Journal of Sound and Vibration (1995) 184(5), 767–799

DYNAMICS OF A PIECEWISE NON-LINEARSYSTEM SUBJECT TO DUAL HARMONIC

EXCITATION USING PARAMETRICCONTINUATION

C. P R. S

Acoustics and Dynamics Laboratory, Department of Mechanical Engineering,The Ohio State University, 206 West 18th Avenue, Columbus, Ohio 43210-1107, U.S.A.

(Received 27 October 1993, and in final form 4 April 1994)

The dynamic behavior of piecewise linear and/or non-linear vibratory systems subject toa single harmonic excitation has been recently analyzed by using a one- or multi-termharmonic balance (or Galerkin’s) method, piecewise linear techniques, analog simulationand direct numerical integration (digital simulation). Instead, this paper proposes to utilizethe technique of parametric continuation to study the steady state response and globaldynamics of a two-degree-of-freedom piecewise non-linear system with backlash ormulti-valued springs and impact damping. The physical system is under the influence ofa mean load and is subject to a dual harmonic external excitation, where the secondfrequency is either twice or three times the fundamental excitation frequency. The effectsof various parameters such as impact damping, mean and alternating loads, the piecewisestiffness ratios and the relative phase between the excitation harmonics on the systemresponse are thoroughly examined. System or excitation parameter regimes that exhibitsub-harmonic, super-harmonic quasi-periodic and chaotic solutions are obtained efficientlyand systematically by using the proposed scheme. This investigation also clarifies thedifferences between viscous and non-linear impact damping models, which have beenproposed earlier in the literature for studying clearance non-linearities. Limited experimen-tal data validate our modelling strategy. Finally, selected predictions from this techniquematch very well with a multi-term harmonic balance method and with direct numericalintegration, wherever applicable.

7 1995 Academic Press Limited

1. INTRODUCTION

Many machine elements or assemblies such as automotive clutches, splines, gears andbearings have clearances or multi-valued springs that lead to piecewise linear or non-linearrestoring force (torque) characteristics. Most of these systems are generally excited byexternal periodic forces or torques, while under the influence of a mean force or torque.The modelling and analysis of such systems have been the subject of numerous investi-gations. Among the early investigators of such systems, Dubowsky and Freudenstein [1, 2]modelled a single-degree-of-freedom (SDOF) system with clearance based on the theoryof Hertzian compliance and included viscous damping. Analytical solutions were obtainedby using both piecewise-linear technique and describing function [3] approaches. Thedescribing function approach was unable to exhibit the force amplification induced by theclearance. Furthermore, their linear viscous damping model yielded an unrealistic hys-teresis diagram, as evident from Figure 1(a), where a compressive force still exists eventhough the impacting bodies have separated. This motivated Hunt and Crossley [4] tointroduce the concept of non-linear impact damping. The new hysteresis diagram

767

0022–460X/95/300767+33 $12.00/0 7 1995 Academic Press Limited

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Figure 1. A comparison of the impact force models: (a) by Dubowsky and Freudenstein [1]; (b) by Hunt andCrossley [4].

corresponding to their model, shown in Figure 1(b), removes the anomaly noted earlier.This model was used by Veluswami et al. [5, 6] to study the impacts of a ball between twoplates. Their study, based on analog and digital simulations, demonstrated the presenceof multiple steady state responses, which were a function of the clearance and excitationamplitude. The correlation between experiment and simulation was found to be good.Furthermore, Azar and Crossley [7] used the same model to represent Hertzian complianceand dissipation in a spur gear mesh, which yielded reasonable correlation betweensimulation and experiment. This model was later modified by Yang and Lin [8], whodemonstrated the need to include tooth bending stiffness, based on a comparison of thestrain energy due to bending with that of the Hertzian deformation. Subsequently, Herbertand McWhannell [9] compared the results predicted by Hunt and Crossley’s [4] model withthose corresponding to Dubowsky and Freudenstein’s [1] model. When the systemresponse could be adequately represented by a single harmonic, they found significantchange in the frequency composition of impact forces, predicted by the two models, onlyat higher frequencies. Nonetheless, notable differences existed between the two modelswhen the impacting phenomenon was more complex, leading to a larger number of impactsin one time period of the excitation. This issue will be revisited in this paper.

More complex systems with piecewise linear and/or non-linear characteristics have beeninvestigated, but most of them subject to a single harmonic excitation. The describing

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Figure 2. The reduced order model of an automotive transmission: (a) the three-degree-of-freedom torsionalmodel; (b) the clutch torque profile.

function approach [3] was used by Comparin and Singh [10] to study the frequencyresponse characteristics of multi-degree-of-freedom systems with clearances. Padmanab-han and Singh [11] developed the same technique to examine the issue of spectralinteractions between clearance non-linearities in a two-degree-of-freedom system. Bothinvestigations were based on the viscous damping model of [1] and were limited by theassumption of a single harmonic term in the response. A piecewise linear approach wasproposed by J. Shaw and S. W. Shaw [12] to analyze a two-degree-of-freedom impactingsystem. Their analysis involved taking a Poincare section at one of the rigid walls, andemploying the piecewise-linear solution along with the velocity change at the wall duringimpact (expressed through a coefficient of restitution) to examine various periodicsolutions and their possible bifurcations. This approach however, limited its applicationto piecewise-linear systems.

Kahraman and Singh [13] have analyzed multi-degree-of-freedom (MDOF) gearedsystems with backlashes using numerical integration, and established the occurrence ofperiod doubling bifurcations, quasi-periodic and chaotic solutions in the context of gearwhine problems. The other non-linear geared system phenomenon deals with gear rattlein manual automotive transmissions. For this problem, a number of different gearedsystems with backlashes and multi-stage clutch stiffnesses have been analyzed [14–16]. Forinstance, Pfeiffer and Kunert [14] and Karagiannis and Pfeiffer [15] have developed gearrattle models based on the principles of multi-body dynamics. Solutions were obtained bya patching scheme while utilizing a coefficient of restitution term for impact conditions.

