Expe imen al s udy o he use o a ans e unc ion o ind ail co uga ion
om axle-box accele a ions
Xinxin Yu
a,d,*
, Se gio Mu˜
noz
b
, Ped o U da
a
, Ja ie F. Acei uno
c
,
Miguel Rod íguez G´
omez
a
, Jos´
e L. Escalona
a
a
Dep . o Mechanical and Manu ac u ing Enginee ing, Uni e si y o Se ille, Spain
b
Dep . o Ma e ials and T anspo a ion Enginee ing, Uni e si y o Se ille, Spain
c
Dep . o Mechanical and Mining Enginee ing, Uni e si y o Ja´
en, Spain
d
Au oma ion Technology and Mechanical Enginee ing, Facul y o Enginee ing and Na u al Sciences, Tampe e Uni e si y, Finland
ARTICLE INFO
Keywo ds:
Vehicle ib a ion
No malized accele a ion
Signal p ocessing
T ack i egula i y
ABSTRACT
This in es iga ion uses a scale ehicle- ack expe imen al acili y o s udy he calcula ion o ail co uga ion
using e ical accele a ions measu ed in he axle-box o ail ehicles and a ans e unc ion (TF). The ail
co uga ed p o ile is machined in he ail heads o he scale ack ollowing a pe iodic unc ion wi h ou ha -
monics. Expe imen s a e pe o med wi h a scale bogie-like ehicle a di e en o wa d eloci ies in he ange
inspec ion eloci ies. Two simple analy ical o ms o he TF a e s udied: he kinema ic TF, ha assumes ha he
axle box ollows he ail p o ile, and he TF o a 2-do model o he ehicle- ack sys em. Fo he ehicle esponse
analysis, his wo k p oposes o no malize he measu ed accele a ion wi h he squa e o he o wa d eloci y o
he ehicle, ha is assumed o be app oxima ely cons an . This no malized accele a ion educes he e ec o he
o wa d eloci y on he TF. Expe imen al esul s show ha he kinema ic TF can be used o measu e he ack
co uga ion o mode a e o wa d eloci ies p o iding easonable bu no accu a e esul s. The limi a ion o he
kinema ic TF is mainly due o ee ligh s and wheel ail cu a u e incompa ibili y. The measu ed axle-box
accele a ions may include equency peaks ha a e no exci a ion equencies and can dis o he ail p o ile
measu emen . Resul s show ha linea elas ic models like he assumed 2-do model do no explain he
appea ance o hese non-exci a ion peaks.
1. In oduc ion
Rail co uga ion is a wa e- ype wea along he ail wi h a ange o
wa eleng hs be ween 10 and 1000 mm [1]. I o en appea s in me o
lines, u ban ailways, and high-speed ailways, esul ing in ema kable
ib a ion and noise ha a ec he ope a ing pe o mance o ail ehi-
cles. Co uga ion o ma ion is a complex p ocess ha is a ec ed by he
ehicle- ack dynamic in e ac ions [2]. Li e al. [3] p oposed ha lon-
gi udinal comp ession modes and co esponding longi udinal ack dy-
namics a e esponsible o co uga ion ini ia ion, and his s udy u he
is alida ed by using he 1/5 scaled V-T ack es ig [4].
Due o he sho wa eleng hs and ampli udes, i s measu emen has
been adi ionally done using walking-speed olleys, a a ound 1 m/s,
pushed by human ope a o s [5]. This echnique is commonly known as a
di ec measu ing me hod. Besides hei low speed o ope a ion, he
di ec measu ing me hod a e obus wi h high accu acy (abou he o de
o mic ons) and can be used no only o demons a e he se e i y o he
co uga ion bu also o quan i y he smoo hness o he e-p o iling
p ocess. Howe e , some o hei main disad an ages a e he ac ha
hey a e equi ed o s op he line a ic due o hei limi a ions on speed
and he ac ha only one ail pe passing is measu ed. Tha is why
manu ac u e s, and he esea ch communi y ha e deeply wo ked
h ough he las decades in he de elopmen o echniques ha can be
used onboa d a comme cial eloci ies.
The echniques, which a e commonly known as indi ec measu ing
me hods, make use o measu emen s o noise [5], imaging p ocessing
algo i hms [6] and axle-box accele a ion (ABA) measu emen s. In [7,8]
i was p oposed ha he co uga ion o he ack can be ob ained om
he measu emen o en i onmen al noise, making i a con enien way
o he indi ec measu emen o ail co uga ion. In he wo k o Liu e al.
* Co esponding au ho a : Dep . o Mechanical and Manu ac u ing Enginee ing, Uni e si y o Se ille, Spain.
E-mail add esses: [email p o ec ed], [email p o ec ed] (X. Yu), [email p o ec ed] (S. Mu˜
noz), [email p o ec ed] (P. U da), [email p o ec ed] (J.F. Acei uno),
[email p o ec ed] (M.R. G´
omez), [email p o ec ed] (J.L. Escalona).
Con en s lis s a ailable a ScienceDi ec
Measu emen
jou nal homepage: www.else ie .com/loca e/measu emen
h ps://doi.o g/10.1016/j.measu emen .2025.117058
Recei ed 11 Oc obe 2024; Recei ed in e ised o m 30 Janua y 2025; Accep ed 18 Feb ua y 2025
Measu emen 249 (2025) 117058
A ailable online 20 Feb ua y 2025
0263-2241/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
[7], an indi ec me hod is p oposed ha employs wa ele packe
decomposi ion (WPD) o analyze he ene gy le els o wheel- ail noise,
which is cap u ed by a mic ophone moun ed on he bogie. Wei e al. [8]
de eloped an inspec ion algo i hm based on in e io noise o iden i y he
posi ion, cha ac e is ic wa eleng h, and se e i y o ail co uga ion. The
me hod u ilizes he measu emen da a om a mic ophone ins alled on
he ca iage loo o high-speed ailway (HSR) ains and ABA.
