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Transition to chaotic motion in mechanical systems with impacts
Journal of Sound and Vibration (1992) 154(l), 95-115
TRANSITION
TO CHAOTIC MOTION IN MECHANICAL SYSTEMS
WITH
IMPACTSt
F. PETERKA Institute of Thermomechanics of the Czechoslovak Academy of Sciences, DoleJ%kova 5, 182 00 Praha 8, C.S.F.R.
J. VAC~K Institute of Information Theory and Automation of the Czechoslovak Academy of Sciences, Pod vodbrenskou &ii 4, 182 08 Praha 8, C.S.F.R.
(Received 8 November 1988, and accepted in revised form 18 February 199 1)
The aim of this work is to give a short explanation of the laws of a simple mechanical impact system, to elucidate the transitions to chaotic motion and to illustiate the periodic and chaotic motions of more complicated systems. In view of the nature of this work and
the broad scope of the problems involved, only the results of theoretical and analogue studies are presented here, with references given to more detailed analyses.
1. INTRODUCTION The performance of some types of machines and equipment is accompanied by the impacts of bodies. The principle of operation of vibration hammers, machinery for driving, compacting, milling and forming, vibratory conveyers, platforms and shaking grizzlies, shock absorbers, etc., is based on the impact action for moving bodies. With other equipment, e.g., mechanisms with clearances, heat exchangers, fuel elements of nuclear reactors, etc., impacts also occur, but they are undesirable as they bring about failures, strain, shorter service life and increased noise levels. Investigations of the dynamics of mechanical systems with impacts were launched in the middle of the present century with the aim of analyzing the existence and stability of periodic motions, optimization of system parameters, parametric vibrations, transient phenomena and processes, self-excited oscillations, random vibration, protection against impact effects, etc. A survey of publications and results of research can be found in several monographs [l-8]. This work deals with periodically excited impact systems. Mechanical systems with impacts are strongly non-linear due to the existence of one or more impact pairs of bodies. In the case of rigid impacts it is often possible to use the simple impact pair model (see Figure 1), where the force connection F between the bodies in its dependence on the relative displacement x2--xl contains a sudden change of the derivative dF/d& - xi). According to Newton’s elementary theory of direct and centric impact the before-impact velocities f,_, i2- of the bodies change by jumps and acquire the values of the after-impact velocities R,+ , i2+ according to the following relations, in t This paper is a revised summary lecture “Causes of Chaotic Motion in Mechanical Systems with Impacts” presented at the EUROMECH 242 Colloquium “Application of Chaos Concepts to Mechanical Systems”, Wuppertal, F.R.G., 26-29 September 1988. 95 0022-460X/92/070095 f21 $03.00/O
0 1992 Academic Press Limited
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Figure 1. Model of the impact pair.
which ml, m2 are the masses of the impacting bodies and R = -(i,+ - i2+)/(i, _ -&) the coefficient of restitution: k,+ =ip-
m2(i, - - i2-) ml
+m2
R,
&+ =ip+
ml(fl_ -a,_) ml
is
R.
+m2
Here ip=(mli~-+m~i2_)/(mI +m2). The impacts persistently introduce a component of eigenvibrations of the system into the motion (the solution of the homogeneous differential equations of the motion), which together with the component due to forced vibration (the particular solution of the equations of motion) cause ambiguity of the response of a mechanical system with impacts to periodic excitation. Until about 1970 the theoretical analysis of periodic motions and their stability was largely limited only to the simplest impact motions and their behavior in relation to small disturbances, In the 1970s the development of methods of impact system motion simulation by means of analogue computers [5] yielded a more general view of the regularities of mechanical systems with internal impacts and also connections between periodic and chaotic motions. It was necessary to evaluate the chaotic motions by means of statistical methods. For this purpose a hybrid computer system (EAI PACER 600) was used as an optimal tool. In the past ten years, with the increasing speed and availability of digital computers, interest in detailed impact motion investigations has been revived, especially in the field of bifurcation and chaotic motions [lo-l 51.
2. LAWS OF IMPACT
MOTION
OF THE ONE-DEGREE-OF-FREEDOM
SYSTEM
The theoretical, analogue and experimental investigation of the forced impact vibration was most detailed for a vibrating body striking against a fixed stop (see Figure 2). The equation of motion and its solution can be written in the form mli:+cx=FOcos(ot+cp),
x(t)=Acos(Rt+t,v)+Bcos(wt+q~),
(2)
where A and IJJare constants depending on the initial conditions of the motion, B=x,,/ (1 - q2) is the amplitude of forced vibration, r,r= w/G is the relative excitation frequency, Q=G is the eigenfrequency of the system and x,, = F,/c is the static deflection of the body. 2.1. PERIODIC IMPACT MOTIONS Let us assume the existence of a periodic motion, in which the impact is repeated after a period T= 2nn/w which is an integer multiple (n = 1,2, , . .) of the excitation force period (see Figure 2). From the periodicity conditions x(0) =x(T) = -r,
i(O) = f+ = -Ri( T) = -RL
,
(3)
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it is possible to derive [5] the body motion in the time interval 0 < t G T and the value of velocity .Y_of the body immediately before the impact in the form X 1 ---=p cos qcos wl-sin x,, 1 - v2
sin Ot_
1+ R tl cosP-1-R
Wrl) sin (~n/r~)
--
’
(4)
where cpis the excitation force phase at the instant of impact,
[cosp],,,,=-[rf KJE2(1 + K2) [sin v)]~,~,=[-~K*JB~(~
r2]/[R( 1 + K2)],
+K2)-?]/[B(l
+K2)],
r> 0 is the static clearance between the body and the rigid stop, while for r
Azi+ = /?.A ,.i+ ,
(5)
between differences at successive impacts. By linearization of the solution it is possible to derive the equation P2+p(U-1-R2)+R2=0,
(6)
where U= (1 - R2)( 1 -K/tan p) sin2 (nn/r~), for determination of the dependence of /3 on the system parameters. The stability condition of impact motion can be written in the form IPI
(7)
At the stability boundary the system parameters will satisfy the condition ]/I]= 1. As the roots p of equation (6) are in general complex, it is possible-according to the position of the roots /3 on the unit circle-to distinguish three kinds of impact motion stability boundaries s+~ s_] and s,: one has (1) s+, . . . p= 1 and (2) sWl. . . /I= -1 when equation (6) has real roots; one has (3) sg. . . IpI = 1 when equation (6) has complex roots. The existence condition of the impact motion solution excludes the possibility of the onset of an additional impact at the time interval between the periodic impacts, which can be expressed as x(t) > -r
for O
(8)
The existence boundary p of the given periodic impact motion corresponds to the state when the trajectory of the body’s motion at some instant t of the interval (0, T) will touch
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i+
Figure 2. Schematic of the simple mechanical vibro-impact system and of its fundamental impact motion (FIM).
O-6
0.7
0.8
0.9
I.0
1.1
Frequency,
Figure 3. Region of existence and stability of the (;=O+l).
FIM (2 = 1, R = 0.6)
d-.tLl
-0.5 07
0.9
1.1
1.3
I.2
1.5
01 07
I.3
I.4
I.5
I.6
7)
: impactless motion (z = 0) ; beat motions
1 0,9
1.1
I.3
I,5
Frequency, 7
Figure 4. Frequency characteristics of the impact motion; P= 3, R = 0.6. - -, Solution I of the FIM ; - - --, solution II of the FIM; -, analogue simulation. (a) Excitation force phase at instant of impact; (b) beforeimpact velocity; (c) motion amplitudes .Y,,,,,/.v,,-theoretical results; simulated amplitudes with increasing (d) and decreasing (e) frequency, n.
F3
F4
Figure 5. Time (a) and phase (b) trajectories of the system motion, r’=3. FI, Impactless motion, n =0.79, n= 1, -_=O; F2 chaotic motion, n=O.8, z= 0 + 1; F3, periodic beat motion, n = 0.87, n =4, z = I /2; F4, chaotic motion, n=O.91, z= O-+ I ; F5, chaotic motion, n =0,95, I= 0 i 1; F6, FIM, n = z = I ; F7, stability loss of the FIM and development of the impactless motion, q = 1.48, n = 1, 3 = I + 0; F8, s-bifurcation and development of the FIM from the impactless motion, n = 1’15, n = 1, z = 0 --t I.
the stop, and then the following condition will be satisfied: x(t) = -r,
i(t)
= 0.
(9)
It is not possible to express the boundary p in explicit form even for the simple system mentioned and it is necessary to find this boundary by iterative methods. 2.3. REGIONS OF EXISTENCE AND STABILITY OF PERIODIC MOTIONS The stability and existence conditions define the regions of stability and existence of the periodic impact motions in the system parameters space. The system motion is mainly influenced by the relative clearance i and the relative tuning n of the excitation force. We will therefore demonstrate the results of the stability and existence analysis of different types of impact motions in the plane (F, n). It is possible to characterize the periodic impact motion by
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where p is the number of impacts and n is the number of excitation force periods in the motion period T (n is called the subharmonic number). The main group of moths is formed by those motions of which the period is identical with the period of excitation (T= 27r/w, n = 1). These motions create series of z = 0, 1,2, . . .-impact motions, including the impactless one (z=O). The time and phase trajectories of these motions are denoted F9(a, b)-Fl 1(a, b) and Fl2(a, b, c) in Figure 7. The regions of existence and stability are hatched obliquely in Figure 6. With decreasing clearance V,the region of z-impact motion approaches the boundary of existence p, where the so far impactless trajectory of body motion begins to touch the stop (see Figure 7 F9 for z = 0 and F12(b) for z= 2) and the quantity z gradually grows. On the contrary, with increasing clearance i;, the region of zimpact motion reaches the stability boundary s. The boundaries p, s between the particular regions of z-impact motions alternately intersect at points X and so create two kinds of transient zones. There are zones partly of hysteresis (cross-hatched in Figure 6(a)) and zones partly of beat z+z+ l-impact motions (hatched horizontally in Figure 6(a)). Both neighbouring z- and z + l-impact motions are stable and can exist in the hysteresis zones. On the segments of the boundaries p, s, which define the hysteresis zones, one neighbouring motion switches into another. Along the segment p, the quantity z increases by one and along the segments s the quantity z decreases by one. In the zones of beat motions no motion with excitation force period is stable or can exist and consequently periodic n-multiple impact motions (n = 2, 3, . . .) and chaotic ones appear. From theoretical analysis (see the Appendix) of the group of z= (l/n)-impact motions (n=l,2,. . .), the character of the boundaries p, s can be explained (see Figure 6) between the impactless (z =0) motion and the z= l-impact motion (fundamental impact motion (FIM),p=n=z= 1). The boundary p. of existence of the impactless motion and boundaries pI of existence and $+I, S-I , sq of stability of the FIM in the vicinity of resonance 17= 1 are shown in. Figure 3. The transition from the existence and stability regions of the neighbouring motions is demonstrated in the characteristics of system motion along the line q in Figure 3 (r=3, 0.5~n~l.6). On the boundary p. (at points G, K=J) the body with impactless forced vibration begins to touch the stop (see Figure 5(Fl)); it can be considered as the FIM, the impact velocities of which degenerate to bare touching. Thus along the boundary p. both neighbouring motions change continuously. This transition is called s-bifurcation (see the splitting of the amplitude characteristics at points G, KU= J in Figure 4(c)). The stability analysis of the FIM has shown [5] that this transition is never stable, and the stability boundary sg,is in agreement with it. Along the boundary po=s, the relation sin q=O holds (see Figure 4(a) at points GI, KII): i.e., the impact velocity f- is zero (see equation (4) and points G,, K,,= J in Figure 4(b)). At point G the impactless motion develops into solution I of the FIM, but it is unstable in the interval between points G, S and the system passes through the transient zone of the beat motions, the characteristics of which are shown in Figures 4(a, b, d, e). Several time- and phase-trajectories of the chaotic beat motions are shown in Figures 5 F2, F4, F5 and periodic beat motion in Figure 5 F3 (for a more detailed description, see the next section). Point S lies on the stability boundary se1 of the FIM, where the differences Aq, Ai+ of the quantities p, i+ from the corresponding values of the FIM start to grow and alternate in sign at impacts. However, their increase becomes stable in a certain limit cycle and the body motion alternates in the vicinity of the unstable FIM. In this way a motion with two different impacts occurs, and is repeated after two periods of excitation (see, e.g., the motion z=2/2= 1, p=n= 2 in Figure 7 FlO(c). Thus the period doubling bifurcation is seen along the stability boundary s-1 . The two-way transition across the
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boundary s-, at point S in Figure 3 and across the whole zone of the beat motions between points G, S is shown in Figures 4(d, e). The FIM, which has been stabilized (see Figure 5 F6) at point S, continues up to point F (see Figure 4(d)), at which the FIM transits by jump to point L of the impactless motion (see Figure 3 and Figure 5 F7). Point Flies on the stability boundary s+~ (see Figure 3), where the differences dq, di+ grow with the same sign and the impact vanishes. At point F both solutions I and II of the FIM coincide (see point F,,,, in Figures 4(a-c)) and beyond this point no solution of the FIM is real. The saddle-node bifurcation of the motion corresponds to the state along the stability boundary s+ , It is evident from Figures 4(c, d, e) that in the transient hysteresis zone between points K=J and F= L (Figure 3) both neighbouring motions can exist and are stable. The zones of attraction of the above-mentioned motions have been described in reference [16(e)]. By the s-bifurcation at point J a jump from the unstable solution II to the stable solution I of the FIM is seen (see Figure 5 F8 and points K,,=.I and K, in Figures 4(a, b, c, e)). The transition between the stability and existence regions of the multiple impact motions (z = 1,2, 3, . .) has a similar character. It follows from this description that by continuously changing the parameters ?, n of this system the transition from one neighbouring motion into another is accompanied either by the jump effects on the boundaries of type p, s+~ of the transient hysteresis zones or by the rise of the series of beat motions on the boundaries of type p, se1 (see Figure 6(a)). A continuous transition between the neighbouring motions can occur only through points X in the vicinity of which the impact motion with nearly zero impact velocities is stable.
