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Rochester Institute of Technology Rochester Institute of Technology
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Theses
5-1-1989
Analysis of high-speed rotating systems using Timoshenko beam Analysis of high-speed rotating systems using Timoshenko beam
theory in conjunction with the transfer matrix method theory in conjunction with the transfer matrix method
Beth Andrews O'Leary
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ANALYSIS OF HIGH-SPEED ROTATING SYSTEMS USING
TIMOSHENKO BEAM THEORY IN CONJUNCTIONWITH
THE TRANSFERMATRIX METHOD
Beth Andrews O'Leary
A Thesis Submitted in Partial Fulfillment
of the Requirements for the Degree of
Master of Science
in
Mechanical Engineering
Approved by: Professor. s>.crn^i
J.S. Torok (Thesis Advisor)
CDoctor
G.H. Garzon
Professor
H.A. Ghoneim
Professor
Professor
R.B. Hetnarski
B.V. Karlekar (Department Chairman)
Department ofMechanical Engineering
College ofEngineering
Rochester Institute of Technology
Rochester, New York
May 1989
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ANALYSIS OF HIGH-SPEED ROTATING SYSTEMS USING
TIMOSHENKO BEAM THEORY IN CONJUNCTIONWITH
THE TRANSFERMATRIXMETHOD
I, Beth Andrews O'Leary, hereby grant permission to theWallaceMemorial Library of the
Rochester Institute of Technology to reproduce my thesis in whole or in part. Any
reproductionwill not be for commercial use or profit.
&
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ROCHESTER INSTITUTE OF TECHNOLOGY
- This volume is the property of the Institute, but the
literary rights of the author must be respected. Passages must
notv be copied or closely paraphrased without the previous
written consent of the author. If the reader obtains any
assistance from this volume he must give proper credit in his
own work.
This thesis has been used by the following persons, whose
signatures attest their acceptance of the above restrictions.
Name and Address Date
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ACKNOWLEDGEMENTS
Numerous people have supported and encouraged me during the research and
development of this thesis. In particular, I would like to thank
My advisor, Dr. Joseph S. Torok, for challenging me to excel beyond my
expectations. His insight and guidance helped to produce a successful thesis.
My first mentor in Rotor Dynamics, Dr. Guillermo Garzon, for our many
conversations and his continual encouragement.
My committee members, Dr. Garzon, Dr. Ghoneim and Dr. Hetnarski for their
detailed review ofmy thesis and for their constructive comments and challenging
questions.
My parents, Thorp and Dorothy Andrews, for their faith in my determination and
their continual support.
My husband, Kevin O'Leary, for drawing many of the figures in the thesis and
for his patience and encouragement throughout theMaster's program.
n
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ABSTRACT
Higher operating speeds and increased sensitivity ofmodern electro-mechanical systems
require improved methods for the computation of critical speeds and system response of
flexible rotating shafts. Many high-speed systems generally contain disks with masses
approaching the mass of the shaft. These observations emphasize the importance of
including the effects of rotatory inertia and shear deformation of the shaft in the analysis.
Traditional theory, which models a massless shaft, would be inaccurate for these
systems.
An analysis of flexible rotor systems has been performed using the Transfer Matrix
Method. Although the method is well known, the present study utilizes Timoshenko
Beam Theory in the construction of field matrices, which relate state vectors at adjacent
nodes of the system. This approach takes into consideration the effects of transverse
shear and rotatory inertia. Also included in the model are gyroscopic effects of the
spinning disks. These effects are generally neglected in classical rotor dynamic theory.
A general model was developed for the analysis of typical configurations in which the
shaft is simply supported, and can carry an arbitrary number of disks. Numerical
simulations were performed for comparision with classical results. These case studies
show agreement with what is to be expected by introducing the greater flexibility of
Timoshenko Beam Theory and the stiffening effects of gyroscopic couples.
111
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TABLE OF CONTENTS
List of Figures vi
List of Symbols ix
1 INTRODUCTION 1
2 BACKGROUND 3
3 THEORY AND PROGRAM DEVELOPMENT 6
3.1 Transfer Matrix Method 6
3.1.1 Natural Frequencies 20
3.1.2 Mode Shapes 22
3.1.3 Forced Response 27
3.1.4 Description ofGeneral Point and Field Matrices 31
3.2 Shaft Motion 34
3.2.1 Formulation of Field Matrices Using 35
Bernoulli-Euler Beam Theory
3.2.1.1 Field Matrix [FXJ 35
3.2.1.2 Field Matrix [FyJ 39
3.2.2 Formulation of FieldMatrices Using Timoshenko 44
Beam Theory
3.2.2.1 Field Matrix [Fx] 44
3.2.2.2 Field Matrix [Fy] 61
3.3 Disk Motion 71
3.3.1 Moment Equilibrium Equations Relating to 72
Development ofGyroscopic Couple and
Rotatory Inertia
3.3.1.1 Transition Matrix 75
3.3.2 Mass Unbalance 82
3.3.3 Formulation of Point Matrices 85
3.3.3.1 Gravitational Force 93
3.3.3.2 Mass Moments of Inertia 94
IV
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4 RESULTS 97
4.1 Figures 101
5 CONCLUSIONS 108
6 RECOMMENDATIONS 110
References Ill
Appendix A : Implementation of the Transfer Matrix Method 113
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List of Figures
3.1.1 Typical System Configuration 1
3.1.2 Simple Spring-Mass System Utilized to Illustrate the 6
TransferMatrixMethod
3.1.3 Free Body Diagram of Spring-Mass System 7
3.1.4 Relationship Between StationsWith Respect to Masses 12
3.1.5 Description of FieldsWith Respect to Masses 13
3.1.6 Relationship Between FieldsWith Respect to Masses 13
For a Simply Supported ShaftWithout an Overhang
3.1.7 Identification of Stations For aMulti-Disked Simply 15
Supported ShaftWith Overhangs
3.1.8 Multi-Disked, Simply Supported Shaft 22
3.1.9 First Bending Mode For Figure 3.1.8 22
3.1.10 Second BendingMode For Figure 3.1.8 23
3.1.11 Third Bending Mode For Figure 3. 1.8 23
3.2.1 Typical System Configuration With Coordinate Axes 35
3.2.2 Free Body Diagram of Shaft Element in X-Z Plane 36
Using Bemoulli-Euler Beam Theory
3.2.3 Free Body Diagram of Shaft Element in Y-Z Plane 39
Using Bemoulli-EulerBeam Theory
3.2.4 Free Body Diagram of Shaft Element in X-Z Plane 45
Using Timoshenko Beam Theory
3.2.5 Shaft Element in X-Z Plane Subjected to Bending Moment 46
3.2.6 Shaft Element in X-Z Plane Subjected to Bending 47
Moment and ShearDeformation
3.2.7 Free Body Diagram ofDifferential Element of Shaft 48
in theX-Z Plane
3.2.8 Free Body Diagram of Shaft Element in Y-Z Plane 62
Using Timoshenko Beam Theory
vi
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3.2.9 Shaft Element in Y-Z Plane Subjected to Bending Moment 63
3.2.10 Shaft Element in Y-Z Plane Subjected to Bending 64
Moment and ShearDeformation
3.2. 1 1 Free Body Diagram ofDifferential Element of Shaft in Y-Z Plane 65
3.3.1 Local and Global Coordinate Systems ForWhirling Disk 73
3.3.2 Relationship Between Global and Local Coordinate 76
Systems in the X-Z Plane
3.3.3 Relationship Between Global and Local Coordinate 77
Systems in the Y-Z Plane
3.3.4 Rotatory Inertias and Gyroscopic Couples Acting on Disk 81
3.3.5 Relationship Between Center ofGravity of a Disk 82
and Center ofRotation
3.3.6 Forces Acting on Disk Due to Mass Unbalance 84
3.3.7 Free Body Diagram of Forces Acting on Disk in the X-Z Plane 85
3.3.8 Free Body Diagram of Forces Acting on Disk in the Y-Z Plane 86
3.3.9 Free Body Diagram ofMoments Acting on Disk in X-Z Plane 87
3.3.10 Free Body Diagram ofMoments Acting on Disk in Y-Z Plane 88
3.3.11 Gravitational Force Acting on Angled System 93
3.3.12 Mass Moments of Inertia For a Disk WithoutMass Unbalance 94
3.3.13 Location ofMass Unbalance on a Disk 95
4.1.1 Simply Supported ShaftWith Three Nested Disks 97
4.1.2 Simply Supported ShaftWith Overhanging Mass 98
4.1.3 Case Study Utilizing Bernoulli-Euler Beam Theory and the 101
Transfer Matrix Method Without Gyroscopic Couple
4.1.4 Case Study Utilizing Timoshenko Beam Theory and the 102
Transfer MatrixMethod Without Gyroscopic Couple
4. 1.5 Natural Frequencies for Timoshenko and Bernoulli-Euler Beam 103
TheoriesWith andWithout Gyroscopic Couple
For an Overhanging Disk
vu
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4.1.6 Non-SynchronousMotion For an Overhanging Disk 104
Utilizing Timoshenko Beam Theory
4.1.7 Case Study Utilizing Bernoulli-Euler and Timoshenko 105
Beam Theories to Determine Forced Response Due to
Mass Unbalance Without Gyroscopic Couple
4.1.8 Forced Response ofOverhanging Disk Due to 106
Mass UnbalanceWith Gyroscopic Couple UtilizingTimoshenko Beam Theory
4.1.9 Effect ofGravitational Force on Overhanging Disk 107
Vs. Orientation of Shaft
A.l System Configuration 113
A.2 Flowchart Deriving Global Transfer Matrix 114
A.3 Flowchart Deriving Natural Frequencies From 116
Global Transfer Matrix
A.4 Flowchart Deriving Mode Shapes 1 17
A.5 Flowchart Deriving Forced Response 118
vin
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List of Symbols
A cross-sectional area of the shaft
[A] overall transfermatrix for left overhanging section of shaft
[A'] [A] matrix modified to include boundary conditions at station a
[A(j] displacement terms from [A] matrix
[ASJ slope terms from [A] matrix
a station containing left bearing
[B] overall transfermatrix formiddle section of shaft nested between the
bearings
[B'] [B] matrix modified to include [A'] matrix
[B"] [B'] matrix modified to include the boundary conditions at station b
b station containing right bearing
(3j angle between center of gravity and Y axis at station i
[C] overall transfermatrix for right overhanging section fo shaft
D shaft diameter
{d0 } displacement vector at station 0
E elastic modulus of shaft
q distance from center of rotation to center of gravity of disk at station i
Fg gravitational force acting on disk
[F]j fieldmatrix for shaft element Lj
[FXJ field matrix forX-Z plane
[Fy] field matrix forY-Z plane
G shear modulus of shaft
ggravitational constant
H angular momentum of disk
I area moment of inertia of shaft
Isx> Isv mass moments of inertia of shaft about the X and Y axes, respectively
It 1 , It 2 transverse mass moments of inertia of the disk about its center ofmass
I_ radial mass moment of inertia of the disk about its center ofmass
IX
Page 15
i station label
ks shaft form factor
Lj length of shaft element
\\ , ^2, ^-3, A.4 roots to characteristic equation
M couples acting on disk
M(j mass of disk without mass imbalance
Mj unbalanced mass at station i
Mx moment acting aboutX axis
My moment acting about Y axis
mj total mass of disk at station i including mass due to imbalance
mj+i total mass of disk at station i+1 including mass due to imbalance
m-_i total mass of disk at station i-1 including mass due to imbalance
ms mass of shaft element
n last station in rotating system
Q. rotating speed of disk
(0 whirl frequency; natural frequency of the sytem
coxvz angular velocity vector of disk in XYZ coordinate system
fxvz)' absolute velocity vector of disk in(XYZ)'
coordinate system
Pa reaction force at bearing'a'
Pjj reaction force at bearing'b'
[P]j point matrix formass mj
Qx> Qv parameter substitutions utilized in Timoshenko Theory for X and Y
directions
R parameter substitution utilized in Timoshenko Beam Theory
r radius of disk
S parameter substitution utilized in Timoshenko Beam Theory
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{ S }Lj state vector to the left of station i
{ S }^i state vector to the right of station i
{ S }L-
. i state vector to the left of station i- 1
{ S }Rj_ i state vector to the right of station i- 1
{S'
} state vector { S } modified to include reaction force at bearing'a'
{ S"
} state vector { S'
} modified to include the reaction force at station b
{ Sx } state vector describing X direction
{ Sy } state vector discribingY direction
t time at which forced response will be determined
[Tx], [Ty] transition matrices relating(XYZ)'
coordinate system to XYZ
coordinate system
x angle between the shaft and horizontal surface
6X slope in X-Z plane
9y slope in Y-Z plane
9X angular velocity of disk in Y direction
6V angular velocity of disk in X direction
0 angular acceleration of disk in Y direction
9 angular acceleration of disk in X direction
Uy mass unbalance in Y direction
Ux mass unbalance in X direction
[U] global transfermatrix
Vx force in X direction
Vy force in Y direction
w displacement utilized to explain the TransferMatrix Method
x displacement in X direction
ydisplacement in Y direction
XI
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Chapter 1
INTRODUCTION
A model is developed for a rigid bearing, flexible shaft, nonsynchronous, rotating system
defined in both transverse directions. The general model can be described as a simply
supported shaft with overhangs, on which any multiple number of disks can be attached.
The analysis allows for the variation of the shaft diameter along the length of the shaft.
Nonsynchronous motion, which is generic, takes place when the whirl velocity and spin
velocity are not equal. Synchronous motion can be derived from information obtained
for nonsynchronous motion.
/////> /<w/7 IFigure 3.1.1: Typical System Configuration
The model is analyzed for the natural frequencies along with their corresponding mode
shapes and/or the forced responses using the Transfer Matrix Method. This method
consists of the calculation of a series of relationships between the field and point
matrices. The field matrix describes the motion of the shaft and the point matrix
describes the motion of the disk or mass. Boundary conditions relating to discontinuities
in the shaft (ie. bearings, the ends of the shaft) are taken into account in the general
procedure.
The derivation of field matrices is outlined, for comparison, using two different beam
theories, the Bemoulli-Euler and the Timoshenko. The Bemoulli-Euler Beam Theory,
Page 18
which is incorporated in the traditional analysis, relates the stiffnesses between various
sections of the shaft, but considers the shaft to be massless. The Timoshenko Beam
Theory includes the centrifugal force, rotatory inertia and shear deformation of the shaft,
along with the stiffnesses of the shaft. The flexibility inherent to this theory tends to
lower the natural frequencies, since it reduces the overal stiffness of the shaft. Such a
modification becomes very important for high-speed rotating shafts, in which the mass of
the shaft approaches the mass of the disk.
