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JOURNAL OFSOUND ANDVIBRATION

0022-460X/$ - s

doi:10.1016/j.js

Abbreviations

freedom system�CorrespondE-mail addr

Journal of Sound and Vibration 294 (2006) 503–528

www.elsevier.com/locate/jsvi

Influence of harmonically varying normal load on steady-statebehavior of a 2dof torsional system with dry friction

Chengwu Duan, Rajendra Singh�

Acoustics and Dynamics Laboratory, Department of Mechanical Engineering, The Center for Automotive Research,

The Ohio State University, Columbus, OH 43202, USA

Received 4 November 2004; accepted 22 November 2005

Available online 23 February 2006

Abstract

A nonlinear dry friction problem with harmonically varying normal load is formulated, in the context of a two-degree of

freedom torsional system, since virtually all of the prior literature focuses on the topic of time-invariant normal load. First,

pure stick, pure slip and stick–slip motions are computationally and analytically determined when excited by a sinusoidal

torque, in the presence of harmonically varying saturation torque; mean terms are included in both. These analyses yield

both transient and steady-state time histories under various conditions. Second, the effects of time-varying normal load on

steady-state responses have been investigated and nonlinear spectral maps (including super-harmonics) are developed.

Results show that the actuation system parameters could affect steady-state stick–slip motions in different ways over the

lower and higher frequency regimes, as a result of time-delay in slip motions with respect to the torque excitation. In

particular, the negative slope characteristics in the friction law exaggerate the stick–slip vibration problems, and it is the

major cause of bifurcations and quasi-periodic or chaotic motions. Around the super-harmonic peak frequencies, the

nonlinear system tends to lose stability as abrupt jumps in the spectral maps take place. An equivalent viscous damping

model is considered to analytically investigate the instability mechanism. Further, the periodicity of the system response

under harmonically varying actuation is conceptually by employing the harmonic balance method. Finally, steady-state

behavior is examined for the nonlinear, time-varying dry friction problem.

r 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Dry friction elements are encountered in many mechanical and structural systems, under a variety ofoperational conditions [1–10]. First, consider the classical friction damper example of Fig. 1(a) where a single-degree-of-freedom (sdof) torsional system is shown. Assuming a time-invariant normal load N, the equationof motion is I €yþ Tcðy; _yÞ þ Tf ðN; _yÞ ¼ TeðtÞ where Tcðy; _yÞ is the constraint torque, TeðtÞ is the externallyapplied torque excitation, I is the inertia, y is the angular displacement and Tf is the friction torque giventime-invariant N; one could employ one of the several Tf ð

_yÞ relationships [3,6,8–10]. Indeed, there is a

ee front matter r 2006 Elsevier Ltd. All rights reserved.

v.2006.01.001

: LHS, left-hand side; max, maximum value; NLTV, nonlinear time-varying; RHS, right-hand side; sdof, single-degree-of-

; TCC, torque converter clutch; 2dof, 2-degree-of-freedom system

ing author. Tel.: +1614 292 9044; fax +1614 292 3163.

ess: singh.3@osu.edu (R. Singh).

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Nomenclature

A area of actuation pressureC torsional viscous damping coefficientI torsional inertiaK torsional stiffnessN normal forceP pressureR moment armt timeT torquea exponentially decaying factord relative angular displacemento angular frequencyc phase lagy absolute angular displacements conditioning factorO angular speedz various damping ratiom friction coefficient

Subscripts

1, 2, 3 inertial element indicesc constraint

d dampede enginef frictionk kineticm meanmax maximumn natural frequencyp fluctuating component or perturbations static or saturationss steady statet transmission

Superscripts

– normalized value. first derivative with respect to time.. second derivative with respect to time

Operators

j j absolute valueo 4t time-average operator

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528504

substantial body of literature on time-invariant friction torque or force, especially when the saturation forcesor torques are small [1–8]. Second, assume that N varies with time, say intentionally through an activelycontrolled actuation mechanism that applies time-varying pressure PðtÞ on an area A. Thus the nonlinear,time-varying (NLTV) friction torque Tf ðNðtÞ; _yÞ is given by mNðtÞR ¼ mPðtÞAR, where R is a moment arm andm is the coefficient of friction. Of course, one may find yet physical processes where the contact loads maychange periodically anyway, such as in gear pairs [11,12]. To the best of our knowledge, no prior researcherhas addressed the harmonically varying nonlinear friction force or torque issue, with the exception of slidingfriction in gears [11,12].

2. Problem formulation

2.1. Physical system

In this article, we investigate the effect of harmonically varying normal load NðtÞ on the dynamics of a two-degree-of-freedom (2dof) torsional system with nonlinear dry friction path. Example systems include theslipping torque converter clutch (TCC) in an automotive driveline system as schematically illustrated in Fig.1(b). Other applications may include smart clutch [13] and dual clutch transmission [14,15] concepts. Unlike apure dry friction damper (such as the one in Fig. 1(a)), the dry friction element in Fig. 1(b) functions as a keypower transmission path. For example, in a vehicle with automatic transmission, the fuel economy can beimproved by applying a slipping TCC to avoid the power loss within the fluid torque converter [16,17]. Asshown in Fig. 1(b), the nonlinear friction torque Tsf transmitted within the TCC is applied by a hydraulicactuation pressure PðtÞ, which is controlled by a pulse-width modulated solenoid valve [18]. Accordingly, atime-varying normal load is created assuming a fixed pressure area for a given physical system, i.e.NðtÞ ¼ PðtÞA. By controlling P or N, a target slip speed (Oe � Ot) is achieved, ensuring a best compromise

ARTICLE IN PRESS

Te(t)

Te(t)

Tf(t)

I

N

θ

θ1θ2

Tc

Tf

I1

I2

P(t)K23

C23

(a)

(b)

Fig. 1. Torsional systems with dry friction element: (a) classical sdof dry friction damper system; (b) example case of this article: 2dof

driveline system with time-varying normal load in the dry friction path.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 505

between the fuel efficiency and ride quality [18,19]. Several researchers have studied the effect of the feedbackcontrol systems and hydraulic actuation system dynamics [18,20]. However, the essential dry frictionnonlinearity has been either ignored or linearized around the operating point assuming small motions.