2.4. ZONES OF THE BEAT IMPACT MOTIONS A more detailed view of one of the beat motion zones (ztz+ 1) is presented in Figure 6(b) which is a detail of Figure 3. The zone z = 0 -+ 1 on the right side of point X3 is limited at the top by the existence boundary p. of the impactless motion (po- B) and at the bottom by the stability boundary s-i of the FIM. By means of a theoretical analysis [16(a)] (see also the Appendix) it is possible to find in this zone the existence and stability subregions of the z = (I /n)-impact motions. These subregions are alternately obliquely hatched in Figure 6(b) and limited by the stability boundaries se1 and the existence boundaries p2,. . . , p5. The trajectories of motion in these subregions are shown in Figures 7 F13(a, b)-Fl6(a, b). There are adjoining subregions of z = (1 /n)-impact motions (n > 6) for higher values of clearance ?. The impact motion is invariably found below the boundary p. (r
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0.7
0.8
0.9
I.0
Frequency, 7
Figure 6. Existence and stability regions of the z-impact motions in the plane (I:, q). (a) Schematic diagram, (b) theoretical results; (c) results of analogue computer simulation for R=0.6.
2.5. CHAOTIC IMPACT MOTIONS Chaotic motions can occur, as in other non-linear systems, as a result of successive motion period doubling. This was described for the impact system being described here by Isomlki 1141. The process of sequential period doubling is possible if the existence condition of type (8) is always satisfied. The quantity z holds its value since z = 2kp/(2kn), where k=O, 1,2,. . . is the order of the z = (p/n)-impact motion splitting. When, in connection with the transition to the chaotic motion, the value of z decreases, then one impact in the motion period vanishes due to the stability loss of the previous split-up motion. The loss of stability must be connected with the saddle-node bifurcation, which is accompanied by a jump to the impact motion with a smaller value of z. If the periodic impact motion with a smaller value of z can exist and is stable, then it becomes stabilized. An example is the transition from point FlO(a, b) through point FlO(c) to point Fl3(a, b) in Figure 6(c). The z= (l/l)-impact motion (see Figure 7 FlO(a, b) splits to the z=2’.1/ (2’.1)-impact motion (see Figure 7 FlO(c); see also point 4 in Figure 8(a), a weaker impact of which vanishes on the stability boundary of type s+~ and the z = (l/2)impact motion becomes stabilized by a jump (see Figures 7 F13(a, b); see also points 5 in Figure 8(a)).
CHAOS
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Fl4
F22
Fl5
F23’
Fl6
F24
IMPACTS
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Figure 7. Several kinds of periodic and chaotic impact motions: (a) time trajectories; (b, c) phase trajectories. F9, Impactless motion, q=O.7, r’=2.1, n=l, z=O; FlO(a, b), FIM, q=O.75, r:= 1.5, n=z=l; FlO(c), split-up FIM, q=O.8, P=2.1, n=2, z=2/2; Fll(a, b), two-impact motion, q=O.7, i=O,8, n= 1, 2=2; Fll(c), q=O.8, r=0,04, n=2, z=4/2; Fl2(a, b), two-impact motion, q=O.62, P=O.O3, n= 1, z=2; F12(c), three-impact motion, q=O.62, F=0.02, II= 1, z=3; F13(a, b), q=O.83, P=2.5, n=2, z=1/2; F13(c), q=O.9, f=2.8, n=4, z=2/4; F14(a, b), q=O.88, T=3.4, n=3, z= l/3; F14(c), q=O.9, r=4.15, n=6, z=2/6; F15(a, b), q=O.895, r=4.65 n=4, 2=1/4; F15(c), q=O.92, r:=5.2, n=8, z=2/8; F16(a, b), q=O.915, f=57, n=5, z= l/5; F16(c) ;=0.93, v‘=6.3, n= 10 , z=2/10; F17(a, b), q=O.61, r‘=2, n= 1, z=2; F17(c), r:= 1.7, n=2, z=4/2; F18(a, b), q=O.76, F=3.5, n=2, 2=2/2; FlS(c), r=3.19, n=4, z=4/4; F19(a, b), q=O.82, i;=4.85, n=3. z=2/3, Fl9(c), f=4.67, n=6, z= 4/6; F20(a, b), q=O.845, P=6.85, n=4, z=2/4; F20(c) r:=6.6, n=8, z=4/8. Chaotic motions: F21, q=O.65, i=I.25, z=lt2; F22, q=O.78, ~=2.3, z=1/2+1; F23, 17=0,95, ?=5, z=l/3-1; F24, q=O.95, f=6.5, 2=1/4+-l.
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Chaotic motion appears when such a stable periodic motion cannot exist. An example of this is provided by the z = 1 f 2- and z = (l/2) f l-impact motions in Figures 7 F21(a, b) which develop from the split-up z=2-impact motions (see Figure 7 F17) after the first splitting (~=2~.2/(2~.1)=2, k= 1) and from z = l-impact motions (see Figures 7 FlO(a, b, c) and FlS(a, b, c)) after the second splitting (z = 2k.1/(2k. 1) = 1, k = 2). Such an origin of chaotic impact motions can be generalized and the transition from the existence and stability regions of the z + l-impact motions (see Figure 6(a)) into the z t z + l-impact beat motions along the stability boundaries of type se1 (period doubling bifurcation) can be explained. However, chaotic impact motion can be caused by the fact that the existence condition of type (8) of the periodic impact motion is not met. This is directly related to the transition of z-impact periodic motions into the regions of zt z+ l-impact motions along the segments of existence boundaries of type p = s+,(see Figure 6(a)). After crossing these segments of the boundaries p either some periodic impact motion will become stabilized if it can exist and is stable (e.g., the transition of impactless motion in Figure 7 F9 into the z= (1 /n)-impact motions for n = 2, . . . , 5 in the vicinity of points X2, X6, X, and X, in Figures 6(b, c) and Figures 7 F13(a, b)-F16(a, b)) or more complex periodic motions (of type F18(a, b, c)-F20(a, b, c) in Figure 7) occur. In the opposite case the chaotic z+z+ limpact motion will appear directly (e.g., see horizontally hatched regions z= (l/n) + 1 in Figure 6(c) for n = 2, 3, . . . , cc and see also Figures 7 F22-F24). In any case the quantity z increases after crossing the existence boundary p. A similar case of the origin of chaotic motions occurs when the process of period doubling of the beat impact motions is discontinued due to the fact that the existence condition (8) is not met. In such a way the laws of the changes of the impact motions inside the beat motion zones can be explained. As an example the changes of impact motions along line p in Figure 6(c) will be described. For constant frequency r~=0*92 and a continuously changed clearance P the characteristics of the impact velocities -L/Rx,, were obtained theoretically and by way of simulation [5], and are presented in Figures 8(a, b). Thanks to the theoretical analysis of the group of z = (2/n)-impact motions (n = 132,. . ,) which was carried out by Kotera [ 16d], it was possible to describe both the jump transitions connected with saddle-node bifurcation (see the jump from points 2 of the z = motion in Figure 8(a) and, (2/l)-impact motion into point 3 of the z= (l/l)-impact similarly, the jump from points 5 of the z = (2/2)-impact motion into the z = (l/2)-impact motion) and the splitting of the z = (1 /n)-impact motions into the z= (2/2n)-impact motions (see the splitting of characteristics in points 4, 6 and 8 in Figure 8(a)) connected with period doubling bifurcation. Only in the vicinity of points 6 and 8 in Figure 8(a) is it possible to observe the development of split-up motion discontinued at points 7 and 9 due to the failing of the existence condition of type (8). The body’s motion up to the existence boundary (near points 7 and 9 in Figure 8(a)) is shown in Figures 7 F13(c) and F14(c), where one of the impactless loops almost touches the stop. With increasing clearance ? and further development of splitting, the chaotic impact motion (z = (l/2) +- 1 and z = (l/3) + 1) develops at the moment of additional impact. If we exclude the existence condition for the impactless loops, which can be done on the analogue computer [16c], then the split-up motion will develop along the dashed branches, starting from points 7 and 9 in Figure 8(a). Such a situation is demonstrated in Figure 9(a) where, for large splitting of the ;= (l/3)-impact motion, even two primary impactless loops (see Figure 7 F14(c)) penetrate the stop and the z = “2/6”-impact motion is stable. Actually the existence condition causes the onset of the chaotic motion (see Figure 9(b)). The autocorrelation function R,,(1.2~w) in Figure 9(c) shows evidence of the random character of the chaotic motion.
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I
I
I
WITH I
I
3
4
I
4
3
2
2
I
.;: J
0
-I
0
2
2
I
0
-I
0
I
2
3
4
5
6
7
Cleoronce,l
Figure 8. Impact velocity characteristics along the line p in Figure 6(c). (a) Theoretical results; (b) results of the impact motion simulation.
Figure 9. (a) Non-real periodic z= “2/6”-impact motion; (b) chaotic z= l/3+ l-impact motion; (c) autocorrelation function of chaotic motion, R = 0.6, q =0.92, f=4.5.
