Computing Natural Frequencies and Mode Shapes of a Reddy Beam

This paper presents an algorithm to find natural frequencies and mode shapes of an uniform Reddy beam. In this method, spatial state equations are developed. Then, the spatial state transition matrix is computed, which is independent of the boundary conditions of the beam. Then, natural frequencies and mode shape equations are easily derived for any boundary conditions in terms of elements of the state transition matrix evaluated at the right end of the beam. Here, these equations are derived for pinned-pinned and clamped-clamped beams. Numerical difficulties in implementing this method are recognized, and a reduced order spatial state space model is developed by analyzing the nature of roots of the sixth order characteristic equation. Numerical results are presented for rectangular beams and compared to those for Timoshenko beams.