PIECEWISE ANALYSIS OF OBLIQUE VIBRO-IMPACTING

Transcription

1 ACTA MECHANICA SINICA, Vol.19, No.6, December 2003 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A. ISSN PIECEWISE ANALYSIS OF OBLIQUE VIBRO-IMPACTING SYSTEMS* JIN Dongping (,~,~) HU Haiyan (~)t ( Nanjing University of Aeronautics and Astronautics, Nanjing , China) Steven R. BISHOP (Centre for Nonlinear Dynamics and its Applications, University College London, Gower Street, London WC1E 6BT, UI 0 ABSTRACT: A great number of studies have shown the complex nonlinear dynamics of mechanical systems with repeated normal impacts. An oblique frictional impact introduces even more complicated dynamics such as stick-slip motions to those systems. Henee~ the dynamics of oblique vibro-impacting systems with possible sliding motion is an open problem. Based on a hybrid analysis of vibro-impact dynamics~ kinematics and complementary conditions, a piecewise analysis method is developed in the paper to describe the sliding motion during an oblique impact. Thereby, a parametrically excited planar pendulum between two parallel rigid walls is studied as an illustrative example. The example, together with the corresponding numerical results, shows that the sliding impacts occur in such a system with a set of properly selected parameters. KEY WORDS: nonlinear dynamics, oblique impact, sliding motion, piecewise analysis 1 INTRODUCTION Abundant nonlinear phenomena can often be observed in practical vibro-impacting systems, such as rattling gears and restrained heat transfer pipes. As reviewed by Bishop [I] and Brogliato [2], the nonlinear dynamics of vibro-impacting systems has drawn a considerable attention over recent years. To gain an insight into the complicated dynamical phenomena of vibro-impacts, a great number of studies have been carried out on impacting pendulums since their dynamics without impacts is relatively simple. For instance, Kane and Levinson [3] conducted a numerical study on two pendulums coupled with each other, one of which impacts a fixed rigid plane at the end. Moore and Shaw [4] experimentally studied an inverted pendulum subjected to normal impacts. They found that the pendulum under a given harmonic excitation might undergo one of i0 distinct steady-state motions, among which an impacting motion can coexist with a non-impacting motion. Bayley and Virgin [5] made another experimental study on an impacting pendulum, the swing amplitude of which was not restricted to a small angle, so that both periodic and chaotic phenomena were observed. Friction has to be taken into account in modeling an impact if any relative slip occurs between rough contacting surfaces, or two impacting bodies collide at a non-zero relative tangential velocity. Ratner [6], Lu [7] and Babitsky[ 8] ~tudied the friction characteristics of oblique impacts. In most previous studies on the vibro-impacts, the duration of impacts was assumed to be negligibly short, compared with the time scale of overall dynamics of the system. This assumption results in the concepts of instantaneous impact, Newton's law and Poisson's law as well. For instance, Stronge [9] studied an impact between two rough rigid bodies on the basis of Newton's law and Poisson's law, and showed the difference between the two laws. Subsequently, Wang and Matthew [I~ presented the analysis of a two-dimensional rigid-body collision with dry friction. They used the graphical Received 15 August 2001, revised 14 March, 2003 * The project supported by the National Natural Science Foundation of China ( ) and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education, China t hhyae@nuaa.edu.cn