The point matrix can be assembled by calculating the centrifugal force of the disk along
with the moment and gyroscopic couple acting on the disk. The gyroscopic couple is
determined from the moment equations and is a function of the radial mass moment of
inertia of the disk, the whirl velocity and the spin velocity. The gyroscopic couple raises
the natural frequencies of the system due to its tendency to straighten the shaft.
Analyzing a rotating system for natural frequencies, mode shapes and forced responses is
essential in determining a proper design configuration for the system and in
troubleshooting existing systems. The natural frequency of the system must not be in the
proximity of forcing frequencies that drive the various components of the system in order
to avoid resonant behavior. A forced response curve indicates the frequency range
surrounding the natural frequency that produces unacceptable displacements and indicates
the severity of the displacements when passing through this range.
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Chapter 2
BACKGROUND
The earliest reference to vibration of rotating shafts was by Rankine [1], who in 1869
defined the critical speeds or natural frequencies of such a system. His model consisted
of a uniform shaft that displaced from static equilibrium and was considered stable only
up to its first critical speed, an undetermined stability at critical speed and unstable above
critical speed. In 1894 Rayleigh [2] developed an approximate energy method to
determine the fundamental frequency due to lateral vibration of a non-rotating shaft. This
method, which is the basis of the Finite Element Method, minimizes the total energy of
the system by first assuming a single shape function and then obtaining the fundamental
frequency. The success of the method depended upon choosing a proper shape function
that correspondedwith the mode shape and matched its boundary conditions.
Timoshenko [3] applied Rayleigh's Method to rotating shafts and investigated the effect
of shear deformation on the natural frequencies of a shaft. Jeffcott [4] first described
whirl with his rotating system of a single unblanced mass situated on a massless shaft
between two rigid bearings, now known as a Jeffcott rotor. He considered the shaft to
displace in a plane and to precess at an angular velocity equal to the rotational speed of the
shaft. He concluded that the whirl amplitude increased while approaching the critical
speed and decreased beyond the critical speed; thereby, claiming that the system was
stable above the critical speed.
Prohl [5] first proposed the use of the Transfer Matrix Method for lateral dynamic
analysis. He divided the rotor into discrete masses and thereby, considered the shaft
massless. Gyroscopic moments were also included in the analysis. Computations were
quite tedious without computers. Therefore, the models had to be kept as basic as
possible. Pestel and Leckie [6] described the formulation of the field matrix in the
transverse direction using Timoshenko Beam Theory. In the Transfer Matrix Method,
the field matrix describes the motion of the shaft.
Several more sophisticated methods were soon developed. Tse, Morse and Hinkle [7]
used the Influence Coefficient Method to determine the natural frequencies of amassless,
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simply supported, overhanging shaft with gyroscopic moments. The flexibility
coefficients were determined by defining the displacement or rotation at one station using
Bernoulli-Euler Beam Theory due to a unit force or moment acting on the system at an
adjacent station. This method can be used for systems with multiple disks, but it
becomes cumbersome for more than three disks. Examples are charted describing
forward synchronous motion. Eshleman and Eubanks [8] studied the effect of axial
torque on the critical speeds of a simple system using partial differential equations.
Included in the study were the effects of transverse shear, rotatory inertia and gyroscopic
couple. The mathematical model was kept simplified in order to analyze the effects of the
various parameters.
Using Bernoulli-Euler Beam Theory, which refers to a massless shaft, Ruhl [9] studied
the stability of rotating shafts due to a mass unbalance using the Transfer Matrix and
Finite Element Methods. Ruhl was the first to study the use of the finite element method
for modeling rotating systems. Bearing effects were included in the model, but
gyroscopic couple rotatory inertia and shear deformation were not included. The effect
of residual shaft bow on the unbalanced response of a Jeffcott rotor was analyzed using
differential equations by Nicholas, Gunter and Allaire [10]. Residual bow may be due to
various effects, such as a gravitational force. Damping forces were included in the study.
The study was conducted to determine possible improvements to the balancing technique.
Nelson [11] was the first to study the use ofTimoshenko Beam Theory to determine the
shape function of a rotating shaft, which was then utilized in the Finite Element Method
to determine the natural frequencies of the system. Previous analyses had included the
study of the effects of rotatory inertia and gyroscopic couple using finite elements, but
had not included shear deformation. His results were compared to classical Timoshenko
Beam Theory analyses for non-rotating and rotating shafts. Rao [12] published an
analysis of critical bending speeds and forced responses of a simply supported shaft
using the TransferMatrix Method. He assumed the shaft to be massless and without a
gyroscopic couple.
Benson [13] modeled a clamped overhung disk with"active"
and"passive"
gyroscopic
couples. Active gyroscopic couples are forcing functions due to disk skew. Passive
gyroscopic couples, which are considered in this analysis, result from the interaction
between the change in slope of the shaft, which is due to whirl, and the resulting change
Page 21
of angularmomentum of the disk. Rieger [14] described the non-synchronous motion of
a clamped, overhung disk using theMethod of Influence Coefficients and Bernoulli-Euler
Beam Theory. Rao [12] analyzed a simplified model of the non-synchronous motion of a
clamped, overhung disk using Bemoulli-Euler Beam Theory in conjunction with the
Transfer Matrix Method. Both authors plotted the natural frequencies as a function of
whirl frequency parameter versus rotational speed parameter.
The present analysis utilizes the typical setup of a simply supported shaft with multiple
overhanging disks as well as disks nested between the bearings. The analysis, which
applies the gyroscopic couple to each disk and assumes non-synchronous motion, utilizes
the power of the Transfer Matrix Method and the much improved computational speed
and ability of the computer. Finally, the analysis includes the mass, rotatory inertia and
shear deformation of the shaft, which is of practical importance for systems driven at
very high speeds. These high speed systems have disks and shafts of comparable mass,
and therefore itmay be erroneous to consider the shaft as being massless.
Page 22
Chapter 3
THEORY AND PROGRAM
DEVELOPMENT
3.1 Transfer Matrix Method
The Transfer MatrixMethod is a discretization principle that can be used to determine the
natural frequencies, mode shapes and forced responses of a vibrating system. The
method consists of defining the boundary conditions at one end and appending to it
information pertaining to the system at each defined increment along the length of the
shaft, until the opposite end is reached. The system information, referred to as the state,
is the displacement, slope, moment and shear force at each boundary, shaft section and
disk. This information is transferred from one section to the next adjoining section until
an overall transfermatrix has been formulated.
The Transfer Matrix Method can be applied to any linear system. The method is
demonstrated using a simple spring-mass system with a forcing function. This system is
presented in figure 3.1.2. Vierck [17] presents an analysis of a spring-mass system
without a forcing function.
I
i> R
vV,
R R
l-H^- ? F cosl t
TTT)^/sss^ssJ>jZi/s^J DZH0
y
y
Figure 3. 1 .2: Simple Spring-Mass System Utilized to Illustrate TransferMatrixMethod
Page 23
Stations are located at changes in equilibrium. The terms'R'
andL'
refer to the right
and left of each station. Three stations, whose locations are indicated by the numbers 0,
1 and 2, are represented in figure 3.1.2. A state vector is a column vector containing, in
this case, the displacement and force on the right and left of each station. State vectors
are related to one another through point and field matrices, which describe the motion of
the masses and springs, respectively. A free body diagram of the system is developed in
figure 3.1.3 to facilitate formulation of the point and fieldmatrices.
,R
^->VW^- v>W/uF cos 2 1
% X, % X,
Figure 3.1.3: Free Body Diagram of Spring-Mass System
The forces and displacements acting on each component are summarized from the free
body diagram and expressed in matrix form in the following steps.
1) Sprine 1:
xV x0 +
tf = Fc
(3-1.1)
(3.1.2)
Equations 3.1.1 and 3.1.2 can be re-written as
Page 24
1
0 1
r I RX
IpJ(3.1.3)
which is of the form
{S}\ = [F]*{S}?
R
(3.1.4)
2) Mass 1:
xl-
xl
-R
F^ = m-xlM
(3.1.5)
(3.1.6)
where in the case of sinusoidal motion ofm\
Xj= Ai sin cot
"2 2x- =
-A! co sin cot = -co xx
(3.1.7)
(3.1.8)
Substituting x^ into equation 3.1.6 obtains
F*
=-mjco^j
+ F^ (3.1.9)
Equation 3.1.5 and 3.1.9 can be re-written as
1 0
-tr^co 1 f),(3.1.10)
which is of the form
{S}? = [Ph {S}\ (3.1.11)
8
Page 25
3) Spring 2:
*5t xi +F?
Fo = FR
(3.1.12)
(3.1.13)
Equations 3.1.12 and 3.1.13 can be re-written as
*
X
F
L
2
"ik2
1.
x
F
R
(3.1.14)
which is of the form
{S)L2 = [F]2{S}Ri (3.1.15)
4) Mass 2:
x!
b2
x^
F^ = m2x2
(3.1.16)
(3.1.17)
where in the case of harmonic motion ofm2
x2= A2 sin cot
2 2
x2=-A2 co sin cot = -co x2
(3.1.18)
(3.1.19)
Substituting X2 intoequation 3.1.17 obtains
F? =-m2co2x2
+ F^ (3.1.20)
Page 26
Equation 3.1.16 and 3.1.20 can be re-written as
1 0
-m2co 1 Fl
(3.1.21)
which is of the form
{S}2 = [P]2 (S>2 (3.1.22)
The state vectors are related through the point and fieldmatrices in the following manner:
{S}?
{S}?
= [FMSh
[PMS}^
[F]2{S}!
[P]2(S}2
[P]i[F]i{S}0
[F]2[P]i[F]i{S}0
[P]2[F]2[P]i[F]i{S}0
(3.1.23)
(3.1.24)
(3.1.25)
(3.1.26)
Equation 3.1.26 can be written as
XR
1 "lk2
1i -L
ki
.F, 2-m2co 1 1
.
-mjco 1 1
(3.1.27)
A global transfermatrix is formulated uponmultiplication of the point and fieldmatrices.
RX
< . =
F 2
m-co
1-
i (m2 + m1)o)
01201-0)
k2ki
k- k-k2 k2
mi mi mi
+ - +
Ui k2k-j
b2+ 1
(3.1.28)
10
Page 27
The boundary condition of xR0 = 0 is substituted into state vector { S }R0 to obtain
xR
2 m2miC0
k2ki
_1_
ki4 (
mjco i
kik2 k2
m2 m2 vt\\
\ *!
co2+ 1
1 /
(3.1.29)
For the homogeneous solution, F^ equals zero, while for the particular solution, FR2 is
equal to the applied force FcosQt. The natural frequency, co, of the system is determined
from the homogeneous solution of the equation
0 =
m 2m i co
k2ki
m
V M
m2 mi
k
coz
+ 1
l /
(3.1.30)
For the nontrivial solution, F0 cannot equal zero; therefore, the terms in the bracket must
be zero. The natural frequency, co, can now be solved using the quadratic formula.
To determine the forced response, the forcing function, for instance Fcos2t, is
substituted in equation 3.1.29 for FR2-
F cos Qt =
m2m1co
k2ki
^m2 m2m-^
+ +
k2 ka*
coz
+ 1
l /
F0 (3.1.31)
The force, F0, can be calculated from equation 3.1.31 utilizing the natural frequency
determined from equation 3.1.30. Now that F0 is known, the forced responses xR2 and
xRj can bedetermined from equations 3.1.26 and 3.1.24.
11
Page 28
In general, for a rotating shaft, the system information is written in matrix form defined
as field matrices and point matrices. A station is located at any change in equilibrium
such as a mass, bearing, coupling or boundary (see figure 3.1.4). A state vector is a
column vector containing displacement, slope, moment and shear force for station i.
Adjoining state vectors are related to one another through either field matrices or point
matrices. The field matrix[F]-
for the shaft element Li describes the equilibrium
conditions for L{ or, in the traditional theory, the stiffness matrix for the shaft section.
The point matrix [P]-describes the equilibrium conditions for mass i or, in general, the
mass matrix for the disk.
In general,
SRii-1
m(i-l)
t Li
SLjSR
mi
tm(i+l)
Figure 3. 1 .4: Relationship Between StationsWith Respect toMasses
where L refers to the left of a station and R refers to the right of a station.
The relationship between the state vectors {S}^ and[S)i_iR is given by
{S}iL =[F]i{S}i_iR
(3.1.32)
The relationship between the state vectors{S}-L
and{S)iR is given by
{S}iR =[P]i{S}iL
(3.1.33)
The point and fieldmatrices can also be related to each other in the following manner:
12
Page 29
A) If the length of the shaft begins with a mass such as in the case of an
overhanging shaft
mass 1 mass 2 mass 3
k- field 1 JCfield 2 H* field 3
Figure 3.1.5: Description of FieldsWith Respect toMasses
let { S } jL = { S }0 = starting boundary conditions (3.1.34)
{S)lR = [P]i{S}0 (3.L35){S}2L = [F]i
{S}iR= [F]i [P]i {S}0 (3.1.36)
{S}2R = [P]2{S)2L = [P]2[F]l[P]l {S}0 (3.1.37)
{S}3L = [F]2{S}2R = [F]2[P]2[F]i[P]i [S}0 (3.1.38)
This transfer of state information is continued until a bearing is encountered. The state
vector is modified at the bearing according to the required boundary conditions of zero
displacement, continuous slope and moment and a change in shear force due to the
reaction force of the bearing.
B) If the shaft does not contain a mass at { S }0 such as is the case of a simply
supported shaft without an overhang as in figure 3.1.6, then
mass 1 mass 2
field 1 [ field 2 + field 3
Figure 3. 1.6: Relationship Between FieldsWith Respect toMasses
For a Simply Supported ShaftWithout an Overhang
13
Page 30
{S)iL = [F]i{S}0
{S)iR= [P]i{S}iL = [P]1[F]1{S}0
(S}2L = [F]2{S)iR = [F]2 [P]i [F]! {S}0
{S)2R= [P]2
{S}2L= [P]2 [F]2 [P]l [F]i {S}0
(3.1.39)
(3.1.40)
(3.1.41)
(3.1.42)
This transfer of state information is continued until a bearing is encountered, at which the
state vector is modified according to the boundary conditions specified at the bearing.
Specifically, the boundary state vector {S }0 can be defined as
{S}0 =
w0 w0
e0. .
e0
0
V0. .
o.
for an overhanging shaft
(3.1.43)
(S0} =
w0 0
e0
M0
. <
e0>
0
.V0. V0.
for a simply supported shaft
(3.1.44)
{S0> =
W0 0
e0
M0
=
0
.V0..V0.
for a clamped end
(3.1.45)
where w is the displacement, 0o is the slope, M is the moment and V is the shear force.
