Generalized Orthogonality Condition for Beams with Intermediate Lumped Masses Subjected to Axial Force

Abstract

The orthogonality of the modes of vibration of a distributed parameter system plays an important role in the study of the dynamic behavior of that system. The definition of orthogonality for beams with classical boundary conditions is well known. However, it has been shown that the exact mode shapes of beams that carry one or more attached lumped masses are not orthogonal to each other under the classic condition. In this paper, the effect of axial force on the orthogonality condition of the exact mode shapes of beams with several attached lumped masses, as well as translational and rotational springs, is investigated. It has been shown that, in contrast to the mode shapes themselves, the orthogonality condition remains unchanged when an axial force is applied. Furthermore, the generalized orthogonality condition is employed to study the dynamic behavior of a beam—mass system under different boundary conditions. It has been shown that for the precise investigation of the dynamic response, application of exact mode shapes is not adequate enough, and the orthogonality condition associated with the problem must also be used. Finally, a discussion of the effect of the orthogonality condition on the damping matrices is presented. It has been shown that using the exact mode shapes and generalized orthogonality condition may result in non-modal and non-diagonal damping matrices, which, in turn, may increase the computation time.