research article driveline torsional analysis and clutch damper...

Research ArticleDriveline Torsional Analysis and Clutch DamperOptimization for Reducing Gear Rattle

Huwei Wu1 and Guangqiang Wu12

1College of Automotive Studies Tongji University 4800 Caorsquoan Road Jiading District Shanghai 201804 China2Institute of Industrial Science The University of Tokyo Tokyo 153-8505 Japan

Correspondence should be addressed to Huwei Wu 1133054tongjieducn

Received 6 July 2015 Accepted 27 October 2015

Academic Editor Miguel Neves

Copyright copy 2016 H Wu and G Wu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper describes a research work on driveline modeling torsional vibration analysis and clutch damper parametersoptimization for reducing transmission gear rattle on the vehicle creeping condition Firstly major driveline components includingquasi-transient engine multistage stiffness clutch damper detailed manual transmission and differential mechanism and LuGretire are modeled respectively Secondly powertrain system modeling adopting a two-stage stiffness clutch damper is constructedand analyzed Transient responses predicted by the model show that the driveline undergoes severe torsional vibration andtransmission gear rattle phenomenon By analysis it is concluded that the clutch damper works jumping between the first- andsecond-stage stiffness which results in this problem for the creeping condition Then a three-stage stiffness clutch damper isproposed innovatively to solve this problem It is shown that severe driveline vibration and gear rattle phenomenon are inhibitedeffectively Finally it draws a conclusion that clutch damper parameters could have a great effect on driveline vibration and gearrattle phenomenon and a three-stage stiffness clutch damper could be utilized to solve gear rattle phenomenon efficiently on thevehicle creeping condition

1 Introduction

Vibroimpacts in manual transmission (MT) are of criticalconcern to vehicle manufacturers based on noise vibrationand reliability consideration Gear rattle is a typical gearnoise that is generated under the existence of torsionalfluctuations which in turn leads to gear teeth impact ofunloaded gears fluctuating within tooth lash The impactcollision is transmitted to the transmission housing via shaftsand bearings and then converted into an audible rattle noisewhich is broadband in the frequency spectrum Rattle noisehas a distinct sound quality that differentiates it from othernoises produced by other sources in the vehicle whichmakespassengers usually annoyed by this noise and attribute it tosome vehicle companies So a better understanding of thedynamic behavior of drivelines and transmission gear rattlemechanism is in urgent need and has drawn many scholarsrsquoattention

Gear rattle phenomenon is a comprehensive problem ofthe driveline that includes many nonlinearities of multistage

clutch damper gear meshing stiffness gear backlash dragtorque and so on These nonlinearities make it difficult toanalyze the mechanism for this phenomenon Some attemptson numerical simulation and experiment studies are con-ducted in some literature

In terms of numerical simulation initial research on gearrattle focused on one gear pair Nakamura firstlymodeled onestraight spur gear pair in which the time-varying meshingstiffness was equivalent to square wave function and statictransmission error was the sum of harmonic Fourier seriesIt gave the moment of gear rattling clearly through thenumerical simulation method [1] Since then many researchscholars paid more attention to solving algorithms of math-ematical models Comparin and Singh utilized harmonicbalance method to solve one gear pair rattling model whicharrived at the fact that there was two-side impact one-sideimpact or no impact with some parameters changing [2]Kahraman and Singh found that one gear pair nonlinearproperty involved subharmonic response and chaos responseby the numerical simulation method and harmonic balance

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 8434625 24 pageshttpdxdoiorg10115520168434625

2 Shock and Vibration

method [3] As the study continued research object wastransferred from one simple gear pair to complicated geartransmission system Based on the four-degree-of-freedommodel of one gear pair Bozca et al proposed empirical modeland torsional vibration model based optimization of a 5-speed gearbox design parameters to reduce rattle noise in anautomotive transmission Despite the geometric parametersoptimization overall rattle noise level was reduced and alloptimized geometric design parameters also satisfied allconstraints [4 5] Besides gear rattle problem is regarded asa comprehensive problem of the driveline system Most ofthe drivelinemodels used for the driveline torsional vibrationanalysis are lumped discrete models with a few degreesof freedom Wang et al described a model most early fortorsional vibration of automotive manual transmission (MT)to analyze and predict gear rattle of all speeds Accordingly arattle index was used to compare the rattle levels producedby different gear pairs But in that model gear meshingstiffness was constant and self-excited vibration of time-varying stiffness was ignored [6]Wu and Luan paid attentionto the impact of gearmeshing stiffness on the vehicle drivelinetorsional vibration and gave a comparison of simulationbetween variable meshing stiffness and averaged stiffness ofloaded gear pairs based on overall powertrain system [7]Robinette et al developed a representative model for a frontwheel drive (FWD) vehicle with MT by lumped parameteranalysis and presented functional relations for torque lossesassociated with shafts gears seals lubricating oil flow andbearing clearance as a function of basic design parame-ters [8] Drag torque including bearing friction torque oilshearing torque or oil churning torque was then validatedby experimental results [9] De La Cruz et al considereddifferent lubrication states influence of gear pairs on the rattlephenomenon and proposed a rattle index in considerationof lubrication state [10] Fietkau and Bertsche proposed asimulation approach for loaded and unloaded gear con-tacts which include oil films and elastic deformations Thisapproach is validated with experiment measurement and itis concluded that lubricant condition could not be ignored[11] Theodossiades et al took into consideration the effectof lubrication during engine idle conditions and examinedthe influence of lubrication in torsional vibration It is shownthat the lubricant film behaved as a time-varying nonlinearspring-damper element and could have a great influence onthe gear rattle problem [12 13] Crowther et al put for-ward 6-degree-of-freedom (DOF) model using a frequencysweep with engine excitation derived from measured datawith two-stage gear meshing and an unloaded gear pairIt is found that the gear rattling is more severe when theengine speed passes the resonation frequency region of thesystem natural model It concluded that an effective dynamicengine model is needed in order to get transient drivelinecomponent motion and then rattle phenomenon actually[14] Bhagate et al put forward a 6-DOFmathematical modelfor the torsional vibrations of front wheel drive automotivedrivetrain and developed the optimization of sensitive systemparameters for reducing the driveline rattle [15] So as fortransmission modeling various factors such as gear pairtime-varying stiffness gear friction bearing friction and gear

oil churning loss are in urgent need in the future modelingwork

In terms of rattle experiments Couderc et al designedand built early an experimental setup of a vehicle driveline forthe prediction of the dynamic behavior of vehicle drivelinesIt is concluded that the simulation model validated by theexperimental setup could provide transient response truly[16] Bellomo et al analyzed the contribution of individualsound source to the overall rattle noise by means of noise-source analysis and proposed a pareto-optimal solution toreduce the rattle noise emission utilizing a rattle test bench[17] This improved test bench reproduced the brancheddriveline system rather than the one-string driveline systemin [16] Forcelli et al set up a virtual engine simulator for auto-motive transmission and conducted a parametric sensitivitystudy for amplitude of the torsional vibration Moreover arelationship between the vibroacoustics measurements andthe human perception was found [18] Barthod et al analyzedthe rattle threshold and the rattle noise evolution for dif-ferent multiharmonic excitation parameters and mechanicalgearbox parameters through a bench test [19] Crowther andRozyn introduced a gear rattle test rig in which the electricmotor drives the transmission at a steady mean speed via adouble telescoping Hookersquos joint By changing the angle ofthe joints the amplitude of the vibration could be adjusted[20] Baumann and Bertsche built one gear pair test rigfor rattle research and compared the rattle intensity underdifferent lubricant oil condition It is found that adopting highviscosity oil could inhibit rattle phenomenon when angularacceleration of the input shaft is larger [21] Brancati et alset up a specific test rig for one lightly loaded gear pairwhich is able to acquire the relative rotation motion ofgears by two high resolution incremental encoders Basedon measurement data from this test rig a gear rattle metricbased on the wavelet multiresolution analysis was proposed[22]

The clutch damper is a component of drivelines thatcould have significant influence on the torsional dynamicbehavior of drivelines Gear rattle phenomenon can be greatlyreduced by opportunely setting some clutch parameters suchas the multistage torsional springs Steinel examined theinfluence of the twin-mass flywheel on the driveline naturalcharacteristics and transient responses It was shown that thetwin-mass flywheel was the ideal solution for drivetrains ofwhich the vibrations could not be reduced sufficiently if therewas no need for the consideration of costs [23] Prasad etal found that elimination of gear rattle could be achievedby maximizing the hysteresis of clutch thereby absorbing theenergy being transferred through the subjective and objectiveevaluation in the passenger bus experiment [24] But it isobvious that maximizing the hysteresis of the clutch damperwould reduce the transmission efficiency of the powertrainsystem Xu et al introduced a novel clutch damperwith three-stage stiffness and solved the rattle phenomenon effectively inlow torque condition compared with the damper with two-stage stiffness by vehicle experiments [25] Similarly manyresearch scholars found that the clutch damper property playsan important role in reducing driveline vibration and rattlephenomenon [26 27]

Shock and Vibration 3

1

6 6

77

7

7

72

5 5

3

4

Figure 1 RWD vehicle powertrain system A four-cylinder andfour-stroke engine B clutch damper C manual transmission Ddifferential mechanismE half axleF wheel andG power flow

This paper presents a lumped parameters model capableof predicting the driveline vibration the onset of loose gearrattle and the clutch damper optimization for reducingloose gear rattle Firstly a description of the driveline andmodeling of major components are presented Then thedriveline model is used to perform transient analysis ofcurrent systems and provide a comprehensive understandingof a four-cylinder and four-stroke engine excitation thestrong nonlinearities of the driveline elements (includingmultistage clutch stiffness and frictional hysteresis) andparameter excitations of loaded gear pair meshing stiffnessThe drivelinemodel is divided into the baseline vibration andthe rattling vibration The baseline vibration is taken as theexcitation to the rattling vibration and it is neglected thatthe rattling vibration has an effect on the baseline vibrationA detailed manual transmission modeling could reproducethe onset of rattle phenomenon of unloaded gear pairsFinally a comparison of the baseline vibration and the rattlingvibration between using a two-stage stiffness clutch damperand using an improved three-stage stiffness clutch damperis studied on the vehicle creeping condition which showsthat it is achievable to optimize clutch damper parameters forreducing driveline vibration and gear rattle

2 Description and Modeling ofPowertrain System

A classical front wheel drive (FWD) vehicle is a researchobject Major components of powertrain system composedof an inline four-cylinder and four-stroke engine the clutchdamper a 5-speedMT the differential mechanism half axlesand wheels are as shown in Figure 1

Effective modeling of powertrain components which isdiscussed in this section is vital to driveline vibration andmanual transmission rattle phenomenon analysis Quasi-transient engine torque is as a power source to the drivelineand applicable enginemodel should consider dynamic outputtorque rather than steady output torque in order to study

TDC

BDC

s=2r

r

l

AB

O

120572

M1

M2

M

120573

s kl

cos 120573

rco

s 120572

Figure 2 Kinematic relation of the crank and connecting rodmechanism

transient response The clutch damper in consideration ofelastic torque and hysteresis torque is modeled so that clutchdamper parameters affecting the driveline vibration and gearrattle could be analyzed A detailed 5-speed manual trans-missionmodel based on lumped parametersmethodwill alsobe explained Simultaneously the differential mechanism andthe tire property are taken into consideration Furthermoretime-varying meshing stiffness of loaded gear pairs is as aninner excitation in the driveline and accurate and effectivecalculation method of it could enhance simulation efficiency

21 Quasi-Transient Engine Model

211 Kinematic Relations of a Single Cylinder Kinematicdiagram of the crank and connecting rod mechanism whichis shown in Figure 2 is calculated by

120572 = 120596 sdot 119905

120582119901 =119903

119897

119904119896 = 119903(1 +

120582119901

4

minus cos120572 minus120582119901

4

cos 2120572)

119904119896 = 120596119903(sin120596119905 +120582119901

2

sin 2120596119905)

119904119896 = 1205962119903 (cos120596119905 + 120582119901 cos 2120596119905)

(1)

where 120572 is the crankshaft angle 120596 is the crankshaft rotationangle speed 119905 is the time 119903 is the crank radius 119897 is theconnecting rod length 119904119896 is the length between the top deadcenter and the piston center and 119904119896 119904119896 are the translationalvelocity and acceleration of the piston respectively