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Figure 3. A comparison between viscous and impact damping models at V=1·5: (a) acceleration q01 ; (b)impact force. R, Viscous damping; ----, impact damping.

Chaos was observed in their simulations, which was further examined by means of theFokker–Planck equations. Kataoka et al. [16] used the method of harmonic balance topredict torque impulses and its dependence on the two-stage clutch stiffness ratio. Theytoo assumed rigid body type of impact, and patched solutions at the impact boundaries.

The multi-term harmonic balance method, called the Galerkin’s technique and originallyproposed by Urabe and Reiter [17], has been used by Choi and Lou [18] to study a SDOFoscillator with piecewise non-linear stiffness. The presence of chaotic solutions was,however, established using numerical integration. Kim and Noah [18, 19] have used thesame technique to analyze MDOF rotor dynamic systems with bearing clearances. Limitedbifurcation studies were conducted. The shooting method has also been used by a fewinvestigators [21–23] to investigate the forced vibrations of non-linear systems. Forinstance, Ling [21] examined several single- and two-degree-of-freedom systems with cubicnon-linearities using this technqiue. Sato et al. [22] analyzed the pitchfork and tangentbifurcations of a SDOF gear pair model with backlash and time-varying gear meshstiffness. Awrejcewicz and Someya [23] tracked pitchfork and Hopf bifurcations in asystem of coupled oscillators. The continuation or parameter imbedding method [24–27]has been used recently in conjunction with the shooting method. This powerful techniqueaffords a systematic way in which to study the global dynamics of any non-linear systemand pinpoint regimes exhibiting complex phenomena due to bifurcations. However, most

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Figure 4. Frequency response comparison for a heavily loaded oscillator: (a) with impact damping only; (b)with viscous damping only. ----, Stable; R, unstable (−1); – · – · – ·, unstable (+1); ×, unstable (Hopf).

applications of this technique have been restricted to autonomous non-linear systems ornon-autonomous systems with continuous type non-linearities. In this paper, we remedythis situation by considering the dynamics of a non-autonomous piecewise non-linearvibratory system.

2. PROBLEM FORMULATION

Based on the literature reviewed, it is clear that several research issues remainunresolved. As mentioned earlier, most of the prior investigations of piecewise non-linearsystems have been limited to only a single harmonic excitation case. Most practicalmechanical systems are, however, subject to a periodic excitation and their responses maynot be obtained from single harmonic studies. For instance, it has been established forDuffing oscillators [28] that the presence of additional higher harmonics in the excitationcan cause significant change in the global system dynamics. Furthermore, there is a needto explore the effect of various parameters on the system response in a systematic manner.Hence, the main focus of this paper is to examine such issues by using a two-degree-of-free-dom model with piecewise non-linear characteristics, under the influence of a mean loadand subject to multi-harmonic external excitation. We propose to use the continuationtechnique towards this purpose. Concurrently, an attempt is made to re-examine the

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Figure 5. The response of the model at V=0·17 with impact damping only: (a) impact force versus q1; (b)steady state time history of the impact force; (c) the FFT of the time response.

viscous damping [1] and the non-linear impact damping [4] models, in terms of timedomain and frequency domain solutions, with a view towards establishing the suitabilityof the impact damping model for vibro-impact problems. The automotive drive train isused as the practical example, with an emphasis on the clutch. Our prediction will becompared with limited experimental data [29]. Some of the other results are compared witha multi-term harmonic balance and with direct numerical integration in order to validatethe proposed continuation scheme.

In Figure 2 is shown a three-degree-of-freedom semi-definite torsional model which, forexample, could represent an automotive drive train without the meshing gears, with I�1

representing the clutch disk and flywheel inertia, I�2 representing the clutch hub inertia andI�3 representing the reflected inertia of the rest of the drive train. The governing equationsfor the model are as shown below, where q1=u1−u2, q2=q

.1, q3=u2−u3 and q4=q

.3; also

refer to the Appendix for a list of symbols:

I�1u� 1 +C�1(u� 1 − u� 2)+K�1(u1 − u2)+ f�(q1, q.1)=T�m +T�e (V�t), (1)

I�2u� 2 −C�1(u� 1 − u� 2)−K�1(u1 − u2)− f�(q1, q.1)+C�2(u� 2 − u� 3)+K�2(u2 − u3), (2)

I�3u� 3 −C�2(u� 2 − u� 3)−K�2(u2 − u3)=−T�m . (3)

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Figure 6. The response of the model at V=0·17 with viscous damping only: (a) impact force versus q1; (b)steady state time history of the impact force; (c) the FFT of the time response.

The non-linear torque, f�(q1, q�1), shown in Figure 2(b) for instance, can be thought ofas representing the secondary stage of a multi-staged clutch. We use this representationsince this model closely matches clutch torque profiles [29]. Equations (1)–(3) are insemi-definite form and are transformed to definite form by using the relative displacements,q1 = u1 − u2 and q3 = u2 − u3. Furthermore, the transformed equations are non-dimension-alized by defining t=vt and V=V�/v, where V� is the fundamental excitation frequency,and qi = qi /bci , i=1, 3, thus yielding in state space form,

q'=F(q; Vt), F=Aq+ h(q)+T(Vt), (4, 5)

where the vectors, matrices and other parameters are defined below:

q=gG

G

F

f

q1

q2

q3

q4

hG

G

J

j; A=G

G

G

K

k

0−K11

0K21

1−C11

0C21

0K12

0−K22

0C12

1−C22

GG

G

L

l, h= f(q1, q2)g

G

G

F

f

0−n1

0n2

hG

G

J

j, (6–8)

T=gG

G

F

f

0Tm1

0Tm2

hG

G

J

j+g

G

G

F

f

0Te1 cos (Vt)+Te2 cos (pVt+f)

00

hG

G

J

j, p=2 or 3, (9)

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Figure 7. A comparison of the viscous and impact damping models at V=0·17 with high damping constants:(a) time history of impact force; (b) FFT of the time histories; (c) acceleration q01 for impact damping model;(d) acceleration q01 for viscous damping model. ----, Impact damping; r, ×, viscous damping.

