## 2 Dynamic Model and Parameters Optimization
### The Analytical Solution  
We introduce the following parameter transformations  
\[\omega_{1}=\sqrt{\frac{k_{1}}{m_{1}}},\,\omega_{2}=\sqrt{\frac{k_ {2}}{m_{2}}},\,\ddot{\xi}=\frac{c}{2m_{2}\omega_{2}},\,\mu=\frac{m_{2}}{m_{1}}\] \[\alpha_{1}=\frac{k_{3}}{k_{2}},\,\alpha_{2}=\frac{k_{4}}{k_{2}}, \,f=\frac{F}{m_{1}},\,\beta=\frac{b}{m_{2}}\]  
where \(\omega_{1}\) and \(\omega_{2}\) represent the natural frequencies of the primary system and the DVA, respectively. \(\xi\) represents the damping ratio of the DVA. \(\mu\) represents the mass ratio of the DVA to the primary system. \(\alpha_{1}\) and \(\alpha_{2}\) represent the corresponding stiffness ratios. \(f\) represent the ratio of the external excitation amplitude to the mass of the primary system. \(\beta\) represent the ratio of the inerter coefficient to the mass of the DVA.  
Equation (1) becomes  
\[\dot{x}_{1}+\omega_{1}^{2}x_{1}+\mu L\omega_{2}^{2}(Lx_{1}-x_{2} )=f\cos{(\omega t)}\] \[(1+\beta)\ddot{x}_{2}+\omega_{2}^{2}(x_{2}-Lx_{1})+2\xi\omega_{2 }(\dot{x}_{2}-\dot{x}_{3})+\alpha_{2}\omega_{2}^{2}x_{2}=0 \tag{2}\] \[2\xi\omega_{2}(\dot{x}_{3}-\dot{x}_{2})+\alpha_{1}\omega_{2}^{2 }x_{3}=0\]  
Suppose the form of steady state solutions is  
\[x_{i}=X_{i}\mathrm{e}^{i\omega t},\quad i=1,2,3 \tag{3}\]  
Figure 1: The viscoelastic Maxwell-type DVA model with lever, inerter, and grounded stiffness.  
Substituting Equation (3) into Equation (2) yields  
\[X_{1}=\frac{f(B_{2}C_{3}-B_{3}C_{2})}{\det\Bigl{(}\sum_{i=1}^{3}\big{(}\partial_{1,i}^{3,3}(A_{i})+\partial_{2,i}^{3,3}(B_{i})+\partial_{3,i}^{3,3}(C_{i})\big{)} \Bigr{)}} \tag{4}\]  
where the symbol \(\partial_{p,\,q}^{m,\,n}(\mathbf{M})\) represents \(m\times n\) block matrix, with \((p\,,\ q)\)-th block \(M\), and all other blocks are zero matrices [45]. We have  
\[A_{1} =-\omega^{2}+\omega_{1}^{2}+\mu L^{2}\omega_{2}^{2},\quad A_{2}=- \mu L\omega_{2}^{2},\quad A_{3}=C_{1}=0\] \[B_{1} =-\omega_{2}^{2}L,\quad B_{2}=-(1+\beta)\omega^{2}+2j\omega \omega_{2}\xi+(1+\alpha_{2})\omega_{2}^{2} \tag{5}\] \[B_{3} =C_{2}=-2j\omega\omega_{2}\xi,\quad C_{3}=2j\omega\omega_{2}\xi+ \alpha_{1}\omega_{2}^{2}\]  
For the convenience of derivation, we introduce the following parameters  
\[\lambda=\frac{\omega}{\omega_{1}},\quad\
u=\frac{\omega_{2}}{\omega_{1}},\quad \delta_{st}=\frac{F}{k_{1}}\]  
The amplitude amplification factor \(A\) of the primary system can be written as  
\[A=\left|\frac{X_{1}}{\delta_{st}}\right|=\sqrt{\frac{D_{1}^{2}+D_{2}^{2}\xi^{ 2}}{D_{3}^{2}+D_{4}^{2}\xi^{2}}} \tag{6}\]  
where  
\[D_{1} =\alpha_{1}\
u[-(1+\beta)\lambda^{2}+(1+\alpha_{2})\
u^{2}],\quad D _{2}=2\lambda[h_{1}\
u^{2}-(1+\beta)\lambda^{2}]\] \[D_{3} =\alpha_{1}\
u\Bigl{\{}(1+\beta)\lambda^{4}-[(1+\beta)\Bigl{(}1+ \mu\
u^{2}L^{2}\Bigr{)}+(1+\alpha_{2})\
u^{2}]\lambda^{2}+h_{2}\Bigr{\}}\] \[D_{4} =2\lambda[(1+\beta)\lambda^{4}-\Bigl{(}1+\beta+h_{3}\
u^{2} \Bigr{)}\lambda^{2}+h_{1}\
u^{2}+\mu(\alpha_{1}+\alpha_{2})\
u^{4}L^{2}]\] \[h_{1} =1+\alpha_{1}+\alpha_{2},\quad h_{2}=(1+\alpha_{2})\
u^{2}+\alpha _{2}\mu\
u^{4}L^{2}\] \[h_{3} =h_{1}+r,\quad r=\mu(1+\beta)L^{2} \tag{7}\]  
By setting the undamped denominator part of Equation (6) equal to zero, the dimensionless natural frequencies \(\Omega_{1}\) and \(\Omega_{2}\) of the coupled system can be obtained as  
\[\Omega_{1,2}=\frac{\sqrt{2}}{2}\frac{1}{\sqrt{1+\beta}}\sqrt{ \Phi\mp\sqrt{\Phi^{2}-\Psi}}\] \[\Phi=\omega_{1}^{2}(1+\beta)+\omega_{2}^{2}(1+\alpha_{2}+r) \tag{8}\] \[\Psi=4\omega_{2}^{2}(1+\beta)[(1+\alpha_{2})\omega_{1}^{2}+\mu L ^{2}\alpha_{2}\omega_{2}^{2}]\]  
As can be seen from the Equation (8), the natural frequencies of the coupled system are affected by the stiffness ratio \(\alpha_{2}\). The inappropriate stiffness ratio \(\alpha_{2}\) will make the natural frequencies imaginary numbers, which further leads to the instability of the coupled system.