A Tutorial on Complex Eigenvalues

This tutorial examines the nature of natural frequencies and damping ratios for systems with nonproportional damping from first principles. Computational schemes are discussed for the numerical determination of complex eigenvalues. A common misunderstanding of the nature of the natural frequency of a nonproportionally damped system is discussed and illustrated through numerical simulations. 1 . I N T R O D U C T I O N This manuscript provides a tutorial on methods of computing the damping rat ios and natural frequencies for underdamped mechanical systems with complex eigenvalues. Complex eigenvalues occur when systems have underdamped modes. A system may have underdamped modes, and hence complex eigenvalues and be proportionally damped or nonproportionally damped. Generally, nonproportionally damped systems have complex mode shapes while, proportionally damped systems can be represented with real modes. Less well known is the fact that the undamped natural frequencies determined from complex eigenvalues are different from the natural frequencies calculated from the same system with zero damping. The goal of this manuscript is to examine this difference in “natural frequency” and hence interpret the concept of undamped natural frequency from nonproportionally damped structures as well as to provide a review of calculations for complex eigenvalues. 2 . C O M P U T A T I O N O F C O M P L E X EIGENVALUES The standard lumped parameter or multiple-degree-offreedom model of a” undamped, autonomous vibrating system is given by the vector differential equation Mi(t) + ‘42(t) = 0 (1) A TUTORIAL ON COMPLEX EIGENVALUES Daniel J. Inman Department of Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061 USA where t denotes the time; M is a” n x n mass matrix, K is a” ” x n stiffness matrix and z is a” n x 1 vector of displacements, or generalized, coordinates, each element of which corresponds to a degree of freedom. The over dots denote time derivatives so that i: represent the n x 1 vector of accelerations. Often in vibration analysis it is useful to model energy dissipation, or damping, by including the term C? where C is a” n x n matrix of damping coefficients and & is the n x 1 vector of velocities. Then equation (1) becomes Ali + c%(t) + IiZ(t) = 0 (2) Equation (1) is referred to as the undamped system while equation (2) is referred to as the damped system. A complete discussion of these two models can be found in a variety of texts from introductory [l-4] to advanced [5-a]. Knowledge of equation (1) and (2) and the types of systems they model is assumed at the level of the beginning chapters found in any of ref. [l-4]. Usually, but no always, the matrices M, C, and It’ have special properties. For instance M, C and IC are generally symmetric and positive definite or at least semi-definite (see [l], [4] or [9] for definitions). I” addition, if the structure or machine being modeled is spa&ally repetitive or periodic in nature, these matrices may be banded or sparsely populated. For problems with a large number of degrees-of-freedom, taking advantage of any special structure of the coefficient matrices M, C and Ii’ could be essential. The algorithms presented in [8] and available in MATLAB are intended to capitalize on any special structure as needed. Computation of the modal information associated with equations (1) and (2) is presented here. The mode shapes, natural frequencies and darnping rat&, collectively called modal data., form the backbone of much vibration analysis arid design. Vibration modal data is however strongly related to the algebraic eigcnvalue problems, a discipline that