It was evaluated from samples ui= x,, ing to the formula
of the motion amplitudes (see Figure 9(b)) accord-
(11) where 5= (Ci”, ai)/N is the mean value of the motion amplitudes, and Z.~W/Wis the time delay expressed as the excitation period multiple (O
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It should be mentioned that the chaotic impact motion can occur due to the failing of the existence condition of the non-split-up z= (l/n)-impact motions (see, e.g., point 10 in Figure 8(a) where a decrease in clearance ! after the impactless loop of the z = (1/3)impact motion-Figure 7 Fl4(b)-has touched the stop results in the chaotic z = I /2 + 1impact motion; similarly the next z= (l/n)-impact motions-Figures 7 F15(b, F16(b), etc.-transit into the chaotic z = 1/(n - 1) t 1-impact motions-Figure 8(b)). This explanation can be generalized and allows us to elucidate the system behaviour in the regions of beat z+ (z + I)-impact motions between the existence and stability regions of the neighbouring z- and z+ l-impact motions (see Figure 6(a)). The occurrence of different types of impact motions, including the chaotic ones, depends on the system damping. The smaller the damping, the more diverse is the system response to the periodic excitation [5, 16f].
3. IMPACT MOTIONS OF MORE COMPLEX SYSTEMS The more complex the mechanical system is (with a higher number of degrees of freedom and impact pairs) the more difficult it is to find system parameters for which some periodic impact motion will satisfy the stability and existence conditions. Therefore for such systems the development of chaotic motions is more frequent than the development of periodic ones. As an example we will consider the five-mass system shown at the top of Figure 11. This is a model of transverse vibration of a beam placed between two rigid stops. The beam is supported at its ends and is excited in the centre by a periodic force F, cos wt. The beam model has five degrees of freedom, and each of the substituting masses can impact on stops. The system contains five impact pairs with two-sided impacting. The regions of the necessary onset of the impact motion are limited by the amplitude characteristics p of the forced vibrations. The regions are hatched in Figure 10 for the vicinity of the resonances n = 1, 4, 9, 16, 25 (where n = o/f2, ; 0, is the first eigenfrequency of transverse vibration). Due to the system’s symmetry and the point of excitation, the even resonances are suppressed. Chaotic motion exists in all regions under the boundary p and by increasing the relative clearance 7= r/y,, it can be maintained up to the dashed boundaries where it transits into impactless motion. Periodicmotions were found only near the fourth eigenfrequency (0,/L!, = 16) and their existence and stability regions are hatched obliquely in Figure 11. The results were obtained by analogue computer simulation. The relations used for simulation can be expressed as yi =
YilYsr
3
dY, dY, dt YI=~=-d-~=~il(Y,ss21),
(12)
where y,, is the static deflection of the beam centre caused by the force also acting at the centre, r= 5fJ,t is the time transformation of the simulation, Yi, Yl are the simulated deflections and velocities of particular bodies (i = 1,2, . . . , 5), and yi, ji are the actual deflections and velocities. For the characteristic points A-Q (see Figure 11) the corresponding cross-sections of the Poincare map (see Figure 12) as well as the phase trajectories Y/T Yf (see Figure 13) are added. In all of the motions described below only the centre of the beam (i= 3) impacts. The points representing the phase nr of the excitation force and velocity Y$_ at the instant of impact are plotted on the polar diagram (nr, Y;-). The systems of co-ordinates (r~r, Y;_) and ( Yi, YI) are described in Figure 12(a) and Figure 13 H. The largest region corresponding to the z = 2-impact motion is denoted in Figure 11 by oblique and horizontal hatching. There are three kinds of two-impact motion, as follows.
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-‘;_,‘rr_-rF
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-?Fr~,-Fry-r----~-~
,,-,,,,,~,.,.,,-,,,.,.,~ 3.8
4.0
4-2
4-4
46
9.0
9-2
94
9.6
16 2
16.-l
166
25.2
25.4
25.6
_:
o-o
8.6
8.8
ok‘=-=-----~-------~------1 15.6
158
P
I.5
16 0
WI
I.0 0.5 0.0 24.6
24.8
25.0 Frequency.
7)
Figure IO. Existence regions of the chaotic impact motion of a beam placed between rigid stops.
(1) The first symmetrical periodic motion in which the impacts of equal intensity alternately occur on the opposite stops after time intervals T/2 = K/W. The trajectories Yi, Yl are symmetric in the phase plane (see Figure 13H). (2) Second is the non-symmetrical periodic motion, the period T of which is unevenly divided between the successive impacts of equal intensity. This is expressed by the central non-symmetry (except for the impact points) in the phase plane (see Figure 13 F). Since the system is symmetric and the non-symmetric motion has the same relation to each of the two stops, there must be two different non-symmetric motions. We will differentiate them as type a and type b. They are mutually symmetric in the phase plane (see Figure 13 F). (3) Finally there is the non-periodic motion which chaotically oscillates around one of the two non-symmetric motions (see Figures 13 D, C). The periodic motions exist in the obliquely hatched region z=2 in Figure 11. The symmetric (non-symmetric) motion transits into the non-symmetric (symmetric) one with decreasing (increasing) clearance J or frequency n on the dashed curve passing through point G. The non-symmetric motions transit into the chaotic motion (e.g., for type a see Figures 13 E, D, C) along the boundary passing through point E, and this motion exists in the horizontally hatched region with internal points’ D, K (see Figure 11). The z = 24mpact motion is described in detail by means of the impact characteristics in Figure 12 obtained by changing the clearance 7 along the lines u( q = 16) and o( q = 15.25). When the clearance F increases from the value ?=0.018 (point B in Figure 11) the nonsymmetric chaotic motion, the attracting set (AS) of which creates the narrow closed
108
F. PETERKA
z>4
0.00 13
14
-Y 15
AND
J. VACiK
” 16 Frequency,
I?
18
19
20
q
Figure I I, Existence regions of periodic (///I/) and chaotic (g) motions of a beam placed between rigid stops (R=0.6).
curves (see Figures 12(c, d)-the arrows C), starts to deviate at point C (r’=O.O193) from the large developed chaotic z=2+ 3-impact motion, the AS of which is shown in Figure 12(b). During the following increase of clearance r: the AS of chaotic motion contracts the area into two points E (see Figures 12(c, d)-the arrows D, E-and Figures 13 C, D, E) which are the AS of the non-symmetric periodic motion. In the interval between points E and G, in Figure 11 the AS branches continuously travel from points E (see Figure 12(c)) until they merge at point G which is the AS of the symmetric impact motion. This motion is stable up to point I (see Figures 11 and 12(c, d)). The series of the AS from Figure 12(c) is shown for discrete values of I: in Figure 12(d). The symmetric motion loses its stability at point I by the saddle-node bifurcation and changes by a jump into the impactless motion. The points of the transition process are recorded along the dot and dash arrow in Figure 12(d). Point G is important because it lies on the boundary of motion symmetry and corresponds to the pitchfork bifurcation. There is the same probability of the onset of each type (a, b) of the non-symmetric motion on this boundary. The two branches of the characteristics between points E and G in Figure 12(c) describe both types a, b of the non-symmetric motion because the absolute values of the impact velocities Y;_ are identical for both stops. The difference is in the fact that with type a a certain branch EG (see Figure 12(c)) corresponds to the impact on one stop and to the impact on the other stop for type b.
CHAOS
(a),
IN
MECHANICAL
SYSTEMS
WITH
109
IMPACTS
..,
T=0~011-
(f
Ih)
h0*003-
(0-J)
’
I
0.04
\
Figure 12. Cross-section of the Poincark map of the periodic and chaotic motions
The change of the AS throughout the interval 0.003 < rcO.036 (for q = 16) is shown in Figure 12(c). For small values of r the AS of the z 2 3-impact motions occurs in two dark regions designated by arrows A in Figure 12(c). The chaotic z=2 + 3-impact motion (between points A, C in Figure 11) is suppressed in Figure 12(c) so that the AS of z=2impact motions would not be superimposed. This chaotic motion is shown at full size in Figure 12(a). Weak impacts also appear in this motion, and the small dark region near the centre of the polar diagram is in agreement with it. The phase trajectories of the beam centre are shown in Figures 13 A and B for points A and B. The cross-section of the regions along the line D in Figure 11 was chosen to demonstrate the AS of the z>2-impact motions in a better way. The motion at points J, K and L is similar to the motion at points E, D and C (see Figures 13 E, D and C) but the ASS are more expressive. The AS of the motion at point K (q= 15.25, F=0.0326) is represented in Figure 12(i) by two closed curves into which the points at two-sided impacts are plotted chaotically, while one of the two types a or b of the non-symmetric motion is maintained. These AS curves expand with decreasing clearance into an AS with a larger area (see the AS in Figure 12(h) for point L) while the chaotic motion remains throughout as either of
110
F. PETERKA
AND
J. VACiK
I //////////////////,/////////////////////~~,,~~~,,,~~
////
///////////
1///////////////////////////////////////~/////////~/~//////////////
F
&-a6
/
?=0.02?
Qij4-*- -,A
-
‘//////////////////////////////////////////////////////////////////
i=3
i=3
i=3 0
E 7~0.023
E ?=0.023
a
i50.0105 /////u//// /II /
b I \
Figure 13. Phase trajectories of the periodic and chaotic motions.
the two types a, b of non-symmetric motion (see, e.g., Figures 13 E, D and C). Both areas of the AS will merge only after crossing the boundary which passes through point L (see the AS in Figure 12(g) for point M in Figure 11) while both types a and b of nonsymmetric motion will alternate during chaotic motion. This is shown by the accumulation of the phase trajectories in Figure 13 B. The continuous change of AS with changing clearance r along the line D in Figure 11 is demonstrated in Figure 12(f). The samples of this continuous spectrum are shown in Figures 12(g, h, i) for clearances ? which correspond to arrows M, L and K in Figure 12(f). The band of the chaotic z=2+ 3-impact motion is again suppressed, to illustrate the triad of the z = 3- and z = 3 +4-impact motion AS (see arrows N and P showing one part of the AS in Figure 12(f)). The ASS of more-impact motions are shown in Figure 12(e). The periodic z = 3-impact motion develops at point N in Figure 11 from the chaotic z = 2 t 3-impact motion. There are again two types, a and b, of the z = 3-impact motion, the phase trajectories of which are also mutually symmetric (see Figures 13(0a, b)). Two impacts on one stop and one impact on the.other are repeated during this motion. The triad of the attractors shifts along the segment NO of the characteristics in Figure 12(e) when the clearance changes from point N to point 0 in the z = 3-impact motion region in Figure 11. Point 0 is important again because it is here that the pitchfork bifurcation occurs, just as it does along the boundary passing through point G in the z2-impact motion region, The z= 3-impact motion changes on the boundary passing through point 0 into one of two subtypes, a and /I, with equal probability. One branch of the characteristics between points 0 and P in Figure 12(e) corresponds to every subtype of z = 3-impact motion. The phase trajectories of the type a - a, /I are illustrated in Figure 13 P. The z = 3-impact motion loses its stability at point P and chaotic z = 2 + 3-impact motion appears. Its phase trajectories are shown in Figure 13 at the bottom right for parameters n = 15.25, ?=0.006. It is here that the region of z =Cimpact motion (see point Q in
CHAOS IN MECHANICAL
SYSTEMS WITH IMPACTS
111
the region z =4 in Figure 11 and see Figure 13 Q) and the regions of more-impact motions (z > 4) for small values of clearance i: are found. The group of four fixed pointsattractors corresponds to the periodic z=4-impact motion at point Q (see the arrows Q in Figure 12(e)). This explanation is of a definite value for further research, because it illustrates the heterogeneity of the periodic motion bifurcations of more complex systems and of the chaotic motions.