14
Page 31
The clamped end condition would exist in a system that had two bearings very close
together, thereby, causing the slope of the shaft at that point to be essentially zero.
All matrices are shown as 4 x 4 and column vectors as 4 x 1 for ease of explanation. The
analysis was actually carried out for the general case of a 17 x 17 matrix and a 17 x 1
vector to be discussed in section 3.1.4.
Further discussion on the specifics of the Transfer Matrix Method pertains to a simply
supported shaft with overhanging disks, the same system that serves as the general
model.
Simply Supported Shaft With Overhangs:
1 2a a+1 b b+l n
7\ 7\
Figure 3.1.7: Identification of Stations For aMulti-Disked Simply SupportedShaftWith Overhangs
In general, it follows from the above that
{S}aL = [A][S}0 (3.1.46)
{S}bL =[B]{S}aR
(3.1.47)
{S}n =[C]{S}bR
(3.1.48)
where [A], [B] and [C] are the overall transfer matrices for the left, middle and right
sections, respectively. For example,
[A] = [F]2[P]2[F]l[P]l (3.1.49)
in the case where only two masses are on the left overhang. The only unknown in matrix
[A] is co, the whirl frequency.
15
Page 32
The state vector and the matrix [A] are reformulated at the first bearing. The state vector
{ S }0 is redefined to be a function of displacement only, through the following steps:
{s = [A] {S}0 =
0
6a
Ma
? = -
.Va.
An A12 A13 A14
A21 A22 A23 A24
A31 A32 A33 A34
A41 A42 A43 A44j
w.
e.
0
0 J
(3.1.50)
The first row of equation 3.1.50 shows that the displacement vanishes at the bearing,
thereby resulting in the relation
Anwo + A129o= 0 (3.1.51)
which can be rewritten as
e0 =
fAnl
K^XIJ
wr (3.1.52)
This can be substituted into { S }0 to give
(S}0 =
w0 <
11
A12
0
0
(3.1.53)
which will then produce amodified [A] matrix called [A'].
16
Page 33
{si =
0
fAn^22
l32
>42
VM2/
(W0} =
0
A'2i
A'
31
A'
41
(w0>
(3.1.54)
Now {S}aR={S}aL + Pa (3.1.55)
where Pa is the reaction force at the bearing. The slope and moment are continuous
across the bearing and are therefore not modified.
{ S }aR can be written as
{S}*
=
0 0 0 w0
A'21 0 0 0 0
A'31 0 0 0
< >
0
A'41 0 0 1 ^a.
= [A'] {S*} (3.1.56)
This equation is substituted into the equation for { S }5L to give
{S}bL = [B][S}aR = [B] [A] {S'} = [B'] {S'} (3.1.57)
A similar procedure is carried out for the second bearing as was done at the first bearing.
The state vector {S'} and the matrix [B'] will be modified at the second bearing.
{S }bL can be written as
17
Page 34
(S)b =
f \
0
eb
Mb
. =
[vbj
B'n B'l2 B13 B14 W
B21 B22 B23 B24 0
B31 B32 B33 B 34 0
B'41 B?42 B>43 B44_ P,
(3.1.58)
indicating that the displacement at the second bearing is zero. From equation 3. 1 .58, it
follows that
B'nwo + B'14Pa = 0 (3.1.59)
which can be rewritten as
Pa ="
a11
VB'l4y
Wn (3.1.60)
This can be substituted into {S'} as
IS'} =
wc
1
0
0
B'nB'
14J
(3.1.61)
which will then produce a modified [B'] matrix called [B"].
{S)t =
0
\D 14/
{w0) =
0
B21
B31
B41j
{w0> (3.1.62)
18
Page 35
Now {S}bR={S}bL + Pb (3.1.63)
where Pb is the reaction force at the bearing. The slope and moment are continuous
across the bearing and therefore, are notmodified.
{ S)bR
can be written as
(S)b =
0 0 0 Wo
ln 0 0 0 0
$3! 0 0 0
"
0
J41 0 0 1 Pb.
[B"] {S"} (3.1.64)
This equation can be substituted into equation 3.1.48 for {S}n
{S]n = [C]{S}bR
= [C] [B"] {S") = [U] {S"} (3.1.65)
The ntb disk is at the end of an overhanging shaft, whose boundary conditions reflect
zero moment and shear force and continuity of slope and displacement. Thus,
{S}n =
Wr
en
o
o
u u12 u13 u14
U2i u22 u23 u24
u31 u32 u33 u34
u41 u42 u43 u44_
wc
0
0
Pk
(3.1.66)
The [U] matrix is a global matrix that describes the motion of the entire rotating system in
the transverse directions. The matrix was developed by relating adjoining state vectors
and appropriate boundary conditions through point matrices and field matrices in a
systematic approach that begins at station 1 and ends at station n. The only unknown in
the [U] matrix is CO, the whirl frequency. Whirl frequencies that maintain equality in
equation 3.1.66 are known as natural frequencies.
19
Page 36
3.1.1 Natural Frequencies
In a typical environment, disturbances are transferred to the rotating system from its
surroundings. These vibrations, for example, can result from an operating frequency of
an adjoining component or from floor vibrations. When a forcing frequency, caused by
the disturbance, equals the natural frequency, resonance will occur. An undamped,
resonant mode will displace with an arbitrarily large amplitude. It is important, therefore,
to define the natural frequencies of a system to determine if resonance, and the
subsequently large amplitudes, will be avoided. This analysis is most usefully conducted
during the design of the rotating system.
The overall transfer matrix [U] is utilized to find the natural frequencies of the system.
Since the moment and shear force are equal to zero in the state vector, {S}n, the
following equations can be written:
woU3i+PbU34= 0 (3.1.67)
w0 U41 + Pb U44 = 0 (3.1.68)
This is written in matrix form as:
U31 u34
u41 u44_
w.
(3.1.69)
For the system to be physically meaningful, w0 and P\j cannot both equal zero. Thus for
a nonzero solution of equation 3.1.69, the determinant of the coefficient matrix must be
zero:
detU31 u34
u41 u44_
= 0 (3.1.70)
For a 2 x 2 matrix, this requires
U3iU44-U34U4i=0(3.1.71)
20
Page 37
Equation 3.1.71 is referred to as the frequency equation. The only unknown in this
equation is CO, the whirl velocity (this is shown in later derivations). Every solution of
the frequency equation will be a natural frequency.
Due to the complexity of the frequency equation 3.1.71, the natural frequencies cannot be
solved for directly, but must be determined numerically through an iterative process.
This process entails making an initial guess for co and then checking if the solution of the
frequency equation is zero within some tolerance. If it is not zero, another guess is made
and the solution is again checked to be zero. If the solution to the frequency equation had
been approaching zero, but then changed directions resulting in increasing magnitudes,
then the solution to the frequency equation is within this range of guesses. Linear
interpolation is utilized within this range to determine the final solution. A new guess for
the next natural frequency is begun at a small increment away from this region. The
process continues as before, beginning with incremental steps in the new guess toward a
new minimum. This scenario continues until the entire frequency range of interest has
been examined.
21
Page 38
3.1.2 Mode Shapes
The mode shape describes the displacement configuration of the system. The shape can
be utilized to determine if the performance of the system will be acceptable. The first
bending motion in forward whirl produces the most severe displacement on the system.
This severity decreases as the number of bends increase in the shape. For the following
multi-disked, simply supported, symmetric system the bending mode shapes are shown
below:
I ~?F 1 I 1^ I
Figure 3.1.8: Multi-Disked, Simply Supported Shaft
Figure 3. 1 .9: First Bending Mode For Figure 3.1.8
22
Page 39
Figure 3.1.10: Second Bending Mode For Figure 3.1.8
Figure 3.1.11: Third Bending Mode For Figure 3.1.8
The TransferMatrixMethod will be utilized to determine the mode shapes associatedwith
the natural frequencies. Each mode shape represents a collection of relative
displacements between disks and not actual displacements. Therefore, the initial
23
Page 40
displacement, w0, at station zero can arbitrarily be set to unity. The only other
unknowns in the state vectors at the boundaries are Pb, Pa and 0o which can be solved in
terms of w0. At this point, it is assumed that a natural frequency has already been
determined for the only unknown, co, in the transfer matrices. This value of co will be
utilized in equations 3.1.66, 3.1.60 and 3.1.52 to find Pb, Pa and 9, respectively, and in
the point and field matrices to find the displacements at each station. It should be pointed
out that the displacements at the bearings are already specified to be zero.
Since the shear force is zero at {S }n, the last row of equation 3.1.66 can be written as
woU4i+PbU44= 0 (3.1.72)
or
Pk = -
'LV
vU44 Jw (3.1.73)
Also, equations 3.1.52 and 3.1.60 can be written as
e0 = -
t A "\All
wr
V^12/
(3.1.74)
Pa = "
'bV
VB'l47
wn (3.1.75)
With w0 set equal to unity,these equations become
24
Page 41
Pk = -
'LV
vU44y
(3.1.76)
e0 = -
All
12/
(3.1.77)
Pa ="
VB'l4V
(3.1.78)
The mode shapes are determined by first substituting the values for Pa, Pb, 0o, and w0
into the state vectors and substituting co into the field matrices and point matrices and then
solving for the displacement at the left (for consistency) of each station.
For example,
{S}2L = [F]i[P]i {S}0 (3.1.79)
Denote [F]i[P]i as [FP]i so that
w
9
M
V
= [FPli
w0
e0
o
o
(3.1.80)
and
25
Page 42
1 1 LW
9
M
V
= [FPL
f \
1
An
A12
0
.0
.
> wr (3.1.81)
from which
w2 FPn-
'AiT
vAi2J
FP12 Wr (3.1.82)
This procedure is then repeated for each station. The displacement values can then be
renormalized to the largest displacement in the system.
26
Page 43
3.1.3 Forced Response
The actual displacement due to applied forcing on the system can be determined using the
TransferMatrix Method. This displacement is referred to as the forced response. The
investigated model has been developed to determine the forced response due to a mass
unbalance and a gravitational force. A similar analysis can be carried out for any force
application since the overall Transfer Matrix was developed as a general 17 x 17 matrix.
The specific values of the load would remain to be defined in the 17th column of the
respective field or point matrices.
A system force is incorporated in the transfer matrix as an extra column appended to the
point matrix, if it is a force on the disk, or the field matrix, if it is a force on the shaft.
Unity is appended to the state vector to compensate for this extra column. For example,
the point matrix and its associated state vectors for a 4 x 4 matrix is
{S)iR =[P]i{S}iL
(3.1.83)
-
wR
9
< M =
V
.
1. i
Pll Pl2 Pl3 Pl4 0
P21 P22 P23 P24 0
P31 P32 P33 P34 0
P41 P42 P43 ?44Uco2
0 0 0 0 1
w
9
M
V
1
(3.1.84)
m whichUco2
refers to the centrifugal force due to a mass unbalance.
Equation 3. 1 .84 is of the form of a standard linear system
[A]{x)=[B] (3.1.85)
where [A] contains the mass and stiffness matrices, {x}is the state vector and [B] is the
forcing function. Solving for {x} defines the particular solution for this equation.
Natural frequencies are determined from the homogenous solution of the equation [A]
{x} = [0].
27
Page 44
The overall transfermatrix can now be written as
{S}n =
wE
en
o
o
1
u u12 un u14 u15
U21 u22 u23 u24 u25
U31 u32 u33 u34 u35
u41 u42 u43 u44 u45
0 0 0 0 1
W0
0
0
Pb
1
(3.1.86)
The displacement, w0, and the bearing reaction force, P^ can be deduced directly from
3.1.86. Since the moment and shear force are zero at station n, the following equations
can be written:
U31wo + U34Pb + U35 = 0 (3.1.87)
Equation 3.1.87 can be rewritten as
Pk =
'-U35-U3iw0^
u34 J
(3.1.88)
and U41 w0 + U44 Pb + U45 = 0 (3.1.89)
which can be rewritten as
(-UuPk-IL,^wc
=
44 rb~
u45
u41
(3.1.90)
Equation 3.1.88 can be substituted in equation 3.1.90 to find w0:
28
Page 45
w0= -
U44 1
vU41y
u35 u3177- ~ 77- (w0)u34 u34
'U45^
vU4iy
(3.1.91)
fUu-UU,^ (VAA
w =
vU4iy vU34y vU41y
1 +44fu
\U4iy
A fTTu31
vU34y
(3.1.92)
Equation 3.1.92 can then be substituted back into equation 3.1.88 to find Pb.
Now 0o and Pa can be determined from equations 3.1.74 and 3.1.75.
9rfAnl
vAi2y
w (3.1.93)
Pa ="
a11
VB'l4/
wr (3.1.94)
The forced responses are now determined in the same manner as the mode shapes. The
values for Pa, Pb, 9Q and w0 are substituted into the state vectors and the corresponding
29
Page 46
whirl frequency, co, is substituted into each field and point matrix. The displacement or
forced response at the left (for consistency) of each station can be determined from the
relationships that comprise the TransferMatrix Method. For example, the displacement
at station 2 can be determined from equation 3.1.82.
30
Page 47
3.1.4 General Description of Point and Field Matrices
A rotating system can be described in general terms by a 17 x 17 matrix formulated from
point matrices and field matrices. The state vector can be expanded to include motion in
the X and Y directions, respectively. These directions are orthogonal to one another and
also to the shaft. (See figure 3.2.1) The motion in the X and Y directions can most
generally be described by terms containing sin(cot) and cos(cot) factors. The state vectors
for the general matrix are defined as
{S,.c} =
Xc
'x,c
My.c
-Vx,c
and {Sx,s} =
xc
'x,s
M
-V
y.s
x,sj
(3.1.95)
<Sy,c>
-Yr
'y,c
Mx,c
y.c
and *Sy,s>9y.s
M
V
x,s
y.sJ
(3.1.96)
where'c'
and's'
represent cosine and sine. The negative signs associated with Vx and Y
are convention that yields positive coefficients in the field and point matrices and are
carried throughout the entire analysis. The moment My is in the state vector, {Sx},
because themoment is acting on the system in the X-Z plane, but its direction is along the
Y axis. This symbol convention also applies to the momentMx being in the state vector
{Sv}- These state vectors, along with a unity row to compenstae for forcing functions,
can be combined to form a single 17x1 state vector:
31
Page 48
{S} =
Xc
9x,c
My.c
-Vx,c
Xs
9x,s
My.s
-Vx,s
-Y, (3.1.97)
9y.c
Mx,c
Ty.c
"Ys
9,'y.s
Mx,s
Vy.s
1
The overall transfermatrix for the general solution is derived in a similarmanner as the 4
x 4 matrix. The general solution requires transferring all information in the form of
matrices instead of equations. For example, equation 3.1.51 in the 4 x 4 matrix is written
as
Anw0 + A1290= 0 (3.1.98)
In the general solution, this same step in the Transfer MatrixMethod is written as
[Ad]{do} + [As]{So} = {0} (3.1.99)
where {do} is the 4 x 1 displacement vector at station 0 and {S0} is the
4x1 slope vector at station 0.