Recently, Duan and Singh investigated system dynamics of a torsional system with a dry friction controlledpath [21,22]. Significant stick–slip motions are found under harmonic torque excitation. Nonlinear frequencyresponse characteristics have been studied using analytical or semi-analytical methods. However, a constant N

was assumed in the previous studies. In this article, we assume the normal load N to be sinusoidal with a mean(dc) term, as a result of the oscillations within the hydraulic control circuit [18]. We also consider thepossibility of a negative slope in the friction characteristics (the Stribeck effect of the lining material). Both ofthese effects could induce quasi-periodic or chaotic responses and consequently pose difficulty for the systemcontrol and introduce objectionable noise and vibration problems to the vehicle systems.

2.2. Governing equations of 2dof torsional system

The vehicle driveline system can be represented by a simplified 2dof torsional model as in Fig. 1(b). Here, I1represents the combined torsional inertia of flywheel, I2 is the inertia of friction shoe and pressure plate andthe wheel and vehicle sub-system I3 is considered as ground. The governing equations are

I1 €y1 þ Tf ð_y1 � _y2; tÞ ¼ TeðtÞ ¼ Tm þ Tp sinðotÞ, (1a)

I2 €y2 þ C23_y2 þ K23y2 ¼ Tf ð

_y1 � _y2; tÞ. (1b)

Here, y1 and y2 are absolute angular displacements; C23 and K23 are the lumped viscous damping and stiffnessassociated with the automotive driveline. Further, we reformulate the equations in terms of relative motions

ARTICLE IN PRESSC. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528506

where d1 ¼ y1 � y2 andd2 ¼ y2:

I1 €d1 �I1

I2C23

_d2 �I1

I2K23d2 þ 1:0þ

I1

I2

� �Tf ð

_d1; tÞ ¼ Tm þ Tp sinðotÞ, (2a)

I2 €d2 þ C23_d2 þ K23d2 ¼ Tf ð

_d1; tÞ. (2b)

The engine torque excitation TeðtÞ is composed of mean Tm ¼oTe4t and pulsating TpðtÞ components, whereo 4t is the time-average operator. Using the Fourier series expansion, express it as TeðtÞ ¼ TmþP

nTpn sinðopntþ fpnÞ, where n is the harmonic order of the firing torque sequence, opn ¼ ðNe=2ÞnOe, Ne isthe number of engine cylinders [23], Tpn is the amplitude for the nth harmonic and fn is the associated phaselag. In this study, only the dominant harmonic component (op1 ¼ o) is considered for the sake of simplicity,i.e. TeðtÞ ¼ Tm þ Tp sinðotÞ.

2.3. Friction torque formulation

The nonlinear friction torque Tf ð_d1; tÞ is carried by the clutch and then it acts as an equivalent torque

excitation to the downstream system. In a realistic automotive system, a pulse-width modulated solenoid valvewould generate a time-varying P by changing the command value or duty ratio [18]. That results in a nonlineartime-varying (NLTV) friction torque formulation, Tf ð

_d; tÞ ¼ mð_dÞNðtÞR. Note that R is assumed to be timeinvariant. Assume NðtÞ as a sinusoidal signal with mean pressure Nm, amplitude Np and the actuation loadfrequency of :

NðtÞ ¼ Nm þNp sinðof tþ cÞ. (3)

Here, c is the phase lag between the actuation load (pressure) and TeðtÞ. Further, NðtÞ is assumed to bepositive-definite to ensure that no separation occurs across the frictional interface, i.e. Np=Nm 2 ½0; 1Þ. Weemploy the following friction formulation mð_d1Þ to examine the phenomenological dynamic behavior [9]:

mð_d1Þ ¼mk þ ðms � mkÞe

�a _d1j jh i

sgnð_d1Þ; _d1�� ��40;

½0�ms�; _d1 ¼ 0:

8<: (4)

Here, a is a positive constant that controls the gradient of m with respect to _d1. In our study, a ¼ 2 is chosen forthe sake of illustration. In addition, a multi-valued regime exists at _d1 ¼ 0. To facilitate parametric studies, weincorporate some parameters such as mk and R into NðtÞ to yield the following NLTV friction torque:

Tf ð_d; tÞ ¼ mð_d1ÞTsðtÞ; TsðtÞ ¼ mkNðtÞR ¼ Tsm þ Tsp sinðof tþ cÞ. (5a,b)

Here, mð_d1Þ is a normalized friction coefficient with respect to mk:

mð_d1Þ ¼1:0þ ms

mk� 1:0

� �e�a

_d1j jh i

sgnð_d1Þ; _d1�� ��40;

0� ms

mk

h i; _d1 ¼ 0:

8><>: (6)

2.4. Objectives

The first objective is to analytically determine the nature of steady-state responses of the torsional systemunder harmonically varying dry friction. Analytical solutions under assumed motion conditions will also bedeveloped to understand the dynamic behavior. The second objective is to analytically and numericallyexamine the steady-state responses under the effect of NðtÞ and compare with the time-invariant case withnonlinear dry friction. Both periodic and chaotic responses will be identified via results in time and frequencydomains. Further, bifurcation diagram and Poincare sections will be constructed by cascading the steady-statetime-domain responses. Interaction between the clutch actuation parameters such as c, of and Tsp, and thenegative slope friction characteristics will be examined.

