# Time Domain Damage Identification Approach of the Newly Hollow Floor Structure with TRMD
## 2 Theoretical Formulation  
### Simplified Model and Motion Equation of a Controlled Structure with TRMD  
The model of the single-degree-of-freedom structure with TRMD is shown in Figure 2, where \(M\), \(K\), and \(C\) are the mass, stiffness, and damping of the main structure, respectively; \(m\) is the ball mass of the TRMD oscillator; and \(F(t)\) is the external ambient excitation. In the equation \(F(t)=(M+x)\tilde{x}_{g}\), \(\tilde{x}_{g}\) is the acceleration of the external excitation. The motion model of the TRMD oscillator is shown in Figure 3, where \(R\) and \(r\) are the radii of the arc path and oscillator, respectively, \(\theta\) and \(\psi\) are the rotation angles of the oscillator with respect to the center of the path and of the oscillator ball itself, and \(\omega=\dot{\psi}\) represents the angular velocity of the oscillator. The equations for the motion of the controlled system with TRMD can be derived by using [PERSON]'s equation, and a detailed derivation of the equation is proposed by the authors [10]:  
\[\frac{\mathrm{d}}{\mathrm{d}t}(\frac{\partial T}{\partial q_{i}})-\frac{ \partial T}{\partial q_{i}}+\frac{\partial V}{\partial q_{i}}=Q_{i}^{nc},i=1,2 \tag{1}\]  
where \(T\) and \(V\) donate the kinetic energy and potential energy of the controlled structure, respectively, \(q_{i}\) is the _ith_ generalized coordinate, and \(Q_{i}^{,nc}\) is the nonconservative force corresponding to the generalized coordinate \(q_{i}\). The kinetic energy of the controlled structure \(T\) includes the kinetic energy of the main structure, the translational motion of the TRMD oscillator, and the kinetic energy of rotation around its own spherical center. Therefore, the kinetic energy and potential energy of the controlled structure can be expressed as the following:  
\[T=\frac{1}{2}M\dot{x}^{2}+\frac{1}{2}m(\dot{x}+\rho\dot{\theta}\cos\theta)^{2}+ \frac{1}{2}m(\rho\dot{\theta}\sin\theta)^{2}+\frac{1}{5}m\Big{(}\rho\dot{ \theta}\Big{)}^{2} \tag{2}\]  
\[V=\frac{1}{2}Kx^{2}+mg\rho(1-\cos\theta) \tag{3}\]  
where \(x\), \(\dot{x}\), and \(\ddot{x}\) are the displacement, velocity, and acceleration response of the main structure. \(\rho=R-r\) represents the radius difference between the arc track and oscillator.  
It is assumed that the arc track surface is smooth without considering the rolling friction of the oscillator. Based on the virtual work principle, the virtual work generated by nonconservative forces, including external forces and the damping force, can be expressed as  
\[\delta W=(F(t)-C\dot{x})\delta x \tag{4}\]  
Considering the angular motion of the oscillator \(\theta\) is very small, the motion equation of the TRMD controlled structure can be written as:  
\[(M+m)\ddot{x}+m\rho\ddot{\theta}+C\dot{x}+Kx=F(t) \tag{5}\]  
\[\ddot{\theta}+\frac{5\ddot{x}}{7\rho}+\frac{5g\theta}{7\rho}=0 \tag{6}\]  
Equations (5) and (6) can be rewritten in a more compact form:  
\[\begin{bmatrix}M+m&m\rho\\ \frac{5}{7\rho}&1\end{bmatrix}\begin{Bmatrix}\ddot{x}\\ \ddot{\theta}\end{Bmatrix}+\begin{bmatrix}C&0\\ 0&0\end{bmatrix}\begin{Bmatrix}\dot{x}\\ \dot{\theta}\end{Bmatrix}+\begin{bmatrix}K&0\\ 0&\frac{5g}{7\rho}\end{bmatrix}\begin{Bmatrix}x\\ \theta\end{Bmatrix}=\begin{Bmatrix}F(t)\\ 0\end{Bmatrix} \tag{7}\]  
From Equation (6), it is shown that the natural frequency of the oscillator, which can be calculated as \(\omega_{d}=\sqrt{\frac{5g}{7\rho}}\), depends on the radius difference between the arced track and the oscillator. In order to obtain the optimal control effect, the natural frequency of the TRMD is tuned near the first modal frequency of the main structure, which is similar to the vibration control of the traditional TMD [21].