4. CONCLUSIONS
The discussion and the results of the analogue simulation presented above confirm that chaotic motions accompany the periodic motions of periodically excited mechanical systems with impacts. The s-bifurcation appears at the moment when the existence condition of the periodic impact motion is not satisfied. It is connected with an increase in the quantity z which expresses the average number of impacts related to one period of excitation. When the stability condition of the periodic impact motion is not satisfied, one of the following situations occur: (a) saddle-node bifurcation with a decrease in the z value; (b) period doubling bifurcation with the z value retained, but the period of the periodic motion increased; (c) pitchfork bifurcation, when both the z value and the period are retained; this bifurcation is connected with the onset of non-symmetry of the motion in systems with symmetrically arranged stops. The mechanism for transition from the periodic impact motions into chaotic ones after the crossing of the existence and stability boundaries in the beat motion zones is also explained.
REFERENCES 1.
2. 3. 4. 5.
KOBRINSKII 1964 Mechanisms with Elastic Links. Moscow: Nauka. A. E. KOBRINSKII and A. A. KOBRINSKII 1973 Vibro-impact Systems. Moscow: Nauka. V. L. RAGULSKENE 1974 Vibro-impact Systems. Vilna: Mintis. V. I. BABITSKII 1978 Theory of Vibro-impact Systems. Moscow: Nauka. F. PETERKA 1981 Introduction to Vibration of Mechanical Systems with Internal Impacts. Prague:
A. E.
Academia.
6. R. BANSEVITCHIUS and K. RAGULSKIS 1981 Vibromotors. Vilna: Moskals. 7. V. I. BABITSKII and V. L. KRUPENIN 1985 Vibration in Strongly Non-linear Systems. Moscow: Nauka. 8. G. SILAS and L. BRINDEU 1986 Vibro-impact Systems. Bucharest: Technical Editions. 9. F. PETERKA 1973 Laws of Impact Motion (short film). Prague: Institute of Thermomechanics, CSAV. 10. P. J. HOLMES 1982 Journal of Sound and Vibration 84, 173-189. The dynamics of repeated impacts with a sinusoidally vibrating table. 11. S. W. SHAW and P. J. HOLMES 1983 Journal of Sound and Vibration 90, 129-I 55. A periodically forced piecewise linear oscillator.
12. M. I. FEIGIN 1978 PMM 42, 820-829. About structure of s-bifurcation boundaries in piecewise
continuous systems. 13. T. KOTERA and H. YAMANASHI 1986 Transactions of the Japan Society of Mechanical Engineers 52, 1883-1886. Chaotic behaviour in an impact vibration system, 2nd report. Influence of damping coefficient and coefficient of restitution. 14. H. M. ISOMKKI, J. VON BOEHM and R. R;~TY 1987 Proceedings of the XZth ICNO, Budapest, 656-659. Chaotic oscillations and the fractal basin boundaries of an impacting body. 15. M. S. HEIMAN, P. J. SHEARMAN and A. K. BAJAJ 1987 Journal of Sound and Vibration 114, 535-547. On the dynamics and stability of an inclined impact pair.
112
F. PETERKA
AND
J.VACiK
set of companion papers on the laws of impact motion of a mechanical system with one degree of freedom.) (a) F. PETERKA 1974 Acfa Technica &A V 19(4), 462-473. Part I-Theoretical analysis of nmultiple (l/n)-impact motions. (b) F. PETERKA 1974Acta Technica es.4 V 19(S). 569-580. Part II-Results of analogue camputer modelling of the motion. (c) F. PETERKA and J.VAC~K 1981 Acta Technica &AV26(2), 161-184. Part III-Statistical characteristics of beat motions. (d) T. KOTERA and F. PETERKA 1981Acta Technicu &41/26(6), 747-758. Part IV-Analytical solution of the 2/n-impact motions and its stability. (e) F. PETERKA and T. KOTERA 1982 Actu Technicu CSAV 27(l), 92-l 17. Part V-Regions of existence and stability and domains of attraction of different kinds of impact motion. (f) T. KOTERA and F. PETERKA 1984 Actu Technicu k$AV 29(3), 255-279. Part VIAnalytical and analogue solution of the multiple-impact motion and its stability. 17. F. PETERKA and J.VAC~K 1987 Proceedings of the XIth ICNO, Budapest, 707-710. Chaotical and periodical forced transversal vibrations of the beam type mechanical systems with impacts. 16. (A
APPENDIX: BRIEF THEORETICAL ANALYSIS OF PERIODIC (l/n)-IMPACT MOTIONS AND THEIR STABILITY OF A ONE-MASS SYSTEM A.I.
CONDITIONS
NECESSARY
FOR THE RISE OF IMPACT MOTION
(a) For r > 0 the amplitude B of forced vibration must be higher then the static clearance r (see equation (2) and Figure 2). (b) For r
Fp= (rlc. According to equation (2) the necessary conditions can be expressed using the formula p,~-l
(Al)
(e.g., in the region between the boundaries p. and pP in Figure 3 some impact motion must exist). A.2. ANALYSIS OF SOLUTION OF PERIODIC (l/n)-IMPACT MOTIONS When the case of absolutely elastic impacts (R= 1) is excluded, the solution (4) of the group of (l/n)-impact motions (Figure 2) is undetermined, due to the function K in equation (4), at the frequency values (A2)
q = n/l,
is the subharmonic number and I= 1,2,3, . . . is the ultraharmonic wheren=l,2,3,... number. The frequency interval 0 < 9 < co is divided by the values (A2) into the sub-intervals of the definiteness of the solution (4). It is necessary to search for a region of real periodic impact motion existence and stability in every sub-interval. The results of the analysis will be-explained in the sub-interval O-52 v < 1 (for n = 1 and I= 1,2; see Figure Al). A.3. MATHEMATICAL The condition
REALITY OF THE SOLUTION
(4)
-p” < F< p n= poJiTz
follows from the necessary positivity of the discriminant in solution between the boundaries fp” in Figure Al(a)).
(A31 (4) (see the region
CHAOS
A.4.
PHYSICAL
REALITY
IN
MECHANICAL
OF THE
SOLUTION
SYSTEMS
WITH
113
IMPACTS
(+-CONDITION
OF
IMPENETRABILITY
The conditions of impenetrability of the impacting bodies can be expressed in two wajrs, as follows. (a) The before-impact velocity must be negative, (or CC+= -RL>O),
L
CA4)
for the instants (r=O, 2lrn/w) of the impact. According to equations (4) the condition (A4) has the form
-rKf
B*(l +K')-r*>O.
645)
The analysis of this condition for the solution I (the + sign) and the solution II (the sign in the relation (A5)) of the periodic impact motion depends on the polarity quantity K (equation (4)). The zero-points of the function K, which correspond to the frequency values 17= fll(k + 1PI,
k=O, 1,2,.
.)
(A61
divide the frequency sub-intervals according to the polarity K in respect to the next two sub-intervals (e.g., for the sub-interval (l/2) < 17< 1 the quantity K is negative in the subinterval l/2 < q < 2/3 and positive in the sub-interval (2/3) < 17< 1; see Figure Al. The summary results of the condition (A4) analysis can be expressed as follows: For K
For K>O
-p()
Solution I Solution II
-f
(A71
-p”
8
6
4 r
r
F6a S-I
2
2
P.
0
-2
-4
-5
?
1
2
.A14)
3
3
u>O 2(I+R2)-U>0
(Al21 tA,4j2(l+R2)-
U>O
(A121
U=-0
(Al4)
X,--r
(0)
(b)
Cc) Figure
A.
1
id)
18)
114
F. PETERKA
AND
J. VACiK
These results are shown by hatching in Figure Al(a). Between the boundaries fp, the condition (A4) is satisfied only by solution I in the whole sub-interval n. Between the boundaries po, p” and -pa, -p”, for K< 0 and K>O, respectively, the condition (A4) is satisfied by both solutions 1, II of the periodic (1 /n)-impact motion. (b) The trajectory x(t) of motion must satisfy the condition (8) for the time interval 0 < t < 2rrn/o between impacts. It is possible to express this condition as r’>Pl,
(A8)
where the boundary pI (see Figure Al(d)) must be found AS.
CONDITIONS
OF
STABILITY
OF
PERIODIC
numerically.
(l/n)-IMPACT
MOTIONS
For the analysis of equation (6) according to condition (7) it is useful to transform this equation by means of the relation JI=(w+l)/(w-I)
(A9
into the equation tiw2+2(1 -R2)W+2(1
+R2)-
U=O.
(AIO)
The stability condition (7) corresponds to the condition w,
(AI I)
where w, is the real part of the roots of equation (AlO). The Hurwitz conditions u>o,
2(1 +R2)-
2(1 -R2)>0,
U>O
(A12-A14)
must be satisfied in order to satisfy condition (Al 1). (a) Condition (A12) can be transformed as follows: f
B2(1+K2)-r2/{-rKf
B2(1+K2)-r2}>0.
(A15)
Condition (A15) is identical with conditions (A3) and (A5) for the solution I (+ signs) and therefore the stability region of this solution is described by conditions (A7). The stability boundaries s+, up”
for Kc0
and
S+, zs -p”
for K>O
(Al@
correspond to the U= 0 (see conditon (A12) and Figure Al (a)). These boundaries were named s+~ for the value J? = + 1 which the quantity /? achieves along them. The solution II satisfies the stability conditon (A15) only in the regions in which condition (A4) is not satisfied, and therefore further analysis concerns only solution I. (p) Condition (A13) is satisfied for the whole interval O< R < 1 of the restitution coefficient R. The stability boundary corresponds to motion with the absolutely elastic impacts (R= 1). ( y) The analysis of condition (A14) determines the boundaries of type s-1 (/I = - 1 along them) as (A17)
S-l = -PoJWl, where L= (1 + K2- V)/dm and V=2(1+ R2)/[(1 -R)2sin2 (an/q)]. stability regions are determined by the conditions f>s_,
for Kc0
and
as graphically shown, e.g., in Figure Al(b).
?
for K>O,
The
6418)
CHAOS
IN MECHANICAL
SYSTEMS
WmH
IMPACTS
115
The stability conditions (A12) and (A14) in summary have the form s-r
for K-CO
and
s+i
for K>O
(A19)
as shown in Figure Al(c). It is necessary also to find the stability boundaries s,, along which the imaginary roots /l of equation (6) lie on the unit circle. This circle transits into the imaginary axis of the w plane and the boundaries sp,were found from analysis of the condition w,=O.
(A20)
This equation is satisfactory for R # 1 when sin q= 0. It follows from equation (4) that equation (A20) is satisfied along the boundaries p. where the impactless body motion with touching of the stop is identical with the degenerated impact motion with zero-velocity of impacts. The solutions I and II satisfy equation (A20) along the boundaries (sJ~,,~ according to Kc0 GO
i
(.Q,f = PO (sp),= -po
K>O
(%?)I”PO &J,1= -po
6421)
as shown, e.g., in Figures Al(c), (d). The stability boundaries s+~, set and sP were obtained under the assumption of small disturbances of the periodic impact motion. The analogue simulation explained the system motion development after crossing the stability boundaries: (a) the saddle-node bifurcation appears along the boundaries s+~ and the impact motion transits by a jump into another motion; (p) the period-doubling bifurcation of the impact motion appears along the boundaries sWl; (y) the beat impact motion (periodic or chaotic) arises from the impactless motion along the boundaries (s,),=po; (6) the jump transition from impactless motion into the (I/n)-impact motion (solution I) arises if this motion satisfies the existence condition (8) along the boundaries (s~),,EP~. It follows from the relations (A21) and items (y) and (6) that continuous transition from the impactless motion into the stable (l/n)-impact motions is impossible (the excep tion is the situation shown around points X3, X2, X6, X7, X8, . . . in Figures 6(b, c)). The resulting conditions (A8) and (A19) give only the hatched region in Figure Al(d) in which the one-impact motion can exist and is stable. By means of similar analysis it is possible to find the resulting existence and stability regions of every (l/n)-impact motion (see Figure 6(a)) in the whole frequency interval (see, for more detail, reference [5]). The analysis of the (2/n)-impact motions is similar and more complicated [ 16d], but it provides the possibility of constructing the bifurcation diagrams (see Figure 8(a)).
