Chaotic motion of a horizontal impact pair

Journal of Sound and Vibration (1995) 181(2), 231–250

CHAOTIC MOTION OF A HORIZONTALIMPACT PAIR

R. P. S. Han and A. C. J. Luo

Department of Mechanical and Industrial Engineering, University of Manitoba,Winnipeg, Manitoba, Canada R3T 2N2

and

W. Deng

Department of Mechanical Engineering, Sichuan Institute of Light Industry andChemical Technology, Zigong, Sichuan 643033, People’s Republic of China

(Received 22 April 1992, and in final form 6 December 1993)

A theory for a system with discontinuities and applied to the impact analysis of ahorizontal impact pair is presented. Mappings for four switch-planes are defined and fromthese several impact models are developed. As a case of special interest, the case of a steadystate, periodic two-impacts/N-cycles motion is studied in greater detail. Numericalsimulations of the various models are also given. The results show that the ensuing chaoticbehavior can be either regular with period-doubling bifurcation or random with other typesof bifurcation. The former refers to chaos, where its mathematical structure is regular, whilethe latter refers to one with a random mathematical structure.

1. INTRODUCTION

In recent years, there has been a rapid development in the understanding and modelling

of collisions or impacts caused by discontinuous motion. Such motions exist in a wide

variety of engineering applications, particularly in mechanisms and machines where they

occur in the clearances of these mechanical systems. The physical process during impact

is complex and highly non-linear, but it is necessary to be able accurately to model the

dynamics of an impacting system, so as to minimize adverse effects such as pitting, scoring

and high noise levels. The most basic model of an impacting system available for

investigation is an impact pair [1–4]. Other impact models include the acceleration or

impact dampers [6–11] and the impact oscillators [13–15].

This paper is concerned with the dynamical modelling of a horizontal impact pair,

subjected to a periodic base excitation. Using the theory for discontinuous motion,

mappings for four switch-plane are defined, and from these, four impact models are

developed: Model I, Model II, Model III and the generalized Model III. The most

studied impact model, that is steady state, periodic two-impacts in exactly N cycles of

base motion, represents a generalization of Model I. Starting conditions for this situation

are given. Numerical simulations for all the formulated models are also presented.

The results indicate that the chaos can be either regular with period-doubling bifurcation

or random with other types of bifurcation. The former refers to chaos, where its

mathematical structure is regular, while the latter refers to one with a random mathemati-

cal structure.

231

0022–460X/95/120231+20 $08.00/0 � 1995 Academic Press Limited

r. p. s. han et al.232

2. SYSTEM DESCRIPTION

In Figure 1 is shown an impact pair consisting of a secondary mass (rigid ball) Ms ,

moving freely within a gap d in a primary mass Mp , where Mp �m. Assume the excitation

of the primary mass E(t, Ai ) is harmonic, namely,

E(t, Ai )= �n

i=1

Ai sin (�it+ �), (1)

where Ai , �i and � are the ith amplitude, the ith frequency and the phase angle,

respectively. A list of symbols used is given in Appendix B. Let (x, t) denote the absolute

displacement of the secondary mass and (y, t) its displacement relative to Mp . If

(·)=d( )/dt refers to the time derivative, we have

x= y+E, x= y+E� , x= y+E� . (2)

2.1. basic theory for systems with discontinuities

Consider a non-linear dynamical system described by

x= f(x, t; p), (3)

where x= x(t) � Rn is a vector function of time t and p characterizes the system parameters

Ai , �i , �, etc. All results concerning the existence, stability and bifurcation of periodic

solutions of equation (3) can be applied to a system with discontinuities by partitioning

the state space TM (�Rn) into the open subsets of Xi�TM [16]. The vector field f,

f: TM�TM, (4)

must be Ck-continuous. The mechanical model on each subset Xi is described by a vector

field fi (for example, the process between two impacts of the ball in the impact pair). Hence,

f(x, t; p)=��

�

�

�

f1(x, t; p),

f2(x, t; p),...

fN (x, t; p),

� x � X1,

� x � X2,...

� x � XN .

(5)

The boundary �Xi of subset Xi is the discontinuous plane, or switch-plane, of codimension

1. In this case, the state space is

TM= �N

i=1

Xi . (6)

Figure 1. The mechanical model of an impact pair.

horizontal impact pair 233

This switch-plane can be described by the solution set of non-linear algebraic equations

given by

gi (x)=0, gi : TM�R. (7)

For the trajectories, the switch-planes are the Poincare sections. The trajectory in the state

space is unique and thus a law for the discontinuities (an impact law) is necessary. That

is,

�i : g−1i (0)�g−1

i (0), x−(k) �x+

(k), (8)

where �i maps a point x−(k) before the discontinuity to a point x+

(k) immediately after the

discontinuity. This impact law is usually determined by the physics of the problem.

