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DOI: 10.1177/0954407013493267

originally published online 9 July 2013 2013 227: 1273Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering

Jong-Yun Yoon and Rajendra Singhengine conditions

Effect of the multi-staged clutch damper characteristics on the transmission gear rattle under two

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Original Article

Proc IMechE Part D:J Automobile Engineering227(9) 1273–1294� IMechE 2013Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0954407013493267pid.sagepub.com

Effect of the multi-staged clutchdamper characteristics on thetransmission gear rattle under twoengine conditions

Jong-Yun Yoon1,2 and Rajendra Singh1

AbstractThe scope of this article is limited to the torsional driveline system of a front-engine front-wheel type of vehicle. Areduced-order model of a manual transmission (with the third gear engaged and the fifth gear unloaded) is developed onthe basis of the modal characteristics. A simplified nonlinear mathematical model is then proposed with the focus on themulti-staged clutch damper properties such as the asymmetric transition angles and the pre-loads. The drag torque isestimated under two engine conditions (wide-open throttle and coast) by assuming that the vehicle is in the steady-statecondition. Finally, the gear rattle criteria are investigated, and illustrative examples for three real-life clutch dampers aredescribed. In particular, methods for solving the real-life gear rattle phenomenon are suggested.

KeywordsVehicle driveline dynamics, nonlinear system simulation, clutch damper design, drag torque calculation

Date received: 15 January 2013; accepted: 16 May 2013

Introduction

The gear rattle phenomenon in vehicle driveline systemshas received much attention, and both numerical simu-lation and experimental methods have been adopted.1–21

For example, Barthod et al.9 have conducted an experi-ment on a simplified gearbox and studied the roles ofkey parameters. Some semi-analytical dynamic modelswith a reduced order have been suggested; these employthe harmonic balance method.22–25 The impact dampingmodel26,27 has been utilized as well. However, most ofthese studies have examined the system using simplifiedclutch damper models.22–27 Further, previous work hasusually been restricted to the neutral-gear rattle issues.In practice, vibro-impacts depend upon the dynamicconditions of both the engine and the vehicle systems.For example, the wide-open throttle (WOT) and idlingconditions are defined by the engine firing status. Theacceleration or coast conditions pertain to the mode ofvehicle driving. Thus, this article will examine thedynamic characteristics of gear rattle as affected by real-life clutch damper design (as it also operates as adynamic isolator in addition to acting as a torque-transmitting device). The previous work on a two-stagedsymmetric clutch damper22–34 will be extended to amulti-staged asymmetric clutch damper. Also, the drag

torques on the input and output shafts and on the tireunder alternate vehicle driving conditions will beestimated.35

Problem formulation

The scope of this article is limited to the driveline of afront-engine front-wheel type of vehicle. A manualtransmission (with the third gear engaged and the fifthgear unloaded) is analyzed. Figure 1 shows a schematicdiagram of a six-degree-of-freedom (6DOF) torsionalsystem model with relevant nonlinear functions. Thereduced-order system is developed from the originalsystem35 where most of the speed gears are unloadedand rotating except for one mesh that is engagedthrough the synchronizer and gear-shifting mechanism.

1Acoustics and Dynamics Laboratory, Smart Vehicle Concepts Center,

The Ohio State University, Columbus, Ohio, USA2Machinery System Research Team, STX Heavy Industries Co., Seoul,

Republic of Korea

Corresponding author:

Rajendra Singh, Acoustics and Dynamics Laboratory, Smart Vehicle

Concepts Center, The Ohio State University, 201 West 19th Avenue,

Columbus, OH 43210, USA.

Email: [email protected]



The key components in Figure 1 such as the flywheel,the clutch hub, the input shaft, the gear pairs, the axlesand the tire–vehicle system are described by the lumpedtorsional inertia I or stiffness k. The synchronizerassembly and fixed gears are lumped to the input andoutput shafts. The values used in this study are sum-marized in Table 1. The values of the gear mesh stiff-nesses will be given later. In the engaged-gear status,the dynamic torque from the engine induces rattlesymptoms at small mean loads.

In order to simulate single-sided or double-sidedvibro-impacts, several assumptions can be made.

1. Gear rattle is associated with torsional vibro-impacts.27,33,35

2. The dynamic torque from the engine fluctuationcan excite the loaded and unloaded gear pairs.

3. Nonlinearities of interest include the characteristics ofthe multi-staged clutch damper and gear backlashes.

The chief objectives of this research are as follows.First, a mathematical model for the multi-staged clutchdamper for the general case is developed. Figure 2shows nonlinear torque–displacement profiles of amulti-staged clutch damper including the transitionangles and hysteresis levels. Here, TC and d1 are definedas the clutch torque and the relative displacementbetween the flywheel and clutch hub respectively.

Second, the drag torques under both the WOT condi-tion and the coast condition are estimated. The frictionmodel between the tire–vehicle torque and the inertialtorque under different driving conditions will also beconsidered. Third, the dynamic characteristics of gearrattle are compared with three real-life multi-stagedclutch dampers under both the WOT condition and thecoast condition.

Development of linear system models

Based upon the schematic diagram shown in Figure 1,a linear time-invariant (LTI) system model is developedfor the following conditions.

1. The rattle phenomenon occurs in the highest-acceleration regime.

2. The natural mode of gear motion is correlated withthe rattle phenomenon.

3. The most severe rattle with the third gear engagedand the fifth gear unloaded occurs at 1800 r/minand 3000 r/min for the WOT condition and thecoast condition respectively.

Accordingly, several assumptions are made.

1. The system is considered as a torsional undampedsemidefinite system.

Figure 1. Example case: 6DOF nonlinear torsional system model. Here, the third gear is engaged and the fifth gear is unloaded.

Table 1. Employed parameters for the linear and nonlinear simulation models.

Inertia (subsystem) Value (kg m2) Radius (subsystem) Value (mm)

If (flywheel) 1.38 3 1021 Rie (engaged gear on the input shaft) 35.5Ih (clutch hub) 5.76 3 1023 Roe (engaged gear on the output shaft) 46.0Iie (input shaft) 4.53 3 1023 Riu (unloaded gear on the input shaft) 45.9IOG (ouput shaft) 7.80 3 1023 Rou (unloaded gear on the output shaft) 35.6Iou (unloaded gear) 5.23 3 1024

IVE2 (vehicle) 3.28

1274 Proc IMechE Part D: J Automobile Engineering 227(9)



2. Nonlinearities such as clutch hysteresis and gearbacklash are ignored.

3. Gears are always in contact on the driving side,resulting in a rattle-free system.27,33–35

4. This LTI system model has one clutch stiffnessvalue, although the clutch has multi-staged proper-ties. Here, the following stiffness values are consid-ered: stiffness of the clutch, kC = 1838.0N m/rad;stiffness of the input shaft, ki = 1.0 3 104 Nm/rad;

gear mesh stiffness between the gear pairs, kg =2.7 3 108 N/m; torsional stiffness between theoutput shaft and the vehicle, kVE2 = 6.63 3

