Mode Evolution of Cyclic Symmetric Rotors Assembled to Flexible Bearings and Housing

Abstract

This paper is to study how the vibration modes of a cyclic symmetric rotor evolve when it is assembled to a flexible housing via multiple bearing supports. Prior to assembly, the vibration modes of the rotor are classified as “balanced modes” and “unbalanced modes.” Balanced modes are those modes whose natural frequencies and mode shapes remain unchanged after the rotor is assembled to the housing via bearings. Otherwise, the vibration modes are classified as unbalanced modes. By applying fundamental theorems of continuum mechanics, we conclude that balanced modes will present vanishing inertia forces and moments as they vibrate. Since each vibration mode of a cyclic symmetric rotor can be characterized in terms of a phase index (Chang and Wickert, “Response of Modulated Doublet Modes to Travelling Wave Excitation,” J. Sound Vib., 242, pp. 69–83; Chang and Wickert, 2002, “Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks,” J. Sound Vib., 250, pp. 379–400; Kim and Shen, 2009, “Ground-Based Vibration Response of a Spinning Cyclic Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects,” ASME J. Vibr. Acoust. (in press)), the criterion of vanishing inertia forces and moments implies that the phase index by itself can uniquely determine whether or not a vibration mode is a balanced mode as follows. Let N be the order of cyclic symmetry of the rotor and n be the phase index of a vibration mode. Vanishing inertia forces and moments indicate that a vibration mode will be a balanced mode if n≠1,N−1,N. When n=N, the vibration mode will be balanced if its leading Fourier coefficient vanishes. To validate the mathematical predictions, modal testing was conducted on a disk with four pairs of brackets mounted on an air-bearing spindle and a fluid-dynamic bearing spindle at various spin speeds. Measured Campbell diagrams agree well with the theoretical predictions.

Modal properties and stability of centrifugal pendulum vibration absorber systems with equally spaced, identical absorbers

“…Thus, the net force and moment that the absorbers exert on the rotor both vanish for the absorber modes. This agrees with the result derived by Kim et al [25] showing that a mode with phase index ka0; 1,NÀ1 is a balanced mode, i.e., a mode with no net force or moment on the rotor. The net rotor moment is not balanced (zero) for the rotational modes, and the net rotor force is not balanced for the translational modes.…”

“…The question arises whether this mode, which has the form of both Eqs. (25) and (34), should be regarded as a translational mode or an absorber mode. The answer comes from the phase index.…”

“…(14) and the eigenvectors of the three mode types in Eqs. (16), (25), and (34), the modal forces / T F are calculated. From these, the rotational modes can be excited only by external rotor torque but not force.…”

Vibration mode structure and simplified modelling of cyclically symmetric or rotationally periodic systems

“…Substitution of u kl in equation (2.5) for a chosen k into the equation that governs the motion of each substructure in equation (2.1) (that is, the L equations defined by each A The eigenvectors in equation (2.5) show that all vibration modes of a cyclically symmetric system with non-vibrating central components consist of identical motions for each substructure (i.e. the v kl ), but the substructures vibrate out of phase with the phase difference determined by the integer k, known as the phase index of the vibration mode [2,3,[23][24][25], and the spacing angles β i . (a) An example system with cyclic symmetry…”

Effects of Bearings and Housing on Mode Localization of a Nearly Cyclic Symmetric System

“…For example, Chen and Shen [15] identified that mode localization in a mistuned cyclic symmetric system will only occur when the two following conditions are met: (a) there is a group of vibration modes of the tuned cyclic symmetric rotor whose natural frequencies are nearly identical and the mode shapes have similar characteristics and (b) the vibration modes in the group contain a wide range of wave numbers. Kim et al [16] showed that the vibration modes of a rotor will evolve in a certain manner when bearings are introduced to support the rotor. Predictions of the FEA should be consistent with the conclusions drawn from these prior publications.…”