Proc IMechE Part D: Analysis of vibration amplification in a ......1408 Proc IMechE Part D: J Automobile Engineering 229(10) Downloaded from pid.sagepub.com at OHIO STATE UNIVERSITY

Original Article

Proc IMechE Part D:J Automobile Engineering2015, Vol. 229(10) 1406–1418� IMechE 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0954407014563362pid.sagepub.com

Analysis of vibration amplification in amulti-staged clutch damper duringengine start-up

Laihang Li and Rajendra Singh

AbstractTransient vibration amplifications of torsional systems passing through critical speeds have been of interest for a consid-erable amount of time. However, previous investigations on the piecewise linear system have focused mostly on numeri-cal methods, and thus a reliable analytical method is not available for predicting transient amplification events. Thisarticle overcomes this void by developing and utilizing the closed-form solution of a linear single-degree-of-freedom tor-sional system, given a motion input under a constant acceleration rate, to approximate the transient responses of a pie-cewise linear system. This system represents a simplified vehicle powertrain system with a multi-staged clutch damperduring the engine start-up process under an instantaneous motion input from the flywheel. First, the utility of a single-degree-of-freedom system and the motion input for the start-up process are experimentally and numerically illustratedby vehicle start-up measurements. Second, a closed-form solution of a linear damped torsional oscillator, giveninstantaneous-frequency excitation, is successfully developed and numerically verified. Finally, the proposed analyticalsolution of a linear system is utilized to predict the approximate peak-to-peak value of the displacement of a piecewiselinear system during transient amplification for a rapid variation in speed.

KeywordsEngine start-up, speed-dependent vibration amplification, driveline transients, analytical methods

Date received: 25 April 2014; accepted: 17 November 2014

Introduction

The speed-dependent behavior of linear and nonlineartorsional systems passing through critical speeds hasbeen of interest for several decades.1–9 These transientvibrations result from a nonstationary process whichhas time-varying or instantaneous excitation frequen-cies. This concept was first introduced by Lewis1 in1932 where an approximate solution for a single-degree-of-freedom (SDOF) linear system under a uni-form acceleration rate was proposed. Several numericalstudies have been conducted to investigate variousrotor dynamics problems.2–8 In particular, Newland9

stated that an analytical solution is not possible, andthus he numerically examined the amplification for anSDOF linear model for both acceleration cases anddeceleration cases with a uniform rate. The responseamplitudes under an instantaneous-frequency excita-tion were compared with the steady-state harmonicresponse, and the frequencies corresponding to peakresponse amplitudes were identified in a book byNewland.9 Yet another practical problem where such

transient vibration judder responses are observed is inthe braking systems of vehicles.10–13 In a recent study,Sen et al.12 have experimentally found resonant amplifi-cation in a torsional system, with clearance nonlinearityand with a uniform deceleration rate. Similar vibrationamplification problems are seen during the engine start-up process; such powertrain torsional systems usuallyinclude a nonlinear multi-staged clutch damper.14 Thevibration mode corresponding to transient amplifica-tion is usually controlled by the clutch damper whoseproperties may be approximated by piecewise linearstiffness and damping components.15,16 However, even

Acoustics and Dynamics Laboratory, NSF Smart Vehicle Concepts Center

Department of Mechanical and Aerospace Engineering, The Ohio State

University, Columbus, Ohio, USA

Corresponding author:

Rajendra Singh, Acoustics and Dynamics Laboratory, NSF Smart Vehicle

Concepts Center Department of Mechanical and Aerospace Engineering,

The Ohio State University, Columbus, Ohio 43210, USA.

Email: [email protected]

at OHIO STATE UNIVERSITY LIBRARY on August 24, 2015pid.sagepub.comDownloaded from

http://pid.sagepub.com/

the latest studies regarding the transient torsionalamplification in vehicle driveline systems focusedmainly on numerical methods.17–20 An analyticalmethod for predicting the transient amplification levelis not available.

The characteristics of a piecewise linear system for astationary process have been usually examined in thefrequency domain by using a numerical integrationtechnique or semi-analytical methods such as the har-monic balance method, the describing function method,and the stochastic linearization method.21–23

Computational or approximate analytical methodshave been utilized to study vibro-impacts in the timedomain.24 For a nonstationary process, numerical orexperimental methods have been applied to a piecewiselinear system,25–28 but their behavior has yet to be fullyunderstood. To overcome such a void in the literature,this article proposes to find analytically a closed-formsolution of a linear torsional oscillator (an engine–fly-wheel–clutch-damper–transmission system) with amotion input (from flywheel) and to approximate thetransient amplification level of a piecewise linear sys-tem under a high acceleration rate. The practical prob-lem of interest is the transient vibration amplificationassociated with the engine start-up.14

Problem formulation

The first torsional mode of vehicle powertrain systems(say, from 6Hz to 15Hz) is often dominant during theengine start-up;14,27,28 it is usually controlled by a dryclutch damper. Note that, during the engine start-up,the downstream driveline subsystem consisting of thetransmission main shaft, the propeller shaft, the differ-ential, and the axles is decoupled. Therefore, the focusis the powertrain subsystem up to a lumped transmis-sion in which the flywheel is the dominant inertiacomponent. Accordingly, an engine–flywheel–clutch–transmission system could be described by a linearfour-degree-of-freedom (4DOF) semidefinite system asdisplayed in Figure 1(a), where I1, I2, I3, and I4 repre-sent the inertias (kg m2) of the engine, the flywheel, theclutch, and the lumped transmission respectively. Theconstant torque TD (N m) provides the lubricant-induced drag in the transmission box. The torque TE(t)(N m) for a multi-cylinder internal-combustion engineis expressed via a Fourier series29,30 as

TE(t)=Tm +Xn

Tan sin n O0 +12at

� �t+cn

� �ð1Þ

where Tm (N m) and Tan (N m) are the mean torqueand the alternating torque of nth order respectively.Here, O0 +at is the mean crankshaft speed (rad/s)which is accelerated with a constant rate a (rad/s2), andn O0 +atð Þ is the instantaneous firing frequency (rad/s).Note that the numerical values of O0 +at and a areusually given in units of r/min and (r/min)/s respec-tively. In general, the dominant firing order ndom is

calculated as ndom= ncyl�2, where ncyl is the number of

cylinders. Since internal-combustion engines are cur-rently downsized to enhance the fuel consumption,such as in hybrid electrical vehicles, a two-cylinder gas-oline engine is used as an example and, thus, ndom is 1.For clarity, only the dominant order is considered firstand, thus, equation (1) is rewritten as

TE(t)=Tm +Ta sin O0 +12 at

� �t

� �ð2Þ

where c is assumed to be zero without losing anygenerality.