2 580 ACTA MECHANICA SINICA 2003 method of Routh to describe the impact and determine the impulse of the frictional force. Stronge [11'12] investigated a lumped parameter model of contact between two impacting bodies to obtain the tangential force, the energy dissipated by friction during oblique impacts and the swerve of the relative velocity during a three-dimensional impact between two rough rigid bodies. Galvanetto, et alfl 31 studied the friction induced stick-slip vibration. In the field of multi-body dynamics, the frictional impacts due to unilateral contacts have also drawn much attention since the 1990's. For instance, Glocker and Pfeiffer[ 141 suggested the use of algebraic relations to avoid analyzing the change in degrees of freedom during a stick-slip impact. Pfeiffer [15] dealt with the complementarity problem of the stick-slip vibration of a multi-body system with many frictional contacts. Van de Vrande, et alfl 6] investigated the qualitative behavior of dry-friction-induced stick-slip vibrations by using a bounded smooth function. In this paper, based on a hybrid analysis of vibro-impact dynamics, kinematics and complementary conditions, a piecewise analysis is made first to describe the sliding motion in an oblique impact. Thereby, a parametrically excited planar pendulum between two parallel rigid wails is studied. Because the pendulum is subject to a vertical excitation, the horizontal impact between the pendulum and any wall may not be an instantaneous process, but a timedependent one. The analysis predicts the possible sliding impacts of such a system with a set of properly selected parameters. Finally, a numerical study is outlined to support the analytical findings9 2 EQUATION OF MOTION DURING AN IMPACT The mechanical system of concern consists of n particles with their positions given by n generalized coordinates as a vector q(t) = [ql(t),'",qn(t)] T in an inertial frame, and has n~ fixed constraints, The dynamic equation of the system during a free flight (before or after any impact) yields mi~ + f(q,//, t) = 0 (1) where rn represents the inertial matrix associated with the mass of n particles, f(q,//, t) the sum of all forces acting on the particles, and the dot over q the derivative with respect to time t. If nc constraints of the system are active when an impact begins, ttie impulse forces are applied at corresponding nc constraint surfaces. The study is confined to the impacts occurring in a two-dimensional space. If t0i represents the initial moment of an impact, the impact of ith particle starts from certain values of the generalized coordinates and velocities, which yields h~ = hni(q, t) C R 1 t > to~ (2a) Yyi = Vyi(q, q, t) C R 1 t > toi (2b) where hni is the displacement of the ith particle measured from the equilibrium position of the system, and Vyi the tangential velocity component of the ith particle, i = 1,. 9 n~. Thus, the equation of motion in the contact phase reads my1 + f(q'q't) -- E icn~ (H,~iF,~i v~f~) = 0 t > t0~ (3) where F.i represents the normal impulse force and Fy~ the tangential impulse force9 Hni and Vyi are the Jacobian matrix of the constraint defined by = (0hn T (4a) Hni \ Oq / = (Ovyi '~ T (4b) Vyi \ 0// / The study in this paper will focus on the oblique impact, which tends to produce a sliding motion. It is easy to see that, for an impact process with a sliding motion, the acceleration component perpendicular to the constraint surface during the slide keeps to be Zero. Differentiating hni with respect to the time, with help of Eq.(4a), gives W.. Hniq + Wni = 0 V~i = H~i q + (5@ 9 W. 02hni (5b) Ot----- F- The generalized can be eliminated from Eqs.(3) and (5). Then, one obtains icnr W~=0 t > t0~ (6) According to Coulomb's law, the magnitude of the tangential impulse force during the slide phase is proportional to that of the normal impulse force such that = (7) f vv~ [ where #i is the coefficient of the sliding friction.

3 Vo1.19, No.6 Jin DP & Hu HY: Analysis of Oblique Vibro-lmpacting Systems It should be emphasized that for a nonautonomous vibro-impacting system, the slip duration cannot be directly determined from the impulsemomentum equations. Hence, the additional complementary conditions for the constraint and motion are required to arrive at a set of algebraic equations for solving the duration and the velocity before and after an impact. 