Journal of Sound and Vibration (1988) 122(1), 119-130 MODAL ANALYSIS OF A DISTRIBUTED PARAMETER ROTATING SHAFF C. W. LEEr, R. KATZ, A. G. ULSOV AND R. A. SCOTT Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, Michigan 48109-2125, U.S.A. (Received 31 March 1987, and in revised form 9 June 1987) Forced response analysis of an undamped distributed parameter rotating shaft is investi- gated by using a modal analysis technique. The shaft model includes rotary inertia and gyroscopic effects, and various boundary conditions are allowed (not only the simply supported case). Presented here is a study of the resulting non-self-adjoint eigenvalue problem and its characteristics in the case of rotor dynamics. In addition to the modal analysis, Galerkin's method is applied to analyze the forced response of an undamped gyroscopic system. Both methods are illustrated in a numerical example and the results are compared and discussed. 1. INTRODUCTION The transverse vibration of rotor systems with distributed mass has attracted the attention of many investigators during the past few decades. In order to gain deeper insight into the dynamic behavior of rotor systems, the equations of motion have been formulated with account being taken of rotary inertia, gyroscopic moments [1], shear deformation [2, 3] and their combined effects [4]. However, most of these studies are concerned only with the resonant frequency, critical speed calculations and stability analysis. On the other hand, the mode shapes associated with the natural frequencies and the forced response analysis of distributed mass rotor systems have seldom been discussed in the literature. In reference [5] modal analysis is applied to an Euler-Bernoulli rotor model (without the gyroscopic terms). The difficulties in modal analysis of rotor systems arise from the fact that the resulting eigenvalue problems are characterized by the presence of skew-symmetric matrices with differential operators as elements, due to rotation and/or damping, resulting, in general, in a non-self-adjoint eigenvalue problem [6]. Symmetric rotor systems including the rotary inertia and gyroscopic effects are often characterized by self-adjoint inertia and stiffness matrix operators and a skew symmetric gyroscopic matrix operator. The equation of motion of such systems when using a complex notation becomes a second-order partial differential equation with real self-adjoint inertia (positive definite) and stiffness (positive semidefinite) operators, and a non-self-adjoint gyroscopic operator due to the presence of the pure imaginary multiplier. The resulting eigenvalue problem then becomes a standard non-self-adjoint eigenvalue problem when the equation of motion is written in state space rather than in configuration space [6]. In this paper, the differential eigenvalue problems associated with symmetric undamped rotor systems and their adjoint problems are analyzed to yield system eigenvalues and associated system eigenfunctions (or mode shapes) and adjoint eigenfunctions. Based on these quantities, modal equations of motion, representing an infinite set of independent tVisiting Professor, Korea Advanced Institute of Science and Technology. 119 0022-460X/88/070119+ 12 $03.00/0 9 1988 Academic Press Limited