resistance force decreases during the braking process due to the
declining vehicle velocity and reaches the value R1for v= 0.
The vehicle velocity
v
(
t
)is obtained by integration of Equation 2.18 with respect to time.
If
v0
represents the initial velocity at the start of the brake application, the following
initial value problem arises:
v(t) = v0+Za(t)dt =v0−1
mtrain
n
X
j=1
rfric,i
rwheel,i ZFC,i ·µidt +ZFRdt
(2.20)
For the traveled distance s(t)during the brake application, it follows
s(t) = s0+Zv(t)dt =s0+v0·t−1
mtrain
n
X
j=1
rfric,i
rwheel,i Z Z FC,i ·µidtdt +Z Z FRdtdt
,
(2.21)
where s0represents the starting point.
As an example, Figure 2.9 shows measurements recorded during an emergency brake
process of a disc-braked multiple unit conducted on a straight and level track with
an initial velocity of
v0
= 160
km
h
. The upper plot depicts the measured instantaneous
longitudinal deceleration
|a|
(
t
)of the train and the progression of the clamping forces
FC,i
(
t
)of all brake units. Note that the level of the clamping forces depends on the
axle load and is not the same for all disc brake units. Once the clamping forces have
reached their demand values, they remain constant. However, the deceleration is clearly
changing afterwards. Primarily, this behavior is caused by the characteristics of friction
prevailing in the disc brake units. Additionally, decreasing resistance forces contribute
to the changing deceleration during braking. As indicated by the deceleration recorded
during the first second, the maximum resistance forces are however small compared to
the total deceleration.
The vehicle velocity
v
(
t
)and traveled distance
s
(
t
)are shown in the lower plot of Figure
2.9. The total distance traveled from the initiation of the braking process until standstill
is the so-called stopping distance
sst
, which is about 900
m
in this case. As shown,
the friction braking process of an emergency brake application can be separated into
atransient phase (0
s
-5
s
), until the calipers are fully acting, and the subsequent actual
braking phase until standstill (5
s−
38
s
). During the transient phase the deceleration
is ruled by the delay and progression of the clamping forces, i.e. the dynamics of the
pneumatic or electro-pneumatic actuation. During the braking phase the behavior of
COF and the resistance forces are driving the course of the deceleration. The stopping
23
2 State-of-the-art in railway brake technology
distance of an emergency brake process
sst
is thus composed of the response distance
s1
traveled during the transient phase (
s1
= 200
m
) and the so-called braking distance
s2traveled with fully actuated brakes (s2= 700m)
sst =s1+s2.(2.22)
The higher the initial velocity and the shorter the response time, as it is the case for
disc-braked passenger trains, the larger the contribution of
s2
and thus the role of
instantaneous pad friction forces regarding the stopping distance. For smaller initial
velocities and longer actuation phases, as it is the case for freight trains, the contribution
of s1and thus the role of the actuation increases.
Figure 2.9:
Instantaneous deceleration
|a|
(
t
), velocity
v
(
t
), stopping distance
s
(
t
)and clamping
forces
FC,i
(
t
)of a train measured during brake applications with disc brake units .
24
2.3 Brake performance assessment
2.3 Brake performance assessment
In this section, the brake performance assessment of rail vehicles is discussed. Initially, a
brief overview of assessment methods is given with regard to an operation with ETCS.
Subsequently, state-of-the-art methods for the deterministic and probabilistic prediction
of the brake performance are discussed.
2.3.1 Brake assessment with regard to operation with ETCS
The role of the brake performance assessment is to determine the brake capability of
a rail vehicle. Figure 2.10 gives an overview of different assessment methods and their
interface to an operation with ETCS.
Brake Assessment
ETCS Operation
Brake performance
Deviations from nominal behavior
Nominal behavior
Deviations related to
brake system
•UIC B126/ DT 414/421
Correction factors Kdry, Kwet
Braking Curve
Infrastructure related parameters
•Emergency Brake Confidence Level (EBCL)
•Available Adhesion Weighting Factor (AVADH)
•Track gradient
Save deceleration
Deceleration profile
Verify Deterministic
Calculations
•EN 14531, ISO 20138
•NNTR
Experimental
Testing
•EN 16834, UIC 544-1
•NNTR
Stopping distance, Deceleration, Braked weights
Degraded adhesion
conditions
•EN 15595
Speed & distance
monitoring
Vehicle measurements und location
•Position
•Velocity
•Acceleration
|a| |a| v
vvs
γ-Trains γ-Trains
Figure 2.10:
Methods for the brake performance assessment of rail vehicles and derivation of
ETCS braking curves for γ-trains.
25
2 State-of-the-art in railway brake technology
2.3.1.1 Nominal performance
The nominal brake performance corresponds to the average behavior on a
straight and level track with respect to mean in-service conditions of the brake
equipment and dry rails [63]. The TSI requires both calculation and experimental
testing to determine the nominal brake performance, while the results of both methods
need to coincide [14][15]. EN 15431 is the referred european standard for the calculations,
which will be discussed in detail in the remainder of this section [16]. ISO 20138 represents
the corresponding international standard [68]. EN 16834 and UIC 544-1 specify the
brake assessment based on experimental brake tests [16][69]. To consider country-specific
operational aspects in the brake assessment, there are additional national specifications
that are defined by the so-called Notified National Technical Regulations (NNTR) [70].
The standards apply three methods to characterize the nominal brake performance,
namely stopping distance,deceleration and braked weights. The deceleration-
based assessment characterizes the brake capability of the vehicle by means of a speed-
dependent deceleration profile. This method is necessary for high-speed rail applications
and signalling systems with a permanent velocity control, such as ETCS [63]. Hence, the
deceleration-based assessment is applied for disc-braked vehicles with fixed configuration,
such as multiple units, which are also referred to as γ-Trains within ETCS [7]4.
The usage of braked weights has a long tradition in railway applications. This measure
allows to estimate the brake capability of an assembled train with a variable composition
in operation, such as freight trains or locomotive-hauled coaches, with respect to a
particular reference performance [63][25]. Within ETCS, these vehicles are called
λ
-trains
[7]
5
. Since the braked weight is not a physical quantity, a conversion is necessary for the
operation with ETCS. This assessment method is not further discussed.
2.3.1.2 Deviations from nominal performance
As indicated by Figure 2.10, deviations from the nominal behavior have two major
sources. Firstly, the brake torques are subject to a certain scatter during operation,
which is caused by the inherent stochastic behavior of the brake components as well as
wear-induced changes during the service life. Typical parameters contributing to this
brake system-related scatter are varying cylinder pressures, mechanic efficiencies and,
in particular, the friction coefficients between pads and discs [72][7]. In addition, the
brake torques may be affected by potential failures of brake components during operation.
On the other hand, degraded adhesion conditions due to wet or contaminated rails
affect the transmissible contact forces between wheel and rail. In particular, the presence
4The term γ-trains is based on the usage of the symbol γfor the deceleration in France [71].
5
The term
λ
-trains is based on the symbol
λ
, which defines the so-called braked weight percentage of
an assembled train [63].
26
2.3 Brake performance assessment
of wet compacted leafs on the rails may result in a drastic reduction of the adhesion and
thus brake forces (fx<0.05) [73].
Figure 2.11 visualizes how these two sources affect the dispersion of the brake deceleration
according to [72]. Deviations caused by the brake system typically result in a Gaussian-
like distribution because they are the superposition of a number of different uncertainties
that affect the brake torque. In contrast, performance deviations caused by degraded
adhesion are strongly related to the environmental conditions as well as the efficiency of
the WSP and sanding equipment applied in the vehicle [7]. In addition, the conditions
on the rail surface change due to cleaning effects as the vehicles pass over it [46]. The
resulting probability density of the brake deceleration thus strongly deviates from typical
distributions and may be subject to rare events with a very low brake performance. To
ensure a safe rail operation, the different types of deviations need to be considered within
the brake assessment. Therefore, ETCS applies two correction factors,
Kdry
and
Kwet
, as
discussed in the next section.
Deceleration
Probability density
Deviations related to brake system (→ 𝑲𝒅𝒓𝒚)Deviations related to degraded adhesion (→ 𝑲𝒘𝒆𝒕)
Nominal braking performance (dry rails)
Figure 2.11:
Probability density of brake deceleration caused by deviations related to brake
system and degraded adhesion conditions according to [72].
2.3.1.3 ETCS braking curves of γ-Trains
ETCS supervises the velocity of a rail vehicle to guarantee that it stops before the next
danger point or supervised location [7]. If a vehicle exceeds the permitted velocity, the
system intervenes by initiating a forced emergency brake application. This so-called
parachute function requires that the on-board control system is able to estimate the
brake capability of the vehicle. The so-called Emergency Brake Deceleration Curve
(EBD) describes the velocity decrease with respect to the distance during an emergency
brake application, as indicated by Figure 2.10. The curve is permanently calculated
by the control system based on information regarding the track-gradient, the available
brake capability and the measured velocity. The EBD is deduced by superimposing the
save brake deceleration profile on level track
|a|brake_safe
and a profile representing the
27
2 State-of-the-art in railway brake technology
deceleration caused by the track gradient [7]. For
γ
-Trains, the save or also referred to as
the guaranteed brake deceleration is obtained by down-scaling the nominal deceleration
profile |a|n(v)with the correction factors Kdry and Kwet according to6
|a|brake_safe =|a|n(v)·Kdry(v, EBCL)·{Kwet(v) + AV ADH ·(1 −Kwet(v))}.(2.23)
Typically,
Kdry
and
Kwet
take values between 0 and 1 and are generally defined individually
for each velocity interval of the nominal deceleration profile.
Kdry
addresses deviations
related to the brake system, i.e. the potential reduction of the brake performance on dry
rails. The value of
Kdry
has to be selected in a manner that the product
|a|n
(
v
)
·Kdry
yields
the minimum available brake potential in operation during normal adhesion
conditions with respect to a certain probability. The term
|a|n
(
v
)
·Kdry
is thus
called the safe brake deceleration on dry rails. The so-called Emergency Brake Confidence
Level (EBCL) is the safety target set by the infrastructure manager, which defines the
probability that the deceleration of a train is at least equal to the safe deceleration. As
shown in the Table A.1 of Appendix A.1.5, the EBCL takes values between 0 and 9,
which corresponds to probabilities between 50% and 99.9999999%. In Germany, the
EBCL is 7, which is related to a probability of 99.99999% [75]. However, the EBCL can
vary widely between countries, as also shown in Table A.1. The ERTMS/ETCS system
specification proposes to apply a probabilistic analysis based on Monte-Carlo Simulation
(MCS) to derive
Kdry
with respect to a certain probability [74][7]. The method originates
from the UIC working group B126 and will be discussed in Section 2.3.3.
Kwet
represents the correction factor related to degraded adhesion conditions. The
derivation of
Kwet
with respect to a certain safety target, in analogy to the procedure
proposed for
Kdry
, requires detailed information regarding the current conditions of the
rails, which is usually not available. Consequently, the distribution shown in Figure
2.11 would lead to extremely small safety factors, which would result in unrealistically
large distances between subsequent trains [46]. Therefore, it is proposed to retrieve the
correction factor
Kwet
from field testing, as discussed in EN 15595 [76][7]. Nevertheless,
this procedure does not allow to quantify the actual safety level during
degraded adhesion conditions [77]. This explains why the influence of
Kwet
on
Equation 2.23 may be adapted by the Available Adhesion Weighting Factor (AVADH). If
AV ADH
= 1, as it is the case for infrastructure operated by the DB Netz AG [77], the
influence of Kwet is eliminated in Equation 2.23 and the safe deceleration is given by
|a|brake_safe =|a|n(v)·Kdry(v, EBCL).(2.24)
In this case, the influence of degraded adhesion conditions needs to be considered in
other ways during operation.
6The acronyms applied in Equation 2.23 slightly differ from the original specification in [74].
28
2.3 Brake performance assessment
2.3.2 Deterministic brake calculations according to EN 14531
Predicting the nominal brake performance of a disc-braked rail vehicle based on the
derived kinetics requires to solve Equations 2.20 and 2.21. The choice of method depends
on the assumptions made for the integrands
FC
(
t
),
µ
(
t
)and
FR
(
t
)as well as the purpose
of the prediction. As shown in Figure 2.12, the standard EN 15431 proposes two methods
for the calculations, namely the application of mean values or step by step calculations.
Calculations based on mean values allow for a straightforward estimation of the brake
performance based on a simplified model of the braking process. This is in particular of
interest during the layout of the brake system and for the brake assessment with regard
to operation. Step by step calculations allow for a more detailed analysis of the braking
process and provide an estimation of the instantaneous brake capability of a vehicle
during the entire braking process. Both methods are briefly introduced for rail vehicles
equipped with disc brake units.
Calculate brake performance according to EN 14531
14531-2: Step by step calculations
𝑎 𝑡𝑗
14531-1: Mean value calculations
𝑡𝐸
|𝑎|𝐸
𝑡
|𝑎|
|𝑎|
𝑡|𝑎|𝐸=1
𝑠2න
0
𝑠2|𝑎| 𝑠 𝑑𝑠
Average over
braking distance 𝒔𝟐
Figure 2.12: Methods proposed by EN 14531 to calculate the brake performance [16][17].
2.3.2.1 EN 14531-1: Mean value calculations
The use of mean values is based on transforming the instantaneous braking process into
a step function characterized by the equivalent response time
tE
and the constant mean
deceleration
|a|E
, as indicated by the left sketch in Figure 2.12. This simplification allows
to quantify the brake capability by two single parameters and strongly facilitates the
solution of the kinetics, as shown in Appendix A.1.2.
|a|E
represents the equivalent
deceleration, which yields the same braking distance as the instantaneous deceleration.
The approach requires to provide the corresponding equivalent forces
FBE
and
FRE
for
the applied brake equipment and the acting resistance forces. The derivation of equivalent
forces based on a speed-dependent force characteristic is given in Appendix A.1.3. In
case of disc brake units, FBE is calculated according to
FBE =FC·µm·rfric
rwheel
.(2.25)
29
2 State-of-the-art in railway brake technology
In this equation,
FC
corresponds to the demand value of the clamping force, i.e. fully
actuated. µmis the instantaneous COF averaged over the braking distance
µm=1
s2
s2
Z
0
µ(s)ds. (2.26)
The derivation of Equation 2.26 is shown in Appendix A.1.4.
µm
is the essential
parameter characterizing the brake potential of a frictional brake unit. Therefore, EN
14531-1 requires to derive
µm
based on data obtained from experimental brake tests on a
full-scale dynamometer test-rig according to UIC Code 541-3 or EN 15328.
Brake systems can be equipped with a function to temporary reduce the brake forces
during braking at high speed in order to limit the frictional heat input, which results in
systematic changes of the deceleration during the braking process. Furthermore, certain
brake forces, such as those of electro-dynamic brakes, are characterized by strongly
varying behavior in different speed ranges. To account for these changes in the brake
assessment, it is proposed to use a speed-dependent deceleration profile composed of
several equivalent deceleration values, whereby each value applies to a particular speed
interval (see Equation A.5 in Appendix A.1.2).
From a mathematical perspective, this equivalent profile corresponds to the nominal
deceleration profile
|a|n
(
v
)applied in ETCS. However, the nominal profiles ultimately
used for ETCS are derived from experimental vehicle brake tests based on EN 16834 [63].
In contrast to mean value calculations, which approximate the friction characteristics of
a braking process by a single value (
µm
), the profiles gained from vehicle tests can take
into account systematic changes of the COF during the braking process.
2.3.2.2 EN 14531-2: Step by step calculations
Part 2 of EN 14531 represents the solution of Equations 2.20 and 2.21 resulting from
instantaneous brake and resistance forces based on numerical integration with respect
to time. This means that the corresponding force characteristics are approximated by
discrete time series. In case of disc brake units, this requires to provide discrete time
series for the clamping force
FC
(
tj
)and the COF
µ
(
tj
), where
tj
represents the discrete
time instants. The step by step method makes it possible to take into account mutual
dependencies of the applied characteristics when calculating the brake performance.
Accordingly, the COF may be an arbitrary function of different variables calculated
during the numerical integration, such as µ(tj, FC(tj), v(tj), ...).
As shown in [78], there is a large number of implicit and explicit methods to be applied for
numerical integration. Regardless of the selected method, it needs to be guaranteed that
the approximated solution converges against the exact solution by setting a sufficiently
30
2.3 Brake performance assessment
small time step. The results are time series of the instantaneous brake force
FB
(
tj
),
deceleration
a
(
tj
), velocity
v
(
tj
)and distance
s
(
tj
). As indicated by Figure 2.12, the
output of the step by step calculations can be transformed into an equivalent braking
process based on mean values via
|a|E=1
s2
s2
Z
0|a|(s)ds. (2.27)
The derivation of Equation 2.27 is equivalent to the one of
µm
conducted in Appendix A.1.4.
Hence, the equivalent deceleration
|a|E
corresponds to the instantaneous deceleration
averaged over the braking distance.
2.3.3 Stochastic analysis of vehicle brake applications
2.3.3.1 Principle procedure
In the early 2000s, the UIC working group B126 laid the foundation for the probabilistic
analysis of brake applications with rail vehicles to explicitly derive safety margins for rail
operation based on a prescribed safety target [45][46][47][48]. The developed method is
illustrated in Figure 2.13 and represents the basis for the derivation of the correction
factor Kdry in the context of ETCS.
P( |𝑎|
|𝑎|𝑛)
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
1
0.9
0.80.7
0.6 11.1
Correction factor 𝑲𝒅𝒓𝒚 =|𝒂|
|𝒂|𝒏
P𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚
𝟏−𝑷(𝑬𝑩𝑪𝑳)
Cumulative probability
density function
Brake model
𝜇
Monte-Carlo
Simulation
Distributed brake parameters
p( |𝑎|
|𝑎|𝑛)
Probability density of
normalized deceleration
Probabilistic analysis Derivation of correction factor
Reliability of brake components
Figure 2.13:
Derivation of correction factor
Kdry
based on probabilistic analysis and EBCL.
Initially, a probabilistic analysis is conducted to estimate the dispersion of the deceleration
to be expected in operation. Inputs for this analysis are assumptions regarding the
uncertainty of the brake parameters, which are subject to a non-negligible scatter in
operation, such as distributions of cylinder pressures, caliper efficiencies and the pad
31
2 State-of-the-art in railway brake technology
friction. In addition, the fault rates of the single brake components are taken into account
of the analysis. The working group proposes to use normalized distributions, i.e. to
characterize the dispersion of each brake parameter with respect to its nominal value
[45]. This procedure simplifies the brake model used for the probabilistic analysis and
allows to avoid the application of absolute values in the calculation.
The output of the analysis is a probability density function of the normalized deceleration,
which defines how often and to which extent the actual deceleration
|a|
will deviate
from the nominal deceleration
|a|n
. Based on the corresponding cumulative probability
density function
P
(
|a|
|a|n
), the correction factor
Kdry
may be simply deduced with respect
to a certain probability
P
= 1
−P
(
EBCL
), as shown in the right sketch of Figure 2.13,
while
P
(
EBCL
)corresponds to the values given in Table A.1. In simple words,
Kdry
corresponds to the smallest deceleration with respect to the nominal value that can be
expected in one of 10EBCL braking applications7.
2.3.3.2 Monte-Carlo Simulation
The working group proposes to use Monte-Carlo Simulation (MCS) for the probabilistic
analysis. MCS is a common numerical procedure to solve complex problems based
on a large number of artificially generated samples [79]. In the case of stochastic
problems, this means to determine the distribution of a random output variable, here
the brake deceleration, that is usually non-linearly dependent on a number of random
input variables, here the brake parameters [79]. The applied brake model defines the
deterministic dependency between brake deceleration and brake parameters.
According to this method, random input samples with sample size
n
of all uncertain brake
parameters are artificially generated with respect to defined distributions. Subsequently,
n
calculations, also referred to as simulations, are conducted based on theses input
samples. This means that during each simulation, the parameters of the brake model
take values according to one set of the input samples. This yields a sample of
n
output
values, which is applied to deduce a discrete probability density of the deceleration. To
provide trustworthy estimates for the outer ranges of this discrete distribution, i.e. very
rare events, such as
P≤
10
−6
, it is necessary to conduct a sufficiently large number of
simulations, e.g.
n
= 10
·
10
EBCL
. This meas that the mathematical model used for the
MCS needs to be computational efficient.
Malvezzi et al. also derive an analytic solution for
P
(
|a|
|a|n
)[48]. However, this solution is
limited to normally distributed brake parameters. In contrast, MCS allows to take into
account arbitrary types of distributions as well as logical parameters, which are necessary
to simulate failures of components [45].
7Note, this simplified definition is only valid for EBCL ≥1.
32
2.3 Brake performance assessment
2.3.3.3 Uncertainty of friction within probabilistic analysis
For trains equipped with disc or tread brakes made of composite material the UIC
working group proposes to express the ratio of actual to nominal COF by
µ
µn
=µmed
µn·µ
µmed
,(2.28)
•where µncorresponds to the nominal COF of the applied pad material and
•µmed
is the so-called mean COF, which represents the friction properties with
respect to a certain braking scenario and material type8[48].
Accordingly, the ratio
µmed
µn
may interpreted as deviations due to deterministic influences
with respect to the nominal behavior. The ratio
µ
µmed
represents deviations caused by
actual stochastic phenomena observed for identical brake applications with a single vehicle.
This interpretation explains why the UIC proposes a uniform distribution for
µmed
µn
with
a dispersion of ±13% −14% and a Gaussian distribution for µ
µmed with 1σ=±2.5%9.
Horn and Pavlovic suggest to apply sophisticated brake models based on the actual
architecture of the train [49]. This allows to address the propagation of component
failures with respect to the topology of the system. Moreover, it is possible to introduce
stochastic phenomena at certain layers of the system hierarchy. They demonstrate that
the uncertainty of the overall braking process is increasing, if the COF of all brake units
is subject to a common systematic variance, e.g. due to a deviation caused by the batch
of the friction material applied for all pads of the vehicle [49]. In contrast, if the scatter
is simulated on component level, i.e. the COF at each brake unit is an independent
random variable, the vulnerability of vehicle with regard to uncertain friction properties
is less prominent. However, they do not provide values for the applied distributions.
Jaenichen and Kroenert suggest a brake model based on EN 14531-1 [50]. They assume
that the COF of each brake unit is an independent random variable and that the total
variance is thus reduced with an increasing number of brake units. For a passenger train
with disc brake units, they assume that the standard deviation of
µm
is 6% with respect
to a mean value of 0.34. For freight trains, they additionally consider the possibility that
different material qualities or batches may be applied in the vehicle. In this case, the
total variance results from a weighted sum of the single variances with respect to the
percentage of each material [50].
8The author assumes that the material type corresponds to the influence of the batch.
9In contrast, Malvezzi et al. propose a Gaussian distribution for µmed
µnwith 1σ=±5%.
33
2 State-of-the-art in railway brake technology
2.4 Friction in disc brake units
2.4.1 Fundamental laws of friction
Amontons laws of friction represent two fundamental observations related to the friction
between to two contacting bodies [80]
10
. First, friction forces are directly proportional to
the applied normal load FN
Ffric =µ·FN,(2.29)
where the COF, denoted by
µ
, represents the ratio between normal force and the tangential
friction force
Ffric
[80]. Moreover, friction forces are independent of the nominal contact
area [80]. Coulomb confirmed Amontons laws by extensive experimental investigations
[80]. In addition, he differentiates between static and kinetic friction and observes that
friction forces are independent of the sliding velocity. The fundamental friction laws of
Amontons and Coulomb are an exceptionally simple representation of the highly complex
friction process. This makes them very valuable. However, they represent only a very
rough approximation. Many engineering applications, such as disc brake units, may
deviate from these general laws.
2.4.2 The origin of friction in disc brake units
In fact, the development of brake pad materials is often described as an elaborate trial
and error procedure seeking for an ideal compromise of stable friction behavior and the
reduction of wear [81][30][82]. Furthermore, friction forces are related to undesirable
effects, such as brake squeal or the evolution hot spots on the disc surface [83][84].
Hence, increasing attention has been drawn in the past decades to understand the actual
processes acting in the frictional boundary layer. An extract of results based on tribologic
investigations are briefly discussed. To avoid going beyond the scope, the following results
focus on dry friction.
Based on a comprehensive analysis of pad surfaces topographies during experimental
brake tests, Erikkson and Jacobson introduced the theory of so-called contact plateaus
[83][58][85]. As indicated by the sketch in Figure 2.14, plateaus or also called patches
protrude from the pad surface and represent comparatively hard and grinded areas, which
are established between disc and pad during the braking process. The patches represent
about 10
−
20% of the nominal pad area. According to topography measurements, the
dimensions of the plateaus takes values between 50
−
1000
µm
, while they are only a
few microns high [85][86][87]. It is understood that the areas of true contact are located
10These investigations have already been conducted by Leonardo da Vinci in 15th century[80].
34
2.4 Friction in disc brake units
on the surfaces of these plateaus and that they carry the mechanical load between the
contacting bodies during braking [83][86]. The evolution of transmissible friction forces
and thus the resulting COF is therefore driven by the formation and disintegration of
plateaus as well as their properties.
Primary Plateau/Patch
Hard fiber (e.g. Steel)
Hard particel (e.g. Quarz)
Wear debris from disc and pad
Pad Binder Matrix
(e.g. phenolic resin)
Secondary Plateau/Patch
Sliding direction
Figure 2.14: Simplified sketch of pad surface topography based on [58] and [81].
Size and occurrence of plateaus result from micro- and macroscopic processes affected by
normal load, relative velocity, contact temperatures, wear debris and the properties of
contacting materials [85][81][87]. So-called primary plateaus originate from wear resistant
reinforcement additives, such as Quarz particles or metallic fibers, protruding at the
friction surface, as indicated by Figure 2.14. During the braking process wear particles
may agglomerate in front of primary plateaus or due-to self locking processes and form
so-called secondary plateaus based on thermo-mechanical processes [83][87][86]. It is
assumed that the temperature acting in the areas of true contact exceed 1000
◦C
and
evoke sinter processes leading to particle reinforcement [81]. The generation of secondary
plateaus increases the capability of the surface to transfer friction forces. Conversely,
these structures may disintegrate due to mechanical loads and wear. The resulting
gradual destruction or instantaneous brake-ups of plateaus reduce the amount and size
of stable contact areas [81][85]. Furthermore, compacted secondary plateaus might peel
off as soon as the normal load on the contact is reduced, e.g. during the release of the
caliper.
Primary and secondary plateaus may be covered with a thin top-layer. The so-called
tribofilm, also referred to as friction layer or third body, is primarily formed of oxidized
and compacted wear particles [58][88][82]. It is found that magnitude and stability
of friction forces are affected by the type and properties of the tribofilm [88][86]. As
discussed in [89], these properties are strongly related to the applied frictional additives
and the tribo-chemistry taking place in the frictional boundary layer. Although the
aforementioned investigations stem from organic pad materials, the findings also apply
for sintered brake pads applied in rail vehicles, as discussed in [90],[91],[92] and [93].
35
2 State-of-the-art in railway brake technology
In conclusion, the frictional properties between pad and disc are driven by the amount
and size of patches detectable on a mesoscopic scale as well as properties of microscopic
structures, such as tribofims. Accordingly, the actual material composition including
additives and reinforcement particles has a large influence on the resulting properties.
Moreover, the load case characterized by relative velocity, contact pressure and tempera-
ture evolution strongly influence the amount, sizes and properties of plateaus as well as
third bodies between pad and disc.
2.4.3 Friction phenomena observed in disc brake units
Based on topography investigations, assumptions have been made to describe common
phenomena related to friction. When comparing friction results, it needs to be consid-
ered that the investigated materials, the type of test equipment, such as Pin-On-Disc
Dynamometer (POD) or Full-Scale Dynamometer (FSD), as well as the investigated
load cases and braking scenarios, such as stopping brake or drag braking applications,
influence the observed friction behavior. Nevertheless, the following paragraphs try to
structure typical observations related to dry friction in disc brake units documented in
literature by the phenomena illustrated in Figure 2.15. Note that sticktion and stick-slip
effects, occurring at the very end of the braking process, will not be discussed.
Dynamic phenomena Load history phenomena
𝜇
Load Consecutive brake applications
Loadcase A Loadcase B
𝜇𝑚
Scatter
Instantaneous phenomena
𝜇
Time
Fade
Reset
Figure 2.15:
Simplified sketch of friction phenomena observed for brake units according to
[85],[30],[87],[86] and [91].
2.4.3.1 Instantaneous phenomena
Instantaneous phenomena outline the transient behavior of the COF during a single
brake application. For many brake applications, a continuous growth of friction forces is
observed [94][85][87]. Since the velocity decreases during stopping brake applications,
this in-stop friction increase is often related to a reciprocal correlation of the COF and
36
2.4 Friction in disc brake units
the sliding speed
µ
(
v
)[29][25][34]. Erikkson et al. presume that increasing friction forces
are evoked by the contamination and cleaning processes of the plateaus, thermal induced
changes of the surface properties, such the generation of secondary plateaus, as well as
the shape adaption between disc and pad during braking [83][58][85]. However, these
adaptions and structures are not necessarily stable and disintegrate as soon as the brakes
are released and the contact pressure vanishes. In consecutive brake applications, these
process need to restart and the COF at the beginning is smaller than at the end of the
previous brake application, as indicated by the Reset in Figure 2.15. If the established
topography changes are stable, the initial COF may rise from one to the next brake
application, as described in Section 2.4.3.3.
Ostermeyer and Wilkening show that the in-stop friction increase is less pronounced, if
initial velocity and normal load are increasing [87]. For high energy brake application, this
effect can lead to a decrease of instantaneous COF during braking, as shown Figure 2.15.
This behavior is also known as fading. From a tribological perspective, fading is commonly
explained by the pyrolytic disintegration of organic composites, such as the binder matrix
or organic fibers, and glass transition due to high temperatures (
≈
350
◦C
) [95][30][60][96].