TRANSITION
TO CHAOTIC MOTION IN MECHANICAL SYSTEMS
WITH
IMPACTSt
F. PETERKA Institute of Thermomechanics of the Czechoslovak Academy of Sciences, DoleJ%kova 5, 182 00 Praha 8, C.S.F.R.
J. VAC~K Institute of Information Theory and Automation of the Czechoslovak Academy of Sciences, Pod vodbrenskou &ii 4, 182 08 Praha 8, C.S.F.R.
(Received 8 November 1988, and accepted in revised form 18 February 199 1)
The aim of this work is to give a short explanation of the laws of a simple mechanical impact system, to elucidate the transitions to chaotic motion and to illustiate the periodic and chaotic motions of more complicated systems. In view of the nature of this work and
the broad scope of the problems involved, only the results of theoretical and analogue studies are presented here, with references given to more detailed analyses.
1. INTRODUCTION The performance of some types of machines and equipment is accompanied by the impacts of bodies. The principle of operation of vibration hammers, machinery for driving, compacting, milling and forming, vibratory conveyers, platforms and shaking grizzlies, shock absorbers, etc., is based on the impact action for moving bodies. With other equipment, e.g., mechanisms with clearances, heat exchangers, fuel elements of nuclear reactors, etc., impacts also occur, but they are undesirable as they bring about failures, strain, shorter service life and increased noise levels. Investigations of the dynamics of mechanical systems with impacts were launched in the middle of the present century with the aim of analyzing the existence and stability of periodic motions, optimization of system parameters, parametric vibrations, transient phenomena and processes, self-excited oscillations, random vibration, protection against impact effects, etc. A survey of publications and results of research can be found in several monographs [l-8]. This work deals with periodically excited impact systems. Mechanical systems with impacts are strongly non-linear due to the existence of one or more impact pairs of bodies. In the case of rigid impacts it is often possible to use the simple impact pair model (see Figure 1), where the force connection F between the bodies in its dependence on the relative displacement x2--xl contains a sudden change of the derivative dF/d& - xi). According to Newton’s elementary theory of direct and centric impact the before-impact velocities f,_, i2- of the bodies change by jumps and acquire the values of the after-impact velocities R,+ , i2+ according to the following relations, in t This paper is a revised summary lecture “Causes of Chaotic Motion in Mechanical Systems with Impacts” presented at the EUROMECH 242 Colloquium “Application of Chaos Concepts to Mechanical Systems”, Wuppertal, F.R.G., 26-29 September 1988. 95 0022-460X/92/070095 f21 $03.00/O
0 1992 Academic Press Limited
96
F. PETERKA
AND
J. VACiK
Figure 1. Model of the impact pair.
which ml, m2 are the masses of the impacting bodies and R = -(i,+ - i2+)/(i, _ -&) the coefficient of restitution: k,+ =ip-
m2(i, - - i2-) ml
+m2
R,
&+ =ip+
ml(fl_ -a,_) ml
is
R.
+m2
Here ip=(mli~-+m~i2_)/(mI +m2). The impacts persistently introduce a component of eigenvibrations of the system into the motion (the solution of the homogeneous differential equations of the motion), which together with the component due to forced vibration (the particular solution of the equations of motion) cause ambiguity of the response of a mechanical system with impacts to periodic excitation. Until about 1970 the theoretical analysis of periodic motions and their stability was largely limited only to the simplest impact motions and their behavior in relation to small disturbances, In the 1970s the development of methods of impact system motion simulation by means of analogue computers [5] yielded a more general view of the regularities of mechanical systems with internal impacts and also connections between periodic and chaotic motions. It was necessary to evaluate the chaotic motions by means of statistical methods. For this purpose a hybrid computer system (EAI PACER 600) was used as an optimal tool. In the past ten years, with the increasing speed and availability of digital computers, interest in detailed impact motion investigations has been revived, especially in the field of bifurcation and chaotic motions [lo-l 51.
2. LAWS OF IMPACT
MOTION
OF THE ONE-DEGREE-OF-FREEDOM
SYSTEM
The theoretical, analogue and experimental investigation of the forced impact vibration was most detailed for a vibrating body striking against a fixed stop (see Figure 2). The equation of motion and its solution can be written in the form mli:+cx=FOcos(ot+cp),
x(t)=Acos(Rt+t,v)+Bcos(wt+q~),
(2)
where A and IJJare constants depending on the initial conditions of the motion, B=x,,/ (1 - q2) is the amplitude of forced vibration, r,r= w/G is the relative excitation frequency, Q=G is the eigenfrequency of the system and x,, = F,/c is the static deflection of the body. 2.1. PERIODIC IMPACT MOTIONS Let us assume the existence of a periodic motion, in which the impact is repeated after a period T= 2nn/w which is an integer multiple (n = 1,2, , . .) of the excitation force period (see Figure 2). From the periodicity conditions x(0) =x(T) = -r,
i(O) = f+ = -Ri( T) = -RL
,
(3)
CHAOS
IN MECHANICAL
SYSTEMS
WITH
97
IMPACTS
it is possible to derive [5] the body motion in the time interval 0 < t G T and the value of velocity .Y_of the body immediately before the impact in the form X 1 ---=p cos qcos wl-sin x,, 1 - v2
sin Ot_
1+ R tl cosP-1-R
Wrl) sin (~n/r~)
--
’
(4)
where cpis the excitation force phase at the instant of impact,
[cosp],,,,=-[rf KJE2(1 + K2) [sin v)]~,~,=[-~K*JB~(~
r2]/[R( 1 + K2)],
+K2)-?]/[B(l
+K2)],
r> 0 is the static clearance between the body and the rigid stop, while for r
Azi+ = /?.A ,.i+ ,
(5)
between differences at successive impacts. By linearization of the solution it is possible to derive the equation P2+p(U-1-R2)+R2=0,
(6)
where U= (1 - R2)( 1 -K/tan p) sin2 (nn/r~), for determination of the dependence of /3 on the system parameters. The stability condition of impact motion can be written in the form IPI
(7)
At the stability boundary the system parameters will satisfy the condition ]/I]= 1. As the roots p of equation (6) are in general complex, it is possible-according to the position of the roots /3 on the unit circle-to distinguish three kinds of impact motion stability boundaries s+~ s_] and s,: one has (1) s+, . . . p= 1 and (2) sWl. . . /I= -1 when equation (6) has real roots; one has (3) sg. . . IpI = 1 when equation (6) has complex roots. The existence condition of the impact motion solution excludes the possibility of the onset of an additional impact at the time interval between the periodic impacts, which can be expressed as x(t) > -r
for O
(8)
The existence boundary p of the given periodic impact motion corresponds to the state when the trajectory of the body’s motion at some instant t of the interval (0, T) will touch
98
F. PETERKA
AND
J. VACiK
i+
Figure 2. Schematic of the simple mechanical vibro-impact system and of its fundamental impact motion (FIM).
O-6
0.7
0.8
0.9
I.0
1.1
Frequency,
Figure 3. Region of existence and stability of the (;=O+l).
FIM (2 = 1, R = 0.6)
d-.tLl
-0.5 07
0.9
1.1
1.3
I.2
1.5
01 07
I.3
I.4
I.5
I.6
7)
: impactless motion (z = 0) ; beat motions
1 0,9
1.1
I.3
I,5
Frequency, 7
Figure 4. Frequency characteristics of the impact motion; P= 3, R = 0.6. - -, Solution I of the FIM ; - - --, solution II of the FIM; -, analogue simulation. (a) Excitation force phase at instant of impact; (b) beforeimpact velocity; (c) motion amplitudes .Y,,,,,/.v,,-theoretical results; simulated amplitudes with increasing (d) and decreasing (e) frequency, n.
F3
F4
Figure 5. Time (a) and phase (b) trajectories of the system motion, r’=3. FI, Impactless motion, n =0.79, n= 1, -_=O; F2 chaotic motion, n=O.8, z= 0 + 1; F3, periodic beat motion, n = 0.87, n =4, z = I /2; F4, chaotic motion, n=O.91, z= O-+ I ; F5, chaotic motion, n =0,95, I= 0 i 1; F6, FIM, n = z = I ; F7, stability loss of the FIM and development of the impactless motion, q = 1.48, n = 1, 3 = I + 0; F8, s-bifurcation and development of the FIM from the impactless motion, n = 1’15, n = 1, z = 0 --t I.
the stop, and then the following condition will be satisfied: x(t) = -r,
i(t)
= 0.
(9)
It is not possible to express the boundary p in explicit form even for the simple system mentioned and it is necessary to find this boundary by iterative methods. 2.3. REGIONS OF EXISTENCE AND STABILITY OF PERIODIC MOTIONS The stability and existence conditions define the regions of stability and existence of the periodic impact motions in the system parameters space. The system motion is mainly influenced by the relative clearance i and the relative tuning n of the excitation force. We will therefore demonstrate the results of the stability and existence analysis of different types of impact motions in the plane (F, n). It is possible to characterize the periodic impact motion by
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where p is the number of impacts and n is the number of excitation force periods in the motion period T (n is called the subharmonic number). The main group of moths is formed by those motions of which the period is identical with the period of excitation (T= 27r/w, n = 1). These motions create series of z = 0, 1,2, . . .-impact motions, including the impactless one (z=O). The time and phase trajectories of these motions are denoted F9(a, b)-Fl 1(a, b) and Fl2(a, b, c) in Figure 7. The regions of existence and stability are hatched obliquely in Figure 6. With decreasing clearance V,the region of z-impact motion approaches the boundary of existence p, where the so far impactless trajectory of body motion begins to touch the stop (see Figure 7 F9 for z = 0 and F12(b) for z= 2) and the quantity z gradually grows. On the contrary, with increasing clearance i;, the region of zimpact motion reaches the stability boundary s. The boundaries p, s between the particular regions of z-impact motions alternately intersect at points X and so create two kinds of transient zones. There are zones partly of hysteresis (cross-hatched in Figure 6(a)) and zones partly of beat z+z+ l-impact motions (hatched horizontally in Figure 6(a)). Both neighbouring z- and z + l-impact motions are stable and can exist in the hysteresis zones. On the segments of the boundaries p, s, which define the hysteresis zones, one neighbouring motion switches into another. Along the segment p, the quantity z increases by one and along the segments s the quantity z decreases by one. In the zones of beat motions no motion with excitation force period is stable or can exist and consequently periodic n-multiple impact motions (n = 2, 3, . . .) and chaotic ones appear. From theoretical analysis (see the Appendix) of the group of z= (l/n)-impact motions (n=l,2,. . .), the character of the boundaries p, s can be explained (see Figure 6) between the impactless (z =0) motion and the z= l-impact motion (fundamental impact motion (FIM),p=n=z= 1). The boundary p. of existence of the impactless motion and boundaries pI of existence and $+I, S-I , sq of stability of the FIM in the vicinity of resonance 17= 1 are shown in. Figure 3. The transition from the existence and stability regions of the neighbouring motions is demonstrated in the characteristics of system motion along the line q in Figure 3 (r=3, 0.5~n~l.6). On the boundary p. (at points G, K=J) the body with impactless forced vibration begins to touch the stop (see Figure 5(Fl)); it can be considered as the FIM, the impact velocities of which degenerate to bare touching. Thus along the boundary p. both neighbouring motions change continuously. This transition is called s-bifurcation (see the splitting of the amplitude characteristics at points G, KU= J in Figure 4(c)). The stability analysis of the FIM has shown [5] that this transition is never stable, and the stability boundary sg,is in agreement with it. Along the boundary po=s, the relation sin q=O holds (see Figure 4(a) at points GI, KII): i.e., the impact velocity f- is zero (see equation (4) and points G,, K,,= J in Figure 4(b)). At point G the impactless motion develops into solution I of the FIM, but it is unstable in the interval between points G, S and the system passes through the transient zone of the beat motions, the characteristics of which are shown in Figures 4(a, b, d, e). Several time- and phase-trajectories of the chaotic beat motions are shown in Figures 5 F2, F4, F5 and periodic beat motion in Figure 5 F3 (for a more detailed description, see the next section). Point S lies on the stability boundary se1 of the FIM, where the differences Aq, Ai+ of the quantities p, i+ from the corresponding values of the FIM start to grow and alternate in sign at impacts. However, their increase becomes stable in a certain limit cycle and the body motion alternates in the vicinity of the unstable FIM. In this way a motion with two different impacts occurs, and is repeated after two periods of excitation (see, e.g., the motion z=2/2= 1, p=n= 2 in Figure 7 FlO(c). Thus the period doubling bifurcation is seen along the stability boundary s-1 . The two-way transition across the
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boundary s-, at point S in Figure 3 and across the whole zone of the beat motions between points G, S is shown in Figures 4(d, e). The FIM, which has been stabilized (see Figure 5 F6) at point S, continues up to point F (see Figure 4(d)), at which the FIM transits by jump to point L of the impactless motion (see Figure 3 and Figure 5 F7). Point Flies on the stability boundary s+~ (see Figure 3), where the differences dq, di+ grow with the same sign and the impact vanishes. At point F both solutions I and II of the FIM coincide (see point F,,,, in Figures 4(a-c)) and beyond this point no solution of the FIM is real. The saddle-node bifurcation of the motion corresponds to the state along the stability boundary s+ , It is evident from Figures 4(c, d, e) that in the transient hysteresis zone between points K=J and F= L (Figure 3) both neighbouring motions can exist and are stable. The zones of attraction of the above-mentioned motions have been described in reference [16(e)]. By the s-bifurcation at point J a jump from the unstable solution II to the stable solution I of the FIM is seen (see Figure 5 F8 and points K,,=.I and K, in Figures 4(a, b, c, e)). The transition between the stability and existence regions of the multiple impact motions (z = 1,2, 3, . .) has a similar character. It follows from this description that by continuously changing the parameters ?, n of this system the transition from one neighbouring motion into another is accompanied either by the jump effects on the boundaries of type p, s+~ of the transient hysteresis zones or by the rise of the series of beat motions on the boundaries of type p, se1 (see Figure 6(a)). A continuous transition between the neighbouring motions can occur only through points X in the vicinity of which the impact motion with nearly zero impact velocities is stable.
