AIAA JOURNAL Vol. 40, No. 10, October 2002 Derivative of Eigensolutions of Nonviscously Damped Linear Systems Sondipon Adhikari ¤ University of Cambridge, Cambridge, England CB2 1PZ, United Kingdom Derivatives of eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters are presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy any orthogonality relationship. It is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems. Numerical examples are provided to illustrate the derived results. Nomenclature C = viscous damping matrix C = space of complex numbers c = damping constant D.s / = dynamic stiffness matrix G.s / = damping function in the Laplace domain G.t / = damping function in the time domain I = identity matrix =() = imaginary part of ( ) i = unit imaginary number, p ¡1 K = stiffness matrix k 1 ; k 2 ; k 3 = spring constants M = mass matrix m = order of the characteristic polynomial N = degrees of freedom of the system O = null matrix p = design parameter R = space of real numbers s = Laplace domain parameter t = time u j = j th eigenvector of the system N u.s / = Laplace transform of u.t / u.t / = response vector ± jk = Kroneker delta function ±.t / = Dirac delta function μ j = normalization constant for j th mode ¸ j = j th eigenvalue of the system ¹ 1 ;¹ 2 = parameters of the Golla–Hughes–McTavish (see Refs. 23 and 32) damping model º .s / = diagonal matrix containing º j .s/ º j .s / = j th eigenvalue of D.s / U.s / = matrix containing Á j .s/ Á j .s / = j th eigenvector of D.s / ! j = j th undamped natural frequency j./j = l 2 norm of the vector ./ Subscripts e = elastic modes n = nonviscous modes Received 12 January 2001; revision received 5 July 2001; accepted for publication 15 April 2002. Copyright c ° 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose- wood Drive, Danvers, MA 01923; include the code 0001-1452/02 $10.00 in correspondenc e with the CCC. ¤ Research Associate, Department of Engineering, Trumpington Street; sa225@eng.cam.ac.uk. Superscripts T = matrix transpose ¡1 = matrix inverse ¢ = derivative with respect to t ¤ = complex conjugate I. Introduction D URING the design process of a structure, it is often required to make changes in the design parameters so that the design is optimal. When dynamic problems are considered, the interest of designers lies in understanding the changes in natural frequencies and mode shapes due to the changes in the system parameters. The sensitivity of eigensolutions, or more precisely, the derivative of eigensolutions with respect to the design parameters has an impor- tant role in such studies because it helps to avoid repeated calcu- lations. Also the eigensensitivity analysis plays a major role in the system identication problems and in the analysis of stochastically perturbed dynamic systems. Because of such widespread applica- tions, the calculation of derivative of eigenvalues and eigenvectors has emerged an important area of research over past four decades. In one of the earliest works, Fox and Kapoor 1 gave exact ex- pressions for the derivative of eigenvalues and eigenvectors with respect to any design variable. Their results were obtained in terms of changes in the system property matrices and the eigen- solutions of the structure and have been used extensively in a wide range of application areas of structural dynamics. The expressions derived in Ref. 1 are valid for symmetric undamped systems. Later, many authors 2¡5 extended Fox and Kapoor’s 1 approach to determine eigensolution derivatives for more general asymmetric systems. For these kinds of systems, Nelson 6 proposed an efcient method to cal- culate the derivative of eigenvectors that requires only the eigenvalue and eigenvector under consideration. A review on calculating the derivatives of eigenvalues and eigenvectors associated with general (non-Hermitian) matrices may be found by Murthy and Haftka. 7 The cited works do not explicitly consider the damping present in the system. To apply these results to systems with general nonpro- portional (viscous) damping, it is required to convert the equations of motion into state-space form (for example, see Ref. 8). Although exact in nature, state-space methods require signicant numerical effort as the size of the problem doubles. Moreover, these methods also lack some of the intuitive simplicity of the analysis based on N space. For these reasons some authors have considered the prob- lem of the calculation of derivatives of eigensolutions of viscously damped systems in N space. One of the earliest work to consider damping was by Cardani and Mantegazza 9 in the context of utter problems. Note that, unlike undamped systems, in damped systems the eigenvalues and eigenvectors, and consequently their deriva- tives, become complex in general. Adhikari 10 derived an exact ex- pression for the derivative of complex eigenvalues and eigenvectors. 2061