Ritz Series Analysis of Rotating Shaft System: Validation, Convergence, Mode Functions, and Unbalance Response

A shaft with attached rigid disks is modeled as a rotating Timoshenko beam supported by nonconservative, flexible bearing supports. The continuous shaft-disk system is described with kinetic and potential energy functionals that fully account for transverse shear, translational and rotatory inertia, and gyroscopic coupling. Ritz series expansions are used to describe the flexural displacements and cross-sectional rotations about orthogonal fixed axes. The equations of motion are derived from Lagrange’s equations and placed in a state-space form that preserves the skew-symmetric gyroscopic matrix as well as the full effects of the bearings. Both the general and adjoint eigenproblems for the nonsymmetric equations are solved. Bi-orthogonality conditions lead to the ability to evaluate dynamic response via modal analysis. Whirl speeds and logarithmic decrements calculated with the present model are verified with a finite element analysis. The present work provides two ways of evaluating the convergence of results to demonstrate an advantage of the Ritz method over other discretization methods. Natural mode functions and unbalance response are calculated for an example system.