Compu e ision echnique is one ype o noncon ac echnologies
ha can be used o he ack quali y inspec ions. Chen e al. [9] com-
bined he adi ional cho d wi h he lase senso and came as o ob ain a
p ecise measu emen o ail co uga ion. Howe e , hei sys em can only
un a he maximum speed o 6 km/h. A compu e ision-based iden i-
ica ion app oach is p o ided in [6] o assess he se e i y o ail co u-
ga ion. Gaza udi [10] de eloped a high-speed ail co uga ion
measu emen sys em based on lase iangula ion p inciple and image
p ocessing echniques. Bu he measu emen sys em in [10] has only
been es ed on he manu ac u ed co uga ed ails by CNC wi h a con-
s an wa eleng h o 50 mm and ampli udes om 0.01 mm o 0.1 mm.
Fu he mo e, Lee e al. [11] es ima ed he co uga ion wa eleng h on
ail heads using he compu e ision echnique and a ea u e desc ip o
app oach.
ABA measu emen has he ad an age o low cos , easy main enance,
and comme cial speeds. In his echnology, he accele a ion o he axle
box is assumed o be equal o he accele a ion o a igid wheelse .
Bocciolone e al. [12] s udied he ail s a us based on he co ela ion
be ween he oo mean squa e (RMS) o he ABA measu emen s and he
co uga ion. Simila esea ch was conduc ed by Tanaka e al. [13] whe e
he de ec ion accu acy be ween he leading and ailing ABA measu e-
men s is compa ed using a co uga ion-ABA measu emen co ela ion.
In addi ion, Sal ado e al. [14] sugges ed he op imal sampling and
il e ing equencies and he loca ion o accele ome e s on he axle-box
o he de ec ion o co uga ion and ib a ion modes. Hassanieh e al
[15] de eloped a machine lea ning (ML) model o c ea e he mo ing
RMS o he co uga ion using he ABA signals measu ed om a com-
me cial ain. The esul s show ha he es ima ion based on he ML
model [15] is mo e accu a e when compa ing he one based on he
model-based TF [16]. A ime-domain app oach is de eloped in [17] o
es ima e he co uga ion using a wheel- ail in e ac ion model wi h he
inpu o ABA signals. A one-dimensional con olu ion neu al [18] and a
da a-d i en me hod [18,19] using a ehicle- ack coupling simula ion
model and op imiza ion app oaches, a e employed o es ima e he
co uga ion wa eleng h and dep h om he ABA signals. S ill, he
esea ch in [17–19] is ca ied ou unde ideal condi ions in a simula ed
en i onmen .
I he wa eleng h o a ack de ec is ela i ely long, anging om 3
o 200 m, i is ypically classi ied as ack i egula i y. T ack i egula i ies
can be measu ed h ough bo h con ac and non-con ac me hods. Con-
ac me hods include app oaches such as walking-speed olleys, while
non-con ac me hods in ol e he use o inspec ion ains equipped wi h
lase s and came as. De ailed e iews o ack i egula i y measu emen
echniques a e p o ided in [20]. The ollowing pa ag aph in oduces
ecen de elopmen s in he es ima ion o ack i egula i ies. The Kal-
man il e (KF) algo i hm is used in [21] o iden i y ack i egula i ies
on ailway b idges using ehicle accele a ion as inpu s. The esul s
shows ha he de eloped KF can p oduce accu a e esul s e en in he
p esence o measu emen noise, non-s a iona y ehicle ope a ing con-
di ions, pa ame e unce ain ies, and bounda y sp ing e ec s. S ano
e al. [22] es ima ed he la e al wheel- ail con ac o ces and ack i -
egula i ies using a Cen al Di e ence Kalman Fil e wi h he eloci y
and accele a ion measu emen s o he bogie ame as inpu s. Simila ly,
an unknown inpu obse e is cons uc ed o es ima e he ack i egu-
la i ies [23] using a ehicle suspension model wi h mul i-senso accel-
e a ion measu emen s as inpu . Howe e , he model-based app oach
de eloped in [23-5 a e only demons a ed unde he simula ion
en i onmen .
The dynamic in e ac ion be ween he ehicle and ack is complex,
making he ela ionship be ween ack i egula i ies and ehicle
esponse equally in ica e. Lei e al. [24] examines he e ec o he
spa ial cohe ence o ack i egula i ies on ailway ehicle dynamics
using a mul ibody simula ion model, e ealing ha he cohe ence e ec
dec eases as he i egula i y’s wa eleng h sho ens, he eby jus i ying
he assump ion o incohe en exci a ions a high equencies. Fu he -
mo e, Ka is e al [25] s udied he co ela ion be ween ehicle esponse
and ack i egula i ies using simula ions and measu emen s om a
passenge ca . Howe e , a s ong co ela ion is only obse ed in he
simula ion en i onmen , indica ing ha he eal-wo ld ack s i ness
a ec s he ABA dynamics. The s udy o [26] in oduces a simple and
e icien p edic ion scheme o ain-induced g ound and building i-
b a ions. The ib a ions a e p edic ed in he equency domain by
conside ing h ee key s ages: ’emission’ (exci a ion om ailway
a ic), ’ ansmission’ (wa e p opaga ion h ough he soil), and
’immission’ ( ans e o buildings). Xu and Zhai [27] examined he
impac o spa ial a iabili y in he geome y, physical, and mechanical
p ope ies o ailway acks on ain- ack in e ac ion. Simula ion esul s
show ha he spa ial a ia ion o ack pa ame e s signi ican ly a ec s
wheel- ail in e ac ion and ack ib a ions, highligh ing he impo ance
o conside ing hese unce ain ies in ain- ack dynamic analysis. A
no el d i e-by sys em is p oposed by [28] o high-speed ailways o
de ec esonan b idges using he di e ence o he measu ed ack i -
egula i ies be ween he i s and las ehicles o a ain. Nume ical and
expe imen al esul s demons a e ha his me hod can accu a ely de ec
esonan b idges wi h spans anging om 20 o 60 m. Xu and Liu [29]
p opose a coupled sleepe - ehicle- ack model o analyze he impac o
ack geome y and i egula i ies on wheel- ail in e ac ion. The simu-
la ion esul s indica e ha he a ia ion in ail geome y a he c ossing
sec ion has a mo e signi ican e ec on wheel- ail in e ac ion han ail
i egula i ies. This phenomenon is u he alida ed using he mul ibody
dynamics p og ams VI-Rail and Simpack by [30].
