Iterative Methods for Eigenvalues of Viscoelastic Systems

Abstract

This paper proposes a new iterative approach for the calculation of eigenvalues of single and multiple degree-of-freedom viscoelastic systems. The Biot model of viscoelasticity is assumed. With this model, the viscoelastic forces depend on the past history of motion via convolution integrals over exponentially decaying kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the order of the eigenvalue problem to be solved and can become computationally expensive for large systems. The method proposed in this paper is aimed to address this issue. In total, five iterative algorithms for the real and complex eigenvalues of single and multiple degree-of-freedom systems have been proposed. The results are obtained in terms of explicit closed-form expressions. This enables one to approximately calculate the eigenvalues of complex viscoelastic systems using the eigenvalues of the underlying elastic systems. Representative numerical examples are given to verify the accuracy of the derived expressions.