In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors that contain slight mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be linearly combined to form localized modes when the mistune is present. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh–Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we develop an effective visual method—through use of the deviatoric component and the rotor mistune—to precisely identify those modes needed to form localized modes. Finally, we show that curve veering is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. Numerical examples on a disk–blade system with mistune confirm all the findings above.
This technical brief is to study how flexible bearings and housing affect mode localization of a nearly cyclic symmetric system with mistuning. This study is conducted via finite-element analyses and deductive reasoning. A reference system studied is a bladed disk with two groups of 24 localized modes. When bearings and housing are introduced into the reference system, their presence changes natural frequencies, mode shapes, and the number of the localized modes. Moreover, the mistuning causes bearing forces to surge for all the localized modes. A deductive reasoning based on the existing literature supports the observation from the finiteelement analyses.
In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors with mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be easily reoriented when the mistune is present to form localized modes. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh-Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we can develop an effective visual method — through use of the deviatoric component and the rotor mistune — to precisely identify those modes needed to form localized modes. Finally, we show that curve veering with respect to engine orders is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. A numerical example confirms the findings above.
This paper is to study how flexible bearings and housing affect mode localization of a nearly cyclic symmetric system with mistune. A finite element analysis is first conducted on a reference system that consists of a circular disk and 24 blades with mistune. The disk is annular with an inner rim and an outer rim. A fixed boundary condition is imposed at the inner rim, while the 24 blades with mistune are evenly attached to the outer rim and subjected to a free boundary condition. As a result of the mistune, the reference system presents 26 localized torsional modes as well as 24 localized in-plane modes in its blade vibration. When the fixed inner rim is replaced by a bearing support (i.e., an elastic boundary condition), not only the localized torsional modes can change their natural frequencies and mode shapes but also the number of the localized torsional modes may be increased to 28 in some range of bearing stiffness. Similarly, when the bladed-disk reference system is mounted on a stationary housing via a bearing support, the number of the localized in-plane modes can change from 24 to 33 modes. Moreover, localized mode shapes change significantly, and some of them involve significant housing deformation. To understand this phenomenon theoretically, we first demonstrate that the presence of bearing and housing provides additional degrees of freedom, which, in turn, allow the bladed-disk system to have additional disk modes. When the bearing and housing stiffness is properly tuned, some of these additional disk modes may possess significant torsional or in-plane displacement components in the blades. If these additional modes happen to have a natural frequency that is close to those of the localized modes of the reference system, these additional modes will join the localized modes to form new localized modes. As a result, the number of localized modes increases and the mode shapes change significantly.
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