2.4. ZONES OF THE BEAT IMPACT MOTIONS A more detailed view of one of the beat motion zones (ztz+ 1) is presented in Figure 6(b) which is a detail of Figure 3. The zone z = 0 -+ 1 on the right side of point X3 is limited at the top by the existence boundary p. of the impactless motion (po- B) and at the bottom by the stability boundary s-i of the FIM. By means of a theoretical analysis [16(a)] (see also the Appendix) it is possible to find in this zone the existence and stability subregions of the z = (I /n)-impact motions. These subregions are alternately obliquely hatched in Figure 6(b) and limited by the stability boundaries se1 and the existence boundaries p2,. . . , p5. The trajectories of motion in these subregions are shown in Figures 7 F13(a, b)-Fl6(a, b). There are adjoining subregions of z = (1 /n)-impact motions (n > 6) for higher values of clearance ?. The impact motion is invariably found below the boundary p. (r
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0.7
0.8
0.9
I.0
Frequency, 7
Figure 6. Existence and stability regions of the z-impact motions in the plane (I:, q). (a) Schematic diagram, (b) theoretical results; (c) results of analogue computer simulation for R=0.6.
2.5. CHAOTIC IMPACT MOTIONS Chaotic motions can occur, as in other non-linear systems, as a result of successive motion period doubling. This was described for the impact system being described here by Isomlki 1141. The process of sequential period doubling is possible if the existence condition of type (8) is always satisfied. The quantity z holds its value since z = 2kp/(2kn), where k=O, 1,2,. . . is the order of the z = (p/n)-impact motion splitting. When, in connection with the transition to the chaotic motion, the value of z decreases, then one impact in the motion period vanishes due to the stability loss of the previous split-up motion. The loss of stability must be connected with the saddle-node bifurcation, which is accompanied by a jump to the impact motion with a smaller value of z. If the periodic impact motion with a smaller value of z can exist and is stable, then it becomes stabilized. An example is the transition from point FlO(a, b) through point FlO(c) to point Fl3(a, b) in Figure 6(c). The z= (l/l)-impact motion (see Figure 7 FlO(a, b) splits to the z=2’.1/ (2’.1)-impact motion (see Figure 7 FlO(c); see also point 4 in Figure 8(a), a weaker impact of which vanishes on the stability boundary of type s+~ and the z = (l/2)impact motion becomes stabilized by a jump (see Figures 7 F13(a, b); see also points 5 in Figure 8(a)).
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Fl5
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Figure 7. Several kinds of periodic and chaotic impact motions: (a) time trajectories; (b, c) phase trajectories. F9, Impactless motion, q=O.7, r’=2.1, n=l, z=O; FlO(a, b), FIM, q=O.75, r:= 1.5, n=z=l; FlO(c), split-up FIM, q=O.8, P=2.1, n=2, z=2/2; Fll(a, b), two-impact motion, q=O.7, i=O,8, n= 1, 2=2; Fll(c), q=O.8, r=0,04, n=2, z=4/2; Fl2(a, b), two-impact motion, q=O.62, P=O.O3, n= 1, z=2; F12(c), three-impact motion, q=O.62, F=0.02, II= 1, z=3; F13(a, b), q=O.83, P=2.5, n=2, z=1/2; F13(c), q=O.9, f=2.8, n=4, z=2/4; F14(a, b), q=O.88, T=3.4, n=3, z= l/3; F14(c), q=O.9, r=4.15, n=6, z=2/6; F15(a, b), q=O.895, r=4.65 n=4, 2=1/4; F15(c), q=O.92, r:=5.2, n=8, z=2/8; F16(a, b), q=O.915, f=57, n=5, z= l/5; F16(c) ;=0.93, v‘=6.3, n= 10 , z=2/10; F17(a, b), q=O.61, r‘=2, n= 1, z=2; F17(c), r:= 1.7, n=2, z=4/2; F18(a, b), q=O.76, F=3.5, n=2, 2=2/2; FlS(c), r=3.19, n=4, z=4/4; F19(a, b), q=O.82, i;=4.85, n=3. z=2/3, Fl9(c), f=4.67, n=6, z= 4/6; F20(a, b), q=O.845, P=6.85, n=4, z=2/4; F20(c) r:=6.6, n=8, z=4/8. Chaotic motions: F21, q=O.65, i=I.25, z=lt2; F22, q=O.78, ~=2.3, z=1/2+1; F23, 17=0,95, ?=5, z=l/3-1; F24, q=O.95, f=6.5, 2=1/4+-l.
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Chaotic motion appears when such a stable periodic motion cannot exist. An example of this is provided by the z = 1 f 2- and z = (l/2) f l-impact motions in Figures 7 F21(a, b) which develop from the split-up z=2-impact motions (see Figure 7 F17) after the first splitting (~=2~.2/(2~.1)=2, k= 1) and from z = l-impact motions (see Figures 7 FlO(a, b, c) and FlS(a, b, c)) after the second splitting (z = 2k.1/(2k. 1) = 1, k = 2). Such an origin of chaotic impact motions can be generalized and the transition from the existence and stability regions of the z + l-impact motions (see Figure 6(a)) into the z t z + l-impact beat motions along the stability boundaries of type se1 (period doubling bifurcation) can be explained. However, chaotic impact motion can be caused by the fact that the existence condition of type (8) of the periodic impact motion is not met. This is directly related to the transition of z-impact periodic motions into the regions of zt z+ l-impact motions along the segments of existence boundaries of type p = s+,(see Figure 6(a)). After crossing these segments of the boundaries p either some periodic impact motion will become stabilized if it can exist and is stable (e.g., the transition of impactless motion in Figure 7 F9 into the z= (1 /n)-impact motions for n = 2, . . . , 5 in the vicinity of points X2, X6, X, and X, in Figures 6(b, c) and Figures 7 F13(a, b)-F16(a, b)) or more complex periodic motions (of type F18(a, b, c)-F20(a, b, c) in Figure 7) occur. In the opposite case the chaotic z+z+ limpact motion will appear directly (e.g., see horizontally hatched regions z= (l/n) + 1 in Figure 6(c) for n = 2, 3, . . . , cc and see also Figures 7 F22-F24). In any case the quantity z increases after crossing the existence boundary p. A similar case of the origin of chaotic motions occurs when the process of period doubling of the beat impact motions is discontinued due to the fact that the existence condition (8) is not met. In such a way the laws of the changes of the impact motions inside the beat motion zones can be explained. As an example the changes of impact motions along line p in Figure 6(c) will be described. For constant frequency r~=0*92 and a continuously changed clearance P the characteristics of the impact velocities -L/Rx,, were obtained theoretically and by way of simulation [5], and are presented in Figures 8(a, b). Thanks to the theoretical analysis of the group of z = (2/n)-impact motions (n = 132,. . ,) which was carried out by Kotera [ 16d], it was possible to describe both the jump transitions connected with saddle-node bifurcation (see the jump from points 2 of the z = motion in Figure 8(a) and, (2/l)-impact motion into point 3 of the z= (l/l)-impact similarly, the jump from points 5 of the z = (2/2)-impact motion into the z = (l/2)-impact motion) and the splitting of the z = (1 /n)-impact motions into the z= (2/2n)-impact motions (see the splitting of characteristics in points 4, 6 and 8 in Figure 8(a)) connected with period doubling bifurcation. Only in the vicinity of points 6 and 8 in Figure 8(a) is it possible to observe the development of split-up motion discontinued at points 7 and 9 due to the failing of the existence condition of type (8). The body’s motion up to the existence boundary (near points 7 and 9 in Figure 8(a)) is shown in Figures 7 F13(c) and F14(c), where one of the impactless loops almost touches the stop. With increasing clearance ? and further development of splitting, the chaotic impact motion (z = (l/2) +- 1 and z = (l/3) + 1) develops at the moment of additional impact. If we exclude the existence condition for the impactless loops, which can be done on the analogue computer [16c], then the split-up motion will develop along the dashed branches, starting from points 7 and 9 in Figure 8(a). Such a situation is demonstrated in Figure 9(a) where, for large splitting of the ;= (l/3)-impact motion, even two primary impactless loops (see Figure 7 F14(c)) penetrate the stop and the z = “2/6”-impact motion is stable. Actually the existence condition causes the onset of the chaotic motion (see Figure 9(b)). The autocorrelation function R,,(1.2~w) in Figure 9(c) shows evidence of the random character of the chaotic motion.
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I
I
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I
3
4
I
4
3
2
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I
.;: J
0
-I
0
2
2
I
0
-I
0
I
2
3
4
5
6
7
Cleoronce,l
Figure 8. Impact velocity characteristics along the line p in Figure 6(c). (a) Theoretical results; (b) results of the impact motion simulation.
Figure 9. (a) Non-real periodic z= “2/6”-impact motion; (b) chaotic z= l/3+ l-impact motion; (c) autocorrelation function of chaotic motion, R = 0.6, q =0.92, f=4.5.
