NEW LIMITATION CHANGE TOdtic.mil/dtic/tr/fulltext/u2/470315.pdfNEW LIMITATION CHANGE TO ... s1...
Transcript of NEW LIMITATION CHANGE TOdtic.mil/dtic/tr/fulltext/u2/470315.pdfNEW LIMITATION CHANGE TO ... s1...
UNCLASSIFIED
AD NUMBER
AD470315
NEW LIMITATION CHANGE
TOApproved for public release, distributionunlimited
FROMDistribution authorized to U.S. Gov't.agencies and their contractors;Administrative/Operational Use; May 1965.Other requests shall be referred to AirForce Aero Propulsion Lab., Research &Technology Div., Wright-Patterson AFB, OH45433.
AUTHORITY
AFAPL ltr dtd 21 Jul 1971
THIS PAGE IS UNCLASSIFIED
AFAPL-TR-65-45
S4Part V
s1 ROTOR-BEARING DYNAMICS DESIGN TECHNOLOGYPart V : Computer Program Manual for Rotor
Response and StabilityJ. W. Lund
CaMechanical Technology Incorporated
C/ TECHNICAL REPORT AFAPL-TR-65-45, PART VC. J
may 1965
A:,? Force AWre Propion Lalu'atu'j,Research aid Tocknolhl Division
Air Force System Commal-Wright-Pattersi. Air Force Base, OWii
Best Available Copy
NOTICES
When U.S. Government drawings, specifications, or other data are used for any pur-
pose other than a definitely related Government procurement operation, the Gover-
nment thereby incurs no responsibility nor any obligation whatsoever, and the fact
that the government may have formulated, furnished, or in any way supplied the
said drawings, specificattons, or other data, is not to ba regarded by implication
or otherwise, or Ln any manner licensing the holder or any other person or corp-
Oration, or conveying a.ny rights ot permission to manufacture, use, or sell any
patent invention that may in any way be related thereto.
Qualified users may obtain copies of this report from the Defense Documentation
Center.
Defense Documentation Center release to the Clearinghouse for Federal Scientific
and Tiechnical Inforn.ation (formerly OTS) is not authorized. Foreign antounce-
ment and dissemination by the Defense Documentation Center'is not authorized.
Release to foreign nationals is not authorized.
DDC release to OTS is not authorized in order to prevent foreign announcement and
distribution of this report. The distribution of this report is limited because
it contains technology identifiable with items on the strategic embargo lists
excluded from export or re-exports under U. S. Export Control Act of 1949
(63 Stat. 7), as amended (50 U.S.O. App. 2020, 2031), as implemented by AFt 400-10.
Copies of this report should not be returned to the Research and Technology Div-
ision unless return is required by security considerations, contractual obligations
or notice on a specific document.
Non
AFAPL -TR- 65-45Part V
ROTOR-BEARING DYNAMICS DESIGN TECHNOLOGYPart V : Computer Program Manual fop Rotor
Response and StabilityJ. W. Lund
Mechanical Technology Incorporated
TECHNICAL REPORT AFAPL-TR-65-45, PART V
May 1965
Air Force Aero Propulsioe LaboratoryResearch and Technology Division
Air Force Systems CommandWright-Patterson Air Force Base, Okio
FOREWORD
This report was prepared by Mechanical Technology Incorporated, 968 Albany-Shaker Road, Latham, New York 12110 under USAF Contract No. AF 33(615)-1895.The contract was initiated under Project No. 3145, "Dynamic Energy ConversionTechnology" Task.No. 314511, "Nuclear Mechanical Power Units." The workwas administered under the direction of the Air Force Aero Propulsion Labora-tory, Research and Technology Division, with Mr. John L. Morris (AxYFL) actingas project engineer.
This report covers work conducted from 1 April 1964 to 1 April 1965.
This report was submitted by the author for review on 7 April 1965. Prior toassignment of an AFAPL document number, this report was identified by thecontractor's designation MTI-65TR15. This report is Part V of final doc-umentation issued in multiple parts.
The assistance of Miss Joyce A. Freeman, Mr. Kenneth W. Bauman, andMr. Ralph S. Graham of the Digital Computation Division (SESCD) at Wright-Patterson Air Force Base during the review and final preparation of thecomputer programs is gratefully acknowledged.
This technical report has been reviewed and is approved.
ARTHiUR V. CHURCHILL, ChiefFuels, Lubrication and Hazards BranchSupport Technology DivisionAir Force Aero Propulsion Laboratory
ABSTRACT
This report 13 a maniial for using the two computer programs:
I. "Unbalance Response of a Rotor in Fluid Film Bearings"
2. "The Stability of a Rotor in Fluid Film Bearings"
The report gives the analysis on which the programs are based, and the in-
structions for preparing the computer input and for interpreting the computer
output.
Lii
TABLE OF CONTENTS
Page
ABSTRACT iii
ILLUSTRATIONS v
SYMBOLS vi
INTRODUCTION 1
DISCUSSION 3
a. General 3b. Special Considerations in Performing the Numerical Calculations 5c. Analysis and Dimensionless Equations .......................... 6
COMPUTER PROGRAM: "UNBALANCE RESPONSE OF A ROTOR IN FLUID FILMBEARINGS" 11
Theoeticl Aalyss---------------------------------------------1Theoretical Ana lysis ......................... 11........ IComputer Input ............. -------------- 27Computer Output 36
COMPUTER PROGRAM: "THE STABILITY OF A ROTOR IN FLUID FILM BEARINGS" 41
Theoretical Analysis ---------------------------------------------- 41Computer Input ---------------------------------------------------- 49Computer Output --------------------------------------------------- 51
ACKNOWLEDGMENT 53
REFERENCES 54/
FIGURES 55
APPENDIX A: 57
Listing of Unbalance Response Program ----------------------------- 59Sample Calculations . I 82Input Forms for Unbalance Response Proram......7----------------------
APPENDIX B: 103
Listing of Stability Program -------------------------------------- 104Sample Calculations 112Input Forms for Stability Poa 117
iv
ILLUSTRATIONS
Figure Shaft Section Between Two Mass Stations-------------------
Figure 2 Convention and Nomenclature for Rotor Calculation 55
Figure 3 Bearing and Pedestal System ................................ 55
Figure 4 Gyroscopic Moment Catculation 56
Figure 5 Elliptical Whirl Path 56
V
SYMBOLS
A Cross sectional area of shaft, in 2
46 •Major and minor axis of ellipse, in (or: lbs, lbs.in)
0GqbI,4 41% Influence coefficients for shaft section, see Eqs.(4) to (7)
C Radial bearing clearance, inch
C"A"AC",4~4 Bearing damping coefficient for translatory whirl, lbs.sec/in
Bearing damping coefficientus for conical whirl, lbs.in.sec/radian
E Youngs modulus, lbs.in2
0. Rotor mass eccentricity, inch
Fu Ff x-and y-components of bearing reaction, lbs.
