A Unified Approach to Analyze Vibration of Axisymmetric Rotating Structures with Flexible Stationary Parts

This paper demonstrates a unified approach to analyze linear vibration of rotating machines with arbitrary geometry and complexity. In this formulation, the rotating machine consists of three components: a rotating part (rotor), a stationary part (stator or housing), and multiple bearings. The rotor is assumed axisymmetric and spinning at constant speed. Moreover, the rotor and the housing have arbitrary geometry and complexity. The bearings connecting the rotor and housing could be rolling-element bearings or hydrodynamic bearings. The paper consists of three major sections: mathematical modeling, integration with finite element analysis (FEA), and experimental verification. For the mathematical modeling, a stationary rotor with free boundary conditions is first discretized to obtain its normal vibration modes and modal parameters. In the meantime, the housing with its actual boundary conditions (but no bearings) is also discretized. The discretization can be achieved, for example, through FEA to accommodate arbitrary and complex geometry of the rotor and the housing. Because these vibration modes are complete, modal response of each mode can serve as a generalized coordinate to describe vibration of the actual spinning rotor and housing system. With these generalized coordinates, gyroscopic effects of the spinning rotor can be derived through material derivatives for a ground-based observer. As a result, application of Lagrange equation leads to a set of gyroscopic equations of motion with constant coefficients. These coefficients, however, contain complicated volume integrals of the mode shapes and their spatial derivatives. Therefore, algorithms are developed to calculate these coefficients explicitly from FEA. For the experimental verification, a ball-bearing spindle carrying a cylinder closed at one end is used to validate the mathematical model. Frequency response functions of the spindle/cylinder system are measured for spin speed ranging from 0 to 6000 rpm. Natural frequencies measured from the experiments agree very well with the theoretical predictions from the unified approach up to 2 kHz.