148 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533
T uck ib a ions caused by unbalanced o a ing sha
Pa el Kuče a1, Václa Píš ěk2
B no Uni e si y o Technology, Technicka 2896/2, 616 69, B no, Czech Republic
1Co esponding au ho
E-mail: 1kuce a@ me. u b .cz, 2pis ek. @ me. u b .cz
(Recei ed 29 July 2016; accep ed 31 July 2016)
Abs ac . This a icle p esen s an analysis o ib a ions in d i e ain o an 8×8 uck caused by an
unbalanced o a ing d i e sha . This uck is also used in mili a y and i e sphe e. The aim was o
educe he ib a ions a ec ing he d i e s, he e o e, a ansien compu a ional model o analysis
o d i e ain sha de lec ion was c ea ed in Ma lab so wa e. This model was compa ed o he
simula ion o he ansien beha iou o a d i e ain sha 3D model wi h he use o FEM. To
e i y he compu a ional models, measu emen s o he o a ing sha de lec ion and di ec ional
oscilla ions on he uck d i eline we e ca ied ou in he o iginal e sion and hen also wi h he
designed modi ica ions o he componen s. The conclusion p esen s he in e p e a ion o he
esul s.
Keywo ds: uck, d i e sha , ib a ion, d i eline, Ma lab, unbalance, FEM.
1. In oduc ion
The ib a ions which make an impac on he d i e also a ec he d i e com o and may also
a ec he heal h o he d i e . The e o e, an analysis o he ib a ions has become impo an o e
he yea s, and no only wi h he passenge ca s bu also wi h ucks. The ib a ions in ucks may
be mo e in ensi e and may gene a e noise which makes he d i e unpleasan o he d i e . The
8×8 d i e ain o a uck is a complex dynamic sys em whe e he ib a ions may come in o being
in any o i s pa s. This a icle ocuses on he ib a ions caused by an unbalanced o a ing sha in
a d i eline. This uck showed p oblems wi h ib a ions and noise a e eaching a ce ain speed,
documen ed ab oad whe e he limi s o he Eu opean legisla ions a e no alid. The e o e, i was
assumed ha i was caused by he e olu ions o he d i e ain sha . To analyse he de lec ion o
he longes sha he ansien compu a ional models we e c ea ed using analy ical and di e en ial
equa ions and Fini e Elemen Me hod (FEM). The alida ion o he compu a ional models was
ca ied ou by se e al measu emen s on he 8×8 and 8×4 uck d i esha while d i ing on he
highway. Vib a ions and de lec ion o he longes sha we e measu ed. The conclusion p esen s
in e p e a ion o he esul s ob ained om bo h he compu a ional models and measu ed da a. The
u he u iliza ion o he compu a ional models is also included.
2. Compu a ional models
As men ioned in he in oduc ion, o analysis wo compu a ional models o he sha de lec ion
while o a ing we e p epa ed. They a e desc ibed u he in he ex . The compu a ional model
analysed by FEM is desc ibed only ma ginally, since i is used only as a ce ain moni o ing
indica o o he compu a ional model c ea ed wi h he use o analy ical and di e en ial equa ions.
Fig. 1 shows he 3D model o he d i e sha used o he FEM analysis.
Fig. 1. 3D model o he d i e sha and he measu ing posi ion
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2.1. T ansien model o he sha de lec ion – model I
To be able o e i y he p esupposed o ma ion o ib a ions on he uck caused by he
unbalanced o a ing sha , a compu a ional model o a o a ing sha wi h equa ions based on
[1, 2] had o be c ea ed. The esul s o he calcula ions we e subsequen ly compa ed o he
measu emen .
The compu a ional model o he d i e sha includes he pa o he sha be ween he wo
bea ings. The i s bea ing is loca ed on he pa o he axial di e en ial o he on axle and he
second bea ing is loca ed on he second on axle on he le (see Fig. 1). This pa o he sha is
di ided in o  mass poin s. By he measu emen on he uck he ini ial de lec ion was se in he
place o he de lec ion senso . The sha de lec ion is supposed o be based on hese equa ions:

=


sin󰇡
󰇢, (1)
=

sin󰇡
󰇢, (2)
whe e  is de lec ion in he middle o he analysed sha leng h,  – de lec ion in he measu ed
poin ,  – loca ion o measu emen ,  – sha leng h,  – ini ial sha de lec ion in he indi idual
poin s based on he gi en segmen s o he sha , and  – coo dina es o he gi en sha
c oss-sec ion.
Fig. 2. D i e sha compu a ional model and i s bounda y condi ions
Cen i ugal o ces ac on he indi idual mass poin s o he sha . Thei alues we e de e mined
in each s ep o he simula ion because de lec ion and sha speed a e inc eased. The ollowing
equa ion was used:
=

