On ensuring solution accuracy of model analysis problems

Аuthors

Korovaytseva E. A.

Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia

e-mail: katrell@mail.ru

Abstract

By the example of modal analysis problems formulation, introduction of three simplest canonical forms is suggested, allowing structuring the description of problems solution algorithms and reasonably minimize their spectrum. For the basic algorithm, supposing the method of initial parameters application, the use of integral approach to the problem solution is proposed.

This approach supposes orthogonality of normalized integral matrices of the initial and conjugate differential equation systems condition checking at each integration step at the problem preprocessing step and segmentation of integration interval at the points in which this condition is not met. This approach efficiency is illustrated by the examples of free oscillations of hinged beam and cylindrical shell problems solution analysis. It is shown, that tenth and higher natural frequencies of the hinged beam cannot be calculated correctly without using segmentation according to the approach suggested. The article compares several numerical integration methods combined with segmentation. They are Runge-Kutta method, realized in authorial program, and Runge-Kutta, Adams-Bashforth, Gear and trapezoids with free interpolation methods realized as built-in MATLAB functions. Runge-Kutta method realized in authorial program demonstrated the best accuracy and computational speed. Besides, comparison of Runge-Kutta method of the fourth and tenth orders of accuracy shows that the method of the higher order is less accurate.

Calculations of the first natural frequency and mode of cylindrical shell oscillations were performed according to the general moment theory with account for inertia in three directions. Godunov’s orthogonalization method was used. The numbers of orthogonalization and integration steps were calculated according to segmentation methodology suggested in the paper and common approximate methodology based on numerical experiments of the prior authors. The studies revealed that applying common approach to integration interval segmentation cold lead to wrong results of natural frequencies and modes calculation, while application of the approach suggested in the paper allows obtaining results with the required accuracy.

Keywords:

boundary problems, modal analysis, initial parameters method, conjugate equations

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