- Email: [email protected]
Response regimes of integrable damped strongly nonlinear oscillator under impact periodic forcing
Chaos, Solitons and Fractals 32 (2007) 405–414 www.elsevier.com/locate/chaos
Response regimes of integrable damped strongly nonlinear oscillator under impact periodic forcing D. Meimukhin, O.V. Gendelman
*
Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Technion City, Haifa 32000, Israel Accepted 10 May 2006
Abstract We investigate the response regimes of integrable strongly nonlinear damped oscillator in the conditions of periodic impact loading. Integrability is achieved by special choice of the coefficients of the model. Stable and unstable response regimes corresponding to single-period responses are revealed analytically. Numeric simulations are undertaken in order to verify the extent to which the single-period solutions describe the global dynamics of the system. For some regions in the space of parameters and initial conditions it is the case; for others, co-existence of different response regimes for different initial conditions is revealed and investigated. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Investigations of vibro-impact oscillations in various types of linear and nonlinear dynamical systems have been accomplished by many researchers [1–8]. Objects of the investigation varied from single-DOF forced oscillators [2,4] to continuous structures [8]. As for the methods used, they included direct numerical simulations [2], exact and approximate analytic approaches [4], as well as analytic and numeric solutions of appropriately stated boundary-value problems and nonsmooth time transformations [1,3,5–8]. Majority of the papers mentioned above dealt with prediction and investigation of some peculiar periodic or chaotic response regimes and did not attempt to describe the variety of possible responses for given set of parameters in the whole space of initial conditions. It is well-known that for forced nonlinear systems many different regimes may exist simultaneously [9] but no way besides straightforward numeric simulation is known to investigate them. One may construct a number of exact or approximate solutions and even check their stability, but any such result does not guarantee that the system will not respond in different way if, for instance, the initial conditions are not chosen exactly on the solution found. In other terms, there are some ways to construct the solutions for vibro-impact systems but there is no way to verify that the list of the solutions obtained is complete.
*
Corresponding author. E-mail address: [email protected] (O.V. Gendelman).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.028
406
D. Meimukhin, O.V. Gendelman / Chaos, Solitons and Fractals 32 (2007) 405–414
This paper deals with investigation of the response of strongly nonlinear damped oscillator to periodic vibro-impact excitation. Important peculiarity of the model chosen is that due to special choice of parameters the strongly nonlinear damped oscillator is integrable [10,11], i.e. the solution for its free motion may be expressed explicitly in terms of elliptic functions despite strong nonlinearity and damping. If there was no damping, then the integrability of the unforced system would be trivial, but the lack of the damping prohibits dissipation of transients; therefore the damping is necessary to allow the attractors in the state space. Integrability of the damped system allows complete analytic investigation of single-period response regimes and evaluation of their stability; if stable, they are expected to serve as attractors for certain regions in the space of initial conditions. After that, for given set of parameters it is instructive to investigate the whole variety of real responses in the complete space of initial conditions by means of direct numeric simulation. Comparison of these two results allows one to decide whether the regimes predicted analytically adequately represent the multitude of possible responses of the system. Strongly nonlinear oscillators have recently attracted a lot of attention since it is possible to use them as attachments to linear systems for the sake of targeted energy transfer [12–14] or efficient vibration absorbers with nontrivial response [15]. Understanding of possible behavior of such strongly nonlinear oscillators under vibro-impact excitations may pave new ways for their applications. The structure of the paper is as follows. Section 2 deals with the description of the model and analytic computation of the responses. Section 3 contains numeric simulations of possible responses in the space of the initial conditions and comparison with analytic results. It is followed by concluding remarks.