MFNj

120573

FLj

FLj

Fj

FrjFtj

120572

Figure 3 Force analysis of the crank and connecting rod mecha-nism

212 Force Analysis of a Single Cylinder Force analysis of thecrank and connecting rod mechanism in Figure 3 is derivedin

119865119895 =

119875119892 (120572) sdot

120587119889119901

2

4

119895 = 119892

minus119898119901 sdot 119904119896 119895 = 119868

119879119892 = 119865119892119903 (sin120572 +120582119901

2

sin 2120572)

119879119868 = 119865119868119903 (sin120572 +120582119901

2

sin 2120572)

(2)

where 119875119892(120572) is the cylinder pressure with the change of crankangle 119889119901 is the piston diameter119898119901 is the reciprocating massincluding piston piston ring piston pin and connecting rodmass 119865119892 is the gas pressure force on the piston 119879119892 is the gaspressure torque 119865119868 is the reciprocating mass force and 119879119868 isthe reciprocating mass torque

213 Transient Engine Friction Model of a Single CylinderEngine friction modeling is a key step in the quasi-transientengine model Transient engine friction model of Rezeka-Henein model is adopted here and engine friction torque 119879119891is yielded by the following equation [28]

119879119891 = 1198791198911 + 1198791198912 + 1198791198913 + 1198791198914 + 1198791198915 + 1198791198916 (3)

where

1198791198911

= 1198881 [120583 (119903120596 |119885|) (119875119903 + 119875119892)119908119900]

05

119889 (119899119900 + 04119899119888) 119903 |119885|

1198791198912 = 1198882120587119889119899119888119908119888 (119875119903 + 119875119892) (1 minus |sin120572|) 119903 |119885|

1198791198913 = 1198883 (120583120596119903119885

ℎ119900

)119889119871 119904119903119885

Gas ring

Oil ring

d

Ls

wo

wc

Pr

Figure 4 Some parameters for transient engine friction model

1198791198914 = 1198884119899V119865119904119903 |119885| 120596minus05

1198791198915 = 1198885120583120596

1198791198916 = 1198886

1205871198892

4

119903119895119887119875119892 |cos120572| 120596minus05

119885 = sin120572 +120582119901 sin120572 cos120572

radic1 minus 120582119901

2sin2120572

(4)

where 119888119894 (119894 = 1 2 6) are fitting coefficients 120583 is thekinematic viscosity of lubricant oil 119875119903 is the contact pressurebetween piston ring and cylinder wall 119908119900 is the thickness ofoil ring 119889 is the inner diameter of cylinder wall 119899119900 is thenumber of oil rings 119899119888 is the number of gas rings 119908119888 is thethickness of gas ring ℎ119900 is the thickness of lubricating oil film119871 119904 is the length of piston skirt 119899V is the number of valves 119865119904is the force of valve spring and 119903119895119887 is the average radius ofjournal bearing Some parameters are as shown in Figure 4

214 Effective Output Torque of an Inline Four-Cylinder andFour-Stroke Engine For an inline four-cylinder and four-stroke engine effective output torque 119879119890 results from thegas torque reciprocating inertia torque and friction torquecomprehensive in

119879119890 =

4

sum

119895=1

(119879119892119895 + 119879119868119895 minus 119879119891119895) (5)

On the condition of vehicle creeping engine speed isabout 800 rpm and each engine cylinder gas pressure is asseen in Figure 5 Accordingly effective output torque of four-cylinder and four-stroke engine is as shown in Figure 6

22 The Clutch Model The clutch plays an important role indriveline vibration especially in transmission rattle impact


0 180 360 540 7200

05

1

15

2

Crankshaft angle 120572 ( ∘)

1st cylinder2nd cylinder

3rd cylinder4th cylinder

Gas

pre

ssur

ePg

(MPa

)

Figure 5 Each cylinderrsquos gas pressures of the engine

0 180 360 540 720

0

50

100


minus100

minus50Effec

tive t

orqu

eTe (N

m)

Figure 6 Effective output torque of the engine

The clutch is composed of two parts or masses when it isengagedThe primary mass is attached to the flywheel rigidly(called the first mass together) and the secondary mass isconnected to the input shaft of MT through spline teethMultistage springs are placed between the primary mass andthe secondary mass

For an asymmetric two-staged clutch damper inFigure 7(a) the clutch torque 119879119862 is expressed as a function ofthe relative displacement 120579119903 = 120579119891minus120579119862 and the relative velocity120579119903 =

120579119891 minus

120579119862 and is defined by the sum of elastic torque 119879119878 in

Figure 7(b) and hysteresis torque 119879119867 in Figure 7(c) [26]

119879119862 (120579119903120579119903) = 119879119878 (120579119903) + 119879119867 (120579119903

120579119903) (6)

The elastic torque 119879119878 is calculated in

119879119878

=

11989611 (120579119903 minus 1206011199011) + 119896121206011199011 120579119903 gt 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199011

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(7)

where 11989612 is the first-stage stiffness 11989611 is the second-stagestiffness of the drive side 11989621 is the second-stage stiffness ofthe coast side 11989622 is the third-stage stiffness of the coast sideand 1206011199011 1206011199012 and 1206011199013 are the corresponding transition angles

The hysteresis torque 119879119867 is defined in

119879119867

=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(8)

where1198672 is the first-stage hysteresis torque1198671 is the second-stage hysteresis torque of the drive side1198673 is the second-stagehysteresis torque of the coast side and 1198674 is the third-stagehysteresis torque of the coast side

For a three-staged clutch damper in Figure 7(d) theelastic torque 1198791015840

119878and the hysteresis torque 1198791015840

119867are defined in

(9) and in (10) respectively Consider

1198791015840

119878

=

11989611 (120579119903 minus 1206011199011) + 11989610 (1206011199011 minus 1206011199010) + 119896121206011199010 120579119903 gt 1206011199011

11989610 (120579119903 minus 1206011199010) + 119896121206011199010 1206011199010 lt 120579119903 le 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199010

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(9)

1198791015840

119867=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0 120579119903 gt 1206011199011

1198670

2

+

1198670 minus 1198672

2

sgn (120579119903 minus 1206011199010) 120579119903 gt 0 120579119903 lt 1206011199011

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(10)

where 11989610 is the second-stage stiffness of the three-stagedclutch damper1198670 is the corresponding hysteresis torque and1206011199010 is the corresponding transition angles


TC

120579r120601p1

120601p2120601p3

(a)

120579r120601p1

120601p2120601p3

TS

k11

k12

k21

k22

(b)

120579r120601p1

120601p2120601p3

TH

H1H2

H3H4

120579r gt 0

120579r lt 0

(c)

TC

120579r120601p1

120601p2120601p3

k10

120601p0

H0

(d)

Figure 7 Nonlinear characteristics of a multistage clutch damper (a) nonlinear characteristics of a two-stage clutch damper (b) piecewisestiffness characteristics of the two-stage clutch damper (c) piecewise hysteresis characteristics of the two-stage clutch damper and (d)nonlinear characteristics of a three-stage clutch damper

23 Modeling of 5-Speed MT and Loose Gear Drag Torque

231 MT Mechanism and Equivalent Physical Model Forthe transverse 5-speed and two-axis design MT in Figure 8which includes five forward gear ratios and one reverse gearratio input and output shafts are mounted on tapered rollerelement bearings The 1st driven gear 2nd driven gear 3rddriving gear 4th driving gear and 5th driving gear rotate onthe input or output shaft through needle bearings 1st drivingand 2nd driving gear are integrated on the input shaft while3rd driven 4th driven and 5th driven gear are splined on theoutput shaft The 1st driven gear and 2nd driven gear utilizethe same triple cone synchronizer which is supported by onehydrodynamic journal bearing 3rd driving and 4th drivinggears utilize one and 5th driving gear utilizes another one

Based on lumped parameter modeling method everygear and synchronizer are equivalent to rotational inertiasThe inertia of the segment shaft between two gears orbetween one gear and one synchronizer is divided into twoparts averagely and they will be added on adjacent inertiasrespectively Simultaneously the segment shaft is equivalentto one rotational stiffness and one rotational damping Each

inertia of one gear pair couples through meshing stiffnessmeshing damping and backlash and drag torques are appliedon loose gears The coupling between the input shaft and theoutput shaft is obtained by the power transmitting gear pairThe equivalent physical model of 5-speed MT consisting ofan arrangement of discrete inertias and stiffness is as shownin Figure 9

232 Calculation of Loose Gear Drag Torque In Figure 9drag torques 119879119863119894 (119894 = 1 2 5) acting on 1st driven gear2nd driven gear 3rd driving gear 4th driving gear and 5thdriving gear are generated through bearing friction torqueoil shearing torque or oil churning torque Gear windagelosses are ignored since gear speeds are relatively low andloose gears on the input shaft are splash lubricated

For the 1st speed driven and 2nd speed driven gearrotating on the output shaft 1198791198631 in (11) and 1198791198632 in (12) areapplied on the gears respectively

1198791198631 = 1198791199031198871 + 119879sh1 + 119879ch1 (11)

1198791198632 = 1198791199031198872 + 119879sh2 + 119879ch2 (12)


Bearing

Input shaft

Output shaft

1st driving gear

1st driven gear 2nd driven gear 3rd driven gear

4th driven gear 5th driven gear

2nd driving gear

3rd driving gear4th driving gear 5th driving gear

Synchronizer

Reverse gear

Figure 8 Mechanical structure of 5-speed MT

JP1 JP0

JP2JP3 JP4

JP5

k10

k1

k02

k2

k13 k34 k45

k4k3 k5

c10c1

c2c3

c02

c5c4

c45c34c13

kssTD4

TD3

TD2

TD1

TD5

JG2JG1 JG3 JG4

JG5

b3b2 b4 b5

b1

JS2

JS1

JS3

TI

TO

k2s

k1s

c2s

c1s

css

Figure 9 Equivalent physical model of 5-speed MT

Bearing frictional torque 119879119903119887119894 is defined in the followingequation [29]

119879119903119887119894 = 1031198910 (]119873)

231198893

119898]119873 ge 2 times 10

minus3

119879119903119887119894 = 1611989101198893

119898]119873 lt 2 times 10

minus3

(13)

where 119873 is the bearing rotation speed 119889119898 is the bearingaverage diameter1198910 is a lubrication factor and ] is lubricationoil kinematic viscosity

Oil shearing torque 119879sh119894 (119894 = 1 2) is defined in thefollowing equation [8]

119879sh119894 =4120587

2120583119871119877

3Δ119873

30119895119895

(14)

where 120583 is the lubrication oil absolute viscosity 119871 is the gearlength 119877 is the pitch radius of the gear Δ119873 is speed differen-tial between the gear and synchronizer or its bounding shaftand 119895119895 is the radial clearance of the bearing

Oil churning torque 119879ch119894 (119894 = 1 2) is defined in thefollowing equation [30]

119879ch119894 =1

2

1205881205962

119892119878119898119877

3119862119898

(15)

y

x

JFD2

120579FD2

120579D4

120579D2

120579D3

JD4 JD3

JD2

120579D1JD1

Figure 10 Structure diagram of the differential

where 120588 is the lubrication oil density 120596119892 is the gear oilchurning angle velocity 119878119898 is the oil-submerged surface areaand 119862119898 is the oil churning coefficient

For the unloaded 3rd driving gear 4th driving gear and5th driving gear rotating on the output shaft affected bybearing friction drag torque 1198791198633 in (16) drag torque 1198791198634 in(17) and drag torque 1198791198635 in (18) are applied on the gearsrespectively

1198791198633 = 1198791199031198873 (16)

1198791198634 = 1198791199031198874 (17)

1198791198635 = 1198791199031198875 (18)

24 The Differential Model The bevel gear differential mech-anism assembly and kinetic relation of each part are as shownin Figure 10 Rotational angle relation is defined in

21205791198651198632 = 1205791198633 + 1205791198634

1205791198631 =119894119889

2

(1205791198634 minus 1205791198633)

1205791198632 =119894119889

2

(1205791198633 minus 1205791198634)

