## 8 Numerical Example
### MDOF Nonlinear Mass Spring System  
In this section, a series of a new mass spring system is considered and its details are shown in Figure 19. The mass of the system is chosen as \(m_{i}~{}=~{}10\) kg and the stiffness is \(k_{i}~{}=~{}10^{9}\Big{(}1-10\sqrt{|u_{i}-u_{i-1}|}\Big{)}\text{N}/\text{m}\) Due to the nonlinear term, the stiffness that will decrease after deformation is assumed. In order to check the computational efficiency, the DOFs of the system are specified as \(n=500\), \(n=1000\), \(n=2000\) and \(n=3000\). The system is excited by the \(\sin\) load, and the loading details are given in Table 1. The time integration data and the lowest and highest natural frequencies of the system are given in Tables 2 and 3, respectively. It is noted that the highest natural frequency is 20,000 rad/s. In this dynamic analysis, the average acceleration method (AAM) with the time steps of 0.010, 0.015, 0.020 and 0.025 are taken as the reference solution for the 500-DOF, 1000-DOF, 2000-DOF and 3000-DOF, respectively. Each analysis is carried out for a total number of 100 steps (N = 100). As a result, the two ends and center displacement responses of the system are shown in Figure 20a-d, corresponding to the 500-DOF, 1000-DOF, 2000-DOF and 3000-DOF. It seems that the \(\Delta t\) = 0.050, 0.075, 0.100 and 0.125 s are the maximum permissible time steps to obtain the reliable solution, corresponding to the 500-DOF, 1000-DOF, 2000-DOF and 3000-DOF at the two ends and center of the new spring system.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**N-DOF** & **Ground Acceleration (m/s\({}^{2}\))** \\ \hline
500 & \(200\sin(5t)\) \\
1000 & \(150\sin(3t)\) \\
2000 & \(100\sin(1.5t)\) \\
3000 & \(50\sin(t)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Ground acceleration.  
Figure 18: Seismic responses of the top-storey of the eight-storey building subjected to the earthquake record CHY028 (a) \(\sigma\)= 1.0 (b) \(\sigma\)= 2.0.  
The consumed CPU time for each nonlinear dynamic analysis is noted and listed, in Table 4. It is seen that there is a consequential difference between the CPU time taken by AAM and CVM. This is because of nonlinear iterations are needed in each time step for AAM. However, CVM can integrate the explicit formulations and unconditional stability together. In the last column of Table 4, R is given. It shows that R decreases with the increase of \(n\). It infers that the computational efficiency of the CVM will increase as the total number of DOF of the system increases.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**N-DOF** & **CPU\({}^{(\text{AAM})}\)** & **CPU\({}^{(\text{CVM})}\)** & **R-\(\frac{\text{CVM}}{\text{AAM}}\)** \\ \hline
500 & 74.59 & 1.73 & 0.023 \\
1000 & 288.02 & 6.29 & 0.022 \\
2000 & 1262.46 & 21.99 & 0.017 \\
3000 & 3195.88 & 52.06 & 0.016 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Time integration data.  
Figure 19: A new mass spring system.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**N-DOF** & \(\Delta t(\mathbf{s})\) & \(\Delta t(\mathbf{s})\) & \(\mathbf{t_{d}(s)}\) & **N** \\ \hline
500 & 0.05 & 0.010 & 10 & 200 \\
1000 & 0.075 & 0.015 & 15 & 200 \\
2000 & 0.100 & 0.020 & 20 & 200 \\
3000 & 0.125 & 0.025 & 25 & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Time integration data.  
Figure 20: _Cont._