Effects of initial shapes on the natural frequencies of a beam with initial curvature and axial constraints (Q1804391)

The linearized equation of motion in matrix form of an Euler-Bernoulli beam with initial curvature is derived based on the Lagrangian approach and the assumed mode method. The partial axial constraint is imposed in the form of a linear spring at one end of the beam, modeling the relative axial mobility of the two supports. It has been shown in the derivation of the equations that the effect of initial curvature is contained only in the potential energy of the spring and in the strain energy due to the extensibility of the beam. A major feature of the present formulation is its relative simplicity in generating natural frequencies for a beam under various boundary conditions and with different prescribed initial shapes for the undeformed state. It is shown that the natural frequencies strongly depend on the initial shape as well as on the axial constraints of the beam.