Vol.:(0123456789) 1 3 Acta Mechanica Sinica https://doi.org/10.1007/s10409-020-00984-x RESEARCH PAPER Subharmonic resonance of single‑degree‑of‑freedom piecewise‑smooth nonlinear oscillator Jiangchuan Niu 1,2  · Wenjing Zhang 2  · Yongjun Shen 1,2  · Shaopu Yang 1,2 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 afects the amplitude-frequency response of subharmonic resonance in the form of equivalent negative stifness. Through a demonstration example, the accuracy of approximate analytical solution is verifed by numerical solution, and they have good consistency. Based on the approximate analytical solution, the infuences 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 efective analytical method for solving the subhar- monic resonance of piecewise-smooth nonlinear system. And it provides an efective 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 nons- mooth factors often induce unique dynamic phenomena. For examples, the stick–slip vibration induced by dry fric- tion [1], the vibro-impact caused by rigid stop [2], and so on. The piecewise-smooth models also appear in the sys- tems 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. Maka- renkov and Lamb [4] discussed current research felds of nonsmooth system dynamics, and focused on aspects of dynamics involving bifurcations. Pfeifer [5] reviewed the nonsmooth dynamics in multibody systems, and discussed equations of motion for nonsmooth multibody systems and possible numerical solution procedures. Angulo et al. [6] showed several applications of piecewise-smooth bifur- cation theory to electronics, energy markets and popu- lations. Hetzler [ 7] investigated the efect of Coulomb friction on the single-degree-of-freedom system with self-excitation induced by negative damping, including the steady-state stability and bifurcation behavior. Dimentberg and Iourtchenko [8] summarized the analytical methods, results and literature available for random vibrations of vibro-impact systems. Flores et al. [ 9] investigated the infuence of the joint clearances in the response of models of mechanical systems, where the joints with clearance and lubrication were modeled to refect the actual char- acteristics. Tang et al. [10] studied the nonlinear dynamic behavior of a four-bar linkage with a clearance at the coupler-rocker connection. Ding [11] studied the stable steady-state periodic responses of a belt-drive system * Jiangchuan Niu menjc@163.com 1 State Key Laboratory of Mechanical Behavior and System Safety of Trafc Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China 2 School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China