# Equivalent Linearization Methods for a Control System with Clutching Inerter Damper
## 3 ELM1 Based on the Adaptation of FRF  
In this section, an ELM for a SDOF system with CID is proposed based on the adaptation of FRF of the CID system and an equivalent linear system (ELS). As shown in Figure 3, the CID can reduce not only the frequency of the structure but also its peak response. Therefore, it is reasonable to consider that the CID is equal to a combination of an IMD and a viscous damper.  
The equation of motion of the ELS can be written as  
\[\left(m+b_{eq}\right)\ddot{x}+\left(c+c_{eq}\right)\dot{x}+kx=-m\ddot{x}_{g} \tag{11}\]  
in which \(b_{eq}\) and \(c_{eq}\) are the equivalent inertance and equivalent damping of the CID, respectively. The FRF of the ELS is given by  
\[H(\Omega)=\frac{m}{k-\left(m+b_{eq}\right)\Omega^{2}+i\Omega\left(c+c_{eq} \right)} \tag{12}\]  
The amplitude of \(H(\Omega)\) written as \(|H(\Omega)|\) is given by  
\[|H(\Omega)|=\frac{1}{1+\mu_{eq}}\frac{1}{\sqrt{\left(\omega_{eq}^{2}-\Omega^{ 2}\right)^{2}+\left(2\xi_{eq}\Omega\omega_{eq}\right)^{2}}} \tag{13}\]  
where \(\omega_{eq}\) and \(\xi_{eq}\) are defined as  
\[\omega_{eq}=\sqrt{\frac{k}{m+b_{eq}}}=\frac{1}{\sqrt{1+\mu_{eq}}}\omega \tag{14}\]  
\[\xi_{eq}=\frac{c+c_{eq}}{2\sqrt{k\left(m+b_{eq}\right)}}=\left(1+\frac{c_{eq} }{c}\right)\frac{1}{\sqrt{1+\mu_{eq}}}\xi \tag{15}\]  
According to the theory of resonance reaction, \(|H(\Omega)|\) reaches the peak at the frequency  
\[\Omega=\omega_{eq}\sqrt{1-2\xi_{eq}^{2}} \tag{16}\]  
Substituting Equation (16) into Equation (13), the maximum amplitude of \(|H(\Omega)|\) is given by  
\[|H(\Omega)|_{max}=\frac{1}{\omega^{2}}\frac{1}{2\xi_{eq}\sqrt{1-\xi_{eq}^{2}}} \tag{17}\]Considering Figure 3b, the ordinate is the steady-state displacement amplitude of the CID systems with different inertance to mass ratios under sinusoidal input of unit amplitude. The steady-state reaction of the ELS under the input of \(\sin(\Omega t)\) can be expressed as  
\[x_{p}(\Omega,t)=\left|H(\Omega)\right|\sin(\Omega t-\theta) \tag{18}\]  
\[x_{p}(\Omega,t)_{\max}=\left|H(\Omega)\right| \tag{19}\]  
\[U_{\max}=\left|H(\Omega)\right|_{\max} \tag{20}\]  
where \(x_{p}(\Omega,t)\) is the steady-state displacement response under the input of \(\sin(\Omega t)\); \(\theta\) is the phase angle with which the response lags behind the applied loading; \(x_{p}(\Omega,t)_{\max}\) is the maximum steady-state displacement response, which is equal to the amplitude of \(H(\Omega)\); and \(U_{\max}\) is the resonance response, which is equal to the peak of \(\left|H(\Omega)\right|\).  
For any \(\mu\) in Figure 3b, \(U_{\max}\) and its corresponding \(\Omega\) is easy to find. Substituting them into Equation (16), Equation (17) and Equation (20), the equivalent linearization parameters are obtained:  
\[\xi_{eq}=sin(\frac{1}{2}arcsin(\frac{1}{\omega^{2}U_{\max}})) \tag{21}\]  
\[\omega_{eq}=\frac{\Omega}{\sqrt{1-2\xi_{eq}^{2}}} \tag{22}\]  
\[\mu_{eq}=\left(\frac{\omega}{\omega_{eq}}\right)^{2}-1, \tag{23}\]  
\[b_{eq}=\mu_{eq}m=\left(\left(\frac{\omega}{\omega_{eq}}\right)^{2}-1\right)m \tag{24}\]  
After substituting \(\xi_{eq}\) and \(\mu_{eq}\) into Equation (15), \(c_{eq}\) is finally obtained:  
\[c_{eq}=c\bigg{(}\frac{\xi_{eq}}{\xi}\sqrt{1+\mu_{eq}}-1\bigg{)} \tag{25}\]  
**4. ELM2 Based on Period and Energy**  
In this section, an ELM based on the equal decay period and conservation of energy in one circle of free vibration is discussed. This method was discussed by [PERSON] et al. (2016). In this paper, the principle to specify the equivalent damping is revised and better results can be obtained.