102 N m/rad.

As mentioned, a reduced 6DOF system model ofFigure 1 is investigated since a large dimension maycause problems for nonlinear analysis.7,23,27,33,35 Thus,all inertia and stiffness values of the gear pairs are mul-tiplied by (R1/R2)

2 based on the original system val-ues,35 where R is the gear radius.27,33,34,36 The modelreduction procedures and the original 14-degree-of-freedom (14DOF) system model have been previouslyinvestigated.35

The governing equations are placed in the matrixform

M €u tð Þ+C _u tð Þ+K u tð Þ=T tð Þ ð1Þ

where C is the viscous damping matrix, T tð Þ is the tor-que excitation vector,

u= uf uh ui uou uOG uv½ �T ð2ÞM =diag If, Ih, Iie, Iou, IOG, IVE2

� �ð3Þ

and

K=

kC �kC 0 0 0 0�kC kC + ki �ki 0 0 00 �ki ki + kg R2

iu+R2ie

� �kgRiuRou kgRieRoe 0

0 0 kgRiuRou kgR2ou 0 0

0 0 kgRieRoe 0 kgR2oe + kVE2 �kVE2

0 0 0 0 �kVE2 kVE2

26666664

37777775

ð4Þ

From equations (1) to (4), the eigensolutions are found.The first three natural modes include the driveline sur-ging mode (7.6Hz), the clutch spring mode (60.6Hz),and the clutch and input shaft mode (273Hz). If thesystem has a severe gearbox rattle problem around60Hz, its mode shape could be altered by selecting theclutch or axle stiffness. The 6DOF model should beassessed by the frequency response functions of equa-tion (1) which are calculated by applying unity harmo-nic torque at the engine to give

T tð Þ= 1 0 0 0 0 0½ �Teivot ð5Þ

Then, the viscous damping matrix is constructed byusing the normal modal matrix and assumed dampingratios (say, 5% at each mode).7,27,33,34,35 Now, wedefine the torsional mobility asYaj voð Þ=

_ua

Tjvoð Þ

Figure 2. Torque TC(d1) profiles for three clutch dampers based on real-life designs: - - -, clutch A; - � -, clutch B; ——, clutch C.

Yoon and Singh 1275



where the subscripts a and j are the excitation (orinput) locations at the relevant subsystem and where vo

is the excitation frequency; the subscript a (or j) = f, h,i, ou, OG, v. Figure 3 shows that the mobility spectrafor the 6DOF system match well with the results fromthe original 14DOF system over the frequency range ofinterest.35 Here, Yff

�� , Yhf

�� , and Yif

�� are defined as themobility of the flywheel, the mobility of the clutch huband the mobility of the input shaft respectively.Resonances are observed at 7Hz, 60Hz, and 273Hz, asexpected.

In general, the dynamic characteristics of the accel-eration under the coast condition are different fromthose under the WOT condition. First, the maximumvalues of acceleration (assumed as resonances) underthe WOT condition are moved into a higher range ofengine speeds between 3000 r/min and 3500 r/min.Second, the dynamic characteristic of the engine accel-eration is also changed since there is no tip-in (at theacceleration pedal) under the coast condition. In thisstudy, the coast condition is assumed to be at 3000 r/min (firing frequency, 100Hz).

The effective modal properties are evaluated byassuming that the system is affected by only the clutchspring mode. The default clutch stiffness value of1838.0N m/rad is increased by up to four times, andnow the natural frequency at the clutch spring mode is102Hz (close to the firing frequency of 100Hz). This

suggests that clutch springs may have large excursionsand may hit the stopper, which will be investigated inthe nonlinear simulations later.

Nonlinear system model

Again, the system of Figure 1 includes nonlinear ele-ments such as the multi-staged clutch torque, the gearbacklash torque, and the drag torque. Practical systemswould include even more nonlinear components, suchas a backlash between the clutch hub and the inputshaft, thermal effects in the transmission, and the wind-induced reaction force.27,33 However, this study is lim-ited to only a few nonlinear elements, as shown inFigure 1. The governing equations in terms of the rela-tive motions and state space model are derived as

x tð Þ= uf uh ui uou uo½

uVE2_uf

_uh_ui

_uou_uo

_uVE2 �T ð6Þ

d tð Þ= d1 tð Þ d2 tð Þ d3 tð Þ d4 tð Þ d5 tð Þ½ �T ð7aÞd1 tð Þ= uf tð Þ � uh tð Þ ð7bÞd2 tð Þ= uh tð Þ � ui tð Þ ð7cÞd3 tð Þ=Rieui tð Þ+Roeuo tð Þ ð7dÞd4 tð Þ=Riuui tð Þ+Rouuu tð Þ ð7eÞd5 tð Þ= uo tð Þ � uVE2 tð Þ ð7fÞ

where the state vector x tð Þ describes the absolute displa-cements uj (where j is the index of the subsystems) andthe absolute velocities _uj (where j is the index of the sub-systems). Next, the state vector d tð Þ is defined in termsof the relative displacements according to

d tð Þ=Px tð Þ ð8aÞ

and the relationship between x tð Þ and d tð Þ is defined bythe transformation matrix P given by

P=

1 �1 0 0 0 00 1 �1 0 0 00 0 Rie 0 Roe 00 0 Riu Rou 0 00 0 0 0 1 �1

266664

377775 ð8bÞ

Then, a new state vector xr tð Þ for the relative displace-ments is defined as

xr tð Þ= d1 tð Þ d2 tð Þ d3 tð Þ d4 tð Þ d5 tð Þ½_d1 tð Þ _d2 tð Þ _d3 tð Þ _d4 tð Þ _d5 tð Þ �T ð9aÞ

xr tð Þ=Prx tð Þ ð9bÞ

Pr =P 0

0 P

� �ð9cÞ

The relationship between x tð Þ and xr tð Þ is defined fromequation (9) and is given by

x tð Þ=P+r xr tð Þ ð10aÞ

P+r = PT

rPr

�1PT

rð10bÞ

Figure 3. Mobility spectra of the reduced-order and originaltorsional systems: ——, mobility of the original 14DOF system;- - -, mobility of the 6DOF system.




