Термомеханічна поведінка товстостінної в’язкопружної циліндричної панелі

Abstract

Дано постановку просторової задачі про вимушені резонансні коливання й дисипативний розігрів в’язкопружної товстостінної циліндричної панелі з шарнірно обпертими торцями. Непружна поведінка матеріалу описується концепцією комплексних характеристик. Вважається, що механічні й теплофізичні властивості матеріалу не залежать від температури. Задача розв’язується методом Фур’є і зводиться до крайових задач для систем звичайних диференціальних рівнянь. Для розв’язування цих систем запропоновано використовувати ефективний метод дискретної ортогоналізації. Дано порівняння результатів розрахунків із використанням просторової постановки й класичної та уточненої теорій оболонок.

The statement of three-dimensional problem on the forced resonance vibrations and dissipative heating of the viscoelastic thick-walled cylinder panel with simply supported edges is presented. The nonelastic material behavior is described by conception of complex characteristics. It is supposed that mechanical and thermophysical material properties do not depend on temperature. The problem is solved by the method of variable separation and is reduced to the boundary-value problems for the systems of ordinary differential equations. These systems are solved by the method of discrete orthogonalization. The comparison of calculation results taking advantage of three–dimensional and classical shell theories is presented. To describe non-elastic mechanical behavior the concept of complex characteristics is used. Panel edges are considered to be pressed by the hinge. It makes possible to reduce the problem to the system of general differential equations of high order using the method of variables distribution, for solving of which the effective numerical method of discrete orthogonalisation is proposed to be used, which provides high enough accuracy. The characteristics of the viscoelastic material are accepted not to be dependent by temperature. Then the problem is divided into some separate problems: 1) the problem on the induced resonance vibrations of the viscoelastic cylindrical panel; 2) the problem on calculation of the dissipation; 3) the problem on calculation the temperature field with known heat source. Thus, for independent on the temperature properties of materials the problem was reduced to the solving of boundary problems for the system of general differential equations, which describes mechanical vibrations of the cylindrical panel and solving of boundary problems for general differential equations, which describe heat conduction. To obtain accurate analytical solutions of given boundary problems is, in fact, impossible. Thus, to solve them the numerical method of discrete orthogonalisation has been proposed. As the example the problem on vibration of the cylindrical panel, made of polyethylene, has been analyzed. To estimate the shear strain effect the problem on the vibration of the cylindrical shell using the theory of more precise definition has been analyzed. Amplitude – frequency characteristics calculated taking advantage of the discrete orthogonalisation method, have been presented.