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Modeling, analysis and experimental validation of flexible rotor systems with water-lubricated rubber bearings
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Modeling, analysis and experimental validation of flexible rotor systems with water-lubricated rubber bearings
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Content
MODELING, ANALYSIS AND EXPERIMENTAL VALIDATION OF
FLEXIBLE ROTOR SYSTEMS WITH WATER-LUBRICATED RUBBER
BEARINGS
by
Shibing Liu
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2015
Copyright 2015 Shibing Liu
ii
To my parents, my elder brother and my wife
iii
Acknowledgements
First, I wish to express my deepest gratitude to my Ph.D thesis advisor, Professor
Bingen Yang, for his smart guidance, insights, encouragements, excellent advice and
supports throughout my Ph.D study. One important thing I learned from him is how to
find problems and solve problems and "Do anything you are interested in". I would also
like to thank my committee members, Professor Geoffrey Shiflett and Professor Carter
Wellford for their generous assistance.
I would like to express my appreciation to ITT Corporation for supporting the design
and test of the experiments. My sincere thanks go to Mr. Paul Behnke, Dr. Lily Ding and
Mr. Scott Liscum from ITT Corporation for their guidance and advice throughout the
research.
There is no word that can express my thanks to my beloved parents, my elder brother
and my wife for their support, patience and understanding. They are, and will be, always
the motive of my life.
iv
Table of Contents
Acknowledgements ............................................................................................................ iii
Table of Contents ............................................................................................................... iv
List of Figures ................................................................................................................... vii
List of Tables ..................................................................................................................... xi
Abstract ............................................................................................................................. xii
Chapter 1 Introduction..................................................................................................... 1
1.1. Motivation and Purpose ....................................................................................... 1
1.2. Literature and Background ................................................................................... 4
1.3. Scope and Summary of Study ............................................................................ 10
Chapter 2 The Distributed Transfer Function Method .............................................. 12
2.1. Formulation of the General Problem .................................................................. 12
2.1.1. Shaft Segments............................................................................................ 14
2.1.2. Rigid Disks.................................................................................................. 15
2.1.3. Short Bearings ............................................................................................. 17
2.1.4. Couplings .................................................................................................... 18
2.2. Transfer Function Formulation .......................................................................... 19
2.3. Dynamic Analysis .............................................................................................. 22
2.3.1. Eigensolutions ............................................................................................. 22
v
2.3.2. General Steady State Response ................................................................... 23
2.3.3. Steady State Response to Periodic Excitations ........................................... 24
2.4. Numerical Examples .......................................................................................... 26
2.4.1. Example I: Critical Speeds of a Uniform Shaft with a Rigid Disk ............. 26
2.4.2. Example II: Unbalanced Mass Response of a Flexible Rotor System ........ 27
Chapter 3 Experimental Identification of Dynamic Stiffness of WLRBs .................. 29
3.1. Approach ............................................................................................................ 31
3.2. End Conditions of Rotating Shaft ...................................................................... 34
3.3. Results ................................................................................................................ 35
3.3.1. Dynamic Stiffness of WLRBs .................................................................... 35
3.3.2. Comparison of Theoretical and Experimental Unbalanced Mass
Response .................................................................................................................... 41
3.3.3. Post-Test Examination ................................................................................ 42
3.4. Conclusions ........................................................................................................ 43
Chapter 4 A New Model of Water-Lubricated Rubber Bearings .............................. 44
4.1. Boundary Lubrication ........................................................................................ 46
4.2. Elastohydrodynamic Lubrication ....................................................................... 50
4.2.1. Determination of h(p) by Compliance Operator Method ........................... 52
4.2.2. Steady State and Dynamic Characteristics ................................................. 59
4.2.3. Implementation of Finite Element Method in Solution of Steady State
Pressure ..................................................................................................................... 66
4.3. Mixed Lubrication via the Method .............................................................. 67
vi
4.4. Parameter Study on Dynamic Coefficients of WLRBs ...................................... 74
4.4.1. Effect of Eccentricity .................................................................................. 74
4.4.2. Effect of Land Region Angle ...................................................................... 77
4.4.3. Effect of Number of Grooves ..................................................................... 80
4.5. Conclusions ........................................................................................................ 82
Chapter 5 Vibration Analysis of a Flexible Rotor System with WLRBs ................... 84
5.1. Effect of Model of WLRBs ................................................................................ 86
5.2. Effect of Length of WLRBs ............................................................................... 89
5.3. Effect of Location of WLRBs ............................................................................ 90
5.4. Effect of Number of WLRBs ............................................................................. 93
5.5. Conclusions ........................................................................................................ 96
Chapter 6 Conclusions .................................................................................................... 97
References ....................................................................................................................... 100
Appendix A List of State Matrices ................................................................................. 107
Appendix B Complex Variable Filtering and Full Spectrum Method ............................ 111
Appendix C Geometry and Operation Parameters of the WLRBs ................................. 114
vii
List of Figures
Figure 1.1 A water-circulated vertical pump (from ITT Corporation) ............................... 2
Figure 1.2 A typical water-lubricated rubber bearing: (a) 3D view; (b) cross-section
view ..................................................................................................................................... 3
Figure 2.1 Schematic of a flexible multistage rotor-bearing system ................................ 13
Figure 2.2 Basic elements of the rotor-bearing system..................................................... 13
Figure 2.3 Free body diagram of a rigid disk at node i ..................................................... 17
Figure 2.4 Free body diagram of a short bearing at node i ............................................... 17
Figure 2.5 Free body diagram of a stepped shaft at node i ............................................... 19
Figure 2.6 A uniform shaft with a rigid disk .................................................................... 26
Figure 2.7 A flexible rotor system with three disks .......................................................... 27
Figure 2.8 Unbalanced mass response of a flexible rotor system ..................................... 28
Figure 3.1 Water-lubricated rubber bearing test device.................................................... 29
Figure 3.2 Schematic of the water-lubricated rubber bearing test device ......................... 30
Figure 3.3 Experimental data processing procedure ......................................................... 33
Figure 3.4 Top end of the rotating shaft and its spring constraint model ......................... 34
Figure 3.5 Unbalanced mass response for the 101.6 mm rubber bearing ......................... 37
Figure 3.6 Dynamic stiffness of the 101.6 mm rubber bearing (a) Direct stiffness; (b)
Cross-coupled stiffness ..................................................................................................... 39
viii
Figure 3.7 Comparison of direct stiffness for 50.8 mm and 101.6 mm rubber bearing ... 40
Figure 3.8 Comparison of theoretical and experimental unbalanced mass response at
the lower proximity probe at 800rpm the case with one disk and 101.6 mm rubber
bearing............................................................................................................................... 41
Figure 3.9 The rotating shaft with polished areas (in squares) ......................................... 42
Figure 4.1 Examples of contacts between the rotating shaft and rubber staves ............... 45
Figure 4.2 Nonlinearity of dynamic stiffness of WLRBs ................................................. 45
Figure 4.3 Rubber bearing in boundary lubrication: (a) A stave of WLRBs; (b)
rectangular rubber block as approximation....................................................................... 46
Figure 4.4 Model used in Abaqus - Contact ..................................................................... 48
Figure 4.5 Stiffness of rubber staves in boundary lubrication .......................................... 49
Figure 4.6 Coordinates in elastohydrodynamic lubrication .............................................. 51
Figure 4.7 Bering model used in Abaqus - compliance operator ..................................... 53
Figure 4.8 Compliance operator vs. applied strain ........................................................... 53
Figure 4.9 Compliance operator by multi-region exponential approximation ................. 56
Figure 4.10 Rubber deformation for parabolic-shape pressure distribution (a)
Parabolic distribution of pressure; (b) Comparison of rubber deformation ...................... 58
Figure 4.11 Elastohydrodynamic coefficients of the 101.6 mm rubber bearing (a)
Direct stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d)
Cross-coupled damping coefficients ................................................................................. 64
ix
Figure 4.12 Elastohydrodynamic coefficients of the 50.8 mm rubber bearing (a)
Direct stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d)
Cross-coupled damping coefficients ................................................................................. 66
Figure 4.13 Procedure to determine steady state pressure with FEM implementation .... 68
Figure 4.14 Circumferential pressure distribution of the 101.6 mm bearing with
and at rpm ........................................................................................... 69
Figure 4.15 Mixed lubrication of long WLRBs ................................................................ 70
Figure 4.16 Comparison of theoretical and experimental values of .............................. 71
Figure 4.17 Comparison of theoretical and experimental direct stiffness ........................ 71
Figure 4.18 Procedure for determination of dynamic coefficients of WLRBs ................. 73
Figure 4.19 Elastohydrodynamic coefficients with respect to for various (a)
Direct stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d)
Cross-coupled damping coefficients ................................................................................. 76
Figure 4.20 Rubber bearing dynamic stiffness with respect to for various ................ 76
Figure 4.21 Stiffness of rubber staves with respect to land region angle (
) .................. 77
Figure 4.22 Compliance operator for various land region angle ...................................... 78
Figure 4.23 Elastohydrodynamic coefficients with respect to for various
(a)
Direct stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d)
Cross-coupled damping coefficients ................................................................................. 79
Figure 4.24 Rubber bearing dynamic stiffness with respect to for various
.............. 79
x
Figure 4.25 Elastohydrodynamic coefficients with respect to for different number
of grooves (a) Direct stiffness; (b) Cross-coupled stiffness; (c) Direct damping
coefficients; (d) Cross-coupled damping coefficients ...................................................... 81
Figure 4.26 Rubber bearing dynamic stiffness with respect to for different number
of grooves.......................................................................................................................... 81
Figure 5.1 A multistage rotor system with long WLRBs ................................................. 84
Figure 5.2 Distribution of dynamic stiffness of WLRBs (a) Case I: Uniform dynamic
stiffness; (b) Case II: Stepped dynamic stiffness .............................................................. 87
Figure 5.3 Effect of model of WLRBs on the unbalance mass response of the
multistage system at disk 1 ............................................................................................... 88
Figure 5.4 Effect of length of WLRBs on the unbalance mass response of the
multistage system at disk 1 ............................................................................................... 89
Figure 5.5 Effect of the bearing locations on the unbalance mass response at disk 2 of
the multistage system (a) First bearing; (b) Third bearing; (c) Fifth bearing ................... 91
Figure 5.6 Whirl motion at rpm at disk 2 of the multistage system with
different location of bearing 1........................................................................................... 92
Figure 5.7 Effect of number of bearings on the unbalance mass response at disk 3 of
the multistage system ........................................................................................................ 93
Figure 5.8 Whirl motion at rpm of the multistage system at (a) Disk 1; (b)
Disk 2; (c) Disk 3 .............................................................................................................. 95
xi
List of Tables
Table 2.1 Data of a flexible rotor system .......................................................................... 27
Table 3.1 Theoretical and experimental natural frequency for 101.6 mm rubber
bearing............................................................................................................................... 38
Table 3.2 Theoretical and experimental natural frequency for the 50.8 mm rubber
bearing............................................................................................................................... 41
Table 4.1 Bearing geometry and operation parameters in numerical studies ................... 74
Table 5.1 Parameters of a multistage flexible rotor-bearing system ................................ 85
Table 5.2 Critical speeds of a multistage rotor system ..................................................... 88
xii
Abstract
Flexible rotor systems that are supported or guided by water-lubricated rubber
bearings (WLRBs) have a variety of engineering applications. Vibration analysis of this
type of machinery for performance and duality requires accurate modeling of WLRBs
and related rotor-bearing assemblies. In this study, a new model of WLRBs, with the
focus on determination of bearing dynamic coefficients, is presented. Due to its large
length-to-diameter ratio, a WLRB cannot be described by conventional pointwise bearing
models with good fidelity. The bearing model considered in this work considers spatially
distributed bearing forces, and for the first time, addresses the issue of mixed lubrication,
which involves interaction effects of shaft vibration, elastic deformation of rubber
material and fluid film pressure, and validates the WLRB model in experiments.
Additionally, with the new bearing model, vibration analysis of WLRB-supported
flexible rotor systems is performed through use of a distributed transfer function method
(DTFM), which delivers accurate and closed-form analytical solutions of steady-state
response without discritization.
