## 4 Vibration Control Analysis of Track-Bridge Structure Based on TIDs  
### TID-Based Vibration Control Model for Track-Bridge Structures  
In the four-layer track-bridge structure, Tuned Inertia Dampers (TIDs) are added to further analyze their vibration control effectiveness under high-frequency excitation caused by short-wavelength irregularities. A total of \(N_{t}\) TIDs with identical parameters are installed at equal intervals between the rail and the track slab, as shown in Figure 4.  
Figure 3: Vertical vibration response. (**a**) Vertical displacement of the rail. (**b**) Vertical acceleration of the rail. (**c**) Vertical displacement of the track slab. (**d**) Vertical acceleration of the track slab.  
In the figure, the parameters \(b_{t}\), \(c_{t}\), and \(k_{t}\) represent the inertance, damping coefficient, and stiffness of the TID, respectively, and \(q_{t1},q_{t2},\cdots,q_{tN_{t}}\) denotes the displacement of the inertance mass.  
Let the distance from the first TID to the bridge node be \(x_{0}\), and the spacing between TIDs be \(\Delta x\). The coordinate of the \(i\)th TID is then defined as  
\[x_{i}=x_{0}+(i-1)\Delta x\ (i=1,2,\cdots,N_{t}) \tag{36}\]  
According to the working principle of TIDs, the output of each device depends on the relative acceleration between the rail and the track slab at the connection points [23; 24], i.e., \(F_{ti}=b_{t}[\tilde{U}_{t}(x_{i},t)-\tilde{U}_{s}(x_{i},t)]\), where \(\tilde{U}_{t}(x_{i},t)\) and \(\tilde{U}_{s}(x_{i},t)\) are the accelerations of the TID and track slab at the respective contact points \(x_{i}\). By introducing the TID displacement into the displacement vector of Equation (35), the new displacement vector becomes  
\[Q_{t}(t)=[q_{c1}\ \ q_{c2}\cdots q_{c10}\ \ q_{t1}\ \ q_{t2}\cdots q_{tN_{t}}\ q_{t1} \ \ q_{t2}\cdots q_{tN_{t}}\ q_{s1}\ \ q_{s2}\cdots q_{sN_{t}}\ q_{f1}\ \ q_{f2}\cdots q_{fN_{f}}\ q_{b1}\ \ q_{b2}\cdots q_{bN_{b}}] (1\times(N_{z}+N_{t}))\]  
The total force generated by all TIDs is given by  
\[F_{bt}=\sum_{i=1}^{N_{t}}b_{t}(\tilde{U}_{ti}(x_{i},t)-\tilde{U}_{s}(x_{i},t))= b_{t}N_{st}N_{st}{}^{T}\tilde{Q}_{t}(t)=M_{t}\tilde{Q}_{t}(t) \tag{37}\]  
where  
\[N_{ts}=\begin{bmatrix}&\mathcal{O}_{((10+N_{t})\times N_{t})}\\ &I_{(N_{t}\times N_{t})}\\ -\phi_{1}(x_{1})&-\phi_{1}(x_{2})&\cdots&-\phi_{1}(x_{N_{t}})\\ -\phi_{2}(x_{1})&-\phi_{2}(x_{2})&\cdots&-\phi_{2}(x_{N_{t}})\\ \cdots&\cdots&\cdots&\cdots\\ -\phi_{N_{t}}(x_{1})&-\phi_{N_{t}}(x_{2})&\cdots&-\phi_{N_{t}}(x_{N_{t}})\\ &\mathcal{O}_{((N_{f}+N_{b})\times N_{t})}\\ \end{bmatrix}_{((N_{z}+N_{t})\times N_{t})}\]  
\(I\) is the identity matrix, and \(M_{t}\) is the effective inertial mass matrix generated by the TIDs.  
The stiffness and damping matrices of the TIDs are calculated similarly to those of discrete elastic supports and are derived from equations  
\[K_{t}=k_{t}\sum_{i=1}^{N}N_{rt}^{i}N_{rt}^{iT}\text{ and }C_{t}=c_{t}\sum_{i=1}^{Nt}N_{rt}^{i}N_{rt}^{iT} \tag{38}\]  
where \(N_{rt((N_{z}+N_{t})\times 1)}^{i}=[\mathcal{O}_{1\times 10}\ \ \phi_{t1}(x_{ti})\ \ \phi_{t2}(x_{ti})\cdots\phi_{tN_{t}}(x_{ti})\ \ \mathcal{O}_{(1\times(i-1))}\ \ -1\ \mathcal{O}_{(1\times(N_{z}-N_{r-i}))}]^{T}\).  
Figure 4: TID control analysis model.  
By incorporating the effective inertial mass, stiffness, and damping matrices of the TIDs into Equation (35), we obtain  
\[\left\{\begin{array}{l}M_{ut}\ddot{Q}_{l}(t)+C_{ut}\dot{Q}_{l}(t)+K_{l1}Q_{l}(t )=F_{u}\\ \\ M_{ut}=\left[\begin{array}{cc}M_{u(N+10)\times(N_{t}+10)}&\\ &O_{N_{t}\times N_{t}}&\\ &M_{u((N_{t}+N_{f}+N_{b})\times(N_{t}+N_{f}+N_{b}))}\\ &O_{N_{t}\times N_{t}}&\\ K_{l1}=\left[\begin{array}{cc}K_{(N_{t}+10)\times(N_{t}+10)}&\\ &O_{N_{t}\times N_{t}}&\\ &K_{((N_{t}+N_{f}+N_{b})\times(N_{t}+N_{f}+N_{b}))}\\ \end{array}\right]+K_{t}\\ \end{array}\right]+C_{t} \tag{39}\]  
Solving this equation yields the vibration response of the track-bridge structure with TID control.