## 2 System Description  
In this section, the two-degrees-of-freedom system with an encapsulated nonsmooth dynamic vibration absorber under low-frequency excitation is proposed based on the NES model established by [PERSON] and [PERSON] [17], which is modeled using the six-dimensional piecewise smooth system and shown in Figure 1.  
Figure 1 depicts a two-degrees-of-freedom primary system coupled with an encapsulated nonsmooth dynamic vibration absorber. The external excitation is \(y(t)\) acting on the base. Compared with Reference [17], the difference is that the model uses linear springs. The piecewise stiffness and piecewise damper of the system are constructed by using the gap. According to [PERSON]'s second law, the governing equation is described as  
\[m_{1}\dot{x}_{1}+c_{1}(\dot{x}_{1}-\dot{y})+k_{1}(x_{1}-y)=c_{2}( \dot{x}_{2}-\dot{x}_{1})+k_{2}(x_{2}-x_{1}),\] \[m_{2}\dot{x}_{2}+c_{2}(\dot{x}_{2}-\dot{x}_{1})+k_{2}(x_{2}-x_{1} )=c_{N}(\dot{x}_{3}-\dot{x}_{2})+k_{4}(x_{3}-x_{2})+g(x_{3}-x_{2})+h(\dot{x}_{ 3}-\dot{x}_{2}),\] \[m_{3}\dot{x}_{3}+c_{N}(\dot{x}_{3}-\dot{x}_{2})+k_{4}(x_{3}-x_{2} )+g(x_{3}-x_{2})+h(\dot{x}_{3}-\dot{x}_{2})=0, \tag{1}\]  
where \(\cdot\) is the derivation of \(t\). \(x_{1},x_{2},x_{3}\) and \(m_{1},m_{2},m_{3}\) are displacements and masses of the two-degrees-of-freedom system and attached mass. \(c_{1},c_{2},c_{3},c_{N}\) are the damping coefficients and \(k_{1},k_{2},k_{3},k_{4}\) are the stiffness coefficients. \(y(t)=a\sin(\omega t)\) is the external excitation and \(L_{c}\) is the gap displacement. The piecewise elastic restoring force is  
\[g(x_{3}-x_{2})=\left\{ \begin{array}{cc}k_{3}(x_{3}-x_{2}-L_{c}),&x_{3}-x_{2} >L_{c}\\ 0,&-L_{c}\leq x_{3}-x_{2}\leq L_{c}\\ k_{3}(x_{3}-x_{2}+L_{c}),&x_{3}-x_{2}<-L_{c}\end{array} ,\right.\]  
Figure 1: The nonsmooth dynamic vibration absorber model. and the piecewise damping force is  
\[h(\dot{x}_{3}-\dot{x}_{2})=\left\{ \begin{array}{cc}c_{3}(\dot{x}_{3}-\dot{x}_{2} ),&x_{3}-x_{2}>L_{c}\\ 0,&-L_{c}\leq x_{3}-x_{2}\leq L_{c}\\ c_{3}(\dot{x}_{3}-\dot{x}_{2}),&x_{3}-x_{2}<-L_{c}\end{array} \right..\]  
The following dimensionless parameters are used:  
\[\rho_{1}=\frac{m_{2}}{m_{1}},\rho_{2}=\frac{m_{3}}{m_{2}},\omega_{1 }=\sqrt{\frac{k_{1}}{m_{1}}},\omega_{2}=\sqrt{\frac{k_{2}}{m_{2}}},\omega_{3}= \sqrt{\frac{k_{3}}{m_{3}}},\zeta_{1}=\frac{c_{1}}{2m_{1}\omega_{3}},\zeta_{2}= \frac{c_{2}}{2m_{1}\omega_{3}},\] \[\zeta_{3}=\frac{c_{2}}{2m_{2}\omega_{3}},\zeta_{4}=\frac{c_{N}}{2 m_{2}\omega_{3}},\zeta_{5}=\frac{c_{3}}{2m_{2}\omega_{3}},\zeta_{6}=\frac{c_{N}}{2 m_{3}\omega_{3}},\zeta_{7}=\frac{c_{3}}{2m_{3}\omega_{3}},\
u_{1}=\frac{\omega_{2}}{ \omega_{3}},\
u_{2}=\frac{\omega_{1}}{\omega_{3}},\
u=\frac{k_{4}}{k_{3}},\] \[u=\frac{k_{4}}{k_{2}},\dot{x}_{1}=\frac{x_{1}}{D},\dot{x}_{2}= \frac{x_{2}}{D},\dot{x}_{3}=\frac{x_{3}}{D},L=\frac{L_{c}}{D},\dot{t}=\omega_ {3}t,w_{1}=\frac{a}{D}\sin\frac{\omega}{\omega_{3}}\dot{t},w_{2}=\frac{a\omega }{D\omega_{3}}\cos\frac{\omega}{\omega_{3}}\dot{t}.\]  
For convenience, the over bar of \(\ddot{t}\) is dropped. The dimensionless equations of system (1) are shown as follows. \[\ddot{x}_{1}+2\zeta_{1}(\dot{x}_{1}-w_{2})+\
u_{2}^{2}(\ddot{x}_{1 }-w_{1})=2\zeta_{2}(\dot{x}_{2}-\dot{x}_{1})+\
u_{1}^{2}\rho_{1}(\ddot{x}_{2} -\ddot{x}_{1}),\] \[\ddot{x}_{2}+2\zeta_{3}(\dot{x}_{2}-\dot{x}_{1})+\
u_{1}^{2}( \ddot{x}_{2}-\dot{x}_{1})=2\zeta_{4}(\dot{x}_{3}-\dot{x}_{2})+u\
u_{1}^{2}( \ddot{x}_{3}-\dot{x}_{2})\] \[+\rho_{2}\ddot{g}(\ddot{x}_{3}-\ddot{x}_{2})+2\zeta_{5}\ddot{h}( \dot{x}_{3}-\dot{x}_{2}),\] \[\ddot{x}_{3}+2\zeta_{6}(\dot{x}_{3}-\dot{x}_{2})+v(\dot{x}_{3}- \dot{x}_{2})+\widetilde{g}(\ddot{x}_{3}-\ddot{x}_{2})+2\zeta_{7}\ddot{h}(\dot {x}_{3}-\dot{x}_{2})=0. \tag{2}\]  
where  
\[\tilde{g}(\ddot{x}_{3}-\ddot{x}_{2})=\left\{ \begin{array}{cc}\dot{x}_{3}- \dot{x}_{2}-L,&\dot{x}_{3}-\ddot{x}_{2}>L\\ 0,&-L\leq\dot{x}_{3}-\ddot{x}_{2}\leq L,\\ \dot{x}_{3}-\dot{x}_{2}+L,&\dot{x}_{3}-\ddot{x}_{2}<-L\end{array} \right.\]  
\[\tilde{h}(\dot{x}_{3}-\dot{x}_{2})=\left\{