# DYNAMIC ANALYSIS OF A BAR WITH END RESTRAINT UNDER LONGITUDINAL VIBRATION

## Keywords:

Lagrange’s equations, discrete masses, lumped mass, stiffness, natural frequency## Abstract

Lagrange equations allow structural systems to be modeled as an assemblage of discrete masses connected by mass-less elements. The solution presented by the Lagrange equations is exact for such systems, but when a continuous system is modeled as having discrete masses connected by mass-less elements the results become approximate. Mass discretization as seen in the use of Lagrange equations for the analysis of continuous systems introduces an error in the inertia matrix. This error can be corrected by making a corresponding modification in the systems’ stiffness matrix. To achieve this, the force equilibrium equations of discrete elements of the continuous system were formulated for such systems under free vibration (using the Hamilton’s principle and the principle of virtual work) and the inherent forces causing vibration obtained. This was then equated to the corresponding equation of motion of the lumped massed (with discrete masses) system and the stiffness matrix of the system necessary for such equality obtained as a function of a set of modification factors. This was used to generate a table of stiffness modification factors for segments of the fixed-fixed beam under longitudinal vibration. By employing the Lagrange equations to lumped massed beams using these modification factors, we were able to predict accurately the fundamental frequency of the beam irrespective of the position or number of lumped masses introduced.