a unified matrix formulation for the unbalance response of a flex
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Unified matrix approach for rotordynamicsBy RIT Scholar WorksTRANSCRIPT
Rochester Institute of TechnologyRIT Scholar Works
Theses Thesis/Dissertation Collections
1974
A Unified Matrix Formulation for the UnbalanceResponse of a Flexible Rotor in Fluid-Film BearingsCharles Thomas Jr
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Recommended CitationThomas, Charles Jr, "A Unified Matrix Formulation for the Unbalance Response of a Flexible Rotor in Fluid-Film Bearings" (1974).Thesis. Rochester Institute of Technology. Accessed from
A UNIFIED MATRIX FORMULATION FOR THE UNBALANCE
RESPONSE OF A FLEXIBLE ROTOR IN FLUID-FILM BEARINGS
Approved by,
by
Charles B. Thomas Jr.
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
. in
Mechanical Engineering
Prof. Name Illegible (Thesis Advisor)
Prof. Name Illegible (External Reviewer)
Prof. Name Illegible
Prof. William L. Halbleib
Prof. Name Illegible (Department Head)
DEPARTMENT OF MECHANICAL ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
July, 1974
.$
-3
.ft ACKNOWLEDGEMENTS
The author takes this opportunity to express his apprecia
tion to those who have assisted him in the course of this thesis
research and in particular:
To Professor N.F. Rieger, the author's thesis advisor, for
his continued interest, insight, and direction of the author's
thesis research. Also, for his contributions to the development
of the author's professional career.
To the Department of Mechanical Engineering for the support
ing funds, as a research assistant, which made this thesis pos
sible.
To Professor W. Halbleib, member of the author's thesis
committee, for his insight into the theoretical presentation of
this thesis and for his encouragement and guidance in preparing
the final manuscript.
To Professor W. Walters, member of the author's thesis
committee, for his insight, and suggestions concerning this
investigation.
To Professor B. Karlekar, Mech. Eng. Dept., RIT, for his
interest and suggestions concerning the final manuscript.
To Professor J. F. Booker, Mech. Eng. Dept., Cornell
University, for his interest and time in reviewing this
investigation.
To the author's wife, Sandra, for her continued encourage
ment during the author's graduate program and for her outstanding
patience and professional work in the typing of this thesis.
ABSTRACT
An analysis and a computer program for determining
the steady-state response of a general rotor-bearing system, based
on the concept of a dynamic stiffness matrix, are presented in
this thesis. The rotor is idealized as an axial assemblage of beam
elements that have continuous mass and isotropic, elastic properties.
These properties are developed by using the Bernoulli-Euler beam
theory equations to form the dynamic stiffness matrix. Transverse
shear effects are neglected. Gyroscopic coupling effects, asym
metric linearized bearing properties, and unbalanced loading are
represented as optional end effects on the beam elements. The
development necessitated the use of complex variables to account
for the coupling of motion in the two coordinate bearing planes.
From the above development, a computer program was written
and was applied to four test cases in order to identify the ad
vantages and limitations of this technique. Test case one inves
tigated the effects of support stiffness on the critical speeds of
a uniform elastic rotor. The rotor response, up to and through the
third critical speed agreed with theoretical results within 2%.
Test case two involved a uniform elastic rotor supported at its
ends in fluid-film bearings and demonstrated the program's ability
to predict elliptical whirl orbits. Critical speed investigations
of several overhung shaft-disk combinations, presented in test
case three, predicted results within 2% of the experimental values
observed by Dunkerley. Presented in test case four are the
unbalance response curves for two overhung rotor configurations, a
one disk and a three disk model. Correlations with the experimental
and analytical results of Lund and Orcutt are also presented for
these two models.
TABLE OF CONTENTS
PAGE
i LIST OF FIGURES i
iii LIST OF TABLES iii
iv NOMENCLATURE iv
1.0 INTRODUCTION 1
2.0 STATEMENT OF THE PROBLEM 4
3.0 LITERATURE SURVEY 8
4.0 DEVELOPMENT OF A GENERAL ROTOR ELEMENT l6
4.1 Equation of Lateral Vibration for a Uniform 16
Elastic Beam Element
4.2 The Dynamic Stiffness Matrix [k] , in the x-z 23
and y-z Plane, due to End Shearing Forces
and Bending Moments.
4.3 The Dynamic Stiffness Matrix for a Uniform 33
Elastic Beam in Two Dimensions Based on
Closed Form Exact Solutions
4.4 Axial Assembly of a System of Uniform Elastic 36
Beam Elements
4.5 Effects on the Dynamic Stiffness Matrix due 40
to Fluid-Film Bearings at the Ends of the
Beam Element
4.6 Effects on the Dynamic Stiffness Matrix due 48
to Disks at the Ends of the Beam Element
4.7 A General Unbalance Force Vector 58
4.8 Shear and Moment Balance at a General Node 62
PAGE
5.'0 PRESENTATION OF EXAMPLE PROBLEMS 69
5.1 Critical Speed Map for a Uniform Elastic Rotor 73
5. 2 Unbalance Response of a Uniform Elastic Rotor 81
Supported in Fluid-Film Bearings
5.3 Overhung Disk on a Uniform Elastic Shaft 94
Supported in Rigid Bearings
5.4 Lund and Orcutt Test Rotor (MTI Rotor) One 103
and Three Disk Models
6.0 DISCUSSION OF RESULTS 117
7.0 CONCLUSIONS 122
8.0 RECOMMENDATIONS 124
9.0 REFERENCES 125
10.0 APPENDIX A - THE DYNAMIC STIFFNESS MATRIX 128
11.0 APPENDIX B - EQUATIONS FOR THE ELLIPTICAL 135
WHIRL ORBIT
12.0 APPENDIX C - WHIRL RADIUS FOR SYMMETRIC ONE 137
MASS MODEL
13.0 APPENDIX D - BEARING DYNAMIC STIFFNESS AND DAMPING l4l
COEFFICIENTS
14.0 APPENDIX E - COMPUTER PROGRAM"ROTOR"
145
LIST OF FIGURES
Figure Title PAGE
1 Free Body Diagram of a Differential Length 18
of an Element
2 Applied End Forces and Moments 25
3. Numbering Convention for Axial Assembly of Beams 37
4. Free Body Diagram of an Assembly of Two Beam 37
Elements
5 Free Body Diagram of Beam with Bearing Forces 41
6 Free Body Diagram of Disk 50
7 Free Body Diagram of Nodes with Disk Forces 51
and Moments Acting
8 Unbalance Force 59
9 Steady-State Unbalance Force 59
10 Forces and Moments Acting at a General Node 62
11 Example Rotor-Bearing System 65
12 Rotor Model for Test Case 1 76
13 Critical Speed Map for Uniform Elastic Rotor 77
14 Mode Shapes for Uniform Elastic Rotor 78
15 Typical Unbalance Response Curve for Test Case 1 79
16 Rotor Models for Test Case 2 87
17 Unbalance Response- no Cross-Coupling Test Case 2
ROTOR, 1MASS
88
18 Distributed and Consistant Mass Models- Test Case 2 89
19 Response Curves for 2, 4, and 6 Element Solutions 90
Test Case 2- FINITE5
PAGE
20 Response Curves for 2, 4, and 6 Element Solutions 91
Test Case 2- ROTOR
21 Unbalance Response- no Cross-Coupling Test 92
Case 2- FINITE 5, ROTOR
22 Unbalance Response- with Cross-Coupling 93
Test Case 2- FINITE 5, ROTOR
23 Model and Idealization for Test Case 3 98
24 Mode Shapes for the Overhung Disk Models 100
25 Critical Speed vs. Overhung Length for 101
Models I and II of Test Case 3
26 Typical Unbalance Response Curve for Overhung 102
Rotor
27 Models for Test Case 4- Lund, Orcutt Rotor 110
28 Model Idealization for Test Case 4 111
29 Theoretical and Equivalent Bearing Stiffness 112
and Damping Properties for Test Case 4
30a Unbalance Response of One Disk Rotor 113
Center Position- Test Case 4
30b Unbalance Response of One Disk Rotor 114
End Position- Test Case 4
31a Unbalance Response of Three Disk Rotor 115Center Position- Test Case 4
31b Unbalance Response of Three Disk Rotor 116
End Position- Test Case 4
Bl Elliptical Whirl Orbit Dimensions 136
Cl One Mass Model 137
Dl Fluid-Film Journal Bearing 141
D2 Dynamic Representation of Bearing Forces by Spring and
144
Damping Coefficients
153El Sample Input-Output Problem for ROTOR
154
E2 Program Listing- ROTOR
ii
LIST OF TABLES
Table Title PAGE
1 Critical Speed Results for Test Case 1 80
2 Model Description of Test Case 2 83
3 Results for the 2, 4, and 6 Element Idealization 85for Test Case 2- FINITE5 , ROTOR
4 Disk Properties for Test Case 3 94
5 Critical Speeds of Overhung Rotor 99
6 Calculated Critical Speeds by Prohl Method 109Test Case 4
ii:
NOMENCLATURE
2A - cross-sectional area in.
A^- unknown coefficients
a -
eccentricity of unbalance in.
B. - unknown coefficients
C - length of overhanging portion of shaft in.
D - shaft diameter in.
D|DX
- bearing damping coefficients lb. -sec.xx y
in.
D ,Dyx'
yy
E - Young's modulus for shaft section lb./in.2
F^-F1Q- transendental frequency functions
H - moment of momentum vector for spinning disk
c,g*with respect to its center of gravity
g- acceleration of gravity, 386.4 in/sec.
I - cross-sectional transverse moment ofin.4"
inertia of the shaft section with respect
to its center of gravity
2I - polar mass moment of inertia of the disk lb. -m. -sec.
IT- transverse mass moment of inertia of the lb. -in. -sec.
disk with respect to its center of gravity
i --/-l
i2*k - unit vectors in x,y,z direction respectively
K_ , K- bearing spring coefficients lb. /in.
K ,Kyx'
yy
- length of shaft section in.
MD - mass of disk lb. -sec.
in.
- eccentric unbalance mass lb. -sec.
in.
iv
Mo
12M
, etc-
bending moment? first subscript refers toy
the normal of the surface on which it acts,
second refers to the direction in which it
acts when referred to the coordinate system j
the first superscript refers to the end
position (1- left end 2-right end), the
second refers to the element number.
n - left to right indexing number for an axial
assembly of shaft sections
Nc- critical speed rpm.
|^( ) - denotes the real part of
sl2,etc - shear force { subscripts and superscripts lb.zx
have same meaning as for the bending moment
t - time sec.
u - rotor displacement in x-direction in.
U=UR-ti.U-r- steady-state rotor displacements in x-
direction; UR, Uj are the real and
imaginary parts
v - rotor displacement in the y-direction in.
V=VR+i Vj- steady-state rotor displacement in y-
directionj VR,Vj- are the real and
imaginary parts
x,y- direction of rotor displacements
z - axial coordinate
ol- angle between x-axis and major semi-axis of
elliptical orbit
Q - rotation about y-axis i.e. rotor slope in rad.
x-z plane
(2) - steady-state rotor slope in x-z plane rad.
- -
Ugrj
P - weight density of shaft section lb. /in.
to- rotation about x-axis i.e. rotor slope rad.
in y-z plane
"W - steady-state rotor slope in y-z plane rad.
***- angular speed of rotor rad. /sec.
GV,Oy- precession velocity about x and y axis rad. /sec.
(angular velocity of disk with respect
to x and y axes) .
JF I - general unbalance force vector
fF-A
- general nodal point force vector for a structure
C^xzl ~ dynamic stiffness matrix in x-z plane
[K^] - dynamic stiffness matrix for a shaft section in
the x-z and y-z planes
LKeJ"* dynamic stiffness matrix for a shaft section
in the x-z and y-z planes
[Kg 1 - matrix of left end bearing effects
Tk| J - matrix of right end bearing effects
\kX J- matrix of left end disk effects
fKpJ - matrix of right end disk effects
Tk ^1 - structure dynamic stiffness matrix
- column vector of applied end forces and
moments on the shaft section in the x-z plane
- column vector of applied end forces and moments
in the y-z plane
- general vector of applied end forces and
moments in the x-z and y-z planes
- column vector of bearing forces at left end
of the shaft
- column vector of bearing forces at right end
of the shaft
- column vector of disk forces at the left end
of the shaft
- column vector of disk forces at the right end
of the shaft.
M- general vector of nodal point displacements
and rotations
vi
1
1.0 INTRODUCTION
All rotors will deflect and whirl under the influence of
unbalanced forces. Even the strictest of balancing procedures
fail to totally eliminate the rotor residual unbalance. For a
high speed flexible rotor running near any of its critical speeds,
the unbalanced forces are capable of causing large amplitude build
up. Unless dissipation of energy is allowed (i.e. damping), the
response at a critical speed theoretically would be infinite.
Experimental observations by Lund and Orcutt [15 J and other
investigators, have shown that the conventional critical speed
calculations, which includes the bearing flexibility but not
the bearing damping, tend to give values for the critical speeds
which are lower than the ones actually observed. Inclusion of
the bearing flexibility in the calculations lowers the critical
speed, while bearing damping usually has the opposite effect. An
unbalance response analysis involves the calculations of rotor
response, due to unbalance, at specified speeds throughout a
speed range. The actual critical speeds can be determined from
a series of unbalance response calculations by plotting rotor
response vs. speed to locate the response peak.
High speed rotor applications have generated the need for a
more accurate analysis of the total rotor-bearing dynamic problem.
Such an analysis would include investigation of critical speeds,
unbalance response, balancing and instability. Each of these
areas play an important role in the overall rotor vibration prob
lem-.- Although problems in each of these areas may existsimul-
2
taneously in a single high speed rotor application, the complexity
of a combined analysis makes the solution useless for practical
purposes.
The current trend in the published literature (i.e. Lund and
Orcutt [15], Rieger [18] and Ruhl and Booker [17], etc.) is to
combine certain aspects of the rotor-bearing problems into a single
analysis. This is made possible with the new generation digital,
analog, and hybrid computers available. The engineer can formulate
a more complex and accurate mathematical representation of the
physical system, without fear of how to handle the ensuing equations.
Critical speed and unbalance response calculations are an
integral step in the design analysis of any rotating system. The
critical speeds of a proposed design must be known so that none
occur in the operating speed range. A critical speed may be de
fined as ;
"The rotor speed at which local maximum-amplitude
occurs. Where no gyroscopic effect occurs, the
critical speeds coincide with the systems natural
frequencies."
[22.1
In the general case, machine run up and run down would
require the rotor to pass through several of its critical speeds.
However, the rotor system is a uniform elastic structure which
theoretically possesses an infinite number of critical speeds.
Therefore, to be mathematically correct in representing the rotor,
a model should be chosen which reflects the effects of the higher
modes, even at the lower speeds. Also, by choosing a model which
includes the bearing forces with their associated stiffness and
damping coefficients, the effects of bearing damping may be inves
tigated.
3
With the information obtained from the unbalance response
investigation of a proposed rotor-bearing design, the sensitivity
of the system to unbalance may be determined, and the damped
critical speeds of the system may be located. Generally, this
information will then be combined with a balancing procedure to
minimize the rotor amplitudes in the operating speed range.
This thesis is concerned with the development of a proce
dure , based on the dynamic stiffness matrix concept, for the
unbalance response of a general rotor-bearing system. Several
general procedures and computer programs are available in the
open literature for design use, as discussed by Rieger [39] .
The procedure and computer program developed in this thesis are
not meant to supersede the existing response programs, but merely
to introduce an alternative approach to the problem. The use
of the dynamic stiffness matrix concept allows the derivation
of one stiffness matrix which includes shaft dynamic stiffness,
distributed mass effects, lumped mass effects, rotary inertia,
and gyroscopic stiffening effects directly. The procedures
ability to accurately calculate the rotor response, due to un
balance is the key point which is investigated. In depth inves
tigation of the relative efficiency of this procedure compared
to the existing procedures is left as a recommendation for future
work.
2.0 STATEMENT OF THE PROBLEM
In this thesis, a unified matrix formulation for the steady-
state unbalance response of a flexible rotor in fluid-film bear
ings is presented. The term unified refers to the use of a right-
handed set of Cartesian coordinates and the sign conventions, as
used in elasticity theory. The procedure developed is based on
the dynamic stiffness matrix concept and involves the axial assem
bly of "rotorelements"
to represent a true rotor-bearing configu
ration. A "rotorelement"
refers to a uniform section of an elas
tic beam with optional end bearings and disks. A "rotorelement"
may be constructed from the following components;
1. A uniform section,, elastic beam which is continuous,
homogeneous, isotropic, and satisfies the Bernoulli-
Euler beam theory. Young's modulus E, weight density f^
cross-sectional area A, and transverse inertia I, are
all assumed to be constant along the beam element of
length^. nock , />,r,/,l(nj
J~
1(f>i0Symbol ;
-lu-J j< -
2. End disks posessing mass MD, polar mass moment of in
ertia Ip, with respect to its neutral axis, and trans
verse mass moment of inertia IT ,with respect to a
transverse axis through its center of mass.
pi Moyr? ,iT
Symbol; \vread*
3. End bearings, represented by 4 stiffness coefficients
(K,
K , etc.) and four damping coefficients (D,
aa**,y
.A.A.
D , etc.) of the type derived by Lund [30],xy
_
Jr- M r. < t /-r( IA n n A
Symbol?Ep-
V , , ly . I ''-
f/// /V/',r
5
4. An unbalanced force-eccentricity combination (expressed
in oz.-in.) at the ends of the beam element. The unbal
ance is assumed to arise from a small mass M0 which exists
off the neutral axis with an eccentricity, a.
Symbol}
A general "rotor element", (n), with all of the optional end
effects acting at the nodes n and (n-i-1) would appear as follows,
node (n) " I fl
Kxv~
nV̂
~eTe *.,'* * i-t.
t^nbalan cc
rtocfe(ntO
J?ri-
/ /ss'/rss;-
ZT D<*,
/ / /s.
In the axial assembly of several elements it may be necessary
to connect just the right end of a beam section to the left end
of another beam section without any disk, bearing, or unbalance
acting. This is accounted for in the computer program by allowing
the programmer to first build each individual element (with any
combination of end effects acting) and then provides an assembly
procedure to build the structure dynamic stiffness matrix.
The isotropic linear elastic beam element is developed with
distributed mass and elasticity, so that the effects of all the
system modes are included in each calculation. A dynamic stiff
ness matrix for the beam element is derived.
The gyroscopic coupling and concentrated. mass effects of a
disk are derived in matrix form and treated as point effects on
the system (i.e. they act at the ends of a beam element).
The bearing forces are represented by the following linear
equations, as derived by Lund in reference [30],
Fx~
Kx* X f Kxy Y f-Dxx / * DxyY'
fy- KyX X + Kyy V f Dy* J? + Dyy ?
No attempt shall be made here to derive these equations or the
stiffness and damping coefficients since their derivation is
quite lengthy and beyond the scope of this thesis. However, a
basic description of the derivations is given in Appendix D to
provide some background information. The bearing effects are put
into matrix form for easy handling on the computer and are treated
as point effects in the system.
The general force vector of applied loads is derived for the
unbalanced forces that act at the element ends.
A computational method is developed so that the beam elements
may be axially assembled and the effects of a bearing, a disk, and
unbalanced forces acting at any nodal point (beam end or junction
point) may be represented. A computer program is written for the
element set up and assembly. The dynamic stiffness matrix of the
structure [K^m] is generated and it is inverted for each speed,
such thatj
or {*$t}~
LKst]"'
[ Fn]
where j F I - is the general vector of nodal point
loads for the structure.
\ K J - is the structure dynamic stiffness matrix.
fAST ]- is the general vector of nodal point dis
placements and rotations for the structure.
Thus, the nodal displacements are obtained. The whirl el
lipse information is then calculated and plotted vs. speed to
obtain the unbalance response curves. The peak amplitudes of the
response curves locate the systems critical, speeds.
Therefore, the direct calculation of a value for the critical
speeds are not performed in an unbalance response analysis, as is
done in a conventional critical speed calculation.
The thesis problem was formulated in this manner for several
reasons.
1. To investigate the dynamic stiffness matrix concept as
a possible approach in the analysis of rotor-bearing dyna
mics.
2. Matrix notation was used to provide an easy assembly
procedure for idealization of true rotor-bearing systems.
3. To investigate the distributed mass formulation.
4. To investigate the effects of rotary inertia and gyro
scopic stiffening due to disks.
5. To allow any bearing type, for which the eight dynamic
coefficients are available, to be investigated.
8
3.0 LITERATURE SURVEY
A rotor supported in fluid-film journal bearings is a com
plex dynamical system which exibits a variety of physical char
acteristics such asj critical speeds, instability and unbalance
vibrations. Several general surveys of published work and
state-of-the-art commentaries on Rotor-Bearing System Problems
hav$ been presented in the open literature. Bishop [34], in
1959, discussed the unbalance response literature concerning
rotors having distributed mass and elastic properties (29 refer
ences). Dimentberg [35] , in 1961, outlined many of the foreign
contributions on unbalance response and rotor stability (52
references). In 1965, Rieger C29]presented a comprehensive
review of the American and British literature and collated the
major results on critical speed, unbalance response,rotor-
balancing and other important aspects of rotor-bearing system
performance (162 references). Bishop and Parkinson [36] , in
1968, presented a review paper covering the publications on
unbalance response, stability and flexible rotor balancing t93
references). In 1973 t Rieger [19J reviewed the published lit
erature on unbalance response and balancing of flexible rotors.
Some of the major topics discussed were $ nonsynchronous whirl,
dissimilar rotor stiffness, coupled bending-torque and bending-
axial effects, computer analysis of rotor-bearing system, balanc
ing principles and criteria, and foreign language contributions.
This thesis is concerned with rotor response which is due to
unbalance and with the critical speeds of a rotor-bearing system.
9
The published work related to these two fields will be discussed.
Furthermore, since the literature is so extensive, and several
general surveys have been made, only major contributions will
be reviewed and no consideration will be given to the literature
on either rotor stability or flexible rotor balancing.
Rankine [ ll, in I869, was the first to perform a dynamical
analysis of a rotating shaft. He derived a formula, which gave
the critical speed as an eigenvalue, for the cases of a simple
shaft in end bearings and for an overhung shaft supported in a
shoulder at one end. Although his analysis was correct, the
fact that the mechanics of shaft whirl was not completely un
derstood at this time led to the erroneous conclusion that criti
cal speeds were dynamically unstable conditions, beyond which
shaft operation would cause excessive amplitude build-up.
In I895, Dunkerley [ 2 ] performed extensive experimental
work on the measurement of critical speeds of a number of shaft-
disk combinations. After observing the critical speed, calcu
lations using Reynolds theory for shaft-disk combinations were
performed and good correlation was obtained.