3 PIECEWISE ANALYSIS OF SLIDING IMPACT To gain an insight into an impact process, an analysis is made for the entire process of the impact.step by step as shown in Fig.1. In what follows, the initial moment of impact is denoted by t0~, the final moment of impact by tfi, and the duration of slip by At~i = tfl - toi. If these parameters are in hand, the impact process can be analyzed to obtain the velocities before and after the impact such that the succeeding motion of the system can be studied. It is easy to see from Figs.l(a) and l(b) that the normal acceleration *)~i0 and the normal impulse force undergo different changes during the impact, namely /'~i0 -~ 0 -+ ~5~a (8a) 0 -~ F~(t~ _< t _< t~) -~ 0 (8b) where/&i~ is the normal acceleration at the end of an unidirectional slip, to + the upper limit of t0i and t~ the lower limit of tfi. As shown in Fig.l(c), the normal velocity V~io gives rise to a jump from position C to position A. That is V~io e~iv~io (9) where 0 _< eni <_ 1, and eni is the coefficient of the normal restitution. The comparison of Figs.l(b) and 1 (c) indicates that the impacting bodies leave the constraint surface at the end of the unidirectional slip as soon as the normal impulse force vanishes. At the moment tfi, the normal velocity becomes -enivxio. In this case, Eqs.(6) and (7) enable one to see that T -1 -H,~im f(q'il't)lt tf~+w~i[t=t,, =0 t > t0i (10) ~C (a) 0 ' BJ T toi, tfi F; (b) B ~ C O' (c) toil m B! t~i --eniv xio VyiO I ~N~J.. [d ~, Vyif i ~ ~ J AL...- V yis ~ f " ~ 0 5 B]-eylvyio t~ I t. Fig.1 Piecewise analysis of an impact process This is the first equation associated with the duration of the slip. Also, the initial tangential velocity vyi0 travels in a path of CSBA shown in Fig.l(d). As analyzed above, one has VyiO --} Vyis --'} Vyi -'-} --eyivyio ---} Vyif < t _< 0 (11) where vyif is the tangential velocity jumping out of the constraint surface. According to the theorem of impulse[ 17], the tangential velocity vyia at the end of an unidirectional slip is vyi~ = -eyivyio (12) which models the tangential restitution. The quantity eyi is a coefficient of the tangential restitution and -1 _< eyi <_ O. Here eyi = -1 implies no tangential effects, and ey i a sticking impact. Using Eq.(2b), one finds that vyi~(q, il, t)lt=tf = -ey~vy~o(q,q,t)it=~o ' (13) This is the second equation for solving the duration of the slip. Now, two of the three equations required are ready. The remaining one can be found by considering the geometry of the constraint configuration during the impact. 4 EXAMPLE OF IMPACTING PENDULUM This section further analyzes the oblique vibroimpacts through an illustrative example shown in

4 582 ACTA MECHANICA SINICA 2003 Fig.2, where a parametrically excited pendulum oscillates between two parallel rigid wails. The pendulum consists of a small, elastic ball of mass rn and a weightless, rigid rod of length l. The pivot of the pendulum is subjected to a harmonic excitation of frequency w and velocity amplitude V0. For certain excitation amplitudes (at low frequencies for instance), the pendulum oscillates in the plane around its stable hanging equilibrium while the basement is moving up and down. If the excitation is adjusted, the amplitude of the oscillations may grow sufficiently large such that the ball of the pendulum collides a wall or both walls. Once any impacts occur, the subsequent dynamics becomes very complicated with a nonlinear nature. The type of vibro-impacting motion depends on the frictional impact between the bail and the walls, as well as the system parameters. 4.1 Sliding Impact The ball undergoes a slip as soon as the impact starts if the initial vertical velocity component vyo # O. During the slip, the magnitude of tangential impulse force Ft is proportional to that of normal impulse force F~ according to Coulomb's law, and can be denoted by j#f~, where # is the coefficient of sliding friction and j = +1 an index of sign used to indicate the direction of the frictional force. For instance, j = 1 implies that the ball is sliding down in the positive y direction and the direction of the frictional force is upward. Similarly, ~ = +1 is used to indicate the direction of the normal impact force for vx0 > 0 and v~0 < 0, respectively. According to Eqs.