The TF o equency esponse unc ion (FRF) is commonly used in
s uc u al dynamics o cha ac e ize he dynamic esponse o a physical
sys em in he equency domain [31]. I de ines he ela ionship be ween
inpu (such as loads o exci a ions) and ou pu (such as dynamic e-
sponses in e ms o displacemen , eloci y, o accele a ion). The ela-
ionship be ween he ABA and he ail co uga ion can be conside ed as
a single inpu –single ou pu sys em and he TF is assumed o ha e he
capabili y o de ec co uga ion. Based on he ABA spec um and he TF
o he wheel- ail sys em, Liu e al. [32] de eloped a me hodology o
de e mine he main enance limi o ail co uga ion. Howe e , he
p oposed es ima ion o he main enance limi is es ic ed o he e-
quency band o in e es and he ack ype.
The esea ch ques ion o his pape is he use he TF o he de i a-
ion o ail co uga ion using he ABA accele a ion measu emen s as he
inpu s. The objec i e o he pape is no o ind an accu a e TF associa ed
wi h he ehicle and ack used in he expe imen s. As i will be shown in
he con en o he pape , he calcula ion o he speci ic TF o he p oblem
a hand is no needed o check i he concep o he TF, ha is associa ed
wi h linea esponse a he ope a ion condi ions, wo ks easonably well
when measu ing co uga ion. Two simple app oaches, he kinema ic TF
and he TF o a 2-do model, a e compa ed. In he expe imen s, a 1:10
scaled expe imen al ack and ehicle a e used. The co uga ion is
machined on he ail head and measu ed wi h lase o wo k wi h a well-
known exci a ion. Resul s ob ained unde labo a o y condi ions canno
be di ec ly ex ended o eal scale ehicles and ack, bu hey can be
help ul o he de elopmen o a de ec ion p ocedu e o ail co uga ion
based on ABA signals. This pape is o ganized as ollows. Sec ion 2
shows he ounda ion o he wo simple TFs conside ed in his wo k.
Sec ion 3 p o ides he de ails o he expe imen al se up: he ack, he
ehicle and he machined co uga ion p o ile. Sec ion 4 shows he
expe imen al esul s and includes a discussion abou he use o he TFs
o he measu emen o co uga ion. Summa y and conclusions a e
p o ided in Sec ion 5.
X. Yu e al.
Measu emen 249 (2025) 117058
2
2. Fundamen als o he accele a ion o pa h-p o ile ans e
unc ion
2.1. Kinema ic ans e unc ion
The simples model ha can be used o ind he TF conside s a igid
wheel mo ing wi h a cons an o wa d eloci y V on an i egula p o ile,
as shown in Fig. 1. This model is called he e he kinema ic model.
The e ical posi ion z
w
( ) o he igid wheel is gi en by:
zw( ) = u(s)|s=V +R,(1)
whe e s is he a c-leng h coo dina e along he ack, u(s)is he ail
p o ile and R is he adius o he wheel. Equa ion (1) means ha he
ajec o y o he cen e o he wheel is a ansla ed copy o he ail
p o ile. Fo his assump ion o be ue, he ul ilmen o he igid body
assump ion is no enough. Two o he condi ions mus apply:
The e a e no ee ligh s o he wheel. The wheel s ays in con ac wi h
he ail. In o he wo ds, he con ac o ce is always comp essi e.
The e is cu a u e compa ibili y be ween he wheel and he ail p o-
ile. In o he wo ds, he cu a u e o he wheel is la ge han he
cu a u e o he ail p o ile, 1/R≥uʹʹ, whe e uʹʹ is he second space-
de i a i e o he ail p o ile, o an app oxima ion o he p o ile
cu a u e.
I he model applies, he accele a ion o he cen e o he wheel can
be ob ained by di e en ia ing Eq. (1) using he chain ule, as ollows:
¨
zw=V2uʹʹ,(2)
whe e he ‘do ’ ep esen s ime de i a i e and ‘p ima’ he space de i -
a i e. Taking he Fou ie ans o m o bo h sides o Eq. (1) shows ha
Zw( ) = U( ), whe e is ime equency, in cycles/s, Zw( )is he Fou ie
ans o m o zw( ), and U( )is he Fou ie ans o m o u(V ). This
equali y is alid o all equencies excep a =0, due o he dis ance R
be ween he pa allel ajec o ies. The Fou ie ans o m o he e ical
accele a ion o he wheel, ¨
Zw( ),is ela ed o he Fou ie ans o m o he
displacemen by ¨
Zw( ) = − (2
π
)2Zw( ). Using his ela ionship, he
ollowing TF is ob ained:
Tkin
ABA( ) =
¨
Zw( )
U( )= − (2
π
)2,(3)
whe e his unc ion is called he e kinema ic ans e unc ion om accel-
e a ion o p o ile. TFs a e usually de ined as he ans o m o he inpu
(p o ile) di ided by he ans o m o he ou pu (accele a ion). Howe e ,
in Eq. (3) he de ini ion is aken as ou pu di ided by inpu . Wi hou
losing gene ali y, his de ini ion is used h oughou his pape o con-
enience, because he esul ing exp essions a e simple and mo e
amilia in he con ex o he heo y o mechanical ib a ions.