(19)

where 1205791198651198632 is the assembly rotational angle of the final gearthe differential housing and the planetary-gear pin aroundthe 119909-axis 120579119863119894 (119894 = 3 4) is the rotational angle of the half axlegear around the 119909-axis 120579119863119894 (119894 = 1 2) is the rotational angle ofthe planetary gear around the 119910-axis and 119894119889 is the speed ratioof the planetary gear to the half axle gear

Defining 1205791198651198632 and 1205791198634 as generalized coordinates otherrotational angles could be presented by these two coordinates

[1205791198651198632 1205791198634 1205791198633 1205791198631 1205791198632]119879

= [

1 0 2 minus119894119889 119894119889

0 1 minus1 119894119889 minus119894119889

]

119879

[1205791198651198632 1205791198634]119879

(20)


120596

Wr

r

LFx

O120577

d120577

zFz

Figure 11 The LuGre tire model

Now the kinetic energy T119889119891 of the differential assembly iscalculated by

T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)

where 1198691198651198632 is the rotational inertia of the assembly rotationalangle of the final gear the differential housing and theplanetary-gear pin around the 119909-axis 119869119863119894 (119894 = 3 4) is therotational inertia of the half axle gear around the 119909-axis and119869119863119894 (119894 = 1 2) is the rotational inertia of the planetary geararound the 119910-axis

25TheLuGreTireModel For the LuGre tiremodel the forceanalysis and the motion diagram are as shown in Figure 11

The force analysis of the average lumped LuGre tiremodelis given by the following equation [31]

119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road

Wet asphalt pavement roadAsphalt pavement road

Figure 12 The LuGre tire property for different road

119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903

drivingV minus 120596119903

V=

V119903V

braking

120583 =

119865119909

119865119911

(22)

where 119911 is the average deformation of brush V119903 is the relativevelocity between the tire and the ground 1205900 is the normalizedrubber longitudinal lumped stiffness 1205901 is the normalizedrubber longitudinal lumped damping 1205902 is the normalizedviscous relative damping 120583119888 is the normalized coulombfriction 120583119904 is the normalized static friction V119904 is the Stribeckrelative velocity 120582 is the Stribeck effect index 119871 is the lengthof the contact patch119891119899(120577) is the distribution density functionof the longitudinal pressure 119865119909 is the longitudinal force of thetire 119865119911 is the vertical force of the tire 119904 is the tire slip rate 120596is the rotational velocity of the tire 119903 is the rolling radius ofthe tire and 120583 is the longitudinal road friction coefficient

By the LuGre model the relation between the longitu-dinal road friction coefficient 120583 and the tire slip rate 119904 ondifferent ground condition is obtained in Figure 12

26 Calculation of Gear Pair Time-Varying Meshing StiffnessFinite element analysis (FEA) is themost effectivemethod forhelical gear pair time-varying meshing stiffness The helicalgear meshing stiffness is defined as

119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)


where 119896 is the gear pair meshing stiffness 119865119899 is the normalforce of the contact force 120576 is the comprehensive deformationof gear pair 1205761198871199041 is the bending and shear deformation ofone gear on the contact point 1205761198871199042 is the bending and sheardeformation of the other gear on the contact point and 120576119888 isthe contact deformation of the gear pair on the contact point

Simon got the bending and shear deformation 120576119887119904119894 (119894 =1 2) computational formula of (24) based on large amountsof FEA results through regression analysis [32] Therefore

120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)

where 119864 is the elastic modulus 119898119899 is the normal module1198911 is the coefficient of normal force load point 1198912 is thecoefficient of the relative radial position between load pointand deformation point 1198913 is the coefficient of the relativeaxial position between load point and deformation point 119911is the teeth number 120572119899 is the normal pressure angle 1205730 is thespiral angle in base on base circle 120594119901 is the gear modificationcoefficient ℎ119891 is the addendum ℎ119896 is the dedendum 119903fil is thetooth root fillet radius and 119887 is the tooth width

As for the contact deformation 120576119888 Cornell derived thefollowing equation [33]

120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)

where Δ119911 is the piece length along the tooth width Δ119865 is thepiece force applied on the piece length Δ119911 1199041 is the tooththickness of one gear 1199042 is the tooth thickness of the othergear 1205921 is Poissonrsquos ratio of one gear 1205922 is Poissonrsquos ratio ofthe other gear 1198641 is the elastic modulus of one gear and 1198642 isthe elastic modulus of the other gear

Through (23) to (25) the time-varying meshing stiffness1198961 of the 1st gear pair (as shown in Figure 9) and the final drivegear pair 119896119891 (as shown in Figure 16) for a two-tooth cycle areshown in Figures 13 and 14

0 02 04 06 08 1

3

4

5

6

Rotational angle 120579P1 (rad)

Mes

hing

stiff

nessk1

(Nm

)

times108

Figure 13 The meshing stiffness of 1st gear pair

0 02 04 06 08

5

6

7

8

Rotational angle 120579JD1 (rad)

Mes

hing

stiff

nesskf

(Nm

)

times108

Figure 14 The meshing stiffness of final drive gear pair

3 Numerical Modeling andSimulation Algorithm

31 Modeling Framework The 1st shift of MT on the vehiclecreeping condition when gear rattle noise could be perceivedclearly by passengers on the researched vehicle is used as anexample Gear rattle phenomenon is comprehensive resultsof complex interactions between the baseline vibration forthe loaded driveline system and the rattling vibration forunloaded gear pairs in Figure 15 The baseline vibrationconsists of the engine the clutch the 1st gear pair gearsintegrated on the input shaft gears splined on the outputshaft final drive gear pair the differential the haft shaft andthe tire while the rattling vibration concludes lightly loadedgear pairs namely the 2nd the 3rd the 4th and the 5th gearpair

It has beenwidely recognized in literature that the rattlingvibration has little effects on the motion of the baselinevibration [6 14] which could be utilized to study the overallsystem behavior more efficiently The pinion gear motionsof lightly loaded gear pairs in the baseline vibration become


Engine

Clutch

Working shiftInput shaft integrated gears

Output shaft splined gears

Final drive

Differential

Tire

Baseline vibration

Unloaded gearsUnloaded gears

MT modeling

Rattling vibration

Vehicle body

Half shaft

Figure 15 Modeling framework for driveline vibration and gearrattle phenomenon

excitations to loose gear pairs in the rattling vibration Thenthe rattle force of loose gear pairs could be obtained

32The Baseline Model of Vehicle Driveline System DynamicFWD driveline model based on the branched model isdescribed in Figure 16when the 1st gear pair is engagedTheseloaded gear pairs namely the 1st gear pair and the final drivegear pair are considered to be always in contact with a time-varying meshing stiffness respectively which is calculatedin Section 26 Those unloaded gear pairs with lighted loadmay be driven across the backlash causing impacts and rattlenoise The driveline model consists of the two-stage stiffnessclutch damper model and the detailed MT model considersthe differential property and utilizes the average lumpedparameters LuGre tire model The input power of drivelinesystem is the effective output torque of the four-cylinderand four-stroke engine Accordingly the longitudinal forceanalysis of the vehicle and the torsional force analysis of thetire are as shown in Figure 17 assuming that vertical left andright tires load of the front or rear axle are equivalent

In the branched model the simplified factors include(1) ignoring the oil shearing torque and the oil churningtorque applied on the 1st gear pair in the power flow and(2) neglecting dynamic property influence of bearings on theinput shaft and the output shaft in Figure 8 and final drivegear bearings

By the Lagrange equation the baseline system vibrationdynamics is placed in the matrix form

J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879

T = [119879119890 minus 119879119888 119879119888 0 0 0 0 0 0 0 0 0 0 0 0 0 0 minus119879119892119897 minus119879119892119903 119865119909119891119897 + 119865119909119891119903 minus 119865119909119903 minus 119865119908]119879

J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751

2minus11989610 minus1198961 lowast 1198771198661 lowast 1198771198751

minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904

minus1198961 lowast 1198771198661 lowast 1198771198751 1198961119891 + 11989613 + 1198961 lowast 11987711986612minus11989613

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631

2minus119896119891 lowast 1198771198651198631 lowast 1198771198651198632

minus119896119891 lowast 1198771198651198631 lowast 1198771198651198632 119896119891 lowast 1198771198651198631

2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904

0 minus1198882119904 1198882119904 + 119888119904119904 minus119888119904119904 0minus119888119904119904 119888119904119904

minus1198881 lowast 1198771198661 lowast 1198771198751 1198881119891 + 11988813 + 1198881 lowast 1198771198661

2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631


minus119888119891 lowast 1198771198651198631 lowast 1198771198651198632 119888119891 lowast 11987711986511986312+ 4119888119903119863 minus2119888119903119863 minus2119888119903119863

minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)

where diag expresses the diagonal matrix 120579119891 is the angulardisplacement (AD) of engine (namely the flywheel andclutch) 120579119862 is the AD of clutch hub 1205791198751 is the pinion gear ADof the 1st gear pair and 12057911986611198781 is the wheel gear of the 1st gearpair and the corresponding synchronizer AD 1205791198752 and 1205791198662 arethe AD of the 2nd gear pair 1205791198753 and 1205791198663 are the AD of the 3rdgear pair 1205791198754 and 1205791198664 are the AD of the 4th gear pair 1205791198755 and1205791198665 are theADof the 5th gear pair 1205791198782 is theADof the 3rd and4th gear pair synchronizer 1205791198783 is the AD of the 5th gear pairsynchronizer 1205791198651198631 and 1205791198651198632 are the AD of the final drive gearpair 1205791198634 is the AD of a half axle gear about its own rotationalaxis 120579119904119897 and 120579119904119903 are AD of left and right half axle 120579119905119897 and 120579119905119903are the AD of left and right tire 119909119904 is the vehicle longitudinaldisplacement 119869119891 is the inertia of flywheel and clutch 119869119862 isthe inertia of clutch hub 1198691198751 is the pinion gear inertia of the1st gear pair and 11986911986611198781 is the sum of the wheel gear inertia ofthe 1st gear pair and the corresponding synchronizer 1198691198752 and1198691198662 are the inertia of the 2nd gear pair 1198691198753 and 1198691198663 are theinertia of the 3rd gear pair 1198691198754 and 1198691198664 are the inertia of the4th gear pair 1198691198755 and 1198691198665 are the inertia of the 5th gear pair1198691198782 is the inertia of the 3rd and 4th gear pair synchronizer1198691198783 is the inertia of the 5th gear pair synchronizer 1198691198651198631 is thepinion gear inertia of the final drive gear pair 1198691198651198632 is the suminertia of differential ring gear differential shell planetarygear and axis pin 1198691198631 and 1198691198632 are the inertia of a planetarygear about its own rotational axis 1198691198633 and 1198691198634 are the inertiaof a half axle gear about its own rotational axis 119869119904119897 is the suminertia of the left half axle wheel hub wheel rim and brakedisc 119869119904119903 is the sum inertia of the right half axle wheel hubwheel rim and brake disc 119869119905119897 is the inertia of the left-front

tire 119869119905119903 is the inertia of the right-front tire 119898 is the vehiclemass 119896119895 (119895 = 1 2 5) is themeshing stiffness of gear pairs119888119895 (119895 = 1 2 5) is the meshing damping of gear pairs and119887119895 (119895 = 1 2 5) are the backlash of unloaded gear pairsOther 119896 and 119888 are torsional stiffness and torsional dampingrespectively

Here in these matrices of J K C 120579 and T someparameters are formulated

1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5

JP1 JP0JP2 JP3 JP4 JP5

b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f

c1fcfkf JG1S1 JG2 JG3 JG4 JG5

Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl

Figure 16 The torsional vibration model of vehicle driveline withdetailed MT model for the first speed

r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La

Figure 17 The half vehicle longitudinal motion and force analysisdiagram

V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)

where 1205731198871 is the helical angle on base circle of the piniongear on the 1st gear 120573119887119891 is the helical angle on base circle ofthe pinion gear on the final drive gear 119903 is the tire dynamicradius 119891 is the rolling resistance coefficient 119911119894 (119894 = 119897 119903) is thetire bristle average deformation in LuGre tire model 119894 (119894 =119897 119903) is the tire bristle average deformation rate in LuGre tiremodel 119871119886 is the distance from the mass center to the frontaxle 119871119887 is the distance from the mass center to the rear axle119867 is the mass center height 119909119886 is the vehicle longitudinalacceleration 119909V is the vehicle longitudinal velocity 119862119863 is theair resistance coefficient119860 is the vehicle frontal area and 120588 isthe air density

