Characteristic equations of longitudinally vibrating rods carrying a tip mass and several viscously damped spring-mass systems in-span

Abstract

This paper deals with the determination of two alternative approximate formulations for the frequency equation of a longitudinally vibrating fixed-free elastic rod carrying a tip mass (primary system) to which several spring-mass-damper systems (secondary systems) are attached in-span. The first approximate formulation presented in this study is based upon the assumed-mode method in conjunction with the Lagrange multiplier method. The result is a simple analytical formula for the characteristic equation of the system. Hence, the eigenfrequency parameters of the system are determined by solving this non-linear equation. In this method, the beam is treated as one component and the spring-mass-damper systems that are attached to the beam are treated as separate components. The dynamics of the beam and spring-mass-damper systems are initially expressed in terms of component modes; then the total system dynamics are evaluated by invoking the constraint equations that described the attachments of the spring-mass-damper system to the beam. Lagrange multipliers are used to include the constraint equations in the Lagrange equations of motion for the total system. The second form of the characteristic equation presented in this study follows directly from the formalism of the Lagrange equations where the displacements of the attachment points of the spring-mass-damper systems to the rod are expressed in terms of the generalized coordinates. This method is essentially based upon the classical Rayleigh-Ritz modal method. The formulation leads to a standard eigenvalue problem, the solution of which gives the eigenfrequency parameters of the system. Afterwards, for comparison purposes, ‘exact’ characteristic equations for one secondary system and two secondary systems are established separately via a boundary value problem formulation. All characteristic equations are then numerically solved for various combinations of physical parameters and the results are collected in tables. The comparison of the numerical results obtained via boundary value problem formulations justifies the approximate approaches used here.