Mathematical Insights Into Linear Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

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.

Mathematical Insights of Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune

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.

Frequency Clustering in Cyclic Symmetric Rotors Considering Geometry Nonlinearity

Motivated by finite element results of a bladed disk model, a discrete model and numerical analysis are developed and presented in this paper with view toward understanding effect of large geometric design factors on the vibrations of a cyclic symmetric integrally bladed rotor (IBR). It is found that nonlinear eigenvalue-loci crossing for split frequency doublet modes and eigenvalue-loci veering for repeated frequency modes having the same wavenumber content can occur when large blade span distribution and blade to disk inertia ratio are considered in the discrete IBR model. Clustering of eigen-loci is found and characterized in blade span parameter study when blade’s inertia is significant large. These three findings are new to the literature and have engineering implications on structure design and vibration control of engine components in turbine and automobile applications.