Because he p o ile is a space unc ion, i makes mo e sense o u n i
in o a unc ion o he space equency , in cycles/m. This is e y simple
because =V . Subs i u ing yields:
Tkin
ABA( ) = − (2
π
V )2.(4)
Ob iously, he TF depends on he o wa d eloci y o he wheel V. I he
measu ed ABA accele a ion is no malized by he squa e o he o wa d
eloci y, he kinema ic TF becomes eloci y independen , as ollows:
ano =
¨
zw
V2=uʹʹ⟹T
kin
no ( ) = Ano ( )
U( )= − (2
π
)2,(5)
whe e ano is he no malized accele a ion, wi h uni s o cu a u e (m
−1
),
and Ano i s Fou ie ans o m.
2.2. Simple dynamic ans e unc ion
The kinema ic model is no gene ally alid because in he wheel- ail
sys ems he e a e many sou ces o lexibili y:
1. The local elas ici y in he wheel- ail in e ace, some imes called
He zian s i ness [33].
2. The lexibili y o he bea ing in he axle-box.
3. The s uc u al lexibili y o he ail.
4. The s uc u al lexibili y o he wheel.
The simples model o he wheel- ail sys em ha accoun s o lexi-
bili y is he 2-do dynamic model shown in Fig. 2. The e ical
displacemen o he cen e o he wheelse is zw, and mw,cwand kw a e i s
gene alized mass, damping and s i ness cons an s. The e ical
displacemen o he ail sec ion unde he wheelse is z , and m ,c and k
a e i s gene alized mass, damping and s i ness cons an s. The equa ions
o mo ion o he sys em abou he e ical s a ic equilib ium posi ion a e
gi en by:
[mw0
0m ][¨
zw
¨
z ]+[cw−cw
−cwcw+c ][˙
zw
˙
z ]+[kw−kw
−kwkw+k ][zw
z ]
=[kwu+cw
˙
u
−kwu−cw
˙
u]⇒M¨
q+C˙
q+Kq =F( )(6)
whe e m, c, and k a e he mass, equi alen iscous damping coe icien
and s i ness cons an o he sys em, espec i ely. Rep esen ing he ack
dynamics using concen a ed elemen s (poin mass, dashpo , sp ing)
ha un wi h he ehicle is a common modeling echnique in ail oad
dynamics, as done o example in he model used in he Manches e
Benchma k [34]. The esea ch g oup o he au ho s o his pape has
de eloped a compu a ional me hodology [35], called Mo ing Modes
Me hod, ha can be used o ob ain he “ unning” cons an s associa ed
wi h he ack lexibili y using a de ailed ini e elemen model.
As i is gene ally he case, assume ha kw≫k . Le us de ine he
s i ness a io N=kw/k ≫1. Assume also ha he a io o damping
cons an s ul ils N=cw/c . The mass a io is de ined as:
α
=m /mw.
No e ha in N he ail cons an is in he denomina o while in
α
he
wheelse cons an in in he denomina o . This is done on pu pose
because k and mw a e easie o measu e han kw and m , espec i ely.
The assump ion ha N=cw/c implies s i ness-p opo ional damp-
ing. In his case, he eigen alue analysis based on he mass and s i ness
ma ices p o ide he na u al equencies and modes o ib a ion,
Fig. 1. (a) Kinema ic wheel- ack model. (b) Cu a u e incompa ibili y. Fig. 2. Two-do ehicle- ack model.
X. Yu e al.
Measu emen 249 (2025) 117058
3
yielding:
K−
ω
2M=0⇒
ω
=
ω
n1,
ω
n2
[K−
ω
ni2M]ϕ=0⇒ϕ=ϕ1,ϕ2⇒ Φ = [ϕ1ϕ2],(7)
whe e
ω
ni and ϕi, i=1,2, a e he na u al equencies and modes o
ib a ion and Φ is he modal ma ix. These equa ions can be sol ed
symbolically, yielding:
ω
n1≅
k
mw(1+
α
)
√,
ω
n2≅
Nk (1+
α
)
α
mw
√,Φ
≅⎡
⎢
⎣1+1
N(1+
α
)−
α
(1−1
N(1+
α
))
1 1 ⎤
⎥
⎦,(8)
whe e he assump ion N≫1 has been used o he simpli ica ion. Fo he
s anda d modal ans o ma ion, he new se o modal coo dina es p=
[p1p2]T, such ha q=Φp, a e subs i u ed in o he equa ions o
mo ion, yielding:
ΦTMΦ¨
p+ΦTCΦ˙
p+ΦTKΦp=ΦTF( )⇒m¨
p+c˙
p+kp = ( ),(9)
whe e he modal mass, damping and s i ness ma ices and he modal
o ce a e gi en by:
m≅[mw(1+
α
)0
0
α
mw(1+
α
)],c≅[c 0
0Nc (1+
α
)2],k
≅[k 0
0Nk (1+
α
)2], ( ) ≅ ⎡
⎢
⎣
1
N(1+
α
)
− (1+
α
)
⎤
⎥
⎦(kwu+cw
˙
u).(10)
The uncoupled equa ions o mo ion in e ms o he modal coo dina es
a e gi en by:
mw(1+
α
)¨
p1+c
˙
p1+k p1=1
(1+
α
)(k u+c
˙
u),
m
¨
p2+cw(1+
α
)˙
p2+kw(1+
α
)p2= − (kwu+cw
˙
u).(11)
As shown in Fig. 3, he sys em beha es a a modal le el as wo suspended
ehicles mo ing on i egula acks. The mass, suspension p ope ies
and le el o i egula i ies ha hese ehicles “see” can be obse ed in he
igu e. Taking he Fou ie ans o m o hese equa ions and eo ganizing
yields:
P1(
ω
) = (k +i
ω
c )
k −
ω
2mw(1+
α
)+i
ω
c
1
(1+
α
)U(
ω
) = 1
(1+
α
)T1(
ω
)U(
ω
),
P2(
ω
) = − (kw+i
ω
cw)
kw(1+
α
)−
ω
2m +i
ω
cw(1+
α
)U(
ω
) = − 1
(1+
α
)T2(
ω
)U(
ω
),
(12)
whe e i=
−1
√, P1(
ω
), P2(
ω
), U(
ω
)a e he Fou ie ans o ms o p1( ),
p2( ), u(s/V), espec i ely, and T1(
ω
)and T2(
ω
)a e modal ans-
missibili y unc ions. The ansmissibili y unc ions can be ew i en
using he usual non-dimensional pa ame e s in he heo y o ib a ions,
as ollows:
T1(
ω
) = (1+2iξ1
τ
1)
1−
τ
12+2iξ1
τ
1
,T2(
ω
) = (1+2iξ2
τ
2)
1−
τ
22+2iξ2
τ
2
,(13)
whe e ξ1=c /(2
ω
n1mw(1+
α
)), ξ2= (1+
α
)cw/(2
ω
n2m )a e he
damping ac o s associa ed wi h each mode, and
τ
1=
ω
/
ω
n1,
τ
2=
ω
/
ω
n2 a e he non-dimensional ack i egula i y equencies.