Because of the massive torsional inertia of aflywheel, u2 tð Þ and _u2 tð Þ are assumed to be unaffectedby u3 tð Þ and _u3 tð Þ; this suggests that u2 tð Þ and _u2 tð Þmay be used as motion excitation terms for a three-degree-of-freedom (3DOF) semidefinite system, asshown in Figure 1(b). Further, since only the clutchmode is of interest, I2 and I3 are lumped together, andthe SDOF definite system is formed as depicted inFigure 1(c).

The chief objectives of this article are as follows:

1. to compare the transient vibration amplificationlevel between the linear and piecewise linear tor-sional systems describing the engine–flywheel–clutch-damper–transmission system using a numer-ical method (here the clutch damper is modeled bya dual-staged spring and viscous dampingelements);

2. to develop and verify a new closed-form solutionfor a linear damped torsional oscillator, given aninstantaneous frequency (speed) input;

3. to utilize the proposed closed-form solution toapproximate the transient amplification of a piece-wise linear torsional oscillator (limited experimen-tal validation is included).

This study will also extend the work of Sen et al.13

who considered the torque input with an instantaneousfrequency; this article will consider motion excitationwith both instantaneous frequency (speed) and instan-taneous amplitude at various orders of the speed.

Comparison of linear torsional models andexperimental validation

In order to illustrate the utility of an SDOF system thatis excited by flywheel motion, the torsional natural fre-quencies of three linear semidefinite systems are firstcalculated. The first clutch mode frequency fclutch isfound as follows: 13.9Hz for the 4DOF semidefinitesystem, 13.5Hz for the 3DOF semidefinite system, and13.5Hz for the 2DOF semidefinite system. The corre-sponding (normalized) modes at fclutch for three modelsare compared as follows: [0.01 1 0.99 0.99]T for the4DOF semidefinite system, [0 0.99 1]T for the 3DOFsystem, and [0 1]T for the 2DOF system where thesuperscript T indicates the transpose of a column

Singh and Li 1407



vector. The natural frequency calculations suggest thatthe 2DOF semidefinite system or the SDOF definitesystem can fully represent the clutch mode of the4DOF semidefinite system with focus on the relativemotion u23 tð Þ= u3 tð Þ � u2 tð Þ.

The transient responses are numerically examinednext with a=15 (r/min)/s, Tm =27:4 N m, Ta =300:0N m, TD = � 20 N m, and O0 =400 r/min.

First, the absolute velocities _u2 tð Þ of the flywheeland the absolute velocities _u3 tð Þ of the clutch hub ofthe 4DOF system are compared. As shown in Figure 2,both _u2 tð Þ and _u3 tð Þ have a rotational component (themean crankshaft speed O0 +at), which is acceleratedfrom 400 r/min to 1000 r/min. However, _u3 tð Þ exhibits atransient amplification when the clutch mode is excited

by the instantaneous firing frequency O0 +at; conver-sely, _u2 tð Þ has only a bare minimum amplificationwhich could be neglected. This implies that the flywheelmotion may be used as a system excitation.

Second, the u23 tð Þ values from these three systemsare compared. The flywheel responses u2 tð Þ and _u2 tð Þ ofthe 4DOF semidefinite system are numerically recordedduring the calculation process, and then the u2 tð Þ and_u2 tð Þ time histories are directly applied as the motioninput to a 2DOF (or an SDOF) system. Figure 3 showsthat u23 tð Þ from an SDOF model is sufficiently close tou23 tð Þ from a 4DOF system. Also, both a 4DOF systemmodel and an SDOF system model yield similaranswers for _u23 or €u23; these are not included here,however.

Figure 1. Powertrain system (consisting of an engine–flywheel, a clutch damper and a transmission) described by three lineartorsional models: (a) a 4DOF semidefinite system under an engine torque input TE (t); (b) a 3DOF semidefinite system underflywheel motion inputs u2 tð Þ and _u2 tð Þ as obtained from (a); (c) a two-degree-of-freedom (2DOF) semidefinite (or SDOF definite)system under the flywheel motion inputs u2 tð Þ and _u2 tð Þ as obtained from (a).

1408 Proc IMechE Part D: J Automobile Engineering 229(10)



Finally, a vehicle start-up experiment is conductedon a medium-duty pickup truck with a four-cylindergasoline engine and a six-speed manual transmission.The schematic diagram of the test rig is illustrated inFigure 4; other downstream driveline componentsincluding the propeller shaft, the axles, and the differ-ential are not shown as they are decoupled during theengine start-up. The absolute torsional velocities _u2 tð Þand _u3 tð Þ (as measured by two magnetic sensors) andkey parameters (estimated from the physical dimen-sions or vehicle measurements) are then applied to themodels in Figure 1(b) and (c). Experimental measure-ment of _u23 p�p (the peak-to-peak value of _u23 tð Þ) iscompared with the corresponding predictions from lin-ear models. The comparisons in Table 1 further justifythe utility of an SDOF powertrain system with aninstantaneous motion input (from the flywheel).

