# Equivalent Linearization Methods for a Control System with Clutching Inerter Damper
## 3 ELM1 Based on the Adaptation of FRF
### Equivalent Damping  
Considering one circle of free vibration, the elastic potential energy of the structure decreases due to the decay of the displacement response. During the circle, when the structure attains its peak velocity, the CID reaches its peak kinetic energy and will be separated from the whole system at the next moment. The damping of the structure also dissipates a certain amount of energy. However, the kinetic energy of the structure does not change during the circle.  
For the CID system, taking the first circle in Figure 4 as an example, the system was given an initial displacement \(x_{0}\) when \(t=0\). The energy during the circle is balanced. That is  
\[\Delta E_{s}=\Delta E_{k}^{CID}+\Delta E_{d}^{s} \tag{35}\]  
\[\left\{\begin{array}{c}\Delta E_{s}=\frac{1}{2}k\big{(}x_{0}{}^{2}(t)-x_{4} {}^{2}(t)\big{)}\\ \Delta E_{CID}=\frac{1}{2}\mu\ m\big{(}\dot{x_{1}}{}^{2}(t)+\dot{x_{3}}{}^{2}( t)\big{)}\\ \Delta E_{d}^{s}=\int_{0}^{t_{4}}c\dot{x}{}^{2}(t)dt\end{array}\right. \tag{36}\]  
where \(\Delta E_{s}\) represents the reduced elastic potential energy; \(\Delta E_{k}^{CID}\) is the kinetic energy obtained by the CID; \(\Delta E_{d}^{s}\) is the damping energy dissipated by the structure; \(x_{0}(t)\), \(x_{4}(t)\) and \(\dot{x}_{1}(t)\), \(\dot{x}_{3}(t)\) represent the displacement at time 0, \(t_{4}\) and the velocity at time \(t_{1}\), \(t_{3}\) respectively.  
For the ELS, the kinetic energy of the system also does not change during one circle. Thus, the energy equation in the ELS during a circle becomes  
\[\Delta E_{s}=\Delta E_{d}^{ELS}\ =\Delta E_{d}^{s}+\Delta E_{d}^{CID} \tag{37}\]  
where \(\Delta E_{s}\) represents the reduced elastic potential energy, and \(\Delta E_{d}^{ELS}\) is the damping energy dissipated in the ELS, which consists of the damping energy dissipated by the structure, namely \(\Delta E_{d}^{s}\), and the equivalent damping of the CID, namely \(\Delta E_{d}^{CID}\).  
From Equations (35) and (37), the following relationship can be obtained  
\[\Delta E_{k}^{CID}=\Delta E_{d}^{CID} \tag{38}\]  
which indicates that the energy dissipated by equivalent damping \(c_{eq}\) is equal to the kinetic energy obtained by the CID. Equation (38) can be expanded as  
\[\frac{1}{2}\mu\ m\Big{(}\dot{x_{1}}{}^{2}(t)+\dot{x_{3}}{}^{2}(t)\Big{)}=c_{ eq}\int_{0}^{t_{4}}\dot{x}{}^{2}(t)dt \tag{39}\]  
As shown in Figure 5, the area enclosed by solid lines and the coordinate axes is equal to the area enclosed by the dotted lines and coordinate axes for a sine curve. Therefore,  
\[\int_{0}^{t_{4}}\dot{x}{}^{2}(t)dt=\frac{1}{2}\bigg{(}\frac{T_{CID}}{2}\Big{(} \dot{x_{1}}{}^{2}(t)+\dot{x_{3}}{}^{2}(t)\Big{)}\bigg{)} \tag{40}\]Substituting Equation (40) into Equation (39), the equivalent damping \(c_{eq}\) is expressed by  
\[c_{eq}=\frac{2\mu m}{T_{CID}} \tag{41}\]  
Substituting Equation (30) into Equation (41), the equivalent damping \(c_{eq}\) is finally obtained:  
\[c_{eq}=\frac{2\mu\sqrt{mk}}{\pi\big{(}1+\sqrt{1+\mu}\big{)}} \tag{42}\]  
Substituting Equation (42) and Equation (34) into Equation (15), the equivalent damping ratio is obtained:  
\[\xi_{eq}=\frac{2\mu}{\pi\big{(}1+\sqrt{1+\mu}\big{)}^{2}}+\frac{2\xi}{\big{(}1 +\sqrt{1+\mu}\big{)}} \tag{43}\]