Using he modal ans o ma ions om Eq. (8), he Fou ie ans o m
o he o iginal coo dina es yields:
[Zw(
ω
)
Z (
ω
)]=Φ[P1(
ω
)
P2(
ω
)]
=⎡
⎢
⎣(1+1
N(1+
α
))P1(
ω
)−
α
(1−1
N(1+
α
))P2(
ω
)
P1(
ω
)+P2(
ω
)
⎤
⎥
⎦
≅[P1(
ω
)−
α
P2(
ω
)
P1(
ω
)+P2(
ω
)].(14)
Subs i u ing Eq. (12) in o Eq. (14) yields:
Zw(
ω
) ≅ (1
(1+
α
)T1(
ω
)+
α
(1+
α
)T2(
ω
))U(
ω
) = Tw(
ω
)U(
ω
).(15)
Tha means ha he ansmissibili y o m ack co uga ion o axle-box
displacemen , Tw(
ω
), is a linea combina ion o T1(
ω
)and T2(
ω
). When
he ehicle goes e y slow, he equency o he exci a ion
ω
end o ze o
and bo h, T1(
ω
)and T2(
ω
), end o one, he e o e:
V⟶0⟹
ω
⟶0⟹T1(
ω
)⟶1,T2(
ω
)⟶1⟹Tw(
ω
)⟶1
(1+
α
)+
α
(1+
α
)
=1.
(16)
I he co uga ion o axle-box ansmissibili y ends o one, he axle-box
pe ec ly ollows he ail co uga ion o low o wa d eloci y V, as
expec ed.
The dynamic ans e unc ion om ail co uga ion o axle-box
no malized accele a ion, ha is de ined as ¨
zw( )/V2, is gi en by
Tdyn
no ( ) = 1
(1+
α
)Tno ,1( )+
α
(1+
α
)Tno ,2( ),Tno ,1( )
= − (2
π
)2(1+2iξ1
τ
1)
1−
τ
12+2iξ1
τ
1
,Tno ,2( ) = − (2
π
)2(1+2iξ2
τ
2)
1−
τ
22+2iξ2
τ
2
,
(17)
The dynamic TF om ail co uga ion o axle-box accele a ion is gi en
by
Tdyn
ABA( ) = V2Tdyn
no ( ).(18)
Apa om speed V, he dynamic TF based on a 2-do sys em depends on
i e pa ame e s: he na u al equencies n1 and n2, he damping ac o s
Fig. 3. Modal esponse o wo-do ehicle- ack model.
X. Yu e al.
Measu emen 249 (2025) 117058
4
ξ1 and ξ2, and he mass a io
α
. Assuming he alues: n1=332 Hz, n2=
1035 Hz, ξ1=0.0295, ξ2=0.0173,
α
=0.11, Fig. 4 shows Tdyn
ABA( )
( op) and Tdyn
no ( )(bo om) o di e en o wa d eloci ies o he
wheelse anging om 0.5 o 2.5 m/s. Sec ion 3.2 will show ha he
assumed alued o he i e pa ame e s a e ac ually he expe imen ally
iden i ied alues o he wheel- ail sys em used in his pape .
The plo on op is he TF based on he ABA accele a ion and he plo
a he bo om is he TF based on he no malized accele a ion. As is can be
obse ed, he use o a no malized accele a ion is s ill con enien in he
dynamic case because o low equencies all he TF’s collapse o he
kinema ic one, ha is a s aigh line wi h slope 2 when plo ed in log-
a i hmic scale, as ollows:
log(Tkin
no )=2[log( )+log(2
π
)],(19)
The esonance peaks mo e o smalle space equencies when he o -
wa d eloci y inc eases. I can be in e p e ed ha , he lowe he o wa d
eloci y V, he wide is he ange o space equencies whe e he kine-
ma ic model is alid. In p ac ice, his means ha he lowe he inspec-
ion eloci y he wide is he ange o measu able co uga ion
wa eleng hs a oiding esonance e ec s, ha clea ly in oduce unce -
ain y in o he measu emen . I he wa eleng h o he ack de ec is
ela i ely long (3––200 m), hen hey a e called “i egula i y” ins ead o
“co uga ion”. Con a y o he p oblem o esonance e ec s, measu ing
i egula i ies using ABA a low inspec ion eloci y may ha e a sensi-
i i y p oblem, because he e ical accele a ion induced in he axle-box
may be oo small.
Mo e complex and de ailed dynamic model o he wheel- ack sys-
em can be de eloped o yield a mo e accu a e TF ha in u n depends
on a la ge se o pa ame e s. Anyway, he concep o he TF wo ks only
i he sys em dynamics is linea . The pu pose o his pape is o check he
alidi y o he TF expe imen ally using a scaled ehicle- ack sys em
wi h co uga ion ha is desc ibed in nex sec ion.
