# An Improved Differential Quadrature Time Element Method
## 4 Numerical Analysis
### Comparison of Dynamic Responses  
For evaluating the performance of the improved DQTEM in dynamic analysis, this section investigates the transient problem of a fixed-free uniform rod subjected to an impact at free end by a rigid body of mass \(1.5\rho Sl\). Initial conditions are zero except for the initial velocity of the free end colliding with the body is \(v\). The relative error used in this section is defined by  
\[\text{RelErr2}=\frac{\text{abs}(\text{Exact}-\text{Approximate})}{\|\text{ Exact}\|_{\infty}} \tag{41}\]  
where \"Exact\" denotes the exact displacements from the analytical mode superposition method at each time sampling point; the infinity norm of \"Exact\" denotes the maximum absolute value of a vector, and the elements of this vector are the exact values at all time nodes; and \"Approximate\" denotes the displacements of DQTEM0 or the improved DQTEM at each time sampling point. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**Order**} & \multirow{3}{*}{**Exact**} & \multicolumn{3}{c}{\(M=20\)} & \multicolumn{3}{c}{\(M=100\)} \\ \cline{3-8}  & & \multicolumn{2}{c}{**DQTEM0**} & \multicolumn{2}{c}{**Present**} & \multicolumn{2}{c}{**DQTEM0**} & \multicolumn{2}{c}{**Present**} \\  & & \multicolumn{2}{c}{\(NE=10,m=4\)} & \multicolumn{2}{c}{\(NE=1,m=22\)} & \multicolumn{2}{c}{\(NE=50,m=4\)} & \multicolumn{2}{c}{\(NE=1,m=102\)} \\ \cline{3-8}  & & \(\lambda\) & **RelErr1** & \(\lambda\) & **RelErr1** & \(\lambda\) & **RelErr1** & \(\lambda\) & **RelErr1** \\ \hline
1 & 3.1416 & 3.1416 & 0.0003 & 3.1416 & \(3.0125\times 10^{-10}\) & 3.1416 & \(6.9242\times 10^{-7}\) & 3.1416 & \(1.2103\times 10^{-11}\) \\
2 & 6.2832 & 6.2835 & 0.0054 & 6.2832 & \(2.7142\times 10^{-11}\) & 6.2832 & \(8.7025\times 10^{-6}\) & 6.2832 & \(2.2619\times 10^{-12}\) \\
3 & 9.4248 & 9.4273 & 0.0267 & 9.4248 & \(1.1085\times 10^{-12}\) & 9.4248 & \(4.4115\times 10^{-5}\) & 9.4248 & \(9.8517\times 10^{-12}\) \\
4 & 12.5664 & 12.5678 & 0.0882 & 12.5664 & \(1.5574\times 10^{-11}\) & 12.5664 & \(8.0001\) & 12.5664 & \(8.2115\times 10^{-13}\) \\
5 & 15.7080 & 15.7389 & 0.1972 & 15.7080 & \(2.5245\times 10^{-11}\) & 15.7080 & 0.0003 & 15.7080 & \(6.6817\times 10^{-13}\) \\
6 & 18.8496 & 18.9243 & 0.3963 & 18.8496 & \(1.1124\times 10^{-10}\) & 18.8497 & 0.0007 & 18.8496 & \(2.4812\times 10^{-14}\) \\
7 & 21.9911 & 22.1465 & 0.7064 & 21.9911 & \(2.7856\times 10^{-7}\) & 21.9914 & 0.0013 & 21.9911 & \(8.5338\times 10^{-13}\) \\
8 & 25.1327 & 25.4206 & 1.1453 & 25.1327 & \(6.4210\times 10^{-7}\) & 25.1333 & 0.0022 & 25.1327 & \(6.4821\times 10^{-13}\) \\
9 & 28.2743 & 28.7486 & 1.6773 & 28.2746 & 0.0008 & 28.2753 & 0.0035 & 28.2743 & \(1.1581\times 10^{-14}\) \\
10 & 31.4159 & 33.0975 & 5.3526 & 31.4163 & 0.0012 & 31.4176 & 0.0054 & 31.4159 & \(3.9475\times 10^{-12}\) \\ \hline \hline \multicolumn{8}{c}{Ner5} & \multicolumn{3}{c}{45\%} & \multicolumn{3}{c}{70\%} & \multicolumn{3}{c}{49\%} & \multicolumn{3}{c}{70\%} \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Frequency comparisons of beam with the same spatial DOFs (Degrees of Freedom) for both methods. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{**Order**} & \multirow{3}{*}{**Exact**} & \multicolumn{3}{c}{**DQTEM0**} & \multicolumn{2}{c}{**Present**} & \multicolumn{2}{c}{**DQTEM0**} & \multicolumn{2}{c}{**Present**} \\  & & \multicolumn{2}{c}{\(NE=20,m=4\)} & \multicolumn{2}{c}{\(NE=1,m=22\)} & \multicolumn{2}{c}{\(NE=50,m=4\)} & \multicolumn{2}{c}{\(NE=1,m=102\)} \\ \cline{3-8}  & & \(\lambda\) & **RelErr1** & \(\lambda\) & **RelErr1** & \(\lambda\) & **RelErr1** & \(\lambda\) & **RelErr1** \\ \hline
1 & 3.1416 & 3.141  
In order to capture the physical phenomena of impact clearly, a few dimensionless parameters are introduced as  
\[\widetilde{u}=\frac{u}{l}\frac{c}{v},\ \tau=t\frac{c}{l},\ \widetilde{v}=\frac{ \mathrm{d}\widetilde{u}}{\mathrm{d}\tau} \tag{42}\]  
where \(u\) is the longitudinal displacement.