A new two-eigenfunctions theory, using theamplitude deflection and the generalized curvature as twofundamental eigenfunctions, is proposed for the freevibration solutions of a rectangular Mindlin plate. The threeclassical eigenvalue differential equations of a Mindlin plateare reformulated to arrive at two new eigenvalue differentialequations for the proposed theory. The closed form eigen-solutions,which are solved from the two differential equationsby means of the method of separation of variables areidentical with those via Kirchhoff plate theory for thin plate,and can be employed to predict frequencies for any combinationsof simply supported and clamped edge conditions.The free edges can also be dealt with if the other pair ofopposite edges are simply supported. Some of the solutionswere not available before. The frequency parameters agreeclosely with the available ones through pb-2 Rayleigh-Ritzmethod for different aspect ratios and relative thickness ofplate.
Yufeng Xing Bo Liu The Solid Mechanics Research Center, Beihang University,100191 Beijing, China
The separation of variables is employed to solveHamiltonian dual form of eigenvalue problem for transversefree vibrations of thin plates,and formulation of the naturalmode in closed form is performed.The closed-form natu-ral mode satisfies the governing equation of the eigenvalueproblem of thin plate exactly and is applicable for any typesof boundary conditions.With all combinations of simply-supported(S)and clamped(C)boundary conditions appliedto the natural mode,the mode shapes are obtained uniquelyand two eigenvalue equations are derived with respect to twospatial coordinates,with the aid of which the normal modesand frequencies are solved exactly.It was believed that theexact eigensolutions for cases SSCC,SCCC and CCCC wereunable to be obtained,however,they are successfully foundin this paper.Comparisons between the present results andthe FEM results validate the present exact solutions,whichcan thus be taken as the benchmark for verifying differentapproximate approaches.
Y. Xing B. Liu The Solid Mechanics Research Center, Beihang University, 100083 Beijing, China