33 The Rattling Vibration Model of Unloaded Gear PairsThe rattling impact is the source of rattle noise The impactcollisions through their gear backlash are transmitted to thetransmission housing via shafts and bearings The vibrations

pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p

Figure 18 A simplified model of a rattling gear pair

1

1

X

f(X)

minusb2

b2

Figure 19 Backlash function

are then converted into an audible rattle So rattling force isthe focus of dynamic study of each gear pair

For one rattling gear pair the mechanical model is asshown in Figure 18 Each gear is equivalent to a lumpedinertia As the motion of the pinion gear 119901 which is obtainedin the baseline model is taken as an excitation to the systemfor 1st shift pinion gears include the 2nd driving gear the 3rddriven gear the 4th driven gear and the 5th driven gear inFigure 8 So rattling force of unloaded gear pair is deduced

119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)

Here 119883 denotes the relative displacement along the lineof conjugate action of unloaded gear pairs Therefore eachrattling gear pair is then reduced to a single degree of freedomsystem 119891(119883) and

119891(119883) are the backlash function as shownin Figure 19 and its derivative function respectively whichare defined as

119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)

where 120579119875 is the driving gear AD 120579119892 is the driven gear AD119877119875 is the base circle radius of the driving gear 119877119892 is the basecircle radius of the driven gear 119868119892 is the inertia of the drivengear 119879119863 is the drag torque applied on the driven gear 119865119898 isthe rattling force 119896119898 is the average meshing stiffness of thegear pair 119888119898 is the average meshing damping of the gear pairand 119887 is the gear backlash

34 Simulation Method and Numerical Algorithm As equa-tions of the baseline vibration and the rattling vibration arederived the driveline vibration includes highly nonlinearfactors and the condition number of the systemmatrix whichis the ratio of its maximum to minimum eigenvalue is veryhigh As MATLAB is taken as our numerical simulation toola ldquostiffrdquo problem for ordinary differential equation (ODE) isusually difficult to solve on hand

MATLAB provides kinds of solvers for stiff ODE whichconsist of ODE15s ODE23s ODE23t and ODE23tb Amongthem ODE15s is a variable order solver based on thenumerical differentiation formulas Optionally it uses thebackward differentiation formulas and it is also known asGearrsquos method which are usually less efficient ODE23s isbased on a modified Rosenbrock formula of order 2 Becauseit is a one-step solver it ismore efficient thanODE15s at crudetolerances and it could solve some kinds of stiff problems forwhich ODE15s is not effective [34 35] ODE23s is used forthe stiff problem on hand and it is found that the efficiency isacceptable

4 Simulation Results Analysis

41TheDrivelineVibrationAnalysis In the numericalmodelrequired parameters are from a mass production vehicle Aproper and accurate driveline model could insure a prac-tical result Firstly a two-stage stiffness clutch damper (seeFigure 20) is adopted in the baseline model And the two-stage stiffness clutch damper characteristics including elasticand hysteresis property adopted in the original drivelinesystem are described in Figure 21 in the solid line

According to (26) in the time domain the vehicle velocityand the engine speed are obtained in Figures 22 and 23respectively From Figure 22 it is found that the vehiclemoves forward slowly at the speed between 18ms and1815ms namely the vehicle creeping speed In Figure 23the engine rotates at about 800 rpm and the speed fluctuationamplitude is nearly 80 rpm while the clutch hub rotates atabout 800 rpm and the speed fluctuation amplitude is about10 rpm Accordingly the angular acceleration amplitude ofthe clutch hub in Figure 25 is much smaller than the accelera-tion amplitude of the engine in Figure 24 As seen the clutchdamper plays a role in attenuating the fluctuation amplitude

(a)

(b)

Figure 20 A two-stage stiffness clutch damper adopted in themodel (a) 3D model (b) physical model

0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50

minus5minus10minus15minus20

minus100

minus150

Working angular displacement (∘)

Figure 21 Nonlinear characteristics curve of a two-stage clutchdamper

of the engine speed in the driveline But Figure 25 shows thatthe clutch hub fluctuates remarkably about the mean speed

On this special condition it was found that the trans-mission rattle was severe through the driver subjective


0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)

Figure 22 The vehicle velocity on the creeping condition

evaluation Now from the simulation results in Figure 26it is concluded that the clutch damper works at the angulardisplacement from 57∘ to 86∘ between the first and secondclutch damper mass namely the actual working area inthe dot-line ellipse in Figure 21 The clutch damper worksjumping between the first-stage stiffness and the second-stagestiffness of the drive side and it excites severer driveline tor-sional vibration which results in drastic fluctuation vibrationof the clutch hub and transmission rattle impact noise thatcould be perceived by the driver or the passenger

Besides in the frequency domain frequency spectrum ofthe engine speed (see Figure 27) shows that primary frequen-cies include 1343Hz 2686Hz and 531 Hz which are one-time frequency double frequency and four-time frequencyrespectively Correspondingly primary frequencies of theclutch hub speed (see Figure 28) include 1343Hz 2686Hzand 531 Hz as well Moreover amplitudes of eight-time fre-quency (1062Hz) the twelve-time frequency (1593Hz) andother frequencies which are comparedwith those amplitudesof 1343Hz 2686Hz and 531 Hz are considerable Throughtheoretical analysis amplitudes of higher frequencies aresmaller than those of lower frequencies The two-stage stiff-ness clutch damper working between the first-stage stiffnessand the second-stage stiffness could be explained for theresults in Figure 28

42 Rattle Force Analysis of Unloaded Gear Pairs Asexplained in Section 33 pinion gear motions which areobtained from the baseline vibration are excitations to therattling vibration The pinion gear motions of the 2nd 3rd4th and 5th gear pairs are as shown in Figure 29 Accordinglythe 2nd gear pair rattling force 1198651198982 the 3rd gear pair rattlingforce 1198651198983 the 4th gear pair rattling force 1198651198984 and the 5thgear pair rattling force 1198651198985 are as shown in Figure 30

In Figure 29 pinion gears fluctuate at the mean speedand pinion gear motions of 3rd gear pair 4th gear pair and5th gear pair are nearly consistent with each other FromFigure 30 it is found that two-side rattling impacts happenand larger rattling force is excited in all unloaded gear pairs

0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)

Figure 23 Comparison between engine speed and clutch hub speed

Maximum amplitude of the 3rd and 4th gear pair rattlingforce could be nearly up to 2000N while rattle force of the2nd gear pair is about 1000N and the 4th gear pair rattleforce is about 500N So the 3rd and 4th gear pair undergosevere rattle phenomenon Furthermore although piniongear motions of 3rd gear pair 4th gear pair and 5th gear pairare nearly consistent rattling forces of those three gear pairsare completely different which proves that it is essential toestablish a detailed MT model

5 Clutch Damper Parameters Optimizationfor Reducing Gear Rattle

51 The Driveline Vibration Analysis after ImprovementAs concluded in Section 41 the two-stage stiffness clutch


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)

Figure 24 Angular acceleration of the engine

0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400

Figure 25 Angular acceleration of the clutch hub

0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )

Figure 26 Working AD of the clutch damper

0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365

Figure 27 Frequency spectrum of engine speed

0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571

Figure 28 Frequency spectrum of clutch hub speed

damper works jumping between the first- and the second-stage stiffness of the drive side which excites severer drivelinevibration and gear rattle phenomenon So a three-stage stiff-ness clutch damper for adding one-stage stiffness for low loadtorque between the first-stage and the second-stage stiffnessis proposed innovatively in Figure 31 As seen other stageproperty parameters of the two-stage clutch damper are notrevised except the added stage property parameters and thisthree-stage clutch damper could play a good performanceoriginally for other vehicle driving conditions except thevehicle creeping condition Nonlinear characteristics of thethree-stage clutch damper are as shown in Figure 32 in thesolid line

According to (26) in the time domain the engine fluctu-ates at about 800 rpm and speed fluctuation amplitude isnearly 80 rpm in Figure 33 which is similar to the result inFigure 23 But it is obviously found that fluctuation degreeof the clutch hub is reduced and the fluctuation amplitude isless than 10 rpm Similarly angular acceleration of the clutchhub in Figure 35 is much smaller than that in Figure 25 while


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)

Figure 29 Pinion gear speed of all unloaded gear pairs (a) the 2nd gear pair (b) the 3rd gear pair (c) the 4th gear pair and (d) the 5th gearpair

the angular acceleration of the engine in Figure 34 is similarto that in Figure 24

Further analysis of the three-stage clutch damperworkingAD in Figure 36 shows that it works at the angular displace-ment from 5∘ to 8∘ namely the actual working area in the dot-line ellipse in Figure 32 Now after adopting the three-stageclutch damper jumping phenomenon between the first-stagestiffness and the second-stage stiffness is eliminated

Besides in the frequency domain frequency spectrum ofthe engine speed (see Figure 37) is similar to that in Figure 27and primary frequencies consist of 1343Hz 2686Hz and531 Hz as well Correspondingly primary frequencies of theclutch hub speed include 1343Hz 2686Hz and 531 Hz aswell in Figure 38 But in Figure 38 amplitudes of 1062Hzand 1593Hz and other frequencies in Figure 28 amplitudesof which could not be neglectful are reduced to a muchlower valueThrough comprehensive analysis of those resultsjumping phenomenon elimination between the first-stagestiffness and the second-stage stiffness could be explained for

the result in Figure 38 after adopting the three-stage stiffnessclutch damper

52 Rattle Force Analysis of Unloaded Gear Pairs after Opti-mization Through the baseline model pinion gear motionsof the 2nd 3rd 4th and 5th gear pair after optimizationare as shown in Figure 39 Compared with the result inFigure 29 speed fluctuations of all pinion gears are apparentlymuch lower change trends of which are the same with theclutch hub Then pinion gear motions are as excitations tounloaded gear pairs and rattle forces of unloaded gear pairsare calculated in Figure 40 Rattle intensities of all unloadedgear pairs are obviously improved and one-side rattle impactsare dominant in all unloaded gear pairs Maximum rattleforce of the 2nd gear pair is less than 150N and rattle force ofthe 3rd gear pair is less than 50N while rattle force of the 4thand 5th gear pair is similarly less than 100N It is concludedthat all unloaded gear pairs undergo rattle vibration butthe intensity of rattle impacts is much weaker So MT rattle


0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)

Figure 30 Rattle force of all unloaded gear pairs (a) the 2nd gear pair (b) the 3rd gear pair (c) the 4th gear pair and (d) the 5th gear pair

(a) (b)

Figure 31 A three-stage stiffness clutch damper adopted in the model (a) 3D model (b) physical model


0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N

m) Actual working area

minus150

minus100

minus50



Figure 32 Nonlinear characteristics of a three-stage clutch damper

0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )

Figure 36 Working AD of the three-stage stiffness clutch damper

0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)

Figure 39 Pinion gear speed of unloaded gear pairs after optimization (a) the 2nd gear pair (b) the 3rd gear pair (c) the 4th gear pair and(d) the 5th gear pair


0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)

Figure 40 Rattle force of unloaded gear pairs after optimization (a) the 2nd gear pair (b) the 3rd gear pair (c) the 4th gear pair and (d) the5th gear pair

phenomenon (or rattle force) is improved after adopting thethree-stage stiffness clutch damper on the vehicle creepingcondition

6 Conclusions

Based on the branched model including quasi-transientengine model multistage clutch damper model detailedMT model differential model and LuGre tire model andconsidering time-varying stiffness of the 1st speed gear pairand final drive gear pair 19-DOF model of the baselinevibration is established on the vehicle creeping conditionTherattling vibration is then obtained as the baseline vibrationis as an excitation The baseline vibration and the rattlingvibration reproduce a comprehensive study of the drivelinesystem and MT rattle phenomenon It is concluded that

