A methodology is presented to study the resonance and stability for a single-degree-of-freedom (SDOF) system with a piecewise linear-nonlinear stiffness term (i.e., one piece is linear and the other is weakly nonlinear). Firstly, the exact response of the linear governing equation is obtained, and a modified perturbation method is applied to finding the approximate solution of the weakly nonlinear equation. Then, the primary and 1/2 subharmonic resonances are obtained by imposing continuity conditions and periodicity conditions. Furthermore, Jacobian matrix is derived to investigate the stability of resonance responses. Finally, the results of theoretical study are compared with numerical results, and a good agreement is observed. 1. Introduction Vibration isolation and shock absorbing have always been a hot research topic in engineering practice. A solid and liquid mixture (SALiM) vibration isolator was developed to isolate vibrations and shocks induced by heavy machines [1–3]. The stiffness of the SALiM isolator is piecewise linear-nonlinear in a quasi-static test . That is, the isolator exhibits linear stiffness in some displacement region, beyond which the nonlinear stiffness is observed. Therefore, the elastic restoring force is continuous, but its first-order derivative at the turning point is discontinuous. In the past two decades, the majority of researches focused on the dynamics response, stability, bifurcation, and chaos of piecewise linear systems, such as piecewise bilinear systems or piecewise trilinear systems [4–8]. However, piecewise linear-nonlinear or even nonlinear-nonlinear systems have not received much attention, but many physical systems in fields of aerospace engineering, electric circuit, and so forth appear to be piecewise linear-nonlinear systems [9–12]. For nonsmooth stiffness systems, most approaches finding their steady state responses could be sorted into three groups including harmonic balance method (HBM) and its modified form, increment harmonic balance method (IHBM) [9, 10], classical approximate analytical methods like average method [4, 13–15], and the matching method [6, 7, 11, 16]. For instance, using HBM, Jin et al. investigated the response and stability of an unsymmetrical multiple-degree-of-freedom system with piecewise linear elastic elements , and for a more complex oscillator possessing a periodically time-varying and piecewise binonlinear restoring force function HBM is still applicable to its periodic motion [9, 17]. But like all harmonic balance techniques, the accuracy of the results obtained by
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