The decomposition impairs the stability of the plateaus and thus their load carrying
capacity. Although sintered pads withstand higher temperatures (
≈
600
◦C
), they are
subject to temperature fading as well when they reach their performance limit, as
discussed by Zhang et al. [91][93].
2.4.3.2 Dynamic phenomena
Erikkson observed a hysteresis effect of the COF [58][85]. For certain pad materials
the COF is increasing during brake applications with decreasing normal load and vice
versa, i.e. the direction of the load change affects the change of the COF. It is assumed
that the resulting hysteresis is caused by dynamic nature of the topography changes, i.e.
the change of friction properties lags behind the rate of change of the external load [85].
This is a very important observation emphasizing that friction changes are related to
dynamic processes and an immediate change of external loads is generally not causing an
immediate change of instantaneous friction forces. Based on these findings, Ostermeyer
considers friction as a closed-loop dependency between heat, wear, the number and size
of patches and the resulting sliding resistance [38]. During braking the frictional contact
generates heat and wear. The resulting temperature increase and generated abrasion
particles drive the growth and destruction of contact patches, which in turn affect the
sliding resistance and thus the generation of heat and wear. According to Ostermeyer
et al., this tribo-system represents an equilibrium of flow driving the dynamics of the
pad topography [38][81][39][60]. If the rate of growth and patch destruction is equal,
the tribo-system is within a dynamic equilibrium and yields stable friction properties.
37
2 State-of-the-art in railway brake technology
This stationary state represents a constant coverage of the pad surface by plateaus with
consistent properties.
2.4.3.3 Load history phenomena
Load history phenomena are related to changes of the friction properties observed for
consecutive brake applications by applying the mean COF over multiple braking processes,
as showcased by the sketch on the very right in Figure 2.15. A common observation is
that the COF changes over a number of identical braking processes with equal initial
conditions until a more or less stable state is achieved [58][87][86]. This observation
is related to the adaption of the surface topography with respect to the load case. In
other words, the repetition of identical brake applications establishes a certain surface
structure, which corresponds to certain friction characteristics.
This grinding-in effect is exploited during so-called bedding or conditioning procedures.
Before applying newly manufactured pads in service or after extreme events, which
induce significant changes of the friction properties, brake pads are exposed to a certain
sequence of bedding brake applications to generate stable frictional properties suitable
for operation. Ostermeyer and Wilkening showed that the rate of the surface adaption
depends on the change of the load case [87]. They investigated that a switch from higher
to lower velocities yielded a longer adaption phase, i.e. more brake applications are
necessary to receive a stable behavior, than vice versa. Hence, they postulate that the
change of the surface structure proceeds faster for higher mechanical loads [87].
Neis et al. observed that the stability of the established surface structures depends on
the pad material. They found that a material with less metallic fibers tends to form
a smoother homogeneous surface based on the formation of secondary plateaus. The
corresponding variance of the COF observed after the bedding is smaller than the one
observed for a pad material with a larger content of reinforcement fibers, which act
aggressive in the frictional contact [86]. However, the material with a larger amount
of fibers yielded a larger COF compared to the material with less fibers. This reveals
opposing friction properties regarding the goal to increase the COF, while minimizing its
variance.
The aforementioned findings suggest that the stochastic nature of friction observed for
disc brakes comprise a steady-state scatter depending on the stability of the surface
topography as well as a dispersion evoked by the load history, i.e. the change of the
topography with respect to the diversity of applied load cases. Both effects generally
depend on the pad material.
38
2.5 Modeling the coefficient of friction
2.5 Modeling the coefficient of friction
As indicated by Figure 2.16, there are various parameters, which influence the COF and
lead to the previously discussed friction phenomena. The choice of the model approach
to describe a change of the COF during braking is thus driven by the phenomena to be
replicated and the influencing variables to be considered. Figure 2.16 names different
model approaches, while the selected classification is not adequately sharp and some
models may belong to several groups. In the following, static and dynamics approaches
as well as neural networks are briefly discussed. Since state observers are applied for
online prediction, they will not be further discussed. The same applies for distributed
models, which aim at estimating local friction properties along the pad surface and
require a large computational effort. For details to these approaches the reader is referred
to [97],[98],[61] and [30].
An extract of the discussed static and dynamic models is additionally given in Table
2.2. As shown in the table, the formulations originate from different applications, such
as automotive or rail vehicles, and brake types, such as tread, drum or disc brakes.
Furthermore, the parameters were identified for particular load cases and material types
as well as different experimental set-ups, such as FSD or POD. Hence, it needs to be
considered that the models are not generally valid.
Influences
Load history
Wear
Clamping force
Contact temperatures
Pressure
distribution
Relative
velocities
Moisture and
humidity
Impurities and contamination
Pad material composition
Disc material
Coefficient
of friction
In-stop increase
Observation
Grind-in
Scatter
Friction
phenomena
Fading
Dynamic
behavior
Prediction
Modelling
approaches
Static models
Dynamic models
Neural networks
Instantaneous
behavior
Time lagHysteresis
Load history
effects
State observers
Distributed models
Figure 2.16: Influences, phenomena and model approaches for the COF.
39
2 State-of-the-art in railway brake technology
2.5.1 Static models
Static models represent a deterministic relation between the COF and the influencing
variables expressed by an ordinary mathematical function. The formulation of Karwatzki
from 1955 is a common approach, which is still largely applied today [22][23][24][25][26].
He proposes a rational function to describe the reciprocal relation between
µ
and the
velocity as well as the normal load (see the first line in Table 2.2). Although the model was
originally developed for gray cast blocks, parameters have also been identified for different
materials and geometries as well as disc brake units [25][24]. In addition, there are
diverse modifications to consider additional influencing factors [99][26][23]. For instance,
Saumweber adds the rotor temperature as influencing variable [23]. The influence of the
temperature on the COF is also addressed by Rhee [100]. In comparison to Karwazki, he
proposes an exponential approach, as shown in Table 2.2.
Polynomial regression is another common static approach to correlate the instantaneous
behavior of the COF with its influencing variables. The application of the method
for railway brakes is discussed in [29],[35],[24] and [32]. An example of Cantone and
Ottati is given in Table 2.2. All regression approaches have in common that high degree
polynomials are necessary to estimate the complex behavior of the COF observed for
composite friction materials.
2.5.2 Dynamic models
Dynamic models consider friction as a state of a dynamic system governed by differential
equations. Classical approaches, e.g. the Dahl model or the LuGre model, originate
from control problems, such as friction compensation [101][102]. During the last decades
dynamic modeling techniques have also emerged for the performance assessment of brake
units. Loh et al. discuss the use of growth models to predict the in-stop friction increase
[94]. They derive an exponential function with respect to time
µ
(
t
). Yuan et al. take up
the dynamic approach and derive an exponential solution with respect to the velocity
µ
(
v
)[34]. Lee and Kang additionally consider the exponential influence of the disc surface
temperature within their model
µ
(
v, T
). Even though the latter is not explicitly specified
to be a dynamic model, it may be considered as the solution of a differential equation.
The models of Loh, Yuan and Lee are shown in Table 2.2.
As discussed in Section 2.4.3.2, Ostermeyer considers friction, or rather the sliding
resistance, as part of an equilibrium of flow within a dynamic tribo-system [38]. Based
on this theory, he proposes a generic and phenomenological model approach for the
derivative of the COF with respect to time [38][39][60]
dµ
dt =fg(t, ξ1(t), ξ2(t), ...)−fd(t, ξ3(t), ξ4(t), ...).(2.30)
40
2.5 Modeling the coefficient of friction
Equation 2.30 relates the change of the COF to growth and destruction processes of
patches during braking represented by the functions
fg
and
fd
, respectively. The processes
are driven by modulating variables
ξi
(
t
), such as temperature or wear stream. Each
influencing state is a dynamic quantities itself, which is coupled to other states of the
tribo-system. Hence, the dynamics of friction is part of a system of coupled differential
equations with respect to time. In contrast to Loh, Yuan and Lee, Ostermeyers model is
not stipulated to growth or decay, but allows to describe any kind of progression with
respect to the system states. It is therefore one of the most sophisticated models applied
for the friction of brake units. In [38] and [39], Ostermeyer proposes different forms of his
generic approach. A model, taking friction power and contact temperature into account
is given in Table 2.2, while the parameter identification was performed by Ricciardi et al..
Wan et al. propose an improved approach to predict the friction behavior observed for
repeated brake applications [40]. They also derive a closed solution for certain braking
scenarios. Ricciardi et al. also propose a sophisticated phenomenological approach, as
shown in Table 2.2 [41]. In addition to the contact area, they also include the dynamics
of the adhesion properties in their model.
2.5.3 Artificial Neural Networks
In the course of ever-increasing available computational performance, Artificial Neural
Networks (ANN) are also exploited to predict the friction properties of brake units. The
relation between output, i.e.
µ
, and input variables, i.e.
FC, v, T
, is captured by a
network of interconnected so-called neurons. From a simple mathematical perspective,
each neuron may be considered as a weighted sum, which transforms multiple inputs into
an output value. A neural network is thus a large system of coupled weighted sums, while
the weights represent the model parameters. Training the network means that a large set
of, usually experimentally gained, input/output data is exploited to tune the parameters
in a manner that the output of the network fits best to the measured output.
Ćirović and Aleksendrić compare different ANN architectures to predict the brake torque
with respect to initial speed and temperature [37]. The application of ANN to predict
the COF and the wear rate of a disc brake unit with respect to disc temperature, normal
pressure and sliding velocity is discussed by Yin et al. [103]. Senatore et al. correlate the
COF with the normal pressure, sliding speed and the sliding acceleration [104]. Mutlu
uses ANN to estimate the mean COF and its standard deviation based on the composition
of the pad material [105]. The latter work exemplifies, that ANN is a powerful method to
predict the COF with respect to any influence shown in Figure 2.16, if the corresponding
input/output data is available. As discussed by Ricciardi et al., a downfall of ANN is that
it is a black box, mapping input and output without any phenomenological background
[97]. The prediction quality of ANN is therefore strongly related to the amount and
quality of training data as well as the selected network architecture.
41
2 State-of-the-art in railway brake technology
Table 2.2:
Extract of deterministic friction models to predict the instantaneous COF of friction brakes applied in rail and
automotive vehicles. The limits of applicability of the models are given in the references. Column Dyno. specifies
whether the given parameters were identified using measurements from POD or FSD test rigs. - means that there
was no specification in the reference. Note that the symbols in the formulas of the table correspond to the ones
applied in the references, which may deviate from the glossary of this work.
Reference Application Friction material
Formula Influencing Variables
Dyno
Karwatzki-
Wende [25]
Railway
tread brakes
Grey cast block
Types P10/P14 µ= 0.5·1+0.016·FBL
1+0.032·FBL ·1+0.01·v
1+0.05·v
Vehicle velocity v[km
h]
Block force FBL[kN]-
Karwatzki-
Saumweber
[23]
Railway
tread brakes
Grey cast block
Types P10/P14 µ= 0.53 ·1+0.001·pBL
1+0.003·pBL ·1+0.01·v
1+0.03·v·1+0.001·T
1+0.004·T
Vehicle velocity v[km
h]
Block pressure pBL[N
cm2]
Wheel temperature T[K]
FSD
Karwatzki-
Wende [25]
Railway
disc brakes Pad type 5-6-60 µ= 0.415 ·1−0.041·FC
1−0.037·FC·1+0.025·v
1+0.03·v
Vehicle velocity v[km
h]
Clamping force FC[kN]-
Cantone
[35]
Railway
tread brakes
Composite block
Type LL (2xBg)
µ
= 0
.
128
−
2
.
467
·
10
−2·V
+3
.
589
·
10
−3·V2
+5
.
349
·
10
−2·V3
+4
.
921
·
10
−2·V4−
3
.
748
·
10
−2·V5−
3
.
684
·
10
−2·V6
+7
.
889
·
10
−3·V7
+7
.
251
·
10
−3·V8−
4
.
175
·
10−4·V9
Centralized vehicle velocity
V=v[km
h]−60
37.5
FSD
Yuan [34] Railway
disc brakes Organic pad µ= 0.365 + 0.0323 ·e−0.0502·vVehicle velocity v[km
h]FSD
Lee [33] Railway
disc brakes Organic pad µ
= 0
.
38
·
(0
.
184
·e−0.1·v
+ 1)
·
(0
.
105
·e−0.014·T
+ 1)
Sliding velocity v[m
s]
Disc temperature T[◦C]-
Rhee [100] Automotive
drum brakes
Organic pad µ=µ0·Fa
N·vb
with a(T)∈[0.8,1.25] and b(T)∈[−0.25,+0.25]
Sliding velocity v[rpm]
Normal Load FN[lb]
Drum temperature T[◦F]
POD
Loh [94] Automotive
disc brakes Low metallic pad
µ=a+b·e
−t
c
with a(FN) = e−1.16−0.0003·FN,
b(FN) = e−2.6−0.0081·FNand c(FN) = −0.54 −653
FN
Time t[s]
Normal Load FN[lbs]
POD
Ostermeyer-
Ricciardi
[41]
Automotive
disc brakes -dµ
dt =−1.123 ·10−5·FC·v·µ−40.473 ·µ+ 0.04 ·T
Sliding velocity v[m
s]
Clamping force FC[N]
Contact temperature
T
[
K
]
FSD
Ricciardi
[41]
Automotive
disc brakes -
µ=κ·α
with κ= 3.075 ·10−4·T+ 0.2687
and
dα
dt
= 10
−9·κ·α·σzz ·v
+ 2
.
302
·
10
−2·dv
dt −
1.226 ·10−10 ·σzz ·T
Sliding velocity v[m
s]
Sliding acceleration dv
dt [m
s2]
Normal stress σzz[N]
Disc temperature T[K]
FSD
42
3 Analysis of experimental data
The goal of this chapter is to characterize the frictional properties of a typical brake pad
material applied for disc brake units of rail vehicles within a representative operating
spectrum. Therefore, experimental data obtained from almost 2000 brake applications
conducted with a disc brake unit on a Full-Scale Dynamometer (FSD) test rig is exploited.
In the beginning, the test rig and measurement procedures are described. The main part
of this chapter comprises the presentation and discussion of measurement results, i.e. the
instantaneous COF with respect to time and velocity as well as the COF averaged over
the braking distance. Moreover, measurements of the surface temperature of the disc
will be analyzed and correlated to the friction properties. Finally, the key findings will
be summarized.
3.1 Description of experiments
3.1.1 Dynamometer test rig
The data analyzed in this work is provided by the Knorr-Bremse SfS GmbH. The company
operates test rigs to perform various braking scenarios with full-scale brake units under
operating loads, as shown in Figure 2.5. Figure 3.1 shows a sketch of the applied
dynamometer test rig to illustrate its working principle. As indicated, the brake unit is
attached to a rotating shaft representing the axle of the wheel-set. The shaft is connected
to an electric motor and an inertia dynamometer with adjustable fly wheels, which allow
to adapt the dynamic masses braked by a single brake unit during testing. The motor
accelerates the disc brake and contributing masses to the desired initial velocity of the
braking process to be simulated. Furthermore, the motor can be used to simulate an
additional rotating inertia or to act as an additional electro-dynamic brake. During the
simulation of a stopping brake process the pressurized caliper presses the pads against
the rotating disc to decelerate the dynamic masses until standstill. Detailed information
to dynamometer test rigs can be found in [106],[107] and [28].
43
3 Analysis of experimental data
Ineratia
Dynamometer
Adjustable
Flywheels
Electric motor
Clutches
Telemetry
system
Air supply
Air outlet
Bearing
Force
transducer
Frame Shaft
Test
chamber
Brake Unit
and Caliper
Figure 3.1: Simplified sketch of a full-scale dynamometer test rig FSD.
3.1.2 Friction measurement
Referring to Equation 2.6, the instantaneous COF
µ
(
t
), with respect to a predefined
friction radius
rfric
, is given by the ratio of the instantaneous brake torque
MB
(
t
)and
the clamping force FC
µ(t) = MB(t)
FC(t)·rfric
=FM(t)·lF
(pC(t)·AC−FS)·ηC·iT·rfric
.(3.1)
To measure the brake torque, the caliper unit is attached to a fixed frame. During
braking a load cell integrated within the frame records the counteracting force
FM
(
t
).
The brake torque is thus the product of lever lF, i.e. the distance between the load cell
and rotational axis of the disc, and
FM
. The clamping force is either directly measured
or indirectly deduced from measurements of the cylinder pressure
pC
. In this case, the
indirect method is applied. Since sensors and caliper characteristics are calibrated, larger
systematic measurement errors of µ(t)may be excluded.
However, parameters and quantities of Equation 3.1 are subject to an irreducible scatter
related to random errors of the measurement equipment and the uncertainty of the brake
parameters, which influence the repeatability of the friction measurements. To estimate
the resulting measurement-related scatter of the COF, fault propagation is conducted in
Appendix A.2.2. Since the same caliper unit and test-rig are applied for the entire testing
procedure,
AC
,
iT
and
lF
are excluded from the propagation
1
. The measurement range
of the cylinder pressure is directly deduced from the data (1σ=±0.0023 bar). For FM,
ηC
,
FS
and
rfric
, assumptions are made regarding their standard deviations, as shown in
1
In addition, the uncertainty of measurement amplifiers is not respected, since the necessary details
regarding the measurement chain were not available.
44
3.1 Description of experiments
Table A.3
2
. As discussed in Section 2.2, friction radius and COF represent integrated
values averaged over the pad surface. Hence, it is not possible to clearly distinguish
whether a change of friction forces is related to a change of the friction radius or the
actual frictional properties. Therefore, the propagation is conducted with and without
considering
rfric
as individual source of uncertainty (1
σrfric
= [0; 3
.
5%]). Based on the
assumption that the parameters of Equation 3.1 are independent random variables, the
fault propagation yields 1
σµ,error
= 0
.
006, if the friction radius is excluded as source of
uncertainty, and 1
σµ,error
= 0
.
014, if the scatter of the friction radius is included in the
fault propagation. This corresponds to 2% - 5% with respect to the nominal value of
µ
between 0.3and 0.4.
3.1.3 Additional measurements
In addition to the friction measurement, the test rig is equipped with sensors to measure
the evolution of temperature of the brake components as well as the rotational speed,
which is related to the simulated vehicle velocity according to Equation 2.12. To simulate
similar convection conditions prevailing during the movement of the vehicle, the brake
unit is vented by an air stream with respect to the simulated vehicle velocity, as specified
by EN 14535-5 [56]. The temperature evolution in both disc surfaces is measured by
three thermoelements positioned according to the left sketch in Figure 3.2 on each side of
the disc. As indicated by the right sketch, the elements are drilled approximately 1
mm
beneath the surface and connected to a telemetry system, which transmits the data from
the rotating shaft to a recording unit.
Measurement positions on disc surfaces
3
1,2
3,4
5,6
120°
Middle
circle
Integration of thermoelements within disc
Thermoelement
Telemetry
System
1mm
Thermoelement
Shaft
Wheel
Disc
Figure 3.2: Position of temperature sensors [E2].
2
Note, that the assumed uncertainties are related to this testing environment and specimen. In
particular within an operational environment, the values might strongly differ.
45
3 Analysis of experimental data
3.1.4 Brake disc and pad specimen
All measurements investigated in this chapter result from brake applications conducted
with a single disc brake unit assembled of caliper, a WMD made from gray cast
equipped with cooling fans on the backside and composite pads made from an
identical material. Geometries of the investigated brake pad and disc are in line with
the specifications given in EN 15328 and EN 14535-5 [55][56]. The parameters of the
investigated brake unit are given in Table 3.1.
The data basis comprises 1950 brake stopping applications resulting from a test
campaign with five different pad specimen all made from the same composite
material, hereinafter called Material A. As shown in Table 3.2, the brake applications
comprise 12 different load cases specified by initial velocities ranging between 100
km
h
and 160
km
h
, clamping forces between 33
kN
and 44
kN
as well as braked masses between
7
tons
and 9
tons
. Consequently, the resulting energy to be transformed varies between
2
MJ
and 9
MJ
, which corresponds to Class B1 according to Table 2.1. Prior bedding-in
brake applications conducted for each specimen are not considered in Table 3.2.
Table 3.1: Pad, disc and caliper parameters.
Parameter Value Parameter Value
Outer disc diameter 0.68m ηC0.97
Inner disc diameter 0.39m iT8.58
rwheel 0.42m AC0.01427m2
rfric 0.273m FS500N
Table 3.2: Number of brake applications performed for each load case and specimen.
Loadcase v0mdyn FCNumber of brake applications per specimen (Spec.)
[km
h] [kg] [kN]Spec.1 Spec.2 Spec.3 Spec.4 Spec.5 all
V0100_Case1 100 7250 33 40 20 40 40 40 180
V0100_Case2 100 8250 38 40 20 40 40 20 160
V0100_Case3 100 9050 44 40 20 40 39 20 159
V0120_Case1 120 7250 33 40 20 40 40 40 180
V0120_Case2 120 8250 38 40 20 40 40 20 160
V0120_Case3 120 9050 44 40 20 40 40 20 160
V0140_Case1 140 7250 33 36 18 36 36 37 163
V0140_Case2 140 8250 38 36 18 36 36 18 144
V0140_Case3 140 9050 44 36 18 36 36 18 144
V0160_Case1 160 7250 33 40 20 40 40 40 180
V0160_Case2 160 8250 38 40 20 40 40 20 160
V0160_Case3 160 9050 44 40 20 40 40 20 160
All load cases and specimens 1950
46
3.1 Description of experiments
3.1.5 Testing procedure and braking scenario
The test campaign is based on the performing an identical testing procedure multiple times
for each specimen. During each run of the procedure, 40 dry stopping brake applications
are conducted in a predefined order with respect to the initial velocity, while clamping
force and braked mass are the same for all 40 brake tests. Appendix A.2.1 exemplifies the
procedure for a run of Case 1 brake applications. As shown, the initial velocity changes
latest after two brake applications. To guarantee similar initial thermal conditions, the
brake disc is cooled by rotating with a constant velocity after each brake application
until Tfric falls below 60◦C.
All brake tests are stopping brake applications conducted with a constant cylinder pressure
and without an additional electro-dynamic brake, as exemplified in Figure 3.3. The
velocity decreases strictly monotonous to zero, while the COF and thus the deceleration
exhibits a highly non-linear progression during braking. At the beginning, transient
phenomena are observed for the clamping force and the COF, which are caused by the
dynamic response of the cylinder pressure due to the sudden pressure built-up. The
phenomena observed for the COF at the end of the braking process are probably related
to stick-slip effects. The temperature measurements (
T1−6
) correspond to the positions
illustrated in Figure 3.2. The records in Figure 3.3 demonstrate that the heating of the
disc surface strongly differs in radial direction. To quantify the disc temperature by a
single quantity, the empirical mean value of all sensors is calculated according to
Equation A.32, hereinafter called Tfric(t).
Figure 3.3:
Instantaneous progression of COF
µ
, clamping force
FC
, velocity
v
, local disc tem-
peratures
T1−6
and mean temperature
Tfric
during a stopping brake application.
47
3 Analysis of experimental data
3.2 Instantaneous coefficient of friction
3.2.1 Data processing
In the following, the instantaneous coefficient of friction is analyzed, while it is not
differentiated between the single specimen but the load case only. Due to the large
amount of friction curves to be compared, statistical moments are calculated for each
load case and time instant
tj
, as exemplified by Figure 3.4 for load case V0140_Case2.
The set of colored curves in the left plot visualizes all 144
µ
(
t
)-curves measured for this
load case. To quantify their average behavior, the temporary means values are calculated
according to
¯µ(tj) = 1
n
n
X
i=1
µi(tj).(3.2)
µi
(
tj
)represents the COF measured during the
ith
stopping brake application at time
instant
tj
.
n
corresponds to the sample size given by the column on the very right of
Table 3.2. The bolt black line in the right plot of Figure 3.4 represents the temporary
mean value
¯µ
(
t
)calculated for load case V0140_Case2. As shown, the single curves
µi
(
t
)strongly deviate from each other and the mean progression, although clamping
force, braked mass and initial velocity are identical. Whereas some curves show strong
fading within the first 10s, others increase in this time interval. Consequently,
µ
ranges
at
t
= 10
s
between 0.27 and 0.39. This corresponds to a deviation of -20% and +15%.
Figure 3.4:
Comparison of
µ
(
t
)of all 144 brake applications (left) measured during load case
V0140_Case2 and temporary mean value and range of variation (right).
48
3.2 Instantaneous coefficient of friction
To quantify this dispersion, the instantaneous empirical variance is additionally calculated
at each time instant according to
σµ2(tj) = 1
n−1
n
X
i=1
(µi(tj)−¯µ(tj))2,(3.3)
while
σ
represents the empirical standard deviation. The resulting temporal range of
variation is quantified by
¯µ
(
t
)
±
3
σ
[E2]. These envelope curves are depicted by thin
black lines in the right plot of Figure 3.4. As shown, most of the gray curves are located
within this interval.
Figure 3.5 illustrates mean values and envelope curves with respect to time of all 12 load
cases. Each row of plots corresponds to the same initial velocity. Each column contains
load cases with the same clamping force and braked mass. Comparing the plots reveals
that the initial velocity strongly influences the progression of the COF during braking.
For brake applications with 100
km
h
,
µ
(
t
)is subject to the typical in-stop friction increase
often observed in literature and discussed in Section 2.4.3. With increasing initial velocity,
the rise weakens and fading occurs at the beginning of the braking process. The observed
fading behavior coincides with the investigation made by Cordes and Zhang et al. for
composite pads of railway disc brakes, as discussed in Section 2.4.3. Compared to the
initial velocity, the influence of the clamping force and braked mass on the evolution of
the instantaneous COF seems to be less prominent. As indicated by the envelope curves,
µ
(
t
)is subject to a strong dispersion for all load cases. Hence, the braking processes are
subject to significant differences, although the braking scenario is identical.
Figure A.2 in Appendix A.2.3 shows the progression of friction curves with respect to the
velocity, while mean values and standard deviations are calculated for each velocity instant
vj
. The figure demonstrates that the non-linear and scenario-dependent progression of
the COF as well as its strong dispersion also prevails within the velocity domain.
In conclusion, Figure 3.5 and Figure A.2 reveal that the observed friction curves are a
superposition of a deterministic behavior and stochastic phenomena. The mean values
render the deterministic behavior and represent the scenario-dependent trend of the
friction curves. The stochastic phenomena are characterized by instantaneous standard
deviation and distribution. In the following, the instantaneous mean values, the standard
deviations and the distributions are analyzed in detail.
49
3 Analysis of experimental data
Figure 3.5:
Comparison of instantaneous COF of all brake applications measured for all load
cases including temporary mean values and ranges of variation [E2].
3.2.2 Instantaneous mean values
The plots in Figure 3.6 allow for a detailed comparison of the instantaneous mean
values, i.e. the deterministic behavior. Although the initial values
¯µ0
of each load case
are similar, the curves subsequently follow a certain non-linear characteristic behavior
primarily ruled by the initial velocity. During brake applications from 100
km
h
to standstill
¯µ
(
t
)progressively increases from 0.38 to 0.51. This corresponds to a change of 38% with
50
3.2 Instantaneous coefficient of friction
respect to the initial value. As discussed in Section 2.4.3.1, the growth indicates the
establishment of friction promoting secondary plateaus and tribofilms during the braking
process. The right plot reveals that the increase of friction forces with respect to the
velocity is similar for all load cases and starts as soon as the velocity falls below 100
km
h
.
This indicates a reciprocal correlation between velocity and patch growing processes.
Figure 3.6: Comparison of mean values ¯µ(t)of all load cases.
With increasing initial velocity, i.e. kinetic energy to be transformed, fading occurs at
the beginning of the braking process. For
v0
= 160
km
h
,
¯µ
(
t
)decreases between 15% and
20% within the first 20s. Due to the high relative velocity and thus large friction power
transferred within the contact zones, the frictional boundary layer is exposed to large
shear forces. Moreover, the heating of the materials due to the dissipated energy during
the braking process might affect the stability of the patches. It is presumed that the
resulting large thermo-mechanical loads promote patch destruction processes and thus
the fading of
¯µ
. After about on third of the braking process, the curves start to rise
again, i.e. plateau growing processes and the generation of friction-promoting tribofilms
are becoming predominant.
Figure 3.6 also demonstrates that higher clamping forces yield a reduction of the COF.
This observation can be explained by the resulting increasing thermo-mechanical loads in
the contacting areas, which evoke destruction processes. As shown, this effect becomes
more severe with higher initial velocities and applies in particular for Case 1 and Case 2,
whereas only minor differences between the mean values of Case 2 and Case 3 are visible,
although the differences of the clamping force are similar. This non-linearity additionally
emphasizes the essential role of the friction power regarding the change of the COF.
51
3 Analysis of experimental data
3.2.3 Instantaneous standard deviation
Figure 3.7 illustrates the standard deviation over time and velocity. The figure reveals
significant changes of the instantaneous dispersion during the braking process. The
standard deviation of the initial values ranges between 1
σµ0
= 0
.
008
−
0
.
012. From
there, it increases up to 0
.
03 in the time domain. This range corresponds to 3%-8% with
respect to the initial mean value.
σµ
(
v
)takes somewhat smaller values. For
v0≥
140
km
h
,
the standard deviation is subject to a sudden increase at the beginning of the braking
process within a velocity interval between 150
km
h
and 100
km
h
. Afterwards the dispersion
attenuates in the time domain or even declines in the velocity domain. For load cases
with
v0≤
120
km
h
, it follows a continuous increase throughout time and velocity domain.
Figure 3.8 lists potential sources of uncertainty, which are assumed to contribute to the
observed dispersion. As indicated, it is differentiated between epistemic sources, also
referred to as reducible uncertainties, and aleatoric sources, also referred to as irreducible
uncertainties. In the following, the impact of the different types is discussed.
