Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA MECCA 01 2013 PAGE 16
10.2478/mecdc-2013-0003
Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA
1. INTRODUCTION
Knowledge o momen o ine ia, ob ained ei he by
expe imen al measu emen o compu a ional es ima es, is an
impo an ac o du ing he design and cons uc ion o all mo o
ehicles. I is a c ucial inpu o a ious ehicle ma hema ical
models, and he yaw momen o ine ia in pa icula is an
in eg al pa o di ec ional dynamics in es iga ions. These mass
p ope ies o a ehicle a e also used by he con ol algo i hms
o elec onic d i e assis ance sys ems, such as ESP (elec onic
s abili y p og am). I a su icien ly de ailed ehicle model is o
be cons uc ed, i is necessa y o ensu e ha i has inpu s o
adequa e accu acy, including momen o ine ia alues.
Mos yaw momen o ine ia measu emen me hods a e based
on he physical pendulum p inciple. One such me hod is he
suspension me hod, p esen ed in de ail in Re . [1–4]. This
me hod uses he oscilla ing mo ion o an objec (au omobile,
ai c a , e c.) which is suspended by cables. Va ious e sions o
his me hod a e possible using di e en numbe s o cables. The
o he me hod uses a o sion pendulum, whe e he o a ional
oscilla ion o an objec placed on a ig is achie ed using o sional
o coil sp ings. Addi ional in o ma ion abou his me hod can
be ound in [5, 6]. The ela i e accu acy o all hese app oaches
a ies om 0.3 % o 1% [3, 5 & 11], and one o he main issues
is p ecise posi ioning o he ehicle on he de ice.
The de ice desc ibed he e is based on he o sion pendulum
p inciple. The accu acy o acqui ed momen o ine ia is he e o e
dependen , in pa icula , on he oscilla ion pe iod measu emen
e o [7]. This a icle u he p esen s he p ocedu e o yaw
momen o ine ia measu emen o a ehicle using a es
ig and da a analysis p ocess, bo h designed a he Ins i u e
o Au omo i e Enginee ing o he Facul y o Mechanical
Enginee ing, B no Uni e si y o Technology.
2. TEST RIG DESIGN
The basic p inciple o yaw momen o ine ia de e mina ion
is de i ed om physical pendulum heo y. Expe imen al
de e mina ion o momen o ine ia is pe o med on a o a ional
ig which oscilla es a ound he e ical axis, he e u ning mo ion
p o ided by coil sp ings. The o mula de i ed om he physical
pendulum momen o ine ia de e mina ion has he ollowing
o m [6]:
PETR HEJTMÁNEK, ONDŘEJ BLAŤÁK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA
Ins i u e o Au omo i e Enginee ing, B no Uni e si y o Technology, Technická 2, CZ 616 69 B no, Czech Republic
Tel.: +420 541 142 272, Fax: +420 541 143 354, E-mai: hej manek@ me. u b .cz
SHRNUTÍ
Článek popisuje pos up u čení momen u se ačnos i ozidla ke s islé ose. Je na žen měřicí s a p o měření momen u se ačnos i
založený na p incipu yzikálního ky adla a je analyzo án li chyby měření pe iody kmi u na celko ou chybu u čení momen u
se ačnos i. V p og amu Lab iew je ses a en ýpoč o ý pos up p o přesné u čení pe iody kmi a ého pohybu z naměřených da .
Funkčnos celého měřicího pos upu je o ěřena s ano ením přesnos i a opako a elnos i měření.
KLÍČOVÁ SLOVA: MOMENT SETRVAČNOSTI VOZIDLA KE SVISLÉ OSE, PERIODA KMITU, FYZIKÁLNÍ KYVADLO
ABSTRACT
This a icle se s ou a me hod o he de e mina ion o a ehicle’s yaw momen o ine ia. A es ig o momen o ine ia measu emen s
based on he physical pendulum p inciple has been designed and he e ec o he oscilla ion pe iod measu emen e o on he o al
measu emen e o was analysed. The compu a ional p ocedu e o accu a e oscilla ion pe iod de e mina ion was de eloped in
he LabVIEW so wa e en i onmen , and he unc ion o he comple e measu emen p ocedu e was hen e i ied by e alua ing he
accu acy and p ecision o he measu emen s.