Here, the pseudo-inverse matrix P+r is used since Pr

has n 3 m (n6¼m) dimensions. Thus, the state equationspertaining to xr tð Þ are derived as

_xr tð Þ=S�xr tð Þ+T�non tð Þ+T�E tð Þ ð11aÞ

S�=PrSP+r ð11bÞ

T�non tð Þ=PrTnon tð Þ ð11cÞ

T�E tð Þ=PrTE tð Þ ð11dÞ

S=0 I

KS CS

� �ð12aÞ

KS =

0 0 0 0 0 0

0 � kiIh

kiIh

0 0 0

0 kiIie

� kiIie

0 0 0

0 0 0 0 0 0

0 0 0 0 � kVE2IOG

kVE2IOG

0 0 0 0 kVE2IVE2

� kVE2IVE2

26666666664

37777777775

ð12bÞ

TE tð Þ=

000000

TE tð ÞIf

00000

26666666666666666664

37777777777777777775

ð12dÞ

Tnon tð Þ=

000000

� TC d1, _d1ð ÞIf

TC d1, _d1ð ÞIh

� RiuFgu ruð Þ+RieFge reð Þ+TDi

Iie�RouFgu ruð Þ+TDu

Iou�RoeFge reð Þ+TDo +TDu

IOG

� TDVE2

IVE2

266666666666666666666664

377777777777777777777775

ð12eÞ

Here, Fgu(ru) and Fge(re) are the gear mesh force onthe unloaded gear pair and the gear mesh force onthe engaged gear pair respectively, ru and re are therelative displacement between the gears in theunloaded gear pair and the relative displacementbetween the gears in the engaged gear pair respec-tively, Riu and Rie are the unloaded gear radius andthe engaged gear radius respectively on the inputshaft, and Rou and Roe are the unloaded gear radiusand the engaged gear radius respectively on the outputshaft. The mathematical description of Fgu(ru) andFge(re) will be explained later. Also, the viscous damp-ing matrix CS is estimated from the assumed modaldamping ratio (say, 5%) in the nonlinear model. Inorder to investigate the system responses using equa-tion (11), a modified Runge–Kutta method, suitablefor numerically stiff problems as suggested byDormand and Prince,37 is used.

Nonlinear characteristics of themulti-staged clutch damper and the gearbacklash

Figure 4 shows the typical static characteristics of aclutch damper in terms of a TC versus d1 profile includ-ing the hysteresis (due to dry friction resulting from thefriction assembly inside the driven plate of the clutch).Figure 4(a) illustrates the properties of clutch B whichcontains asymmetric transition angles and pre-load.Figure 4(b) illustrates the torque induced by the stiff-ness TS with the asymmetric transition angles (�fn1 onthe negative side and fp1 on the positive side).Figure 4(c) and (d) shows the clutch torque TH (orTSPr) due to the hysteresis (or the pre-load), where fPr

is the transition angle corresponding to the pre-load.Also, TPr1 and TPr2 are the positive pre-load and thenegative pre-load respectively. A general mathematicalmodel should include all piecewise linear stages andasymmetric cases, as illustrated in Figure 4.

First, consider a two-staged elastic torque (with thesubscript S) as

TS d1ð Þ=kC1fp1 + kC2 d1 � fp1

� �, d1 . fp1

kC1d1, � fn14d14fp1

�kC1fn1 + kC2 d1 +fn1ð Þ, d1 \ � fn1

8<:

ð13Þ

CS =

� chIf

chIf

0 0 0 0chIh

� ch + ciIh

ciIh

0 0 0

0 ciIie

� ci + cgeR2ie + cguR

2iu

Iie� cguRiuRou

Iie� cgeRieRoe

Iie0

0 0 � cguRiuRou

Iou� cguR

2iu

Iou0 0

0 0 � cgeRieRoe

IOG0 � cgeR

2ie + cVE2IOG

cVE2IOG

0 0 0 0 cVE2IVE2

� cVE2IVE2

26666666664

37777777775

ð12cÞ

Yoon and Singh 1277



where fp1 and 2fn1 are the first transition angle on thepositive side and the first transition angle on the nega-tive side respectively. Thus, TS d1ð Þ is approximated interms of kC1, kC2, fp1 (or �fn1) and d1 using the hyper-bolic tangent function to give

TS d1ð Þ= kC1d1 + kC2 d1 � fp1

� �+ kC1fp1 � kC1d1

� �12 tanh sC d1 � fp1

� �� +1

� �

+ kC2 d1 +fn1ð Þ � kC1fn1 � kC1d1½ �12 �ftanh sC d1 +fn1ð Þ½ �+1f g

ð14Þ

where sC is the smoothing of the regularizing factor forthe clutch torque.27,33,35,38 By manipulating equation(14), TS d1ð Þ is expressed as

TS d1ð Þ= kC1d1 +12 kC2 � kC1ð Þ d1 � fp1

� �tanh sC d1 � fp1

� �� +1

� �� 1

2 kC2 � kC1ð Þ d1 +fn1ð Þ tanh sC d1 +fn1ð Þ½ � � 1f gð15Þ

Second, TS d1ð Þ for a three-staged clutch damper iswritten, on the basis of equations (13) to (15), as

TS d1ð Þ= kC1d1 +12 kC3 � kC2ð Þ d1 � fp2


� �� +1

� �+1

2 kC2 � kC1ð Þ d1 � fp1


� �� +1

� �

� 12 kC3 � kC2ð Þ d1 +fn2ð Þ tanh sC d1 +fn2ð Þ½ � � 1f g

� 12 kC2 � kC1ð Þ d1 +fn1ð Þ tanh sC d1 +fn1ð Þ½ � � 1f g

ð16Þ

Finally, TS(d1) for the multi-staged clutch with asym-metric transition angles is derived, on the basis of equa-tions (14) to (16), as

TS d1ð Þ= kC1d1 +12

XNi=2

kC ið Þ � kC i�1ð Þ� �

Tsp i�1ð Þ � Tsn i�1ð Þ� �

ð17aÞ

Tsp ið Þ= d1 � fp ið Þ

tanh sC d1 � fp ið Þ

h i+1

n o

ð17bÞ

Tsn ið Þ= d1 +fn ið Þ

tanh sC d1 +fn ið Þ

h i� 1

n o

ð17cÞ

Here, some of the properties are denoted as follows:kC(N) (or kC(i)), Nth stage (or ith stage) of the clutchstiffness (with the subscript N or i); Tsp(i) (or Tsn(i)),clutch torque induced by the stiffness at the ith stageon the positive side (or on the negative side) (with thesubscript p or n); fp ið Þ(or 2fn ið Þ), ith transition angleon the positive side (or on the negative side). As a nextstep, the clutch torque TH induced by hysteresis is con-sidered, as shown in Figure 4(c).

First, if the system has only one stage, simpler for-mulation where TH d1, _d1

� �is the function of d1 and _d1

is used and is given by

δ1

p1

n1

TH

H1

H2

n1

p1 δ1

TS

kC1

kC2

kC2

φ

δ1

TSPr

Pr

TPr1

TPr2 φ

φ

φ–

φ–

0 0.1 0.2-0.2 -0.1δ1 (rad)

TC(N m)

-200

200

-100

100

0

(a)

(c) (d)

(b)

Figure 4. Nonlinear characteristics of a typical multi-staged clutch damper: (a) profile of a real-life clutch damper; (b) piecewisestiffness characteristics with asymmetric transition angles; (c) piecewise hysteresis characteristics with asymmetric transition angles;(d) pre-load characteristics.




