# Study on Dynamic Characteristics of the Bistable Nonlinear Damper
## 3 Analysis of Dynamic Parameters  
In order to investigate the dynamic response of the bistable vibration absorber and the main system under harmonic excitation, the dimensionless Equations (3) and (4) were solved, using Simulink simulation to sweep frequency and amplitude, in terms of fundamental excitation frequency and the amplitude of the system. The simulation parameters were \(\beta=1.0\), \(\mu=0.3,f=1.0\), \(\gamma_{1}=0.05\), \(\gamma_{2}=0.05\), and the initial value of the system [\(x\ x^{\prime}\ y\ y^{\prime}\)] was set to [0 0 0 0]. Effects of the Excitation Frequency on the Response Characteristics of the Vibration Absorber and the Main System  
Figure 4 shows the bifurcation phenomenon of the vibration absorber and the main system at an excitation amplitude of \(p=0.01\), \(p=0.15\) and \(p=0.5\). As shown in Figure 4a (excitation amplitude \(p=0.01\)), the absorber never overcame the potential barrier in the whole excitation frequency range, while there was only slight motion around the equilibrium point \(x=-1\). In Figure 4b (excitation amplitude \(p=0.15\)), when excitation frequency \(\omega=0\)-0.5, the absorber performed slight motion at and around the equilibrium point \(x=-1\); when \(\omega=0.7\)-1.7, the absorber could overcome the potential barrier and showed large amplitude motion between two equilibrium points, while the main system also performed large amplitude motion; when \(\omega>1.7\), the system  
To visually display the change in the system motion characteristics, the corresponding system phase diagrams at excitation frequency \(\omega=2\) and excitation amplitude \(p=0.1\) and \(0.5\) are plotted in Figure 5. It can be observed that, at a constant excitation frequency, the vibration absorber and main system changed from slight periodic motion to large amplitude chaotic motion with excitation amplitude increasing. Figure 4: Bifurcation of the vibration absorber (on the left side) and the main system (on the right side) along excitation frequency at different excitation amplitudes. (**a**) Excitation amplitude \(p=0.01\); (**b**) Excitation amplitude \(p=0.15\); (**c**) Excitation amplitude \(p=0.5\). Figure 5: _Cont._  
Effects of the Excitation Amplitude on the Response Characteristics of the Vibration Absorber and the Main System  
Figure 6 shows the bifurcations of the vibration absorber and the main system with the excitation amplitude when the excitation frequency is \(\omega\) = 0.1, \(\omega\) = 0.5, \(\omega\) = 1, \(\omega\) = 2 and \(\omega\) = 3.5. As shown in Figure 6a (\(\omega\) = 0.1), even at a quite low excitation frequency, the absorber can cross the potential barrier from an equilibrium point to another equilibrium point, while the main system executes slight motion around the equilibrium point. In contrast, as shown in Figure 6b-d, as the excitation frequency increased, the absorber performed slight motion around the equilibrium point, and the main system executed a slight motion around the zero-equilibrium point. As the excitation amplitude increased, both the absorber and the main system performed large amplitude chaotic or period motions. Moreover, as the excitation frequency increased, the required excitation amplitude for the absorber and the main system to perform large amplitude motions first decreased and then increased. As shown in Figure 6e (\(\omega\) = 3.5), as the excitation frequency increased to a certain value, the absorber failed in overcoming the potential barrier and executed only slight motion around the equilibrium point \(x\) = \(-\)1, while the main system also performed slight motion at any excitation amplitude. Figure 5: Phase diagrams of the vibration absorber (on the left side) and the main system (on the right side) at different excitation amplitudes. (**a**) \(\omega\) = 2, \(p\) = 0.15; (**b**) \(\omega\) = 2, \(p\) = 0.5. Figure 6: _Cont_. Figure 6: Bifurcation of the vibration absorber (on the left side) and the main system (on the right side) along excitation amplitude at different excitation frequencies.