Ground-Based Vibration Response of a Spinning, Cyclic Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects

2008 ◽  
Author(s):  
I. Y. Shen ◽  
Hyunchul Kim
Keyword(s):  
Theoretical Analysis ◽  
Fourier Expansion ◽  
Vibration Response ◽  
Modal Testing ◽  
Mode Shapes ◽  
Secondary Resonances ◽  
Spinning Disk ◽  
Phase Index ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to study ground-based vibration response of a spinning, cyclic symmetric rotor through a theoretical analysis and an experimental study. The theoretical analysis consists of three steps. The first step is to analyze vibration characteristics of a stationary cyclic symmetric rotor with N identical substructures. For each vibration mode, we identify a phase index n and derive a Fourier expansion of the mode shape in terms of the phase index n. The second step is to predict rotor-based vibration response of the spinning, cyclic symmetric rotor based on the Fourier expansion of the mode shapes and the phase indices. The rotor-based formulation includes gyroscopic and centrifugal softening terms. Moreover, rotor-based response of repeated modes and distinct modes is obtained analytically. The third step is to transform the rotor-based response to ground-based response using the Fourier expansion of the stationary mode shapes. The theoretical analysis leads to the following conclusions. First, gyroscopic effects have no significant effects on distinct modes. Second, the presence of gyroscopic and centrifugal softening effects cause the repeated modes to split into two modes with distinct frequencies ω1 and ω2 in the rotor-based coordinates. Third, the transformation to ground-based observers leads to primary and secondary resonances. In general, the ground-based response presents resonance branches in the Campbell diagram at ω1 ± kω3 and ω2 ± kω3, where k is phase index n plus an integer multiple of cyclic symmetry N. When the gyroscopic effect is significantly greater than the centrifugal softening effect, two of the four resonance branches disappear. The remaining resonances take the form of either ω1 + kω3 and ω2 − kω3 or ω1 − kω3 and ω2 + kω3. To verify these predictions, we also conduct a modal testing on a spinning disk carrying 4 pairs of brackets evenly spaced in the circumferential direction with ground-based excitations and responses. The disk-bracket system is mounted on a high-speed, air-bearing spindle. An automatic hammer excites the spinning disk-bracket system and a laser Doppler vibrometer measures its vibration response. A spectrum analyzer processes the hammer excitation force and the vibrometer measurements to obtain waterfall plots at various spin speeds. The measured primary and secondary resonances from the waterfall plots agree well with those predicted analytically.

Ground-Based Vibration Response of a Spinning, Cyclic, Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Hyunchul Kim ◽  
I. Y. Shen
Keyword(s):  
Theoretical Analysis ◽  
Fourier Expansion ◽  
Vibration Response ◽  
Modal Testing ◽  
Mode Shapes ◽  
Stationary Mode ◽  
Spinning Disk ◽  
Phase Index ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to study ground-based vibration response of a spinning, cyclic, symmetric rotor through a theoretical analysis and an experimental study. The theoretical analysis consists of three steps. The first step is to analyze the vibration characteristics of a stationary, cyclic, symmetric rotor with N identical substructures. For each vibration mode, we identify a phase index n and derive a Fourier expansion of the mode shape in terms of the phase index n. The second step is to predict the rotor-based vibration response of the spinning, cyclic, symmetric rotor based on the Fourier expansion of the mode shapes and the phase indices. The rotor-based formulation includes gyroscopic and centrifugal softening terms. Moreover, rotor-based response of repeated modes and distinct modes is obtained analytically. The third step is to transform the rotor-based response to ground-based response using the Fourier expansion of the stationary mode shapes. The theoretical analysis leads to the following conclusions. First, gyroscopic effects have no significant effects on distinct modes. Second, the presence of gyroscopic and centrifugal softening effects causes the repeated modes to split into two modes with distinct frequencies ω1 and ω2 in the rotor-based coordinates. Third, the transformation to ground-based observers leads to primary and secondary frequency components. In general, the ground-based response presents frequency branches in the Campbell diagram at ω1±kω3 and ω2±kω3, where k is phase index n plus an integer multiple of cyclic symmetry N. When the gyroscopic effect is significantly greater than the centrifugal softening effect, two of the four frequency branches vanish. The remaining frequency branches take the form of either ω1+kω3 and ω2−kω3 or ω1−kω3 and ω2+kω3. To verify these predictions, we also conduct a modal testing on a spinning disk carrying four pairs of brackets evenly spaced in the circumferential direction with ground-based excitations and responses. The disk-bracket system is mounted on a high-speed, air-bearing spindle. An automatic hammer excites the spinning disk-bracket system and a laser Doppler vibrometer measures its vibration response. A spectrum analyzer processes the hammer excitation force and the vibrometer measurements to obtain waterfall plots at various spin speeds. The measured primary and secondary frequency branches from the waterfall plots agree well with those predicted analytically.

