Qualitative Properties of Frequencies and Modes of Vibrating Multi-Span Beam

Abstract

We begin in this paper by introducing the finite-difference model and the corresponding stiffness matrix for an arbitrarily supported multi-span beam. Next, we derive the condition under which the stiffness matrix becomes sign-oscillatory for a two-span beam. The work is then expanded to cover a three-span beam. In the subsequent discussion, we show that the more generalized case of a multi-span beam can be obtained by applying the method of mathematical induction. Based on the theory of oscillatory matrices, we establish several qualitative properties of natural frequencies and normal modes for the multi-span beam. Finally, a numerical example on a three-span overhanging beam is given to demonstrate the qualitative properties of vibration.