KEYWORDS: VEHICLE YAW MOMENT OF INERTIA, OSCILLATION PERIOD, PHYSICAL PENDULUM
MEASURING THE YAW MOMENT OF INERTIA
OF A VEHICLE
Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA MECCA 01 2013 PAGE 17MECCA 01 2013 PAGE 16
2
2
4л×
×= T
C
TI (1)
Whe e:
I… yaw mass momen o ine ia
T…oscilla ion pe iod
CT… o al o sional s i ness gene a ed by coil sp ings
This momen o ine ia equa ion desc ibes undamped oscilla ions,
while in he eal wo ld he mo emen is damped by ic ion.
Howe e , his damping can be igno ed, as i s in luence on he
oscilla ion pe iod magni ude is negligible, which is p o ed
u he in he a icle.
FIGURE 1: Comple e ig o measu ing a ehicle’s yaw momen o ine ia
OBRÁZEK 1: Komple ní s a p o měření momen u se ačnos i ozidla
The measu emen de ice consis s o wo main pa s: a s a ic pa
and a o a ional pa . The s a ic pa is ep esen ed by a igid
s eel c oss a ached o a s eel base wi h sc ews. This s eel base
is a pla e embedded in o he labo a o y loo , wi h a weigh
o 14.5 onnes, an uppe su ace o 15 m2 and a la ness o
0.1 mm - p ecise ho izon al posi ioning is necessa y o high
measu emen accu acy.
The o a ional pa including ca amps (da k colou ed pa
in Figu e 1) is a ached o he s a ic pa using low- ic ion
axial bea ings, which allow o a ion a ound he e ical axis.
Fou coil sp ings, which p o ide he oscilla ing mo ion, a e
moun ed be ween bo h pa s, wo on each side pe pendicula
o he o a ion axis a he dis ance li. To al o sional s i ness
CT gene a ed by coil sp ings wi h s i ness ci can he e o e be
calcula ed as:
Σ×=
i
iiT lcC 2 (2)
The ca amps no only p o ide a means o d i ing on and o he
ig - hey a e also able o ock, so ha he ehicle’s longi udinal
cen e o g a i y can be placed in he middle o he amps. This
mo emen along he la e al axis o he ig is p o ided by adial
ball bea ings. A e posi ioning he ehicle app op ia ely, bo h
amps a e ixed using s u ods.
The main ad an ages o he designed ig a e i s simple and
igid cons uc ion wi h low space equi emen s, and abo e all
i s abili y o enable isola ed oscilla ions o he ehicle a ound
he desi ed axis o o a ion wi hou he addi ional unwan ed
mo emen s ha occu when using he suspension me hod.
FIGURE 2: Measu emen de ice wi hou amps o illus a e sp ing placemen
OBRÁZEK 2: Měřicí zařízení bez nájezdů se znázo něním umís ění p užin
3. MEASUREMENT PROCEDURE
In he i s s ep, he ehicle is d i en on o he amps o he es
ig and es ablished in equilib ium, bu only in he longi udinal
di ec ion. The ehicle cen e o g a i y may be displaced om he
o a ional axis in he la e al di ec ion. Nex , he amps a e ixed
using ods and he es ig wi h he ehicle can s a oscilla ing,
and he oscilla ion pe iod o he sys em TPV can be measu ed.
