## 3 Optimal Calibration of the TLIMD
### \(H_{2}\) Norm Optimization  
Similar to (5), a few parameters for the approximate un-damped SDOF system are introduced, including the frequency ratio \(\overline{\beta}\) and the damping ratio \(\overline{\zeta}_{2}\). Two more non-dimensional parameters, i.e., the external excitation frequency \(\overline{\phi}\) and the time parameter \(\overline{\tau}\) are also included, defined as  
\[\overline{\beta}=\frac{\omega_{2}}{\omega_{1e}},\ \overline{\zeta}_{2}=\frac{c_{2}}{2 m_{2}\omega_{1e}},\ \overline{\phi}=\frac{\omega}{\omega_{1e}},\ \overline{\tau}=\omega_{1e}t \tag{17}\]They are connected to the parameters of the primary damped SDOF system, given in (5) through the following relations:  
\[\beta=\overline{\beta}coe,\ \zeta_{2}=\overline{\zeta}_{2}coe,\ \phi=\overline{\phi}coe,\ \tau=\frac{\overline{\tau}}{coe} \tag{18}\]  
Thus, the EOMs for the primary analogue TLIMD-SDOF system with the damped SDOF system represented by an approximate un-damped one, i.e., the approximate TLIMD-SDOF system, can be written as  
\[ \begin{array}{l}\tilde{z}_{1}+z_{1}+\mu\overline{\beta}^{2}(z_{1}-z_{2})+2 \overline{\zeta}_{2}u\overline{\beta}(\dot{z}_{1}-\dot{z}_{2})=-e^{i\overline{ \phi}\overline{\tau}}\\ \left[\alpha_{1}^{2}+\frac{1}{3}u_{32}(1-\alpha_{2}+\alpha_{2}^{2})\right]\dot{ z}_{2}+\overline{\beta}^{2}(z_{2}-z_{1})+2\overline{\zeta}_{2}\overline{\beta}( \dot{z}_{2}-\dot{z}_{1})=\alpha_{1}e^{i\overline{\phi}\overline{\tau}}+\frac{ 1}{2}u_{32}(\alpha_{2}-1)e^{i\overline{\phi}\overline{\tau}}\end{array}  \tag{19}\]  
Note that (19) and (6) possess a unified expression, i.e., (19) can be directly obtained from (6) by ignoring the structural damping \(\zeta_{1}=0\) with substitutions of the frequency ratio \(\beta=\overline{\beta}\), the damping ratio \(\zeta_{2}=\overline{\zeta}_{2}\), the excitation frequency \(\phi=\overline{\phi}\) and the time parameter \(\tau=\overline{\tau}\). The structural responses in (19) can be expressed as  
\[z_{1}=z_{1m}e^{i\overline{\phi}\overline{\tau}},\ z_{2}=z_{2m}e^{i\overline{ \phi}\overline{\tau}} \tag{20}\]  
By inserting (20) into (19), the complex amplitudes for the host structure \(z_{1m}\) and the lever \(z_{2m}\) can be obtained, given as  
\[z_{1m}=\frac{A_{1}+B_{1}\overline{\zeta}_{2}i}{C+D_{52}^{5}z^{2}i},\ z_{2m}= \frac{A_{2}+B_{2}\overline{\zeta}_{2}i}{C+D_{52}^{5}z^{2}i} \tag{21}\]  
where  
\[ \begin{array}{l}A_{1}=\overline{\beta}^{2}(1-\mu\mu_{2})-\mu_{1}\overline{ \phi}^{2},\ B_{1}=2\overline{\beta}\overline{\phi}(1-\mu\mu_{2})\\ A_{2}=\mu_{2}\left(\overline{\phi}^{2}-1\right)+\overline{\beta}^{2}(1-\mu\mu_ {2}),\ B_{2}=2\overline{\beta}\overline{\phi}(1-\mu\mu_{2})\\ C=\mu_{1}\overline{\varphi}^{2}\left(1-\overline{\phi}^{2}+\overline{\beta}^{2 }\mu\right)-\overline{\beta}^{2}\left(1-\overline{\phi}^{2}\right),\ D=2 \overline{\beta}\overline{\phi}\left(\overline{\phi}^{2}+\mu\mu_{1}\overline {\phi}^{2}-1\right)\\ \mu_{1}=\alpha_{1}^{2}+\frac{1}{3}u_{32}(1-\alpha_{2}+\alpha_{2}^{2}),\ \mu_{2}= \alpha_{1}+\frac{1}{2}u_{32}(\alpha_{2}-1)\end{array}  \tag{22}\]  
For the white noise external excitation with a constant spectral density \(S_{0}\) applied on the host structure, the variance of the host structure response can be obtained from the integral operation:  
\[\sigma_{z_{1}}^{2}=\int_{-\infty}^{+\infty}\left|z_{1m}\right|^{2}S_{0}d\omega =S_{0}\omega_{1}\int_{-\infty}^{+\infty}\left|z_{1m}\right|^{2}d\phi \tag{23}\]  
The performance index is defined by  
\[I_{H}=\frac{\sigma_{z_{1}}^{2}}{2\pi}=\frac{1}{2\pi}\int_{-\infty}^{+\infty} \left|z_{1m}\right|^{2}d\phi \tag{24}\]  
where the integral part inside (24) can be calculated by  
\[\int_{-\infty}^{+\infty}\left|z_{1m}\right|^{2}d\phi=\int_{-\infty}^{+\infty} \frac{\psi(\phi)}{\Gamma(\phi)\Gamma^{*}(\phi)}d\phi=\frac{\pi}{a_{4}}\frac{ \mathrm{Det}[G]}{\mathrm{Det}[F]} \tag{25}\]where  
\[