Timoshenko frequencies and the Wittrick-Williams algorithm: some simple exact solutions to complex problems

An exact fourth order differential equation governing the motion of a Timoshenko beam-column on a uniformly distributed, bi-parametric foundation is presented in non-dimensional form. Two related problems are then solved simply and efficiently. In the first of these, the governing differential equation is transformed into a standard quadratic equation that provides the core of a simple procedure that yields the exact natural frequencies of the member for the simply supported case. In the second problem, the same quadratic equation is used to solve the complementary problem of predicting the number of simply supported natural frequencies of the member passed by any given trial frequency.