TH d1, _d1

� �=

H1

2tanh sC

_d1

� �ð18Þ

Second, TH for the two-staged clutch is derived onthe basis of the illustration in Figure 4(c) according to

TH d1, _d1

� �=

H2

2tanh sC

_d1

� �

+H2

4�H1

4

�tanh sC d1 � fp1

� ��

1+ tanh sC_d1

� ��

+H2

4�H1

4

�tanh sC d1 +fn1ð Þ½ �

1� tanh sC_d1

� ��

ð19Þ

Third, TH for the three-staged clutch is derived in a sim-ilar manner according to

TH d1, _d1

� �=

H3

2tanh sC

_d1

� �

+H2

4�H1

4


� ��

1+ tanh sC_d1

� ��

+H2

4�H1

4

�tanh sC d1 +fn1ð Þ½ �

1� tanh sC_d1

� ��

+H3

4�H2

4


� ��

1+ tanh sC_d1

� ��

+H3

4�H2

4

�tanh sC d1 +fn2ð Þ½ � 1� tanh sC

_d1

� ��

ð20Þ

Finally, TH for the multi-staged clutch case withasymmetric transition angles is derived from equations(18) to (20) and is defined as

TH d1, _d1� �

=H Nð Þ2

tanh sC_d1

� �

+XNi=2

H ið Þ4�H i�1ð Þ

4

�THp i�1ð Þ+THn i�1ð Þ� �

ð21aÞ

THp ið Þ=tanh sC d1 � fp ið Þ

h i1+ tanh sC

_d1

� �� ð21bÞ

THn ið Þ=tanh sC d1 +fn ið Þ

h i1� tanh sC

_d1

� �� ð21cÞ

Here, the properties are denoted as follows: HN (orH(i)), Nth stage (or ith stage) of hysteresis (with thesubscript N or i); THp(i) (or THn(i)), positive part (ornegative part) of the clutch torque induced by hyster-esis at the ith stage (with the subscript p or n). In orderto include the pre-load effect as illustrated in Figure4(d), d1 is first modified to d1Pr= d1 � fPr. Then, the

pre-load TPr is calculated as a function of d1Pr accord-ing to

TSPr d1pr

� �=1

2TPr1 tanh sCd1Prð Þ+1½ �+1

2TPr2 � tanh sCd1Prð Þ+1½ �ð22Þ

When the clutch torque is affected by the pre-load, therelative displacement d1 should be modified tod1Pr= d1 � fPr, together with equations (13) to (21).The total clutch torque corresponding to Figure 4 isfound as

TC d1Pr, _d1Pr

� �=TS d1Prð Þ+TH d1Pr, _d1Pr

� �+TSPr d1Prð Þ

ð23Þ

where the value of the smoothing factor sC is 1 3 103

in this study.Likewise, the gear mesh force Fg and the relative dis-

placement r between the driving gear and the drivengear (in the translation units) are written as27,33,35

Fgk rkð Þ= kgrk

+ kgrk � b=2ð Þ tanh sg rk � b=2ð Þ

� �� rk + b=2ð Þ tanh sg rk+ b=2ð Þ

� �2

ð24aÞ

and

rk =Rikui +Rokuo, k = e or u ð24bÞ

respectively where the gear mesh stiffness is kg andwhere Ri and Ro are the gear radii of the input shaftand the output shaft (or unloaded gear) respectively, uiand uo are the torsional displacements of the inputshaft and the output shaft (or unloaded gear) respec-tively, and b is the gear backlash. Again, the smoothingfactor sg is assumed to be 1 3 1010 for the calculationof the gear mesh forces. The following three contactconditions are numerically checked. When the gearsalways maintain contact on the drive side, the linearsystem behavior is found as the relative displacementexceeds + b/2. When the gears are on the no-contactzone or on the driven side, single-sided and double-sided vibro-impacts will take place.

Estimation of the drag torque under thewide-open throttle condition

For gear rattle analysis, knowledge of the engine tor-ques (the mean torque TM and the dynamic torqueTpi(t)) and the various drag torques is important. Forexample, the pulsating engine torque may have differ-ent spectral contents according to the driving (accelera-tion) conditions. Also, the throttle angles may be atspecific different values such as 30%, which implies theWOT condition, suggesting a significant alternatingtorque.26,27,33–35 Figure 5 shows a typical engine torqueprofile under the WOT condition. Here, the dot-dashedline is the mean engine torque TM, and to is the periodof the firing frequency. The torque spectrum describing

Yoon and Singh 1279



the dominant harmonics is shown in Figure 5(b).Therefore, the engine torque is given as27,33–35

TE tð Þ=TM +XNi=1

Tpi cos ivpt+fpi

� �ð25Þ

Here, Tpi is the amplitude of the alternating parts ofthe ith harmonic, fpi is the phase of the ith harmonic,and vp is the firing frequency. In this study, the follow-ing values under the WOT condition are assumed:TM = 168.9N m; Tp1 = 251.5N m; Tp2 = 106.9 N m;Tp3 = 29.8N m; Tp4 = 24.2N m; Tp5 = 18.0N m;fM= 0 rad; fp1= 21.93 rad; fp2= 22.58 rad; fp3=2.16 rad; fp4= 0.72 rad; fp5= 20.09 rad.

The drag torque TDo on the output shaft, the dragtorque TDu on the unloaded gear, and the drag torqueTDVE2 from the vehicle (lumped to the input shaft) areestimated next. Overall, the total drag torque (the sumof TDi, TDoe, TDue and TDVEe) is assumed to be equal tothe mean engine torque TM under the steady-state con-dition (with a constant speed). Further assumptionsinclude the following.

1. The drag torque TDVEe on the vehicle is muchlarger than the sum of the drag torques on the gearand the shaft.

2. The drag elements associated with all the subsys-tems are the same (designated as c), except for thedrag damping on the unloaded gear (assumed to be0.1c).

3. The alternating component of the drag torques isignored.

4. The slope of the road is 0�.

Figure 6 describes the drag torque from the vehicleunder the driving condition by considering the relation-ship between the static friction coefficient ms and thedynamic friction coefficient ms1 of the road. Here, thesymbols are as follows: Otire is the angular velocity ofthe tire; Vvehicle is the vehicle velocity, Wvehicle_front is thedistributed weight of the vehicle for the front side, andWvr is the reaction force from the road. From the sche-matic diagram in Figure 6, the original drag torqueTDve from the vehicle can be estimated. Let the distrib-uted weight ratio on the front side of vehicle be equalto 70% of the total vehicle weight. The mass of thevehicle on the front side is assumed to be mvehicle_front

= 1500 kg 3 0.7. Thus, Wvehicle_front = 9.8130.7mvehicle

and Wvehicle_front = Wvr. The static frictional force fromthe tire is considered to be msWvr. The dynamic fric-tional force ms1Wvr induced by the vehicle dynamicsshould be less than msWvr. Also, the values of ms1 andms are assumed to be 0.35 and 0.7 respectively undersevere driving conditions such as the WOT condition.A more precise estimate of the drag torque (under themultiple driving and road conditions) is beyond thescope of this study.39 Let the radius of the tire be RT =17 in/2 = 0.2159m. Thus, TDve is estimated, by assum-ing that ms1Wvr = TDve/RT, to be

TDve =ms1WvrRT ð26aÞ

TDVEe =Rie

Roe

Rod

RdTDve ð26bÞ

TDVE2 =Rod

RdTDve ð26cÞ

Thus, the drag torques are calculated as

TM =TDi +TDue+TDoe+TDVEe ð27aÞ

TDue=Riu

RouTDu ð27bÞ

Figure 5. Engine torque TE under the WOT condition at1800 r/min: (a) time history of the engine torque; (b) amplitudeand phase spectra for the dominant firing orders.

