Upper and Lower Bounds for the Eigenvalues of Vibrating Beams with Linearly Varying Axial Load

Abstract : Previous investigations have demonstrated the importance of the effect of linearly varying axial or in-plane loading on the vibration characteristics of beams and flat plates. It has already been established that the problem reduces to solving for the eigenvalues of a fourth order, variable coefficient differential equation that can not be solved in closed form. Beginning with a variational representation of the eigenvalue problem, methods are discussed by which both upper and lower bounds for the eigenvalues may be formed, The true eigenvalues may thus be estimated as being bracketed by the upper and lower bounds which are shown to approach each other. The bounds for the eigenvalues may also be estimated by an averaging procedure which may or may not compare favorably with the true values depending on the values of the loading parameters. Finally, numerical values for upper bounds, lower bounds, and average lumped end-load eigenvalues are computed on an IBM 7090 Computer.