I Cross-sectional moment of inertia of shaft,0k between stations n and (n+l)l in 4
Ip Polar mass moment 2 of inertia of a rotor mess, lbs.in.sec2
(in input: lbs.in )
Transverse mass moment of inertia of a rotor mass, lbs.in.sec2
(in input: lbs.in2 )
IVVVI1 Bearing spring coefficients for translatory whirl, lbs/in.
14. Rotor stiffness, lbs/in
Lot Length of shaft section between station n and (n+l), inch
L Bearing length, inch
1 Rotor length, inch
M Bending oment (Mn to the left, M' to the riLght of station n),lbs.in.
mram x and y-components of pedestal mass, lbs.uc 2/in (in input: lbe).
My Total rotor mass, lbs.sec 2/in
MO3,MImgp'tM" Bearing spring coefficients for conical whirl, lbs.in/radian.2
Mass at rotor station n, lbs.aec /in (in Input: lbs)
P• xz and y-compofents for force transmitted to base, lbs.
vi
t Time, seconds
U•, U, Ccsine and sine-components of unbalance, lbs.sec2 (in input:oz.ir.)
V Shear force (Vn to the right of station n),lbs.
W Bearing reaction, lbs.
Y Ccmnponents of the rotor amplitude, inch (in output: mils)
Coordinate along length of rotor, inch.
t Phase angle between amplitude radius vector and unbalancesee Fig. 5, radians
Angle between major axis and x-axis, see Fig. 5, radians
0', Pedestal damping coefficients for conical whirl, lbs.in.sec/rad.
fel f •z . components of the slope of the deflected rotor,rad.
OWIX Pedestal spring coefficients, lbi/in.
duo 4Pedestal damping coefficients, lbs.sec/in.
Ca) Angular speed of rotor, radians/sec.
Wa. Critical rotor speed, radians/sec.
Subscripts and Superscripts
X in x-direction
4 in y-direction
XXX'K'Y9KVV first index gives force direction,second index gives'amplitudedirection.
C cosine component
3 sine component
#1 applies to station n
p pedestal
relative between Journaland pedestal
vii
INTRODUCTION
A rotor supported in fluid film journal bearings is a complex dynamical syst
which exhibits a variety of physical characteristics: critical speeds, insta
ility, unbalance vibrations, etc. In designing a rotor-bearing system for a
given application it is necessary to have methods available from which these
performance characteristics of the system can be predicted and thereby ensur
that the design is adequate for the specified operational condiftions. It ii
the purpose of the present, report to describe two computer programs by which
a particular rotor-bearing system can be investigated. The first program:
"Unbalance Response of a Rotor in Fluid Film Bearings" calculates the whirl
amplitudes induced by a specified unbalance. The second program: "The Stabi
of a Rotor in Fluid Film Bearings", calculates the threshold of instability
for the rotor-bearing system.
In the dynamics of a rotor-bearing system the fluid film journal bearings pl
a very importatit role. They are normally the predominant source of damping
such that without this source it would be impossible to run the rotor throul
any of its critical speeds. Secondly, the bearing jilm is flexible and the:
it may lower the critical speeds drastically. The film flexibility also cau
the bearing to act as a vibration isolator, attenuating the dynamical forco
transmitted to the pedestals. Finally, if the speed gets sufficiently high
the bearing film looses its ability to dampen out any transient motion of tk
rotor and transfers instead energy from the rotation of the rotor into a wk
ing motion of the rotor mass. This is called fractional frequency whirl
("oil whip') and is a self-excited instability of the rotor bearing system.
speed at the onset of the instability is called the threshold speed and as
rotor speed is increased beyond the threshold speed the whirl amplitude inct
rapidly, preventing further operation of the machine. It is, therefore, neo
at the design stage to ensure that the selected rotor-bearing syit'm does nc
experience instability in the operating speed range. Likewise, it is elsn 4
sirable to evaluate the magnitudg of the rotor amplitude due to a residual
balance such that too large amplitudes will not be encountered in the actual
machine. The two computer programs described in the present report provide
method for performing these calculations.1
'1i
- '- ~
The unbalance response program is very general. It calculates the rotor whirl
amplitude and the force transmitted to the base due to a given rotor unbalance.
The rotor is flexible and may have any arbitrary geometry. Also, there can be
splined couplings in the rotor and several bearings. The bearing pedestals can
be assigned both flexibility and damping. Since the bearing film forces are
not the same in all directions the whirl motion of the rotor is treated as Lwo-
dimensional such that it becomes an orbit around the equilibrium position. The
orbit is elliptical and its dimensions and orientation vary along the length of
the rotor. The computer program calculates the whirl orbits for a number ')f
points along the rotor and gives also the components of the force transmitted
to the foundations.
The program for investigating the stability of the rotor-bearing system applies
to an arbitrary rotor geometry. There may be several bearings and the stiffness
and damping of the bearing pedestals can be included. The program calculates
the speed at onset of instability (the threshold speed) and the corresponding
whirl frequency.
In both programs, the dynamic properties of a fluid film bearing are expressed
in terms of 8 coefficients: 4 spring coefficients and 4 damping coefficients.
The coefficients depend on the bearing type, the bearing dimensions, the vis-
cosity of the lubricant, the bearing load and the rotor speed. The values of
the coefficients are given in a previous report (Ref. 6) for a widi variety
of bearing types, geometries and operating conditions.
The report sets forth the analyses on which the computer programs are based.
Detailed instructions are given for preparing the computer input and for inter-
preting the output..
S,•.. • -. = , ... . .. , • I l III Ill I2
ao
DISCUSSION
a. General
The computer programs determine the interaction between the bearings and the
rotor. The unbalance response program is concerned with the synchronous
amplitude of the system under the action of unbalance forces and the stability
program is concerned with the free, self-excited motion at the onset of hydro-
dynamic instability ("oil whip"). Whereas the dynamic properties of the
bearings derive from lubrication theory (Ref. 4), the analysis of the rotor it-
self derives in its principle from the Myklestad-Prohl method (Refs.1, 2, 3).