Nonlinear torsional model (with piecewiselinear clutch damper)

A generic piecewise linear torsional oscillator problemis formulated with the angular motion inputs f tð Þ and_f tð Þ, as shown in Figure 5(a). Given the numericalsimulation of _u2 tð Þ and available measurement of theflywheel speed during the engine start-up process,28 theanalytical forms of f tð Þ and _f tð Þ are assumed to be

_f(t)= _fr(t)+_fa(t) ð3aÞ

_fr(t)=O0 +at ð3bÞ

_fa(t)=XNn=1

gnn O0 +atð Þ cos nO0t+n

2at2 +un

� �ð3cÞ

f(t)=fr(t)+fa(t) ð4aÞ

fr(t)=O0t+12at

2 ð4bÞ

fa(t)=XNn=1

gn sin nO0t+nat2

2+un

� �ð4cÞ

It is assumed that O0 +at� gnn O0 +atð Þ orgn � 1rad since the speed fluctuations _fa(t) are usuallymuch smaller than the mean speed _fr(t);

28–30 these arealso found in the _u2 tð Þ calculation. In addition, the gn

values are selected on the basis of the classical paper byPorter30 which provided dimensionless Fourier coeffi-cients for a multi-cylinder internal-combustion engine.

The real-life clutch dampers usually include multi-staged elastic and dissipative properties.15,16 For simpli-city, this component is approximated in this article byonly elements with a two-staged piecewise linear stiff-ness K(d(t)) and piecewise linear damping C(d(t)), asdisplayed in Figure 5(b) and (c). Here, assume thatK=v2

1 and C=2zv1 for the normalized unity tor-sional inertia, where z is the viscous damping ratio andu(t) is the angular displacement. Further, v1 is selectedto be 82.9 rad/s (13.2Hz) to represent the clutch mode.

Figure 3. Speed-dependent relative displacement between theflywheel and the clutch hub of three powertrain system modelswith a = 15 (r/min)/s: (a) 4DOF semidefinite system (Figure1(a)); (b) 3DOF semidefinite system (Figure 1(b)); (c) 2DOFsemidefinite (or SDOF definite) system (Figure 1(c)).rpm: r/min.

Figure 2. Speed-dependent absolute velocity from the linear4DOF semidefinite system (consisting of an engine–flywheel, aclutch damper and a transmission as shown in Figure 1(a))described by three linear torsional models and under an enginetorque input TE (t) with a = 15 (r/min)/s: (a) flywheel velocity; (b)clutch hub velocity.rpm: r/min.

Singh and Li 1409



The governing equation of the nonlinear SDOF systemis given as

€u(t)+T(d(t), _d(t))=TD ð5Þ

where the relative angular displacementd(t)= u(t)� f(t) is of interest.

The nonlinear torque T(d(t), _d(t)) is given as

T(d(t), _d(t))

=

hKx +K½d(t)� x�+C _d(t), d(t). x

hKd(t)+mC _d(t), � x4d(t)4x

�hKx +K½d(t)+ x�+C _d(t), d(t)\ � x

8><>:

ð6Þ

where the transition angle is x (rad), and m and h arethe dimensionless damping ratio and the dimensionlessstiffness ratio respectively between stage I and stage II.

The transition angle x is assumed to be 0.5 rad, andthe m or h range is given as 04m=h41 in this study.Equation (6) is rewritten as

T(d(t))=Kd(t)+ (1� h)Kd(t)� xj j � d(t)+ xj j

2+C _d(t)+ (1� m)C _d(t)

sgn d(t)� x½ � � sgn d(t)+ x½ �2

=Kd(t)+ (1� h)Ktanh s d(t)� x½ �f g d(t)� x½ � � tanh s d(t)+ x½ �f g d(t)+ x½ �

2

+C _d(t)+ (1� m)C _d(t)tanh s d(t)� x½ �f g � tanh s d(t)+ x½ �f g

2

ð7Þ

where sgn d(t)6x½ � is the sign function. The discontinu-ity in equation (7) is smoothened using the hyperbolicfunction tanh s d(t)6x½ �f g where s (say, 106–109) is aregularizing factor.28 A numerical method is utilized tosolve equation (5) with a single order (n=1) as anexample. Transient responses of a piecewise linearpowertrain system (with h=m=0:3) are examined,given v1 =13:2Hz=792 r/min, a=5 (r/min)/s,O0 =400 r/min, z =0:001, g1 =0:01, and u1 =0:Since the drag torque TD affects the operating point ofa piecewise linear system, TD is adjusted to ensure thatthe operating point (red full circle in Figure 5(b)) islocated in the transition region from stage II to stage Iso as to activate both stages. As shown in Figure 6,d tð Þ, _d tð Þ, and €d tð Þ exhibit similar transient amplifica-tion trends between 700 r/min and 800 r/min. Thisimplies that only dp�p (rad) may be used to evaluate theresonant amplification level, where the subscript p–pindicates the peak-to-peak value of d (t). Further, rapidvariations in the speed are considered with a up to175 r/min/s. As shown in Figure 7, dp�p for two cases

Figure 4. Schematic diagram of the start-up experiment for a medium-duty truck with a four-cylinder gasoline engine and a six-speed manual transmission.

Table 1. Comparison between the _u23 p�p value (the peak-to-peak value of _u23(t)) for the vehicle measurement (during the enginestart-up process) and the predictions of the linear models of the engine–flywheel, the clutch damper and the transmission torsionalsystem.

Parameter (units) Value for the following

Experiment Linear 3DOF semidefinite model Linear SDOF definite model

_u23 p�p(r=min) 513 526 525

SDOF: single-degree-of-freedom.




h=m=0:3 and h=m=1:0 are compared. It is clearthat, as a increases up to 25 r/min/s, the dp�p value of alinear system (h=m=1:0) is very close to the dp�pvalue of a nonlinear system. Accordingly, it is reason-able to assume that a linear torsional oscillator may be

utilized to approximate a nonlinear system when0:3\ h=m \ 1:0.

