## 3 Space--Time Dynamic Analysis Method Based on DCM
### Forced Vibration Analysis of Beams
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(t\)} & \multicolumn{3}{c}{_x/L_ = 0.067} & \multicolumn{3}{c}{_x/L_ = 0.25} & \multicolumn{3}{c}{_x/L_ = 0.5} \\ \cline{2-9}  & \multicolumn{1}{c}{**DC**} & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**Error(\%)**} & \multicolumn{1}{c}{**DC**} & \multicolumn{1}{c}{**Solution**} & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**Error(\%)**} \\  & \multicolumn{1}{c}{**Solution**} & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**Error(\%)**} & \multicolumn{1}{c}{**DC**} & \multicolumn{1}{c}{**Solution**} & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**Error(\%)**} \\  & \multicolumn{1}{c}{**Solution**} & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**Error(\%)**} & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**Error(\%)**} & \multicolumn{1}{c}{**SD**} & \multicolumn{1}{c}{**Solution**} & \multicolumn{1}{c}{**Exact**} & \multicolumn{1}{c}{**Error(\%)**} \\ \hline
0.5 & \(-17.8585\) & \(-17.8586\) & \(-0.0006\) & \(-60.4521\) & \(-60.4507\) & \(0.0023\) & \(-85.4954\) & \(-85.4902\) & \(0.0061\) \\
1.0 & \(4.4188\) & \(4.4173\) & \(0.0340\) & \(14.9579\) & \(14.9522\) & \(0.0381\) & \(21.1544\) & \(21.1457\) & \(0.0411\) \\
1.5 & \(11.9420\) & \(11.9414\) & \(0.0050\) & \(40.4245\) & \(40.4211\) & \(0.0084\) & \(57.1711\) & \(57.1641\) & \(0.0122\) \\
2.0 & \(-4.0400\) & \(-4.0345\) & \(0.1363\) & \(-13.6576\) & \(-13.6567\) & \(0.1384\) & \(-19.3410\) & \(-19.3135\) & \(0.1424\) \\
2.5 & \(-4.3708\) & \(-4.3747\) & \(-0.0891\) & \(-14.7953\) & \(-14.8083\) & \(-0.0878\) & \(-0.9244\) & \(-20.9421\) & \(-0.0845\) \\
3.0 & \(-2.2223\) & \(-2.2274\) & \(-0.2290\) & \(-7.5225\) & \(-7.5398\) & \(-0.2294\) & \(-10.6388\) & \(-10.6629\) & \(-0.2260\) \\
3.5 & \(0.4370\) & \(0.4372\) & \(-0.0457\) & \(1.4553\) & \(1.4799\) & \(-1.6623\) & \(2.0940\) & \(2.0929\) & \(0.0526\) \\
4.0 & \(10.8579\) & \(10.8553\) & \(0.0240\) & \(36.7545\) & \(36.7448\) & \(0.0264\) & \(51.9807\) & \(51.9650\) & \(0.0302\) \\
4.5 & \(-2.3860\) & \(-2.3960\) & \(-0.4174\) & \(-8.0769\) & \(-8.1104\) & \(-0.4130\) & \(-11.4229\) & \(-11.4698\) & \(-0.4089\) \\
5.0 & \(-16.4010\) & \(-16.3886\) & \(0.0757\) & \(-55.5183\) & \(-55.4748\) & \(0.0784\) & \(-78.5177\) & \(-78.4532\) & \(0.0822\) \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of DC solution and analytic solution for time-history displacement of Timoshenko beam. Figure 8: Time-history displacement of beam midpoint (T = 5 s). Figure 7: Time-history displacement of beam midpoint (T = 2.5 s). relationship between \(\omega\) and load variation frequency \(s\) determines the displacement time-history shape of these six figures. When the value of \(EI\) is small and \(\omega\) is small relative to \(s\), as in case 1 and 2, the main trend of displacement is determined by the natural frequency of the foundation, and external load \(F(x,\,t)\) has only a small effect on the time-history displacement, which is reflected in small fluctuations in Figure 9a,b. When \(EI\) increases, the effect of small fluctuations caused by \(F(x,\,t)\) becomes obvious. When \(EI\) continues to increase so that \(\omega\) reaches about 0.4 times of s, as in case 3, the effect of load frequency \(s\) on time-history displacement is very obvious, completely disturbing the dominant trend of natural frequency on time-history displacement. When natural frequency \(\omega\) is close to load frequency \(s\), as shown in case 4 of Figure 9d, resonance occurs, and the displacement amplitude increases with time. When the increase of \(EI\) makes \(\omega\) much larger than \(s\), as shown in cases 5 and 6 of Figure 9e,f, the trend of time-history displacement is mainly determined by the load frequency. In addition, the bending stiffness not only affects the natural frequency of the structure, but also has a dominant influence on the maximum displacement of the structure. In structural mechanics, the stiffness is defined as the force required for the unit displacement of the structure.