3. Expe imen al se up
3.1. Vehicle- ack sys em
The 1:10 scaled ack is loca ed on he oo o he School o Engi-
nee ing a he Uni e si y o Se ille [36]. I is 90 m long and is o med by
s aigh segmen s, wo cu es wi h 24 m and 6 m adii, and ansi ion
segmen s, as shown in he plan iew o Fig. 5 (a). The ails a e manu-
ac u ed using ec angula s eel beams, whe e a scaled e sion o he
UIC-54 ail p o ile has been machined jus in he ail head a ea. Fig. 5 (b)
shows he geome y o he cen eline o he ack and he loca ion o he
co uga ed segmen s.
The scaled ehicle, shown in Fig. 6, is a bogie wi h wo igid
wheelse s. The p ima y suspension includes eigh helical sp ings con-
nec ing bo h wheelse s wi h he bogie ame. The ehicle ins umen a-
ion includes:
Two piezoelec ic accele ome e s (b and is PCB Piezo onics and
e e ence is 352C33 wi h 50 g- ange) ins alled on he axle-boxes in
he on wheelse and a hi d one in he cen al pa o he bogie
ame. All accele ome e s measu e e ical accele a ions.
A high p ecision encode (b and is Kuble and e e ence is
05.2400.1122.0360) ha egis e s he o a ion o he on wheelse .
A da a acquisi ion sys em (b and is Na ional Ins umen s and e e -
ence is NI myRio) wi h acquisi ion a e o 5 KHz.
The ehicle is d i en wi h a Maxon elec ic mo o using conical gea s
in he ansmission.
3.2. Expe imen al modal analysis
The expe imen al modal analysis o he wheelse on he ack was
pe o med as obse ed in Fig. 7. An impac hamme was used in he
es s. Two accele ome e s we e used. One was ins alled on he axle-box
while he second was ins alled in he oo o he ail. Loca ions we e
selec ed o measu e he accele a ions associa ed wi h he 2-do model
shown in Fig. 2. Fig. 8 shows he measu ed accele a ions. The accele -
a ions a e app oxima ely a w0-ha monic signal, whe e he high-
equency ha monic, wi h an app oxima e equency o 1 MHz, dies
ou app oxima ely a 8 ms. A e wa ds, only he low equency ha -
monic, wi h an app oxima e equency o 300 Hz, emains. The high
equency has li le e ec on he accele a ion o he ail. I can also be
obse ed ha o he low equency ib a ion he axle-box and he ail
ib a e in phase, wi h highe ampli ude on he axle-box. Howe e , o
he high equency ha monic, ib a ion happens a app oxima ely 180◦
o phase di e ence and much highe ampli ude on he axle-box. All
hese ema ks a e consis en wi h he 2-do model p esen ed in Sec ion
2.2 and i s analy ical modal analysis.
Fig. 4. Top: dynamic TF based on accele a ion. Bo om: dynamic TF based on no malized accele a ion.
X. Yu e al.
Measu emen 249 (2025) 117058
5
Fig. 9 shows he ecep ance o he axle-box ob ained wi h he powe
spec al densi ies o he con ac o ce a he impac hamme and he
accele a ion a he axle-box and hei c oss-spec al densi y. The cu e is
i ed o he esul s o he modal analysis o he 2-do model shown in
Sec ion 2.2. The modal p ope ies, needed o ind he ansmissibili y
unc ions T1(
ω
)and T2(
ω
)gi en in Eq. (13) a e:
ω
n1=2
π
×332 ad/s,
ω
n2=2
π
×1035 ad/s,ξ1=0.0295,ξ2=0.0173
Φ=[1.5787 −2.1438
1 1 ].(20)
Numbe s gi en in Eq. (20) coincide wi h hose used o plo he TFs in
Fig. 4. The e o e, he TFs shown in Fig. 4 a e no jus an example, bu he
ac ual unc ions associa ed wi h he scale wheel- ack sys em used in
his esea ch acco ding o he expe imen al modal analysis.
In ailway dynamics, he P2 equency is he lowes na u al e-
quency o he e ical ib a ion o a wheelse unning on he ack. An
app oxima ion o his equency can be ob ained assuming a simple
mass-sp ing sys em in which he wheelse mass and e ical s i ness o
he ack a e used o se he model pa ame e s. The na u al equency
ω
n1, whose analy ical o mula is gi en in Eq. (8) and has an expe i-
men ally measu ed alue o 332Hz, can be conside ed as he P2 e-
quency o he scaled ack. Conside ing ha in eal acks his equency
lies in he in e al 30 – 100 Hz, i can be concluded ha he scale ack is,
in ela i e e ms, much s i e han a eal ack [37].
3.3. Co uga ed ails
Fou ail segmen s o 1.8 m, which a e ins alled wo on he le side
and wo on he igh side, ha e been machined o c ea e a co uga ed
ailhead p o ile. The co uga ed a ea co e s mo e han 3.6 m because
he s a ing and end poin s in he le and igh sides do no coincide.
The e o e, he e a e a eas a he en ance and exi wi h co uga ion only
in one side and a cen al a ea wi h bo h ails co uga ed.
The co uga ion p o ile is buil by adding ou ha monic unc ions
wi hou phase di e ence, as ollows:
z (s) = ∑
4
i=1
Bisin(2
π
λi
s),(21)
whe e he ampli udes Bi and he wa eleng hs λi a e gi en in Tab 1.
Acco ding o he s anda d EN-13231–2 [38], and conside ing he
scale, he selec ed wa eleng h lies wi hin he anges 30–100 mm and
100–300 mm. Howe e , he ampli udes Bi a e no scaled bu exagge -
a ed due o he di icul y o machine a p o ile wi h ampli ude o jus a
ew mic ons.
Machining was done wi h a CNC model LAGUN L-650 wi h a
sphe ical 2-mm end-milling cu e . This machine, shown in Fig. 10 has a
Fig. 5. (a): Plan iew o he scaled ack: ae ial pho og aph and (b) scheme o he ack cen e line.
