*4.3. Five-Degree-of-Freedom Structure*

In conclusion, a structure with five degrees of freedom (5DOF) is analyzed with the LGM method, both under free vibration with and without damping.


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-β 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 13.** Free vibration analysis of a 5DOF system. Comparison of *u*1(*t*) as a function of time, as obtained using the LGM method or the linear Newmark-β method.

**Figure 14.** Free vibration analysis of a 5DOF system. Comparison of *u*2(*t*) as a function of time, as obtained using the LGM method or the linear Newmark-β method.

**Figure 15.** Free vibration analysis of a 5DOF system. Comparison of *u*3(*t*) as a function of time, as obtained using the LGM method or the linear Newmark-β method.

**Figure 16.** Free vibration analysis of a 5DOF system. Comparison of *u*4(*t*) as a function of time, as obtained using the LGM method or the linear Newmark-β method.

**Figure 17.** Free vibration analysis of a 5DOF system. Comparison of *u*5(*t*) as a function of time, as obtained using the LGM method or the linear Newmark-β 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.

#### **5. Conclusions**

The Legendre–Galerkin matrix (LGM) method was developed in this study to solve systems of differential equations of motion. As shown, the advantage of this spectral method is that it converts the governing differential equation of a given problem to a system of algebraic equations, based on a set of orthogonal Legendre polynomials. The final solution leads to a good estimate of the solution for a system of differential equations. In the present research study, the selected differential equations were typical of single degree (SDOF) and multi-degree-of-freedom (MDOF) structural systems.

In order to prove the accuracy of the proposed method in the response calculation of SDOF and MDOF structures, a number of numerical examples for damped and undamped structural systems under free or forced vibrations were developed and discussed. When available, exact solutions were taken into account for the comparative analysis of LGM predictions, otherwise, the results of the numerical linear Newmark-β method were used to verify the estimates from the developed LGM method. The overall comparative data showed that the LGM method is of high accuracy in estimating the response of SDOF and MDOF systems and can be thus effectively employed in the solution of fundamental motion equations of structures.

**Author Contributions:** Conceptualization, M.M., M.R.B., B.J., C.B. and M.A.N.; methodology, M.M., M.R.B.; software, M.M., M.R.B. and C.B.; validation, M.M., M.R.B. and M.A.N.; formal analysis, C.B., M.A.N., M.A.H. and S.M.D.; investigation, C.B., M.A.H., M.A.N. and S.M.D.; writing—original draft preparation, M.M. and B.J.; writing—review and editing, C.B., M.A.N., B.J., S.M.D. and M.A.H.; visualization, M.M., M.R.B., B.J. and C.B.; supervision, C.B., M.A.N., S.M.D. and M.A.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Supporting data will be made available upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

