On the transverse vibrations of non-uniform beams with axial loads and elastically restrained ends.

The dynamic behaviour of slender tapered beams is examined, in the presence of conservative axial loads, and lower and upper bounds on the free vibration frequencies are obtained. Two di!erent approaches are employed, in order to obtain a narrow range to which the frequencies belong. In the "rst case a Rayleigh}Ritz method is used, with displacement trial functions given by linearly independent orthogonal polynomials. In the latter case the structure is reduced to rigid bars, connected together by means of elastic hinges, and lower bound to the true frequencies is obtained. It is well known that the Rayleigh}Ritz approach leads to upper bounds, and therefore a (narrow) range is obtained for the exact frequencies. The paper ends with some numerical examples which con"rm the usefulness of the proposed methods, and are in good agreement with some previously known results. ( 2000 Elsevier Science Ltd. All rights reserved.