Analytical and numerical investigation of beam-spring systems with varying stiffness: a comparison of consistent and lumped mass matrices considerations

This study examined the vibration behavior of a beam with linear spring attachments using finite element analysis. It aims to determine the natural frequency with both consistent/coupled mass and lumped mass matrices. The natural frequencies and corresponding mode shapes were correctly determined which formed the basis of any further noise vibration and severity calculations and impact or crash analysis. In order to obtain eigenfrequencies subject to the attached spring, the characteristic equation was obtained by eigenfunctions expansion whose roots were extracted using the root-finding technique. The finite element method by coupled and lumped mass matrices was then used to determine complete mode shapes against various eigenfrequencies. The mode shapes were then analyzed subject to supports with varying stiffness thereby comparing the analytical and numerical results in case of consistent and lumped masses matrices so as to demonstrate how the present analysis could prove more valuable in mathematical and engineering contexts. Utilizing a consistent mass matrix significantly enhanced accuracy compared to a lumped mass matrix, thereby validating the preference for the former, even with a limited number of beam elements. The results indicated that substantial deflection occurred at the beam's endpoints, supporting the dynamic behavior of the spring-beam system.