Figure 8. A comparison of acceleration (u� 3) time histories obtained from (a) simulation using only impactdamping and (b) experimental measurement [29].

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Figure 9. A frequency response comparison of the predictions yielded by the proposed continuation schemewith a multi-term harmonic balance scheme: (a) a heavily loaded oscillator with k=0·1; (b) a moderately loadedoscillator with k=0·4. ----, proposed scheme; r, harmonic balance.

K11 =K�1

I�av2 , K12 =

K�2

I�2v2

bc3

bc1

, K21 =K�1

I�2v2

bc1

bc3

, K22 =K�2

I�bv2 , (10a–d)

C11 =C�1

I�av, C12 =

C�2

I�2v

bc3

bc1

, C21 =C1

I�2v

bc1

bc3

, C22 =C�2

I�bv, (11a–d)

n1 =K�b

I2v2 , n2 =

K�b

I2v2

bc1

bc3

, (12a, b)

I�a = I�1I�2/(I�1 + I�2), I�b = I�2I�3/(I�2 + I�3), (13a, b)

Tm1 =T�m

I�1bc1v2 , Tm2 =

T�m

I�3bc3v2 , Te1 =

T�e1

I�1bc1v2 , Te2 =

T�e2

I�1bc1v2 , (14a–d)

f(q1, q2)= g(q1){1+ bq2}, b= b�/bc1v, (15, 16)

g(q1)= 8(1− k)(q1 + b0/bc1),0,

(1− k)(q1 − b0/bc1),

q1 E−b0/bc1,−b0/bc1 Q q1 Q b0/bc1,

q1 e b0/bc1.k=K�1/K�b , (17, 18)

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Figure 10. The effect of reducing the clutch second stage stiffness, K�b : (a) frequency response for a high valueof K�b ; (b) frequency response for a low value of K�b . ----, Stable; r, unstable (−1); – · – · – ·, unstable (+1);and ×, unstable (Hopf). Note that some of the subsequent figures do not show P2 and higher period loops forthe sake of clarity. However, some figures do show the sub-harmonic loops of the fixed point, q1(0). The entirebifurcation diagram can be pieced together from parts (a) and (b) of those figures. See Figure 13 as an example.

Note that only two excitation terms are included; however, Te2(t) could represent secondor third harmonic of Te1(t), i.e., p=2 or 3. Non-integer values of p leading to aperiodicexcitation are beyond the scope of this study.

3. PARAMETRIC CONTINUATION

We now solve equation (4) using a parametric continuation scheme, based on theshooting method [21–27]. The steps involved in this scheme are briefly described below;for a more detailed description, refer to our earlier paper [30].

3.1.

We begin by choosing arbitrary initial conditions, qi (0)= hi , i=1, . . . , 4. The systemof equations described by equation (4) can be solved from t=0 to t=mt0, t0 =2p/V,by using any numerical integration scheme. Since we are interested in finding solutions theperiod of which is an integral multiple or sub-multiple of the periodic excitation’sfundamental period t0 we need qi (0) such that qi (0)= qi (mt0), i=1, . . . , 4, where m is apositive integer. For instance, m=1 would solve for period-1 (P1) solutions, while m=2

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Figure 11. The effect of reducing the spacing between the resonances: (a) frequency response for DV=0·6;(b) frequency response for DV=0·80. ----, Stable; r, unstable (−1); – · – · – ·, unstable (+1); and × unstable(Hopf).

would solve for period-2 (P2) sub-harmonic solutions. The periodicity boundary con-dition, say for m=1, requires that the following non-linear equations be satisfied, for afixed value of any parameter, say a:

Gi (h; a)=Fi (h; a)− hi =0, Fi (h; a)= qi (t0), i=1, . . . , 4. (19, 20)

This can be solved iteratively by using a Newton–Raphson technique. However itrequires the formation and computation of the 4×4 Jacobian matrix, J, the elements ofwhich are given by

Jij =1Gi

1hj=61Fi

1hj− dij7, i, j=1, . . . , 4; dij =61,

0,i= ji$ j

. (21, 22)

In order to obtain 1Fi /1hj additional variables must be defined, and the equationsdescribing them are derived from an infinitesimal variation of the governing differentialequations (6–9):

mij (t)=1qi (t)1hj

, i, j=1, . . . , 4, (23)

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Figure 12. The frequency response of a heavily loaded oscillator: ----, Stable; – · – · – ·, Unstable (+1); ×unstable (Hopf).

Figure 13. The response of a moderately loaded oscillator (a) The frequency response: ----, Stable; r,unstable (−1); – · – · – ·, unstable (+); ×, unstable (Hopf). (b) the sub-harmonic solution loop, q1(0): ----, stableP2; – · – · – ·, unstable P2 (+1); · · · ·, unstable P2 (Hopf); r, unstable P1 (−1); q, stable P1. Note that parts(a) and (b) should be viewed together. Also refer to Figure 10.