Chapter 1
Introduction
1.1. Motivation and Purpose
Flexible rotor systems with water-lubricated rubber bearings (WLRBs) have many
industrial applications. Take water-circulated vertical pumps as an example, which, as
large and expensive industrial equipments, are commonly found in power generation,
mining dewatering, water waste treatment, oil and gas industries and industrial processes.
In a vertical pump (see Figure 1.1), WLRBs guide the rotating shaft that carries multiple
rigid bodies (impellers). Due to high flexibility of the shaft, dynamic modeling and
vibration analysis is essentially important to the operation and duality of this kind of rotor
systems.
A typical WLRB is a long cylindrical metal shell that hosts multiple rubber staves
separated by axial grooves; see Figure 1.2. The usage of the bearing has three major
advantages: (i) pumped water going through the bearing is conveniently used as a
lubricant, which reduces the pump operation cost; (ii) water flow takes away heat and
fine particles through the bearing grooves; and (iii) the natural resilience of rubber gives
the bearing good properties for shock and vibration absorption and wear resistance. The
2
knowledge on the dynamic coefficients (stiffness and damping coefficients) of WLRBs is
necessary in vibration analysis and optimal design of flexible rotor systems with WLRBs.
However, there is no experimentally validated model for WLRBs in both literature and
industries.
Figure 1.1 A water-circulated vertical pump (from ITT Corporation)
3
(a) (b)
Figure 1.2 A typical water-lubricated rubber bearing: (a) 3D view; (b) cross-section view
Due to lack of a faithful WLRB model, R&D in industries nowadays is still without
reliable computer-aided design tools. Existing commercial software codes on rotor
dynamics are not suitable for vibration analysis of flexible rotor systems with WLRBs
because they usually compute the dynamic coefficients of bearings by plain journal
bearings models, and ignore the elastic deformation of rubber material. As found out
from our technical communication with the industry, usage of commercial rotor dynamics
software codes can yield significant errors, and in some cases give totally wrong results.
Therefore, these software codes are not reliable in analysis and design of flexible rotor
systems with WLRBs in terms of safe and efficient operation of these rotor systems.
The purpose of this dissertation is to study the vibration of flexible rotor systems
with WLRBs. Special attention is given to development and experimental validation of a
new model of WLRBs. The objectives of the investigation are as follows.
(1) Develop a distributed transfer function method (DTFM) for vibration analysis of
flexible rotor systems.
(2) Develop a new model of water-lubricated rubber bearings.
4
(3) Design an experiment to validate the new model of WLRBs.
(4) Study the vibration of flexible rotor systems with WLRBs using the distributed
transfer function method with the new model of WLRBs.
1.2. Literature and Background
The subject of vibration of rotor-bearing systems has been extensively studied in the
past and is well documented in references [Rao, 1991; Childs, 1993; Lalanne and Ferraris,
1998; Chen and Gunter, 2007; Friswell et al., 2010]. Various numerical methods have
been developed for dynamic analysis of rotor-bearing systems, including the finite
element method [Kang et al., 1992; Oncescu et al., 2001; Hili et al., 2007], transfer
matrix method [Rao, 1991; Yim and Ryu, 2011], Rayleigh-Ritz method [Lalanne and
Ferraris, 1988], Galerkin method [Yim ect., 1986; Lee and Yun, 1993], modal analysis
method [Lee et al., 1988], and assumed mode method [Lee, 1995]. These numerical
methods, especially finite element method, have been widely used to predict the vibration
of flexible rotor-bearing systems. However, analytical methods are always desirable as
analytical methods provide more information and deeper physical insights and offer more
accurate solutions. Tan and Kuang [Tan and Kuang, 1995] studied the vibration of a
rotating discontinuous shaft and obtained the exact and closed form solutions for the free
and forced response by the distributed transfer function method (DTFM) and a
generalized displacement formulation. They also studied the influence of boundary
conditions on mode shape. However, disks, which are also important components of
5
flexible rotor-bearing systems, are not included in the analysis. Fang and Yang [Fang
and Yang, 1998] further developed the distributed transfer function method for complex
flexible rotor systems. They calculated the eigensolutions and forced response of flexible
rotor-bearing systems based on a global dynamic stiffness matrix which has dimension n-
by-n for a complex system with n nodes. However, the computation load increases as the
number of nodes increases.
Yang [Yang, 2008] developed a new distributed transfer function method
formulation methodology to solve conduction heat transfer problems in multilayer
composites. In this study, the DTFM formulation methodology developed by Yang [Yang,
2008] will be first time used to model the vibration of flexible rotor-bearing systems. The
DTFM in this thesis delivers closed-form analytical solutions and only involves
manipulation of eight-by-eight matrices. Thus, it is highly efficient in computation.
The dynamic stiffness coefficients of bearings play an important role in vibration
analysis of flexible rotor-bearing systems. Modeling and analysis of bearings have been
studied for many years by researchers; for instance see [Someya, 1989; Childs, 1993;
Hamrock, 2004; Chen and Gunter, 2007; Zakharow, 2010; San André s, 2010; Szeri,
2011]. This research is interested in fluid lubricated bearings. For different purposes,
there are many different types of fluid lubricated bearings, such as plain journal bearings,
squeeze film dampers, tilting pad bearings, water-lubricated bearings, etc. Experimental
methods to identify dynamic coefficients of bearings are reviewed by Tiwari [Tiwari et
al., 2004] and Dimond [Dimond et al., 2009]. The different force excitation methods,
including incremental loading, sinusoidal, pseudorandom, impulse, known/additional
6
unbalance, and non-contact excitation, were discussed. Data processing methods in the
time and frequency domains were also presented. However, no experimental
investigation on the dynamics stiffness coefficients of water-lubricated rubber bearings
has been reported.
The tribological properties of WLRBs have been experimentally studied by some
researchers. Wang and Wang [Wang and Wang, 2011] showed that the friction
coefficient decreases with increasing velocity; the friction coefficient increases at first
and then decreases with increasing load; and the friction coefficient increases sharply
with the rising temperature. Cheng [Cheng et al., 2012] tested the friction coefficient
changes of WLRBs due to the effect of temperature under different pressures. They found
that the friction coefficient increases along with temperature rising, especially when the
bearing is at low speed. Wang et al. [Wang et al., 2013] measured the circumferential
pressure distribution for water-lubricated rubber bearings with concave and flat staves.
They found that WLRBs with concave staves have continuous water film in the load
carrying region; but independent film pressure peak on each stave occurs for WLRBs
with flat staves. However, they only carried out experiments for very high eccentricities.
Cabrera et al. [Cabrera et al., 2005] also measured the circumferential pressure
distribution in a WLRB and compared the experimental data with the simulation results
obtained by using commercial CFD software. It was shown that the film pressure profiles
were very different from those of conventional rigid bearings. However, the contribution
of the fluid film to load support is small, which suggests that WLRBs do not operate in
completely hydrodynamic manner. They concluded that WLRBs operate under
7
conditions of mixed lubrication. So, to theoretically model WLRBs, the fluid film
lubrication, the elastic deformation of the rubber bearing staves, and the contact between
the rotating shaft and rubber bearing staves must be considered.
Pai et al. [Majumdar, 2004; Pai and Pai, 2008; Pai and Hargreaves, 2012] derived a
model for water-lubricated journal bearings with multiple axial grooves. Based on
Reynolds equation, they obtained load capacity, bearing dynamic coefficients, and
stability criteria. The results were compared with experimental and simulation results.
However, the rubber deformation was not considered in the bearing model.
Using the hydrodynamic (HL) and elastohydrodynamic (EHL) lubrication theories,
Molka et al. [Molka et al., 2010] studied the bearing elastic deformation on the
performance characteristics of a cylindrical journal bearing. They find that the
deformation of the bearing liner has a significant influence on the performance of the
cylindrical journal bearing. The pressure area in the bearing is extended. The deformation
of the bearing liner increases the minimum film thickness, although the dynamic
coefficient, load capacity and attitude angle decrease. Skotheim and Mahadevan
[Skotheim and Mahadevan, 2005] provided a method for modeling soft lubrication but
without grooves. Vrande [Vrande, 2001] developed a model for a compliant journal
bearing. With this model, he performed experiments on a rotor-rubber bearing system
with a flexible shaft. While a nonlinear phenomenon was observed, the experimental data
did not support the theoretical model of the rotor-bearing system. Lahmar et al. [Lahmar
et al., 1998; Lahmar et al., 2010] theoretically investigated the effects of static and
dynamic deformation of thin elastic bearing liners on the dynamic performance
8
characteristics and stability of a water-lubricated, rubber-lined journal bearing operating
under small harmonic vibrations.
Water lubricated rubber bearings in flexible rotor systems usually are made of rubber,
which is incompressible material with Poisson's ratio almost equal to 0.5. The
mathematical models developed by above researchers cannot be applied to water
lubricated rubber bearings, because all of them model the rubber liner as compressible
material. In addition, they did not consider grooves in WLRBs.
As Cabrera et al. [Cabrera et al., 2005] found in their experiment, WLRBs operate
under the condition of mixed lubrication. Mixed lubrication is a mix of
elastohydrodynamic lubrication and boundary lubrication. In mixed-lubrication, the fluid
film is very thin and some of the opposing asperities touch. Wang et al. [Wang et al.,
1997] studied the mixed-lubrication of journal bearing conformal contacts considering
the effects of bearing deformation, surface roughness and asperity interaction. Park and
Kim [Park and Kim, 1998] numerically solved the elastohydrodynamic lubrication of the
finite line contact problem. Liu et al. [Liu et al., 2004] presented a finite element model
for mixed-lubrication of journal bearing systems operating in adverse conditions. They
considered the elastic deformation due to both hydrodynamic and contact pressure. And
their results are in good agreement with the work by Wang et al [Wang et al., 1997].
However, no studies have been reported on experimental validation of dynamic
coefficients of WLRBs, considering the mixed-lubrication.
In literature, all bearings are modeled as pointwise springs and dampers. But, for a
typical WLRB, the length-to-diameter ratio is larger than two. It is not reasonable to
9
model long bearings as pointwise springs and dampers. Zhang et al. [Zhang et al., 2013]
studied the dynamic characteristics of large ship rotor-bearing system based on an
advanced water lubricated rubber bearing model. In the advanced WLRB model, they
divided the whole stern bearing into four sections equidistantly along the bearing length
direction. And the water film force is determined for each small section. Using the new
WLRB model, they found that the predicted shaft centerline locus of the nodes located at
the propeller and the left WLRB rise significantly. Obviously, even they modeled a long
bearing as four pointwise springs and dampers, the model cannot fully describe the
spatial distributions of stiffness, damping and pressure along the axis of a rotating shaft.
In summary of previous researches, several open issues regarding the dynamic of
flexible rotors with WLRBs are as follows.
(i) There exists a big gap between theoretical and experimental results in modeling
WLRBs. In fact, no theoretical model of WLRBs has ever been well validated by
experiments, considering both grooves and mixed lubrication operation of WLRBs.
(ii) Most previous works treat bearings as pointwise springs and dampers. WLRBs
used in flexible rotor systems have a large length-to-diameter ratio, indicating a large area
of application of fluid-film pressure on a rotating shaft. Therefore, standard pointwise
bearing models are not sufficient enough to capture the physics of WLRB-supported
vertical pumps in vibration.
(iii) Those models for WLRBs accounting for elastic deformation of bearings are not
applicable to incompressible materials like rubber.
(iv) There is no experimentally validated model of the entire assembly of a flexible
10
rotating shaft with WLRBs.
(v) More accurate and efficient analytical methods for vibration analysis of flexible
rotor systems are always desirable.
1.3. Scope and Summary of Study
Chapters of this dissertation are organized in the following order.
Chapter 2 introduces the distributed transfer function method (DTFM) for vibration
analysis of flexible rotor systems. With the DTFM formulation, various dynamic
problems of flexible rotor systems, such as eigenvalue problem and steady-state response
problem, can be conveniently solved. Compared with previous numerical and analytical
methods, the DTFM in this dissertation delivers analytical solution and is more efficient
in computation. In addition, instead of commonly used pointwise bearing models, a
distributed visco-elastic bearing model is introduced for long bearings like WLRBs.