Fig. 6. Scaled ehicle: a) le iew, b) igh iew.
X. Yu e al.
Measu emen 249 (2025) 117058
6
Fig. 7. Expe imen al modal analysis o wheel on ack.
Fig. 8. Accele a ions measu ed wi h expe imen al modal analysis.
Fig. 9. Axle-box ecep ance measu ed wi h expe imen al modal analysis.
X. Yu e al.
Measu emen 249 (2025) 117058
7
1-µm p ecision in Ca esian displacemen and a o al ope a ing leng h o
600 mm. Tha means ha machining he 1.8 ail segmen s equi es 3
phases, wi h he esul ing inaccu acies a he connec ing sec ions.
Including he ime equi ed o calib a ion and alignmen , a pe iod o 61
h was used in he machining p ocess, being 2 h and 40 min he ime
equi ed o machine each 600 mm-segmen .
The ins alled co uga ed ails, shown in Fig. 11, a e loca ed a he
ack dis ance s =54.9 m in he le side and a he ack dis ance s =
55.6 m in he igh side.
A e machining, he ailhead p o iles we e measu ed wi h a lase
p o ilome e . Fig. 12 shows he le and igh ail p o iles along 1.8 m. I
can be obse ed ha :
•In he i s 0.8 m he le ail is co uga ed, bu he igh one is no .
•Co uga ion, his is, sho wa eleng h i egula i y, appea s supe -
imposed o e a longe wa eleng h i egula i y, as i happens in eal
acks.
•The machined co uga ion is no pe ec ly pe iodic, as expec ed.
Fig. 13 shows he measu ed e ical p o ile o he igh ail a e
de ending, ha is he one ha will be used in he esul s p esen ed in
his pape . Fig. 14 shows he spec a o his e ical p o ile. The op plo
o Fig. 14 shows he RMS spec um, and he bo om plo shows he PSD.
Bo h spec a a e shown because bo h will be used in he calcula ion o
he TF in la e sec ions. Peaks a he design wa eleng hs (5, 10, 20, 30
mm) a e clea ly obse ed. The peak alues o he RMS spec a coincide
wi h he alues gi en in he ou h ow o Table 1 di ided by
2
√.
To analyse he esul s ha will be shown in he nex sec ion, i is
impo an o obse e ha he p o ile unc ion gi en in Eq. (21) can be
conside ed as a pe iodic unc ion whose pe iod is he leas common
mul iple o he wa eleng hs gi en in he hi d ow o Table 1, his is,
λlcm =60mm. The e o e, he p o ile unc ion can be w i en as a Fou ie
se ies, as ollows:
z (s) = a0+∑
∞
i=1
aisin(i2
π
0s),(22)
being 0=1
λlcm =16,61/m he undamen al pe iod, and a2=B1,a3=
B2,a6=B3,a12 =B4,and ai=0 o all o he i.
3.4. F ee ligh s and cu e compa ibili y
In he expe imen s made in his in es iga ion he ehicle a elled
wi h app oxima e o wa d eloci ies o 0.5, 1.0, 1.5, 2.0 and 3.0 m/s. I
he kinema ic wheel- ail model shown in Fig. 1 applies, a simple o ce
balance shows ha he wheel ail no mal con ac o ce is gi en by:
Fc=m(g+¨
zw).(24)
The e o e, he no mal con ac o ce will end o change sign whene e
he e ical accele a ion o he wheel is nega i e and equal in no m o
he accele a ion o g a i y. Using he kinema ic assump ion, and
conside ing he equa ion o he p o ile gi en in Eq. (21), he e ical
accele a ion o he wheel yields:
¨
zw( ) = ∑
4
i=1−Bi(2
π
λi
V)2
sin(2
π
λi
V ).(25)
The e o e, he ampli ude o he e ical accele a ion associa ed wi h
each o he ou ha monics is gi en by Bi(2
π
λiV)2
,i=1,2,3,4.
A simple o ce balance shows ha he wheelse sepa a es om he
ack when he ampli ude o he ine ia o ce equals he s a ic o ce
ansmi ed by he wheelse , his is, i s own weigh plus hal he weigh
o he es o he ehicle, as ollows:
Fig. 10. (a): CNC machine model LAGUN L-650. (b): Machining o a co uga ed segmen .
Fig. 11. (a): Ins alla ion o le co uga ed ail on he scaled ack. (b): De ail o co uga ed ail.
X. Yu e al.
Measu emen 249 (2025) 117058
8
munBi(2
π
λi
V)2
=(mun +msp
2)g⟹Bi(2
π
λi
V)2
=(1+msp
2mun)g=2.63g,
(26)
whe e mun is he unsp ung mass o he ehicle (1.964 kg, mass o he
wheelse , axle-boxes and bea ings) and msp is he sp ung mass (6.397 kg,
mass o he es o he ehicle). Fig. 15 shows he alue o he ampli udes
Fig. 12. Co uga ed ail p o iles measu ed wi h lase p o ilome e .
Fig. 13. Righ ail e ical p o ile a e de ending.
Fig. 14. Righ ail spec a. Top: RMS spec um. Bo on: PSD.
Table 1
Design and eal p ope ies o pe iodic co uga ed p o iles.
Wa e Numbe 1 2 3 4
Space equency 1/λ (cycles/m) 33 50 100 200
Wa eleng h λ (mm) 30 20 10 5
Ampli ude B (
μ
m) 44.7 51.6 25.6 24.7
Maximum cu a u e (1/m) 2.0 5.1 10.1 39.0
X. Yu e al.
Measu emen 249 (2025) 117058
9
TF and i s applica ion. Machining in he ail head a p o ile ha can be
conside ed as Gaussian whi e noise would ha e been mo e app op ia e
o s udy he TF. Howe e , his ype o p o ile would be o ally di e en
o he co uga ed p o iles in eal ack. The solu ion adop ed in his
in es iga ion can be conside ed as a ade-o solu ion.
