## 3 Influence Factors of Higher-Order CSI Effect
### Influence Analysis of Structural Parameters  
The force equilibrium of a SDOF system can be described as:  
\[m\ddot{x}(t)+c^{\prime}\dot{x}(t)+k^{\prime}x(t)=-u(t)+\ddot{x}_{g}(t) \tag{22}\]  
where \(m\), \(c^{\prime}\), and \(k^{\prime}\) are the nominal mass, damping, and stiffness of the SDOF system, respectively.  
Since the uncertainties of structural damping or stiffness exist, the transfer functions of an AMD system from the control voltage to the control force can be obtained. The change in the stiffness and damping are described as:  
\[k^{\prime}=\beta k,\ c^{\prime}=\alpha c\ (\alpha,\beta=1,2,\cdots,5) \tag{23}\]  
where \(\beta\) and \(\alpha\) are the amplification factors of stiffness and damping, respectively.  
Substituting Equation (23) into Equation (20) leads to:  
\[\dot{\dot{Z}}=\left[\begin{array}{ccc}0&I&0\\ -\dfrac{\beta k}{m}&-\dfrac{\alpha c}{m}&\dfrac{1}{m}\\ 0&\dfrac{K_{b}K_{i}K_{g}^{2}}{L_{a}r_{m}^{2}}B_{s}^{T}&-\dfrac{R_{a}}{L_{a}} \end{array}\right]\dot{Z}+\left[\begin{array}{ccc}0&0\\ \dfrac{1}{m}&0\\ 0&\dfrac{K_{i}K_{g}}{L_{a}r_{m}}\end{array}\right]\left[\begin{array}{c} \ddot{x}_{g}\\ V\end{array}\right] \tag{24}\]  
Its observation equation is:  
\[Y=\left[\begin{array}{ccc}I&0&0\\ 0&I&0\\ -\dfrac{\beta k}{m}&-\dfrac{\alpha c}{m}&\dfrac{1}{m}\\ 0&0&1\end{array}\right]\dot{Z}+\left[\begin{array}{ccc}0&0\\ 0&0\\ \dfrac{1}{m}&0\\ 0&0\end{array}\right]\left[\begin{array}{c}\ddot{x}_{g}\\ V\end{array}\right] \tag{25}\]  
The changes in the transfer functions from the control voltage to the control force of the AMD system with different stiffnesses and dampings are shown in Figures 9 and 10.  
From Figure 9, with the increase of structural stiffness, its fundamental frequency becomes larger. When the frequency of the control voltage is consistent with the structural frequency, a sudden change in the transfer function is generated. Meanwhile, mutation positions gradually shift to the right side, which is shown in Figure 9a. Furthermore, as the frequency of the control voltage is smaller than the fundamental frequency, the system has a long time-delay (negative phase). Therefore, the chosen frequency should be greater than the fundamental structural frequency. For example, a suitable frequency of the control voltage of the experimental system is 10 Hz, which is larger than its fundamental frequency (0.162 Hz).  
From Figure 10, because structural damping cannot change the structural fundamental frequency, the mutation positions of these transfer functions remain unchanged when the damping increases. Additionally, when the chosen frequency is greater than the fundamental frequency, the phase (negative value) of the transfer function becomes smaller with the increase in structural damping. This indicates that the CSI effect becomes more significant.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Structure** & & & & **The AMD System** & & \\ \hline
**Mass \(m\) (kg)** & **Rigidity \(k\) (N/m)** & **Damping Ratio** & **AMD Mass \(m_{a}\) (kg)** & **AMD Rigidity \(k_{a}\) (N/m)** & \(Q\) & \(R\) \\ \hline \(2\times 10^{6}\) & \(5\times 10^{7}\) & 0.05 & 250 & 0.001 & \(10^{8}\) & \(10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the control system.  
Figure 10: _Cont._  
Figure 9: Transfer functions from control voltage to control force with different stiffnesses, (a) Magnitude; (b) Phase.