Chapter 3 describes a newly-built experiment device for identification of dynamic
stiffness of water-lubricated rubber bearings (WLRBs). The device can be used to
identify the rubber bearing dynamic stiffness using a flexible rotor-WLRB system. This
study is the first one that identifies the WLRB stiffness via a flexible rotor-WLRB system
in experiments.
In Chapter 4, a new model of WLRBs will be presented. This bearing model
describes mixed lubrication through combination of a fluid lubrication model and an
elastic rubber model, and treats spatial distributions of stiffness and pressure of WLRBs
11
as a visco-elastic foundation. For the first time in the literature, the WLRB model is
validated in experiments.
Chapter 5 studies the vibration of flexible rotor systems with WLRBs. The new
model of WLRBs is implemented in the DTFM to study the effect of bearing model,
bearing length, location and number of WLRBs on the vibration of flexible rotor systems.
This analysis is very useful for the vibration analysis and optimal design of flexible rotor
systems with WLRBs.
Chapter 6 concludes the results and contributions of this dissertation.
12
Chapter 2
The Distributed Transfer Function Method
The distributed transfer function method (DTFM) is a powerful method for modeling,
analysis and control of flexible dynamic systems [Yang, 2005; Yang, 2010; Yang and
Noh, 2012]. One obvious advantage of the DTFM over traditional finite element methods
(FEM) is that it can deliver highly accurate solutions of dynamic problems without the
need for discretization. As another advantage, the DTFM is convenient for computer
coding, and efficient in numerical simulation [Fang and Yang, 1998; Yang, 2005]. The
current formulation is different from the previous results [Fang and Yang, 1998] in that it
adopts a global state formulation that is convenient in analysis and efficient in simulation.
2.1. Formulation of the General Problem
A schematic of a flexible rotor system is shown in Figure 2.1. In the DTFM
formulation, divide the shaft into a number of segments by nodes
. These
nodes include the locations where rigid disks (e.g. impellers of vertical pumps) are
mounted, short bearings, the boundaries of long bearings, and the two ends of the shaft.
Unlike in finite element method (FEM) modeling, the segment length
does not have to
13
be small. The rotor-bearing system has five basic elements as shown in Figure 2.2: shaft
segment with or without long bearings, mounted disks, short bearings and couplings. In
this work, long bearings are modeled as distributed springs and dampers along the
bearing length, which shall be called "distributed model" for bearings.
Figure 2.1 Schematic of a flexible multistage rotor-bearing system
Figure 2.2 Basic elements of the rotor-bearing system
14
2.1.1. Shaft Segments
Using Hamilton's principle and Rayleigh beam theory, the transverse displacement
and
of the th element described in a fixed frame xyz are governed by the
partial differential equations
4 3 4 2
4 2 2 2 2
4 3 4 2
4 2 2 2 2
2 Ω
2 Ω
i i i
y
i
xx
i i i i
y
u v u u
EI I I A f q
z z t z t t
v u v v
EI I I A f
z z t z t t
q
(2.1)
where and are the Young's modulus, moment of inertial, density, rotation
speed of the shaft and cross-section area respectively;
and
external forces
applied on the shaft; and
and
the long bearing force given by
ii
x xx xy xx i xy i
ii
y yx yy yx i yy i
uv
q C C K u K v
tt
uv
q C C K u K v
tt
(2.2)
with
being the stiffness and damping coefficients of a long bearing. The inner
shear force and bending moments of the component are given by
(2.3)
in which the positive and negative signs correspond to
and
, respectively.
15
Now, take Laplace transform of Eq. (2.1) and cast into the spatial state form as
follows
(2.4)
where
and is an eight-by-eight state matrix formed (See Appendix A). The over-bar
represents the Laplace transformation with respect to time and s is the Laplace transform
parameter. The boundary conditions of the system can be written in the matrix form
(2.5)
where matrices
and
can be formed for different boundary conditions (see
Appendix A) and represents the boundary disturbances.
Because the rotating shaft is a continuum, two adjacent shaft segments must meet the
matching conditions at the interconnecting node: (i) continuity of displacement and slope,
and (ii) force and moment balance. Matching conditions for mounted disks, short
bearings and couplings are derived in the following sections.
2.1.2. Rigid Disks
Consider a node (say node i in Figure 2.3) with a mounted rigid disk of negligible
thickness and axisymmetric geometry. The rigid disk can be viewed as an impeller of a
pump. The governing equations of the disk with respect to its geometric center are given
by
16
(2.6)
where
is the diametral moment of inertial and
is the polar moment of inertial of
the disk.
The matching conditions at node i are given by
(2.7)
with the internal forces of the shaft segments given in Eq. (2.3). By Laplace
transformation, Eq. (2.7) can be cast into the state form
(2.8)
where matrices
and
consist of the coefficients in Eq. (2.7).
17
Figure 2.3 Free body diagram of a rigid disk at node i
Figure 2.4 Free body diagram of a short bearing at node i
2.1.3. Short Bearings
Short bearings are modeled as pointwise springs and dampers; see Figure 2.4. The
spring forces at node i are
(2.9)
where
are the dynamic coefficients of short bearings.
18
The matching conditions at node i are
(2.10)
Taking Laplace transform of Eq. (2.10) and they can be cast into spatial state form in
-domain as
(2.11)
2.1.4. Couplings
Couplings connect two shafts with either same or different diameters. For coupling,
the matching conditions at node i are given by
(2.12)
which can be written in the state form as follows
(2.13)
19
Figure 2.5 Free body diagram of a coupling at node i
In general, the matching conditions at nodes (i.e. mounted disks, short bearings,
couplings) can be written as
(2.14)
where
(See Appendix A).
With the above derivation, the time-domain governing equations of the flexible
multistage rotating system are converted to the equivalent s-domain state Eq. (2.4)
subject to boundary conditions (2.5) and the matching conditions (2.14).
2.2. Transfer Function Formulation
Eq. (2.4) is solved using the state transition matrix of the shaft, which is
a unique solution of
(2.15)
subject to the condition , where is the eight-by-eight identity matrix.
The state transition matrix has the properties
20
(2.16)
The state transition matrix can be expressed by
(2.17)
where is any fundamental matrix that is a nonsingular solution of
(2.18)
To satisfy the matching condition Eq. (2.14), the fundamental matrix must be continuous
at all nodes.
Eq. (2.18) is solved using the eigenvalue matrix ( ) and eigenfunction matrix ( )
of . Then the fundamental matrix of the element is
(2.19)
where is any constant matrix.
To satisfy the matching condition, we need
(2.20)
Since the matrix can be any constant matrix, to satisfy Eq. (2.20), we choose
(2.21)
Then
(2.22)
At node
,
which is an identity matrix. So, Eq. (2.20) is satisfied.
According to the above analysis, the fundamental matrix of the system is selected as
following
21
(2.23)
where
and
is defined in Eq. (2.14). [U] and are the eigenfunction and eigenvalue matrix
of
, respectively.
Substitute Eq. (2.23) into Eq. (2.17) yields
(2.24)
Because
then
(2.25)
Once the transition matrix is obtained by Eq. (2.17), the unique solution to Eq. (2.4) with
boundary conditions (2.5) is
(2.26)
where
22
and the state transition matrix is given by Eq. (2.17) and the fundamental
matrix is given by Eq. (2.23).
See [Yang, 2008] for the proof of Eq. (2.26). It was proved that the fundamental
matrix is continuous at all elements and nodes. So the solution of Eq. (2.4)
automatically satisfies the matching conditions (2.14). The spatial state representation
and distributed transfer function formulation derived in this section lay a foundation for
development of exact and closed-form solutions for eigensolutions and steady-states
response.
2.3. Dynamic Analysis
2.3.1. Eigensolutions
Once the transition matrix is computed, the unique solution to the homogenous Eq.
(2.4) can be taken as
(2.27)
Substitute Eq. (2.27) into boundary conditions (2.5) to obtain
(2.28)
with {
} being a vector to be determined. This gives the transcendental
characteristic equation of the flexible rotor-bearing system
(2.29)
whose roots
, are the eigenvalues of the rotor-bearing system and it is
composed of
23
(2.30)
where . If the system is undamped,
.
According to Eq. (2.29), the characteristic equation of a complex flexible rotor system
can be written as
(2.31)
where
is defined in Eq. (2.14). A Campbell diagram between rotation speeds and
natural frequencies can also be obtained using Eq. (2.31). As the eigenvalue is
determined, the corresponding eigenvector is solved using Eqs. (2.27) and
(2.28).
Setting the imaginary part of the Laplace domain variable equal to the rotation
speed, the critical speed of the rotor-bearing system can be determined by Eq. (2.31). If
the system is undamped, the critical speed coincides with its natural frequency, which can
be determined by
(2.32)
The root relates to forward (backward) critical speed.
2.3.2. General Steady State Response
The governing equations for the steady state response are derived from Eq. (2.26) by
dropping the time derivative term in the time dependence of all the quantities. The
general exact and closed-form solution is given by Eq. (2.26) with , which is
24
(2.33)
where
(2.34)
with
(2.35)
2.3.3. Steady State Response to Periodic Excitations
Periodic excitations applied on a shaft include the unbalanced mass, asynchronous
force, and harmonic excitations. The periodic excitation applied on shaft at
is written as
(2.36)
where
is the Dirac delta function; is the excitation frequency.
According to Eq. (2.26), by setting , the steady state response due to the
periodic excitation is
(2.37)
where
25
In general, above equations can be written as
(2.38)
where
and
are the imaginary and real parts of , respectively.
Using above equations,
, which is the 1st and 5th element of
respectively, can be obtained as
(2.39)
where
and
are the elements of matrices
and
in Eq. (2.38), respectively. If the
applied force is a periodic function, expand it in Fourier series and obtain a solution for
each harmonic component in the series. Then by the principle of superposition, the steady
state response of the periodic excitation is the sum of these harmonic solutions. This is
exactly same with Fang and Yang’s results [Fang and Yang, 1998]. See [Fang and Yang,
1998] for further information on whirl direction and whirl orbit.
As can be seen from above analysis, the DTFM yields exact and closed-form
solutions for eigensolutions and steady state responses. Compared with Fang and Yang’s
formulation methodology [Fang and Yang, 1998], the DTFM method in this dissertation
only involves eight-by-eight matrices, instead of n-by-n matrices. Thus, this DTFM is
much more efficient in computation.
26
2.4. Numerical Examples
Figure 2.6 A uniform shaft with a rigid disk
2.4.1. Example I: Critical Speeds of a Uniform Shaft with a Rigid Disk
An example from Fang and Yang [Fang and Yang, 1998] is used to validate the
DTFM introduced in this chapter. A simply supported, undamped, and uniform shaft with
a rigid disk is shown in Figure 2.6. The parameters of the rotor are
. Using Eq. (2.32), the first
three critical speeds are: 254.1 rad/s, 306.1 rad/s and 506.7 rad/s. The results are exactly
same with Fang and Yang's results [Fang and Yang1988].
27
2.4.2. Example II: Unbalanced Mass Response of a Flexible Rotor System
Figure 2.7 A flexible rotor system with three disks
Table 2.1 Data of a flexible rotor system
Shaft
Disks
Short Bearings
Long Bearings (Distributed Model)
28
Figure 2.7 shows a flexible rotor system with three disks and supported by two
bearings. Table 2.1 lists all the data of the system. Assume an unbalanced mass
is situated on the second disk. By Eqs. (2.37) and (2.39), the unbalanced
mass response at the second disk is shown in Figure 2.8. It is seen that the system has
seven critical speeds in the range from 0 rpm to 10000 rpm of rotation speed. The
operation speed should be away from the critical speeds.
Figure 2.8 Unbalanced mass response of a flexible rotor system
29
Chapter 3
Experimental Identification of Dynamic Stiffness of
WLRBs
The experimental identification of the dynamic stiffness coefficients of WLRBs is
performed on a water-lubricated rubber bearing test device. Figure 3.1 shows two photos
of the test device and Figure 3.2 gives a schematic of the test device.
Figure 3.1 Water-lubricated rubber bearing test device
30
Figure 3.2 Schematic of the water-lubricated rubber bearing test device
The test device can host one water-lubricated rubber bearing with diameter ranging
from 50.8mm (2 inches) to 152.4 mm (6 inches) and for a length-to-diameter ratio of 2.
