The Fundamental Frequencies of Plates with a Core

a₁ arbitrary constants asymptotic expansion asymptotic series averaging radius B₁ B₁Jo(kor B₂lo(kor B₂Yo(cko Bessel functions Biharmonic equation boundary perturbation method Chapter characteristic equation circumferencial waves clamped and simply Clamped Boundary Condition clamped circular plate clamped elliptic plate clamped hexagonal plate coefficients concentric circular core Condition We perturb core of radius Cramer's rule determinant equation following boundary conditions frequency of clamped frequency of simply frequency values fundamental frequency gives the fundamental Imposing the boundary Io(ko Jo(ko Ko(ko let W(r Natural frequencies O(e³ order O(e perturb the solution perturbed boundary Poisson ratio polar coordinates polygonal plates Rayleigh-Ritz method Ritz method Simply Supported Boundary simply supported circular simply supported hexagonal solution of type solution W(r subject to following Supported Boundary Condition supported circular plate supported hexagonal plate type W2(r Un(r unit normal vector Up(r W₁ W₁(r wavy boundary plate Worr Worrr Yo(ko zeroth order equation ᎧᎳ