Analytical solution for a damped lineartorsional oscillator

An analytical solution of the linear damped system isfirst sought by setting h=m=1 to give

€u(t)+2zv1_u(t)+v2

1u(t)=2zv1_f(t)+v2

1f(t) ð8Þ

Since the operating point of the linear system doesnot affect dp�p, TD is assumed to be zero for conveni-ence. Also, u(t) is divided into two parts: one is therotational part ur(t) which is caused by fr tð Þ, and thesecond is the alternating part ua(t) which is induced byfa tð Þ. Then, ur(t) is solved as

€ur(t)+v21ur(t)+2zv1

_ur(t)

=v21 nO0t+

n

2at2

� �+2zv1 nO0 + natð Þ

ð9Þ

First, the system is assumed to rotate with a con-stant speed O0 and, thus, the initial conditions are asfollows: ur(0)=0 rad, _ur(0)=O0, ua(0)=0 rad, and_ua(0)=0 rad/s. Then, ur(t) is found by independentlysolving equation (9) with the assumed initial condi-tions. The general response in the Laplace domain31 toan arbitrary excitation T(t) with the initial conditionsu(0)= u0 and _u(0)= v0 is found to be

Q(s)=T(s)

s2 +2zv1 +v21

+s+2zv1

s2 +2zv1 +v21

u0

+1

s2 +2zv1 +v21

v0

ð10Þ

Let

T(s)=L v21 nO0t+

n

2at2

� �+2zv1 nO0 + natð Þ

h ið11Þ

Figure 6. Speed-dependent relative motion of the piecewiselinear SDOF system (Figure 5(a)) with h = m = 0.3: (a) d tð Þ; (b)_d tð Þ; (c) €d tð Þ.

Figure 5. Powertrain torsional system (piecewise linear SDOFsystem): (a) schematic diagram of the piecewise system underthe motion input; (b) piecewise linear stiffness K(d(t)) for theclutch damper where d(t) = u(t)� f(t) and K = v2

1(d, operatingpoint); (c) piecewise linear damping C(d(t)) for the clutchdamper where C = 2zv1. Here, stage I is given by �x, x½ �, andstage II is characterized by �‘, � xð Þ [ x, + ‘ð Þ.

Figure 7. Comparison of the peak-to-peak values dp-p from thelinear SDOF system and the nonlinear SDOF system (Figure5(a)): , h = m = 0:3; , h = m = 1:0.

Singh and Li 1411



Now employ the inverse Laplace transformationtogether with the convolution theorem to yield

ur(t)=O0t+12a

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

qt2 ð12Þ

Then, ua(t) is solved as

€ua(t)+v21ua(t)+2zv1

_ua(t)

=2zv1

Xn

gn(nO0 + nat) cos nO0t+n

2at2 +un

� �" #

+v21

Xn

gn sin nO0t+n

2at2 +un

� �" #

ð13Þ

Like equation (12), the convolution theorem is againemployed to yield

ua(t)=L�11

s2 +2zv1s+v21

�(2zv1

Xn

gn nO0 + natð Þ cos nO0t+n

2at2 +un

� �" #

+v21

Xn

gn sin nO0t+n

2at2 +un

� �" #)ð14Þ

where * represents the convolution product. With theorder of integration and summation switched, equation(14) is rewritten as

ua(t)=Xn

ua An+ ua Bnð Þ ð15Þ

ua An(t)=v21gn

vd

ðt0

sin vd(t� u)½ � e�zv1(t�u)

sin nO0u+n

2au2 +un

� �du

ð16Þ

ua Bn(t)=2zv1gn

vd

ðt0

sin vd(t� u)½ � e�zv1(t�u)

nO0 + nauð Þ cos nO0u+n

2au2 +un

� �du ð17Þ

where vd1 =v1


p=vd is the damped natural

frequency. First, equation (16) is solved by employingthe trigonometric identities and is expanded into twoterms as

ua An(t)= ua An 1(t)+ ua An 2(t) ð18Þ

ua An 1(t)= � v21gn

2vdðt0

cosn

2au2 + nO0 � vdð Þu+vdt+un

h ie�zv1(t�u) du

ð19Þ

ua An 2(t)=v21gn

2vdðt0

cosn

2au2 + nO0 +vdð Þu� vdt+un

h ie�zv1(t�u) du

ð20Þ

Second, the general solution of an indefinite integralof the product of cos (ax2 + bx+ c) and ef(g�x) overdomain x is given as

ðcos ax2 + bx+ c� �

ef(g�x) dx

=Re

ðei(ax

2 + bx+ c) ef(g�x) dx

=Re

ðei(ax

2 + bx+ c)+ fg�fx dx

ð21Þ

Re

ðei(ax

2 + bx+ c)+ fg�fx dx

=Re �

ffiffiffiffiffiffiffi�14p ffiffiffiffi

pp

e�i(b+if)2=4a+ ci+ fgerf (� 1)3=4b.2ffiffiffiap�

ffiffiffiffiffiffiffi�14p

f�2ffiffiffiap

+(� 1)3=4ffiffiffiap

xh i

2ffiffiffiap

8<:

9=;

ð22Þ

where theÐcosx dx=Re

Ðeix dx

� �relationship is utilized32 and where Re is the real part of a complex quantity.

Based upon the above solution, ua An(t) is calculated by separately solving ua An 1(t) and ua An 2(t) according to

ua An 1(t)= � v21gn

2vdRe �


pp

e�i(nO0�vd�zv1i)2=2na+(vdt+un)i�zv1t ERFAn 1(t)� ERFAn 1(0)½ �

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffin=2ð Þa

p( )

ð23Þ

where erf(x)= 2=ffiffiffiffipp

ð ÞÐ x0 e�t

2

dt is the error function32 and where

ERFAn 1(u)= erf(� 1)3=4 nO0 � vdð Þ


p �ffiffiffiffiffiffiffi�14p

�zv1ð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffin=2ð Þa

p +(� 1)3=4ffiffiffiffiffiffiffin

2a

ru

" #ð24Þ




ua An 2(t)=v21gn

2vd

Re �ffiffiffiffiffiffiffi�14p ffiffiffiffi

pp

e�i(nO0 +vd�zv1i)2=2na+(�vdt+un)i�zv1t ERFAn 2(t)� ERFAn 2(0)½ �


p( )

