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.

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 Asymmetric Rotors Assembled to Flexible Bearings and Housing

2009 ◽  
Author(s):  
Hyunchul Kim ◽  
I. Y. Shen
Keyword(s):  
Boundary Conditions ◽  
Vibration Mode ◽  
Fluid Dynamic ◽  
Modal Testing ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Fixed Boundary ◽  
Practical Applications ◽  
Asymmetric Rotor ◽  
Surface Integrals

This paper is to study how vibration modes of a stationary asymmetric rotor evolve when it is assembled to a flexible housing via multiple bearing supports. Prior to the assembly, 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.” In this paper, we first develop two mathematical criteria to identify balanced modes. For the first criteria, the rotor is subjected to fixed boundary conditions at the bearings prior to assembly. In this case, a vibration mode will be a balanced mode if the reactions at the fixed boundary vanish. For the second criterion, the rotor is subjected to free boundary conditions (including the bearing points) prior to assembly. In this case, a vibration mode will be a balanced mode if the bearing locations are nodal points of the vibration mode. These mathematical criteria are then applied to a rotor consisting of a rigid hub supporting a flexible structure, which appears in many practical applications. For balanced modes, the criteria lead to a conclusion that surface integrals of modal forces and moments at the flexible-rigid rotor interface will vanish. Moreover, these surface integrals can be conveniently calculated using finite element methods. To validate the mathematical criteria, modal testing was conducted on a disk with 4 pairs of brackets mounted on a rigid spindle, a ball-bearing spindle and a fluid-dynamic bearing spindle.

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.

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.

Identifying VIV Vibration Modes by Use of the Empirical Orthogonal Functions Technique

2002 ◽  
Author(s):  
Gudmund Kleiven
Keyword(s):  
Model Test ◽  
Vibration Mode ◽  
Multivariate Data ◽  
Empirical Orthogonal Functions ◽  
Mode Shapes ◽  
Data Sets ◽  
Vibration Modes ◽  
Ocean Engineering ◽  
Orthogonal Functions ◽  
Related Technique

The Empirical Orthogonal Functions (EOF) technique has widely being used by oceanographers and meteorologists, while the Singular Value Decomposition (SVD being a related technique is frequently used in the statistics community. Another related technique called Principal Component Analysis (PCA) is observed being used for instance in pattern recognition. The predominant applications of these techniques are data compression of multivariate data sets which also facilitates subsequent statistical analysis of such data sets. Within Ocean Engineering the EOF technique is not yet widely in use, although there are several areas where multivariate data sets occur and where the EOF technique could represent a supplementary analysis technique. Examples are oceanographic data, in particular current data. Furthermore data sets of model- or full-scale data of loads and responses of slender bodies, such as pipelines and risers are relevant examples. One attractive property of the EOF technique is that it does not require any a priori information on the physical system by which the data is generated. In the present paper a description of the EOF technique is given. Thereafter an example on use of the EOF technique is presented. The example is analysis of response data from a model test of a pipeline in a long free span exposed to current. The model test program was carried out in order to identify the occurrence of multi-mode vibrations and vibration mode amplitudes. In the present example the EOF technique demonstrates the capability of identifying predominant vibration modes of inline as well as cross-flow vibrations. Vibration mode shapes together with mode amplitudes and frequencies are also estimated. Although the present example is not sufficient for concluding on the applicability of the EOF technique on a general basis, the results of the present example demonstrate some of the potential of the technique.

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].

