Vibration Analysis Skeletal Systems using a Mixed Formulation with an Arnoldi-Based Nonlinear Eigensolution in Technique

Vibration analyses of structural systems are concerned with accurately predicting the natural frequencies and mode shapes of the vibrating system. This analysis process involves two general parts: the dynamic finite element model of the physical system and the numerical algorithm for determining the frequencies and mode shapes from the model. The finite element model establishes the number of equations of motion (degrees-of-freedom) needed to accurately define the behavior of the vibrating system, and the numerical algorithm extracts the frequencies and mode shapes of the system from the resulting eigenproblem. To optimize the effectiveness of a dynamic analysis procedure, both the finite element model and the eigensolution technique must be chosen such that the desired accuracy can be obtained with the most efficient use of computer resources.

The research presented here is in four parts, addressing separate aspects of free vibration analysis. Part one of this study presents a frequency dependent finite element modeling procedure that is more accurate than the conventional finite element models, and a nonlinear eigensolver for determining the natural frequencies and mode shapes from finite element models. The new solution methodology is based upon the ability to evaluate a specific set of parameterized nonlinear eigenvalue curves through an implicitly restarted Lanczos technique. Numerical examples illustrate 'that this method coupled with a secant based zero finder accurately evaluates the exact natural frequencies and modes of the nonlinear eigenproblem and is more computationally efficient than the determinant search technique.

Part two presents a performance comparison of the frequency dependent mixed finite element formulation with the h- and p-formulations of the conventional finite element method for free vibration analysis. Performance is evaluated through a detailed study of accuracy and computational efficiency in obtaining the natural frequencies and modes of skeletal systems. Additional studies compare the computational effectiveness of the mixed finite element formulation with the dynamic element method which involves use of a quadratic eigenvalue problem. Results show that for lower accuracy requirements, the h-formulation is more efficient than the p-formulation but this trend is reversed for higher accuracy requirements. When very high accuracy is needed, especially for higher-order modes, the mixed formulation is more computationally efficient than both the h- and p-formulations.

Part three formulates the interior eigenvalue problem that is used to extract frequencies higher than a specified shift. The eigensolver is modified to solve the nonpositive definite eigenproblem and to extract eigenvalues from the interior of the spectrum. A numerical example is presented to demonstrate the effectiveness of this method. It is also shown that beyond a threshold it is economical to solve the shifted eigenproblem despite the increased cost of factoring a nonpositive definite matrix.

The last part of this study extends the mixed finite element method to model nonclassically damped structures. A new form of frequency dependent damping matrix is presented that better models the nonhomogenous nature of the energy loss mechanisms in structures. The eigenproblem resulting from this model is unsymmetric; the k-step Arnoldi method with implicit restart is used to extract the complex frequencies and mode shapes without resorting to costly complex arithmetic.