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Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities. (English) Zbl 1268.34088
Summary: Based on temporal rescaling and harmonic balance, an extended asymptotic perturbation method for parametrically excited two-degree-of-freedom systems with square and cubic nonlinearities is proposed to study the nonlinear dynamics under 1:2 internal resonance. This asymptotic perturbation method is employed to transform the two-degree-of-freedom nonlinear systems into a four-dimensional nonlinear averaged equation governing the amplitudes and phases of the approximation solutions. Linear stable analysis at equilibrium solutions of the averaged equation is done to show bifurcations of periodic motion and homoclinic motions. Furthermore, analytical expressions of homoclinic orbits and heteroclinic cycles for the averaged equation without dampings are obtained. Considering the action of the damping, the bifurcations of limit cycles are also investigated. A concrete example is further provided to discuss the correctness and accuracy of the extended asymptotic perturbation method in the case of small-amplitude motion for the two-degree-of-freedom nonlinear system.

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
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[1] Yang, J., Shen, H.S.: Vibration characteristics and transient response of shear deformable functionally graded plates in thermal environment. J. Sound Vib. 255, 579–602 (2002) · doi:10.1006/jsvi.2001.4161
[2] Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, New York (2004) · Zbl 1075.74001
[3] Chen, C.S.: Nonlinear vibration of a shear deformable functionally graded plate. Compos. Struct. 68, 295–302 (2005) · doi:10.1016/j.compstruct.2004.03.022
[4] Hao, Y.X., Chen, L.H., Zhang, W., Lei, J.G.: Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. J. Sound Vib. 312, 862–892 (2008) · doi:10.1016/j.jsv.2007.11.033
[5] Chen, H.K., Zhang, Z.H., Wang, J.L., Xu, Q.Y.: Global bifurcation and chaotic dynamics in suspended cables. Int. J. Bifurc. Chaos 19, 3753–3776 (2009) · Zbl 1182.34067 · doi:10.1142/S0218127409025092
[6] Huang, J.L., Su, R.K.L., Lee, Y.Y., Chen, S.H.: Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities. J. Sound Vib. 330, 5151–5164 (2011) · doi:10.1016/j.jsv.2011.05.023
[7] Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-Interscience , New York (1979)
[8] Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981) · Zbl 0449.34001
[9] Malhotra, N., Sri Namachchivaya, N.: Global bifurcations in externally excited two-degree-of-freedom nonlinear systems. Nonlinear Dyn. 8, 85–109 (1995)
[10] Maccari, A.: Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation. Nonlinear Dyn. 15, 329–343 (1998) · Zbl 0909.34037 · doi:10.1023/A:1008235820302
[11] Maccari, A.: The asymptotic perturbation method for nonlinear continuous systems. Nonlinear Dyn. 19, 1–18 (1999) · Zbl 0944.74031 · doi:10.1023/A:1008304701252
[12] Maccari, A.: Multiple resonant or non-resonant parametric excitations for nonlinear oscillators. J. Sound Vib. 242, 855–866 (2001) · Zbl 1237.34063 · doi:10.1006/jsvi.2000.3386
[13] Feng, Z.C., Sethna, P.R.: Global bifurcation and chaos in parametrically forced systems with one-one resonance. Dyn. Stab. Syst. 5, 201–225 (1990) · Zbl 0727.34027 · doi:10.1080/02681119008806098
[14] Kovačič, G., Wiggins, S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation. Physica D 57, 185–225 (1992) · Zbl 0755.35118 · doi:10.1016/0167-2789(92)90092-2
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