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

2015 ◽  
Vol 137 (4) ◽  
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.

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

2013 ◽  
Vol 135 (5) ◽  
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.

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.

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 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].

Dynamic Analysis of a Turbocharger Centrifugal Compressor Impeller

1991 ◽  
Author(s):  
V. Ramamurti ◽  
D. A. Subramani ◽  
K. Sridhara
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Centrifugal Compressor ◽  
Element Model ◽  
Simplified Model ◽  
Cyclic Symmetry ◽  
Compressor Impeller ◽  
Cyclic Symmetric ◽  
The Finite Element Model

Abstract Stress analysis and determination of eigen pairs of a typical turbocharger compressor impeller have been carried out using the concept of cyclic symmetry. A simplified model treating the blade and the hub as isolated elements has also been attempted. The limitations of the simplified model have been brought out. The results of the finite element model using the cyclic symmetric approach have been discussed.

scholarly journals Analytical and Experimental Investigation of the Stability of the Rotor-Bearing System of a New Small Turbocharger

1987 ◽  
Author(s):  
H. R. Born
Keyword(s):  
Experimental Investigation ◽  
Dynamic Stability ◽  
Squeeze Film ◽  
Test Results ◽  
Flow Capacity ◽  
Squeeze Film Dampers ◽  
Rotor Bearing ◽  
The Stability ◽  
Turbine Design ◽  
Bearing System

This paper presents an overview of the development of a reliable bearing system for a new line of small turbochargers where the bearing system has to be compatible with a new compressor and turbine design. The first part demonstrates how the increased weight of the turbine, due to a 40 % increase in flow capacity, influences the dynamic stability of the rotor-bearing system. The second part shows how stability can be improved by optimizing important floating ring parameters and by applying different bearing designs, such as profiled bore bearings supported on squeeze film dampers. Test results and stability analyses are included as well as the criteria which led to the decision to choose a squeeze film backed symmetrical 3-lobe bearing for this new turbocharger design.

Dynamic modelling of a rotor-bearing-housing system including a localized fault

2017 ◽  
Vol 232 (3) ◽  
pp. 385-397 ◽  
Author(s):  
Jing Liu ◽  
Yajun Xu ◽  
Yimin Shao
Keyword(s):  
Dynamic Characteristics ◽  
Ball Bearing ◽  
Contact Stiffness ◽  
Great Influence ◽  
Fault Detection And Diagnosis ◽  
Time Dependent ◽  
Housing System ◽  
Stiffness Coefficient ◽  
Rotor Bearing ◽  
Vibration Responses

An in-depth understanding of the dynamic characteristics through rotor-bearing-housing systems is very valuable for fault detection and diagnosis applications of rotating machines such as high-speed spindle, roll mill, gearbox, engines, etc. A new vibration model of a rotor-bearing-housing system considering the rotor compliance, elastic interface between the housing and outer race, housing compliance, and time-dependent excitations introduced by a localized fault on the inner and outer races of an inherent ball bearing is proposed in this work. An analytical method for calculating the time-dependent excitations including the time-dependent displacement excitation and contact stiffness coefficient between the ball and fault edges is presented. Differences between vibration responses of a rotor-bearing-housing system from the proposed model and the previous model without the rotor compliance in the literature are discussed. The presented model is used to discuss the influences of all the rotor compliance, housing compliance, and fault sizes on the races of the inherent ball bearing on the vibration responses and vibration transmission characteristics through the rotor-bearing-housing system, which cannot be formulated by the current dynamic models in the listed references. An experimental study is introduced to validate the presented model. The results show that the rotor compliance and time-dependent contact stiffness coefficient caused by the fault have great influence on the dynamic characteristics through the rotor-bearing-housing system. It also seems that the developed method can provide a new vibration modelling method for the vibration analysis for a rotor-bearing-housing system with and without the faults.

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.

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.

Influence of Bearing Alignment on Vibration of Multi-Span Rotor

1997 ◽  
Author(s):  
Toshio Hirano ◽  
Tatsuo Yamashita
Keyword(s):  
Vertical Direction ◽  
Vertical Position ◽  
Test Results ◽  
Rotor Vibration ◽  
Turbo Generator ◽  
Generation Plant ◽  
Rotor Bearing ◽  
Power Generation Plant ◽  
Film Characteristics ◽  
Bearing System

Abstract This paper describes the influence of bearing alignment (vertical position) on vibration of multi-rotor-bearing system. The rotor-bearing system used in this study consists of three rotors and six bearings. It is designed to have the equivalent dynamic characteristics to the turbo-generator-systems for utility power generation plant. And it is equipped with alignment setting devices which can set the bearing alignment in vertical direction precisely without disassembling the test rig. It was confirmed that bearing alignment affects the rotor vibration. Some critical speeds and their peak amplitudes change. When the bearing alignment changes, the load of the bearing changes and it affects oil film characteristics. The calculated results considering this agree with the test results. The effect of the bearing alignment on the rotor vibration becomes large when the bearing load is about zero.

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