## 3 Space--Time Dynamic Analysis Method Based on DCM
### Forced Vibration Analysis of Beams  
In the previous section, the stability of the DC method in analyzing dynamic problems is analyzed. Next, the displacement response of a [PERSON] shear beam under dynamic load \(F(x,t)\) is presented in this paper. The expression of dynamic load is  
\[F(x,t)=Q\sin\frac{\pi x}{L}\sin st \tag{29}\]  
where \(\mathbf{Q}\) is the peak load and \(s\) is the time-domain variation frequency of load. Considering that the boundary condition is simply supported at both ends, and the initial condition is that the displacement and velocity at \(t\) = 0 are zero. The displacement analytical solution of forced vibration of the [PERSON] shear beam under load \(F(x,t)\) is  
\[y(x,t)=\frac{Q}{\rho\Big{[}1+\zeta(\pi/L)^{2}\Big{]}}\sin\Bigl{(}\frac{\pi x} {L}\Bigr{)}\frac{\sin st-(s/\omega)\sin\omega t}{\omega^{2}-s^{2}} \tag{30}\]  
where \(\omega\) is the foundation natural frequency of the beam, and the expression is  
\[\omega=\pi^{2}\sqrt{\frac{EI}{\rho L^{4}\Big{[}1+\zeta(\pi/L)^{2}\Big{]}}} \tag{31}\]  
The attribute parameters of the beam remain unchanged, and the relevant parameters of the load are: the peak load \(\mathbf{Q}\) = 10\({}^{7}\) N; \(s=2\pi/0.28335\); and Time step \(\Delta T\) = 0.25 s. The values of the total time \(T\) are taken at 2.5 s and 5 s respectively. According to the value of \(\Delta T\), the total time \(T\) is divided into 10 and 20 time steps. The orthogonal grid of CGL distribution with \(m\) = 7, \(n\) = 9 is used for calculation, and the DC solution is compared with the analytical solution obtained by Equation (30). Figures 7 and 8 give the displacement of DC solutions and the analytical solutions of the nodes on the beam under the whole-time history, in which the curves of the DC solution and the analytical solution are always perfectly consistent, with no phenomenon of detachment over time. Table 2 shows the detailed data comparison between the displacement DC solution and the analytical solution of the beam node at each moment. It can be seen that the errors between DC solutions and analytical solutions are almost kept within one percent, more than half of the data errors remain within one thousand, and there is no sign of error accumulation and amplification. By comparing the data in Figures 7 and 8 and Table 2, it is found that highly accurate results can be obtained by applying the DCM for calculating the forced vibration of the beam. Then, this paper will continue to use the space-time DCM to analyze the influence factors of time-history displacement form of beam under dynamic load. The displacement form of the beam under forced vibration is mainly determined by the relative relationship between the natural vibration frequency \(\omega\) of the beam and the change frequency \(s\) of the load in time. According to Equation (31), the fundamental natural frequency \(\omega\) of the beam is determined by the bending stiffness \(EI\), distributed mass \(\rho\), beam length \(L\) and shear coefficient \(\zeta\). Keep the other three factors unchanged to study the effect of \(EI\) on time history displacement, as shown in Table 3. Figure 9 shows the time history displacements in six cases in Table 3. As an influencing factor, bending stiffness \(EI\) affects the value of foundation natural frequency \(\omega\), and the relative