(1) on the creeping condition the two-stage stiffnessclutch damper tends to work jumping between thefirst- and second-stage stiffness and it causes severer

driveline vibration and disturbing rattle noise per-ceived by passengers Larger rattling force of two-sideimpact is excited in all unloaded gear pairsMaximumrattle force of the 3rd and 4th gear pair is up to about2000N while rattle force of the 2nd gear pair is about1000N and rattle force of the 4th gear pair is nearly500N

(2) a three-stage stiffness clutch damper is adopted andit could obviously improve the driveline vibrationand MT rattle phenomenon on the vehicle creepingcondition One-side impacts are dominant in allunloaded gear pairs Maximum rattle force of the 4thand 5th gear pair is less than 100N while rattle forceof the 2nd gear pair is smaller than 150N and rattleforce of the 3rd gear pair is less than 500N

(3) achievements of numerical simulation developed inthis research could be utilized for the design ofdriveline system and practical strategies for solvingMT rattle phenomenon Currently all results are


mainly obtained from numerical modeling and simu-lation and they are indispensable to be validated withfurther experimental results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The research leading to these results has received fundingfrom the National Natural Science Foundation of China(Grant no 51175379)

References

[1] K Nakamura ldquoTooth separations and abnormal noise onpower-transmission gearsrdquo Bulletin of JSME vol 10 no 41 pp846ndash854 1967

[2] R J Comparin and R Singh ldquoNon-linear frequency responsecharacteristics of an impact pairrdquo Journal of Sound and Vibra-tion vol 134 no 2 pp 259ndash290 1989

[3] A Kahraman andR Singh ldquoNon-linear dynamics of a spur gearpairrdquo Journal of Sound and Vibration vol 142 no 1 pp 49ndash751990

[4] M Bozca ldquoTorsional vibration model based optimization ofgearbox geometric design parameters to reduce rattle noise inan automotive transmissionrdquo Mechanism and Machine Theoryvol 45 no 11 pp 1583ndash1598 2010

[5] M Bozca and P Fietkau ldquoEmpirical model based optimizationof gearbox geometric design parameters to reduce rattle noisein an automotive transmissionrdquoMechanism ampMachineTheoryvol 45 no 11 pp 1599ndash1612 2010

[6] M YWangW Zhao and RManoj ldquoNumerical modelling andanalysis of automotive transmission rattlerdquo Journal of Vibrationand Control vol 8 no 7 pp 921ndash943 2002

[7] GWu andW Luan ldquoThe impact of gearmeshing nonlinearitieson the vehicle launch shudderrdquo SAE Technical Paper 2015

[8] D Robinette R S Beikmann P Piorkowski et al ldquoCharac-terizing the onset of manual transmission gear rattle part IIanalytical resultsrdquo SAE Technical Paper 2009

[9] D Robinette R S Beikmann P Piorkowski and M PowellldquoCharacterizing the onset of manual transmission gear rattlepart I experimental resultsrdquo SAE Technical Paper 2009

[10] M De La Cruz S Theodossiades and H Rahnejat ldquoAninvestigation of manual transmission drive rattlerdquo Proceedingsof the Institution of Mechanical Engineers Part K vol 224 no 2pp 167ndash181 2010

[11] P Fietkau and B Bertsche ldquoInfluence of tribological andgeometrical parameters on lubrication conditions and noise ofgear transmissionsrdquo Mechanism and Machine Theory vol 69pp 303ndash320 2013

[12] S Theodossiades O Tangasawi and H Rahnejat ldquoGear teethimpacts in hydrodynamic conjunctions promoting idle gearrattlerdquoNoiseampVibrationWorldwide vol 303 pp 632ndash658 2007

[13] O Tangasawi S Theodossiades and H Rahnejat ldquoLightlyloaded lubricated impacts idle gear rattlerdquo Journal of Sound ampVibration vol 308 no 3ndash5 pp 418ndash430 2007

[14] A R Crowther C Halse and Z Zhang ldquoNonlinear responsesin loaded driveline rattlerdquo SAE Technical Paper 2009

[15] R Bhagate A Badkas and K Mohan ldquoDriveline torsionalanalysis and parametric optimization for reducing drivelinerattlerdquo SAE Technical Paper 2015

[16] P Couderc J Callenaere J Der Hagopian et al ldquoVehicledriveline dynamic behaviour experimentation and simulationrdquoJournal of Sound and Vibration vol 218 no 1 pp 133ndash157 1998

[17] P Bellomo N De Vito C H Lang and L Scamardi ldquoIndepth study of vehicle powertrains to identify causes of loosecomponents rattle in transmissionsrdquo SAETechnical Paper 2002-01-0702 2002

[18] A Forcelli C Grasso and T Pappalardo ldquoThe transmissiongear rattle noise parametric sensitivity studyrdquo SAE TechnicalPaper 2004-01-1225 2004

[19] M Barthod B Hayne J-L Tebec and J-C Pin ldquoExperimentalstudy of gear rattle excited by a multi-harmonic excitationrdquoApplied Acoustics vol 68 no 9 pp 1003ndash1025 2007

[20] A R Crowther andM K Rozyn ldquoDesign and analysis of a gearrattle test rigrdquo SAE International Journal of Passenger CarsmdashMechanical Systems vol 2 no 1 pp 1431ndash1439 2009

[21] A Baumann and B Bertsche ldquoExperimental study on transmis-sion rattle noise behaviour with particular regard to lubricatingoilrdquo Journal of Sound amp Vibration vol 341 pp 195ndash205 2015

[22] R Brancati E Rocca and S Savino ldquoA gear rattle metricbased on the wavelet multi-resolution analysis experimentalinvestigationrdquo Mechanical Systems and Signal Processing vol50-51 pp 161ndash173 2015

[23] K Steinel ldquoClutch tuning to optimize noise and vibrationbehavior in trucks and busesrdquo SAE Technical Paper 2000-01-3292 2000

[24] J S Prasad N C Damodar and T S Naidu ldquoClutch hysteresismaximization for elimination of gear rattle in a passenger busrdquoSAE Technical Paper 2013-26-0100 2013

[25] X Xu W Fang F Ge X Chen J Wang and H Zhou ldquoThedevelopment and application of a novel clutch torsional damperwith three-stage stiffnessrdquo Automotive Engineering vol 35 no11 pp 1011ndash1022 2013

[26] J-Y Yoon and R Singh ldquoEffect of the multi-staged clutchdamper characteristics on the transmission gear rattle undertwo engine conditionsrdquoProceedings of the Institution ofMechan-ical Engineers Part D Journal of Automobile Engineering vol227 no 9 pp 1273ndash1294 2013

[27] S Jadhav ldquoPowertrain NVH analysis including clutch and geardynamicsrdquo SAE Technical Paper 2014-01-1680 2014

[28] C D Rakopoulos D T Hountalas A P Koutroubousis and TZannis ldquoApplication and evaluation of a detailed frictionmodelon a DI diesel engine with extremely high peak combustionpressuresrdquo SAE Technical Paper 2002-01-0068 2002

[29] A Palmgren Ball and Roller Bearing Engineering SKF Indus-tries Philadelphia Pa USA 1959

[30] C Changenet X Oviedo-Marlot and P Velex ldquoPower losspredictions in geared transmissions using thermal networks-applications to a six-speedmanual gearboxrdquo Journal ofMechan-ical Design vol 128 no 3 pp 618ndash625 2006

[31] C Canudas-de-Wit P Tsiotras E Velenis M Basset and GGissinger ldquoDynamic friction models for roadtire longitudinalinteractionrdquo Vehicle System Dynamics vol 39 no 3 pp 189ndash226 2003

[32] V Simon ldquoLoad and stress distributions in spur and helicalgearsrdquo Journal of Mechanical Design vol 110 no 2 pp 197ndash2021988


[33] R W Cornell ldquoCompliance and stress sensitivity of spur gearteethrdquo Journal of Mechanical Design vol 103 no 2 pp 447ndash4591981

[34] L F Shampine and M W Reichelt ldquoThe MATLAB ODE suiterdquoSIAM Journal on Scientific Computing vol 18 no 1 pp 1ndash221997

[35] R Ashino M Nagase and R Vaillancourt ldquoBehind andbeyond the Matlab ODE suiterdquo Computers amp Mathematics withApplications vol 40 no 4-5 pp 491ndash512 2000

International Journal of

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Submit your manuscripts athttpwwwhindawicom

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



method [3] As the study continued research object wastransferred from one simple gear pair to complicated geartransmission system Based on the four-degree-of-freedommodel of one gear pair Bozca et al proposed empirical modeland torsional vibration model based optimization of a 5-speed gearbox design parameters to reduce rattle noise in anautomotive transmission Despite the geometric parametersoptimization overall rattle noise level was reduced and alloptimized geometric design parameters also satisfied allconstraints [4 5] Besides gear rattle problem is regarded asa comprehensive problem of the driveline system Most ofthe drivelinemodels used for the driveline torsional vibrationanalysis are lumped discrete models with a few degreesof freedom Wang et al described a model most early fortorsional vibration of automotive manual transmission (MT)to analyze and predict gear rattle of all speeds Accordingly arattle index was used to compare the rattle levels producedby different gear pairs But in that model gear meshingstiffness was constant and self-excited vibration of time-varying stiffness was ignored [6]Wu and Luan paid attentionto the impact of gearmeshing stiffness on the vehicle drivelinetorsional vibration and gave a comparison of simulationbetween variable meshing stiffness and averaged stiffness ofloaded gear pairs based on overall powertrain system [7]Robinette et al developed a representative model for a frontwheel drive (FWD) vehicle with MT by lumped parameteranalysis and presented functional relations for torque lossesassociated with shafts gears seals lubricating oil flow andbearing clearance as a function of basic design parame-ters [8] Drag torque including bearing friction torque oilshearing torque or oil churning torque was then validatedby experimental results [9] De La Cruz et al considereddifferent lubrication states influence of gear pairs on the rattlephenomenon and proposed a rattle index in considerationof lubrication state [10] Fietkau and Bertsche proposed asimulation approach for loaded and unloaded gear con-tacts which include oil films and elastic deformations Thisapproach is validated with experiment measurement and itis concluded that lubricant condition could not be ignored[11] Theodossiades et al took into consideration the effectof lubrication during engine idle conditions and examinedthe influence of lubrication in torsional vibration It is shownthat the lubricant film behaved as a time-varying nonlinearspring-damper element and could have a great influence onthe gear rattle problem [12 13] Crowther et al put for-ward 6-degree-of-freedom (DOF) model using a frequencysweep with engine excitation derived from measured datawith two-stage gear meshing and an unloaded gear pairIt is found that the gear rattling is more severe when theengine speed passes the resonation frequency region of thesystem natural model It concluded that an effective dynamicengine model is needed in order to get transient drivelinecomponent motion and then rattle phenomenon actually[14] Bhagate et al put forward a 6-DOFmathematical modelfor the torsional vibrations of front wheel drive automotivedrivetrain and developed the optimization of sensitive systemparameters for reducing the driveline rattle [15] So as fortransmission modeling various factors such as gear pairtime-varying stiffness gear friction bearing friction and gear

oil churning loss are in urgent need in the future modelingwork

In terms of rattle experiments Couderc et al designedand built early an experimental setup of a vehicle driveline forthe prediction of the dynamic behavior of vehicle drivelinesIt is concluded that the simulation model validated by theexperimental setup could provide transient response truly[16] Bellomo et al analyzed the contribution of individualsound source to the overall rattle noise by means of noise-source analysis and proposed a pareto-optimal solution toreduce the rattle noise emission utilizing a rattle test bench[17] This improved test bench reproduced the brancheddriveline system rather than the one-string driveline systemin [16] Forcelli et al set up a virtual engine simulator for auto-motive transmission and conducted a parametric sensitivitystudy for amplitude of the torsional vibration Moreover arelationship between the vibroacoustics measurements andthe human perception was found [18] Barthod et al analyzedthe rattle threshold and the rattle noise evolution for dif-ferent multiharmonic excitation parameters and mechanicalgearbox parameters through a bench test [19] Crowther andRozyn introduced a gear rattle test rig in which the electricmotor drives the transmission at a steady mean speed via adouble telescoping Hookersquos joint By changing the angle ofthe joints the amplitude of the vibration could be adjusted[20] Baumann and Bertsche built one gear pair test rigfor rattle research and compared the rattle intensity underdifferent lubricant oil condition It is found that adopting highviscosity oil could inhibit rattle phenomenon when angularacceleration of the input shaft is larger [21] Brancati et alset up a specific test rig for one lightly loaded gear pairwhich is able to acquire the relative rotation motion ofgears by two high resolution incremental encoders Basedon measurement data from this test rig a gear rattle metricbased on the wavelet multiresolution analysis was proposed[22]

