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Analysis of damped linear dynamic systems and application of component mode synthesis

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Title: Analysis of damped linear dynamic systems and application of component mode synthesis
Author: Muravyov, Alexander
Degree Master of Applied Science - MASc
Program Mechanical Engineering
Copyright Date: 1994
Abstract: The analysis of nongyroscopic damped (viscous) linear dynamic systems is presented. Discrete systems having symmetric mass, stiffness and damping matrices are considered. Discretization of the systems is accomplished by application of the finite element proce dure. The general case of classically damped systems is considered, and the necessary and sufficient condition for classical damping is given. For a system of this type, it is possible to specify the damping matrix that will result in each mode having either a prescribed decay factor or a damped eigenfrequency. This may be accomplished with only a knowl edge of the undamped eigenfrequencies. The equations required to accomplish this task are presented. Graphical results are presented that illustrate the effect of damping for mass-proportional, stiffness-proportional and for Rayleigh damping. Nonclassically damped systems are considered and the formulation of a component mode synthesis (CMS) method for the solution of the free vibration problem is described. The component mode synthesis method is a procedure in which the exact solution is approximated by one constructed from some basis vectors (e.g., mode shapes) of sub systems (components of the original system). This method allows significant reduction of the eigenvalue equation size due to the use of a limited number of basis vectors. The approximate solution found for the lower eigenvalues and eigenvectors is very close to the exact one due to the proper selection of the basis vectors and the procedure followed (e.g., Galerkin’s method) that determines the best approximation. The use of CMS methods is especially advantageous in the case of large systems, subjected to numerous modifications. In this work the formulation of the CMS method was developed for the general case of nonclassically damped systems. It was tested for different cases of noriclassically damped systems and the excellent agreement with the exact results derived from nonsubdivided systems was found. Also a new method to treat an unconstrained component for the purpose of stiffness matrix inversion is presented. The selection procedure of component modes is generalized from the undamped system case to the damped one. Some examples of forced responses are considered, particularly, the case of sinusoidal excitation and the influence of the damping factor is analyzed. The experimental part of this study consists of the designing and testing of a vibration rig designed to simulate the behaviour of a rigid engine resting on isolators that in turn are supported on flexible beams. Free and steady-state responses of the rig are experimentally determined. Comparison of analytical results with experimental ones show good agreement for eigenquantities and steady-state forced response.
URI: http://hdl.handle.net/2429/5087
Series/Report no. UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/]

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