2.2. impact analysis

The mechanical model of the horizontal impact pair is shown in Figure 1. For simplicity,

a one-term expression is assumed and the external excitation becomes

E=A sin (�t+ �). (9)

Substituting equation (9) into equation (2) and neglecting friction yields the equation of

motion in relative co-ordinate system: that is,

y=A�2 sin (�t+ �). (10)

Integrating equation (10) leads to

y=−A� cos (�t+ �)+ y0 +A� cos (�t0 + �),

y=−A sin (�t+ �)+ [y0 +A� cos (�t0 + �)](t− t0)+A sin (�t0 + �)+ y0.(11)

Thus, the equation of motion for state subset Xi+1 is

y=−A� cos (�t+ �)+ y+i +A� cos (�ti + �),

y=−A sin (�t+ �)+ [y+i +A� cos (�ti + �)](t− ti )+A sin (�ti + �)+ y+

i ,(12)

in which t � [ti , ti+1], y(ti+1)= y−i+1, y(ti+1)= y−

i+1. Note that ( )− and ( )+ denote,

respectively, before and immediately after an impact. The switch-plane or the

discontinuous subset �Xi+1 is

y−i+1 − y+

i =−A sin (�ti+1 + �)+ [y+i +A� cos (�ti + �)](ti+1 − ti )

+A sin (�ti + �). (13)

An impact is deemed to occur whenever

�y+i �= �y−

i �= d/2. (14)

Assume the simplest impact law for �i ; namely, one that considers the impact process as

instantaneous and employs the concept of a constant coefficient of restitution e to model

the energy loss during impact. The relative velocities before and after an impact are related

by

y+i =−ey−

i . (15)

In view of equations (14) and (15), we have the discontinuous plane, or switch-plane,

yi+1 =−A� cos (�ti+1 + �)+ [−ey−i +A� cos (�ti + �)],

y−i+1 − y−

i =−A sin (�ti+1 + �)+ [−ey−i +A� cos (�ti + �)] (16)

×(ti+1 − ti )+A sin (�ti + �). (17)


For convenience, the minus superscript used for the relative co-ordinates will be dropped

in the ensuing derivation: that is, y− and y− will simply be denoted by y and y, respectively.

The switch-plane is now defined as

�= {(ti , yi )��yi �= d/2, t mod (2�/�)}. (18)

Using equation (6), for every discontinuity we have

�=�+ ��−, (19)

where

�+ = {(ti , yi )�yi = d/2, t mod (2�/�)} (20)

and

�− = {(ti , yi )�yi =−d/2, t mod (2�/�)}. (21)

In terms of the absolute co-ordinate reference frame, the corresponding expression for

equation (18) is

�= {(x, x)��yi �= d/2}=�+ ��−, (22)

or

�= {(t, x)��yi �= d/2, t mod (2�/�)}=�+ ��−, (23)

where �+ and �− can be similarly defined.

3. POSSIBLE IMPACT MODELS

Based on the formulation developed previously, four new mappings can be defined.

These are

P1: �+��−, P2: �−��+, (24, 25)

P3: �+��+, P4: �−��−. (26, 27)

The switch-plane �+ (or �−) can be interpreted as the Poincare section. Hence, the

Poincare mapping P can be defined for several models.

3.1. model i

The commutative diagram for Model I is depicted in Figure 2(a), with the physical

interpretation given in Figure 2(b). As shown, this model corresponds physically to the

situation of just one impact per side, or more commonly known as two alternating impacts.

It could be either an impact on the left side, followed by an impact on the right (LR) or

Figure 2. Model I—LR impact sequence: (a) commutative diagram; (b) physical model.