Figure 3.7: Comparison of standard deviation σµ(t)and σµ(v)of all load cases.
3.2.3.1 Epistemic sources
Measurement and estimation errors are potential sources of epistemic uncertainties. In
fact, the standard deviation of the initial value
σµ0
corresponds well to
σµ,error
estimated
by the fault propagation (see Appendix A.2.2). This suggests that the scatter observed
at the beginning of the braking process is partially related to the uncertainty of the
52
3.2 Instantaneous coefficient of friction
measurement equipment. However, this source does not explain the strong increase of
the variance during braking. Due to the large sample size, the impact of estimation error.
i.e. the confidence interval related to the estimation of the empirical standard deviation
(see Appendix A.2.6), is not of significance for the investigated data. However, for testing
procedures with less repetitions, such as the program suggested by EN 15328, this source
becomes an important factor to be considered.
Uncertainty observed on test-rig 𝝈𝝁
Epistemic uncertainty (reducible) Aleatoric uncertainty (irreducible)
Measurement uncertainty Estimation uncertainty Difference of specimen Stochastics of friction
Accuracy of measurement
equipment
Number of brake tests
per load case
Production tolerances and
composition of batches
Load history and
material type
Figure 3.8: Potential causes for the dispersion of the COF observed on the test rig.
3.2.3.2 Aleatoric sources
The difference of the specimen due to individual composition of the material and tolerances
of the manufacturing process represents a source of aleatoric uncertainty. In particular,
different batches can result in significant differences of the friction properties [49][50]. The
figures A.3 - A.6 in Appendix A.2.4 compare instantaneous mean values and standard
deviations of the single specimen for all load cases. As shown, the mean values of the
single specimen coincide mostly very well. Apart from minor deviations for load cases
with 100
km
h
, the standard deviation gained from all specimen at once is not exceeding
the ones of the single specimen. Hence, this source of uncertainty is hardly contributing
to the observed dispersion. Moreover, the general good accordance suggests that there
are no larger differences between material compositions of the single specimens. This
indicates that they originate from the same batch3.
Since the previously discussed epistemic and aleatoric sources of uncertainty play a
subordinate role, it is assumed that the stochastic nature of the friction process is mainly
responsible for the strong rise of the dispersion during the braking process shown in
Figure 3.7 and thus represents the major contributor to the observed dispersion. The
stochastic nature of the friction process is also reflected in the left plot of Figure 3.4. As
illustrated, each friction curve follows a mostly smooth and unique course, although the
braking scenario is identical. In particular within the first 10 seconds, the curves are
3However, this assumption could not be proved based on the provided testing documentation.
53
3 Analysis of experimental data
subject to large differences, which results in the sudden increase of the dispersion visible
in Figure 3.7 for load cases with
v0≥
140
km
h
. This implies that especially the friction
processes related to fading are subject to a strong scatter.
3.2.3.3 Interpretation
The findings propose to consider the mechanisms acting in the frictional boundary layer
as random processes. According to the results of the literature study presented Section
2.4.3.3, the extent of scatter related to these processes is strongly correlated to the pad
material and the load history, i.e. the dispersion of the friction curves depends
on the sequence of the test program. In this case, the initial velocity changes
latest after every second brake application, as shown in Appendix A.2.1. Hence, the pad
topography is subject to permanent modifications, which amplify the variance between
the friction curves. Conversely, the dispersion is expected to decreases, if a series of
identical brake tests is conducted, due to the establishment of a stable topography.
It needs to be mentioned that the brake tests were conducted within an laboratory
environment. Under actual operating conditions, moisture, air humidity, impurities
and wear are additional sources that can contribute to the dispersion of the frictional
properties. Nevertheless, the findings emphasize that friction forces are subject to
significant scatter even within an idealized test environment.
3.2.4 Instantaneous distribution
The aforementioned findings motivate to take a closer look at the distribution of the
instantaneous friction curves. Therefore, samples of the friction values are analyzed at
different time instants
tj
= 5
,
15
,
25 and 35
s
, as exemplified by Figure 3.9 for load case
V0140_Case2 and
tj
= 25
s
. The right plot shows a histogram representing the scaled
relative frequency of the 144
µi
-values at this time instant. In addition, the theoretical
probability density function of a Gaussian distribution based on the empirical mean value
¯µ(tj= 25s)and standard deviation σµ(tj= 25s)is shown.
Comparing the histogram with the density function suggests that the values are normally
distributed. To verify this assumption, a
χ2
-test (Chi-squared-test) is performed. The null
hypothesis of this statistical test states that the sample follows a Gaussian distribution
based on the estimated parameters
¯µ
(
tj
= 25
s
) = 0
.
364 and 1
σµ
(
tj
= 25
s
) = 0
.
023. The
level of significance chosen to reject the null hypothesis is
α
= 0
.
05 (5%). By categorizing
the data into the 10 bins shown in the histogram, the test statistic yields
Pχ2
= 0
.
47.
Since this so-called P-value is larger than
α
, the null hypothesis is confirmed and the
assumption of a Gaussian distribution is valid.
54
3.2 Instantaneous coefficient of friction
Figure 3.9:
Comparison of
µ
(
t
)of all 144 brake applications (left) measured during load case
V0140_Case2 and temporary histogram of µ(tj= 25s)(right).
Applying
χ2
-tests for the samples of all load cases at
tj
= 5
,
15
,
25 and 35
s
yields the
P-values summarized in Table 3.3. As demonstrated, the majority of samples is normally
distributed. Exceptions are found for
v0
= 160
km
h
and
t >
15
s
. A closer look at these
samples reveals that they are subject to a skewed distribution, as exemplified by Figure
A.7 in Appendix A.2.5. It is remarkable, that the findings regarding the distributions in
the time domain are also true for the velocity domain, as demonstrated by the analysis of
samples at
vj
= 10
,
50
,
90 and 130
km
h
in Table A.4 and Figure A.8 of Appendix A.2.5.
Table 3.3:
P-values resulting from
χ2
-tests conducted for samples of
µi
(
t
)at
tj
= 5
,
15
,
25 and
35
s
gained from all load cases. Grey cells indicate that
Pχ2<
0
.
05, i.e. the null
hypothesis is rejected with respect to α= 5%.
Loadcase P-values of µi(t)samples at
tj= 5s tj= 15s tj= 25s tj= 35s
V0100_Case1
0.34 0.49 - -
V0100_Case2
0.06 0.13 - -
V0100_Case3
0.06 0.20 - -
V0120_Case1
0.12 0.77 0.35 -
V0120_Case2
0.76 0.65 0.53 -
V0120_Case3
0.20 0.10 0.53 -
V0140_Case1
0.40 0.33 0.57 0.88
V0140_Case2
0.00 0.16 0.47 0.06
V0140_Case3
0.07 0.10 0.18 0.68
V0160_Case1
0.46 0.00 0.00 0.00
V0160_Case2
0.69 0.00 0.00 0.00
V0160_Case3
0.33 0.07 0.00 0.00
55
3 Analysis of experimental data
3.3 Friction coefficient averaged over braking distance
In the following, it is analyzed how the instantaneous behavior affects the COF averaged
over the braking distance
µm
. Initially,
µm,i
is calculated for each brake application
i
according to Equation 2.26. This yields 1950 single values. To characterize the influence
of initial velocity, clamping force and braked mass, the
µm
values are divided into samples
according to each load case. A comparison of the resulting samples is depicted by the
box-plots in Figure 3.104.
Figure 3.10: Comparison of µmsamples by box plots for each load case.
The plot reveals the large influence of the load case on the samples of
µm
. As shown,
the medians decrease with increasing initial velocity as well as increasing clamping force
and braked mass. Furthermore, the samples are subject to a strong dispersion, although
the load case is identical. As indicated by the interquartile ranges and whiskers, the
dispersion tends to increase for larger initial velocities. The notches of the boxes reveal
that most of the medians significantly differ from each other. These observations suggest
that
µm
is a random variable characterized by a unique distribution for each load case.
4
The gray boxes of each box plot envelope 50% of all observations, i.e. the so-called interquartile range
between 25th and 75th percentile. The central red line marks the median of a sample and the dashed
whiskers represent the maximum and minimum normal observations. Non-normal outliers marked
as red +signs are those values, which are located more than 1.5 times the interquartile range with
respect to the 25th and 75th percentiles. If the notches of two samples do not overlap, the medians
differ from each other with respect to a level of significance of 5% [108].
56
3.3 Friction coefficient averaged over braking distance
Therefore, central statistical moments, i.e. empirical mean values
¯µm
and variances of
σµm
2
, are calculated for each load case according to Equation A.32 and Equation A.34 in
Appendix A.2.6. The sample sizes correspond to the column on the very right of Table
3.2. In addition,
¯µm
and
σµm
are calculated from a sample comprising all 1950
µm
values,
i.e. all brake applications and load cases. The results are visualized in Figure 3.11 and
summarized in Table 3.4.
Figure 3.11:
Comparison of empirical mean values
¯µm
and standard deviations
σµm
from
samples of µmvalues of each load case.
3.3.1 Mean values
As shown by the left plot in Figure 3.6,
¯µm
ranges between 0.31 and 0.41. The extent
of the interval corresponds to
−
11% and +15% with respect to mean value of all brake
applications (
¯µm
= 0
.
364)
5
. The extent of this interval demonstrates the strong relation
between load case and mean friction forces. To take account for the systematic behavior
of
¯µm
within the brake assessment, it is possible to apply regression models, as proposed
by EN 15328 and Sawczuk [55] [109]. Based on the data given in Table 3.4, it is found
that the correlation
¯µm
(
v0
)with respect to a certain clamping force and braked mass is
very well rendered by a polynomial regression curve of the second order, as shown by
the upper plot of Figure A.9 in Appendix A.2.7. For higher clamping forces and braked
masses, the regression curve is shifted along the y-axis to lower values.
5
Note, that this value is also affected by the number of brake applications performed for each load case.
57
3 Analysis of experimental data
Table 3.4:
Comparison of statistical quantities calculated for
µm
samples for all load cases.
Gray cells in the last column indicate that
Pχ2<
0
.
05, i.e. the sample is not
normally distributed with respect to α= 5%.
Load case v0[
km h]mdyn[kg]FC[kN]Sample size n¯µm1σµm
1σµm
¯µm[%] Pχ2
V0100_Case1 100 7250 33 180 0.405 0.010 2.5 0.40
V0100_Case2 100 8250 38 160 0.398 0.008 2.0 0.02
V0100_Case3 100 9050 44 159 0.393 0.009 2.3 0.16
V0120_Case1 120 7250 33 180 0.391 0.011 2.9 0.27
V0120_Case2 120 8250 38 160 0.378 0.012 3.2 0.87
V0120_Case3 120 9050 44 160 0.373 0.013 3.5 0.20
V0140_Case1 140 7250 33 163 0.366 0.017 4.7 0.20
V0140_Case2 140 8250 38 144 0.352 0.016 4.5 0.20
V0140_Case3 140 9050 44 144 0.343 0.015 4.4 0.24
V0160_Case1 160 7250 33 180 0.336 0.019 5.5 0.00
V0160_Case2 160 8250 38 160 0.318 0.017 5.2 0.03
V0160_Case3 160 9050 44 160 0.310 0.016 5.2 0.00
All load cases 1950 0.364 0.034 9.2 0.00
3.3.2 Standard deviation
As shown by the right plot of Figure 3.11,
σµm
increases for higher initial velocities and
ranges between 2% and 6% with respect to the corresponding mean values (see also
Table 3.4). The increase is explained by the sudden rise of the instantaneous standard
deviation at the beginning of braking processes with
v0≥
140
km
h
, as shown in Figure 3.7.
In addition, the bars illustrate that load cases with smaller clamping forces and braked
masses tend to higher standard deviations (except for load cases with
v0
= 120
km
h
). This
could be caused by the larger dispersion of the initial value
µ0
, shown in Figure 3.7. The
correlation
σm
(
v0
)with respect to a certain clamping force and braked mass can also be
captured by a polynomial regression curve of the second order, as shown in the lower
plot of Figure A.9 in Appendix A.2.7.
Moreover, the dark gray bar in the right plot of Figure 3.11 illustrates that
σµm
of the
sample considering all load cases at once is significantly larger than the bars of the single
load cases. This is caused by the large systematic differences between the single samples.
Hence, the characterization of
µm
by a single random variable irrespective of the load case
requires to consider a standard deviation of 1
σµm
=
±
9
.
2%. Conversely, the distinction
between load cases strongly reduces the standard deviation to be considered for µm.
58
3.4 Surface temperature of the disc
3.3.3 Distribution
To analyze whether the samples depicted in Figure 3.10 follow a Gaussian distribution
χ2-tests are conducted. The results are shown in the column on the very right of Table
3.4. As indicated by the colors of the cells, most samples do not significantly differ from
a Gaussian distribution. Exceptions are found for samples with
v0
= 160
km
h
and the
sample comprising all load cases. A comparison of histograms for each sample is given in
Figure A.10 of Appendix A.2.8. The histograms reveal that the distribution of
µm
tend
to a skewed distribution for
v0
= 160
km
h
. This observation corresponds to the analysis
of the instantaneous curves in Section 3.2.4. Due to the systematic differences between
the means of the single load cases, the distribution of the sample comprising all brake
applications strongly deviates from a normal distribution, as shown by the subplot at
the very bottom of Figure A.10.
3.4 Surface temperature of the disc
3.4.1 Mean values and variance
Figure 3.12 displays the average temperature of the six sensors
Tfric
(
t
)in analogy to
the illustration in Figure 3.5. The figure reveals the strong relation between the power
consumption of the load case and the temperature evolution at the surface of the disc.
During high speed brake applications with
v0
= 160
km
h
the average temperature rises up
to 350
◦C
. The instantaneous mean values
¯
Tfric
(
t
)follow a similar nonlinear progression
with a maximum in the second half of the braking process. At the beginning of the
braking process, the temperature at the disc surface strongly increases due to the large
heat input resulting from the high friction power. The evolving temperature gradient
stimulates conduction and convection processes, which dissipate the heat within in disc
and pad. Simultaneously, the incoming friction power decreases during the stopping
brake process due to the decrease of the velocity. Consequently, the temperature rise at
the surface weakens and finally fades towards the end of the braking process.
The large ranges of variation
¯
Tfric
(
t
)
±
3
σTfric
(
t
)depicted in Figure 3.12 reveal significant
differences regarding the temperature evolution, although the load case is identical and
similar initial thermal conditions prevail
6
. For load case V0160_Case2, the interval at 10s
ranges between 140
◦C
and 290
◦C
. This corresponds to a deviation of
±
35%. However,
the spreading is strongly reduced towards the end. Consequently, the average surface
temperature is also subject to a superposition of deterministic and, in particular at the
beginning of the braking process, stochastic effects.
6
Note, that the single brake applications begin as soon as the mean temperature of all sensors falls
below 60 ◦C, as shown in Appendix A.2.1.
59
3 Analysis of experimental data
Figure 3.12:
Comparison of averaged surface temperature
Tfric
(
t
)of all brake applications
including mean values and range of variation [E2].
Tfric
represents the average of the surface temperatures measured at the six sensor
positions described in Figure 3.2. The local temperatures might strongly differ from
Tfric
.
Figure A.11 and Figure A.12 in Appendix A.2.9 illustrate mean values and standard
deviations of the signals recorded by each sensor. It is found that the radial position
has a large influence on local temperature evolution. According to Figure A.11, Sensor
3 and 4, i.e. the sensors positioned in the middle circle, tend to a significantly lower
temperature compared to the ones measured at the inner and outer positions. This
observation indicates the occurrence of so-called heat bands or hot spots during braking,
60
3.4 Surface temperature of the disc
as described in [110] and [111]. Figure A.12 reveals that the standard deviations of the
local temperatures calculated with respect to all conducted brake applications is also
strongly affected by the radial position of the sensor. In particular, the temperatures
measured in the middle circle of the disc are subject to a large dispersion. This observation
indicates that the heat bands and hot spots move along the surface during repeated
brake applications. The investigations suggest that the large dispersion observed for
Tfric
has two major sources, namely the radial temperature distribution at the disc surface
as well as the continual modification of this distribution evoked by the alternating load
history.
3.4.2 Correlation of surface temperature and friction
Figure 3.13 illustrates the correlation between friction properties and the surface temper-
ature of the disc. A linear reciprocal trend is recognizable as long as the temperature
rises, in particular for brake applications with higher initial velocities. This represents
the previously described fading effect. However, as soon as the temperature decreases,
¯µ
strongly increases and does not follow the same trend backwards, i.e. a strong hysteresis
is emerging. Moreover, the shape of the hysteresis depends on the load case. It is
remarkable that the COF starts to increase as soon as the temperature decreases. This
behavior indicates that friction and surface temperature are mutual coupled states within
a dynamic system. The finding coincides with the investigation of Ostermeyer, who
considers the tribo-system as closed loop dependency between heat, wear, the number
and size of patches and the resulting sliding resistance, as discussed in Section 2.4.3.2.
Figure 3.13: Comparison of ¯µ(¯
TFric)for each load case.
61
3 Analysis of experimental data
3.5 Key findings of the data analysis
The data analysis of the test rig measurements demonstrates that both friction forces and
temperature are ruled by a superposition of deterministic and stochastic phenomena. The
characterization of the investigated pad material is briefly summarized by the following
items. Note that the findings are related to stopping brake scenarios conducted with a
single disc brake within the alternating test program given in Appendix A.2.1.
•
Instantaneous friction forces follow a non-linear progression in the time
and velocity domain:
–
For
v0
= 100
km
h
, a large progressive in-stop friction increase yields a growth of
µ
(
t
)of nearly 40%. In contrast, for
v0≥
140
km
h
, fading occurs within the first
20 seconds, which leads to a temporary reduction of µ(t)by up to 20%.
–
For
v0≥
120
km
h
, increasing clamping forces yield a reduction of
µ
(
t
). This
effect is saturated for FC>38 kN.
–µm
varies between 0.41 and 0.31 and decreases with increasing initial velocity,
clamping force and braked mass. The correlation of
µm
(
v0
)may be rendered
by polynomial regression of the 2nd order.
•
Friction forces are subject to a strong dispersion, although the load case
is identical:
–
The instantaneous standard deviation is time- and velocity-variant and 3
σµ
takes values between 5% and 25% with respect to the initial value.
–
The diversity of the fading behavior is a major contributor to the dispersion
of the COF, in particular for high energy brake applications. Moreover, it is
presumed that the extent of the variance is correlated to the load history.
–
In general, the instantaneous distributions of friction curves follow a Gaussian
Distribution in time and velocity domain. This observation is also true for
µm-samples sorted according to each load case.
–
The standard deviation of
µm
grows for load cases with larger initial velocities,
while 3
σµm
ranges between 6% and 17% with respect to
¯µm
. A sample
comprising all
µm
-values irrespective of the load case is subject to a significantly
larger dispersion and strongly deviates from a Gaussian Distribution.
•
The investigated pad specimen show identical deterministic and stochas-
tic friction behavior.
•
The data analysis suggests a correlation between friction and average
disc surface temperature.
62
4 Development of a brake model for
disc brake units
The measurements presented in the previous chapter emphasize that friction forces
applied by a single disc brake unit are strongly related to the load case specified by
initial velocity, clamping force and braked mass. Moreover, this deterministic behavior
is superimposed by stochastic phenomena. Accounting for these phenomena within the
brake performance prediction requires a friction model that is capable to render
•
the deterministic behavior, i.e. instantaneous mean friction forces with respect to
the load case, as well as
•the stochastic behavior, i.e. the dispersion of the friction curves.
In the following, a new friction model is derived based on the findings of the literature
study and the comprehensive data analysis conducted in Chapter 2 and 3. To apply the
friction model for the brake performance prediction, an additional temperature model is
developed that estimates the temperature prevailing at the surface of the disc during
braking. Subsequently, friction and temperature model are coupled to simulate brake
scenarios on a virtual test rig. By comparing the simulated results to experimental data,
the deterministic brake model is verified regarding its prediction accuracy. To additionally
consider the dispersion of friction forces in the prognosis, a stochastic model approach
based on uncertain parameters is derived and identified. A MCS is conducted to verify
the results of this stochastic brake model. Finally, the results are concluded.
4.1 Deterministic friction model
4.1.1 Selection of model approach
To render the deterministic phenomena of the investigated pad material within a repre-
sentative operative spectrum, a suitable model needs to map both
•the friction increase observed for initial velocities ≤120km
has well as
•
the fading at the beginning of high energy braking processes with larger velocities.
63
4 Development of a brake model for disc brake units
Accordingly, models which solely predict an in-stop friction increase during braking, such
as the approaches of Karwazki, Yuan or Loh presented in Section 2.5, are not suitable,
since they only cover the lower part of the load spectrum. In contrast, the models of
Saumweber, Lee and Rhee suggest a correlation between rotor temperature and friction
forces to additionally render temperature-related fading [23][33][100]. However, only the
model of Lee was identified for disc brakes, whereas the others were developed for tread
and drum brake units, as shown in Table 2.2. Accordingly, their applicability for the
investigated friction pairing has to be questioned. A downfall of Lees approach is that it
has not been validated by vehicle brake tests.
Due to the lack of any phenomenological background, neural networks shall not be
applied in this work. On the one hand, these models do not contribute to a deepened
understanding of friction processes. On the other hand, their pure imitative approach
makes them prone to errors when interpolating and extrapolating to load cases not used
for identification. This also applies for polynomial regression. As shown by Cantone,
Bing and Hendrichs, complex and high degree polynomials are required to render the
non-linear characteristics of friction brakes used in rail vehicles [35][24][29]. Moreover,
Cantone and Hendrichs propose to use a single set of parameters for each load case.
These shortcomings complicate the model-based characterization of a brake pad within
its entire operative spectrum.
With respect to the results of the data analysis and the thorough literature review
on friction theory, it is presumed that a phenomenlogical modeling approach is more
promising to predict the friction behavior observed on the test rig than the purely
empirical models just mentioned. In [39], Ostermeyer and Bode propose a differential
equation to simulate the dynamics of processes prevailing within the frictional boundary
layer during braking:
dµ
dt =−c1·µ·FC·vfric −c2·µ−c3·µ·TD+c4·ψ+c5·TP.(4.1)
According to this equation, the derivative of the COF with respect to time. i.e. the
change of
µ
, is related to different influencing variables, i.e. dynamic states, of the
tribo-system. These are
•friction power µ(t)·FC(t)·vfric(t),
•disc temperature TD(t),
•the amount of wear particles ψ(t),
•the patch temperature TP(t)and
•µ(t)itself.
64
4.1 Deterministic friction model
The constants
ci
in Equation 4.1 represent the weighting of these influences. As indicated
by the positive signs, the presence of wear and the patch temperature, i.e. the temperature
prevailing in the areas of true contact, are considered to promote the growth of contact
patches and thus the rise of
µ
(
t
), due to the thermally induced reinforcement of particles.
The other terms describe processes driving the reduction of the COF. Firstly, it is
presumed that larger plateaus are less stable and an increasing COF is related to
reinforced decomposition of patches. This saturation effect is expressed by the term
−c2·µ
. Moreover, it is suggested that larger relative velocities and surface pressures
prevailing in the contact evoke larger mechanical loads on the load carrying structures.
Therefore, the term
−c1·µ·FC·vfric
relates the destruction of patches with the magnitude
of the friction power. Finally, the model considers the temperature fading caused by the
heating of load carrying components in the composite material by the term
−c3·µ·TD
.
Note, that Ostermeyer and Bode assume a dynamic coupling between disc temperature
and COF, since they observe a lag behavior of the temperature fading [39].
The underlying theory of Equation 4.1 coincides very well with the observations presented
in the previous chapter. In particular, the correlation between friction power as well
as disc temperature and the COF is confirmed by the results of the data analysis (see
Section 3.2.2 and 3.4.2). Moreover, the differential approach of Ostermeyer and Bode is
not stipulated to map rising or fading of the COF during braking, but allows to predict
either behavior depending on the course of the influencing variables. Due to these reasons,
it is decided to derive a new friction model suitable for railway disc brakes based on
Equation 4.1. This requires to adapt the equation with respect to the available input
data and observations on the test rig as well as to to identify a set of model parameters,
which is valid for the investigated friction pairing and load spectrum. Model synthesis
and identification are presented in the following.
4.1.2 Model synthesis
To transfer the selected model approach to the railway friction pairing and load cases
investigated in this work, the following adaptions of Equation 4.1 are proposed:
1.
The influencing variables are scaled by a reference value to get similar magnitudes
for the coefficients and the same units for each term.
2.
It is assumed that the amount of wear particles and patch temperature is propor-
tional to the magnitude of the clamping force:
ψ
+
TP∼FC
. This means that an
increasing clamping force promotes the establishment of patches.
3. It is assumed that the influence of each term is in general non-linear.
4.
It is assumed that the temperature fading is proportional to the average temperature
of the disc surface Tfric, as visualized by Figure 3.13.
65
4 Development of a brake model for disc brake units
Applying these adaptions yields the following governing equations [E1]:
dµ
dt =−c1· µ·FC·vfric
Pref !e1
−c2·µe2−c3· µ·TFric
Tref !e3
+c4· FC
Fref !e4
(4.2)
The input variables are
FC
(
t
)[
N
],
vfric
(
t
)[
m
s
]and
Tfric
(
t
)[
◦C
], while the coefficients
c1
,
c2
,
c3
and
c4
with the unit [
1
s
]as well as the unitless exponents
e1
,
e2
,
e3
and
e4
constitute the
parameters of the model. The coefficients
ci
weight each term regarding its contribution
to
dµ
dt
, while the exponents
ei
express the extent of non-linearity.
Pref
,
Tref
and
Fref
are
the scaling constants. The prediction of the instantaneous COF based on Equation 4.2
requires to solve the initial value problem
µ(t) = µ0+Z˙µdt, (4.3)
while
µ0
represents the initial value at the beginning of the braking process. Consequently,
µ0
is a further model parameter. In [40], Wan et al. derive a solution for Equation 4.3
based on a similar differential friction model. However, they investigate drag braking
scenarios, i.e brake applications with constant loads and friction velocity. In case of the
stopping brake scenarios investigated in this work, input power and friction velocity as
well as disc temperature are time-variant and non-linear quantities, as shown in Figure
3.3. Therefore, a general closed solution of Equation 4.3 is not existing and numerical
integration needs to be applied.
4.1.3 Model identification
To predict the mean behavior
¯µmeas
(
t
)observed for the investigated pad material and
brake scenarios illustrated in Figure 3.6, a suitable set of model parameters
p
p= (c1, c2, c3, c4, e1, e2, e3, e4, µ0)T(4.4)
needs to be identified
1
. The time series of the input variables required for the identification
are obtained from the experimental data. Therefore, instantaneous mean values of
clamping force, friction velocity and disc surface temperature are calculated for each of
the 12 load cases in analogy to Equation 3.2 based on the data recorded at the test rig.
For load case i, the input uiis thus
1
To ease the comparison between measurement and simulation, the subscripts meas and sim are
introduced.
66
4.1 Deterministic friction model
ui(t) =
¯
FC,meas,i(t)
¯vfric,meas,i(t)
¯
Tfric,meas,i(t)
.(4.5)
The reference values of the model are set according to Table 4.1. The selection of these
values is mainly influenced by the magnitudes of the input variables with respect to the
investigated load cases. Hence, they are not related to a certain pad material2.
Table 4.1: Reference values of the friction model.
Scaling constant Value
Pref 105W
Tref 200◦C
Fref 5.5·104N
Equations 4.2 and 4.3 are implemented by the physical modeling language Modelica
(Version 3.2.3 [112]) within the numerical simulation environment Dymola (Version 2020
[113] [114]). The simulated instantaneous COF with respect to a certain set of parameters,
model input and load case
µsim,i
(
p,ui
(
t
)
, t
)is calculated by numerical integration of
Equation 4.3. This is conducted by applying the DASSL solver implemented in Dymola
with an error tolerance of 0.001. Details of the solver are given in [115].
The identification of model parameters is conducted by fitting the simulated time series
of the COF
µsim,i
(
t
)to the measured mean values
¯µmeas,i
(
t
). This means to find the set
of parameters, which yields the minimal deviation between measurements and simulation
results. Initially, the coefficients and exponents are manually tuned by comparing the
qualitative progressions of measured and simulated COF for all load cases. As exemplified
by Figures A.13 in Appendix A.3.1, the manually tuned model already provides a good
agreement with the measurements, in particular for load case V0100_Case1. The
manually adjusted parameter values are shown in Table 4.2 (see column
pstart).
To obtain the best possible match between measurement and simulation, a multi-case
optimization is conducted, while the manually identified parameters values serve as
first guess. The goodness of fit for a single load case is quantified by the optimization
criterion
cf,i(
p) = 1
tst ·
tst
Z
t95 h¯µmeas,i(t)−µsim,i(
p,ui(t), t)i2·¯vmeas,i(t)dt. (4.6)
2In contrasts to the model parameters ci,eiand µ0.
67
4 Development of a brake model for disc brake units
cf,i
integrates the weighted quadratic deviation of measured and simulated COF
3
. As
denoted by
t95
, the integration starts as soon as the brakes are actuated by 95% to
exclude the influence of non-meaningful deviations occurring during the transient built-up
phase. From thereon, the integrated failure quantifies the agreement of instantaneous
friction forces at each point of time. Since deviations at high velocities have a stronger
impact on the deviation of the resulting braking distance, the quadratic deviation is
weighted by the instantaneous velocity
¯vmeas,i
(
t
). The latter is gained from the friction
velocity with respect to wheel and friction radius
¯vmeas,i(t) = ¯vfric,i(t)·rwheel
rfric
.(4.7)
The criterion aims to improve both, the agreement of instantaneous friction forces as
well as the resulting braking distance. Hence, deviations of the brake force are accepted,
if their impact on the braking distance is not of significance. Note that the criterion is
normalized by the time period of the braking process
tst
. This allows for a quantitative
comparison of single criteria obtained from different load cases. Since the input data of
the model
ui
(
t
)as well as
¯µmeas,i
(
t
)and
¯vmeas,i
(
t
)are discrete time series, the integral in
Equation 4.6 becomes the sum over all time steps.
