Keywords: dynamic structural analysis, plates, moving load, spectral method, matrix operators, strong formulation.

This paper shows an application of the spectral method in dynamics of structures for the special case of a thin plate under the action of a moving load. The strong form (partial differential equation) is usually discretized using finite differences. The accuracy of finite differences can be improved if spectral analysis is applied to the strong form. The spectral differential matrix operator is fast and simple to construct; it is sufficient to construct a one dimensional operator and expand it into two or more dimensions using the formalism of the matrix Kronecker product. The resulting matrix is rather dense and small in size since spectral methods achieve high accuracy with a modest number of points.

Boundary conditions are expressed by using Lagrange multipliers and can be of the Dirichlet or Neumann type (or any combination of them). This approach also allows for nonholonomic conditions (i.e. conditions involving velocity).

A plate with two free ends requires more elaborated boundary conditions [1]. The reaction and the moment along the free boundary y are required to vanish. That is the additional reason for the application of the Lagrange multipliers in enforcing the boundary conditions.

The main characteristic of the moving load problem is the right hand side of differential equation. It is convenient to perform loading discretization prior to the time integration procedure. After it has been discretized in space and time, the solution of the dynamic equation can proceed using any time integration scheme. Integration of the dynamic equations is performed using a modified Newmark method [2].

Mass and "stiffness" matrices could be rather stiff for some boundary conditions. Matrix stabilization can be obtained by removal of rigid body modes [3]). In order to assess the quality of the solution, a condition number for spectral operators has been calculated. Among many possibilities we have chosen the L1 norm based on singular values of a matrix. Without regularization actual matrices can be nearly singular and as a result of the limited computer calculation capabilities the practical performance can be similar to singular matrices.

The proposed procedure has been tried on a dynamic example of a moving load analysis with very good results and the realistic behaviour of a plate under a moving load. The matrix operator formalism of the spectral method produces accurate results while retaining the small size of the problem. Hence it is very suitable for integration of all strong forms of engineering problems.

1

J. Blaauwendraad, "Plates and FEM. Surprises and Pitfalls", Springer, 2010.

2

I. Kozar, "Security aspects of vertical actions on bridge structure", Engineering Computations, 26(1/2), 145-165, 2009. doi:10.1108/02644400910924843

3

A. Ibrahimbegovic, S. Mamouri, "Energy conserving/decaying implicit time-stepping scheme for nonlinear dynamics of three-dimensional beams undergoing finite rotations", Comput. Methods Appl. Mech. Engrg., 191, 4241-4258, 2002. doi:10.1016/S0045-7825(02)00377-8

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