In experiments, bearings with diameters 50.8 mm (2 inches) and 101.6 mm (4 inches)
were tested. For these rubber bearings, the liner material is Nitrile; the rotating journal
sleeves are made of polished stainless steel and they are mounted on a shaft that is
adjustable to be concentric with the centerline of the rubber bearing. The lower end of the
shaft is connected to a disk with four symmetric holes, which is used to introduce
31
unbalanced mass to the rotor. To lubricate the WLRB, city water flows through the
running clearance by injection to the water inlet and collection from the water outlet; see
Figure 3.2. Because the water inlet is lower than the water outlet, the bearing housing is
always completely filled with water, making water fully functioning as a lubricant during
the operation of the test device.
The bearing housing is sealed by two face-type mechanical seals (modified type
5610Q from John Crane, see Figure 3.1), whose radial forces applied onto the shaft are
negligible. Four proximity probes (3300XL NSv proximity transducer system) that are
mounted above and below the bearing housing are used to measure the transverse
displacement of the rotating shaft in x and y directions. A data acquisition system
records pressure and vibration measurements.
3.1. Approach
In this research, the dynamic stiffness coefficients of the WLRB shown in Figure 3.1
are obtained by matching unbalanced mass responses of the rotor system (test device)
that are measured experimentally and predicted theoretically.
In experiments, unbalanced responses of the rotor-bearing system in both time and
frequency domains were recorded by the four proximity probes shown in Figure 3.2. As
mentioned previously, unbalanced mass is introduced by mounting a pair of bolt and nut
in one of the holes of the disk (see Figure 3.2). Theoretically, unbalanced mass response
of a rotor system only consists of the frequency that equals to 1X running speed.
32
However, from the experimental results, the recorded frequency response always shows
higher harmonic frequencies, which may result from shaft misalignment. But, the
amplitude of the higher harmonic response is significantly small, compared with that at
the frequency equal to 1X running speed. Because the lower two proximity probes are
closer to the unbalanced mass, the signal recorded by the lower proximity probes is much
cleaner. So, the unbalanced mass response recorded by the lower proximity probes is
used in the analysis. To obtain the pure 1X running speed frequency from the recorded
response, the complex variable filtering method explained by Muszynska [Muszynska,
2005] was applied. The purpose of using complex variable filtering is to separate the
frequency components contained in a rotor orbit into circular, forward and reverse
frequency components. Following the procedure of the complex variable filtering method
(see Appendix B), the experimental 1X-filtered orbit can be expressed by
(3.1)
where and are the experimental vibration amplitudes in the and directions;
and
are the coefficients obtained from the complex variable filtering method;
is the rotation speed of the shaft.
In this effort, theoretical unbalanced mass response is derived through use of the
distributed transfer function method (DTFM) introduced in Chapter 1. In the DTFM
formulation, distributed model of long WLRBs is used (see Figure 2.2). By the DTFM,
the unbalanced mass response of a flexible rotor-bearing system can be written as
33
(3.2)
where and are the transverse vibration amplitudes of the shaft in and directions;
and
are coefficients computed by the DTFM.
Figure 3.3 Experimental data processing procedure
Finally, comparison of Eq. (3.1) and Eq. (3.2) yields
1 1 1 1 2 2 2 2
, , , a A b B a A b B (3.3)
34
Solution of these equations eventually gives the stiffness coefficients of the bearing,
namely,
. A flow chart of the related data processing procedure is
shown in Figure 3.3.
It should be pointed out that damping also plays an important role in vibration of
rotor systems. However, damping in a rotor system has little effect on the values of its
whirl frequencies even though it significantly affects the decay rate of system response
[Genta, 2009]. Because this effort is focused on dynamic stiffness of WLBRs under the
influence of mixed lubrication, identification of damping coefficients of this type of
bearings will be left as a topic of future investigation.
3.2. End Conditions of Rotating Shaft
Figure 3.4 Top end of the rotating shaft and its spring constraint model
To obtain accurate results, the end conditions of the rotating shaft in the test device
(see Figure 3.2) need to be properly prescribed. At the bottom end of the shaft, the
boundary conditions are those as a free end with a mounted disk. At the top end, the shaft
is connected to the motor by a nut and a key, as shown in Figure 3.4. Due to the elastic
properties of the nut and key, the shaft end is neither simply supported nor clamped. In
35
this work, the top end of the rotating shaft is modeled as a hinged end that is constrained
by translational and torsional springs. The constraint forces of the springs are of the form
[Friswell et al., 2010]
;
x uu u y vv v
UV
f k U k f k V k
zz
(3.4)
where
and
are the equivalent spring coefficients. By Euler-Bernoulli
beam theory, these equivalent spring coefficients are found as
32
12 6
;
uu vv u v
bc bc
EI
kk
EI
k k
ll
(3.5)
where and are the Young's modulus and moment of inertial of the shaft, respectively;
is the effective length of the shaft at the top boundary (see Figure 3.4). For given
geometric and material parameters of the nut and key, the
can be computed by the
finite element method.
The aforementioned end conditions of the rotating shaft will be implemented in the
DTFM formulation for the WLRB-supported rotor system.
3.3. Results
3.3.1. Dynamic Stiffness of WLRBs
As stated previously, WLRBs of two different sizes were used in experiments: a
101.6 mm rubber bearing and a 50.8 mm rubber bearing. The parameters of these
bearings are given in Appendix C. In experiments, a constant supply pressure is
maintained. The unbalanced mass response of the flexible rotor-bearing system is
36
recorded in three disk-mounting cases: the rotating shaft carrying one disk, two disks, and
three disks, with unbalanced mass of 36.3 g (0.08 lb), 43.1 g (0.095 lb) and 45.4 g (0.1 lb),
respectively. The shaft rotation speed varies from 600 rpm to 1600 rpm. Due to the
limitation of the data acquisition system, the maximum vibration amplitude that can be
recorded is 0.4mm (0.015 inches) and experimental data can only be obtained with the
shaft rotation speed up to 1350 rpm in the two-disk case, and 1150 rpm in the three-disk
case. The Reynolds number (
, with being fluid density) at rotation speed
600rpm and 1600rpm for the 101.6mm rubber bearing is 767.8 and 2047.5, respectively.
For 50.8mm rubber bearing at rotation speed 600rpm and 1500rpm, the Reynolds number
is 170.7 and 426.8, respectively.
The experimental results on the larger (101.6 mm) rubber bearing are first presented.
The unbalanced mass response of the rotor system in the three disk-mounting cases is
shown in Figure 3.5. where the 1X running speed and 2X running speed components of
vibration are obtained separately through use of complex variable filtering method. As
expected, the 1X running speed response increases as the rotation speed increases. The
response at the 2X running speed, however, is seen to have peak amplitudes at 1050 rpm,
950 rpm and 850 rpm for one-, two- and three-disk cases, respectively. Indeed, resonant
vibration was observed at these speeds during the tests, which indicates that the natural
frequencies of the rotor system equal the 2X running speeds at the peak response. It
follows that the natural frequencies of the rotor system with one, two and three disks, are
35.00 Hz, 31.67 Hz and 28.33 Hz, respectively. Because the amplitude of the 1X running
37
speed response is much larger than that of the 2X running speed response, the unbalanced
mass response of the rotor system is almost the same as the 1X running speed response.
Figure 3.5 Unbalanced mass response for the 101.6 mm rubber bearing
The dynamic stiffness coefficients of the rubber bearing are identified by Eqs. (3.1)
to (3.3). Plotted in Figure 3.6 are the dynamic stiffness coefficients of the 101.6 mm
rubber bearing versus the shaft rotation speed. As can be seen from the figure,
and
at most points of measurement. Figure 3.6a shows that the direct
stiffness coefficients change in an approximately parabolic manner as the rotation speed
increases. Furthermore, Figure 3.6a and Figure 3.5 suggest that, as the rotation speed of
the shaft increases, so does its vibration amplitude, which causes more contacts between
38
the rotating shaft and the rubber staves. However, Figure 3.6b does not show obvious
correlation between the cross-coupled stiffness and the rotation speed. As stated by
Vance, et al. [Vance et al., 2012], the cross-coupled dynamic stiffness coefficients that
produce forces tangential to the whirl orbit have little effect on system natural
frequencies, but they affect stability and amplification factors at a critical speed. As such,
the natural frequencies of the rotor system can be predicted with or without the cross-
coupled stiffness coefficients.
The natural frequencies of the rotor system can be predicted by using the determined
dynamic stiffness coefficients with the DTFM. In Table 3.1, the theoretical predictions
are compared with the experimentally-determined natural frequencies in the three disk-
mounting cases. It is seen that the theoretical and experimental natural frequencies are in
good agreement.
Table 3.1 Theoretical and experimental natural frequency for 101.6 mm rubber bearing
Cases
Theoretical
Natural Frequency
(
, Hz)
Experimental
Natural Frequency
(
, Hz)
Error
One disk at 1050 rpm 32.77 35.00 6.3%
Two disks at 950 rpm 30.70 31.67 3.1%
Three disks at 850 rpm 29.27 28.33 3.3%
39
(a)
(b)
Figure 3.6 Dynamic stiffness of the 101.6 mm rubber bearing (a) Direct stiffness; (b)
Cross-coupled stiffness
40
Now the results on the smaller (50.8 mm) rubber bearing are presented. The
unbalanced mass response of the rotor system with this bearing is only recorded in the
case of the shaft carrying one disk. All other conditions are the same as for the larger
(101.6 mm) bearing. Figure 3.7 compares the direct stiffness parameters of the two
bearings, which show the similar parabolic patterns as the shaft rotation speed increases.
Also, Table 3.2 lists the results on the first natural frequency of the rotor system with the
50.8 mm rubber bearing. It is seen from the above-mentioned results on the 50.8 mm and
101.6 mm bearings that the proposed experimental method can give fairly accurate
prediction of dynamic stiffness of WLRBs.
Figure 3.7 Comparison of direct stiffness for 50.8 mm and 101.6 mm rubber bearing
41
Table 3.2 Theoretical and experimental natural frequency for the 50.8 mm rubber bearing
Cases
Theoretical Natural
Frequency
(
, Hz)
Experimental
Natural Frequency
(
, Hz)
Error
One disk at 1050 rpm 32.24 35.00 7.9%
Figure 3.8 Comparison of theoretical and experimental unbalanced mass response at the
lower proximity probe at 800rpm of the case with one disk and 101.6 mm rubber bearing
3.3.2. Comparison of Theoretical and Experimental Unbalanced Mass Response
The theoretical unbalanced-mass response (2-D trajectory) is calculated through use
of the determined rubber bearing dynamic stiffness coefficients and the DTFM. The
experimental and theoretical unbalanced mass response at the lower proximity is
42
compared in Figure 3.8. At most points, the theoretical and experimental unbalanced
mass response match with each other very well.
Figure 3.9 The rotating shaft with polished areas (in squares)
3.3.3. Post-Test Examination
In a post-test examination, several polished areas on the rotating shaft were found in
the regions that are covered by the inner surface of the WLRB; see Figure 3.9. These
polished areas indicate that during the experiments on the test stand, the rotating shaft
and the rubber bearing were in contact with each other from time to time, which is in line
with the prediction by Cabrera et al. [Cabrera et al., 2005]. It is therefore reasonable to
assume that the WLRB operates under the influence of mixed lubrication, which involves
the interactions among the shaft vibration, the elastic deformation of rubber material and
the fluid film pressure by the lubricant (water in the current case). In other words, under
43
mixed lubrication, the contact between the rotating shaft and the surface of the bearing
causes elastic deformation of the rubber staves.
3.4. Conclusions
The newly-built experimental device is shown to be useful for investigation of the
behaviors of long WLRBs. Since the theoretical and experimental natural frequency and
unbalanced mass response match with each other very well, it can be concluded that the
determined dynamic stiffness coefficients of water lubricated rubber bearings are very
precise. From the post-test examination, it is concluded that the WLRBs operates under
the influence of mixed lubrication, which shall be considered in the theoretical model of
WLRBs.
44
Chapter 4
A New Model of Water-Lubricated Rubber Bearings
As states in Chapter 3, mixed lubrication occurs in water-lubricated rubber bearings
(WLRBs). However, it is difficult to know when and where the shaft is in touch with the
rubber staves of the bearing because the vibration of the shaft is unknown in the first
place. Also, the shaft and bearing do not engage in full contact all the time (see Figure
4.1), indicating that the shaft-bearing interaction is nonlinear in natural (see Figure 4.2).