Wvehicle_front

Wvr

μsWvr-μs1Wvr

Vvehicle

Ωtire

Figure 6. Drag torque from the vehicle under the WOTcondition.




TDoe=Rie

RoeTDo ð27cÞ

TDi = c _ui ð28ÞTDu =0:1c _uou � _uo

� �

=0:1cRiu

Rou� Rie

Roe

�_ui

ð29Þ

TDo = c _uo

= cRie

Roe

_ui

ð30Þ

Thus, the net drag coefficient is found as

c=TM � TDVEe

1+ Rie=Roeð Þ2 +0:1 Riu=Rouð Þ2 � Riu=Rouð Þ Rie=Roeð Þh in o

_ui

ð31Þ

where the value of _ui is 2p 1800 r=min=60minð Þ at188.4956 rad/s in this study.

Drag torque assumptions under the coastcondition

Table 2 lists the torque amplitudes and phases when thevehicle driving condition is changed by an abrupt tip-out process. Here, to2 and to+ refer to the momentswhen the driving condition of the vehicle is under theWOT (with tip-in) condition and then the coast (withtip-out) condition respectively. The effective drag tor-que from all the subsystems to the flywheel is denotedTDT(to2) (with tip-in) or TDT(to+) (with tip-out).40 Inorder to develop the simulation model, several assump-tions are made.

1. The torsional system is under the steady-state con-dition with a constant input torque when theengine speed is 3000 r/min (WOT).

2. The engine speed is changed abruptly by the tip-out process. Thus, the vehicle is mostly driven bythe inertial torque under the same firing frequency(100Hz or 3000 r/min). The only difference is that

now the torque profiles at 3000 r/min are changedinto those at 700 r/min under the idling condition.

3. The moment when the vehicle system undergoesan abrupt change in the input torque is defined asvoto2!voto+ where vo = 2p100 rad/s.

Thus, the effective input torque is

TEc tcð Þ=TE to+ð Þ+TDT to+ð Þ ð32aÞ

TDT to+ð Þ’TDT to�ð Þ’� TE to�ð Þ ð32bÞ

TE to�ð Þ=TE 3000 r=minð Þ tð Þ ð32cÞ

TE to+ð Þ=TE 700 r=minð Þ tð Þ ð32dÞ

Here, the subscripts c, 700 r/min and 3000 r/min refer tothe coast condition, the 700 r/min condition and the3000 r/min condition respectively. From equation(32b), the total drag torque TDT(to+) on the flywheel isequal to TDT(to2) by assuming that the inertial torqueremains with the same magnitudes of TE(3000 r/min)(t)right after tipping out. Thus, the profile of TDT(to+) isthe same as TE(to2) except for the effective direction ofthe torque. Also, the other drag torques TDi, TDo, TDu,and TDVE2 are assumed to have the same values as theproperties at to2. Therefore, only the input torque pro-files are changed, as described in Table 2, and the effec-tive torque TEc(tc) is given by equations (32a) to (32d).The relevant harmonic terms of TEc(tc) are estimated as

TEc tð Þ=TMc +XNi=1

Tpci cos ivpt+fpci

� �ð33aÞ

TMc =TM to+ð Þ � TM to�ð Þ ð33bÞ

~Tpci = ~Tpi to+ð Þ � ~Tpi to�ð Þ ð33cÞ

~Tpi to6ð Þ= ~Tpi cos fpi to6ð Þ� �

+i ~Tpi sin fpi to6ð Þ� �

ð33dÞ

where TMc and ~Tpci are the effective mean torque andthe alternating torque respectively.

Table 2. Fourier decomposition of the engine torque under the coast condition at to + and to2.

Torque component Value for the following engine speeds Effective component of the input torque

700 r/min 3000 r/min

Magnitude (N m) TM 22.0 192.3 2170.3Tp1 41.5 589.6 548.9Tp2 14.4 143.3 129.7Tp3 4.3 53.3 50.7Tp4 2.7 24.2 24.2Tp5 1.8 14.1 14.6Tp6 1.0 8.1 8.6

Phase (rad) fp1 21.90 21.71 1.45fp2 22.69 22.37 0.81fp3 2.25 3.14 0.06fp4 0.64 2.18 20.85fp5 20.38 1.40 21.62fp6 21.37 0.72 22.32

Yoon and Singh 1281



Results and discussion

Table 3 lists the properties of a four-staged clutch withasymmetric transition angles. Figures 7 to 9 show thenumerical results in terms of the clutch torque (or thegear forces) versus the relative angular displacementand the relative translational motions of the engaged(or unloaded) gear pair under the WOT condition at1800 r/min. Here, the gear backlash regime (from b/2 to–b/2) is marked by the dashed lines. First, vibro-impacts are indeed dependent upon the clutch designs,as shown in Figures 7 to 9. For example, severe vibro-impact is observed in the unloaded gear pair with clutchA. As shown in Figure 7(a) and (c), the double-sidedimpacts are seen on the unloaded gear pair.27,33,35 Also,the simulated peak-to-peak (P–P) accelerations on boththe engaged gear pair and the unloaded gear pair aremuch higher than the P–P accelerations with dampers,as shown in Figures 8 and 9. Table 4 compares the P–Paccelerations of three clutch damper designs. The P–Paccelerations with clutch A are about 770 m/s2 and4845 m/s2 for the engaged gear pair and the unloadedgear pair respectively. On the other hand, the P–P

accelerations with clutch B and clutch C are less than

4m/s2 and 0.6m/s2 for the engaged gear pair and the

unloaded gear pair respectively. Second, the vibro-

impact regimes with clutch A are clearly different. For

example, the dynamic TC passes through the pre-load

regime between the third and fourth stiffness values, as

shown in Figure 7(a). Thus, impulsive motions occur

when the dynamic TC is located between the pre-load

regime and the stiffness transition regime. When clutch

B or clutch C is examined, no dynamic torque is seen.Figures 10 to 12 show the results under the coast

condition. The results anticipate the dynamic motions

well. For example, the dynamic clutch torque

TC d1Pr, _d1Pr� �

is located on the negative side since the

mean effective torque has a negative value from equa-

tions (32) and (33) and Table 2. The relative motions of

the engaged gear pair agree well with the dynamic

direction since the engaged gear on the input shaft is

driven by the inertial torque. However, the unloaded

gear pair still remains under the driving condition as

the unloaded gear is always driven by the input shaft.

As expected from the previous linear analysis,

Table 3. Properties of three multi-staged clutch dampers.