The clutch damper is a component of drivelines thatcould have significant influence on the torsional dynamicbehavior of drivelines Gear rattle phenomenon can be greatlyreduced by opportunely setting some clutch parameters suchas the multistage torsional springs Steinel examined theinfluence of the twin-mass flywheel on the driveline naturalcharacteristics and transient responses It was shown that thetwin-mass flywheel was the ideal solution for drivetrains ofwhich the vibrations could not be reduced sufficiently if therewas no need for the consideration of costs [23] Prasad etal found that elimination of gear rattle could be achievedby maximizing the hysteresis of clutch thereby absorbing theenergy being transferred through the subjective and objectiveevaluation in the passenger bus experiment [24] But it isobvious that maximizing the hysteresis of the clutch damperwould reduce the transmission efficiency of the powertrainsystem Xu et al introduced a novel clutch damperwith three-stage stiffness and solved the rattle phenomenon effectively inlow torque condition compared with the damper with two-stage stiffness by vehicle experiments [25] Similarly manyresearch scholars found that the clutch damper property playsan important role in reducing driveline vibration and rattlephenomenon [26 27]


1

6 6

77

7

7

72

5 5

3

4






TDC

BDC

s=2r

r

l

AB

O

120572

M1

M2

M

120573

s kl

cos 120573

rco

s 120572





120572 = 120596 sdot 119905

120582119901 =119903

119897

119904119896 = 119903(1 +

120582119901

4

minus cos120572 minus120582119901

4

cos 2120572)

119904119896 = 120596119903(sin120596119905 +120582119901

2

sin 2120596119905)

119904119896 = 1205962119903 (cos120596119905 + 120582119901 cos 2120596119905)

(1)



MFNj

120573

FLj

FLj

Fj

FrjFtj

120572



119865119895 =

119875119892 (120572) sdot

120587119889119901

2

4

119895 = 119892

minus119898119901 sdot 119904119896 119895 = 119868

119879119892 = 119865119892119903 (sin120572 +120582119901

2

sin 2120572)

119879119868 = 119865119868119903 (sin120572 +120582119901

2

sin 2120572)

(2)



119879119891 = 1198791198911 + 1198791198912 + 1198791198913 + 1198791198914 + 1198791198915 + 1198791198916 (3)

where

1198791198911

= 1198881 [120583 (119903120596 |119885|) (119875119903 + 119875119892)119908119900]

05

119889 (119899119900 + 04119899119888) 119903 |119885|

1198791198912 = 1198882120587119889119899119888119908119888 (119875119903 + 119875119892) (1 minus |sin120572|) 119903 |119885|

1198791198913 = 1198883 (120583120596119903119885

ℎ119900

)119889119871 119904119903119885

Gas ring

Oil ring

d

Ls

wo

wc

Pr


1198791198914 = 1198884119899V119865119904119903 |119885| 120596minus05

1198791198915 = 1198885120583120596

1198791198916 = 1198886

1205871198892

4

119903119895119887119875119892 |cos120572| 120596minus05

119885 = sin120572 +120582119901 sin120572 cos120572


2sin2120572

(4)



119879119890 =

4

sum

119895=1

(119879119892119895 + 119879119868119895 minus 119879119891119895) (5)




0 180 360 540 7200

05

1

15

2




Gas

pre

ssur

ePg

(MPa

)


0 180 360 540 720

0

50

100


minus100

minus50Effec

tive t

orqu

eTe (N

m)




120579119891 minus



119879119862 (120579119903120579119903) = 119879119878 (120579119903) + 119879119867 (120579119903

120579119903) (6)


119879119878

=

11989611 (120579119903 minus 1206011199011) + 119896121206011199011 120579119903 gt 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199011

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(7)



119879119867

=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(8)






1198791015840

119878

=

11989611 (120579119903 minus 1206011199011) + 11989610 (1206011199011 minus 1206011199010) + 119896121206011199010 120579119903 gt 1206011199011

11989610 (120579119903 minus 1206011199010) + 119896121206011199010 1206011199010 lt 120579119903 le 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199010

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(9)

1198791015840

119867=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0 120579119903 gt 1206011199011

1198670

2

+

1198670 minus 1198672

2

sgn (120579119903 minus 1206011199010) 120579119903 gt 0 120579119903 lt 1206011199011

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(10)



TC

120579r120601p1

120601p2120601p3

(a)

120579r120601p1

120601p2120601p3

TS

k11

k12

k21

k22

(b)

120579r120601p1

120601p2120601p3

TH

H1H2

H3H4

120579r gt 0

120579r lt 0

(c)

TC

120579r120601p1

120601p2120601p3

k10

120601p0

H0

(d)








1198791198631 = 1198791199031198871 + 119879sh1 + 119879ch1 (11)

1198791198632 = 1198791199031198872 + 119879sh2 + 119879ch2 (12)


Bearing

Input shaft

Output shaft

1st driving gear



2nd driving gear


Synchronizer

Reverse gear


JP1 JP0

JP2JP3 JP4

JP5

k10

k1

k02

k2

k13 k34 k45

k4k3 k5

c10c1

c2c3

c02

c5c4

c45c34c13

kssTD4

TD3

TD2

TD1

TD5

JG2JG1 JG3 JG4

JG5

b3b2 b4 b5

b1

JS2

JS1

JS3

TI

TO

k2s

k1s

c2s

c1s

css



119879119903119887119894 = 1031198910 (]119873)

231198893

119898]119873 ge 2 times 10

minus3

119879119903119887119894 = 1611989101198893

119898]119873 lt 2 times 10

minus3

(13)



119879sh119894 =4120587

2120583119871119877

3Δ119873

30119895119895

(14)



119879ch119894 =1

2

1205881205962

119892119878119898119877

3119862119898

(15)

y

x

JFD2

120579FD2

120579D4

120579D2

120579D3

JD4 JD3

JD2

120579D1JD1




1198791198633 = 1198791199031198873 (16)

1198791198634 = 1198791199031198874 (17)

1198791198635 = 1198791199031198875 (18)


21205791198651198632 = 1205791198633 + 1205791198634

1205791198631 =119894119889

2

(1205791198634 minus 1205791198633)

1205791198632 =119894119889

2

(1205791198633 minus 1205791198634)

(19)



[1205791198651198632 1205791198634 1205791198633 1205791198631 1205791198632]119879

= [

1 0 2 minus119894119889 119894119889

0 1 minus1 119894119889 minus119894119889

]

119879

[1205791198651198632 1205791198634]119879

(20)


120596

Wr

r

LFx

O120577

d120577

zFz



T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)




119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road



119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903


V=

V119903V

braking

120583 =

119865119909

119865119911

(22)




119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)




120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

6 6

77

7

7

72

5 5

3

4






TDC

BDC

s=2r

r

l

AB

O

120572

M1

M2

M

120573

s kl

cos 120573

rco

s 120572





120572 = 120596 sdot 119905

120582119901 =119903

119897

119904119896 = 119903(1 +

120582119901

4

minus cos120572 minus120582119901

4

cos 2120572)

119904119896 = 120596119903(sin120596119905 +120582119901

2

sin 2120596119905)

119904119896 = 1205962119903 (cos120596119905 + 120582119901 cos 2120596119905)

(1)



MFNj

120573

FLj

FLj

Fj

FrjFtj

120572



119865119895 =

119875119892 (120572) sdot

120587119889119901

2

4

119895 = 119892

minus119898119901 sdot 119904119896 119895 = 119868

119879119892 = 119865119892119903 (sin120572 +120582119901

2

sin 2120572)

119879119868 = 119865119868119903 (sin120572 +120582119901

2

sin 2120572)

(2)



119879119891 = 1198791198911 + 1198791198912 + 1198791198913 + 1198791198914 + 1198791198915 + 1198791198916 (3)

where

1198791198911

= 1198881 [120583 (119903120596 |119885|) (119875119903 + 119875119892)119908119900]

05

119889 (119899119900 + 04119899119888) 119903 |119885|

1198791198912 = 1198882120587119889119899119888119908119888 (119875119903 + 119875119892) (1 minus |sin120572|) 119903 |119885|

1198791198913 = 1198883 (120583120596119903119885

ℎ119900

)119889119871 119904119903119885

Gas ring

Oil ring

d

Ls

wo

wc

Pr


1198791198914 = 1198884119899V119865119904119903 |119885| 120596minus05

1198791198915 = 1198885120583120596

1198791198916 = 1198886

1205871198892

4

119903119895119887119875119892 |cos120572| 120596minus05

119885 = sin120572 +120582119901 sin120572 cos120572


2sin2120572

(4)



119879119890 =

4

sum

119895=1

(119879119892119895 + 119879119868119895 minus 119879119891119895) (5)




0 180 360 540 7200

05

1

15

2




Gas

pre

ssur

ePg

(MPa

)


0 180 360 540 720

0

50

100


minus100

minus50Effec

tive t

orqu

eTe (N

m)




120579119891 minus



119879119862 (120579119903120579119903) = 119879119878 (120579119903) + 119879119867 (120579119903

120579119903) (6)


119879119878

=

11989611 (120579119903 minus 1206011199011) + 119896121206011199011 120579119903 gt 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199011

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(7)



119879119867

=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(8)






1198791015840

119878

=

11989611 (120579119903 minus 1206011199011) + 11989610 (1206011199011 minus 1206011199010) + 119896121206011199010 120579119903 gt 1206011199011

11989610 (120579119903 minus 1206011199010) + 119896121206011199010 1206011199010 lt 120579119903 le 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199010

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(9)

1198791015840

119867=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0 120579119903 gt 1206011199011

1198670

2

+

1198670 minus 1198672

2

sgn (120579119903 minus 1206011199010) 120579119903 gt 0 120579119903 lt 1206011199011

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(10)



TC

120579r120601p1

120601p2120601p3

(a)

120579r120601p1

120601p2120601p3

TS

k11

k12

k21

k22

(b)

120579r120601p1

120601p2120601p3

TH

H1H2

H3H4

120579r gt 0

120579r lt 0

(c)

TC

120579r120601p1

120601p2120601p3

k10

120601p0

H0

(d)








1198791198631 = 1198791199031198871 + 119879sh1 + 119879ch1 (11)

1198791198632 = 1198791199031198872 + 119879sh2 + 119879ch2 (12)


Bearing

Input shaft

Output shaft

1st driving gear



2nd driving gear


Synchronizer

Reverse gear


JP1 JP0

JP2JP3 JP4

JP5

k10

k1

k02

k2

k13 k34 k45

k4k3 k5

c10c1

c2c3

c02

c5c4

c45c34c13

kssTD4

TD3

TD2

TD1

TD5

JG2JG1 JG3 JG4

JG5

b3b2 b4 b5

b1

JS2

JS1

JS3

TI

TO

k2s

k1s

c2s

c1s

css



119879119903119887119894 = 1031198910 (]119873)

231198893

119898]119873 ge 2 times 10

minus3

119879119903119887119894 = 1611989101198893

119898]119873 lt 2 times 10

minus3

(13)



119879sh119894 =4120587

2120583119871119877

3Δ119873

30119895119895

(14)



119879ch119894 =1

2

1205881205962

119892119878119898119877

3119862119898

(15)

y

x

JFD2

120579FD2

120579D4

120579D2

120579D3

JD4 JD3

JD2

120579D1JD1




1198791198633 = 1198791199031198873 (16)

1198791198634 = 1198791199031198874 (17)

1198791198635 = 1198791199031198875 (18)


21205791198651198632 = 1205791198633 + 1205791198634

1205791198631 =119894119889

2

(1205791198634 minus 1205791198633)

1205791198632 =119894119889

2

(1205791198633 minus 1205791198634)

(19)



[1205791198651198632 1205791198634 1205791198633 1205791198631 1205791198632]119879

= [

1 0 2 minus119894119889 119894119889

0 1 minus1 119894119889 minus119894119889

]

119879

[1205791198651198632 1205791198634]119879

(20)


