LUMPED MASS AND STICK APPROACH TO DYNAMIC ANALYSIS USING MODIFIED ELEMENT STIFFNESSES

Abstract

Lagrange equations allow a structure to be modeled as an assembly of discrete masses connected by mass-less elements. This is known as the lumped mass and stick model. The result obtained from the application of Lagrange equations is exact for such systems, but when a system with a continuous distribution of mass is modeled as a lumped mass and stick system the results become approximate. Mass discretization as seen in the Lumped mass and stick approach to dynamic approach for the analysis of continuous systems introduces an error due to the mass distribution. An attempt was made at making a corresponding modification in the systems’ stiffness matrix.  The force equilibrium equations of discrete elements of the vibrating continuous system were formulated under free vibration using the Hamilton’s principle and the principle of virtual work and the inherent forces causing vibration obtained. These were then equated to the corresponding equation of motion of the lumped mass and stick system and the stiffness matrix necessary for such equality obtained and expressed as a function of a set of stiffness modification factors. By employing the Lagrange equations to lumped mass beams using these modification factors, we were able to predict accurately the fundamental frequencies of the beams irrespective of the position or number of lumped mass introduced. From this work we can infer that in order to obtain an accurate dynamic response from a lumped mass and stick beams we can modify the stiffness composition of the system. To get a lumped mass beam to be dynamically equivalent to a continuous beam there is need to carry out a nonlinear modification of the structure’s stiffness distribution.

KEYWORDS: Lagrange equations, stiffness matrix, inertia matrix, lumped mass, natural frequency