2. Description and analytical treatment of the model Let us consider the following damped nonlinear oscillator with odd power nonlinearity m (m P 3) and linear viscous dissipation 1 €y þ y_ þ x20 y þ Cy m ¼ 0 s
ð1Þ
where y = y(t) denotes the displacement, s1 is the viscous damping coefficient (s may be characterized as characteristic time of damping), and x0 the linearized natural frequency. The coefficients s and C are nonnegative scalars, and all coefficients are considered to be O(1) quantities, therefore both the nonlinearity and the damping may be strong. In general, this system is nonintegrable, but we are interested in the particular case where the integrals of motion exist and the exact solution of motion can be expressed analytically. According to [10,11] the exact integrability condition is x20 s2 ¼
2ðm þ 1Þ
ð2Þ
ðm þ 3Þ2
Thus the exact analytical solution of (1) is given by ! rffiffiffiffiffiffiffiffiffiffiffiffi 2t m1 t ðmþ3Þs ðmþ3Þs 2C mþ3 m1 ð Þ sA 2 e camðmÞ þ/ yðtÞ ¼ A e mþ1 m1 where A and u are related to initial conditions of the problem and Z dcamðmÞ du ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmþ1Þ 1 camðmÞ In the simplest case m = 3 the cam function is reduced to well known elliptic functions of Jacobi: pffiffiffi pffiffiffi! Z dcamð3Þ 2 2 cn u; du ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) camð3Þ ¼ 2 2 4 1 cam
ð3Þ
ð4Þ
ð5Þ
ð3Þ
We assume C = 1. This assumption does not affect the generality of the treatment due to possibility of scaling of the dependent variables. The solution of (1) is presented as
D. Meimukhin, O.V. Gendelman / Chaos, Solitons and Fractals 32 (2007) 405–414
yðtÞ ¼
pffiffiffi pffiffiffi pffiffiffi! t t 2 2 2 A eð3sÞ cn 3 s A eð3sÞ þ /; 2 2 2
407
ð6Þ
We would like to investigate this system under periodic impact loading; let us formulate this kind of input to the system. Elastic impact leads to a change in body’s velocity in infinitely small period of time. If the change of the velocity DV occurs with period T, then single-period stationary response regime of the system should obey the following conditions of continuity and smoothness: ( yðtÞ ¼ yðt þ T Þ ð7Þ yðtÞ _ ¼ y_ ðt þ T Þ þ DV In order to compute the response according to (7), one can choose t to be zero (different choice will lead to phase shift only). After substituting expression (6) to the first equation of (7), one obtains: ! rffiffiffi pffiffiffi 1 2 ð0=3sÞ ð0=3sÞ 3sAe Ae cn þ/ 2 2 ! rffiffiffi pffiffiffi 1 2 ðð0þT Þ=3sÞ ð0þT =3sÞ 3sAe Ae cn þ/ ¼ 2 2 ! rffiffiffi 1 3sAþ/ ) cn 2 ! rffiffiffi 1 ðT =3sÞ ðT =3sÞ 3sAe cn þ/ ð8aÞ ¼e 2 The second equation of (7) yields: " ! " !# rffiffiffi rffiffiffi 1 1 1 1 pffiffiffi A eðt=3sÞ cn 3 s A eðt=3sÞ þ / þ A eðt=3sÞ sn 3 s A eðt=3sÞ þ / 3s 2 2 2 # ! rffiffiffi 1 3s 1 eðt=3sÞ DV 3 s A eðt=3sÞ þ / A pffiffiffi dn 2 3s 2 " ! rffiffiffi 1 1 1 ðtþT =3sÞ ðtþT =3sÞ e cn þ / þ A eðtþT =3sÞ 3sAe ¼ pffiffiffi A 3s 2 2 " !# ! # rffiffiffi rffiffiffi 1 1 3s 1 ðtþT =15Þ ðtþT =3sÞ ðtþT =3sÞ e þ / dn þ / A pffiffiffi 3sAe 3sAe sn 2 2 3s 2 After regrouping and substituting t = 0, one obtains: ! ! rffiffiffi rffiffiffi 1 1 A 1 cn 3 s A þ / þ pffiffiffi sn 3sAþ/ ) 3s 2 2 2 ! rffiffiffi pffiffiffi 1 2 DV dn 3sAþ/ 2 A ! rffiffiffi 1 1 A ðT =3sÞ ðT =3sÞ e 3sAe cn þ / þ eð2T =3sÞ pffiffiffi ¼ 3s 2 2 " !# ! rffiffiffi rffiffiffi 1 1 3 s A eðT =3sÞ þ / 3 s A eðT =3sÞ þ / dn sn 2 2
ð8bÞ
Two parameters of external excitation – T (period between impacts) and DV (magnitude of impact) together with single free parameter of the system s (relaxation time of the system) completely determine the structure of solutions of System (8a) and (8b). Different solutions for u and A correspond to single-period response regimes of the system
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with different initial conditions. System (8a) and (8b) of transcendent equations is solved numerically. In order to describe the structure of its solutions it is convenient to introduce the new variables pffiffiffi 3 2 s A e0 þ u x 2 ffiffiffi ð9Þ p 3 2 T =3s sAe þu y 2 and to present Eqs. (8a) and (8b) as lines at x–y plane. Intersection points of these curves will correspond to exact solutions of System (1) with boundary conditions (7) and thus to single-period response regimes of System (1) with periodic vibro-impact loading. Values of A and u are easily computed for every intersection point from Eq. (9). In order to illustrate the method and possible bifurcation scenarios for single-period response regimes, computations for selected values of T and DV are presented below for fixed s (Figs. 1 and 2). Only one parameter is varied in each pair of computation. In both cases, the increase of the impact intensity or the decrease of its period leads to formation of new pair of single-period responses. Let us take a deeper look at the correlation between period, magnitude of impact and number of solutions of the system: when increasing DV or decreasing T the ‘‘neck’’ on the solid curve becomes thinner until a couple of new solutions is born when it intersects the dashed one (Fig. 3).
