The derivation of eigenvalues and mode shapes for the bending motion of a damped beam with general end conditions

Abstract

The eigenvalues and mode shapes for the bending motion of a beam with restricted classes of end support, and generally undamped, have been derived separately by a number of authors for use in particular applications. A general method is presented here for the derivation of the complex eigenvalues and eigenvectors for a uniform beam governed by the classical fourth order bending wave equation, the constraints of the supports to shear and rotation being expressed as complex impedances with any desired frequency dependence and damping (with any necessary frequency dependence) being included in the beam material as well as in the supports. The matrix describing the motion is written in a form which can be handled numerically on a computer to provide finite, complex, solutions for any set of dimensonless parameters (describing the magnitude of the support constraint in terms of the beam characteristics) each permitted to vary over the whole range from zero to infinity. The distinction between the effects of damping in the beam material and in the supports is considered, and it is noted that the effect of material damping is dependent on the end conditions.