The misconception of critical speed instability was finally
resolved by Jeffcott [ 3]# in 1919. He performed a thorough anal
ysis of the response of a damped flexible rotor in rigid bearings.
due to a specified unbalance. Jeffcott's model demonstrated
the important features of shaft whirl up to and through the
first critical speed, bending mode. However, the model did not
include disk inertia effects, accurate bearing representation,
or the higher criticalspeeds'
effects.
10
Following Jeffcott, several investigators examined the
problem in more detail. Smith [4], in 1933, presented a com
prehensive analysis of the unbalance whirl and stability of a
flexible rotor in flexible bearings, thus, taking the first
step toward a true rotor-bearing analysis. Robertson [53, in
1934, examined the effects of damping on the unbalance response
and critical speeds ofJeffcott'
s model.
The first study intended for design use was presented by
Prohl [6], in 1945. He devised a method whereby a rotor could
be represented by discrete masses joined by massless flexible
springs. The analysis included bearing flexibility but no damp
ing and was formulated as a set of recurrence equations. This
procedure was only capable of calculating critical speeds and the
corresponding mode shapes, and not for the unbalance response.
The next major contribution came from Green [8], in 1948.
He investigated the whirling of several shaft-disk systems in
which he included the gyroscopic effects in the critical speed
calculations. It was shown that the gyroscopic action tended
to stiffen the rotor and thus raise the critical speed. Also,
when the disk motion could be described by pure translation
(i.e. centrally located disk) no gyroscopic action was present.
Hagg [7J# in 1947, putJeffcott'
s flexible single-disk
rotor in bearings with identical radial stiffness and damping
properties and performed an unbalance response analysis. The
influence of bearing stiffness and damping properties on the
calculation of critical speeds was shown to be significant.
11
This study encouraged the more rigorous investigation of Linn
and Prohl [9 J, in 1951. They analyzed the effect of bearing
flexibility on the critical speed of flexible rotors and pre
sented their results in the form of a critical speed map
(i.e. critical speed vs. support stiffness).
In the years 1953-1963, extensive work was published by
Yamamoto and was collected into a single manuscript in reference
[383. Yamamoto presents comprehensive analytical and experimen
tal studies of the synchronous forward and backward precessional
whirling of a flexible rotor in rigid and flexible bearings.
Emphasis was placed on studies of the critical speeds which
arise when double row and single row rolling-element bearings
were employed. Forced response of sub-harmonic and "summed
and differentialharmonic"
oscillations are included. Also,
nonsynchronous whirling of asymmetrical rotors and the zones
of instability due to bearing pedistals and various aspects of
the rolling-element bearings are discussed in detail. This
series of papers presents a very detailed treatment of rolling-
element bearing rotordynamic problems.
Warner CLO], in 19&2, extended Hagg's model to a two-disk
flexible rotor, without disk inertia effects supported in two
bearings having identical stiffness and damping properties,
thereby extending the analysis through the second critical
speed.
Lund and Sternlicht [ll] , in I962 presented the first
analysis to include the direct and cross-coupled stiffness and
12
damping properties of the bearing, which were obtained by
solving the linearized Reynold's equation. They studied the
response of a flexible single disk rotor in several types
of fluid-film bearings and found that the bearing properties
greatly influenced the rotor response and attenuation of
transmitted bearing forces.
Morrison [12], in 1962, made a similar analysis of an
elastic rotor that was supported in fluid-film bearings. The
dynamic stiffness and damping properties of the bearings were
derived from the short (Ocvirk) bearing theory. Equations
for critical speed and rotor response are derived and experimental
verification is given. When full account was taken of the
dynamic bearing characteristics, two critical speeds were
calculated and observed. This was attributed to the asymmetric
properties of the bearing coefficients.
Lund [13], in 1965 published a computer program for the
unbalance response of flexible rotors supported on several
fluid-film bearings. Splined couplings and massive bearing
pedestals could also be accounted for in this program. The
analysis is an extension of the Myklestad -Prohl method, where
the rotor is divided into a number of discrete mass stations and
connected by weightless flexible bars. Gyroscopic stiffening
effects are also included at the mass stations. Unlike the
Myklestad-Prohl method, this analysis holds for any speed and
the rotor response, due to unbalance, may be calculated through
out a speed range. No experimental varification or check out
13
of the computer program is given.
Morton [14], in 1965-66, presents experimental and theo
retical data on the unbalance whirl of generator rotors. The
analysis is presented in a matrix formulation for easy handling
on the computer, using a receptance formulation developed by
Bishop, [25] , in 1955. An experimental procedure, for finding
the linearized receptances of both the bearing oil film and
of the pedestals is described. The analysis shows that three
flexible modes are adequate for identifying the flexural char
acteristics of a rotor. Experimental tests showed that the
rotor may be considered as an undamped structure whose char
acteristics may be predicted with good accuracy by established
techniques, such as the Myklestad technique or the Receptance
approach developed by Gladwell and Bishop [37]. The character
istics of supporting structure were not as completely defined.
Lund and Orcutt [15] , in 19&7- presented an exhaustive
analytical and experimental investigation of the unbalance
response of a flexible rotor. The analysis is an extension of
Lund's earlier work using Prohl *s transfer matrix method. The
rotor is represented by cylindrical bar sections connected at
stations along the axis of the rotor. At each station disk
gyroscopic effects and bearing reaction forces may be added. The
bearings are represented by eight speed dependent stiffness and
damping coefficients. In general the unbalance response will
be elliptical, but due to the bearing type used the orbits are
actually circular. Rotor unbalance, rotary inertia and gyroscopic
14
moments in the bar itself are ignored, for simplicity. Data
for three rotor test configurations are presented, one disk,
two disk and three disk assemblies. Good agreement is ob
tained between test results and calculated results for all
three configurations,
Rieger [18], in 1971, presented an unbalance response an
alysis for a uniform flexible rotor in plain cylindrical fluid-
film bearings for speeds up to twenty tines the lowest rigid-
bearing critical speed. The mass and elastic properties are distri
buted along the length of the rotor, thus the effects of all the
modes are felt in each rotor calculation. Influence of rotor
speed, bearing operating eccentricity, relative stiffness of
rotor and bearings, and unbalance location along the rotor is
investigated. Results are presented as dimensionless parameters
so that a wide range of rotor-bearing configurations may be
covered. Charts of the rotor maximum whirl amplitude and
the transmitted bearing force vs. speed are presented. No
provision for the addition of disks was given in the analysis. The
results obtained were verified by using a discrete mass rotor
bearing program which accepted direct and cross-coupled bearing
coefficients. Rieger 's distributed mass-elastic model presents
new parametric insight into the problem and has stimulated the
dynamic stiffness matrix approach which is developed in this
thesis.
Ruhl [16], in 1970, and Ruhl and Booker [17], in 1971, de
veloped a finite element model for stability and unbalance re
sponse analyses of rotor systems. The finite element model is
developed with a consistent mass matrix thus giving a more
15
accurate representation of mass throughout the system. The
eight bearing coefficients are derived from the Ocvirk short
bearing theory. A comparison of the finite element model to
the lumped mass pregression technique of Lund [13]in calculating
the response due to unbalance, indicates a more accurate so
lution with fewer degrees of freedom using the finite element
model,
Rieger [39], in June, 1974, presented a state-of-the-art
review of the nature and functioning of computer programs for
rotor-bearing dynamic analysis. Current program approaches to
critical speed, unbalance response, stability, torional analysis,
and balancing are reviewed. The strengths and weaknesses of
these present capabilities for rotor-bearing dynamic analysis
are identified. The greatest strengths of the critical speed
and unbalance response programs are the generality with which
they are written (i.e. number of rotor sections permitted,
number of bearings, and number of substructure levels) and user
convenience options. The lack of accurate bearing dynamic
coefficients, seal coefficients, foundation data, neglection of
shear effects, and experimental validation, are some of the
major weaknesses associated with the general rotor-bearing dy
namic calculations. The best documented and most efficient
program for critical speed analysis was CADENSE 26 developed by
J. Lund. The most comprehensive rotor-bearing system analysis
program was identified as GIBERSON, developed by M. Giberson,
but no user's manual is available for the novice programmer.
Program details and comments on the state-of-the-art capabilities
for each program category are enumerated in reference [38] .
16
4.0 DEVELOPMENT OF A GENERAL ROTOR ELEMENT
4.1 Equation of Lateral Vibration for a Uniform Elastic
Beam Element
The free body diagram of a differential length of an
element is shown in figure 1. For convenience, the free
body diagram is divided into two component diagrams-one in
the x-z plane and the other in the y-z plane. The sign con
ventions and subscripts that are used in the following der
ivations are those used in the theory of elasticity. The
derivations of the lateral motion of the beam element in
the two respective coordinate planes have included the
normal assumptions used in the Bernoulli-Euler beam theory
of beam bending. These assumptions are:
1. The element is assumed to be a straight beam in its
undeformed state, and therefore, the Bernoulli-Euler
beam theory applies.
2. The radius of curvature of the deformed beam is large
in comparison to its length, that is, the respective
curvature of the beam in each of the planes is equal
to the second partial derivative of the respective
displacements with respect to the z coordinate.
3. Plane sections remain plane after bending.
4. Deformation due to shearing of one cross-section
relative to an adjacent one is negligible.
5. The beam is free from longitudinal force, gravity
forces and distributed static forces.
17
In addition, the following assumptions, which are associated
with the motion of the beam, will also be made.
6. The mass is distributed along its neutral axis, and
therefore, rotary inertia of the element is neglected.
7. The gyroscopic or Coriolis effects of the beam element
are negligible with respect to those same effects asso
ciated with any lumped masses in the system, and there
fore, they are neglected.
The shear forces and bending moments on the positive face
(right hand end) of the differential element are related to the
respective shear forces and bending moments on the negative face
(left hand side) by assuming that the shear force and bending
moment functions are continuous in z and that terms containing
the elemental length dz to the second and higher powers are
zero in the limit.
An example of the notation used in the following derivations
is:
M - bending moment
Szx- shear force
The first subscript associated with these forces and moments
refers to the direction of the normal of the surface on which
they act. The second subscript refers to the direction in which
they act when refered to the coordinate system. The forces and
moments are positive by definition when they act in a positive
direction on a positive surface or when they act in a negative
direction on a negative surface. Conversely, they are negative
by definition when they act in a negative direction on a posi-
18
tive surface or when they act in a positive direction on a
negative surface. Throughout the derivations, right handed
cartesian coordinate systems will be used where i, j_t and
k are the unit vectors in the x, yf and z directions
respectively.
Figure 1 Free Body Diagram of a Differential Length of an Element
L v
(Szy+^yli) f
(my VhLcii) jl
yt-
2. W
Roto* ---7-
Z <
"
(/)!& r MllcIi) I
(S& 1- )$M-clh) jk
hi
X, IL
19
Consider an element of the shaft subjected to end forces and
moments as shown in figure 1. Applying Euler's first equation
of motion:
^ Iitnba/. '
*K Qz c.a. (1)
For motion in the x-direction,
Z Fh,ibal. y=
cfaL ilK (2)*y-
ty >t
where : dm = fAdz
g
(3)
u - is displacement in the x-z plane (Figure 1)
^ - is the weight densitylb/in-5
A - is the cross-sectional area in
g- is the acceleration due to gravity
= 386.4in/sec2
Equation 3 reduces to,
*fht1
hi (4)
Applying Euler's second equation of motion:
^
lUc.^VrtbJ- = ti e.p. (5)
For rotation about the y axis,
^ Illc p'lot bad. <J_
-
IT y&(6)
It }J& -
- %s + 0>lzy ^>JlUz/ch)+ $*<d* - (5*x +)yycii)ch (7)
20
where O - rotation about y axis (figure 1)
Ij - transverse mass moment of inertia with respect
to the transverse axis through the center of
gravity.
Neglecting the rotary inertia of the element, Ij-jt*. and
terms of the second order and higher in dz, equation (7)
reduces to the following:
S"=
"^ (8)
Substitution of equation (8) into equation (4) gives,
ft ^ ^
"
(9)
Using figure 1, with all the sign conventions that are shown,
and the Bernoulli-Euler beam theory equations, the following
relationship between the applied moment, in the x-z plane,
and the corresponding curvature in the x-z plane is:
P1*y~- El $L (10)
Substituting equation (10) into equation (9) and assuming
E and I are constant along the length of the element, we
obtain,
(id
as the equation of lateral motion for the element in the x-z
plane .
Applying Euler's first equation of motion, equation (1), for
motion in the y-direction,
-^ I linbcxl- '4." dnt y*
0 i/-- (12)
21
hy n d3)
or $- ^V- j-^ __ o (14)
where v is displacement in the y-z plane (figure 1), and
all other terms are as previously defined.
Applying Euler's second equation of motion, equation (5),
for rotation about the x-axis,
z. Mo unbai.y IT ^p, (15)
dLj-i x ^i x (16)
where; <f - is the rotation about the x-axis (figure 2) and
all other terms are as previously defined.\ T-
Neglecting the rotary inertia of the element, It ju , and
the
terms of the second order and higher in dz, equation (16)
reduces to the following;
^"-JT-
(17)
(18)
Substitution of equation (17) into equation (14) gives,
tA fy- yjlzA zz
O
Using figure 1 with all of the sign conventions that are
shown, and the Bernoulli-Euler beam theory equations, the
following relationship between applied moment, in the y-z
plane, and the corresponding curvature in the y-z plane is;
!Aik = ' EZ
J^ (19)
Substituting equation (19) into equation (18) and assuming
E and I are constant along the length of the element, we
obtain;
as the equation of lateral motion for the element in the
y-z plane.
22
ftyb?
t m yvclo^X
Hft
ht^
(20)
23
4.2 The Dynamic Stiffness Matrix CK3, in the x-z and y-z
plane, due to End Shearing Forces and Bending Moments
The equations of lateral motion derived in section 4.1
are used to develop a method for calculating the steady-
state response of a uniform elastic beam subjected to shear
ing forces and bending moments concentrated at its ends.
Since the end forces and moments act only at the ends of the
beam, they may be accounted for in the end conditions and
therefore do not enter into the equations of motion. The
forces and moments at each section will be assumed to vary
as a harmonic function of time, with a common angular fre
quency SL, For convenience, the complex notation, Re1
,
will be used instead of cosiit, where i= (~y ,-&-= angular
frequency, t-time, and R denotes the real part. This allows
a more general and efficient derivation to obtain the steady-
state solution.
The steady-state solution for the lateral displacement u,
of the beam, may be obtained by the standard separation of
variables technique as discussed by Timoshenko L28] . The
displacement u is assumed to be equal to the product of a
spatial function, U(z), and a time function, T(t). The dis
placement u may be written as,
IA-=
L/C*j (21)
where, U(z) is a function of z alone ande1 ^
is the assumed
form of T(t). Substitution of equation (21) into equation (11)
gives;
24
dill _
MA1-
U x o
dl1* fl (22)
from which the shape of the normal mode of vibration in the
x-z plane for any particular end condition may be found.
Letting,
$EI (23)
and noting that sin/)z, cos ^) z, sinh/^z, cosh/)z, are all
solutions to equation (22), we obtain,
as a general solution to equation (22). Therefore the dis
placement u is,
11 =(A,s/>i?,2 tAt cosh ^ tAj smhte t-Av CoskAz) (25)
Since only the steady-state solution is sought we are only
concerned with the solution to the spatial part of this func
tion, U(z).
Figure 2 shows the directions for applied end forces and
moments, on an element of length^, and the assumed positive
directions for displacement and rotation, in each of the
respective coordinate planes, where the superscripts indicate
the end position.
Figure 2 Applied End Forces and Moments
25
'Y,V
"May 4.
X,U
2 .7^
Equations (8) and (10) in section 4.1 are the expressions
for shear force and bending moment in the x-z plane and are
restated here for convenience,
(8)
^ ZN
(10)
expressing equation (8) in terms of displacement u gives,
SZ*= "tX
(8a)
26
These equations are used here to obtain the following end
conditions, noting that only the spatial part of the function
u, U(z), needs to be considered;
at end 1 <)& ^
z=0
-w-
-"^'f
(26)
at end 2 <^
Differentiating U(z) three times yields,
TJCi) =
( A, S/>i /^* +Ai cos/iz r A3 sinh At + At cosli /)%)
\HQlI z (AjAtosDe -Axhsi'ih* +Ay) cosJ, ,)z t-Avlislniute) (27)
yUii - (^/s/}M? -At. fcosAz +Aj/y$>))hy + Affects lite)
yjJ(M - (rA.I)3ca5/)Z tAz/YsmM tAsA3
Cosk /)t -MyJ>3
sMi)hi3
Applying the end conditions from equation (26) and using
equation (27), and assuming the left end of the beam is at
z=0 and the right end of the beam is atz=|
a relationship
between the applied end shear forces and bending moments and
the coefficients A. can be found and written in matrix form;
27
S4,
< > -EI
I w*7 J Ha
o V
^ ! a
o
7)ucosKMl_
r
/
hi
A>
Ay
?
Jy*/
(28)
which may be written as ,
[f--.] ^ [o] [a] (29)
where {Fxz} is a column vector of applied end loads, in the
x-z plane, and [AJ= [AltA2, A-., AJ , and [d] is as shown
above .
_ <) Ul&)Noting that Q> =
s-^~i), and using the first two relations
in equation (27), displacements and rotations as related to
the coefficients A. can be found and written in matrix form;l
<e,
>
Hki
o
sin), I
d
i
COS/) L
0
O j /
>> a
Acos AJL -hsinhl 7) co^ihi -. 7)s/r,h)>l4XH
fA^
<Al
Ai
>
-Vki
(30)
where the subscripts on 17 and & refer to the ends of the beam
(i.e. 1 indicates left end, 2 indicates right end).
Equation (30) may be rewritten as,
[->} - ['] IM (3D
where (AxA is a column vector of end displacements and rota
tions, in the x-z plane, and a^ is as previously defined
28
and [C J is as shown above.
Solving equation (31) for [_k\. we obtain,
Substitution of equation (32) into equation (29) yields,
[Fxz] = Lk] [^Xil
where [*K J is the dynamic stiffness matrix for the beam in
the x-z plane.
From Appendix A it is seen that [k ~] takes the following
(32)
(33)
(34)
xz
form;
Fj
-Tift hFi - F, hu
-Fi -f/u FJ/A **//.
F, *//> ^/A
(35)
-VMry
where F. -
F.0 are transendental equations given in the appen
dix using the notation of Bishop [_25] .
The steady-state solution for the v displacement may
also be obtained by the standard separation of variables
technique. The displacement v is assumed to be equal to the
product of a spatial function V(z) and a time function T(t).
The displacement v may be written as,
where V(z) is a function of z alone ande1^
is the assumed
form of T(t). Substitution of equation (36) into equation (20)
(36)
29
gives;
of1
lAaj __ lA_yy Va) = a
cit* ft* (37)
from which the shape of the normal mode of vibration in the
y-z plane for any particular end conditions may be found.
Noting that $*i r^ from equation (23) and that sin/\ z,
cos/)z, sinh/)z, and cosh/) z, are all solutions to equation
(37) we obtain,
Vl*) ~
C &, 5/S,7)Z t6xC05 77)i: +83 S//l/l/)i r fy cosMiXjS)
as a general solution to equation (37). Therefore the displace
ment v is,
lr = C/3/S//I 7)i tSi.cosAi *&} s/'ihAl. +/3v c osJc/) 2 J e*at
(39)
Since only the steady-state solution is sought we are only
concerned v/ith the solution to the spatial part of this
function, V(z). Again, figure 2 shows the direction for appl
ied end forces and moments and assumed positive displacement
and rotation.
Equations (17) and (19) in section 4.1 are the expressions
for shear force and bending moment in the y-z plane and are
restated here for convenience,
5*v =7T~
(17)
M2X =-ETL <ClT (19)
restating equation (17) in terms of displacement v gives;
S^V=
,_j
tx*h? (17a)c>2
These equations are used here to obtain the following
end conditions, noting that only the spatial part of the
function v, V(z) need to be considered;
30
at end 1
2=o
-
5^i
Z
^L,
2
BI hlVO) ^
-
ET h 3Va) j
-ei yvci) ^
(40)
Differentiating V(z) three times yields,
~\{l) - (fi/S/i.Ai iBzcosAi -h C>J 5/ilhAl +61 cost hi)
YVW = ( B,Accs/)i 'bxAsn.Ai +63/) coshte + #? A s/nhAz) (4i)
TVC}) - (-S,f^hi 7)1 -BzAXCOS))t t
0JI)*-
StAkte 1-By frcoSJ'te)
Mf) -.(-ey3caste +BiAl5S*.Ai + Bj^coskA* i-
n?AJ
sf^hH)
Applying the end conditions from equation (40) and using
equations (41), and assuming that the left end of the beam
is at z=0 and the right end of the beam is at z=%, a relation
ship between the applied end shear forces and bending moments
and the coefficients B. can be found and written in matrix
form;
<
-si, ^ 0A3
O Hi
?= -yc0y 1
0
-A3Sir.U -tfcos/M
0A1-
<
63
tit* J4*1
K a.itn A^cos/ihd -ti'sirihM -fcoshhi
Y
1--VJ
(42)
>
Vxi
31
which may be rewritten as,
where [F "] is a column vector of applied end loads in the
y-z plane, and [b]T= (B-_, B2, B3, B^] , and[.E} is as stated
above ,
Noting that T -
~
, and using the first two relations
in equation (41), displacements and rotations as related to
coefficients B. are written in matrix form as follows;
(W
<
V,
Vf
Ix
EI
-Vx/
0 1 a 1
'
SiriAl Cos /) s/AHH cosh,)I<
8,>
-V0
'>)0
c
db cosM /) 5//J /) i ->)COS/)M ->)syi)iL . By
(44)
^Xl
Yxy
where the subscripts on V and T have the same meaning as the
subscripts for XJ and Q.
Equation (44) may be rewritten as;
[***} - ]_&][*] (45)
where~[&yi\
is a column vector of end displacements and
rotations in the y-z plane, and [b] is as previously defined
and g] is as shown above.
Solving equation (45) for | BJ , we obtain;
Substitution of equation (46) into equation (43) yields;
(47)
(48)
32
where j^K J is the dynamic stiffness matrix for the beam
in the y-z plane.
From Appendix A, it is seen that flC ~) takes the following
form;
yz-
[Kyt] =Ft
-AFg AF-, Fi ~Fio
AF? -A Fb Fib ~Fi
Fi Fin fy* | "xTyFio y, '*/> yj
(49)
f<i
where F-j_-
F^q are transendental equations given in
Appendix A,
33
4'. 3 The Dynamic Stiffness Matrix for a Uniform Elastic Beam
in Two Dimensions Based on Closed Form Exact Solutions
For an isotropic homogeneous uniform elastic beam, it
was shown in the previous section that the dynamic stiffness
matrices on the two orthogonal planes were symmetric, as
is expected. The two stiffness matrices may be combined to
give one symmetric stiffness matrix for the beam, relating
end forces and moments to end displacements and rotations.