Fig. 1. Formation of pair of new solutions with change of parameters (decrease of impact period), s = 5. Left figure is computed for T = 4, DV = 4, right figure is computed for T = 2, DV = 4. Solid line corresponds to numeric solution of Eq. (8a), dashed line to Eq. (8b).
Fig. 2. Formation of pair of new solutions with change of parameters (increase of impact intensity), s = 5. Left figure is computed for T = 4, DV = 4, right figure is computed for T = 4, DV = 8. Solid line corresponds to numeric solution of Eq. (8a), dashed line to Eq. (8b).
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409
Fig. 3. Parameters of solutions for the case T = 2, DV = 4, s = 5. Solid line corresponds to numeric solution of Eq. (8a), dashed line to Eq. (8b). Parameters A and u are computed for each particular solution.
Fig. 4. Numerical simulation of single-period solutions for the case T = 2, DV = 4, s = 5. Solution 1: A = 5.777840068, u = 64.31023165. (a) Plot of the displacement versus time; (b) Poincare section; (c) frequency content of the solution – fast Fourier transform.
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One can conclude that the number of solutions of Eqs. (8a) and (8b) is determined by number of intersections between two families of curves, which are either closed or go to infinity. From simple geometric reasons, it is clear that the single-period solutions are born or disappear in pairs (at least one solution always exists; therefore the overall number of the solutions is odd). Period of the external forcing generates natural Poincare map of the state space on itself and appearance of a pair of the ‘‘newborn’’ solutions corresponds to generic saddle-node bifurcation of this map. Therefore one of these solutions should be stable and the other-unstable. However the variety of possible response regimes is not restricted by the single-period solutions. Once the single-period solution is created, it may be subject to generic period-doubling bifurcations, with subsequent transition to chaos. Besides, the Poincare maps generated by multiple periods of the external forcing may generate the saddle-node bifurcations of their own, not manifested at the single-period map–multiple-period solutions also may be created without previous creation of the single-period ones. Of course, all this variety of solutions may be in principle revealed analytically by solution of system of equations equivalent to (8a) and (8b). However, even for the investigation of double-period response one should already solve a system of four transcendent equations with four unknowns; this problem is extremely complicated even for numeric solution. In the following section, only the single-period responses are computed exactly and the other responses (as well as stability of the single-period ones) are determined by direct numerical simulation.
Fig. 5. Numerical simulation of single-period solutions for the case T = 2, DV = 4, s = 5. Solution 2: A = 5.387553133, u = 57.75301971. (a) Plot of the displacement versus time; (b) detailed view of displacement versus time for period 5 solution; (c) Poincare section; (d) frequency content of the solution – fast Fourier transform.
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3. Examples and numerical simulation We will examine now a case of DV = 4 and T = 2 (Fig. 3). There are three solutions, two are supposed to be stable, while the third is not. Let us verify this conclusion by direct numeric simulation of the single-period solutions with parameters defined at Fig. 3. The results are presented at Figs. 4–6. It is obvious that the solutions 1 and 3 are stable, while 2 is not. Due to computation errors, the unstable solution switches from single-period response to the response of period 5. The latter is not related to any of the stable singleperiod responses and has been created by another mechanism. Hitherto the analytical model gave some good ‘‘forecast’’ for number of solutions and their stability comparing to numerical simulations, but there is still a question if this ‘‘forecast’’ is exhaustive. Unfortunately the answer is negative. In order to reveal that, we turn to direct simulation of the system with wide set of initial conditions; this task is assessable since the space of IC is only two-dimensional. The result is presented in Fig. 7. The simulation is run for different initial conditions with the step of 0.01 in each direction. The initial conditions that converge to the same attractor are marked with the same color or background. From the picture it is clear that there are four different periodic response regimes within the region [2.5 2.5; 2.5 2.5]. Thus it is clear that there are more stable response regimes that we just can predict with the single-period analytical model.