Figure 3. Model I—RL impact sequence: (a) commutative diagram; (b) physical model.

an impact on the right side, followed by an impact on the left (RL). The latter is depicted

in Figure 3. Hence, there are two possible cases and since these two cases are identical,

we will consider only the LR impact sequence for further discussion. For this situation,

we have �+�P�+, so that the mapping becomes

P=P2 � P1. (28)

This classical, two alternating impacts per cycle of the base motion is perhaps the most

studied steady state impact motion. A more general investigation involving two impacts

in exactly N cycles of base motion will be presented later. For any x � �− we have

Px= x, (29)

so that

P2 � P1 = I, (30)

where I is an identity mapping, i.e., mappings P1 and P2 are the inverse mapping of each

other. If the period of the Poincare mapping is 2, i.e., the new steady state motion has

twice the period of the old one, we have a period-doubling bifurcation. Its commutative

diagram is shown in Figure 4. This new steady state motion is now given by

P(2)x= x; (31)

that is,

P(2) =P2 � P1 � P2 � P1. (32)

A period-doubling bifurcation is attained at P(2) = I. Similarly, for a period-3 bifurcating

motion, we have

P(3)x= x; (33)

that is,

P(3) =P2 � P1 � P2 � P1 � P2 � P1. (34)

Figure 4. The commutative diagram for Model I with a period-doubling bifurcation.


Again, P(3) = I determines the period-3 bifurcation. If the motion defined by equation (34)

exists, the system will exhibits chaos. The most general motion is given by

P(k)x= x; (35)

that is,

P(k) = (P2 � P1) � · · · � (P2 � P1), (36)

��k terms

As before, bifurcation occurs at P(k) = I. The stability of the resulting new motion is

presented next and the discussion will be limited to codimension 1 bifurcations. For a more

general and detailed description, one could refer to books on non-linear vibrations by

Guckenheimer and Holmes [17] and Wiggins [18]. If DP denotes the Jacobian of Pevaluated at (ti , yi ), the following relationships are applicable:

DP=DP2 · DP1,

DP(2) =DP2 · DP1 · DP2 · DP1,

DP(3) =DP2 · DP1 · DP2 · DP1 · DP2 · DP1,

.

.

.

.

.

.

DP(k) = (DP2 · DP1) · · · (DP2 · DP1), k=1, 2, 3, . . . , (37)

��k terms

in which DP is defined in Appendix A. the eigenvalues of DP(k) are

�1,2 = {tr [DP(k)]��tr [DP(k)]2 −4 det [DP(k)]}/2, (38)

where tr [DP(k)] and det [DP(k)] represent the trace and determinant of DP(k). The type of

bifurcation depends on the sign of the eigenvalues. If one of the eigenvalues is −1, then

only a period-doubling bifurcation can occur. The condition for this kind of bifurcation

is

tr [DP(k)]+det [DP(k)]+1=0. (39)

On the other hand, if one of the eigenvalues is +1, then one of the three kinds of

bifurcation can occur; a saddle-node bifurcation, a transcritical bifurcation or a pitchfork

bifurcation. The condition for any of these three types of bifurcation is

tr [DP(k)]=1+det [DP(k)]. (40)

If the eigenvalues form a complex conjugate pair exiting the unit circle, then we have a

Poincare–Andronov–Hopf bifurcation. However, it is quite easy to show that no Poincare–

Andronov–Hopf bifurcation to doubly periodic motion can occur in an impact pair.

Finally, if the eigenvalues are �1, the motion is a stable one.

3.2. model ii

For this model, the commutative diagram is given in Figure 5(a) and the corresponding

physical model in Figure 5(b). Physically, the impact process is one impact per side, per

cycle. Since the impact can be on the right side (R) or on the left side (L), we again have

two possibilities. However, the discussion will be restricted to just one case, as these two

situations are the same. As before, the motion is described by

Px= x=P3x (41)


Figure 5. Model II: (a) commutative diagram; (b) physical model.

and its periodic solution is characterized by

t1 =NT+ t0, y1 = y0 =0, �t0 + �= �/2, (42)

where T=2�/� is the period. From the result of Mapping P3 listed in Appendix A for

t � (t0, t1), the solution simplifies to

y=−A� cos (�t+ �), �y�= �−A sin (�t+ �)+A+ d/2�� d/2; (43)

that is,A� 0. (44)

However, E=A sin (�t+ �), where A is always positive. This conflicting result implies

that this model is physically unattainable, and therefore constitutes a non-viable solution.

The ball actually stops before reaching the sides of the impact pair. Similarly, for the

period-doubling bifurcation motion, the commutative diagram is given in Figure 6. Again,

we haveP(2)x=P(2)

3 x=P3 � P3x= x, (45)

where from Appendix A, for Mapping P3, we have

(1+ e)(y1 + y0)=0, t1 =NT/2+ t0, t2 =NT+ t0. (46)

Since y1, y0 � �+ and e� 0, we obtain y1 = y0 = y2 =0. Once again, this constitutes a

non-viable solution and, therefore, the motion is physically unattainable. This conclusion

of a non-viable motion remains valid, even for the general case of P(k)x= x.