A e d i ing he ehicle o he amps, he de ice is se in o he
measu ing posi ion again and he oscilla ion pe iod o he ig
wi hou he ehicle TP is measu ed. Thus he yaw ine ia alue
o he ehicle can be calcula ed acco ding o ollowing o mula:
( )
22
2
4PPV
TTT
C
I-×
×
=л (3)
Finally, o de e mine he yaw momen o ine ia o he ehicle
a i s cen e o g a i y i is necessa y o iden i y he dis ance
be ween he la e al posi ion o he cen e o g a i y and he
longi udinal cen e line o ehicle - , and nex ans o m he
acqui ed alue using S eine ’s heo em:
2
mIIz×-=
(4)
Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA MECCA 01 2013 PAGE 18
4. MEASUREMENT ERROR ANALYSIS
The es ig measu emen e o is de e mined om he accu acy o he oscilla ion pe iod measu emen . The gene al equa ion o he
ela i e e o o an indi ec ly measu ed cha ac e is ic is as ollows [7]:
( )
I
x
x
I
x
x
I
I
I
xxI
...
,...,
2
2
2
2
1
1
21
+Δ
д
д
+Δ
д
д
=
Δ
=ξ
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
(5)
I CT is a cons an aking in o accoun he es ig’s p ope ies, he equa ion o he absolu e e o o he yaw momen o ine ia
measu emen ΔI is:
( )( ) ( )( ) 2
22
2
22
Δ
д
-×д
+Δ
д
-×д
=Δ P
P
PPV
PV
PV
PPV T
T
TTcons
T
T
TTcons
I
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛ (6)
Al hough he absolu e e o o he oscilla ion pe iod may a y o di e en con igu a ions o he es ig, he accu acy o he oscilla ion
pe iod is conside ed cons an . P o ided ha he ig’s oscilla ion absolu e measu emen e o is he same bo h wi h and wi hou he
ehicle, he equa ion can be simpli ied o:
( ) ( ) 22
22 222 PPVPPV TTTcons TTcons TTcons I+×Δ××=Δ×××+Δ×××=Δ (7)
Fo he ela i e measu emen e o , he ollowing equa ion applies:
( )
22
2
2
22
22
2
2
PPV
PPV
PPV
PPV
TT
TT
T
TTcons
TTTcons
I-
+
Δ×=
-×
+×Δ××
=ξ (8)
Tab. 1 illus a es he ela ionship be ween he ela i e e o o yaw momen o ine ia measu emen and he absolu e e o o
oscilla ion pe iod measu emen . Values a e calcula ed o a ig oscilla ing pe iod o TPV = 1.3256s wi h he ehicle and TP = 0.5324s
wi hou i .
TABLE 1: The absolu e oscilla ion pe iod e o - ela i e momen o ine ia
e o ela ionship
TABULKA 1: Vz ah absolu ní chyby pe iody kmi ání a ela i ní chyby
momen u se ačnos i
ΔT [s] ξI [%]
0.1 19.39
0.05 9.69
0.01 1.94
0.005 0.97
0.001 0.19
0.0005 0.10
0.0001 0.02
I was es ablished ha he esul ing momen o ine ia e o
ises linea ly wi h he oscilla ion pe iod measu emen e o , and
ha o achie e a ela i e accu acy o ehicle momen o ine ia
measu emen o 2%, i is necessa y o measu e he pe iod wi h
an e o magni ude equal o o less han 0.01s. A su icien ly
accu a e me hod o de e mining he oscilla ion pe iod is shown
u he in he a icle.
O e all ehicle mass m and he dis ance measu emen e o s
a e no o high signi icance, as o a s anda d au omobile
he ela i e de ia ion be ween IZ and I does no ise abo e
app oxima ely 0.05% ( he dis ance a ies om 0 o 0.04 m).
5. OSCILLATION PERIOD DETERMINATION
As men ioned ea lie , he momen o ine ia measu emen
p ocedu e u ilizes he physical pendulum p inciple. The p e ious
chap e demons a es he impo ance o accu a e measu emen
Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA MECCA 01 2013 PAGE 19
o he oscilla ion pe iod. Hence a new sophis ica ed measu emen
sys em o oscilla ion pe iod de e mina ion was de eloped
and designed. This sys em consis s o a da a acquisi ion ca d
NI DAQPad-6015, an angula eloci y senso (in eg a ed in o
MBOX – 3 accele ome e s, 3 angula eloci y senso s) and
a powe supply. The eco ded angula eloci y signal ollows he
oscilla ions and o ms he pe iodic unc ion. The comple e da a
eco ding and p ocessing so wa e uses he LabVIEW so wa e
package om Na ional Ins umen s. The p og am s uc u e is
di ided in o h ee phases: calib a ion, da a eco ding and da a
p ocessing.