ð25Þ

ERFAn 2(u)= erf(� 1)3=4 nO0 +vdð Þ


p �ffiffiffiffiffiffiffi�14p

�zv1ð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffin=2ð Þa

p +(� 1)3=4ffiffiffiffiffiffiffin

2a

ru

" #ð26Þ

Then, ua An(t) is found as

ua An(t)= ua An 1(t)+ ua An 2(t)

= � v21gn

2vdRe �


pp

e�i(nO0�vd�zv1i)2=2na+(vdt+un)i�zv1t ERFAn 1(t)� ERFAn 1(0)½ �


p( )

+v21gn

2vdRe �


pp

e�i(nO0 +vd�zv1i)2=2na+(�vdt+un)i�zv1t ERFAn 2(t)� ERFAn 2(0)½ �


p( ) ð27Þ

Further, ua Bn(t) is expanded as

ua Bn(t)= ua Bn 1(t)+ ua Bn 2(t) ð28Þ

ua Bn 1(t)=zv1gn

vdðt0

sinn

2au2 + nO0 � vdð Þu+vdt+un

h in o

(nO0 + nau) e�zv1(t�u) du

ð29Þ

ua Bn 2(t)=zv1gn

vdðt0

sin � n

2au2 � nO0 +vdð Þu+vdt� un

h in o

(nO0 + nau) e�zv1(t�u) du

ð30Þ

The general solution of the indefinite integral of theproduct of sin (ax2 + bx+ c), ef(g�x) and h+ kx overdomain x is given byð

sin (ax2 + bx+ c) ef(g�x) (h+ kx) dx

=Im

ðei(ax

2 + bx+ c) ef(g�x) (h+ kx) dx

� �

=Im

ðei(ax

2 + bx+ c)+ fg�fx (h+ kx) dx

� �

=Im

e(f

2�b2)i=4a+i bx+ cð Þ+ f g�xð Þ

4 aið Þ3=2�2ffiffiffiffiaip

k e i2 4a2x2 + b2ð Þ+ f2½ �=4ai �ffiffiffiffipp

ebf=2a+x(f�bi)

erfif� i(2ax+ b)

2ffiffiffiffiaip

� �2ah� bkð Þi+ fk½ �

�!ð31Þ

where erfi(x)= � ierf(ix) is the imaginary error func-tion and the

Ðsinx dx=Im

Ðeix dx

� �relationship is

employed, and where Im is the imaginary part of a com-plex variable.

Based on the above solution, ua Bn 1(t) and ua Bn 2(t)are calculated as

ua Bn 1(t)=zv1gn

vd

Im � 1

4 na=2ð Þi½ �3=2Fa Bn 1(t)�Fa Bn 1(0)½ �

( ) ð32Þ

Fa Bn 1(u)=FIa Bn 1(u) FII

a Bn 1(u)+FIIIa Bn 1(u)

� �ð33Þ

FIa Bn 1(u)

=e�(nO0�vd)2i=2na�(�zv1)

2=2nai+ (nO0�vd)iu+(vdt+un)i+ (�zv1) t�uð Þ

ð34Þ

FIIa Bn 1(u)

= � 2

ffiffiffiffiffiffiffiffina

2i

rna e i2 nað Þ2u2 + nO0�vdð Þ2½ �+ �zvð Þ2f g=2nai

ð35Þ

FIIIa Bn 1(u)= e�zv1 nO0�vdð Þ=na+ u �zv1� nO0�vdð Þi½ �

ffiffiffiffipp

na �zv1 +vdið Þ erfi �zv1 � i nau+ nO0 � vdð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffina=2ð Þi

p" #

ð36Þ

ua Bn 2(t)=zv1gn

vd

Im � 1

4 �na=2ð Þi½ �3=2Fa Bn 2(t)�Fa Bn 2(0)½ �

( )

ð37Þ

Fa Bn 2(u)=FIa Bn 2(u) FII

a Bn 2(u)+FIIIa Bn 2(u)

� �ð38Þ

FIa Bn 2(u)

= e(�nO0�vd)2i=2na+(�zv1)

2=2nai+ (�nO0�vd)iu+(vdt�un)i+ (�zv1) t�uð Þ

ð39Þ

Singh and Li 1413



FIIa Bn 2(u)

= � 2

ffiffiffiffiffiffiffiffiffiffiffiffi�na

2i

rna e i2 �nað Þ2u2 + �nO0�vdð Þ2½ �+ �zv1ð Þ2f g= �2naið Þ

ð40Þ

FIIIa Bn 2(u)

= ezv1 �nO0�vdð Þ=na+ u �zv1 + nO0 +vdð Þi½ �

ffiffiffiffipp

na �zv1 +vdið Þ erfi �zv1 + i nau+ nO0 +vdð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�na=2ð Þi

p" #

ð41Þ

Finally, ua Bn(t) is found by combining ua Bn 1(t) andua Bn 2(t) as

ua Bn(t)

=zv1gn

vdIm

�14 na=2ð Þi½ �3=2

Fa Bn 1(t)�Fa Bn 1(0)+Fa Bn 2(t)�Fa Bn 2(0)½ �( )

ð42Þ

Then, ua(t) is found by summing ua An(t) and ua Bn(t)over all orders, and u(t) is finally determined by com-bining ur(t) and ua(t) according to

ua(t)=Xn

ua An(t)+ ua Bn(t)½ � ð43Þ

u(t) = ur(t)+ ua(t)

=O0t+12a


pt2 +

Pn

ua An(t)+ ua Bn(t)½ �

ð44Þ

The analytical solution of d(t) is then derived fromequation (44) as

d(t)= u(t)� u(t)

= O0t+1

2a


qt2 +

Xn

ua An(t)+ ua Bn(t)½ �( )

� O0t+1

2at2 +

Xn

gn sin nO0t+n

2at2 +un

� �h i( )

ð45Þ

Then, since z and gn assume small values, equation (45)is further simplified to

d(t)’Xn

ua An(t)+ ua Bn(t)½ � ð46Þ

Much simpler expressions of the closed-form solu-tions of u (t) and d (t) are found for an undamped

SDOF system; these are summarized in Appendix 1 forcompleteness.

