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 ((&omega;) over bar (R)) and the imaginary part ((&omega;) over bar (l)) of a complex eigenvalue ((&omega;) over bar = (&omega;) over bar (R) + (i) over bar(&omega;) over bar (1)) to derive the mathematical expressions, this paper directly employs the implicit-form of the complex eigenvalue ((&omega;) over bar) to formulate the problem. In the FENII, 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 inatrix. Good agreement between the natural frequencies obtained from FEM2 and the corresponding ones from FEM1 confirms the reliability of the presented theory. (C) 2003 Elsevier Ltd. All rights reserved.
effective stiffness matrix finite element method complex eigenvalue natural frequency degree of freedom (dof)