The calib a ion phase equalizes he measu ed signal ampli ude
symme ically a ound he ze o alue.
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FIGURE 3: The oscilla ion pe iod measu ed da a – signal om he angula
eloci y senso
OBRÁZEK 3: Da a z měření pe iody kmi ání – signál snímače úhlo é
ychlos i
In he da a eco ding phase, da a is eco ded a a speci ied
sampling a e. The o e all measu emen ime is usually 40s and
he sampling equency 100Hz. The oscilla ion pe iod changes
sligh ly du ing he aking o measu emen s due o he non-linea
o sional s i ness o he de ice (caused by a mino o ce ec o
o a ion as he coil sp ings o a e sligh ly wi h he mo emen o
he o a ional pa o he ig).
To e alua e he in luence o o sional s i ness non-linea i y on
he calcula ion esul s o he oscilla ion pe iods, an analysis
o oscilla ion pe iod change du ing he eco ding o angula
eloci y was pe o med. Resul s o his analysis show ha he
ela i e de ia ion o he oscilla ion pe iod does no ise abo e
0.5%, which co esponds o a ela i e e o o 1.2% o he
ehicle momen o ine ia.
Al hough he sys em is damped, as was sugges ed ea lie ,
he oscilla ion is conside ed o be undamped. The damping
i sel admi edly in luences he oscilla ion pe iod alue, bu
i s signi icance is negligible. Acco ding o conduc ed analysis,
he di e ence o pe iod be ween damped and undamped
mo ion is less han 0.01% and he damping coe icien alue is
app oxima ely 0.03 – 0.04.
Using he eco ded da a, he calcula ion o he oscilla ion
pe iod using a non-linea eg ession based on he Le enbe g-
Ma qua d (LM) algo i hm is pe o med [8,9]. The eg ession
analysis in es iga es he ela ionship be ween he independen
a iable X ( ime) and he dependen a iable Y ( he measu ed
p ope y). The Nonlinea Cu e Fi LM is used in he LabVIEW
so wa e package o he non-linea eg ession. The model
unc ion is shown he e:
(9)
The unc ion includes 5 pa ame e s, and hei alues mus be
es ima ed o s a wi h. This is done wi h he i s analysis o he
measu ed da a. The a o emen ioned pa ame e s a e: ampli ude
A, damping coe icien α, ime o se 0, oscilla ion pe iod T and
ampli ude o se y0 (examples o alues eco ded o a ehicle:
A=0.221; α=-0.035; 0=0.861s; T=1.471s; y0=0.003). Ano he
LabVIEW unc ion used is he Hanning Window, which se es o
e ining he e alua ion [10], applying he weigh unc ion in he
eg ession. This unc ion wi h he Hanning Window is shown in
Figu e 4.
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OBRÁZEK 4: Uži í áho é unkce Hanningo a okna
The algo i hm con e ges o he inal esul quickly a e he
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Resul s a e shown in Figu e 5 (black – measu ed da a, ed –
eg ession, g een – weigh ing unc ion).
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FIGURE 5: Resul s o angula eloci y measu emen
OBRÁZEK 5: Výsledky měřen úhlo é ychlos i
These pa ame e s con ain he oscilla ion pe iod ha is used
in he nex s ep o he analy ical calcula ion o he momen
o ine ia. The iden i ica ion o he esul ing yaw momen o
Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA MECCA 01 2013 PAGE 20
ine ia is based on he s eps desc ibed abo e. To ob ain he
s a is ical p edica i e esul s, i is necessa y o conduc he
whole p ocedu e ( he measu emen and he oscilla ion pe iod
de e mina ion) epea edly.