ARTICLE IN PRESSC. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 507

3. Computational methodology

For a piece-wise linear or nonlinear element such as dry friction, two numerical schemes, namely thediscontinuous and continuous solutions, can be employed. The first one finds the solutions for different statesand then assembles them [3,8,21]. This method can give an ‘‘exact’’ solution of the non-analytical system butenormous time is often required in the iterative matching process. Recently, Leine successfully employedHenon’s scheme but it is not convenient for a multi-degree-of-freedom system [6]. The second method employsa stiff differential equation solver but one must first condition the discontinuous nonlinearity. Typicalconditioning functions include arctangent, hyper-tangent and the like [6,21,25]. This method iscomputationally efficient but an artificial uncertainty is introduced by the smoothening factor s. Based onDuan and Singh’s study [21], the user must exercise caution and use the discontinuous solution as abenchmark. Several researchers have recently used the harmonic balance method to study the nonlinearsystems. For example, Blankenship and Kahraman developed a harmonic balance method to study the gearedsystem under parametric excitation [26]. However, their method cannot be conveniently implementedespecially when many harmonics must be included in the expansion. More recently, Kim et al. [27] and Duanand Singh [22] have proposed refined multi-term harmonic balance but both formulations are not suitable fora system with time-varying parameters because the procedure that converts the residue from time domain tofrequency domain by factoring out the discrete Fourier transform matrix is no longer feasible. Further,analytical methods such as multi-term harmonic balance always look for periodic steady-state solutions.Quasi-period or chaotic responses that are of particular importance from the vehicle performance standpointcannot be obtained.

To efficiently obtain a complete map of the system behavior including both periodic and aperiodicresponses, an explicit Runge–Kutta 4th(5th) order numerical integration routine with adaptive step sizecontrol (designate as RK45) due to Dormand and Prince is employed [28]. This numerical solution will bevalidated, by comparing our predictions with those obtained by a popular ordinary differential solver Xppaut

[24,30]. To facilitate the direct numerical integration, the discontinuous friction law of Eq. (6) is conditionedemploying a hyper-tangent function with a smoothening factor s ¼ 50:

mð_d1Þ ¼ 1:0þms

mk

� 1:0

� �e�a

_d1j j

� �tanhðs_d1Þ. (7)

Fig. 2 illustrates the typical steady-state stick–slip behavior. First note that our numerical solution RK45matches quite well with the discontinuous method of Ref. [31] and Gear’s method (Gear) that is recommendedby the Xppaut software for stiff nonlinear problems [24,30]. Differences are seen when the classicalRunge–Kutta fourth order routine (RK4) of the Xppaut software is selected, possibly due to the numericalstability problem and as a result of the stiff nature computational problem at hand.

4. Nature of steady-state responses based on approximate analytical solution

For the physical system of Fig. 1(b), three kinds of steady-state responses across the frictional interface arepossible following the initial transients: pure stick, pure slip and stick–slip. While the stick–slip is essentiallycontrolled by the strong nonlinearity of dry friction element, pure stick or pure slip can be determined aheadof a numerical simulation by using simplified analytical solutions. For example, Duan and Singh [22] haveproposed a procedure to determine the pure stick response regime for a time-invariant dry friction torque; thisstill holds for a time-varying dry friction formulation. Assume the pure stick motion (_d1 ¼ 0) that wouldconsolidate I1 and I2 into a single inertial body. The nonlinear system of Fig. 1(b) would now behave as alinear system. The internal friction force Tfi across the interface is determined by ðTeðtÞ � I1 €y1Þ, where €y1 ¼ €d2under the condition _d1 ¼ 0:

d2 ¼Tm

K23þ

TpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK23 � ðI1 þ I2Þo2Þ

2þ ðC23oÞ

2q sinðotþ jÞ, (8a)

ARTICLE IN PRESS

1.8 1.9 20

10

20

30

40

50

0

0.05

0.1

0.15

0.2

0.25

t, s

1.8 1.9 2t, s

δ 1, rad

/sδ 2, r

ad

.

Fig. 2. Typical steady-state response for the 2dof torsional system given o ¼ 150 rad/s, Tm ¼ 300, Tp ¼ 250, Tsm ¼ 500, Tsp ¼ 0.2 Tsm,

of ¼ o, c ¼ 0, mk ¼ 0.8mk and Oe ¼ 150 rad/s: ___ , RK45, y , gear, _ _ _ , RK4 methods, _U_, discontinuous solution.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528508

j ¼ �tan�1C23o

K23 � ðI1 þ I2Þo2. (8b)

Thus,

TfiðtÞ ¼ Tm þ Tp sinðotÞ þo2Tpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðK23 � ðI1 þ I2Þo2Þ2

q sinðotþ jÞ. (9)

When the internal friction torque is asymptotically less than the time-varying dry friction, pure stick motion issatisfied:

TfiðtÞpmsNðtÞR 8t 2 ½0;1Þ. (10)

Thus, we define a difference variable DðtÞ ¼ TfiðtÞ � msNðtÞR and given its periodicity, the following criterion isdefined to satisfy the pure stick condition:

DðtÞp0 8t 2 ½0; PperiodÞ; Pperiod ¼2po

. (11a,b)

As noted, when the amplitudes of excitation and friction torques are fixed, the lower and higher transitionfrequencies from pure stick to stick–slip can be determined via upward and downward frequency sweeps,respectively [22]. However, it should be pointed out that this process is based on the piece-wise nonlinearfriction law (6). In contrast, when that smoothened friction law (7) is used, the following two effects arebrought into the system dynamics. First, the assumption of only pure stick motions may not be valid. Second,the maximum friction coefficient or torque decreases with mk given the same ms value as a result of thesmoothening process as shown in Fig. 3. Thus, the resulting ‘‘transition’’ frequency would be affected by thevalue of mk, as explored in a subsequent section.

Further, pure slip type steady-state motions are possible as reported by Duan and Singh [29]. As a result ofthe speed difference between I1 and I2 during the initial engagement, a pure slip transient would exist [29]. Ifwe were to assume pure positive slip motion (_d140) and that mk ¼ ms since the negative gradient would

ARTICLE IN PRESS

-2 0 2

-0.3

0

0.3 0.3

µ µ

δ1

.

δ1

.

δ1

.

µk decreased

(a) (b)

1

0.5

-0.5

-1

0

0.2 0.4

∂µ/∂

δ 1.

