Application of the Orthogonal Differential Scan Method in the Analysis of Stationary Dynamic Problems in Heterogeneous Structures
Year: 2025
Type: article
Abstract: Purpose. It is development of a numerical algorithm for solving a wide range of stationary dynamics problems on the interaction of plates and cylindrical shells with elastic inertial foundations, the mechanical parameters of which vary in thickness according to arbitrary continuous or discrete laws. Solution of specific problems by the obtained algorithm and assessment of the influence of the variability of the medium characteristics on the main dynamic characteristics using three stationary dynamics problems as an example. Research methods. Timoshenko-type equations and equations based on the Kirchhoff-Love hypothesis are used to describe the motion of plates and shells. The behavior of the foundation is described by dynamic equations of elasticity theory in stresses, from which, taking into account the changeability of Young's modulus and density, equations in displacements with variable coefficients are derived, which in the general case cannot be integrated analytically. For the boundary value problem for a system of ordinary differential equations obtained as a result of the transition to amplitudes and recording the contact conditions between a thin-walled structure and a non-uniform foundation, the orthogonal differential sweep method is used, which uses the Runge-Kutta method at each orthogonalization step to solve the problem. Thus, in problems of free waves and natural frequencies, the eigenvalue problem is solved, and in problems of moving loads, it is necessary to use the reverse move and then apply the superposition method. To assess the accuracy, as well as the optimal choice of orthogonalization nodes in the limiting cases of all the considered problems with a homogeneous foundation, analytical solutions are found and a comparison of the results is carried out. Results. A software package for numerical solution of homogeneous and heterogeneous boundary value problems for systems of ordinary differential equations with variable coefficients based on the orthogonal differential sweep method has been developed. In this study, this software package is used for problems from three main classes of stationary dynamics problems: propagation of free waves in a cylindrical shell – heterogeneous elastic filler system, determination of natural frequencies in a three-layer cylindrical shell with a heterogeneous middle layer, and motion of a normal load along a plate on a foundation heterogeneous in thickness. Each problem shows the influence of the foundation heterogeneity on phase velocities, natural frequencies, and plate displacement. The efficiency of the developed program for more complex practical problems has been tested using boundary transitions to corresponding homogeneous materials, and recommendations have been made on the use of this algorithm for media with a significant number of layers. Scientific novelty. An algorithm based on the orthogonal differential sweep method has been developed for problems of stationary dynamics of plates and shells interacting with a three-dimensional heterogeneous inertial medium. Three problems from three different classes of this section of solid mechanics have been solved with its help. The ways of further use of the developed package for problems of such classes are indicated. Practical value. The developed software package and the results obtained with its help can be used in the practice of design organizations engaged in the development of elements of new technology, construction, and the construction of modern highways. In addition, the mathematical part of this publication can be useful not only in the mechanics of a deformable solid, but also in those sections of science where boundary value problems arise for systems of ordinary differential equations.
Language: English
Subfield: Mechanics of Materials
Field: Engineering
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4409214068