Using Operator Method and Efros Transform for Taking Into Account Dissipative Properties of Deformable Structural Elements

Abstract

An approach for taking into account the dissipative properties of deformable structural elements (and in particular internal friction in the material) is proposed. This approach, based on the operator method and Efros transformation, is applied to analytical solutions obtained in the framework of elasticity theory, which can be represented as integrals of convolution type. The direct Laplace transform is applied to the operator equation. It is shown in the paper that changing to the Laplace variable enables one to pass from the relations taking into account dissipation energy to those for an elastic plate in the image space. The Efros theorem is used to perform the inverse Laplace transform of the complex function obtained. The posed problem is reduced to finding an unknown transforming function. Numerical calculations showed that in the problem considered (for small values of the time variable) the law of this function variation is very close to the law of normal distribution (Gaussian) with variable dispersion. Since the integral equations are discretized the kernel numerical modification is proposed by the Efros theorem. This modification is carried out by multiplying the discrete analogs of the original kernels by the special matrix obtained on the basis of the Gauss integral. The example of computing elastic and viscoelastic plate deflection is given, kernels of convolution integrals are presented for the "elastic" solution taking into account dissipation energy (internal friction in the material of the plate). Transversal impulse action on rectangular isotropic plates of medium thickness is considered. The plates are assumed to be hinged along the whole boundary. The plate deforming is simulated by the Timoshenko’s refined theory. It is possible to relatively quickly recalculate the deflections and deformations of the plate, taking into account internal friction for different values of the dissipation coefficients.