(c)

Fig. 3. Effect of smoothening on the friction coefficient vs. velocity relations: (a) smoothened friction law; (b) zoomed part; (c) calculated

qm=q_d1. Key: ___ , mk ¼ ms; y , mk ¼ 0.8ms; _ _ _, mk ¼ 0.75ms, _U_, mk ¼ 0.677ms.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 509

dominate only when _d1 approaches zero, Eqs. (2a) and (2b) can be approximated as

I1 €d1 �I1

I2C23

_d2 �I1

I2K23d2 þ 1:0þ

I1

I2

� �ðTsm þ Tsp sinðof tþ cÞÞ ¼ Tm þ Tp sinðotÞ, (12a)

I2 €d2 þ C23_d2 þ K23d2 ¼ Tsm þ Tsp sinðof tþ cÞ. (12b)

Together with the following initial conditions as assumed according to the physical system behavior:

d1ð0Þ ¼ 0; _d1ð0Þ ¼ Oe; d2ð0Þ ¼ 0; _d2ð0Þ ¼ 0. (13a2d)

Through mathematical manipulation, Duan and Singh [29] obtain the following functional form where thecoefficients a0, a1, a21, a22 and a3 are determined by Eq. (13):

_d1ðtÞ ¼ a0 þ a1tþa21 sinðotþ j21Þ

þa22 sinðof tþ j22Þ

( )þ a3e

�Bont sinðod tþ j3Þ. (14)

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Table 1

Parameters and excitation amplitude used for simulating the driveline system of Fig. 1b

Parameters and excitation Value(s)

Torsional inertias (kgm2) I1 ¼ 0.20; I2 ¼ 0.02

Torsional viscous damping (Nmrad/s) C23 ¼ 0.6

Torsional stiffness (Nm/rad) K23 ¼ 3000

Engine torque: mean and excitation amplitude (Nm) Tm ¼ 300; Tp ¼ 250

Static (ms) and kinetic (mk) friction coefficients ms ¼ 0.3; mk ¼ 0.15–0.30

Actuation pressure area (m2) A ¼ 0.08

Moment arm (m) R ¼ 0.1

Actuation pressure (kPa) Pm ¼ 200–400; Pp ¼ 0–Pm

Phase lag (radian) c ¼ 0, p/2, p

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528510

As reported previously [29], a1 is determined by ðTm � TsmÞ=I1. When it is greater than zero, the motionacross the frictional interface will be pure slip type although no steady-state responses can be defined since therelative velocity keeps on increasing. However, when a1 ¼ 0 (Tm ¼ Tsm), steady-state pure slip motions couldalso take place and the resulting response can be analytically obtained by eliminating the exponentiallydecaying term in Eq. (14):

_d1ssðtÞ ¼ a0 þ a21 sinðotþ j21Þ þ a22 sinðof tþ j22Þ. (15)

Nonetheless, the above-mentioned condition alone cannot warrant pure slip motions. Assuming pure positiveslip motions, another condition has to be added to avoid a crossing of zero velocity:

a0j j4 a21j j þ a22j j. (16)

Although a detailed description of the a0, a21 and a22 expressions would be lengthy for the time-varying dryfriction case, analytical results for the time-invariant dry friction are discussed below for the sake ofillustration. When Tsp ¼ 0 and Tm ¼ Tsm, Eq. (15) will assume the following expression:

_d1ðtÞ ¼ V m �Tp

I1ocosðotÞ þ

Tsm

K23

2Bon

e�Bont cosðod tÞ þTsm

K23

2B2 � 1

od

e�Bont sinðodtÞ, (17a)

on ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK23=I2

p; B ¼ C23=ð2

ffiffiffiffiffiffiffiffiffiffiffiffiK23I2

pÞ; od ¼ on

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� B2

p. (17c2e)

Substitute the initial condition _d1ð0Þ ¼ Oe to obtain V m, the steady-state response of _d1ssðtÞ is found as

_d1ðtÞ ¼ ðOe �Tsm

K23

2Bon

Þ þTp

I1oð1� cosðotÞÞ. (18)

Typical parameters of Table 1 suggest that the term ðTsm2BÞ=ðK23onÞ is negligible compared to the enginespeed Oe, and thus _d1ðtÞ ¼ Oe þ ðTp=I1oÞð1� cosðotÞÞ. Thus, the positive definite condition of _d1ðtÞ(_d1ðtÞ40 8t 2 ½0; 1Þ) is also satisfied and correspondingly pure positive slip motions are guaranteed. Finally,note that the dynamic amplitude will be dictated by Tp, I1 and o.

5. Effect of negative slope friction characteristics with a time-invariant normal load

Duan and Singh had earlier studied the dry friction path problem [22] by assuming mk ¼ ms. In this section,we focus more on the steady-state response when mkpms. Two spectral maps are constructed for examining theeffect of mk in the frequency domain. The first one is the maximum response vs. frequency ratio map bypicking the maximum (max) response amplitude of _d1 at each excitation frequency o. Further, to illustrate thenonlinear frequency maps, o is normalized with respect to the linear sub-system natural frequencyon ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK23=I2

p. The second one is the bifurcation diagram (with o=on or mk=ms as a parameter) by picking the

values of d2 corresponding to starting time of each excitation cycle (i.e. t ¼ 2pn=o; n ¼ integer). The reason touse d2 in constructing the bifurcation diagram is that the variation in d2 is relatively smaller than _d1 at the fixed

ARTICLE IN PRESS

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

-0.5

0

0.5

-0.5

0

0.5

-0.5

0

0.5

-0.5

0

0.5

-0.5

0

0.5

δ 1max

(ra

d/s)

.δ 1m

ax (

rad/

s).

δ 1max

(ra

d/s)

.δ 1m

ax (

rad/

s).

δ 1max

(ra

d/s)

.

ω /ωn

ω /ωn

µ k =

1.0

µs

µ k =

0.8

3 µ s

µ k =

0.8

0 µ s

µ k =

0.7

5 µ s

µ k =

0.6

7 µ s

µ k =

1.0

µs

µ k =

0.8

3 µ s

µ k =

0.8

0 µ s

µ k =

0.7

5 µ s

µ k =

0.6

7 µ s

(a)

δ 2, ra

dδ 2,

rad

δ 2, ra

dδ 2,

rad

δ 2, ra

d

(b)

Fig. 4. Effect of mk given a time-invariant friction torque: (a) d1max maps; (b) bifurcation diagrams with o/on as a bifurcation parameter.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 511

phase point. Since a deterministic relationship between d2 and _d1 exists, a bifurcation in d2 should also indicatea similar response for _d1. To ensure that the steady-state response has been obtained, the first 50 transientcycles in the numerical solution are discarded.