Referring to figure 2 in section 4.2 it is seen that the
steady-state motion of the beam is completely expressed in
terms of the displacements and rotations at its ends. Thus,
8 degrees of freedom (two displacements and two rotations
in each plane at each end) are needed to specify the beam
motion. Equations (34) and (48) in section 4,2 are express
ions relating applied end loads to end displacements,
equation (34) holds for the x-z plane and equation (48)
holds for the y-z plane. These equations are restated here,
for convenience:
[F] ^ U<*1 {*} (34)
{Fyi\~~
L Ky] { Mij (48)
These equations may be combined as follows:
*z- k*2: o AXi
(50)
LFY* \ K^
Expanding equation (50) gives:
34
r
\
-y
s_l,.
Mil} ~
-s^
'ZN
"fHM/.
m tx
1 r
FI(J\ vy.
^j-^ko'^ 17a.
-ftr/jJ^'^: e,
F3
ov^v J v
AF7 -AFu f>o y, y\
F, Fl0 /a //. X
\ lo -F,
'
Ft,% k
<f^r
^
<f</
(5D
Rearranging equation (51) so that the 4 degrees of freedom
at each end of the beam are together, and so that the stiff
ness matrix remains symmetric, we obtain;
r
{
1
> ZL
<Kl
7^
6
-/)/V -Z5 O O'
AF) Fj0, ,
o
-A ^ o o |-/:/0 % o o
a-AFL Fi
,o
'
o 7}F? -p./o
Fs, FiiFi "% o 0 F/a r%
F/a % o o ; F, % 0 o
y7 Fn o-vfol-f,
o o:FJo % \ o y, %
i
Hr
r ^
U.
v,
XJ-t-
B-.
>
Hi
(52)
35
or equation (52) may be written as:
If] * Lke]{a} (53)
where: f Ft - is a general vector of applied end forces
and moments
rKgl- is the dynamic stiffness matrix for the beam
(a) - is a general vector of end displacements and
rotations
The dynamic stiffness matrix, [_KE] , includes the mass
effects directly as well as the stiffness effects. Therefore,
a separate mass matrix, either lumped or consistent, need
not be developed. The force and displacement vectors in
equation (52) are arranged as shown for ease in axial
assembly of a system of beams, also it was found that this
arrangement gave the smallest bandwidth for the structural
dynamic stiffness matrix of an axial assembly of elements.
36
4'.4 Axial Assembly of a System of Uniform Elastic Beam Elements
Equation (52) in the previous section is a general express
ion for force and corresponding displacement relations of a
single element. Consider two general elements to be joined
together in the axial direction. At the junction between the
n and n+1 beam surfaces, the sum of the internal forces must
equal the external forces, thus insuring force equilibrium.
Also, geometric continuity must be preserved, thus insuring
displacement compatibility. This implies that the displace
ments at the common junction for beams n and n+1 must be
equal. Figure 3 shows the convention for axial assembly of
the elements. The subscripts on 17,,"^ , and ^ have the follow
ing meaning; first subscript refers to the beam end (i.e.
1 - left end, 2 - right end), the second subscript refers to
the beam number. A left to right sequencing of numbers is
used for both beam number and the node number. (The term node
here refers to the ends of a beam or a common junction point
and should not be confused with the natural nodes or zero
deflection points of a vibrating system.) Here the system
consists of two general beam elements, n and n+1 (or beam n=l
and n+1=2) .
A free body diagram of the forces acting at the nodes in
the two respective coordinate planes, is shown in figure 4.
Figure 3 Numbering Convention for Axial Assembly of Beams
V/T"
fi,ZL
Mtc
/ t, yy i /Vic-tiuA. / I l(/?^X ^"4 (4^^ | 0\
'xCfitOZi
/
y^7~
s\( jTh
2<l^H-
(lit 1 1
A
x
-? Z.Ttr
/
QiOinlj.
Xj &2(nt/)J
37
Figure 4 Free Body Diagram of an Assembly of Two Beam Elements
V,T"
IO.H)
'HIx^L
0W)t ***t
fit*
Ql*)
,lCntO
^f/ / ltn+1)
r^^r//js:z01U)
>- 2,V
in,L\<
-3***1
ft- D^ cir^n'j)
XLnu)
7-M.
I
llMtl)Mn + 1)
A *-
X, u
38
Using equation (52) and the free body diagram, shown in figure
4 the force and moment balance equations for the assembly of
the two general elements may be written as,
r
-s
in
-M*
-s
IK.
y
IK
2\ 'Wh)
A*. KjuO
liy. lUtti\
Sax
02y
K E /L
r1
i Ke_ t KeCrjto'
J
KE(n fi)
f -\
Via
eu
fMM.
Vu\Vi(nti)
QxnrGi(fat)
Uo,ti)
SuntO
(54)
/
where the area labeled Kg is 8x8 and contains terms from the
dynamic stiffness matrix [[Kg"] for beam n=l, and the area label
ed Kw +1\ is 8x8 and contains terms from the dynamic stiffness
matrix Kp] for the beam n+l=2. In the 4x4 area labeled K+
KE(n+i) "the sum ^e common stiffness terms exist.
39
It can be seen that for an N element system the matrix
equation relating force and displacement can be written as;
where; T - 4(N+1) and N is the number of beams assembled.
is a column vector of loads applied at the
nodal points,
L-^J" ^s "^e structural dynamic stiffness matrix.
[^ 1 - is a column vector of nodal displacements and
rotations.
N-
40
4.5 Effects on the Dynamic Stiffness Matrix due to Fluid-Film
Bearings at the Ends of the Beam Element.
The previous sections were concerened with developing a
method for obtaining the steady-state solution of a system of
beams, assembled in the axial direction. Section 4.4 provided
the following matrix equation (54),
where JKSy] is a 12x12 matrix of real numbers and relates applied
end forces and moments to end displacements and rotations of
the assembly of beam elements. What is sought in this section
is the effects on the structure stiffness matrix [KsyJ due to
the addition of a fluid-film bearing at a node between the n
and n+1 beam elements.
Consider a thin massless bearing, acting at the node between
the n and n+1 beam elements. The bearing cannot create an addi
tional moment and serves to transmit shear forces and bending
moments between adjacent beam elements.
Figure 5 shows the free body diagram of the general beam
element assembly. The bearing forces which arise are due to the
positive beam displacements u, v and velocities u, v , at node
ntl. The dynamic bearing forces are represented by 4 linear
stiffness and 4 linear damping coefficients which are speed
dependent, Kxx, Kx , ...and Dxx, Dxy, ..., such as those
derived by Lund in reference [30] and take the form,
$RK~
Ku U1-
Kxy lr t Dxx K +0Ky l
(55)
fey - KyX IC + ftyy v + Pyx ^ f Dyy ir
41
Figure 5 Free Body Diagram of Beam with Bearing Forces.
y,ir
(nn)
-Fay f
/
/// * //jY//J7/y'///~^>
v
-5*
v,<^t<j
-Unrt)
f\i<j.
X,U.
(.t\)
vn\n
Sz*, LvtIA-
I) lil
,,'Cmi) .
(mi)
"5/X ^
-.ir- fuU-nt"-aa
i) (tmi) m - m aH
-7z<rui)
~
Fax 4*
where ^ and 4y are ----e dynamic bearing forces. Assuming the
forces and the displacements vary harmonically and representing
the bearing forces in complex variable form to account for the
phase angle between spring forces and damping forces, we obtain,
:at
^BnL~
( Fit* f^ Fdk^ ^~ Fbk
Lilt
'at_
hy^ C FKs, +* F0y)
<fc
'- Fey e
Equation (55) may be put into the form,
CSLt
Fe*eiXt
- (f^r.0 Ftx)^^- [[k ^JiDx-jU v {k^
r^SL^V\M
(56)
Fty(2Lat
-
(Fv U Fv)e*at'-L[^-i^yx3l7*-[Ky> t <7 su dyy}V]cy-t
42
equating the real and imaginary parts we obtain,
Fk<c
K<* V + K *- V
Fd<-
L Dkx: ~0 +-fl-Dx*,
,v
Fkv- -
Kyx V f Kvy V (57)
?o^- Si. Oy- LJ f il D7>V
where; Fkx'Fdx Fkv' Fdv" are ^he magni'tudes of "t^16 bearing
stiffness and damping forces.
Kxx, Kxy, Kyx, Kyy- are the bearing dynamic stiffness
coefficients.
Dxx, D,
D,Dyy- are the bearing dynamic damping
coefficients.
and all other notation is as previously defined. This type of
bearing representation assumes that no additional moment effects
are created by the bearing, i.e. no significant restraint is
offered by the bearing to the shaft slope. The following analysis
will allow any type of fluid-film bearing, for which the eight
dynamic coefficients are available, to be investigated. Rigid
bearings may also be represented by assuming large direct
stiffness terms (Kxx, K^) and all cross- coupled stiffness and
all damping terms to be zero.
Referring to figure 5 and applying Eulers first and second
equations of motion (i.e. equations (1) and (5) in section 4.1)
to the massless bearing, node (n+1), we obtain,
^(nt"y^
Hzm4 e-
r\y i c-
tvi cn^\
M'Lm,f, ,,sit
1 ' 2X aO
.
iJLt-
2>iK H e= O
.<** t.t
Kay Itc
1-Ka LSlt
Hyj*
- o
,llL- '. iLt
-
o
m?-jy
- o
43
Also applying Eulers first and second equations of motion to the
massless nodes n and n+2, we obtain,
at node n =>* AJ
V\iy <
Ml IL. ,
C7JU
uAJ:
o
o
'
a
-o
(57b)
at node n+2
-
Szx ^, C
'"
sy i e
IIIe
'/Vl
nan)4
i JLt
= o
-
o
(57c)
Multiplying these equations thru by a minus sign and arranging
these equations in the same order as those in the matrix form
of equation (54), we obtain,
3 2X
~*\
s*y
5
<
*?x
Me-ycr-7-
r
V.
Itrltl)
Inn)
Kzx -Mb/
_>.Cnt<)
^ ex
-Knnj
IY\ X ^
>,vS
= ^
O
o
o
o
"
Pex
o
inn;- fBy
O
O
O
O
O
"\
(58)
>)cct
or I F- 1cut
i-y2 sut
(59)
44
Substituting this relationship into the left hand side of
equation (54) we obtain,
{-FeW^~-U/U^yW^
(60)
Expressing the 12x1 column vector, i-Fgj , as the product of a
12x12 matrix of complex bearing dynamic stiffness coefficients
and a 12x1 column vector of complex nodal point displacements
and rotations (i.e. ^sy}), we obtain,
L-K6]l*^] = [K5y] [a5>] (6l)
or, bringing the bearing effects to the right side of the equation,
[o] = [LK-] f [K6}]{ASvi (62)
where rKs-J Is as defined by equation (54). Using equations (^6)
to expand |FgY , as stated above, we obtain,
[F.!^>Df-]fc,]eil=
0
Iy,x,
i
I
T~
I
4
O-r
ii-
[asJyt
(63)
IMIL
where ; xl
x2
X3
( Kxx + iQ.Dxx )
( Kxy + i^Dxy )
( Kyx + iiLDvv )
( Kyy tiaD^ )
'yx
}yy
(64)
^5
It is seen that the matrix [kg] contains complex terms.
Therefore, the displacements and forces will now be complex
and must be treated as such. Rather than break this matrix up
into two real matrices, to work with on the computer, the complex
matrix will be retained and all force and displacement vectors
will be treated as complex ,( i.e. F=FR+iFj , K=KR+iKj,
and a =
^1-
Z^x ) .
Equation (63) represents the terms which must be added to
the structure stiffness matrix [_KSyJ , as shown in equation (62),
and reflects the effects on the structure due to the addition
of a bearing at a general node between the n and n+1 beam
elements.
For the development to be completely general, provision
must be made so that the effects of a bearing may be added at
one end of the element and not necessarily the other. The
following sample problem will illustrate this point.
Sh* +-\<
rrrrrrrrr ' / / /-
/ f /
Because of the different cross-sectional areas of the shaft, at
least 3 beam element sections must be used to accurately
represent the model using the method developed in this thesis.
46
Model idealization would look as follows;
3-
?/,??/?>'/?, >
(D
_f
7T7T7T77
elt"i e.nf 5
It is seen that the effects of a bearing on the left end of
element 1 and on the right end of element 3, 'must be taken into
consideration. From figure 5 and equation (63) it is seen that
if element n were not there, the bearing effects could be
associated with the left end of element n+1. Conversely if
element n+1 were not there, the bearing effects could be associate:
with the right end of element n. Also, by eliminating one of the
elements the matrix reduces to 8x8. Two matrices, one to reflect
the effects of a bearing added at the left end of an element,
the other to reflect the effects of a bearing added at the
right end of an element, may be written as follows;
<
o
rev
o
o
o
0
o
ix/
D '
ia._M, Ua-bcj).
o o
o o 0
0 0
0 0
0 0
0 0
00
0 0
0 0
o
a
O
o
O
o
o o
o
o
{
\J,_
y
V",
x
Ux
>(65)
/<^i-x/
or tFi] - tKil^i
47
(66)
where; f^ - is a column vector of bearing forces at left end
jj<k]- is the effects on the stiffness matrix due to
the addition of bearings,
$^- is the column vector of nodal point displacements
and rotations.
<
o
o
D
O
o
r6y
O
y
H\
a<J o
o o o
o a
o a
o o
O
o
o
o o
(Kxv;o o
O
O
O
o D o
(Kyx-t
^-Dyx)O
(Kyy t
O O o a
<
u,
e,
a
a
>(6?)
or [f.M = Iff W<fv<r
<P*/
(68)
The complex matrices [_Kg J and [Kg] will be used in the
computer program to represent the effects on the dynamic stiffness
matrix Kg due to the addition of bearings at the ends of
beam elements. On the computer real matrices may be added
directly to complex matrices yielding a complex matrix. Therefore,
[KjjJ and [kJ may be added directly to [ke] giving a complex
structural stiffness matrix. This implies that the force and
displacement vectors must now be complex and therefore treated
as such on the computer.
48
4.6 Effects on the Dynamic Stiffness Matrix due to Disks
at the Ends of the Beam Element.
In the previous section the effects, on the stiffness
matrix [ke] , due to the addition of bearings to the system
were found, A similar procedure will be applied in this section
to find the effects on the stiffness matrix due to the addition
of disks to the system.
When a disk, whose diameter is large in relation to its
thickness, is to be included in the structure, it is necessary
to take into account the rotary inertia and gyroscopic or Coriolis
effects, as well as the concentrated mass effects, when calcu
lating the response of the structure, as discussed by Green
in reference L8j . The gyroscopic or Coriolis effects come about
from the rate of change of the angular momentum of the rotating
disk when the structure flexes and causes the angular momentum
vector of the rotating disk to change direction.
It will be assumed here that these effects are concentrated
at the e.g. of the disk and that the disk e.g. coincides with
a node of the beam. Figure 6 is a free body diagram of the
disk, showing the forces and moments acting on the disk, which
would be necessary to maintain the motion.
These are the elastic forces and moments of the shaft
acting on the disk. The sign convention for the positive direc
tions of angular motion are as shown and the small set of stationary
axes are parallel to the large set of axes,with its origin fixed
at the e.g. of the disk. Assume the disk is rotating at a constant
speed CTL about the oz axis. The equations of angular motion of the
49
disk with respect to its center of gravity can be obtained
by using the principle of angular momentum which states;
(with respect to the e.g.)
"The rate of increase of the total moment of momentum
of any moving system about any axis through the e.g.
is equal to the total moment of the external forces
about this axis."
[28]
The principal axes of inertia are such that they form a
Cauchy InertiaEllipsoid'
of revolution; that is, due to
symmetry of the disk there are an infinite number of principal
axes of inertia in the x-y plane. The polar mass moment of inertia
associated with the neutral axis z is defined as Ip, and the
transverse mass moment of inertia associated with any axis
in the x-y plane is defined as L. It is assumed that the disk
rotates about the neutral axis, z, at a constant angular velocity,
Si. The rotations of the disk about the x and y axis are
assumed to be small as defined by linear elasticity.
Referring to the free body diagram of figure 6, the moment
of momentum of the disk about its center of gravity is stated as
follows, with respect to the x, y, z axes,
He.}. =
Hc,j -f
^yy-f' iHc^"'h^'
(70)
' A
(71)
y- O 0
f0
7/<-' \ e
0 o J>J U
where ;
and :j_'
,
j' ii'are uni"t vectors which do not rotate with the
disk about the z axis but do rotate with the disk about the x
and y axes. Equation (70) can be expanded and written as follows;
li... - lrtf *!<>*(72>
Figure 6 Free Body Diagram of the Disk
50
Vt"V
OKIS^xx,
(Jl)
n-
W*y
X, ^
kki::
U/Oiii) -
tf
A*.
$ 2 ,-ur
H
/
ftCOyf'H+-
0tt
J>--0 -i^b in*1)
IH'
Ui-lYlt-l)
j.-v i 'tnt'J
Mzyf'
jM*
5^toil-**
-st,\ilnt-I
'*r
0.afc
4/
51
Figure 7 Free Body Diagram of Nodes with Disk Forces and
Moments Acting
EH\
IH.?V
**1
-F
y.}iu
ICHh)
.linn J
(/u
<7J yy
\\ c I) cf
_ iCimi
S*X^lAA'inH)
(/-')
Jcnt>)
,Mnti)
{/iH)
\H\
innl
Xtnti)
S2rX^
Euler gave us the fact that;
ftc.'j=
ILlaj- J'r ^y
'
* zT & i-i-
lfi (oJ k
tITt Ik. rXT Q H tlpl). cjjl
dt'
citr
dt
The time derivatives of the unit vectors ,
j.'
,
k'are
defined as follows; oj, =. H2 + &4.'
52
(73)
iiri
dl( a- / < /- ')
/
ciyfi
( ^i X * ')
1
J rJ
t
e o
I 0 0
f'J'f
H: & o
o i o
'
j
'
67
\{i G O
-
Cr. t^
'f K
C--i-
- I-
(74)
Equation (73) can be written as follows when substituting the
time derivatives of the unit vectors derived in equation (74) j
hc.<r=
Ir il'
* lr ^j'
-It
*'
&x'
+
lrie*'
/-
I-p sl o A
'-
Xp y t +'
Dotting equation (75) withi'
and j/ gives the following two
scalar equations,
(75)
A,.; x
/Vcty
=
xr v +iP & ~y
-- xT &-tp
i- y-(76)
53
be simplified to obtain,
#c.t.x=
IT % +IP jzl. ^'(77)
Hep y^ IT &
~
Ip St ^_
Assuming harmonic displacements, where the frequency of ex
citation is equal to the rotational frequency, , equation (77)
may be rewritten as ;
LyA - c-Zjsyr + * i? sii&)***-*
'
,
(78)
fic.Jy '- C-ITSLX& '*Ifj^Y)^^
where; Ip- is the polar mass moment of inertia of the disk.
I.J- is the transverse mass moment of inertia of the
disk with respect to the e.g.
0 - is the angular rotation in the x-z plane.
If - is the angular rotation in the y-z plane.
SL- - is the frequency of rotation.
Referring to figure 7, and applying Euler's first and second
equations of motion; ("i.e. equation (1) and (5) in section 4.1)
to the end nodes and the middle node where the disk is acting,
and arranging the equations in the same order as those in the
matrix form of equation (54), we obtain,
= ci> ZX JU
m2V f
M^ i-
o
SZy^- <szi
-
MD" ^,Un*t)
xk.. ?i
i'"{?J *C*U (79)
-
sJOl+"
= o
IAAXlnU>*
O
HI ,
U
o
o
54
Multiplying equation (79) through by a minus sign and
arranging the equations as two column vectors we obtain,
F^ e IK-
-> ^X
Cn ft.)
<
-
*Vl *y
cIre
~
->
y'
1*1 2*C l\ c 'Cit-i)
S&- S&T>
Mix- M ex
SXCn-
C UrU-. )
2>^yvtfCnYl)
V <
o
o
-Y\\bcf-
O
o
o
a
>(80)
or
[Fs,] [-Pol(81)
Substituting this relationship into the left hand side of
equation (54) we obtain,
[-fD] * lKJy]{^y]<82)
Expressing the 12x1 column vector, [-fA , as the product of a
12x12 matrix of complex disk dynamic stiffness coefficients
and a 12x1 column vector of complex nodal point displacements
and rotations,
L-kH-y1' Lk,v] {Aiyj
(83)
or bringing the disk effects to the right side of the equation,
lo]IUi
= LLKsv)^ [],,J {~.yL,(81t)
55
whereLKsyJ ^
is as defined by equation (54). Using equation(78)
to expand ?p0\ , as stated above, we obtain,
[h] '- Lkd] Kl
4-
i
fO'
%
i h '
1If J<A
10
[*v]
nxti.
(85)
where ;
x4
X5
x6
-- MDIL
= - I-piL
- -llp-ft-
-
-md sy
= - I--^
(86)
It is seen that the matrix L K^l contains complex terms. Therefore,
the displacements and forces will be complex in the same manner as
in section 4.5, where a bearing was added to the system.
Equation (85) represents the terms which must be added
to the structure stiffness matrix [Ksy^J , as shown in equation(84) ,
and reflects the effects on the structure due to the addition
of a thin disk at a general node between the n and n+1 beam
elements.
56
For the development to be completely general, provision
must be made so that the effects of a disk may be added at one
end of the element and not necessarily the other, as was
pointed out in section 4,5, where the addition of bearings to the
system was considered. As discussed in section 4.5, the effects
may be associated with either the right end of element n or the
left end of element n+1. Two matrices , one to reflect the effects
of a disk added at the left end of an element, the other to
reflect the effects of a disk added at the right end of an
element, may be written as follows,
[fo]
Xi
Xfhl
Ii_-_.
o
"V,,-
Uxk
VI
W///
or
where ;
If;] - iKii {a] (87)
xl-xg
- are as defined in equation (86).
is a column vector of disk loads at the
left end.
is the effect on the stiffness matrix due
tothe addition of a disk
is the column vector of nodal point dis
placements and rotations.
fi]
Similarly,
57
or
where ;
M
S :
Xi Xj
XT
<
r
X7M
>-
Jr*-
[fl] [Ki] I -j (88)
Xjl-x^- are as defined in equatio (86).
is a column vector of disk loads at the
right end.
is the effect on the stiffness matrix due
to the addition of a disk.
is as previously defined.
Matrices [K-- | and Lkd -i
are seen ^ contain complex expressions,
thus causing the stiffness matrix to become complex. The same
complex formulation is used here as was described in section 4.5
for the bearing matrices.
y\
58
4.7 A General Unbalance Force Vector
In section 4.3 the dynamic stiffness matrix for a uniform
elastic beam was developed. Sections 4.5 and 4.6 developed ad
ditional stiffness matrices which could be added directly to the
beam dynamic stiffness matrix to account for the presence of
either a bearing or a disk added at either the left or right
end of a beam section. All of the above mentioned effects are
terms on the right hand side of the general equation.
