Vibration of a Timoshenko Beam: Clarifications on Second Spectrum and Fourth-Order Single Partial Differential Equation

Purpose: This paper investigates into the existence of the second frequency spectrum of an uniform Timoshenko beam, which has been debated in the literature. The equivalence of the fourth order partial differential equation to two second-order partial differential equations of an uniform Timoshenko beam from the perspective of boundary conditions, another topic of debate in the literature, is examined by using the concept of observability of a state space model. Methods: Natural frequencies and mode shapes are computed using the new approach developed by the author in his previous papers, which is based on the spatial state space analysis. Results: Numerical results are presented for pinned-pinned, clamped-free, clamped-clamped and clamped-pinned Timoshenko beams. Conclusions: It has been found that the second frequency spectrum exists irrespective of boundary conditions. Also, boundary conditions for the fourth-order partial differential equation can be derived from the boundary conditions for two original second-order partial differential equations.