CHAOTIC BEHAVIOR OF MECHANICAL VIBRO IMPACT SYSTEM WITH TWO DEGREES OF FREEDOM AND POSSIBILITIES OF CHAOTIC BEHAVIOR OF QUARTER VEHICLE MODEL

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1 REGIONAL WORKSHOP TRANSPORT RESEARCH AND BUSINESS COOPERATION IN SEE 6-7 Deceber 00, Sofia CHAOTIC BEHAVIOR OF MECHANICAL VIBRO IMPACT SYSTEM WITH TWO DEGREES OF FREEDOM AND POSSIBILITIES OF CHAOTIC BEHAVIOR OF QUARTER VEHICLE MODEL M.Sc. Aleksandar Kostic, Prof. Dr Milan Kosevski, Prof. Dr. Ljupco Kocarev, Dr. Darko Danev, Dr. Igor Gjurkov Faculty of Mechanical Engineering Skopje Karpos II PO Box 6, 000 Skopje, R. Macedonia Abstract In this paper, a echanical vibro ipact syste with two degrees of freedo is analyzed. The dynaical behavior of the syste is deterined using different tools fro dynaical systes theory, including bifurcation diagras, largest Lyapunov exponent, and Poincare cross sections. Depending on the angular velocity of the excitation, the syste shows both periodic and chaotic behavior.the sae procedure is than used to exaine the possibility of chaotic behavior of quarter vehicle odel. Key words: echanical vibro ipact syste, bifurcation diagra, largest Ljapunow exponents, attractors, Poincare sections. Mechanical vibro ipact syste with with two degrees of freedo Mechanical oscillator systes, in which ipact loads occur because of utual collision of its parts or because of collision with rigid obstacles, are called ipact oscillators or vibro ipact systes [], []. A Vibro ipact phenoenon, itself, is present in different areas of applied echanics and engineering, and often is unwanted. Its presence leads to weakening of the syste, noise, even to syste destruction. Regardless the fact, weather the ipact loads in the syste is pleasant or not, research of such systes akes certain interest. Knowing that the differential equations of otion of a vibro ipact syste are with disturbed continuity, it is ore convenient to observe the systes behavior with the chaos theory tools, than to do it directly through the differential equations of otion. F =a sin(ωt) F =a sin(ωt)

2 b b B c c c x x Fig.. Model of vibro ipact syste with two degrees of freedo A odel of vibro ipact syste with two degrees of freedo is shown on fig. []. The syste oscillates under the extern excitations F and F. During the oscillations, the ass ipacts the rigid obstacle and so the continuity of the oveent of the syste changes. Depending on the paraeters of the syste, the distortion of the continuity of the oveent of the syste can lead to destruction of the periodical oveent and appearance of chaos [8], [9]. The syste equations of otion are given below. The ipact load is odeled with additional spring c, with variable stiffness depending on the position of the ass. In other words, when there is no ipact, the stiffness of the spring c is zero, and at the sae oent when the ipact occurs, the stiffness of the spring c is enorously big and siulates rigid obstacle. That is: && x && x = y = + c x + c x + c c ( x x ) + b x& + b ( x& x& ) = a sin( ωt) ( x x ) b ( x& x& ) = a sin( ωt) ( a sin( ωt) ( c y + c ( y y ) + b y + b ( y y ))) = y = ( a sin( ωt) c y + c ( y y ) + b ( y y )) Where: y = x; y = x& ; y = x; y = x&. Analyses of the dynaical behavior of the syste Vibro ipact syste with the following paraeters is being analyzed: c c = 00 kg; = 50 kg; = N = 969 N ; ; c b 0, = 000 c = 000 Ns ; ( x B), ( x > B) ; b a a = 0; = 0; = 00 N; B = 0.0. The syste equations of otion show that its attractor lies in a four diensional phase space [], [5]. A bifurcation diagra of the syste is shown on fig.. It shows the relation between the ass oveent and angular velocity of the excitation ω, which is taken as a bifurcation paraeter.

