###### Abstract  
The dynamic vibration absorber (DVA) is widely used in engineering models with complex vibration modes. The research on the stability and periodic motions of the DVA model plays an important role in revealing its complex vibration modes and energy transfer. The aim of this paper is to study the stability and periodic motions of a two-degrees-of-freedom system coupled with an encapsulated nonsmooth dynamic vibration absorber under low-frequency forced excitation. Based on the slow-fast method, the model is transformed into a six-dimensional piecewise smooth system coupling two time scales. The existence and stability of the admissible equilibrium points for the model are discussed under different parameter conditions. Based on the first integrals, the Melnikov vector function of the nonsmooth dynamic vibration absorber model is calculated. The existence and number of periodic orbits bifurcated from a family of periodic orbits under different parameters are discussed. The phase diagram configuration of periodic orbits is given based on numerical simulation. The results obtained in this paper offer a new perspective for vibration analysis and parameter control for nonsmooth dynamic vibration absorbers. nonsmooth dynamic vibration absorber; stability; periodic motions; Melnikov function +
Footnote †: journal: Applied sciences  
0000-0002-4000-3999-3899-  
efficiently. [PERSON] et al. [15] studied the energy harvesting and targeted energy transfer (TET) of a piecewise linear NES through numerical simulation. [PERSON] and [PERSON] [16] proposed an enhanced NES with piecewise linear stiffness and considered the influence of the stiffness and the gap displacement on the vibration absorption. For a two-degrees-of-freedom primary system coupled with a nonsmooth DVA, [PERSON] and [PERSON] [17] designed an encapsulated NES to control two resonance modes. With the aid of a harmonic balance method and experimental methods, the vibration control under different parameters of the NES was investigated. [PERSON] et al. [18] discussed the vibration suppression and modal energy transfer for a vibro-impact nonlinear energy sink model. Low-frequency forced excitation, referred to as a slow-varying parameter due to its frequency being much lower than the natural frequency of the system, is crucial to analyze the periodic motions and other dynamical behaviors of DVAs [19]. Regarding the dynamic behavior of oscillator systems, in [20] and [21] the authors focused on the dynamical motion for two classes of three-degrees-of-freedom (DOF) auto-parametric systems. [PERSON] et al. [22] analyzed the dynamical motion of a new three-degrees-of-freedom nonlinear damped dynamical system. [PERSON] et al. [23] investigated the planar motion of a dynamical model with a two-degrees-of-freedom dynamical tuned absorber system under a harmonic excitation moment. For the stability and bifurcation of DVAs, [PERSON] et al. [24] investigated the stability and bifurcation of equilibrium points for a piecewise linear aeroelastic system for the cases of adding a vibration absorber to the system and not adding a vibration absorber, respectively. [PERSON] et al. [25] studied the vibration reduction effect of a Van der Pol oscillator coupled with the nonlinear energy sink under low-frequency forced excitation, which was described using a four-dimensional nonlinear smooth system. The results obtained by the finite element method are consistent with the numerical simulation results. To our knowledge, few studies have considered the stability of a two-degrees-of-freedom system coupled with an encapsulated nonsmooth DVA. For periodic motions of a nonsmooth DVA, [PERSON] et al. [26] explored the periodic orbit, saddle-node bifurcation and TET of a vibro-impact NES using bifurcation diagrams, frequency-energy plots and Poincare maps. [PERSON] and [PERSON] [27] investigated the vibration absorption for a nonlinear energy sink model with piecewise quadratic damping.