However, in its original form, the Myklestad-?rchl method is set up only for
calculating the critical speeds of the rotor and is further limited to plane
vibrations. In t0e present analysis the motion is tryated as two-dimensional,
damping is included in the bearingsin addition to stiffness and the analysis
is valid for any spee4, not just the critical speed.
In general a rotor's cross-sectional dimensions and its ass distribution varies
along the length of the rotor. Thus, fur calculation purposes it is conven-
ient to break the rotor up into short sections, each section having a constant
cross-section. Furthermore, when there are many sections, the mass of each
section can be divided into two parts and lumped at the end points of the sectic
Concentrated masses like wheels, impel1ers, etc., can be made to coincide with
an end point of a section. In this way the rotor is replaced by an idealized
model consi-sting of a number of mass points connected by weightless, flexible
bars. The model can be brought as close to the actual rotor as desired by mak-
ing the subdivisions small but in practice only a limited number of divisions
is needed to obtain a very good accuracy.
Since the bearing film properties to a large extent control the whirl motion
and the stability of the rotor, it is necessary to represent the dynamical bear-
ing film forces as accurately as possible. The method of representation is
based on the assumption that the whirl amplitude is small compared to the bear-
ing clearance such that the dynamical forces can be replaced by their gradients
around the steady stqte journal center position. In this way the dynamical
3
forces become proportionil to the whirl amplitude and to the corresponding
velocity, and the factors of proportionality are called spring and damping
coefficients. They differ from conventional mechanical spring-dashpot systems
by also containing cross-coupling terms in addition to direct-coupling terms, i.e.,
the dynamical force in a given direction (say the x-direction) is not only pro-
pTrtional to the amplitude ar.J velocity components in that direction but is also
proportional to the amplitude and velocity components in the mutually perpen-
dicular direction (i.e., the y-direction). Hence, in an arbitrary reference
coordinate system with x and y-exds the two dynamical force components can be
expressed by:
where x and y are the amplitude components, i and y are the velocity componentsK and K are the direct coupling spring coefficients, C and C are the
xx yy xx yydirect-coupling damping coefficients, K and K are the cross-coupling springxy yxcoefficients, and C and C are the cross-coupling damping coefficients. Thesexy yx8 coefficients are functions of the bearing Sommrfeld number defined through
the rotor speed, the steady state bearing reactions, the lubricant viscosity and
the bearing dimensions (for gas bearings the coefficients are functions of the
compressibility number, the bearing eccentricity ratio and the whirl frequency).
Thus, the coefficients vary with speed. A method for calculating the coefficients
is given in Refs. 4 and 5 and values of the coefficients for several bearing
types may be found in Ref. 6.
Frequently the pedestals, on which the bearings are monted, are as flexible as
the bearing film. In such cases, the pedestal stiffness saut be included in the
calculations. For completeness the analysis allow for both stiffness, damping
and inertia in the pedestals. Furthermore, as the rotor bends under the in-
fluence of the unbalance forces, the journals become cocked in their bearings.
The fluid film resists the tilting and this can be expressed by a set of B spring
and damping coefficients in analogy to the previously discussed coefficients.
The unbalance response analysis includes this effect, both in the beartins aud in
the pedestals. The resistance to tilt normally affects the rotor motion only at
speeds above the second or third critical speed but if the pedestals are made
4
tt
soft for alignment purposes resonance conditions may exist which can only be
explored if the effect of tilt is included. For the stability analysis this
effect is in almost all cases very small and it has been ignored.
Occasionally the rotor is not a single member but consists of several rotors
connected by splined couplings (e.g., a turbine-generator set connected by
a splineC coupling). The unbalance response program allows for ircluding
splined couplings anywhere in the rotor and assumes that no bending moment is
transferred through the coupling. This feature is not included in the stability
program.
In the unbalance response program the whirling motion of the rotor is generated
by unbalances built into the rotor. In general the unbalance varies in magnitude
and circumferential location along the rotorsuch that under speed the unbalance
forces may bend the rotor into complicated shapes (e.g., resembling a "cork-screw')
The bend rotor whirls around its steady state position (i.e., the position the
rotor would occupy if there were no unbalance forces) with each point of the rotor
axis describing an elliptical path. The dimensions and orientation of the ellipse
varies along the length of the rotor.
If the rotor runs at high apeed and has large disks (e.g., turbine wheels, etc.)
mounted on the shaft the gyroscopic moment becomes important, especially if a
wheel is overhung at one end of the rotor. The gyroscopic moment is proportional
to the mass-moment of inertia of the wheel, the square of the speed and the de-
flection angle of the rotor. If the rotor motion is considered as a transverse
vibration of a beam (i.e., the whirl orbit is a straight line) the gyroscopic
moment tends to "soften" the rotor and lower the critical speed. On the other
hand, if the bearing spring and damping coefficients are the same in the vertical
and the horizontal direction the rotor whirl orbit becomes a circle and the gyro-
scopic moment stiffens the rotor. Actually, the whirl orbit is elliptical, i.e.,
somewhere between a straight line and a circle, and the effect of the gyroscopic
moment can only be assessed by performing the complete rotor analysis. It is a
non-linear effect since it depends on the dimensions of the elliptical whirl orbit.
In the present analysis the gyroscopic moment is takse into account in the un-
balance response program and is calculated by an iteration procedure.
5
b. Special Considerations in Performins the Numerical Calculations
The greatest difficulty encountered in performing the numerical calculations
is the magnitude of the numbers and the loss of significant figures. These
difficulties become pronounced when: a) there is an excessive number of rotor
mass stations, b) the rotor is very stiff and, c) the bearings are very stiff.
There is no universal remedy for the problem but if trouble arises two possi-
bilities may be tried: a) reduce the number of rotor stations to the essential
minimum and, b) apply a scale factor.
Let the scale factor be t.. Then:
multiply the speed by o.
multiply (EI) by ((e g., multiply 9 by 4)
multiply the bearing spring and damping coefficients by
(i.e., multiply K xxM, & w etc. by Cx )
multiply the pedestal stiffness by 40 and the pedestal
damping coefficients by OL.
(i.e., multiply X, and %1g by de iand 4y by 4 )leave the rotor mosses, the rotor length, the pedestal masses
and the unbalance unchanged.