Fig. 6. Numerical simulation of single-period solutions for the case T = 2, DV = 4, s = 5. Solution 3: A = 2.531297498, u = 28.86903629. (a) Plot of the displacement versus time; (b) Poincare section; (c) frequency content of the solution – fast Fourier transform.
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Fig. 7. Domains of attraction of different response regimes at the plane of initial conditions are marked by different color or background. T = 2, DV = 4, s = 5.
Fig. 8. A number of solutions for different parameters of the loading for T = 2 and s = 1. (a) DV = 40; (b) DV = 4.
D. Meimukhin, O.V. Gendelman / Chaos, Solitons and Fractals 32 (2007) 405–414
413
Fig. 9. A number of single-period solution as a function of s/T and DV.
This region of parameters is interesting for analysis, but is of a small engineering interest because it is very difficult to predict the system’s behavior. For instance, an unnoticeable error in initial conditions will result in completely different system’s response regime. So if any application of the strongly nonlinear oscillator is planned (for instance for mitigation of periodic impacts), it is desirable to work in the region of parameters with well-predictable behavior. It can be done by changing the system’s time of relaxation s. Making s small will result in greater damping in the system. If one will choose s = 1 instead of s = 5 with the same conditions of loading T = 2, DV = 4, one will get a homogenous picture with one single-period response, without any unexpected multiple-period solutions. It is instructive also to investigate the sensitivity of the system for changing DV (Fig. 8). It is obvious that no different single-period response regimes appear even if the change of the impact intensity is immense (10 times). In order to assess the effect of the relaxation time it is instructive to investigate a possible number of the single-period solutions as function of time of relaxation – s, period of impact – T and impact intensity – DV. Since s is the parameter of the system, we use the dimensionless ratio T/s in order to characterize the influence of DV (Fig. 9). It is evident that the system’s behavior differs a lot for the different values of s/T and DV. Small T/s corresponds to multiple solutions region, so it corresponds to previously made assumption that oscillating system with large time of relaxation will not absorb the energy of impact completely, thus giving rise to birth of new response regimes. The global picture is rather complicated but it is clear that there is a limit of T/s 2 above which for all values of the impact intensity only one single-period response regime will exist.
4. Concluding remarks We developed an analytical model for systems with strongly nonlinear attachment under periodic impact loading with one degree of freedom. This model manages to describe attracting single-period response regimes. However set of attractor of the system is in general broader than predicted by the single-period response model. Strong sensitivity of domains of attraction with respect to initial conditions is hardly tolerable if any applications of the system are considered; therefore some criterion for avoiding this situation may be of interest. We have demonstrated that the system has no multiplication of the single-period response regimes if the period of external forcing is at least twice as the relaxation time. Of course, even if it is the case, some other periodic regimes may appear, but this criterion may be useful for the system design.
Acknowledgement The authors are grateful to Israel Science Foundation (grant 486/05) for financial support.
