## 3 Optimal Calibration of the TLIMD
### Dual-Equivalent Linearization for the Primary Damped SDOF System  
The original EOM for the primary damped SDOF system (shown in Figure 1a) can be written as  
\[\ddot{x}_{1}+{\omega_{1}}^{2}x_{1}+2\zeta_{1}\omega_{1}\dot{x}_{1}=0 \tag{7}\]  
where \(\omega_{1}\) and \(\zeta_{1}\) denote the natural frequency and damping ratio (defined in (5)), respectively. The EOM for the approximate un-damped SDOF system (shown in Figure 2) can be constructed as  
\[\ddot{x}_{1}+{\omega_{1e}}^{2}x_{1}=0 \tag{8}\]  
where \({\omega_{1e}}^{2}={k_{1e}}/{m_{1}}=\gamma+{\omega_{1}}^{2}\) is the natural frequency squared of the approximate un-damped SDOF system. The parameter \(\gamma\) indicates the contribution from the structural damping (of the primary damped SDOF system) to the equivalent stiffness term \(k_{1e}\) (of the approximate un-damped SDOF system), and it shall be determined by the following criterion [42]. \[P=\frac{1}{T}{\int_{0}^{T}}\left[(2\zeta_{1}\omega_{1}\dot{x}_{1}-\gamma x_{1} )^{2}+(\gamma x_{1}-2\lambda\zeta_{1}\omega_{1}\dot{x}_{1})^{2}\right]dt \tag{9}\]  
where \(T\) is a random period and \(\lambda\) is the so-called dual parameter. The coefficients \(\gamma\) and \(\lambda\) can be determined by  
\[\frac{\partial P}{\partial\gamma}=0,\ \frac{\partial P}{\partial\lambda}=0 \tag{10}\]  
where \((\partial/\partial)\) denotes the partial derivative operator. By inserting (9) into (10), a set of equations in terms of \(\gamma\) and \(\lambda\) can be developed:  
\[ \begin{array}{l}\gamma{\int_{0}^{T}}{x_{1}}^{2}dt-(\lambda+1){\int_{0}^{T}} \zeta_{1}\omega_{1}x_{1}\dot{x}_{1}dt=0\\ 2\lambda{\int_{0}^{T}}\zeta_{1}\omega_{1}\dot{x}_{1}^{2}dt-\gamma{\int_{0}^{T} }x_{1}\dot{x}_{1}dt=0\end{array}  \tag{11}\]  
Figure 2: The approximate un-damped SDOF system. The above equations lead to the solutions for \(\gamma\) and \(\lambda\), given as  
\[ \begin{split}\gamma&=2\zeta_{1}\omega_{1}\frac{\int_{0}^{ T}x_{1}\dot{x}_{1}dt}{\int_{0}^{T}x_{1}{}^{2}dt}\frac{\int_{0}^{T}x_{1}{}^{2}dt \int_{0}^{T}\dot{x_{1}}{}^{2}dt}{2\int_{0}^{T}x_{1}{}^{2}dt\int_{0}^{T}\dot{x_{ 1}}{}^{2}dt-(\int_{0}^{T}x_{1}\dot{x}_{1}dt)^{2}}\\ \lambda&=\frac{(\int_{0}^{T}x_{1}\dot{x}_{1}dt)^{2}}{2 \int_{0}^{T}x_{1}{}^{2}dt\int_{0}^{T}\dot{x_{1}}{}^{2}dt-(\int_{0}^{T}x_{1} \dot{x}_{1}dt)^{2}}\end{split}  \tag{12}\]  
By replacing the period parameter T with \(\Phi/\omega_{1e}\) (i.e., \(T=\Phi/\omega_{1e}\)), (12) can be rewritten as  
\[ \begin{split}\gamma&=2\zeta_{1}\omega_{1}\frac{\int _{0}^{\Phi}x_{1}\dot{x}_{1}d\varphi}{\int_{0}^{\Phi}x_{1}{}^{2}d\varphi}\frac{ \int_{0}^{\Phi}x_{1}{}^{2}d\varphi\int_{0}^{\Phi}\dot{x_{1}}{}^{2}d\varphi}{2 \int_{0}^{\Phi}x_{1}{}^{2}d\varphi\int_{0}^{\Phi}x_{1}{}^{2}d\varphi-(\int_{0} ^{\Phi}x_{1}\dot{x}_{1}d\varphi)^{2}}\\ \lambda&=\frac{(\int_{0}^{\Phi}x_{1}\dot{x}_{1}d \varphi)^{2}}{2\int_{0}^{\Phi}x_{1}{}^{2}d\varphi\int_{0}^{\Phi}\dot{x_{1}}{}^ {2}d\varphi-(\int_{0}^{\Phi}x_{1}\dot{x}_{1}d\varphi)^{2}}\end{split}  \tag{13}\]  
Further, the structure response corresponds to the approximate un-damped SDOF system (given in (8)) can be expressed by a general equation, given as  
\[x_{1}=a\cos(\omega_{1e}t+\varphi_{0}) \tag{14}\]  
where a is the amplitude and \(\varphi_{0}\) is a phase angle. Substituting (14) into (13) and using the definition of \(\omega_{1e}{}^{2}=\gamma+\omega_{1}{}^{2}\) leads to a quadratic equation of the equivalent natural frequency \(\omega_{1e}\), given as  
\[\omega_{1e}{}^{2}+\frac{2(1-\cos(2\Phi))(2\Phi-\sin(2\Phi))}{8\Phi^{2}-2\sin^ {2}(2\Phi)-(1-\cos(2\[PERSON]))^{2}}\zeta_{1}\omega_{1e}-\omega_{1}{}^{2}=0 \tag{15}\]  
Thus, the natural frequency \(\omega_{1e}\) of the approximate un-damped SDOF system, as shown in Figure 2, can be obtained, given by  
\[