Calculation of eigenvalue and eigenvector derivatives for nonlinear beam vibrations

A beam with immovable ends under large amplitude oscillations is axially subjected to an amplitudedependent stretching force which is induced by the geometrical nonlinearity. The design sensitivity analysis for such a beam is investigated in this study. Two variational formulations pertaining to nonlinear beam vibration based on two different approximations of nonlinear stretching force are presented. The analytical equations of design sensitivity analysis for these nonlinear equations are then derived. Finally, a computational scheme using the finite element method is developed to calculate the design sensitivity derivatives of the eigenvalue and eigenvector numerically. The numerical results show that the proposed algorithm for calculating design derivatives of nonlinear eigenvalues and eigenvectors is valid, and, compared with the direct finite difference method, the proprosed algorithm is efficient.

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