EXACT BOUNDARY-CONDITION PERTURBATION SOLUTIONS IN EIGENVALUE PROBLEMS

A perturbation method is developed for linear, self-adjoint eigenvalue problems with perturbation operators confined to the boundary conditi ons. Results are derived through third order perturbation for distinct eigensolutions of the unperturbed problem and through second order pe rturbation for degenerate eigensolutions, where splitting of the degen erate eigensolutions from asymmetry is identified. A key feature, demo nstrated for the plate vibration and Helmholtz equation problems on an nular domains, is that the solutions of the perturbation problems are determined exactly in closed-form expressions, The approximation in th e eigensolutions of the original problem results only from truncation of the asymptotic perturbation series; no approximation is made in the calculation of the eigensolution perturbations. Confinement of the pe rturbation terms to the boundary conditions ensures that the exact sol utions can be calculated for any combination of unperturbed and pertur bed boundary conditions that render the problem self-adjoint. The exac t solution avoids the common expansion of the solution to the perturba tion problems in infinite series of the unperturbed eigenfunctions. Th e compactness of solution in this formulation is convenient for modal analysis, system identification, design, and control applications. Exa mples of boundary asymmetries where the method applies include stiffne ss nonuniformities and geometric deviations from idealized boundary sh apes such as annuli and rectangles.