Free vibration analysis of beams carrying a number of two-degree-of-freedom spring-damper-mass systems

摘要

This paper presents a technique to replace a two-degree-of-freedom (two-dof) spring-damper-mass (SDM) system by a set of effective masses, so that one may obtain the natural frequencies of a beam carrying any number of elastically mounted two-dof lumped masses from the same beam carrying a number of rigidly attached effective masses. Due to the existence of damping, the characteristic equation is in the form of complex numbers. Instead of the conventional approach of using both the real part and the imaginary part of an eigenvalue (ω_R +- ω_I), this paper directly uses the implicit complex eigenvalue (ω) to derive the mathematical expressions. Therefore, the formulation of this paper is as compact as that for a beam carrying multiple one-dof SDM systems. To validate the reliability of the presented theory, all the numerical results obtained from the effective-mass method (EMM) are compared with the corresponding ones obtained from the conventional finite element method and good agreement is achieved. Since all the degrees of freedom for each two-dof SDM system are suppressed by the effective masses, the total degree of freedom for the entire vibrating system does not increase with increasing the total number of the two-dof SDM systems attached to the beam and some computer storage memory may be saved. This advantage of the EMM will be predominant if the total number of the two-dof SDM systems is large.