# Study on Dynamic Characteristics of the Bistable Nonlinear Damper
## 2 System Vibration Model  
### Dynamic Model  
Figure 1 displays two steady-state forms of the bistable unit model. The model consists of two mutually exclusive permanent magnets and a bistable cantilever beam vibrator. The cantilever beam is made up of Aluminum (Al) materials. A magnet is fixed at the end of the cantilever, which is opposite to the other fixed magnet. The repulsive force between two magnets forms a bistable structure. Because of the repulsion between the two magnets, when the whole system is at rest, three equilibrium positions are formed. By following the horizontal straight line that passes through the center of the fixed magnet on the right as the datum, the three equilibrium positions lie along this horizontal straight line, above and below the straight line, respectively. Among these three positions, the equilibrium position on the horizontal straight line is the unstable equilibrium position. This is because some tiny turbulence can cause the shift of the system due to large repulsive force. Excluding the unstable equilibrium position, the other two positions are referred to as stable equilibrium positions, which constitute the bistable structure of the cantilever beam. The frequency resonance bandwidth of the bistable vibration system exceeds that of the linear vibration system. This is due to the fact that only when the natural frequency of the system, \(\omega_{n}\), is equal to or close to the external excitation frequency, \(\omega\), can the linear system produce the forced vibration. Bistable vibration can not only generate forced vibration at a frequency of \(\omega_{n}\) but also produce sub-resonance at a frequency of \(\omega/n\) or ultra-harmonic vibration at a frequency of \(n\omega\) and even aperiodic vibration. The broadband frequency of the bistable resonance can effectively overcome the problem of too narrow resonant frequency bandwidth in the traditional linear vibration system. Figure 2 displays the mechanical model of the main system with a bistable vibration absorber, in which \(m_{1}\) and \(m_{2}\) are the equivalent masses of the permanent magnets and the main system, respectively; \(c_{1}\) and \(c_{2}\) are the damping coefficients of the absorber and the main system, respectively; \(k_{2}\) is the stiffness of the main spring; \(Y\) and \(Z\) are the absolute displacements of the main system and the absorber, respectively; \(X=Y-Z\) denotes the relative displacement between the absorber and the main system; \(dU/dX=-k_{1}X+k_{3}X^{3}\) denotes the bistable spring restoring force of the absorber (\(U\) is the elastic potential energy of the nonlinear spring, \(k_{1}\) is the linear stiffness coefficient, \(k_{3}\) is the nonlinear stiffness coefficient of the bistable absorber); \(P\) denotes the excitation force, applied to the main system. The governing equation of the system can be written as:  
\[m_{1}(\ddot{X}+\ddot{Y})+c_{1}\dot{X}-k_{1}X+k_{3}X^{3}=0 \tag{1}\]  
\[m_{2}\ddot{Y}+c_{2}\dot{Y}+k_{2}Y-c_{1}\dot{X}+k_{1}X-k_{3}X^{3}=P_{0}\sin( \Omega\tau) \tag{2}\]  
By conducting nondimensionalization on Equations (1) and (2), the following expression can be obtained:  
\[x^{\prime\prime}+2(1+\mu)f\gamma_{1}x^{\prime}-(1+\mu)f^{2}x+(1+\mu)f^{2}\beta x ^{3}-2\gamma_{2}y^{\prime}-y=-p\sin(\omega t) \tag{3}\]  
\[y^{\prime\prime}+2\gamma_{2}y^{\prime}+y-2\mu f\gamma_{1}x^{\prime}+\mu f^{2} x-\mu f^{2}\beta x^{3}=p\sin(\omega t) \tag{4}\]  
where \(x^{\prime}=\dot{X}/\omega_{2}\); \(x^{\prime\prime}=\ddot{X}/\omega_{2}^{2}\); \(y^{\prime}=\dot{Y}/\omega_{2}\); \(y^{\prime\prime}=\dot{Y}/\omega_{2}^{2}\); \(\omega_{1}^{2}=k_{1}/m_{1}\); \(\omega_{2}^{2}=k_{2}/m_{2}\); \(t\) denotes the dimensionless time (\(t=\tau\omega_{2}\)); \(p\) denotes the excitation amplitude (\(p=P_{0}/k_{2}\)); \(\omega\) denotes the excitation frequency (\(\omega=\Omega/\omega_{2}\)); \(\mu\) denotes the mass ratio (\(\mu=m_{1}/m_{2}\)); \(f\) denotes the tuning frequency ratio (\(f=\omega_{1}/\omega_{2}\)); \(\beta\) denotes the nonlinear intensity  
Figure 1: The bistable structural unit.