Fig. 4(a) shows the effect of mk on d1max. When mk ¼ ms, the map of d1max is a smooth curve and super-harmonic resonant peaks are not very active. As mk is decreased, more super-harmonic resonant peaks around

ARTICLE IN PRESSC. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528512

o=on ¼ 0:167, 0.20, 0.25 and 0.33 appear along with jumps. An upward shift in peak frequencies with adecrease in mk can be attributed to the ‘‘negative damping’’ effect. Further, the ‘‘transition’’ frequency frompure stick to stick–slip regimes (around o=on ¼ 0:4) change with respect to the value of mk; this is due to thesmoothened friction law as discussed in the previous section. As noted in Fig. 4(b), a jump in d1max yields acorresponding bifurcation point. For example, a jump at o=on ¼ 0:2 with mk ¼ 0:83ms changes the period-onemotions in to a period-two type. As mk is decreased further, the period-two type motions become quasi-periodic or even chaotic responses.

The underlying instability mechanism can be theoretically explained by re-considering the nonlinear frictiontorque given essentially as a nonlinear damping term. Expand the nonlinear friction torque using theTaylor series and neglect the higher order terms to yield the following where Tf ð

_d0Þ is the instantaneous meantorque:

Tf ð_d1Þ ¼ Tf ð

_d1cÞ þqT

q_d

����_d¼_d1c

ð_d� _d1cÞ, (19)

qT

q_d1¼ mkNR

qm

q_d1, (20a)

@m

q_d1¼ s 1:0þ

ms

mk

� 1:0

� �e�a

_d1j j

� �½1:0� tanh2ðs_d1Þ� � a

ms

mk

� 1:0

� �e�a

_d1j jsgnð_d1Þ tanhðs_d1Þ. (20b)

The negative values of qT=q_d1 are shown in Fig. 3(c). The minimum value decreases with a reduction of mk=ms.This justifies our choice of mk=ms as the bifurcation parameter. Conventional approaches to address thestability issues have relied on the absolute value of qT=q_d1 and have assumed small oscillations [32]. However,this approach would not work for our system because the vibrating amplitude is quite large for a system withsignificant stick–slip motions as shown in Fig. 2. Instead, we define an equivalent viscous damping (Cf )element by equating the cyclic energy dissipation:

Cf

Z G

0

_d2

1 dt ¼

Z G

0

qT

q_d1_d2

1 dt. (21)

Re-write the governing Eqs. (1a) and (1b) in the following autonomous form, excluding the excitation and theinstantaneous mean friction torque:

I1 €y1 þ Cf ð_d1 � _d1cÞ ¼ 0, (22a)

I2 €y2 þ ðC23 þ Cf Þ_y2 þ K23y2 � Cf

_y1 ¼ 0. (22b)

Observe in the second-order Eq. (22b), that unstable solutions of y2ðtÞ (or d2ðtÞ) shall take place whenever thefollowing negative damping condition is satisfied. Under such a condition, _y1 and _d1 also lose stabilityprovided a definite torsional system exists:

Cf o� C23. (23)

Fig. 5 shows the correlation between the calculated Cf and the bifurcation diagram in the frequency range of0.08–0.30. Since we have essentially employed a straightforward ‘‘brute force’’ linearization method,it is of course true that we can only partially explain the mechanism of the bifurcations. But, as evidentfrom our numerical results, whenever the Cf crosses the threshold value of �0.6, a bifurcation is observed. Inother bifurcation regimes where Cf does not cross the threshold, we may need a more sophisticatedmethod. However, this article focuses on the effect of harmonically varying normal loads and therefore adetailed examination of the role of negative damping and associated bifurcations is left for a futurestudy.

On the other hand, a decrease in the value of mk=ms further reduces the minimum value of qT=q_d1. Thus, theresulting equivalent damping Cf will decrease. Fig. 6 illustrates the tendency by employing mk=ms as a

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0

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-0.6

-1.2

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-2.4

ω /ωn

δ 2C

f

Fig. 5. Correlation between instability regimes and equivalent viscous damping value given mk ¼ 0:667ms.

0.2

0.15

0.05

-0.05

-0.10.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0

0.1

µk / µ

s

δ 2, ra

d

Fig. 6. Bifurcation diagram employing mk=ms as a bifurcation parameter given o=on ¼ 0:36.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 513

bifurcation parameter. As observed, when mk=ms is closer to unity, only period-one motions are seen. Whenmk=ms is decreased to around 0.92, a sudden bifurcation yields period-two type motions. Then as mk=ms isdecreased further down to 0.7, the period-two motions become quasi-periodic or chaotic. Fig. 7 shows thesample time histories of _d1, d2 and the Poincare sections corresponding to three values of mk=ms.

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-5

-10

-15

0

δ 2

δ2 δ2 δ2

. δ 2. δ 2.

2π/ω

2(2π/ω)

(a) (b) (c)

Fig. 7. Sample time histories and Poincare sections given o=on ¼ 0:36: (a) mk=ms ¼ 0:94, (b) mk=ms ¼ 0:74, (c) mk=ms ¼ 0:64.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528514

6. Effect of harmonically varying normal load on steady-state behavior

6.1. Effect of actuation parameters c, of and Tsp with mk ¼ ms

Under the condition of mk ¼ ms, the effect of phase lag c in the actuation pressure is first studied withof ¼ o. Fig. 8 shows the max spectral map of _d1 and the corresponding bifurcation diagram with o=on as abifurcation parameter. At lower frequencies (o=ono0:30 here) the c ¼ 0 value yields the best attenuation ofstick–slip motions compared with the c ¼ p=2 and p values. This is consistent with our analysis of thetransient stick–slip motion as reported in the previous article [29]. But when the frequency is increased, thisattenuation effect is reversed since the time-delay in the positive slip response increases. Fig. 9 shows thecoupled excitation torque and _d1 response at several frequencies with Tsp ¼ 0. When the excitation frequencyis low (o=on ¼ 0:15 and 0.18), the positive slip starts and persists during the first half-cycle of engine torque asshown in Figs. 9(a) and (b). But when the excitation frequency is higher (o=on ¼ 0:37 and 0.40), a positive slipis initiated and it continues during the second half of the engine torque cycle as shown in Figs. 9(c) and (d).Differences in the attenuations over different frequency regimes can be analytically explained by assumingsemi-positive stick–slip motions. First re-examine Eqs. (8) and (9) of the pure stick case under time-invariantdry friction. Note that the interfacial friction force of Eq. (9) can be also written as follows where d2ðtÞis givenby Eq. (8):

TfiðtÞ ¼ I2 €d2ðtÞ þ C23_d2ðtÞ þ K23d2ðtÞ, (24)

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ω /ωn

ω /ωn

δ 1max

, rad

/s.

