厚圆板轴对称自由振动的三维理论解

Three-Dimension Theory Solution of AxisymmetricallyFree Vibration of a Thick Circular Plate

本文用三维弹性动力学理论研究厚圆板在柱面约束ｕ＝τ＿（ｒｚ）＝０下的轴对称自由振动问题，给出位移和应力振型的解析表达式，求得固有频率的精确值。研究结果表明：在板的厚度方向存在着不同半波数的反对称形式和对称形式的位移振型，就是说介质之间在厚度方向具有不同的运动形式，这是各种厚板理论都无法反映的。但是对于圆板的轴对称基本振动来说，Ｈｅｎｃｋｙ、Ｒｅｉｓｓｎｅｒ和Ｍｉｎｄｌｉｎ理论求得的固有频率与本文的弹性动力学理论结果都比较接近，仍不失其实用价值。

n this paper, the axisymmetrically free vibration of a tick circular plate with the restrainedcylindrical surface of u= τ_rz = 0 is investigated in the theory of three-dimensional elastodynamics.The analytical expressions of the displacement and strese model shapes and the exact values of the natu-rel frequencies are obtained.The model shapes of the placement components as well as their partial dirivatives can be expandedin the theel series in the radial direction by means of the H transformation. Introducing the series intothe elastedynamical soverning equations, the analytical seneral solutions are obtained. To satisfy theboundary conditions of the circular plate , the frequency equations are derived from which the exactvalues of various natural frequencies are calculated.It is found that there are asymmetric and symmetric medel shapes of displacements with differentwave numbere along the thickness of the plate , and there are various motion modes among the mediumin the thicknesa direction. It is more possible for the thick circular plate to vibrate in the symmetricmedel shapes. The thick-wall engineerins structures are provided with the similar properties which aremore important for the high-frequency vibration of the structure. These characteristics of the vibrationare impossibly shown from any thick plate theory beeause in thcse theories the placement models alongthe thickness are assumed in advance.The natural frequencies of the plates with different thicknesses are calculated by use of the elasto-dynamics theory and the thick plate theories of Henkey , Reisaner and Mindlin. The numerical resultsshow that for the fundamental vibration of the plate, the results of the theories of Henkey. Reisanerand Mindlin are all approaching to ones of the elastodynamics theory in the pal)er and the thick platetheories are practicable.