The goal of the multi-case optimization is to identify a set of parameters
pident
within
a given tuning interval [
p−
;
p+
], which minimizes
cf,i
(
p
)of all 12 load cases at once.
Therefore, the objective function csum is introduced
csum(
p) = 103·
12
X
i=1
cf,i(
p).(4.8)
The task of the optimization is thus to minimizes the objective function csum
min
p−≤
p≤
p+
csum(
p).(4.9)
csum
is weighted by 10
3
to avoid the occurrence of very small values, which might be
disadvantageous for the optimization. The multi-case optimization task is conducted
by implementing the problem formulated in Equation 4.9 within the DLR Optimization
Library [116]. This commercial tool provides different algorithms for numerical optimiza-
tion within the simulation environment Dymola. Different methods have been applied for
this optimization task, while an algorithm based on the Pattern-Search method yielded
the best results, i.e. the smallest value for
csum
. The algorithm is a gradient-free and
thus direct search method, which is known to be robust in particular for locating minima
on hypersurfaces with sharp valleys [117][118].
3Note the analogy to the derivation of µmgiven in Equation A.23 of Appendix A.1.4.
68
4.1 Deterministic friction model
The identified parameters values as well as their relative change with respect to the
start values are given in Table 4.2. Figure 4.1 shows the criteria
cf,i
resulting from
the simulation of all 12 load cases with optimized parameters and compares them to
the start criteria. As shown, the numerical optimization drastically reduces
cf,i
of the
load case 4-12 and the criteria of the load case 1-3 remain on a similar low magnitude.
Consequently, the identification strongly enhances the accordance within the entire load
spectrum, while cf,i ≤0.001 for all load cases. Accordingly, csum is reduced by 79%.
Table 4.2:
Comparison of start and identified parameter values resulting from numerical
optimization of the friction model valid for Pad Material A.
Model parameter
Unit
pstart
p−
p+
pident
pident−
pstart
pstart
Coefficients
c1s−10.01 0.005 0.05 0.0110 +10%
c2s−10.01 0.005 0.05 0.0080 -20%
c3s−10.02 0.005 0.05 0.0256 +28%
c4s−10.04 0.005 0.1 0.0397 -1%
Exponents
e1- 1 0.2 2 0.7852 -21%
e2- 0.5 0.2 2 0.3383 -32%
e3- 1 0.2 2 0.8300 -17%
e4- 0.5 0.2 2 0.2747 -45%
Initial value
µ0- 0.38 0.37 0.39 0.3715 -2%
Figure 4.1: Comparison of optimization criteria cf,i of all 12 load cases.
69
4 Development of a brake model for disc brake units
4.1.4 Discussion of results
4.1.4.1 Instantaneous COF
Figure 4.2 demonstrates that the general behavior of the instantaneous friction curves
resulting from the model with optimized parameters coincides very well with mean values
of the measurements. For all 12 load cases, the instantaneous relative error ϵ
ϵ(t) = |¯µmeas(t)−µsim(t)|
¯µmeas(t)(4.10)
remains smaller 5% during large parts of the braking process. Merely at the end, larger
deviations up to 20% occur. As shown,
µsim
(
t
)keeps increasing for very low velocities,
while
¯µmeas
(
t
)declines in this interval
4
. Recalling that
cf,i
weights deviations stronger for
larger velocities, an adaption of the optimization criterion could reduce the relative error
close to standstill. However, as the comparison of the curves indicates, the deviation at
small velocities is structural in nature. An explanation for the decline of
¯µmeas
(
t
)at the
very end may be related to the transition between dynamic and static friction, which
is not covered by the dynamics of the model. It is well understood that the adhesion
between pad and disc changes during the transition from slipping to sticking. Moreover,
oscillation phenomena emerge that influence the transmissible friction forces [36]. These
stick-slip phenomena not covered by the model contribute to the change of the COF at
very low velocities. A proper solution for this problem would be to implement a further
equation for ˙µ(t)valid for the transition region by introducing a case differentiation.
Nevertheless, the model excellently renders the friction forces acting in the velocity
interval from 20
km
h
to 160
km
h
, as shown in the left plot of Figure 4.3. Even the small
differences between load cases of the same initial velocity are captured by the model
in this interval. Moreover, the right plot in Figure 4.3 demonstrates that deviations
occurring for velocities smaller 20
km
h
hardly contribute to the final braking distance.
This means that the model is well suited to predict the friction forces acting during the
significant part of the braking process and a more detailed description for
v <
20
km
h
is
not assumed to significantly improve the prediction accuracy of the braking distance.
With respect to this task, the overestimation of friction forces at the end of the braking
process is graded as insignificant. Nevertheless, the results reveal that the identified
model is not valid for the prediction of brake forces acting at the very end of the stopping
brake process.
4
Note, that
¯µmeas
(
t
)may contain non-physical effects at the end of the braking process due to the
synchronization of the single curves with different lengths.
70
4.1 Deterministic friction model
Figure 4.2:
Comparison of measured and simulated instantaneous COF over time based on
identified model parameters including relative error [E2].
4.1.4.2 Model interpretation
Figure 4.4 compares simulated and measured derivatives of the COF with respect to
time for load case V0100_Case1 and V0160_Case3 (bolt black and red lines). Apart
from deviations at the beginning and the end,
˙
¯µmeas
and
˙µsim
match very well. The peak
of
˙
¯µmeas
(
t <
3
s
)is physically not meaningful, since it is caused by singularities resulting
from the numerical estimation of
˙
¯µmeas5
. The deviations at the end are presumably
5Note, that a filter is applied to reduce numerical scatter of ˙
¯µmeas for sake of presentation.
71
4 Development of a brake model for disc brake units
Figure 4.3:
Comparison of measured and simulated COF based on identified model parameters
for each load case with respect to velocity and distance.
related to the previously discussed stick-slip effects. Figure 4.4 additionally illustrates
the single terms of the identified model. According to Equation 4.2,
˙µsim
(
t
)is the sum of
these terms, i.e. the bolt red line is the sum of all colored thin lines. The purple curve
represents friction promoting processes related to the magnitude of the clamping force.
The other curves are smaller zero and related to destruction processes, i.e. a reduction
of the COF. For V0100_Case1, the purple curve dominates, which yields a continuous
rise of
µsim
(
t
). For V0160_Case3, destruction processes predominate during the first 20
seconds and µsim(t)declines in this phase, since ˙µsim(t)is smaller zero.
In general, Figure 4.4 exhibits a strong correlation of friction power (blue thin line)
and the shape of
˙µsim
(
t
)(bolt red line). The similarity is the reason that the friction
power term receives a high weighting by the numerical optimization (compare blue thin
line to yellow and red thin line). The influence of the surface temperature of the disc
Tfric
is not initially present but becomes larger during the braking process. Hence, the
dynamics related to this influencing factor are slower. This lag behavior, also observed
by Ostermeyer and Bode in [39], might be related to the dynamics of chemical processes
taking place during the thermal degradation of the composite material. The term related
to the COF itself has the smallest impact on
˙µsim
(
t
)according to the identification (see
thin red line). This indicates that the observed changes of the COF do not lead to a
significant modification of patch stability. However, the term influences the vertical
position of ˙µsim(t), which is crucial for the shape of µsim(t).
In conclusion, the interpretation suggests that the processes acting in the frictional
boundary layer are manifold and each related to a certain dynamic. Moreover, it becomes
clear that the role of each process is related to the braking scenario. In case of drag-
72
4.1 Deterministic friction model
braking on long downhill tracks, the influence of the temperature on the COF will be
presumably larger compared to the investigated stopping brake scenarios due to the large
heat input prevailing during a long period of time. In total, the results are in line with
the theory worked out in the literature study and the good agreement between model
and measurements is substantiated by a profound phenomenological reasoning.
Figure 4.4:
Comparison of measured and simulated derivative of the COF with respect to
time as well as progression of single terms based on identified parameters.
4.1.4.3 Comparison to state-of-the-art approaches
In the following, the identified model is compared to the approaches of Karwazki,
Saumweber, Yuan, Lee and Rhee given in Table 2.2. For sake of comparability, the
parameters of these models are adapted using a multi-case optimization in analogy to
the procedure described in Section 4.1.3, while the start values correspond to the values
found in literature (see Table 2.2). The identified parameters are given in Table A.5 of
Appendix A.3.1.
Figure 4.5 compares the resulting criteria
cf,i
with respect to all load cases. It is
demonstrated that the model developed and identified in this work, also referred to as
Ehret, yields the smallest criteria for all load cases. The model of Saumweber is the
best reference. However,
csum
of the Ehret model is still three times smaller than the
one of Saumwebers model, as shown in the legend. The large criteria of Yuans and
Karwatzkis models indicate that the investigated load spectrum is beyond the limits of
their applicability. As exemplified in Figure 4.6, these models are not able to render the
73
4 Development of a brake model for disc brake units
observed fading of the COF. In contrast, the approaches of Saumweber, Lee and Rhee
are suitable to predict fading for
v0≥
140
km
h
, since they consider the disc temperature
as additional influencing variable. However, both figures reveal that none of them
provides the good agreement as the dynamic model developed in this work. This supports
the hypothesis that multiple processes contribute to the observed fading behavior, as
postulated by the Ehret model.
Figure 4.5: Comparison of optimization criteria cf,i of different model approaches.
Figure 4.6:
Comparison of measured and simulated
µ
(
t
)based on different model approaches.
74
4.2 Deterministic temperature model
4.2 Deterministic temperature model
To apply the friction model developed in the previous section for the brake performance
prediction, it is necessary to estimate the instantaneous temperature prevailing at the
surface of the disc. Therefore, a temperature model is derived in this section, which
calculates
Tfric
(
t
)based on the frictional heat input dissipated during the braking
process. Initially, the dynamic equations ruling the heat transfer in the disc are derived.
Subsequently, the model parameters are identified with respect to the friction pairing
investigated on the test rig. Finally, the results are discussed.
4.2.1 Model synthesis
As discussed in [119] and [120], different approaches exist to predict the temperature
evolution in pad and disc during braking. The selection of the model approach depends
on the scope of the analysis as well as the type of brake system and scenario to be
analyzed. In this case, the average temperature prevailing on the surface of the disc
during a single stop shall be estimated, i.e. the temperature distribution along the disc
surface is not of interest. In general, the temperature increase close to the surface of
the disc is significantly higher than in the interior of disc during a single stop, i.e. a
temperature gradient evolves along the thickness of the disc [120][121][122]. As shown
in [120],[122] and [123], the application of lumped capacity models is a suitable and
computational efficient method to estimate the temperature gradient. Following this
approach, the disc can be segmented in axial direction into nlumped capacities each
representing a slice of the brake disc. By using thermal resistances between each capacity,
the temperature gradient along the thickness is discretised by ntemperature states.
For this work, a simple model based on only two thermal capacities is proposed. As
illustrated in Figure 4.7, the ring of the brake disc is segmented into two lumped masses,
namely Friction Zone and Bulk Disc with the thermal capacities
Cfric
and
Cdisc
as well
as their temperatures
Tfric
and
Tdisc
, respectively. Presuming that both sides of the disc
consume the same amount of energy, it is sufficient to model only one ring of the disc.
Since the thermal resistance of the pad is significantly larger than the one of the disc,
the disc consumes the vast majority of the heat input during a single stop. Hence, the
thermal capacity of the pad is neglected within the model. Based on this simplification,
the frictional heat input of one ring
˙
Qfric
corresponds to the half of the frictional power
consumed by the friction pairing during braking [E2]:
˙
Qfric =1
2·FC·vfric ·µ. (4.11)
75
4 Development of a brake model for disc brake units
ሶ
𝑄𝑓𝑟𝑖𝑐
ሶ
𝑄𝑣𝑒𝑛𝑡
ሶ
𝑄𝑠𝑢𝑟𝑓
Zoom of disc: B
Wheel-mounted brake disc Cut: A-A
A
A
B
ሶ
𝑄𝑣𝑒𝑛𝑡
ሶ
𝑄𝑓𝑟𝑖𝑐
ሶ
𝑄𝑠𝑢𝑟𝑓
ሶ
𝑄𝑐𝑜𝑛𝑑
Capacity: Friction Zone Capacity: Bulk Disc
𝑇
𝑓𝑟𝑖𝑐, 𝐶𝑓𝑟𝑖𝑐 𝑇𝑑𝑖𝑠𝑐,𝐶𝑑𝑖𝑠𝑐
ሶ
𝑄𝑐𝑜𝑛𝑑
Figure 4.7: Two capacity model for one ring of WMD including heat flows [E2].
For load cases with
v0
= 160
km
h
,
˙
Qfric
exceeds 200
kW
at the beginning of the braking
process. During a single-stop the heat losses due to convection and radiation are small
compared to the large amount of heat stored in the disc [120]. For sake of simplification,
only forced convection processes are taken into account, whereas radiation processes are
neglected in this model. Accordingly, the temperature dynamics in the friction zone are
ruled by the frictional heat input
˙
Qfric
, convective heat dissipation to the ambient air
˙
Qsurf as well as conductive heat dissipation ˙
Qcond to the bulk capacity [E2]:
Cfric ·˙
Tfric =˙
Qfric −˙
Qsurf −˙
Qcond.(4.12)
The dynamics of the bulk temperature is driven by the conductive heat flow
˙
Qcond
and
the dissipation to the ambient air via the fans at the inner side of the ring
˙
Qvent
[E2]:
Cdisc ·˙
Tdisc =˙
Qcond −˙
Qvent.(4.13)
The conductive heat transport between friction zone and bulk disc depends on the
temperature gradient in the disc as well as the thermal conductance G[E2]:
˙
Qcond =G·(Tfric −Tdisc).(4.14)
As discussed in [124] and [125], the conductivity of gray cast depends on the temperature.
Therefore,
G
(
T
)is approximated by a Taylor Series of the first order developed at a
reference point with respect to the mean temperature of disc and friction zone [E2]:
76
4.2 Deterministic temperature model
G(T)≈Gref +∂G
∂T ·Tfric +Tdisc
2−273.15K.(4.15)
For a mean temperature of 273
.
15
K
, the conductance corresponds to the value
Gref
. The
derivative of the conductance with respect to the mean temperature
∂G
∂T
represents the
slope of
G
at the reference point. Note that the heat capacity of a material also depends
on the temperature. For sake of simplification, this dependency is however neglected
in the model. As soon as the temperature in the friction zone rises, forced convection
processes between disc surface area
Asurf
and ambient air are evoked by the temperature
difference and the rotation of the disc [E2]:
˙
Qsurf =hsurf ·Asurf ·(Tfric −T∞).(4.16)
Similarly, forced convection occurs along the area Avent of the venting fans at the inner
part of the ring [E2]:
˙
Qvent =hvent ·Avent ·(Tdisc −T∞).(4.17)
The convectional heat transfer depends on the surface geometry as well as the airflow
along the friction surface and vanes of the fan [57]. Investigations of disc brakes reveal a
linear or degressive correlation between the specific heat transfer coefficient
h
and the
rotational speed [57] [126]. Therefore, a simplified model is used in this work, which
relates the parameter hto the friction velocity according to [E2]:
h(vfric) = h0+ (href −h0)· vfric
27.7m
s!ex
.(4.18)
The exponent
ex
is assumed to take values
≤
1. The function yields a degressive
(
ex <
1) or linear (
ex
= 1) progression of
h
through the points
h0
=
h
(
vfric
= 0) and
href
=
h
(
vfric
= 27
.
7
m
s
). Since the geometries of the two convection surfaces strongly
differ, Equation 4.18 is implemented for each convention process individually.
4.2.2 Model identification
In total, the thermal model contains 10 parameters to be identified:
p= Cdisc, Cfric, Gref ,∂G
∂T , exvent, exsurf , href,vent, href,surf , h0,vent, h0,surf !T
(4.19)
77
4 Development of a brake model for disc brake units
In analogy to the friction model, a single set of parameter
p
shall be identified by applying
numerical optimization, which allows to predict the mean behavior of the temperature
¯
Tfric,meas
(
t
)illustrated in Figure 3.12. Initially, Equations 4.12 - 4.18 are implemented
with Modelica in the simulation environment Dymola. The input variables
ui
(
t
)of the
thermal model are the mean values of measured clamping force, friction velocity and
COF
ui(t) =
¯
FC,meas,i(t)
¯vfric,meas,i(t)
¯µmeas,i(t)
,(4.20)
calculated for each of the 12 load cases according to Equation 3.2. The instantaneous
temperature
Tfric,sim,i
(
p,ui
(
t
)
, t
)is determined by numerical integration of Equation
4.12 and 4.13 with respect to a certain set of parameters and load case. The initial
temperatures of both capacities are set according to the initial value of the measurements
¯
Tfric,meas,0≈
60
◦C
.
T∞
= 20
◦C
represents ambient temperature prevailing during the
test. The solution of the differential-algebraic system with respect to the initial values is
conducted by applying the DASSL solver with an error tolerance of 0.001.
4.2.2.1 Derivation of start values and tuning intervals for optimization
Before applying numerical optimization, start values and tuning interval need to be
derived for the parameters to be identified. Table A.7 sums up the material properties of
gray cast, geometry data of the disc and values of convective heat transfer coefficients
based on literature study
6
. The absolute values of the lumped capacities and the
conductance need to be derived with respect to material and geometry properties. The
thermal capacity of a lumped body is the product of the specific heat capacity
cp
and its
mass. As indicated by Figure 4.7, the mass can be estimated from the volume of a ring
segment and the density of gray cast. For a segment with the thickness
l
and area
Asurf
,
it follows
C=cp·ρ·Asurf ·l. (4.21)
If the friction zone represents 5
mm
and the bulk disc 30
mm
of the ring thickness,
Cfric
and
Cdisc
correspond to 4356
J
kgK
and 30494
J
kgK
for
cp
= 500
J
kgK
and
ρ
= 7150
kg
m3
.
Analogously, the conductance between both capacities is derived from a ring segment
with thermal conductivity λaccording to
6The exact composition of the disc material is not available.
78
4.2 Deterministic temperature model
G=λ·Asurf
l.(4.22)
For a 15
mm
thick ring segment with
λ
= 50
W
mK
, the conductance is 813
W
mK
. Presuming
that
∂λ
∂T
=
−
0
.
02
W
mK2
, it follows that the derivative of the conductance with respect to
the temperature related to the segment is ∂G
∂T =−0.3W
K2.
Based on these consideration, the start values for the parameter identification
pstart
and the corresponding tuning intervals [
p−
;
p+
]are set according to Table 4.3. It is
important to mention that the temperature model represents a strong simplification
of the actual physics. The model parameters should therefore not be considered as
actual physical properties but rather averaged values with respect to the lumped model
approach. Consequently, the limits of the tuning intervals are generously selected.
4.2.2.2 Numerical optimization
To identify the set of parameters, which yields the best possible match between model and
measurements, an optimization problem is formulated. The goodness of fit for each load
case is quantified by the quadratic deviation of measured and simulated temperature:
cf,i =1
tst ·
tst
Z
t=0 h¯
Tfric,meas,i(t)−Tfric,sim,i(
p,ui(t), t)i2dt. (4.23)
In contrast to Equation 4.6, this criterion is not weighted by the velocity, since the
temperature does not directly affect the instantaneous deceleration. Based on this
criterion, a multi-case optimization is conducted equivalent to the one presented in
Section 4.1.3. For the optimization task, the pattern search algorithm is applied. The
identified parameters
pident
are given in Table 4.3. Figure 4.8 compares the criteria of
the identified model with the start criteria. As shown, the optimization strongly reduces
the criteria of all load cases and the objective function csum is reduced by 97%.
4.2.3 Discussion of results
Figure 4.9 compares the measured average surface temperature with simulated results
from the temperature model based on the identified parameter values. The plot reveals
that the two-capacity model excellently matches with the instantaneous mean values
observed on the test rig. As shown, the relative error remains below 5% for all load
cases. According to Table 4.3,
Cdisc
is reduced and
Cfric
is enlarged with respect to their
initial values. Applying Equation 4.21, yields that the identified values correspond to
79
4 Development of a brake model for disc brake units
Table 4.3:
Comparison of start and identified parameter values resulting from numerical
optimization of the friction model valid for the investigated friction pairing.
Model parameter Unit
pstart
p−
p+
pident
pident−
pstart
pstart
Capacities Cdisc J
K25000 15000 35000 17600 -30%
Cfric J
K4500 1000 10000 7098 58%
Conduction Gref W
K800 200 2000 697 -13%
∂G
∂T
W
K2-0.3 -2 0 -1.25 318 %
Convection
exvent - 0.8 0.4 1 0.4 -50%
exsurf - 0.8 0.4 1 0.4 -50%
href,vent W
m2K90 50 120 120 33%
href,surf W
m2K90 50 120 53 -41%
h0,vent W
m2K5 1 15 15 200%
h0,surf W
m2K5 1 15 15 200%
Figure 4.8:
Comparison of criteria
cf,i
of all 12 load cases. Due to the strong reduction of
criteria with respect to the initial values, the y-axis is logarithmically scaled.
a friction zone with a thickness of about 8
mm
and a bulk disc of about 20
mm7
. The
identified conductance roughly corresponds to 18
mm
with respect to Equation 4.22. It
should be noted that the gradient
∂G
∂T
=
−
1
.
25
W
K2
is quite small and therefore probably
not physically meaningful. Hence, the modeled conductance between the capacities is
considered as theoretical value required to render the surface and bulk the temperature
by two capacities only. This emphasizes that the proposed two capacity model is a
strong simplification of the actual physics. The heat transfer coefficients take reasonable
values with regard to the literature study, while the identified heat transfer coefficient at
the vanes
href,vent
is larger than the one at the disc surface
href,surf
. This observation
corresponds to the results in [126] and may be related to the strong differences of
7
Note, that the volume reduction of the disc due to the cooling vans is neglected in this rough estimation.
80
4.2 Deterministic temperature model
the airflow prevailing in the vanes compared to the flat surface of the disc. It needs
to be mentioned that a proper identification of the convection parameters requires to
investigate the cooling phases after braking. Due to the tangential role of convection
within the stopping brake process and the good results, this additional identification is
however not expected to significantly enhance the prediction quality. In conclusion, the
results demonstrate that the proposed simple approach based on two lumped capacities
is sufficient to predict average surface temperature prevailing during stopping brake
scenarios with an error smaller 5%.
Figure 4.9:
Comparison measured average temperatures at the disc surface with simulated
temperature based on identified parameter values [E2].
81
4 Development of a brake model for disc brake units
4.3 Deterministic brake model
4.3.1 Dynamometer model
The prediction of instantaneous friction forces requires to couple the dynamics of friction,
temperature and mechanics. A corresponding brake model related to a single disc brake
unit is shown in Figure 4.10. The scheme displays a virtual test environment to simulate
brake applications on the dynamometer test rig. The input of the model is the time
series of the clamping force
FC
(
t
), which serves as input of all three submodels. Friction
and temperature model correspond to the equations derived in Section 4.1 and 4.2 and
are are mutually coupled, since they exchange the states
µ
(
t
)and
Tfric
(
t
). Temperature
and friction model are linked to a mechanic model representing the deceleration of inertia
mdyn.
Dynamometer brake model
Friction model
Equation 4.2 –4.3
Temperature model
Equations 4.11 –4.18
Mechanic model
Equations 4.24 –4.28
𝜇(𝑡)
𝜇(𝑡) 𝑇
𝑓𝑟𝑖𝑐(𝑡)
𝑣𝑓𝑟𝑖𝑐(𝑡)
𝑣𝑓𝑟𝑖𝑐(𝑡)
Input: Clamping force
𝐹𝐶𝑡
𝐹𝐶(𝑡)
𝐹𝐶(𝑡)
𝐹𝐶(𝑡)
Figure 4.10:
Structure and interaction of submodels within dynamometer brake model to
conduct virtual stopping brake scenarios with a single disc brake unit [E2].
The instantaneous brake torque
MB
(
t
)results from the current friction forces evoked by
µ(t)and the clamping forces FC(t)applied at the radius rfric [E2]:
MB(t) = FC(t)·µ(t)·rfric.(4.24)
Applying Equation 2.13 for the longitudinal deceleration of mdyn yields
a(t) = −MB(t)
mdyn ·rwheel
.(4.25)
82
4.3 Deterministic brake model
The integration of
a
(
t
)with respect to time yields the instantaneous velocity
v
(
t
)based
on the initial velocity v0:
v(t) = v0+Za(t)dt. (4.26)
The distance
s
(
t
)traveled during the virtual brake test follows from the integration of
the vehicle speed with respect to time:
s(t) = s0+Zv(t)dt. (4.27)
Since the friction velocity is required as input for friction and temperature model, it is
calculated based on the ratio of friction radius and wheel radius according to
vfric(t) = v(t)·rfric
rwheel
.(4.28)
The solution of this system of coupled differential equations requires to solve an initial
value problem based on the initial states
µ0
,
T0
,
v0
and
s0
= 0. For this task, numerical
integration is applied. The prediction of the brake performance based on the brake model
thus corresponds to the step by step method presented in Section 2.3.2.2.
4.3.2
Deterministic analysis of stopping-brake scenarios with a single
disc brake unit
To verify the accuracy of the brake model, it is deployed to simulate the mean, i.e.
deterministic, behavior of the investigated friction pairing investigated in Chapter 3.
Therefore, the parameters of friction and temperature models are set according to the
identified values
pident
in Tables 4.2 and 4.3. The input of the model is the mean measured
clamping force
¯
FC
(
t
)recorded for each load case, while initial velocity and braked mass
correspond to the values given in Table 3.2.
4.3.2.1 Coefficient of friction averaged over the braking distance µm
In order to assess the accuracy regarding the prediction of friction forces,
µm
is calculated
according to Equation 2.26 for all 12 load cases based on the simulated friction curves
and braking distances resulting from the virtual brake tests. Figure 4.11 compares
µm,sim
with the mean values from the measurements
¯µm,meas
for each load case, while Table A.6
in Appendix A.3.2 contains the underlying values. As shown, the gray and red bars in
Figure 4.11 match very well for all load cases. According to Table A.6, the maximum
83
4 Development of a brake model for disc brake units
relative prediction error is found for load case V0120_Case1 and accounts for 1.2%. The
error bars illustrated in Figure 4.11 represent the confidence intervals related to the
estimation of the mean values
¯µm,meas
with a confidence level of 95% (see Appendix
A.2.6). For 7 of 12 load cases the deviation between measurement and deterministic
simulation lies within this interval, which underlines the very good performance of the
brake model.
Figure 4.11:
Comparison of COF averaged over braking distance gained from deterministic
simulation
µm,sim
with mean value from measurements
¯µm,meas
including confi-
dence intervals with a confidence level of 95% (see Appendix A.2.6) [E2].
4.3.2.2 Stopping distance
To assess the model regarding the prediction accuracy of the braking distance, the
measured stopping distance
sst,meas,i
is calculated based on the integration of the recorded
velocity data for each of the 1950 brake applications
8
. To eliminate errors caused by
slight deviations of the initial velocities from the nominal velocity, the final stopping
distances are additionally corrected, as proposed in EN 16834 [63]. Subsequently, the
nominal stopping distance
¯sst,meas
related to each load case is estimated by calculating
the empirical mean value of the measured and corrected stopping distance according to
¯sst,meas =1
n
n
X
i=1
sst,meas,i.(4.29)
8
Note, that stopping distance and braking distance hardly differ from each other in case of the braking
scenarios on the test rig due to the sudden actuation of the brake.
84
4.3 Deterministic brake model
Table 4.4 compares
¯sst,meas
to the stopping distance resulting from the deterministic
simulation
sst,sim
. The largest error is observed for load case V0160_Case2 and accounts
for 2%, which corresponds to 21
m
with respect to a nominal stopping distance of about
1050
m
. For the other load cases, the relative error is less than 10
m
in any case. For
six load cases, the prediction error is even smaller than the half length of confidence
intervals related to estimation of ¯sst,meas.
In conclusion, the comparison reveals that the combination of friction and temperature
model, both developed in this work, allows for a very accurate prediction of the nominal
braking distance. Moreover, the results demonstrate that the deviations of the identified
friction model regarding the brake forces at very low velocities do not significantly impair
the prediction accuracy. Finally, it needs to be highlighted that the good results are
gained by applying a single set of model parameters valid for all load cases.
Table 4.4:
Comparison of stopping distances from deterministic simulation
sst,sim
with mean
values of measurements
¯sst,meas
including length of confidence intervals ∆
L
related
to the estimation of
¯sst,meas
based on a confidence level of 95% (see Appendix
A.2.6).
Load case sst,sim
[m]
¯sst,meas
[m]
∆L
2
[m]
¯sst,meas −sst,sim
[m]
|¯sst,meas−sst,sim|
¯sst,meas
V0100_Case1 326 329 1.2 3.4 1.0%
V0100_Case2 327 332 1.0 4.5 1.4%
V0100_Case3 315 318 1.2 2.3 0.7%
V0120_Case1 491 491 2.1 0.1 0.0%
V0120_Case2 497 501 2.4 4.4 0.9%
V0120_Case3 482 481 2.5 -0.6 0.1%
V0140_Case1 707 710 5.2 2.4 0.3%
V0140_Case2 722 730 5.5 7.7 1.0%
V0140_Case3 707 708 5.2 1.3 0.2%
V0160_Case1 1000 1006 8.1 6.4 0.6%
V0160_Case2 1032 1052 8.2 20.8 2.0%
V0160_Case3 1017 1022 8.0 4.6 0.4%
85
4 Development of a brake model for disc brake units
4.4 Stochastic brake model
4.4.1 Stochastic modeling approach
The goal of this section is to find a suitable approach, which enables to predict the
stochastic behavior related to braking processes with disc brake units. In general, each of
the four white boxes in Figure 4.10 may be considered as a potential source of uncertainty
within the brake model. As discussed in Section 3.2.3, the vast majority of the scatter
observed on the test rig is assumed to be aleatory and caused by the stochastic nature of
the friction process. To simplify the derivation and identification of a stochastic brake
model, it is therefore proposed to treat the the scatter related to the measurement not
as an independent source of uncertainty, but as part of the stochastic friction process.
