Modal projections for synchronous rotor whirl

Abstract

We consider the synchronous whirl of arbitrary axisymmetric rotors supported on rigid bearings. Prior computational treatments of this problem were based on adding elementlevel gyroscopic terms to the governing equations. Here, we begin with a direct continuum formulation wherein gyroscopic terms need not be added on separately and explicitly: all gyroscopic effects are captured implicitly within the continuum elastodynamics. We present two new methods for obtaining the whirl speed, where we project the dynamic equilibrium equations of the rotor on to a few of its non-spinning vibration mode shapes. The first modal projection method is direct and more accurate, but requires numerical evaluation of more demanding integrals. The second method is iterative and involves a small approximation, but is simpler. Both the methods are based on one new insight: the gyroscopic terms used in other treatments are essentially the result of a prestress in the rotor caused by the non-zero spin rate, and may be incorporated as such in the continuum formulation. The accuracy of the results obtained, for several examples, is verified against detailed calculations with a commercial finiteelement package, against our own nonlinear finite-element code or against analytical estimates. For further verification and illustration, a closed-form analytical solution for a simple problem, obtained using our method, matches the results obtained with explicit gyroscopic terms.

Flexible spinning and precessing rotor–stability analysis based on different analytical and finite element models

“…Three-dimensional finite element analysis of rotors has been attempted by Neogy (2001, 2005). A rigorous general continuum formulation has also been reported for non-axisymmetric rotors on fixed bases (Mahadevan et al, 2008). Lin and Meng (2003) were the first to report on the dynamics of an elastic rotor on a maneuvering support.…”

Stability Analysis of a Flexible Spinning and Precessing Rotor with Non-symmetric Shaft

“…Many sophisticated analyses have been attempted for spinning, symmetric and nonsymmetric shaft-disk systems on fixed base (Nelson and McVaugh, 1976, Nelson, 19801 Ozguven and Ozkan, 19841 Greenhill et al, 19851 Genta and Gugliotta, 19881 Vest and Darlow, 19891 Stephenson and Rouch, 1989, 19931 Gmur and Rodrigues, 19931 Nandi and Neogy, 2003. Apart from numerous finite element analyses of circular and non-circular shafts with multiple disks and three-dimensional analysis of rotors Neogy, 2001, Nandi et al, 2005), a rigorous general continuum formulation has also been recently reported for non-axisymmetric rotors on fixed bases (Mahadevan et al, 2008). It was Lin and Meng (2003), who first reported on dynamics of an elastic rotor on a maneuvering support.…”

Nonlinear secondary whirl of an overhung rotor

“…However, if we ignore gravity and seek synchronous whirl solutions, then we must use det(C) = 0, showing that some gyroscopic effects are included in C as well. Mahadevan et al (2008) discussed this issue for continuum rotors. Moving on, following , we introduce…”