## 2 Dynamic Model and Parameters Optimization
### Parameters Optimization
 \tag{22}\]Solving the above equation, we can obtain  
\[\xi_{\text{opt}}\simeq\frac{\sqrt{\mu L^{2}}(1+\beta)}{\sqrt{1+\alpha_{2}+\sqrt{ r}}} \tag{23}\]  
Because the grounded stiffness in the model of this paper may be negative stiffness, according to the characteristics of negative stiffness devices, a pre-compression spring or pre-compression connecting rod is a common mechanism to realize negative stiffness characteristics, and pre-loading will cause pre-displacement of the system. We can use the pre-displacement as the amplitude at the fixed points. We further adjust the response values at the three fixed points to be equal to the response values when \(\lambda=0\), namely  
\[\left|\frac{X_{1}}{\delta_{st}}\right|_{\lambda=0}^{2}=\left|\frac{X_{1}}{ \delta_{st}}\right|_{P,Q,R}^{2} \tag{24}\]  
where  
\[\left|\frac{X_{1}}{\delta_{st}}\right|_{\lambda=0}^{2}=\frac{(1+\alpha_{2})^{2 }(-1-\alpha_{2}+r-\sqrt{r})^{2}}{[r-(1+\alpha_{2})\sqrt{r}-(1+\alpha_{2})^{2} ]^{2}}\]  
Solving Equation (24), all possible optimal stiffness ratios can be obtained  
\[\alpha_{2a} =-1+r-\sqrt{r} \tag{25}\] \[\alpha_{2b} =-1+\sqrt{r}\] \[\alpha_{2c} =-1-\sqrt{r}\] \[\alpha_{2d} =-1+(-1-\sqrt{2})\sqrt{r}\] \[\alpha_{2e} =-1+(-1+\sqrt{2})\sqrt{r}\]  
The appropriate grounded stiffness ratio should be selected on the premise of ensuring the stability of the system. Taking these five grounded stiffness into Equations (8), (21) and the first part of Equation (19), it is found that all these grounded stiffness ratios, except \(\alpha_{2b}\)will make the natural frequency of the coupled system or the optimal natural frequency ratio \(\
u_{a}\) as imaginary number. So, \(\alpha_{2b}\) is choosen as the optimal grounded stiffness ratio, namely  
\[\alpha_{2 opt}=\alpha_{2b}=-1+\sqrt{r} \tag{26}\]  
By analyzing the formula of the optimal grounded stiffness, we discover that the optimum grounded stiffness ratio \(\alpha_{2 opt}\) will exhibit three different kinds of stiffness characteristics (positive, zero, and negative). \[ \begin{array}{ll}\alpha_{2 opt}<0,&\text{if}\quad 0<r<1\\ \alpha_{2 opt}=0,&\text{if}\quad r=1\\ \alpha_{2 opt}>0,&\text{if}\quad r>1\end{array}  \tag{27}\]  
For the case 1, when \(\
u_{a}\) is selected as the optimal frequency ratio, all optimal parameters of the model are obtained as  
\[ \begin{array}{l}\
u_{opt}=\sqrt{\frac{1+\beta}{2\sqrt{r}-r}}\\ \alpha_{1 opt}=2\sqrt{r}\\ \alpha_{2 opt}=-1+\sqrt{r}\\ \xi_{\text{opt}}\cong\left[\frac{r(1+\beta)^{2}}{4}\right]^{\frac{1}{4}}\end{array}  \tag{28}\]  
Similarly, for the case 2, when \(\
u_{b}\) is selected as the optimal frequency ratio, all optimal parameters of model are obtained as follows  
\[ \begin{array}{l}\
u_{opt}=\sqrt{\frac{1+\beta}{\sqrt{2r}-r}}\\ \alpha_{1 opt}=-2\sqrt{r}\\ \alpha_{2 opt}=-1+(1+\sqrt{2})\sqrt{r}\\ \xi_{\text{opt}}\cong\left[\frac{r(1+\beta)^{2}}{2}\right]^{\frac{1}{4}}\end{array}  \tag{29}\]  
Obviously, there are three possibilities for the value of \(\alpha_{2 opt}\), the grounded stiffness ratio can be negative, zero, or positive, namely  
\[ \begin{array}{l}\alpha_{2 opt}<0,&\text{if}\quad 0<r<3-2\sqrt{2}\\ \alpha_{2 opt}=0,&\text{if}\quad r=3-2\sqrt{2}\\ \alpha_{2 opt}>0,&\text{if}\quad r>3-2\sqrt{2}\end{array}  \tag{30}\]