(a)

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

δ 2, rad

δ 2, rad

δ 2, rad

(b)

ψ = 0

ψ = π/2

ψ = π

Fig. 8. Effect of phase c on the steady-state response given of ¼ o and Tsp=Tsm ¼ 0:25: (a) maximum frequency response map for _d1with: yJy , c ¼ 0; y�y , c ¼ p=2; y+y , c ¼ p; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 515

TfiðtÞ ¼ Tm þTp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK23 � I2o2Þ

2þ ðC23oÞ

2q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK23 � ðI1 þ I2Þo2Þ

2þ ðC23oÞ

2q sinðotþ jþ WÞ, (25a)

W ¼ tan�1C23o

K23 � I2o2. (25b)

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e(t) (

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)

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) (ra

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.

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t, s t, s

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25

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2

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10

(a) (b)

(c) (d)

Fig. 9. Time delay in the positive slip motion with respect to the engine torque excitation: (a) o=on ¼ 0:15, (b) o=on ¼ 0:18, (c)o=on ¼ 0:37, (d) o=on ¼ 0:40.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528516

Given the system parameters in Table 1, we find that C23o5ðK23 � I2o2Þ in the frequency range of interest.Thus, W � 0 and

TfiðtÞ � Tm þTp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK23 � I2o2Þ

2þ ðC23oÞ

2q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðK23 � ðI1 þ I2Þo2Þ

2þ ðC23oÞ

2q sinðotþ jÞ. (26)

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Now the interface friction torque rises with a phase lag j to the excitation TeðtÞ. And j increases from 01 to1801 and crosses the 901 value at the system natural frequency under the pure stick condition

osn ¼ K23=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI1 þ I2

p. (27)

And note the following relationship between osn and on given I1 ¼ 0:20 and I2 ¼ 0:02:osn

on

¼

ffiffiffiffiffiI2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI1 þ I2p � 0:3. (28)

This clearly explains the specific boundary dividing the two frequency regimes as illustrated in Fig. 8. Asevident from the results, this critical frequency is mainly determined by the system parameters I1and I2.

On the other hand, under the time-varying condition, the following relationships hold:

I1 €d1 �I1

I2C23

_d2 �I1

I2K23d2 ¼ T eqðtÞ, (29a)

T eqðtÞ ¼ Tm � 1:0þI1

I2

� �sm

� �þ Tp sinðotÞ � 1:0þ

I1

I2

� �Tsp sinðotþ cÞ

� �. (29b)

Note a reduction in the equivalent torque excitation leads to a different phase lag c. This is similar to thediscussion in the previous study on transient responses [29]. As noted, when the positive slip tends to occurduring the first-half of the engine torque cycle, c ¼ 0 provides the maximum decrease in the ‘‘effective’’excitation T eq and vice versa. Further, all curves in Fig. 8(a) are relatively smooth for the case of mk ¼ ms andno jumps are seen. Accordingly, no bifurcation is observed in Fig. 8(b).

Second, the effect of of is shown in Figs. 10 and 11. As seen in Figs. 10(a) and 11(a), best attenuation of theslip motion occurs at lower frequencies as of ¼ o. This effect is however reversed when o is increased. Anexplanation can be found in a manner similar to the case discussed above, but with different values of c. Also,no bifurcations seem to take place in Figs. 10(b) and 11(b). Let us assume that of ¼Mo=N where both M

and N are integers and o is the engine excitation frequency. It is noted that it does not matter whether M andN are commensurable or not, the resulting motions will possess a period identical to o=N. For example, whenM =N is 1=2, then the motions are of period-two type; when M =N is 2/1, the motions are of period-one typeas shown in Fig. 10(b); when M=N ¼ 2=3, the motions are of period-three type; and when M=N ¼ 4=3, themotions are also of period-three type as shown in Fig. 11(b). Time histories in Fig. 12 confirm such results.This phenomenon can be explained via the harmonic balance analysis. For a period—N (NX1) time-varyingactuation, assume periodic-u (uX1) solution of _d1ðtÞ as a truncated Fourier series where o0 ¼ o=u is thefundamental frequency component:

_d1ðtÞ ¼XNh

n¼1

an cosðno0tÞ þ bn sinðno0tÞ: (30)

Thus, the NLTV friction torque can be written in the following form where cn and dn expression are Fouriercoefficients of mð_d1Þ. Further, assume M ¼ 1 without a loss of generality:

Tf ðtÞ ¼ Tm þ Tp sin1

No

� �� � XNh

n¼1

cn cosðno0tÞ þ dn sinðno0tÞ

!. (31)

Substitute Eqs. (30) and (31) into Eq. (1) to yield the following, where the coefficients f n, gn, rn and sn areundetermined constants:

XNh

n¼1

f n cosðno0tÞ þ gn sinðno0tÞ

þ sin1

Not

� �XNh

n¼1

rn cosðno0tÞ þ sn sinðno0tÞ½ � ¼ Tm þ Tp sinðotÞ. (32)

The following trigonometric relationships hold:

sin1

Not

� �cosðno0tÞ ¼

1

2sin

n

uþ

1

N

� �ot

� �� sin

n

u�

1

N

� �ot

� �� �, (33a)

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ω /ωn

ω /ωn

δ 1max

, rad

/s.

