# PTMD Control on a Benchmark TV Tower under Earthquake and Wind Load Excitations
## 3 Modeling of Structure Installed with PTMD  
Assuming the host structure has \(n\) degree of freedoms (DOFs), a controlled structure coupled with a PTMD has the equation of motion as:  
\[\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\dot{\mathbf{X}}+\mathbf{K}\mathbf{X}=\mathbf{E}\mathbf{F}+\mathbf{D}\mathbf{P} \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\) are the mass, damping, and stiffness matrices of the coupled system, which has \(n+1\) DOFs; \(\mathbf{F}\) is the excitation force and \(\mathbf{P}\) is the control force acting between the added mass and the host structure; matrices \(\mathbf{E}\) and \(\mathbf{D}\) are the index matrix indicating the DOFs of the excitation and the control forces, respectively. By reordering the DOFs of the controlled system, making \(n\) represent the DOF of the host structure connected with PTMD, and \(r\) to represent the DOF of the added mass, the coupled equation is rewritten as:  
\[\left[ \begin{array}{ccc}m_{1}&&\\ &\ddots&\\ &&m_{n}\\ &&m_{r}\\ \end{array} \right]\left[ \begin{array}{c}\ddot{x}_{1}\\ \vdots\\ \ddot{x}_{n}\\ \ddot{x}_{r}\\ \end{array} \right]+\left[ \begin{array}{ccc}c_{1}&\ldots&c_{1n}&0\\ \vdots&\ddots&\vdots&\vdots\\ c_{n1}&\ldots&c_{n}+c_{r}&-c_{r}\\ 0&\ldots&-c_{r}&c_{r}\\ \end{array} \right]\left[ \begin{array}{c}\dot{x}_{1}\\ \vdots\\ \vdots\\ \dot{x}_{n}\\ \dot{x}_{r}\\ \end{array} \right]+\left[ \begin{array}{c}k_{1}&\ldots&k_{1n}&0\\ \vdots&\ddots&\vdots&\vdots\\ k_{n1}&\ldots&k_{n}+k_{r}&-k_{r}\\ 0&\ldots&-k_{r}&k_{r}\\ \end{array} \right]\left[ \begin{array}{c}x_{1}\\ \vdots\\ \vdots\\ x_{n}\\ x_{r}\\ \end{array} \right]=\left[ \begin{array}{c}f_{1}\\ \vdots\\ f_{n}\\ f_{r}\\ \end{array} \right]+\left[ \begin{array}{c}0\\ \vdots\\ \vdots\\ p\\ -p\\ \end{array} \right] \tag{2}\]  
where \(m_{r}\) represents the mass of the PTMD, \(c_{r}\) and \(k_{r}\) are, respectively, the connecting damping and stiffness of the PTMD, and \(p\) is the bounding force between the attached mass and the VE layer. If decoupling the system, one is able to obtain the internal forces, which is the control force exerted on the host structure. Extracting the last line in Equation (2), the equation of motion referring to the dynamic response of added mass is expressed as:  
\[m_{r}\ddot{x}_{r}-c_{r}\dot{x}_{n}+c_{r}\dot{x}_{r}-k_{r}x_{n}+k_{r}x_{r}=f_{r }-p \tag{3}\]  
Figure 1: Schematic model and prototype of the bounding tuned mass damper (PTMD). By moving those terms related to \(x_{n}\) to the right side of the equation yields the governing equation of the added mass as:  
\[m_{r}\ddot{x}_{r}+c_{r}\dot{x}_{r}+k_{r}x_{r}=k_{r}x_{n}+c_{r}\dot{x}_{n}+f_{r}-p \tag{4}\]  
and \(k_{r}x_{n}+c_{r}\dot{x}_{n}\) in above equation can be considered as the external force applied to the PTMD mass. Further extracting those equations corresponding to the original system from Equation (2), the governing equation of the system is rewritten as:  
\[\left[ \begin{array}{cc}\mathbf{M}_{n}&0\end{array} \right]\left[ \begin{array}[ ]{c}\dot{\mathbf{X}}_{n}\\ \dot{x}_{r}\end{array} \right]+\left(\left[ \begin{array}{cc}\mathbf{C}_{n}&0 \end{array} \right]+\left[ \begin{array}{cc}0&\cdots&0&0\\ \vdots&\ddots&\vdots&\vdots\\ 0&\ldots&c_{r}&-c_{r}\end{array} \right]\right)\left[ \begin{array}{c}\dot{ \mathbf{X}}_{n}\\ \dot{x}_{r}\end{array} \right]+\left(\left[ \begin{array}{cc}\mathbf{K}_{n}&0 \end{array} \right]+\left[ \begin{array}{cc}0&\cdots&0&0\\ \vdots&\ddots&\vdots&\vdots\\ 0&\ldots&k_{r}&-k_{r}\end{array} \right]\right)\left[ \begin{array}{c} \mathbf{X}_{n}\\ x_{r}\end{array} \right]=\mathbf{F}+\left[ \begin{array}{c}0\\ p\end{array} \right] \tag{5}\]  
where \(\mathbf{K}_{n}\), \(\mathbf{M}_{n}\), and \(\mathbf{C}_{n}\), respectively, represent the stiffness, mass, and damping matrices of the host structure. Reconstruction of the above equation yields:  
\[\mathbf{M}_{n}\ddot{\mathbf{X}}_{n}+\mathbf{C}_{n}\dot{\mathbf{X}}_{n}+\mathbf{ K}_{n}\mathbf{X}_{n}=\mathbf{F}+\mathbf{C}_{d}\left[ \begin{array}{c}\dot{\mathbf{X}}_{n} \\ \dot{x}_{r}\end{array} \right]+K_{d}\left[