An Efficient Numerical Method for Computing Dynamic Responses of Periodic Piecewise Linear Systems

HE Dongdong; GAO Qiang; ZHONG Wanxie

doi:10.21656/1000-0887.390055

Volume 39 Issue 7

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2018 > 39(7): 737-749

HE Dongdong, GAO Qiang, ZHONG Wanxie. An Efficient Numerical Method for Computing Dynamic Responses of Periodic Piecewise Linear Systems[J]. Applied Mathematics and Mechanics, 2018, 39(7): 737-749. doi: 10.21656/1000-0887.390055

Citation:

PDF( 2556 KB)

An Efficient Numerical Method for Computing Dynamic Responses of Periodic Piecewise Linear Systems

doi: 10.21656/1000-0887.390055

State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology); Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

Funds: The National Natural Science Foundation of China(11572076；914748203)；The National Basic Research Program of China(973 Program)（2014CB049000）

Received Date: 2018-02-04
Publish Date: 2018-07-15

Abstract

Abstract

An efficient method based on the parametric variational principle (PVP) was proposed for computing the dynamic responses of periodic piecewise linear systems with multiple gap-activated springs. Through description of gap-activated springs with the PVP, the complex nonlinear dynamic problem was transformed to a standard linear complementary problem. This method can avoid iterations and updating the stiffness matrix in the computing process and can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method (PIM), an efficient numerical time-integration method was developed to obtain the dynamic responses of the system. This method indicates that there are a large number of identical elements and zero elements in the matrix exponents of a periodic structure, and saves computation load and computer storage by avoiding repeated calculation and storage of these elements. Numerical results validate the proposed method. The dynamic behaviors of a 5-DOF piecewise linear system under harmonic excitations were analyzed, including the stable periodic motion, the quasi-periodic motion and the chaotic motion. In comparison with the Runge-Kutta method, the proposed method has satisfactory correctness and efficiency.
- piecewise linear system,
- dynamic response,
- parametric variational principle,
- periodicity,
- precise integration method

FullText(HTML)

References(22)

References

[1]	INABA N, SEKIKAWA M. Chaos disappearance in a piecewise linear Bonhoeffer-Van der Pol dynamics with a bistability of stable focus and stable relaxation oscillation under weak periodic perturbation[J]. Nonlinear Dynamics,2014,76(3): 1711-1723.
[2]	GOUZ J, SARI T. A class of piecewise linear differential equations arising in biological models[J]. Dynamics & Stability of Systems,2002,17(4): 299-316.
[3]	KALMR-NAGY T, CSIKJA R, ELGOHARY T A. Nonlinear analysis of a 2-DOF piecewise linear aeroelastic system[J]. Nonlinear Dynamics,2016,85(2): 739-750.
[4]	MOTRO R. Tensegrity: Structural Systems for the Future[M]. London: Kogan Page Science, 2003.
[5]	PUN D, LIU Y B. On the design of the piecewise linear vibration absorber[J]. Nonlinear Dynamics,2000,22(4): 393-413.
[6]	滕云楠, 韩明. 振动压实-土系统非线性滞回模型研究[J]. 真空, 2016,53(4): 61-64.(TENG Yunnan, HAN Ming. Study on nonlinear hysteretic model of vibration compaction-soil system[J]. Vacuum,2016,53(4): 61-64.(in Chinese))
[7]	ZHANG C. Theoretical design approach of four-dimensional piecewise-linear multi-wing hyperchaotic differential dynamic system[J]. Optik-International Journal for Light and Electron Optics,2016,127(11): 4575-4580.
[8]	CHOI Y S, NOAH S T. Forced periodic vibration of unsymmetric piecewise-linear systems[J]. Journal & Sound Vibration,1988,121(1): 117-126.
[9]	HUDSON J L, ROSSLER O E, KILLORY H C. Chaos in a four-variable piecewise-linear system of differential equations[J]. IEEE Transaction on Circuits & Systems,1988,35(7): 902-908.
[10]	NAYFEH A H. Introduction To Perturbation Techniques [M]. New York: John Wiley & Sons, 1981.
[11]	吴志强, 雷娜. 分段线性系统的振动性能分析[J]. 振动与冲击, 2015,34(18): 94-99.(WU Zhiqiang, LEI Na. Vibration performance analysis of piecewise linear system[J]. Journal of Vibration and Shock,2015,34(18): 94-99.(in Chinese))
[12]	钟顺, 陈予恕. 分段线性非线性汽车悬架系统的分岔行为[J]. 应用数学和力学, 2009,30(6): 631-638.(ZHONG Shun, CHEN Yushu. Bifurcation of piecewise-linear nonlinear vibration system of vehicle suspension[J]. Applied Mathematics and Mechanics,2009,30(6): 631-638.(in Chinese))
[13]	NARIMANI A, GOLNARAGHI M E, JAZAR G N. Frequency response of a piecewise linear vibration isolator[J]. Journal of Vibration Control,2004,10(12): 1775-1794.
[14]	DESHPANDE S, MEHTA S, JAZAR G N. Optimization of secondary suspension of piecewise linear vibration isolation systems[J]. International Journal of Mechanical Sciences,2006,48(4): 341-377.
[15]	MOUSSI E H, BELLIZZI S, COCHELIN B, et al. Nonlinear normal modes of a two degrees-of-freedom piecewise linear system[J]. Mechanical Systems & Signal Processing,2015,64(S1): 266-281.
[16]	XU L, LU M W, CAO Q. Nonlinear vibrations of dynamical systems with a general form of piecewise-linear viscous damping by incremental harmonic balance method[J]. Physics Letters A,2002,301(1): 65-73.
[17]	ZHONG W, ZHANG R. Parametric variational principles and their quadratic programming solutions in plasticity[J]. Computers & Structures,1988,30(4): 887-896.
[18]	YU S D. An efficient computational method for vibration analysis of unsymmetric piecewise-linear dynamical systems with multiple degrees of freedom[J]. Nonlinear Dynamics,2013,71(3): 493-504.
[19]	ACARY V, DE JONG H, BROGLIATO B. Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems[J]. Physica D: Nonlinear Phenomena,2014,269(2): 103-119.
[20]	ZHONG W, WILLIAMS F. A precise time step integration method[J]. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical,1994,208(63): 427-430.
[21]	ZHONG W X. On precise integration method[J]. Journal of Computational and Applied Mathematics,2004,163(1): 59-78.
[22]	SHA D, SUN H, ZHANG Z, et al. A variational inequality principle in solid mechanics and application in physically non-linear problems[J]. International Journal for Numerical Methods in Biomedical Engineering,1990,6(1): 35-45.