32
Page 49
In expanded form, equation 3.1.99 becomes:
Ai,i A15 A19 A1>13 'Xc Ai,2 A1>6 A1>10 A1>14 6x,c
A5,i A5>5 A5>9 A5ii3 A5,2 A56 A5)10 A5)U 6x,s
Ag.i Ac, 5 A99 A913 -Yc A9,2 A96 A910 A9U 6y.c
^13,1 Ai3)5 A]3>9 A13i3_ ."YSj 0 .Ai3,2A13>6 A1310 A1314_ ,6y,s.
= {0}
(3.1.100)
Further complexity arises in the solution of the equations. For example, to solve for
{ S }0 in equation 3. 1 .99, the inverses of the matrices must be found.
{S0}=-[As]-l[Ad]{d0} (3.1.101)
This complexity is carried through the model until the overall transfer matrix is found.
To obtain the natural frequencies of the general system, a value must be found that will
ensure that the determinant of an 8 x 8 subset of the overall transfer matrix [U] is zero.
The 8x8 matrix is obtained by requiring that four moment equations and four shear force
equations are zero at station n.
33
Page 50
3.2 Shaft Motion
The motion of the shaft is determined by the natural frequencies and the forced responses
are deduced from the field matrix as part of the TransferMatrix Method. The mechanics
that govern the motion can be described by either the Bernoulli-Euler Beam Theory or the
Timoshenko Beam Theory. The Bemoulli-Euler Beam Theory is the traditional theory
used in rotor dynamic analysis due to its simplicity. It neglects the mass of the shaft and
only relates shaft sections to one another through stiffness values. The Timoshenko
Beam Theory includes, along with the shaft stiffnesses, shear deformation, rotary inertia
and centrifugal force. These additional features increase the complexity of the problem
by producing, and requiring the solution of, a fourth order differential equation. Each of
these methods will be described in detail.
34
Page 51
3.2.1 Formulation of Field Matrices Using Bernoulli-Euler
Beam Theory
In this section, the general 17 x 17 field matrix (see equation 3.1.97) will be defined
using the Bernoulli-Euler Beam Theory. The mechanics will be described in terms of the
state vector for the X direction, { Sx } , and the state vector for the Y direction, { Sy } . The
coordinate system will be defined as having its origin at the farthest left point in the
system with the Z axis being positive to the right along the length of the shaft, the X axis
being positive in the upward vertical direction and the Y axis being positive out of the
page. In figure 3.2.1, the coordinate axis is shown on a typical system.
H
/
'
H-.-y^ /\
Figure 3.2. 1: Typical System Configuration With Coordinate Axes
3.2.1.1 Field Matrix [Fx]
The relationship between displacement x, the slope 9X, the moment My and the shear
force Vx acting on a shaft element is embodied in a free body diagram.
35
Page 52
*>z
Figure 3.2.2: Free Body Diagram of Shaft Element in X-Z Plane UsingBemoulli-Euler Beam Theory
The sign convention for moment is defined as positive and for shear is defined as
negative for this shaft element. This corresponds to traditional matrix notation for ease of
solution. This is valid if a negative Vx shear force is maintained throughout the entire
transfermatrix. The parameters on this free body diagram that have not been defined in
this section are
i = station at location i
i-1 = station located before station i
R = the right section adjoining a station
L = the left section adjoining a station
Lj = length of shaft located between stations i-1 and i
z = distance along the shaft from the origin
36
Page 53
The following general relations describe the relative displacement and relative slope of a
beam element
x =
9 =
ML2
2EI+
VL3
3EI
ML+
VL2
EI 2EI
(3.2.1)
(3.2.2)
These equations are from Bemoulli-Euler Beam Theory for a cantilever beam. The E
represents the elastic modulus of the shaftmaterial and the I represents the area moment
of inertia of the shaft. For a round shaft (which is the only cross-section considered in
this analysis), I equals n D^/64, in which D is the shaft diameter. Using these beam
theory equations, plus compensating for a change in slope at x^.j, the following
equations for displacement and slope can be formulated.
X; =
X?-l +6*
-_1 Li + M^j
'l?^
2EI+ V
X.l
\J
( L3 "
3EI-(3.2.3)
9V i'X,l ex,i-l + Ml r Li.y,i
I Eli+ V
f L2
X,l
2EI:\J(3.2.4)
The shaft diameter can vary along its length; thereby, the area moment of inertia must be
defined for each shaft section 'i'. In a rotating system, a change in the shaft diameter
may be required to accommodate such components as motors, bearings and collars.
Tapered beams are analyzed by discretizing the shaft into numerous sections. Each
section would contain a subsequently smaller diameter, thereby approximating the
continuity of the taperedbeam.
The equations for shear and moment are written based on the equilibrium conditions
embodied in the free body diagram.
37
Page 54
M^i = M^i_i -
V^i_i ^ (3.2.5)
yL. vR-i or -VL- = 1v
x,ivx,i-l
U1 Yx,i
yx,i-l (3.2.6)
The moment and shear equations are substituted into the displacement and slope
equations to obtainx-L
and 9X-L
as a function of the right section of station'i-1'
only.
These equations become
xi= 4-i + 0x,i-i Li+ (MRi_!
-
v;ti_! L^
^L2 ^
v2EIiy
+ VR
x,i-l
f j2 \
X; = 4-1 + li-i L; +MR
i_i
v2EIiy
- VR
x,i-l
v6EIiy
UElJ
(3.2.7)
(3.2.8)
and
ei,i = eVi + (My.i-i VR,i_i L;)
r t . \
vEIi,
+VR
1T vx,i-l
v2EIiy
(3.2.9)
eiti = eRi_i + MRi_ivEIiy
VR- -
vx,i-l
( L2 ^
2EIJ(3.2.10)
In matrix form, these equations become
XL
6x ? =
My
[-VxJ i
1 Li
0 1
< L2 ^ ( L3 ^
v2EIiy
IEli J1
v6EIiy
^ ,2 >
0 0
0 0 0
2EI-
Li
1
< vx
M3-V
R
xj i-1
(3.2.11)
38
Page 55
Equation 3.2. 1 1 can be represented in the form:
{SxhL = [FJi {SX}RM (3.2.12)
3.2.1.2 Field Matrix [Fy]:
The relationships between displacement y, the slope 9y, the moment Mx and the shear
force Vy acting on a shaft element are embodied in a free body diagram depicted in figure
3.2.3.
Y
4
VR
y.i-i
i-1 i,^
<s> +
tY
iX
R-? Z
l-l'i-1
Li-
Figure 3.2.3: Free Body Diagram of Shaft Element in Y-Z Plane UsingBemoulli-Euler Beam Theory
The sign convention for moment and shear is defined as positive for this shaft element.
The parameters on this free body diagram have the same definition as those described in
39
Page 56
the X-Z diagram. Note that a positive retation of the shaft produces a displacement in the
negative direction. In the X-Z plane, a positive rotation produces a positive
displacement.
From Bemoulli-EulerBeam Theory, the general equations can be written as
y = -
6V =
ML/
2EI
ML
EI
VLJ
3EI
VL2
2EI
(3.2.13)
(3.2.14)
Using these beam theory equations, in conjunction with the equilibrium equations derived
from the free body diagram, the following equations for displacement, slope, moment
and shear force can be written. These have the form
y\=
y*-i- <i-i ^ -
Miti
( t2 ^
2EIV^M/
+VL
y.i
(l}
^
ey,i ~ 6y,i-l + Mx,ivEIiy
- VL-
2EIi,
& = MRi_i + VRi_i Lj
VL
=
VR
^vy,i vy,i-l
(3.2.15)
(3.2.16)
(3.2.17)
(3.2.18)
The shear and moment equations can be substituted into the displacement and slope
equations, which will result in the displacement and slope in the left section of station i
being a function of the right section of i-1 only.
0y,i
9y.i ~
ey,i-i + (MRi_i + VRi_i Lj)fLi
vEIi,
- Vy.i
( L2 ^
v2EIiy
e?fi-i + MRi_ivEIiy
+ VR-1^ v
y,i-i
( L2 ^
2EIi
(3.2.19)
(3.2.20)
40
Page 57
In matrix form these equations, along with the moment and shear equations, become:
f *\
YL
9y =
Mx
IVy. i
1 -Li-
=
0 1
0 0
0 0
'L2 1
.2ElJ
( L ^
Eli
1
0
v^M/
( L3 1
,6ElJ
2ElJ
Li
1
M,
R
(3.2.21)
i-1
The displacement equation is multiplied by a negative one to produce positive
coefficients for ease of solution. This is valid if a negative y displacement is maintained
throughout the entire transfer matrix. The matrix equation 3.2.21 becomes:
-Y
Mv
( t2 A ( T3 ^
1 Lj
0 1
0 0
0 0
L
2EIiy
Eli,
1
0
6EI
Li2 ^
2EIi
Li
1
-Y
9,
M,
yJ i-1
(3.2.22)
Equation 3.2.22 can be written in the form
{Sy}-L = [F]y>i {Sy}Ri_i (3.2.23)
The displacement, slope, moment and shear can be expanded and written in the general
terms
41
Page 58
x = Xc cos cot + xs sin cot (3.2.24)
ex = 6xc cos m + 9xs sin m (3.2.25)
My=Myccoscot + Myssincot (3.2.26)
vx = Vxc cos cot + Vxs sin cot (3.2.27)
and
y=
yc cos cot + ys sin cot(3.2.28)
9y=
9yc cos cot + 9ys sin cot (3.2.29)
Mx =Mxc cos cot + Mxs sin cot (3 2 30)
Vy= Vyc cos cot + Vys sin cot (3.2.31)
in which'c'
refers to cos(cot) and's'
refers to sin(cot).
Natural frequencies of simplified models, that contain equivalent stiffnesses in the X and
Y directions, can be determined from 4x4 field and point matrices formulated in one
direction only. The simplified model assumes that the slope and displacement of each
disk and shaft element are equivalent in both the X and Y directions. Therefore, one
direction can be eliminated from the analysis.
The general model presented in this investigation, whose state vector is represented by
equation 3.1.99, would be required to determine the natural frequencies of a system with
anisotropic bearings and to determine the response due to a forcing function. Bearing
forces can be modeled in this analysis by modifying the field matrices surrounding the
bearing station to accommodate for the stiffness of the bearing. Utilizing the general
model for analyzing a system does not impose restrictions that would limit the scope of
the problem.
The [Fx] and [Fy] field matrices will be formulated into an overall 17x17 general field
matrix that will be of the form
42
Page 59
^X.CR
^x,s
Sy.c=
Sy,s
.1
. i-1
[Fx] 0 0 0 0
0 [Fx] 0 0 0
0 0 [FyJ 0 0
0 0 0 [Fy] 0
0 0 0 0 1
^x,c
Sx,s
sy.c
^y.s
[ 1
(3.2.32)
where'c'
represents cos(cot) and's'
represents sin(cot).
43
Page 60
3.2.2 Formulation of Field Matrices Using Timoshenko Beam
Theory
A general 17 x 17 field matrix will be defined from the mechanics associated with
Timoshenko Beam Theory. This theory includes, in addition to the stiffness of the shaft,
the effects of shear deformation, rotary inertia and centrifugal force. The coordinate
system will be the same as that defined for the Bemoulli-Euler Theory shown in figure
3.2. 1. The origin will be located at the furthest left point in the system with the Z axis in
the direction of the shaft, the X axis oriented vertically and the Y axis oriented
horizontally. The field matrix will be derived in terms of the state vector for the X
direction, {Sx}, and the state vector for the Y direction, {Sy }.
3.2.2.1 Field Matrix [Fx]
The displacement x, the slope 9X, the momentMy and the shear force Vx are related as
depicted in the free body diagram in figure 3.2.4. Also shown in the diagram are the
centrifugal force, msco^xdz, and the rotary inertia, Isyco29x dz. The sign convention in
the free body diagram defines the moment as positive and the shear as negative. This
convention is consistent with the conventions associated with the Bemoulli-Euler
Method.
44
Page 61
Y
Figure 3.2.4: Free Body Diagram of Shaft Element in X-Z Plane UsingTimoshenko Beam Theory
For a circular shaft section, Is y is calculated as
IS)y=(l/12)mS)i(3a2 + Li2) (3.2.33)
in which'a'
is the radius of the shaft [18].
To define and understand the equation for shear deformation, the following free body
diagrams are presented:
45
Page 62
+- z
Figure 3.2.5: Shaft Element inX-Z Plane Subjected to BendingMoment
Figure 3.2.5 shows a shaft element subjected to pure bending only, due to the moment
My= EI (d9x/dz). This is the fundamental relation in Bernoulli-Euler Beam Theory.
Line a', that passes through the end of the bent shaft, is perpendicular to the
cross-sectional face of the end of the shaft. Line b', which indicates the position of an
unbent shaft, is parallel to the Z axis.
A negative shear force on the shaft element produces a positive displacement at z-L,
which, along with the displacement due to a bending moment, produces a net positive
displacement.
46
Page 63
<>
Figure 3.2.6: Shaft Element in X-Z Plane Subjected to BendingMoment and Shear
Deformation
The orientation of the centerline of the shaft, along line a", changes without any rotation
occurring at x. Lineb'
remains parallel to the Z axis. 9X is the slope due to a bending
moment ,VX/GA is the slope due to shear force and dx/dz is the total slope of the
centerline of the shaft.
The parameters in figure 3.2.6 are defined as
G = shearmodulus
GAS = shear stiffness
As = A/Kg
A = cross-sectional area of the shaft
iq = form factor that depends on the shape of the cross-section
As defined byWashizu [15], Kg equals.851 for circular cross-section beams.