(a)

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

δ 2, rad

δ 2, rad

δ 2, rad

ωf = 0.5ω

ωf = ω

ωf = 2 ω

(b)

Fig. 10. Effect of of on the steady-state response given c ¼ 0 and Tsp=Tsm ¼ 0:25: (a) maximum frequency response map for _d1 with:

yJy , of ¼ 0:5o; y�y , of ¼ o; y+y , of ¼ 2:0o; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528518

sin1

Not

� �sinðno0tÞ ¼

1

2cos

n

u�

1

N

� �ot

� �� cos

n

uþ

1

N

� �ot

� �� �. (33b)

Eq. (32) can be further simplified as follows where l1n, l1n and l3n are the Fourier coefficients and Yn, Cn andUn are corresponding phase angles:

XNh

n¼1

l1n sinn

uotþYn

� �h iþXNh

n¼1

l2n sin nu �

1N

�otþCn

�þl3n sin n

u þ1N

�otþ Un

�" #

¼ Tm þ Tp sinðotÞ. (34)

On the left-hand side (LHS) of Eq. (34), the first term is contributed by the mean term of friction torque andthe second term by the alternating TspðtÞ. The right-hand side (RHS) is the periodic excitation. According to

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ω /ωn

ω /ωn

δ 1max

, rad

/s.

(a)

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

δ 2, rad

δ 2, rad

δ 2, rad

ωf = 0.67ω

ωf = ω

ωf = 1.33 ω

(b)

Fig. 11. Effect of of on the steady-state response given c ¼ 0 and Tsp=Tsm ¼ 0:25: (a) maximum frequency response map for _d1 with:

yJy , of ¼ 0:667o; y�y , of ¼ o; y+y , of ¼ 1:33o; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 519

Eq. (34), given an arbitrary choice of integer N, the following relationship must hold:

u ¼ N. (35)

To prove the above conclusion, first assume u and N are incommensurable, to satisfy the periodicity and likeharmonic terms on the LHS and RHS of Eq. (34). The second term on the LHS needs to be self-balanced tozero. However, that would indicate that the alternating TspðtÞ has no effect on the overall system response but

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m

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m

300

Te(t

), N

m

t, s

2.6 2.7 2.8 2.9 3.1 3.2 3.33t, s

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t, s

t, s

δ 1(t),

rad

/s.

60

40

20

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δ 1(t),

rad

/s.

40

20

0

δ 1(t),

rad

/s.

2(2π/ω)

2(2π/ω)

3(2π/ω)

3(2π/ω)

3(2π/ω)

3(2π/ω)

(a)

(b)

(c)

Fig. 12. Sample time histories for different of given o ¼ 150, Tsp=Tsm ¼ 1=4: (a) of ¼ 0:5o, (b) of ¼ 0:67o, (c) of ¼ 1:33o.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528520

this is not true from the physical perspective. Second, assume u and N are commensurable (but not identical,i.e. N ¼ Ku where K is an integer), the same requirement as for the incommensurable case is applied to thesecond term on the LHS of Eq. (34). The other situation corresponding to u ¼ KN is disregarded here sinceonly the minimum period defines the periodicity. This reveals an interesting phenomenon: under a time-varying actuation pressure with period—N, the resulting nonlinear response exhibits the same frequency as thesystem would under a dual frequency (1 and M=N) excitation.

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ω /ωn

δ 1max

, rad

/s.

(a)

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

δ 2, rad

δ 2, rad

δ 2, rad

δ 2, rad

(b)

Tsp

/ Tsm

= 0

Tsp

/ Tsm

= 0.125

Tsp

/ Tsm

= 0.25

Tsp

/ Tsm

= 0.5

Fig. 13. Effect of of on the steady-state response given c ¼ 0 and of ¼ o: (a) maximum frequency response map for _d1 with: ___ ,

Tsp=Tsm ¼ 0; yJy, Tsp=Tsm ¼ 0:125; y�y , Tsp=Tsm ¼ 0:25; _._, Tsp=Tsm ¼ 0:50; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 521

Last, the effect of Tsp is studied given of ¼ o and c ¼ 0. Fig. 13 shows the corresponding max spectralmaps and bifurcation diagram. Consistent with the above-mentioned analyses, a higher value of Tsp providesthe best attenuation of the slip motions at lower frequencies, but this effect is reversed at higher frequencies.Again, no bifurcation and quasi-periodic or chaotic motions are generated.

6.2. Interaction between negative slope mð_d1Þ and harmonically varying NðtÞ

First, the interaction between mð_d1Þ and c is investigated under the condition of ¼ o and Tsp=Tsm ¼ 0:25.Fig. 14 shows the resulting max spectral maps and bifurcation diagram with c ¼ 0, p=2 and p, given

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ω /ωn

δ 1max

, rad

/s.

(a)

0.2

0.1

-0.1

0

0.2

0.1

-0.1

0

0.2

0.1

-0.1

0

δ 2, rad

δ 2, rad

δ 2, rad

(b)

ψ = 0

ψ = π /2

ψ = π

Fig. 14. Interaction between mkð_d1Þ and c given Tsp=Tsm ¼ 0:25, of ¼ o and mk ¼ 0:75ms: (a) maximum frequency response maps of _d1max

with: yJy , c ¼ 0; y�y , c ¼ p=2; y+y , c ¼ p; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528522

mk ¼ 0:75ms. The combined effects of the negative damping phenomenon and varying phases (or time delays)are clearly observed in Fig. 14(a). The negative damping introduces active super-harmonic peaks, similar toFig. 4(a). Further, three values of c seem to produce the same attenuation effect in the slip motion as the casemk ¼ ms yielded in Fig. 8(a). The c ¼ 0 phase lag has the best attenuation at lower o but it acts in an oppositeway at higher o. When c ¼ 0, only the pure stick motions exist when o=ono0:15. Consequently, only theperiod-one type motions occur in this regime. Note that the transition occurs at a lower value of o=on thanthat observed (around 0.18) in Fig. 8(a). This could be attributed to an amplification that is induced by thenegative damping. As o is increased further, super-harmonic peaks appear with jumps. Such jumps would

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δ 1max

, rad

/s.

