VIBRATION OF A SPINNING, CYCLIC SYMMETRIC ROTOR ASSEMBLED TO A FLEXIBLE STATIONARY HOUSING VIA MULTIPLE BEARINGS
Tai, wei che
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A spinning cyclic symmetric rotor mounted on a stationary housing via multiple bearings is a very common platform used in modern rotary machinery. Representative examples include propellers, wind turbines, bladed turbine disks, and compressors. Nowadays, there are two main industrial trends in designing machines with the cyclic symmetric rotors. The first is to use larger rotors and lighter housing in order to increase efficiency and reduce costs. The second is to employ ground-based measurements. Motivated by the industrial trends, this research is to develop a reduced-order formulation that accommodates arbitrary geometry of the spinning rotor and the stationary housing. Such a formulation is universal and will be valid for various cyclic symmetric rotors, ranging from wind turbines to bladed turbine disks. Mathematically, the governing equation of motion with reduced order takes the form of a set of ordinary differential equations with periodic coefficients associated with the spin speed. Characteristics drawn from this formulation can then be applied to any cyclic symmetric rotor mounted on a flexible housing. Furthermore, unstable response (e.g., parametric resonances) and stable response (e.g., rotor-based and ground-based response) of the rotor-bearing-housing system will be studied analytically, numerically, and experimentally, as follows. In the analytical study, the system is first shown to have instabilities in terms of combination resonance of the sum type as a result of the periodic coefficients. All the resonance peaks and corresponding bandwidth within a proper range of spin speeds, where the system remains positive definite, are analytically predicted by the method of multiple scales. As a result of combination resonance of the sum type, the instability occurs at extremely high spin speeds Next, the stable response of the stationary and spinning system is studied analytically. The stationary system has two types of modes: rotor-dominant modes and housing-dominant modes. For the spinning system, two types of stable response are studied: the rotor-based response and the ground-based response. Both responses of rotor-dominant modes are similar to the case with rigid housing. The rotor-based response of housing-dominant modes, however, possesses a specific frequency splitting due to dominant vibration of the housing. For a housing-dominant mode with natural frequency ω<super>(H)</super> obtained from the stationary system, when the system is spinning at spin speed ω<sub>3</sub>, the rotor-based response splits into a forward and backward frequency branch equal to ω<super>(H)</super>±ω<sub>3</sub>. Such frequency splitting is defined as gyroscopic splitting. The gyroscopic splitting is analytically predicted via a perturbation analysis. Subsequently, the ground-based response is theoretically predicted. The theoretical prediction is briefly summarized as follows. The rotor-based response of a housing-dominant mode has frequency components ω<super>(H)</super>±ω<sub>3</sub> due to gyroscopic splitting. Furthermore, if a rotor-based, cylindrical coordinate (r,θ, z) is employed to describe the vibration mode shape of a cyclic symmetric rotor, the mode shape is circumferentially modulated by the exponential function e<super>jk</super>, where k is the harmonic number which follows the identity k = n + M(N). In this identity, n is the phase index governed by the cyclic symmetry of the rotor while M(N) is multiples of numbers of identical substructures N. When the response is viewed from a ground-based observer, the circumferential harmonics kθ gives rise to additional frequency splitting -kω<sub>3</sub>. Together with the gyroscopic splitting, the ground-based response splits into multiple forward and backward frequency branches following the rule ω<super>(H)</super>-(k&plumn;1)ω<sub>3</sub>. To confirm the results from the analytical study, a benchmark numerical model consisting of a cyclic symmetric rotor, a stationary housing, and two bearings is developed. The rotor is a circular disk with four evenly spaced radial slots and a rigid hub. The stationary housing is a square plate with a central shaft subjected to fixed boundary conditions on the displacements at four corners. Based on this model, a numerical integration of the equation of motion ad use of the Floquet theory confirms the parametric resonance frequency and the instability bandwidth obtained from the method of multiple scales. Through the benchmark model, the gyroscopic splitting is also numerical confirmed for the rotor-based response. Moreover, ground-based response at various speed in the form of waterfall plots confirms that a housing-dominant mode splits following the rule (k &plumn; ω<sub>3</sub>). In order to verify the theoretical prediction of the ground-based response, a series of experiments on a stationary and spinning test rig is carried out. First of all, frequency response functions (FRFs) of the stationary rig are measured. Two FRFs are obtained using two excitation mechanisms. The first is to use an automatic hammer while the second is to use a piezoelectric (PZT) actuator. Two housing-dominant modes are identified by comparing the FRFs. Their mode shapes are characterized by one-nodal diameter and one-nodal line on the rotor and housing, respectively. Next, ground-based response of the spinning rig is measured to obtain waterfall plots. For the waterfall plot obtained form the hammer excitation, both housing-dominant modes reveal forward frequency branches which agree very well with the theoretical prediction. Only one housing-dominant mode presents a backward frequency branch. Nonetheless, the backward branch also agrees well with the theoretical prediction. Lastly, a closed-form solution of rotor-bearing-housing systems with a special class of cyclic symmetry is derived. Specifically, the equation of motion can be transformed into a set of ordinary differential equations with constant coefficients, when the hub is rigid and the flexible portion of the rotor has only out-of-plane vibration motion. The transformed equation of motion appears as a time invariant gyroscopic system, whose closed-form solution is hence readily available. Both the original and transformed equation of motion are shown to have identical instabilities and rotor-based response through numerical simulations via the benchmark model.
- Mechanical engineering