Parameter Stage Value for the following multi-staged clutch dampers

A B C

Torsional stiffness kCi (linearized in a piecewise manner) (N m/rad) 1 10.1 500.2 9.02 61.8 854.3 28.13 595.8 – 281.04 1838.0 – 567.7

Hysteresis Hi (N m) 1 0.98 19.6 0.882 1.96 26.5 0.983 19.6 – 19.64 26.5 – 26.5

Transition angle fpi on the positive side (d1 . 0) (rad) 1 0.05 0 0.112 0.16 0.236 0.263 0.30 – 0.264 0.39 – 0.64

Transition angle fni on the negative side (d1 \ 0) (rad) 1 20.04 20.19 20.092 20.05 20.24 20.093 20.09 – 20.264 20.15 – 20.46

Pre-load (N m) Magnitude of the positive value – – 7.65 –Magnitude of the negative value – – 21.67 –Transition angle fPr – – 0 –

Table 4. Comparative evaluations of three clutch dampers for peak-to-peak accelerations for two engine conditions.

Engine condition and speed Peak-to-peak acceleration Value for the following clutch dampers

A B C

WOT at 1800 r/min Engaged gear (m/s2) 769.5 3.9 3.7Unloaded gear (m/s2) 4845.0 0.6 0.5Impact type Double-sided impact No impact No impact

Coast at 3000 r/min Engaged gear (m/s2) 1052.2 808.5 4.7Unloaded gear (m/s2) 5687.1 3961.1 0.7Impact type Double-sided impact Single-sided impact No impact

WOT: wide-open throttle.




Figure 10 clearly shows that the clutch torque is

impacting the stopper close to the transition angle.

Also, the unloaded gear pair shows double-sided

impacts. The results for clutch B and clutch C in

Figure 11 and Figure 12 respectively show that the

wide range of transition angles reduces vibro-impacts.

For example, the dynamic behaviors with clutch B and

clutch C exhibit only the single-sided impact regime

and the no-impact regime respectively. As seen in

Figures 11(b) and 12(b), clutch B has a stopper effect

similar to clutch A, but clutch C has a wider range of

transition angles, sufficient to avoid hitting the stopper.

In order to simulate the stopper effect, another clutch

stiffness value without a hysteresis level is employed at

the last stage of the transition angle for clutch A and

clutch B, such as 20.15 rad and 20.24 rad respectively.

Here, the stiffness values are 4 3 1838.0N m/rad as

used in the previous linear analysis. Table 4 compares

the P–P accelerations on the engaged gear pair and the

unloaded gear pair together with the impact type. The

P–P accelerations of clutch A are much higher thanthose of clutch B and clutch C for both the engagedgear pair and the unloaded gear pair.

Development of a five-degree-of-freedomlinear time-invariant system model

Figure 13 illustrates the five-degree-of-freedom (5DOF)torsional system model with the nonlinear elementsreduced from the 6DOF model shown in Figure 1.36

Here, Iie2 and IVE3 are the effective inertia of the inputshaft and the effective inertia of the vehicle respectively.kVE3 is the effective stiffness of the drive shaft. Also,TD1, TD2, and TD3 are defined as the drag torque on theinput shaft, the drag torque on the unloaded gear andthe drag torque from the vehicle respectively. In orderto reduce the number of degrees of freedom (DOFs),the following assumptions for the system are made:first, gear rattle does not occur on the engaged gear pairby considering the stable condition of the vehicle

Figure 7. Nonlinear model results for clutch A under the WOT condition at 1800 r/min: (a) clutch torque (or gear mesh forces)versus relative displacement; (b) relative motions of the engaged gear pair; (c) relative motions of the unloaded gear pair.

Yoon and Singh 1283



system; second, the minimum number of DOFs toretain the dynamic characteristics of the original sys-tem35 is 5. Thus, equation (2) is simplified on the basisof the LTI system model to

u5 = uf uh ui uou uVE3½ �T ð34Þ

M5 =diag If, Ih, Iie2, Iou, IVE3� �

ð35Þ

K5 =

kC �kC 0 0 0�kC kC + ki �ki 0 00 �ki ki + kgR

2iu+ kVE3 kgRiuRou �kVE3

0 0 kgRiuRou kgRou2 0

0 0 �kVE3 0 kVE3

266664

377775

ð36Þ

Iie2 = Iie +Rie

Roe

�2

IOG ð37aÞ

IVE3 =Rie

Roe

�2

IVE2 ð37bÞ

kVE3 =Rie

Roe

�2

kVE2 ð37cÞ

u5, M5, and K5 are the absolute displacement vector,the inertia matrix, and the stiffness matrix respectivelyof the 5DOF LTI system model.

Based upon the given nonlinear system model shownin Figure 13, the LTI system model can be consideredby linearizing the nonlinear elements as

If€uf tð Þ+ kCuf tð Þ � kCuh tð Þ=TE tð Þ ð38aÞTC d1ð Þ= kC uf tð Þ � uh tð Þ

� �ð38bÞ

Ih€uh tð Þ � kCuf tð Þ+ kC + kið Þuh tð Þ � kiui tð Þ=0 ð39ÞIie2€ui tð Þ � kiuh tð Þ+ ki + kVE3 + kgR

2iu

� �ui tð Þ

+ kgRiuRouuou tð Þ � kVE3uVE3 tð Þ= � TD1

ð40Þ

Iou€uou tð Þ+ kgRiuRouui tð Þ+ kgR2ouuou tð Þ=TD2 ð41Þ

IVE3€uVE3 tð Þ � kVE3ui tð Þ+ kVE3uVE3 tð Þ= � TD3

ð42Þ

Figure 8. Nonlinear model results for clutch B under the WOT condition at 1800 r/min: (a) clutch torque (or gear forces) versusrelative displacement; (b) relative motions of the engaged gear pair; (c) relative motions of the unloaded gear pair.




Here, the drag torques are estimated asTD1 =TDi + Rie=Roeð ÞTDo, TD2 =TDu, andTD3 = Rie=Roeð ÞTDVE2. Let the relative displacementsbe

d5 tð Þ= d51 tð Þ d52 tð Þ d53 tð Þ d54 tð Þ½ �T ð43aÞ

d51 tð Þ= uf tð Þ � uh tð Þ ð43bÞ

d52 tð Þ= uh tð Þ � ui tð Þ ð43cÞ

d53 tð Þ=Riuui tð Þ+Rouuou tð Þ ð43dÞ

d54 tð Þ= ui tð Þ � uVE3 tð Þ ð43eÞ

Here, d5 tð Þ is the relative displacement vector of the5DOF LTI system model, and d5i tð Þ (i = 1, 2, 3, 4) areits elements. Thus, the relative motions are

€d51 tð Þ+ kC1

If+

1

Ih

�d51 tð Þ � ki

Ihd52 tð Þ= 1

IfTE tð Þ

ð44Þ

€d52 tð Þ � kCIh

d51 tð Þ+ ki1

Ih+

1

Iie2

�d52 tð Þ � kgRiu

Iie2

d53 tð Þ � kVE3Iie2

d54 tð Þ= 1

Iie2TD1

ð45Þ

€d53 tð Þ � kiRiu

Iie2d52 tð Þ+ kg

R2iu

Iie2+

R2ou

Iou

�

d53 tð Þ+ kVE3Riu

Iie2d54 tð Þ= � Riu

Iie2TD1 +

Rou

IouTD2

ð46Þ

€d54 tð Þ � kiIie2

d52 tð Þ+ kgRiu

Iie2d53 tð Þ

+ kVE31

Iie2+

1

IVE3

�d54 tð Þ= � 1

Iie2TD1

+1

IVE3TD3

ð47Þ

Figure 9. Nonlinear model results for clutch C under the WOT condition at 1800 r/min: (a) clutch torque (or gear forces) versusrelative displacement; (b) relative motions of the engaged gear pair; (c) relative motions of the unloaded gear pair.

