Subharmonic resonance of single-degree-of-freedom piecewise-smooth nonlinear oscillator
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RESEARCH PAPER
Subharmonic resonance of single‑degree‑of‑freedom piecewise‑smooth nonlinear oscillator Jiangchuan Niu1,2 · Wenjing Zhang2 · Yongjun Shen1,2 · Shaopu Yang1,2 Received: 14 April 2020 / Revised: 23 June 2020 / Accepted: 29 July 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Take the single degree of freedom nonlinear oscillator with clearance under harmonic excitation as an example, the 1/3 subharmonic resonance of piecewise-smooth nonlinear oscillator is investigated. The approximate analytical solution of 1/3 subharmonic resonance of the single-degree-of-freedom piecewise-smooth nonlinear oscillator is presented. By changing the solving process of Krylov-Bogoliubov–Mitropolsky (KBM) asymptotic method for subharmonic resonance of smooth nonlinear system, the classical KBM method is extended to piecewise-smooth nonlinear system. The existence conditions of 1/3 subharmonic resonance steady-state solution are achieved, and the stability of the subharmonic resonance steady-state solution is also analyzed. It is found that the clearance affects the amplitude-frequency response of subharmonic resonance in the form of equivalent negative stiffness. Through a demonstration example, the accuracy of approximate analytical solution is verified by numerical solution, and they have good consistency. Based on the approximate analytical solution, the influences of clearance on the critical frequency and amplitude-frequency response of 1/3 subharmonic resonance are analyzed in detail. The analysis results show that the KBM method is an effective analytical method for solving the subharmonic resonance of piecewise-smooth nonlinear system. And it provides an effective reference for the study of subharmonic resonance of other piecewise-smooth systems. Keywords Subharmonic resonance · Piecewise-smooth system · Approximate analytical solution · Asymptotic method
1 Introduction In engineering applications, many mechanical models of systems contain discontinuity factors or sudden change, thus forming a piecewise-smooth system. These nonsmooth factors often induce unique dynamic phenomena. For examples, the stick–slip vibration induced by dry friction [1], the vibro-impact caused by rigid stop [2], and so on. The piecewise-smooth models also appear in the systems under switching control and other switching systems [3]. The research on the dynamics of piecewise smooth or nonsmooth systems has always been a hot topic. Makarenkov and Lamb [4] discussed current research fields of * Jiangchuan Niu [email protected] 1
State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2
nonsmooth system dynamics, and focused on aspects of dynamics involving bifurcations. Pfeiffer [5] reviewed the nonsmooth dynamics in multibody systems, and discussed equa
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