## 2 Theoretical Background  
The structure of a PSPP can be regarded as a multiple-degree-of-freedom (MDOF) system, and the governing equation of the structure under external excitation is as follows:  
\[[M]\Big{[}\ddot{x}\Big{]}+[C]\Big{[}\dot{x}\Big{]}+[K]x=-[M][I]\ddot{x}_{g} \tag{1}\]  
where \([M]\), \([C]\), and \([K]\) are the mass matrix, decoupled damping matrix, and stiffness matrix of the structure, respectively; \(\{x\}\) is the displacement vector; \(\{l\}\) is the load indicator vector, where the element along the vibration direction is 1 and the rest is 0; and \(\ddot{x}_{g}\) is the ground acceleration. For a damping structure TMD system with N DOF and a TMD placed on the \(j\)th floor of the structure, the dynamic equation can be written as follows:  
For a general \(i\)th floor mass, assuming that the TMD is installed at a certain position of the structure, the corresponding is the \(i\)th degree of freedom of the structure. When the TMD is acting, the equation of motion of the structure is:  
\[M_{i}\ddot{x}_{i}+C_{i}\dot{x}_{i}+K_{i}x_{i}-\delta_{ij}\Big{\{}c_{t}\Big{(} \dot{x}_{t}-\dot{x}_{i}\Big{)}+k_{t}(x_{t}-x_{i})\Big{\}}=-M_{i}\ddot{x}_{g} \tag{2}\]  
where \(\delta_{ij}=\left\{ \begin{array}{ll}0&\quad i\
eq j\\ 1&\quad i=j\end{array} \right.\) and the equation of motion of the TMD is:  
\[m_{t}\ddot{x}_{t}+c_{t}\Big{(}\dot{x}_{t}-\dot{x}_{i}\Big{)}+k_{t}(x_{t}-x_{i })=-m_{t}\ddot{x}_{g} \tag{3}\]  
where \(M_{i}\), \(C_{i}\), and \(K_{i}\) are the mass, damping, and stiffness of the \(i\)th floor of the structure, respectively; \(m_{t}\), \(c_{t}\), and \(k_{t}\) are the mass, damping, and stiffness of the TMD; \(x_{t}\) is the displacement of the TMD; and \(x_{i}\) is the displacement of the \(i\)th floor of the structure. For high-frequency vibration of the plant structure, the high-frequency excitation causes the \(r\)th mode vibration of the structure, and the TMD is designed to be the \(r\)th mode of the control structure. Assuming that the modal mass of the \(r\)th-order main mode is \(M_{r}\), the modal stiffness is \(K_{r}\), and the modal damping is \(C_{r}\), the relevant equations of motion can be written as [26]:  
\[M_{r}\ddot{q}_{r}+C_{r}\dot{q}_{r}+K_{r}q_{r}-\phi_{rj}\Big{\{}c_{t}\Big{(} \dot{x}_{t}-\dot{x}_{j}\Big{)}+k_{t}\Big{(}x_{t}-x_{j}\Big{)}\Big{\}}=-\Phi_{r} M_{r}\ddot{x}_{g} \tag{4}\]  
\[m_{t}\ddot{x}_{t}+c_{t}\Big{(}\dot{x}_{t}-\dot{x}_{j}\Big{)}+k_{t}\Big{(}x_{t} -x_{j}\Big{)}=-m_{t}\ddot{x}_{g} \tag{5}\]Then the design problem is essentially similar to that of designing a TMD for a single-degree-of-freedom (SDOF) structure. Relative to the SDOF structure TMD system, there are the terms \(q_{r}\) and \(\phi_{rj}\) in the equation. However, if the \(r\)th mode shape vector of the structure is normalized with respect to its \(j\)th element, which corresponds to the TMD location, \(\phi_{rj}\) becomes unity and \(x_{j}=\phi_{rj}\ q_{r}=q_{r}\). If \(\phi_{rj}\) is unity, the expressions for calculating the \(r\)th modal response for the MDOF structure TMD system will be exactly the same as those for responses in the SDOF structure TMD system. In this way, the vibration control of higher modes for the powerhouse structure can be simplified to the TMD design problem of the corresponding modal coordinates. Closed-form expressions of optimum damper parameter \(\gamma_{\text{opt}}\) were developed by [PERSON] [27] and [PERSON] [28], which minimize the steady-state response of the main mass subjected to a harmonic main mass excitation. The expressions for calculating the optimum damper parameters are given as:  
\[\gamma_{\text{opt}}=\sqrt{\frac{1}{1+\mu}} \tag{6}\]  
\[\zeta_{\text{opt}}=\sqrt{\frac{3\mu}{4(2+\mu)(1+\mu)}} \tag{7}\]  
where \(\gamma_{opt}=\omega_{t}/\omega_{r}\) denotes the optimal ratio of TMD frequency to the structure's r-order frequency; \(\zeta_{\text{opt}}\) denotes the optimal TMD damping; and \(\mu=m_{t}/M_{r}\) denotes the mass ratio of the TMD to the structure's r-order modal mass.