Free vibration of a beam subjected to large static deflection

Considered in this paper is the problem of free vibration of a beam that has undergone a large static deflection. The nonlinear equations of motion for the beam are derived first. The equations are then decomposed into a set of nonlinear differential equations for the static deflection and a set of linear differential equations for the vibrational motion of the beam. The coefficients of the vibration equations consist of the beamʹs static deflection. The nonlinear differential equations are solved analytically to obtain the static deflection. The vibration equations are solved by expanding the displacements in a power series. Coefficients of the power series are constructed analytically through a recursive relationship. The natural frequencies of the beam under large static bending are determined by solving a 3×3 eigenvalue problem. Substitution of the eigenvalues and eigenvectors into the power-series expansion of the displacements yields the corresponding modes of vibration. Several numerical examples are given to illustrate the solution procedure.