# Influence Analysis of Traveling Wave Effect on Rail Interaction of Long-Span Suspension Bridge
## 2 Solving the Equations of Inconsistent Excitation by Large Mass Method  
With the continuous development of theory and the maturity of computer technology, there are more earthquake response analysis methods mainly including the following: static analysis method, response spectrum method, random vibration method, and dynamic time history analysis method. The static analysis method uses the ground motion as an inertial force to statically load the structure, takes less consideration of the factors affecting the ground motion, and ignores the dynamic characteristics of the structure itself, so its application is limited. The response spectrum method and the random vibration method can only obtain the peak response of the structure under seismic excitation, and both have limitations in solving nonlinear problems. The time-history analysis method divides the entire stress process of the structure into many sub-steps for calculation, so the seismic response of the structure under each sub-step can be obtained. The time-history analysis method can overcome the limitations of the response spectrum method in dealing with long-period structures and correct the errors caused by ignoring some high-order mode shapes. In addition, the time-history analysis method calculates the dynamic response of the structure through the determined earthquake records that have occurred, and has a huge advantage in calculation accuracy.  
Set a large-mass point M at the foundation support (generally twice the mass of the structure), release the constraints on the movement direction of the large-mass, and apply an external load to make the large-mass point consistent with the earthquake motion [13]:  
\[P_{b}=\text{M}\ddot{u}_{g} \tag{1}\]where \(P_{b}\) is the coercive force; M is the mass of large mass points; and \(\ddot{u}_{g}\) is the acceleration of large mass point.  
The dynamic balance equation of the structure can be written as:  
\[\begin{bmatrix}M_{aa}&M_{ab}\\ M_{ba}&M_{bb}\end{bmatrix}\begin{vmatrix}\ddot{u}_{a}\\ \ddot{u}_{b}\end{vmatrix}+\begin{bmatrix}C_{aa}&C_{ab}\\ C_{ba}&C_{bb}\end{bmatrix}\begin{vmatrix}\dot{u}_{a}\\ \dot{u}_{b}\end{vmatrix}+\begin{bmatrix}K_{aa}&K_{ab}\\ K_{ba}&K_{bb}\end{vmatrix}\begin{vmatrix}u_{a}\\ u_{b}\end{vmatrix}=\begin{vmatrix}0\\ P_{b}\end{vmatrix} \tag{2}\]  
where \(M\) is the mass matrix; C is the damping matrix; \(K\) is the stiffness matrix; \(a\) indicates non-supporting place; \(b\) indicates the support place; \(\ddot{u}\) is the acceleration; \(\dot{u}\) is the speed; and \(u\) is the displacement.  
Let:  
\[\begin{vmatrix}u_{a}\\ u_{b}\end{vmatrix}=\begin{vmatrix}u_{as}\\ u_{bs}\end{vmatrix}+\begin{vmatrix}u_{ad}\\ 0\end{vmatrix} \tag{3}\]  
where \(s\) is the quasi-static item, \(d\) is the dynamic response item, \(u_{a}\) is the displacement of the degree of freedom at the non-structural support, and \(u_{bs}\) is the displacement of the degree of freedom at the support. Expand the second item in (2) to obtain:  
\[M_{ba}\ddot{u}_{a}+M_{bb}\ddot{u}_{b}+M\ddot{u}_{b}+C_{ba}\dot{u}_{a}+C_{bb} \dot{u}_{b}+K_{ba}\ddot{u}_{a}+K_{bb}\ddot{u}_{b}=P_{b} \tag{4}\]  
Considering that the system is a concentrated mass, \(M_{ba}=0\), then the damping force can be regarded as an infinitesimal amount of forcing force, and (4) can be simplified as:  
\[\ddot{u}_{b}=\ddot{u}_{a} \tag{5}\]  
Expand the first term of (2):  
\[M_{aa}\ddot{u}_{a}+M_{ab}\ddot{u}_{b}+C_{aa}\dot{u}_{a}+C_{ab}\dot{u}_{b}+K_{ aa}u_{a}+K_{ab}u_{b}=0 \tag{6}\]  
Let all the dynamic items in (2) be zero, and the balance of the static force items in the equation still holds, so we can obtain:  
\[u_{as}=K_{as}^{-1}K_{ab}u_{bs}=R_{1}u_{bs} \tag{7}\]  
where \(R_{1}=K_{as}^{-1}K_{ab}\), and when considering that the system is a concentrated mass, write (6) as:  
\[M_{aa}\ddot{u}_{ad}+C_{aa}\dot{u}_{ad}+K_{aa}u_{ad}=M_{aa}K_{as}^{-1}K_{ab} \ddot{u}_{bs}+C_{aa}K_{as}^{-1}K_{ab}\dot{u}_{bs} \tag{8}\]  
Equations (1) to (8) are the process of solving structural non-consistent excitation by the large mass method.