Verification of analytical solution

Both single-order (n=1) and multi-order (n=1.5, 3,4.5, 6, 7.5, 9) excitation cases are examined with a=5(r/min)/s and z =0:001 so as to verify the new analyti-cal solution. The solutions of d(t) for n=1 are dis-played in Figure 8 by using the parameters in thesection entitled ‘Nonlinear torsional model (with piece-wise linear clutch damper)’; a reasonable matchbetween theory and computation verifies equation (44).Resonant amplifications from both solutions begin atOs (760 r/min) and reach the same peak-to-peak ampli-

fication value at Op (805 r/min). Note that Op

(805 r/min) is slightly higher than the critical speed Oc

(792 r/min).The multi-order excitation case is examined next for

a six-cylinder internal-combustion engine case withn=3 as the dominant firing order. The typical valuesof gn and un for n=1.5, 3, 4.5, 6, 7.5, 9 reported inTable 2 (as extracted from the literature25,26) are

Figure 8. Verification of the analytical solution for the dampedlinear SDOF system (Figure 5(a)): (a) analytical solution; (b)numerical solution.rpm: r/min.

Table 2. Values of gn and un calculated for a typical six-cylinder internal-combustion engine.

Parameter (units) Value for the following n

1.5 3 4.5 6 7.5 9

gn (rad) 0.0150 0.0600 0.0075 0.0300 0.0050 0.0100un (rad) –0.01 0.01 –0.02 0.02 –0.03 0.03




utilized to construct the flywheel motion input in equa-tions (3) and (4). Then equation (8), given the con-structed motion input, is solved via the proposedanalytical and numerical methods. The solutions of d(t)from both methods are shown in Figure 9, where a rea-sonable agreement is achieved between the analyticalsolution and the numerical method. Both transientresponses exhibit the same four peaks at these speeds:Op1 (92 r/min), Op2 (137 r/min), Op3 (271 r/min), andOp4 (540 r/min). The peak-to-peak values dl p�p ofamplification are summarized in Table 3, where a com-parison for the n=1 case is also included. For themulti-order excitation case, both Figure 9 and Table 3indicate that d(t) has the highest amplification at n=3,as expected; other amplifications occur at the ordersn=1:5, n=6, and n=9. Additionally, the speeds Op

corresponding to the local peaks are slightly higherthan the corresponding critical speeds Oc. Both

Figure 9 and Table 3 suggest that the new analyticalsolution is valid for the multi-order excitation case.

The time–frequency domain representation of thenumerically obtained d(t) is examined for the speedorder analysis during the acceleration process.Specifically, the short-time Fourier transform techniqueis employed, and the result is displayed in Figure 10.Here, the n=3 order has the highest amplitude (withthe darkest color), and the other three dominant ordersare n=1.5, 6, and 9. The horizontal line at 13.2Hzrepresents the natural frequency v1. As the speedincreases with a=5 (r/min)/s, the four order linesintersect with v1 at four speeds; the four critical speedsOc are observed as darker regions.

Finally, in order to examine the application of thelinear-system-based theory, the peak-to-peak valuedp�p found from the closed-form solution is comparedwith the results of a nonlinear SDOF powertrain sys-tem (with h=m=0:3) by using the parameters in thesection entitled ‘Nonlinear torsional model (with

Table 3. Comparison of the analytical method and the numerical method for the transient response d(t) of a damped linear SDOFsystem given z = 0.001 and a = 5 (r/min)/s.

Excitation order n Critical speed Oc (r/min) Analytical closed-form expression Numerical solution

Op (r/min) dp–p (rad) Op (r/min) dp–p (rad)

Single-order (n = 1) excitation 792 805.1 2.8 805.1 2.8

Excitation order n Critical speed Oc (r/min) Analytical closed-form expression Numerical solution

Op (r/min) dl_p–p (rad) Op (r/min) dl_p–p (rad)

Multiple-order excitation 88 (n = 9) 93.2 2.3 92.8 2.2132 (n = 6) 138.4 3.1 137.7 2.8264 (n = 3) 272.9 11.6 272.1 10.4528 (n = 1.5) 540.8 3.7 538.9 3.6

Figure 9. Verification of the analytical solution for the dampedlinear SDOF system (Figure 5(a)) with multi-order motionexcitation, given z = 0.001 and a = 5 (r/min)/s: (a) analyticalsolution; (b) numerical solution. See Table 2 for the n, gn, and un

values.rpm: r/min.

Figure 10. Short-time Fourier transform of d(t) for the lineardamped SDOF system (Figure 5(a)) with multi-order motionexcitation, given z = 0.001 and a = (r/min)/s. The colored areasrepresent the amplitude in decibels with reference to 1.0 rad.rpm: r/min.

Singh and Li 1415



piecewise linear clutch damper)’ for n=1 only. Asshown in Figure 11, dp�p from the closed-form solutiond tð Þ is able to predict dp�p of the piecewise linearpowertrain system successfully under a rapid accelera-tion rate during the start-up process (a525 (r/min)/s).

Conclusion

The chief contribution of this article is the developmentof a new closed-form solution that approximates thetransient vibration amplification of a nonlinear multi-staged clutch damper during the engine start-up; adamped linear torsional oscillator with instantaneous-frequency excitation is utilized to find this solution. Theresults of an SDOF powertrain system with a flywheelmotion input for the start-up process is numerically ver-ified and experimentally validated by comparing thepeak displacements. The previous analytical work bySen et al.12,13 with an instantaneous torque input isextended by developing a closed-form solution with amotion input that includes instantaneous frequency andamplitude terms. The chief limitation is the applicabilityof the error function algorithm, since it loses high accu-racy in specific regions; this has obviously hindered his-torical calculations as well. Future work should focuson the transient analyses of multi-degree-of-freedomnonlinear driveline systems for both conventional vehi-cles and hybrid vehicles.14

Acknowledgements

The authors acknowledge the Eaton CorporationClutch Division for providing support for the appli-cation aspects of this research. Individual contribu-tions from L Pereira, B Franke, P Kulkarni, J Dreyer,and M Krak are gratefully appreciated. Theauthors would also like to acknowledge the SmartVehicle Concepts Center33 and the National Science

Foundation Industry & University CooperativeResearch program34 for supporting the fundamentalaspects of this work.