Modal Analysis for Swing-Arm of LED Die Bonder

2011 ◽  
Vol 422 ◽  
pp. 379-382
Author(s):  
Wei Chuang Quan ◽  
Mei Fa Huang ◽  
Zhi Yue Wang ◽  
Da Wei Zhang
Keyword(s):  
Modal Analysis ◽  
Vibration Mode ◽  
Three Dimensional ◽  
Element Model ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Lead Frame ◽  
Swing Arm ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

Led die bonder used for bond lead frame and chip is one of the key equipment of led production line. The swing-arm is an important component of led die bonder and its dynamic characteristics will directly affect the piece accuracy. At present, the accuracy and efficiency of led die bonder are limited because of the vibration of the swing-arm. In solving this problem, a three-dimensional finite-element model for swing-arm is built to provide analytical frequencies and vibration modes. Then the modal distribution and vibration mode shapes for swing-arm are obtained after analyzing the modal by ansys10.0. Finally the dynamics effects of this structure by modal frequency and vibration mode are analyzed. The modal analysis of structural would provide the reference to dynamics analysis and structural optimization for swing-arm in practical use.

Excitation of Arbitrary Displacement/Velocity Conditions in Rotationally Periodic Structures

1995 ◽  
Author(s):  
P. Schmiechen ◽  
D. J. Ewins ◽  
I. Bucher
Keyword(s):  
Natural Frequencies ◽  
Periodic Structures ◽  
Search Algorithm ◽  
Simulated Data ◽  
Travelling Wave ◽  
Modal Testing ◽  
Mode Shapes ◽  
Wave Form ◽  
Structure Interaction ◽  
Wave Patterns

Abstract For an investigation into the structural interaction between rotating and non-rotating rotationally periodic turbine components, it was required to be able to generate experimentally prescribed response conditions. In more descriptive terms, conditions were sought to excite wave-patterns such as travelling and standing waves, and to suppress certain modes. In the paper these conditions are derived from modal properties. Simulated data are presented to demonstrate some of the phenomena and to highlight the practical difficulties. For rotationally periodic structures, most natural frequencies are of multiplicity two, and are sometimes called ‘double modes’. Their associated mode shapes can rotate in the plane of symmetry. The responses due to the two modes can be combined and expressed in a wave form, which can be split into travelling and standing wave components. Theoretically, it is possible to excite a pure travelling wave in a perfectly rotationally periodic structure, but there are limits to this in practice as real structures will always exhibit some degree of imperfection. These structures are said to be mistuned, and the imperfection splits the double modes into pairs of close modes. Simulations show the predicted vibration phenomena. In particular, the case of discrete excitations relevant to modal testing is investigated. The simulations show clearly that in this case components of other modes will generally be present. In an experiment, the results for driving the excitations will not give the theoretically expected response due to non-linearities of the shaker-structure interaction. However, the effects can be reduced by employing a computerised search algorithm.

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.

Measurement of Fan Vibration Using Double Pulse Holography

1978 ◽  
Vol 100 (4) ◽  
pp. 655-663 ◽  
Author(s):  
B. S. Hockley ◽  
R. A. J. Ford ◽  
C. A. Foord
Keyword(s):  
Vibration Mode ◽  
Mechanical Vibration ◽  
Propagation Speed ◽  
Double Pulse ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Television System ◽  
Temporal Phase ◽  
Frequency Split ◽  
Laser Holography

Supersonic unstalled flutter in gas turbine fans is a self-excited instability in which mechanical vibrations give rise to unsteady aerodynamic forces which drive the mechanical vibration. The phenomenon is very sensitive to the deflected shapes of the blades and to the spatial and temporal phases of the blades’ responses. This paper is concerned with the measurement of vibrational behavior on static fans and relating it to flutter. Accurate detailed data on the blade and disk vibration mode shapes of fans up to 2.2 m diameter has been measured using double pulse laser holography. Both axial and tangential components of the blade mode shape are obtained by taking holograms from two directions. The analysis of the holograms is performed with the aid of a computer linked television system which generates the required blade mode shapes directly from the photographs of the hologram reconstructions. The disk mode measurements on real fans have shown the existence of pairs of spatially orthogonal vibration modes which have similar shapes (e.g. both 4D) but slightly different natural frequencies. This frequency split between modes means that the flutter wave will experience a cyclic variation in amplitude and propagation speed as it travels round the fan. In addition, the temporal phase angle between twist and flap in a single blade, which is generally assumed to be 90 deg, will vary from blade to blade.

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