3.3. model iii

In Figure 7 is shown Model III: its commutative diagram in Figure 7(a) and the

corresponding physical interpretation in Figure 7(b). The impact process as represented

here contains four impacts per cycle, comprising of either an RLLR or an LRRL impact

sequence. Since these two cases are the same, the discussion here will be limited to just

RLLR sequence. The mapping and the linearized matrix are

P=P2 � P4 � P1 � P3, DP=DP2 · DP4 · DP1 · DP3 (47, 48)

and, for the general model, it is

P(k) = (P2 � P4 � P1 � P3) � · · · � (P2 � P4 � P1 � P3), k=1, 2, 3, . . . . (49)

��k terms

Figure 6. The commutative diagram for Model II with a period-doubling bifurcation.


Figure 7. Model III—RLLR impact sequence: (a) commutative diagram; (b) physical model.

It should be mentioned that in going from an LR sequence (Model I) to an RLLR sequence

(Model III) it is possible to generate two intermediate impact models: an LLR impact

sequence and an RRL impact sequence. These are sketched in Figure 8. Their mappings

are P=P2 � P4 � P1 and P=P1 � P3 � P2, respectively. A catastrophe exists in going from

Model I into these two intermediate models, and from the intermediate models to Model

III. Since the bifurcation condition is similar to that of Model I, no further discussion is

necessary. The general mapping for the LLR motion is

P(k) = (P2 � P4 � P1) � · · · � (P2 � P4 � P1), k=1, 2, 3, . . . , (50)

��k terms

and for the RRL motion we have

P(k) = (P1 � P3 � P2) � · · · � (P1 � P3 � P2), k=1, 2, 3, . . . . (51)

��k terms

3.4. generalized model iii

It is desirable to generalize Model III, to handle the situation of uneven multiple impacts

per side. This is shown in Figure 9, where we have m impacts on the right side and nimpacts on the left, resulting in an impact sequence described by

R · · · R L · · · L R.

��m n

Figure 8. Intermediate models for Model III: (a) LL5 motion; (b) RRL motion.


Figure 9. Generalized Model III—R · · · Rm

L · · · Ln

R impact sequence: (a) commutative diagram; (b) physicalmodel.

The motion is defined by

P=P2 � P(n)4 � P1 � P(m)

3 , DP=DP2 · DP(n)4 · DP1 · DP(m)

3 . (52, 53)

This model could have also been equally described an

L · · · L R · · · R L��

m n

impact sequence. The total number of impacts per cycle of the base motion for either

description is (m+ n+1). The model bifurcation is given by

P(k) = (P2 � P(n)4 � P1 � P(m)

3 ) � · · · � (P2 � P(n)4 � P1 � P(m)

3 ), k=1, 2, 3, . . . , (54)

��k terms

and for the most general case of a varying, uneven multiple impacts per side, we have

P(k) = (P2 � P(n1)4 � P1 � P(m1)

3 ) � · · · � (P2 � P(nk )4 � P1 � P(mk )

3 ). (55)

in which mi and ni consist of a set of positive integers, i=1, 2, . . . , k, that can be randomly

selected. Equations (54) and (55) have similar bifurcation conditions, as discussed

previously.

4. STEADY STATE, PERIODIC TWO-IMPACTS/N-CYCLES MOTION

As a special case of Model I, the steady state, periodic two-impacts in exactly N cycles

of base motion is discussed here. A considerable amount of research for this type of motion

has already been conducted, particularly for the case of symmetric two equispaced impacts

per cycle motion (i.e., N=1) [1–12, 15]. Note that the term symmetric as used here implies

that N is an old integer. It will be shown in the following derivation that it is not necessary

to invoke this assumption. This is in contrast to previous work for the two impacts per

N cycles of motion [1, 2], where it is assumed from the outset of their analysis that N is

an odd integer. Also, from the bifurcation analysis for Model I, it is clear that under

appropriate starting conditions, the solutions begin as equispaced solutions which then

experience either a period-doubling bifurcation or some other types of bifurcation, such

as the saddle-node bifurcation, a transcritical bifurcation or a pitchfork bifurcation, into

unequispaced solutions enroute to chaos.


