Ritz series analysis of rotating machinery

A unified method for modeling rotating machinery with the Ritz method is presented. Kinetic and potential energy functionals that fully account for transverse shear, translational and rotatory inertia, and gyroscopic coupling describe the continuous shaft-disk system. 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 bearing supports. 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. Two simple examples that use idealized bearings verify critical speeds calculated by the present work. A finite element analysis validates the calculation of whirl speeds and logarithmic decrements. Two ways of evaluating the convergence of results with a Ritz analysis demonstrate an advantage of this method over discretization methods. Natural mode functions and unbalance response are also calculated for an example system.; A procedure for combining equations of motion of substructures with those of the rotating shaft is introduced. Equations constraining the rotating shaft model to generalized coordinates of substructures are enforced with Lagrange multipliers. A thick disk with two rotating shafts attached at each of its sides serves as an application of the substructuring method. A distributed parameter approach is also implemented for modeling a thick disk on the continuous shaft. An example, devised to have a thick disk off mid-span of a shaft supported by fluid film bearings, allows comparison of three methods for modeling the thick disk system: substructuring, distributed parameters, and lumped parameters. A parametric study investigates how changing the disk diameter and length affects critical speeds and stability. Results indicate that the most significant effects occur from changes in the disk length.