CHARACTERIZATION OF NON-IDEAL OSCILLATORS IN PARAMETER SPACE

Transcription

1 CHARACTERIZATION OF NON-IDEAL OSCILLATORS IN PARAMETER SPACE Silvio L. T. de Souza 1, Iberê L. Caldas 2, José M. Balthazar 3, Reyolando M. L. R. F. Brasil 4 1 Universidade Federal São João del-rei, Campus Alto Paraopeba, Ouro Branco, MG, Brazil, thomaz@if.usp.br 2 Instituto de Física, Universidade de São Paulo, São Paulo, SP, Brazil 3 DEMAC, IGCE, Universidade Estadual Paulista, Rio Claro, SP, Brazil 4 Departamento de Engenharia Estrutural e de Fundações, Escola Politécnica, Universidade de São Paulo, São Paulo, SP, Brazil Abstract: The dynamics of the systems with non-ideal energy source, represented by a limited power supply, are investigated. As eamples of non-ideal problems, we analyze chaotic dynamics of an impact oscillator and a Duffing oscillator. We characterize these systems in two-dimensional parameter space by using the largest Lyapunov eponents identifying self-similar periodic sets, such as Arnold tongues and shrimp-like structures. In addition, we observe multistabilities associated with fractal basin boundaries. Keywords: Non-ideal oscillators, Lyapunov eponents, Arnold tongues. 1. INTRODUCTION Most studies of mechanical systems consider oscillators with energy source described by a harmonic function whose amplitude and frequency can be arbitrarily chosen [1]. However, sometimes for practical situations, the dynamics of the driving system cannot be considered as given aprioribecause the forcing source has a limited available energy supply. In other words, the dynamics of the driving system is influenced by the oscillator that is being forced [2 4]. This kind of system has been called non-ideal oscillator [5]. A common eample of non-ideal oscillator appears when the driving comes from an unbalanced rotor linked to a cart. As a consequence of this mechanical coupling, the rotor dynamics may be heavily influenced by the oscillating system being forced [6 9]. Futhermore, in the engineering contet, mathematical models describing mechanical impacts have been intensively studied [1 13]. The impact oscillators appear in a wide range of practical problems, such as impact dampers, gearrattling system, percussive drilling tools, print hammers, and vibro-impact moling systems [14 18]. Here, we investigate the dynamics of an impact oscillator and a Duffing oscillator with limited power supply. We characterize these systems in two-dimensional parameter space by using the Lyapunov eponents identifying periodic structures such as Arnold tongues [19 21] and shrimps [22 25]. This paper is organized as follows. In Section 2 we present the model and the equations of motion for the nonideal oscillator with impacts. For this oscillator, we investigate coeistence of attractors and their basins of attraction. In Section 3, the non-ideal Duffing oscillator is treated. We characterize the Duffing oscillator in the two-parameter space by using the Lyapunov eponents. The last section contains our conclusions. 2. NON-IDEAL IMPACT OSCILLATOR 2.1. Mathematical Description The considered oscillator with piecewise-linear restoring force is shown schematically in Figure 1. It is composed of a cart of the mass M connected to a fied frame by a linear spring (coefficient k 1 ) and a dash-pot (viscous coefficient c) and of another linear spring (coefficient k 2 k 1 ). Motion of the cart is due to an in-board non-ideal motor driving the rotor. Figure 1 Schematic model of a non-ideal oscillator with impacts. The equations of motion are obtained by using Lagrangian approach and are given by: Serra Negra, SP - ISSN

