# Dynamics Analysis of a Variable Stiffness Tuned Mass Damper Enhanced by an Inerter
## 4 Results and Discussion
### Impulse Response  
Figure 11 illustrates the transient responses of \(u_{1}\) with different initial velocity values applied on the mass \(m_{1}\). The solid black line and the dashed red line express the response of the primary without the vibration reduction and with the VS-TMD-I. It can be shown that the VS-TMD-I can improve the energy absorption rate with different initial impulse values. Moreover, the VS-TMD-I can reduce the time for the primary structure to reach steady state. In addition, the VS-TMD-I is effective for an extremely small initial impulse \(X\) = 0.01.  
Figure 11: The transient responses of \(u_{1}\) with various initial impulses: (**a**) \(X\) = 0.01; (**b**) \(X\) = 0.05; (**c**) \(X\) = 0.2; (**d**) \(X\) = 0.5.  
Figure 10: The root-mean-square of \(u_{1}\) based on the values of the optimal parameter: (**a**) \(\beta_{4}\); (**b**) \(\beta_{5}\).  
Figure 12a shows the response of the relative displacement \(u_{3}\)-\(u_{2}\) between the VS-TMD-I and the primary structure with an initial impulse \(X=0.01\). The corresponding wavelet transform spectra is shown in Figure 12b. It can be seen that the 1:1 internal resonance occurs in the VS-TMD-I system. It should be noted that obvious multifrequency resonance captures appear in the VS-TMD-I system as the initial impulse increases. Figure 12d shows both 1:1 and 1:2 internal resonance exhibiting the transient dynamics process with the initial impulse \(X=0.05\). Initially, 1:1 and 1:2 internal resonances are excited in the VS-TMD-I system. After 150 s, the vibration energy dissipation is transferred to a lower level, and the VS-TMD-I system evolves to 1:1 internal resonance. It can be seen that the VS-TMD-I can absorb the vibration energy from multiple frequency ranges. As shown in Figure 13b, 1:3 internal resonances may appear in the VS-TMD-I system. The 1:1 and 1:3 internal resonances have always existed together in the transient dynamics process. Figure 13d shows low-frequency internal resonance exhibiting the transient dynamics process with the initial impulse \(X=0.5\). Initially, low-frequency internal resonance is excited, and the VS-TMD-I quickly oscillates at a low frequency. After 100 s, the VS-TMD-I system evolves to 1:1 internal resonance. It can be concluded that the VS-TMD-I engages in 1:1 internal resonance, 1:2 internal resonance, 1:3 internal resonance, and low-frequency resonance and accelerates the dissipation of the vibration energy.  
Figure 12: Responses of the relative displacement \(u_{3}\)-\(u_{2}\) and the corresponding wavelet transform spectra with various initial impulses: (**a,b**) \(X=0.01\); (**c,d**) \(X=0.05\).  
It can be seen from numerical simulations that the proposed VS-TMD-I can enhance vibration reduction performance under both harmonic and impulse excitations. With the introduction of the inerter, the additional mass is significantly reduced. Furthermore, the VS-TMD-I exhibits various nonlinear phenomena, such as unstable responses and inerter resonances. The main limitation of the proposed VS-TMD-I is the complicated structure; it requires more space for installation. Moreover, this work only considers a simple controller of PI control, and the performance of the system is expected to be further improved with a more complex control method.