The numerical results will give the amplitude unchanged whereas the bending
moment and the transmitted force must be divided by 4e to obtain the actual
values.
c. Analysis and Dimensionless Eauations
Referring to the sign convention given in Fig. 2 and conaidering first a con-
tinuous rotor the three basic equations for determining the rotor motion are:
(l-a) Force balance for a shaft increment, d, j 'u p•A•(xva)
(2-s) Moment balance for a shaft increet, Jz 2w'ýy
(3-a) Shaft deflection: Er "6
where:x - amplitude in vertical direction, Inch
y - amplitude in bocisontal directiLo, Imah
/
z - coordinate along the rotor length, inch
a - eccentricity between mass center and shaft center, inch
A - cross-sectional area of shaft, in 2
1 - cross-section moment of inertia, in 4
E - Youngs modulus, lbs/in2
' - mass density, lbs.sec 2/in4
(iP-i) - mass moment of inertia per unit length, which is effective in
gyroscopic moment, lbs.sec
w - angular speed, radians/sec
H - bending moment, lbs An
V - shear force, lbs.
For the stability analysis set a - 0 and redefine a) to man the whirl frequenc)
These three equations may be combined to give the familiar 4th-order differen-
tial equation governing the unbalance vibrations of a rotor:
(4-a) (+O
and the same for the y-directton.(see Ref. 3, page 330)
For a circular whiri orbit:
For a straight line orbit:
For an elliptical whirl orbit, see Eq.( 2 8) and (29) in this report.
At the bearings there is an abrupt change ir the shear force and the bending
moment due to the bearing reactions. Let :e bearing be at a - zo. Then:
(5-a) V -v =
7
- -.. .. r
where X x , D etc., are the bearing spring and damping coefficients.
Actually, the effect of the pedestal should be included in the above equations
as shwon in.Eq. (12) and (13). in the analysis.
The numerical method uses Eqs. (1-a), (2-a) and (3-a) by rewriting them into
finite difference form:
W &U ~(A7) (x+a)
Together with Eqs. (5-s) and (6-a) these ,quatiots form a sot of recurrence
relationships which can be solved step by step, starting from one end of the
rotor until reaching the other end. The details are given later.
Occasionally it is desired to perfrrm a dimensionless analysis. The two
governing quantities are:
(7-a) a Mr
(8-") ()
where:(1I) - reference value of 21, lbe.1n 2
A - rotor spen between bearinp, inch
N I S - 1P :dz, total rotor mess lbe.soc 2/in
Kr - rotor stiffness, lbs/in
(D n - equal to or proportional to a critical rotor speed, cad/sec.
For a uniform shaft (3! - constant, A - Constant):
Mf "'KAfI aA r iii J
S!
N
where n designates the order of the critical speed. Thus, for the first mode:
n=I i.e.,
(EI)O= !4±'tr =2.464.Er (Uniform shaft, first mode)
However, it is not necessary that w be a critical speed but Eq. (7-a) mustnbe satisfied.
The dimensionless parameters become:X': X/a.x I= x,/'O
zi': z/e(En': Er/(Gr)*
M': M/a.,e
(?A): " tA/M1..-Cof. UP-io)/•r•where:
a - reference value for the rotir mass eccentricity, inch
C - reference value for the radial bearing clearance, inch0C - actual radial bearing clearance, inch
W - reference value for the bearing reaction, lbs.0
W - actual bearing reaction, lbs.
L - bearing length, inch
The dimensionless bearing coefficients are given the form above since the values
obtained from lubrication theory are CK /W, OaWC /W, etc. Normally, a dimen-xx xx
sionless analysis is only performed for a simple system where all bearings are
identical, i.e. C - C and H - W . In that. case the basic dimensionless0 0
parameters are:
speed ratio:
dimensionless rotor stiffness: K' = CK /Wr r '-2'C VLecdimensionless bearing coefficients: = /,O, (0 1VmtL). etc.
xx x x
Thus, to perform a dimensionless calculation for a given value of K' use asre
input to the unbalance response computer program:
Speed - (')/.10471976n f
Mass at station m = 3.86069,105 (m-station weight, lbs;E.1 n-number of stations)
(IP-IT) at station i . PI 3.86069"1O5
MT'
Cross-sectional moment of inertia for section i-(i+l): 1000-1/1
0
Young modulus - 1
Length of section i-(i+l) - C
Bearing spring coefficient 1 ,-r
Bearing damping coefficient -
rn n
Unbalance such that: E U (on.in) -6177.1 E U (o:.in) - 6177.11x Y
Then the computer output will give:
amplitude - &- and L..a a0 0
abending moment - H' - H/aoKrJ - H/•--%gIjK
transmitted force - (actual force)/a K - (actual force)/l -a ;o r C r
10
COr•PUTER PROGRAM: UNBALANCE RESPONSE OF A ROTOR IN FLUID FILM BEARINGS
This section of the report describes the basic analysis and the detailed in-
structions for using the computer program: PNO011: "Unbalance Response of a
Rotor in Fluid Film Journal Bearings" for the Im4 704 digital computer. The
program calculates the rotor deflection and bending moment, the pedestal de-
flection and the transmitted force resulting from a specified rotor unbalance.
It differs from conventional programs by taking into account the variation of
support flexibility and damping along the whirl path of the rotor.
The supports for the rotor consist of a fluid film bearing on a pedestal, both
members possessing flexibility and damping for translatory and rotational motion.
The flexibility and damping are linear in displacement and velocity respectively,
the proportionality factors denoted as spring and damping coefficients. The
fluid film is represented by 4 spring coefficients and 4 damping coefficients
for translatory motion and similarly for rotational motion, thus allowing for
coupling between the motion in two mutually perpendicular directions. The
pedestal has no such coupling and is represented by 2 6pring and 2 dmping
coefficients for both translatory and rotational motion with corresponding
pedestal mass and mass moment of inertia. Hence, each point of the rotor will
whirl in an elliptic path around its steady state position.
In addition, the program includes the effect of gyroscopic. moment and provides
for couplings in the rotor.
THEORETICAL ANALYSIS
The analysis is an extension of the Myklestad-Prohl method, see Ref. 1, 2 and
3. The rotor, which is actually a continuous system with an infinite number
of degrees of freedom, is replaced by a finite number of lumped masses conn-
ected by weightless springs. The computer program calculates the vibrational
response of this equivalent system exactly.
11
Thus the accuracy of the results depends only on how closely the
idealized system rese:.bles the actual rotor.