−,(3)
whe e  is cen i ugal o ce in he loca ion o he indi idual mass poin s,  – mass o he gi en
mass poin ,  – sha de lec ion in he loca ion o he gi en mass poin , and  – sha angula
speed. The bounda y condi ions a e an impo aspec o he sha . I may be assumed ha he
bounda y condi ions will be be ween he hinged and ixed suppo because he e a e di e en sha
suppo s and clea ance in he spline sha . The e o e, h ee models we e c ea ed whe e he mos
impo an one is p esen ed in Fig. 2. I desc ibes he sha wi h p esc ibed o a ional s i ness in
he suppo . ,  a e he momen eac ions and ,  he o ce eac ions. The bounda y
condi ions o ixed suppo ha e o be based on he ac ha i he beam is in ixed suppo on
bo h sides i is a wice s a ically inde e mina e ask. The e o e, de o ma ion condi ions need o be
added o gain o ce eac ions and momen ac ing on he sha . In ixed suppo he de o ma ion
condi ions a e de ined a e a pa ial elease whe e he gi en end o he sha is assigned ze o
displacemen and o a ion. Subsequen ly, 4 equa ions o 4 unknown a e a ailable – o ce
equa ion, momen equa ion and equa ion o o a ion and displacemen calcula ed using he
Maxwell-Moh a ian o he Cas igliano’s heo em:
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=()
()∂()
∂=0,


(4)
=()
()()


=0, (5)
whe e  is displacemen in poin ,  – o a ion in poin ,  – bending momen ,  –
modulus o elas ici y, and  – quad a ic momen o he c oss sec ion o he sha .
Fo he beam wi h he de ined s i ness in he suppo he o ce equa ion, momen equa ion and
wo equa ions o o a ion in poin s  and  will be used. Howe e , hese o a ion equa ions will
no equal ze o bu hey will be modi ied o he o m o he ollowing equa ions:
=()
()()
=−
,


(6)
=()
()()


=−
,(7)
whe e ,  a e o a ions in poin s  and , ,  – s i ness o he sha suppo . Fo he
hinged suppo i is a s a ically de ini e ask, he e o e, he eac ions a e de e mined based on he
o ce and momen equa ions.
To gain alues o he de lec ions in he indi idual s eps, he equa ion o he de lec ion line is
used whe e he o a ion and displacemen in he gi en place is de e mined by a g adual in eg a ion.
The dis ance be ween he mass poin s is he same as he dis ance be ween he o ces. The equa ions
o he de lec ion line a e de ined as ollows:

