# Dynamic Response Analysis of Structures Using Legendre-Galerkin Matrix Method
###### Abstract  
The solution of the motion equation for a structural system under prescribed loading and the prediction of the induced accelerations, velocities, and displacements is of special importance in structural engineering applications. In most cases, however, it is impossible to propose an exact analytical solution, as in the case of systems subjected to stochastic input motions or forces. This is also the case of non-linear systems, where numerical approaches shall be taken into account to handle the governing differential equations. The Legendre-Galerkin matrix (LGM) method, in this regard, is one of the basic approaches to solving systems of differential equations. As a spectral method, it estimates the system response as a set of polynomials. Using [PERSON]'s orthogonal basis and considering [PERSON]'s method, this approach transforms the governing differential equation of a system into algebraic polynomials and then solves the acquired equations which eventually yield the problem solution. In this paper, the LGM method is used to solve the motion equations of single-degree (SDOF) and multi-degree-of-freedom (MDOF) structural systems. The obtained outputs are compared with methods of exact solution (when available), or with the numerical step-by-step linear Newmark-\(\beta\) method. The presented results show that the LGM method offers outstanding accuracy. differential equation of motion; Legendre-Galerkin matrix (LGM) method; algebraic polynomials; single degree of freedom (SDOF); multi degree of freedom (MDOF)  
1  
M. B.  
past, to this aim, different integration algorithms in the time domain have been developed using various methods. The same algorithms are widely used to solve the equation of systems under dynamic loading. Several well-known algorithms have been presented by researchers, among which the Newmark-\(\beta\)[2], Wilson-Theta [3], Runge-Kutta [4], etc., can be pointed out. Detailed explanations can be found in structural dynamics textbooks [1; 5; 6]. Since 1994, the Chebyshev polynomials [7], [PERSON] polynomials [8], Bessel polynomials [9], Hermite polynomials [10], Laguerre polynomials [11], and matrix methods have been used in many research studies to solve linear and nonlinear equations with high orders including partial differential equations, hyperbolic partial differential equations, delay equations, integral and integro-differential equations, SDOF and MDOF systems, etc. One of the approaches to find the solution to an initial value problem is taking semi-analytical procedures such as the differential transform method (DTM) [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. The nature of dynamic equations of motion of SDOF systems makes them differential equations with initial values; hence, DTM has been also applied to solve non-linear SDOF problems [24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. Alternatively, Equation (1) can also be solved through spectral methods of discretization [34; 35; 36], which are strategic for the numerical solution of differential equations. The main advantage of these methods lies in their accuracy for a given number of unknowns. For smooth problems with simple geometries, these methods offer exponential rates of convergence/spectral accuracy. In contrast, methods such as finite difference and finite element yield only algebraic convergence rates. Three spectral methods, namely the Galerkin [37], Collocation [38], and Tau [39] are extensively used in the literature. Spectral methods are preferable in numerical solutions of partial differential equations due to their high-order accuracy [40; 41]. The Standard Spectral and Galerkin methods have been extensively investigated to handle different types of problems [42; 43; 44; 45]. Several numerical methods have been also developed to solve different types of differential equations. Some of these methods can be used to solve more practical equations.