Moreover, it is suggested that the friction model is the only source of uncertainty within
the brake model. Hence, the scatter of the disc temperature is presumed to play a
secondary role for the dispersion of the COF.
Different approaches exist to superimpose deterministic and stochastic effects within
a friction model, as discussed in [42],[43] and [44]. Figure 3.7 reveals that 1
σµ
(
t
)is
time-variant, non-linear and depends on the load case. Adding a single stochastic scalar
with zero mean to the deterministic model, as proposed by Hinrichs et al., is thus not
sufficient to capture the individual variance of each load case. With regard to the
phenomenological background of the friction model developed in this work, it is assumed
that an approach based on uncertain parameters, as discussed by Ostermeyer et al. and
Zhang et al., is more promising. The results of the data analysis in Section 3.2.3 indicate
that the processes prevailing in the frictional boundary layer need to be considered as
stochastic processes. To transfer this observation to the friction model, it is proposed
that the coefficients ciin Equation 4.2 are stochastic quantities:
˙
˜µ(t) = −˜c1· ˜µ·FC·vfric
Pref !e1
−˜c2·˜µe2−˜c3· ˜µ·TFric
Tref !e3
+ ˜c4· FC
Fref !e4
.(4.30)
Consequently,
˙
˜µ
(
t
)is a stochastic variable and thus its primitive
˜µ
(
t
), as denoted by
the tilde operator. Due to the mostly smooth progression of the single friction curves,
illustrated by the left plot of Figure 3.4, it is additionally proposed that the stochastic
coefficients do not change from time instant to time instant, but only from brake
application to brake application. This means that each coefficient takes a constant
random value for each simulated stopping brake application. Analogously, the initial
value is considered as stochastic quantity, which takes a random value for each simulated
brake application. This yields
86
4.4 Stochastic brake model
˜µ(t) = ˜µ0+Z˙
˜µdt. (4.31)
According to the results of the statistical analysis summarized in Table 3.3, the friction
curves are mostly normally distributed at the beginning of the braking process, i.e.
˜µ0∼ N(¯µ0, σµ0).(4.32)
Except for samples taken from brake applications with 160
km
h
, this observation also
applies for the distribution of the single friction curves during the braking process, as
discussed in Section 3.2.4. Consequently, it is hypothesized that the underlying processes
acting in the frictional boundary layer are also normally distributed and that the uncertain
coefficients of the stochastic friction model follow a normal distribution:
˜c1∼ N(¯c1, σc1),˜c2∼ N(¯c2, σc2),˜c3∼ N(¯c3, σc3),˜c4∼ N(¯c4, σc4).(4.33)
The nominal values
¯µ0
and
¯ci
correspond to the most probable evolution of
˜µ
(
t
)for
each load case. Accordingly, the values which were identified by fitting the deterministic
friction model to the mean behavior observed on the test rig
¯µmeas
(
t
)represent estimates
for these nominal values (see Table 4.2). It remains to identify the standard deviation of
the coefficients
σµ0
and
σci
. This means to determine the extent of scatter, that leads to
the observed dispersion of the model output ˜µ(t).
4.4.2 Stochastic identification
The identification of
σµ0
is straight forward, since it can directly be derived from the
measurement data recorded at the beginning of the braking process by simply looking at
the left plot in Figure 3.7. Consequently, a value of 1
σµ0≈
0
.
01 is proposed.
˜µ0
is thus
assumed to be distributed according to
˜µ0∼ N(¯µ0= 0.3715,1σµ0= 0.01).(4.34)
The identification of the standard deviation of the coefficients
σci
is more complicated
and represents an inverse problem, which requires to solve two related problems:
1.
It is not possible to derive an analytic relation between output dispersion and
dispersion of the model parameters.
87
4 Development of a brake model for disc brake units
2.
The problem is in general inconclusive. This means that the output dispersion may
be a result of a single uncertain parameter or different combinations of uncertain
parameters.
From a mathematical perspective, the solution of these problems demands to apply
methods developed for the quantification of uncertainty, such as Bayesian approaches
or Polynomial Chaos Expansions [127]. However, the non-linear differential systems of
equation describing the underlying friction model strongly complicate the application
of these methods. Alternatively, numerical methods can be applied, for instance a
combination of MCS and numerical optimization, which are however related to large
computational effort as well as a tedious implementation. Due to the aforementioned
reasons, a more practical approach is proposed to address the problems above. From a
physical perspective, the identification of
σci
may be interpreted as the question: Which
of the processes modeled by Equation 4.30 significantly contributes to the observed
scatter of the
˜µmeas
(
t
)? Based on this phenomenological viewpoint, a Sensitivity Analysis
is suggested to estimate how and to which extent each coefficient affects the model
output. The idea of the analysis is to fit the friction model to the outer margins
¯µmeas ±
3
σµ,meas
(
t
)observed in the measurements in analogy to the identification of the
friction model presented in Section 4.1.3. Based on the identified sets of parameters,
conclusions regarding the uncertainty of each coefficient are drawn.
To quantify the goodness of the fit for each load case with respect to both margins, the
criteria cf,i+and cf,i−are introduced
cf,i±=1
tst ·
tst
Z
t95 h¯µmeas,i(t)±3σµ,meas,i(t)−µsim(
p,ui(t), t)i2·¯vmeas(t)dt. (4.35)
Apart from the target values
¯µmeas ±
3
σµ,meas
(
t
), the criteria correspond to the one
introduced in Section 4.1.3. The parameters to be identified are the coefficients of the
model
p= (c1, c2, c3, c4)T.(4.36)
Since the model is fitted to both upper and lower margins, two sets of parameters
pident
are identified. To investigate the influence of each coefficient individually, both fits are
conducted by tuning only one coefficient, whereas the values of the other three coefficients
remain constant. This means that the fits to upper and lower margins are conducted each
four times. The start values for each fit correspond to the identified values in Table 4.2,
i.e. the mean behavior observed on the test rig. Moreover, parameter
µ0
takes the values
¯µ0±
3
σµ0
= 0
.
3715
±
0
.
03 depending on the current margin to be fitted. Analogously to
Section 4.1.3, the resulting eight multi-case optimization tasks are solved numerically by
88
4.4 Stochastic brake model
applying the Pattern search algorithm. Table 4.5 summarizes the results. As shown,
csum
of the fits to the upper boundary are subject to larger values than the fits to the lower
boundary, i.e. larger deviations between measurement and simulation (compare 5th and
8th column). Consequently, column 5 contains the decisive values to assess the quality of
fit. The smallest value belongs to coefficient
c1
(49). Hence, an adaption of parameter
c1
yields the best fit to both ranges of variation when each coefficient is adapted individually.
The absolute values of the changes with respect to the start value ∆
p
are similar when
fitting to both sides. This indicates a symmetric transformation of
µ
(
ci
)and supports
the hypothesis that the coefficients are normally distributed.
|¯
∆p|
may be interpreted as
measure that quantifies the sensitivity of the model output with regard to a change of
each coefficient. The smaller
|¯
∆p|
, the higher the sensitivity. Accordingly,
c1
yields the
largest sensitivity, whereas parameter
c3
the lowest. Note that the high sensitivity of the
model output regarding coefficient
c1
coincides with the results of model analysis and
identification, which reveal a significant influence of the friction power on the change of
the COF.
Table 4.5:
Results of sensitivity analysis based on 8 optimization problems. During each
optimization only one coefficient is individually tuned, whereas the others remain
at their start values. ∆
p
represents the change of the fitted value with respect to
the start value.
|¯
∆p|
is the mean of the absolute changes of each coefficient for both
fits. csum is the final value of the objective function according to Equation 4.8.
.
Fit to ¯µmeas + 3σµ,meas Fit to ¯µmeas −3σµ,meas
Coefficient
pstart
pident ∆pcsum
pident ∆pcsum |¯
∆p|
c10.0110 0.0082 -0.0027 49 0.0145 0.0035 43 0.0031
c20.0080 0.0004 -0.0076 59 0.0164 0.0084 45 0.0080
c30.0256 0.0130 -0.0126 55 0.0434 0.0178 33 0.0152
c40.0397 0.0460 0.0062 54 0.0334 -0.0063 39 0.0063
With regard to these findings, it is suggested to simplify the proposed stochastic model
approach. It is assumed that the instantaneous dispersion of friction curves is primarily
driven by the uncertainty of
µ0
and stochastic processes related to the friction power
[E2]. The uncertainty of the other processes is assumed to play a secondary role. i.e.
σc2=σc3=σc4= 0.(4.37)
The mean values of the absolute changes
|¯
∆p|
in the column on the very right quantify
to which extent each coefficient needs to vary in order to render the limits of observation
¯µmeas ±
3
σµ
(
t
)by simulation. Accordingly, it is assumed that
|¯
∆p|
is an estimate for 3
σci
,
if only one coefficient is considered as uncertain. Applying this estimation for
˜c1
yields
˜c1∼ N(¯c1= 0.011,1σc1≈0.001),(4.38)
89
4 Development of a brake model for disc brake units
while the nominal value
¯c1
corresponds to the value of
c1
in Table 4.2 and 1
σc1
corresponds
to
|¯
∆p|
3
with respect to the first row in Table 4.5. The assumptions based on the
sensitivity analysis strongly facilitates the solution of the inverse problem. Due to the
lack of mathematical proof, the resulting stochastic model needs to be verified. This is
conducted in the following chapter.
4.4.3 Stochastic analysis of stopping-brake scenarios with a single
disc brake unit
In the following, the proposed stochastic friction model is applied to predict the perfor-
mance dispersion related to stopping brake applications based on a single disc brake unit.
Therefore, the experimental brake tests presented in Chapter 3 are conducted virtually
by applying a Monte-Carlo Simulation (MCS) based on the brake model presented in
Section 4.3.1. To obtain a similarly large sample size as in the experimental brake tests,
each load case is simulated 200 times. During each simulated brake application, the
parameters
˜µ0
and
˜c1
of the friction model each take a constant random value according
to the distributions given in Equations 4.34 and 4.38. The two random values applied
for each simulation are neither correlated to each other, nor to the load history, i.e. the
values of the previous simulations. All other parameters of the brake model remain
deterministic quantities and correspond to the identified values given in Table 4.2 and
Table 4.3 for all simulations. The mean clamping force
¯
FC
(
t
)measured for each load
case is the input of the brake model. Initial velocity and braked mass correspond to the
values given in Table 3.2.
4.4.3.1 Instantaneous coefficient of friction
Figure 4.12 compares the results of the MCS conducted for load case V0140_Case2,
in analogy to the illustration in Figure 3.4. The left plot visualizes all 200 simulated
friction curves. Each simulated friction curve follows a unique and smooth progression, as
postulated by the stochastic friction model. To quantify the deterministic and stochastic
behavior, mean value and standard deviation are calculated at each time instant according
to Equations 3.2 and 3.3. The right plot visualizes the resulting instantaneous values
based on the MCS of this load case. Apart from the lack of smaller oscillations observed
for the measured friction curves, Figure 4.12 and Figure 3.4 look similar. To compare
the distributions, Figure 4.13 superimposes the histograms of samples resulting from
the simulated friction curves at
t
= 25
s
of this load case with its equivalent from the
measurements. The histograms of both samples as well as the corresponding theoretical
probability densities based on temporary mean value and standard deviation match very
well. Hence, the figure demonstrates that the application of two normally distributed
parameters yields a correct output distribution for this time instant and load case.
90
4.4 Stochastic brake model
Figure 4.12:
Comparison of 200 simulated brake applications during MCS of load case
V0140_Case2 (left) and instantaneous mean values and range of variation
(right).
Figure 4.13:
Comparison of measured (black) and simulated (red) distributions of friction
curves at t= 25sduring load case V0140_Case2.
Figure 4.14 expands the comparison to all load cases by plotting instantaneous mean
values and ranges of variation resulting from measurement against the values of the MCS.
As shown, the ranges of dispersion illustrated by the thin black and red lines coincide
very well. Minor deviations are found for load case V0160_Case1 and V0160_Case3. The
source of these deviations may be related to the not normally distributed measurement
data identified for these load case, as discussed in Section 3.2.4. The good accordance
proofs that the estimation of 1
σci
based on the sensitivity analysis conducted in the
previous section yields satisfactory results regarding the extent of the instantaneous
91
4 Development of a brake model for disc brake units
scatter. Furthermore, the assumption is confirmed to apply only two uncertain model
parameters
˜µ0
and
˜c1
to render the observed dispersion. Finally, the good accordance
of instantaneous mean values demonstrates that the average behavior of the MCS
corresponds to the deterministic model. In total, Figure 4.13 and 4.14 demonstrate
that both deterministic and stochastic friction phenomena observed on the test-rig are
predicted very well by a MCS based on the proposed stochastic brake model.
Figure 4.14:
Comparison of instantaneous mean values and ranges of variations from mea-
surements and MCS [E2].
92
4.4 Stochastic brake model
4.4.3.2 Coefficient of friction averaged over braking distance
To quantify the accordance, mean values
¯µm,sim
and standard deviations
σµm,sim
of the
COF averaged over the braking distance resulting from the simulations are calculated
in analogy to Section 3.3. Figure 4.15 opposes empirical the standard deviations of the
MCS 1
σµm,sim
with the values from the measurements 1
σµm,meas
. The underlying values
are given in Table A.8 of Appendix A.3.4. As shown, the bars in Figure 4.15 coincide
well for all load cases as well as the samples containing all brake applications. As it is
the case for the values gained from the measurements, 1
σµm,sim
tends to increases for
load cases with higher initial velocities. According to Table A.8, the maximum error of
1
σµm,sim
with respect to 1
σµm,meas
is 0.004. Figure 4.15 illustrates that the confidence
intervals related to the estimation of 1
σµm
overlap for most cases. This means that there
are no significant differences between measured and predicted standard deviations for 10
of 13 samples. Note that the good accordance of the bars related to All load cases is
primarily caused by the good agreement of the mean values of the single samples. As
shown in Table A.8, the largest relative error of
¯µm
is 1
.
1%. In total, the comparison
emphasizes that the stochastic brake model excellently reproduces the deterministic and
stochastic behavior of µm, if the sample size of the MCS is sufficiently large.
Figure 4.15:
Comparison of 1
σµm
resulting from measurements and MCS including confidence
intervals based on a confidence level of 95% (see Appendix A.2.6) [E2].
93
4 Development of a brake model for disc brake units
4.4.3.3 Stopping distance
Figure 4.16 compares the samples of stopping distances resulting from the experiments
with the samples gained from the MCS
9
. Apart from load case V0100_Case2 and
V0160_Case2, the medians of the corresponding samples do not significantly differ,
i.e. the prediction of the nominal stopping distances is correct. The similar lengths of
whiskers and boxes additionally indicate that the distribution of measured and simulated
samples coincide well.
Figure 4.16:
Comparison of stopping distance samples resulting from measurements and MCS
by box plots (Explanation of box plot is given in Section 3.3).
Based on the samples illustrated in Figure 4.16, empirical mean values
¯sst
and standard
deviations
σsst
are calculated in analogy to Equations 3.2 and 3.3 for each load case. The
results are listed in Table 4.6. Figure 4.17 additionally opposes the values of the standard
deviation resulting from the measurement 1
σsst,meas
with the ones resulting from the
MCS 1
σsst,sim
by bar plots. As shown, 1
σsst,meas
increases for larger initial velocities
and takes values between 7
m
and 55
m
. This corresponds to 2% to 5% with respect to
the nominal stopping distance. For an initial velocity of 160
km
h
, the observed range of
variation is
¯sst,meas ±
3
σsst,meas ≈
1000
m±
165
m
. As visualized by Figure 4.17, 1
σsst,sim
coincides very well with the standard deviation observed in the measurement data for all
load cases, while the confidence intervals overlap for 10 of 12 load cases. According to
Table 4.6, the maximum error of 1
σsst,sim
is 12
m
. This corresponds to 1% with respect
to the nominal stopping distance. This is a very good result.
9
Note that that stopping distance
sst
is almost the same as the braking distance
s2
in this case, since
the actuation of the clamping force on the test rig is very fast.
94
4.4 Stochastic brake model
Table 4.6:
Comparison of mean values and standard deviation of stopping distance resulting
from MCS and measurements.
Load case ¯sst,sim
[m]
¯sst,meas
[m]
|¯sst,meas−¯sst,sim|
¯sst,meas
1
σsst,sim
[m]
1
σsst,meas
[m]
1σsst,meas −1σsst,sim
[m]
V0100_Case1
326 329 1.1% 8 8 0
V0100_Case2
326 332 1.6% 9 7 2
V0100_Case3
315 318 0.9% 10 7 2
V0120_Case1
491 491 0.0% 15 14 1
V0120_Case2
498 501 0.6% 17 15 2
V0120_Case3
482 481 0.2% 19 16 3
V0140_Case1
707 710 0.4% 28 34 -6
V0140_Case2
721 730 1.3% 32 34 -1
V0140_Case3
706 708 0.3% 31 32 -1
V0160_Case1
1000 1006 0.7% 43 55 -12
V0160_Case2
1037 1052 1.5% 52 53 -1
V0160_Case3
1020 1022 0.2% 47 51 -4
Figure 4.17:
Comparison of 1
σsst
resulting from measurements and MCS including confidence
intervals based on a confidence level of 95% (see Appendix A.2.6).
95
4 Development of a brake model for disc brake units
4.5 Key findings of the model development
The results of this chapter demonstrate that the friction model developed in this work
allows to render both the deterministic behavior of
¯µ
(
t
)as well as the time-variant
standard deviation 1
σµ
(
t
). As shown, the error related to the prediction of
¯µ
(
t
)is
smaller 5% for the significant part of the braking process regarding the prediction of
the braking distance. Moreover, the comparison to five state-of-the-art friction models
reveals that the phenomenological model approach developed in this work yields the best
results in case of the investigated friction pairing and load cases, while using only a single
set of model parameters. Consequently, the newly developed friction model strongly
enhances the prediction capability of instantaneous friction forces evoked by disc brake
units applied in rail vehicles. Analogously, it is shown that a model based on two single
thermal capacities is well suited to predict
¯
Tfric
(
t
)during stopping brake scenarios, while
the instantaneous prediction error remains below 5%.
The coupling of friction and temperature model within a brake model allows to simulate
brake tests on a virtual test rig. Applying this virtual test rig for the investigated friction
pairing and load cases yields a prediction accuracy of the nominal COF averaged
over the braking distance
¯µm
of 1.2%. The prediction accuracy of the nominal
stopping distance is 2%. It needs to be noted that these values correspond to the
investigated friction pairing and load cases. The application of friction and temperature
models for other friction pairings requires to identify the model parameters based on
corresponding data analogously to the procedure proposed in Section 4.1.3 and 4.2.2.
In addition to the estimation of the nominal behavior, this work firstly proposes a
stochastic approach to consider instantaneous stochastic effects related friction pairings
applied in rail vehicles in the prognosis. As shown, the introduction of two uncertain
parameters within the friction model allows to render the range of variation
¯µ
(
t
)
±
3
σµ
(
t
)
observed on the test rig very well when applying a MCS of the investigated braking
scenarios. The resulting prediction accuracy of the standard deviation 1
σµm
is
0.004. This is as accurate as possible with respect to the estimation of the empirical
standard deviation based on the sample size.
As shown by the experimental brake tests conducted on the test rig, the standard deviation
of the stopping distances related to brake applications with a single disc brake unit
increases for larger initial velocities and takes values from 7
m
to 55
m
. This corresponds
to 2% to 5% with respect to the braking nominal distance. For brake applications with
160
km
h
, the range of variation is thus
¯s±
3
σs≈
1000
m±
165
m
. The distributions of the
stopping distances predicted by a MCS based on the proposed stochastic model coincide
well with ones of all load cases observed on the test rig. The resulting prediction
accuracy of the standard deviation of the stopping distance 1
σsst
is 1% with
respect to the nominal stopping distance.
96
5 Analysis of vehicle brake applications
In this chapter the brake model developed in the previous chapter is applied to simulate
braking applications of a disc-braked vehicle. In contrast to test rig experiments, multiple
disc brake units contribute to the deceleration of a rail vehicle on track during emergency
brake applications. Moreover, the load cases in the field usually differ from the ones
investigated on the test rig and the brakes are not applied within idealized laboratory
conditions. These disparities affect the brake performance of the rail vehicle and raise
the question of whether and to what extent the information obtained on the test rig
can be used to predict vehicle behavior. Consequently, the goal of this chapter is to
validate the developed brake model regarding the prediction of nominal as
well as stochastic brake performance of a disc-braked rail vehicle. Moreover,
the investigations aim to deepen the understanding regarding the deterministic and
stochastic behavior of disc-braked rail vehicles.
For the validation data from experimental vehicle tests of a multiple unit is available,
which was provided by the Knorr-Bremse SfS GmbH. Initially, vehicle configuration,
braking scenarios as well as measurement set-up and post-processing of the data gained
during the corresponding test campaign are presented. Subsequently, a simulation model
is set up to conduct the vehicle brake tests virtually. The results of the virtual tests
are then compared to the measurements to assess the prediction accuracy of the model.
Furthermore, the stochastic friction model identified in the previous chapter is applied
to analyze the dispersion of the vehicle brake performance. Two different approaches
regarding the scatter of the singe brake units in the vehicle are compared and verified by
the experimental data.
97
5 Analysis of vehicle brake applications
5.1 Experimental field tests
5.1.1 Vehicle configuration and braking scenarios
Figure 5.1 illustrates the multiple unit and its frictional brake system, which was applied
to conduct experimental brake tests on track. The train is composed of four bogies with
each two wheel sets, i.e. eight wheel sets in total. Each wheel set is equipped with two
identical WMD units, i.e. 16 brake units in total.
Figure 5.1: Configuration of friction brakes applied in investigated multiple unit.
Table 5.1 sums up the nominal parameter values of the vehicle and the applied brake units.
The total dynamic mass of the train varies between 121
tons
and 149
tons
with respect
to the three investigated load conditions (Empty, Nominal and Maximum). Friction
radii and wheel radii are the same for all axles and brake units. The brake units applied
in the vehicle slightly differ from the one investigated on the test rig (see Table 3.1).
However, the same pad material type is used in the vehicle as the one analyzed
on the test rig. Moreover, the pad geometry is identical. This is crucial to conduct a
meaningful comparison between the results derived from the test rig data and the data
of the field tests. It is important to mention that none of the 16 pad pairs applied in the
vehicle corresponds to the five specimen investigated on the test rig. In fact, it needs
to be assumed that the pads applied in the vehicles result from different batches than
the ones of the test rig. Finally, Table 5.1 includes resistance parameters
Ri
, which were
identified during coasting tests of the train on a flat and level track.
Table 5.1: Parameter values of multiple unit and applied brake units.
Train parameter Brake unit parameter
nb16 Outer disc diameter 0.355 m
nrot 8 Inner disc diameter 0.210 m
mtrain(Empty) 121176 kg rfric 0.286 m
mtrain(Nominal) 143892 kg ηC0.99
mtrain(Maximum) 148678 kg iT8.51
rwheel 0.42 m AC0.01427 m2
R12500 N FS500 N
R20.0216 Ns
mPad Material Material A
R35.4173 Ns2
m2Disc Material Grey cast
98
5.1 Experimental field tests
In total, the data of 88 stopping brake applications with an initial velocity between
100
km
h
and 160
km
h
is available, as shown in Table 5.2. All brake scenarios are conducted
on a straight and level track, while the friction brakes are the only active brake
system. Moreover, the brake applications were conducted during dry conditions.
The disc brake units are either actuated by the Indirect Pneumatic Brake (PN) or the
Direct Electro-Pneumatic Brake (EP). This leads to varying built-up times, as shown in
Figure 2.4. Moreover, different brake demands (Emergency and Service) are investigated,
which affect the magnitudes of the clamping forces. During the test campaign brake
demands and initial velocity are subject to frequent changes, i.e. the brake applications
belonging to each scenario in Table 5.2 are not conducted in series.
The clamping forces at each brake unit depend on the axle load. To allow for a comparison
with the load cases investigated on the test rig, clamping forces and the brake masses in
Table 5.2 correspond to the mean values of all brake units. As the comparison to Table
3.2 reveals, the brake units equipped in the vehicle are subject to similar loads during
the field tests as the one analyzed on the test rig.
In conclusion, the vehicle experiments represent a broad spectrum of brake demands,
modes and configurations expected during operation and the resulting set of data is
regarded as very valuable for the validation of the developed brake model.
Table 5.2:
Number of brake applications conducted during field tests with respect to braking
scenario and initial velocity
v0
[E2]. Act. type defines if brakes are actuated by EP
or PN. Values of mdyn and FCrepresent mean values of all 16 brake units.
Brake
demand Load ¯
FC¯mdyn Act.
type
Number of brake appl. per v0
100
km
h
120
km
h
140
km
h
160
km
hP
Emergency
Empty 36 kN 7574 kg EP 2 7 1 2 12
PN 2 5 2 2 11
Nominal 44 kN 8993 kg EP 1 5 1 1 8
PN 2 3 1 2 8
Maximum 45 kN 9292 kg EP 2 4 1 1 8
PN 1 1 1 1 4
Service Empty 33 kN 7574 kg EP 3 9 2 3 17
Nominal 36 kN 8993 kg EP 3 13 2 2 20
All load cases 16 47 11 14 88
99
5 Analysis of vehicle brake applications
5.1.2 Post-processing
The multiple unit is equipped with various sensors to analyze its behavior during braking.
The records comprise speed measurements of the vehicle
v
(
t
)as well as data from an
acceleration sensor to measure its longitudinal deceleration
|a|
(
t
). Figure 2.9 exemplifies
acceleration and speed measured during one of the 88 brake applications. Since the
oscillations of the deceleration signal are small compared to the overall progression, the
vehicle may be considered as a single lumped mass and a single acceleration sensor is
sufficient.
Additional measurements of the rotational speeds of all axles prove that the limits of
adhesion between wheel and rail are not exceeded during braking. Integrating the
vehicle velocity with respect to time yields the distance
s
(
t
)traveled during the braking
process. Moreover, the velocity data is used to estimate the driving resistance force
FR
(
t
)
based Equation 2.19 and the provided values for R1−3(see Table 5.1).
Records of the cylinder pressure at all axles
pC,i
(
t
)allow to deduce the clamping forces
FC,i
(
t
)acting in each brake unit based on Equations 2.2 and 2.5. As shown in Figure 2.9,
the cylinder pressures and thus clamping forces remain constant as soon as their target
values are reached. Since the axle loads of the two outer bogies and the ones of the inner
bogies are almost identical, only two pressure levels are visible in Figure 2.9.
To estimate the thermal loads on the disc brakes during braking, the surface temperature
of one of the brake discs is recorded at a single position based on a sliding contact
measurement. In addition, the ambient temperature is recorded.
The experimental set up does not allow to estimate the COF of each brake unit in the
vehicle
µi
(
t
). However, it is possible to estimate the effective COF
µeff
(
t
)related to the
sum of all frictional brake forces based on the data of
pC,i
(
t
),
|a|
(
t
)and
FR
(
t
). Inserting
Equation 2.6 in Equation 2.18 and solve for µyields1
µeff (t) = mtrain ·|a|(t)−FR(t)
rfric
rwheel ·nb
P
i=1
FC,i(t)
=mtrain ·|a|(t)−FR(t)
rfric
rwheel ·nb
P
i=1[(pC,i(t)·AC−FS)·ηC·iT]
.(5.1)
The relation between µeff and µiis given by2:
µeff =
nb
P
i=1
FC,i ·µi
nb
P
i=1
FC,i
.(5.2)
1E2.
2
If friction and wheel radii are not identical for all brake units and axles of the vehicle, these values
need to be considered in Equation 5.2.
100
5.2 Deterministic analysis of field tests
5.2 Deterministic analysis of field tests
5.2.1 Synthesis of the train model
Based in the previously described vehicle configuration and the brake model developed
in Chapter 4, a train model is derived to simulate the field tests virtually. Therefore, two
assumptions are made with regard the longitudinal dynamics of the rail vehicle:
•The vehicle is considered as single lumped mass.
•
No macroscopic slip is occurring, i.e. the contact between wheel and rail is
approximated by pure rolling.
These assumptions strongly simplify the kinetics and allow to use the equations derived in
Section Chapter 2.2.2. The scheme of the resulting train model is illustrated in Figure 5.2.
The references of the required equations applied for the virtual train are given in the
white boxes. Each of the 16 brake units applied in the multiple unit is modeled by
an individual brake model, which is composed of a friction, temperature and wheel set
dynamics model. The clamping forces
FC,i
(
t
)serve as input for each brake model. The
sum of all brake forces
PFB,i
(
t
)and the resistance force
FR
(
t
)decelerate the dynamic
mass of the train. The driving resistance
FR
(
t
)is calculated by Equation 2.19 with
respect to the current velocity of the vehicle. The train model is implemented with the
physical modeling language Modelica within the simulation environment Dymola. This
yields a system of 131 differential and 65 algebraic equations.
5.2.2 Parameters and simulation scenarios
All 88 stopping brake applications conducted during the experimental field tests are
simulated. The model inputs correspond to the clamping forces
FC,i,j
(
t
)calculated from
the measured cylinder pressures, while
i
=1-16 represents the index of the brake unit
and
j
=1-88 the index of the brake application. The initial velocity corresponds to the
velocity recorded during the vehicle brake tests at beginning of the brake applications.
In analogy, the temperature models are initialized by the temperature measured at the
surface of the disc.
