Analytical treatment of in-plane parametrically excited undamped vibrations of simply supported parabolic arches

Abstract

The present work offers a simple and efficient analytical treatment of the in-plane un-damped vibrations of simply supported parabolic arches under parametric excitation. After thoroughly dealing with the free vibration characteristics of the structure dealt with, the diffierential equations of the forced motion caused by a time dependent axial loading of the form P = P-0 + P-t cos thetat are reduced to a set of Mathieu-Hill type equations. These may be thereafter tackled and the dynamic stability problem comprehensively discussed. An illustrative example based on Bolotin's approach produces results validating the proposed method.