Vibration of Cantilever Beam with Time-Varying Length

Abstract

Free vibration of a cantilever beam with time varying length is analyzed. An equation of motion described by a partial differential equation cannot be solved by the method of separation of the variables. In the present paper an equation of motion of a beam with a constant length is described by a partial integro differential equation and then the vibration of the beam with a moving support where unknown load and bending moment act is analyzed. From the condition that the displacement and its derivative of the beam at the support should be zero, two integral equations of Volterra type of first kind with respect to unknown load and moment are obtained. These integral equations are solved approximately and numerical examples of free vibration are shown.