[Fich L,<5r~- ^s^ (89)
where iFkJ- is a general vector of applied nodal point
loads for the whole structure.
LKjtJ _ is -t^g structure dynamic stiffness matrix con
taining any beam, bearing or disk effects on the
structure.
(. STi - is a vector of nodal point displacements and rotati;
This section is concerned with the development of a general
load vector for the left hand side of equation (89). The only
external forces which are assumed to be acting on the nodal points
are due to a specified unbalance in the system. The unbalance forc:
are assumed to arise from a small mass MQ , which has
eccentricity, a, from the geometric center of the beam as shown
in figure 8. The eccentric mass gives rise to a centrifugal
force about the shaft axis which rotates with the shaft speeds .
Referring to figures 8 and 9, and applying Euler's first
and second equations of motion(i.e. equations 1 and 5 in section
4.1) to the end nodes and the middle node where the unbalance
weight is added, and arranging the equation in the same order
Figure 8 Unbalance Force
59
-jra Ve7,-iUt
ty-uy^
dCOiH-t
O-W^t
iV
Figure 9 Steady-State Unbalance Forces
^A^y
60
as in the matrix equation (54), we obtain,
~>2X ^-
o
M,e7 -~
a
$ ^ ff-
o
S%Ti-
SXI*<2-~ Wo & <*- ^ "J^;
f(t-t). <~2l Iii"
M'iTl '^a2 '- O (90)ciCntil
The right side of equation (90) is evaluated as,
filo JiJ-tlA. f-cu /w^ sidr)- A1oT4
"
/Ho a.su*-
cosjy-t
M ft^Cir -h a^sui)- MP- M, a, sy^^iut
Assuming- that the terms /%%! and /}ja^ will contribute very
little effect to the mass acceleration terms in comparison
to similar terms due to the rotoror"
disk mass, they may be
neglected. Also, representing cosslI - ty***) and sinTU -j^_-<7e Ji.n
complex variable notation, the nonzero terms on the right
hand side of equation (90) become,
%1-Aha.sI e^l (91)
Substituting theje expressions into the right side of equation
61
(90) and multiplying the set of equations through by a minus
sign, we obtain,
or
r
A
L
-
HA zy-
S ay
**&
S zx
/ Cut.)
-Sax
c iCHh)~
>3y
hi:- ^r'
"Cnt-.l
S*x
^u7
^
w*;Cnf(_)
V
yy\yy
r
<
^
p
o
O
0
Moa.Sb'-
o
-UttfoCLy
o
0
0
o
o
-\
>
J
(92)
(93)
The unbalance forces at any node may be accounted for as point
effects which are described by equation (92) and in this manner
a general force vector may be built for any system.
62
4; 8 Moment and Shear Balance at a general node
Assume that at a general node in the system, a bearing, a
disk, and an unbalance exist and their effects are as previous
ly derived. Figure 10 shows the forces and moments acting at
the general node.
Figure 10 Forces and Moments acting at a general node
XV"
-
Fali
a s^At
syinj /%,
*&>,
1".
\) Ci CH.)
a*
-flfeyj
) (1
acoial
t
I) Cf
FfixttL
CHH) Cl
+>\
1c
63
where j_p
. . r- are the bearing forces in the
J
x-z and y-z planes respectively.
__ C _ M A^j
->*xx>
i-jvy ^ _ are ^he shear forces and bend in
y 'moments m the x-z plane.
-ciK
n/i*n <
- are the shear forces and bending>iy+ )"M?X:
S'^f'+ j 01 i*"^ moments in* the y-z plane.
Referring to figure 10, and applying Euler's first and second
equations, we obtain,
JgrXL U
MiK
y t
?O
Ji? . .
y \ -W1./
7
s*yf 5zy f 6? j-mow, *v /710<lso i
/>1 sx ^ /^**^
-
Hc.j*
~
S7>x-
o
Way ^
0?* -c
Rearranging equation (94) so that only the unbalance force terms
are on the left hand side, we obtain,
64
r
<
o
o
O
O
WUolSiI
o
-aOIM0Cl.I
o
o
0
0
o
\
>
- c:,tv
34X
-
/-Uy
5z.y-
rvi*xAn ich+)
b sx "Sax
-^y ->-z-y
w*M
-
c"(H*'l
->*x
HI""?''J
l-V| --y1L1 +W
5yl/yl aCntO
/ 1
M
o
'Vy > + <MOV
k.j-x
o
o
o
6>
o
o
o
l"6y>
o
0
o
o
o
(95)
or [r]= i^j " IM f I F83 (96)
[fk]" L^H^vj ^ LKHA^J f La) i/^i (97)
A]Wl-- LLk,vJ +U.] ^UeJ]^ U.vL, (98)
Equation (98) is the general equation for shear and moment balance
at the nodes in the structure. All of the terms in the equations
were previously derived in matrix form in equations (5^), (63),
(84"), and (92).
With these expressions any general rotor system may be
divided up into a number of elements. At the ends of each element
any or all of the following conditions may exist.
65
a), a bearing may be acting.
b). a disk may be acting.
c), an unbalance force may be acting.
d). another element may be acting.
In this manner the elements may be axially connected to represent
a total system. As an example consider the simple rotor-bearing
system shown in Figure 11.
Figure 11 Example Rotor Bearing System.
PisK
fill to~a
T
t77,
b Fluid- fii,
Qeurinc.S
3-
I'i- <**^Sa
The system consists of a uniform shaft supported at its ends in
Fluid-Film Bearings, a disk centrally located on the shaft and
an unbalance assumed to be acting in the center plane. The pro
cedure developed may be applied here and the system may be ideal
ized as follows.
06)
c/m'-
RoHi
CS)
UnboJa^ct
l Senrin'j
77-7777V7
C3) no/m
e les*.**/-
1
////Va? ->S
66
The system is described by two uniform elastic beam elements
and three nodes. At nodes 1 and 3 the bearings are assumed to
be acting, thus equations (65) and (67) can be used to describe
their effects on the system. At node 2 an unbalance force and
a disk are assumed to act (these components may be associated
with either the right end of element 1 or the left end of el
ement 2, since this node is common to both elements). Equations
(87), (88), and (92) can be used to describe their effects on the
system. The resulting structural matrix to be solved is ?
r ~\
o
<
o
0
o
Fu*
o
Fi-v-
6
o
0
0
o
v.Uil
Xi1 '
1\
~
r -\
-h <i *B 0 *l X, 0 0_..
0 0 a u,.
A >-. O 0, X, *. ;
0 0 a 0<8>
*B 0X/ +
*6
i
*lj
a
'
*(. -
6> O
j
O Vi.
0 0 x, X,'0 0 x, */
0 u 0 0 r
12. *'..0
FxT+Xt+ 0 0 ^ Xi. 0 0 VH
x,
0
Xi
0
6
x>
0 '?
*'l
X,t*-
X, + Xv
0f*o
0
0 0
0
!*-
6
Xv-
< &y-9,\
Zi *V,u
0
0
0
0
X\
0
1
*,l
O1
yCx
1
"1
^ it
xY 0
0
x-
xo
0 *>-i
!
xe 0VI
0 0 0
I
0 1 Xi Xv tf > xT-
Xv 0 6 G^
0 0
!
0
\
0 1 0'
j
c- x- XL Xa 0 Xv VI
0
16 6
10 | 0 0 A X,
i0 ix. tfl-
1
11X12.
ny
67
where the x's indicate locations in the structural stiffness
matrix where nonzero terms appear.
where ; x.. - represents terms from the beam element dynamic
stiffness matrix, kd^] for element number 1,
as described in equation (52).
X- represents similar terms for element 2.
Xg- represents location of bearing terms as described
by equations (65) and (67).
Xq- represents location of disk terms
as described by equations 587) and (88).
The terms in the force vector are as described in equation
(92). In order to solve for the nodal displacements the structural
stiffness matrix must be inverted. A general purpose computer pro
gram has been written for the assembly of up to seven elements
and eight nodes. (limited only by the computer core available).
The program constructs the structural dynamic stiffness matrix,
which is in complex form due to the addition of bearing and
disk terms. The program also constructs the complex force
vector. The stiffness matrix is then inverted and postmultiplied
by the complex force vector to give the complex displacement vector.
The displacement vectoris then used, as discussed in Appendix B,
to calculate the steady-state whirl orbit information, i.e. the
major and minor semiaxis of the ellipse and the ellipse angle
from the positive x direction to the major axis of the ellipse.
The above sample problem could have been solved using center-
line symmetry, thus reducing the problem to an 8x8 matrix as follows,
68
<
( ^
O
o
a
o
>"kx
0
[yy
o
>
+ A |* o *l *l tf 0 Vi,
<i *l o U * v,0 0 O,
x, 0
v4
X+G*l 0 o >-! X| 'Vi,
0 0 *i *i 0 o A \ *\<
A !-. O ;u i o | o
i. j
! -l//.
<i *, 0i*
+ <oXo e^a-
6 0 x. X, c1
X, M XT,
v;=7
0 c; *l
1o Xo
Xi
Xl ,
Xitx'i.
+Xq >= ifJ
y
This may be done for any symmetrical rotor, but , in general
for an unsymmetrical system (i.e. over hung disk, out of
phase unbalance forces etc. ) the matrix cannot be reduced
and the method developed offers an easy assembly procedure
for analysing the steady-state response-
of:'a general rotor
bearing system.
69
5.0 PRESENTATION OF EXAMPLE PROBLEMS
Based on the element developed a computer program (ROTOR)
was written for the matrix computation procedure outlined. Two
other programs, based on alternate techniques, were also used
for the comparison of results in some of the example problems
presented. One of the programs was FINITE5, a finite element
program written for the steady-state unbalance response of rotors.
This program was written by Ruhl in reference [l6] and is used
here for a comparison of results in test case two. The second
program was 1MASS, a program based on the equations developed
in Appendix C for the unbalance response of a lumped one mass
model in identical end bearings. This program was written by
the author and was also used for a comparison of results in test
case two.
Four test cases are presented to provide insight into the
capabilities and limitations of the program developed. The four
test cases to be presented are;
Case 1. Critical Speed Map for a Uniform Elastic Rotor.
A simple rotor with uniform cross-section supported in
simple end bearings (no cross-coupling stiffness or damp
ing coefficients) was investigated in test case one. This
test case was selected to investigate the effect of support
stiffness on the critical speeds of the rotor system. Also,
the developed procedures capability of representing the
continuous mass and elastic properties of the rotor is
investigated. By applying a series of unbalance response
70
calculations, varying the support stiffness with each
calculation, a critical speed map for the model is gen
erated. The map is obtained by plotting the predicted
critical speeds vs. support stiffness. This critical speed
map is compared with the theoretical results of Linn and
Prohl in reference [9 J ,and the results agree within 2^.
This test case demonstrates the programs capability of
accurately predicting the critical speeds of a rotor-
bearing system.
Case 2. Unbalance Response of a Uniform Elastic Rotor Supported
in Fluid-Film Bearings.
A simple rotor in complex end bearings (all eight bearing
coefficients) was investigated in test case two. This test
case was selected for investigation to determine the
effects of the bearing asymmetric stiffness and damping
properties on the elliptical whirl orbits of the rotor.
Results are compared with those obtained from Ruhl's
program and the one mass model program. The results
demonstrate the programs capability to predict the
elliptical whirl orbits of a rotor-bearing system.
Case 3. Overhung Disk on a Uniform Elastic Shaft in Rigid Bearings.
In test case three a shaft supported in rigid bearings
with an overhung disk was investigated. This model was
selected to analyse the effects on the fundamental critical
speed due to varying the length of the overhung portion
of the shaft, due to gyroscopic coupling and different
concentrated masses of the disk. Results are compared
71
with the experimental results obtained by Dunkerley in
reference [ 2J. The results demonstrate the programs capa
bility to analyse overhung rotors. No added difficulties
were encountered in applying the procedure to this more
complex rotor-bearing configuration.
Case 4. Lund and Orcutt Test Rotor (MTI rotor) .
Presented here are the results of two overhung rotor con
figurations, a one disk model and a three disk model.
The rotor is supported in flexible bearings. This model
was selected, for investigation since it tests all of the
capabilities of the developed program in a single con
figuration. Results are compared with the analytical
and experimental work of Lund and Orcutt in reference [15] .
The results demonstrate the programs capability to repre
sent a complex rotor bearing configuration where, disks,
bearings, and unbalance forces are working in the system.
Each test case gives full details of the models elastic and
geometric properties and the idealization used to obtain the
results. All results are presented in either tabular or graphical
form. When applying the program in the test cases the following
problem areas were identified.
1. At each speed increment bearing stiffness and damping
coefficients had to be entered. This created extensive
computer input decks for each run.
2. Material properties for the beam elements were assumed
to be the same, thus composite material problems could
72
not be investigated.
3. In generating the critical speed map, an unbalance
response curve had to be calculated for each point
on the map.
4. The overhung disk investigated in test case three
was an unsymmetrical system. Thus, the bearing
forces should have been different for each of the
bearings represented. This was not possible to re
present since the program assumes identical bearings.
5. To excite the modes which were not symmetrical about
the mid-span, other than mid-span unbalance was
necessary. Although this involved only forming a
new idealization, it brings out the important fact
that the axial location and out of phase application
of the unbalance forces dictates to what extent the
higher modes will effect the rotor response.
73
5.1 Critical Speed Map for a Uniform Elastic Rotor
The critical speeds of a high speed rotor are a primary
concern for the rotor-bearing design engineer, since at these
speeds the rotor amplitudes and transmitted bearing forces are
at a local maximum. This test case is presented to demonstrate
that the dynamic stiffness matrix technique can be used to define
the critical speeds of a rotor-bearing system. The effect of rotor
support stiffness is also investigated and presented in the form
of a critical speed map (speed vs. support stiffness). By plotting
rotor response vs. speed, the critical speeds may be found at
the speeds where the rotoa? amplitudes are at a maximum. Having
found the critical speeds for a particular rotor system, other
design parameters may be investigated for their effects on the
system critical speeds.
A rotor experiencing synchronous whirl (unbalance whirl) near
a critical speed will assume the characteristic mode shape
associated with that critical speed. At low speeds and for low
values of support stiffness, the rotor will adopt the rigid mode
shapes (i.e. translatory and conical whirl modes). As the bearing
stiffness is raised the rotor becomes more restrained at the ends
and will start to bend. When the bearing stiffness becomes large
in comparison to shaft flexibility, the rotor becomes pinned in its
bearings and as discussed by Rieger in reference [29] the bending
critical speeds are defined by,
74
where ^L takes on values ft , 2 /? , 3 7? , ...etc.
Figure 12a shows the rotor-bearing model used in this analysis.
The rotor mass and elasticity are uniformly represented as derived
in section 4.3. The value of E used here was 28.5 x 106psi and
the value ofj>
was .283 lb. /in. 3 . The bearing supports are rep
resented as derived in section 4.5. For this test case the cross-
coupled stiffness and all damping coefficients are assumed to be
zero so as not to unnecessarily complicate the analysis. To excite
the first and third critical speeds, mid-span unbalance was needed
so a model idealization as shown in figure 12b was used. The rotor
was divided into two uniform elastic elements and 3 nodal points.
The bearings were acting at nodes 1 and 3 and the mid-span unbal
ance at node 2. An arbitrary unbalance of .33 oz-in. was used so
that resonable rotor amplitudes could be calculated and plotted.
To excite the second mode a model idealization as shown in figure
12c was used. The rotor was divided into four uniform elastic
elements and 5 nodal points. At nodes 1 and 5 the bearings were
acting.
An unbalance of .33 oz-in was placed at node2 and .33 oz-in
180
out of phase from the unbalance at node 2 was placed at node
4. This was done since node 3 (mid-span) is a true node (zero
deflection point) for the second mode.
The solid lines in figure 13 show the theoretical critical
speed map for this particular rotor model, this was calculated from
Linn and Prohl reference [9]. The circles are points found using
the method developed in this thesis. It is seen that good correla-
75
tion is obtained. Typical mode shapes corresponding to specified
end conditions (low, medium, and high values of support stiffness)
are shown in figure 13. Figure 14 a-f show the mode shapes found
from this analysis and are seen to correspond with the expected
mode shapes. Figure 15 shows .typical unbalance response curve
from which the critical speed is determined.
This test case has shown that the technique developed is
capable of defining accurately the critical speeds of a rotor-
bearing system. Table 2 gives a comparison of theoretical critical
speeds vs. those found in this analysis. It is seen that the
largest percent difference is 2,0%.
It should be noted that few real bearings retain constant
stiffness with speed change. But, as noted by Rieger in reference
[22] it is only necessary to plot bearing stiffness characteristic
over the rotor critical speed lines, then the point of intersection
are the critical speeds for the rotor-bearing system. Also, the
damping effects are neglected but may be taken into accout by
calculating an effective stiffness and plotting this over the rotor
critical speed map to obtain a closer approximation to the true
critical speeds.
In subsequent test cases it will be seen that the actual rotor-
bearing system critical speeds may be found directly by plotting
the unbalance response of the system vs. speed, since the analysis
includes all the speed dependent stiffness and damping character
istics of the bearing.
76
a).
Kxx Kyy
<h 100
EAIf
-*
12"
TKxx Kyy
b).
Kxx
33 oz-in
EAIf EAIf
node no.
kxx, ...
////// /77777
0> <3> <8> d) node no.
c).
^x*
I,.16 oz-in
EAI ./ \I- EAI/ { EAI/
1.16 Oz-in v
. EAI f ,
<K.
^^xx'
//////
Fig. 12 Rotor Model for Test Case .J .
77
-crnx>r-<x>m ^r co caj -UKOh to m ** ro oj
v\o
WdH peadso
CM
Fig, 14 Mode Shapes for Uniform Elastic Rotor78
a), lst rigid mode ( trans latory), K = 10 lb. /in., N 149 rpm.c
-
o
---.<r
X 10 ir,.
b). 2nd rigid mode (conical), K = 10 lb. /in, , N. = 255 rpm.
-f- 1.0
o
-I-
-1-0
X 10 IK
c). 3rd (free-free bending mode), K = 10 lb. /in., N = 12650 rpm.c
+ zo
-?,
O X!0 ir.
-2,0
o
d). lst bending mode, K = 10 lb. /in. , Nc= 5550 rpm.
- MO
- O
---MO
x lo'^i.-.
o
e). 2nd bending mode, K = 10 lb. /in. , Nc= 21750 rpm.
-- H.O
-
o
--
-M-Q
x 10 *, -.
8
f). 3rd bending mode, K * 10 lb. /in., N = 47500 rpm.
-- i.o
O X. IO in.
4 -M.o
100
Fig. 15 Typical Unbalance Response Curve for Test Case 1
1st mode K = 10 lb. /in.
A
79
10.
1.0
Amplitude
in.x
.1
Whirl radius at midspan in mils
(10"
vs. speed in rpm.
01
5000
Speed , rpm.
5200 5400 5600 5800 6000 6200
80
Table 1
Critical Speed Results for Test Case 1
Support Stiffness
(K lb. /in.)Crit. Speed(rpm)(Linn and Prohl)
Crit. Speed(rpm)
(Predicted)% Difference
lst mode
103
IO5
610
IO7
io8
150
475
1427
3637
5204
5484
149
470
1435->
3675
5300
5500
.66
1.0
.6
1.04
1.84
.3
2nd mode
103
10*
IO5
io6
107
io8
254
801
. 2490
8002
17627
22104
256
810
2545
7850
17750
21700
.79
1.10
2.2
1.9
.7
1.8
3rd mode
IO3
410
lo5
io6
io7
io8
12703
12703
12803
16116
32177
47566
12650
12700
13000
15900
31500
47500
.4
.02
1.5
1.3
2.1
.14
81
5.2 Unbalance Response of a Uniform Elastic Rotor Supported
in Fluid-Film Bearings.
All rotors retain some degree of residual unbalance, even
after balancing [29] . This unbalance causes a rotor to whirl,
and, for the undamped case, the whirling will be maximum when
the rotor speed is coincident with any of the systems natural
frequencies. The unbalance whirl is a stable whirl, since at
successive rotations the whirl orbits traced out are identical,
under steady state conditions. The unbalance response of a
rotor- bearing system involves the calculations of rotor amplitudes
at specified points along the rotor axis and at specified speeds
throughout a speed range.
The mechanics of shaft whirling have been explored by many
researchers as discussed in the literature survey. The early
investigations were generally carried out with a one mass model
and were concerned with establishing the nature of the whirl
motions and investigating the effects of certain parameters;hystere-
damping, flexible bearings, massive pedestals, etc.. This present
test case demonstrates that the method developed accurately
predicts the unbalance response of a simple rotor in complex
bearings(all eight coefficients present). As discussed by
Lund [13J ,the whirl orbits are, in general, elliptical due to
the asymmetric properties of the bearings. The elliptical orbits
will also be tilted with respect to the fixed load line due to
the cross-coupling terms in the bearing representation.
82
Figure 16-a shows the rotor model under concideration in this
analysis. This model is presented by Ruhl [16] and is used here
so a comparison could be made with a previously investigated case.
Appendix C gives the derivation for the response radius at midspan
and at the bearings for the one mass model representation shown
in figure l6-b. This simple model was developed to give a guide
line for the results from both Ruhl's finite element representation
and the dynamic stiffness matrix representation developed in this
thesis. Also the shortcomings of the one-mass model are investigated.
The test rotor had a total weight of 100 pounds and was
supported in identical short (Ocvirk) bearings. As discussed by
Ruhl [16] , the operation of the rotor bearing system was assumed
to hold the"static"
eccentricity ratio equal to .5throughout
the speed range. Physically this is not true as shown by Lund [3C'J ,
and many other experimenters on the dynamic properties of
fluid-film bearings. This assumption was made to simplify the input
to the computer programs and provides a qualitative example
for comparison between the one-mass model, Ruhl's finite element
model and the dynamic stiffness matrix model. Specific information
about the model is listed in Table 2.
Figure 17 shows the comparison of the one-mass model results
with those found by the dynamic stiffness matrix formulation,
assuming no cross-coupling terms acting. It is seen that the
one-mass model adequately identifies the first bending critical
speed at approximately 5700rpm, but unsatisfactorily represents
thr rotor response, especially at the bearings. Looking at the
83
Table Z
Test Case 2 - Model Description
Rotor
Length = 50"
Diameter=3"
E = 30 xIO6
psi
f -
.283
lb/in3
Midspan Unbalance =1.0 0z-in.
Bearings
eccentricity-
.5
clearance=.0005
in
kvv= 283300.0 lb/in
kxy= 400000.0 lb/in
k = -83000.0 lb/in
k = 216600.0 lb/in
(caLa^y- 658OOO.O lb/inXA
(W0xy)= 227000.0 lb/in
(qjdvy)= 227000.0 lb/inyx
(wi>yy)= 300000.0 lb/in
Speed Range
1000 rpm- 14000 rpm
1 st. bending critical5720
rpm
84
bearing response it is seen that after the critical speed is
passed, the distributed mass model predicts higher response values
than does the one mass model. This is resonable since the
distributed mass model by the nature of its formulation includes
the effects of all the system modes in each calculation as dis
cussed by Rieger U.8J . Thus the effects of the higher modes,
will continue to represent the rotor response after passing
through the critical speed.