The mechanical and geometrical parameters of the model correspond to the nominal
values given in Table 5.1, while
mtrain
corresponds to the current load condition. Since
the same pad material is applied in the vehicle as the one analyzed on the test rig, the
parameters of all 16 friction models are set to the identified values
pident
given in Table
4.2, .i.e. they correspond to the mean behavior of Pad Material A observed on the
test rig. Analogously, all 16 temperature models are identically parameterized based on
the identified values given in Table 4.3. It is noted that the brake discs applied in the
101
5 Analysis of vehicle brake applications
Train Model
Input i
Clamping force
𝐹𝐶,𝑖 𝑡
𝑣(𝑡)
Train dynamics
Equation 2.18
Equation 2.20
Equation 2.21
∑
Resistance model
Equation 2.19
Brake unit i
Friction model
Equations 4.2 –4.3
Temperature model
Equations 4.11 –4.18
Wheel set dynamics
Equation 2.14
Equation 4.28
𝜇𝑖(𝑡)
𝜇𝑖(𝑡) 𝑇𝑓𝑟𝑖𝑐,𝑖(𝑡)
𝑣𝑓𝑟𝑖𝑐(𝑡)
𝑣𝑓𝑟𝑖𝑐(𝑡)
𝐹𝐶,𝑖(𝑡)
𝐹𝐵,𝑖(𝑡)
𝐹𝐵,𝑖(𝑡)
𝐹𝑅(𝑡)
𝑣(𝑡)
𝑖 = 1,2…16
𝑖 = 1,2…16
Figure 5.2:
Structure of train model to simulate stopping brake applications of multiple unit.
train are slightly larger than the one analyzed on the test rig. To take this difference into
account of the simulations, the values of the thermal capacities, conductance and gradient
of the conductance are scaled by 1.056. This number corresponds to the approximate
mass ratio of the discs applied in the vehicle to the one of the test rig.
For each simulation the DASSL solver is applied with an error tolerance of 0.001 [115]. The
computation time for a single brake application takes only a few seconds. Consequently,
the virtual field tests are completed in less than five minutes. In contrast, several days
are necessary to conduct the 88 brake applications experimentally.
5.2.3 Discussion of results
5.2.3.1 Brake deceleration
Figure 5.3 exemplifies measured and simulated deceleration profiles of the rail vehicle
for emergency brake applications with empty load condition and electro-
pneumatic actuation. The number of brake tests conducted for this brake scenario
are given in the first row of Table 5.2. Each grey curve in Figure 5.3 represents the
measured deceleration recorded during one of the tests, while the light red curves depict
the corresponding results of the deterministic simulations. In addition, the mean values
102
5.2 Deterministic analysis of field tests
of the single curves are illustrated by bold red and black lines. The latter are calculated
in analogy to Equation 3.2 and represent an estimate of the nominal deceleration profile
related to each initial velocity. The purple curves illustrate the relative error between
the mean values of measurement and simulation3.
Figure 5.3:
Comparison of measured and simulated deceleration for emergency brake
applications conducted with PN and empty load condition.
The figure reveals that the deceleration of the vehicle strongly changes during the stopping
brake process. According to the measurements, it takes values between 1.1m
s2and 1.6m
s2.
3The relative error is calculated analogously to Equation 4.10.
103
5 Analysis of vehicle brake applications
In case of brake applications with
v0
= 100
km
h
, an increase of more than 20% is observed,
whereas for
v0
= 160
km
h
a decline up to 15% is detected within the first 20 seconds.
This characteristic behavior underlines the strong relation between vehicle deceleration
and the friction characteristics observed on the test rig (see Section 3.2). Moreover, the
deceleration of the real train is subject to a large variation, even though the measurements
result from identical brake applications. In case of stopping brake tests conducted with
v0
= 120
km
h
, the deceleration recorded after 10s takes values between 1.3
m
s2
and 1.4
m
s2
.
The two brake applications conducted with
v0
= 160
km
h
exhibit even larger differences.
The dispersion of the simulated curves give an indication to which extent the dispersion
observed in the experiments can be attributed to fluctuations of initial conditions and
cylinder pressures. In particular for
v0
= 120
km
h
, the variance of the simulated curves is
however much smaller than the one of the measured curves. Consequently, there must be
additional sources of uncertainty prevailing during the experimental tests. This explains
why international standards, such as EN 16834 and UIC 544-1, require at least four valid
tests to estimate the nominal brake behavior based on experimental brake tests [63] [69].
Regarding the experimental data, this prerequisite is only fulfilled for brake applications
with 120km
h(see Table 5.2).
The subplots demonstrate that the simulations basically reproduce the characteristic
behavior of the vehicle deceleration recorded for each initial velocity. In particular for
v0
= 120
km
h
and
v0
= 140
km
h
, the nominal progressions coincide very well, while the
relative error remains below 5% during the braking process. For
v0
= 100
km
h
, the error
slightly increases within the last 5 seconds, since one of the two measured curves decreases
within this time interval. In case of
v0
= 160
km
h
, the mean of the simulations declines
stronger than the measured mean and the relative error reaches almost 10% at
t
= 15
s
.
Due the large dispersion observed for the measurements, it is however not possible to
determine if the mismatches observed for these initial velocities result from a systematic
model error or are simply caused by the small number of brake tests conducted for these
load cases. Accordingly, a quantitative assessment of the predicted deceleration based
on the available experimental data is only possible for brake scenarios performed with
v0
= 120
km
h
. For the other initial velocity, the meaningfulness of the comparison is limited
due to the small sample size (see Table 5.2).
The results of all brake scenarios listed in Table 5.2 are shown in Figures A.14 - A.22
in Appendix A.4. For
v0
= 120
km
h
, the relative errors between measured and simulated
deceleration remain smaller than 5%. Exceptions are found for emergency brake appli-
cations with empty load and indirect actuation (PN). It is striking that for this load
case two of the five experimental brake tests show a significantly smaller deceleration,
which shifts the average of the measurements downwards. However, the relative error
does not exceed 7% during the significant part of the braking process. For the load cases
conducted with
v0
= 100
km
h
,140
km
h
and 160
km
h
, the maximum relative error is 15%. With
regard to the limited sample size and the large dispersion of the experimental curves,
this result suggests that there are no significant deviations between measurement and
104
5.2 Deterministic analysis of field tests
simulation. To quantify the prediction quality of the model for these initial velocities,
a closer look at resistance forces, temperatures and brake forces is conducted in the
following.
5.2.3.2 Driving resistance
The measured and simulated deceleration within the first second of the braking process
is shown for each initial velocity in Figure A.23 of Appendix A.4. Until
t
= 0
.
4
s
, only
driving resistance forces act on the train. The subplots show the data of all brake
applications conducted with each initial velocity, i.e. the number of brake tests
corresponds to last row of Table 5.2. In analogy to Figure 5.3, the bold black and red
lines correspond to the mean values of measurement and simulation.
The means of the measured curves take values between 0
.
05
m
s2
and 0
.
1
m
s2
. This corresponds
to 4%-8% with respect to the total deceleration when the brake forces are fully developed
(about 1
.
3
m
s2
). Obviously, the means of the simulation coincide well with the measurements.
The maximum difference is about 0
.
02
m
s2
, i.e. a relative error of 1.5% with respect to a full
deceleration of 1
.
3
m
s2
. Accordingly, the are no significant errors regarding the prediction of
the mean driving resistance and the application of Equation 2.19 has proven its suitability
for the investigated multiple unit. Moreover, the parameters values of
R1,2,3
given in
Table 5.1 are confirmed. Nevertheless, the plots in Figure A.23 demonstrate that the
deceleration of the real vehicle caused by resistance forces is also subject to some scatter,
which is not covered by the deterministic simulation.
5.2.3.3 Disc temperature
Figure 5.4 compares the temperatures measured at the surface of one of the discs with
the simulation results. The illustration corresponds to the representation of data shown
in Figure 5.3. The subplots show the data of all brake applications conducted with
each initial velocity.
In general, measurements and simulations show a similar temperature evolution with
respect to time. Apart from the first 5 seconds, the relative error remains smaller than
15%. It is striking that the mean temperature values of the simulations are slightly
larger than the mean values of the measurements. Moreover, the measurements exhibit a
significantly larger dispersion.
It is important to mention that the measured curves represent the temperatures
recorded at a single position at the disc surface, whereas the simulated temperature
estimates the average surface temperature. As discussed in Section 3.4, the temperature
values at single positions may strongly differ from the average of all positions and are
subject to a high variability. These systematic differences presumably contribute to the
105
5 Analysis of vehicle brake applications
observed bias between measurements and simulations as well as the differences regarding
the dispersion of the single curves. Apart from this, the progressions of mean values from
simulation and experiment agree well and the comparison suggests that the simulation
provides a representative estimation of the thermal loads prevailing at the surface of the
brake discs during braking.
Figure 5.4:
Comparison of measured and simulated disc temperatures. Note that measure-
ments represent a single position at the disc, while simulations represent the
estimated average temperature of the surface Tfric.
106
5.2 Deterministic analysis of field tests
5.2.3.4 Effective coefficient of friction
Figure 5.5 illustrates the effective COF with respect to the traveled distance of all brake
applications conducted with each initial velocity. The plots demonstrate that
COF is subject to strong changes, in particular towards the end of stopping distance.
This behavior strongly resembles the progressions observed on the test rig, which is
illustrated in the right plot of Figure 4.3.
Figure 5.5:
Comparison of
µm,eff
(
t
)resulting from measurements and deterministic simula-
tions with respect to the stopping distance.
107
5 Analysis of vehicle brake applications
The measured progressions are generally well captured by the simulations, but the curves
of the experimental brake tests are subject to a larger dispersion in comparison to the
simulations. The dispersion of the simulated curves results from brake applications with
differing actuation types, clamping forces and braked masses, which affect the output of
the friction model. But it is obvious that these deterministic effects do not explain the
large dispersion of the experimental curves.
The mean curves of the simulations match well with the ones of the measurements.
Apart from the beginning and very end, the relative error remains smaller than 5 % for
v0
= 100
km
h
,120
km
h
and 140
km
h
. In case of
v0
= 120
km
h
, it is even smaller than 2%. For
v0
= 160
km
h
, the error increases up to 10% within the second half of the stopping distance.
As shown above, the friction forces are slightly underestimated for this initial velocity.
The large relative error observed at the very beginning of all subplots is not meaningful,
since pads and discs are not in contact. The deviations at the end are presumably related
to stick-slip phenomena arising at low velocities, as discussed in Section 4.1.4. However,
these deviations do not significantly contribute to the final stopping distance.
Mean values and standard deviations of the effective COF averaged over the braking
distance are depicted in Figure 5.6. The underlying samples are gained by applying
Equation 2.26 for each of the single curves illustrated in Figure 5.5. Subsequently, the
empirical moments are calculated for each initial velocity using Equation A.32 and
Equation A.34. The values corresponding to Figure 5.6 are given in Table A.9.
Figure 5.6:
Comparison of empirical mean values and standard deviations of
µm,eff
-samples
resulting from measurements and deterministic simulations. Confidence intervals
correspond to a confidence level of 95%.
108
5.2 Deterministic analysis of field tests
The mean values of the experimental samples decrease with increasing initial velocity
and take values between 0.34 and 0.4, which corresponds to the observations made at
the test rig (see Figure 3.10). As demonstrated by the bars in the left plot of Figure
5.6, the systematic behavior is well captured by the simulations, while the relative error
between the mean values is smaller than 1.5% for brake applications with
v0
= 120
km
h
and
v0
= 140
km
h
(see Table A.9). For
v0
= 100
km
h
and
v0
= 160
km
h
, the relative errors
are 3.3% and 5.5%, respectively. This indicates small systematic differences between
measured and simulated friction forces for these two initial velocities.
The samples of the experimental brake tests gained for each initial velocity exhibit a
standard deviation between 0.009 and 0.016. This range corresponds to 2%
−
5% with
respect to the mean values, which coincides with results found in literature (see Section
2.3.3). Obviously, the measured samples are subject to a larger standard deviation than
the samples gained from the simulation. This indicates that the frictional properties of
the disc brake units in the real vehicle are subject to a stochastic behavior, which is not
captured by the deterministic simulation model.
In conclusion, the results in Figure 5.5 and Figure 5.6 prove that the brake model
developed in this work provides a good estimation of the mean frictional brake forces to
be expected for each load case. The maximum error related to the prediction of ¯µeff (t)
is smaller 10% and smaller 6% for the prediction of ¯µm,eff , respectively.
5.2.3.5 Stopping distance
The stopping distances of all of the 88 brake applications resulting from experiments
and simulations are compared in Figure 5.7. The black line represents the ideal case
where measured and simulated values are identical. In addition, the relative errors are
illustrated by purple crosses. As shown, the maximum relative error is 11 %. For 16
cases, the errors take values between 5% and 10%. For 70 of 88 cases, the relative errors
between measurement and prediction are less than 5%.
In order to quantify the actual prediction accuracy of the model regarding the stopping
distance, it is necessary to assess the nominal stopping distance related to a certain load
case. Therefore, Figure 5.8 illustrates empirical mean values and standard deviations
of the stopping distances obtained from identical braking scenarios, which were
conducted at least four times during the experimental field tests. The mean
values illustrated in the upper plot of Figure 5.8 and given in Table 5.3 represent an
estimate of the nominal stopping distance. As shown, they take values between 450
m
and 550
m
depending on load, brake demand and actuation. The bar plot reveals that
the nominal stopping distance of measurements and simulations coincide very well, while
the confidence intervals related to the estimation of the empirical means overlap in most
109
5 Analysis of vehicle brake applications
Figure 5.7: Comparison of measured and simulated stopping distances.
cases. The maximum deviation is 28
m
, which corresponds to a relative deviation of 5%.
For three of seven load cases, the deviation is 1%. This is an excellent agreement.
The lower plot in Figure 5.8 displays the empirical standard deviation of the samples
resulting from measurements and simulations. The standard deviation of the measured
stopping distances takes values between 8
m
and 25
m
. This corresponds to a range of
variation of 3
σsst
= 6% -15% with respect to the nominal stopping distance
4
(see also
Table 5.5). Since the maximum relative error illustrated in Figure 5.7 is within this
range, there are no significant errors regarding the predicted stopping distance. However,
it is not possible to quantify the actual prediction accuracy of the nominal braking
distance for load cases with
v0
= 100
km
h
,140
km
h
and 160
km
h
, since the sample size of the
experimental data is to small.
Finally, the lower plot in Figure 5.8 shows that the standard deviations resulting from
the simulation are significantly smaller than those of the measurements. This confirms
the previous results and indicates that the deterministic model is not sufficient to predict
the variance of the brake performance, i.e. a stochastic approach is necessary for this
task. Nevertheless, the deterministic model has proven its suitability to estimate the
nominal braking distance related to a certain load case.
4
Note that the stopping distance has been corrected according to [63], in order to exclude additional
scatter caused by fluctuations of the initial velocity.
110
5.2 Deterministic analysis of field tests
Table 5.3:
Nominal stopping distances resulting from det. simulations
¯sst,sim
and measure-
ments
¯sst,meas
including length of confidence intervals ∆
L
related to the estimation
of ¯sst,meas with respect to a confidence level of 95%.
Load case ¯sst,sim
[m]
¯sst,meas
[m]
∆L
2[m] ¯sst,meas −¯sst,sim
[m]
|¯sst,meas−¯sst,sim|
¯sst,meas
V0120_empty_emergency_EP
439 436 8 -3 1%
V0120_empty_emergency_PN
482 509 22 28 5%
V0120_empty_service_EP 501 510 9 9 2%
V0120_nominal_emergency_EP
464 454 10 -10 2%
V0120_nominal_service_EP 528 531 23 3 1%
V0120_maximum_emergency_EP
470 455 23 -15 3%
V0120_maximum_service_EP 537 541 19 4 1%
Figure 5.8:
Comparison of empirical mean values and standard deviations of stopping distances
resulting from measurements and deterministic simulations for brake applications
with v0= 120km
h. Confidence intervals correspond to a confidence level of 95%.
111
5 Analysis of vehicle brake applications
5.2.4 Interpretation of results
Overall, the previously presented results demonstrate a good agreement between experi-
mental and virtual brake tests regarding the mean behavior of each load case. Nevertheless,
the comparison reveals minor deviations between measured and simulated frictional brake
forces for
v0
= 100
km
h
and
v0
= 160
km
h
. Since the applied friction model has proven to
reproduce the instantaneous friction behavior observed on the test rig with an accuracy
smaller than 2%, it is assumed that differences between laboratory experiments and
vehicle brake tests contribute to the small discrepancies between vehicle measurements
and simulations. To identify the root-causes, potential sources of deviations between test
rig experiments and vehicle brake tests are briefly discussed with regard to the investi-
gated data sets. As shown in Figure 5.9, it is differentiated between environment-related,
component-related and test-related sources.
Deviation of frictional brake forces
Environment-related sources
Air humidity
Ambient temperature Moisture/ice
Contaminations
Air flow
Component-related sources
Geometry Material Load history
Test-related sources
Load cases
Batch
Wear
Bedding
Figure 5.9:
Classification of potential sources of deviations between test rig experiments and
vehicle brake tests.
5.2.4.1 Environment-related sources
In general, the ambient conditions regarding temperature, moisture, air humidity and
contamination may be subject to strong changes in operation and impair the frictional
properties. In this case, the influence of moisture and ice as well as contaminated pads
may be neglected, since all vehicle brake tests were conducted during dry and clean
conditions. However, ambient temperature and relative humidity cannot be controlled
during the experimental vehicle tests, in contrast to the idealized conditions at the test
rig. As illustrated in Figure A.24 in Appendix A.4, the temperature varies between 2
◦C
and 22
◦C
during the experimental vehicle test campaign, while the corresponding relative
humidity takes values between 50% and 92%, respectively. Whereas the influence of the
ambient temperature is taken into account of the prediction, the relative humidity is not
considered within the friction model as an additional influencing factor (see Equation
4.2). Eriksson et al. investigated that the COF of friction materials decreases, if the
relative humidity of the surrounding air exceeds 80% [128]. However, they also found
that the COF is hardly affected by the relative humidity, if it varies within an interval
from 20% to 80%. As illustrated in Figure A.24, only 10 of the 88 brake applications
112
5.2 Deterministic analysis of field tests
were conducted at a relative humidity larger 80%. The influence of the relative humidity
is thus excluded as main source for systematic deviations between measurement and
simulation. Nevertheless, it needs to be assumed that fluctuations of the relative humidity
contribute to the large dispersion of the frictional brake forces observed during the vehicle
tests.
As the brake discs are mounted underneath the vehicle, it is presumed that the air
flows around the brake discs differ from the conditions in the laboratory. This can
lead to different thermal loads during braking. The good agreement of measured and
simulated disc temperatures demonstrated in Figure 5.4 suggests that potential deviations
caused by differing conventional heat flows play a subordinate role for the investigated
stopping brake applications. Nevertheless, during other brake scenarios, such as repeated
braking without cooling phases or drag-braking on a long downhill track, the influence of
convection processes is assumed to increase significantly.
5.2.4.2 Test-related sources
As discussed in Section 5.1, the load cases investigated during test rig and vehicle
experiments are similar. In addition, the friction pairings applied in the vehicle and on
the test rig were grinded-in before the actual brake tests by conducting prior bedding
brake applications. Hence, they were in a conditioned state during testing. Moreover,
both test campaigns were conducted with a similar load history, i.e. the brake tests
were performed with frequent load case changes and cooling periods between each brake
application. Accordingly, test-related sources are excluded as root causes of systematic
deviations between measurement and simulation. However, the results indicate that the
alternating load histories strongly contribute to the large dispersion of frictional brake
forces observed for both vehicle and test rig experiments.
5.2.4.3 Component-related sources
Despite a slightly differing geometry of the brake disc, the friction pairings applied in
vehicle and on the test rig are the same. This means that material and geometry of
the brake pads are identical. However, it cannot be excluded that the compositions
of the pad materials slightly differ from each other due to production tolerances, even
though they are nominally the same material. These batch-related differences could be
an explanation for the small differences between simulation and vehicle brake tests. As
discussed in [49] and [50], the influence of the batch of friction materials on the brake
forces is known in the railway branch. This emphasizes that a friction model identified
on a test rig may only be as a accurate as the comparability of pad properties allows.
This is an important finding with regard to the transferability of results gained on the
component test rig to the entire vehicle.
113
5 Analysis of vehicle brake applications
5.3 Stochastic analysis of field tests
The goal of this section is to retrace the large dispersion of the brake performance
observed during the experimental vehicle tests and to draw conclusions regarding the
stochastic phenomena occurring during brake applications of rail vehicles equipped with
friction brakes.
5.3.1 Monte-Carlo Simulation
The previous results demonstrated that a deterministic model approach is not sufficient
to reproduce the dispersion of the vehicle brake performance. Therefore, it is proposed
to apply MCS based on a stochastic train model. This means that certain sub-models
illustrated in Figure 5.2 are considered stochastic models. With respect to the previous
findings it is proposed that
•the COF between pads and discs as well as
•the driving resistance of the multiple unit
are the main drivers for the observed scatter of the vehicle brake performance. Accordingly,
all 16 pad/disc friction models and the resistance model of the train model
illustrated in Figure 5.2 are considered as stochastic models with uncertain model
parameters. To obtain a sufficiently large sample size without drastically increasing the
computational effort, each of the 88 brake scenarios is simulated 50 times during the
MCS. This yields a total of 50 ·88 = 4400 simulations to be conducted.
5.3.1.1 Distribution of resistance parameters
To simplify the problem, only
˜
R1
is considered as random parameter of the resistance
model, whereas
R2
and
R3
remain deterministic quantities (see Equation 2.19). The
distribution of
˜
R1
is deduced from the scatter of the deceleration data recorded at the
beginning of the braking process, which is illustrated in Figure A.23. Taking into account
the respective mass of the train, it is concluded that
˜
R1
follows a Gaussian distribution
with
˜
R1∼ N(¯
R1= 2500N, 1σR1≈1600N).(5.3)
The mean value
¯
R1
corresponds to the nominal value given in Table 5.1. During each
simulated brake application
R1
takes a random value according to this distribution, i.e.
the velocity depended resistance forces FR(v)are shifted along the vertical axis.
114
5.3 Stochastic analysis of field tests
5.3.1.2 Distribution of friction parameters: Two approaches
To simulate the uncertainty of the COF between pads and discs, the stochastic friction
model derived in Section 4.4.2 based on two uncertain parameters
˜c1
and
˜µ0
is applied.
In contrast to the investigation conducted on the test rig, the train is equipped with
multiple disc brake units. This fact rises the question, whether the frictional properties
of the brake units applied in the vehicle are either
1.
uncorrelated, i.e. the COF of each brake unit is a independent random quantity, as
proposed in [99] and visualized in the left plot of Figure 5.10, or
2. partially correlated, i.e. the COF of all 16 brake units have a global behavior with
respect to each brake application, as visualized in the right plot of Figure 5.10.
The second hypothesis bases on the fact that all brake units of the vehicle have a common
load history and are applied under similar environmental conditions. This promotes the
establishment of similar topographies and frictional properties at all 16 brake pads. Due
to the continual topography adaptions evoked by the subsequent brake applications and
changes of ambient conditions, such as the relative air humidity shown in Figure A.24,
the resulting global friction properties change from brake application to brake application.
This additional global scatter contributes to the dispersion of
µm,eff
, as visualized in
the right plot of Figure 5.10. Since the test rig experiments reveal a large dispersion of
friction properties despite constant ambient conditions, it is assumed the extent of the
global scatter is primarily driven by the alternating load history.
Uncorrelated COF (MCS100)
𝜇𝑚
Brake appl. 1 Brake appl. 2 Brake appl. 3
Brake
unit 1
Brake
unit 16
Brake
unit 1
Brake
unit 16
Brake unit 1
4
3
2
5
8
7
6
9
12
11
10
13
16
15
14
Correlated COF (MCS75/25)
𝜇𝑚
Brake appl. 1 Brake appl. 2 Brake appl. 3
Brake
unit 1
Brake
unit 16
Brake
unit 1
Brake
unit 16
𝝁𝒎,𝒆𝒇𝒇
𝝁𝒊
𝝁𝒎,𝒆𝒇𝒇
𝝁𝒊
Figure 5.10: Illustrative comparison of correlated and uncorrelated COF.
115
5 Analysis of vehicle brake applications
To investigate the influence of both stochastic models on the variance of the brake
performance two different MCS, namely MCS100 and MCS75/25, are conducted. In
case of MCS100, the random parameters
˜c1,i
and
˜µ0,i
of each of the
i
= 1
−
16 friction
models take a unique random value during each simulated brake application, i.e. they are
100% individual for each brake unit. The distributions correspond to the ones identified
in Section 4.4.2:
˜c1,i ∼ N(¯c= 0.011,1σc1= 0.001) (5.4)
˜µ0,i ∼ N(¯µ0= 0.3715,1σµ0= 0.01) (5.5)
In case of MCS75/25, the scatter of the frictional behavior is a superposition of global
and individual scatter, as illustrated in the right plot in Figure 5.10. Since the load cases
of the subsequent brake applications are subject to frequent changes, it is assumed that
the extent of global scatter is larger than the individual scatter. To simulate global
scatter, two additional random parameters
˜c1,g
and
˜µ0,g
are introduced, which take a
random value for each simulated brake application according to
˜c1,g ∼ N(¯c= 0.011,1σc1= 0.75 ·0.001),(5.6)
˜µ0,g ∼ N(¯µ0= 0.3715,1σµ0= 0.75 ·0.01).(5.7)
Their mean values correspond to the ones identified on the test rig (see Section 4.4.2).
The standard deviation corresponds to 75% of the identified standard deviation. This
implies that the major but not entire part of the scatter observed on the test rig is related
to the load history of the testing procedure (see A.2.1). In analogy to MCS100, the
individual scatter is modeled by two random parameters, which take a unique random
value for each brake unit i= 1 −16 and simulated brake application:
˜c1,i ∼ N(¯c= ˜c1,g,1σc1= 0.25 ·0.001),(5.8)
˜µ0,i ∼ N(¯µ0= ˜µ0,g,1σµ0= 0.25 ·0.01).(5.9)
In contrast to MCS100, their mean values correspond to the global values
˜c1,g
and
˜µ0,g
of the current simulation. In doing so, the individual scatter is superimposed to the
global behavior. The standard deviations of the individual scatter are set to 25% of the
identified values. Consequently, the ratio of global to individual parameter scatter is 3:1,
i.e. the global dispersion predominates.
116
5.3 Stochastic analysis of field tests
5.3.2 Discussion of results
5.3.2.1 Effective coefficient of friction
Figure 5.11 illustrates mean values and standard deviation of
µm,eff
resulting from
measurements, deterministic simulation, MCS100 and MCS75/25, in analogy to the
illustration in Figure 5.6. The comparison in the upper plot demonstrates that the
mean values of the MCS corresponds to the ones of the deterministic simulation. This
proves that the expected values of the COF resulting from the MCS converge against
the deterministic values, if the number of simulations is sufficiently large.
Figure 5.11:
Comparison of empirical mean values and standard deviations of
µm,eff
samples
resulting from measurements, deterministic simulation as well as MCS100 and
MCS75/25 [E2]. Confidence intervals correspond to a confidence level of 95%.
117
5 Analysis of vehicle brake applications
The lower subplot compares the standard deviations. The corresponding values are listed
in Table 5.4. Obviously, there are significant differences between MCS100 and MCS75/25
(compare yellow and purple bars). The values of MCS100 are hardly larger than the ones
of the deterministic simulation (compare yellow and red bars). This is caused by the
mutual compensation of the independent random values for each brake unit. This effect
leads to a comparatively low variance of µm,eff .
In contrast, the standard deviations of MCS75/25 are significantly larger than the ones
of MCS100. As indicated by the right plot in Figure 5.10, the global scatter increases
the dispersion of
µm,eff
. The values of MCS75/25 are in very good accordance with the
experiments (compare purple and grey bars). According to Table 5.4, the maximum
deviation with respect to the samples of identical initial velocities is 0.003 for
v0
= 100
km
h
.
For the samples of v0= 140km
hand v0= 160km
hthe values are nearly identical.
Table 5.4:
Comparison of empirical standard deviation of effective COF averaged over the
braking distance 1
σµm,eff
resulting from samples of experimental vehicle brake tests,
deterministic simulation, MCS100 and MCS75/25 (error=1σµm,meas −1σµm,sim).
Measurement Det. Sim. MSC100 MSC75/25
Load case ¯µm1σµm
1σµ,m
¯µm1σµmerror 1σµmerror 1σµmerror
v0= 100km
h0.390 0.009 2% 0.006 0.003 0.007 0.002 0.011 -0.003
v0= 120km
h0.383 0.013 3% 0.008 0.005 0.010 0.003 0.014 -0.001
v0= 140km
h0.361 0.015 4% 0.009 0.006 0.011 0.005 0.015 0.000
v0= 160km
h0.345 0.016 5% 0.010 0.007 0.012 0.005 0.016 0.000
v0
= 100
−
160
km
h0.376 0.020 5% 0.026 -0.005 0.026 0.006 0.028 -0.008
5.3.2.2 Stopping distance
Figure 5.12 compares standard deviations of the stopping distance resulting from experi-
mental brake tests with the ones obtained by deterministic and stochastic simulations. In
analogy to Figure 5.8, the comparison only considers load cases, which were conducted at
least four times during the experimental testing. As shown, the values of MCS75/25 are
in good accordance with the experimental results, whereas the samples of MCS100 are
subject to a significantly smaller standard deviation. Table 5.5 contains the corresponding
values as well as the errors between measured and simulated standard deviations. It
is found that MCS75/25 provides the smallest errors for six of seven load cases, while
the largest deviation of MCS75/25 is 11
m
. Accordingly, it is necessary to consider the
presence of global scatter of friction forces within the MCS in order to predict of the
standard deviation of the stopping distance observed for the investigated multiple unit.
