## 4 Numerical Examples
### Long Span Planar Skeletal Structure  
A 25 m (length) \(\times\) 1 m (height) large span truss model is considered to demonstrate the numerical stability and dissipation of the proposed algorithm. The model shown in Figure 12 is composed of 101 member bars, and each bar is a single element with hinged constraints at both ends. As for the mechanical properties of the model, the elastic modulus is \(E\) = 2.0 \(\times\) 10\({}^{5}\) MPa, the density is 7800 kg/m\({}^{3}\), and the cross-section area of each bar is 100 mm\({}^{2}\). An initial velocity \(v\) = 100 m/s is set on the points A and B located near the middle of the span, as shown in Figure 12.  
Setting \(\Delta t=0.01\) s, the total solution time \(t_{f}\) = 10 s, parameters of the proposed method = 0.1, \(\beta\) = 0.5, \(c_{1}=c_{3}=1/4\), and the parameter of the GGW method \(\alpha\) = 0, the responses of point A obtained from different methods are shown in Figures 13-16. The response calculated by the trapezoidal rule starts to diverge after 3 s, and at around 8 s, it fails to give a reasonable result. However, other methods can all obtain convergent solutions with such a length of time step. Figure 11: Results of the trapezoidal rule with different time steps \(\Delta t\) = \(T\)/10,000, \(T\)/20,000, \(T\)/50,000, \(T\)/100,000. Figure 12: Long span planar skeletal structure. Figure 13: Displacement responses of Point A with the [PERSON] method, [PERSON] method, GGW method, trapezoidal rule, and the proposed method: (**a**) responses from 0 to 10 s; (**b**) partial enlarged drawing of responses from 9 to 10 s.  
As shown in Figure 14, the velocity response calculated by the GGW method is larger than that calculated by other methods, i.e., the GGW method does not filter the effects of high-frequency oscillations. In order to demonstrate this effect more clearly, the velocity response from 3 to 8 s in Figure 14 is replotted in Figure 16. The presented method has much improved the performance of the GGW method on high-frequency oscillation problems. Figure 16 also shows that after filtering out high-frequency effects, the velocity responses  
Figure 16: Partial enlarged drawing of velocity responses from 3 to 8 s of Point A with the [PERSON] method, [PERSON] method, GGW method, and the proposed method. Figure 14: Velocity responses of Point A with the [PERSON] method, Bathe method, GGW method, trapezoidal rule, and the proposed method: (**a**) responses from 0 to 10 s; (**b**) partial enlarged drawing of responses from 9 to 10 s.  
Figure 15: Acceleration responses of Point A with the [PERSON] method, [PERSON] method, GGW method, trapezoidal rule and the proposed method: (**a**) responses from 0 to 10 s; (**b**) partial enlarged drawing of responses from 9 to 10 s.  
To verify the decaying speed of the numerical solution with the proposed method, the time steps are set to be \(\Delta t=0.001\), \(0.005\), \(0.01\), and \(0.02\)\(s\), respectively, and the corresponding results are shown in Figures 20 and 21, where \(\alpha=2/7\) so that the corresponding final spectral radius is \(0.5\), and low frequencies are retained while high frequencies are filtered out. It can be found that when the time step increases, the high-frequency part of the numerical solution gradually disappears while the low-frequency part is retained. This conclusion shows that the numerical result obtained from the proposed method gradually attenuates, but the accurate and effective low-frequency solution can still be obtained. This result can help to improve the computational efficiency. When filtering out high-frequency numerical solutions, it is also necessary to consider the influence of numerical damping on low-frequency solutions. Due to the rapid numerical damping and amplitude decay, the low-frequency solution of some algorithms will gradually lose its value during calculation. When using the Wilson method, with the passage of time, the amplitude of numerical solution gradually decays.