# A Multi-Degree of Freedom Tuned Mass Damper Design for Vibration Mitigation of a Suspension Bridge
## 2 Formulation of the Bridge-TMD System
\[u_{2}=k_{2}\Big{(}x_{1}-B_{2}^{\rm T}\mathbf{X}+\theta_{1}d_{1}\Big{)}+c_{2}\Big{(} \dot{x}_{1}-B_{2}^{\rm T}\dot{\mathbf{X}}+\dot{\theta}_{1}d_{1}\Big{)}, \tag{6}\]  
where \(\mathbf{X}\) is the vector of global coordinates of the finite element model, \(\mathbf{B}_{2}\) and \(\mathbf{B}_{3}\) are the input vector of this two TMDs, respectively. The equations can govern the vibration of the coupled system can be decomposed into:  
\[m_{d1}\ddot{x}_{1}=-u_{1}-u_{2}-u_{3}, \tag{7}\]  
\[m_{d2}\ddot{x}_{2}=-u_{4}-u_{5}-u_{6}, \tag{8}\]  
\[I_{d1}\ddot{\theta}_{1}=u_{1}d_{1}-u_{2}d_{1}, \tag{9}\]  
\[I_{d2}\ddot{\theta}_{2}=u_{4}d_{2}-u_{5}d_{2}, \tag{10}\]  
The governing equations can then be written as  
\[\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\dot{\mathbf{X}}+\mathbf{K}\mathbf{X}=\mathbf{B}_{1}f_{d}+\mathbf{B}_{2}( u_{1}+u_{2}+u_{3})+\mathbf{B}_{3}(u_{4}+u_{5}+u_{6}), \tag{11}\]  
where \(f_{d}\) is the external disturbances, \(\mathbf{B}_{1}\) is the input vector of the external disturbances. We can express Equations (7)-(11) in matrix form as:  
\[\left[ \begin{array}{cccc}\mathbf{M}&0&0&0&0\\ 0&m_{d1}&0&0&0\\ 0&0&m_{d2}&0&0\\ 0&0&0&I_{d1}&0\\ 0&0&0&0&I_{d2}\\ \end{array} \right]+\left[ \begin{array}{cccc}\mathbf{C}&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{array} \right]+\left[ \begin{array}{cccc}\mathbf{K}&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{array} \right]\left[ \begin{array}{cccc}\mathbf{X}\\ x_{1}\\ x_{2}\\ \theta_{1}\\ \theta_{2}\\ \end{array} \right] \tag{12}\]  
\[=\left( \begin{array}{cccc}\mathbf{B}_{1}\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{array} \right)f_{d}+\left[ \begin{array}{cccc}\mathbf{B}_{2}&\mathbf{B}_{2}&\mathbf{B }_{2}&\mathbf{B}_{3}&\mathbf{B}_{3}&\mathbf{B}_{3}&\mathbf{B}_{3}\\ -1&-1&-1&0&0&0\\ 0&0&0&-1&-1&-1\\ d_{1}&-d_{1}&0&0&0&0\\ 0&0&0&d_{2}&-d_{2}&0\\ \end{array} \right]\left( \begin{array}{cccc}\mathbf{u}_{1}\\ u_{2}\\ u_{3}\\ u_{4}\\ u_{5}\\ u_{6}\\ \end{array} \right)\]  
or  
\[\mathbf{M}_{p}\ddot{\mathbf{p}}+\mathbf{C}_{p}\dot{\mathbf{p}}+\mathbf{K}_{p}\mathbf{p}=\mathbf{B}_{f}f_{d }+\mathbf{B}_{u}\mathbf{u}, \tag{13}\]  
where \(\mathbf{p}=[\mathbf{X},\,x_{1},\,x_{2},\,\theta_{1},\,\theta_{2}]^{\rm T}\) and \(\mathbf{u}=[u_{1},\,u_{2},\,\ldots,\,u_{6}]\) and T denote the complex conjugate matrix transpose. The matrices \(\mathbf{M}_{p}\), \(\mathbf{C}_{p}\), \(\mathbf{K}_{p}\), \(\mathbf{B}_{f}\), and \(\mathbf{B}_{u}\) can be obtained from Equation (12) directly. Defining the state variables of the system as:  
\[\mathbf{x}=\left[ \begin{array}{c}\mathbf{p}\\ \dot{\mathbf{p}}\\ \end{array} \right], \tag{14}\]  
The governing equations are written in first-order form as:  
\[\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}_{11}\mathbf{w}+\mathbf{B}_{12}\mathbf{u}, \tag{15}\]  
where \(\mathbf{w}=f_{d}\) and:  
\[\mathbf{A}=\left[ \begin{array}{cc}0&\mathbf{I}\\ -\mathbf{M}_{p}^{-1}&\mathbf{K}_{p}\\ \end{array} \right],\quad\mathbf{B}_{11}=\left[ \begin{array}{c}0\\ M_{p}^{-1}\mathbf{B}_{f}\\ \end{array} \right],\quad\mathbf{B}_{12}=\left[ \begin{array}{c}0\\ M_{p}^{-1}\mathbf{B}_{u}\\ \end{array} \right], \tag{16}\]The cost output can be taken as the absolute or relative displacement, velocity, or acceleration of the primary system, which can be expressed in the form:  
\[\mathbf{z}=\mathbf{C}_{1}\mathbf{x}+\mathbf{D}_{11}\mathbf{w}+\mathbf{D}_{12}\mathbf{u}, \tag{17}\]  
For the displacement response of the primary system, the cost output can be written as:  
\[\mathbf{z}=\mathbf{X}=\mathbf{C}_{1}\mathbf{x}, \tag{18}\]  
where:  
\[\mathbf{C}_{1}=\Big{[} \begin{array}{ccccc}\mathbf{I}_{n\times n}&\mathbf{O}_{n\times 4}& \mathbf{O}_{n\times n}&\mathbf{O}_{n\times 4}\end{array} \Big{]}, \tag{19}\]  
To complete the state-space description, we rewrite the control force given by Equations (1)-(6) as a static feedback gain \(\mathbf{F}\) multiplied by the \"measurement output\" \(\mathbf{y}\):  
\[\mathbf{u}=\mathbf{F}\mathbf{y}=\left[