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Journal of Vibration Testing and System Dynamics 6(3) (2022) 297--315 | DOI:10.5890/JVTSD.2022.09.003

Yu Guo

McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA

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Abstract

In this paper, the complete bifurcation trees of period-1 motions to chaos are predicted through discrete implicit maps for a periodically excited spring pendulum. Such discrete maps are used to construct mapping structures that describe any periodic motions in the system. Analytical bifurcation trees are obtained through the nonlinear algebraic equations of such implicit maps. The corresponding stability and bifurcations are achieved through eigenvalue analysis. For a better understanding of the motion complexity, the corresponding frequency-amplitude characteristics are presented through harmonic responses. Finally, simulation results of periodic motions are illustrated in compare to the analytical predictions.

References

1. [1]	Popov, A.A. (2003), The application of Spring pendulum analogies to the understanding of nonlinear shell vibration, Computational Fluid and Solid Mechanics, Elsevier, 590-594.

2. [2]	Eissa, M. and Sayed, M. (2008), Vibration reduction of a three DOF non-linear spring pendulum, Communications in Nonlinear Science and Numerical Simulation, 13, 465-488.

3. [3]	Wu, Y.P., Qiu, J.H., Zhou, S.P., Ji, H.L., Chen, Y., and Li, S. (2018), A piezoelectric spring pendulum oscillator used for multi-directional and ultra-low frequency vibration energy harvesting, Applied Energy, 231, 600-614.

4. [4]	Lee, W.K. and Park, H.D. (1999), Second-order approximation for chaotic responses of a harmonically excited spring-pendulum system, International Journal of Nonlinear Mechanics, 34, 749-757.

5. [5]	Eissa, M., EL-Serafi, S.A., EL-Sheikh, M., and Sayed, M. (2003), Stability and primary simultaneous resonance of harmonically excited non-linear spring pendulum system, Applied Mathematics and Computation, 145, 421-442.

6. [6]	Alasty, A. and Shabani, R. (2006), Chaotic motions and fractal basin boundaries in spring pendulum system, Nonlinear Analysis: Real World Applications, 7, 81-95.

7. [7]	Amer, T.S. and Bek, M.A. (2009), Chaotic responses of a harmonically excited spring pendulum moving in circular path, Nonlinear Analysis: Real World Applications, 10, 3196-3202.

8. [8]	Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP/L\&H Scientific: Beijing/Glen Carbon.

9. [9]	Luo, A.C.J. and Huang, J.Z. (2012), Analytical dynamics of period-m flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, Article No. 1250093.

10. [10]	Luo, A.C.J. and Huang, J.Z. (2012), Analytical routines of period-1 motions to chaos in a periodically forced Duffing oscillator with twin-well potential, Journal of Applied Nonlinear Dynamics, 1, 73-108.

11. [11]	Luo, A.C.J. (2015), Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems, International Journal of Bifurcation and Chaos, 25(3), Article No. 1550044 (62 pages).

12. [12]	Luo, A.C.J. and Guo, Y. (2015), A semi-analytical prediction of periodic motions in Duffing oscillator through mapping structures, Discontinuity, Nonlinearity, and Complexity, 4(2), 121-150.

13. [13]	Guo, Y. and Luo, A.C.J. (2015), On complex periodic motions and bifurcations in a periodically forced, damped, hardening Duffing oscillator, Chaos, Solitons & Fractals, 81, 378-399.

14. [14]	Luo, A.C.J. and Guo, Y. (2016), Periodic motions to chaos in pendulum, International Journal of Bifurcation and Chaos, 26(9), 1650159.

15. [15]	Guo, Y. and Luo, A.C.J. (2017), Routes of periodic motions to chaos in a periodically forced pendulum, International Journal of Dynamics and Control, 5(3), 551-569.

16. [16]	Guo, Y. and Luo, A.C.J. (2017), Complete bifurcation trees of a parametrically driven pendulum, Journal of Vibration Testing and System Dynamics, 1(2), 93-134.

17. [17]	Luo, A.C.J. and Yuan, Y.G. (2020), Bifurcation trees of period-1 to period-2 motions in a periodically excited nonlinear spring pendulum, Journal of Vibration Testing and System Dynamics, 4(3), 201-248.