47
Page 64
The relationship including shear deformation can now be written as
dx Vx
dz GAC(3.2.34)
Rearranging, the shear force is deduced as
VY = GACdz
(3.2.35)
The general free body diagram for the field matrix in the X-Z plane can be more simply
depicted on a differential element of the shaft:
MsCO2
XdZ
X
A
<2>
IsyCG2edZ
My+dMy
Vx+dVx
-? Z
Figure 3.2.7: Free Body Diagram ofDifferential Element of Shaft in the X-Z Plane
48
Page 65
The summation of the moments acting on the element is written as
+
J X M = :"My
+ My + dMy + Isy co29xdz + V,
dz
I 2 .
MtMti-0 (3.2.36)
The last term in the equilibrium equation is dropped since it is infinitesimal.
Equation3.2.36 is then rewritten as
dMj~dz" = -Vx-Isycoz9 (3.2.37)
The summation of the forces acting on the element is written as
+ t ^F = 0: -Vx + Vx + dVx + msco2
x dz = 0 (3.2.38)
which can be rearranged as
dz
=
-ms co x (3.2.39)
The derivative of the shear equation 3.2.35, with respect to z, results in
dVx
dz
( *i
= GA<d2x de,
dz2 dz
(3.2.40)
The equation
My_
d9x"e7~
dz(3.2.41)
49
Page 66
can be substituted into equation 3.2.40 to obtain
dV,
dz
= GA.MyEI
(3.2.42)
Now, the derivative of the shear equation 3.2.42 can be equated to the force equilibrium
equation 3.2.39.
-ms
co2
x d2x MyGAC dz' EI
(3.2.43)
Rearranged, this becomes
d2x msco2
x
dz' GAC
MyEI
= 0 (3.2.44)
Differentiating the moment equilibrium equation 3.2.37 with respect to z,
d2M,
dz'
dV
dz
xT 2
-
ISy w'dO
V dz(3.2.45)
The force equilibrium equation
dV,
dz
= -
mc co x(3.2.46)
and the moment equation
My_
d9xHz~
(3.2.47)
50
Page 67
can be substituted into equation 3.2.45 to obtain
d2M
dz
-
msco2
x + Isyco2'M/
EI j
= 0 (3.2.48)
Equation 3.2.44 can be rearranged in terms ofM,
l^+mmico^x
dz' GAC(3.2.49)
and then substituted into equation 3.2.48 to obtain
dz'
EId^x
dz2
TTT 2 AEIms co x
rtij co x +(3.2.50)
Isy0>'
EIEI
d2x EIms co x
dz' GA,
= 0
Simplifying, equation 3.2.50 becomes
dz4
ms co
GAC
.2^
dzx
dz2
ms co x
EI
(
ISy'
I EI
f
d^x
dz2
Isy m
I EI
l\ms co x
GA J
(3.2.51)
= 0
That is,
d^x
dz4
(h me "l d2x+ co
^sy ms
EI GA
msco Isy m
s/dz' EI V GAe
- 1 = 0 (3.2.52)
51
Page 68
Equation 3.2.52 is a fourth order differential equation describing the motion of the shaft.
Further simplification is attained by defining the parameters
Qx =co2
IsyL2
EIR =
co2
msL2
gTS =
ms co L"
EI(3.2.53)
resulting in the differential equation
d*x_dz4
^Qx +R'
I L2 Jdzx
dz2
RQx-s^
V
x = 0 (3.2.54)
Equation 3.2.54 is an ordinary differential equation with constant coefficients. The
solutions, therefore, are of the form
x = C e-^/L
(3.2.55)
in which C is a constant.
Successively differentiating equation 3.2.55 with respect to z gives
dx= ^
Xz/L
dz II
d2^dz2
^,2^
= cXz/L
IL'
(3.2.56)
(3.2.57)
d3^dz3
d4^dz4
r,3^i= c
= c
Az/L
Az/L
x";
(3.2.58)
(3.2.59)
52
Page 69
and then substituting equations 3.2.55, 3.2.57 and 3.2.59 into the ordinary differential
equation, the following equation is obtained.
CAz/L
,^.(Qx
+RV^e^'1-
+ C
VL' IL'
eXz/L
+c,
RQx-sN
K.L4 J
eXz/L
= 0
(3.2.60)
Upon reduction of equation 3.2.60, the characteristic equation can be written as
X4
+ (Qx + R)X2
+ (RQX - S) = 0 (3.2.61)
The roots of the characteristic equation 3.2.61 can be determined by noting that equation
3.2.61 is quadratic in X , with roots
X2
=
-(Qx + R) V(Qx + R)2-4(RQx-S)
(3.2.62)
X2
=
-
(Qx + R) ^0?+ 2RQX +R2
-
4RQX + 4S
(3.2.63)
X2
=
- (Qx + R) ->/oi- 2RQX+R2+4S
(3.2.64)
X2
=
- (Qx + R) V(Qx "
R)2
+ 4S
(3.2.65)
2 1X =-j(Qx +R)4\ (Qx-R)+S (3.2.66)
53
Page 70
Further, by taking square roots,
^=V~2(Qs+r)Vi (Qx-R) +S (3.2.67)
The roots of the characteristic equation 3.2.61 are expressed as
Xi and i X2
and thus the solution to equation 3.2.54 has the form
= C\e^z/L
+ C'2&^zlL
+ C3ea2z/L
+ C'4e_iX2z/L
(3.2.68)
The followingmathematical substitutions can be made
+ ae~
= cosh a sinh a
e* ia= cos a i sin a
(3.2.69)
(3.2.70)
resulting in the expression
x = Q cosh Xi !) C2 sinh
(
xlL,!? c3 cos
f
X2V
1?
L,+ C4 sin
f
X27^
L,
(3.2.71)
where
c2 =
c3 =
C'l-
c3 +
c2
C4
(3.2.72)
(3.2.73)
(3.2.74)
c4 = i(C3 -C4) (3.2.75)
54
Page 71
The expressions for x, 9X, My and Vx will all be of the same form, therefore, the shear
force can be written as
-Vx= A- cosh \Xi -1+ A2 sinh \x i- 1+ A3 cos
A4 sin \X2
(;)*
A
(3.2.76)
in which Ai, A2, A3 and A4 are unknown constants.
Differentiating equation 3.2.76 with respect to z,
dVx
dzAi [ -j-j-J
sinh \Xij-j
+ A2 cosh X-,VL J I
*L
A,l^
\
sin X-)Vl
L.
a 1 XA+ AA I
L J
\
cos X.?I 2l;
(3.2.77)
Relating equation 3.2.46 to equation 3.2.77, the derivative ofVx, the following equation
for x is obtained.
x =
1
ms co
(X, \
VL7
sinh Xi \+ Ao cosh X.-
L J {.1
LJ
A3 | -jj-Jsin \X2
j-j+ A4
(X-
cos
Vl
L)
Differentiating equation 3.2.78 with respect to Z, gives(3.2.78)
55
Page 72
dX
dZms co KL*J
cosh
\ L+ A-,
KL2J
sinh XiI LJ
l2;
cos |X2 j-
A4 sm
V L(3.2.79)
A relationship for the slope, 9X, can be obtained from the shear equation
Vx = GASdx
^dz-9,
from which
(3.2.80)
GAS dz(3.2.81)
The equations for shear force, 3.2.76, and the derivative of X, 3.2.79, are substituted
into the slope equation 3.2.81 to result in
6v =
GA,A! cosh fx*l
ms co
'*TIL2
cosh
+ A2 sinh
Z\
fx *!( z\ ( z)
+ A3 cosX2 + A4 sin X2
v LJ v U
(,2\
LJ+ Ai
\L2)
sinh
( 7\
PL- A3
f,2\
vl2;
cosZ^
V LJ
KL2J
sm
V L
To simplify equation 3.2.82, the followingparameter substitution is made.
(3.2.82)
R =
2 2co ms L
gX(3.2.83)
56
Page 73
e* =R
2 0
co mtL
A- cosh
V L+ A2 sinh
( 7\
+ A3 cos X2 + A4 sin
\ Lj , LJ
co2
msL2
A- coshr z
v Ly+ A2 sinh
2 2co ms L
A3 cos
v 2L;
A4 sin (3.2.84)
so that
9Y =
co msL2
(R + Xf) AY cosh X: + A2 sinh X,
V LJ v Lj+
( R -
Xl ) A3 cos
I 2lJ+ AA sin X9-
(3.2.85)
The derivative of equation 3.2.85 with respect to Z is
de,
dZ 2 T 2co ms L
(R-X|)
A 1 sinh(R+Xf )
- A3 an X2-
L v Ly
^7v Ly
x-
+ A7 cosh
L V Ly
^ AX2 (.
Z^
+ A4 cos X2(3.2.86)
The moment equation 3.2.41 then becomes upon substitution of 3.2.86
M = EIdex H |(R + X2)
(R-X22)^
(
Ax sinh
V
-
A3 sin
Z^
Lj
f ZX2-
V 2L.
+ A2 cosh
+ A4 cosJ
( Z\l
^7V. L)_
{
Z\lX2-
dZ 2 T 2co ms L
+
(3.2.87)
57
Page 74
The equations for displacement, slope, moment and shear force can be written as the
system
58
Page 75
< < < <
/_^ N|J
1
n|j N|JIN
^ GO
OCO B N|_JO
/
"N.
u CO cs
* "N' ~\ r-*
J"CN
3^
CNCN
ri
1
CM
-J
tN
3^
CNCN
i
CN
C
CO
gs. >
5
n|j
N|J n|j ^^ a
s- *so
O
CO
n|jc
COu
1
CN
y s
J
3^
<N<N
<
1
CN
3
jCN
3^
VI
Ou
ia:
^ > 5i
N|J N|_l
<<j=
GA
j=JS o N|J
OS E
CO
u
O ^v
-J
3_
(S-H
+
CN
-J
CN
3^
CN
+
-J
CN
3GO
s as gA
E-*
H
N|J
n|j n|j.T
<<rT
J5
JS-C
Cn|-j
a o ?"
~N<
*BB
rn
'
^ CN J3
t<
<n
3m
fi
+
CS
3^
CN
+
cn
3
CO
OCJ
ocE
H
II
XcT 2
H
>1
oo
00
ts
59
Page 76
Equation 3.2.88 is of the form {X(z)} = [B(z)] [A]. This equation must be formulated
into the form of a field matrix relating station'i'
to station 'i-1', so that it can be utilized
in the Transfer Matrix Method. This reformulation is done by applying boundary
conditions on the shaft element. At Z equals zero, in the local coordinate system of the
shaft element, the matrix becomes
9,
M,
-V,xJi_i
0
(R + Xf)
2T 2msco L
0
1
msco L
0
EKR + X^X,
msco2L3
0
0
R-Xj
2T 2msco L
0
1
x2 1"Ai"
msco L
0 A2
EI(R -
Xf)X2 A3
msco L
0 LaJ
The location Z equals zero corresponds to station {X}j_i.
(3.2.89)
Equation 3.2.89 is of the form
{X}Ri.! = [B(0)][A] (3.2.90)
Solving for [A] in equation 3.2.90 and then substitution into
{X(z)}=[B(z)][A] (3.2.91)
results in the the following equation
{X(z)} = [B(z)][B(0)]-l{X}Ri.i (3.2.92)
At the position specified by Z equal to L, (X(z)}= {X}L-. Making this substitution into
equation 3.2.92,
(3.2.93){X}Li = [B(L)][B(0)]-MX}Ki_i-lnnR-
60
Page 77
The field matrix for the X-Z plane is
[Fx] = [B(L)] [B(0)]-1 (3.2.94)
Equation 3.2.88, which contains expressions for X, 9X, My and -Vx in terms of the
coefficients A\, A2, A3 and A4, was reformulated into a field matrix by applying
boundary conditions relating to station T and station 'i-1'. These boundary conditions,
along with subsequent substitutions, eliminated the coefficients Aj and thereby resulted in
an equation that relates station{X}Li to {X}R-_i through the field matrix [Fx].
3.2.2.2 Field Matrix [Fy]
The displacement Y, slope 9y, moment Mx and shear force Vy, along with the
centrifugal force msco^YdZ and rotatory inertia Isxco^9ydZ, are shown on the free body
diagram in figure 3.2.8. The sign convention in the free body diagram defines the
moment as positive and the shear force as positive. This is the same convention that was
used in the Bernoulli-Euler system.
61
Page 78
Y
MsiCO2
YdZ
^xCO^dZ
? Z
Figure 3.2.8: Free Body Diagram of Shaft Element in Y-Z Plane Using Timoshenko
Beam Theory
For a circular shaft section, Isy is calculated as
2 , T 2
I.y= ms'i(3a +Li } (3.2.95)
To better understand the kinematics of shear deformation, the following diagrams of
differential beam elements are presented.
62
Page 79
Y
A
b'
X& -? z
Figure 3.2.9: Shaft Element in Y-Z Plane Subjected to Bending Moment
Figure 3.2.9 depicts a shaft element subjected to the bending moment Mx = EI(d9y/dZ).
Lineb'
is parallel to the Z axis and is along the line of an undeformed shaft element. Line
a'
is perpendicular to the cross-sectional area of the shaft element. Using Bemoulli-Euler
Beam Theory, only the bending of the shaft is considered. When shear deformation is
included, as in the Timoshenko Beam Theory, the deformation is depicted in figure
3.2.10.
63
Page 80
Y
X& + Z
Figure 3.2.10: Shaft Element in Y-Z Plane Subjected to BendingMoment and Shear
Deformation
The orientation of the cross-sectional area of the beam is rotated 9y due to bending.
Shear deformation changes the orientation of the beam, but does not change the
orientation of the cross section of the beam. The cross section will remain unchanged at
angle 9y. Vy/GAS is the change in slope of the shaft element due to shear deformation.
Note that the displacement resulting from shear deformation is assumed positive. The
following relationship can be deduced from figure 3.2.10.
^ -
+ e,GAC dZ
dY
Vv = GAJ + 9yy HdZ y.
(3.2.96)
(3.2.97)
The equilibrium conditions that exist for a differential element can be depicted as in figure
3.2.11.