Figure I8a-b show examples of the model idealizations
used in the analysis by ROTOR and FINITE5. Idealizations of
2, 4, and 6 elements were calculated using Ruhl's finite element
model and the dynamic stiffness matrix model. This was done in
an attempt to investigate the effect of shear deformation
on element refinement. It was found that this was not an
adequate test of the shear effects, since no calculations for
comparison, including the shear terms, were performed. The author
recommends that the shear terms be retained in the equations
of motion developed in section 4.1. Then the corresponding terms
may be developed for inclusion in the dynamic stiffness matrix.
The computer program, FINITE5 , given by Ruhl in reference [l6]
was punched up by the author for use in this analysis. Basically
the results show that both the finite element (FINITE5) and
dynamic stiffness matrix (ROTOR) idealizations converge with the
2 element model and futher model refinement (4 and 6 element
solution) gave practically identical results. Table 3 presents
the results for the 2, 4, and 6 element solutions at two speeds.
Figures 19-20 give the response curves.
85
Table 3
Results for the 2. 4, and 6 Element Idealizations of Test Case 2
at 5500rpm and l4000rpm Using ROTOR and FINITE 5
(values for Major Orbit Radius at Midspan)
Speed No. el. FINITE 5
(in)
ROTOR
(in)
% Diff.
(rpm)
5500 2 .14754
10"1
.15452
IO"1 4.73
4 .14754
IO"1
.15452
IO"1
4.73
6 .14754
IO"*1
.15327
io'1
3.88
14000 2 .13706
10"2
.12797
10"2
6.63
4 .13706
IO"2
.12797
10"2
6.63
6 .13706
IO"2
.12807
10"2
6.56 |
% Diff.
.8
.08
At 5500rpm (near the first critical) the response calculated
by ROTOR is 4.73% higher than that calculated by FINITE5. At
l4000rpm the results are 6.63% lower. Two possible explanations
for this result are;
1. It is known that the finite element method, based on the
displacement formulation, provides an upper bound to the
true stiffness of the structure. Thus the calculated
response is lower.
2. The distributed mass formulation gives a better distri
bution to the mass, thus predicting a lower critical speed
i.e. the response curve is shifted to the left thereby
giving higher amplitudes for the same speed.
86
Looking at the results for the 2, 4, and 6 element solutions,
it is seen that those calculated by FINITE5 are the same for
each idealization. Those calculated by ROTOR are the same for the
2 and 4 element solution but vary for the 6 element solution.
At 5500rpm the % difference is .8% and at l4000rpm it is .08%.
The 6 element solution had a length to diameter ratio of 8.33/3. 0.
Thus for this configuration it is indicated that the shear effects
are negligable. However, nothing definite can be said about
the shear effects, especially at the higher speeds, since a
thourough investigation was not carried out.
A comparison of the finite element model results to those
found in this analysis for the two element solution is shown
in figure 21. The cross-coupling stiffness and damping terms were
not included in the analysis. It is seen that both formulations
give practically identical results. It is seen from the graphs
that both models define the two criticals introduced from the
fact that the bearing stiffnesses in the x and y direction are
unsymmetrical(only a few points are shown on the graph in this
critical zone so that the shape of the curve may be visualized,
but computer runs were made with fine speed increments to
completely investigate this area of the curve and the results
do substantiate the curve). The inclusion of the cross-coupling
terms tended to increase the response at the bearings and had
little effect on the response at the midspan. This is shown in
figure 22 for both the finite element model and the distributed
mass model. Neglecting the cross-coupling terms predicted responses
at the bearings which were approximately 50% lower.
KxxKxy
DT,Dv,r,..xx"xy'
}//////
87
50"
\1.0 oz-in
E A I w 3.0
T\ ii
KxxKxy'"
xx--- xy'
S77777
a) . Uniform Elastic Rotor Model
E = 30x10 psi
w =.283
lb/in3
<&-
W,a
K,
KC-
yyy/f
50"
1.0 oz-in
Ke
K O D
//////
b) . One mass model
W = 49.2 lb.
a = 1.27xl0"3in.
K = 22944 lb. /in.
Fig. l6 Rotor Models for Test Case 2
100. 88
2
Fig. 17 Unbalance Response- no Cross Coupling
Test Case 2- ROTOR. 1MASS
1 mass model
-o-o- Distributed mass
model
10.
Amplitude
in. x 10
1.0
.10
Orbit radius in,mills (10 ) vs.speed in rpm
01iitt.irltliniir
10 lu III,' I,n I,
2/
3 ii 5
Speed rpm. x 10J
, I t
6 7 9 10 il 12 13 14
89
>':::Ul..; r\:j
50"
25"
E A I w
1.0 oz-in.
E A I w
Kxx,Kxy#'"^ im
Dxx&xy L~y
j_3.0"
T Kls>xxrvxy#
DxxDxy*
2 Kxx'Kxy
ty
//////
a) . Idealization for Uniform Elastic Model
Distributed mass formulation
50"**Jf-
25"
KxxKxy *a\
DxxDxy *1
E A I w
1.0 oz-in.
E A I w
*I
3.0"
I K
//////
xxDxy
S77777
b). Idealization for Finite Element Model
Consistant mass formulation (Ruhl's model)
Fig. 1.8 Distributed mass and Consistant mass Model
100. v.90
4
3
Fig. 19 Response Curves for 2.4 and 6 Element
Solutions- Test Case 2- FINITE5
2 element s
l\ elements
6 elements
10.0
Amplitude
in. x 10
1.0
.10
ius in mils (10 ) vs.
rpm.
,01
Speed rpm x 10
6 7 8 9 lo 11 12 13 14
100.0 91
j
-I
!-
- i
Fig. 20 Response Curves for 2.4 and 6 Element
Solutions- Test Case 2- ROTOR
2 element s
^ elements
6 elements
10.0
Amplitude
in. x 10
1.0
.10
.,01
mi,Speed rpm x
1Q-
56789 13 14
100.
Fig. 21 Unbalance Response- no Cross CouplingTest Case 2- FINITE* ROTOR
92
10.
Amplitude
in. x 10
1.0
10
01
Finite Element Model
Distributed Mass Model
Major and Minor Orbit
radius at midspan
Major and Minor Orbit
radius at bearings
-Cr
3
Orbit radius in mills (10 )speed xlO3 rpm
vs.
Speed rpm x10-
5 6"? 8 10 11 12 13
100. 93
Fig. 22 Unbalance Response- with Cross Coupling
Test Case 2 - FIN ITE 5. ROTOR
Finite Element Model
Distributed Mass Model
10.
Amplitude
in. x 10
1.0
.10
01
Speed rpm x10'
56 7 8 9 10 11 12 13 J.t
III 1,1 ll,, In. I,
94
5.3 Overhung Disk on a Uniform Elastic Shaft supported in
Rigid Bearings
This test case is presented to investigate the developed
procedure's ability to analyse shafts with overhung disks.
The model choosen was one of the clasic experimental cases
performed by S. Dunkerley in 1894. The specific set of
experiments are refered to as Case XI in reference {fZ J and
is described as follows.
"Shaft resting on two bearings and overhung on one side,
loaded with a pulley at end of overhanging portion". O"]
The model is shown in Fig. 23. The shaft length L, was
32.18 in. and the diameter D, was .2488 in. . The shaft
weighed .4414 lb. and the experimentally determined value
of Young's modulus E was 27-974x 106psi. The value of weight
density j , was .282 lb. /in.3.
Dunkerley presents experimental results for two different
pulley (disk) sizes and for each pulley size six cases
are presented by varying the ratio of the overhanging portion
of the span. Thus 12 different configurations for the same
basic model are presented. The physical properties of the
disks used in the two models are listed in Table 4.
Table 4.
Disk properties for Test Case 3
Model Diam. (in.) Thickness (in. ) Weight (lb. )
I 3.005 .0497.1216
II 3.5134 .0882 .2735
95
This problem was investigated in order to analyse the effects
on the fundamental critical speed due to the following three
parameter changes.
1. varying the length of the overhanging portion of the
shaft (C).
2. Different concentrated mass effects due to overhanging
disk.
3. Effect of gyroscopic coupling due to overhung disk.
The model idealization used in the analysis is shown in Fig. 2.1
The length L remains constant while the overhung portion C
will vary. The model is idealized as four beam elements, with
the bearings, disks and unbalance force acting at the proper
node. The bearings in this test case are assumed to be rigid,
therefore only large equal stiffness values need be entered
as input to the computer program. The damping and all cross-
coupling terms were set to zero. Although the bearing reactions
in the physical problem will be different due to the unsymmetri-
cal nature of the problem, the model may still be represented
accurately by two identical bearings since the displacements at
the bearings are substantially smaller than at any point on
the shaft, properly indicating a rigid support.
Table 5 lists the results obtained for the twelve cases. It
is seen that in all cases the % difference between Dunkerley's
experimentally observed results and the predicted values (with
gyroscopic coupling included) is within 2%, Fig. 24 shows
the mode shapes, at the critical speeds, for several of the
test cases. As the overhanging portion is increased the
96
fundamental mode shape assumes that of a cantilivered beam
with an end mass. The effect on the critical speed due to
varying the overhung length C is shown in Fig. 25. As the
overhung portion C, is increased the critical speed first
increases, hits a peak and then rapidly decreases. This same
trend is evidenced by both disk models (I and II). The author
belives this trend may be explained by noting that the shaft-
disk system is made up of two distinct parts; a section
spanned between two bearings and a overhung section free
at one end. For a light disk and low values of C, the dynamic
characteristics of the span between the bearings will
predominate. As the length of this section is decreased the
natural frequency of the section will increase. This will
continue until at a certain value of C where the dynamic
characteristics of the overhung portion will predominate
and cause the natural frequency to be lowered. Thus with
futher increase in C, the critical speed value drops rapidly.
The same trend appears for the heavier disk but at substantially
lower values for the critical speeds. Intuitively this is
expected since the mass of model II is larger.
After the twelve cases had been completed and good
correlation established, the investigation was continued, this
time neglecting the gyroscopic coupling effects. The concentrated
mass of the disk was allowed to act at the end of the shaft,
but the polar and transverse mass moments of inertia of the
disk were assumed to be zero, thus no gyroscopic coupling.
Table 5 gives the results for the 12 cases with no gyroscopic
coupling. Comparison to the original calculations indicate
verv litte change. This was an unexpected result , which
97
initiated a discussion with Rieger L^O . It was decided
that the results were not all that unreasonable. First, the
general trend of a reduced value for the critical speed
is shown. By neglecting the gyroscopic coupling the system
is made less stiff indicating a lower critical speed value.
This was observed. Second, a hand calculation of the gyroscopic
coupling terms for a speed of 1400 rpm was made and the values
were compared to the proper terms of the dynamic stiffness
matrix to which they would be added. This comparison
indicated that at this speed the gyroscopic terms were three
orders of magnitude less than the beam dynamic stiffness
terms. A reasonable explanation may now be stated as follows.
Since the disks are light and the speeds are low (lower than
1400 rpm) , the above observations hold for all twelve test
cases and substantiate the predicted results that neglecting
the gyroscopic coupling terms has little effect on the critical
speed value for this problem.
Fig. 26 shows a typical response curve, used to identify
the critical speed. It is seen that the response peak is sharp
and well defined. The fact that no damping was present in this
system explains this result.
This example case has successfully demonstrated the ability
to analyse complex rotor configurations. The ease in problem
set up using this technique is as valuable as the accuracy
of its results.
98
i-H -
(H
c *
BTA
Shaft
Rigid
Bearings1727
Disk
a). Dunkerley*s 2 Overhung Disk Model (case XI)
*L 4L
Kxx Kyy
S77777
he ic - MD, ip, itI'
f | { .001 oz. in.
Kxx Kyy o
S77777
b). Model idealization; E, A, I constant in each element
Fig. 23 Model and Idealization for Test Case 3P
99
Table 5
Critic al Speeds of Overhung I^otor
Disk I W 1216 lb. Diam. 3. 005 in.
% Diff. 1
Length .0497 in.
L(in) C(in) Dunkerley's Predicted & 2 Predicted
results 1 gyro. 2 no gyro. 3
a) 30.7 -U00 1223 1225 -16 1225
b) 29.1 2.61 1329 1335 .451330
c) 28.0 3.69 1384 1390 .43 1385
d) 26.66 5.02 1407 1410 .21 1405
e) 24.0 7.69 1224 1240 1.30 1235
f) 21.33 10.35 968 975 .72 975
Disk II W 2735 lb. Diam. 3.5134 in. Length .0882 in.
a) 30.63 1.00 1227 1230 .24 1225
b) 29a 2.54 1276 1300 j 1.80 1300
c) 28.0 3.63 1281 1305 1.80 1295
d) 26.66 4.96 1215 1230 1,20 1220
e) 24.0 7.63 928 945 1.8a 940
f) 21.33 10.29 712 720 1.10 720
100
'.Af.'
i .4.(-U t-i -L.uy.
ZT
30.7
Ttf
-*-
4- -of
--
,oz.
-- o in.-02
-OH
^,02
1.0
a). Model Ia, Nc 1225 rpm., C 1.0 in.
A"
7T^
28.0
0?
.00
ov
02. .
-- O'O in,Ol
- ov
-.0 2
3.69
b). Model Ic, Nc 1390 rpm., C 3.69 in.
H
21.33-
7^"
9I0.35**"
--
.OS?
--
-ot
--
-01
--
01 .
--
00 in.
,ov
---.oc.
---.08
c). Model If, Nc 975 rpm., C 10.35 in,
Fig. 24 Mode Shapes for the Overhung Disk Models
101
1400
1300 "..
1200
1100
CRITICAL
SPEED
(rpm)
1000
900
800
700
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Overhung Length C (in.)
Fig. 25 Critical Speed vs. Overhung Length for
Models I & II of Test Case }.
102
4
3
1.0
i Test Case 3, Model Ha, Nc= 1230 rpm.
Responce at Disk Location, .001 oz. in. unbalance
Mills (10"-5) vs. Speed rpm.
.10
Amplitude
in.x 10
.01
Fig. 26 Typical Unbalance Response Curve for
Overhung Rotor
Speed , rpm.
1180 1200 1220 1240 1260 1280
103
5.4 Lund - Orcutt Rotor (MTI Rotor)
Lund and Orcutt [15j in I967 presented an extensive
analytical and experimental investigation of the unbalance
response of a rotor in fluid-film bearings. This
investigation was chosen as the final test case to be
presented since it combined all of the special features
which have been developed (i.e. overhung disks with
gyroscopic coupling, complete bearing representation,
unbalance forces and shafts with uniform mass and elastic
properties). In reference 15} "the general analysis and
computational method is presented and applied to calculate
the unbalance response of each of three experimental
rotor configurations supported by tilting-pad bearings.
The analysis is an extension of the Prohl method for cal
culating the critical speeds of a rotor. Lund and Orcutt
included the anisotropic stiffness and damping characteristics
of the fluid-film bearings which couple the rotor motion
in the horizontal and vertical direction. The rotor was
represented by a series of stations connected by shaft
sections of uniformly distributed mass and elastic proper
ties. The equations of motion for the rotor stations, together
with the equations for the shaft sections, established a
set of recurrance formulas by which a step by step calcu
lation of the rotor can be performed. Successive application
of the recurrence formulas allows computing the amplitude,
slope, bending moment, and shear force along the rotor
104
in terms of the amplitude and slope of the first station.
By back substitution, the amplitude at each station can
finally be calculated. It is the purpose of this test
case to compare the procedure developed to the analytical
and experimental work of Lund and Orcutt.
There are some basic similarities and differences
between Lund andOrcutt*
s developement and that presented
in this thesis, they are;
1. Although Lund and Orcutt included, in the procedure they
presented, the effects of shear deformation in developing
the dynamic influence coefficients, it was not clearly
indicated whether they included the shear effects in their
response calculations, since at the end of the development
they assume that shear is negligable. In the analysis
presented here the shear effects were neglected.
2, The Lund and Orcutt procedure requires a set of re
currence formulas to be solved and by back substitution
the amplitudes are found. In the analysis presented
here, a closed form solution dynamic stiffness matrix
has been developed. The solution requires only to use
this matrix to build up the structural dynamic stiffness
matrix and bearing reactions, disk effects and unbalance
forces are provided for as point effects. The analysis
solves directly for the displacements and rotations at
the stations or nodal points.
3. The bearings, disks and unbalance force representations
are the same for both procedures.
105
What will be presented here is the unbalance response anal
ysis of two of the three rotor configurations presented
by Lund and Orcutt. The one disk model and the three disk
model will be presented. The two disk model will not be
analysed since it is an unsymmetrical model which causes
the bearing reactions to be different indicating the need
for two different bearing configurations. As noted by
Lund and Orcutt \_)-5~\ the one and two mass models are
greatly influenced by the bearing properties, thus in its
present form the computer program developed can not be
used to accurately represent the two mass model. (Test case
3 was also an unsymmetrical model, but since the bearings
were assumed rigid they did not play an important role
in the overall rotor response).
The two basic models are shown in Fig. 27. Model two
(3 disk rotor) was obtained from model one (one disk rotor)
by attaching two end disks. Model idealizations are shown
in Fig. 28. It is seen that five shaft sections are required
since a node must appear at ends of the rotor, at all
bearing and disk locations and where ever a change in
diameter occurs. The test rotor was a cylindrical steel
shaft(a value of E = 30x10 psi was used, although Lund and Orcutt
never stated) with overall length 4lin. and a diameter of
2.5 in. except for the central integral disk of 6 in. diam
eter and 6 in. length. The rotor weighed 881b. not including
106
the detachable end masses. The center disk not including
the inner 2.5 in. diameter section weighed 36lb. and the
end disks weighed 181b. each. This information was used
to calculate a value of weight density f ~.256 lb/in^.
This low value of density reflects the fact that machining
of holes in the center and end masses for unbalance weights
was performed. The rotor was supported in fluid film bearings
whose theoretical bearing stiffness and damping properties
are given by Lund and Orcutt [15']. Also the flexibility of
the bearing pad pivots was introduced as a stiffness in
series with the bearing film stiffness. The values for both
theoretical and equivalent bearing stiffness and damping
properties are shown in Fig. 29.
Figs. 30a and b show the theoretical and experimental
results as obtained by Lund and Orcutt for the one mass
model, and the predicted results using the technique devel
oped in this thesis (graphs are for the center and end
positions of the rotor) . It is seen that the predicted
results follow the general trend of the response but are
closer to the experimental results than to the theoretical
results of Lund and Orcutt. Several possible explainations
can be given for this discrepency.
1. The fact that shear deformation was neglected in the
developement in this thesis could explain the higher
predicted value of critical speed. Neglecting the shear
deformation would tend to stiffen the rotor thus indi
cating a higher critical speed.
107
2. The bearings are flexible and the dynamic properties
might not be correctly specified by the information
provided and as evidenced by Lund and Orcutt 151 these
properties play a big role in the overall rotor response
for this model. It was not explained clearly enough in
reference [151 exactly what bearing coefficients were
used in the analysis.
3. The material properties were not explicitly stated,
therefore reasonable values were assumed which this
author felt closely represented what was being analysed.
4. The exact positions of the measurement planes were not
indicated. It was assummed that measurement planes were at
the ends of the rotor and at the center plane of the rotor.
An interesting result was observed when the damped critical
speed was soyght. Figures 30a and b give these results. For
the one disk model the predicted results were 18.5$ (l6,000rpm)
higher than the experimental results (13,500rpm), and Lund
and Orcutt *s theoretical results were 18,5$ (ll.OOOrpm) lower
than the experimental. Many different configurations were run
to investigate changes in weight density f , bearing dynamic
properties, location of measurement planes and the disk effects
of the center mass. All the parameters investigated had little
effect on the rotor response except for the bearing properties,
and location of the measurement planes.
108
The response curves for the center and end positions for
the three disk rotor are shown in Fig. 31a and b. Good
correlation is obtained between the experimental and theore
tical results given by Lund and Orcutt. The response curves
are seen to contain a sharp peak around ll,000rpm. An
independent calculation of the third critical speed,free-
free mode, of the three disk rotor system using the Prohl
transfer technique was given by Lund and Orcutt as ll,000rpm.
At this third critical speed the bearings are very close
to the natural nodal points of the rotor and the bearing
properties have little effect on the rotor response. The
good agreement between calculated critical speed and rotor
response in this case verifies the accuracy of the rotor
response program in calculating the characteristics of
the rotor, since the bearing only plays a minor role
in the response. When the bearings do exhibit control
over the rotor response, as with the one disk rotor model,
accurate bearing data must be supplied to obtain good
correlation.
Lund and Orcutt Cl5l calculate values of the undamped
critical speeds for the two rotor bearing systems, using
the Prohl Technique together with the theoretical bearing
stiffness data. These values are given in Table <o .
Comparison of these values to the theoretical and exper
imental response data (both Lund - Orcutt data and data obtained
using the thesis technique) shows that the undamped
109
critical speed calculations do not accurately identify the
speeds at which maximum amplitudes occur, except for
configurations where the bearings play little role in the
response. The results presented here again affirm that the
rotor response calculations provide valuable information
which is not given by standard critical speed calculations.
Table t0
Calculated Cr:Ltical Speeds by Prohl Me thod (rpm) L15]
lst 2nd 3rd
One Disk 7200 17,800 24,500
Three Disk 3000 5,800 11,000
110
Kxx Kxy* 7 ,V
5xx Dxy//////
9.5"- 9.5'1
U- 8"
12.5"
$ Kxx Kxy
Dxx Dxy
:). Model One - one disk model ^153
3"
r
XX,
xx'
//////
It
41"
~A3"
f.
i
y Kxx,Dxx
S77777
b). Model Two - three disk model [l 53(same as model two, plus end disks)
Fig. 27 Models for test case 4 Lund, Orcutt Rotor
111
!-u>H !