Declaration of conflict of interest

The authors declare that there is no conflict of interest.

Funding

This work was partially supported by the EatonCorporation Clutch Division and the National ScienceFoundation Smart Vehicle Concepts Center (Industry& University Cooperative Research Program).

References

1. Lewis FM. Vibration during acceleration through a criti-

cal speed. Trans ASME 1932; 54: 253–261.2. Gasch R, Markert R and Pfutzner H. Acceleration of

unbalanced flexible rotors through the critical speeds. J

Sound Vibr 1979; 63: 393–409.3. Hassenpflug HL, Flack RD and Gunter EJ. Influence of

acceleration on the critical speed of a Jeffcott rotor.

Trans ASME, J Engng Power 1981; 103: 108–113.4. Plaut RH, Andruet RH and Suherman S. Behavior of a

cracked rotating shaft during passage through a critical

speed. J Sound Vibr 1994; 173: 577–589.5. Sekhar AS and Prabhu BS. Transient analysis of a

cracked rotor passing through critical speed. J Sound

Vibr 1994; 173: 415–421.6. Irretier H and Balashov DB. Transient resonance oscilla-

tions of a slow-variant system with small non-linear

damping – modeling and prediction. J Sound Vibr 2000;

231: 1271–1287.7. Darpe AK, Gupta K and Chawla A. Transient response

and breathing behavior of a cracked Jeffcott rotor. J

Sound Vibr 2004; 272: 207–243.8. Roques S, Legrand M, Cartraud PC et al. Modeling of a

rotor speed transient response with radial rubbing. J

Sound Vibr 2010; 329: 527–546.9. Newland DE. Mechanical vibration analysis and computa-

tion. New York: Dover Publications, 2006.10. Jacobsson H. Disc brake judder considering instanta-

neous disc thickness and spatial friction variation. Proc

IMechE Part D: J Automobile Engineering 2003; 217(5):

325–342.11. Duan C and Singh R. Analysis of the vehicle brake judder

problem by employing a simplified source–path–receiver

model. Proc IMechE Part D: J Automobile Engineering

2011; 225(2): 141–149.12. Sen OT, Dreyer JT and Singh R. Order domain analysis

of speed-dependent friction-induced torque in a brake

experiment. J Sound Vibr 2012; 331: 5040–5053.13. Sen OT, Dreyer JT and Singh R. Envelope and order

domain analyses of a nonlinear torsional system deceler-

ating under multiple order frictional torque. Mech Sys-

tems Signal Processing 2012; 35: 324–344.14. Ledesma R, Keeney C and Shih S. Transient analysis of

an integrated powertrain and suspension system. In:

12th European ADAMS Users’ Conference, Marburg,

Germany, uc970007, 18–19 November 1997. Santa Ana,

California: MSC Corporation.

Figure 11. Comparison of dp-p from the proposed closed-formsolution and a nonlinear SDOF system (Figure 5(a)):

,h = m = 0:3; , closed-form solution for a lineardamped oscillator.rpm: r/min.




15. Gaillard CL and Singh R. Dynamic analysis of automo-

tive clutch dampers. Appl Acoust 2000; 60: 399–424.16. Shaver RSCS Committee. Manual transmission clutch

systems. Warrendale, Pennsylvania: SAE International,

1997.17. Sugimura H, Takeda M, Takei M et al. Development of

HEV engine start-shock prediction technique combining

motor generator system control and multi-body dynamics

(MBD) models. SAE paper 2013-01-2007, 2013.18. Wellmann T, Govindswamy K and Tomazic D. Integra-

tion of engine start/stop systems with emphasis on NVH

and launch behavior. SAE paper 2013-01-1899, 2013.19. Kuang M. An investigation of engine start-stop NVH in

a power split powertrain hybrid electric vehicle. SAE

paper 2006-01-1500.20. Chen JS and Hwang HY. Engine automatic start–stop

dynamic analysis and vibration reduction for a two-mode

hybrid vehicle. Proc IMechE Part D: Journal of Automo-

bile Engineering 2013; 227(9): 1303–1312.21. Comparin RJ and Singh R. Non-linear frequency

response characteristics of an impact pair. J Sound Vibr

1989; 134: 259–290.22. Kim TC, Rook TE and Singh R. Super- and sub-

harmonic response calculations for a torsional system

with clearance nonlinearity using the harmonic balance

method. J Sound Vibr 2005; 281: 965–993.23. Kim TC, Rook TE and Singh R. Effect of nonlinear

impact damping on the frequency response of a torsional

system with clearance. J Sound Vibr 2005; 281: 995–1021.24. Babitsky VI and Krupenin VL. Vibration of strongly non-

linear discontinuous systems. Berlin: Springer, 2001.25. Kononenko VO. Vibrating systems with a limited power

supply. London: Iliffe, 1969.26. Dresig H and Holzweissig F. Dynamics of machinery: the-

ory and applications. Berlin: Springer, 2010.27. Crowther AR, Singh R, Zhang N and Chapman C.

Impulsive response of an automatic transmission system

with multiple clearances: formulation, simulation and

experiment. J Sound Vibr 2007; 306: 444–466.28. Li L and Singh R. Analysis of speed-dependent vibration

amplification in a nonlinear driveline psystem using Hil-

bert transform. SAE paper 2013-01-1894, 2013.29. Den Hartog JP. Mechanical vibrations. New York: Dover

Publications, 1985.30. Porter FP. Harmonic coefficients of engine torque curves.

