Apparently first closed-form solution for vibrating: inhomogeneous beams

Free vibration of non-uniform beams, which possess non-homogeneous material density and elastic modulus along their axis, are studied under various boundary conditions. Closed-form expressions for the fundamental natural frequency are derived. It is shown that there is an infinite number of beams that share the same natural frequency. Moreover, it is proved that some coefficients describing the density and elastic modulus functions can be deterministic or random, yet, remarkably, in special circumstances, the fundamental natural frequencies turn out to be deterministic quantities. Extensive numerical analysis is performed to substantiate this seemingly paradoxical finding by the Monte Carlo method, Boobnov-Galerkin method and the finite-element method. (C) 2001 Elsevier Science Ltd. All rights reserved.