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

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Hyunchul Kim ◽  
Nick Theodore Khalid Colonnese ◽  
I. Y. Shen
Keyword(s):  
Vibration Mode ◽  
Fluid Dynamic ◽  
Travelling Wave ◽  
Modal Testing ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Phase Index ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Inertia Forces

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.

A Numerical Study on Ground-Based Vibration Response of a Spinning Cyclic Symmetric Rotor With Cracks

2010 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Numerical Simulation ◽  
Radial Crack ◽  
Numerical Study ◽  
Crack Size ◽  
Vibration Response ◽  
Variable Depth ◽  
Spinning Disk ◽  
Circumferential Crack ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to study how presence of cracks affects ground-based vibration response of a spinning cyclic symmetric rotor via a numerical simulation. A reference system used in this study is a spinning disk with four pairs of brackets, representing a 4-fold cyclic symmetric rotor. A crack with a variable depth is introduced at one of the eight disk-bracket interfaces. Both radial and circumferential cracks are simulated. The ground-based vibration response of the spinning disk-bracket system is simulated using an algorithm introduced by Shen and Kim [1]. Compared with a perfectly cyclic symmetric rotor, the crack introduces additional resonances when the crack size is large enough. Frequencies of these additional resonances can be predicted accurately and may be used as a way to detect presence of cracks. In addition, the additional resonances are more prominent for the circumferential crack than the radial crack.

Vibration of a Spinning Rotationally Periodic Rotor

2007 ◽  
Author(s):  
Hyunchul Kim ◽  
I. Y. Shen
Keyword(s):  
Theoretical Analysis ◽  
High Speed ◽  
Experimental Studies ◽  
Mode Shapes ◽  
Integer Multiple ◽  
Numerical Studies ◽  
Theoretical Predictions ◽  
Symmetric Rotor ◽  
Unified Algorithm ◽  
Cyclic Symmetric

This paper studies the vibrations of a spinning, rotationally periodic (also known as cyclic symmetric) rotor through theoretical analysis and experimental studies. The theoretical analysis consists of two parts. The first part is Fourier analysis of mode shapes of a stationary rotor with periodicity N. A periodic mapping of the n-th mode shape shows that its k-th Fourier coefficient is generally zero, except when k ± n is an integer multiple of N. The second part is to apply the derived mode shapes through a unified algorithm developed by Shen and Kim [1] to predict primary and secondary resonances of spinning, rotationally periodic rotors. The experimental study focuses on vibration measurements of a spinning disk carrying 4 pairs of evenly spaced brackets mounted on a high-speed air-bearing spindle. Initially, experimentally measured waterfall plots do not agree well with those from theoretical predictions. Further numerical studies show that mistune of rotationally periodic rotors could substantially change their waterfall plots. After the mistune is modeled, experimental and theoretical results agree very well with a difference of only 0.8% in natural frequencies observed in the ground-based coordinates.

A Numerical Study on Ground-Based Vibration Response of a Spinning Cyclic Symmetric Rotor With Cracks

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Hyunchul Kim ◽  
I. Y. Shen
Keyword(s):  
Numerical Simulation ◽  
Reference System ◽  
Numerical Study ◽  
Crack Size ◽  
Vibration Response ◽  
Variable Depth ◽  
Spinning Disk ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to study how presence of cracks affects ground-based vibration response of a spinning cyclic symmetric rotor via a numerical simulation. A reference system used in this study is a spinning disk with four pairs of brackets, representing a fourfold cyclic symmetric rotor. A crack with a variable depth is introduced at one of the eight disk-bracket interfaces. Both radial and circumferential cracks are simulated. The ground-based vibration response of the spinning disk-bracket system is simulated using an algorithm introduced by Shen and Kim published in 2006. Compared with a perfectly cyclic symmetric rotor, the crack introduces additional resonances when the crack size is large enough. Frequencies of these additional resonances can be predicted accurately and may be used as a way to detect presence of cracks.