(3) and (14a), (14c), the motion of the pendulum in contact can be described by rnl20 + do + ml(g - Vow cos wt) sin0+ vy # 0 l(~fn cos 0 - jft sin 0) = 0 (16a) (16b) J lb' F~,~ 2b Fig.2 A parametrically excited oblique vibro-impacting system For simplicity, the subscript i is omitted hereinafter. The velocity of the ball during its free flight can be decomposed into the horizontal component v~ perpendicular to the wall and the vertical component vy parallel to the wall as follows h~ = I sin0 (14a) vx = 10 cos 0 vy = V0 sin wt - lo sin 0 (14b) (14c) where hn is the displacement of the ball and t) the angular velocity of the pendulum. If the pendulum collides one of the wails at the angular velocity ~)0 when t = to, Eq.(14) can be used to give the horizontal velocity component vx0 and vertical velocity component vy0 of the ball as follows Vxo = loo cos 00 Vyo = Vosinwto - IOosinOo (15a) (15b) where 00 is the maximal swing angle of the pendulum limited by the walls. where d is the coefficient of the linear viscous damping contributed by the pivot of the pendulum, air resistance, etc. The additional complementary conditions for the sliding impact are where e+x = e(~ + ~F~) <_ 0 ef~ _< 0 (17a) (17b) F~x = 0 (17c) a = It~ 2 sin 0 + ml cos O + (g - V0co cos wt) sin 0 cos 0 (is) = l(e cos 0 - jp sin 0) cos 0 (19) m If the sliding motion occurs during an impact, Eq.(10) holds. With the aid of Eq.(17a), one arrives at g - V0a~ cos wtf = 0 (20) where tf is the final moment at the end of the impact. In addition, it is easy to see from Fig.l(d) that the tangential motion immediately after the slip stops takes the form --eyvyo -=- Vo sinwtf (21) where ey is the coefficient of the tangential restitution. Using Eqs.(20) and (21), one finds that V~ - = a 2 a = --eyvyo (22)

5 Vo1.19, No.6 Jin DP & Hu HY: Analysis of Oblique Vibro-lmpacting Systems 583 This is the necessary and sufficient condition for a sliding impact, which can be illustrated in Fig.3. Substituting vx~ = -envxo and Vy a ~- --eyvy 0 into Eq.(25) gives 3 OB : Vo = g/~.i B ~;7 sliding impact VyO 1 4-e~ -- -@~ (g tan 0o) (26) ~Vxo 1 t ey It can be seen immediately that an instantaneous impact occurs while the ball is moving up under the condition eyr -1. Otherwise, the impact involves a sliding motion. 5 CASE STUDIES 0 A Fig.3 Sliding impact conditions As indicated by OB and curve AC in Fig.3, a difficulty arises in solving such a problem because the system parameters are not known in advance and must be found by trial and error. In accordance with the pieeewise analysis in Section 3, the sliding velocity gives rise to a jump from -eyvy0 to Vyf at the moment tf, and one arrives at V0 Vyf : V 0 sin wtf + envxo tan 00 (23) where en is the coefficient of the normal restitution. Hence, the vibro-impacts can be traced by repeating this procedure through the use of the piecewise analysis provided that tfi, q(tfi), and//(tfi) are given. 4.2 Instantaneous Impact In this case, the duration of impact can be assumed to be negligibly short, compared with the duration of the overall dynamics of the system. This assumption results in the concept of instantaneous impact. According to the theorem of impulse [s], the impulse-momentum equations describing a change in the velocity components of a ball during the short interval At = t -- to are as follows m(vx~-vxo) = -epn cos 20o+JPt sin 200/2 m(vy~-vyo) =epn sin 20o/2-jPt sin 2 00 (24a) (24b) where Pt and Pn represent the impulse of the tangential friction and normal force, respectively. The condition for an instantaneous impact can be determined as follows Vy~ - Vyo --_ tan00 (25) Vxa -- VxO In the case study on the system in Section 4, the mass rn of the ball was taken as 0.5 kg, the length l of the pendulum as 0.7m, the distance 2b between two walls as 0.I m, the coefficient # of friction as 0.6, the coefficient en of the normal restitution as 0.3, the coefficient ey of the tangential restitution as -0.i, the coefficient d of damping d as 0.01 N.s/m, respectively. The initial states of the system were set as 0(0) = 0 and ~)(0) = 0.1rad/s. The system dynamics was recorded after the first 200 impacts since they were unsteady-state motions. Because the initial velocity Vyo for the sliding impact cannot be solved exactly, it is necessary to properly choose the excitation parameters so that the initial impacts yield the sliding condition consistent with Eq.