3 x Fig.. Bifurcation diagra The projections of the Poincare sections on the x axis, for different values of the angular velocity ω, are given on the ordinate axis. The bifurcation paraeter ω which is in between 0,5 and 7 rad/s, with step of 0,0 rad/s, is given on the abscissa axis. The bifurcation diagra iplies on a variable nature of the syste behavior, depending on the angular velocity of the excitation. At sall values of the angular velocity, the syste behavior is periodical. On the other hand, at greater values of the angular velocity, the syste behavior is chaotic. Additional proof for the chaotic behavior of the syste is the positive value of the largest Ljapunow exponent for certain values of the angular velocity [8], [9]. The variation of the largest Ljapunow exponent in relation with the angular velocity of the excitation, which refers to the bifurcation diagra fro fig., is shown on fig.. ω λax Fig.. Largest Ljapunow exponent λ ax in relation with the angular velocity of the excitation ω ω

4 Cobined analyses of the bifurcation diagra and the variation of the largest Ljapunow exponent λ ax in relation with the angular velocity of the excitation ω, exactly defines the zones of periodic and chaotic behavior of the syste. Syste attractors and Poincare projections for different values of ω are shown on the following figures. For the ω interval (0.) rad/s, the syste behavior is periodic. The projection of the periodical syste attractor on ( x, x & ) plane and projection of the Poincare cross section on the sae plane, for ω=. rad/s, are shown on fig.. It can be noticed, fro the figure, that the otion of the syste is with the sae period as the excitation of the syste. x' x' a) b) x x Fig.. Projection of the syste attractor (a) and Poincare cross section (b) for ω=. rad/s All Poincare projections, in this part, are obtained at the oents when the excitation reaches its axiu value. Crossover fro periodic to chaotic behavior happens in ω interval (..) rad/s. With bifurcation paraeter increasing, the syste wonders between chaotic and periodic behavior. This wondering lasts till ω=.6 rad/s. A projection of a periodic syste attractor and Poincare cross section fro the zone with doinant chaotic syste behavior, for ω=.6 rad/s, are shown on fig.5. x' x' a) x b) x Fig.5. Projection of the syste attractor (a) and Poincare cross section (b) for ω=.6 rad/s

5 Projections of chaotic syste attractors and Poincare cross sections for different ω values within the entioned zone with doinant chaotic syste behavior are shown on fig.6. The largest Ljapunow exponents are given within the shown projections. ω=. λ ax =0.88 x' x' ω=. x x ω=. ω=. λ ax =0.789 x' x' ω=. x x' x' x ω=. λ ax =0.905 a) b) x x Fig.6. Projection of the syste attractor (a) and Poincare cross section (b) for ω=., ω=. and ω=. rad/s

6 With bifurcation paraeter increasing over ω=.6 rad/s, the behavior of the syste changes fro periodic to chaotic and stays chaotic till the end of the observed ω interval. Projections of chaotic syste attractors and Poincare cross sections for different ω values within the zone defined with ω>.6 rad/s, are shown on fig.7. ω=.6 λ ax =0.98 x' x' x x ω=5.6 x' x' λ ax =0.89 x x ω=6 x' x' λ ax =0.95 a) b) x x Fig.7. Projection of the syste attractor (a) and Poincare cross section (b) for ω=.6, ω=5.6 and ω=6 rad/s

7 Quarter nonlinear vehicle dynaical odel with two degrees of freedo Quarter vehicle dynaical odel with real suspension characteristics is a nonlinear and dissipative syste. Considering this fact, the analyses of its dynaical behavior can be conducted with the chaos theory tools. Because we are considering a echanical syste, with stable linear version, at first it sees that the chaos theory does not overtakes the classical vehicle dynaical behavior research ethods. On the other hand, the chaos theory will at least show the sae results as the classical ethods, and will point out all hidden phenoena in the syste behavior, if they exist. Quarter nonlinear vehicle dynaical odel with two degrees of freedo is shown on fig.8 [], [], [6]. The equations of otion are given below []. b c z (t) b c z (t) z 0 (t) Fig.8. Quarter nonlinear vehicle dynaical odel with two degrees of freedo && z && z + c + c ( z z0 ) + c ( z z ) + b ( z& z& ) ( z z ) + b ( z& z& ) = 0 = 0 That is: ( c ( y a ( ωt) ) + c ( y y ) + b ( y y )) = sin = y = = y ( c ( y y ) + b ( y y )) Where: y = z& ; y = z; y = z& ; y = z. The excitation of the syste z 0 (t) is introduced through haronic sine function that siulates the road surface. The velocity of the vehicles odel is iplicitly presented by the angular velocity of the excitation []. In the equations of otion, the duping in the tyre because of its insignificant value is oitted [], [5].

