Use of equivalent-damper method for free vibration analysis of a beam carrying multiple two degree-of-freedom spring–damper–mass systems

Although the dynamic characteristics of a beam carrying multiple two degree-of-freedom (dof) spring–damper–mass systems is one of the important topics in engineering, the information in this aspect is rare. The object of this paper is to replace the effect of each 2-dof spring–damper–mass system, composed of two springs, two dashpots and one lumped mass, by a set of equivalent dampers, so that the natural frequencies of a beam carrying any number of 2-dof spring–damper–mass systems may be solved from a beam supported by the same number of sets of equivalent dampers. Instead of using both the real part ( ω ¯ jR ) and the imaginary part ( ω ¯ jI ) of a complex eigenvalue, this paper uses the implicit-form complex eigenvalue, ω ¯ j , to derive the mathematical expressions, therefore, much more compact formulations were obtained. To confirm the reliability of the presented theory, all the numerical results obtained from the equivalent-damper method (EDM) were compared with those obtained from the conventional finite element method (FEM) and good agreement was achieved. Since the order of the overall property matrices for the equations of motion of the entire structural system derived from the EDM is much less than that derived from the FEM, the computer time required by the EDM is much less than that required by the FEM, particularly in the forced vibration analysis of a structural system using the step-by-step integration method, where the CPU time consumed is proportional to the total number of time steps. In addition, the EDM also provides a simple approach for evaluating the damping effect of each spring–damper–mass system. Furthermore, the presented equivalent dampers will provide an alternative choice for the effective vibration absorbers, because the damping effects of the equivalent dampers are dependent on the physical properties of their constituent parts (i.e., the springs, the dashpots and the lumped masses) and will be more flexible (or adjustable) than the damping effects of the classical dampers.