6. ACCURACY AND PRECISION
OF MEASUREMENTS
The de e mina ion o accu acy and p ecision o he measu ed
da a is c ucial o each measu ing de ice. Wi hou his knowledge
i would be di icul o assess he quali y o measu ed alues.
E alua ion o measu ed alues (momen o ine ia) agains he
calcula ed ones is one o he possible me hods – in his case
i was achie ed using lead blocks o known dimensions and
weigh as a ballas . These blocks a e calib a ed o a p ede ined
momen o ine ia wi h maximum allowed de ia ion o 0.7 %.
Thei simple shape ensu es ha subsequen calcula ions o
ine ia will be s aigh o wa d. To de e mine he epea abili y i
is jus necessa y o pe o m he momen o ine ia measu emen s
se e al imes.
In he i s s ep, yaw momen o ine ia o he ig’s o a ional
pa s IP was measu ed expe imen ally, eaching he alue o
413.06 kg·m2. The ea e , lead b icks we e placed a each co ne
o he ig (Figu e 7), making he ballas ’s cen e o g a i y
in e sec he ig’s e ical o a ion axis. The posi ion o each cube
was measu ed and no ed. The measu emen was epea ed i e
imes, he esul s o which a e p esen ed in Table 2, wi h he
ballas ’s yaw momen o ine ia deno ed as IV.
The co esponding calcula ion was ca ied ou using he CAD
so wa e P o/Enginee (Figu e 6). B icks we e modelled and
placed in acco dance wi h he expe imen ; he ballas ’s momen
o ine ia alue can hen be calcula ed om he dimensions,
olume, densi y and spa ial dis ibu ion o all lead b icks.
A compa ison o measu ed and calcula ed alues is p esen ed
in Table 3.
TABLE 2: The esul s o he epea ed measu emen s o he ballas ’s yaw
momen o ine ia
TABULKA 2: Výsledky opako aného měření momen u se ačnos i zá ěže
Measu emen
numbe
IP + IV
[kg·m2]
IV
[kg·m2]
Δ IV
[kg·m2]
|δ IV |
[-]
12718.86 2305.80 0.60 0.03%
22719.01 2305.5 0.75 0.03%
32718.62 2305.56 0.37 0.02%
42718.18 2305.12 -0.08 0.00%
52716.62 2303.56 -1.64 0.07%
a e age -2305.19 - -
( ) %04.0
12
=
Δ
=
Σ
i
I
I
n
IRSD (10)
TABLE 3: Momen o ine ia – expe imen s. CAD model compa ison
TABULKA 3: Momen se ačnos i – s o nání měření a CAD modelu
IV - measu ed
[kg·m2]
IV - CAD
[kg·m2] ela i e e o
alue 2305.19 2283.85 0.93%
The esul s ob ained show ha he accu acy and p ecision o
he p oposed de ice o he momen o ine ia measu emen a e
easonably high. Based on he pe o med expe imen s, he e o
o he measu emen - accu acy is less han 1% and he RSD
( ela i e s anda d de ia ion) - p ecision is no highe han 0.04 %.
FIGURE 6: CAD model o he measu ed ballas
OBRÁZEK 6: CAD model měřeného zá aží
FIGURE 7: De ailed pho o showing he loca ion o qua e s o he b icks
OBRÁZEK 7: De ailní o og a ie zob azující umís ění jedné č iny kos ek

Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA MECCA 01 2013 PAGE 21
The nex expe imen was a ge ed a analysing he p ecision o
ac ual passenge ca measu emen s. The ehicle yaw momen
o ine ia measu emen was ca ied ou on a small amily ca
(C-segmen ) wi h a weigh o 1350 kg. The ehicle con igu a ion
o he measu emen included: no passenge s, no luggage o
ballas , ully illed uel ank. Ty es we e in la ed o a p essu e ha
is highe han he alue ecommended by he manu ac u e wi h
he aim o a oiding excessi e de o ma ion o he i es du ing
he measu emen . Addi ional mo emen s o sp ung pa s o he
ehicle du ing oscilla ions, which could ha e an ad e se e ec
on he measu emen , a e supp essed by ying he ehicle’s side
sills up o he amps. In he pe iod be ween ballas and ehicle
measu emen s, he ig was sligh ly modi ied, which educed i s
momen o ine ia alue o 403.5 kg·m2. The ehicle momen
o ine ia was measu ed only ela i e o he cen e o he ig
and was no ans o med in o he ehicle’s cen e o g a i y.