It was evaluated from samples ui= x,, ing to the formula
of the motion amplitudes (see Figure 9(b)) accord-
(11) where 5= (Ci”, ai)/N is the mean value of the motion amplitudes, and Z.~W/Wis the time delay expressed as the excitation period multiple (O
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It should be mentioned that the chaotic impact motion can occur due to the failing of the existence condition of the non-split-up z= (l/n)-impact motions (see, e.g., point 10 in Figure 8(a) where a decrease in clearance ! after the impactless loop of the z = (1/3)impact motion-Figure 7 Fl4(b)-has touched the stop results in the chaotic z = I /2 + 1impact motion; similarly the next z= (l/n)-impact motions-Figures 7 F15(b, F16(b), etc.-transit into the chaotic z = 1/(n - 1) t 1-impact motions-Figure 8(b)). This explanation can be generalized and allows us to elucidate the system behaviour in the regions of beat z+ (z + I)-impact motions between the existence and stability regions of the neighbouring z- and z+ l-impact motions (see Figure 6(a)). The occurrence of different types of impact motions, including the chaotic ones, depends on the system damping. The smaller the damping, the more diverse is the system response to the periodic excitation [5, 16f].
3. IMPACT MOTIONS OF MORE COMPLEX SYSTEMS The more complex the mechanical system is (with a higher number of degrees of freedom and impact pairs) the more difficult it is to find system parameters for which some periodic impact motion will satisfy the stability and existence conditions. Therefore for such systems the development of chaotic motions is more frequent than the development of periodic ones. As an example we will consider the five-mass system shown at the top of Figure 11. This is a model of transverse vibration of a beam placed between two rigid stops. The beam is supported at its ends and is excited in the centre by a periodic force F, cos wt. The beam model has five degrees of freedom, and each of the substituting masses can impact on stops. The system contains five impact pairs with two-sided impacting. The regions of the necessary onset of the impact motion are limited by the amplitude characteristics p of the forced vibrations. The regions are hatched in Figure 10 for the vicinity of the resonances n = 1, 4, 9, 16, 25 (where n = o/f2, ; 0, is the first eigenfrequency of transverse vibration). Due to the system’s symmetry and the point of excitation, the even resonances are suppressed. Chaotic motion exists in all regions under the boundary p and by increasing the relative clearance 7= r/y,, it can be maintained up to the dashed boundaries where it transits into impactless motion. Periodicmotions were found only near the fourth eigenfrequency (0,/L!, = 16) and their existence and stability regions are hatched obliquely in Figure 11. The results were obtained by analogue computer simulation. The relations used for simulation can be expressed as yi =
YilYsr
3
dY, dY, dt YI=~=-d-~=~il(Y,ss21),
(12)
where y,, is the static deflection of the beam centre caused by the force also acting at the centre, r= 5fJ,t is the time transformation of the simulation, Yi, Yl are the simulated deflections and velocities of particular bodies (i = 1,2, . . . , 5), and yi, ji are the actual deflections and velocities. For the characteristic points A-Q (see Figure 11) the corresponding cross-sections of the Poincare map (see Figure 12) as well as the phase trajectories Y/T Yf (see Figure 13) are added. In all of the motions described below only the centre of the beam (i= 3) impacts. The points representing the phase nr of the excitation force and velocity Y$_ at the instant of impact are plotted on the polar diagram (nr, Y;-). The systems of co-ordinates (r~r, Y;_) and ( Yi, YI) are described in Figure 12(a) and Figure 13 H. The largest region corresponding to the z = 2-impact motion is denoted in Figure 11 by oblique and horizontal hatching. There are three kinds of two-impact motion, as follows.
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-‘;_,‘rr_-rF
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-?Fr~,-Fry-r----~-~
,,-,,,,,~,.,.,,-,,,.,.,~ 3.8
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46
9.0
9-2
94
9.6
16 2
16.-l
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25.2
25.4
25.6
_:
o-o
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8.8
ok‘=-=-----~-------~------1 15.6
158
P
I.5
16 0
WI
I.0 0.5 0.0 24.6
24.8
25.0 Frequency.
7)
Figure IO. Existence regions of the chaotic impact motion of a beam placed between rigid stops.
(1) The first symmetrical periodic motion in which the impacts of equal intensity alternately occur on the opposite stops after time intervals T/2 = K/W. The trajectories Yi, Yl are symmetric in the phase plane (see Figure 13H). (2) Second is the non-symmetrical periodic motion, the period T of which is unevenly divided between the successive impacts of equal intensity. This is expressed by the central non-symmetry (except for the impact points) in the phase plane (see Figure 13 F). Since the system is symmetric and the non-symmetric motion has the same relation to each of the two stops, there must be two different non-symmetric motions. We will differentiate them as type a and type b. They are mutually symmetric in the phase plane (see Figure 13 F). (3) Finally there is the non-periodic motion which chaotically oscillates around one of the two non-symmetric motions (see Figures 13 D, C). The periodic motions exist in the obliquely hatched region z=2 in Figure 11. The symmetric (non-symmetric) motion transits into the non-symmetric (symmetric) one with decreasing (increasing) clearance J or frequency n on the dashed curve passing through point G. The non-symmetric motions transit into the chaotic motion (e.g., for type a see Figures 13 E, D, C) along the boundary passing through point E, and this motion exists in the horizontally hatched region with internal points’ D, K (see Figure 11). The z = 24mpact motion is described in detail by means of the impact characteristics in Figure 12 obtained by changing the clearance 7 along the lines u( q = 16) and o( q = 15.25). When the clearance F increases from the value ?=0.018 (point B in Figure 11) the nonsymmetric chaotic motion, the attracting set (AS) of which creates the narrow closed
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” 16 Frequency,
I?
18
19
20
q
Figure I I, Existence regions of periodic (///I/) and chaotic (g) motions of a beam placed between rigid stops (R=0.6).
curves (see Figures 12(c, d)-the arrows C), starts to deviate at point C (r’=O.O193) from the large developed chaotic z=2+ 3-impact motion, the AS of which is shown in Figure 12(b). During the following increase of clearance r: the AS of chaotic motion contracts the area into two points E (see Figures 12(c, d)-the arrows D, E-and Figures 13 C, D, E) which are the AS of the non-symmetric periodic motion. In the interval between points E and G, in Figure 11 the AS branches continuously travel from points E (see Figure 12(c)) until they merge at point G which is the AS of the symmetric impact motion. This motion is stable up to point I (see Figures 11 and 12(c, d)). The series of the AS from Figure 12(c) is shown for discrete values of I: in Figure 12(d). The symmetric motion loses its stability at point I by the saddle-node bifurcation and changes by a jump into the impactless motion. The points of the transition process are recorded along the dot and dash arrow in Figure 12(d). Point G is important because it lies on the boundary of motion symmetry and corresponds to the pitchfork bifurcation. There is the same probability of the onset of each type (a, b) of the non-symmetric motion on this boundary. The two branches of the characteristics between points E and G in Figure 12(c) describe both types a, b of the non-symmetric motion because the absolute values of the impact velocities Y;_ are identical for both stops. The difference is in the fact that with type a a certain branch EG (see Figure 12(c)) corresponds to the impact on one stop and to the impact on the other stop for type b.
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..,
T=0~011-
(f
Ih)
h0*003-
(0-J)
’
I
0.04
\
Figure 12. Cross-section of the Poincark map of the periodic and chaotic motions
The change of the AS throughout the interval 0.003 < rcO.036 (for q = 16) is shown in Figure 12(c). For small values of r the AS of the z 2 3-impact motions occurs in two dark regions designated by arrows A in Figure 12(c). The chaotic z=2 + 3-impact motion (between points A, C in Figure 11) is suppressed in Figure 12(c) so that the AS of z=2impact motions would not be superimposed. This chaotic motion is shown at full size in Figure 12(a). Weak impacts also appear in this motion, and the small dark region near the centre of the polar diagram is in agreement with it. The phase trajectories of the beam centre are shown in Figures 13 A and B for points A and B. The cross-section of the regions along the line D in Figure 11 was chosen to demonstrate the AS of the z>2-impact motions in a better way. The motion at points J, K and L is similar to the motion at points E, D and C (see Figures 13 E, D and C) but the ASS are more expressive. The AS of the motion at point K (q= 15.25, F=0.0326) is represented in Figure 12(i) by two closed curves into which the points at two-sided impacts are plotted chaotically, while one of the two types a or b of the non-symmetric motion is maintained. These AS curves expand with decreasing clearance into an AS with a larger area (see the AS in Figure 12(h) for point L) while the chaotic motion remains throughout as either of
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I //////////////////,/////////////////////~~,,~~~,,,~~
////
///////////
1///////////////////////////////////////~/////////~/~//////////////
F
&-a6
/
?=0.02?
Qij4-*- -,A
-
‘//////////////////////////////////////////////////////////////////
i=3
i=3
i=3 0
E 7~0.023
E ?=0.023
a
i50.0105 /////u//// /II /
b I \
Figure 13. Phase trajectories of the periodic and chaotic motions.
the two types a, b of non-symmetric motion (see, e.g., Figures 13 E, D and C). Both areas of the AS will merge only after crossing the boundary which passes through point L (see the AS in Figure 12(g) for point M in Figure 11) while both types a and b of nonsymmetric motion will alternate during chaotic motion. This is shown by the accumulation of the phase trajectories in Figure 13 B. The continuous change of AS with changing clearance r along the line D in Figure 11 is demonstrated in Figure 12(f). The samples of this continuous spectrum are shown in Figures 12(g, h, i) for clearances ? which correspond to arrows M, L and K in Figure 12(f). The band of the chaotic z=2+ 3-impact motion is again suppressed, to illustrate the triad of the z = 3- and z = 3 +4-impact motion AS (see arrows N and P showing one part of the AS in Figure 12(f)). The ASS of more-impact motions are shown in Figure 12(e). The periodic z = 3-impact motion develops at point N in Figure 11 from the chaotic z = 2 t 3-impact motion. There are again two types, a and b, of the z = 3-impact motion, the phase trajectories of which are also mutually symmetric (see Figures 13(0a, b)). Two impacts on one stop and one impact on the.other are repeated during this motion. The triad of the attractors shifts along the segment NO of the characteristics in Figure 12(e) when the clearance changes from point N to point 0 in the z = 3-impact motion region in Figure 11. Point 0 is important again because it is here that the pitchfork bifurcation occurs, just as it does along the boundary passing through point G in the z2-impact motion region, The z= 3-impact motion changes on the boundary passing through point 0 into one of two subtypes, a and /I, with equal probability. One branch of the characteristics between points 0 and P in Figure 12(e) corresponds to every subtype of z = 3-impact motion. The phase trajectories of the type a - a, /I are illustrated in Figure 13 P. The z = 3-impact motion loses its stability at point P and chaotic z = 2 + 3-impact motion appears. Its phase trajectories are shown in Figure 13 at the bottom right for parameters n = 15.25, ?=0.006. It is here that the region of z =Cimpact motion (see point Q in
CHAOS IN MECHANICAL
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the region z =4 in Figure 11 and see Figure 13 Q) and the regions of more-impact motions (z > 4) for small values of clearance i: are found. The group of four fixed pointsattractors corresponds to the periodic z=4-impact motion at point Q (see the arrows Q in Figure 12(e)). This explanation is of a definite value for further research, because it illustrates the heterogeneity of the periodic motion bifurcations of more complex systems and of the chaotic motions.
4. CONCLUSIONS
The discussion and the results of the analogue simulation presented above confirm that chaotic motions accompany the periodic motions of periodically excited mechanical systems with impacts. The s-bifurcation appears at the moment when the existence condition of the periodic impact motion is not satisfied. It is connected with an increase in the quantity z which expresses the average number of impacts related to one period of excitation. When the stability condition of the periodic impact motion is not satisfied, one of the following situations occur: (a) saddle-node bifurcation with a decrease in the z value; (b) period doubling bifurcation with the z value retained, but the period of the periodic motion increased; (c) pitchfork bifurcation, when both the z value and the period are retained; this bifurcation is connected with the onset of non-symmetry of the motion in systems with symmetrically arranged stops. The mechanism for transition from the periodic impact motions into chaotic ones after the crossing of the existence and stability boundaries in the beat motion zones is also explained.
