Dynamic analysis of asymmetric piecewise linear systems

doi:10.1007/s10483-025-3234-9

Abstract

Abstract:

Piecewise linear systems are prevalent in engineering practice, and can be categorized into symmetric and asymmetric piecewise linear systems. Considering that symmetry is a special case of asymmetry, it is essential to investigate the broader model, namely the asymmetric piecewise linear system. The traditional averaging method is frequently used for studying nonlinear systems, particularly symmetric piecewise linear systems, with the harmonic response of the oscillator serving as a key prerequisite for calculating steady-state solutions. However, asymmetric systems inherently exhibit non-harmonic, asymmetric responses, rendering the traditional averaging method inapplicable. To overcome this limitation, this paper introduces an improved averaging method tailored for an oscillator characterized by asymmetric gaps and springs. Unlike the traditional method, which assumes a purely harmonic response, the improved averaging method redefines the system response as a superposition of a direct current (DC) term and a first harmonic component. Herein, the DC term can be regarded as the offset induced by model asymmetry. Furthermore, the DC term is treated as a slow variable function of time, with its time derivative assumed to be zero when calculating the steady-state solution, akin to the amplitude and phase in the traditional averaging method. Numerical validation demonstrates that the responses computed in both time and frequency domains with the improved averaging method exhibit greater accuracy compared with those derived from the traditional method. Leveraging these improved results, the study also examines the parameter effect, stability, and bifurcation phenomena within the amplitude-frequency responses.

Key words: nonlinear vibration, piecewise linear system, asymmetry, averaging method, offset

2010 MSC Number:

O322

Ruiliang ZHANG, Yongjun SHEN, Xiaotong YANG. Dynamic analysis of asymmetric piecewise linear systems. Applied Mathematics and Mechanics (English Edition), 2025, 46(4): 633-646.