Before the rotating shaft touches the rubber staves, the fluid film hydraulic stiffness (
)
dominates the overall rubber bearing stiffness (
). After the rotating shaft touches the
rubber staves, the rubber stiffness (
) will take control. Therefore, considering the
nonlinearity of WLRBs, the overall rubber bearing stiffness is modeled as
(4.1)
where represents the eccentricity and is the bearing radial clearance. Because the
rubber stave can deform, rotating shaft will not touch rubber staves even when the
eccentricity is equal to the bearing radial clearance. Thus, a small number ( ) is
added to the radial clearance.
45
(a) Axial Direction
(b) Circumferential Direction
Figure 4.1 Examples of contacts between the rotating shaft and rubber staves
Figure 4.2 Nonlinearity of dynamic stiffness of WLRBs
46
In this work, investigation on mixed lubrication in WLRBs takes two steps: (i) study
the effects of boundary lubrication and elastohydrodynamic lubrication separately; and (ii)
study the mixed lubrication through combination of the effects of boundary lubrication
and elastohydrodynamic lubrication.
4.1. Boundary Lubrication
(a) (b)
Figure 4.3 Rubber bearing in boundary lubrication: (a) A stave of WLRBs; (b)
rectangular rubber block as approximation
In boundary lubrication, it is assumed the rotating shaft is in full contact with the
surface of the rubber bearing. Because small amount of fluid always exists between the
asperities of the contact surfaces, the shaft rotation speed has little effect on the stiffness
in boundary lubrication. As such, the dynamic stiffness of the WLBR bearing in
boundary lubrication is only related to the stiffness of rubber staves of the bearing that
are in elastic deformation.
47
To derive a simple analytical expression for the above-mentioned stiffness, a rubber
stave is approximated as a rectangular rubber block as shown in Figure 4.3, where the
length and height of the rectangular block are the same as those of the stave. Also, it is
assumed that the shaft is only in touch with the land region of the rubber block (the
slightly-curved top area of the stave in Figure 4.3a), which is 38% of the total width
of the stave. Matching the stiffness of the stave computed by FEM to that of the
rectangular rubber block gives the effective width of the block as half of
; see Figure
4.3b. Under the assumptions, the shape factor method, which was first developed by Gent
and Meinecke [Gent and Meinecke, 1970] and Lindley [Lindley, 1981] and later on
documented by Hill and Lee [Hill and Lee, 1989], is applied to obtain the stiffness
of
the rectangular rubber block as follows
2
0
3
()
(1 ) (1 )
w s s r r r rh
r
s r r
W L E C S E
K
T e e
(4.2)
In the previous equation,
is the percentage width of the rectangular block
compared to the rubber stave (namely,
);
is the
length of the rubber block;
is the linear Young's modulus;
is a constant given by
() 4 11
2
3 10
w s w s
s
r
s
W W
C
LL
(4.3)
is the original shape factor of the rubber stave, which is
0
(
)
2
ws
w
s
s
r
ss
W
W
L
S
LT
(4.4)
is t e ‘ omogenous’ compression modulus defined by
48
2
22
22
() 1
1
3 ( )
s w s
s w s
rh r
LW
LW
EE
(4.5)
and
is the applied strain, which is
s
r
s
T
e
T
(4.6)
with
being the thickness of the stave and
the deformation displacement of the
stave surface. For small applied strain (say
),
in Eq. (4.2) is negligible, rubber
thus deforms linearly [Harris and Piersol, 2002], and the stiffness of rubber staves is
expressed by
2
0
()
s
r rh r r r
s
ws
L
K E E C
T
W
S
(4.7)
Figure 4.4 Model used in Abaqus - Contact
49
Figure 4.5 Stiffness of rubber staves in boundary lubrication
The stiffness of rubber staves given by Eq. (4.2) is compared with that computed by
the nonlinear FEM software Abaqus. To this end, an FEM model for use in Abaqus is
built; see Figure 4.4, where the entire rubber body of the bearing is considered. In
simulation, Neo-Hookean model for rubber material is used, a constant pressure is
applied on the half shaft surface, and the bottom of the shaft is in full contact with the
rubber stave. The stiffness of rubber staves versus applied strain for the two bearings
used in the experiments is plotted in Figure 4.5, where the solid lines represent the
Abaqus results and the dashed lines are for the predictions by Eq. (4.2). It is seen from
the figure that the stiffness by Abaqus is always larger than that given by Eq. (4.2). This
is due to the fact that the entire body of the rubber bearing is engaged in elastic
50
deformation in the FEM model. If the FEM results are used as reference solutions, the
maximum error in the stiffness predictions by Eq. (4.2) is around 3.5%, for both the
bearings. As such, the analytical expression given in Eq. (4.2) can be used to accurately
predict the stiffness of rubber staves of WLRBs in boundary lubrication.
4.2. Elastohydrodynamic Lubrication
In elastohydrodynamic lubrication, the rotating shaft is always separated from the
rubber surface by the fluid. In this research, the following assumptions about
elastohydrodynamic lubrication are made: (i) the fluid is Newtonian fluid with constant
viscosity; (ii) inertial and body force terms are negligible compared to the viscous term;
(iii) the variation of pressure across the film thickness is negligibly small; and (iv) the
flow is laminar. Under these assumptions, the elastohydrodynamic lubrication can be
described by Reynolds equation [Pai and Hargreaves, 2012]
2
33
2
6 12
p p h h
h h R
x x z x t
(4.8)
where is the distribution of the fluid pressure; is the circumferential coordinate
in the direction of rotation as shown in Figure 4.6a; is the shaft rotation speed; is the
fluid viscosity; and is the fluid film thickness, with being the
radial clearance, the eccentricity, and the rubber deformation caused by the fluid
pressure.
By introducing dimensionless parameters
51
'
; ' ; ; ; ' ;
p
s
x p h z D
p t h z
R p C L L
where
and
are the supply pressure and whirling frequency, and and are the
length and diameter of the bearing, Eq. (4.8) is reduced to
(4.9)
(a)
(b)
Figure 4.6 Coordinates in elastohydrodynamic lubrication
52
Furthermore, dropping the prime in Eq. (4.9) yields the dimensionless Reynolds equation
(4.10)
where bearing number
with
.
With the same token, non-dimensional boundary conditions for the rubber bearing can be
written as (i) at at staves; (ii) in the region at
grooves; and (iii) in the cavitation zone, Reynolds boundary conditions are used and they
are
.
4.2.1. Determination of h(p) by Compliance Operator Method
To solve Eq. (4.10), the elastic deformation of the rubber caused by the fluid
pressure must be known in advance. As stated in the Introduction (Chapter 1), the
available models in the literature do not deal with for rubber material. In this
research, a model of the pressure-induced rubber deformation is proposed. Two methods
are used to obtain : the compliance operator method and the finite element method
(FEM). In the compliance operator method, a compliance operator ( ) is used to relate
the rubber deformation to the fluid pressure, namely
() h p p (4.11)
where
in Eq. (4.11) with being the dimensional representation of . Because a
compliance operator can be easily implemented in Reynolds equation, the compliance
operator method has been popular in studies on relevant topics [Hooke et al., 1966;
Lahmar et al., 1988; Lahmar et al., 2010; Zhang et al., 2013].
53
Figure 4.7 Bering model used in Abaqus - compliance operator
Figure 4.8 Compliance operator vs. applied strain
54
In this investigation, the compliance operator method is applied to determine the
rubber deformation. A relationship between rubber deformation and applied strain is first
determined. To this end an Abaqus model of the rubber bearing body is built; see Figure
4.7, where a varied pressure ( ) is applied at one land region
and a constant supplied
pressure (
) is applied elsewhere in the bearing. The FEM simulation for bearings with
different diameters, different length and thickness results in plots of the compliance
operator against the strain
of the land region of the bearing; see the solid lines in
Figure 4.8. In the figure, the compliance operator, which is determined by Eq. (4.11), is
equal to
, and the strain equals to
.
Two observations can be made from the FEM simulation results: (i) Operator only
depends on the strain
of the land region of the rubber bearing; and (ii) Operator can
be approximately expressed by an exponential function
(4.12)
A typical range of strain in WLRB applications is
. To obtain an
accurate formula of in this strain range, parameters
and
in Eq. (4.12) are
determined in the following two cases.
Case I: At
, can be approximated by Winkler’s ypot esis [Conway and
Lee, 1975; Zhang et al., 2013] as
2
2
1
1
sr
r
T
C
p
E
(4.13)
where is the Poisson ratio of the rubber material.
55
Case II: At
, the compliance operator is approximated by
sL
r
pA
KC
(4.14)
where
is the area of the land region shown in Figure 4.7, the radial clearance of the
rubber bearing, and
the stiffness of rubber staves given by Eq. (4.7). Equation (4.14)
describes the rubber stave as a linear spring of coefficient
. The dimensional
compliance operator is
. The dimensionless representation of
thus yields Eq. (18). Hence, matching the compliance operator by Eq. (4.14) to
that obtained by Abaqus gives
.
With Eqs. (4.13) and (4.14), parameters
and
in the exponential expression in
Eq. (4.12) can be obtained. The above-described approach to the determination of shall
be called the two-point method. The dashed lines in Figure 4.8 are the results predicted
by the two-point method. It is seen from the figure that the two-point method has a
maximum deviation of 5% from the Abaqus results for
, and a
maximum deviation of 10% for
. Because a typical range of strain for
WLRBs is
, the two-point method proposed herein gives fairly
accurate results without having to rely on computationally intensive numerical methods.
The accuracy of analytical estimation of can be further improved by a method of
multi-region exponential approximation. According to the FEM (Abaqus) simulation
results, the compliance operator can be described in multiple regions of strain. Take the
101.6-mm (4-inch) rubber bearing with length 203.2 mm (8 inches) for example. Select
56
five pairs of parameters
) from the Abaqus data: (0.2%, 0.078), (0.4%, 0.11), (0.8%,
0.136), (1.6%, 0.155) and (3.2%, 0.162). Application of the two-point method to these
pairs of parameters yields the following multi-region function for
ln 0.4103
0.002
ln 0.2364
0.004
ln 0.1397
0.008
ln 0.0452
0.016
0.1323 1 e for 0% 0.4%
0.1441 1 e for 0.4% 0.8%
0.1581 1 e for 0.8% 1.6%
0.1623 1 e for 1.6%
r
r
r
r
e
r
e
r
e
r
e
r
e
e
e
e
(4.15)
Figure 4.9 Compliance operator by multi-region exponential approximation
57
As shown in Figure 4.9, the multi-region exponential approximation has a maximum
deviation of 2% from the Abaqus results for
. For the strain range of
in WLRB applications, the multi-region exponential approximation has a
maximum deviation of 1.5%.
It should be pointed out that, in the two-point method, Winkler's model is not valid
for = 0.5. The aforementioned multi-region exponential approximation, however, is
applicable to any values of Poisson ratio. In fact, even the two-point method does not
have to rely on Winkler's hypothesis if Eq. (4.13) is replaced by a relation that is obtained
by matching the FEM result at
. Therefore, the proposed formulas for
estimation of are accurate enough for WLRB applications, and are applicable to both
compressible and incompressible materials.
(a)
58
(b)
Figure 4.10 Rubber deformation for parabolic-shape pressure distribution (a) Parabolic
distribution of pressure; (b) Comparison of rubber deformation
In the above analysis, uniform pressure distribution is applied at the land region to
determine the compliance operator shown in Figure 4.8. However, for a WLRB, the
pressure distribution on the land region is typically parabolic or alike. To check the
accuracy of predicted by the two-point method, a parabolic pressure distribution shown
in Figure 4.10(a), is applied at one of the land regions and the fourth land region (
in Figure 4.6(b)). With the parabolic pressure distribution, the dimensionless
elastic deformation of the rubber bearing by the two-point method is obtained by
, where is given in Eq. (4.12). Figure 4.10(b) compares the results on the
distribution of the elastic deformation of the rubber bearing along the annual coordinate
59
see Figure 4.6), where the solid line is the prediction by the two-point method with a
uniform distribution of equivalent resultant force, and the dashed line is the result by the
FEM (Abaqus) with the parabolic pressure distribution given in Figure 4.10(a).