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Figure 14. The response of moderately loaded oscillator with a higher mean load Tm2. (a) The frequencyresponse: ----, stable; r, unstable (−1); – · – · – ·, unstable (+1); ×, unstable (Hopf). (b) The sub-harmonicsolution loop, q1(0): ----, stable P2; – · – · – ·, unstable P2 (+1); r, unstable P1 (−1); q, stable P1.

m'ij (t)=1

1t 61qi (t)1hj 7=

1

1hj 61qi (t)1t 7= s

4

k=1

1Fi

1qkmkj . (24)

We then integrate the original governing equations along with equation (24), usingappropriate initial conditions (mij (0)= dij ) from t=0 to t= t0; the Jacobian elements, Jij ,are given by mij (t0)− dij , i, j=1, . . . , 4. This process is repeated until the New-ton–Raphson procedure converges to h0 =F(h0; a*). The solution h0

i , which could bestable or unstable, is then a fixed point of the iteration process described above; i.e., itrepresents the solution of the Poincare mapping of the system [24–26]. The stability of thefixed point is determined by the eigenvalues of the linearized mapping F, called themonodromy matrix, B:

B=$1Fi

1hj%h0

= [mij (t0)]= {J+ I}=h0, i, j=1, . . . , 4. (25)

The periodic solution is stable if the magnitude of the eigenvalues =li = is less than unity[31]. The stability of the periodic solution changes when some li crosses the unit circle.There are three different types of instabilities (i) li =−1, period-doubling or flipbifurcation occurs; (ii) li =1, saddle-node or symmetry breaking bifurcation occurs; and

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Figure 15. The frequency response of a lightly loaded oscillator: (a) the complete frequency response; (b) anexpanded view of the inset in (a). ---- Stable; r, unstable (−1); – · – · – ·, unstable (+1); ×, unstable (Hopf).

(iii) a pair of complex conjugates (=l==1) crosses the unit circle, termed as the secondaryHopf bifurcation.

3.2.

Often, we would like to know how the fixed points and/or the periodic solutions changeas this parameter of interest is varied from a*. The trajectory of the solution branch inthe neighborhood of a* can be obtained from equation (19) by a first order Taylor seriesexpansion in a:

s4

k=1 61Fi

1hk− dik7 1hk

1a+

1Gi

1a=0, i=1, . . . , 4. (26)

As long as J is non-singular one can compute 1hk /1a, k=1, . . . , 4, and hence determinehow the solutions change, provided that the 1Gi /1a’s are known. To obtain their values,we define new variables, say ci (t)= 1qi (t)/1a, their associated governing equations andinitial conditions (ci (0)=0, i=1, . . . , 4), to be used with the shooting procedure

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Figure 16. The sub-harmonic loop of q1(0) (a) at high frequency and (b) at low frequency. ----, Stable P2;– ·– ·– ·, unstable P2 (+1); ×, unstable P2 (Hopf); · · · ·, unstable P2 (−1); r, unstable P1 (−1); q, stable P1.

described earlier:

c'i (t)=1

1t 61qi (t)1a 7=

1

1a 61qi (t)1t 7= s

4

k=1

1Fi

1qkck +

1Fi

1a, i=1, . . . , 4. (27)

Then, 1Gi /1a=ci (t0). The procedure outlined above is termed continuation [24–27] andinvolves two further steps to obtain the solution for a1 = a*+Da:

(i) A predictor step, usually based on a first order Euler expansion; it could be moresophisticated if necessary [26, 27]. The Euler predictor is as follows, where the 1hi /1a’s areobtained from equation (14):

h1*i = h0

i +1hi

1a ba*

Da, h1*i = hi (a1), i=1, . . . , 4. (28)

(ii) A corrector step (Newton–Raphson iteration):

JDh(k+1) =−G(hk; a1). (29)

This procedure works well until J becomes singular, which is usually due to a turningpoint (li =+1). Such a situation occurs when 1hk /1a:a for some k, or 1Fk /1a ( R(J),where R(J) denotes the range space of the Jacobian matrix. When li=1, the Jacobian isstill non-singular for the regular Poincare map (m=1), but has a period-doubling turning

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Figure 17. The response of the lightly loaded system at V=0·63 showing a period-3 solution: (a) time historyof q1; (b) FFT of the time history in (a); (c) Poincare section.

point for the double Poincare map (m=2), leading to an eigenvalue li,m=2 = l2i,m=1 =1.

The turning point can be handled quite easily by re-parameterizing using a new parameter,say arc length s, so that hi 0 hi (s); a0 a(s) and i=1, . . . , 4. Then, equation (26) isrewritten as

s4

k=1 61Fi

1hk− dik7 1hk

1s+

1Gi

1a

1a

1s=0, i=1, . . . , 4. (30)

Having introduced one more unknown we define a normalizing equation in order tohave the same number of equations as there are unknowns:

s4

k=1 01hk

1s 12

+01a

1s12

=1. (31)

At the turning point rewrite equation (30) as follows, which renders J* non-singular:

s4

k=1k$m

J*jk1hk

1s+

1Gj

1a

da

dd=−61Fj

1hm− djm7 1hm

1s, J*jk =

1Fj

1hk− djk , J*jm =

1Gj

1a. (32)

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Figure 18. The response of the lightly loaded system at V=0·634 showing a quasi-periodic solution: (a) timehistory of q1; (b) FFT of the time history in (a); (c) Poincare section.

In this manner, starting from an initial fixed point calculated as outlined in section 3.1,the continuation procedure described above can be carried out for any parameter range,with its unique ability of going around the turning points and tracing out the entire branchof fixed point solutions; see references [24–27]. At the turning point there is a change inthe direction of the parameter a. If Da was positive (negative) to begin with, then past theturning point Da becomes negative (positive). Period-doubling branches, say P2 (period2t0), which bifurcates from the P1 (period t0) branch, can be traced in a similar fashiononce the initial P2 fixed point has been obtained using the regular shooting procedure ofsection 3.1 with m=2. A repeated use of this procedure will yield P4 and other higherperiod solutions. A uniform step size can be used over the whole parametric range, butthis makes the process computationally inefficient. Hence, a step size control algorithm,based on an error estimation criterion developed by Den Heijer and Rheinboldt [32], isused in our study.