󰇘
=−
+(−(−1)),



(8)
󰇗=−
2+󰇧
2−(−1)󰇨+,



(9)
=−
6+󰇧
6−(−1)
2󰇨++,



(10)
whe e  is 1, 2, 3, + 2,  is 1, 3, 5, + 2,  is displacemen and  in eg a ion cons an s. These
cons an s ha e o be de e mined om he bounda y condi ions, i.e. he ze o displacemen s in he
suppo . Fu he mo e, he equali y o o a ion alues and de lec ions be ween he indi idual
eleased segmen s has o exis .
Based on he equa ions abo e, he compu a ional model is o med in Ma lab so wa e whe e
he esul s o he measu ed d i e ain in he o iginal design a e compa ed wi h he modi ied
solu ion and wi h he compu a ional model o he o iginal d i e sha . The esul s a e p esen ed in
Fig. 5.
2.2. T ansien model o he sha de lec ion – model II (FEM)
Subsequen ly, analyses o he pa s o he sha and he whole d i e sha wi h he axle
di e en ial we e pe o med using FEM. The ansien solu ion was used simila ly o he p e ious
model. This compu a ional model II wo ks as a check o model I, he e o e, he e is no need o
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de ailed desc ip ion.
3. Measu emen
The measu emen o he o a ing sha de lec ions and di ec ional ib a ions was ca ied ou
wi h he use o IMC CRFX 400. I is a modula swi chboa d whe e he use may add he indi idual
measu emen ca ds as necessa y. In his case, he ca d ICPU2-8 was used o connec he
accele ome e s and he ca d CRFX/ISO2-8 was used o eco d he analogue signals. Two
h ee-axial accele ome e s, a de lec ion senso and a pho oelec ic e lex swi ch o speed
measu emen we e connec ed o he swi chboa d. Du ing he measu emen he speed was
inc eased g adually om he minimal alue o he maximal and he speed gea s we e changed o
gain da a om he whole span o he uck speed. The measu emen was ca ied ou on a d i e ain
o an 8×8 and 8×4 uck on a highway.
Subsequen ly, FFT spec ums [3] o measu ed ib a ions and sha de lec ions we e in e p e ed
depending on he sha speed. Sc ip s o p ocess he da a in Ma lab so wa e we e c ea ed. The
signal o he sha o a ions is a e aged and sc ip sea ches o he sha o a ion wi h a s ep o
10 min-1. Thus, an index o da a loca ion in ma ix is eached. This index looks o he da a o
in e p e a ion o he ib a ions and de lec ions. The measu ing posi ion is illus a ed in Fig. 1.
Fig. 3. Analysis o di ec ional ib a ion in he Z-axis, d i e 8×4, o iginal and new d i e sha e sion
Fig. 4. Analysis o di ec ional ib a ion in he -axis, d i e 8×8, o iginal and new d i e sha e sion
The sha speed was measu ed by a senso loca ed on he main pa o he swi chboa d. The
speed signal was eco ded as a ma k pe o a ion om he pho oelec ic e lex swi ch. The
measu emen o he sha de lec ions was ca ied ou by a BAW M30ME-UAC10B-S04G senso .
By i s pa ame e s his senso ul illed he assump ions o measu emen in he span o de lec ion
up o 10 mm. The esul s o he gained da a a e shown in Fig. 5 whe e he s a es be o e and a e
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d i e ain modi ica ion a e compa ed. To measu e he d i e ain ib a ion, wo h ee-axial
piezoelec ic accele ome e s B üel & Kjæ 4524 we e used. To in e p e he ib a ion, he FFT
analysis om he segmen ep esen ing he gi en speed was used. The sample equency was
20 kHz and he numbe o samples was 8192. F om hese analyses FTT spec ums we e calcula ed
(see Fig. 3 and 4). These spec ums a e illus a ed only o he -axis because he esul s o he
o he axes a e simila . The igu es show he peak which s ands o he 1s sha speed ha monic
o de .
This is he mos signi ican ac o when i comes o he ib a ion a ec ing he d i e . Mo e
peaks also occu mainly wi h he all-wheel d i e which e lec s he oo h equency o he axle.
Bu hese equencies a e highe han 100 Hz [4], he e o e, in e ms o com o hey canno be
educed by he modi ica ion o he d i esha . The compa ison o measu emen s be o e and a e
he modi ica ion o he d i e ain is also shown. A conside able educ ion o ib a ions caused by
he o a ing sha de lec ion was eached, which is illus a ed in Fig. 3 and 4 on he igh .
4. Resul s
Based on he de lec ion simula ion and measu emen i may be concluded ha he
compu a ional models we e chosen app op ia ely o assess he de lec ion, he e o e, he e is no
need o ex end he compu a ional model I o he whole d i e sha .
The esul s a e compa ed in Fig. 5.
Fig. 5. Compa ison o he measu ed alues and compu a ional models
The measu ed da a we e o 6 speed gea s; hus he mos ex ensi e span o he sha o a ions
would be included especially in he a ea o 400-2600 min-1. These alues we e hen a e aged.
Based on he dimensions o he cons uc i e assembly a ound he sha , he sha de lec ion limi
is 3.4 mm. The measu ed da a show ha he sha de lec s o such s a e ha i leans agains ano he
pa o he d i e ain.
The esul s o he de lec ion measu emen , ib a ion analysis and model simula ion show ha
he de lec ion o his sha is he main cause o he d i esha oscilla ion which is ansmi ed o
he whole s uc u e o he uck. To dec ease he uck ib a ion, i is necessa y o minimalize he
sha de lec ion and hus i s unbalance. The e o e, a modi ica ion o he d i esha was designed
and ca ied ou be o e he second ound o measu emen . The modi ica ion was a cons uc i e
solu ion o he sha suppo . The sha was spli in o wo sho e sha s which we e inse ed in o
he socke o a ing in he bea ing.
The ollowing measu emen p o ed ha he sha de lec ion was conside ably dec eased (see
Fig. 5). A he same ime subs an ial dec ease o ib a ions on he uck was measu ed.

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5. Conclusions
To analyze he sha de lec ion, compu a ional models we e o med and hei esul s we e hen
compa ed wi h measu emen s. Based on he esul s, i may be concluded ha he s uc u e o he
compu a ional model is sui able o he analysis o he sha de lec ion and may be used o u he
de elopmen o he d i esha in all ehicles. The ansla ional ib a ions caused by he o a ing
sha de lec ion we e educed by he cons uc ional design. Based on he measu emen ca ied ou
a e he modi ica ion o he d i e ain as seen om Fig. 3 and 4, he suppo o he analyzed sha
led o he dec ease o ib a ions and a nea loss o he 1s sha speed ha monic o de . This
co esponds o he analysis o he sha de lec ion and i may be uni ocally concluded ha he
sha de lec ion and i s unbalance was he main cause o he ib a ions on he uck. The
compu a ional model I may be easily used o he de elopmen o he d i e ain wi h long sha s
whe e he sha de lec ion may occu de o ma ion o unbalance
Acknowledgemen s
This wo k is an ou pu o he coope a ion be ween in e nal BUT Resea ch P ojec Reg.
No. FSI-S-14-2334 and NETME CENTRE PLUS (LO1202) by inancial means om he Minis y
o Educa ion, You h and Spo s unde he “Na ional Sus ainabili y P og amme I”.
Re e ences
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2006.
[2] Walle H., Schmid R. Vib a ion Theo y o Enginee s: Theo y, Simula ion, Applica ions. Fi s
Edi ion, Wissenscha s e lag, Zü ich, 1989.
[3] Tůma J. Vehicle Gea box Noise and Vib a ion. Wiley, Chiches e , 2014.
[4] ISO 2631-1. Mechanical Vib a ion and Shock: E alua ion o Human Exposu e o Whole-Body
Vib a ion – Pa 1. In e na ional O ganiza ion o S anda diza ion, 1997.