REFERENCES 1.
2. 3. 4. 5.
KOBRINSKII 1964 Mechanisms with Elastic Links. Moscow: Nauka. A. E. KOBRINSKII and A. A. KOBRINSKII 1973 Vibro-impact Systems. Moscow: Nauka. V. L. RAGULSKENE 1974 Vibro-impact Systems. Vilna: Mintis. V. I. BABITSKII 1978 Theory of Vibro-impact Systems. Moscow: Nauka. F. PETERKA 1981 Introduction to Vibration of Mechanical Systems with Internal Impacts. Prague:
A. E.
Academia.
6. R. BANSEVITCHIUS and K. RAGULSKIS 1981 Vibromotors. Vilna: Moskals. 7. V. I. BABITSKII and V. L. KRUPENIN 1985 Vibration in Strongly Non-linear Systems. Moscow: Nauka. 8. G. SILAS and L. BRINDEU 1986 Vibro-impact Systems. Bucharest: Technical Editions. 9. F. PETERKA 1973 Laws of Impact Motion (short film). Prague: Institute of Thermomechanics, CSAV. 10. P. J. HOLMES 1982 Journal of Sound and Vibration 84, 173-189. The dynamics of repeated impacts with a sinusoidally vibrating table. 11. S. W. SHAW and P. J. HOLMES 1983 Journal of Sound and Vibration 90, 129-I 55. A periodically forced piecewise linear oscillator.
12. M. I. FEIGIN 1978 PMM 42, 820-829. About structure of s-bifurcation boundaries in piecewise
continuous systems. 13. T. KOTERA and H. YAMANASHI 1986 Transactions of the Japan Society of Mechanical Engineers 52, 1883-1886. Chaotic behaviour in an impact vibration system, 2nd report. Influence of damping coefficient and coefficient of restitution. 14. H. M. ISOMKKI, J. VON BOEHM and R. R;~TY 1987 Proceedings of the XZth ICNO, Budapest, 656-659. Chaotic oscillations and the fractal basin boundaries of an impacting body. 15. M. S. HEIMAN, P. J. SHEARMAN and A. K. BAJAJ 1987 Journal of Sound and Vibration 114, 535-547. On the dynamics and stability of an inclined impact pair.
112
F. PETERKA
AND
J.VACiK
set of companion papers on the laws of impact motion of a mechanical system with one degree of freedom.) (a) F. PETERKA 1974 Acfa Technica &A V 19(4), 462-473. Part I-Theoretical analysis of nmultiple (l/n)-impact motions. (b) F. PETERKA 1974Acta Technica es.4 V 19(S). 569-580. Part II-Results of analogue camputer modelling of the motion. (c) F. PETERKA and J.VAC~K 1981 Acta Technica &AV26(2), 161-184. Part III-Statistical characteristics of beat motions. (d) T. KOTERA and F. PETERKA 1981Acta Technicu &41/26(6), 747-758. Part IV-Analytical solution of the 2/n-impact motions and its stability. (e) F. PETERKA and T. KOTERA 1982 Actu Technicu CSAV 27(l), 92-l 17. Part V-Regions of existence and stability and domains of attraction of different kinds of impact motion. (f) T. KOTERA and F. PETERKA 1984 Actu Technicu k$AV 29(3), 255-279. Part VIAnalytical and analogue solution of the multiple-impact motion and its stability. 17. F. PETERKA and J.VAC~K 1987 Proceedings of the XIth ICNO, Budapest, 707-710. Chaotical and periodical forced transversal vibrations of the beam type mechanical systems with impacts. 16. (A
APPENDIX: BRIEF THEORETICAL ANALYSIS OF PERIODIC (l/n)-IMPACT MOTIONS AND THEIR STABILITY OF A ONE-MASS SYSTEM A.I.
CONDITIONS
NECESSARY
FOR THE RISE OF IMPACT MOTION
(a) For r > 0 the amplitude B of forced vibration must be higher then the static clearance r (see equation (2) and Figure 2). (b) For r
Fp= (rlc. According to equation (2) the necessary conditions can be expressed using the formula p,~-l
(Al)
(e.g., in the region between the boundaries p. and pP in Figure 3 some impact motion must exist). A.2. ANALYSIS OF SOLUTION OF PERIODIC (l/n)-IMPACT MOTIONS When the case of absolutely elastic impacts (R= 1) is excluded, the solution (4) of the group of (l/n)-impact motions (Figure 2) is undetermined, due to the function K in equation (4), at the frequency values (A2)
q = n/l,
is the subharmonic number and I= 1,2,3, . . . is the ultraharmonic wheren=l,2,3,... number. The frequency interval 0 < 9 < co is divided by the values (A2) into the sub-intervals of the definiteness of the solution (4). It is necessary to search for a region of real periodic impact motion existence and stability in every sub-interval. The results of the analysis will be-explained in the sub-interval O-52 v < 1 (for n = 1 and I= 1,2; see Figure Al). A.3. MATHEMATICAL The condition
REALITY OF THE SOLUTION
(4)
-p” < F< p n= poJiTz
follows from the necessary positivity of the discriminant in solution between the boundaries fp” in Figure Al(a)).
(A31 (4) (see the region
CHAOS
A.4.
PHYSICAL
REALITY
IN
MECHANICAL
OF THE
SOLUTION
SYSTEMS
WITH
113
IMPACTS
(+-CONDITION
OF
IMPENETRABILITY
The conditions of impenetrability of the impacting bodies can be expressed in two wajrs, as follows. (a) The before-impact velocity must be negative, (or CC+= -RL>O),
L
CA4)
for the instants (r=O, 2lrn/w) of the impact. According to equations (4) the condition (A4) has the form
-rKf
B*(l +K')-r*>O.
645)
The analysis of this condition for the solution I (the + sign) and the solution II (the sign in the relation (A5)) of the periodic impact motion depends on the polarity quantity K (equation (4)). The zero-points of the function K, which correspond to the frequency values 17= fll(k + 1PI,
k=O, 1,2,.
.)
(A61
divide the frequency sub-intervals according to the polarity K in respect to the next two sub-intervals (e.g., for the sub-interval (l/2) < 17< 1 the quantity K is negative in the subinterval l/2 < q < 2/3 and positive in the sub-interval (2/3) < 17< 1; see Figure Al. The summary results of the condition (A4) analysis can be expressed as follows: For K
For K>O
-p()
Solution I Solution II
-f
(A71
-p”
8
6
4 r
r
F6a S-I
2
2
P.
0
-2
-4
-5
?
1
2
.A14)
3
3
u>O 2(I+R2)-U>0
(Al21 tA,4j2(l+R2)-
U>O
(A121
U=-0
(Al4)
X,--r
(0)
(b)
Cc) Figure
A.
1
id)
18)
114
F. PETERKA
AND
J. VACiK
These results are shown by hatching in Figure Al(a). Between the boundaries fp, the condition (A4) is satisfied only by solution I in the whole sub-interval n. Between the boundaries po, p” and -pa, -p”, for K< 0 and K>O, respectively, the condition (A4) is satisfied by both solutions 1, II of the periodic (1 /n)-impact motion. (b) The trajectory x(t) of motion must satisfy the condition (8) for the time interval 0 < t < 2rrn/o between impacts. It is possible to express this condition as r’>Pl,
(A8)
where the boundary pI (see Figure Al(d)) must be found AS.
CONDITIONS
OF
STABILITY
OF
PERIODIC
numerically.
(l/n)-IMPACT
MOTIONS
For the analysis of equation (6) according to condition (7) it is useful to transform this equation by means of the relation JI=(w+l)/(w-I)
(A9
into the equation tiw2+2(1 -R2)W+2(1
+R2)-
U=O.
(AIO)
The stability condition (7) corresponds to the condition w,
(AI I)
where w, is the real part of the roots of equation (AlO). The Hurwitz conditions u>o,
2(1 +R2)-
2(1 -R2)>0,
U>O
(A12-A14)
must be satisfied in order to satisfy condition (Al 1). (a) Condition (A12) can be transformed as follows: f
B2(1+K2)-r2/{-rKf
B2(1+K2)-r2}>0.
(A15)
Condition (A15) is identical with conditions (A3) and (A5) for the solution I (+ signs) and therefore the stability region of this solution is described by conditions (A7). The stability boundaries s+, up”
for Kc0
and
S+, zs -p”
for K>O
(Al@
correspond to the U= 0 (see conditon (A12) and Figure Al (a)). These boundaries were named s+~ for the value J? = + 1 which the quantity /? achieves along them. The solution II satisfies the stability conditon (A15) only in the regions in which condition (A4) is not satisfied, and therefore further analysis concerns only solution I. (p) Condition (A13) is satisfied for the whole interval O< R < 1 of the restitution coefficient R. The stability boundary corresponds to motion with the absolutely elastic impacts (R= 1). ( y) The analysis of condition (A14) determines the boundaries of type s-1 (/I = - 1 along them) as (A17)
S-l = -PoJWl, where L= (1 + K2- V)/dm and V=2(1+ R2)/[(1 -R)2sin2 (an/q)]. stability regions are determined by the conditions f>s_,
for Kc0
and
as graphically shown, e.g., in Figure Al(b).
?
for K>O,
The
6418)
CHAOS
IN MECHANICAL
SYSTEMS
WmH
IMPACTS
115
The stability conditions (A12) and (A14) in summary have the form s-r
for K-CO
and
s+i
for K>O
(A19)
as shown in Figure Al(c). It is necessary also to find the stability boundaries s,, along which the imaginary roots /l of equation (6) lie on the unit circle. This circle transits into the imaginary axis of the w plane and the boundaries sp,were found from analysis of the condition w,=O.
(A20)
This equation is satisfactory for R # 1 when sin q= 0. It follows from equation (4) that equation (A20) is satisfied along the boundaries p. where the impactless body motion with touching of the stop is identical with the degenerated impact motion with zero-velocity of impacts. The solutions I and II satisfy equation (A20) along the boundaries (sJ~,,~ according to Kc0 GO
i
(.Q,f = PO (sp),= -po
K>O
(%?)I”PO &J,1= -po
6421)
as shown, e.g., in Figures Al(c), (d). The stability boundaries s+~, set and sP were obtained under the assumption of small disturbances of the periodic impact motion. The analogue simulation explained the system motion development after crossing the stability boundaries: (a) the saddle-node bifurcation appears along the boundaries s+~ and the impact motion transits by a jump into another motion; (p) the period-doubling bifurcation of the impact motion appears along the boundaries sWl; (y) the beat impact motion (periodic or chaotic) arises from the impactless motion along the boundaries (s,),=po; (6) the jump transition from impactless motion into the (I/n)-impact motion (solution I) arises if this motion satisfies the existence condition (8) along the boundaries (s~),,EP~. It follows from the relations (A21) and items (y) and (6) that continuous transition from the impactless motion into the stable (l/n)-impact motions is impossible (the excep tion is the situation shown around points X3, X2, X6, X7, X8, . . . in Figures 6(b, c)). The resulting conditions (A8) and (A19) give only the hatched region in Figure Al(d) in which the one-impact motion can exist and is stable. By means of similar analysis it is possible to find the resulting existence and stability regions of every (l/n)-impact motion (see Figure 6(a)) in the whole frequency interval (see, for more detail, reference [5]). The analysis of the (2/n)-impact motions is similar and more complicated [ 16d], but it provides the possibility of constructing the bifurcation diagrams (see Figure 8(a)).