4. COMPARISON OF DAMPING MODELS

Our first objective is to investigate whether the viscous model is proposed by Dubowskyand Freudenstein [1, 2] is different from the impact damping model of Hunt and Crossley[4]. We begin by outlining the procedure used in selecting the value of the impact damping

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T 1

Parameters used in the damping model comparison study with b0 =1

Figure(s) $ K11

−K21

−K12

K22 % $ C11

−C21

−C12

C22 % b n1, n2 Tm1, Tm2 Te1

3 (impact), 4(a) $00 −0·61·1% $00 −0·04

0·06% 0·075 1·0, 0·6 0·5, 0·25 0·25

3 (viscous), 4(b) $00 −0·61·1% $ 0·04

−0·04−0·04

0·06% 0 1·0, 0·6 0·5, 0·25 0·25

5, 7 (impact) $00 −0·61·1% $00 −0·04

0·06% 0·1 1·0, 0·6 0·25, 0·25 0·25

6, 7 (viscous) $00 −0·61·1% $ 0·04

−0·04−0·04

0·06% 0 1·0, 0·6 0·25, 0·25 0·25

8(a) $00 −0·61·1% $00 −0·04

0·06% 0 1·0, 0·6 0·25, 0·25 0·25

Figure 19. The response of the lightly loaded system at V=0·636 showing a chaotic solution: (a) time historyof q1; (b) FFT of the time history in (a); (c) Poincare section.

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coefficient, b, as shown in Figure 1(b). It is to be chosen such that the impact dampingmodel matches the viscous damping model as far as the resulting time response isconcerned. This is done for an excitation frequency at which the system response can beadequately represented in terms of a single harmonic; i.e., the system behaves almost likea linear system. In Figure 3 is shown a comparison of acceleration and impact force timedomain histories between the two models. Observe an excellent agreement for theacceleration profiles even though the impact profiles deviate slightly for the occurrence ofa single impact every two time periods of the excitation. The parameter values used in thedamping comparison study are listed in Table 1. The frequency response is compared inFigure 4 between an impact damping model (as in Figure 4(a)) and a viscous dampingmodel (Figure 4(b)). This case corresponds to a heavily loaded system with k=0, whichrepresents a pure backlash case. Both of the models exhibit saddle-node bifurcations(li =+1) at around the same frequencies, and show the presence of secondary Hopfbifurcations which indicates the possible presence of stable quasi-periodic solutions.However, the peak amplitude predicted by the non-linear damping model is lower thanthat of the linear viscous damping model. In contrast, the regime of flip bifurcationsleading to sub-harmonic solutions is smaller for the viscous damping case. Beyond the firstresonance region the two models yield almost the same results for the heavily loaded case,

Figure 20. The effect of the relative phase f between the two excitation harmonics on the frequency responseof a heavily loaded oscillator: (a) zero phase; (b) f=120°. ----, Stable; r, unstable (−1); – · – · – ·, unstable(+1); ×, unstable (Hopf).

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since the system is almost linear under these conditions. However, when the mean loadTm1 is reduced, there is considerable difference in the response, both in terms of timeand frequency domain results, especially at lower frequencies. For instance, in Figure 5are shown the time history and frequency spectrum of the impact force yielded bythe impact damping model at V=0·17. Here, b has been chosen according to theprocedure described above. It is obvious from Figures 5(a) and (b) tht the systemundergoes double-sided impacts [10, 11], with four impacts on one side followed by onlyone impact on the other side. The frequency spectrum shows that at least ten to twelveharmonics dominate. In Figure 6, where the corresponding results are shown for theviscous damping case, only single-sided impacts are observed, even though the othersystem parameters are identical. Also, the higher harmonics are much smaller, but thesystem exhibits more broadband characteristics than the impact damping case. When bothof the viscous and impact damping values are increased so as to match the impact forcetime histories, note the differences in the amplitude as well as in the nature of the higherharmonic response in Figure 7(b). Also, as seen in Figures 7(c) and (d), the time domainacceleration records look different even though the impact forces of Figure 7(a) have beenmatched.

Finally, we show a sample comparison between an experimentally obtained accelerationprofile [29] with the results of our simulation incorporating an impact damping model.

Figure 21. A close-up view of the differences between the frequency responses in Figure 20: (a) near the firstmajor resonance; (b) around the super-harmonic resonance. ----, Zero phase; – – – –, phase f=120°.

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Good qualitative and quantitative agreement is found in Figure 8 between theory andexperiment. This was for a low frequency of single sine excitation and for very low meanloads, conditions which are typical of automotive rattle problems [10, 11]. However, theviscous damping model could not match the experimental acceleration profiles qualitat-ively or quantitatively. Based on an extensive comparison of both models with measureddata, although not shown here, the impact model was deemed to be more suitable,especially for systems with very low mean loads and high amplitudes of excitation.Consequently, we propose to use the impact damping model to represent the clearancenon-linearity for the rest of this study.

5. RESULTS FOR A SINGLE HARMONIC EXCITATION

The parameter values used for the study of a two-degree-of-freedom system subject toa single harmonic excitation are shown in Table 2. The continuation procedure is validatedby comparing results with those yielded by a multi-term harmonic balance [17]. As canbe seen in Figures 9(a) and (b), an excellent match between the harmonic balance and the

T 2

Parameters used in the study of the two-degree-of-freedom system subject to a singleharmonic excitation, with Te2 =0 and b0 =1