Yoon and Singh 1285



From equations (44) to (47), consider only the first har-monic term of the alternating torque ~TP1 e

ivot byassuming that the first harmonic is dominant comparedwith the higher harmonic terms. Thus, letTE tð Þ=TM + ~TP1 e

ivot, where TEðtÞ, TM, and ~TP1 arethe input torque, the mean torque, and the complex-valued alternating torque respectively of the 5DOF sys-tem model. Also €d5 tð Þ is given by

€d5 tð Þ+K�5d5 tð Þ=TM5 +TP5 tð Þ ð48aÞ

TM5 =

1IfTM

1Iie2

TD1

� Riu

Iie2TD1 +

Rou

IouTD2

� 1Iie2

TD1 +1

IVE3TD3

26664

37775 ð48cÞ

TP5 tð Þ=

1If

~TP1 eivot

000

2664

3775 ð48dÞ

K�5 =

kC1If+ 1

Ih

� ki

Ih0 0

� kCIh

ki1Ih+ 1

Iie2

� kgRiu

Iie2� kVE3

Iie2

0 � kiRiu

Iie2kg

R2iu

Iie2+

R2ou

Iou

kVE3Riu

Iie2

0 � kiIie2

kgRiu

Iie2kVE3

1Iie2

+ 1IVE3

26666664

37777775

ð48bÞ

Figure 10. Nonlinear model results for clutch A under the coast condition at 3000 r/min: (a) clutch torque (or gear forces) versusrelative displacement; (b) dynamic clutch torque at the stopper regime; (c) relative motions of the engaged gear pair; (d) relativemotions of the unloaded gear pair.




Here, TM5, TP5 tð Þ, and K�5 are the mean torque vector,the alternating torque vector, and the stiffness matrixrespectively of the 5DOF system model.

The relative motions d5 tð Þ are evaluated by dividingTM5 and TP5 tð Þ.

First, by considering the mean engine torque, let€d5 tð Þ= 0 and K�5dM5 0ð Þ=TM5, where dM5 0ð Þ is themean relative motion vector. Thus,

dM5 0ð Þ= K�5� ��1

TM5 ð49Þ

Second, the alternating relative motions are esti-mated from equation (48a) as

~dP5 = K�5 � v2oI5

� ��1TP5 ð50Þ

Here, ~dP5 and I5 are the alternating relative displace-ment vector and the identity matrix of the 5DOF sys-tem model respectively. Also, the mean relative motionand the alternating relative motion of the unloaded gearpair are denoted as dM53 and ~dP53 voð Þ respectively. This

analytical solution is based upon the LTI system modelby assuming that the gears in the unloaded gear pairare initially in contact with each other. Thus, the initialvalue of dM53 t0ð Þ= b=2 (= 0.05 mm) under the WOTcondition. Overall, the total mean value of dM53 isdefined as dM53t = dM53 t0ð Þ+ b=2. Therefore, the maxi-mum and minimum of d53 tð Þ are denoted as d53j jmax

and d53j jmin respectively. Their equations are expressedas

d53j jmax = dM53t + ~dP53 voð Þ�� ð51Þ

d53j jmin = dM53t � ~dP53 voð Þ�� ð52Þ

Thus, if d53j jmin . b=2, no impact is predicted.

Investigation of the gear rattle criteria

In this section, the gear rattle criteria will be investi-gated by employing several key parameters such as theclutch stiffness, the drag torque on the unloaded gear

Figure 11. Nonlinear model results for clutch B under the coast condition at 3000 r/min: (a) clutch torque (or gear forces) versusrelative displacement; (b) dynamic clutch torque at the stopper regime; (c) relative motions of the engaged gear pair; (d) relativemotions of the unloaded gear pair.

Yoon and Singh 1287



Figure 12. Nonlinear model results for clutch C under the coast condition at 3000 r/min: (a) clutch torque (or gear forces) versusrelative displacement; (b) dynamic clutch torque at the stopper regime; (c) relative motions of the engaged gear pair; (d) relativemotions of the unloaded gear pair.

Figure 13. 5DOF torsional system model for the case when the third gear is engaged and the fifth gear is unloaded, with nonlinearfunctions.




and the inertia values of the unloaded gear and fly-wheel, as well as the gear rattle behavior together withthe engine speeds. This has been investigated in a previ-ous study34 which has focused on neutral-gear rattlewith a 4DOF system model. In this study, the gear rat-tle criteria are advanced by using the 5DOF systemmodel with a focus on the WOT condition. The rele-vant parameters are employed to examine the vibro-impacts on the unloaded gear pair.34

Based upon equations (43) to (52), the gear rattlecriteria can be constructed by vibro-impacts versus rele-vant parameters (or engine speed). In this section, thegear rattle phenomenon is investigated with a focus onclutch A. Figures 14 to 18 illustrate the linear analysisof the gear rattle criteria by comparing the numericalresults on the unloaded gear pair. First, Figure 14shows the gear rattle behaviors for different excitationconditions. As shown in Figure 14(a), the value ofd53j jmin is higher than b/2 (= 0.05mm) when the enginespeed increases above 2500 r/min. This matches thevibro-impacts described in Table 5 well. Thus, the tor-sional system shows that gear rattle appears in fre-quency ranges less than 3000 r/min. However, noimpact is seen above 3000 r/min. Figure 14(b) and(c) shows the numerical results for 2500 r/min and3000 r/min respectively. Second, gear rattle can beexamined together with the clutch stiffness since thevibro-impacts are normally correlated with the clutchspring mode, as explained previously. Here, theemployed effective clutch stiffness value is kC4 =1838.0N m/rad. Figure 15(a) illustrates the ‘‘rattle’’

area and the ‘‘no-rattle’’ area together with the clutchstiffness. As seen in this relationship, gear rattle isremoved as the stiffness becomes compliant. Figure15(b) and (c) are the numerical results obtained byemploying two different stiffness values such as1124.1N m/rad and 843.1N m/rad. Third, Figure 16shows gear rattle with respect to the drag torque on theunloaded gear. As seen in Figure 16(a), the gear rattlephenomenon can be resolved by increasing the dragtorque on the unloaded gear. In order to predict theboundary value of the drag torque, an effective clutchstiffness value of 1292.8N m/rad is employed. Thenumerical analysis seen in Figure 16(b) and (c) predictsa gear rattle phenomenon well correlated with the rattlearea and the no-rattle area, as seen in Figure 16(a).Fourth, the relationship of gear rattle with the inertiaIou of the unloaded gear is examined in Figure 17. Asseen in Figure 17(a), gear rattle is removed when Iou isreduced. In this study, gear rattle can be resolved forvalues below 0.35Iou. This prediction matches the gearrattle criteria suggested in the previous study well.34 Inorder to predict the gear rattle criteria, a value of1292.8N m/rad is employed as the effective clutch stiff-ness. Finally, gear rattle can be examined with respectto the inertia If of the flywheel. Figure 18 shows thegear rattle criteria together with the inertia of the fly-wheel. Thus, as If is increased, gear rattle can beresolved. This is also found in the numerical results, asshown in Figure 18(b) and (c).