Starting from the left end of the rotor, the program calculates step
by step the bending moment, shear force,, slope and deflection along
the rotor. Neglecting the shear force contribution to the deflection,
we get from Fig. 1:
(1) Mnf Ml= i + L, Vn
(2) 9A)#1÷ 9A + an. Mro + b" V,,
(3) xy f# I A, + CG., 4 dA V11
where:
(4) a., I "-" for Et constant in 0 < L
(5) b f -L4 f -f 4
(6) C * L, ,,b,2h " ,
On L j. -bm1A(7) do L . S
The program assumes El constant between moso points. At the sees
points, the forces acting on the rotor are introduced. Four contrib-
utions exist: (1) inertia force, (2) unbalance forces, (3) bearing
reaction, and (4) gyroscopic moment. In general, not all 4 contrib-
utions apply to each mass point.
Inertia force. The rotor performs harmonic vibratioms at the oI
frequency as the rotational speed. Thus the inertia force is:
12
(8) - r d =z rn Zx
(9) = = lz
Unbalance forces. To allow for change in circumferential position of
the unbalance along the rotor, the unbalance is given two components
Ur and U 1 . This gives rise to an X and Y component of the unbalance
force:
(10) V,!. VX^ U XC s U -OjEUY:5CWt
(11) (V1.6- V1.11*i 4 .4. .dzu CO: j Lt S w W
Bearing reaction. The bearing supports haVe flexibility and damping
for both translatory and rotational motion of the rotor. Since the
equations for the two types of motion are analogous, only the equat-
ions for translatory motion will be derived.
The bearing support is shown in Fig. 3. It consists of a pedestal
with mass ( Max, Y ), supported by springs (hx,•) and dashpots
(d's, ). There is no coupling between the X and U direction, i.e.
no transfer impedance, nor between the translatory and rotational
motion. The pedestal supports the bearing fluid film. which is rep-
resented by 4 springs and 4 damping coefficients. If the relative
motion between the Journal center and the bearing housing is denoted
(X; V'), then the bearing reaction becomes:
(12 -V- ,N-) = -K '- C,,,'- KN '- C
Setting:
X Xe COSUt + X5 %Sflin W
we get from Newton's second law fox the pedestal meso:
13
-MIMI
(KaxtLxLe x +WC +AX~ + K,9 41'c cx (x) tM )s
Solving the equations we obtain:
(v. -V14 ( AVajrXc -A Vjxk jA-A Vcxlc -AVU. V)Cos wt
(14) .f( U,-VX VV o iiW
whore:
A V.x -- "3 +L~a)Camr¶ - +km)$4W~
(15) A VJ6 -K,=L +WC*6-K 1m1t+wC.1$
AVejin KjiAWCALt+aS+ia)Cay't~
a~b - itL jfk ~f w.
14
and: SE_ + FD
FZ+42 *- .
F- PK,,o+uLO.,Ry 6 B-KjR-WC~jQ
gxa~~wc R (ac-- ,, ha Ld
as4
The~~ eqaon for roaioa motion ar analgou to eqL(rexe o
a sign reversal (sign convention, as*e Fig. 2):
(1AXe*)&,"- (AMUeGC +hM6eS + AMXca + .AMwf~) COSWt
(16)
(my'- tm" a(-AMC #A ft. Os AtpMal f + " s COS Wamt
+ 1PAMjO fyf I s3nt
where the coefficients AM",APIW etc. are computed from eq. (15)
as AM,.a'AY&, S AM Ax a AVO etc. by replacing the translatory
spring and damping coefficients by the corresponding rotational co-
efficients.
Since the fluid film coefficients are funct'ons of speed, directly
through the Soerfeld number anJ indirectly through the decrease of
eccentricity ratio with increasing speed, the computer program prov-
ides for expressing the coefficients as a function of speed, e.g.
(18) -K I Kx.xo + Kx,'1 • K= 'a.&
and similarly for the other coefficients. W) is the rotor speed in
radians/sec.
Gyroscopic Mouent. The gyroscopic moment derives from the change of
the angular momentum vector of the rotating rotor mass as it whirls
in an elliptical path around the steady state position of the rotor.
For two special cases the gyroscopic moment is kMn:
circular whirl path: M'5,. a (z-xTh,3 e(19)
straight line (transverse vibrations): Md,. a- ,,3
where 0 is the slope of the rotor deflection madt mndZr are the
polar and transverse mses inometsof inertia. Mor an elliptical path
the gyroscopic moment is no longer linear with respect to the slope of
the rotor, indicating that an elliptical path is actually not possible.
However, in general the effect of the gyroscopic mmet is not too
big and for the present analysis an elliptical pef will be assured.
The coordinate system is shown in Fig. 4, Wue is the steady state
shaft center position and 0 is the whirling shaft center. The moving
coordinate system (9195) is defined by its usiat vetom:
16
V__ / f -a f 7 - O 47g,( 0
(20) ,4, 0)
The angular velocity vector becomes:
(21)
The moment needed to sustain the motion is given by Eulers equations:
where I d.notes mass moment of inertia and I , rr ad Irr., .
Let us first assume that (XIj) corresponds to the directions of the
major and minor axis in the elliptical variation of the rotor slope.
Their.
6,- E o (Wt..)(23)
Combining eq. (20), (21) and (22):
(24)
which c•.early shows that the gyroscopic mut ti not linar with
respect to tht rotor slope. lowever, only the first hamouic can do
17
9L &
work on the rotor. Hence a Fourier analysis vwil be performed. The
folioving integrals apply:
Is[2 srwx CO.Sx = 1 z
E sewli •+ '.) 6j S.WEhtCOsLon 6'$irc b(:*6)
at
Then the first harmonic becomes:
(2•)(24) M ;'%r )W~MV o- x,) go'e
In the limit, eqs. (24) agree with eqs. (19).
Eqs. (24) must be transformed back to the actual (%O)-coordinate
system. Setting
19 GCASWt #*Off 51dI o(25)
Oc Cd&8 4t 4'~s ia 04
16
describing an elliptical variation of slope, we get:
ition in the same "direction as •.). Then:
G, - 0s-e9 + i
(27)~, i,.-# ,;n•+q'A,
Substituting &q. (26)-(27) into eqs. (24) gives:
(28) Co 2 (gi, 01),: -A
"iethere
(30) 6A (o,.¢L)(#5 + s+fc. ),
Since eq. (28) and eq. (29) are not linear, an iteasti: mthod is
used. For each rotor speed, the program perfot.. a nmber of iterat-
ions. The first iteration is done without gyroscopic moment. Afterthe first iteration, the gyroscopic moment ia calculated from eq. (28)-(29) and these values are used in the second iteration and so on. Thecalculation has converged when the relative change in rotor mitude,
and slope between t (t iterations is vmellorthan a specified limit.
19.
