Use of effective stiffness matrix for the free vibration analyses of a non-uniform cantilever beam carrying multiple two degree-of-freedom spring–damper–mass systems

This paper investigates the free vibration characteristics of a non-uniform cantilever beam carrying multiple two degree-of-freedom (dof) spring–damper–mass systems by means of two finite element methods, FEM1 and FEM2. Where the FEM1 is the conventional finite element method (FEM) with each two-dof spring–damper–mass system being considered as a finite element possessing stiffness, damping and mass matrices, while the FEM2 is an alternative approach with each two-dof spring–damper–mass system being replaced by an effective stiffness matrix composed of four massless effective springs. Instead of using both the real part (ω̄R) and the imaginary part (ω̄I) of a complex eigenvalue (ω̄=ω̄R+īω̄I) to derive the mathematical expressions, this paper directly employs the implicit-form of the complex eigenvalue (ω̄) to formulate the problem. In the FEM1, since each spring–damper–mass system has two dof, the total dof of the entire system increases two if the beam carries one more two-dof spring–damper–mass system. However, in the FEM2, the total dof of the entire system remains unchanged, because all dof of each spring–damper–mass system are suppressed by the effective stiffness matrix. Good agreement between the natural frequencies obtained from FEM2 and the corresponding ones from FEM1 confirms the reliability of the presented theory.