Choosing the initial Poincare section to be defined by �−, which implies yi � 0, we have

from equation (29) for periodic motion:

ti+2 =2N�/�+ ti , yi+2 = yi , yi , yi+2 � 0. (56)

Using the results given in Appendix A, the solutions for t= ti+1 are

yi+1 =−A� cos (�ti+1 + �)+ [−eyi +A� cos (�ti + �)], (57)

d=−A[sin (�ti+1 + �)− sin (�ti + �)]+ [−eyi +A� cos (�ti + �)](ti+1 − ti ), (58)

and, for t= ti+2, they are

yi+2 =−A� cos (�ti+2 + �)+ [−eyi+1 +A� cos (�ti+1 + �)], (59)

−d=−A[sin (�ti+2 + �)− sin (�ti+1 + �)]+ [−eyi+1+A� cos (�ti+1 + �)](ti+2 − ti+1).

(60)

Some interesting observations can be obtained from the set of solutions depicted in

equations (57)–(60). Adding equations (57) and (59), and simplifying via equation (56)

yields

(1+ e)(yi + yi+1)=0 � yi =−yi+1, (61)

Substituting equation (61) into equation (57) and for e� 1, we have

y1 =A�

1− e[cos (�ti+1 + �)− cos (�ti + �)]. (62)

Similarly, adding equation (58) and equation (60) leads to

[−eyi +A� cos(�ti + �)](ti+1 − ti )+ [−eyi+1 +A� cos(�ti+1 + �)](2N�/�+ ti − ti+1)

=0.

(63)

Substituting equation (62) into equation (63) enables ti+1 to be evaluated numerically and

the parameter manifold can then be obtained in the parameter space. Consider the case

of an even valued N, i.e., N=2, 4, 6, . . . . For equispaced periodic motion we have

ti+1 =N�/�+ ti , (64)

Substituting into equation (62) yields

yi = yi+1 =0. (65)

Observe that the initial conditions represented by equation (65) are identical to those of

Model II in the second of equations (42). Hence, for this case of an even valued N, the

resulting model is that of Model II, a non-viable model indeed! This proves what has been

suspected all along that it is physically impossible to have the type of motion characterized

by an even valued N. Next, we will examine the motion for the case of when N is odd.

For this situation we have, from equation (62),

yi =−[2A�/(1− e)] cos (�ti + �), (66)

and, from equation (58),

d=2A sin (�ti + �)− [(1+ e)N�/2�]yi . (67)

Solving for the Nth amplitude AN from equations (66) and (67) we obtain

AN =(1/2)�[(1− e)yi /�]2 + [d+(1+ e)N�yi /2�]2, yi � 0 N=1, 3, 5, . . . . (68)


For the case of two alternating impacts per cycle, namely at N=1,

A1 = (1/2)�[(1− e)yi /�]2 + [d+(1+ e)�yi /2�]2, yi � 0. (69)

For a perfectly elastic impact, i.e., e=1, the results are (N=1, 3, 5, . . .)

yi =−yi+1, d=2A sin (�ti + �)−N�yi /�,

cos (�ti+1 + �)= cos (�ti + �), cos (�ti+1 + �)=−cos (�ti + �). (70)

If one further assumes that the impact is equispaced, namely equation (64) applies, then

equation (70) simplifies to (N=1, 3, 5, . . .)

yi =−yi+1, d=�2A−N�yi /�. (71)

If �+ is chosen as the initial Poincare section which implies yi � 0 (in order to impact on

�+), equation (68) changes to

AN = 12�[(1− e)yi /�]2 + [−d+(1+ e)N�yi /2�]2, yi � 0, N=1, 3, 5, . . . . (72)

Note that equations (68) and (72) can be summarized into one general equation; namely,

AN = 12�[(1− e)yi /�]2 + [d−(1+ e)N��yi �/2�]2, �yi �� 0, N=1, 3, 5, . . . . (73)

We shall now discuss some implications of the equation derived here. From equation

(66), because yi � 0, cos (�ti + �)� 0: that is, −�/2+2m�� (�ti + �)� 2m�+ �/2,

m=0, 1, 2, . . . . From equation (67) for 2m�� (�ti + �)� 2m�+ �/2, d+(1+ e)N�yi /

(2�)� 0; otherwise, d+(1+ e)N�yi /(2�)� 0.

In Figure 10 are shown the parameter manifolds of the excitation amplitude A versus

the excitation frequency � for varying cycles of base motion N. The following parameters

are assumed: e=0·8, d=20·0 and an initial impact velocity yi =−20. Note that d and

yi are normalized quantities. The solid curves correspond to the condition

2m�� (�ti + �)� 2m�+ �/2, while the dashed curves represent the situation −�/

2+2m�� (�ti + �)� 2m�. The directions of the relative acceleration and relative

velocity after impact are the same for the former situation, and are opposite for the latter

case.

Figure 10. The parameter manifolds of the excitation amplitude A versus the excitation frequency � fore=0·8, d=20·0 and yi =−20.