ERCArIONS FOR RoroR CALC'lLArION
The bending wmoent, the @hear force, the slope and the deflection are
expressed by:
Mxf Mir, Cos wt + Mt, sin wt
VXfc. 05wt + Vis Simj"A
8C e Cos wt4- 99 S44'
X 4C03 WtL+ X% iwtA
and 4gmilarly for the V -direction. Then sq. (1), (2), (3), (8), (9)
(10). (11), (14), (16), (17) , and (29) may be combined Pto Sive *theequations used in the rotor calculation (see fig. 2):
*.A Myw. ~ dA *i * 6$t+MWm4%ft +A t~Psif t- (MX'Cnj MACA)60Vf.
mrseh x~-A ixic~
t"CAS Mjf +A M&~CAIVGC
MI M~440 -A o &.i +A McyA G9xAmbyRIh+Mer4sti(M~s%-l, L,.
(32) Vase Voj. + AVb X4cA + fro. W 12 A Vow.%] Ys. +A Vja4&-AV~ft Uj.I :' W L.
MYA.. M;.i Lu Vv.
20
my,,,,, Fm 1,.4 + LtVim
Oct.," •.+ 0.4 M• C* +-bv"
GqV* as + a. f , %stIVA
fe0,. +, • Mv 41" +bkk•,m
44, • .+ LA, ( -.is Myu +e dlVw,
t; LJ'"=r •"+ L% Go" + 6 Km~f +d' Von
In the above equations do p 4 4 At are given by eq. (4), (5) and (7),AMOl..AM .. .---- iA1A 1 and iVmu, AV4# .... Aid^ by eqs. (15) and
m -N&.. (Mmit-M),e by eq. (28)-(29).
Boundary Conditions. The rotor io assmed to have free ends. So loss in: .generality occurs by this condition since It say be changed by letting
the end points have bearing support. A proper choice of support coef-ficients will then allow for any type of end conditions.
For a rotor with free ends the beadingu ~mm t and the hear force arezero at the end:
(33) MVU Mal| wMiea s s, •Val t V" s &vs jo a 0
(34) ,
21
Starting from the left end of the rotor (see Fig. 2), eq. (33) is used. -
However, the slope and the deflection are unknown. Using the super-
position principle, each unknown is applied separately. A smmation
gives the combined effect. Ten calculations are performed, using eqs.
(32).
I. Gv&, g *(i x -,0, . • X-I,,=•fe~,= s 1,= U,= fO=
z. b,. *I 9e * *&,-g, .Kux==•c, = 4g A,=•- u 4e=3. aw G.ea9•,, t, ,X==Y,=q,,==2. $9 Ot ?L a~ At m" Xf 4WsC4, 4u=u,-
3. 0, & =AL, aft, =,,=X,, = %,,=q,, =t, = 0o5. x* I
a. -141 I ' gXa)s
9. 90 2 txft L414U~j Aso %'A me4, =f =4 i4=40 4410. Gyroscopic moment applied &1 =----•1 -, ,X4 1 X.-- -- 6 14'.-
For each calculation eqs. (32) are used to calculate the bending moa-
ent, the shear force, the slope and the deflection along the rotor. At
the right rotor end, station r, eq. (34) must be satisfied, i.e.
fv¶., /h4 ---- M.. "m _M ,-• ,
M+'I Mx.&----- MA.' -M;Kr -Ma0, tII I " 1
(35),, - ,*------.,,.
e let g4,a........ v, X a. 10,Kir., yi.------ V•s•, I, -Ae, i,-VOOP,,to
Eqs. (35) are then solved for &,A-----,, and the actual values of
bending moment, shear force etc. along the rotor can be determined.
AP a given rotor speed, eqs. (35) are first solved without gyroecopic
22
moment, i.e. Vxyo.rO. Then the gyroscopic moment isapplied according to eq. (28)-(29) and new values are found fore,,ll 90511--- -- from eqs. (35). This process is repeated until atthe k'th iteration:
(K) ((I 9X(36) 1' - - -- -- 4-1
where E&4, is the convergence limit specified by the computer input.If the calculation does not converge within a specified number of iter-ations, the program goes on to a-new rotor speed.
In the computer output, the rotor deflection is given by ti-e dimensionsof the elliptical whirl path. We have:
X- X5 coswt + .X4 s;1 Wt(37)
q. woswt + qs;,,,wt
As shown in Fig. 5, the (Xl)-coordinate system is rotated an angle iin the same direction as J to become (X,'f,). Then
x, ac, (," + a.+.,,)(38) X-~~sw~
where a and 6 are the major and minor axis respectively of the ellipse.From Fig. 5:
X- , + x ys im3*cosh
~~~' ~+ LiXi)3LCOSP
Then:
23
Here it is necessary to allow b to become negative. The reason is that the trans-
formation from the x-y-coordinates to the ellipse must be able to discern be-
tween forward and backward whirl (i.e. the shaft center may trav*l in the same
direction or in the opposite direction of the direction of rotation depeneing
on the values of xc, xs, YC and y), Let th2 angle between the x-axis ard
the instantaneous radius vector be :
Then: 0
i.e.
(kj,-xrC.) >o : 4ft wi)rl
Therefore:
To find d and • expand Sq.(38)
(a) 4CO a X- c sA-i/
(b) -dLSIS,t Y1, (OSA +, 'k S''P
(c) ,SiWW',,d.XYS;.AP ,CCOSA
(d) Weuii -Ys s51+, (espThen:•an: (a)'+€)(b)'÷(c)'4 ( : a'÷'. ki*m+ '+,
Next: V-W(04))- (c).(4) : I ("'-•,) •4i•A--- (ýX$ t,,g,)
Thus in total:
24
(40) to.nZ (' . , L. .+ .03 z
(41) VW+cq '
Thus • is the angle from the positive X-axis to the major axis of the
ellipse (positive vitho) and a is the phase angle for the radius
vector, measured. positive from the maj r axis in the direction of to
The computer output gives OLb, 0 and 4 for both the deflection ard the
bending moment.
