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 bo undary conditions. Closed-form expressions for the fundamental natural freq uency are derived. It is shown that there is an infinite number of beams th at share the same natural frequency. Moreover, it is proved that some coeff icients describing the density and elastic modulus functions can be determi nistic or random, yet, remarkably, in special circumstances, the fundamenta l natural frequencies turn out to be deterministic quantities. Extensive nu merical analysis is performed to substantiate this seemingly paradoxical fi nding by the Monte Carlo method, Boobnov-Galerkin method and the finite-ele ment method. (C) 2001 Elsevier Science Ltd. All rights reserved.