Computer Solution for Determining Transverse Dynamic Displacements of Thick Elastic Plates Under Uncertainty Conditions

Abstract

The article poses the task of finding the function of dynamic transverse displacements of thick elastic plates under the action of axisymmetric external dynamic disturbances acting on the surface of such plates. When setting the problem, it is indicated that some initial-boundary observations are known for the state of the plate, but their number is insufficient to solve this problem by classical methods of mathematical physics or numerical integration. Therefore, it is possible to apply to this problem the method of mathematical modeling of externally distributed dynamic processes under conditions of uncertainty, i.e. under insufficient number of initial and boundary conditions. For such a method, it is necessary to have an integral mathematical model and its main components - kernels, that is, integral functions for which there is a finding method. Or they can be calculated using numerical methods of computer algebra systems.

With the built integral model of the dynamics of thick elastic plates, taking into account its kernels, the method of mathematical modeling of the dynamics of spatially distributed processes leads to the result - a set of solutions that accurately satisfy the integral and differential models and agree with the initial boundary conditions according to a certain criterion.

The article selects one of the many solutions to the problem of finding the function of transverse dynamic displacements, which is found according to the methodology of mathematical modeling of the dynamics of spatially distributed processes and thanks to the calculation of the integral function of the mathematical integral model. For the problem, the elastic characteristics and density of the slab corresponding to some material are fixed, some specific external dynamic disturbances and initial-at-the-edge observations are determined, which are represented by certain conditions at specific points. Under such assumptions, graphs of the functions of transverse dynamic displacements are constructed at different values of the transverse coordinate z and at the value 0 of the radial coordinate r of the cylindrical coordinate system.