# Dynamic Response Analysis of Structures Using Legendre-Galerkin Matrix Method
## 4 Worked Examples and Discussion of Results
### Five-Degree-of-Freedom Structure  
In conclusion, a structure with five degrees of freedom (5 DOF) is analyzed with the LGM method, both under free vibration with and without damping.  
* Free vibration, without damping: it is assumed that \(\mathbf{M}=\left[\begin{array}{ccccc}1&0&0&0&0\\ 0&2&0&0&0\\ 0&0&2&0&0\\ 0&0&0&2&0\\ 0&0&0&0&2\end{array}\right]\), \(\mathbf{C}=0\) and \(\mathbf{K}=\left[\begin{array}{ccccc}4&-4&0&0&0\\ -4&12&-8&0&0\\ 0&-8&20&-12&0\\ 0&0&-12&24&-12\\ 0&0&0&12&28\end{array}\right]\), respectively. The initial conditions are \(u(0)=\left[\begin{array}{c}1\\ 4/5\\ 3/5\\ 2/5\\ 1/5\end{array}\right]\) and \(\dot{u}(0)=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]\).
* Free vibration, with damping: it is assumed that \(\mathbf{C}=\left[\begin{array}{ccccc}0.5&0&0&0&0\\ 0&0.5&0&0&0\\ 0&0&0.5&0&0\\ 0&0&0&0.5&0\\ 0&0&0&0&0.5\end{array}\right]\), while the other parameters are the same as in the undamped case.  
The solutions of the equation of motion of the structure using the LGM for damped and undamped states, in terms of \(u_{1}(t)\) to \(u_{5}(t)\), are shown in Figures 13-17 and compared with those by the linear Newmark-\(\beta\) method. As for the previously discussed calculation examples, the comparative plots prove an appropriate consistency and a high level of accuracy of the LGM procedure.  
Figure 12: Forced vibration analysis of a 3 DOF system. Comparison of \(u_{3}(t)\) as function of time, as obtained using the LGM method or the linear Newmark-\(\beta\) method.  
Figure 14: Free vibration analysis of a 5 DOF system. Comparison of \(u_{2}(t)\) as a function of time, as obtained using the LGM method or the linear Newmark-\(\beta\) method.  
Figure 13: Free vibration analysis of a 5 DOF system. Comparison of \(u_{1}(t)\) as a function of time, as obtained using the LGM method or the linear Newmark-\(\beta\) method.  
Figure 16: Free vibration analysis of a 5 DOF system. Comparison of \(u_{4}(t)\) as a function of time, as obtained using the LGM method or the linear Newmark-\(\beta\) method.  
Figure 15: Free vibration analysis of a 5 DOF system. Comparison of \(u_{3}(t)\) as a function of time, as obtained using the LGM method or the linear Newmark-\(\beta\) method.  
It should be noted that, in the case of stochastic analysis that is interpreted classically as a repetition of solution for different input values with defined probability density functions (taking into account uncertainties associated with input parameters), the methods such as Monte Carlo simulation (MCS) [60] can be easily used along with the results of LGM method to investigate the problem in a probabilistic manner. This is due to the fact that, in the LGM method, the solution is approximated by discretized Legendre series that can be used as a state function in reliability analysis where no mathematical closed-form state function can be found (e.g., through finite element method). Furthermore, such a solution obtained from the LGM method can also be utilized with analytical reliability methods (e.g., jointly distributed random variables method [61]) or approximate ones (e.g., first and second-order reliability methods (FORM and SORM), point estimate method (PEM), etc. [62]), with less computational efforts in comparison to the MCS method.