Apparently the first closed-from solution of inhomogeneous elastically restrained vibrating beams

In this paper we consider free vibration of inhomogeneous Bernoulli–Euler beam which is clamped at one end and elastically restrained at the other. The closed-form solution is obtained for the beam of constant material density and constant cross-section but of modulus of elasticity, which varies in a polynomial manner. The semi-inverse method is utilized; namely, the fundamental mode of vibration is postulated as a polynomial too. It turns out that such a formulation leads to infinite number of solutions; one can obtain an unique solution by introducing an additional requirement inherent in vibration tailoring: namely, designing the system that possesses the pre-specified natural frequency. It is shown that if in addition to the fundamental mode shape the natural frequency is also specified, the unique solution is derived.