120596

Wr

r

LFx

O120577

d120577

zFz



T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)




119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road



119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903


V=

V119903V

braking

120583 =

119865119909

119865119911

(22)




119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)




120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










MFNj

120573

FLj

FLj

Fj

FrjFtj

120572



119865119895 =

119875119892 (120572) sdot

120587119889119901

2

4

119895 = 119892

minus119898119901 sdot 119904119896 119895 = 119868

119879119892 = 119865119892119903 (sin120572 +120582119901

2

sin 2120572)

119879119868 = 119865119868119903 (sin120572 +120582119901

2

sin 2120572)

(2)



119879119891 = 1198791198911 + 1198791198912 + 1198791198913 + 1198791198914 + 1198791198915 + 1198791198916 (3)

where

1198791198911

= 1198881 [120583 (119903120596 |119885|) (119875119903 + 119875119892)119908119900]

05

119889 (119899119900 + 04119899119888) 119903 |119885|

1198791198912 = 1198882120587119889119899119888119908119888 (119875119903 + 119875119892) (1 minus |sin120572|) 119903 |119885|

1198791198913 = 1198883 (120583120596119903119885

ℎ119900

)119889119871 119904119903119885

Gas ring

Oil ring

d

Ls

wo

wc

Pr


1198791198914 = 1198884119899V119865119904119903 |119885| 120596minus05

1198791198915 = 1198885120583120596

1198791198916 = 1198886

1205871198892

4

119903119895119887119875119892 |cos120572| 120596minus05

119885 = sin120572 +120582119901 sin120572 cos120572


2sin2120572

(4)



119879119890 =

4

sum

119895=1

(119879119892119895 + 119879119868119895 minus 119879119891119895) (5)




0 180 360 540 7200

05

1

15

2




Gas

pre

ssur

ePg

(MPa

)


0 180 360 540 720

0

50

100


minus100

minus50Effec

tive t

orqu

eTe (N

m)




120579119891 minus



119879119862 (120579119903120579119903) = 119879119878 (120579119903) + 119879119867 (120579119903

120579119903) (6)


119879119878

=

11989611 (120579119903 minus 1206011199011) + 119896121206011199011 120579119903 gt 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199011

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(7)



119879119867

=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(8)






1198791015840

119878

=

11989611 (120579119903 minus 1206011199011) + 11989610 (1206011199011 minus 1206011199010) + 119896121206011199010 120579119903 gt 1206011199011

11989610 (120579119903 minus 1206011199010) + 119896121206011199010 1206011199010 lt 120579119903 le 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199010

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(9)

1198791015840

119867=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0 120579119903 gt 1206011199011

1198670

2

+

1198670 minus 1198672

2

sgn (120579119903 minus 1206011199010) 120579119903 gt 0 120579119903 lt 1206011199011

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(10)



TC

120579r120601p1

120601p2120601p3

(a)

120579r120601p1

120601p2120601p3

TS

k11

k12

k21

k22

(b)

120579r120601p1

120601p2120601p3

TH

H1H2

H3H4

120579r gt 0

120579r lt 0

(c)

TC

120579r120601p1

120601p2120601p3

k10

120601p0

H0

(d)








1198791198631 = 1198791199031198871 + 119879sh1 + 119879ch1 (11)

1198791198632 = 1198791199031198872 + 119879sh2 + 119879ch2 (12)


Bearing

Input shaft

Output shaft

1st driving gear



2nd driving gear


Synchronizer

Reverse gear


JP1 JP0

JP2JP3 JP4

JP5

k10

k1

k02

k2

k13 k34 k45

k4k3 k5

c10c1

c2c3

c02

c5c4

c45c34c13

kssTD4

TD3

TD2

TD1

TD5

JG2JG1 JG3 JG4

JG5

b3b2 b4 b5

b1

JS2

JS1

JS3

TI

TO

k2s

k1s

c2s

c1s

css



119879119903119887119894 = 1031198910 (]119873)

231198893

119898]119873 ge 2 times 10

minus3

119879119903119887119894 = 1611989101198893

119898]119873 lt 2 times 10

minus3

(13)



119879sh119894 =4120587

2120583119871119877

3Δ119873

30119895119895

(14)



119879ch119894 =1

2

1205881205962

119892119878119898119877

3119862119898

(15)

y

x

JFD2

120579FD2

120579D4

120579D2

120579D3

JD4 JD3

JD2

120579D1JD1




1198791198633 = 1198791199031198873 (16)

1198791198634 = 1198791199031198874 (17)

1198791198635 = 1198791199031198875 (18)


21205791198651198632 = 1205791198633 + 1205791198634

1205791198631 =119894119889

2

(1205791198634 minus 1205791198633)

1205791198632 =119894119889

2

(1205791198633 minus 1205791198634)

(19)



[1205791198651198632 1205791198634 1205791198633 1205791198631 1205791198632]119879

= [

1 0 2 minus119894119889 119894119889

0 1 minus1 119894119889 minus119894119889

]

119879

[1205791198651198632 1205791198634]119879

(20)


120596

Wr

r

LFx

O120577

d120577

zFz



T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)




119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road



119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903


V=

V119903V

braking

120583 =

119865119909

119865119911

(22)




119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)




120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 180 360 540 7200

05

1

15

2




Gas

pre

ssur

ePg

(MPa

)


0 180 360 540 720

0

50

100


minus100

minus50Effec

tive t

orqu

eTe (N

m)




120579119891 minus



119879119862 (120579119903120579119903) = 119879119878 (120579119903) + 119879119867 (120579119903

120579119903) (6)


119879119878

=

11989611 (120579119903 minus 1206011199011) + 119896121206011199011 120579119903 gt 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199011

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(7)



119879119867

=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(8)






1198791015840

119878

=

11989611 (120579119903 minus 1206011199011) + 11989610 (1206011199011 minus 1206011199010) + 119896121206011199010 120579119903 gt 1206011199011

11989610 (120579119903 minus 1206011199010) + 119896121206011199010 1206011199010 lt 120579119903 le 1206011199011

11989612120579119903 1206011199012 le 120579119903 le 1206011199010

11989621 (120579119903 minus 1206011199012) + 119896121206011199012 1206011199013 le 120579119903 lt 1206011199012

11989622 (120579119903 minus 1206011199013) + 11989621 (1206011199013 minus 1206011199012) + 119896121206011199012 120579119903 lt 1206011199013

(9)

1198791015840

119867=

1198671

2

+

1198671 minus 1198672

2

sgn (120579119903 minus 1206011199011) 120579119903 gt 0 120579119903 gt 1206011199011

1198670

2

+

1198670 minus 1198672

2

sgn (120579119903 minus 1206011199010) 120579119903 gt 0 120579119903 lt 1206011199011

minus

1198673

2

+

1198673 minus 1198672

2

sgn (120579119903 minus 1206011199012) 120579119903 lt 0 120579119903 gt 1206011199013

minus

1198674

2

+

1198674 minus 1198672

2

sgn (120579119903 minus 1206011199013) 120579119903 lt 0 120579119903 lt 1206011199013

(10)



TC

120579r120601p1

120601p2120601p3

(a)

120579r120601p1

120601p2120601p3

TS

k11

k12

k21

k22

(b)

120579r120601p1

120601p2120601p3

TH

H1H2

H3H4

120579r gt 0

120579r lt 0

(c)

TC

120579r120601p1

120601p2120601p3

k10

120601p0

H0

(d)








1198791198631 = 1198791199031198871 + 119879sh1 + 119879ch1 (11)

1198791198632 = 1198791199031198872 + 119879sh2 + 119879ch2 (12)


Bearing

Input shaft

Output shaft

1st driving gear



2nd driving gear


Synchronizer

Reverse gear


JP1 JP0

JP2JP3 JP4

JP5

k10

k1

k02

k2

k13 k34 k45

k4k3 k5

c10c1

c2c3

c02

c5c4

c45c34c13

kssTD4

TD3

TD2

TD1

TD5

JG2JG1 JG3 JG4

JG5

b3b2 b4 b5

b1

JS2

JS1

JS3

TI

TO

k2s

k1s

c2s

c1s

css



119879119903119887119894 = 1031198910 (]119873)

231198893

119898]119873 ge 2 times 10

minus3

119879119903119887119894 = 1611989101198893

119898]119873 lt 2 times 10

minus3

(13)



119879sh119894 =4120587

2120583119871119877

3Δ119873

30119895119895

(14)



119879ch119894 =1

2

1205881205962

119892119878119898119877

3119862119898

(15)

y

x

JFD2

120579FD2

120579D4

120579D2

120579D3

JD4 JD3

JD2

120579D1JD1




1198791198633 = 1198791199031198873 (16)

1198791198634 = 1198791199031198874 (17)

1198791198635 = 1198791199031198875 (18)


21205791198651198632 = 1205791198633 + 1205791198634

1205791198631 =119894119889

2

(1205791198634 minus 1205791198633)

1205791198632 =119894119889

2

(1205791198633 minus 1205791198634)

(19)



[1205791198651198632 1205791198634 1205791198633 1205791198631 1205791198632]119879

= [

1 0 2 minus119894119889 119894119889

0 1 minus1 119894119889 minus119894119889

]

119879

[1205791198651198632 1205791198634]119879

(20)


120596

Wr

r

LFx

O120577

d120577

zFz



T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)




119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road



119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903


V=

V119903V

braking

120583 =

119865119909

119865119911

(22)




119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)




120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










TC

120579r120601p1

120601p2120601p3

(a)

120579r120601p1

120601p2120601p3

TS

k11

k12

k21

k22

(b)

120579r120601p1

120601p2120601p3

TH

H1H2

H3H4

120579r gt 0

120579r lt 0

(c)

TC

120579r120601p1

120601p2120601p3

k10

120601p0

H0

(d)








1198791198631 = 1198791199031198871 + 119879sh1 + 119879ch1 (11)

1198791198632 = 1198791199031198872 + 119879sh2 + 119879ch2 (12)


Bearing

Input shaft

Output shaft

1st driving gear



2nd driving gear


Synchronizer

Reverse gear


JP1 JP0

JP2JP3 JP4

JP5

k10

k1

k02

k2

k13 k34 k45

k4k3 k5

c10c1

c2c3

c02

c5c4

c45c34c13

kssTD4

TD3

TD2

TD1

TD5

JG2JG1 JG3 JG4

JG5

b3b2 b4 b5

b1

JS2

JS1

JS3

TI

TO

k2s

k1s

c2s

c1s

css



119879119903119887119894 = 1031198910 (]119873)

231198893

119898]119873 ge 2 times 10

minus3

119879119903119887119894 = 1611989101198893

119898]119873 lt 2 times 10

minus3

(13)



119879sh119894 =4120587

2120583119871119877

3Δ119873

30119895119895

(14)



119879ch119894 =1

2

1205881205962

119892119878119898119877

3119862119898

(15)

y

x

JFD2

120579FD2

120579D4

120579D2

120579D3

JD4 JD3

JD2

120579D1JD1




1198791198633 = 1198791199031198873 (16)

1198791198634 = 1198791199031198874 (17)

1198791198635 = 1198791199031198875 (18)


21205791198651198632 = 1205791198633 + 1205791198634

1205791198631 =119894119889

2

(1205791198634 minus 1205791198633)

1205791198632 =119894119889

2

(1205791198633 minus 1205791198634)

(19)



[1205791198651198632 1205791198634 1205791198633 1205791198631 1205791198632]119879

= [

1 0 2 minus119894119889 119894119889

0 1 minus1 119894119889 minus119894119889

]

119879

[1205791198651198632 1205791198634]119879

(20)


120596

Wr

r

LFx

O120577

d120577

zFz



T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)




119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road



119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903


V=

V119903V

braking

120583 =

119865119909

119865119911

(22)




119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)




120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Bearing

Input shaft

Output shaft

1st driving gear



2nd driving gear


Synchronizer

Reverse gear


JP1 JP0

JP2JP3 JP4

JP5

k10

k1

k02

k2

k13 k34 k45

k4k3 k5

c10c1

c2c3

c02

c5c4

c45c34c13

kssTD4

TD3

TD2

TD1

TD5

JG2JG1 JG3 JG4

JG5

b3b2 b4 b5

b1

JS2

JS1

JS3

TI

TO

k2s

k1s

c2s

c1s

css



119879119903119887119894 = 1031198910 (]119873)