Coupling Stations. The programs allow for couplings in the rotor. At
these stations, the bending moment vanishes, i.e. M.'0 (the coupling
point is taken just to the right of the mass station). When the prog-
ram encounters a coupling station, say station L , the following
equations are set up:
(42) Ma1•, 9ý , M - - .--- Nivi's X51 - 4,61 " 'M"s
M4 --- - - /,L,:,t -, _ * +i.
or upon solviag
(43) 8'4 1 X, + 6L
where Q-,• 191 ,*cl& etc. and X#OX4 X%,*tAo etc. Then the bending
moment, shear force, slope and deflection before the coupling station
become functions of XY4#,,X$,1 Yi and qS only. As an example, let
the shear force at a station be:
VXS. W L&. #Iu +44~, #4100 *V9Ad, + I V&+ 14( *Vsf91V AV
Introducing eq.(43) Lives:
25
Ivr 12EV 1 VQii +42 VZ + 43A + a.4 ]x4J)5+[IV$ + 4#,V, + 412 V. +QV3+424(44) + [V7 saI,,V, taa?,V + Q,3V3 +a41V4]yq'[V,4I V, -~s4 4?4 V,+% QV4]4j,,
+[V -*vl+ bV ,+6v.,÷V a V, + V4and similarly for V)1" VV.,,VS P MXV4P,- -- - -$k~.Instead Of %, .',afdqwe have am new variables the slopes Just to the right ot the coupling
station m, i.e. Qc.,,esm 44,., , $Sm.Then the calculation proceeds as be-
fore until either a new coupLing station or the right end of the rotor
is reached.
_Trtnsmitteti Force and Pedestal Motion
Let the force transmitted to the pedestal at station fl be denoted F.From Eq. (12) it is seen:
F• =v,•,ye %, + h, 0)(41a + caUs"(45) Fvs -- 614-1.- VVC. ,h" 0)(S". - , •U,
Denoting the amplitude of the pedestal mamae )1Pand I4p (see FIx. 3)
we get:
X F, - ,i F'xa
or
=P Fyc~ IR (at,-MtLL-M.xj W(M. - +t"04 (&kV
The force transmitted to the base becomes:4P M; ,+ Cifd+.(t
or:
26
IXC FC +Mx~hepcP, S Fs -+ M, d'rs
Enerav Balance
Let the relative amplitude between rotor and pedestal mass be Xe=X-Jyp
and Lj' V-lp a.t a bearing station. Then the energy dissi-
pated in the bearing and the pedestal per revolution becomes:
Energy Dissipated -
(48) ItrcoCO, 0 e,2+(,Z.) +GC~1~ +9 ) +4C+C),#X I
+1• xI(cD:,ei+e') r +49 , (Te" + fs' a) + ('4..D +,, +)(eq;+e ,)- (M.,7-H.+)' (o•-e ',p) + cu,<i, (4 .e:)+ •,4 (';P;.t p'A)
A summation over all bearings gives the total dissipated energy.
At each unbalance station there is an energy input:
(49) Energy In~ut: 71-f CJLý(X )-9,Jbee1) +J Y(,Summing over all unbalance stations gives the total energy input
which must equal the dissipated energy.
COMPUTER INPIUT
The input data is prepared according to the following instructions.
Note that, unless specifically stated, no input card may be omitted.
Card 1 and 2: (72 cola. Rollerith) Identification:- Any descriptive
text may be punched in cols. 2-72. These two cards must always be
included.
Card 3: (1015) Control parameters -
Word 1. Number of rotor mass stations - The nusiber of mass stations is
determined by the above considerations. Also, there most be a mass
station at each rotor and, at each bearing, at each unbalance and at
27
-
each coupling point. The mass at a station may be zero. The
maximum number of mass stations is 80.
Word 2. Number of bearings - This integer denotes the total number of
bearings along the rotor. A maximum of 25 bearings is possible.
Word 3:. Number of unbalance stations - This intqger gives the total
number of mass stations at which unbalance is applied. A maximum of
80 unbalance stations is possible.
Word 4. Number of coupling stations - This integer gives the total
number of coupling points. It cannot exceed 20.
Word 5. Pedestal flexible/rigid - If this integer is zero, the program
assumes the pedestal to be rigid for both translatory and rotational
motion and no pedestal data is included. If the integer is 1, the
pedestal has flexibility and damping and pedestal data must be provided.
Word 6. Support tilting - If this integer is zero, neither the bear-
ings nor the pedestals resist rotation. In that case, neither the
input for the bearing dynamic coefficients for rotational notion nor
the pedestal data for rotational motion can be included. If the
integer is 1, the bearings and the pedestals have flexibility and
danping for rotational motion.
Word 7. Gyroscopic moment - If this integer is zero, no gyroscopic
moment is included in the calculation. If gyroscopic moment is
desired, the integer should be 1.
Word 8. Number of computations - It was indicated above that the eight
bearing parameters were dynamic coefficients and so could account for
the variation of parameters with running speed in an approximate manner.
However, if a more precise representation of these parameters is des-
ired, these values can be entered each time a mor running speed is
designated. In order to facilitate this, there is provision in the
28
J %• t t a
prosram for entering only the bearing or bearing and pedestal data
and the corresponding running speed without re-entering the rotor,
coupling or unbalance data. Then this word 8 of the control para-
meters designates the number of sets of parameters which are to be run.
If this value is 1, the program assumes that the bearing data is being
entered as coefficients of quadratic equations in a. . Note below that
the input format of the bearing data differs depending on whether this
value is equal to or greater than one.
Word 9. Diagnostic - If this integer is zero, no diagnostic will be
performed. A value of I will provide the diagnostic output: the
diagnostic increajes the amount of output a considerable amount and
is provided primarily for use in trouble-shooting the program and so
this value should alwaý be zero.
Word 10. Input - If this integer is zero, the program will return to
read in anw set of input upon completion of the computation. For
the last suet of input this value should be 1.
Card 4. (1P4E15.7)
2Word I is Young' modulus E in lbs/in . It is constant throughout the rotor.
Since the program never uses E by itself but always in the product El
(I=cross-sectional moment of inertia) any actual variation in 3 can be
absorbed by changing I accordingly.
Word 2 is the scale factor for the determinant in the simultaneous equation
subroutine. In general this item is 1.0. It is a factor by which the
determinant is multiplied to control computer over/underflow. The simul-
taneous equation subroutine is used 4 places in the program: oncr when
solving for the unknown end defiections (i.e.Sq.(35)) and 3 tima.s when solv-
ing for the unkown slopes in the coupling calculation (i.e. Eq. (42)). If
an over/underflow occurs during the calculation the program output will con-
tain: 'OVER/UNDERFLOW IN XSIMEQF AT - - (integer)" where the value of the
integer is I to 4. If it is 1, 2 or 3 the error is in the coupling cal-
culation. If it is 4 the error is in solving Eq. (35). Changing the scale-
factor may eliminate the trouble.