The s udied TFs a e he simples possible unc ions. The kinema ic TF
assumes ha he axle box o he wheel, whe e he accele ome e is
ins alled, ollows a ajec o y ha is a ansla ed copy o he ailhead
p o ile. This TF is alid i he wheel-axle box sys em beha es as a igid
body in he e ical di ec ion, he wheel keeps con ac wi h he ail (no
ee ligh s) and he e is no cu a u e incompa ibili y. I he ehicle
beha es as a de o mable 2-do sys em, he TF akes an analy ical o m
ha depends on a ew pa ame e s ha can be iden i ied expe imen ally.
Expe imen al modal analysis has been used in his wo k o ind he
app op ia e pa ame e s o he 2-do model. The calcula ion o he TF
using he 2-do model, being e y simple, can be used o explain eso-
nance e ec s in he measu emen s. A simple ule o a oid esonance
e ec s in he measu emen s o co uga ion using ABA is o keep he
inspec ion eloci y V as low as possible.
The analy ical o ms o he men ioned TFs, and he linea ans-
o ma ion o space- equencies in o ime- equencies when he ehicle
mo es wi h di e en o wa d eloci ies V, sugges s ha no maliza ion o
he measu ed accele a ions wi h V can help in he in e p e a ion o he
esul s. This pape sugges s p ocessing a no malized accele a ion ha is
ob ained di iding he measu ed accele a ion by he squa e o he o -
wa d eloci y, ha is assumed o be app oxima ely cons an . I his
no malized accele a ion is used, he kinema ic TF is o wa d eloci y
independen and he 2-do TF oo, bu jus o equencies below he
esonance peak. This esonance peak mo es o wa d in he equency
axis linea ly wi h he in e se o he o wa d eloci y. The expe imen al
esul s shown in his pape con i ms he bene i s o p ocessing he
no malized accele a ions o educe he e ec o he o wa d eloci y
used du ing he measu emen s. Expe imen al esul s show ha he
measu ed axle box accele a ion inc eases wi h he o wa d eloci y, bu
he no malized accele a ion dec eases wi h he o wa d eloci y.
The analysis o he measu ed accele a ions in equency domain
shows ha he axle-box e ical mo ion has he same equency con en
han he ail p o ile, bu i also includes peaks a equencies ha a e
mul iples o he undamen al equency o he ail p o ile bu a e no
exci a ion equencies. This phenomenon has no been explained. One
possible explana ion, no checked in his in es iga ion, is ha his
phenomenon is he esul o nonlinea dynamics e ec s due o ee
ligh s and cu e incompa ibili y o he wheel ail ela i e mo ion.
Al hough he expe imen al condi ions a e no a ou able, he TF has
been a emp ed o be ob ained by signal p ocessing o he inpu ( ail
p o ile) and he ou pu (axle box accele a ion). The kinema ic TF ap-
p oxima es well he expe imen ally ob ained TF a he exci a ion e-
quencies, i equencies a e no oo high so ee ligh s and cu a u e
incompa ibili y do no appea . The H1 app oxima ions o he TF a e e y
noisy, because he selec ed exci a ion is no app op ia e o his ype o
es ima ion. The pa abolic o m o he kinema ic TF can be sligh ly
obse ed. Using his H1 app oxima ion, i has been shown ha he high
peaks o he accele a ions a non-exci a ion equencies a e p obably no
due o esonance o he ehicle sys em. These esul s sugges ha mo e
complex TFs based on mo e ad anced bu linea models o he ehicle
would no help o imp o e he measu emen s.
The ail p o ile has been econs uc ed using he axle-box accele a-
ions and he kinema ic TF. Resul s show ha o ela i ely low and
ela i ely low space equencies he econs uc ed esul s p o ide
inaccu a e bu easonable ep esen a ion o he eal ail p o ile.
CRediT au ho ship con ibu ion s a emen
Xinxin Yu: W i ing – e iew & edi ing, W i ing – o iginal d a ,
Visualiza ion, Valida ion, So wa e, In es iga ion, Fo mal analysis.
Se gio Mu˜
noz: In es iga ion, Funding acquisi ion, Concep ualiza ion.
Ped o U da: Resou ces, In es iga ion, Da a cu a ion. Ja ie F. Acei-
uno: W i ing – e iew & edi ing, Resou ces, P ojec adminis a ion,
In es iga ion, Funding acquisi ion. Miguel Rod íguez G´
omez: W i ing
– e iew & edi ing, Valida ion, Da a cu a ion. Jos´
e L. Escalona: W i ing
– o iginal d a , Visualiza ion.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inancial
in e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
Acknowledgemen s
This esea ch is suppo ed by he Spanish Depa men o Economy,
Science, En e p ise and Uni e si y o he Andalusian Regional Go e n-
men , unde he PAIDI 2020 p og am wi h p ojec e e ence P18-RT-
1772. I is also suppo ed by he Spanish Minis y o Science, Inno a ion
and Uni e si ies, unde he p og am “P oyec os de Gene aci´
on de
Conocimien o 2023”, wi h p ojec e e ence PID2023-152786OB-I00.
This suppo is g a e ully acknowledged. The i s au ho would like o
acknowledge he suppo om he Academy o Finland (Applica ion No.
357038).
Da a a ailabili y
The au ho s a e unable o ha e chosen no o speci y which da a has
been used.
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(2008) 1231–1237.
[2] S.L. G assie, Rail co uga ion: ad ances in measu emen , unde s anding and
ea men , Wea 258 (7–8) (2005) 1224–1234.
[3] Z. Li, S. Li, P. Zhang, A. Nú˜
nez, R. Dolle oe , Mechanism o sho pi ch ail
co uga ion: ini ial exci a ion and equency selec ion o consis en ini ia ion and
g ow h, In . J. Rail T anspo a ion 12 (1) (2024) 1–36.
[4] P. Zhang, Z. Li, New expe imen al e idences o co uga ion o ma ion due o ail
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