M.5"I
.5 OZ-in
9.5*- 1-6"9.5"- 6.5"-
Kxx *yy ijDxx D,
^^
'yy
c _y
K^, ^Dxx D-yy
//////
a). Idealization for Model one
-6.5"
MD,Ip,It
h-9.5u
.16 oz-in
H-6"- \ 9.5"-6.5'H
i i
1 O.33
oz-in
Kxx KyyPxx Dyy
//////
k
//////
9 MD,Ip,It
.33 oz-in
b). Ldealization for Model two
Fig. 28 Model Idealization for test case 4
] 11 112
Fig. 29. Theoretical and Equivalent Bearing Stiffness
and Damping Properties
3
2.- -
100.0 1-
2
34
56
- Damping, 3 Disk Model lb-sec/in
- Damping, 1 DiskkModel lb-sec/in
- Stiffness, x IO?", 3 Disk Model lb/in
- Stiffness, x 10^", 1 Disk Model lb/in
- Equi. Stiff., x lOT", 3 Disk Model lb/in
- Equi. Stiff., x 104, 1 Disk Model lb/in
10.0
1.04 3
Stiffness x 10 or Damping vs. Speed, rpm x10^
8 10 12 14 16 18 20 22 24 26
10.0 113
3.__
1.0
.10
Amplitude
in. x 10-3
.01
1 - ll,000rpm , peak response
2 - 13,500rpm , peakresponse
3 - I6,000rpm,peak response
Response Orbit Radius, mils/oz-in
vs. Speed, rpm x 10-3
Theoretical Results ref. [15]
A A Experimental Data ref. 15
-0-0- Calculated from Thesis Program
Fig. 30a. Unbalance Response of One Disk Rotor
Center Position
8
Speed rpm x10J
10 12 14 16 18 20 22 24 26
114
Amplitude
in. x 10-3
,,.01
1 - ll,000rpm, peak response
2 - 13,500rpm , peak response
3 - I6,000rpm , peak response
Response Orbit Radius, mils/oz-in
vs. Speed, rpm x 10^
Theoretical Results ref. [15]
A A Experimental Data ref. [15]
-o-o- Calculated from Thesis Program
Fig. 30b. Unbalance Response of One Disk Rotor
End Positions
8 10
gpeed rnm x 10
12 14 16 18 20 22 24 26
115
Theoretical Results ref. 15
10.0
Amplitude
in. x 10-3
1.0
A A Experimental Results ref. 15i i
-0--0- Calculated from Thesis Program
.10
Response Orbit Radius, mils/oz-in
vs. Speed, rpm xIO-'
Fig. 31a. Unbalance Response of Three Disk Rotor
Center Position
8
Speed rpm x 10
10 12 14 16 18 20 22 24 26
116
Theoretical Results ref. 15
10.0
Amplitude
in. x 10
1.0
A A Experimental Data ref. 15
-o-o- Calculated from Thesis Program
.10
Response Orbit Radius, mils/os-in
vs. Speed, rpm x10^
Fig. 31b. Unbalance Response of Three Disk Rotor
End Positions
Speed rpm x 10
i i
8 10 12 14 16 18 20 22 24 26
117
6.0 DISCUSSION OF RESULTS
The development of a "rotorelement"
and computational
procedure for the steady-state unbalance response of a flexible
rotor in fluid-film bearings has been presented. The character
istics of the distributed mass and elastic properties of the
rotor have been successfully demonstrated. Inclusion of the
complex stiffness terms introduced by bearings and disks working
in the system has been investigated and has shown that these
effects play an important role in the overall rotor response. The
dynamic stiffness matrix developed has shown to be an accurate
means of handling the ensuing equilibrium equations.
Throughout the development, right handed Cartesian coor
dinate systems and elasticity theory notation were used. This
approach made it possible for a logical and uniform development
to be presented. The computational procedure was set up as follows.
The rotor was represented by an axially assembled series of beam
elements with uniform cross-section and distributed mass and
elastic properties. Bearings, disks, and unbalance were included
as optional effects at the ends of each beam element.
Although this procedure provided for easy idealization and
set up of a rotor model, it is felt by the author that it would
have been more efficient (mainly a savings in computer core) to
treat the bearing and disk effects as point effects as discussed
by Pestal and Leckie in reference [27 1 . This would have avoided
the need for developing the matrices [KBJ , [kbJ , [kd] , and [kdJ
in equations 66% 68, 87, and 88. Instead, these effects could
have been added directly into the structure stiffness matrix
118
merely by knowing the node numbers at which they act. This would
have realized a savings of1.024x10^
words of computer core
(or 6.5%) since each of the above matrices were 8x8, complex, and
double precision.
The developed procedure was applied to the analysis of a
simple uniform shaft in end bearings in test case one. The
uniform distribution of mass and elastic properties within the
element allowed a single model to successfully predict accurate
critical speed values for three modes. Futher investigation is
required for verification of higher modes. It was shown in
figure 13 that at low values of support stiffness the rotor
acts as a rigid body and at the critical speeds it assumes the
rigid rotor mode shapes. At higher values of support stiffness
the rotor becomes pinned in its bearings causing the critical
speed values to substantially increase and the rotor assumes the
flexible rotor mode shapes. The critical speed values were predicts;
within 2% of theoretical values.
Test case two demonstrates that the whirl orbits are
generally elliptical due to the asymmetric properties of the
bearings. The relative amplitudes at the bearings and along the
length of the rotor depend on the amount of unbalance present
and the ratio of shaft stiffness to bearing stiffness. The fic
ticious model investigated in test case two indicates the developed
procedures ability to represent rotor-bearing interactions. Agree
ment with the finite element procedure developed by Ruhl in
reference Q.6] is established. It is seen that the dynamic
stiffness matrix formulation predicts slightly higher response
(4.73$ at 5500rpm) at the midspan and was discussed in section 5.2.
119
It was also shown that model refinement had little effect on the
predicted response for length to diameter ratios as low as
8.33/3.0 and for speeds up to l4000rpm. Further investigation
is required to establish limitations on model refinement where
shear effects are predominant. This can be done by retaining
the shear terms in the original derivations of the equations
of motion , and carrying them through the analysis to obtain
their effect on the dynamic stiffness matrix. It is seen in test
case two that the one mass model accurately predicts the first
bending critical speed of the system. Thus the application of the
dynamic stiffness matrix model or the finite element model
creates an unnecessary burden on the investigator. However, if
detailed information about the response of the system is required,
the formulation developed here proves to be ideal. Futhermore,
for more complicated shaft-disk systems, the application of this
procedure poses no greater difficulty as evidenced in test cases
three and four. In the last two test cases presented the developed
procedure's ability to analyse shafts with overhung disks was
investigated.
It was shown in test case three that predicted fundamental
critical speeds for various shaft-disk combinations agreed with
Dunkerley's experimental results C 2l within 2$. This experimental
verification adds validity to the developed procedures capabilities.
Again, futher investigation is required to establish the limitations
where this validity ceses to exist.
The critical speeds were seen to be a function of the length
of the overhanging shaft, and the mass and the gyroscopic
120
stiffening effects of the disk. When the gyroscopic coupling
effect was ignored the predicted critical speed values were
lower. However, the influence was small, generally less than
1$, and it is concluded that the gyroscopic action has little
effect on critical speed calculations for this model. Independent
calculations (by the author) of the gyroscopic coupling terms,
indicated that these terms were three orders of magnitude less than
the shaft dynamic stiffness terms, substantiating their insignifi
cance.
The most comprehensive test of the developed procedure's
capabilities came from the investigation of the Lund-Orcutt rotor
[151 presented in test case four. Two models were analysed, a one
disk and a three disk model. Good correlation was obtained
between the predicted results and both the experimental and
analytical results presented by Lund and Orcutt. For this
configuration the response curve indicates a sharp peak around
HOOOrpm. An independent calculation (by Lund and Orcutt) of the
critical speeds, using the Prohl technique, gave HOOOrpm as
the third critical. For the three disk configuration the bearing
locations happen to be close to the natural nodal points for
the third mode, thus the bearings play a minor role in the response
of the rotor. Therefore, the good correlation at the third
critical speed verifies the accuracy of the rotor response program
in calculating the characteristics of the rotor itself.
The trend of the results predicted for the one mass model
follow closely the experimental and analytical results of
121
Lund and Orcutt. The predicted peak amplitude occurs around
l6000rpm which is 18.5$ higher than the experimentally observed
value. The analytical results of Lund and Orcutt indicates a peak
value at HOOOrpm or 18.5$ lower than the experimental value. For
this configuration it was found that the rotor response was sen
sitive to the bearing properties. Based on this observation and
the fact that the three disk model, which was not influenced by
the bearings, was accurately represented, the author believes
that a questionable bearing representation has been used in the
analysis of this model.
The procedure developed has demonstrated its ability to
perform unbalance response calculations. Future work on improving
the procedures capabilities and investigations into the relative
efficiency of computation, as compared to alternate techniques,
are left as recommendations.
122
7.0 CONCLUSIONS
1. A general rotor element and computational procedure
based on the dynamic stiffness matrix formulation
has successfully been demonstrated.
2. Strict adherence to true right-handed coordinate systems
and established notation conventions provides for a
uniform and logical development.
3. Matrix algebra has been shown to be an efficient tool for
analysis of rotor-bearing dynamic systems.
4. Unbalance response calculations provide a clear indication
of the speeds at which peak vibration amplitudes will occur.
5. For flexible bearings the rotor acts as a rigid body and
assumes the rigid rotor mode shapes at the critical speeds.
6. For rigid bearings the rotor becomes flexible and assumes
the flexible rotor mode shapes at the critical speeds.
7. In general the whirl orbits are elliptical when a more
realistic bearing representation is used. This is due
to the asymmetric stiffness properties of the bearings.
8. The gyroscopic stiffening effects tend to raise the criti
cal speed values. However, the relative importance of these
effects must be investigated for each rotor-bearing system,
since the effect will vary for different configurations.
9. In general the shear effects seem to be negligible.
However, a more in depth investigation is needed before
anything definite can be stated about these effects.
123
10. Inclusion of the bearing damping in the response calcula
tions predicts a higher critical speed than that calcula
ted with the Mykelstad-Prohl technique, which includes
only the bearing stiffness properties.
11. By performing various unbalance response calculations, with
the unbalance placed at different locations, along the
length of the rotor, the sensiticity of the rotor to un
balance can be evaluated. Thus, response calculations
may be helpful in choosing the best location for balancing
planes.
12. The dynamic stiffness matrix concept has proven to be an
accurate approach to studying the unbalance response of
rotor-bearing systems.
13. The program ROTOR was more efficient, in the use of comput
er storage, than was the program FINITE5. ROTOR required
15. 8K words of main computer core, with a maximum of 7
rotor elements. FINITE5 required 21. 2K words of main
computer core, with a maximum of 6 rotor elements and no
disks were allowed.
14. A more in depth investigation is needed to compare the
relative efficiencies of the different approaches (i.e.
recurrence formula, finite element, and dynamic stiffness
matrix approaches).
124
8.0 RECOMMENDATIONS
1. A revision to the computer program ROTOR eliminating
the matrices [KB] , [kb] , [Kq3 and [kd*] and treating
the bearing and disk properties as true point effects
is recommended.
2. The program should be revised to accept different
values for weight density and Young's Modulus for
each shaft section.
3. Provision for multi-bearing use (i.e. solution for the
statically indeterminant bearing problem) and couplings is
recommended since the analysis of large turbo-generator sets
would require this type of analysis.
4. Futher investigation should be performed on the effects
of shear deformation at higher speeds. A procedure
for incorporating the shear effects into the dynamic
stiffness matrix should be developed.
5. Application of the dynamic stiffness matrix formulation
to stability analysis of rotors should be investigated.
6. Expansion of the program to handle a large number of el
ements and inclusion of bearing pedestal effects is re
commended. This would require investigations into better
equation solving techniques.
125
9.0 REFERENCES
1 Rankine, W.J., McQ., "On the Centrifugal Force of Rotating
Shafts", Engineer, London, Series A, Vol. 27, p. 249, 1869.
2 Dunkdrley, S. , "On the Whirling and Vibration of Shafts",Phil. Trans. Roy. Soc, London, Series A, Vol. I85, p. 279, 1895.
3 Jeffcott, H.H., "The Lateral Vibration of Loaded Shafts in the
Neighborhood of a Whirling Speed-The Effect of Want of
Balance", Phil. Mag., Ser. 6, Vol, 37, p. 304, 1919.
4 Smith, D.M., "The Motion of a Rotor Carried by a Flexible
Shaft in Flexible Bearings", Proc. Boy. Soc, Series A,Vol. 142, pp. 92-118, 1933.
5 Robertson, D. , "The Whirling of Shafts", The Engineer, Vol.155.
pp. 216-217, 228-231, 1934.
6. Prohl, M.A., "A General Method for Calculating Critical
Speeds of Flexible Rotors", Trans. ASME, Jnl. of Appl. Mech.
Vol. 12, p. A-142, 1945.
7 Hagg, A.C.,"
The Influence of Oil-Film Journal Bearings on
the Stability of Rotating Machines", Trans. ASME, Jnl. Appl.
Mech., pp. 77-78, 1947.
8 Green, R.B. , "Gyroscopic Effects on the Critical Speeds of
Flexible Rotors", Trans. ASME, Jnl. Appl. Mech., p. 369, 1948.
9. Linn, F.C., Prohl, M.A., "The Effect of Flexibility of Support
Upon the Critical Speeds of High-Speed Rotors", Trans. SNAME,
Vol. 59, PP. 536-553, 1951.
10 Warner, P.C, "On the Balancing of Flexible Rotors", MTI
Report 62-TR-26, Feb. 1962.
11 Lund, J.W., Sternlicht, B. , "Rotor-Bearing Dynamics with
Emphasis on Attenuation", Trans. ASME, Jnl. Basic Eng.,
Vol. 84, Series D, 1962.
12 Morrison, D. , "Influence of Plain Journal Bearings on the
Whirling Action of an Elastic Rotor", Proc. Instn. Mech.
Engrs., Vol. 176, No. 22, p. 542, 1962.
13 Lund, J.W., "Rotor-Bearing Dynamics Design Technology PartV :
Computer Program Manual for Rotor Response and Stability",
MTI Technical Report AFAPL-TR-65-45, I965.
14 Mortora, P.G. , "On the Dynamics of Large Turbo -Generator
Rotors", Proc. Instn. Mech. Engrs, Vol. 180, part I, No. 12,
p. 295, 1965-66.
15 Lund, J.W., Orcutt, F.K.,"
Calculations and Experiments
on the Unbalance Response of a Flexible Rotor", ASME
paper 67-VIBR-27, First Vibration Conference, Boston, Mass.,1967
126
16 Ruhl, R.L., "Dynamics of Distributed Parameter Rotor Systems j
Transfer Matrix and Finite Element Techniques", PhD Thesis,Cornell University, Ithaca, New York, Jan., 1970.
17 Ruhl, R.L. , Booker, J.F.,"
A . Finite Element Model for
Distributed Parameter Turborotor Systems", Trans. ASME,Jnl. of Eng. for Indus., pi 126, Feb., 1972.
18 Rieger, N.F., "Unbalance Response of an Elastic Rotor inDamped Flesible Bearings at Super-Critical Speeds", Trans.
ASME, Jnl. Power Div. , Vol. 93, Series No. 2, p. 265,April, 1971.
19 Rieger, N.F., "Flexible -Rotor Bearing System Dynamics,Part III- Unbalance Response and Balancing", ASME MONOGRAPH
PUB., Flexible Rotor Systems Subcommittee, 1973.
20 Morton, P.G., "Influence of Coupled Asymmetric Bearings on
the Motion of a Massive Flexible Rotor", Proc. Instn. Mech.
Engrs, Vol. 182, Part I, No. 13, p. 255, 1967-68.
21 Morton, P.G., "Analysis of Rotors Supported Upon ManyBearings", Jnl. Mech. Eng. Sci., Vol. 14, No. 1, 1972.
22 Rieger, N.F. , "Vibration in Rotating Machinery", Union
College Lecture Notes, 1970.
23 Ekong, I.E., Bonthron, R.J., Eshleman, R.L. , "Dynamics of
Continuous Multimass Rotor Systems", ASME Pub, No. 69-VIBR-5I,1969.
24 McCallion, H. , Rieger, N.F. , "Moment and Shear Equations for
Bar Vibration Analysis", The Structural Eng. Vol. 43, No. 7,July 1965.
25 Bishop, R.E.D., "Analysis of Vibrating Systems which EmbodyBeams in Flexure", Proc Instn. Mech. Engrs., Vol. 169,No. 51, PP. 1031-1055, 1955.
26 McCallion, H. , Vibration of Linear Mechanical Systems.
John Wiley and Sons, Halsted Press, New York, pp. 96-HO, 1973.
27 Leckie, F.A., Pestel, E.C, Matrix Methods in Elastomechanics,McGraw Hill Book Company, New York, I963.
28 Timoshenko, S.P., Vibration Problems in Engineering. D.
Van Nostrand Co., Inc., Princeton, New Jersey, Third Edition,1955.
29 Rieger, N.F., "Rotor-Bearing Dynamics Design Technology, Part I:
State-of-the-Art", MTI Technical Report AFAPL-TR -65-45, May, 1965.
127
30 Lund, J.W., "Rotor-Bearing Dynamics Design Technology,Part III: Design Handbook for Fluid-Film Bearings", MTI
Technical Report AFAPL-TR-65-45, May, 1965.
31 Rieger, N.F., private communication and unpublished notes,
April, 1974.
32 Levy, S., "Computer Programs for Vibration Analysis";Computer Workshop in Structural Dynamics, Schenectady,New York; Union College Graduate and Special Programs,Union College, Aug.,1973.
33 Halbleib, W. , private communication, May, 1974, Rochester,Institute of Technology, Rochester, New York.
34 Bishop, R.E.D., "The Vibration of Rotating Shafts", J.
Mech. Eng. Sci., Vol. 1, No. 1, pp. 50-64, 1959.
35 Dimentberg, F.M., Flexural Vibrations of Rotating Shafts,Butterworth and Co., Ltd., London, 1961.
36 Bishop, R.E.D., "Vibration and Balancing of Flexible Shafts",Applied Mechanics Feview, Vol. 21, No. 5, p. 439, May, 1968.
37 Gladwell, G.M.L., Bishop, R.E.D., "The Receptance of
Uniform and Non-Uniform Shafts", Jour. Mech. Eng. Sci.,
1 (D , 78, 1959.
38 Yamamoto, T. , Colected Works ,AiResearch Manufacturing Co. ,
Garrett Corporation, Phoenix, Arizona. Report G-5019, Nov.,
1964.
39 Rieger, N. F.,"Current Programs for Rotor-Bearing System
Dynamic Analysis", Rochester Institute of Technology,
Rochester New York, June 1974.
128
10. APPENDIX A< THE DYNAMIC STIFFNESS MATRIX.
Following is the derivation for the dynamic stiffness matrix
as formulated by McCallion and Rieger in reference [24] . From
equation (33) in section 4.2 it is seen that [kxJ =[d][c"J,
which means [C""1] must be found. Throughout the derivation the
following short hand notation will be used,
s-sin/^L,
c-cosAL,
sh-sinh?)L,
ch- cosh^L
The inverse of a matrix is defined as
A"1 = ad.j A
I A)
where adj A is the adjoint of A, or the matrix of cofactors
transposed and |A|is the determinant of A,
Noting that sin2 -t-
cos2
= 1 andcosh2
-sinh2
= 1,
the matrix of cofactors is as follows for matrix C;
Af<= [cCA'slOtchCh'-s)] =hx(csk tsch)U)
A,x sh(hlh) +cH^c/i -ALc)l= Als sh -m
-
cch)(-i)
Ai3 '-bcCAL5h)-fchC-^^l~
hxC~csU -
s ch)( I)
A,-, 'Ls(^s)~
cC^cfi^Vc) 4-shC-^s)] ~-Mi -cch -ssh^C I)
kit "- i>>I^ +A*sl '- /T C Sh + s)C-0
A2i = lA*ch - 1)11'- ^Cch -c)Ci)
An =L~ysk -1)1]= tfC-sh -s)C-i)
A2 =[-Ulci-M CC -ch)ti)
Aj/ = C (* 5l^" AdS) (UcM^sslo]- h(-\ tech +55h)iO
A3Z= [(A sch -?\csh)]
=
/) Cscln -c sh)C-l)
A33 ~- [~C/)S Sh~ A Cuh^ t
(~?)SL
-
/)Cl
)J =^-55-' * ^/j-/J(
129
hi= ["(Asd) -/)cs/))]
= A(-sch ush)(-i)
A vi -E(-AcH + Ac)]- A (c -
ch)C-/)
/Wz "-[(As r^5h)]r A(s-sMC/)
AV3 =L-C-/kM+ f-^c)]= A CcU -
c)(-i)
Avy s[-M5-^y] r/>><:sh-s)0)
/A/r C-/) A/z t ("'My
* C-0(?>X(ssh Fl -Cch))F(-0(Ayi- CChS'^A
inf=
AAZ
Ccoh -/J
Introducing Bishop's notation [25] for the transcendental functions
(frequency functions) as follow,
F^= s in A L s inh A L
F2= cos A L cosh A L
Fo cos A L cosh Ah - 1
Fh = cos A L cosh A L* 1
F cos/)L sinh^L - sin^L cosh /) L
F^= cosA L sinh A L + sin/} L cosh /) L
F = sin/i L +sinh P) L
Fo = sin A L - sinh^L
Fq= cos^ L + cosh^ L
F^q= cos A 1> - cosh A L
allowsC"l to be written as
/
(est, fSc/lJ (-S -Sll)(ccJ,fss4-0
A
-/ (cc/i -ss*rO
(-c*/, -sd.)
Cch -c)A 7>
ZF3 C 3 tsMCcch -ssh -/J
*1
Cc~ch)
A
(cc/) ^55*-) (---!]^ i /> _
*xv
130
Now perform the operation [ D~][c~
J , where [dJ is defined in
equation (29) of section 4.2, to obtain [k ],V
,1
Ki = Ell-A3Ccsh +sch)+?\3C-
csh-
sch)l= I?CcshfsOi)EI - 'BIA fy
Z F3 Fj F3
KIX* ^r[-AJC-3 -s//) +A3cs+sh)i =
A3Cs +sh) ei = Z A3_r?_
2. F3 F3 Fj
\Xti-
bil-A1
'ac/i Ms;-
~i) +?>3ccch -s^h -m-->/T 5s/> ei
=-ezIJI
2- f3 h F j Fj
Kiu '- EH-A3(ch-c) i-A3
Cc-ch)J =-tf-c c -ch)1 = FA f^_
2PjA fJ Fj
K21'
Ell&teh +sclH+(-Jts)(cch -ssh-i) +(-7?cb)(-csh-scAh-t(~A3shXcch +ssh -rij
Fj
- ET2jA3
Csh ts)J ^ EIA3
Fj
Fj F3
Hiz =lL/)cC-s~sh) J-C-/fs)(*h-c) rC-AIh)Cstsh)tC-A3sh)Cc~ch)J
ZF3
- BiLA3t-csh -sch)J =-
eiA3
I
Pj Fj
$23-
EllA3c(cchtssh-0-H-A3s)Csll-
5ch) + C-tfchKcd*
*F3A
= BI LAIch ~c)J =-
BIA"
Eo
Fj Fj
KXH- BI[A3<l Cch-c) i-C'A^)(5- si,9) r(-A3ch)( c
-
ch) -F(~AJsh)(sh ~s)J.