118
5.3 Stochastic analysis of field tests
Figure 5.12:
Comparison of standard deviations of stopping distances resulting from mea-
surements, deterministic and stochastic simulations for brake applications with
v0= 120km
h[E2]. Confidence intervals correspond to a confidence level of 95%.
Table 5.5:
Empirical standard deviations of stopping distances resulting from samples of
experimental vehicle brake tests, deterministic simulation, MCS100 and MCS75/25
(error=1σsst,meas −1σsst,sim).
Measurement Det. Sim. MSC100 MSC75/25
Load case ¯sst
[m]
1
σsst
[m]
1σsst
¯sst
1
σsst
[m]
error
[m]
1
σsst
[m]
error
[m]
1
σsst
[m]
error
[m]
V0120_empty_emergency_EP 436 9 2% 2 6 6 3 11 -2
V0120_empty_emergency_PN 509 18 4% 5 13 7 11 12 6
V0120_empty_service_EP 510 12 2% 4 8 7 5 13 -1
V0120_nominal_emergency_EP
454 8 2% 3 5 6 2 13 -5
V0120_nominal_service_EP 531 25 5% 4 21 7 18 14 11
V0120_max_emergency_EP 455 15 3% 4 10 7 7 13 2
V0120_max_service_EP 541 19 3% 7 12 9 9 15 3
5.3.3 Interpretation of results
The stochastic analysis provides valuable indications regarding the dispersion of the brake
performance of a disc-braked rail vehicle. As shown, the assumption of pure individual
scatter, as proposed in [50], is not capable to explain the large variance observed during
the experimental vehicle tests. The results rather suggest that the friction forces of the
investigated multiple unit are subject to a superposition of global and individual scatter,
as simulated by MCS75/25. With respect to the findings, a simplified stochastic model
is proposed to describe the stochastic behavior of the effective COF averaged over the
119
5 Analysis of vehicle brake applications
braking distance:
˜µm,eff =µm,det + ˜µm,glob + ˜µm,ind.(5.10)
µm,det
represents the deterministic behavior, i.e. the systematic dependency between the
friction properties and load case.
˜µm,glob
represents the global scatter. It is a random
variable with zero mean, which changes during each brake application.
˜µm,ind
is an
independent random variable for each brake unit of the vehicle, whose means correspond
to the current value of
˜µm,glob
. Assuming that
µeff
represents the mean COF of all
nb
brake units5, the variance of ˜µm,eff may be approximated by [129]
σ2
µm,eff ≈σ2
µm,det +σ2
µm,glob +σ2
µm,ind
nb
.(5.11)
Accordingly, there are three major contributors to the variance of frictional brake forces
of rail vehicles:
•
The systematic variance
σ2
µm,det
represents the dispersion of the the nominal be-
havior related to all load cases consolidated in one sample. According to the samples
of the deterministic simulation illustrated in Figure 5.11,
1σµm,det ≈0.006 −0.01
(see red bars of each initial velocity). If all velocities are consolidated, the contri-
bution of the systematic variance is strongly increased (see last bars on the very
right).
σµm,det
= 0 represents the ideal case that initial velocity, clamping forces
and braked masses are absolutely identical. Hence, the extent of 1
σµm,det
is driven
by the systematic behavior of the frictional properties as well as the selection of
load cases to be considered.
•
The global variance
σ2
µm,glob
represents the variability of the common behavior of
the brake units in a train, which affects all pads of the vehicle to the same extent.
Based on the results of MCS75/25, it is deduced that
1σµm,glob ≈0.01
. The extent
of the global scatter is strongly related to the load history of the vehicle as well as
the variability of the environmental conditions in operation. During an alternating
load history the topographies of the brake pads are subject to frequent changes,
which yields a large global variance. In contrast, the conduction of consecutive
identical brake applications will establish a certain topography at all pads of the
vehicle. In this case, the share of the global variance would decrease.
•
The individual variance
σ2
µm,ind
represents the scatter between the single brake
units of a vehicle during one brake application. In case MCS75/25,
1σµm,ind ≈0.003
.
As shown in Equation 5.11, the contribution of the individual scatter to the total
variance is scaled by the number of brake units of the vehicle. Accordingly, its
5According to Equation 5.2, this means that clamping forces FC,i are assumed to be equal.
120
5.3 Stochastic analysis of field tests
influence on the variance of the vehicle brake performance decreases with the
number of brake units in the vehicle and thus depends on the configuration of the
brake system.
Note that fluctuations caused by batch-related variabilities of friction materials may also
be included in this classification scheme. Either the influence is respected within the
deterministic model, or added to the global variance.
According to the introduced variances, the extent of each source of uncertainty depends
on the load cases to be consolidated, the operational profile and the brake system
configuration. Hence, the given values are only valid for the investigated multiple
unit and the corresponding testing program. Nevertheless, the findings of this
work allow to draw general conclusions regarding the correlation between the dispersion
of the brake performance and the operational profile as well as the configuration of a rail
vehicle. These conclusions are summarized by the following postulations:
•
If the variance of the load history of a train with fixed configuration increases, the
dispersion of the vehicle brake performance will increase.
•
Conversely, if multiple identical brake applications are conducted in series, the
dispersion of the brake performance will decrease and converge to a certain value.
The number of brake applications, which lead to the establishment of stable
topography and friction properties, depends on the load case and the applied
friction material, as discussed in [87].
•
If the vehicle configuration is subject to frequent changes, as it is the case for freight
trains or coaches, which consist of single wagons with identical friction material
but different load histories, the performance dispersion is smaller. In this case, the
variance decreases by the number of wagons, as also found in [50] and [48].
•
As the common load history of an assembled train increases, i.e. a certain number
of vehicle brake applications is conducted, the friction properties equalize and the
variability of the load history becomes a significant factor driving the dispersion of
the brake performance.
121
5 Analysis of vehicle brake applications
5.4 Key findings of vehicle simulations
The analysis of measurement data gained from experimental field tests reveals that the
brake performance of a disc-braked multiple unit strongly depends on the load case.
For
v0
= 100
km
h
, the deceleration rises up to 20% during braking, whereas for
v0
= 160
km
h
a decline of up to 15% is observed. Moreover, the brake capability is
subject to a strong dispersion, even though the load case is identical. The stopping
distance of brake applications conducted with
v0
= 120
km
h
varies up to 14%
(3σ).
A train model based on a lumped mass and the brake model developed in this work is
applied to predict the nominal behavior of the longitudinal vehicle dynamics. The relative
error of the predicted instantaneous deceleration does not exceed 10%. In particular
for
v0
= 120
km
h
, measured and simulated friction behavior coincide very well with a
relative error of
µeff
(
t
)less than 2%. Minor deviations between measured and simulated
friction forces are found for
v0
= 100
km
h
and
v0
= 160
km
h
. As the comparison of measured
and simulated disc temperatures reveals, the thermal loads are well captured by
the model. It is therefore assumed that the discrepancies of friction forces result from
different material batches of the pads applied in the vehicle and those analyzed on the
test rig. The largest relative error of predicted and measured stopping distance
is 11%. However, this error lies within the range of variation observed for the measured
stopping distance and is thus not of significance. For
v0
= 120
km
h
, the prediction
accuracy of the nominal stopping distance is 5%. This represents an excellent
accordance between deterministic simulation and experimental brake tests.
In conclusion, the brake model developed in this work has proven its suitability to
estimate mean brake performance of a rail vehicle related to each load case. Hence, it is
possible to transfer the knowledge gained on a test rig to the brake performance of a
vehicle. However, a friction model identified on a test rig can only be as a accurate as
the comparability of pad properties allows.
The stochastic analysis suggests that the frictional properties of the brake units
applied in the vehicle correlate to a certain extent. In contrast, if the COF is
considered as independent random variable for each brake unit, the dispersion is strongly
underestimated due to the mutual compensation of all 16 brake units. Applying the
hypothesis of global scatter in combination with the stochastic friction model developed
in this work, leads to a very good estimation of the dispersion of the vehicle brake
performance. Based on the results, it is concluded that the variance of the friction
forces has three major sources namely systematic variance,global variance and
individual variance. Moreover, it is proposed that the extent of each source depends on
the operational brake profile, brake conditions and the configuration of the rail vehicle.
122
6 Suggestions for brake system design
and operation
The goal of this chapter is to transfer the findings of the previous chapters to the brake
system design and rail operation. Figure 6.1 illustrates different design steps and phases
of the product life-cycle of a brake system, which are closely related to the prediction
of pad friction characteristics. As indicated, an identified friction model is supposed to
improve the results obtained during each phase. In this context, the term friction model
is manifold and covers all kinds of characterization, which exceed the usage of a single
value for the COF. In the following, suggestions emerging from the results of this work
are made for each phase including the model identification.
Model identification
Brake tests on
full-scale test rig
•Test program
•Load cases A,B,C
Friction
pairing X/Y
Data processing
•Averaged COF: 𝝁𝒎
•Empirical statistics
for each load case
Sub-model
identification
•Friction/Temperature
•Parameter variance
Brake model
verification
•Deterministic
•Stochastic (MCS)
Friction
model of
pairing X/Y
Update
System Design Virtual testing Probabilistic
analysis
System
operation
•Brake assessment
(EN 14531 1/2)
•Thermal loads
•System
optimization
•commissioning &
homologation
•Monte-Carlo
Simulation (UIC
B126 - 407/ 412/
426)
•Braking Curves
(ERTMS / ETCS
SRS Subset 026-3)
Brake system
Innovations
•Controlled
emergency brake
(DCC)
Figure 6.1:
Identification and application of a friction model during brake system design and
operation.
123
6 Suggestions for brake system design and operation
6.1 Model identification
The parameters of the friction model identified in Chapter 4 are only valid for the
investigated friction material. To apply the friction model developed in this work for
other pad materials, the proposed identification procedure needs to be conducted based
on a corresponding set of experimental data. In this case, a very large amount of data was
available for the identification. In scope of limited time, costs and test rig capacities, an
important question arises: What is the smallest possible number of brake tests required
for a reliable identification of friction characteristics regarding both deterministic and
stochastic behavior?
Figure 6.2 illustrates how the confidence intervals related to the estimation of empirical
mean value and standard deviation change with the sample size. The corresponding
equations are given in Appendix A.2.6. The curves in the left plot are interpreted as
follows. For a Gaussian distributed population with a standard deviation of 2% (e.g.
¯µm
= 0
.
4and 1
σµm
= 0
.
008), at least three brake applications are necessary to receive a
confidence interval related to the estimation of the mean value, which is smaller than
10%, i.e. an estimation error of
±
5%. In contrast, if 1
σµm
= 6% (e.g.
¯µm
= 0
.
34 and
1
σµm
= 0
.
02)at least 8 brake applications are required to receive the same estimation
accuracy. Note, the distribution of the samples in brackets corresponds to the ranges of
statistical properties of µmidentified in Chapter 3 (see Table 3.4).
Figure 6.2:
Relative lengths of confidence intervals related to estimation of empirical mean
value and standard deviation with respect to sample size. Intervals correspond to
a confidence level of 95%.
The right plot illustrates the confidence intervals related to the estimation of the standard
deviation. As shown, at least 12 brake applications are required to receive a confidence
124
6.2 System design
interval that is smaller than 100% with respect to the actual standard deviation
1
. For
a confidence interval smaller than 50%, more than 40 brake applications are necessary.
Hence, a very large sample size is required to precisely quantify the dispersion of a
population.
EN 15328 proposes to conduct three brake applications for so-called Priority 1 scenarios,
i.e. load cases with the largest clamping force and braked mass (see Appendix A.1.1).
According to Figure 6.2, the corresponding confidence interval related to the estimation
of the mean value is 10% for a populations with 1
σµm
= 2%. For a population with
1
σµm
= 6%, it is 28%. Irrespective of the population, the confidence interval of the
standard deviation is larger than 500%, if data of only three brake applications is available.
The estimations are thus subject to large uncertainties. To exploit the data recorded
during the certification programs, such as EN 15328, it is suggested to enlarge the
number of brake applications conducted for Priority 1 scenarios up to at least 10 tests
for each load case. This allows to retrieve a good estimation of mean values and a fair
estimation of the standard deviation without drastically increasing the experimental
effort for testing. In addition, it is recommended to apply an alternating load history
during the test program to estimate the largest dispersion of the COF to be expected
in operation. For instance, a sequence in which the initial velocities of successive brake
applications changes according to v0= 120km
h, 50km
h, 160km
h, 80km
h, 200km
h, 140km
h.
6.2 System design
With respect to the findings of the previous chapters, it is strongly recommended to
consider the dependency of friction forces on the load case from the very beginning of the
brake system design. The use of a single value to characterize
¯µm
irrespective of the load
case may yield large deviations between predicted and actual braking performance. For
instance, the application of the average value of all investigated load cases on the test rig
(
¯µm
= 0
.
364
2
) yields a nominal braking distance of 858
m
for load cases V0160_Case1.
In contrast, if the nominal value identified for this load case is used (
¯µm
= 0
.
309
3
), the
predicted braking distance is 1008
m
. Consequently, the nominal braking distance is
underestimated by 15% when using the average value. Regression models to describe
¯µm
(
v0
), as shown in Appendix A.2.7, are a suitable method to consider the systematic
influences of the load case in the prediction, in particular for brake calculations based on
EN 14531-1 during early design phases. However, the validity of regression models for
¯µm
is limited to certain clamping forces and braked masses. To interpolate to load cases
1
Note that the relative length of the confidence interval
∆L
σµm
is independent of the actual dispersion of
the population. Moreover, the interval is not symmetric with respect to the point estimation.
2see Table 3.4.
3see Table 3.4.
125
6 Suggestions for brake system design and operation
or brake scenarios not investigated on the test rig, the usage of instantaneous friction
models, as presented in Chapter 4, is strongly recommended.
As discussed in [119], a change of friction properties between pad and disc during the
braking process strongly affects the temperature evolution in the brake components
due to the varying frictional heat input. It is therefore recommended to consider the
instantaneous friction characteristics when estimating the thermomechanical performance
limits of the brake components. On the one hand, this reduces the risk of an unsatisfactory
system design. On the other hand, it offers the potential to fully exploit the performance
limits and to avoid an oversized design.
6.3 Virtual testing
The ability to predict instantaneous friction forces is indispensable to conduct meaningful
virtual vehicle brake tests in scope of system optimization, commissioning or homolgation.
However, it is important to guarantee a certain degree of comparability between laboratory
testing and vehicle operation with respect to the potential sources of deviations listed
in Figure 5.9 when applying friction models, which were identified on a component test
rig.
Moreover, the verification of simulation results against results of experimental vehicle
brake tests remains a very important task and should be carried out as soon as data
are available. To allow for a meaningful verification, it is recommended to conduct a
sufficiently large number of experimental tests. Regarding the estimation of the nominal
behavior, the suggestion made by EN 16834 and UIC 544-1 of at least four brake tests is
reasonable. According to the left plot on Figure 6.3, this leads to a relative length of the
confidence interval related to the estimation of the nominal braking distance of 18%, i.e.
an error of
±
9%, if a standard deviation of 6% is assumed. To reduce this interval to
less than 10%, i.e. an error of ±5%, at least 8 brake applications are necessary.
Apart from the prediction of the nominal behavior, it is recommended to additionally
apply virtual testing for the analysis of potential deviations from the average behavior.
The application of MCS based on uncertain model parameters, as presented in Section
5.3, allows to investigate the dispersion of brake forces and vehicle deceleration during
the braking process. To conduct a very large number of simulations within a tolerable
computational effort, as it is necessary for the derivation of safety margins applied in
operation, it is recommended to use brake models based on mean values, as shown in the
next section.
126
6.4 Probabilistic analysis
6.4 Probabilistic analysis
With respect to the findings in Section 5.3, two recommendations are derived regarding
the consideration of friction-related scatter within the probabilistic analysis:
•
The applied brake model should allow for the implementation of global and indi-
vidual scatter of the COF.
•
The probabilistic analysis should be conducted for different initial velocities and
take the systematic changes of the friction forces into account of the prediction.
To demonstrate how these assumptions affect the derivation of safety margins for operation
with ETCS, an exemplary probabilistic analysis is conducted. The procedure for the
derivation of the correction factors follows the one presented in Section 2.3.3. With regard
to the implementation of ETCS, the probabilistic analysis shall only consider fluctuations
expected during good adhesion conditions. To take into account the scenario-dependent
scatter of friction forces in the analysis, the procedure is conducted with respect to
different initial velocities, i.e.
v0
= 100
km
h,
120
km
h,
140
km
h
and 160
km
h
. The vehicle to be
investigated corresponds to the multiple unit presented in Chapter 5. In order to handle
the large amount of simulations during the MCS, a computational efficient brake model
based on mean values is proposed (see Section 2.3.2.1). Applying Equation A.3 and
Equation 2.15 and neglecting additional resistance forces yields:
|a|E=
8
X
j=1
2
X
i=1
rfric,ij
rwheel,j ·(pC,j ·AC,ij −FS,ij)·ηC,ij ·iT,ij ·µm,ij
mdyn,j
(6.1)
Based on this simplified brake model, 10
6
simulation shall be performed during each
MCS, while the model parameters vary according to the distributions given in Table
6.1. All parameters not listed are deterministic quantities and correspond to the values
given in Table 5.1. The assumptions regarding the distributions of
pC
,
ηC
and
mdyn
are adopted from literature [47][50][48]. Note that index
j
in Equation 6.1 represents
the axle, while index
i
represents the brake unit applied at each axle. Accordingly,
pC
and
mdyn
vary with respect to every axle, whereas
ηC
and
µm
take a random value for
every brake unit. The distributions of
pC
,
mdyn
and
ηC
are independent of the initial
velocity. In contrast, nominal value as well as standard deviation of
µm
are affected by
the initial velocity. Moreover, the random values of
µm
are a superposition of global
and individual scatter, in analogy to the right plot in Figure 5.10. 1
σµm,ind
and 1
σµm,glob
are set in a manner that the resulting standard deviations of 1
σµm,eff
correspond to
the values identified in Chapter 5 (see second column of Table 5.4), if Equation 5.11 is
applied. In this case, the share of the deterministic variance is included the values of the
global variance.
127
6 Suggestions for brake system design and operation
For sake of comparison, one additional MCS is conducted, which does not take the
systematic differences between the friction properties into account. In other words:
Regardless of the initial velocity, the nominal value of
µm
corresponds to the average
of the value of the single load cases (see row
v0,all
= 100
−
160
km
h
in Table 6.1). This
corresponds to the reference procedure and means that a larger variance of
µm
has to be
considered for the probabilistic analysis.
Table 6.1:
Characterization of distributed parameters applied during MCS (Nominal values of
pCand mdyn correspond to mean values of all axles).
Parameter Nominal value Distribution Standard deviation / interval
ηC0.95 uniform ±5%
pC4bar normal 1σpC= 2%
mdyn 18000 kg uniform ±3%
µm(v0= 100km
h)0.39 normal 1σµm,ind = 0.8%,1σµm,glob = 2.3%
µm(v0= 120km
h)0.38 normal 1σµm,ind = 0.8%,1σµm,glob = 3.4%
µm(v0= 140km
h)0.36 normal 1σµm,ind = 0.8%,1σµm,glob = 4.2%
µm(v0= 160km
h)0.35 normal 1σµm,ind = 0.9%,1σµm,glob = 4.6%
µm(v0,all = 100 −160km
h)0.37 normal 1σµm,ind = 0.8%,1σµm,glob = 5.3%
Figure 6.3 illustrates the results of the probabilistic analysis. The shaded areas represent
the confidence intervals related to the estimation of
P
(
|a|
|a|n
)based on the results of the
five MCS. The figure exposes large differences between the results of the different MCS
and reveals the strong impact of the friction-related scatter on the derivation of the
correction factor. The larger the assumptions made for 1
σµm,glob
, the larger the dispersion
of |a|
|a|nand thus the smaller Kdry.
To quantify these differences, Table 6.2 lists the nominal deceleration and
Kdry
derived
from Figure 6.3
4
for each initial velocity with respect to a safety target of 10
−4
, i.e.
EBCL
= 4. In addition, the resulting save deceleration according to Equation 2.24
is given
5
. The table shows that the diversification of frictional scatter with respect to
the initial velocity allows to realize significantly smaller correction factors. In case of
v0
= 100
km
h
, the value of
Kdry
is 0.90, which is 14% larger than the value of
v0,all
(0.79).
For the other load cases, the differences to the reference vary between 4% and 9%.
The save deceleration of
v0
= 100
km
h
is 20% larger than the one of
v0,all
. This represents a
substantial rise of brake potential despite an identical safety target. The large difference
results from the fact that both correction factor and nominal deceleration exceed the
reference values. Since the nominal deceleration decreases for larger initial velocities, the
differences of
|a|brake_safe
with respect to the reference mitigate. For
v0
= 160
km
h
, the
save deceleration is almost identical with the value of v0,all.
4
The value of
Kdry
in the table corresponds to the upper confidence interval with respect to the selected
safety target.
5The influence of Kwet is neglected in this investigation, i.e. AVADH=1.
128
6.4 Probabilistic analysis
Figure 6.3:
Cumulative probability
P
(
|a|
|a|n
)resulting from MCS with different assumptions
regarding the scatter of the pad friction. Shaded areas represent confidence
intervals with respect to a confidence level of 95%.
In conclusion, the differentiation of friction properties allows to enlarge the save deceler-
ation in three of four cases, while the extent of additional brake potential is larger for
the smaller initial velocities. It is important to mention that the presented probabilistic
analysis does not take component failures and degraded modes of the brake system into
account. These effects influence the distribution of
P
(
|a|
|a|n
)and thus the derivation of
Kdry
and
|a|brake_safe
. Nevertheless, the exemplary investigation reveals a large potential
for the reduction of safety margins applied in operation.
Table 6.2:
Nominal deceleration, correction factor and save deceleration derived for safety
target of 10−4, i.e. EBCL = 4.
Load case |a|nKdry(EBCL = 4) |a|brake_safe(EBCL = 4)
v0= 100km
h1.28 0.90 1.16
v0= 120km
h1.25 0.86 1.08
v0= 140km
h1.18 0.84 0.99
v0= 160km
h1.15 0.82 0.95
v0,all = 100 −160km
h1.22 0.79 0.96
129
6 Suggestions for brake system design and operation
6.5 ETCS braking curves
6.5.1 Dynamic braking curves
Within the current implementation of ETCS, the nominal brake potential related to the
frictional brake system is a single speed-dependent deceleration profile irrespective of the
load case [74]. According to EN 16834, the nominal profile corresponds to the minimum
deceleration investigated during the brake assessment [63]. However, the findings of this
work suggest that the frictional braking process is strongly affected by the initial velocity
and evolves differently with respect to the load case. Moreover, the braking scenario
affects the extent of the performance variance during normal adhesion conditions. In
order to exploit the potential revealed by the previous analysis, it is suggested to apply
dynamic braking curves. This means that both nominal deceleration as well as
Kdry
and
thus |a|brake_safe depend on the initial velocity:
|a|brake_safe(v, v0, EBCL) = |a|n(v, v0)·Kdry(v0, v, EBCL)(6.2)
In analogy to the general speed dependency of the deceleration profile, it is proposed to
provide braking curves with respect to discrete intervals for
v0
, for instance intervals of
20
km
h
. Hence, the braking curve to be applied in operation is determined based on the
measurement of the current speed:
|a|brake_safe =
|a|n,1(v)·Kdry,1(v, EBCL)for v0<80km
h
|a|n,2(v)·Kdry,2(v, EBCL)for 80km
h≤v0<100km
h
|a|n,3(v)·Kdry,3(v, EBCL)for 100km
h≤v0<120km
h
...
|a|n,7(v)·Kdry,7(v, EBCL)for v0>200km
h
(6.3)
Figure 6.4 compares static and dynamic braking curves to exemplify the potential in
operation. The illustrated curves base on the values of
|a|brake_safe
, which were derived
by the probabilistic analysis conducted in the previous section. The dependency of the
velocity decrease with respect to the braking distance is given by:
v=qv2
0−2·|a|brake_safe ·s(6.4)
The dynamic curves correspond to the values of
|a|brake_safe
with respect to the single
initial velocities given in Table 6.2 (see row 1-4). The static curves base on the value for
v0,all
(see row 5). The comparison reveals that the dynamic curves are significantly shorter
130
6.5 ETCS braking curves
than the static ones, in particular for lower initial velocities. In case of
v0
= 100
km
h
, the
stopping distance is more than 17% smaller than the one of the static braking curve.
Figure 6.4: Comparison of dynamic and static braking curves.
The resulting improvement of headways and capacity utilization resulting from dynamic
braking curves depends on the type of rail operation (high-speed, intercity, regional or
urban rail traffic) as well as the applied signaling system. The quantification of potentials
requires sophisticated operational investigations, as discussed in [72] and [130], which are
out of scope of this work. Nevertheless, the concept of dynamic braking curves represents
a vehicle-related opportunity to enhance the infrastructure capacity without technical
modifications of the actual brake system and additional expense. This is essential, if a
rapid improvement of the infrastructure utilization is aimed at.
6.5.2 Passive safety in braking curves
It should be noted that exceeding the supervised location does not necessarily lead
to a catastrophic accident. The probability of occurrence of a collision with serious
consequences for passengers is therefore not to be equated with the EBCL. The probability
of an accident also depends on which type of hazardous point (normal end of block, switch,
level crossing) is involved [46]. In addition, it needs to be considered that rail vehicles
must meet certain passive safety requirements specified by the european standard EN
15227. The aim of these requirements is to mitigate the consequences of a collision when
all other means of avoiding an accident have failed [131]. Rail vehicles must therefore
be able to absorb the kinetic energy in such a way that no serious consequences occur
for passengers and staff in various collision scenarios with vehicles and obstacles. The
most serious scenario is considered to be the head-on collision of two identical trains. In
131
6 Suggestions for brake system design and operation
this case, the structure of passenger compartments must be able to withstand an impact
with a relative speed of 36km
h.
Taking into account these passive safety requirements, further optimization of the braking
curves is possible, as exemplified in Figure 6.5. As shown, the start of a forced emergency
brake application is shifted to the right, if, for example, 18
km
h
is specified as the final
speed instead of zero
6
. By applying this boundary condition in the dynamic braking
curves, the braking distance to be considered in operation is reduced by further 3%
compared to the static braking curve for
v0
= 100
km
h
, i.e. a reduction of 20% in total.
Figure 6.5:
Comparison of static and dynamic braking curves with and without passive safety
included, while the permissible final speed at the supervised location is 18 km
h.
6.6 Controlled emergency brake applications
The controlled emergency brake is an innovation, which is currently developed by system
suppliers to improve the repeatability of the brake performance. First prototypes have
proven the feasibility based on conventional brake systems [132]. The core element of this
innovation is the so-called Deceleration Control (DCC). The Deceleration Control (DCC)
aims at reducing the dispersion of the brake performance by adapting the clamping forces
of the brake units during an emergency brake application, which are usually constant
as soon as they reach their demand value (see Figure 2.8). In doing so, the vehicle
shall decelerate with a prescribed target value, e.g. 1
m
s2
. Accordingly, the control loop
counteracts the friction-related changes of brake forces during the braking process. The
enhanced repeatability of the brake performance during emergency cases allows to apply
6This leads to a relative speed of 36km
h, if the maximum speed of the opposing train is also 18km
h.
132
6.6 Controlled emergency brake applications
a larger correction factor
Kdry
and thus offers potential to enhance the utilization of
infrastructure, as discussed in [72],[133] and [130].
With respect to the findings in this work and the previously introduced dynamic braking
curves, it is suggested to apply an Adaptive Deceleration Control based on a dynamic
target
|a|traget
(
t
), as illustrated in Figure 6.6.
|a|traget
(
t
)represents the nominal brake
capability according to the current velocity, which is deduced from the expected friction
characteristics and vehicle parameters. Note that the friction model applied for deriving
the target deceleration in operation does not necessarily need to be a time-variant
characteristic. It could also be a simple look-up table based on constant values.
Plant
Target value
Brake
system
Brake
system
Controller Brake
unit i Train
Friction
model
Train
model −
+𝐹𝐶,𝑖(𝑡) 𝐹
𝑓𝑟𝑖𝑐,𝑖(𝑡)
𝑎𝑚𝑒𝑎𝑠(𝑡)
|𝑎|𝑡𝑎𝑟𝑔𝑒𝑡(𝑡)
𝑣𝑚𝑒𝑎𝑠(𝑡)
DCC
𝜇
Figure 6.6: Concept of adaptive DCC with dynamic target value.
The sketch in Figure 6.7 exemplifies how the deceleration resulting from brake applications
with the uncontrolled brake system, the DCC without dynamic target and the adaptive
DCC might evolve during the braking process. To exploit the maximum possible brake
capability with respect to the current load case, the adaptive DCC additionally takes the
scenario-dependent deterministic behavior of the friction forces into account of the control.
This allows to maximize both the nominal deceleration
|a|n
as well as the correction
factor
Kdry
applied in operation. Accordingly, the adaptive DCC represents a control
concept to realize dynamic braking curves based on minimal safety margins.
Conventional
(not controlled)
DCC
(constant target)
Adaptive DCC
(dynamic target)
𝑣
|𝑎|
|𝑎|𝑡𝑎𝑟𝑔𝑒𝑡 = 𝑐𝑜𝑛𝑠𝑡.
𝑣
|𝑎| 𝑎𝑡𝑎𝑟𝑔𝑒𝑡(𝑣0)
𝑣
|𝑎|
Figure 6.7: Deceleration with uncontrolled brake system, DCC and adaptive DCC.
133
7 Conclusion
In the following, the main contributions of this work are briefly recapitulated in relation
to the research questions posed in the introduction. Subsequently, an outlook for future
research emerging from the findings will be given.
7.1 Answers to research questions
How does the braking scenario affect the friction forces of disc brakes?