According to the figure, the maximum deviation of the result by the two-point method
from the FEM result is 3.5%. This comparison result indicates that the two-point method
is accurate enough to describe the actual rubber deformation. The accuracy can be
understood by the fact that the pressure variation is small due to the small angle
in
Figure 4.6.
4.2.2. Steady State and Dynamic Characteristics
With the obtained in the previous section, the steady-state solution of the
elastohydrodynamic lubrication can be determined. Drop the time-dependent term in Eq.
(4.10), to have the steady state characteristic equation
(4.16)
By solving Eq. (4.16) for the steady-state pressure
, the elastohydrodynamic forces are
determined by the following integrals
1
1
1
*
0
0
1,3,...15
1
*
0
0
1,3
0
5
0
,...1
cos
sin
i
i
i
i
r
r
i
s
i
t
s
t
W
W p d dz
LDp
W
W p d dz
LDp
(4.17)
where
and
are the elastohydrodynamic forces in radial and circumferential
directions, respectively;
is the attitude angle at steady state (Figure 4.6a); and
is
60
shown in Figure 4.6b. The load capacity and steady state attitude angle
are thus given
by
2
1
2
0
ta ; n
t
rt
W
W W W
Wr
(4.18)
In this work, both infinitesimal perturbation method [Pai and Hargreaves, 2012; Liu,
1995] and finite perturbation method [Qiu, 1995; Martelli and Manfrida, 1981; Choy et
al., 1991] are used to study the elastohydrodynamic coefficients (stiffness (
) and
damping coefficients (
)) of water-lubricated rubber bearings, which are produced by
the elastohydrodynamic lubrication.
(A) Infinitesimal Perturbation Method (IPM)
For small amplitude whirl motions of frequency
about the steady state position
(
) of the rotating shaft, the pressure and film thickness can be written as:
ii
0 1 1 0 1 2
ee p p p p
(4.19)
i i i
0 1 0 1 1 1 0 1 2
Re( e cos sin ( e e )) h h p p
(4.20)
where
with
and
being the steady
state fluid film thickness and pressure distribution, respectively. The steady state
eccentricity (
) is equivalent to the steady state vibration amplitude at WLRBs of the
rotor system (test device) with an unbalanced mass, which can be determined by the
DTFM method with the experimental dynamic coefficients of WLRBs in Chapter 3.
Substitute Eqs. (4.19) and (4.20) into (4.10) and retaining up to first linear terms, to
obtain the following equations
61
22
3 2 2 3 0 0 0 0 0
0 0 0 0 22
1
: 3 0
4
p h p dh p
h h h
dz
(4.21)
2 22
3 2 2 3 2 00 1 1 1 1
1 0 0 0 1 0 2 2 2
2
2 2 2 0 0 0 0
0 0 0 2
i
e 2i cos
sin cos cos sin cos
1
: 3 3
4
3
3 6 2i 0
4
hp p p p p
h h h p h
z
p h p p
h h h
z
(4.22)
2 22
3 2 2 3 2 00 2 2 2
0 0 0 2 0 2 2 2
2
2 2 2 0 0 0
i 2
0
01
0 0 0 2
e 2i sin
cos cos sin cos sin
1
: 3 3
4
3
3 6 2i 0
4
hp p p p p
h h h p h
z
p h p p
h h h
z
(4.23)
Equations (4.21) to (4.23) are solved via a finite difference method. To this end, the
previous differential equations are firstly written in the standard finite difference form, in
which a rectangular grid with 20 elements between
and
(see Figure 4.6b) and 30
elements in direction is applied to each land region as shown in Figure 4.7. The
resulting finite difference equations, which must satisfy the boundary conditions, are then
determined numerically via the successive over relaxation (SOR) scheme in the
rectangular grid. To use this method, the pressure distribution is determined via an
iterative process, with the initial pressure at all grid points set to zero. In each iteration
step, the pressure at every grid point is modified by an over relaxation factor (say, the
value of the factor can be chosen as 1.7). The solution is reached when the convergence
criteria
is met. More detailed description of the solution procedure
for bearings with multiple grooves is seen in Reference [Pai and Hargreaves, 2012].
With the perturbed pressure solutions
and
, the stiffness and damping
coefficients of the rubber bearing are expressed as follows
62
1 2 1 2
11
0 0 0 0
1 2 1 2
00
2
00
2
Re cos ; Re
Re cos ; Re
sin
sin
rr r
r
K p d dz K p d dz
K p d dz K p d dz
(4.24)
1 2 1 2
11
0 0 0 0
1 2 1 2
00
22
00
Im cos / ; Im /
Im cos / ; Im
sin
sin /
rr r
r
C p d dz C p d dz
C p d dz C p d dz
(4.25)
where
and
. By coordinate transformation, the stiffness and
damping coefficients of the bearing in the coordinate system are given by [San André s]
cos sin cos sin
sin cos sin cos
cos sin cos sin
sin cos sin cos
xx xy rr r
yx yy r
xx xy rr r
yx yy r
K K K K
K K K K
C C C C
C C C C
(4.26)
where is the bearing attitude angle as shown in Figure 4.6a.
(B) Finite Perturbation Method (FPM)
In finite perturbation, the four linear elastohydrodynamic stiffness coefficients of the
rubber bearing are evaluated via a small displacement perturbation around the
equilibrium position (
) in both positive and negative and directions, which is
; ; ;
yy
xx
xx yy xy yx
FF
FF
K K K K
x y y x
(4.27)
Based on the results from Reference [Choy et al., 1991], the perturbation values are
chosen as , with being the bearing radial clearance. The
dimensionless displacement perturbations are then . Therefore, using
63
the four points ( ,xy ) around the equilibrium point, the dimensionless stiffness
coefficients are estimated as below:
(i) Perturbation in positive and negative -direction
x, x, y, y,
;
2 Δ 2 Δ
xx yx
F F F F
KK
xx
(4.28)
(ii) Perturbation in positive and negative -direction
, , , ,
;
2 Δ 2 Δ
y y x x
xy yy
F F F F
KK
yy
(4.29)
The loads
are given by
cos sin
sin cos
x r
y t
FW
FW
(4.30)
where is the bearing attitude angle as shown in Figure 4.6; and
and
are given in
Eq. (4.17).
Application of the finite perturbation method takes the following steps:
(i) Determine the steady state hydrodynamic forces (
) and attitude angle
( );
(ii) Determine the eccentricity in x and y directions:
;
(iii) Apply a finite perturbation on the eccentricity
;
(iv) Update eccentricity and attitude angle after perturbation
;
64
(v) Solve Eq. (4.16) for the steady-state pressure and compute the hydrodynamic
forces (
) by Eq. (4.17), with updated and ;
(vi) Evaluate the dynamic stiffness coefficients using Eqs. (4.28) - (4.30).
(a) (b)
(c) (d)
Figure 4.11 Elastohydrodynamic coefficients of the 101.6 mm rubber bearing (a)
Direct stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d)
Cross-coupled damping coefficients
65
Through use of the aforementioned formulas, the elastohydrodynamic coefficients
(stiffness (
) and damping coefficients (
)) of the two WLRBs that are used in the
experiments are computed. The dynamic stiffness coefficients of the 101.6 mm (4-inch)
rubber bearing are computed by both the infinitesimal perturbation method (IPM) and the
finite perturbation method (FPM), and they are plotted against the shaft rotation speed in
in Figure 4.11. As can be seen from the figure, the cross-coupled dynamic stiffness
coefficients (
) obtained by the IPM perfectly match those by the FPM. However,
the direct dynamic stiffness coefficients (
) obtained by the IPM are larger than
those by the FPM, especially at higher rotation speeds. Because the accuracy of FPM is
closely related to the perturbed values ( ) [Choy et al., 1991], the results obtained
by the IPM seem more trustworthy. It is with this argument that the dynamic coefficients
of the 50.8 mm (2-inch) rubber bearing are computed by the IPM; see Figure 4.12.
(a)
(b)
66
(c)
(d)
Figure 4.12 Elastohydrodynamic coefficients of the 50.8 mm rubber bearing (a) Direct
stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d) Cross-
coupled damping coefficients
4.2.3. Implementation of Finite Element Method in Solution of Steady State
Pressure
As mentioned in Section 4.2.1, two methods can be used to determine the fluid
pressure-induced rubber deformation : the finite element method (FEM) and the
compliance operator method. In Section 4.2.2, the compliance operator method is
implemented in the infinitesimal and finite perturbation method to study the steady and
dynamic characteristics of WLRBs. Although computationally intensive and time
consuming, the FEM delivers more accurate results. In this section, the finite element
method is implemented in the infinitesimal perturbation method (IFM), to obtain steady
state pressure distribution of WLRBs and to determine the fluid pressure-induced rubber
deformation. In an iterative solution procedure, the pressure distribution is computed by
67
MATLAB codes and the nonlinear FEM software Abaqus determines the rubber
deformation; see Figure 4.13.
The above-mentioned iterative FEM solution method is used to compute the fluid
pressure distribution of the 101.6 mm rubber bearing with
and rpm;
see Figure 4.14. The
's in Figure 4.14 are corresponding to the coordinate shown in
Figure 4.6(b). For comparison, the pressure distribution that is computed by the
compliance operator method is also included in the figure, which is in good agreement
with the FEM. This confirms that the compliance operator method described in Section
4.2.1 can accurately predict fluid pressure-induced rubber deformation.
4.3. Mixed Lubrication via the Method
The mixed lubrication of a WLRB involves both boundary lubrication (Section 4.1)
and elastohydrodynamic lubrication (Section 4.2). Physically, the rotating shaft and the
rubber bearing are in direct contact sometimes, but they are separated by the fluid at other
times. Obviously, boundary and elastohydrodynamic lubrications cannot coexist at any
particular time. However, in steady-state vibration, the relative occurrences of these two
types of lubrications are in a relation. Hence, estimation of the overall dynamic
coefficients for the WLRB should consider proper combination of the
elastohydrodynamic coefficients and stiffness of the rubber stave.
68
Figure 4.13 Procedure to determine steady state pressure with FEM implementation
69
Figure 4.14 Circumferential pressure distribution of the 101.6 mm bearing with
and at rpm
With this understanding that the -method is proposed as follows
, , , (1 ) rb d r d h d K K K (4.31)
where
is the overall dynamic stiffness of the WLRB,
is the stiffness of rubber
staves which can be determined by Eq. (4.7);
is the fluid film stiffness given by Eq.
(4.26) or Eqs. (4.28) to (4.30); and is a non-dimensional number between 0 and 1
representing the percentage contribution of the stiffness of rubber staves in boundary
lubrication to the overall rubber bearing stiffness in steady-state vibration of the rotor
system. The -method can be illustrated by the distributed model of long bearings in
70
Figure 4.15, where the overall rubber bearing stiffness is a combination of the stiffness of
rubber staves and elastohydrodynamic stiffness.
Figure 4.15 Mixed lubrication of long WLRBs
The value of can be determined experimentally as follows
,exp ,
exp
,,
-
-
rb h d
r d h d
KK
KK
(4.32)
where
is the experimental dynamic stiffness of the WLRB as presented in Chapter
3. Shown in Figure 4.16 are the plots of versus the shaft rotation speed for the two
WLRBs used in the experiments, where the experimental value of is obtained by Eq.
(4.32), and the theoretical value of is given by the non-dimensional formula
1.8
1.15
0.25
3.9
6
s
D
pL
C
R
(4.33)
Eq. (4.33) is devised based on following properties of : (i) is a non-dimensional value
between 0 and 1; and (ii) depends on bearing parameters (i.e. ) and operation
parameters (i.e.
). All the dimensionless terms in Eq. (4.33), i.e.
, are chosen from the non-dimensional Reynolds equation, Eq. (4.10).
71
Based on the experimental results, the values of the indices in Eq. (4.33) are obtained by
curve fitting.
