Mode Localization in a Nearly Cyclic Symmetric System
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Cyclic symmetric systems have been widely used in industry, such as turbines, propellers, and compressors. An ideal, tuned, cyclic symmetric system consists of multiple substructures with same geometry and material properties. In a real system, however, there are always slight differences among the substructures (known as mistuning) as a result of imperfections from manufacturing, wear, and damage. The presence of the mistuning could cause mode shapes to localize on a few substructures leading to a phenomenon called ``mode localization.'' As the system spins at high speed, force excitations from the spin together with the mode localization will expedite fatigue failure and jeopardize the safety of the system. Due to the mistuning, the system is not perfectly cyclic symmetric and cannot be predicted by a single substructure. To reduce computational costs, researchers have dedicated for years to developing reduced-order models to accurately predict mode localization. Nevertheless, fundamental understanding, such as when and where mode localization will occur, remains open. Many researchers believe that curve veering of eigenvalue loci is the main contributor of mode localization, but mode localization is also found in a frequency range without any curve veering. Moreover, most studies for mode localization only considered rotors as independent components, so fixed boundary condition was typically applied to the inner rim of the rotors. In a real system, a rotor is always connected with other mechanical components. Recently, researchers started to analyzed turbinemachinery in greater detail by modeling multi-stage rotors or a rotor connected to a shaft. Nevertheless, effects of flexible bearings and housing on mode localization remain unknown. In a broader context, implications of the boundary effects and how boundary conditions affect mode localization are still open. The objective of this thesis is to develop a rigorous mathematical formulation to deduce concrete conditions under which mode localization will occur. The mathematical formulation also provides an effective means to significantly reduce computational costs in predicting mode localization. With the understanding of the conditions for mode localization to occur, the effects of flexible bearings and housing are investigated numerically and interpreted with deductive reasonings. When slight mistuning is introduced into a tuned, cyclic symmetric system, occurrence of mode localization requires the localized mode shapes deviate dramatically from tuned mode shapes. Enlightened by perturbation theory and Fourier analysis, this behavior will occur only when the following two conditions are met: (1) there is a group of tuned modes that have nearly identical natural frequencies, and (2) this group of modes contains a wide range of wave numbers. Mode localization (i.e. localized modes) will be formed as linear combinations of this group of tuned modes. Without these two conditions, researchers traditionally would predict localized modes using Rayleigh-Ritz methods by including many tuned modes in a frequency range as trial vectors. To improve computational efficiency, a visual method is developed in this thesis to pinpoint those tuned modes that will contribute significantly to the linear combinations. With the visual method, the tuned modes are screened carefully so that mode localization can be predicted without iterations and redundant tuned modes. Numerical examples verify that the computational cost for mode localization can be reduced 36%. In this thesis, the effects of flexible bearings and housing are studied through use of finite element analyses. Three models are studied: (1) a reference rotor system in the form of bladed disk with a fixed inner rim, (2) a rotor-bearing system with the fixed boundary condition at the inner rim replaced by flexible bearings, and (3) a rotor-bearing-housing system that connects the reference rotor with a stationary housing via the bearings. All the systems are studied with a tuned version and a mistuned version. Mode localization is compared among the three mistuned systems. The simulated results show that not only the number of localized modes changes but also the natural frequencies and mode shapes of existing localized modes. Consequently, forced response of the localized modes will also change. When bearings and housing are present, vibration of the housing may excite localized modes of the rotor to resonance. These simulation results can all be deduced theoretically from the two mode localization conditions and existing knowledge on vibration of cyclic symmetric structures. This thesis not only identifies the criteria governing occurrence of mode localization, but also provides an effective method to significantly reduce the computational costs for predicting mode localization. This thesis provides physical insights and qualitative predictions regarding how bearings and housing may affect mode localization.The results discussed in this thesis are general and are valid for any cyclic symmetric system.
- Mechanical engineering