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References [1] Manevich LI, Mikhlin YuV, Pilipchuk VN. The method of normal modes for essential-non-linear systems. Moscow: Nauka; 1989. [2] Shaw S, Holmes P. A periodically forced linear oscillator with impacts; chaos and long period motions. Phys Rev Lett 1983;51(8):623–6. [3] Mikhlin YuV, Vakakis AF, Salenger G. Direct and inverse problems encountered in vibro-impact oscillations of a discrete system. J Sound Vibr 1998;216(2):227–50. [4] Ivanov AP. Analytical methods in the theory of vibro-impact systems. Pnkl Mat Mekh 1993;57(2):5–21. [5] Zhuravlev VF. Investigation of some vibro-impact systems by the method of non-smooth transformations. Izvestia Akad Nauk SSSR MTT 1977;6:24–8. [6] Pilipchuk VN. Transformation of oscillatory systems using a pair of non-smooth periodic functions. Dokl Akad Nauk UkrSSR, Ser A 1988;4:37–40. [7] Aziz MAF, Vakakis AF, Manevich LI. Exact solutions of the problem of the vibro-impact oscillations of a discrete system with two degrees of freedom. J Appl Math Mech 1999;63(4):527–30. [8] Emaci E, Nayfeh TA, Vakakis AF. Numerical and experimental study of nonlinear localization in a flexible structure with vibroimpacts. ZAMM 1997;77(7):527–41. [9] Ueda Y. Randomly transitional phenomena in the systems governed by Duffing equation. J Statist Phys 1979;20:181–96. [10] Salenger G, Vakakis AF, Gendelman OV, Amdrianov IV, Manevitch LI. Transitions from strongly- to weekly-nonlinear motions of damped nonlinear oscillators. Nonlinear Dyn 1999;20:99–114. [11] Gendelman OV, Manevitch LI. Asymptotic study of damped 1D oscillator with close to impact potential. In: Babitsky VI, editor. Dynamics of vibro-impact systems. Springer; 1998. p. 159–66. [12] Gendelman OV. Transition of energy to nonlinear localized mode in highly asymmetric system of nonlinear oscillators. Nonlinear Dyn 2001;25:237–53. [13] Vakakis AF, Gendelman OV. Energy pumping in nonlinear mechanical oscillators II: resonance capture. J Appl Mech 2001;68(1):42–8. [14] Gendelman OV. Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dyn 2004;37(2):115–28. [15] Gendelman OV, Starosvetsky Yu. Quasiperiodic response regimes of linear oscillator coupled to nonlinear energy sink under periodic forcing. J Appl Mech, in press.
Response regimes of integrable damped strongly nonlinear oscillator under impact periodic forcing D. Meimukhin, O.V. Gendelman
*
Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Technion City, Haifa 32000, Israel Accepted 10 May 2006
Abstract We investigate the response regimes of integrable strongly nonlinear damped oscillator in the conditions of periodic impact loading. Integrability is achieved by special choice of the coefficients of the model. Stable and unstable response regimes corresponding to single-period responses are revealed analytically. Numeric simulations are undertaken in order to verify the extent to which the single-period solutions describe the global dynamics of the system. For some regions in the space of parameters and initial conditions it is the case; for others, co-existence of different response regimes for different initial conditions is revealed and investigated. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Investigations of vibro-impact oscillations in various types of linear and nonlinear dynamical systems have been accomplished by many researchers [1–8]. Objects of the investigation varied from single-DOF forced oscillators [2,4] to continuous structures [8]. As for the methods used, they included direct numerical simulations [2], exact and approximate analytic approaches [4], as well as analytic and numeric solutions of appropriately stated boundary-value problems and nonsmooth time transformations [1,3,5–8]. Majority of the papers mentioned above dealt with prediction and investigation of some peculiar periodic or chaotic response regimes and did not attempt to describe the variety of possible responses for given set of parameters in the whole space of initial conditions. It is well-known that for forced nonlinear systems many different regimes may exist simultaneously [9] but no way besides straightforward numeric simulation is known to investigate them. One may construct a number of exact or approximate solutions and even check their stability, but any such result does not guarantee that the system will not respond in different way if, for instance, the initial conditions are not chosen exactly on the solution found. In other terms, there are some ways to construct the solutions for vibro-impact systems but there is no way to verify that the list of the solutions obtained is complete.
*
Corresponding author. E-mail address: [email protected] (O.V. Gendelman).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.028
406
D. Meimukhin, O.V. Gendelman / Chaos, Solitons and Fractals 32 (2007) 405–414
This paper deals with investigation of the response of strongly nonlinear damped oscillator to periodic vibro-impact excitation. Important peculiarity of the model chosen is that due to special choice of parameters the strongly nonlinear damped oscillator is integrable [10,11], i.e. the solution for its free motion may be expressed explicitly in terms of elliptic functions despite strong nonlinearity and damping. If there was no damping, then the integrability of the unforced system would be trivial, but the lack of the damping prohibits dissipation of transients; therefore the damping is necessary to allow the attractors in the state space. Integrability of the damped system allows complete analytic investigation of single-period response regimes and evaluation of their stability; if stable, they are expected to serve as attractors for certain regions in the space of initial conditions. After that, for given set of parameters it is instructive to investigate the whole variety of real responses in the complete space of initial conditions by means of direct numeric simulation. Comparison of these two results allows one to decide whether the regimes predicted analytically adequately represent the multitude of possible responses of the system. Strongly nonlinear oscillators have recently attracted a lot of attention since it is possible to use them as attachments to linear systems for the sake of targeted energy transfer [12–14] or efficient vibration absorbers with nontrivial response [15]. Understanding of possible behavior of such strongly nonlinear oscillators under vibro-impact excitations may pave new ways for their applications. The structure of the paper is as follows. Section 2 deals with the description of the model and analytic computation of the responses. Section 3 contains numeric simulations of possible responses in the space of the initial conditions and comparison with analytic results. It is followed by concluding remarks.