(a)

0.2

0.1

-0.1

0δ 2, rad

0.2

0.1

-0.1

0δ 2, rad

0.2

0.1

-0.1

0δ 2, rad

(b)

ψ = 0

ψ = π /2

ψ = π

Fig. 15. Interaction between mkð_d1Þ and c given Tsp=Tsm ¼ 0:25, of ¼ o and mk ¼ 0:67ms: (a) maximum frequency response maps of

_d1max: yJy , c ¼ 0; y�y , c ¼ p=2; y+y , c ¼ p; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 523

certainly indicate a loss in the stability and thus bifurcations are found in Fig. 14b. The severity of bifurcationsseems to be more dependent on the abruptness of jumps than the amplitudes of slip motions. As noted inFig. 14(a), a significant jump occurs around o=on ¼ 0:35 given c ¼ 0; consequently the periodic motionsbecome chaotic. In contrast, the slip motions for c ¼ p are higher at lower o, but the peaks now occur inrelatively smooth manner and accordingly no bifurcations are seen. Similar phenomena are observed inFig. 15 except that more bifurcations and quasi-periodic or chaotic motions occur as a result of a furtherdecrease in the value of mk.

Second, the effect of mk with several of values is studied given c ¼ 0 and Tsp=Tsm ¼ 0:25. Resulting maxspectral maps and bifurcation diagrams are shown in Figs. 16 and 17. Again, the combined effects of negative

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ω /ωn

δ 1max

, rad

/s.

(a)

δ 2, rad

δ 2, rad

δ 2, rad

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

(b)

ωf = 0.5 ω

ωf = 2.0 ω

ωf = ω

Fig. 16. Interaction between mkð_d1Þ and of given Tsp=Tsm ¼ 0:25, c ¼ 0 and mk ¼ 0:75ms: (a) maximum frequency response maps of _d1max

with: yJy , of ¼ 0:5o; y�y , of ¼ o; y+y , of ¼ 2o; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528524

damping and mismatched frequencies of are clearly evident. At lower o, the friction torque with of ¼ osuppresses the slip motions better than the of ¼ 2o case does. Numerous jumps take place in the particularcase of of ¼ 0:5o in Fig. 16, and quasi-periodic or chaotic motions seem to prevail over the entire frequencyrange of interest. Similarly, Fig. 18 presents results with several values of Tsp under the of ¼ o and c ¼ 0condition. As a result of increased attenuation with Tsp in Fig. 18(a), initial quasi-periodic or chaotic motionsat lower o become period-one type at the transition from stick–slip to pure stick motions as shown inFig. 18(b). However, enhanced slip motions and enlarged jumps convert the periodic motions back into quasi-periodic or chaotic ones at higher o, say around 0.25 and 0.35.

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00.1 0.15 0.25 0.35 0.4 0.450.30.2

0.1 0.15 0.25 0.35 0.4 0.450.30.2

0.1 0.15 0.25 0.35 0.4 0.450.30.2

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ω /ωn

ω /ωn

δ 1max

, rad

/s.

(a)

δ 2, rad

0.2

-0.2

0

δ 2, rad

0.2

-0.2

0

δ 2, rad

0.2

-0.2

0

(b)

ωf = 0.67 ω

ωf = 1.33 ω

ωf = ω

Fig. 17. Interaction between mkð_d1Þ and of given Tsp=Tsm ¼ 0:25, c ¼ 0 and mk ¼ 0:75ms: (a) maximum frequency response maps of _d1max

with: yJy , of ¼ 0:67o; y�y , of ¼ o; y+y , of ¼ 1:33o; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528 525

7. Conclusion

Effects of harmonically varying normal load on the steady-state responses of a torsional system with dryfriction path have been studied. To the best of our knowledge, no prior researcher has addressed such anNLTV friction force or torque issue, with the exception of sliding friction issues in gears in which theperiodically varying friction acts in the off-line-of-action while the mesh force excitation is along the line-of-action [11,12]. Two further contributions of our article emerge. The nature of steady-state response, forexample, pure stick, pure slip and stick–slip motions, has been analytically defined and determined. Second,the effects of time-varying friction on steady-state responses have been analytically and numerically studied.

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ω /ωn

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δ 1max

, rad

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(b)

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-0.2

0

δ 2, rad

0.2

-0.2

0

δ 2, rad

0.2

-0.2

0

δ 2, rad

0.2

-0.2

0

δ 2, rad

Tsp

/ Tsm

= 0

Tsp

/ Tsm

= 0.125

Tsp

/ Tsm

= 0.25

Tsp

/ Tsm

= 0.5

Fig. 18. Interaction between mkð_d1Þ and Tsp given of ¼ o, c ¼ 0 and mk ¼ 0:75ms: (a) maximum frequency response maps of _d1max with:

___, Tsp=Tsm ¼ 0; yJy , Tsp=Tsm ¼ 0:125; y�y , Tsp=Tsm ¼ 0:25; _._, Tsp=Tsm ¼ 0:5; (b) bifurcation diagram.

C. Duan, R. Singh / Journal of Sound and Vibration 294 (2006) 503–528526

Results show that the actuation system parameters c, o and Tsp can possibly attenuate the system response.Analytical study has clearly shown the different attenuations seen at lower and higher frequency regime andthese have been confirmed by the numerical results. Bifurcation diagram are constructed to detect qualitativechanges in the dynamic behavior. The negative slope in m is the major cause of bifurcations and quasi-periodicor chaotic responses. Around the super-harmonic peak frequencies, the nonlinear system tends to lose stability

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in the form of an abrupt jump in the max spectral maps of _d1 and consequently bifurcations take place. Anequivalent viscous damping term is considered to analytically investigate the instability issues. Further, theperiodicity of the system response under harmonically varying actuation is examined by employing theharmonic balance method. Finally, some suggestions for future work are outlined. A more efficientcomputational solution method for the NLTV problem must be developed since numerical stiffness issues areencountered. A path following or parametric continuation study should assist in systematically identifying thenature of the bifurcations.

Acknowledgment

Financial support through the DaimlerChrysler Challenge Fund is gratefully acknowledged.

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