Experimental Verification of Ground-Based Response of a Spinning, Cyclic Symmetric, Rotor Assembled to a Flexible Stationary Housing via Multiple Bearings

2014 ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Laser Doppler Vibrometer ◽  
Nodal Line ◽  
Spindle Motor ◽  
Housing System ◽  
Mode Shapes ◽  
Coupled Modes ◽  
Spin Speed ◽  
Resonance Frequencies ◽  
Symmetric Rotor ◽  
Cyclic Symmetric

This paper is to present an experimental study that measures ground-based response of a spinning, cyclic, symmetric rotor-bearing-housing system. In particular, the study focuses on rotor-housing coupled modes that are significantly dominated by housing deformation. In the experiments, a ball-bearing spindle motor, carrying a disk with four evenly spaced slots (i.e., the rotor), is mounted onto a stationary housing. The housing is a square plate supported with steel spacers at four corners and fixed to the ground. Two different ways are used to excite the rotor-housing system to measure frequency response functions (FRFs). One is to use an automatic hammer tapping at the disk, and the other is to use a piezoelectric actuator attached to the housing. Vibration of the rotor and housing is measured via a laser Doppler vibrometer and a capacitance probe. The experiments consist of two parts. The first part is to obtain FRFs when the rotor is not spinning. The measured FRFs reveal two rotor-housing coupled modes dominated by the housing. Their mode shapes are characterized by one nodal line in housing and one nodal diameter in the rotor. The second part is to obtain waterfall plots when the rotor is spinning at various speeds. The waterfall plots show that the housing dominant modes split into primary branches and secondary branches as the spin speed varies. The primary branches almost do not change with respect to the spin speed. In contrast, the secondary branches evolve into forward and backward branches. Moreover, their resonance frequencies increase and decrease at four times of the spin speed. The measured results agree well with the predictions found in the authors’ previous theoretical study [1].

Ground-Based Response of a Spinning, Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing via Multiple Bearings

2013 ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Theoretical Analysis ◽  
Coordinate System ◽  
Equation Of Motion ◽  
Periodic Coefficients ◽  
Numerical Example ◽  
Housing Systems ◽  
Rotor Bearing ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Multiple Bearings

This paper is to study free response of a spinning, cyclic symmetric rotor assembled to a flexible housing via multiple bearings. In particular, the rotor spins at a constant speed ω3, and the housing is excited via a set of initial displacements. The focus is to study ground-based response of the rotor through theoretical and numerical analyses. The paper consists of three parts. The first part is to briefly summarize an equation of motion of the coupled rotor-bearing-housing systems for the subsequent analyses. The equation of motion, obtained from prior research [1], employs a ground-based and a rotor-based coordinate system to the housing and the rotor, respectively. As a result, the equation of motion takes the form of a set of ordinary differential equations with periodic coefficients of frequency ω3. To better understand its solutions, a numerical model is introduced as an example. In this example, the rotor is a disk with four radial slots and the housing is a square plate with a central shaft. The rotor and housing are connected via two ball bearings. The second part of the paper is to analyze the rotor’s response in the rotor-based coordinate system theoretically. When the rotor is at rest, let ωH be the natural frequency of a coupled rotor-bearing-housing mode whose response is dominated by the housing. The theoretical analysis then indicates that response of the spinning rotor will possess frequency components ωH ± ω3 demonstrating the interaction of the spinning rotor and the housing. The theoretical analysis further shows that this splitting phenomenon results from the periodic coefficients in the equation of motion. The numerical example also confirms this splitting phenomenon. The last part of the paper is to analyze the rotor’s response in the ground-based coordinate system. A coordinate transformation shows that the ground-based response of the spinning rotor consists of two major frequency branches ωH − (k + 1) ω3 and ωH − (k − 1) ω3, where k is an integer determined by the cyclic symmetry and vibration modes of interest. The numerical example also confirms this derivation.

A Linearized Theory on Ground-Based Vibration Response of Rotating Asymmetric Flexible Structures

2006 ◽  
Vol 128 (3) ◽  
pp. 375-384 ◽  
Author(s):  
I. Y. Shen ◽  
Hyunchul Kim
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Equation Of Motion ◽  
Vibration Response ◽  
Mode Shapes ◽  
Element Analysis ◽  
Secondary Resonances ◽  
Modal Response ◽  
Resonance Frequencies ◽  
Unified Algorithm

