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

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].

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

Volume 1: 24th Conference on Mechanical Vibration and Noise, Parts A and B ◽

10.1115/detc2012-70150 ◽

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.

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

10.1115/1.3025847 ◽

2009 ◽

Vol 131 (2) ◽

Cited By ~ 20

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 ◽

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.

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

10.1115/1.4024760 ◽

2013 ◽

Vol 135 (5) ◽

Cited By ~ 3

Author(s):

W. C. Tai ◽

I. Y. Shen

Keyword(s):

Equations Of Motion ◽

Housing System ◽

Combination Resonance ◽

Linear Elastic ◽

Model Free ◽

Parametric Resonances ◽

Symmetric Rotor ◽

Cyclic Symmetric ◽

Central Shaft ◽

Multiple Bearings

This paper is meant to model free vibration of a coupled rotor-bearing-housing system. In particular, the rotor is cyclic symmetric and spins at constant speed while the housing is stationary and flexible. The rotor and housing are assembled via multiple, linear, elastic bearings. A set of equations of motion is derived using component mode synthesis, in which the rotor and the housing each are treated as a component. The equations of motion take 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. The rotor and the housing are connected via two linear elastic bearings. For the second example, the rotor is cyclic symmetric in the form of a disk with four evenly spaced radial slots. The housing and bearings remain the same. In both examples, instability appears as a combination resonance of the sum type between a rotor mode and an elastic housing mode. The cyclic symmetric rotor, however, has more instability zones. Finally, effects of damping are studied. Damping of the housing widens the instability zones, whereas the damping of the rotor does the opposite.

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

10.1115/1.3147167 ◽

2009 ◽

Vol 131 (5) ◽

Cited By ~ 12

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.

Scanning confocal vibrometer microscope for vibration analysis of energy-harvesting MEMS in wearables

tm - Technisches Messen ◽

10.1515/teme-2017-0042 ◽

2017 ◽

Vol 84 (s1) ◽

Cited By ~ 2

Author(s):

Robert Kowarsch ◽

Jürgen Janzen ◽

Christian Rembe ◽

Hyunjun Cho ◽

Hyuck Choo

Keyword(s):

Energy Harvesting ◽

Dynamic Analysis ◽

Vibration Analysis ◽

Microelectromechanical Systems ◽

Laser Doppler Vibrometer ◽

Mode Shapes ◽

Confocal Laser ◽

Contactless Measurement ◽

Medical Implants ◽

Resonance Frequencies

AbstractWe present a scanning confocal laser-Doppler vibrometer microscope for sensitive, contactless measurement of microelectromechanical systems (MEMS). This systems enables the dynamic analysis up to 3.2 MHz with a lateral resolution of few micrometers. We show measurements on developed MEMS for vocal-energy harvesting in wearables and medical implants. For efficient harvesting a cantilever beam with a serpentine form was designed with a fundamental resonance at 200 Hz. We verified the simulated mode shapes with our vibration measurements. The observed deviations in resonance frequencies between simulation and measurement are due to modelling and manufacturing dissimilarities.

Volume 1: 21st Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

Vibration of a Spinning Rotationally Periodic Rotor

10.1115/detc2007-34888 ◽

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 ◽

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.

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

Volume 5: Structures and Dynamics, Parts A and B ◽

10.1115/gt2008-50727 ◽

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 ◽

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 Response of a Spinning, Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing Via Multiple Bearings

10.1115/1.4029989 ◽

2015 ◽

Vol 137 (4) ◽

Cited By ~ 3

Author(s):

W. C. Tai ◽

I. Y. Shen

Keyword(s):

Dominant Mode ◽

Housing System ◽

Cyclic Symmetry ◽

Test Results ◽

Benchmark Model ◽

Rotor Bearing ◽

Symmetric Rotor ◽

Cyclic Symmetric ◽

Square Plate ◽

Multiple Bearings

This paper is to study ground-based response of a spinning, cyclic symmetric rotor assembled to a flexible housing via multiple bearings. In particular, interaction of the spinning rotor and the flexible housing is manifested theoretically, numerically, and experimentally. In the theoretical analysis, we show that the interaction primarily appears in coupled rotor–bearing–housing modes whose response is dominated by the housing. Specifically, let a housing-dominant mode have natural frequency ω(H) and the spin speed of the rotor to be ω3. In rotor-based coordinates, response of the spinning rotor for the housing-dominant mode will possess frequency splits ω(H)±ω3. In ground-based coordinates, response of the spinning rotor will possess alternative frequency splits ω(H)-(k+1)ω3 and ω(H)-(k-1)ω3, where k is an integer determined by the cyclic symmetry of the rotor and the housing-dominant mode of interest. In the numerical analysis, we study a benchmark model consisting of a spinning slotted disk mounted on a stationary square plate via two ball bearings. The numerical model successfully confirms the frequency splits both in the rotor-based and ground-based coordinates. In the experimental analysis, we conduct vibration testing on a rotor–bearing–housing system that mimics the numerical benchmark model. Test results reveal two housing-dominant modes. As the rotor spins at various speed, measured waterfall plots confirm that the housing-dominant modes split according to ω(H)-(k+1)ω3 and ω(H)-(k-1)ω3 as predicted.

Split beam method for determining mode shapes of cantilever beams under multiple loading conditions

Journal of Vibration and Control ◽

10.1177/10775463211027686 ◽

2021 ◽

pp. 107754632110276

Author(s):

Jun-Jie Li ◽

Shuo-Feng Chiu ◽

Sheng D Chao

Keyword(s):

Operation Mode ◽

Point Contact ◽

Mode Shapes ◽

Cantilever Beams ◽

Loading Conditions ◽

Mechanical Responses ◽

Relative Contribution ◽

Resonance Frequencies ◽

Beam Method ◽

Multiple Loading

We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young’s modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the “dynamical” operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam.

Quantify Resonance Inspection With Finite-Element-Based Modal Analyses

Volume 13: Sound, Vibration and Design ◽

10.1115/imece2010-39497 ◽

2010 ◽

Author(s):

K. Lai ◽

X. Sun ◽

C. Dasch

Keyword(s):

Finite Element ◽

High Stress ◽

Mesh Size ◽

Element Model ◽

Sensitivity Study ◽

Production Level ◽

Mode Shapes ◽

Resonance Spectroscopy ◽

Frequency Spectra ◽

Resonance Frequencies

Resonance inspection uses the natural acoustic resonances of a part to identify anomalous parts. Modern instrumentation can measure the many resonant frequencies rapidly and accurately. Sophisticated sorting algorithms trained on sets of good and anomalous parts can rapidly and reliably inspect and sort parts. This paper aims at using finite-element-based modal analysis to put resonance inspection on a more quantitative basis. A production-level automotive steering knuckle is used as the example part for our study. First, the resonance frequency spectra for the knuckle are measured with two different experimental techniques. Next, scanning laser vibrometry is used to determine the mode shape corresponding to each resonance. The material properties including anisotropy are next measured to high accuracy using resonance spectroscopy on cuboids cut from the part. Then, finite element model (FEM) of the knuckle is generated by meshing the actual part geometry obtained with computed tomography (CT). The resonance frequencies and mode shapes are next predicted with a natural frequency extraction analysis after extensive mesh size sensitivity study. The good comparison between the predicted and the experimentally measured resonance spectra indicate that finite-element-based modal analyses have the potential to be a powerful tool in shortening the training process and improving the accuracy of the resonance inspection process for a complex, production level part. The finite element based analysis can also provide a means to computationally test the sensitivity of the frequencies to various possible defects such as porosity or oxide inclusions especially in the high stress regions that the part will experience in service.