Чисельний аналіз вільних коливань прямокутних пластин на основі різних підходів

Abstract

Визначаються частоти та форми вільних коливань тонкої жорстко закріпленої квадратної пластини сталої товщини на основі двох чисельних методів. Дослідження проводились методом скінченних елементів (МСЕ), який реалізовано на ліцензійному програмному засобі Femap з розв’язувачем NX Nastran і методом Релея-Рітца, який модифіковано та реалізовано на ЕОМ у середовищі Delphi. Достовірність отриманих результатів забезпечується використанням обґрунтованої математичної моделі, коректністю постановки задачі, розв’язуванням тестових задач та практичною збіжністю розрахованих частот методом скінченних елементів і методом Релея-Рітца. У результаті дослідження у формулі методу Релея-Рітца виділено три коефіцієнти, що характеризують геометричні розміри пластини, фізико-механічні властивості матеріалу і форму коливань, зроблено уточнення останнього коефіцієнта, що підвищило точність розрахунку. Створено програму в середовищі Delphi та поширено МСЕ для розрахунку частот і форм вільних коливань. Результати розрахованих частот мають хорошу збіжність з результатами, отриманими іншими авторами експериментальним та чисельним методами. Проведено порівняльний аналіз частот і форм вільних коливань розглянутої пластини, розрахованих двома чисельними методами.

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Plates of various shapes are the most common elements of thin-walled structures in various branches of modern technology. It is necessary to calculate the load-bearing elements of structures for the case of various suddenly arising dynamic loads when various engineering facilities are designing and constructing. Therefore, the study of the dynamic behavior of engineering structures elements taking into account the properties of the material and the impact of the environment during dynamic load is an actual problem. In this connection it is necessary to extend the numerical methods for studying the natural oscillations frequencies and forms of rigidly fixed square plates.

Modern science does not ignore this problem, so in [8] we give an overview of the works devoted to the calculation of the frequencies of free oscillations of a rectangular plate; the results of computational experiments are given. In [9] analytic expressions were constructed for the calculation of eigenfrequencies and forms of oscillations of a square homogeneous plate fixed along the contour. In [10; 11] the solution of the problems of free oscillations of rectangular plates with variable thickness is proposed with the aid of a numerical-analytic spline collocation method in conjunction with the discrete-orthogonalization method, for example in [10], the problem of free oscillations of a square plate with a constant thickness fixed along the edges is solved as a mathematical model. The calculated frequencies are compared with the experimental data obtained by the method of holographic interferometry. In [11], the free oscillations frequencies of anisotropic rectangular plates with different thickness with different boundary conditions are calculated. In [12], orthotropic rectangular plates of variable thickness are considered based on the spline approximation method and the discrete orthogonalization method in conjunction with the step-by-step search method. One of the modern methods for calculating problems of dynamics using computers is the finite element method (FEM). In [13], a thin square plate with different physical-mechanical characteristics with free edges is investigated. The geometry modeling and numerical calculation of the frequencies and forms of free oscillation of plates are carried out by the finite element method, which is implemented using the computer program FEMAP with NASTRAN solving. A comparative analysis of the calculated eigenfrequencies with frequencies obtained numerically and experimentally by other authors is carried out.

Free oscillations frequencies and forms of a thin rigidly fixed square plate with constant thickness are determined based on two numerical methods. The research was carried out by the finite element method which is implemented on the licensed software FEMAP and the Rayleigh-Ritz method which is modified and implemented on a computer in the Delphi environment. The reliability of the results obtained is ensured by using a valid mathematical model, the correctness of the problem statement, the solution of test problems and the practical convergence of the calculated frequencies by the finite element method and the Rayleigh-Ritz method. As a result of the research in the Rayleigh-Ritz method formula three coefficients characterizing the geometric dimensions of the plate, the physical-mechanical properties of the material and the oscillations mode were separated, clarification of the last coefficient increased the accuracy of the calculation. The program was created in the Delphi environment and the finite element method was distributed to calculate frequencies and forms of free oscillations. The results of the calculated frequencies have good agreement with the results obtained by other authors by experimental and numerical methods. A comparative analysis of the frequencies and forms of free oscillations of the considered plate calculated by two numerical methods was carried out.