Exact Closed-Form Solutions for Free Vibration of Double-Beam Systems Interconnected by Elastic Supports Under Axial Forces

Abstract

This paper aims to present the exact closed-form solutions for the free vibration of double-beam systems composed of two parallel beams connected by an arbitrary number of discrete elastic supports. The general solutions of the mode shapes of the double-beam system are derived employing the Laplace transform method from a perspective of the entire domain of beams without enforcement of any segmentation. A unified strategy applied to various boundary conditions is proposed to determine the independent constants involved in the general solutions, as well as the frequency equation. Numerical calculations are performed to verify the present solutions by comparing the results from the previous literature and finite element simulation, and to discuss the effects of support parameters (stiffness, location, and number) on the modal characteristics of the double-beam system in detail. Outcomes show that the support location plays a pivotal role in regulating the modal characteristics of the double-beam system; for each-order mode, there are one or more potential optimal positions to maximize the effect of the elastic support. The mode veering phenomenon is detected as the support parameters change. It is highlighted that, by introducing an amplitude similarity index, the proximity degree for the mode shapes of the two beams influenced by the support parameters can be evaluated quantitatively. The present analysis is greatly helpful to the optimal design, health monitoring, and vibration control of the double-beam system.