Eigenfrequencies of an Arbitrarily Supported Beam Carrying Multiple In-Span Elastic Rod-Mass Systems

Many vibrating mechanical and structural systems consist of a host structure to which a number of continuous attachments are mounted. In this paper, an efficient method is proposed for determining the eigenfrequencies of an arbitrarily supported Euler–Bernoulli beam with multiple in-span helical spring-mass systems, where the mass of the helical spring is considered. For modeling purposes, each helical spring can be modeled as an axially vibrating elastic rod. The traditional approach of using the eigenfunctions of the beam and rod in the assumed modes method often leads to an intolerably slow convergence rate. To expedite convergence, a spatially linear-varying function that corresponds to the static deformed shape of a rod is included in the series expansion for the rod. The proposed approach is systematic to apply, easy to code, computationally efficient, and can be easily modified to accommodate various beam boundary conditions. Numerical experiments show that with the addition of a spatially linear-varying function, the proposed scheme converges very quickly with the exact solution.