Trans ASME, Appl Mech Div 1943; 10: A-33–A-48.31. Meirovitch L. Fundamentals of vibrations. New York:

McGraw-Hill, 2002.32. Abramowitz M and Stegun IA. Handbook of mathemati-

cal functions with formulas, graphs, and mathematical

tables. New York: Dover Publications, 1972.33. NSF Smart Vehicle Concepts Center. About the Center,

www.SmartVehicleCenter.org. (accessed 18 April 2014).34. National Science Foundation,Directorate for Engineer-

ing. Industry & University Cooperative Research Pro-

gram (I/UCRC), www.nsf.gov/eng/iip/iucrc. (accessed 18

April 2014).

Appendix 1

Analytical solution for an undamped linear torsionaloscillator

For an undamped SDOF system (z =0), equation (8) isrewritten as the two equations

€ur(t)+v21ur(t)=v2

1O0t+12 v2

1at2 ð47Þ

€ua(t)+v21ua(t)=v2

1

Xn

gn sin nO0t+n

2at2 +un

� �ð48Þ

First, the system is assumed to rotate with a constantspeed O0, and thus the initial conditions are ur(0)=0rad, _ur(0)=O0, ua(0)=0 rad and _ua(0)=0 rad/s.Then, ur(t) is found by independently solving equation(47) with the assumed initial conditions. The generalresponse in the Laplace domain31 to an arbitrary exci-tation T(t) with the initial conditions u(0)= u0 and_u(0)= v0 is found as

Q(s)=T(s)

s2 +2zv1 +v21

+s+2zv1

s2 +2zv1 +v21

u0

+1

s2 +2zv1 +v21

v0

ð49Þ

Let

T(s)=L v21O0t+

12v

21at2

� �ð50Þ

Now employ the inverse Laplace transformation andconvolution theorem to yield

ur(t )=O0t+1

2at2 +

a

v21

cos v1tð Þ � 1½ � ð51Þ

Two issues are observed with this analytical solutionof ur(t). First, ur(t) follows fr(t) since the O0t+

12at

2

term is present in equation (51). Second, there is anoscillatory part a

�v21

� �cos v1tð Þ � 1½ � even though the

amplitude a�

v21 is relatively small because v1 =82.9

rad/s and a normally ranges from 0.1 rad/s2 to 15 rad/s2

(from 1 (r/min)/s to 150 (r/min)/s). As a consequence,a�

v21

� �cos v1tð Þ � 1½ � is neglected. Next, ua(t) is analyti-

cally solved from equation (48) to give

ua(t)=

ðt0

sin v1(t� u)½ �v1

v21

Xn

gn sin nO0u+n

2au2 +un

� �h idu ð52Þ

where u is a dummy variable.By using trigonometric identities, equation (52) is

rewritten as

Singh and Li 1417



ua(t)=v1

2

Xn

gn ua n1(t)� ua n2(t)½ � ð53Þ

ua n1(t)=

ðt0

cosn

2au2 + nO0 � v1ð Þu+v1t+un

h idu

ð54Þ

ua n2(t)=

ðt0

cosn

2au2 + nO0 +v1ð Þu� v1t+un

h idu

ð55Þ

The order of integration and summation is switchedin equation (53) so as to evaluate separately the convo-lution at each order (n). Thus, ua n1(t) and ua n2(t) areseparately calculated at each n by using the formulas

C(x)+ iS(x)=1+ i

2erf

ffiffiffiffipp

2(1� i)x

� �ð56Þð

cos (ax2 + bx+ c) dx

=

ffiffiffiffip

2

rcos b2

�4a� c

� �C (b+2ax)

� ffiffiffiap ffiffiffiffiffiffi

2pp� �

+ sin b2�4a� c

� �S (b+2ax)

� ffiffiffiap ffiffiffiffiffiffi

2pp� �

ffiffiffiap

ð57Þ

ua n1(t)=Ca n1(t)�Ca n1(0) ð58ÞCa n1(u)

=

ffiffiffiffiffiffip

na

rcos

nO0 � v1ð Þ2

2na� v1t� un

" #(

CnO0 � v1 + nauffiffiffiffiffiffiffiffiffi

napp

+ sin

nO0 � v1ð Þ2

2na� v1t� un

" #

SnO0 � v1 + nauffiffiffiffiffiffiffiffiffi

napp

�ð59Þ

ua n2(t)=Ca n2(t)�Ca n2(0) ð60ÞCa n2(u)

=

ffiffiffiffiffiffip

na

rcos

nO0 +v1ð Þ2

2na+v1t� un

" #(

CnO0 +v1 + nauffiffiffiffiffiffiffiffiffi

napp

+ sinnO0 +v1ð Þ2

2na+v1t� un

" #

SnO0 +v1 + nauffiffiffiffiffiffiffiffiffi

napp

�ð61Þ

where C(x) and S(x) represent the Fresnel integrals,32

which are related to the error function erf(x).By substituting equations (58) and (60) into equation

(53), an analytical solution of ua(t) is found as

ua(t)=v1

2

Xn

gn Ca n1(t)�Ca n1(0)½ �f

� Ca n2(t)�Ca n2(0)½ �g ð62Þ

Combining equations (51) and (62) yields u(t) as

u(t)= ur(t)+ ua(t)

=O0t+1

2at2 +

a

v21

cos v1tð Þ � 1½ �

+v1

2

Xn


� Ca n2(t)�Ca n2(0)½ �g ð63Þ

Then, d (t) is found as

d tð Þ= u tð Þ � f tð Þ

’v1

2

Xn


� Ca n2(t)�Ca n2(0)½ �g ð64Þ

For a lightly damped system, equation (64) may beefficiently utilized to approximate the transient amplifi-cation of a piecewise linear system under a rapid accel-eration rate.




Documents

Proc IMechE Part D: Analysis of vibration amplification in a ......1408 Proc IMechE Part D: J Automobile Engineering 229(10) Downloaded from pid.sagepub.com at OHIO STATE UNIVERSITY