2. Description and analytical treatment of the model Let us consider the following damped nonlinear oscillator with odd power nonlinearity m (m P 3) and linear viscous dissipation 1 €y þ y_ þ x20 y þ Cy m ¼ 0 s
ð1Þ
where y = y(t) denotes the displacement, s1 is the viscous damping coefficient (s may be characterized as characteristic time of damping), and x0 the linearized natural frequency. The coefficients s and C are nonnegative scalars, and all coefficients are considered to be O(1) quantities, therefore both the nonlinearity and the damping may be strong. In general, this system is nonintegrable, but we are interested in the particular case where the integrals of motion exist and the exact solution of motion can be expressed analytically. According to [10,11] the exact integrability condition is x20 s2 ¼
2ðm þ 1Þ
ð2Þ
ðm þ 3Þ2
Thus the exact analytical solution of (1) is given by ! rffiffiffiffiffiffiffiffiffiffiffiffi 2t m1 t ðmþ3Þs ðmþ3Þs 2C mþ3 m1 ð Þ sA 2 e camðmÞ þ/ yðtÞ ¼ A e mþ1 m1 where A and u are related to initial conditions of the problem and Z dcamðmÞ du ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmþ1Þ 1 camðmÞ In the simplest case m = 3 the cam function is reduced to well known elliptic functions of Jacobi: pffiffiffi pffiffiffi! Z dcamð3Þ 2 2 cn u; du ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) camð3Þ ¼ 2 2 4 1 cam
ð3Þ
ð4Þ
ð5Þ
ð3Þ
We assume C = 1. This assumption does not affect the generality of the treatment due to possibility of scaling of the dependent variables. The solution of (1) is presented as
D. Meimukhin, O.V. Gendelman / Chaos, Solitons and Fractals 32 (2007) 405–414
yðtÞ ¼
pffiffiffi pffiffiffi pffiffiffi! t t 2 2 2 A eð3sÞ cn 3 s A eð3sÞ þ /; 2 2 2
407
ð6Þ
We would like to investigate this system under periodic impact loading; let us formulate this kind of input to the system. Elastic impact leads to a change in body’s velocity in infinitely small period of time. If the change of the velocity DV occurs with period T, then single-period stationary response regime of the system should obey the following conditions of continuity and smoothness: ( yðtÞ ¼ yðt þ T Þ ð7Þ yðtÞ _ ¼ y_ ðt þ T Þ þ DV In order to compute the response according to (7), one can choose t to be zero (different choice will lead to phase shift only). After substituting expression (6) to the first equation of (7), one obtains: ! rffiffiffi pffiffiffi 1 2 ð0=3sÞ ð0=3sÞ 3sAe Ae cn þ/ 2 2 ! rffiffiffi pffiffiffi 1 2 ðð0þT Þ=3sÞ ð0þT =3sÞ 3sAe Ae cn þ/ ¼ 2 2 ! rffiffiffi 1 3sAþ/ ) cn 2 ! rffiffiffi 1 ðT =3sÞ ðT =3sÞ 3sAe cn þ/ ð8aÞ ¼e 2 The second equation of (7) yields: " ! " !# rffiffiffi rffiffiffi 1 1 1 1 pffiffiffi A eðt=3sÞ cn 3 s A eðt=3sÞ þ / þ A eðt=3sÞ sn 3 s A eðt=3sÞ þ / 3s 2 2 2 # ! rffiffiffi 1 3s 1 eðt=3sÞ DV 3 s A eðt=3sÞ þ / A pffiffiffi dn 2 3s 2 " ! rffiffiffi 1 1 1 ðtþT =3sÞ ðtþT =3sÞ e cn þ / þ A eðtþT =3sÞ 3sAe ¼ pffiffiffi A 3s 2 2 " !# ! # rffiffiffi rffiffiffi 1 1 3s 1 ðtþT =15Þ ðtþT =3sÞ ðtþT =3sÞ e þ / dn þ / A pffiffiffi 3sAe 3sAe sn 2 2 3s 2 After regrouping and substituting t = 0, one obtains: ! ! rffiffiffi rffiffiffi 1 1 A 1 cn 3 s A þ / þ pffiffiffi sn 3sAþ/ ) 3s 2 2 2 ! rffiffiffi pffiffiffi 1 2 DV dn 3sAþ/ 2 A ! rffiffiffi 1 1 A ðT =3sÞ ðT =3sÞ e 3sAe cn þ / þ eð2T =3sÞ pffiffiffi ¼ 3s 2 2 " !# ! rffiffiffi rffiffiffi 1 1 3 s A eðT =3sÞ þ / 3 s A eðT =3sÞ þ / dn sn 2 2
ð8bÞ
Two parameters of external excitation – T (period between impacts) and DV (magnitude of impact) together with single free parameter of the system s (relaxation time of the system) completely determine the structure of solutions of System (8a) and (8b). Different solutions for u and A correspond to single-period response regimes of the system