(22). For this purpose, the excitation amplitude V0 was set to be 1.5m/s and the impacting velocity vc0 was numerically computed to obtain a set of approximate system parameters for the sliding motion, as shown in Figs.4(a) and 4(b). If vc0 - vy0 is close to zero, the corresponding excitation amplitude may be considered as a parameter providing the possibility for the sliding motion. Hence, the excitation frequency for a possible sliding motion was found to be 6.6~7.08 rad/s from Fig.4(a). To avoid the chaotic motion, the excitation frequency was taken as 7rad/s. Figure 4(b) shows the response vr - vy0 versus the excitation amplitude V0. It indicates that the sliding impact occurs if the excitation amplitude V0 is in 1.5~l.6m/s or 3.971~4.163 m/s. Similarly, the excitation amplitude V0 was taken as 4.163m/s. As a result, the two parameters for a sliding impact were V0 = m/s and w = 7 rad/s. Under this excitation, the motions of the pendulum were computed. Figure 4(c) is the time history of the horizontal displacement component of the ball and Fig.4(d) is the corresponding zoom view. The time history in Fig.4(d) enables one to see the unidirectional slip of the ball on the walls. It should be

6 584 ACTA MECHANICA SINICA 2003 pointed out that the sliding impact occurs immediately after an instantaneous impact. 1.5 I.~ 1.0,~ 0.5?~ o (a) co/(rad.s -1) V0 = 1.5m/s -3,, I, i, i, I, i,-"'i'i, v0/(m s -~ ) (b) w = 7rad/s 0.15 " " " "~" t/s (c) t/s (d) Fig.4 Sliding motion of an oblique vibroimpacting system 6 CONCLUSIONS The analysis of oblique impact of two bodies subjected to external loads and movements requires the hybrid analysis of vibro-impact dynamics, kinematics and complementary conditions, instead of the combination of any impact law and impulse-moraentum eqoation only. The paper presents a pieeewise analysis to describe the oblique impact of two bodies under external loads and movements, through the example of a parametrically excited planar pendulum between two parallel rigid walls. Both analysis and numerical study on the example show that the pendulum undergoes sliding impacts if the system parameters are properly chosen. REFERENCES 1 Bishop SR. Impact oscillators. Philosophical Transactions of Royal Society of London A, 1994, 347: 347~351 2 Brogliato B. Nonsmooth Mechanics. London: Springer-Verlag, Kane TR, Levinson DA. Dynamics: Theory and Applications. New York: McGraw-Hill, Moore DB, Shaw SW. The experimental response of an impacting pendulum system. International Journal of Nonlinear Mechanics, 1990, 25:1~16 5 Bayley PV, Virgin LN. An experimental study of an impacting pendulum. Journal of Sound and Vibration, 1993, 164:364~374 6 Ratner SB, Styller EE. Characteristics of impact friction and wear of polymeric materials. Wear, 1981, 73: 213~234 7 Lu ML. Oa coefficient of friction of oblique impact. Acta Mechanica Solid Siniea, 1987, 8: 282~284 (in Chinese) 8 Babitsky VI. Theory of Vibro-impact Systems and Applications. Berlin Heidelberg: Springer-Verlag, I~73 9 Stronge WJ. Rigid body collision with friction. Proceedings of the Royal Society of London A, 1990, 431: 169~ Wang Y, Matthew TM. Two-dimensional rigid-body collisions with friction. Journal of Applied Mechanics, 1992, 59:635~ Stronge WJ. Planar impact of rough compliant bodies. International Journal of Impact Engineering, 1994, 15: 435~ Stronge WJ. Swerve during three-dimensional impact of rough rigid bodies. Journal of Applied Mechanics, 1994, 61:605~ GMvanetto U, Bishop SR, Briseghella L. Mechanical stick-slip vibration. International Journal of Bifurcation and Chaos, 1995, 5:637~ Glocker CH, Pfeiffer F. Dynamical systems with unilateral contact. Nonlinear Dynamics, 1992, 3: 245~ Pfeiffer F. Complementarity problems of stick-slip vibration. Journal of Vibration and Acoustics, 1996, 118:177~ Van de Vrande BL, van Campen DH, de Kraker A. An approximate analysis of dry-friction-induced stickslip vibrations by a smoothing procedure. Nonlinear Dynamics, 1999, 19:157~ Kobrinsky AA, Kobrinsky AE. Two-dimensional Vibro-impact Systems. Moscow: Naul~, 1981