2FjA
* Bl L7)"C s sh )J = biA2"
Il
Fj Fj
131
Hj, = FILyCcch'Ssh-i) +(-A*)Ccch tssh -/)]'- EltfC-ssh)] - -E+s 1
* Fj Fj
Kn,-IfAxrch-c)y-Al)(c-
ch)l = BIIA*
Cch- c -c tch YJ '-
-Ei X
zp32 Fj
K-r'-BlL**
C c sh -
5c/iJ f(^) C sch-
ash )1.~
ZfjA)
ei a jyFj
BI A fT
Fj
KHt- ElhA'sCcsh t-scl. ) t(-A*~c)(ccli -ssh-i)
t(Xsh)C-
csk-schXfchXccr
af3
BTLlA"
C-ch tc)l
F3
EIA*'
fie
F3
KHX = Fll-AxsC-S'sh)rC-A"c)Cch -Q r(A^sh)(.St sh) i-(Alh)(c -c>i
2/=j
*ETA"
*
/=J
fyj- EIL-A^s Ccch fssh-t) f(-AD(c5h-5ch) + (ALsh ) (cch -ssh-O+Cffcyl--.
- BI lAC- 5h -tS)- EI All
F3 Fj
Ki= BJ2l~/)2s(cri-c) +(-A"c)CS-Sh) +(ALsh)CC-c/.) i(/fch)I:-:
2F3/)
- ElA(c sh- s ch )
Fj
EI A fy
Fj
Therefore [kxz"] is given by;
132
[K^D =: ^
Es
-AF^I A_Fj_
AF]_'
-EL
Zfl'~>o_
F,o 'Fi
-_Fi_'
F,o_
'I'I' Z'
_
^//) I Fs/fi
(Al)
^y
[ Kxz3 is the dynamic stiffness matrix which relates forces
and moments applied at the ends of the beam to displacements and
rotations at the ends of the beam, in the x-z plane. Equation
(34) in section 4.2 becomes,
<St,
r
[*JMKH
-/Xl
Hi
G>,7
(A2)
To find f j\z1 it is seen that matrices [Ej and [g] in
equations (43) and (45) in section 4.2 are similar to [ d] and [cl
in equations (29) and (31). Therefore, equations (42) and (44)
in section 4.2 may be put into a form such that the 4x4
matrix on the right hand side of the equation is identical to
equations (28) and (30) in section 4.2 , by multiplying the
last two equations in each set by a minus sign. Thus, equations
133
(47) and (48) become;
-s*y
<
or
S\y
> -
HGOi
. -
Vv> (A3)
MVXV
Vxl
Vl->
-7>
(A4)
Vx/
where [ KxzJ has the identical form as KX23 in equation (Al).
Equation (A4) can be put back into the form of equation (48) by
first multiplying the bottom two rows of the set by a minus
sign so that the force vector is equal to 1 Fyzj and "then
multiplying the last two columns of [Kxz^) by a minus sign
to put the displacement vector equal to S^yz)* Equation (A4)
may now be written in the following form,
[Ft}= Lvz]{*n] (A5)
where [_Fyz] andc/^2i
are as defined previously and [YyzJ is the
dynamic stiffness matrix in the y-z plane and has the following
form,
Ov]
~)\?t* I /V7 --'Fio
*
TA"
hfi 1 -l)H -Fi
h F. F>o_ Fyi fy
^/0 -Fi r*/A-v-
(A6)
yk i
134
where F^-
F10 are transendental equations given before.
(jCy^ is the dynamic stiffness matrix which relates forces and
moments applied at the ends of the beam to displacements and
rotations at the end of the beam in the y-z plane.
135
11. APPENDIX Bt EQUATIONS FOR THE ELLIPTICAL WHIRL ORBIT
In the computer output the rotor deflection is given by
the dimensions of the elliptical whirl path and the angle of tilt
(o{ ) from the positive x axis to the major axis of the ellipse
(positive with rotational). In order to determine the maximum
and minimum values of the whirl amplitude as shown in figure Bl,
the following analysis is performed as discussed by Lund Q5],
The u and v displacements are given by,
u = R (UR + i Uj)eiAt
v = R (VR+ i Vj)e1^
which are comples due to the bearing and disk representation.
Rejecting the imaginary components (since only the real parts
will apply) gives,
u = Ur cosli-t -
Uj sinJVt
v =Vr cosii-t
-
Vj sinA.t
The axis of the whirl ellipse may then be calculated as follows,
Major axis cl
Minor axis b
}-= jic *'
* * * u;<u;,t fllyyllllll'
These equations were reformed by Rieger 18 as follows,
Major ax:
Minor axi
:is ol) j~
and are the form used in this thesis program. The angle o( from
the positive x-axis to the major semi-axis in the direction of
rotation Cl. is given by,
136
<**
Jr *&*~'
[ -3( V* U* ^"V^ Ur) 1
Figure Bl Elliptical Whirl Orbit Dimensions
_Mi for Seym aXiS
-s-
y
tlal.oc S&Yr.laxis
137
112. APPENDIX C: WHIRL RADIUS FOR A SYMMETRIC ONE MASS MODEL
The following derivation was carried out by Rieger in
reference [31] . The whirl radius at the mass point and at
the bearings are sought. Fig. Cl shows the one mass model
representation.
W, a, X,,
Xj. Ks
I
1 Ki.&
Xi
rrp-r
Jf D Q D
V
7-777-
//,,/'/, ,>f; (cot
The equations of motion at the mass are,
IA X, = " KsCx,-^)-
Ks ( X,-
Aj) tMcc^^OS^t
m-V, Ks(7, -7O "Ks( 7, -y.?) s/nct
A force balance at the bearings give,
KsCX(-
xx)
Ks Cy,-yx)
ks(Xr^)
K3 Cy,-
73)
K Xi t DxL
K 7 1+ D y,.
K a3 + 0/.j
K 73t- D y_j
Since the system is assumed symmetrical,x2-
x^and y2= y^.
Therefore, only 4 unknows remain Xj, y1 x2, y2. The four equations
are,
Mx, 2 K5C><, -*-)= ^ ^ CO
"
COS CO t
HV"
+ Z K5 (y,-
Vx) = n a.c1-
Sin w t
K- Cx,-Xa)= ^t + D A
KsCy, -7^ K 7- + D 7;
(Cl)
(C2)
(C3)
(C4)
138
Assuming a harmonic solution,
the equations may be rewritten as,
^^.,
- , t,-lu)~t
,_
^AJCjt,
a. . - JJ OJ t
Ks CX, -Xx) e^wt
- (K Xi ^^ ^ D jcOetajt
which become,
j>M u^ X. ^K-
( X,- XJ j
-
Mac-1 (Cl)*
{-fUu/^, tZ K-(|-^)j-
-^ Me(C2)*
Ks C X, -X^) = C K + ^ cu n ) X*(C3)*
K5(^'^J= t K t ^ co D) #--. (C4)*
Since the orbits will be circular, only equationsCl*
andC3*
need be solved. The solutions of equationsC2*
andC4*
will
give the same answer.
Solve equation C3*for x2,
X K s X
CK5 r K -f ^ uj b(05)*
Substitute this into equation Cl*,
-fWx, UKSI," I- Ks X, = Maco
i_
K<> + K i-AO u, D
139
Expanding this expression out,
J L\-X\ -McUCajL (K+K-) f ^ w 0 3
Multiply and divide this expression by and set*= ~ and sf =
I -
C > * k ) + 2 ^TT Ci -ji*-) ")t _
Maco2-
,1KslCi-ii)+^_jJJ
Rearranging terms and solving for x-, gives,
fo+k) + aL ^ ]
From this expression the magnitude of the whirlradius1
at the
mass point is found,
co D
\X,I -
cr,_sy /Q+k)~
r
(yy)1-
q-si-) Jc-ka)'
f C^)V
(l'SLL)A
From equation C5*,
(C6)
X-
(Ks + * + I co b)
substituting in the expression for Xj gives,
y-
_ F>5 All Cu Co
\_2K, K- (K+ K^rAII F^o cobtlKs
~
PAI ))
140
Multiply and divide this expression by and setk=and I-
tHi
IKiK
CaJ
JKs
X%-
cu K JLl_
Rearranging terms gives,
2, =cu ^ K
() -^)
0 -*7)k J
From this expression the magnitude of the whirl radius at the
bearing is found,
IXJ - culil J^L.
~*
J (< -&jI I f c^y(C?)
V o-s\7)^
K
Equations C6*and
C7*were programed to give the unbalance
response
of the one mass model.
141
13. APPENDIX Dt BEARING DYNAMIC STIFFNESS AND DAMPINC COEFFICIENTS.
The information provided in this appendix is taken from
Lund [30] and Rieger [29] , and is intended to provide backgrour.:
information about the dynamic bearing forces. For a detailed
derivation of the bearing stiffness and damping coefficients,
the reader is referred to either of these references which are
available from the Defense Documentation Center. Futhermore,
the discussion will be limited to Hydrodynamic Fluid-Film
Bearing types.
Hydrodynamic bearings operate by creating a convergent
wedge of fluid between the bearing and journal surfaces, as
shown in figure Dl.
where ;
Figure Dl Fluid-Film Journal Bearing
(hfMtrlr.<j
VcCiU
CO
OB
OJ
e
R
Journftl
6- e/c
- is rotational speed
- is bearing center
- is the static journal center
- is journal static eccentricity
- is bearing radius
142
The resultant pressure generated by the fluid-film
motion is sufficient to support the bearing load. The fluid-
film stiffness and damping properties are determined by the
bearing geometry and the operating conditions. They are obtained
by solving the Reynold's equation for hydrodynamic lubrication.
The underlying assumptions in the derivation are [29] ;
1. The thickness of the fluid-film (y) is very small
compared with the length (x) and breadth (z).
2. No variation of pressure occures accross the film, Is -o.
3. The flow is laminar. No vortex flow and no turbulance
exists within the film.
4. No external forces act on the fluid film.
5. Fluid inertia is small compared with the viscous shear.
6. No slip occurs at the bearing surface.
7. Velocity gradients in the direction of the film thickness
are negligibla.
Considering an incompressible fluid ( $- const.) and a
motionless bearing (i.e. the wall of the bearing is fixed and not
rotating) ,the Reynold's equation has the form,
1 (Ji 12) +1 Ch? if) '- L ficoCi-^)i!i tuecosa (Oi)u y u n a te
u
h
where; P - is the local pressure
h - is the film thickness
M - is the viscosity
R - is the bearing radius
cu - angular velocity of journal
cX - whirl angular speed
6 - dimesionless radial velocity
143
Introducing the dimensionless parameters,
Vo,
2'
'-i/t.
}f * "2c
)6
- %
y ^y - >
f
and assuming constant viscosity throughout the fluid-film, equation
Dl may be written as,
1 o,3V ) ^ lA)^1a'3^*
^n^h'
+ iiff% cc5& (D2)
ir wL n' te'
)r 0')CO
The resulting fluid-film force is then the integral of the
pressure over the load carrying film . The fluid-film forces
in the radial and tangential directions acting on the rotor are ;
Fr '-
"))coCl~
*^)\f
Ft A co C1-*) Jj fw & d*
'J?1
where ; A~
L p- ' C c J
For rotor bearing dynamic analysis, these equations are linear
ized with respect to displacement and velocity, by a first
order Taylor series expansion about the static equlibrium
position. After transforming to a fixed x-y system these
equations are expressed in the form of displacement and
(D3)
144
velocity coefficients as follows;
F*
y
Kxx. X t Kxy y f Dx* X J-
>Xy V
KyK % F Kyy y+ A y K I F &
yy Y(D4)
where the eight bearing coefficients are determined in terms of
the radial and tangential force components. Thus the dynamic
representation of the bearing forces by spring and damping
coefficients are as shown in figure D2.
Figure D2 Dynamic Representation of Bearing Forces by Spring
and Damping Coefficients.
,|3
taringwall
.fiftinnjFiIt*.. )
*-y
145
14.0 APPENDIX E COMPUTER PROGRAM ROTOR
El. Program Capabilities and Limitations
ROTOR is a general purpose computer program for
the unbalance response analysis of a uniform elastic
rotor supported in fluid-film bearings. The program
is capable of assembling a rotor with any or all of
the following basic components.
1. Beam elements- having distributed mass and elastic
properties and constant E.Aand I throughout the
length.
2. Disks- with concentrated mass MD, polar and
transverse mass moments of inertia Ipand I-.
respectively.
3. Fluid-filmbearings- described by eight speed
dependent dynamic stiffness and damping properties
Kxx Kxy D xx Dxy
* ' *
4. Unbalance forces- represented by the real and
imaginary components of a rotating force vector
at each node.
The maximum number of elements are 7 thus fixing
the maximum number of nodes at 8. At each speed
increment the complex nodal displacements and
rotations are calculated and used in formulating the
major and minor ellipse radii and the ellipse angle
from the positive x axis to the major axis of the ellipse
and the program gives this information as output.
146
Limitations of the program are i
1. Only two identical bearings may be accurately
represented. For multibearing use, the
statically indeterminate support problem must be
solved first.
2. Short stubby sections heavily loaded in shear
may not be accurately represented.
3. The same weight density and Young's Modulus is
used for each beam section.
4. Static deflection due to gravity is not taken
into cone ideration thus limiting the analysis
to the classical "verticalrotor"
problem.
5. Steady state response is only concidered.
147
E2. GENERAL PROGRAMING INFORMATION
ROTOR is written in Fortran IV and was developed using
a Xerox Sigma 6 computer. Data input is from a card reader
whose device unit no. is (105) and output is to a line
printer whose device unit no. is (108). All computations
are executed using complex double precision arithmatic.
The main program plus eleven subroutines contained 475
Fortran statements. The program requires 15.8 K words
of main computer core.
The functions of the main program and of the subroutines
are as follows.
1. MAIN program -
array declarations, reads in and writes
out data input for checking, acts as an executive routine
for calling subroutines, assembles the structural stiffness
matrix, generates the force vector and solves for the
major and minor whirl ellipse axis.
2. ELEM - sets up an 8x8 dynamic stiffness matrix for a
single beam element, all terms in the matrix are real
double precision.
3. EDISKL - sets up an 8x8 complex double precision matrix
which reflects the effects of the addition of a disk
to the left end of the element.
4. EDISKR - same as 3 except for a disk added to the right
end of the element.
148
5. EBEARL - sets up an 8x8 complex double precision matrix
which reflects the effects of a fluid-film bearing added
to the left end of the element.
6. EBEARR - same as 5 except for a fluid film bearing
added to the right end of the element.
7. ZEROM - initializes a real matrix to zero.
8. ZEROMC - initializes a complex matrix to zero.
*
9. CADDM1 - adds a real matrix to a complex matrix.
10. CADDM2 - adds two complex matrices.
.7-
11. CMULTM - multiplies two complex matrices
12. CINV - inverts a complex double precision matrix
using the Gauss Elimination with partial pivoting.
* Subroutines obtained from Levy program j_32]
** Subroutine obtained from Ruhl program [l6]
149
E3 INPUT DATA FORMAT
Data input to ROTOR is in the form of punched cards. Seven
sets of data (1-7) are required, with the number of input
cards per set depending on the particular problem being
solved. The definition of the input parameters, the order in
which they should appear and their format is as follows.
DATA SET 1 - General element information
One card (2I5,3F20.3)
NELEM - total no. of elements used (max. 7)
NODES - no. of nodes (max. 8)
EMOD - Young's Moduluslb/in2
ERHO - weight desitylb/in^
DATA SET 2 - Speed information
One card (3F20.3)
BSPEED - begining speed rpm.
SPEEDI - speed increment rpm.
FSPEED - final speed rpm.
DATA SET 3 - Unbalance forces
One card for each node (3F20.3)
CUBALF - cos component of unbalance oz-in.
SUBALF - sin component of unbalance oz-in.
DATA SET 4 - Control cards and specific element information
Two cards for each element, a total of 2xNELEM cards
card 1 (515)
IELEM - 1 if element exists
0 if element does not exist
150
IDISKL - 1 if left end disk is present
0 if no left end disk
IDISKR - 1 if right end disk is present
0 if no right end disk
IBEARL - 1 if left end bearing is present
0 if no left end bearing
IBEARR - 1 if right end bearing is present
0 if no right end bearing
card 2 (2I5,3P20.3)
NA - left end node of element being specified
NB - right end node of element being specified
EINER - element inertia in*.
2.EAREA - element area in .
ELEN - element lenght in.
DATA SET 5 - Disk information
One card for each node, (3F20.3)
DW - disk weight lb.
DRAD - disk radius in.
DLEN - disk lenght in.
DATA SET 6 - Bearing locations
One card (515)
NBL - left end bearing node
NBR - right end bearing node
DATA SET 7 - Bearing properties
Two cards for each speed increment (4F20.3)
151
card 1
SXX -
bearing stiffness in load direction lb/in.
SXY - cross coupled stiffness lb/in.
SYX - cross coupled stiffness lb/in.
SYY - bearing stiffness perpendicular to load direction lb/in.
card 2
DXX - bearing damping in load direction lb. sec. /in.
DXY - cross coupled damping lb. sec. /in.
DYX - cross coupled damping lb. sec. /in.
DYY - bearing damping perpendicular to load direction lb sec/in.
152
E4. example run with input output data
An example computer run demonstrating the input-output
structure of the program is given in Fig. El . This
run is for the two element solution of test case two
given in section 5.2. The model idealization is shown in
Fig. 17 and the plotted results in Fig. It. A complete
listing of input parameters is not given here since
this information is documented in the computer output.
The first section of computer output contains the input
information. This is provided as a chech. Following the
input information the whirl orbit calculations at each
node are presented. The major and minor axis along with
the ellipse angle are given. This information is printed
out for each speed. Since the printout is rather lengthy,
the sample output is shown for only two speed? The output for
the remaining speed increments would look similar to
this output but with different whirl orbit information.
A novice user of this program may completely set up and
check out this sample problem by refering to the above
stated sections and figs, for input information and plotted
results. Pi complete listing of the computer program
ROTOR is given in Fig. E2.
153
Fig. E 1 Sample Input- Output Problem For ROTOR
I 00 oO
a-t O O O
Fig. E2 Program Listing - ROTOR 154C********************* ****************************
C COMPUTER PROGRAM ROTOR.....
C FOTH1*
-TFAPY S TA TF UNBALANCE RESPOND ANALYSIr
C nr ROTORS IN1FLUTD~rTLM BEARTNP-S. U TO (7) ''OTO"
c flevents MAY BF AXIALLY ASSEMBLFD.th-"
ROTOR
C ELEMENT MASS AND FLASTJC PROPERTIES AF DISTRIBUTED
C UNIFORMLY ALONG THE LrNFTH OF THE ELEMENT.
C UNBALANCE FORCES. rLUTD-*'_TLM BEARINGS AND DISKS
C A"E ALLOWED AT EACH NODAL POINT. AT EACH SPEED THE
C DYNAMIC STIFFNESS MA TR TX IS FORMED AND INVERTED BY
C GAUSS ELTWIMATI"^ WITH DARTIAL PIVOTING. AT EACH
C NODAL POINT THr STEADY STATE WHIRL ORBIT IS
C CALCULATED DUE TO S3ECTrTFD UNBALANCE IN THF SYSTEM.
C TKE UNBALANCE FORCES ARE REPRESENTED PY A ROTATING
C VECTOP. THE FLU7P-FTLM p-ARINSS ARE dfdreSENTED ?Y
C EIGHT SPE-0 DEPr\'DFNT DV"JAMIC STTFFNECC AND DAMDTNG
C COEFFICIENTS. TF1 r DISKS L'OSSESS MASS A"P GYROSCOPIC
C COUPLINGPR-
rR7 Tr S. COMPUTATIONS ADE EPFOPMED IN
C COMPLEX DOU^LFrT,ECIS70\'
ARITHMATIC.
C * * * * **^******** ****** * * ****** ** ** ** ******** ** ** ****
C MASTER of SCTEK"^ THEST^.
C CHARLES B. THOMAS JR.
C DOCiJESTER IVSTI'MTE 0>- "^CHNOLOGY JUNr. l?7t
* * * * ****** ********(******* *'**************** ********
C WAIN p?CGcAv
* * * * ** ** ****** ** **** ** ****'****** **** ****** ** ******
IMPLICIT PEAL*8 (A-H.^-Z)
rEAL*8 FrLK (F,8 ) .CUFALF IP) ,SI
COM CM E^ OD .r "'NER ,E Ar^AELEN.ERHO
C DM MON E^LK ,roKLt FDKP.FBKL. FPKR
COMMON DV,DRA".DLEN
COMt-'CN Sv Y, S^y. SX Y.SXV.DY Y.DYX.DY Y.HXX
**************************************************
C READ IN ANDWRIT*-
OJ T INPUT DATA
r * * * * * * * * ** ** ** ****** ** ** * * ****** ****** **** ** ** ****
READ ( IDS. 1)FLE^
,N 0"_S . E *<'0 D. ER HO
READ < IPS. ?) R "IP EED,Sr,rFDIF SPEED
DO lnOC I-I .NODES
1P0D READUP5.2) rUB AL r ( T ) # S UB ALF f 7 )
DOIDS'1 Trl.N^LEM
RP-
A 0(105. 7) T-"LEMn)-TDTSKL(T).IDTc'/R(I)IFEA0L(T).:BEAPP(I)
irsn reaches. n ^ a ( u .npm -einer d ) -ea<-e a(ij .el-^m I)
DO 2PC0 1=1 .NODES
2PCD READIICS.?) 0 '>'( I) tD
PA-1
( I ) C LEN ( T V
READ (105. "SI N^L.MRP
NSI7'"
-Ha(NrLr " + 1)
V/PITE ( IDS -10)
10. F ORMAT (// .1 X.'NO. E Lr Mr NTS' c X . * NO . N ODES ? 5* .
fYO UNGS MOD*.
15XtVFIGHT PrNS ITY* tr X. *F IR^T S PE ED
't SX . ^F rD
INCREMENT*t
25X.f FINAL EPr'"D')
WPTTF(10Ptl5J NFLEM.'- r'.Df"Strw0D.EoHP.BSPT0.SPt"EDI .E^DEED
15 F0RMAT(/.L-,X.T^tnX.Tr.PXtD11.5.DX.r-.^.l?X.F7.i;i-;yT
1F7. 1 ? 1?X. E7.T1
WRI TF ( 1 PP ,?r,)
20 r0RMAT(//.l X.-ELEMrMT N 0 . . 5X . NO DF .5X.NnPE B . 5X ?
IMPLEMENTINF"
TTA'
. 5 X , ELE "E NT A RE A?. 5 X, E LE Mr NT LENGTH')
DO 7050 I r] .NFL EM
2050 WRITF ( 10R,25r I ,N A ( T)'
. NB ( I)
25 FORMAT (/ 5X,*7, 12X.T7.BX, 12
WRITE I lffl .70)
30 FORMA t (// ,lX."UBALANr<~
FORC
1*C0S COMPONF..'T',5 X.