8 Analyses of the dynaical behavior of the syste Quarter nonlinear vehicle dynaical odel with two degrees of freedo with the following paraeters is being analyzed. Unsprunged ass =8 kg; Sprunged ass =80 kg; Tyre stiffness coefficient c =70000 N/ [5]; Suspension spring stiffness coefficient c =, N/; Duping coefficient at rebound b i =00 Ns/, duping coefficient at copression b z =5 Ns/ [7]. The syste equations of otion, like those of the above observed vibro ipact syste, show that the syste attractor lays in a four diensional phase space. The projection of the systes attractor on ( z, z & ) plane, for the velocity of the vehicles odel v=0 /s, is shown on fig.9. Fig.9. Projection of the systes attractor It is an ideal attractor that points on a periodic oveent of the syste, with the sae period as the syste excitation [8]. The relationship between the oveent of the sprunged ass and the velocity of the syste, at the oents when the excitation reaches its axiu value, is shown on fig.0. z z z' v Fig.0. Moveent of the sprunged ass in relation with the velocity of the syste

9 The diagra on fig.0 is obtained fro the Poincare projections for various syste velocities, and indicates on stable periodic behavior of the syste through out the whole considered velocity interval. CONCLUSIONS The dynaical analyses of the echanical vibro ipact syste with two degrees of freedo through the bifurcation diagra, Ljapunow exponents, syste attractors and Poincare projections iplies on its periodic and chaotic behavior in relation with the angular velocity of the excitation. If the influence of the other syste paraeters on its behavior is taken into consideration during the dynaic analyses, it will lead to full picture of the conditions at which a chaos exists in vibro ipact systes with two degrees of freedo. In this way, with appropriate choice of the syste paraeters during the design process, the ipact load will be set on a level at which it will not lead to an unexpected chaotic syste behavior. On the other hand, the syste attractor and the Poincare projections throw out the possibility of whatever chaotic behavior of the quarter nonlinear vehicle dynaical odel with two degrees of freedo. This stateent opposes soe articles that states chaotic behavior of quarter nonlinear vehicle dynaical odel with two degrees of freedo [0],[]. Besides this fact, the vehicle alone is too coplex syste, just not to consider the idea of chaos eleents existing in its behavior. Considering the fact that the equations of otion of a quarter nonlinear vehicle dynaical odel with two degrees of freedo are siilar to those of the above considered vibro ipact syste, than there is a great possibility for a chaotic behavior to occur if an ipact load is ipleented in the syste. REFERENCES. Dragi Danev, "Teorija na dvi`ewe na otorni vozila", Ma{inski Fakultet-Skopje, 999. Dragi Danev, "Konstrukcija na otornite vozila", Ma{inski Fakultet-Skopje, 000. Dushan Siic "Dinaika otornih vozila ", Masinski fakultet Kragujevac, Gjurkov I., Davčev T., Siulation of a quarter vehicle odel using procedures for on-vehicle testing of the dapers, Proceedings of the Faculty of Mechanical Engineering Skopje, Vol., No., pp 7-, Skopje, Hans B. Pacejka, "Tire and Vehicle Dynaics", SAE International, J. Reipell and H. Stoll, "The Autootive Chassis: Engineering Principles", Butterworth Heineann, John Dixon, "The Shock Absorber Handbook", SAE, John Banks, Valentina Dragan, Arthur Jones, "Chaos: A Matheatical Introduction", Cabridge University Press, T.S. Parker and L.O. Chua, "Practical Nuerical Algoriths for Chaotic Systes", Springer, Q. Zhu, M. Ishitobi, "Chaos and bifurcations in a nonlinear vehicle odel", Journal of Sound and Vibration, Vol. 75, Issues -5, pp. 6-6, 00. Q. Zhu, M. Ishitobi, "Chaotic vibrations of a nonlinear full-vehicle odel", International Journal of Solids and Structures, Vol., pp , 006. Ulrich Parlitz and Werner Lauterborn, "Period-doubling cascades and devil s staircases of the driven van der Pol oscillator", Physical review A6, 8-, 987. C.J. Begley and L.N. Virgin, "On the OGY Control of an Ipact-Friction Oscillator", Journal of Vibration and Control, Vol.7, pp. 9-9, 00. G.L.Wen and J.H.Xie, "Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two Degree of Freedo Vibro Ipact Syste", ASME, Journal of Applied Mechanics, Vol. 68 Nr., Lesley Ann Low, Per G. Reinhall, Duane W. Storti, "An Investigation of Coupled van der Pol Oscillators", ASME, Journal of Vibration and Acoustics, Vol. 5 Nr., 00