Exact Boundary Condition Perturbation Solutions in Eigenvalue Problems

Abstract

A perturbation method is developed for linear, self-adjoint eigenvalue problems with perturbation operators confined to the boundary conditions. Results are derived through third order perturbation for distinct eigensolutions of the unperturbed problem and through second order perturbation for degenerate eigensolutions, where splitting of the degenerate eigensolutions from asymmetry is identified. A key feature, demonstrated for the plate vibration and Helmholtz equation problems on annular domains, is that the solutions of the perturbation problems are determined exactly in closed-form expressions. The approximation in the 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 perturbation terms to the boundary conditions ensures that the exact solutions can be calculated for any combination of unperturbed and perturbed boundary conditions that render the problem self-adjoint. The exact solution avoids the common expansion of the solution to the perturbation problems in infinite series of the unperturbed eigenfunctions. The compactness of solution in this formulation is convenient for modal analysis, system identification, design, and control applications. Examples of boundary asymmetries where the method applies include stiffness nonuniformities and geometric deviations from idealized boundary shapes such as annuli and rectangles.