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with different initial conditions. System (8a) and (8b) of transcendent equations is solved numerically. In order to describe the structure of its solutions it is convenient to introduce the new variables pffiffiffi 3 2 s A e0 þ u x 2 ffiffiffi ð9Þ p 3 2 T =3s sAe þu y 2 and to present Eqs. (8a) and (8b) as lines at x–y plane. Intersection points of these curves will correspond to exact solutions of System (1) with boundary conditions (7) and thus to single-period response regimes of System (1) with periodic vibro-impact loading. Values of A and u are easily computed for every intersection point from Eq. (9). In order to illustrate the method and possible bifurcation scenarios for single-period response regimes, computations for selected values of T and DV are presented below for fixed s (Figs. 1 and 2). Only one parameter is varied in each pair of computation. In both cases, the increase of the impact intensity or the decrease of its period leads to formation of new pair of single-period responses. Let us take a deeper look at the correlation between period, magnitude of impact and number of solutions of the system: when increasing DV or decreasing T the ‘‘neck’’ on the solid curve becomes thinner until a couple of new solutions is born when it intersects the dashed one (Fig. 3).
Fig. 1. Formation of pair of new solutions with change of parameters (decrease of impact period), s = 5. Left figure is computed for T = 4, DV = 4, right figure is computed for T = 2, DV = 4. Solid line corresponds to numeric solution of Eq. (8a), dashed line to Eq. (8b).
Fig. 2. Formation of pair of new solutions with change of parameters (increase of impact intensity), s = 5. Left figure is computed for T = 4, DV = 4, right figure is computed for T = 4, DV = 8. Solid line corresponds to numeric solution of Eq. (8a), dashed line to Eq. (8b).
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Fig. 3. Parameters of solutions for the case T = 2, DV = 4, s = 5. Solid line corresponds to numeric solution of Eq. (8a), dashed line to Eq. (8b). Parameters A and u are computed for each particular solution.
Fig. 4. Numerical simulation of single-period solutions for the case T = 2, DV = 4, s = 5. Solution 1: A = 5.777840068, u = 64.31023165. (a) Plot of the displacement versus time; (b) Poincare section; (c) frequency content of the solution – fast Fourier transform.
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One can conclude that the number of solutions of Eqs. (8a) and (8b) is determined by number of intersections between two families of curves, which are either closed or go to infinity. From simple geometric reasons, it is clear that the single-period solutions are born or disappear in pairs (at least one solution always exists; therefore the overall number of the solutions is odd). Period of the external forcing generates natural Poincare map of the state space on itself and appearance of a pair of the ‘‘newborn’’ solutions corresponds to generic saddle-node bifurcation of this map. Therefore one of these solutions should be stable and the other-unstable. However the variety of possible response regimes is not restricted by the single-period solutions. Once the single-period solution is created, it may be subject to generic period-doubling bifurcations, with subsequent transition to chaos. Besides, the Poincare maps generated by multiple periods of the external forcing may generate the saddle-node bifurcations of their own, not manifested at the single-period map–multiple-period solutions also may be created without previous creation of the single-period ones. Of course, all this variety of solutions may be in principle revealed analytically by solution of system of equations equivalent to (8a) and (8b). However, even for the investigation of double-period response one should already solve a system of four transcendent equations with four unknowns; this problem is extremely complicated even for numeric solution. In the following section, only the single-period responses are computed exactly and the other responses (as well as stability of the single-period ones) are determined by direct numerical simulation.
