# An Improved Differential Quadrature Time Element Method
## 4 Numerical Analysis
### Frequency Comparisons
#### 4.1.2 Euler Beam  
A simply supported Euler beam is considered here, the cubic element involving deflection and slope at each node is used in DQTEM0, but all DOFs of an element in the improved DQTEM are deflections. Non-dimensional frequency parameter \(\lambda=\beta l\) is introduced in comparison, where \(\beta^{4}=\rho S\omega^{2}/EI\). Similar to the rod, DQTEM0 adopts 10 and 50 cubic elements, respectively, whereas the improved DQTEM employs one element with 22 and 102 Gauss-Lobatto-Chebyshev points; apparently, they have the same number of spatial DOFs. It follows in Table 7 that the Nerr5 of DQTEM0 with \(NE=10\), \(m=4\) is nearly 45%, while the Nerr5 of the improved DQTEM with \(NE=1\), \(m=22\) is about 70%, and the RelErr1 of the improved DQTEM are much smaller than DQTEM0. A similar conclusion is achieved when DQTEM0 uses \(NE=50\), \(m=4\) and the improved DQTEM uses \(NE=1\), \(m=102\). Note that, in the cubic beam element, half of the spatial DOFs is deflection, and half is slope, but all DOFs in the improved DQTEM are deflections. This motivates us to compare the case with the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{\(M=10\) (\(NE=1\), \(m=11\))} & \multicolumn{4}{c}{\(M=100\) (\(NE=1\), \(m=101\))} \\ \cline{3-10}
**Order** & **Exact** & \multicolumn{2}{c}{**HOLEM**} & \multicolumn{2}{c}{**Present**} & \multicolumn{2}{c}{**HOLEM**} & \multicolumn{2}{c}{**Present**} \\ \cline{3-10}  & & \(\Lambda\) & **RelErr1** & \(\Lambda\) & **RelErr1** & \(\Lambda\) & **RelErr1** & \(\Lambda\) & **RelErr1** \\ \hline
1 & 1.5708 & 1.5708 & 9.1035 \(\times\) 10\({}^{-7}\) & 1.5708 & 6.3611 \(\times\) 10\({}^{-8}\) & 1.5708 & 5.2441 \(\times\) 10\({}^{-9}\) & 1.5708 & 7.2083 \(\times\) 10\({}^{-11}\) \\
2 & 4.7124 & 4.7124 & 1.1863 \(\times\) 10\({}^{-6}\) & 4.7124 & 1.2722 \(\times\) 10\({}^{-7}\) & 4.7124 & 7.4895 \(\times\) 10\({}^{-9}\) & 4.7124 & 6.4515 \(\times\) 10\({}^{-12}\) \\
3 & 7.8504 & 7.8504 & 2.0513 \(\times\) 10\({}^{-5}\) & 7.8504 & 2.5289 \(\times\) 10\({}^{-6}\) & 7.8504 & 8.6134 \(\times\) 10\({}^{-10}\) & 7.8504 & 7.7932 \(\times\) 10\({}^{-12}\) \\
4 & 10.9596 & 10.9957 & 8.3315 \(\times\) 10\({}^{-4}\) & 10.9956 & 1.3213 \(\times\) 10\({}^{-4}\) & 10.9956 & 1.9500 \(\times\) 10\({}^{-9}\) & 10.9956 & 9.0158 \(\times\) 10\({}^{-13}\) \\
5 & 14.1372 & 14.1436 & 0.4586 & 14.1355 & 0.0117 & 14.1372 & 4.2104 \(\times\) 10\({}^{-11}\) & 14.1372 & 1.4325 \(\times\) 10\({}^{-12}\) \\
6 & 17.2788 & 17.3941 & 0.6675 & 17.2258 & 0.3065 & 17.2788 & 6.5482 \(\times\) 10\({}^{-10}\) & 17.2788 & 5.9357 \(\times\) 10\({}^{-13}\) \\
7 & 20.4204 & 21.2563 & 4.0938 & 20.2612 & 0.7792 & 20.4204 & 3.6833 \(\times\) 10\({}^{-10}\) & 20.4204 & 4.9302 \(\times\) 10\({}^{-13}\) \\
8 & 23.5619 & 27.2485 & 15.6461 & 25.1505 & 6.7420 & 23.5619 & 2.5574 \(\times\) 10\({}^{-11}\) & 23.5619 & 4.2105 \(\times\) 10\({}^{-13}\) \\
9 & 26.7035 & 39.5463 & 48.0939 & 36.1197 & 35.2620 & 26.7036 & 3.5423 \(\times\) 10\({}^{-10}\) & 26.7035 & 3.4865 \(\times\) 10\({}^{-13}\) \\
10 & 29.8451 & 77.5366 & 159.7964 & 70.5585 & 136.4153 & 29.8451 & 1.5918 \(\times\) 10\({}^{-10}\) & 29.8451 & 3.3358 \(\times\) 10\({}^{-13}\) \\ \hline \multicolumn{2}{c}{Nerr5} & \multicolumn{2}{c}{70\%} & \multicolumn{2}{c}{70\%} & \multicolumn{2}{c}{70\%} & \multicolumn{2}{c}{70\%} & \multicolumn{2}{c}{70\%} \\ \hline \hline \multicolumn{2}{c}{CPU/s} & \multicolumn{2}{c}{1.106} & \multicolumn{2}{c}{0.016} & \multicolumn{2}{c}{417.625} & \multicolumn{2}{c}{0.051} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Frequency comparisons of rod using high-order rod elements.