2 ẍ + βẋ + g() = ɛ( ϕ sin ϕ + ϕ 2 cos ϕ), ϕ = ẍ sin ϕ + E 1 E 2 ϕ, for the restoring force { if c g() = σ + c (1 σ) if > c, for X/r with β c/ k1(m + m), ɛ m/(m + m), σ k 2 /k 1,and c X c /r Fractal basin boundaries Numerical simulations of the oscillator with impacts were performed by using the fourth-order Runge-Kutta method with a fied step. The system parameters were fied at β =.2, ɛ =.25, E 1 =3., ande 2 =1.. We will consider for our dynamic investigations the parameters clearance c and spring stiffness ratio σ. Initially, in order to investigate the dynamics of the impact oscillator, we present bifurcation diagram in Figure 2 for the parameter clearance c. To characterize the nature of the behavior observed, we calculate the Lyapunov eponents, as shown in Figure 2, using the algorithm described in the Ref. [26]. If the largest Lyapunov eponent is positive the attractor is chaotic, if not the attractor is periodic. The bifurcation diagram [Fig. 2] composed of a periodbubbling scenario. Decreasing the parameter, as can be seen, for the first bifurcation just one Lyapunov eponent is zero [Fig. 2]. In this case, this bifurcation observed is called grazing [13] and others are period-doubling (two Lyapunov eponents zero). In Figure 3, we show a parameter space diagram obtained using a grid of cells equally spaced. For each cell the largest Lyapunov eponent is calculated and plotted with the appropriately allocated color. Chaotic attractors λ 1 > (the largest Lyapunov eponent) are plotted in blue and periodic attractors λ 1 < according to a color scale ranging from minimum value in red and maimum in green. Zero Lyapunov eponents period-doubling bifurcation points are plotted in blue as well. To eamine more precisely the dynamics shown in the parameter space diagram, we fi the clearance at c =2.and depict in Figure 4 another bifurcation diagram in terms of the spring stiffness σ. This diagram shows a route to chaos via period-doubling bifurcation cascade. As we can note, the chaotic attractor disappears abruptly for a specific value. In this case, we identify, in this region of the control parameter, coeisting attractors (periodic and chaotic) with complicated basins of attraction, instead of a crisis phenomenon associated with the disappearance of the chaotic attractor. As an eample of coeistence of attractors, we depict in Figure 5 the phase portraits of the two coeisting periodic and chaotic attractors for the same parameter of the bifurcation diagram [Fig. 4] c = 2. with σ = 5.. To evaluate the basins structure for these attractors, in Figure 6weobtainbasinsofattractionusingagridofanequally spaced 11 set of initial condition points. In this case, λ 1,2, c Figure 2 Bifurcation diagram showing angular velocity ϕ as a function of clearance c for σ =5.; The three largest Lyapunov eponents. σ c Figure 3 Parameter space plots of the spring stiffness σ versus the clearance c for the impact oscillator. Figure 4 Bifurcation diagram of the local maimum value of angular velocity ϕ as a function of spring stiffness σ for c = 2.. Serra Negra, SP - ISSN

3 the basin boundary with very fine scale is apparently fractal [27, 28] ẋ Figure 5 Phase portraits showing the coeistence of a periodic and a chaotic attractors for c =2. and σ =5.. ẋ ẋ Figure 8 Basins of attraction for the three coeisting solutions shown in Figure 7; Magnification of a small bo Figure 6 Basins of attraction for the two coeisting solutions showninfigure5. As additional eample of coeistence in Figure 7 for the control parameters c =2.with σ =16., weshow the phase portraits of the three periodic attractors. The corresponding basins of attraction are depicted in Figure 8 and a magnification of a small bo in Figure 8 showing fractal basin boundaries. 3. NON-IDEAL DUFFING OSCILLATOR 3.1. Mathematical Description We consider a cart of the mass M connected to a fied frame by a nonlinear spring (spring stiffness k 1 X + k 2 X 3 ) and a dash-pot (viscous coefficient c), as shown in Figure 9. We denote by X the displacement of the cart and by ϕ the angular displacement of the pendulum with the mass m and a massless rod of radius r Figure 9 Schematic model of a Duffing non-ideal oscillator Figure 7 Phase portraits showing the coeistence of three periodic attractors for c =2. and σ =16.. The equations of motion for both the cart and the rotor, considering dimensionless dynamical variables X/r, τ t k 1 /M, are given, respectively, by: ẍ + βẋ + δ 3 ( = ɛ 1 ϕ sin ϕ + ϕ 2 cos ϕ ), ϕ ɛ 2 cos ϕ =ẍ sin ϕ + E 1 E 2 ϕ for β c/ k 1 M, δ (k 2 /k 1 )r 2, ɛ 1 m /M,andɛ 2 Serra Negra, SP - ISSN

