Nonlinear Vibration of an Inextensible Rotating Beam

An inextensible beam model is used to study the nonlinear vibration of a rotating cantilever beam undergoing large deformation. An inextensionality constraint is utilized to reduce two general equations of motion governing the coupled axial bending deformation into one. The axial strain on the neutral fiber is decomposed into static and dynamic components and the dynamic strain is assumed to be zero. Nevertheless, the static strain obtained from the solution of the steady-state axial displacement equation is retained in the equation governing the planar motion of the beam. The method of multiple scales is adopted to derive the expression for frequency response for the first vibration mode. The nonlinear frequency response is plotted at various values of rotation speeds and excitation magnitudes. The results show that the first mode exhibits a hardening type of nonlinearity for lower rotation speeds and softens at higher rotational speeds.