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 Legendre’s orthogonal basis and considering Galerkin’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-β method. The presented results show that the LGM method offers outstanding accuracy.
Highlights
Introduction andState-of-ArtMost structural systems in civil engineering applications, as known, are either discrete or can be estimated as discrete equivalents
The Legendre–Galerkin matrix (LGM) method was developed in this study to solve systems of differential equations of motion
In the present research study, the selected differential equations were typical of single degree (SDOF) and multi-degree-of-freedom (MDOF) structural systems
Summary
Several numerical methods have been developed to solve different types of differential equations. Researchers have developed a number of numerical methods to find the solution for such equations. In this regard, matrix methods including the Euler method [52], Bernoulli collocation method [53], hybrid Legendre block-pulse function method [54], least squares method [55], Bessel colocation method [56], etc., have received much attention since 2010. After selecting the type of polynomials and expressing each of the functions, the differential equation system becomes a system of linear equations with several unknown coefficients, which can be solved by Galerkin and collocation methods. The accuracy of the LGM formulation is highlighted in the discussion of comparative calculations
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