Figure 4.16 Comparison of theoretical and experimental values of
Figure 4.17 Comparison of theoretical and experimental direct stiffness
72
Figure 4.16 shows the plots of versus the shaft rotation speed, which are obtained
by Eqs. (4.32) and (4.33), respectively. The maximum difference between the theoretical
and experimental plots is 8%. With the theoretical value, the theoretical rubber bearing
stiffness is determined by Eq. (4.31). The comparison of theoretical and experimental
rubber bearing stiffness is shown in Figure 4.17. And the maximum deviation is 8%. It is
seen from Figure 4.16 and Figure 4.17 that the theoretical dynamic stiffness matches the
experimental dynamic stiffness very well for both the 50.8 mm (2-inch) and 101.6 mm
(4-inch) rubber bearings.
As a summary of the new WLRB model developed in this work, a flowchart of
determination of rubber bearing dynamic coefficients is given in Figure 4.18.
73
Figure 4.18 Procedure for determination of dynamic coefficients of WLRBs
74
4.4. Parameter Study on Dynamic Coefficients of WLRBs
The new model of water-lubricated rubber bearings obtained from this effort is
useful in bearing design for WLRB-supported rotor systems. In this section, the
effects of eccentricity ( ), land region angle (
), and number of grooves on the
dynamic coefficients of WLRBs are examined. Unless otherwise specified, the
bearing parameters used in numerical studies are given in Table 4.1, which are the
same as those of the 101.6 mm (4 inches) rubber bearing.
Table 4.1 Bearing geometry and operation parameters in numerical studies
Parameter Value Parameter Value
0.5 Stave Length (
) 203.2 mm
0.0047 Land Region Angle (
) 19
o
Stave Thickness
(
)
9.52 mm Rubber Young's Modulus (
) N m
-
2
Stave Width (
) 41.28 mm Poisson's Ratio ( ) 0.47
4.4.1. Effect of Eccentricity
Because the eccentricity does not affect the rubber property, the stiffness of rubber
staves and the compliance operator are same as shown in Section 4.1 and 4.2.1, which are
given by
1.21 r K
ln(0.4507)
0.002
0.142 1-e
r
e
(4.34)
The elastohydrodynamic coefficients are determined by the infinitesimal
75
perturbation method (IPM). Figure 4.19 shows the elastohydrodynamic coefficients with
respect to for different values of . Here
is a non-dimensional parameter,
which is given in the dimensionless Reynolds equation Eq. (4.10). It is seen from the
figure that the elastohydrodynamic coefficients generally increase as the eccentricity
increases. However, with high eccentricity (say ), the elastohydrodynamic direct
stiffness (
) and cross-coupled damping coefficients (
) increase at low
bearing number ( ) and decrease at high bearing number ( ). Also, by the -method, the
overall dynamic stiffness coefficients of the WLRB are plotted in Figure 4.20. It is easy
to see from Figure 4.20 that the stiffness of rubber staves is much larger than the
elastohydrodynamic direct stiffness.
(a) (b)
76
(c) (d)
Figure 4.19 Elastohydrodynamic coefficients with respect to for various (a) Direct
stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d) Cross-coupled
damping coefficients
Figure 4.20 Rubber bearing dynamic stiffness with respect to for various
77
4.4.2. Effect of Land Region Angle
In this study, the eccentricity is chosen as
. By the shape factor method and
the software Abaqus, the stiffness of rubber staves for WLRBs with different land region
angles (see Figure 4.7) is plotted in Figure 4.21. Also, the compliance operator ( ) for
various land region angles (predicted by Abaqus) is plotted in Figure 4.22. It is observed
that the stiffness of rubber staves increases with increasing land region angle; but the
compliance operator decreases.
Figure 4.21 Stiffness of rubber staves with respect to land region angle (
)
The multi-region exponential approximation method, which is described in Section
4.2.1, is used to determine the compliance operator for each case. By using the
infinitesimal perturbation method, the elastohydrodynamic coefficients with respect to
parameter for various land region angles are obtained; see Figure 4.23. It is seen that
78
the elastohydrodynamic coefficients increases as the land region angle increase. Figure
4.24 shows the overall rubber bearing stiffness, which increases as the land region angle
increases.
Figure 4.22 Compliance operator for various land region angle
(a) (b)
79
(c) (d)
Figure 4.23 Elastohydrodynamic coefficients with respect to for various
(a) Direct
stiffness; (b) Cross-coupled stiffness; (c) Direct damping coefficients; (d) Cross-coupled
damping coefficients
Figure 4.24 Rubber bearing dynamic stiffness with respect to for various
80
4.4.3. Effect of Number of Grooves
In this study, the eccentricity is chosen as
and the land region angle (see
Figure 4.7) is
for bearings with different number of grooves. In addition,
assume the properties of the rubber staves are the same for all the cases, e.g. same
compliance operator ( ) and same stiffness of the rubber staves (
). Figure 4.25 shows
the elastohydrodynamic coefficients with respect to for WLRBs with different number
of grooves. It is observed that the elastohydrodynamic coefficients of WLRBs increase as
the number of grooves increases. However, since the stiffness of the rubber staves and the
value of do not change, the overall rubber bearing dynamic stiffness does not change
too much, as shown in Figure 4.26. Therefore, the stiffness of rubber staves and the value
of dominate the overall rubber bearing stiffness.
(a) (b)
81
(c) (d)
Figure 4.25 Elastohydrodynamic coefficients with respect to for different number of
grooves (a) Direct stiffness; (b) Cross-coupled stiffness; (c) Direct damping
coefficients; (d) Cross-coupled damping coefficients
Figure 4.26 Rubber bearing dynamic stiffness with respect to for different number of
grooves
82
In summary of the parameter study, in the design of flexible rotating systems with
water-lubricated rubber bearings (WLRBs), higher operation eccentricity and larger land
region angle of the WLRB can increase the dynamic coefficients of WLRBs; but it could
be a cost of energy, which is induced by higher friction due to possible more contacts. In
addition, the stiffness of rubber staves and the value of play an important role in the
determination of dynamic coefficients of WLRBs.
4.5. Conclusions
This chapter presents a new model of water-lubricated rubber bearings (WLRBs) for
a class of flexible rotor-bearing systems. The main results obtained as summarized as
follows.
(i) A theoretical model of WLRBs is developed, which for the first time, describes
mixed lubrication in the vibration of a WLRB-supported rotor system. The proposed
bearing model combines the stiffness of rubber staves caused by direct contact between
the rotating shaft and the rubber bearing and the stiffness due to elastohydrodynamic
forces of the fluid film in WLRBs. The stiffness of rubber staves is given in an analytical
formula that is derived from a rectangular rubber block; the fluid film stiffness is
determined through solution of Reynolds equation, with either two-point approximation
or multi-region approximation of the compliant operator of the rubber material. The
WLRB model developed is applicable to both compressible and incompressible materials.
83
(ii) With mixed lubrication taken into account, the –method is proposed to compute
the overall dynamic stiffness of WLRBs, in which the non-dimensional parameter
indicates the percentage contribution of the stiffness of rubber staves to the overall
dynamic stiffness of WLRBs. Furthermore, experimental measurement and curve fitting
yields a non-dimensional formula for evaluation of at various shaft rotation speeds.
(iii) The proposed bearing model is validated in experiments. The dynamic stiffness
coefficients of the two bearings used in experiments that are predicted by the proposed
bearing model are in good agreement with those that are experimentally determined as
shown in Figure 4.17.
84
Chapter 5
Vibration Analysis of a Flexible Rotor System with
WLRBs
This chapter studies the vibration of a flexible rotor system with water-lubricated
rubber bearings (WLRBs) by implementing the new model of WLRBs (Chapter 4) into
the DTFM method (Chapter 2). Figure 5.1 shows a multistage rotor system with five
WLRBs and three rigid disks (e.g. impellers). This kind of multistage system is
commonly seen in vertical pumps. The parameters of the multistage rotor system are
presented in Table 5.1. The effect of bearing model (i.e. pointwise and distributed bearing
model), length of bearings, bearing location and number of bearings on the vibration of
the multistage rotor system will be investigated.
Figure 5.1 A multistage rotor system with long WLRBs
85
Table 5.1 Parameters of a multistage flexible rotor-bearing system
Shaft
Diameter ( ) m
Density ( ) 7800 kg m
-3
Young's Modulus ( )
N m
-2
Length ( )
m;
m;
m;
m;
m
Water-Lubricated Rubber Bearings
1st, 2nd, 3rd Bearing 4th, 5th Bearing
Length ( )
m
m
0.0047
Geometry of Stave
m;
m
Rubber Young's
Modulus
N m
-2
Poisson's Ratio
Land Region Angle
Supply Pressure
Pa
Rigid Disks
1st, 2nd Disk 3rd Disk
Mass (
)
kg
kg
Mass Moment of
Inertial (
)
kg m
2
kg m
2
kg m
2
kg m
2
Short Bearings
Stiffness
Damping
Coupling
Diameter ( ) m
Density ( ) 7800 kg m
-3
Young's Modulus ( )
N m
-2
86
5.1. Effect of Model of WLRBs
This section compares the critical speeds and unbalanced mass response of the
multistage rotor system with different bearing model, which includes pointwise and
distributed bearing model.
In distributed bearing model, using the parameters of WLRBs given in Table 5.1 and
following the flowchart shown in Figure 4.18, the hydrodynamic coefficients
(
), stiffness of the rubber stave (
) and rubber bearing dynamic
coefficients (
) can be determined. As shown in Figure 4.18, these
dynamic coefficients satisfy
(5.1)
where is determined using Eq. (4.33).
As shown in Figure 5.2, two types of distribution of dynamic stiffness of WLRBs will be
considered in distributed bearing model case. In both types of distribution of dynamic
stiffness, assume the cross-coupled stiffness and damping coefficients are uniformly
distributed along the bearing length. In another word, only the distribution of direct
stiffness along the bearing length is different; see Figure 5.2.
In pointwise model of WLRBs, the dynamic coefficients are determined by
(5.2)
where
and
are given by Eq. (5.1) and is the length of WLRBs.
87
(a) (b)
Figure 5.2 Distribution of dynamic stiffness of WLRBs (a) Case I: Uniform dynamic
stiffness; (b) Case II: Stepped dynamic stiffness
To study the effect of different bearing model on unbalanced mass response of the
multistage rotor system, assume that an unbalanced mass
kg· m is located at
the first rigid disk. By Eq. (2.39), the vibration amplitude with respect to rotation speed at
the location of disk 1 is determined as shown in Figure 5.2. The first four critical speeds
of the system are summarized in Table 5.2. It is seen that the critical speeds by pointwise
model and uniformly distributed model are exactly the same. But using distributed model
with stepped stiffness gives a little bit different critical speeds. If using the results by
uniformly distributed model as reference, the difference is 0.65%, 1.05%, 0.61% and
0.24% for the first through the fourth critical speed, respectively. From Figure 5.3, it is
observed that all the three models almost yield same unbalanced mass response.
From the comparison of the pointwise and the distributed model of WLRBs, it is
concluded that the pointwise and distributed bearing model can yield almost same results.
88
However, the distributed model can better describe the physical function of water-
lubricated rubber bearings.
In the following studies, uniformly distributed model of WLRBs shown in Figure
5.2a will be used.
Table 5.2 Critical speeds of a multistage rotor system
Critical
Speed
(Unit: RPM)
Pointwise Model
Distributed Model
(Uniform)
Distributed Model
(Stepped)
1st 153 153 152
2nd 478 478 473
3rd 1157 1157 1150
4th 1671 1671 1675
Figure 5.3 Effect of model of WLRBs on the unbalance mass response of the multistage
system at disk 1
89
5.2. Effect of Length of WLRBs
Figure 5.4 shows the effect of length of WLRBs on the unbalanced mass response of
the multistage system at the first disk, in which an unbalanced mass
kg· m
is located at the first rigid disk. It can be seen that, for this multistage rotor system, the
length of the WLRBs does not have much effect on the critical speeds and unbalanced
mass response. The maximum deviation of the critical speeds is 1.5%. Therefore, it is
reasonable to replace the 0.25 m WLRBs with 0.20 m WLRBs, which reduces the cost of
the system.
Figure 5.4 Effect of length of WLRBs on the unbalance mass response of the multistage
system at disk 1
90
5.3. Effect of Location of WLRBs
Assume that an unbalanced mass
kg· m is located at the second rigid
disk. By moving the location of the first bearing between the top boundary and the first
coupling, the unbalanced mass response at the second rigid disk is shown in Figure 5.5a.