64
Page 81
Y
AV-
Ms,iCO2
YdZ
X&
Mx+dMx
Vy+dVy
-* Z
Figure 3.2. 1 1 : Free Body Diagram ofDifferential Element of Shaft in Y-Z Plane
Summing the moments acting on the differential element to zero results in
*T)XM=0: M* + dM*"
M* + Is,xO)2exdZ-
Vyy-
Vyy-
dVyy= 0
(3.2.98)
With the last term neglected, since it is infinitesimal, equation 3.2.98 reduces to
dMx + Is xco2
9X dZ -
Vy dZ = 0
Mi = _i^ex+ vydZ
(3.2.99)
(3.2.100)
Summing the forces acting on thedifferential element to zero results in
65
Page 82
+ T F = 0:-Vy
+ Vy + dVy + ms>ico2
Y dZ =0 (3.2.101)
which can be rearranged as
dVydZ
= -
ms^ co Y (3.2.102)
Following the same procedure used to derive the differential equation 3.2.52, it follows
that the displacement, Y, must satisfy the differential equation 3.2.103.
dZ4
+ COms,i Is.x^l d2Y
GAC EI
msi co
dZ^ EI
2 h n2 "l1,.T CO
s,x. i y = 0
The parameter substitutions
V GA. J
(3.2.103)
Qv =Is,x
co2L2
EIR =
2 T 2
ms co L
GACS =
ms co L
EI
"
can be made, resulting in the equation(3.2.104)
d4Y (R + Qy) d2Y (RQy-S)+
l + - = 0dZ4 L2 dZ2 L4
(3.2.105)
Equation 3.2.105 is also an ordinary differential equation with constant coefficients.
The ordinary differentialequations in X and Y are only distinct from one another in terms
ofmass moments of inertia, Is, and the area moments of inertia, I. These parameters are
the same in both planes for a shaft with circular cross-section.
66
Page 83
The solution to the ordinary differential equation 3.2.105 is of the form
Y = Ce^z/L Upon substitution ofY =Ce^ into the differential equation 3.2.105, the
following characteristic equation is deduced.
X4
+ (R +Qy)X2
+ (RQy- S) = 0 (3.2.106)
The roots to the characteristic equation 3.2.106 are
X3 and iX4
and are found from
-3,4= VI--(R+Qy) VF (R -
Qy)z
+ s (3.2.107)
After numerous substitutions and differentiations, as was conducted for the X-Z plane,
the equations for the displacement, slope, moment and shear can be found.
-Y =
1
msco
X3As sinh
5
L
+ A< cosh
( Z\
V LJ
A^
-An sm'
LKa
, L)+
A^
Ao cos8
L
( AXa
I L)
(3.2.108)
9y=
2 T 2
ms co L
(R + Xf)
(R - XZA)
A5 cosh
I 3lJ+ A6 sinh
An COS X4LJ
+ Ac sin Xa
I 3L.
ZYI
LJ
(3.2.109)
67
Page 84
Mx =
EI
2,2
ms co L
( R + Xf )
( R -
Xl )
a^ . . r. z^
A5r
sinh^ja
** .(. Z^+ A6 cosh IX3-^
+
_A7- sm ^X4
r;
X4+ Ac cos
8L
X4^
I LjL
(3.2.110)
Vy= A5 cosh
v LJ+ A^ sinh
( Z
X37;v Ly
+ An cos
/
Xa + As sin X-^lVhL)
(3.2.111)
The negative Y is required to make the equation 3.2.108 of the same form as the
displacement equation 3.2.78 in the X-Z plane. It is also consistent with sign
conventions in the Bernoulli-Euler Theory. Both beam theories utilize the same point
matrix; therefore, their sign conventions must be consistent.
In matrix form, the displacement, slope, moment and shear equations are expressed as
68
Page 85
1
<C r- OO
< < < <r-
N|_3
n!-j n|j
<<GO
o /,
N
CO c
'go
Un|j
O
U **
*N
<<
-*
<*
-1
CN
3^
CN-*
c<
1
J
3^
CNT
i
-J c
'vi
at
v- >
6
n|-jr ~^ r "
*
n|j n|-j <<
*-a-
* '
*^ *< e* s **. ^
GOs ~s
cCO
O Nl-JS "N
CO u<-<
^
:/*
"N?*
"N**
-J
CN
3^
r-)-*t
<<
1
JCN
3
CNT,
<<
1
CN
3^
GO
ou
u. Bet
J H1
n|_)
N|J n|jen
en
e<
<<^e:
x:CO
, s
.C
J= O n|-jCO c
CO
um
O/*
'S<-i
eny N
/**
"Nx:
CN -m
CN
CNen
<<
CNen-4 c
en
3^ + 3^ + 3*CO
a:E
5
N|J
n|-jen
en
en<<
.C
B
CO
C
CO
O /- V^i
"go o en
ri
' **'
p"" .C
CN
CNO
o*
CNcn
**s
CO
O
en 3
E
+
a:3^ + s
E
u
> s. 1
II
>
1 >
(N
69
Page 86
This matrix is of the form
{Y(z)} = [C(z)] [A]
Applying the boundary conditions in the local coordinate system results in
(3.2.113)
{Y}Li = [C(L)][C(0)]-l{Y}i.1 (3.2.114)
where [C(L)] [C(O)]"1 is the field matrix, [Fy].
The [Fx] and [Fy] field matrices will be formulated into an overall 17x17 general field
matrix that will be of the form
Sx.cR
^x,s
Sy,c=
*-*y,s
.1
. i-1
Fx] 0 0 0 Sx,c
0 [Fx] 0 0 0 ^x,s
0 0 [FyJ 0 0 Sy,c
0 0 0 [Fy] 0 ^y,s
0 0 0 0 1. 1
(3.2.115)
where'c'
represents cos(cot) and's'
represents sin(cot).
This allows applied forces and moments on the system to be placed in the last column in
the most general form as terms containing sin(cot) and cos(cot), respectively.
70
Page 87
3.3 Disk Motion
The dynamics of a disk ormass that will influence the natural frequencies and the forced
response of the system is embodied in the point matrix [P]. The point matrix is
comprised of the gyroscopic couple, whirl and a forcing function. The gyroscopic
couple is deduced using the moment equilibrium equations, whirl is defined by the
centrifugal force and the forcing function is defined by the mass unbalance.
The point matrix is formulated by determining all the forces and moments acting on the
disk. These forces and moments, along with the displacements and rotations of the disk
due to the forces and moments, are placed in a matrix form that relates state vectors { S }Lj
and {S}Rj. These state vectors, which contain displacement, slope, moment and shear,
are of the same form as the state vectors that are related through field matrices. This form
is consistent with the formulations of both the Timoshenko and Bemoulli-Euler field
matrices. The final form of the point matrix will be as a general 17x17 matrix.
Historically, the gyroscopic couple has not always been included in rotor dynamic
analyses, due to the additional complexity that the couple presents in a model. The
general effect of the couple tends to straighten the shaft, thereby stiffening it, resulting in
a higher natural frequency. A greater accuracy will result in determining the natural
frequencies, if the gyroscopic couple is included in an analysis.
71
Page 88
3.3.1 Moment Equilibrium Equations Relating to
Development of Gyroscopic Couple and
Rotatory Inertia
The motion of the disk resulting from rotatory inertia and gyroscopic couple will be
determined using rigid body dynamics. A general study of this motion is presented by
Fliigge [16]. The disk motion will be defined with a positive rotation about the Y axis
and a positive rotation about the X axis. This sign convention is identical to the sign
convention utilized in formulating the field matrices.
Two coordinate systems will be defined to describe the motion of the disk. (See figure
3.3.1) The local coordinate system, which will be referred to as the(XYZ)'
coordinate
system, is attached to the spinning disk and is defined by the unit vectors
el5 $2, and e3
In general terms, the absolute velocity of the disk in this coordinate system is
_AAA
<0(xyz)' =
l ei + w2 e2 + co3 e3 (3.3.1)
or more specifically
(xyz)' = y ei + 6X e2 + Q e3 (3.3.2)
where 9 = angular velocity of the disk about theX'
direction
9 = angular velocity of the disk aboutthe
Y'
direction
CI = spin velocity of thedisk about the Z direction
72
Page 89
Y
Figure 3.3.1: Local and Global Coordinate Systems ForWhirling Disk
The inertial coordinate system, which will be referred to as the XYZ coordinate system to
distinguish it from the local coordinate system, is parallel to the undeformed shaft and is
defined by the unit vectors
i,j and k
The angular velocity of theXYZ coordinate system is
^xvz= ev * + ex J'xyz (3.3.3)
Note that the disk spins about theZ'
axis and whirls about the Z axis.
The angular momentum, H, of the disk is the product of the mass moment of inertia of
the disk and the angular velocity. It can be written as
73
Page 90
H = [I] [co](xyz). =
It.l 0 "cof
0 It.2 0 co2
0 o Ip. co3
(3.3.4)
where Itj, It2 and Ip are principal mass moments of inertia about the center of mass of
the disk.
Simplified, equation 3.3.4 becomes
H = Iu co-&i + It2 co2 e2 + Ip co3 e3 (3.3.5)
It,l> It,2 m& Ip are me principal mass moments of inertia of the disk about the center of
mass of the disk. (See figure 3.3.11)
The moment equation can now be written by taking the derivative of the angular
momentum equation 3.3.5 with respect to time.
M = H = Itl co- e- + It>2 C02 e2 + Ip co3 e3 + Iuco- e- + It2 ^ e2 + Ip co3 e3
(3.3.6)
Equation 3.3.6 relates the time rate of change of the angular momentum about the center
ofmass to the resultant applied moment about the center ofmass.
Previously, the angular velocity components in the(XYZ)'
system had been defined as
coj=
9y <2= ex ($-1 = Q (3.3.7)
The time derivative of the components ofangular velocity are
CO!= 9, co2
= 9X coo = 0 (3.3.8)
74
Page 91
The spin velocity, Q, is a constant, therefore its time derivative is equal to zero.
With these substitutions, the moment equation becomes
M =
Iu 9y li + \2 9X ^ + It>1 9ye- + Iu 9X l2 + Ip Q e3
(3.3.9)
A transition matrix can be developed to relate the unit vectors
el5 e2, and e3
to the unit vectors
i, j and k
The final expression for the gyroscopic couple must be in the XYZ coordinate system,
since the field matrices have already been defined in that system.
3.3.1.1 Transition Matrix
The(XYZ)'
coordinate system is mapped from the XYZ coordinate system by a positive
rotation about the Y axis and a positive rotation about the X axis. The final result
defining the gyroscopic couple is not affected by the sequence of these rotations. To
form the transition matrices the system may first be rotated about either the Y axis or the
X axis. The analysis will arbitrarily rotate about the Y axis first. The following free
body diagram shows the rotation of the(XYZ)'
coordinate system about the Y axis.
75
Page 92
*? z
Figure 3.3.2: Relationship Between Global and Local Coordinate Systems in the
X-Z Plane
Through vector resolution, the following equations can be written
e- =
e3=
e2
cos 9X i - sin 9X k
9X k + sin 9X i= cos 9X
= j
(3.3.10)
(3.3.11)
(3.3.12)
The rotation of the(XYZ)'
coordinate system about the X axis is shown in the following
free body diagram.
76
Page 93
Figure 3.3.3: Relationship Between Global and Local Coordinate Systems in the Y-Z
Plane
Through vector resolution, the following equations can be written
ei = 1
e2= cos 9y j + sin 9y k
A A A
e3= cos 9y k
- sin 9y j
(3.3.13)
(3.3.14)
(3.3.15)
In matrix form, the unit vectors
e-, e2, and e3
can be related to the unit vectors
i,j and k
77
Page 94
as
VZ'J
\ = [Tj [Ty] \ V,
ZJ
(3.3.16)
where [Tx] and [Ty] are the transition matrices relating the(XYZ)'
coordinate system to
the XYZ coordinate system. The order of the multiplication of these matrices must be
strictly observed to remain consistent with the rotational sequence previously chosen on
page 75.
The matrix equation can be expanded by utilizing the equations formulated from the
diagrams in figures 3.3.2 and 3.3.3. These equations define the transition matrices.
Vx' 1 0 0 c9x 0 -s9x
vy 0 C9y S9y 0 1 0
vz.. 0"S9y C9y_ s9x 0 c9x
V,
ZJ
(3.3.17)
Cos9 and sinO are represented by the letters'c'
and 's'.
Multiplying, equation 3.3.17 becomes
[V](xyz)- =
c9, 0 -s9,
s9y-s9x c9y s9y-c9x
c9v-s9x -s9v c9v-c9x
[V](xyz) (3.3.18)
Equation 3.3.18 is the overall transition matrix relating the XYZ coordinate system to the
(XYZ)'
coordinate system
78
Page 95
The time derivative of the unit vectors ]_, &2 and3 must be calculated for substitution
into the moment equations. The overall transition matrix will be utilized to define ej. In
general
1
ei=
^xyz x ei (3.3.19)
coxyz is incorporated in equation 3.3.19 instead of co, ysince the final form of the time
derivatives must be in terms of i, j and k unit vectors.
Substituting equation 3.3.3 into equation 3.3.19 obtains the following expressions for
the time derivatives of the unit vectors ej , 62 and 63.
*
ej=
(9y i + 9X j) x e- =
(8y i + 9X j) x (c9x i -
s9x k) (3.3.20)
e- = 9y s9x j-
9X c9x k -
9X s9x i (3.3.21)
e2= (e i + 9X j) x e2
=
(6y i + 9X j) x (s9y s9x i +c9y j + s9y c0x k)
.. (3.3.22)
e2=
9y c0y k -
9y s0y c9x j-
0X s8y s9x k + 9X sGy c9x i (3.3.23)
e2= 9X s9y c9x i -
0y s9y c9x j + (0y c9y-
8X s9y s9x) k (3.3.24)
e3= (9y i + 9X j) x e3
= (9y i + 9X j) x (c6y s9x i -
s9y j + c9y c0x k)
. . . . (3-3-25)
e3=
-9y s9y k -
9y c9y c0x j-
8X c0y s9x k + 0X c0y c6x 1 (3 3 26)
e3= 9X c9y c9x i -
9y c9y c9x j-
(9y s0y + 9X c9y s9x) k (3.3.27)
The unit vectors and their time derivativescan now be substituted into the moment
equation 3.3.9 to obtain
79
Page 96
M =
Iu 9y (c9x i -
s9x k) + It 2 ex (s9y s9x i + c9y j +s9y c9x k) +
It;1 9y (-9X s9x i + 9y s9x j -
9X c9x k) +
Jt,2 Qx (9x s9y c9x i -
9y s9y c9x j + (9y c9y-
9X s9y s9x) k)
Ip Q (9X c9y c9x i -
9y c9y c9x j-
(9y s9y + 9X c9y s9x) k)
(3.3.28)
Rearranging the terms according to i, j and k, the moment equation 3.3.28 becomes
M = {Iu 9y c0x + Itf2 9X s9y s0x-
It>1 9y 9X s0x + It<2 9X s9y c0x +
Ip Q 9X c9y c9x} i + {It 2 9X c9y+ It>1 0y s9x
-
It 2 9X 9y s9y c0x-
Ip Q. 9y c9y c0x} j + {-Itl 9y s9x + It2 0X s9y c0x-
Il?1 0y 0X c9x +
Iti2 9X 9y c9y-
It 2 9X s9y s9x-
Ip a 9y s9y-
Ip Q. 9X c9y s9x} k
(3.3.29)
For small angles of 9, cosine 9 can be considered to be unity and sine 9 can be
considered to be 9.