4 g/(rk 1 ). The parameters E 1 and E 2 can be estimated from the characteristic curve of the energy source (a DC-motor) Arnold Tongues Numerical simulations of nonlinear oscillator systems were performed by using the fourth-order Runge-Kutta method with a fied step. The system parameters were fied at δ =.1, ɛ 1 =.1, ɛ 2 =1., ande 2 =1.5. We will consider for our dynamic investigations the parameters β (the damping coefficient) and E 1 (the motor parameter). Initially, we present bifurcation diagram in Figure 1 for the motor parameter E 1 showing coeistence of attractors. A magnification is presented in Figure 1 showing a period-adding sequence. 1/ β / E / β / E 1 Figure 11 Parameter space plots for the Duffing oscillator showing Arnold tongues. Magnification of a small bo showing a shrimp-like structure. Figure 1 Bifurcation diagram of the local maimum value of displacement as a function of motor parameter E 1 showing coeistence of attractors and period-adding sequence. In order to identify the structures associated with periodadding, we construct the parameter space diagram for 1/β (inverse of the damping coefficient) versus 1/E 1 (inverse of the motor parameter), as shown in Figure 11. To obtain this diagram, we use a grid of 88 cells. For each cell of the grid, we evaluate the largest Lyapunov eponent and plot using a color scale shown on the right side of the diagram. On eamining this parameter space diagram, we note a vast quantity of periodic structures (in blue and white scale) formed by Arnold tongues. Therefore, the period-adding sequence observed here is associated with Arnold tongues. In addition, for a magnification of small bo of Figure 11, we observe in Figure 11 (for different color scale) a periodic set that is quite similar to structures known as shrimps [22]. 4. CONCLUSION We investigated the dynamics of the oscillators with limited power supply. As eamples of non-ideal systems, we considered an impact oscillator and a Duffing oscillator. For the impact oscillator, we identified several coeistence of attractors, showing a couple of them, with fractal basin boundaries. These kinds of basins structures introduce a certain degree of unpredictability on the final state. In other words, the fractal basin boundary results in a severe obstruction to determine what attractor will be a fine state for a given initial condition with eperimental error interval. For the Duffing oscillator, we identified Arnold tongues and shrimp-like structures in two-dimensional parameter space plots by using the largest Lyapunov eponents. As consequence of Arnold tongues, the period-adding sequence was found. ACKNOWLEDGEMENTS This work was made possible by partial financial support from the following Brazilian government agencies: FAPEMIG, FAPESP, CNPq, and CAPES. REFERENCES [1] T. Kapitaniak, Chaos for Engineers, Springer Verlag, New York-Berlin- Heidelberg, 2. [2] V.O. Kononenko, Vibrating systems with a limited power supply, London: Iliffe Books Ltd; Serra Negra, SP - ISSN