Comple e esul s o he ehicle measu emen s a e p esen ed in
Table 4.
Al hough he RSD o he measu emen has isen o 0.2%, he
de ice can s ill be conside ed e y p ecise. E alua ion wi h
calcula ed ( heo e ical) alues was no possible as ele an CAD
da a o he ehicle a e no a ailable, and e en i hey we e, hey
would no include se ice luids and o he ac o s, which ha e
an impac on he expe imen ally de e mined yaw momen o
ine ia. Fo hese easons, he ela i e accu acy o he ehicle
yaw momen o ine ia measu emen canno be accu a ely
deduced om he p esen ed expe imen al se ies.
6. CONCLUSION
This a icle co e s se e al s eps in he cons uc ion o a es ig
o measu emen o ehicle momen o ine ia. The momen
o ine ia measu emen e o has been analysed and i s close
ela ionship wi h he oscilla ion pe iod accu acy was es ablished.
The demand o accu a e oscilla ion pe iod de e mina ion was
sa is ied by de eloping an algo i hm in he LabVIEW so wa e
en i onmen .
The accu acy and p ecision o measu emen was e alua ed in
wo s eps: i s by compa ing measu emen s and calcula ions
o lead b icks se ing as ballas , wi h esul ing ela i e e o
< 1% and ela i e s anda d de ia ion o 0.04 %, and secondly
by measu ing a C-segmen ehicle, which has shown an RSD
o 0.2%. The ela i e e o o ehicle measu emen was no
de e mined; ne e heless, he e o o ehicle measu emen can
be de i ed om alida ion o lead b icks. The accu acy alue
gi en by he measu emen o he ballas co esponds o he
e e ences ci ed.
The main imp o emen o his app oach o e o he designs is
ha he p ecise posi ioning o he ehicle can be achie ed in only
a ew minu es using he ehicle’s own d i e (no c ane o o he
equipmen needed). Some elemen a y pa ame e s o he de ice
can be easily changed wi h espec o he measu ed ehicle size,
such as o sional s i ness o he ig o amp dis ance o gi en
wheel ack.
Fu he wo k will be aimed a imp o ing he es ig design by
inco po a ing he abili y o measu e oll and pi ch momen o
ine ia.
ACKNOWLEDGEMENTS
Published esul s ha e been achie ed wi h he help o “Modelling
o ehicle dynamics” p ojec , egis a ion No. FSI-J-12-1803,
g an ed by speci ic uni e si y esea ch o B no Uni e si y o
Technology. This suppo is g a e ully acknowledged.