""
TN COM
DO 3000 1=1 .NODES
3D00 WRITEt 108t35) I ,C UB Al'F ( J ) ,c
35 FOPMAK/, 23X.T2.1 2X,r7.3tl2
WRITE(108t?7>
37 FORMAT (//;1 X.'D ISK TNFORMAT
l'DISK WEIGHT*,5Xt
DT-K RAD.
DO 3010 1=1. NODES
3013 WRITE( 108.381 I tD W( T T , DRAD (
38 FORMAT (/, ??X.T2tl CX trfl.4t6X
WRITE ( 10Pt40) NPL.VBP
UP r ORMAT (// ,] X. "L EF T Efjr, REAR
l'PIGMT en p. BrARING NODE!'. I
C* * * * **** * **** ****** ** ** ****** **
C BEGINNING Or SPr<rD LCOP
0************** ******** **********
444 ROMEGA=?S PEEP*7.3P3*T
. 141 59
READ (105.1) S YY ,S vy .FXYtS XX
READ (IPS. 4) "YY .D YX .rXY.DXX
WRITE ( IPS .4 5) BSPErP
r ORMAT (///t IX t? RO TOP SPEED=
WTTE ( lrB .SO) S YY ,S YX.SXY .S
FORMAT (//.l X. ^EA RING INFOR
l'KYXr'.Fl^.l.^X.'KXYz'.FlO.
WRITE ( IOP .55) D YY .0 *X .DXY .0
S5 rOPMHT (/, ->^x, rnYYr'
.rT0. 1 .5
IMP. 1 . 5X.'DXVr* .r ID.1'1
f"**** **.** ** **** ** **** ** ** *;** ** **
C INDIVIDUAL ELEMrf.!T SET U D FOR
C MATRIX USING :\pi)T CCNTPOL
(;********************************
45
^n
.FTME (I) .FARE A ( I) -EL?"N< I)
,BX.D10.4,qX.D10.4.DX.P10.4)
E'.SXt'MGDE NO.'.SX.
PONE NT*)
UBALF(I)
X.F7.3)
ION't3X, 'NODE
NO.'.5Xt
'tSXt'DISK LENGTH')
I) tDLEN(T)
tF8.4t7X.F8.4)
ING NODr :*IC. 5X t
5)
******************
**** ****** ** **** **
7G5DC/FP.PD0
'.<-p .1)
XX
'ATI ON .^X *KYY= 'rl C. 1 .5* .
1 .EX . *KVVr .F1 0. !)
XY
X. 'D YXr *,F 10.1 .FX.
'DXVr',
******************
CYVAMTC STIFFNESS
********** **** ** **
155
10'
200
2P1
300
401
500.
goo
GDI
^00
800
R01
DO 3^50 J=l .NTLE"^
TF(IELEM( J) ) 730.200.100
CONTINUE
CALL ELEM (J.POMEG A)
GO TO 201
CONTINUE
CALL 7ERPM(ErLK .8 .8 )
IFdDISKL (J) ) 400.400.300
CONTINUE
CALL EDISKL (JtRO^EG A)
GO TO 401
CONTINUE
CALL 7ER0MC (rpKL.8.87
IFdDISKP (J) T GOO.GOG. 500
CONTINUE
CALL EDISKR ( J.ROMEG A)
GO TO E01
continue
call 7er0mc (- okp. p.p)
TF( inFARL ( J) ) 8 OO .800*700
CONTTNUr
C ALL rpEfl PL C7 OMFG A)
GO TO 8 01
CONTINUE
CALL 7ER0MC (rr^K L. 8. P)
IF( TBEARR (J) ) 9 00 .Bpn * 85 0
156
BO
0
CAL
GO
CON
CAL
CON
CAL
CAL
CAL
CAL
TF(
* * * * *
NITIA
0 7ER
*'**'
*
IC CPK R, 8, P)
L FBEA RR (IOMEGA)
TO 901
TINUE
L 7FR0
TINUE
L CAD^
L CADD
LCAD!"
L CADD
J.GT.l
4 05
40P
30E
p* * *
C B
C V
r* *
c************ **
C TNITIALI7E T
C TO 7ER0 AND
C*** *'**** ******
L 7ER0
02 T1=4*NA(J
II
II
4000 K
t-n5P L
(TI. JJ
JJM
II
II+l
TINUE
L ZERO
******
THE PA
HE pP0
******
5PGD J
T^r (j)
TES (J)
C (4* J-
C (4* J-
* * * * * *
ION0<~
AND S
t A TION
L CINV
L CWUL
* * * * * *
C CALCULATION
C WHIRL ORBIT
C X AXIS T0 TH
C DT'RECTTiN Or
r **************
DO 5050 K
A=DPr AL (X
BrDT^AG (X
E=DFE AL (X
FrDIMAG (X
AlrA**7 + r<
A2=A1**2
A 7r-4.0* (
IF( ( A7* A3
A4=0.5*DC
GO TO e81
epp A 4 r 0.0
Pfll AMrDSQRT(
A.MM =DEOPT
ALPHA =57.
5 r 0
C* * *
C I
C v
C A
C* * *
CAL
11 =
11 =
JJ=
DO
DO
SDK
0 JJ =
JJ =
0 TI =
^ CON
CAL
* * * **
UIL:^
ITH T
* * * * *
CO
CEN
CEN
rVE
0 F VE
* jk * * *
N VERS
ATRIX
ND 0
CAL
CAL
* * * * *
Ml (E
M2<r
M2(E
M2 (E
) GO
* * * *
HE *
FIT
* * *
Mc (s
)-3
ELKt
DKL,
T^KR,
BKL,
TO
rDKLiet8)
EDKR,8.8)
FBKL'*e-8l
EBKE. 8.8)
90 2
a***'************************
TURAL DYNAMIC STT<-f-NESS MATDIX
WE INDIVIDUAL ELEMENT MATRICTfS
** *** ***********************
** **
TRUC
TN T
* * * *
^K .32,321
= 1 .
= 1 .8
) = Sr *( II . JJT + rgKR( K.L)
MC (r
* * **
rtic
* * * W
= 1 .N
= CU">
= 5Un
3.1)
1.1)
* * * *
TH*-
GLUT
S(NS'
TM (<
** **
UL AR
** **
ODES
ALE(
ALF(
rD CM
=DCV
* * **
STR
TON
7E .S
HK I.
****** **
OF T^r M
AND ANGL
E MAJOR
ROT4T TO
********
rl .f'-nrs
X( 4*V-3.
y ( t|w-3,
X ( 4*v-l.
X(4*X-1.
* * 2+ ^t *?
-A *r+^ *r
) . L T . 1 . 3
OR T( A2+A
32. T*
****** ** ** ********** **** ****
FORCE VECTOR ASEn^IATED
********** ******************
J) *-->DMEG A* *?/ ( 3PG.4pn*\%-
. ocn)
jj *r?OM EG A* *2/(3*E.4D^*lE.r,Dri)
PL XtC^MTES (J) .-CEV,T,rC( J) )
PL X( CENT EC (J) .CENTrc. (J) )
****************************
UCT'JPAL DYNAMIC STTFFNES^
FOR THE NODAL DISPLACEMENTS
DK .rr,KI)
NSir^.NSIZE.EVEC t 1 .XX)
****************************
AJCP AND MINGP AXT^ OF t he
E ALPHA FDOM THE nOSITIVEAVT"~
OF THEELLID,rc
IN THE
N
*** **** ** **************** **
1) )
1) )
1) I
1) )
+ F ** 7
} **
) GO TO 380
3)
0. 5* Al +A
f 0 . 5 * A 1 -
2957^3 *A
4)
A4 )
TAN( 7.0E0* <A*E-n*r)/ (E*E-F*F-A*A-R*3>) /2-OE 0
157WRITrnOPtGOT
J
'
GO FORMAT (// ,1 X. WHT RL ORB T T . 5X ,' NODF NO .
'? 5X .
? MA JC P FLLIPSE CAD'.
lFX.'MTNOR ELl'TPSr RAn
.
', 5 X, 'FLL Ip SE ANGLE')
WRI TE ( 108 .BE) K , A M, A^ M, AL PH A
65 F0RMAT(/,l9X,T2.1OX.^15.8.9X.D15.8.r,X,F8.3)
5050. CONTINUE
BSPEED=BSDEED*SPEEDT
IF(BSPEED.GT.T-SPEED) GO TO 9999
GO TO 444
f******** ********************** ********************
C END OF SPEED LOOP
**************#********a'
ft ***********************
1 F0RMAT(2I5-3r3.7)
2 F0RMAT(3r73.T)
3 F0RMAK5T5)
4 FORMAT (4E 70- 7)
ncoo STOP
END
C SUBROUTINE rLFM...S"TC UP THF PEAL D^U^LE PPErTSI0N
C 8X8 DYNAMIC STH^NESS MATRIX FOR AN ELEMENT158
C******** * ****************'************?*********
SUBROUTTN""
ELrM ( J .0 qy rG A)
IMPLICIT Rr AL*8 (A -H.0^7 )
BAR DYNAMIC S TI FF Nr Sc MATRIX 3X8 PrAL
R^AL-B ETNERJ7) .^ ARE A ( 7 ) , EL EN ( 7 )
REAL* 8 EELK (8,8) t DW (M tDRAD (8) -CLE^'tP )
COMPLEX* IS FrKL (8 t8) *FDKR (8 t 8 ) t EEKL (8 ,8 ) ,EEKR (8 ,8 )
COMMON EMODtrTNER tEAR1- AtELENtERHO
COMMON ErLK.rDKL.EnKP,E8KL,EBKR
COMMON DW,DRA.n, DLEN
COMMON SYYt SVX,SX Y.SXX.DYY.DYXtDXY.OXX
ELAM4r (ERHO*rAREA ( J ) *'<ROM EG A* *2 ) )V ( EM OD *E INER (J)*38G.ODC)
EtAM2 =DS3RTCLAM4 )
ELAMrDSQPKEL AM2)
ELL =ELAM*-LES'
(J)
S<rLL =DSIN (ELL)
CELL = DCOS (ELL)
SHELL=DSTNH (FLL )
CHELL =OCOSH C"LL )
FlrSELL*SuELL
r3rCELL*r'JELL'-l .300
r5 = CELL*SWELL-SELL*C>-,^LL
^ 6 = CE LL * G ue LL *-S EL L * Cu rL L
r7=SELL+SHELL
E3rSELL-GHrLL
FlD=CELL-rHELL
EC0MrFM0P*rINrR(J>*!rLlAM**2/r7
CALL 7ER0M(ErLK .8 .8 )
EELKd
EELK (1
EFLK (1
EELKd
ErLK (2
EELK (7
EELKd
ErLK<7
EELK (3
E-LK(3
EELK (3
EELK (7
EELK (4
EFLK (4
ErLK (4
EELK(4
EELK (5
EELK(5
EELMS
EELK (5
EELK(S
EELK (E
EELK (G
ErLK (6
EELKd
EELKd
E^LK (7
E ELKC7
rFLK ( 8
EELK (8
r^LK ( 8
EELK (8
RETURN
END
1 ) rp Cr v* (-EL AM*ES)
2) rEC'M* (-F1 )
5) rECPM* (ELAM*'r7)
G) rECPM* (F 13 )
1) =EEL'K{ 1.7)
2) =ECG ''* (E 5/cL'Aw)
5) =-ErLK (1 ,G)
5) rECOM* d 8/ rL'AM )
Z) -rr^i 1.1)
4) =EELX( 1-2)
7) =ErLK( 1.5)
3) =rEL'K( 1. G)
3) =EELK(2. D
4) =EFL^<( 7. 7)
7) =EFLV2. 5)
8) =E'rLK(2G)
1) rE^LXt 1. E)
7) rErL'K( 2. r)
5) =EELK( 3. 7)
6) r-r-LK (3 .4)
1) =EEL'KI LG)
2) rEFL'K( 2. G)
5 ) rErL K( 5. G)
G) rE<-L'K( 4.4)
3) =EELK< 3.7)
4) =EFL"X( 4.7)
7) r""EL'V( 5. c>
8) =EFLK ( 5. E)
3) rEELK( 3. 8)
4) =F EL K( 4, 8)
7) rTLK( 7t P)
8) =ErL K( S. G)
159
C LE^T END DISK E-rrCTS Rx COMPL-TX DPUPL^ PRECISION<-***********..***********..,,^^.^^^ ^fc-
SUBROUTINE E11 TS KL ( J .R OMEG A)
IMPLICIT REAL*8 (A-H.O-Z)REAL-8 ETNER(7) .E AREA ( 7) , EL EN (7 )REAL* 8 EELK (3.8) ,DW (R) ,DRAD (8) .DLEN(B)
Fni<L(8'^'T:DKR(9t8).EBKL(8,8)tERKP(8.8)tDCMPLXCOMMON EMODtriNERtEAPFAtELFN.ERHO
. COMMON EELK,rnKL,EDKR,EBKLEBKRCOMMON DWtDRAO.DLENCOMMON SYY,SYy,SXY,SXX,DYY,DYXtDXY.DXXDMASSrDW( J) /785.4DD
TINER-:(DMASS/I2.3 0D).-(3.0D0*DPAD(J)**7 + 0LFN(J1**2)
PINEPr(DMASS/^.0D0)*f^RAD(J)**2)CALL 7ERO MC (^DKLt Pt PT
EDKLd t 1) =DC"PLX(-DMASS*ROMEGA* *2.^.n)EDKL ( 7t 3) rEC-KLt 1, 1 )
EDKL (2.2) =DCM
PL X( -TIN ^R*R OMEGA* *2t 0.0 )
EDKL (4.4) r^ D^L( 2. 2)
rDKL (7.4) rDd' -->L X( O.F. --ptner*Romeoa**? )
rDKL (4.2) =DC*'L X( 0. n , PTNER* PO^E GA ** ? >
RETUPN
END
******** ******
C RI'GHT FND OT
C**** ****** ****
SUBROUT IV
IMPLICIT
REAL*8 ET
REA L* 3 EE
COMPLEX*!
COMMON E'-
COWMON Er
COMMON DW
COMMON SY
DMASSrOW (
TINER= (DM
DINEP= (P-<
CALL 7ER0
EOKP(S.S)
EDKR (7.7)
EDKR (6.6)
EDKR (8.8)
<"DKP (G.8)
EDKR (8.6)
RETURN
END
****'**
SK -rr
* * * * **
E rn TS
RF AL*8
NERd)
LK (P .8
G <^^KL
LK.rDK
.DRAFI,
Y. SVX.
J+l) /3ASS/1~
ASS/-7.
MC (rrK
=DCV#PL
= EDK -*(
=DCMOL
= EDK ?(
=DCMPL
=DCVT,L
******************************
EC TS PXP COMPLEX DDL' n-LE RECEcTON
************ ******************
KR (J, IOMEGA)
( A-H.^-7)
,E AR^A (7) , ELEN (7 )
> . DV d* .DR AD (3) < R )
(8t3)r-DKR(PtP)tEPKL(8t3)tEBKP(8tf)tDCMOLY
ER tE Ar"-A.ELEN.ERHO
L. EPKP , rp,KL. E3KR
PL EN
SX Y. SX X.DY Y. DYX. DX Y.DXX
85 .4pr;
.3 03)-<7.0D0*DPAD(J+l) **7 +DLEN(J+l )**2)
03 0) *( ^PAD (J-l ) * *7)
R, 8, RJ
X(-DMiSS*ROMEGA**?.n.p)
5, E)
X(-TTN~P*R0MEGA**7.D.D)
6t G)
X(O.Oi-PINEP*R0MEGA**7)
X(0.0.DINER*FOMEGA**?J
160
c**** ************ * ***?#***** ********** ****** **w -r -v -w *** -r ^ v ^ v * ^ m ^ m m m ap w
LEFT END REARIN-[r"
fc d 8X8 COMPLEX DOUBLE PRECISION****** *TT.^...~~.^..-/- .c * * * * ******
SUBRO
IMPLI
REIAL*
REAL*
COMPL
COMMO
COMMO
COMMO
COMMO
CALL
EBKL (
EBKL (
EBKL(
EBKL(
RETUP
F ND
******-**. aaaaaa*a*a*a*aia*a****a*amaaUTIN^
EBFARL (ROMFGA)
CIT PEAL*8 (A-H,0^2 )
8 ETNERd) ,EAREA'(7) tELENd)8 EELK (P,8 ) tDW (P-) ,DRAD (8) tDLEN(8)
EX*16 EDKL (8,8) EDKR (8 t 8 ) , EBKL (8 , 8 ) . EB K" (8 ,8 ) ,DC MPLXN EMODtETNERtEAFEA.ELENtERHON ErLKtrDKLtEDKR.E3KL.EBKRN DW tDRADt DLEN
N SYYtSYXtSX Y.SXX.DYY.DYXtDXYtPXXZEPOMC (E BKL, 8t P)
1 t 1) rDCvPL X( SY YROMEGA *OYY )
1*3) =Dd'PL X( SYX.?OMEGA*DYX )
3. 1) -DCMPL X( SX Y,POMEGA*DXY )
3.3) =DCVPLX( EXX.POMEGA*DXX)
c** **********************
c PTGMF '-\z B'ARINC EF EE
c** ** ********** ****** ** **
SUBROUTINE E=<~A RR (E
IMPLICIT RE AL*8 (A-H
c PIGHTENP
BEAMING E
EEAL*8 EI NERd) tE AP
REAL*8 ErLK (8,8 ) t PW
COMPLEX*l 6 rri<L (B ,8
COMMON EMODtETNERtE
COMMON Ef"LKtrOKLtED
COMMON DWtDRADtOLrN
COMMON SYYtSYYtSXY.
CALL 7EP0MC (EpK Rt 3,
EBKR(5t5) =DCVaL X( SY
rBKP(5t7) =DCVPL X( SY
EBKF (7t5) =DCVPL X( SX
f BKP (7,7) =DCVPL X( SX
RETURN
END
**************************
CIS 8X8 COMPLEX DOUELE PRr"'"ESION
****** **.****** ****** ** ****
OM"-GA)
?C-Z )
EFC"CTS PXP COMDLE*
FA (7) .ELEN (7 )
d > tDR AD (R ) tDLEN (P )
)."DKR(.'3),EcKL(R.8).EPK^(8t8).PCMPLX
A-P-A.ELEN.ERHO
KP.E3KLrEEKR
SX X.DY Y. DYX, DXY.nxx
8)
Y,POMEGA*DYY )
X. ROMEGA *D YX )
V, -'CMEGA*DXY )
X. ROMEGA*DXX )
SUBROUTINE 7EPOM( A.T,'.))
INITIALIZES A pr AL MATRIX TO
REAL*8 A ( 1)
II=I*J
DO ID K=l ,11
10. A(K)=O.ODO
RETURN
END
161
ZERO
SUBROUTINE Zr T?0 MC T A , 7 - J)
INITIALIZES A COMPLEX MATRIX TO ZERO
C0MPLEX*1G Ad, J)
DO 10 K=l tl
DO 10 L=l .J
10- A (K.L) =(O.OD"tD.DDO)
RETURN
END
SUBROUTINECADDM1 ( A t r .. I J )
ADDS A REAL MAdTX TO A COMPLEX MATRIX
REAL* 8 A (I, J)
C0MPLEX*16 BdtJ)
DO 10 K=l tl
DO 10 L=l .J
B(KiL) =A (K.LT+5 (K .L )
RETURN
END
10
lp0
SUBROUTINE CVUL Tv ( A , V , 7 , B , J , C )
MULTIPLIES TWO COMPLEX MATRICES
C0MPLrX*15 A f "? ,7 7) ,P <32, 1) tC (321 )
DO 10 L = 1 t K
DO IP Kr! .J
C (L.M) rd.DCC ,3 .300)
0 0 10 N-l .1
C(LtM)rc(LM)+A(LtN)*^(NtM)
10 CONTINUE
00.. RETURN
END
SUBROUTINE CAPDM2 (A tE.I, J)
ADDS TWO COMPLr>f MATRICES
COMPLEXES Ad.J)tB(T.J)
DO d K=l tl
DO 10 L=l t J
10. B (K tL) rA (K,L) *B (K tL )
100 RETURN
r ND
c*
c
c
c
c
c
c*
SUBROUTINECT'NV (N.AtAT)
****************'*******>********************
162
DOUBLE
WITH
S IS
c
c
c
CTNV.. .FINDS THr TNVFRSr OF THE COMPLEX
PPECESION MATRIX S BY GAUSS ELIMINATION
PARTIAL PIVOTIN*".. THE INVERSE OF MATRTX
STORED IN AI
N= ORDER OF MATRIX TO BE INVERTED****** +.mm m m mm mm mm, mm mm mm mm'* m mm m* ** mm mm mm mm mm mm mm mm
IMPLICIT C0MPtEX*16 (A-H,0-Z)
REAL*8 CDABS.SMAX
COMPLEX* 16 Sd2,7 2) AT (32,32) t A (32.^2)
DO 10 1=1 ,N
DO 10 Jrl ,N
ID S (I, J) =A (I.J)
INITIALIZE AI
NMrN-1
DO 100 1=1, NM
AI(I,I) = ( l.D^^.n. ODOT
DO 100 J=T,NM
AI(I,J+l)rd.r,DO,O.O^n)
A I ( J+l . I) =(0.OD0t O.OEF-)
IVi CONTINUE
AI( N.N) r < 1.0^^,0. ODOT
DC 430 Kr?.N
C
C
SEARCH FOR L AR G*~"ST ENTRY TN (K-l)TH CU.'JMN OF S
K M r K - 1
I MA X = KM
rMf XrCDAc S(S(KM ,K VJ )
DO 210 JrKt N
IE(SMAX-CPAB-(S(JtKMn) 200. 2 10. 7d
700 IMAX=J
SMAX= CCA BS d (J .K M) )
210 CONTINUE
IE(IMAX-KM) 703.400.300
SWITCH (K-l)TH AND IMAXTH EQUATIONS
C
c
c
7 0C DO 3 10 J=KM,N
TEMPrS (Kw, J)
S (KM, J) =S (IMAX, J)
710 F (TMAX, J)=TEva
DO 320 J=1,N
TEMP=AI (K w. J)
A I( KM, J) r AI (T tA X, J)
AI( IMAX. J) = Tr MP
320 CONTINUE
ELIMINATE X ( K- 1 ) FROM KTH THRU NTH EQUATIONS
4 On DO 420 I = K.N
RS=S ( I ,KM J /S( XM ,K M)
DO 410 J=K. N
410 S(I.J)=E(I,J)-RS*S(KM.J)
DO 420 J= 1, N
470 AKT. J) =A KI. J) -RS*A'(K^.J)
430 CONTINUE
163
c
c
BACK SUBSTITUTE
DO 500 1=1. N
500 AI(N,I)=AI(N,T) /S (N,N
DO 520 K=*>,N
N'K =N+1-K
DO 520 J=1,N
DO 510 L=2,K
510 AKNK, J) =AI (NK, J) -S (NK ,N+ 2-L ) * A I (N + 7-
AKNK, J) = AI (NK, J) /S (NK.NK)
520 CONTINUE
RETURN
E'ND
L,J)