231198893

119898]119873 ge 2 times 10

minus3

119879119903119887119894 = 1611989101198893

119898]119873 lt 2 times 10

minus3

(13)



119879sh119894 =4120587

2120583119871119877

3Δ119873

30119895119895

(14)



119879ch119894 =1

2

1205881205962

119892119878119898119877

3119862119898

(15)

y

x

JFD2

120579FD2

120579D4

120579D2

120579D3

JD4 JD3

JD2

120579D1JD1




1198791198633 = 1198791199031198873 (16)

1198791198634 = 1198791199031198874 (17)

1198791198635 = 1198791199031198875 (18)


21205791198651198632 = 1205791198633 + 1205791198634

1205791198631 =119894119889

2

(1205791198634 minus 1205791198633)

1205791198632 =119894119889

2

(1205791198633 minus 1205791198634)

(19)



[1205791198651198632 1205791198634 1205791198633 1205791198631 1205791198632]119879

= [

1 0 2 minus119894119889 119894119889

0 1 minus1 119894119889 minus119894119889

]

119879

[1205791198651198632 1205791198634]119879

(20)


120596

Wr

r

LFx

O120577

d120577

zFz



T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)




119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road



119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903


V=

V119903V

braking

120583 =

119865119909

119865119911

(22)




119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)




120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120596

Wr

r

LFx

O120577

d120577

zFz



T119889119891 =1

2

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

119879

[

[

[

[

[

[

[

[

[

1198691198651198632

1198691198634 01198691198633

0 1198691198631

1198691198632

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

1205791198651198632

1205791198634

1205791198633

1205791198631

1205791198632

]

]

]

]

]

]

]

]

]

]

=

1

2

[

1205791198651198632

1205791198634

]

119879

J119889119891 [1205791198651198632

1205791198634

]

J119889119891

= [

1198691198651198632 + 41198691198633 + 1198942

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

(21)




119889119911

119889119905

= V119903 minus1205900

1003816100381610038161003816V1199031003816100381610038161003816

119892 (V119903)119911

119892 (V119903) = 120583119888 + (120583119904 minus 120583119888) 119890minus(V119903V119904)120582

119865119909 = int

119871

0

(1205900119911 + 1205901 + 1205902V119903) 119891119899 (120577) 119889120577

0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

12

14

16

Slip rate ()

Long

itudi

nal f

rictio

n co

effici

ent

Ice roadGravel road



119865119911 = int

119871

0

119891119899 (120577) 119889120577

119904 =

120596119903 minus V120596119903

=

V119903120596119903


V=

V119903V

braking

120583 =

119865119909

119865119911

(22)




119896 =

119865119899

120576

120576 = 1205761198871199041 + 1205761198871199042 + 120576119888

(23)




120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120576119887119904119894 =151537119865119899

119864119898119899

119891111989121198913119911minus10622

(

120572119899

20

)

minus03879

sdot (1 +

1205730

10

)

008219

(1 + 120594119901)

minus02165

(

ℎ119891

119898119899

)

05563

sdot (

ℎ119896

119898119899

)

06971

(

119903fil119898119899

)

000043

(

119887

119898119899

)

minus06040

(24)



120576119888 =2Δ119865

120587Δ119911

1198961 [ln(1199041

119887119890

) minus

1205921

2 (1 minus 1205921)

]

+ 1198962 [ln(1199042

119887119890

) minus

1205922

2 (1 minus 1205922)

]

119887119890 =radic

4Δ11986511990311199032 (1198961 + 1198962)

120587Δ119911 (1199031 + 1199032)

1198961 =

1 minus 1205922

1

1198641

1198962 =

1 minus 1205922

2

1198642

(25)



0 02 04 06 08 1

3

4

5

6


Mes

hing

stiff

nessk1

(Nm

)

times108


0 02 04 06 08

5

6

7

8


Mes

hing

stiff

nesskf

(Nm

)

times108






Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Engine

Clutch



Final drive

Differential

Tire

Baseline vibration


MT modeling

Rattling vibration

Vehicle body

Half shaft






J 120579 (119905) + K120579 (119905) + C 120579 (119905) = T (119905) (26)

where

120579 = [120579119891 120579119862 1205791198751 1205791198750 1205791198752 1205791198782 1205791198783 12057911986611198781 1205791198663 1205791198664 1205791198665 1205791198651198631 1205791198651198632 1205791198634 120579119904119897 120579119904119903 120579119905119897 120579119905119903 119909119904]119879


J =[

[

[

[

[

J1 0 0

0 J2 0

0 0 J3

]

]

]

]

]

K =[

[

[

[

[

K11 K12 0

K21 K22 0

0 0 0

]

]

]

]

]

C =[

[

[

[

[

C11 C12 0

C21 C22 0

0 0 0

]

]

]

]

]

J1 = diag ([119869119891 119869119862 1198691198751 1198691198750 1198691198752 1198691198782 1198691198783 11986911986611198781 1198691198663 1198691198664 1198691198665 1198691198651198631])

J2 = [1198691198651198632 + 41198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632 minus21198691198633 minus 119894

2

1198891198691198631 minus 119894

2

1198891198691198632

minus21198691198633 minus 1198942

1198891198691198631 minus 119894

2

1198891198691198632 1198691198634 + 1198691198633 + 119894

2

1198891198691198631 + 119894

2

1198891198691198632

]

J3 = diag ([119869119904119897 119869119904119903 119869119905119897 119869119905119903 119898])


K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










K12 = K21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198961119891 0

minus11989634 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198960 minus1198960 0

minus1198960 1198960 + 11989610 + 1198961 lowast 1198771198751


minus11989610 11989610 + 11989602 minus11989602

minus11989602 11989602 + 1198962119904 minus1198962119904

0 minus1198962119904 1198962119904 + 119896119904119904 minus119896119904119904 0

minus119896119904119904 119896119904119904


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

K22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11989634 + 11989645 minus11989645

minus11989645 11989645 0

1198961119891 + 119896119891 lowast 1198771198651198631



2+ 4119896119903119863 minus2119896119903119863 minus2119896119903119863

minus2119896119903119863 119896119903119863 + 119896119897119863 minus119896119897119863 119896119903119863

minus119896119897119863 119896119897119863 + 119896119897119897 minus119896119897119897

0 minus2119896119903119863 119896119903119863 119896119903119863 + 119896119903119903 minus119896119903119903

minus119896119897119897 119896119897119897

minus119896119903119903 119896119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C12 = C21119879=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0 0

0

0

0

0 0

0

minus1198881119891 0

minus11988834 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]


C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










C11 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

1198880 minus1198880 0

minus1198880 1198880 + 11988810 + 1198881 lowast 1198771198751


minus11988810 11988810 + 11988802 minus11988802

minus11988802 11988802 + 1198882119904 minus1198882119904



2minus11988813

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

C22 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11988834 + 11988845 minus11988845

minus11988845 11988845 0

1198881119891 + 119888119891 lowast 1198771198651198631



minus2119888119903119863 119888119903119863 + 119888119897119863 minus119888119897119863 119888119903119863

minus119888119897119863 119888119897119863 + 119888119897119897 minus119888119897119897

0 minus2119888119903119863 119888119903119863 119888119903119863 + 119888119903119903 minus119888119903119903

minus119888119897119897 119888119897119897

minus119888119903119903 119888119903119903

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(27)




1198961 = 1198961119905cos2(1205731198871)

119896119891 = 119896119891119905cos2(120573119887119891)

119879119892119897 = 119865119909119891119897119903 +

119903119865119911119891119891

2

119879119892119903 = 119865119909119891119903119903 +

119903119865119911119891119891

2

119865119909119891119897 =

(1205900119911119897 + 1205901119897 + 1205902V119903119897) 1198651199111198912

119865119909119891119903 =

(1205900119911119903 + 1205901119903 + 1205902V119903119903) 1198651199111198912

119865119911119891 =

119871119887119898119892 minus 119898119909119886119867

119871119886 + 119871119887

119865119911119903 =

119871119886119898119892 + 119898119909119886119867

119871119886 + 119871119887


Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Detailed MT model

Te TC

JCk0 k10

c0c1 c2

c3

c34 c45

c4c5

c10

c13

c02

k1

k13

k02

k2 k3 k4

k34 k45

k5


b2b5

b3 b4

JfJS2 JS3

TD5TD3

TD2

TD4

JFD1 k1f


Jtl

Tgl

kll klD

cll clD

JFD2

JD4 JD3

JD1

JD2

krD Jsr krr

crr

Jtr

Tgr

crD

k2s

c2s css

kss

Jsl


r r

Fw

Lb

x

H

G

Cghw

FxfFxrFzr

Tfr

Tjw2 Tjw1

FzfTff

Ttmdxdt

La


V119903119897 = 120579119905119897119903 minus 119909V

V119903119903 = 120579119905119903119903 minus 119909V

119865119908 =1

2

1198621198631198601205881199092

V

(28)



pkm

b

cm

g

O2

TD

120579gRg

O1

Rp

120579p


1

1

X

f(X)

minusb2

b2




119883 = 120579119875119877119875 minus 120579119892119877119892

119868119892

119877119892

minus

119877119875

119877119892

119868119892120579119901 + 119865119898 sdot 119877119892 minus 119879119863 = 0

119865119898 = 119896119898119891 (119883) + 119888119898

119891 (119883)

(29)



119891 (119883) =

119883 minus

119887

2

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 +

119887

2

119883 lt minus

119887

2


119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119891 (119883) =

119883 gt

119887

2

0 minus

119887

2

lt 119883 lt

119887

2

119883 lt minus

119887

2

(30)







(a)

(b)


0 5 10 15 20

0

50

100

150

Tran

sfer t

orqu

e (N

m)

Actual working area

minus50


minus100

minus150






0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 118

1805

181

1815

Time (s)

Vehi

cle v

eloci

ty (m

s)






0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)






0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)


0 02 04 06 08 1

0

200

400

600

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus200

minus400


0 02 04 06 08 15

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3291

X 531Y 5622X 1343

Y 3365


0 100 200 300 4000

05

1

15

2

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 1637

X 531Y 1428

X 1593Y 09256

X 1062Y 06226X 1343

Y 04571





0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)








0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1

0

1000

2000

Time (s)

minus1000

minus2000

Fm2

(N)

(a)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm3

(N)

(b)

0 02 04 06 08 1

0

2000

4000

Time (s)

minus2000

minus4000

Fm4

(N)

(c)

0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Fm5

(N)

(d)


(a) (b)



0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30

0

50

100

150

200

Tran

sfer t

orqu

e (N


minus150

minus100

minus50




0 02 04 06 08 1740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(a)

03 04 05740

760

780

800

820

840

860

Time (s)

Rota

tiona

l spe

ed (r

pm)

EngineClutch hub

(b)


0 02 04 06 08 1

0

500

1000

Time (s)

minus500

minus1000

Engi

ne ac

cele

ratio

n (r

ads2)



0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1

0

20

40

Time (s)

Clut

ch h

ub ac

cele

ratio

n (r

ads2)

minus20

minus40


0 02 04 06 08 14

5

6

7

8

9

Time (s)

Wor

king

angu

lar d

ispla

cem

ent (

∘ )


0 50 100 150 2000

10

20

30

40

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 3215

X 531Y 6058

X 1343Y 2101



0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 100 200 300 4000

02

04

06

08

1

Frequency (Hz)

Am

plitu

de (r

pm)

X 2686Y 09167

X 1343X 531

Y 009349Y 01485


0 02 04 06 08 1790

795

800

805

Time (s)

Rota

tiona

l spe

ed (r

pm)

(a)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(b)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(c)

0 02 04 06 08 1225

230

235

Time (s)

Rota

tiona

l spe

ed (r

pm)

(d)



0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 10

50

100

150

Time (s)

Fm2

(N)

(a)

0 02 04 06 08 10

10

20

30

40

50

Time (s)

Fm3

(N)

(b)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm4

(N)

(c)

0

20

40

60

80

100

0 02 04 06 08 1Time (s)

Fm5

(N)

(d)



6 Conclusions










Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgment


References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article driveline torsional analysis and clutch damper...

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