M = {Iu 9y + It>2 9X 9y 9X-
It>1 0y 8X 9X + It>2 9X 8y + Ip Q 9X} i +
Uu^x + It,10y6x"
It,2ex0y0y"
Ip " 6y} j + {-It>1 9y 0X +
It,2 0x 9y- It,l0yex + Jt,2 0x 0y
_
It, 2 6x 0y 0x_
Ip O 9y 9y-
Ip ^ 6X 0x> k(3.3.30)
Since small slope angles have been assumed, which would be consistent with elastic
deformation of a shaft, any angle multiplied by another angle or its time derivative will be
very small. The non-linear terms in equation 3.3.30 consist of these small terms and
therefore, can be neglected.
The final form of the moment equation 3.3.30 becomes
80
Page 97
M = du ey + ip a ex) 1 + (it>2 9X-
ip a ey) j (3.3.31)
X
' It 1vK.A
t,l y
3 -?z
Y I QGp y
Figure 3.3.4: Rotatory Inertias and Gyroscopic Couples Acting on Disk
The moment, M, is equal to the internal moments acting on the disk. Equal, but
opposite, moments occur on the shaftat the location of a disk. The terms IpQ0xi and
-IpQ9yfare referred to as gyroscopic couples. The terms It,l9yi and It,2exJ are referred
to as rotatory inertia. The gyroscopic couple tends to raise the natural frequency of the
system while the rotatory inertia tends tolower the natural frequency.
81
Page 98
3.3.2 Mass Unbalance
A forcing function will be defined for the system as a mass unbalance on the disk. This
function will be placed in the 17th column of the shear force row in the point matrix.
Other types of forcing functions can also be placed in the 17th column, such as an applied
moment, which would be due to a skewed disk on the shaft [13]. The mass unbalance is
defined in the following diagram of a disk.
Y
Figure 3.3.5: Relationship Between Center ofGravity of a Disk and Center ofRotation
82
Page 99
The parameters are defined as
G = center of gravity of the disk
ej= distance from the center of rotation to the center of gravity at station i
Pi = the angle between the center of gravity and the y axis at station i
mj= total mass of disk, including mass due to imbalance, at station i
Uyj = mass unbalance in the y direction at station i
Ux>j = mass unbalance in the x direction at station i
The mass unbalance can be decomposed as
uy,i =
mi ei cs p\ (3.3.32)
ux,i =
mi ei sin Pi (3.3.33)
The forces acting on a disk due to a mass unbalance and their relationship to the spin
velocity are presented in the following diagram. The response due to these forces can be
determined at time, t.
83
Page 100
AX"
Y"
Figure 3.3.6: Forces Acting on Disk Due to Mass Unbalance
The(XYZ)"
coordinate system is a non-rotating local coordinate system utilized as a
reference for the rotating(XYZ)'
coordinate system. The centrifugal force due to the
mass unbalance and whirl frequency is co^U. The spin speed of the disk is Q. For
synchronous motion, co equals Q.
84
Page 101
3.3.3 Formulation of Point Matrices
The forces acting on the disk in the x direction are defined as follows:
X
A
Y(2>
(Uxicos Qt - UyisinQt) co
tMittX:
t...RL
Vx,i 1V .
x,i
TX:
**? Z
Figure 3.3.7: Free Body Diagram of Forces Acting on Disk in the X-Z Plane
vx,i =-vx,i
+ mi > Xi + (Ux,i cos Qt-
Uy>i sin Qt)co^
(3.3.34)
whereMico2
Xi is the centrifugal force that defines disk whirl.
85
Page 102
The forces acting on the disk in theY direction can be defined as follows:
X<&
(Uv. cos Qt + U sinQt)co2
j i ai
tMi CO
y
t
V. !!V.R
TY,
-? Z
Figure 3.3.8: Free Body Diagram ofForces Acting on Disk in the Y-Z Plane
V?.i = Vy,i~
mim2
Yi-
(Uy.i cos Qt + U".i sin Qt) w (3-3.35)
From themoment equation 3.3.31, the moments acting on the shaft at the disk are
M = (-Iu 9y-
Ip n 9X) i + (-Itt2 0x+ lp Q 9y) j (3.3.36)
86
Page 103
In diagram form, equation 3.3.36 can be presented as
and
M^ = M^--
Ip a 9y + It,2 9X
M*i = Mifi + It>1 9y + Ip Q 9X
(3.3.37)
(3.3.38)
X
A
Y@>-
V0y
My\
i eh x
M.R
-** z
Figure 3.3.9: Free Body Diagram ofMoments Acting on Disk in X-Z Plane
87
Page 104
x<&
. e1ti wy
M,
ip"ex
M
R
-? Z
Figure 3.3.10: Free Body Diagram ofMoments Acting on Disk in Y-Z Plane
The displacement, slope, moment and shear can be written for the point matrix in general
terms as functions of cos cot and sin cot. The sign conventions that have been defined
throughout the analysis are maintained in these equations. They can be expressed as
X = Xc cos cot + Xs sin cot
9X = 9XC cos cot + 9X ssin cot
My= Myc cos cot + Mys sin cot
-Vx=-Vx c
cos cot-
Vx ssin cot
(3.3.39)
(3.3.40)
(3.3.41)
(3.3.42)
88
Page 105
and
-Y =
-Yc cos cot -
Ys sin cot (3.3.43)
9y=
9yc cos cot + 9ys sin cot (3.3.44)
Mx = Mxc cos cot + Mys sin cot (3.3.45)
Vy = Vy,c cos cot + Vys sin cot (3.3.46)
The first and second derivative of the slope equations 3.3.40 and 3.3.44 must be found
for substitution into the moment equation 3.3.36.
9X =
-9X cco sin cot + 9X s
co cos cot (3.3.47)
^x = -^xc*0 cos cot- 9xsco2sincot (3.3.48)
and
9y=
-9y cco sin cot + 9ys co cos cot (3.3.49)
"2 2
9y=
-9y cco cos cot
-
9y sco sin cot (3.3.50)
Substituting these equations into the moment equations 3.3.36, 3.3.41 and 3.3.45
Mx,i = Mxiccoscot + Mxissincot =
(3.3.51)
MX)ic cos cot + Mxis sin cot + Itl(6y,ic^ cos * ~
ey,is2
sin^ +
lp Q. (-6X;ic co sin cot + 6xis co cos cot)
89
Page 106
Myi = My4c cos cot + My is sin cot =
(3.3.52)
My4ccoscot + My^ sin cot -
lp SI (-ey4cco sin cot + 6yis co cos cot)-
I*,2 (-ex,ic cos cot -
6X jsco2
sin cot)
Separating these equations into terms containing sin cot and cos cot and dividing through
by these terms results in
Mx,ic =
M^ ic-
Iuco2
9y>ic + Ip SI co 9Xiis (3.3.53)
Mx,is = M^)is-
Itlco2
9y>is-
Ip SI co 9Xiic (3.3.54)
and
My>ic = My>ic-
Ip SI co 9y>is-
It>2 co2
9x>ic (3.3.55)
My>is = M^>is + Ip Q co 9y>ic-
Itt2co2
9xJs (3.3.56)
Substimting equations 3.3.39 and 3.3.43 for displacement into the force equations
3.3.34, 3.3.35, 3.3.42 and 3.3.46 results in
_V*- =
-Vxiccos cot
- Vxissin cot = (3.3.57)
-Vx ic cos cot
-
Vx is sin cot + mjco2
(Xc cos cot + Xs sin cot) +
(Ux j cosSit - Uy>i sin Cit)
co2
90
Page 107
and
7R 7R
VC i=
V*
- cos cot + Vy.i y,ic y,issin cot =
Vyic cos cot + Vy is sin cot -
mjco2
(-Yic cos cot
(Uyi cos Qt + Ux j sin Sit)co2
(3.3.58)
Yis sin cot)-
Separating these terms into groups containing sin cot and cos cot and dividing through by
these terms results in
-V
R
x,ic
-VR- =vX,1S
,j 2 v tt 2 I COS Qt
-V^)ic + mico2
Xi?c + UX)ico2
Vcos cot
,,l 2 ,. tt 2 f sinQt^
-Vx is + m;
coz
X} s
-
Uv j
coz
X'1S ' ' y'Uin cot ;
(3.3.59)
(3.3.60)
and
vR.vy,ic
VR
y,is
= V^,ic + mico2
YiiC-
Uy,ico2
^2 ( cos Qt
cos cot
= V^is + mico2
YiiS-
UXiico2̂ ( sin Qt
V sin cot J
(3.3.61)
(3.3.62)
Synchronous motion occurs when the whirl frequency equals the spin velocity. To
enforce synchronous motion, which simplifies the analysis, Q is equated to co.
Assuming synchronousmotion imposes restrictions on a system thatmay not be realistic,
thereby producinglimited results. Synchronous motion will be analyzed in this study
(along withnon-synchronous motion) in order to compare results to case studies.
The equations for displacement, slope,moment and shear force can now be placed into a
general 17x17 point matrix given asequation 3.3.63.
91
Page 108
>I
S c
a
oo
o o
3*<
C
o
O O
3
l
6/i
O o c
3*>
1
o c3 O o O o
c3 O o c3 O o
3c
o
<=3
E
oo
33_
T
O
o
^o
3
Eo
~
-o o
-
3
T
o
-3
e
'
. i
r*)a3a.
.
3CM
o
? 1
1
I 1
1
3
E
ii
M n o u uVt n V)_
~
u
ST sT -T i ^ > 1>>
<z> 2 >
92
Page 109
3.3.3.1 Gravitational Force
If a gravitational force, Fg, acts on the system, it would be included as a static force in
equation 3.3.59. Fg is dependent on the angular orientation of the shaft, with respect to a
horizontal surface as shown in the following diagram:
cosT
Figure 3.3.11: Gravitational Force Acting on an Angled System
Equation 3.3.59 can be rewritten as
_V- = -V + m; co X; r + Ux :
co"
VX,1C X,1C 1 i.c x,i
2 ( COS Qt
I COS COt J
-
HI; g COS T
where
Fe=
mi g
(3.3.64)
(3.3.65)
and g is thegravitational constant
93
Page 110
3.3.3.2 Mass Moments of Inertia
The mass moments of inertia for a thin diskwithout any mass unbalance are
Ip=4"miri2
T Lr2I = m:T:
l\ 41 l
I. = m:r:
t2 4i i
(3.3.66)
(3.3.67)
(3.3.68)
Figure 3.3.12: Mass Moments of Inertia For a DiskWithout Mass Unbalance
where It b It 2 an<^ Ip are tne n^ss moments of inertia about the center ofmass of the
disk.
The mass moments of inertia have to be modified to include the mass unbalance. The
mass unbalance will be considered to be a sphere, whose mass moment of inertia for all
three axes is (2/5) m,a2 (
'a'
is the radius of the sphere).
94
Page 111
In diagram form, the location of the mass unbalance, Mi,with respect to the center of
gravity, G, of the disk is
Y
Figure 3.3.13: Location ofMass Unbalance on a Disk
where M^ = mass of disk without imbalance
Mi = mass of imbalance
mi= total mass of disk at station i
The mass moments of inertia about thecenter of rotation can now be written using the
parallel axis theorem:
95
Page 112
Iu
It.2
h =
1 r\
= -
Md if + - Mia2
+ Mi [(ei + t>-) cospj]2
=
J Md if +
j Mja2
+ Mj [(ei + h-) sinp-]2
-
Md if + -
Mia2
+ Mj (ei +bj)2
(3.3.69)
(3.3.70)
(3.3.71)
96
Page 113
Chapter 4
RESULTS
The analysis presented in this investigation was formulated into a Fortran program. The
program can analyze non-synchronous and synchronous motion of a multi-disked,
variable shaft, simply supported system with overhangs. The model incorporates both
Bemoulli-Euler and Timoshenko Beam Theories to determine natural frequencies and
mode shapes of flexible rotating systems. The capabilities of the program also include
analysis of an associated forced response. The response due to mass unbalance and a
gravitational force are presented in the results. Data pertaining to particular examples was
input to the program to obtain the following results.
The mode shapes for the case study of three nested disks between two simple supports
are plotted in figure 4. 1 .3. The configuration of the case study is shown in figure 4.1.1.
h-H7^ 1 1 1 7\
Figure 4. 1 . 1 : Simply Supported ShaftWith Three Nested Disks
The mode shapes were configured using Bemoulli-Euler Beam Theory without
gyroscopic couple. The natural frequencies are 42.61 HZ, 169.28 HZ and 359.47 HZ.
Rao, who also utilized the Transfer Matrix Method, published results of 42.52 HZ,
168.83 HZ and 358.62 HZ for an identical system. The calculated results are within
0.3% of the results published by Rao [12].
The case study of threenested disks between two simple supports is repeated using the
Timoshenko Beam Theory. In this particular simulation, gyroscopic couple is not
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utilized. The mode shapes, which are plotted in figure 4.1.4, correspond to natural
frequencies of 40.05 HZ, 155.61 HZ and 325.10 HZ. As expected, the natural
frequencies are lower than those predicted by Bernoulli-Euler Theory due to the greater
flexibility of a Timoshenko Beam. The difference in magnitude between the
corresponding frequencies increases as the number of analyzed modes is increased. The
moment due to rotatory inertia of the shaft, which is included in Timoshenko Theory and
not in Bernoulli-Euler Theory, is a function of the slope of the shaft. This slope
increases in the higher order mode shapes.
The natural frequencies resulting from the Timoshenko and Bemoulli-Euler Beam
Theories with and without gyroscopic couple are plotted versus the aspect ratio, R/2L,
where R is the radius of the disk and L is the length of the overhang. The model which
corresponds to the data in figure 4.1.5 consists of a simply supported shaft with a single
overhanging mass. This model is depicted in figure 4.1.2.
r-
7\ 7V
T"
R
i_
Figure 4. 1.2: Simply Supported ShaftWith OverhangingMass
Synchronous motion was assumed when the gyroscopic couple was applied. The results
consist of values corresponding to forward whirl.
The results in figure 4.1.5 show that for low aspect ratios, the Timoshenko Theory
lowers the natural frequency due to an increased flexibility of the shaft. The gyroscopic
couple does not have an effect on either beam theories since the couple, which is a
function of the radial mass moment of inertia, is small for low aspect ratios and therefore,
is low compared to the stiffness of the shaft.For high aspect ratios, the gyroscopic
98
Page 115
couple dominates the motion of the system, even overriding the flexibility introduced by
the Timoshenko Beam. The couple raises the natural frequency since the direction of the
couple tends to straighten the shaft and thereby increasing the stiffness of the shaft.