Observe from the solid curves that, for a given �, the number of impacts per cycle is

greater than two if the excitation amplitude A�A1, less than two if A�A1 and exactly

two if A=A1. As for the dashed curves, the opposite is true. The number of impacts per

cycle for this motion is less than, greater than or equal to two for A�A1, A�A1 and

A=A1, respectively. Also, as �� the excitation amplitude An�d/2, regardless of the

number of impacts/N cycles of base motion. This is obvious from the asymptotic character

of these curves at very high frequencies. Mathematically, this behaviour can also be

deduced from equation (73). It should be mentioned that, to ensure periodic motion, the

amplitude A cannot exceed d/2. If A� d/2 the motion will exhibit bifurcation and chaos.

The other asymptotic line is �=0.

In Figure 11, the parameter manifolds of the excitation amplitude A versus the initial

impact velocity are sketched for e=0·8, d=20·0 and �=30. Note the symmetry of the

plot, with the right side representing impacts on the (R) and the left side for impacts on

the (L).

5. NUMERICAL SIMULATIONS

Numerical simulations in the form of time–displacement curves, time–velocity curves,

phase diagrams, switch-planes and Poincare mapping sections for the various formulated

models are now presented. Note that some of the parameters and/or initial conditions in

the figures are computed rather than chosen arbitrarily. This allows us to perform the

numerical simulation in a meaningful manner. In particular, we have computed the initial

conditions y0 and y0 and the gap d from analytical expressions derived previously.

Bifurcation and chaos are then obtained by varying one of the parameters.

In Figure 12 are shown the phase plane plots for the simplest but most studied case for

impact pairs, the steady state, periodic two-impact per cycle model. Note that the

alternating impacts are equispaced, namely, ti+1 − ti =N�/�, and exact solutions are

available, for example in references [1, 4, 9]. All parameters, initial conditions, including

values of the excitation amplitude necessary for initiating this motion must satisfy the

equations derived in section 4.

As the forcing amplitude is increased, the impact motion becomes unequispaced; that

is, one assumes only ti+2 − ti =2N�/�. This is shown in Figure 13. The governing

Figure 11. The parameter manifolds for the excitation amplitude A versus the impact velocity yi for e=0·8,d=20·0 and �=30.


Figure 12. Simulation of Model I: an equispaced, periodic two-impact per cycle motion: �= �, A=1·0,d=9·42497, e=0·5, y0 =−4·712485, y0 =−12·56636 and �=0·0. (a) Time–displacement; (b) time–velocity;(c) and (d) are phase planes.

Figure 13. Simulation of Model I: an unequispaced, quasi-periodic two-impact per cycle motion: �= �,A=2·0, d=9·42497, e=0·5, y0 =−4·712485, y0 =−12·56636 and �=0·0. (a) Time–displacement; (b)time–velocity; (c) and (d) are phase planes.


equations are still those of Model I. The time–displacement curve is plotted in Figure 13(a)

and it is quite evident that the motion is quasi-periodic. Observe that the alternating

impacts are regular as opposed to random impacts shown in the time–displacement curve

of Figure 14(a). The time–velocity plot and phase plane plots are depicted in Figures

13(b)–(d). Approximate stability results for unequispaced impact motion have been

presented by Bapat et al. [1].

Further increase of the forcing amplitude results in chaos. This is shown in Figure 14.

From the phase plane plots in Figures 14(c) and (d), it is clear that the motion is chaotic.

The plots here pertain to the intermediate model, transiting from Model I to Model III.

The time–displacement curve in Figure 14(a) clearly shows the occurrence of random chaos.

The term random chaos, as used here, indicates a random mathematical structure, whereas

regular chaos implies that the mathematical structure is regular. The time–velocity plot,

the switch-plane plots and the Poincare mapping sections are illustrated in Figures 14(b)

and 14(e)–(h). The Poincare sections for the left and right sides of the impact pair are

indicated in Figures 14(g) and (h), as shown.

The motion of the generalized Model III is sketched in Figure 15. Here, the impact

sequence is that of m=2, n=3 impacts, as shown in the time–displacement curve of

Figure 15(a). The impact is regular and the motion is periodic. It is also obvious from the

phase plane plots in Figures 15(c) and (d) that there is no chaos in the motion. The

time–velocity plot is given in Figure 15(b).

