Dynamics of a beam with multiple spring/mass attachments using binary asymptotic admissible functions

Abstract

The dynamics of an Euler–Bernoulli beam, with multiple spring/mass attachments, is investigated in this paper. A novel analytical approach for determining the natural frequencies and mode shapes of the system is presented. The formulation is applicable for any parameter value of the attached mass or spring stiffness. The calculation is performed using a minimal binary set of admissible functions for each spring/mass connection, with extremal values (zero or infinity) of the parameter. For example, to study a beam with n intermediate spring/mass connections, only 2 n admissible functions are required. These functions are used in a Rayleigh–Ritz-based energy formulation. The small number of functions used in the formulation leads to an efficient computational procedure. The results from the proposed formulation are compared with those from a finite element simulation, for different boundary conditions. The results obtained by the two methods are in excellent agreement for all boundary conditions as well as for different parameter ranges. Detailed design studies, on the effect of spring stiffness and lumped mass values, as well as their locations, on the natural frequencies and mode shapes, have been carried out. Design charts have also been generated to aid the designer of such structures.