## 2 Dynamic Model and Parameters Optimization
### Parameters Optimization
\[\left|\frac{X_{1}}{\delta_{st}}\right|_{P}=\left|\frac{(\alpha_{1}+2\alpha_{2} +2)v^{2}-2(1+\beta)\lambda_{P}^{2}}{\alpha_{1}v^{2}(1+\mu v^{2}L^{2}-\lambda_ {P}^{2})}\right|\]  
\[\left|\frac{X_{1}}{\delta_{st}}\right|_{Q}=\left|-\frac{(\alpha_{1}+2\alpha_{2 }+2)v^{2}-2(1+\beta)\lambda_{Q}^{2}}{\alpha_{1}v^{2}(1+\mu v^{2}L^{2}-\lambda_ {Q}^{2})}\right| \tag{14}\]  
\[\left|\frac{X_{1}}{\delta_{st}}\right|_{R}=\left|\frac{(\alpha_{1}+2\alpha_{2 }+2)v^{2}-2(1+\beta)\lambda_{R}^{2}}{\alpha_{1}v^{2}(1+\mu v^{2}L^{2}-\lambda_ {R}^{2})}\right|\]  
The ultimate goal of H\({}_{\infty}\) optimization is to minimize the maximum value of the amplitude-frequency curve, so we expect these three fixed points to be almost at the peak of the amplitude-frequency curve. In order to realize the optimal control effect, the optimal frequency ratio, stiffness ratio and damping ratio should be searched. It takes two steps to adjust the three fixed points to the same height. The first step is to adjust the ordinate of \(P\) and \(R\) to the same height, i.e.,  
\[\left|\frac{X_{1}}{\delta_{st}}\right|_{P}=\left|\frac{X_{1}}{\delta_{st}} \right|_{R} \tag{15}\]  
Because of the value \(\left|\frac{X_{1}}{\delta_{st}}\right|\) is independent of \(\lambda^{2}\), Equation (15) gives rise to  
\[\alpha_{1}=\frac{2[(1+\mu L^{2}v^{2})(1+\beta)-(1+\alpha_{2})v^{2}]}{v^{2}} \tag{16}\]Substituting Equation (16) into Equation (10), and solving Equation (10) for \(\lambda^{2}\), yields  
\[ \begin{split}\lambda_{P}^{2}=& 1+\mu v^{2}L^{2}-\sqrt{(1+\mu v^{2}L^{2})^{2}- \frac{(1+\beta)h_{4}v^{2}-(\alpha_{2}+1)^{2}v^{4}}{(1+\beta)^{2}}}\\ \lambda_{Q}^{2}=& 1+\mu v^{2}L^{2}\\ \lambda_{R}^{2}=& 1+\mu v^{2}L^{2}+\sqrt{(1+\mu v^{2}L^{2})^{2 }-\frac{(1+\beta)h_{4}v^{2}-(\alpha_{2}+1)^{2}v^{4}}{(1+\beta)^{2}}}\end{split}  \tag{17}\]  
where \(h_{4}=\mu v^{2}L^{2}(1+2\alpha_{2})+2(1+\alpha_{2})\). Thus, Equation (14) can be written as follows  
\[ \begin{split}\left|\frac{X_{1}}{\delta_{st}}\right|_{P,R}& =\left|\frac{1+\beta}{(1+\beta)(1+\mu v^{2}L^{2})-(1+\alpha_{2})v^ {2}}\right|\\ \left|\frac{X_{1}}{\delta_{st}}\right|_{Q}&=\left| \frac{(1+\beta)(1+\mu v^{2}L^{2})-(1+\alpha_{2})v^{2}}{\mu v^{4}L^{2}}\right| \end{split}  \tag{18}\]  
In the second step, the ordinates of point \(P\) (or \(R\)) and point \(Q\) are adjusted to the same height, and two different optimal frequency ratio can be obtained. \[ \begin{split}\text{case}\,1:v_{a}&=\sqrt{\frac{1+ \beta}{1+\alpha_{2}-r+\sqrt{r}}}\\ \text{case}\,2:v_{b}&=\sqrt{\frac{1+\beta}{1+\alpha_{2}- r-\sqrt{r}}}\end{split}  \tag{19}\]  
Firstly, we consider \(v_{opt}=v_{a}\). By substituting \(v_{a}\) into Equation (16), the corresponding first optimal stiffness ratio can be obtained as  
\[\alpha_{1 opt}=2\sqrt{r} \tag{20}\]  
Then, we can obtain the response of the primary system at the fixed points, as follows  
\[\left|\frac{X_{1}}{\delta_{st}}\right|_{P,Q,R}^{2}=\frac{[-1-\alpha_{2}+r- \sqrt{r}]^{2}}{r} \tag{21}\]  
Thus, the optimal frequency ratio and possible maximum amplitude are obtained when the three fixed points are adjusted to the same height, as shown in Figure 3. At this time, changing the damping ratio \(\tilde{\xi}\) can change the height of the resonance peak. The optimal damping ratio can be achieved by adjusting the two resonance peaks to the same height. It can be seen from Figure 3 that when the fixed points \(P\), \(Q\), and \(R\) are at the same height, the tangent of the frequency response curve at the \(Q\) point is almost horizontal. The abscissa of \(Q\) point has been calculated, and the approximate optimal damping ratio can be obtained according to the abscissa of \(Q\) point. \[ \begin{split}\frac{\partial A^{2}}{\partial\lambda^{2}}=0\\ \alpha_{1}=2\sqrt{r}\\ \
u=\sqrt{\frac{1+\beta}{1+\alpha_{2}-r+\sqrt{r}}}\\ \lambda_{Q}^{2}=1+\mu v^{2}L^{2}\end{split}