Аналитическое компьютерное построение первых интегралов уравнений движения дискретных механических систем

Abstract

Предложены аналитические алгоритмы для систем компьютерной алгебры, решающих задачи механики структурно сложных плоских и пространственных дискретных моделей. При этом используется метод описания геометрических и механических свойств систем, принятый в программе такого класса КиДиМ. Алгоритмы включают аналитическую диагностику модели, формирование функций кинетической и потенциальной энергии, лагранжиана, выражений первых интегралов уравнений движения.

Analytic algorithms for computer algebra systems solving problems of the mechanics of structurally complex plane and spatial discrete models are proposed. In this case, a method is used to describe the geometric and mechanical properties of systems adopted in a program of this class of KiDiM. The algorithms include analytical diagnostics of the model, the formation of the kinetic and potential energy functions, the Lagrangian, the expressions for the first integrals of the equations of motion. Analytical computer algorithms for diagnostics of system conservatism, presence of cyclic coordinates, Jacobi integral – generalized energy integral and Routh integrals, automatic formation of such integrals are presented. Discrete mechanical systems described in generalized coordinates and pseudo-coordinates with arbitrary constraints are considered. The description of mechanical models used here is based on the discrete elements – power, inertial, dissipative and elastic. Such elements are assigned, respectively, to active forces actingin the system, inertia forces, linear viscous frictionforces and linear elastic forces. This allows one to compactly describe the model and easily change its parameters and structure. Algorithms are proposed that, based on these data, allow the automatic construction of analytical expressions for kinetic and potential energies. After the construction of the kinetic and potential energies, the Lagrange function is constructed. Further, on the basis of an analysis of its explicit dependence on time and on generalized coordinates, the presence of the first Jacobi and Routh integrals is diagnosed. After this, these integrals are constructed.