Figure(s) $ K11

−K21

−K12

K22 % $ C11

−C21

−C12

C22 % b n1, n2 Tm1, Tm2 Te1

9(a) $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·25, 0·5 0·25

9(b) $ 0·4−0·24

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·6, 0·36 0·25, 0·5 0·25

10(a) $00 −0·61·1% $00 −0·04

0·06% 0·075 1·0, 0·6 0·5, 0·25 0·25

10(b) $00 −0·61·1% $00 −0·04

0·06% 0·075 0·80, 0·48 0·5, 0·25 0·25

11(a) $00 0·31·1% $00 −0·02

0·06% 0·075 0·7, 0·3 0·25, 0·50 0·25

11(b) $00 −0·061·1 % $00 −0·04

0·06% 0·075 0·6, 1·0 0·25, 0·50 0·25

12 $ 0·0−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·5, 0·25 0·25

13 $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·25, 0·25 0·25

14 $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·25, 0·5 0·25

15–19 $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·25, 0·25 0·5

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continuation method is evident. Although results are not shown here, when the mean loadsare reduced and/or the excitation amplitudes are increased, we need to include a fairly largenumber of harmonics (about 15–20) in the harmonic balance scheme in order to matchthe amplitudes and to reduce the truncation error caused by the neglect of the higherharmonics, especially at the lower frequencies. This renders the harmonic balance schemecomputationally inefficient. The continuation scheme, however, does not suffer from sucha restriction.

The effect of the stiffness K�b for k=0 is demonstrated in Figure 10. There exists a regionof unstable (li E−1) period-1 (P1) solutions in Figure 10(a), which indicates the presenceof stable sub-harmonic solutions. The stable sub-harmonic solutions (period-2, or P2) arenot shown for clarity. A reduction of the stiffness K�b causes the region of instability todisappear in Figure 10(b), being replaced by stable P1 solutions. This is mainly due to thefact that the impacts taking place are less severe when the compliance of the impactingbodies is increased. The effect of system parameters which reduce the frequency spacingbetween the resonances can be seen in Figure 11. The pitchfork bifurcations around thesecond resonance, as seen in Figure 11(b), are not present in Figure 11(a). Furthermore,the stable P1 solution regime (0·5EVE 0·7) after the saddle-node bifurcation in Figure

Figure 22. The frequency response of a heavily loaded oscillator subject to dual excitation. (a) For f=180°:----, Stable; r, unstable (−1); – · – · – ·, unstable (+1); ×, unstable (Hopf). (b) close-up view around thesuper-harmonic resonance: ----, zero phase; – – – –, phase f=180°.

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11(a) is replaced by Hopf bifurcated (unstable P1) solutions in Figure 11(b). Hence, nearthe resonance there is now a broader regime of stable quasi-periodic solutions. Thisobservation has been confirmed by actual numerical integration, although not shown here.

Next, we examine the effect of the mean loads Tm1 and Tm2 on the system response. InFigure 12 is shown the response of a heavily loaded oscillator, with k=0·10 andTm1 =2Tm2 =0·50. When the mean load Tm1 of Figure 12 is reduced by one-half, we canimmediately notice, from Figure 13(a), the occurrence of flip bifurcations at the secondresonant peak and near the saddle-node bifurcation (0·62EVE 0·85). The sub-harmonicloop of the fixed point q1(0) around the second resonance is shown in Figure 13(b). Notethat the P2 solutions exhibit Hopf bifurcations leading to stable quasi-periodic solutions.We also note that there are both sub- and super-critical pitchfork bifurcations taking place.When the mean load Tm2 is doubled from 0·25 (in Figure 13) to 0·50 while keepingTm1 =0·25, we can see, from Figure 14(a), that the flip instability region near thesaddle-node bifurcation is now replaced by stable P1 solutions. The instability regionaround the second resonant peak remains, even though it is reduced in extent. Also, thesub-harmonic loop in Figure 14(b) does not exhibit any Hopf bifurcations. In both cases(Figures 13 and 14), the period-doubling phenomenon is not observed.

Figure 23. A comparison of the frequency response for a moderately loaded oscillator with k=0·4, for (a)single excitation case and (b) dual excitation with Te2 at frequency 2V. ----, Stable; r, unstable (−1); – · – · – ·,unstable (+1); × unstable (Hopf).

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Finally, we examine the effect of increasing the amplitude of the excitation, Te1.The frequency response, when the amplitude is doubled from the previous case ofFigure 13, is shown in Figure 15. Observe that the dynamic behavior has becomefairly complex, especially at the lower frequencies. A number of super-harmonicresonance occurs, as shown in Figure 15(b). Flip as well as Hopf bifurcations occuralmost throughout the entire low frequency range. The sub-harmonic loop of two flipbifurcation regimes is shown in Figure 16. The loop in Figure 16(b) shows theperiod-doubling effect, leading to stable P4 solutions. These P4 solutions doexhibit further period-doubling bifurcations, even though they are not shown in the figurefor the sake of clarity. Actual numerical integration, although not presented here,confirms the presence of the period-doubling phenomenon leading to chaotic solutions inthis parameter range. We do, however, present sample results demonstrating the occur-rence of chaos, through quasi-periodic bifurcations. First, a period-3 (P3) solution occurswhich breaks into a quasi-periodic attractor when V is perturbed slightly. Then, thequasi-periodic attractor bifurcates into a chaotic solution. The time traces, Fourierspectrum and Poincare sections for three different types of solutions are shown in Figures17–19.

Figure 24. The frequency response for a lightly loaded oscillator with k=0·4, subject to a single excitation:(a) complete response; (b) an expanded view of the inset in (a). ----, Stable; r, unstable (−1); – · – · – ·, unstable(+1); × unstable (Hopf).

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We now summarize the system behavior when subject to a single harmonic excitation.When the mean load Tm1 is high, the system does not exhibit any sub-harmonic solutions.The quasi-periodic solutions are restricted to a small region around the first resonance.However, when Tm1 is reduced, a number of regions exhibiting sub-harmonic solutionsshow up; and the most complex dynamic behavior occurs when the mean loads arevery low and the amplitude of excitation Te1 is high. These conditions are typicallyobserved in practical gear rattle problems [10, 11]. Under such conditions, the systemexhibits sub-harmonic, quasi-periodic and/or chaotic solutions, especially at lowfrequencies.

