Analytical Solution For The Forced Vibration Of Orthotropic Rectangular Thin Plates

Plates, as typical structural elements, are widely applied in aviation, aerospace, automobile, shipbuilding, construction and other fields. Thus, the research for the bending problems of plates is always an important part in the study of solid mechanics. On the analysis of structure, the small deflection bending theory of thin plate, based on the Kirchhoff assumption, is an elementary theory commonly used, and various numerical methods have been developed, such as finite element method, boundary element method, and so on. The successful applications of numerical method never reduce the importance of analytical method. The reasons are as follows:(1) the theoretical basis of numerical approximate solution cannot be taken separately from analytical method; (2) some problems, such as crack tip finite element in fracture mechanics, element in infinite domain, etc, are analytical essentially; (3) the problems with obvious local effect, such as edge effect, free boundary and its edge singularity of composite materials, etc, have rigidity problems, when they are computed using numerical method. However, the study on analytical solution is also desirable.The traditional analytical method is mostly confined in Euclidean space with one kind of variables, and is always faced with the solution of higher-order partial differential equations by eliminating the various unknown variables. Thus the effective methods in mathematical physics, such as variable separation and expansion of eigenfunctions, become inapplicable. In contrast to the traditional method, the symplectic dual method is established in symplectic space with two kinds of variables. Hence a rational solution method can be carried out by using the method of separation of variable and eigenfunction expansion method, so the scope of analytical solutions is expanded.In this thesis, based on the appropriate definition of symplectic inner product, the forced vibration problem of orthotropic rectangular thin plate is introduced into symplectic duality system, and the dual equations are given out. So an analytical approach for the steady state solution of forced vibration is presented by employing separation of variables and eigenfunction expansion.In this thesis, the symplectic eigenproblems for the forced vibration of the orthotropic rectangular thin plates with the typical boundary conditions, i.e. opposite sides simply supported and opposite sides clamped, are discussed. And the transcendental equations of symplectic eigenvalues and the symplectic eigenvectors are given in analytical form. Lastly, analytical solutions of two typical examples are presented by using this method. And the solution of a fully simply supported plate under uniformly distributed harmonic load is chosen to compare with the classical Navier's method, and the result shows that new technique not only has good accuracy but also has better convergence speed than the classical methods, especially on internal forces of plate.