TABLE 4: The ehicle yaw momen o ine ia measu emen –
passenge ca (C-segmen )
TABULKA 4: Měření momen u se ačnos i ozidla –
osobní au omobile ( řídy C)
Measu emen
numbe
IP + IV
[kg·m2]
IV
[kg·m2]
Δ IV
[kg·m2]
|δ IV |
[-]
12915.94 2512.44 2.29 0.09%
22916.32 2512.82 2.67 0.11%
32921.72 2518.22 8.07 0.32%
42899.66 2496.16 -13.99 0.56%
52914.99 2511.49 1.35 0.05%
62914.77 2511.27 1.12 0.04%
72911.39 2507.89 -2.25 0.09%
82914.19 2510.69 0.55 0.02%
92915.36 2511.86 1.71 0.07%
10 2912.13 2508.63 -1.51 0.06%
a e age -2510.15 - -
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Δ IV [kg·m2]
FIGURE 8: The a iance o he measu ed ehicle yaw momen o ine ia
OBRÁZEK 8: Rozdíl měřeného momen u se ačnos i ozidla
MECCA 01 2013 PAGE 22
Measu ing he Yaw Momen o Ine ia o a Vehicle
PETR hEJTMáNEK, ONdřEJ BLAŤáK, PAVEL KUČERA, PETR PORTEŠ, JAN VANČURA
LIST OF ABBREVIATIONS AND SYMBOLS
A angula eloci y ampli ude [ ad·s-1]
ci linea s i ness o coil sp ing i [N·m-1]
CT o sional s i ness [N·m · ad-1]
I momen o ine ia abou a gene al axis [kg·m2]
IP measu emen s and yaw momen o ine ia [kg·m2]
IV ehicle (ballas ) yaw momen o ine ia [kg·m2]
Iz yaw momen o ine ia abou he ehicle z axis [kg·m2]
li pe pendicula dis ance o sp ing i om o a ion axis [m]
m ehicle weigh [kg]
la e al posi ion o he cen e o g a i y [m]
T oscilla ion pe iod [s]
TP oscilla ion pe iod o s and [s]
TPV oscilla ion pe iod o s and wi h ehicle [s]
0 ime o se [s]
y0 angula eloci y ampli ude o se [ ad·s-1]
ΔI yaw momen o ine ia absolu e measu emen e o
[kg·m2]
ΔIV yaw momen o ine ia absolu e de ia ion [kg·m2]
ΔT oscilla ion pe iod absolu e e o [s]
α oscilla ion damping coe icien
δIV yaw momen o ine ia ela i e de ia ion [%]
ξI yaw momen o ine ia ela i e measu emen e o [%]
REFERENCES
[1] Wood ield, A. A. (1969). Measu emen o he yawing
momen and p oduc o ine ia o an ai c a by he
single poin suspension me hod: heo y and ig design.
Repo s and memo anda (Ae onau ical Resea ch Council
(G ea B i ain)), No. 3607.
[2] C ede, C. E. (1948). De e mining Momen o Ine ia,
Machine Design, Augus , pp. 138.
[3] Ja din, M. R., Muelle , E. R. (2009). Op imized
Measu emen s o Unmanned-Ai -Vehicle Mass Momen o
Ine ia wi h a Bi ila Pendulum. Jou nal o Ai c a , Vol. 46,
No. 3, May–June, pp. 163 – 775.
[4] Mille , M. P. (1930). An Accu a e Me hod o Measu ing he
Momen s o Ine ia o Ai planes, NACA TN-351
[5] Heydinge , G. J., Du isek, N. J., Coo e , D. A., Guen he ,
D. A., No ak, S. J. (1995). The Design o a Vehicle Ine ia
Measu emen Facili y, SAE, De oi .
[6] Vlk, F. (2003). Au omobilo á echnická pří učka – Zkoušení
a diagnos ika mo o o ých ozidel, VLK, B no,
ISBN 80-238-9681-4.
[7] Volejník, P. (2006). Ná h měřicího zařízení p o u čení
momen ů se ačnos i. Diploma Thesis, B no: Vysoké učení
echnické B ně, Fakul a s ojního inžený s í, pp. 108
[8] Le enbe g, K. (1944). A Me hod o he Solu ion o Ce ain
P oblems in Leas Squa es, Qua . Appl. Ma h. Vol. 2,
pp. 164–168.
[9] Ma qua d , D. (1963). An Algo i hm o Leas -Squa es
Es ima ion o Nonlinea Pa ame e s, SIAM J. Appl. Ma h.
Vol. 11, pp. 431–441.
[10] Oppenheim, A. V., Scha e , R. W. (1989). Disc e e-Time
Signal P ocessing, P en ice-Hall, pp. 447–448.
[11] P e ia i, G., Gobbi, M., Mas inu, M. (2008). S iluppi nella
misu a del enso e di ine zia e del ba icen o dei co pi
igidi, XXXVII Con egno Nazionale, Uni e si a di Roma.
Tes ing o Mode n Vehicles on a 2WD Rolle Tes Bench
ONdřEJ GOTFRÝd, VOJTěCh KLíR, JIří dRdA