6. RESULTS FOR A DUAL HARMONIC EXCITATION

We now consider the effect of adding a second excitation of frequency 2V with amplitudeTe2 and phase f, where V is the fundamental frequency. The parameter values correspond-ing to the subsequent results that will be presented are listed in Table 3. In Figure 20(a)is shown the response of a heavily loaded system, for k=0, with the presence of the secondexcitation at zero relative phase to the first excitation. The second excitation causes aresonance peak at roughly one-half of the major resonant peak. The region exhibits a jump

T 3

Parameters used in the study of the two-degree-of-freedom system subject to a dual harmonicexcitation with b0 =1, and Te2 at frequency 2V

Figure(s) $ K11

−K21

−K12

K22 % $ C11

−C21

−C12

C22 % b n1, n2 Tm1, Tm2 Te1, Te2, f

20(a), 21 $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·5, 0·25 0·25, 0·1, 0°

20(b), 21 $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·5, 0·25 0·25, 0·1, 120°

22 $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·5, 0·25 0·25, 0·1, 180°

23(a), 26† $ 0·4−0·24

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·6, 0·36 0·25, 0·5 0·25, 0·0, 0°

23(b), 26† $ 0·4−0·24

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·6, 0·36 0·25, 0·5 0·25, 0·1, 0°

24 $ 0·4−0·24

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·6, 0·36 0·25, 0·5 0·5, 0·0, 0°

25 $ 0·4−0·24

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·6, 0·36 0·25, 0·5 0·5, 0·1, 0°

27, 28 $ 0·1−0·06

−0·61·1% $ 0·04

−0·04−0·04

0·06% 0·05 0·9, 0·54 0·25, 0·25 Variable, 0·1, 0°

29 $00 −0·61·1% $00 −0·04

0·06% 0·075 1·0, 0·6 0·25, 0·25 Variable, 0·1, 0°

† For Figure 26 the second excitation frequency is 3V.

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phenomenon due to saddle-node bifurcations. However, when the relative phase f ischanged to 120°, the response around the super-harmonic resonance is now considerablychanged. Flip bifurcations occur, leading to the presence of stable sub-harmonic (P2)solutions that can be seen in Figure 20(b). The regions in which differences in the responseare evident are shown in Figure 21. The phase change causes the saddle-node bifurcationto occur earlier in Figure 21(a) and at a lower frequency in Figure 21(b). The effect in bothcases is to stretch the non-linear regime. If f is now increased to 180°, we notice fromFigure 22(a) that an additional flip bifurcation region occurs around V1 0·8, in additionto the region near the super-harmonic resonance. A further broadening of the non-linearregime, as compared to the f=120° case is shown in Figure 22(b). Hence, one canconclude that the system behavior becomes more complex as f is increased.

Next, we present the results for a moderate k (=0·40) value. The response for a singleharmonic excitation is shown in Figure 23(a). Notice that there are several super-harmonicpeaks, at half, one-third and one-quarter of the major resonant peak frequency. When thesecond excitation Te2 is introduced, the amplitude of the half order resonance peak is muchhigher, but the interesting feature to note is the disappearance of the third and fourthsuper-harmonic peaks. This phenomenon is observed for any magnitude of phase

Figure 25. The frequency response for a lightly loaded oscillator with k=0·4, subject to a dual excitation withTe2 at frequency 2V: (a) complete response; (b) an expanded view of the inset in (a). ----, Stable; r, unstable(−1); – · – · – ·, unstable (+1); ×, unstable (Hopf).

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difference f between the excitations. These results suggest that the presence of theadditional excitation tends to suppress the super-harmonic resonant peaks. The effect isstill observed in Figures 24 and 25, when the excitation amplitude Te1 is double inmagnitude. The introduction of the second excitation Te2 at 2V, reduces the amplitude ofthe response for VE 0·25, as seen in Figures 24(b) and (b). In contrast, when the frequencyof the excitation Te2 is changed to 3V, while maintaining the rest of the parameterscorresponding to Figure 23, observe from Figure 26 that the peaks are enhanced at thelower frequencies, unlike the 2V case. Several super-harmonic resonances appear and theseare at least twice the magnitude of those seen for the single excitation case. Therefore, asymmetric excitation with V and 3V components in a system with a light mean loadenhances the low frequency dynamics.

Finally, we present results of the effect of the amplitude of the fundamental excitationTe1 on the system response, with the second harmonic excitation amplitude Te2 (now backto frequency 2V) fixed at 0·10. In the first case, seen in Figure 27 for V=0·25, observetwo isolated branches of solutions. The larger loop contains two very small regions of Hopfbifurcated solutions. The smaller loop shows the presence of a period-doubling bifurcation,which indicates the possible presence of stable sub-harmonic solutions. However, theregion corresponding to 0·3ETe1 E 0·45 does not converge to any P1 or other higher

Figure 26. A comparison of the frequency response for a moderately loaded oscillator with k=0·4, for (a)the single excitation case and (b) dual excitation with Te2 at frequency 3V. ----, stable; r, unstable (−1);– · – · – ·, unstable (+); ×, unstable (Hopf).

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period solution (stable or unstable). Hence, we can conclude that the only type of solutionsthat can exist are quasi-periodic or chaotic solutions. When the frequency is perturbedslightly to V=0·255, the two loops expand in Figure 28 have an overlapping region.Hence, periodic solutions do exist throughout the parameter range. In all these cases, thenegative Te1 value can be interpreted as a phase shift of f=180°. The last case correspondsto V=0·35, as shown in Figure 29. Stable P1 solutions for small values of Te1, and forintermediate values we see flip bifurcation leading to stable P2 (see Figure 29(b)), as wellas period-doubling bifurcations to P4 and higher period solutions. Hopf bifurcations occurand only stable quasi-periodic solutions are found for high Te1 values. Furthermore, at thisfrequency even moderate values of Te1 can lead to chaotic solutions, through period-dou-bling or quasi-periodic solution bifurcations.