TrendMD

Figures/Tables 16

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16

References 45

[1]	HU, H. Y. Progress in the dynamics of piecewise-smooth mechanical systems(in Chinese). Journal of Vibration Engineering, 8(4), 331–341 (1995)
[2]	DEN HARTOG, J. P. and MIKINA, S. J. Forced vibrations with non-linear spring constants. Transactions of the American Society of Mechanical Engineers, 54(2), 157–162 (1932)
[3]	CHEN, Y. S. and JIN, Z. S. Subharmonic solution of a piecewise linear oscillator with two degrees of freedom. Applied Mathematics and Mechanics (English Edition), 7(3), 223–232 (1986) https://doi.org/10.1007/BF01900702
[4]	NATSIAVAS, S. Periodic response and stability of oscillators with symmetric trilinear restoring force. Journal of Sound and Vibration, 134, 315–331 (1989)
[5]	NATSIAVAS, S. and THEODOSSIADES, S. Piecewise linear systems with time varying coefficients. Proceedings of the ASME 1997 Design Engineering Technical Conferences, ASME, New York (1997)
[6]	NATSIAVAS, S., THEODOSSIADES, S., and GOUDAS, I. Dynamic analysis of piecewise linear oscillators with time periodic coefficients. International Journal of Non-Linear Mechanics, 35(1), 53–68 (2000)
[7]	THEODOSSIADES, S. and NATSIAVAS, S. Nonlinear dynamics of gear-pair systems with periodic stiffness and backlash. Journal of Sound and Vibration, 229, 287–310 (2000)
[8]	CHEN, Y. S. Nonlinear Vibrations(in Chinese), Higher Education Press, Beijing, 109–116 (2002)
[9]	NARIMANI, A., GOLNARAGHI, M. E., and JAZAR, G. N. Frequency response of a piecewise linear vibration isolator. Journal of Vibration and Control, 10(12), 1775–1794 (2004)
[10]	XU, L., LU, M. W., and CAO, Q. Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method. Journal of Sound and Vibration, 264, 873–882 (2003)
[11]	XU, H. D. and XIE, J. H. Bifurcation and chaos of a two-degree-of-freedom non-smooth system with piecewise-linearity(in Chinese). Journal of Vibration Engineering, 21(3), 279–285 (2008)
[12]	ZHONG, S. and CHEN, Y. S. Bifurcation of piecewise-linear nonlinear vibration system of vehicle suspension. Applied Mathematics and Mechanics (English Edition), 30(6), 677–684 (2009) https://doi.org/10.1007/s10483-009-0601-z
[13]	YAN, Z. T., WENG, X. T., ZHU, S. J., and LOU, J. J. Analytical solution of free vibration of systems with piecewise linear stiffness(in Chinese). Noise and Vibration Control, 30(6), 18–22 (2010)
[14]	ROCHA, A. H., ZANETTE, D. H., and WIERCIGROCH, M. Semi-analytical method to study piecewise linear oscillators. Communications in Nonlinear Science and Numerical Simulation, 121, 107193 (2023)
[15]	SHEN, Y. J., ZHANG, R. L., HAN, D., and LIU, X. Y. Problems and corrections of classical mathematical model for piecewise linear system. Communications in Nonlinear Science and Numerical Simulation, 139, 108300 (2024)
[16]	DESHPANDE, S., MEHTA, S., and JAZAR, G. N. Optimization of secondary suspension of piecewise linear vibration isolation systems. International Journal of Mechanical Sciences, 48(4), 341–377 (2005)
[17]	LEI, N. and WU, Z. Q. Analysis of jump avoidance and vibration reduction performances of a piecewise linear single DOF system(in Chinese). Noise and Vibration Control, 34(2), 12–16 (2014)
[18]	GAO, X., CHEN, Q., and LIU, X. B., Nonlinear dynamics design for piecewise smooth vibration isolation system(in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 48(1), 192–200 (2016)
[19]	ZHANG, M. X., WANG, J., SHEN, Y. J., and ZHANG, J. C. Analytical threshold for chaos of piecewise Duffing oscillator with time-delayed displacement feedback. Shock and Vibration, 2022, 9108004 (2022)
[20]	WANG, J., SHEN, Y. J., ZHANG, J. C., and WANG, X. N. Chaos of a class of piecewise Duffing oscillator with fractional-order derivative term(in Chinese). Journal of Vibration and Shock, 41(13), 8–16 (2022)
[21]	HU, H. Y. Semi-active vibration control based on a vibration absorber with adjustable clearance. Transactions of Nanjing University of Aeronautics and Astronautics, 13(2), 135–140 (1996)
[22]	SHUI, X. and WANG, S. M. Investigation on a mechanical vibration absorber with tunable piecewise-linear stiffness. Mechanical Systems and Signal Processing, 100, 330–343 (2018)
[23]	ZHANG, K., SU, Y. F., DING, J. Q., and DUAN, Z. Y. A low-frequency wideband vibration harvester based on piecewise linear system. Applied Physics A, 125, 340 (2019)
[24]	WANG, X., GENG, X. F., MAO, X. Y., DING, H., JING, X. J., and CHEN, L. Q. Theoretical and experimental analysis of vibration reduction for piecewise linear system by nonlinear energy sink. Mechanical Systems and Signal Processing, 172, 109001 (2022)
[25]	MAEZAWA, S. Steady, forced vibration of unsymmetrical piecewise-linear system (1st report, explanation of analytical procedure). Bulletin of JSME, 4(14), 201–212 (1961)
[26]	SHAW, S. W. and HOLMES, P. J. A periodically forced piecewise linear oscillator. Journal of Sound and Vibration, 90, 129–155 (1983)
[27]	NATSIAVAS, S. and GONZALEZ, H. Vibration of harmonically excited oscillators with asymmetric constraints. Journal of Applied Mechanics, 59, S284–S290 (1992)
[28]	CAO, Q., XU, L., DJIDJELI, K., PRICE, W. G., and TWIZELL, E. H. Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise-linearity. Chaos, Solitons and Fractals, 12(10), 1917–1927 (2001)
[29]	JIN, J. D. and GUAN, L. Z. Fourier series solution for forced vibration of multi-degree-of-freedom asymmetric piecewise linear systems(in Chinese). Journal of Vibration Engineering, 16(3), 373–378 (2003)
[30]	REN, C. B. and ZHOU, J. L. Periodic solution and stability analysis for piecewise coupled system(in Chinese). Chinese Journal of Applied Mechanics, 28(5), 527–531 (2011)
[31]	LI, Y. H. Nonlinear Vibration and Bifurcation of Asymmetric Piecewise Systems(in Chinese), M. Sc. dissertation, Tianjin University, 12–16 (2013)
[32]	DAI, W., YANG, J., and SHI, B. Y. Vibration transmission and power flow in impact oscillators with linear and nonlinear constraints. International Journal of Mechanical Sciences, 168, 105234 (2020)
[33]	SIRETEANU, T., MITU, A. M., SOLOMON, O., and GIUCLEA, M. Approximation of the statistical characteristics of piecewise linear systems with asymmetric damping and stiffness under stationary random excitation. Mathematics, 10(22), 4275 (2022)
[34]	JIN, H., ZHANG, J. T., LU, X. H., and WANG, X. Discontinuous bifurcations in an impact oscillator with elastic constraint(in Chinese). Journal of Vibration and Shock, 42(18), 46–53 (2023)
[35]	RAO, S. S. Mechanical Vibrations, 5th ed., Prentice Hall, Upper Saddle River, 62 (2016)
[36]	XU, Z. and CHEUNG, Y. K. Averaging method using generalized harmonic functions for strongly non-linear oscillators. Journal of Sound and Vibration, 174, 563–576 (1994)
[37]	DENG, M. and ZHU, W. Energy diffusion controlled reaction rate of reacting particle driven by broad-band noise. The European Physical Journal B, 59, 391–397 (2007)
[38]	JIANG, W. A. and CHEN, L. Q. Stochastic averaging based on generalized harmonic functions for energy harvesting systems. Journal of Sound and Vibration, 377, 264–283 (2016)
[39]	WU, Y. J. and ZHU, W. Q. Stochastic averaging of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. Journal of Vibration and Acoustics, 130(5), 051004 (2008)
[40]	HU, R. C. and ZHU, W. Q. Stochastic optimal control of MDOF nonlinear systems under combined harmonic and wide-band noise excitations. Nonlinear Dynamics, 79, 1115–1129 (2015)
[41]	ZHU, W. Q. and WU, Y. J. Optimal bounded control of harmonically and stochastically excited strongly nonlinear oscillators. Probabilistic Engineering Mechanics, 20(1), 1–9 (2005)
[42]	WU, Y. J. and WANG, H. Y. First-crossing problem of weakly coupled strongly nonlinear oscillators subject to a weak harmonic excitation and Gaussian white noises. Journal of Vibration and Acoustics, 140(4), 041006 (2018)
[43]	CHEUNG, Y. K. and XU, Z. Internal resonance of strongly non-linear autonomous vibrating systems with many degrees of freedom. Journal of Sound and Vibration, 180, 229–238 (1995)
[44]	HU, R. C. and LÜ, Q. F. Optimal time-delay control for multi-degree-of-freedom nonlinear systems excited by harmonic and wide-band noises. International Journal of Structural Stability and Dynamics, 21(4), 2150053 (2021)
[45]	LIU, Y. Z. and CHEN, L. Q. Nonlinear Vibrations(in Chinese), Higher Education Press, Beijing, 80–82 (2001)