Figure 4.1.6 illustrates non-synchronous motion for a simply supported shaft with a
single overhanging mass using Timoshenko Beam Theory. The whirl frequency and
rotational speed are non-dimensionalized and are plotted on the ordinate and abscissa
axes, respectively, for eight natural frequencies. The values plotted as a positive
rotational speed are physically meaningful. The negative rotational speeds, on the other
hand, are only plotted to aid in the development of the curves. The positive whirl
frequencies represent forward whirl, in which case the shaft spins and whirls in the same
direction. The negative whirl frequencies represent backward whirl, in which case the
shaft spins and whirls in opposite directions.
Information pertaining to synchronous motion can be obtained from figure 4.1.6. Two
lines are drawn to represent the coincidence of rotational speed and whirl frequency
(forward motion). A line is also drawn to indicate the equality of the rotational speed
with the negative of the whirl frequency (backward motion). The intersection of these
two lines with the natural frequency curves represent where synchronous motion will
occur. Forward and backward synchronous motion do not always occur at the same
rotational speed as is shown by the difference of points A and B.
Figure 4.1.7 plots the forced response due to a mass unbalance versus rotational speed
using Timoshenko and Bernoulli-Euler Beam Theories. The graphs are plotted for a
range surrounding the firstnatural frequency. The corresponding model consists of three
nested disks on two simple supports with the mass unbalance placed on the middle disk.
Gyroscopic couples are not included. The Bemoulli-Euler curve compares favorably in
shape and magnitude with a case study presented by Rao [12]. At 60 HZ, the response
of the case study wasapproximated as 0.00026 inches, while the results at 60 HZ were
computed to be 0.00027 inches. The response of the Timoshenko Beam below the
critical speed closely followsthe response of the Bemoulli-Euler Beam, but decays more
quickly than theBernoulli-Euler Beam above the critical speed.
The response due to a mass unbalance is plotted versus whirl frequency in figure 4.1.8
for a simplysupported shaft with an overhanging disk (aspect ratio
= 0.25). The mass
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Page 116
unbalance is placed on the disk. The analysis incorporated the Timoshenko Beam Theory
with gyroscopic couple. A rotational speed, which is needed for non-synchronous
motion, was arbitrarily chosen for the model to be 141 HZ. The graph spans the range
for the backward and forward whirl of the first natural frequency. The forced response
tends to infinity at the backward and forward whirls.
In figure 4.1.9, the forced response due to a static gravitational force acting on a
Timoshenko Beam is plotted versus rotational speed. The plot spans the frequency range
that encompasses the first two natural frequencies of the system. The gyroscopic couple
was not applied to the model. The model consists of a simply supported shaft with an
overhanging disk and an aspect ratio equal to 0.1. The four curves show the effect of the
orientation of the shaft on the forced response. The angle between the shaft and the
horizontal surface was varied from0
to 45. The largest response occurred at0
and the
smallest response occurred at 45. The change in the response from the previous angle
increased as the inclination angle is increased.
The response due to a gravitational force was verified by two analyses. In the first
analysis, the static displacement was calculated due to a gravitational force acting on the
disk. The displacement, as predicted, corresponds to the forced response at zero HZ. In
the second model, the amplitude was calculated for a spring-mass system subjected to a
constant force. This amplitude is a function of 1/co2, where co is the natural frequency,
indicating that the response decreases as the frequency increases. This relationship is
shown in figure 4.1.9, with the presence of a smaller amplitude at 120 HZ than at 10 HZ.
100
Page 117
4.1 Figures
H - 42.6; HZ
0.0
-r
10.0 20.0 30.0
DISTANCE ALONG SHAFT
40. 0
Figure 4.1.3: Case Study Utilizing Bernoulli-Euler Beam Theory andthe TransferMatrix MethodWithout Gyroscopic Couple
101
Page 118
H - 40.35 HZ
D.O 20.0 30.0
DISTANCE ALONG SHAFT
50.0
Figure 4. 1 .4: Case Study Utilizing Timoshenko Beam Theory and theTransferMatrixMethodWithout Gyroscopic Couple
102
Page 119
0.0 0.2 0.4 0.E
ASPECT RATIO, R/2L
o.a
Figure 4. 1.5: Natural Frequencies For Timoshenko and Bemoulli-Euler Beam Theories
With andWithout Gyroscopic Couple For an Overhanging Disk
103
Page 120
-6.0
T
-4.0 -2.0 0.0 2.0
ROTATIONAL SPEED PARAMETER
6.0
Figure 4. 1.6: Non-Synchronous Motion For an Overhanging Disk
Utilizing Timoshenko Beam Theory
104
Page 121
TIMOSHENKO -
BERNOULLI-EULER -
10.3 20.0 30.0 40.0 50.:
ROTATIONAL. SPEED, MI
ez.o
Figure 4. 1.7: Case Study Utilizing Bemoulli-Euler andTimoshenko Beam Theories to
Determine Forced Response Due toMass Unbalance Without Gyroscopic
Couple
105
Page 122
-60.0 -40. 0
-r
-200.0
WHIRL FREQUENCY, HZ
Figure 4 18* Forced Response ofOverhanging Disk
Due toMass Unbalancengmc . .
wuhGyroopicQ^ie UtilizingTimoshenko Beam Theory
106
Page 123
0 DEGREES .
15 DEGREES f { -* +
30 DEGREESA A &-
45 DEGREES-Q
Q-
o.o 20.0 40.0 BD-3
ROTATIONAL SPEED,
100.0 123.3
HZ
Figure 4.1.9: Effect ofGravitational Force on Overhanging Disk Vs.
Orientation of Shaft
107
Page 124
Chapter 5
CONCLUSIONS
A general method for analyzing flexible rotating systems using the Transfer Matrix
Method in conjunction with the Timoshenko Beam Theory was presented. A rotating
system has been completely described in terms of its non-synchronous motion.
Synchronous motion is derived from information on non-synchronous motion, along
with backward and forward whirl frequencies. Forced response due to a mass unbalance
or a gravitational force have shown to provide accurate information about the
displacement of disks within a frequency range spanning the natural frequencies. The
results have been verified to be accurate and have shown improved estimates over
classical theory for determining natural frequencies and forced responses.
At low aspect ratios, the Timoshenko Beam Theory lowers the natural frequency of the
system, compared to the Bernoulli-Euler Beam Theory, due to its greater flexibility. The
increase in inertia, due to the gyroscopic couple, dominates the response of the system at
high aspect ratios, even overriding the flexibility of the Timoshenko Beam. The rotatory
inertia reduces the natural frequency by increasing the flexibility of the system. The net
result of the gyroscopic couple and the rotatory inertia is to increase the natural frequency
of the system at high aspect ratios. The gyroscopic couple does not have an effect on
either beam theory at low aspect ratios due to the small inertias of the disk. Using
Timoshenko Beam Theory, in conjunction with the gyroscopic couple, will provide the
most accurate solution independent of the aspect ratio.
The analytical model, since it was written in general terms using the Transfer Matrix
Method, can encompass numerous rotating systems, aside from those discussed in this
investigation. The formulation can be utilized to analyze a system subjected to disk
skew, which enforces a moment on thesystem This moment, which would be placed in
the last column of the point matrix, is referred to by Benson [13] as the "active
gyroscopic couple". Bearing housings that are not truly simple supportscan be modeled
by modifying the stiffness of the fieldmatrices surrounding the support Systems that do
not have disks in each shaft section can be modeled by placing a single, very small mass
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Page 125
in this section. This small mass will appear analytically as a section of shaft. Systems
that have clamped supports can be effectively modeled by placing the two simple
supports very close together.
Even though the computer program that automates the formulation and analysis is quite
extensive, computer time to run the program is very short. It requires only a few CPU
minutes to determine the natural frequencies and mode shapes of a system. The Nastran
finite element code, which currently cannot analyze a whirling disk, would require
approximately 10 CPU minutes to determine the natural frequencies and mode shapes of
a simplifiedmodel. In addition, Nastran charges licensing fees for the use of its code.
Developing the analysis of a rotating system in terms of a general model provides
complete flexibility in applying the analysis to actual systems. Utilizing the Transfer
Matrix Method allows for modification of stiffness and mass matrices at discrete
locations, as may be required for a particular system. The Timoshenko Beam Theory,
used in combination with the gyroscopic couple, provides realistic results for any system,
independent of the aspect ratio. These features, along with the short run time, positions
this formulation to be a viable and important tool for design and analysis of high-speed
flexible rotating systems.
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Page 126
Chapter 6
RECOMMENDATIONS
Analyzing a rotating system using the Transfer Matrix Method in conjunction with
Timoshenko Beam Theory can be expanded from the present study. The non-linear and
"negligible"
terms in the Timoshenko Beam Theory and the moment equation could be
included in the analysis. These terms would include axial forces imposed by the disk on
the shaft. An analysis of the supports, including flexible bearings and foundation, could
be integrated into this analysis. This information would include not only the stiffness of
the supports, but also the damping of the supports and the foundation.
Also, a study should be conducted to correlate the results from experimental modal
analysis to the results from the analytical modal analysis as formulated in this
investigation.
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Page 127
REFERENCES
1 Rankine, W.J.M., "On the Centrifugal Force of RotatingShafts,"
TheEngineer . April. 1 869
& 6
2 Rayleigh, Lord, Theory of Sound, Dover Publications, New York, 1945.
3 Timoshenko, S., "On the Correction For Shear of the Differential EquationFor Transverse Vibrations of Prismatic
Bars,"
Phil. Mag.. Ser. 6, Vol. 41, p. 744-746,1921.
*" y
4 Jeffcott, H.H., "The Lateral Vibration of Loaded Shafts in the Neighborhoodof aWhirling Speed The Effect ofWant of
Balance."
Phil. Mag. 37 . 1919.
5 Prohl, M. A., "A General Method for Calculating Critical Speeds of FlexibleRotors,"
ASME. 1945, pp. A142-A148.
6 Pestel, E.C. and Leckie, F.A., Matrix Methods in Elastomechanics ,
McGraw-Hill Book Co., 1963.
7 Tse, F.S., Morse, I.E. and Hinkle, R.T., Mechanical Vibrations ,Allyn
and Bacon, Boston, 1963.
8 Eshleman, R.L. and Eubanks, R.A., "On the Critical Speeds of a ContinuousRotor,"
J. Engr. Industry. Trans. ASME. 91 (4B), pp. 1180-1188, Nov., 1969.
9 Ruhl, R.L., "Dynamics of Distributed Parameter Rotor Systems: Transfer
Matrix and Finite ElementTechniques,"
Ph.D. Thesis, Cornell University, 1970,
UniversityMicrofilms No. 70-12, 646.
10 Nicholas, J.C, Gunter, E.J. and Allaire, P.E., "Effect of Residual Shaft
Bow on Unbalance Response and Balancing of a Single Mass FlexibleRotor,"
J. Engr.
for Power ,Vol. 98, No. 2, April, 1976.
11 Nelson, H.D., "A Finite Rotating Shaft Element Using Timoshenko BeamTheory,"
Arizona State University, Sept., 1977, Engineering Research Center No.
-R-77023.
12 Rao, J. S., Rotor Dynamics , Wiley Eastern Limited, New Dehli, 1983.
13 Benson, R. C, "The Steady-State Response of a Cantilevered Rotor With
Skew andMassUnbalances,"
ASME , Oct., 1983, Vol. 105, pp. 456-460.
14 Rieger, Neville F., Vibrations ofRotating Machinery,4th ed.,
The Vibration Institute, Clarendon Hills, Illinois, 1984.
15 Washizu, K., Variational Methods in Elasticity and Plasticity , Pergamon
Press, 1968.
111
Page 128
16 Crandall, S.H., "Rotating and ReciprocatingMachines,"
Handbook of
Engineering Mechanics , Fliigge, W., ed., McGraw-Hill Book Co., New York, 1962.
17 Vierck, Robert K., Vibration Analysis 2nd ed., Harper and Row, 1979.
18 Meriam, J.L., Dynamics. 2nd ed., J. Wiley & Sons, Inc., 1975.
112
Page 129
APPENDIX
Flowcharts for applying the Transfer Matrix Method:
The system configuration is for a multi-disked, simply supported, overhung shaft.
b
TV
I
TK"
m
a+1 a+2 a+3 b+1
Figure A. 1: System Configuration
113
Page 130
apply station 1
boundary
conditions, [S}o
Imultiply by field
matrix [F]
multiply by point
matrix [P]
Imultiply by field
matrix [F]
Irewrite
0=f(w)
Iadd bearing reaction
force, Pa
multiply by field
matrix [F]
Imultiply by point
matrix [P]
J
1continue for 1 to
a-1 disks
J
icontinue for a
to b-1 disks
J
Figure A.2: Flowchart Deriving Global TransferMatrix (continued)
114
Page 131
Imultiply by field
matrix [F]
1rewrite
Pa=f(w)
Iadd bearing reaction
force, Pb
EZmultiply by field
matrix [F]
Imultiply by point
matrix [P]
Ithe global Transfer
Matrix [U] is
obtained
Figure A.2: Continued From Previous Page
115
Page 132
global Transfer
Matrix [U]
Iobtain subset of [U]
formulated from
equations with
moment & shear = 0
Iinitiate incremental
guesses for
natural frequency
IDoes the det[U] subset=0
or has it changed direction?
Iequals
zero
I
Ichanged
direction
found
natural
frequency
I
I
17th column equals
zero; no forcing
function present
co is the only
unknown in
the subset
no
interpolate between last
two guesses for co until
det [U] subset = 0
determine
mode shape
find next
natural frequency
Figure A.3: Flowchart Deriving Natural Frequencies From Global Transfer Matrix
116
Page 133
natural frequency
has been found
Iset displacements
at station 1 =
unity
Icalculate Pb, Pa, 0
all functions of w
Iapply these values
to the Transfer Matrix
calculate displacements
at each station
Ire-normalize the
displacements
Iplot results which
are defined as a
mode shape
Figure A.4: Flowchart DerivingMode Shapes
117
Page 134
global Transfer
Matrix [U]
calculate Wo and
Pb from the
particular solution
Icalculate Pa and 9
as a function of Wo
apply these values
to the Transfer Matrix
calculate displacements
at each station
Iplot results of
displacement
vs. frequency
substitute in frequencyat which forced
response is to be
calculated
repeat for
frequencyrange of interest
Figure A.5: Flowchart Deriving Forced Response
118
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