In Figure 16 it is shown that, under high frequency excitation, the impact motion of

the generalized Model III is chaotic, as displayed by the phase plane plot of Figures 16(c)

and (d). Observe that, from the time–displacement curve of Figure 16(a), a quasi-regular

chaos with bifurcation is obtained. The time–velocity plot together with the accompanying

Fig. 14 (a)–(d)


Fig. 14 (e)–(h)

Figure 14. Simulation of intermediate model in Model III: �= �, A=8·0, d=9·42497, e=0·5,y0 =−4·712485, y0 =−12·56636 and �=0·0. (a) Time–displacement; (b) time–velocity; (c) and (d) phase planes;(e) and (f) switch-planes; (g) Poincare mapping for �−; (h) Poincare mapping for �+.

switch-plane plots and Poincare mapping sections are given in Figures 16(b, e–f and g–h),

respectively.

Random chaos for the generalized Model III is depicted in Figure 17. This can be seen

in the time–displacement curve given in Figure 17(a). Observe that the impact sequence

is random and the motion is chaotic, as shown by the phase plane plots in Figures 17(c–d).

As before, the time–velocity plot together with the accompanying switch–plane plots and

Poincare mapping sections are given in Figures 17(b, e–f and g–h), respectively.

6. SUMMARY AND CONCLUSIONS

A theory for a system with discontinuities, applied to the impact analysis of a horizontal

impact pair, is presented. Mappings for four switch-plane are defined and from these

several impact models are developed: Model I, Model II, Model III and generalized Model

III. As a generalization of Model I, the case of a steady state, periodic

two-impacts/N-cycles motion is given a detailed treatment. Numerical simulations of these

models are given. The simulation results show that under appropriate starting conditions,

the solutions begin as equispaced solutions which then experience either a period-doubling

bifurcation or some other types of bifurcation, such as the saddle-node bifurcation, a

transcritical bifurcation or a pitchfork bifurcation, into unequispaced solutions en route

to chaos. This chaos can be either regular and random; the former refers to chaos where

its mathematical structure is regular, while the latter refers to one with a random

mathematical structure.


Figure 15. Simulation of regular impact in generalized Model III: �=10�, A=2·0, d=1·0, e=0·8,y0 =−0·5, y0 =−14·48 and �=0·0. (a) Time–displacement; (b) time–velocity; (c) and (d) are phase planes.

Fig. 16 (a)–(d)


Fig. 16 (e)–(h)

Figure 16. Simulation of regular chaos in generalized Model III: �=20�, A=10·0; d=1·0, e=0·8,y0 =−0·5, y0 =−2·0 and �=0·0: (a) Time–displacement; (b) time–velocity; (c) and (d) phase planes; (e) and(f) switch-planes; (g) Poincare mapping for �−; (h) Poincare mapping for �+.

Fig. 17 (a)–(d)


Fig. 17 (e)-(h)

Figure 17. Simulation of random chaos in generalized Model III: �=200�, A=10·0, d=10·0, e=0·8,y0 =−5·0, y0 =0·0 and �=1·047196. (a)–(h) as in Figure 16.

ACKNOWLEDGMENTS

This work was jointly supported by funding from the Natural Sciences and Engineering

Research Council of Canada, and the Sichuan Applied Science and Technology

Foundation of the People’s Republic of China.

REFERENCES

1. C. N. Bapat, N. Popplewell and K. McLachlan 1983 Journal of Sound and Vibration 87,19–40. Stable periodic motions of an impact-pair.

2. C. N. Bapat and C. Bapat 1988 Journal of Sound and Vibration 120, 53–61. Impact-pair underperiodic excitation.

3. W. Deng 1964 Chinese Journal of Mechanical Engineering 12, 83–93. Action of the impactdamper and determination of its basic parameters.

4. M. S. Heiman, P. J. Sherman and A. K. Bajaj 1987 Journal of Sound and Vibration 114, 535–547.On the dynamics and stability of an inclined impact pair.

5. M. S. Heiman, A. K. Bajaj and P. J. Sherman 1988 Journal of Sound and Vibration 124, 55–78.Periodic motions and bifurcations in dynamics of an inclined impact pair.

6. P. Lieber and D. P. Jensen 1945 Transactions of the American Society of Mechanical Engineers67, 523–530. An acceleration damper: development, design and some applications.

7. C. Grubin 1956 Transactions of the American Society of Mechanical Engineers, Journal ofApplied Mechanics 23, 373–378. On the theory of the acceleration damper.

8. G. B. Warburton 1957 Transactions of the American Society of Mechanical Engineers, Journalof Applied Mechanics 24, 322–324. Discussion of on the theory of the acceleration damper.