5 [3] T.S. Krasnopolskaya, A.Y. Shvets, Chaos in vibrating systems with a limited power-supply, Chaos, Vol. 3, pp , [4] X. Xu, E. Pavlovskaia, M. Wiercigroch, F. Romeo, S. Lenci, Dynamic interactions between parametric pendulum and electro-dynamical shaker, Z. Angew. Math. Mech, Vol. 87, pp , 27. [5] J. Warminsky, J.M. Balthazar, R.M.L.R.F. Brasil, Vibrations of a non-ideal parametrically and self-ecited model, Journal of Sound and Vibration, Vol. 245, pp , 21. [6] M.F. Dimentberg, L. McGovern, R.L. Norton, J. Chapdelaine, R. Harrison, Dynamics of an unbalanced shaft interacting with a limited power supply, Nonlinear Dynamics, Vol. 13, pp , [7] S.L.T. de Souza, I.L. Caldas, R.L. Viana, J.M. Balthazar, R.M.L.R.F. Brasil, Impact dampers for controlling chaos in systems with limited power supply, Journal of Sound and Vibration, Vol. 279, pp , 25. [8] S.L.T. de Souza, I.L. Caldas, R.L. Viana, J.M. Balthazar, R.M.L.R.F. Brasil, Dynamics of vibrating systems with tuned liquid column dampers and limited power supply, Journal of Sound and Vibration, Vol. 289, pp , 26. [9] S.L.T. de Souza, I.L. Caldas, R.L. Viana, J.M. Balthazar, R.M.L.R.F. Brasil, A simple feedback control for a chaotic oscillator with limited power supply, Journal of Sound and Vibration, Vol. 299, pp , 27. [1] M. Wiercigroch, Modelling of dynamical systems with motion dependent discontinuities, Chaos, Solitons and Fractals, Vol. 11, pp , 2. [11] J. Ing, E. Pavlovskaia, M. Wiercigroch, Dynamics of a nearly symmetrical piecewise linear oscillator close to grazing incidence: modelling and eperimental verification, Nonlinear Dynamics, Vol. 46, pp , 26. [12] S.L.T. de Souza, I.L. Caldas, R.L. Viana, Damping control law for a chaotic impact oscillator, Chaos, Solitons and Fractals, Vol. 32, pp , 27. [13] S.L.T. de Souza, M. Wiercigroch, I.L. Caldas,J.M. Balthazar, Suppressing grazing chaos in impacting system by structural nonlinearity, Chaos, Solitons and Fractals, Vol. 38, pp , 28. [14] S. Chaterjee, A.K. Mallik, A. Ghosh, On impact dampers for non-linear vibrating systems, Journal of Sound and Vibration, Vol. 187, pp , [15] K. Karagiannis, F. Pfeiffe, Theoretical and eperimental investigations of gear-rattling, Nonlinear Dynamics, Vol. 2, pp , [16] S.L.T. de Souza, I.L. Caldas, R.L. Viana, J.M. Balthazar, Sudden changes in chaotic attractors and transient basins in a model for rattling in gearboes, Chaos, Solitons and Fractals, 21, , 24. [17] S.L.T. de Souza, I.L. Caldas, R.L. Viana, A.M. Batista, T. Kapitaniak, Noise-induced basin hopping in a gearbo model, Chaos, Solitons and Fractals, Vol. 26, pp , 25. [18] K-C Woo, A.A. Rodger, R.D. Neilson, M. Wiercigroch, Application of the harmonic balance method to ground moling machines operating in periodic regimes, Chaos, Solitons and Fractals, Vol. 11, pp , 2. [19] M. Ding, C. Grebogi, E. Ott, Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic, Physical Review A, Vol. 39, pp , [2] M.S. Baptista, T.P. Silva, J.C. Sartorelli, I.L. Caldas, Phase synchronization in the perturbed Chua circuit, Physical Review E, Vol 67, 56212, 23. [21] C. Rosa, M.J. Correia, P.C. Rech, Arnold tongues and quasiperiodicity in a prey-predator model, Chaos, Solitons and Fractals, Vol. 4, pp , 29. [22] J. A. C. Gallas, Dissecting shrimps: results for some one-dimensional physical models, Physica A, Vol. 22, pp , [23] C. Bonatto, J. C. Garreau, J. A. C. Gallas, Selfsimilarities in the frequency-amplitude space of a lossmodulated CO 2 laser, Physical Review Letters, Vol. 95, 14395, 25. [24] H. A. Albuquerque, R. M. Rubinger, P. C. Rech, Selfsimilar structures in a 2D parameter-space of an inductorless Chua s circuit, Physics Letters A, Vol. 372, pp , 28. [25] S.L.T. de Souza, I.L. Caldas, R.L. Viana, Multistability and Self-Similarity in the Parameter-Space of a Vibro-Impact System, Mathematical Problems in Engineering, Vol. 29, 29356, 29. [26] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov eponents from a time series, Physica D, Vol. 16, pp , [27] C. Grebogi, S.W. McDonald, E. Ott, J.A. Yorke, Final state sensitivity: an obstruction to predictability, Physics Letters A, Vol. 99, pp , [28] S.W. McDonald, C. Grebogi, E. Ott, J.A. Yorke, Fractal basin boundaries, Physica D, Vol. 17, pp , Serra Negra, SP - ISSN