The analysis of almost 2000 brake applications provides fundamental insights into
the friction phenomena of a typical brake pad material applied in rail vehicles and
allows the characterization of the deterministic and stochastic friction properties
in an unprecedented level of detail:
•
The initial velocity (
v0
) has a strong influence on the behavior of the friction
forces. For
v0
= 100
km
h
, the COF of the investigated pad material increases
by almost 40% during braking to standstill. In contrast, for
v0
= 160
km
h
, a
temporary decrease of 20% is observed. Furthermore, the COF decreases
with increasing clamping force when
v0≥
120
km
h
. These observations are
explained by changes in the topography between pad and disc during braking.
•
The described systematic behavior is superimposed by strong stochastic
phenomena. As a result, the instantaneous friction forces vary up to
±
25%
despite an identical braking scenario. It is assumed that the load history
contributes significantly to the extent of this dispersion.
•
In the interval of
v0
= 100
km
h−
160
km
h
, the COF averaged over the braking
distance (
µm
) takes values between 0.41 and 0.31, while it decreases for larger
initial velocities. In contrast, the standard deviation of
µm
increases for
higher initial velocities and ranges between 2% and 6%.
•
It is remarkable that the COF recorded at specific time instants or velocities
is normally distributed for most of the investigated braking scenarios. This
observation is also true for the COF averaged over the braking distance.
135
7 Conclusion
Is it possible to predict the instantaneous friction forces of a single disc brake unit?
This work introduces a new friction model for disc brake units of rail vehicles.
It allows to predict both the deterministic and, for the first time, the stochastic
behavior of the instantaneous friction forces during stopping brake applications:
•
The model predicts the progression of the instantaneous COF (
µ
(
t
)) with
an accuracy of 5% for 12 different load cases, while using a single set of
parameters. A comparison with state-of-the-art friction models shows that
the new model improves the prediction for all investigated scenarios.
•
In addition, a temperature model is derived to estimate the thermal loads
prevailing on the disc surface during braking. Coupling the friction and
temperature models yields a deterministic brake model, which predicts the
nominal braking distance for a single brake unit with an accuracy of 2%.
•
An approach based on stochastic model parameters is presented, which allows
to consider the uncertainty of the friction process when assessing the brake
performance. Applying Monte-Carlo Simulation (MCS), this stochastic brake
model provides a very good estimate of the instantaneous mean values (
¯µ
(
t
))
and dispersion of the COF (¯µ(t)±3σµ(t)), as visualized in Figure 7.1.
•
Based on this approach, it is possible to predict the standard deviation of
the braking distance related to a single brake unit with an accuracy of 1%
with respect to the nominal braking distance.
Figure 7.1:
Instantaneous mean values
¯µ
(
t
)and dispersion
¯µ
(
t
)
±
3
σµ
(
t
)of the COF gained
from test rig experiments and MCS based on the stochastic brake model.
136
7.1 Answers to research questions
Is it possible to predict the deterministic and stochastic behavior of a disc-braked
rail vehicle?
Data from 88 experimental vehicle brake tests conducted with a multiple unit is
used to validate the brake model developed in this work:
•
The application of the deterministic brake model allows to predict the time-
variant and scenario-dependent deceleration observed for the multiple unit
during emergency and service brake applications, as demonstrated in Figure
7.2. In addition, the brake model provides a representative estimate of the
temperature evolution at the surface of the brake discs.
•
For
v0
= 120
km
h
, the nominal braking distance is predicted with an accuracy
of 5%. The largest error between measured and simulated braking distance
of all 88 brake applications is 11%.
•
The experiments reveal a large dispersion of the vehicle brake performance,
while the braking distances vary up to
±
15%. It is found that this large
variability cannot be explained by treating the friction coefficients of the
individual brake units of the same vehicle as independent random variables.
Rather, it must be assumed that the behavior of the disc brakes is correlated
due to their common load history and similar environmental conditions.
•
Applying this assumption within a MCS based on the stochastic brake model
yields a very good approximation of the observed variance. Therefore, this
finding is an important contribution to a reliable quantification of the disper-
sion of the vehicle brake performance to be expected during operation.
Figure 7.2:
Comparison of measured vehicle deceleration during stopping brake applications
with simulation results based on the brake model developed in this work.
137
7 Conclusion
How does improved friction prediction enrich rail operation and design?
To improve the brake performance prediction in the context of system design and
rail operation, the following suggestions are derived from the results of this work:
•
The use of a single value for the COF between pad and disc leads to significant
errors in predicting the brake performance. It is highly recommended to
consider the systematic and stochastic friction properties of the COF in the
system design, e.g. by using a sophisticated friction model, such as the one
presented in this work.
•
It is proposed to perform at least 10 runs of each Priority 1 braking scenario
on the test rig during pad certification programs to enable a reliable quan-
tification of the deterministic and stochastic friction phenomena. This is
essential to apply the models developed in this work to other pad materials.
•
To obtain more reliable results from the probabilistic analysis of railway
brake applications, it is recommended to assume that the scatter of all
brake units applied in the same vehicle is correlated and to consider the
systematic changes of the friction forces in the analysis. An exemplary
analysis demonstrates that these assumptions can reduce operational safety
margins by up to 14% without changing the safety target.
•
Finally, it is proposed to apply dynamic braking curves for the rail operation
with ETCS, based on an individual brake profile with respect to the current
initial velocity. This allows for less conservative assumptions regarding the
brake potential in operation, as indicated by Figure 7.3.
Figure 7.3:
Braking potential during emergency brake applications with different initial
velocities based on static and dynamic braking curves.
138
7.2 Outlook
7.2 Outlook
With the increasing digitalization and automation of rail operations, as aimed by the
Automatic Train Operation (ATO), the automatic control of rail vehicles is becoming
increasingly important. To ensure a reliable and robust control of the brake system, it is
essential to design and verify the control algorithms used for the automatic actuation
based on physical simulation models. Accordingly, the brake model developed in this work
can be applied in Model-in-the-Loop or Hardware-in-the-Loop environments to consider
the role of pad friction within the brake control, as discussed in [33]. These investigations
are of particular interest with regard to the introduction of the Electro-Mechanic Brake
for rail vehicles, which is expected to significantly improve the actuation of brake units
[134]. However, in order to predict the system behavior at the very end of the braking
process, for instance when precise stopping at a certain point is aimed at, it is necessary
to reinvestigate the physical phenomena prevailing in this velocity range and to adapt
the presented friction model, as discussed in Section 4.1.4.
Moreover, the contributions of this work can be applied to monitoring tasks in order to
survey and assess the condition of brake components during operation. The application
of online-identification methods, such as state observers, as discussed in [98], allows the
brake model to be continuously updated and identified with respect to the previous brake
applications. This digital twin of the frictional brake system can be used to provide
information about the current brake potential of the vehicle with respect to the load
history and operating conditions, e.g. for rail operation or predictive maintenance tasks.
As part of these investigations, it is also conceivable to include and identify the influences
of moisture and humidity within the differential equation of the COF derived in this
work.
In addition to the brake performance prediction, the ability to simulate instantaneous
friction forces offers potential for improving a broad spectrum of investigations related to
the development of disc brakes applied in rail vehicles. For instance, noise pollution is a
critical factor hindering the realization of new rail infrastructure. The model approach
developed in this work can be used in the context of vibration analyses to optimize the
design of friction brakes with respect to noise emissions. This is of particular interest for
the introduction of disc brakes in freight trains, as discussed in [135].
Finally, the results motivate to analyze the potential of dynamic braking curves for the
capacity utilization of the rail infrastructure. This requires to quantify the performance for
different types of rail operation (high-speed, intercity, regional and urban rail transport)
as well as signalling systems, such as ETCS (Level 1,2,3) or ATO over ETCS, as discussed
in [72] and [130]. The investigations are necessary in order to underline the potential
offered by a better understanding and control of the frictional braking process with
regard to the use of the railway infrastructure. This is an essential step towards the
transition to a competitive rail transport within the EU.
139
A Appendix
A.1.2
Brake kinetics based on mean values according to EN 14531-1
The use of mean values is based on transforming the instantaneous braking process into
a step function characterized by the equivalent response time
tE
and the constant mean
deceleration |a|E, as indicated by the left sketch in Figure 2.12, according to
¨x(t) =
0for t<tE
−|a|Efor t≥tE.(A.1)
The stopping distance of the equivalent process
sE
is sum of the traveled distance with
initial velocity
v0
until
tE
and the subsequent braking process with the deceleration
|a|E
until standstill [16]
sE=v0·tE+v02
2·|a|E
.(A.2)
Applying Equation 2.18 for the deceleration of a disc-braked rail vehicle on a straight
and even track, yields
|a|E=
n
P
i=1
FBE,i +FRE
mtrain
.(A.3)
FBE
and
FRE
are equivalent constant forces, which yield the same the braking distance
as the instantaneous force characteristics. For a velocity-dependent force characteristic
FB
(
v
), such as the one of magnetic track brakes or the Davis formula, the equivalent
force of a stopping brake process is given by1
FBE =v02
2v0
R0
v
FB(v)dv
.(A.4)
If
FB
(
v
)is assembled of several different characteristics, each acting in a certain velocity
interval, as it is the case for electro-dynamic brakes, Equation A.4 is solved for each
velocity interval separately. In the general case, the mean deceleration is thus a velocity-
dependent profile and the stopping distance is given by the sum of the intervals
sE=v0·tE+Xv1,i2−v2,i2
2·|a|E,i
.(A.5)
1The derivation of Equation A.4 is shown in Appendix A.1.3.
142
A.1 Appendix to state-of-the-art
A.1.3 Equivalent brake force
Applying the relation
a=dv
dt and v=ds
dt (A.6)
yields
ds =v
adv (A.7)
For a stopping brake process with the initial velocity
v0
to standstill starting at position
s0= 0, it follows
sst =
0
Z
v0
v
adv. (A.8)
Inserting the deceleration of the dynamic mass
mdyn
caused by the instantaneous brake
force FB(v)
a=−FB(v)
mdyn
,(A.9)
yields
sst =mdyn
v0
Z
0
v
FB(v)dv. (A.10)
For a constant brake force FBE =const, the solution of this integral becomes
sE=mdyn ·v02
2·FBE
.(A.11)
Equating the stopping distance of instantaneous braking process with the one based on a
constant deceleration, i.e. sst
!
=sE, yields the equivalent brake force FBE
FBE =v02
2v0
R0
v
FB(v)dv
.(A.12)
143
A Appendix
A.1.4 COF averaged over the braking distance
Applying the relation
a=dv
dt and v=ds
dt (A.13)
yields
ads =vdv. (A.14)
For a stopping brake process with the initial velocity
v0
to standstill starting at position
s0= 0, it follows
sst
Z
0
ads =
0
Z
v0
vdv =−v02
2.(A.15)
Considering that the deceleration of the dynamic mass
mdyn
is caused by the instantaneous
brake force FB
a=−FB
mdyn
(A.16)
yields
sst
Z
0
FB(s)ds =mdyn ·v02
2.(A.17)
According to Equation A.11, the stopping distance of a braking process based on a
constant force FBE is given by
sE=mdyn ·v02
2·FBE
.(A.18)
Presuming that the stopping distance based on the mean force
FBE
is identical to the
one of the instantaneous process, i.e. sst
!
=sE, yields
FBE =1
sst
sst
Z
0
FB(s)ds. (A.19)
144
A.1 Appendix to state-of-the-art
Inserting Equation 2.15 in Equation A.19 yields
FBE =1
sst
sst
Z
0
FC(s)·µ(s)·rfric
rw
ds. (A.20)
If we now only consider the part of the braking process in which the brake unit is fully
actuated, i.e. the transient built-up is excluded and
FC
=
const.
, the following results:
FBE =FC·rfric
rw·1
s2
s2
Z
0
µ(s)ds =FC·rfric
rw·µm(A.21)
Accordingly,
µm
represents the instantaneous COF averaged over the braking distance
µm=1
s2
s2
Z
0
µ(s)ds .(A.22)
Due to the relation v=ds
dt , this formulation is equivalent to
µm=1
s2
tst
Z
t95
µ(t)·v(t)dt. (A.23)
In analogy to the derivation shown in Appendix A.1.3, the equivalent COF may also be
expressed by the term
µm=v02
2v0
R0
v
µ(v)dv
.(A.24)
In this case,
v0
represents the velocity at the point of time when the clamping forces
reach 95% of their demand value.
145
A Appendix
A.1.5 Emergency Brake Confidence Level (EBCL)
Table A.1:
Emergency Brake Confidence Levels (EBCL) and related probabilities P(EBCL)
according to the definition in the ERTMS/ETCS system specifications [74]. Column
Countries indicates what countries use which values according to [75].
EBCL Probability P(EBCL) Countries
0 50%
1 90%
2 99% United Kingdom
3 99.9%
4 99.99% Netherlands
5 99.999% Switzerland
6 99.9999% France
7 99.99999% Germany, Belgium
8 99.999999% Norway
9 99.9999999% Denmark, Italy, Luxembourg
146
A.2 Appendix to data analysis
A.2 Appendix to data analysis
A.2.1 Testing procedure
Table A.2:
Exemplary testing procedure comprising 40 subsequent Case1 brake applications.
Brake Application Load Case v0[km/h]FC[N]mdyn[kg]Tfric,0[◦C]
1 V0140_Case1 140 33000 7250 <60
2 V0140_Case1 140 33000 7250 <60
3 V0120_Case1 120 33000 7250 <60
4 V0120_Case1 120 33000 7250 <60
5 V0160_Case1 160 33000 7250 <60
6 V0160_Case1 160 33000 7250 <60
7 V0100_Case1 100 33000 7250 <60
8 V0100_Case1 100 33000 7250 <60
9 V0120_Case1 120 33000 7250 <60
10 V0120_Case1 120 33000 7250 <60
11 V0100_Case1 100 33000 7250 <60
12 V0100_Case1 100 33000 7250 <60
13 V0160_Case1 160 33000 7250 <60
14 V0160_Case1 160 33000 7250 <60
15 V0140_Case1 140 33000 7250 <60
16 V0140_Case1 140 33000 7250 <60
17 V0160_Case1 160 33000 7250 <60
18 V0140_Case1 140 33000 7250 <60
19 V0100_Case1 100 33000 7250 <60
20 V0120_Case1 120 33000 7250 <60
21 V0140_Case1 140 33000 7250 <60
22 V0140_Case1 140 33000 7250 <60
23 V0120_Case1 120 33000 7250 <60
24 V0120_Case1 120 33000 7250 <60
25 V0160_Case1 160 33000 7250 <60
26 V0160_Case1 160 33000 7250 <60
27 V0100_Case1 100 33000 7250 <60
28 V0100_Case1 100 33000 7250 <60
29 V0120_Case1 120 33000 7250 <60
30 V0120_Case1 120 33000 7250 <60
31 V0100_Case1 100 33000 7250 <60
32 V0100_Case1 100 33000 7250 <60
33 V0160_Case1 160 33000 7250 <60
34 V0160_Case1 160 33000 7250 <60
35 V0140_Case1 140 33000 7250 <60
36 V0140_Case1 140 33000 7250 <60
37 V0160_Case1 160 33000 7250 <60
38 V0140_Case1 140 33000 7250 <60
39 V0100_Case1 100 33000 7250 <60
40 V0120_Case1 120 33000 7250 <60
147
A Appendix
A.2.2 Fault propagation of friction measurement at dynamometer
Derivation of µbased on uncertain quantities y = [FM, lF, pC, AC, FC, ηC, iT, rfric]:
µ(yi) = FM·lF
(pC·AC−FS)·ηC·iT·rfric
.(A.25)
Linear fault propagation at
µ0, y0
assuming
yi
are independent random numbers [129]:
1σµ≈v
u
u
tX ∂µ
∂yi
(µ0, y0)!2
·σ2
yi(A.26)
Partial derivations with respect to quantities with σyi= 0:
∂µ
∂FM
=lF
(pC,0·AC,0−FS,0)·ηC,0·iT,0·rfric,0
(A.27)
∂µ
∂rfric
=−FM,0·lF,0
(pC,0·AC,0−FS,0)·ηC,0·iT,0·rfric,02(A.28)
∂µ
∂pC
=−∂µ ·AC
∂FS
=−FM,0·lF,0·AC,0
(pC,0·AC,0−FS,0)2·ηC,0·iT,0·rfric,0
(A.29)
∂µ
∂ηC
=−FM,0·lF,0
(pC,0·AC,0−FS,0)·ηC,02·iT,0·rfric,0
(A.30)
Table A.3 shows the results of Equations A.25 -A.30 based on assumptions for 1σyi:
Table A.3: Fault propagation for µfor dynamometer test-rig.
Parameter yi,0∂µ
∂yi(µ0, y0)1σyi
yi,01σyir∂µ
∂yi2·σ2
yi
pC380000 [N
m2]-1.1 E-06 [m2
N]0.1% 228 [N
m2]0.0003 [N
m2]
AC0.01427 [m2]-29.3 [1
m2]0 0 0
FS500 [N]7.7 E-05 [1
N]5% 25 [N]0.0019 [N]
iT8.58 -0.044 0 0 0
ηC0.97 -0.392 1% 0.01 0.0038
FM4250.1 [N]8.9 E-05 [1
N]1% 43 [N]0.0038 [N]
lF1[m]0.380 [1
m]0 0 0
rfric 0.273 [m]-1.3919 [1
m]0-3.5% 0-0.01 [m]0-0.0133[m]
µ0.380 1.5-3.8%0.006-0.014
148
A.2 Appendix to data analysis
A.2.3 COF with respect to instantaneous velocity (Pad Material A)
Figure A.2:
Comparison of instantaneous empirical mean values
¯µ
(
v
)and standard deviation
1σµ(v)of all load cases of Pad Material A analyzed in Chapter 3.
149
A Appendix
A.2.4 Instantaneous mean values and standard deviations of COF
gained from different specimen (Pad Material A)
Figure A.3:
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single specimen
and all specimen.
150
A.2 Appendix to data analysis
Figure A.4:
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single specimen
and all specimen.
151
A Appendix
Figure A.5:
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single specimen
and all specimen.
152
A.2 Appendix to data analysis
Figure A.6:
Comparison of instantaneous
¯µ
(
t
)and 1
σµ
(
t
)calculated from single specimen
and all specimen.
153
A Appendix
A.2.5 Instantaneous distribution of COF (Pad Material A)
Figure A.7:
Comparison of
µ
(
t
)of all 160 brake applications (left) measured during load case
V0160_Case2 and temporary histogram of µ(t= 25s)(right).
Figure A.8:
Comparison of
µ
(
v
)of all 160 brake applications (left) measured during load case
V0160_Case2 and temporary histogram of µ(v= 90km/h
)(right).
154
A.2 Appendix to data analysis
Table A.4:
p-values resulting from
χ2
-tests conducted for samples of
µi
(
v
)at
vj
= 10
,
50
,
90
and 130
km
h
gained from all load cases. Gray cells indicate that
Pχ2<
0
.
05, i.e. the
null hypothesis is rejected with respect to α= 5%.
Loadcase P-values of µi(v)samples at
vj= 10km
hvj= 50km
hvj= 90km
hvj= 130km
h
V0100_Case1
0.35 0.08 1.00 -
V0100_Case2
0.82 0.32 0.49 -
V0100_Case3
0.04 0.25 0.46 -
V0120_Case1
0.06 0.81 0.24 -
V0120_Case2
0.09 0.59 0.08 -
V0120_Case3
0.29 0.31 0.17 -
V0140_Case1
0.00 0.36 0.22 0.04
V0140_Case2
0.29 0.10 0.05 0.01
V0140_Case3
0.35 0.80 0.08 0.02
V0160_Case1
0.06 0.00 0.00 0.88
V0160_Case2
0.45 0.00 0.00 0.93
V0160_Case3
0.00 0.00 0.00 0.50
155
A Appendix
A.2.6 Estimation of confidence intervals
Presuming a sample consisting of
n
realizations
yi
of a normally distributed stochastic
quantity
˜y
. If the true values of mean value and standard deviation of the distribution
are unknown, the confidence interval related to the estimation of the mean value with
respect to a confidence level of 1−αis defined as [136]
[¯y−L1−α; ¯y+L1−α],(A.31)
while the empirical mean value ¯yis the corresponding point estimation
¯y=1
n
n
X
i=1
yi.(A.32)
The half length L1−αof the confidence interval is given by [136]
L1−α=σy
√n·ζ1−α
2,(A.33)
where σyis the empirical standard deviation given by [136]
σ2
y=1
n−1
n
X
i=1
(yi−¯y)2.(A.34)
ζ1−α
2
is the quantile of a t-Distribution with respect to a sample size of
n−
1and a
probability of 1
−α
2
. The length of the confidence interval related to the estimation of
the mean value is thus ∆L= 2 ·L1−α.α∈[0; 1] is the so-called level of significance.
The confidence interval related to the estimation of the standard deviation of the
sample is defined as [136]
hL−
1−α;L+
1−αi,(A.35)
while L+
1−αand L−
1−αare upper and lower sides of the interval
L+
1−α=v
u
u
t
(n−1) ·σ2
y
χ2
α
2
, L−
1−α=v
u
u
t
(n−1) ·σ2
y
χ2
1−α
2
.(A.36)
χ2
α
2
and
χ2
1−α
2
are the quantiles of a Chi
2
-distribution with respect to a sample size of
n−
1and a probability of
α
2
and 1
−α
2
, respectively. The length of the confidence interval
related to the estimation of standard deviation is thus ∆L=L+
1−α−L−
1−α.
156
A.2 Appendix to data analysis
A.2.7
Polynomial regression of mean values and standard deviations
of COF averaged over braking distance (Pad Material A).
Figure A.9:
Polynomial regression of mean values and standard deviations of
µm
with respect
to the initial velocity. Data points represent values of
¯µm
and
σµm
listed Table
3.4 (The unit of v0in the fitted polynomials is [km
h]).
157
A Appendix
A.2.8 Distribution of COF averaged over braking distance (Pad
Material A)
Figure A.10:
Comparison of sample distributions of
µm
of each load case and all specimens.
158
A.2 Appendix to data analysis
A.2.9 Temperature evolution measured by single sensors in brake
disc surface
Figure A.11:
Instantaneous mean values of temperatures measured by each sensor during all
brake applications of each load case with respect to time.
159
A Appendix
Figure A.12:
Instantaneous standard deviation of temperatures measured by each sensor
during all brake applications of each load case with respect to time.
160
A.3 Appendix to model development
A.3 Appendix to model development
A.3.1 Deterministic model identification
Figure A.13:
Comparison of measured
¯µmeas
(
t
)and simulated values
µsim
(
t
)for two load
cases based on manually tuned parameter values.
161
A Appendix
Table A.5:
Identified parameters of friction models to predict the mean instantaneous COF of Pad Material A for all investigated
load cases.
Reference Formula Influencing Variables
Ehret dµ
dt
=
−
0
.
011
·µ·FC·vfric
1050.785 −
0
.
008
·µ0.338 −
0
.
026
·µ·TF ric
200 0.83
+0
.
04
·FC
5.5·1040.275 Friction velocity vfric[m
s]
Clamping force FC[N]
Disc surface temp. TF ric[◦C]
Karwatzki [25] µ= 0.369 ·0.1·FC−17.8
0.1·FC−20.7·v+40.76
v+23.3
Friction velocity vfric[km
h]
Clamping force FC[kN]
Saumweber
[23] µ= 0.77 ·1+0.0005·FC
1+0.0039·FC·1+0.0127·v
1+0.0267·v·1+0.000005·TF ric
1+0.0018·TF ric
Friction velocity vfric[km
h]
Clamping force FC[kN]
Disc surface temp. TF ric[◦C]
Yuan [34] µ= 0.362 + 0.168 ·e−0.174·vFriction velocity vfric[m
s]
Lee [33] µ= 0.208 ·(0.663 ·e−0.092·v+ 1) ·(0.977 ·e−0.004·TF ric + 1) Friction velocity vfric[m
s]
Disc surface temp. TF ric[◦C]
Rhee [100] µ= 0.28 ·Fa
C·vb
with a= 10−4·TF ric + 0.04 and b=−9·10−4·TF ric + 0.017
Friction velocity vfric[m
s]
Clamping force FC[N]
Disc surface temp. TF ric[◦C]
162
A.3 Appendix to model development
A.3.2 Deterministic brake model
Table A.6:
Comparison of
µm,sim
resulting from deterministic simulation with mean value
from measurements
¯µm,meas
including length of confidence intervals ∆
L
related to
the estimation of
¯µm,meas
with respect to a confidence level of 95% (see Appendix
A.2.6).
Load case µm,sim ¯µm,meas ∆L
2¯µm,meas −µm,sim |¯µm,meas−µm,sim|
¯µm,meas
V0100_Case1 0.405 0.405 0.001 0.000 0.1%
V0100_Case2 0.400 0.398 0.001 -0.001 0.3%
V0100_Case3 0.392 0.393 0.001 0.001 0.2%
V0120_Case1 0.386 0.391 0.002 0.004 1.1%
V0120_Case2 0.378 0.378 0.002 0.001 0.2%
V0120_Case3 0.368 0.373 0.002 0.005 1.2%
V0140_Case1 0.364 0.366 0.003 0.003 0.7%
V0140_Case2 0.352 0.352 0.003 0.000 0.1%
V0140_Case3 0.340 0.343 0.003 0.003 0.8%
V0160_Case1 0.335 0.336 0.003 0.001 0.4%
V0160_Case2 0.321 0.318 0.003 -0.003 1.0%
V0160_Case3 0.308 0.310 0.003 0.002 0.6%
All load cases 0.363 0.364 0.001 0.002 0.5%
A.3.3 Deterministic temperature model
Table A.7:
Dimension, material properties and heat transfer coefficients of gray cast iron disc
according to geometry data and literature. The intervals correspond to the range
of data found in the references.
Property Symbol Value Unit Reference
Friction surface Asurf 0.244 m2
Ventilation surface Avent 0.75 m2
Thickness ring l0.05 m
Density ρ7000-7300 kg
m3[137][138]
Specific heat capacity (273.15K)cp450-550 J
kgK [137][138]
Thermal conductivity (273.15K)λ40-60 W
mK [137][138][124][125]
Conductivity gradient (273.15 −600K)∂λ
∂T -0.04-0 W
mK2[137][138][124][125]
Heat transfer coefficient (vfric = 0)h00-12 W
m2K[57] [126]
Heat transfer coefficient (vfric = 27.¯
7m
s)href 50-100 W
m2K[57] [126]
163
A Appendix
A.3.4 Stochastic brake model
Table A.8:
Comparison of mean values and standard deviations of
µm
resulting from MCS
and measurements.
Load case ¯µm,sim ¯µm,meas |¯µm,meas−¯µm,sim|
¯µm,meas 1σµm,sim
1
σµm,meas
1
σµm,meas −
1
σµm,sim
V0100_Case1 0.406 0.405 0.1% 0.011 0.010 -0.001
V0100_Case2 0.401 0.398 0.7% 0.011 0.008 -0.003
V0100_Case3 0.393 0.393 0.1% 0.012 0.009 -0.003
V0120_Case1 0.387 0.391 1.0% 0.012 0.011 -0.001
V0120_Case2 0.377 0.378 0.3% 0.013 0.012 -0.001
V0120_Case3 0.368 0.373 1.1% 0.015 0.013 -0.002
V0140_Case1 0.365 0.366 0.4% 0.015 0.017 0.002
V0140_Case2 0.354 0.352 0.5% 0.016 0.016 0.000
V0140_Case3 0.342 0.343 0.4% 0.015 0.015 0.000
V0160_Case1 0.336 0.336 0.2% 0.015 0.019 0.004
V0160_Case2 0.321 0.318 0.7% 0.016 0.017 0.001
V0160_Case3 0.308 0.310 0.6% 0.015 0.016 0.001
All load cases 0.363 0.364 0.4% 0.033 0.034 0.001
164
A.4 Appendix to analysis of vehicle brake tests
A.4 Appendix to analysis of vehicle brake tests
Figure A.14:
Comparison of measured and simulated deceleration based on deterministic
model.
165
A Appendix
Figure A.15:
Comparison of measured and simulated deceleration based on deterministic
model.
166
A.4 Appendix to analysis of vehicle brake tests
Figure A.16:
Comparison of measured and simulated deceleration based on deterministic
model.
167
A Appendix
Figure A.17:
Comparison of measured and simulated deceleration based on deterministic
model.
168
A.4 Appendix to analysis of vehicle brake tests
Figure A.18:
Comparison of measured and simulated deceleration based on deterministic
model.
169
A Appendix
Figure A.19:
Comparison of measured and simulated deceleration based on deterministic
model.
170
A.4 Appendix to analysis of vehicle brake tests
Figure A.20:
Comparison of measured and simulated deceleration based on deterministic
model.
171
A Appendix
Figure A.21:
Comparison of measured and simulated deceleration based on deterministic
model.
172
A.4 Appendix to analysis of vehicle brake tests
Figure A.22:
Comparison of measured and simulated deceleration based on deterministic
model.
173
A Appendix
Figure A.23:
Comparison of average measured and simulated driving resistance. Underlying
data corresponds to all brake applications conducted for each initial velocity.
174
A.4 Appendix to analysis of vehicle brake tests
Table A.9:
Comparison of empirical mean values of effective COF averaged of the braking
distance
¯µm,eff
resulting from samples of experimental vehicle brake tests and
deterministic simulation. Length of confidence intervals ∆
L
are related to the
estimation of empirical mean value of measurements with respect to a confidence
level of 95% (see Appendix A.2.6).
Load case ¯µm,sim ¯µm,meas ∆L
2¯µmeas −µsim |¯µmeas−µsim|
¯µmeas
v0= 100km
h0.403 0.390 0.005 -0.013 3.3%
v0= 120km
h0.382 0.383 0.004 0.002 0.5%
v0= 140km
h0.356 0.361 0.01 0.005 1.4%
v0= 160km
h0.327 0.345 0.01 0.019 5.5%
v0= 100 −160km
h0.374 0.376 0.004 0.002 0.6%
Figure A.24:
Time series of ambient temperature and relative humidity recorded during all
88 vehicle brake tests.
175
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