This paper is to develop a unified algorithm to predict vibration of spinning asymmetric rotors with arbitrary geometry and complexity. Specifically, the algorithm is to predict vibration response of spinning rotors from a ground-based observer. As a first approximation, the effects of housings and bearings are not included in this analysis. The unified algorithm consists of three steps. The first step is to conduct a finite element analysis on the corresponding stationary rotor to extract natural frequencies and mode shapes. The second step is to represent the vibration of the spinning rotor in terms of the mode shapes and their modal response in a coordinate system that is rotating with the spinning rotor. The equation of motion governing the modal response is derived through use of the Lagrange equation. To construct the equation of motion, explicitly, the results from the finite element analysis will be used to calculate the gyroscopic matrix, centrifugal stiffening (or softening) matrix, and generalized modal excitation vector. The third step is to solve the equation of motion to obtain the modal response, which, in turn, will lead to physical response of the rotor for a rotor-based observer or for a ground-based observer through a coordinate transformation. Results of the algorithm indicate that Campbell diagrams of spinning asymmetric rotors will not only have traditional forward and backward primary resonances as in axisymmetric rotors, but also have secondary resonances caused by higher harmonics resulting from the mode shapes. Finally, the algorithm is validated through a calibrated experiment using rotating disks with evenly spaced radial slots. Qualitatively, all measured vibration spectra show significant forward and backward primary resonances as well as secondary resonances as predicted in the theoretical analysis. Quantitatively, measured primary and secondary resonance frequencies agree extremely well with those predicted from the algorithm with mostly <3.5% difference.

Parametric Resonances of a Spinning Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing via Multiple Bearings

2012 ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Housing System ◽  
Mode Shapes ◽  
Element Analysis ◽  
Linear Elastic ◽  
Rotor Bearing ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Inertia Forces

This paper is to present findings from a theoretical study on free vibration and stability of a rotor-bearing-housing system. The rotor is cyclic symmetric and spinning at constant speed, while the housing is stationary and flexible. Moreover, the rotor and housing are assembled via multiple, linear, elastic bearings. For the rotor and the housing, their mode shapes are first obtained in rotor-based and ground-based coordinate systems, respectively. By discretizing the kinetic and potential energies of the rotor-bearing-housing system through use of the mode shapes, a set of equations of motion appears in the form of ordinary differential equations with periodic coefficients. Analyses of the equations of motion indicate that instabilities could appear at certain spin speed in the form of combination resonances of the sum type. To demonstrate the validity of the formulation, two numerical examples are studied. For the first example, the spinning rotor is an axisymmetric disk and the housing is a square plate with a central shaft. Moreover, the rotor and the housing are connected via two linear elastic bearings. Instability appears in the form of coupled vibration between the stationary housing and spinning rotor through three different formats: rigid-body rotor translation, rigid-body rotor rocking, and elastic rotor modes that present unbalanced inertia forces or moments. For the second example, the rotor is cyclic symmetric in the form of a disk with four evenly spaced slots. The housing and bearings remain the same. When the rotor is stationary, natural frequencies and mode shapes predicted from the formulation agree well with those predicted from a finite element analysis, which further ensures the validity of the formulation. When the cyclic symmetric rotor spins, instability appears in the same three formats as in the case of axisymmetric rotor. Number of instability zones, however, increases because the cyclic symmetric rotor has more elastic rotor modes that present unbalanced inertia forces or moments.

Experimental Study of Blade Vibration in Centrifugal Compressors

2011 ◽  
Author(s):  
Andrew H. Lerche ◽  
J. Jeffrey Moore ◽  
Timothy C. Allison
Keyword(s):  
High Cycle Fatigue ◽  
Axial Flow ◽  
Modal Testing ◽  
Mode Shapes ◽  
Cyclic Symmetry ◽  
Centrifugal Compressors ◽  
Compressor Blades ◽  
Blade Vibration ◽  
Phase Cancellation ◽  
Blade Failure

Blade vibration in turbomachinery is a common problem that can lead to blade failure by high cycle fatigue. Although much research has been performed on axial flow turbomachinery, little has been published for radial flow machines such as centrifugal compressors and radial inflow turbines. This work develops a test rig that measures the resonant vibration of centrifugal compressor blades. The blade vibrations are caused by the wakes coming from the inlet guide vanes. These vibrations are measured using blade mounted strain gauges during a rotating test. The total damping of the blade response from the rotating test is compared to the damping from the modal testing performed on the impeller. The mode shapes of the response and possible effects of mistuning are also discussed. The results show that mistuning can affect the phase cancellation which one would expect to see on a system with perfect cyclic symmetry.

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