We now summarize the changes in the system dynamics with the introduction of thesecond excitation Te2. For a heavily loaded system (Tm1/Te1 q 1·25), the second excitationmainly introduces a super-harmonic resonance at roughly one-half of the major resonance.Away from this region the response does not vary significantly between the single andmultiple excitation cases. When the system becomes lightly loaded, however, the secondexcitation influences the response over a wider frequency range, and causes morenon-period-1 (P1) solutions. As seen earlier, in some cases, it can even reduce the response

Figure 27. The influence of varying Te1 on the response of a moderately loaded oscillator with k=0·1 andV=0·25: (a) amplitude variation of q1; (b) variation of q1(0). ----, Stable; r, unstable (−1); – · – · – ·, unstable(+1); ×, unstable (Hopf).

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Figure 28. The influence of varying Te1 on the response of a moderately loaded oscillator with k=0·1 andV=0·255: (a) amplitude variation of q1 (b) variation of q1(0). ----, Stable; r, unstable (−1); – · – · – ·, unstable(+1); ×, unstable (Hopf).

amplitude, especially at lower frequencies. In all of these cases, an increase in the phasedifference between the two harmonics of excitation leads to a broadening of the non-linearregime, with more non-P1 solutions.

7. CONCLUSIONS

In this paper the steady state response of a two-degree-of-freedom piecewise non-linearsystem under single and dual harmonic excitations has been analyzed, by using thetechnique of parametric continuation. We believe that we have developed a systematic andefficient way to find system or excitation parameter regimes in which jump phenomena,sub-harmonic super-harmonic and quasi-periodic and chaotic solutions can occur, whichhas not been possible previously by using some of the other available techniques. To thebest of our knowledge this is the first investigation which has examined the influence ofadditional excitation harmonics on the dynamics of such a system. Furthermore, we haveclarified the choice of a suitable damping mdoel for the clearance non-linearity problem;experimental data confirm our modelling strategy. These are the original contributions ofthis paper. Nevertheless, the proposed continuation procedure has limitations in its

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Figure 29. The influence of varying Te1 on the response of a moderately loaded oscillator with k=0·0 andV=0·35. (a) Amplitude variation of q1: ----, stable; r, unstable (−1); – · – · – ·, unstable (+1); ×, unstable(Hopf). (b) Sub-harmonic loop of q1(0); ----, stable P2; – · – · – ·, unstable P2 (+1); · · · ·, unstable P2 (−1);q, stable P1; w, unstable P1 (+1); r, unstable P1 (−1); ×, unstable (Hopf).

application to non-linear systems of very large dimension, because of the tremendousincrease in the total number of equations, arising primarily due to the Jacobiandetermining equations. Alternatively, the quasi-Newton schemes, which lead to a reductionin size of the equations, do not approximate the Jacobian well enough to yield reasonablyaccurate estimates of the stability transitions [31]. Therefore, there is a clear need to refinethese methods to make them more suitable for studying large dimension systems. Futurework is being directed towards such issues.

ACKNOWLEDGMENTS

The authors wish to acknowledge the primary support from the U.S. ArmyResearch Office (URI Grant DAAL 03-92-G-0120; 1992–97; Project Monitor Dr T. L.Doligalski), as well as additional support from the Nissan Technical Center (Japan)and the Center for Automotive Research at The Ohio State University. Thanks are alsodue to Mr T. E. Rook for his help in obtaining the harmonic balance results used in thispaper.

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8. D. C. H. Y and J. Y. L 1987 Transactions of the American Society of Mechanical Engineers,Journal of Mechanisms, Transmissions, and Automation in Design 109, 189–196. Hertziandamping, tooth friction and bending elasticity in gear impact dynamics.

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APPENDIX: LIST OF SYMBOLS

A linear state matrixb0 transition from first to second stage stiffness of the drive trainbci normalizing displacements for the drive train modelB monodromy matrixC�i dimensional viscous damping for the drive train model; i=1, 2Cij non-dimensional viscous damping for the drive train modelf impact force function for the impact pairF general force vectorg non-linear stiffness function for the impact pairG general non-linear function vectorh general non-linear force vectorI�1 first inertia of the drive train modelI�2 second inertia of the drive train modelI�3 third inertia of the drive train modelI�a combined inertia of I�1 and I�2

I�b combined inertia of I�2 and I�3

J Jacobian matrixJ* modified Jacobian matrix past turning pointK�i dimensional stiffness of the drive train model; i=1, 2K�b second stage stiffness of the clutchKij non-dimensional stiffness of the drive train modelni non-linear torque amplitudes of the drive train modelp harmonic index for the second excitation; f=2 or 3q dimensional state vectorq non-dimensional state vectors arc length parameterT external excitation vectort dimensional timea parametera* parameter value at fixed pointb non-dimensional impact damping coefficientb� dimensional impact damping coefficientd Kronecker deltaF state vector magnitude at t0

f relative phase between the excitation harmonicsk ratio of the first and second stage stiffness for clutchl eigenvalue of monodromy matrixu dimensional torsional displacements

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m derivative matrix of the state vector w.r.t. hh initial condition vectorh0 fixed point vectort non-dimensional timet0 fundamental time periodv normalizing frequency for the drive train modelV non-dimensional fundamental excitation frequencyV� dimensional fundamental excitation frequencyC derivative vector of state vector w.r.t. parameter a

Subscriptsm meane alternating components11, 12, 21, 22 indices for stiffness and damping elementsi, j, k vector or matrix indices; i, j, k=1, . . . , 4

Sub-subscriptsi mean or sinusoidal excitation index; i=1, 2

Superscripts1, 1* corrector step in the continuation schemek iteration step in Newton–Raphson scheme( )' derivative with respect to t( ) derivative with respect to t