It is seen that the location of the first bearing mainly affects the third and fourth critical
speed and the second critical speed increases as the first bearing being closer to the first
coupling. But it has negligible effect on the first critical speed.
(a)
91
(b)
(c)
Figure 5.5 Effect of the bearing locations on the unbalance mass response at disk 2 of the
multistage system (a) First bearing; (b) Third bearing; (c) Fifth bearing
92
Figure 5.6 Whirl motion at rpm at disk 2 of the multistage system with
different location of bearing 1
Figure 5.5b and Figure 5.5c shows the effect of the location of the third and fifth
bearing on the unbalanced mass response at the second rigid disk. As can be seen, the
third bearing mainly affects the third and fourth critical speed of the system; and the fifth
bearing has little effect on all the first four critical speeds. Comparing the effect of the
first, third and fifth bearing location, it is observed that the first bearing has more
influence on the critical speeds of the multistage system. Figure 5.6 shows the whirl
motion at rotation speed rpm at the location of the second disk with different
locations of the first WLRB. The maximum deviation of the amplitude of the whirl
93
motion between the response with
m and
m is 83%. So it is concluded
that the location of the first bearing has significant effect on the amplitude of the
unbalanced mass response of the multistage system.
Figure 5.7 Effect of number of bearings on the unbalance mass response at disk 3 of the
multistage system
5.4. Effect of Number of WLRBs
Assume that an unbalanced mass
kg· m is located at the third rigid disk.
Figure 5.7 shows the effect of number of bearings on the unbalanced mass response at
disk 3 of the multistage system. It can be seen that, if the multistage rotor system is
without bearing 2, only the second critical speed changes; if without bearing 3 or without
both bearing 2 and bearing 3, all the critical speed changes. Assume the operation speed
94
of the system is at 751 rpm and the maximum allowable vibration amplitude of the
system is
m. Figure 5.8 shows the whirling amplitude at disk 1, 2 and 3. It is
observed that all the vibration amplitudes are within the safe range. So, it is allowed to
redesign the system without bearing 2. This analysis is very useful for the optimal design
of multistage rotor systems with WLRBs, which can optimize the number of WLRBs to
reduce the cost of the rotor system.
(a)
95
(b)
(c)
Figure 5.8 Whirl motion at rpm of the multistage system at (a) Disk 1; (b) Disk
2; (c) Disk 3
96
5.5. Conclusions
For the first time in literature, the new model of WLRBs is implemented in the
DTFM to study the vibration of multistage rotor systems with WLRBs. From the
numerical examples, different models (i.e. pointwise model and distributed model) of
WLRBs yield almost same results. However, the distributed model of WLRBs better
describes physical function of bearings. This chapter also studied the effect of bearing
length, bearing location, and number of WLRBs on the unbalanced mass response of a
multistage rotor system. This kind of analysis is very useful for optimal design and
duality of multistage rotor systems with WLRBs.
97
Chapter 6
Conclusions
Modeling, analysis and experimental validation of flexible rotor systems with water-
lubricated rubber bearings (WLBRs) have been presented in this thesis. The main results
of this research are summarized as follows.
(i) A distributed transfer function method (DTFM) is generalized and implemented
for modeling and analysis of flexible rotor systems. The DTFM yields exact and closed-
form solutions for eigensolutions and steady state responses. The current formulation is
different from the previous results in that it adopts a global state formulation, in which it
only involves eight-by-eight matrices. Therefore, the DTFM introduced in this thesis is
convenient in analysis and efficient in simulation.
(ii) Unlike many previous investigations that adopt pointwise bearing models, this
effort treats a WLRB of large length-to-diameter ratio as a distributed visco-elastic
foundation. From the investigation of effect of model of WLRBs on the vibration of a
flexible rotor system, it is concluded that the model of WLRBs has little affect on the
vibration of the flexible rotor system. However, the distributed model of WLRBs better
describes the function of long WLRBs.
(iii) For the first time, a theoretical model of WLRBs is developed to describe the
98
mixed lubrication in the vibration of a WLRB-supported rotor system, which involves
interaction effects of shaft vibration, elastic deformation of rubber material and fluid film
pressure. With mixed lubrication taken into account, the method is proposed to
compute the overall dynamic stiffness of WLRBs.
(iv) The proposed bearing model is validated in experiments. The dynamic stiffness
coefficients used in experiments that are experimentally determined are in good
agreement with those predicted by the proposed bearing model.
(v) This thesis is the first one that reports on an experimentally validated dynamic
model for WLRBs. This new WLRB model and the distributed transfer function
formulation presented in this thesis lay a platform for vibration analysis and optimal
design of a class of flexible multistage rotating systems in engineering applications.
This dissertation is a precursor on an ongoing project on the modeling and analysis
of flexible rotor systems with water-lubricated rubber bearings (WLRB). It is
recommended that further studies address following issues:
(1) Improvement of the model of water-lubricated rubber bearings
In the model of WLRBs, the impact effect and the tangential friction effect of the
rotating shaft on the rubber surface of the WLRB were not considered. The model of
WLRBs would be more accurate by taking into the account of the impact effect and
tangential friction effect.
(2) Improvement of the -method
In the -method, the empirical formula to determine the value was proposed by
fitting the experimental results for two sizes of water-lubricated rubber bearings. This
99
empirical formula can be further improved by more experimental results for different
sizes of WLRBs.
(3) Identification of damping coefficients of WLRBs
In this dissertation, the dynamic stiffness coefficients of WLRBs were identified in
the experiment. Damping coefficients of WLRBs also play an important role in the
vibration of flexible rotor systems with WLRBs. Half-power method may be used to
experimentally identify the damping coefficients of WLRBs.
(4) Optimal design of flexible rotor systems with WLRBs
Using the distributed transfer function method and the model of WLRBs presented in
this thesis, the design of flexible rotor systems with WLRBs can be optimized. This kind
of optimization can improve the system performance and reduce the cost of the flexible
rotor system.
100
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107
Appendix A
List of State Matrices
2 2
2
2
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
2
- 0 0 - 0 0
[ ( , )]
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
2
- 0 - 0 - 0 0
xy xy
xx xx
yx yx yy yy
c s k
As c s k Is s
EI EI EI E
F z s
c s k As c s k
s Is
EI E EI EI
A.1 Matrices for Matching Conditions
Disk
2
2
,1
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 2 0 0
0 2 0 0 0 0
[]
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
di
Is EI s
s Is EI
A
EI
EI
108
22
22
2 ,
2
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 2 0 0
0 2 0 0 0
[]
0 0 0 0 0
0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
i
i
di
Dx Dz
Dz Dx
m s Is EI s
s m s Is EI
B
s I EI I s
I s s I EI
Short Bearing
, 1 , 1
[ ] [ ]
b i d i
AA
2
, , , ,
2
, , , ,
,
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
( ) 0 ( ) 2 0 0
( ) 2 0 0 ( ) 0
[]
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
xx i xx i xy i xy i
yx i yx i yy i yy i
bi
k c s Is EI k c s s
k c s s k c s Is EI
B
EI
EI
Coupling
, 1 , 1
[ ] [ ]
c i d i
AA
2
2
,
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 2 0 0
0 2 0 0 0 0
[]
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
ci
Is EI s
s Is EI
B
EI
EI
109
A.2 Matrices for Boundary Conditions
Supported by Springs and Dampers
2
,0 ,0 ,0 ,0
2
,0 ,0 ,0 ,0
0 2 0 0
2 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
[]
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
xx xx xy xy
yx yx yy yy
b
k sc Is EI k sc s
k sc s k sc Is EI
EI
EI
M
2
, , , ,
2
, , , ,
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[]
0 2 0 0
2 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
b
xx n xx n xy n xy n
yx n yx n yy n yy n
N
k sc Is EI k sc s
k sc s k sc Is EI
EI
EI
Simply Supported
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
[]
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
b
EI
EI
M
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[]
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
b
N
EI
EI
110
Clamped End
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
[]
00000000
00000000
00000000
00000000
b
M
00000000
00000000
00000000
00000000
[]
1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0
b
N
Free End
2
2
0 0 0 2 0 0
0 2 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
[]
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
b
Is EI s
s Is EI
EI
EI
M
2
2
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[]
0 0 0 2 0 0
0 2 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
b
N
Is EI s
s Is EI
EI
EI
111
Appendix B
Complex Variable Filtering and Full Spectrum Method
The goal of complex variable filtering is to separate the 1X running speed frequency
component into a circular, forward and reverse frequency components [Muszynska,
2005]. To accomplish this, the data from two proximity transducers mounted in
orthogonal configuration and measuring rotor lateral vibrations is utilized to create a
vector, the real part of this vector being the distance between the rotor and one of the
transducers and the imaginary part being the distance between the rotor and the other
transducer, at each sample time, which is
. This data is then processed using
a Discrete Fourier Transformation (DFT) algorithm
C.1
where
is the coefficient of the frequency in the harmonic series;
is the time to gather all samples ;
is the vector of the -th sample
It should be pointed out that calculated using above equation is not the actual
vibration amplitude at the frequency . According to the inverse DFT, the vibration
displacement at the interested frequency is
112
C.2
Vibration orbit at 1X running speed can be decomposed into the summation of the two
orbits, forward ( ) and reverse ( ), which can be written as
C.3
Let
and
, then
;
C.4
Figure C.1 1X-filtered orbit
The major axis of the 1X orbit, shown in Figure C.1, is
C.5
The minor axis of the 1X orbit is
113
C.6
The angle between the horizontal probe and the ellipse major axis is
C.7
So, using coordinate transformation, any point on the orbit can be written as
C.8
where
The experimental unbalanced mass response can be determined using Eq. (C.8).
114
Appendix C
Geometry and Operation Parameters of the WLRBs
Parameters
101.6mm Rubber
Bearing
50.8mm Rubber
Bearing
Sleeve Diameter (
) 101.56 mm 49.28 mm
Bearing Inside Diameter ( ) 102.04 mm 49.50 mm
Bearing Length ( ) 203.2 mm 190.5 mm
Radial Clearance ( ) 0.24 mm 0.11 mm
Fluid viscosity ( ) 0.001002 Pa· s 0.001002 Pa· s
Supply Pressure (
) 93079.2 Pa 93079.2 Pa
Stave Thickness (
) 9.52 mm 4.76 mm
Stave Width (
) 41.28 mm 20.64 mm
Land Region Angle (
) 19
o
19
o
Rubber Young's Modulus (
)
N/m
2
N/m
2
Poisson's Ratio ( ) 0.47 0.47
Abstract (if available)
Abstract
Flexible rotor systems that are supported or guided by water-lubricated rubber bearings (WLRBs) have a variety of engineering applications. Vibration analysis of this type of machinery for performance and duality requires accurate modeling of WLRBs and related rotor-bearing assemblies. In this study, a new model of WLRBs, with the focus on determination of bearing dynamic coefficients, is presented. Due to its large length-to-diameter ratio, a WLRB cannot be described by conventional pointwise bearing models with good fidelity. The bearing model considered in this work considers spatially distributed bearing forces, and for the first time, addresses the issue of mixed lubrication, which involves interaction effects of shaft vibration, elastic deformation of rubber material and fluid film pressure, and validates the WLRB model in experiments. Additionally, with the new bearing model, vibration analysis of WLRB-supported flexible rotor systems is performed through use of a distributed transfer function method (DTFM), which delivers accurate and closed form analytical solutions of steady-state response without discritization.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Liu, Shibing
(author)
Core Title
Modeling, analysis and experimental validation of flexible rotor systems with water-lubricated rubber bearings
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
10/06/2015
Defense Date
09/24/2015
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
distributed transfer function method,multistage rotor systems,OAI-PMH Harvest,vibration,water lubricated rubber bearing
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Yang, Bingen (
committee chair
), Shiflett, Geoffrey (
committee member
), Wellford, Carter (
committee member
)
Creator Email
liushibing1988@gmail.com,shibingl@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-189420
Unique identifier
UC11278548
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etd-LiuShibing-3970.pdf (filename),usctheses-c40-189420 (legacy record id)
Legacy Identifier
etd-LiuShibing-3970.pdf
Dmrecord
189420
Document Type
Dissertation
Format
application/pdf (imt)
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Liu, Shibing
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
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Repository Location
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Tags
distributed transfer function method
multistage rotor systems
vibration
water lubricated rubber bearing