Using a Characteristic Force Approach to Determine the Eigensolutions of an Arbitrarily Supported Linear Structure Carrying Lumped Attachments

In this paper a characteristic force approach is developed that can be used to determine the eigenvalues and mode shapes of any arbitrarily supported linear structure carrying various lumped attachments, including a lumped mass, rotary inertia, grounded translational or torsional spring, grounded translational or torsional viscous damper, and an undamped or damped oscillator with or without a rigid body degree of freedom. Using the proposed approach, each lumped element is first replaced by the load, either a force or moment, that it exerts on the linear structure, thus transforming the free vibration problem into a forced vibration one. By expressing the deflection of the linear structure in terms of these forces or moments and enforcing the compatibility conditions at each attachment points, the roots of the characteristic determinant of these loads can be graphically or numerically solved to find the eigenvalues of the combined system. Once the eigenvalues have been found, the corresponding mode shapes can be readily obtained. The proposed method is easy to code, systematic to apply, and can be easily modified to accommodate any arbitrarily supported one- or two-dimensional linear structure carrying various lumped attachments.