9. S. F. Masri and T. K. Caughey 1966 Transactions of the American Society of MechanicalEngineers, Journal of Applied Mechanics 33, 587–592. On the stability of the impact damper.


10. A. E. Kobrinskii 1969 Dynamics of Mechanisms with Elastic Connections and Impact Systems.London: Iliffe.

11. Bapat and S. Sankar 1985 Journal of Sound and Vibration 99, 85–94. Single unit impact damperin free and forced vibrations.

12. A. B. Pippard 1985 Response and Stability: an Introduction to the Physical Theory. Cambridge:Cambridge University Press.

13. M. Senator 1970 Journal of the Acoustical Society of America 47, 1390–1397. Existence andstability of periodic motions of a harmonically forced impacting system.

14. S. W. Shaw and P. J. Holmes 1983 Journal of Sound and Vibration 90, 129–155. A periodicallyforced piecewise linear oscillator.

15. S. W. Shaw 1985 Transactions of the American Society of Mechanical Engineers, Journal ofApplied Mechanics 52, 453–464. The dynamics of a harmonically excited system having rigidamplitude constraints, part 1: subharmonic motions and local bifurcations; part 2: chaoticmotions and global bifurcations.

16. E. Reithmeier 1990 Proceedings of the IUTAM Symposium on Nonlinear Dynamics inEngineering Systems, Stutgart, 1989, 249–256, Berlin: Springer-Verlag. Periodic solutions onnonlinear dynamical systems with discontinuities.

17. J. Guckenheimer and P. Holmes 1983 Nonlinear Oscillations, Dynamical Systems andBifurcations of Vector Fields. New York: Springer-Verlag.

18. S. Wiggins 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York:Springer-Verlag.

APPENDIX A: SOLUTIONS FOR MAPPINGS

The solutions for the four mappings and the evaluation of the Jacobian of P are listed

here.

Mapping P1:

yi+1 =−A� cos (�ti+1 + �)+ [−eyi +A� cos (�ti + �)], (A.1)

−d=−A sin (�ti+1 + �)+ [−eyi +A� cos (�ti + �)](ti+1 − ti )+A sin (�ti + �). (A.2)

Mapping P2:


d=−A sin (�ti+1 + �)+ [−eyi +A� cos (�ti + �)](ti+1 − ti )+A sin (�ti + �). (A.4)

Mappings P3 and P4:


0=−A sin (�ti+1 + �)+ [−eyi +A� cos (�ti + �)](ti+1 − ti )+A sin (�ti + �). (A.6)

The Jacobian of mapping P1, DP1, is defined as follows:

DP1 =��(ti+1, yi+1)

�(ti , yi ) �(ti,yi)

=��

�

�

�ti+1

�ti

�yi+1

�ti

�ti+1

�yi

�yi+1

�yi

��

�

�

�(ti,yi)

, (A.7)

where

�ti+1

�ti=

1

yi+1

[−eyi +A�2 sin (�ti + �)(ti+1 − ti )],

�ti+1

�yi=

eyi+1

(ti+1 − ti ),


�yi+1

�ti=

1

yi+1

A�2 sin (�ti+1 + t)[−eyi +A�2 sin (�ti + �)(ti+1 − ti )]−A�2 sin (�ti + �),

�yi+1

�yi=

eyi+1

(ti+1 − ti )A�2 sin (�ti+1 + �)− e.

The Jacobian for the other mappings, P2–P4, are defined in the same manner. The time

of impact for mapping P1, ti+1 and ti are solved from equations (A.1) and (A.2). Similarly,

the impact times ti+1 and ti for mapping P2, and for mappings P3 and P4, are determined

from equations (A.3) and (A.4) and equations (A.5) and (A.6), respectively. Once all these

quantities have been computed, the Jacobian can then be evaluated.

APPENDIX B: NOMENCLATURE

A amplitude of excitationAN amplitude of steady motion of two impacts for period NCk k order differentiable functiond impact gapDP Jacobian of the mapping Pe coefficient of restitutionE excitation of the primary massf vector fieldf vector functionI identity mappingMp primary massMs secondary massm number of impacts on the right siden number of impacts on the left sideP Poincare mappingRn n-dimensional real spaceT period of excitationt time variableti impact time at the ith impactTM state spacex element of switch-plane �+ or �−

x absolute displacement of impact ballxi absolute displacement of impact ball at the ith impactXi Open subset of TM (see equation (6))y relative displacement of impact ballyi relative displacement of impact ball at the ith impact� switch-plane� phase angle� frequency of excitation�i ith frequency of excitation

Chaotic motion of a horizontal impact pair

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