## 3 Analysis of Calculation Examples
### Effect of Speed of Loads on Dynamic Response of Double-Beam System  
To better understand the overall vibration properties of a simply supported double-beam system, the speed of loads was set at \(v=100\) m/s, and a 3D dynamic graph of vertical deflection of the simply supported double-beam system within the entire span scope was plotted under a load series, as shown in Figure 3.  
Figure 2: The primary-secondary beam system under moving load-groups. With the movement of loads, the amplitude of dynamic deflection of the simply supported double-beam system changed constantly; the vertical dynamic deflection of secondary beam at any moment showed an approximately symmetric distribution relative to the midspan position and had only one extreme value in the vicinity of the midspan. The dynamic deflection distribution of the primary beam at any moment was related to the position of the action point of the successive moving loads, and its dynamic deflection showed several extreme values, all emerging at the action point of the successive moving loads. The dynamic deflection of the primary and secondary beams reached their maximum values near the midspan. Considering that this characteristic was not related to the movement position of the load series, the following analyses uniformly used the dynamic deflection response of the midspan. The analytical calculation method proposed in this paper and the ANSYS finite-element numerical method were used to calculate the dynamic response of the simply supported double-beam system under a load series of four different moving speeds (40 m/s, 100 m/s, 122 m/s, and 180 m/s), and the calculation results of the two methods were compared in terms of the time-history curves and peaks of dynamic deflection response of the midspan. The comparison results are shown in Figure 4 and Table 1 (where \(y_{an}\) and \(y_{fe}\) are the calculation results of dynamic deflection response of midspan obtained by the analytical method proposed in this paper and ANSYS finite-element numerical method, respectively; \(p_{an}\) and \(p_{fe}\) are the calculation results of peak deflection of midspan obtained from the analytical method proposed in this paper and ANSYS finite-element numerical method, respectively; \(e_{p}=\big{(}p_{an}-p_{fe}\big{)}/p_{fe}\) is the calculation error of peak deflection of midspan between the two methods); \(\lambda_{p}\) is the ratio of peak dynamic deflection response of midspan of the primary beam to that of the secondary beam. As shown in Figure 4 and Table 1, under a load series of multiple different speeds, the analytical calculation results of time-history curves and peaks of dynamic deflection response of the midspan for the simply supported double-beam system were consistent with the calculation results obtained from the ANSYS finite-element numerical method, thus demonstrating the rationality of the analytical calculation method proposed in this paper. Compared with the secondary beam, the primary beam had a significantly increased peak dynamic deflection response and a high-frequency component in the time-history curve of the dynamic deflection response of the midspan. Under the four speeds, \(\lambda_{p}\) values were 1.950, 2.093, 1.706, and 2.467; the peak dynamic deflection responses of the midspan of the primary and secondary beams did not increase with the increase in the speed of the loads, indicating that the simply supported double-beam system under successive moving loads had critical speeds. Figure 5 shows the relationship between the peak deflection of midspan \(p_{an}\) and speed of loads \(v\) for the simply supported double-beam system; \(p_{an,1}\) and \(p_{an,2}\) are the peak deflections of the midspan of  
Figure 3: The 3D dynamic graphs of vertical deflection of a simply supported double-beam system for: (**a**) primary beam; (**b**) secondary beam. the primary and secondary beams.