Fig. 5. Numerical simulation of single-period solutions for the case T = 2, DV = 4, s = 5. Solution 2: A = 5.387553133, u = 57.75301971. (a) Plot of the displacement versus time; (b) detailed view of displacement versus time for period 5 solution; (c) Poincare section; (d) frequency content of the solution – fast Fourier transform.
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3. Examples and numerical simulation We will examine now a case of DV = 4 and T = 2 (Fig. 3). There are three solutions, two are supposed to be stable, while the third is not. Let us verify this conclusion by direct numeric simulation of the single-period solutions with parameters defined at Fig. 3. The results are presented at Figs. 4–6. It is obvious that the solutions 1 and 3 are stable, while 2 is not. Due to computation errors, the unstable solution switches from single-period response to the response of period 5. The latter is not related to any of the stable singleperiod responses and has been created by another mechanism. Hitherto the analytical model gave some good ‘‘forecast’’ for number of solutions and their stability comparing to numerical simulations, but there is still a question if this ‘‘forecast’’ is exhaustive. Unfortunately the answer is negative. In order to reveal that, we turn to direct simulation of the system with wide set of initial conditions; this task is assessable since the space of IC is only two-dimensional. The result is presented in Fig. 7. The simulation is run for different initial conditions with the step of 0.01 in each direction. The initial conditions that converge to the same attractor are marked with the same color or background. From the picture it is clear that there are four different periodic response regimes within the region [2.5 2.5; 2.5 2.5]. Thus it is clear that there are more stable response regimes that we just can predict with the single-period analytical model.
Fig. 6. Numerical simulation of single-period solutions for the case T = 2, DV = 4, s = 5. Solution 3: A = 2.531297498, u = 28.86903629. (a) Plot of the displacement versus time; (b) Poincare section; (c) frequency content of the solution – fast Fourier transform.
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Fig. 7. Domains of attraction of different response regimes at the plane of initial conditions are marked by different color or background. T = 2, DV = 4, s = 5.
Fig. 8. A number of solutions for different parameters of the loading for T = 2 and s = 1. (a) DV = 40; (b) DV = 4.
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Fig. 9. A number of single-period solution as a function of s/T and DV.
This region of parameters is interesting for analysis, but is of a small engineering interest because it is very difficult to predict the system’s behavior. For instance, an unnoticeable error in initial conditions will result in completely different system’s response regime. So if any application of the strongly nonlinear oscillator is planned (for instance for mitigation of periodic impacts), it is desirable to work in the region of parameters with well-predictable behavior. It can be done by changing the system’s time of relaxation s. Making s small will result in greater damping in the system. If one will choose s = 1 instead of s = 5 with the same conditions of loading T = 2, DV = 4, one will get a homogenous picture with one single-period response, without any unexpected multiple-period solutions. It is instructive also to investigate the sensitivity of the system for changing DV (Fig. 8). It is obvious that no different single-period response regimes appear even if the change of the impact intensity is immense (10 times). In order to assess the effect of the relaxation time it is instructive to investigate a possible number of the single-period solutions as function of time of relaxation – s, period of impact – T and impact intensity – DV. Since s is the parameter of the system, we use the dimensionless ratio T/s in order to characterize the influence of DV (Fig. 9). It is evident that the system’s behavior differs a lot for the different values of s/T and DV. Small T/s corresponds to multiple solutions region, so it corresponds to previously made assumption that oscillating system with large time of relaxation will not absorb the energy of impact completely, thus giving rise to birth of new response regimes. The global picture is rather complicated but it is clear that there is a limit of T/s 2 above which for all values of the impact intensity only one single-period response regime will exist.
4. Concluding remarks We developed an analytical model for systems with strongly nonlinear attachment under periodic impact loading with one degree of freedom. This model manages to describe attracting single-period response regimes. However set of attractor of the system is in general broader than predicted by the single-period response model. Strong sensitivity of domains of attraction with respect to initial conditions is hardly tolerable if any applications of the system are considered; therefore some criterion for avoiding this situation may be of interest. We have demonstrated that the system has no multiplication of the single-period response regimes if the period of external forcing is at least twice as the relaxation time. Of course, even if it is the case, some other periodic regimes may appear, but this criterion may be useful for the system design.
Acknowledgement The authors are grateful to Israel Science Foundation (grant 486/05) for financial support.
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