Overall, the torsional system with no impact can bedesigned by employing the appropriate properties of

Figure 14. Linear analysis of the gear rattle criteria together with the engine speed, showing (a) relative displacement versusengine speed with kC4 = 1838 N m/rad, (b) relative displacement for the unloaded gear pair in the time domain at 2500 r/min, and (c)relative displacement for the unloaded gear pair in the time domain at 3000 r/min: - - -, b/2; h—–, d53Mt; s��, d53j jmax ; – s� –, d53j jmin.

Yoon and Singh 1289



the relevant parameters, as examined in Figures 14 to18. This can be summarized as follows.34,35

1. The clutch stiffness values must be chosen to avoidthe area of the natural frequency corresponding tothe engine speed.

2. The compliant stiffness must be designed.3. A higher value of the drag torque can remove the

vibro-impact.

4. The inertia of the unloaded gear must be reduced.5. The inertia of the flywheel must be increased.

Conclusion

This article investigated the rattle phenomenon in themanual transmission with specific clutch types underboth the WOT condition and the coast condition. In

Figure 15. Linear analysis of the gear rattle criteria together with the clutch stiffness, showing (a) relative displacement versusclutch stiffness, (b) relative displacement for the unloaded gear pair in the time domain with kC4 = 1124 N m/rad, and (c) relativedisplacement for the unloaded gear pair in the time domain with kC4 = 843 N m/rad: - - -, b/2; h—–, d53Mt; s��, d53j jmax ; – s� –, d53j jmin.

Figure 16. Linear analysis of the gear rattle criteria together with the drag torque with kC4 = 1293 N m/rad, showing (a) relativedisplacement versus ratio of TDu, (b) relative displacement for the unloaded gear pair in the time domain with 2TDu, and (c) relativedisplacement for the unloaded gear pair in the time domain with 3TDu: - - -, b/2; ——, d53Mt t; � � � � � �, d53j jmax ; - � -, d53j jmin.




Figure 17. Linear analysis of the gear rattle criteria together with the inertia of the unloaded gear with kC4 = 1293vN m/rad,showing (a) relative displacement versus ratio of Iou, (b) relative displacement for the unloaded gear pair in the time domain with0.4Iou, and (c) relative displacement for the unloaded gear pair in the time domain with 0.3Iou: - - -, b/2; ——, d53Mt; � � � � � �, d53j jmax ;- � -, d53j jmin.

Figure 18. Linear analysis of the gear rattle criteria together with the inertia of the flywheel with kC4 = 1124 N m/rad, showing (a)relative displacement versus ratio of If, (b) relative displacement for the unloaded gear pair in the time domain with 1.4If, and (c)relative displacement for the unloaded gear pair in the time domain with 1.5If : - - -, b/2; ——, d53Mt; � � � � � �, d53j jmax ; - � -, d53j jmin.

Yoon and Singh 1291



order to examine the linear and the nonlinear dynamiccharacteristics, a real physical driveline was examinedand modeled by employing real-life multi-staged clutchdampers. The contribution of this article is summarizedas follows. First, the mathematical model for the multi-staged clutch damper was developed. This modeladvanced previous studies27,33 which have focused ononly a two-staged symmetric clutch damper. Thus,three real-life clutch dampers were implemented toexamine the torsional vibro-impacts. Second, drag tor-que estimation techniques under both the WOT condi-tion and the coast condition were suggested byassuming that the vehicle is in the steady-state condi-tion. Thus, the vibro-impacts under severe vehicle driv-ing conditions can be simulated by employing thesuggested drag torque models. Finally, the gear rattlecriteria were constructed on the basis of the 5DOF lin-ear system model. This has advanced the previousstudy34 by showing the relationship of the gear toothcontact point to key parameters in the torsional system.Also, the effects of multi-staged clutch designs wereinvestigated, which leads to a better understanding ofthe roles of the transitional angles, the pre-loads, andthe stiffness properties of the clutch damper.

Declaration of conflicting interest

The authors declare that there is no conflict of interest.

Funding

A portion of this work was supported by the SmartVehicle Concepts Center41 (grant number ) and theNational Science Foundation, Industry & UniversityCooperative Research Program42 (0732517).

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Peak-to-peakacceleration

Enginespeed (r/min)

Value for the following clutch dampers

A B C

Engagedgear (m/s2)

1500 310.2 274.8 213.01800 769.3 3.9 3.72000 789.0 4.6 4.22500 381.4 194.0 45.53000 6.8 5.3 5.03500 5.0 4.1 3.9

Unloadedgear (m/s2)

1500 2284.0 (double-sided impact) 1950.1 (double-sided impact) 1317.6 (double-sided impact)1800 4842.9 (double-sided impact) 0.6 (no impact) 0.5 (no impact)2000 5034.5 (double-sided impact) 0.7 (no impact) 0.6 (no impact)2500 2056.8 (single-sided impact) 931.4 (single-sided impact) 251.5 (single-sided impact)3000 1.1 (no impact) 0.8 (no impact) 0.8 (no impact)3500 0.7 (no impact) 0.6 (no impact) 0.6 (no impact)




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Appendix 1

Notation

b backlashc viscous damping coefficientC viscous damping matrixF forceH hysteresis levelI inertiaI identity matrixk torsional stiffnessK stiffness matrixM inertia matrixP transformation matrixR gear radiusS state matrixt time historyT torqueT torque vectorV vehicle speedW weightx state vector for absolute displacementsxr state vector for relative displacements

d relative displacementd relative displacement vectoru absolute displacementu absolute displacement vectorm coefficient of frictionr relative displacement between the gears in

the gear pair

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s smoothening factorf transition anglev excitation frequencyV angular velocity

Subscripts

c coast conditionC clutchD drag torquee equivalent value or engaged gearE enginef flywheelg gear meshh clutch hubH hysteresisi input shaftM mean valuen negative sidenon nonlinear functiono output shaft (or fundamental harmonic

term)

OG output shaft for the 6DOF modelp positive side (or alternating part)Pr pre-loadr relative terms static condition or normalized values1 dynamic conditionS stiffness valueu unloaded gearv vehiclevr vehicle on the roadVE2 vehicle with a six-degree-of-freedom

torsional systemVE3 vehicle with a five-degree-of-freedom

torsional system

Superscripts

~ complex value+ pseudo-inverse value* transformed value




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