Symplectic Solutions Of Annular Sector Plate Bended With Circumferential Coordinate Treated As Time

In classical elasticity mechanics, the solution method is to eliminate the some ofunknown variables and can get simplest equation in a brief form. As the result, the rank ofequation is increased, so separating the variables and expanding eigenfunction methodscannot be applied to solve the equation.In this paper, systematic theory is applied to the mechanics of elasticity. There is analogyrelationship between plane elasticity and plate bending. Their governing equations are bothbiharmonic equation. Strain-displacement relation, stress function-stress relationship andstress-strain relationship in plane elasticity correspond to bending moment-bending momentfunction, deflection-curvature relationship and bending moment- curvature relationship inplate bending respectively. So the symplectic solution system can be applied in plate bendingas plane elasticity. Dual equations and Hamiltonian operator matrix can be derived from thePro-H-R variational principle, and it becomes the transverse eigen problem of Hamiltonianmatrix after separation of variable. There is the adjoint symplectic orthogonality relationshipbetween the eigenfunction vectors, so any solution can be expanded by the eigenfunctionvectors. In this paper, the general form of non-zero eigenfunction vector can be given, andthen be substituted into lateral boundary conditions to get the transcendental equation fornon-zero eigenvalues. So the non-zero eigenvalues and eigenvectors can be presented.According to the adjoint symplectic orthogonality relationship and the expansion theorem, theexpression which satisfies all control equation in the domain and two lateral boundaryconditions is listed. After substituting it into two end boundary conditions, solution of theoriginal problem is obtained.In this paper, several examples are given, and excellent accuracy has been obtainare byusing only several eigenvalues. The new method presents the analytical solutions in annularsector plate via separation of variables and expansion of eigenfunction vector. It breaks thelimitation of traditional semi-inverse solution. The results show that the new method has vastapplication foreground.