29
If the determinant is singular the output gives: VATRUIX IS SINGUlUR
IN XSIMEQF AT - -(integer)". If either of the two ervors occur the
program proceeds with a new rotor speed.
ROTOR DATA
The rotor data will differ depending on whether the effect of the
gyroscopic moment is included in the computation. For the case where
no gyroscopic moment is included; i.e. wlre word 7 of card 3 is zero,
the rotor data is entered as foll,:ws:
Card: (IP3U14.6) - An input card must be given for each mass station.
Each card has 3 items.
Word 1 - the mass at the station in lbs.
Word 2 - the length of the shaft section to the ,ight of the station
in inches.
Word 3 - the cross-sectional moment of inertia of the shaft section to
the right of the station in in4.
For the last mass station the shaft length and the croes-sectional
moment of inertia has no meaning and may be set equal to zero.
If gyroscopic motion is included and word 7, card 3, is not equal to
zero, then each card contains two more items in addition to the 3
itemsa indicated just above. Also for this case, the rotor data cards
are immediately preceded by a card which contains two values defined
as follows:
C3Ld: (15AP323.6). Gyroscopic moment parmters -
WordI - Number of iteracions - For each rotor speed the program first
calculates the unbalance response without gyroecopic momet. Based on
30
•~t'.
the thu4 oitained rotor slopes, the gyroscopic moment is computed and
applied to the rotor, resulting in new values of the slope and the process
is repeated. The program counts the number of iterations, excluding the
La'culation without gyroscopic moment. If the count exceeds this input
item, the results obtained are printed out, the iteration count is reset
to I and a new rotor speed calculation starts.
Word 2 - Convergence limit - After each gyroscopic moment iteration, the
following relative error is calculated:
where 19.iell,oc. and flare the slopes and X4 0,,.., 4. and l, are
the deflections at the left rotor end. The superscript is the iteration
number. Fdr each iteration the computer output gives the iteration number
and the error. When the error is less than or equal to the input conver-
gence limit, the program prints the results, resets the iteration count to
I and proceeds with a new rotor speed.
Following this card are the rotor data cards.
Card: (IP5E14.6). An input card is required for each ass station. Each
card contains 5 items; the first 3 words are the same as those for the non-
gyroscopic moment case above and the remaining two are:Word.._.._ - the polar mess moment of inertia in lbs.1n2
2Word 5 - the transverse mass moment of inertia in lbs.in
LOCATION OF BEARING SUPPORTS
Card: (1415). This list provides the numbers of the mes@ stations at
which there is a bearing.
UNBALANCE DATA
Card: (15, 1P2E15. 7). A card is provided for each of the unbalance
stations. Each card contains 3 values:
Word 1 - an integer which denotes the number of the mess station at whichthe unbalance applies.
Wordt 2 - the cosine component of the unbalance in sa. in. -
Word 3 - the sine component of the unbalance in oz. in.
3 1
Wd 3 - t
B. providing two unbalance components, it is possible to take iato
consideration the circumnferential variation of unbalance alon the
rotor.
COUPLING DATA
If the rotor does not contain couplings, (Word 4, card 3 is zero), then
no coupling data is necessary. If word 4, card 3 is not zero, the
following card must be included.
Card: (1415) - This is a list of integers denoting the wsas stations
at which there is a coupling.
PEDESTAL DATA
If the pedestal is considered to be infinitely rigid, then no pedestal
data is required. In this case word 5, card 3, pedestal flexible/rigid
is zero. Otherwise, pedestal data is required. The pedestal data,
like the bearing data, is separated into translatory and rotational
parameters. Also, as before, the ccntrol parameter is the word 6,..-
card 3, support tilting.
Card: (1P6E12.4) - A card must be provided for each bearing. On it
are 6 values as follows:
Word 1 - the weight of the pedestal in the X coordinate in lbs.
Word 2 - the pedestal stiffness along the X coordinate in lbM/in.
Word 3 - the pedestal damping along the X coordinate in lbs-sec/in.
Word 4 - same as word 1 but for the 4 coordinate.
Word 5 - same as word 2 but for the 4. coordinate.
Word 6 - same as word 3 but for the 4 coordinate.
If word 6, card 3, support tilting, is not zero, then all of the Cards
concerned with the translatory parmerers are followed by the card" for
the rotational parameters. Again there are 6 values om sesh card as
fallows:
Word I - the mass moment of inertia of the pedestal ame, associated
with the Xcoordinate in lbs.in2
32
Word 2 - the pedestal spring coeffLcient for rotational motion, assoc-
iated with the X coordinate in lbs-in/rad.
Word 3 - the pedestal damping coefficient for rotational motion, assoc-
iated with the X coordinate in lbs-in.-sec/rad.
Word 4 - same as word 1 but associated with the 4 coordinate.
Word 5 - same as word 2 but associated with the 4 coordinate.
Word 6 - same as word 3 but associated with the 4 coordinate.
SP'ED AND BEARING DATA
Each bearing is repreaented by 16 dynamic coefficients, 8 for trans-
latory motion and 8 for rotational motion. Of the 8 coefficients, 4
are spring coefficients and 4 are damping coefficients. Since the co-
efficients in general change with speed, each coefficient is expressed
by three components; e.g.
where &) is the rotor speed in rad/sec, (im,o in lbe/in., KJ,o inLbs-sec/in. red. and Kx%,% in lbs-sec 2 /in.rad2., Similar equAtions
hold for the other 15 coefficients. As indicated earlier, if it is
desired to enter the bearing data at each value of frequency, there is
provision for this in the program.
If word 8, card 3 is 1, the program assume the bearing data is prov-
ided as frequency dependent coefficients. In this case, a card is
provided with the speed range and increment and this is followed by
the bearing data. An input card is given for each coefficient at each
bearing. Each card contains three item, namesly the above mentioned
three speed components. The sequence of the input cards is: first
all the cards for the translatory motion and then all the cards for
the rotational motion. The cards for the rotational motion are not
required if word 6, card 3, support tilting, is zero. The cards should
be given in the following order: X* ) CCX, ifor bearing 1, Kvuv Ca.r..."C,, for bearing 2, etc. to the last bear-
ing, then (if word 6, card 3 0). Mw(Qj, IV-&Al1"~jAoHIg 3
.33
C -s
for bearing 1, etc. to and including the last bearing.
Card: (OP3E14.6) - Speed data.
Word I - initial speed.
Word 2 - final speed.
Word 3, - speed increment.
Card: (IP3Fl.".6) - Bearing data - in the order defined abcte with
three values on each card as follows:
Word I - the cc'-icient Ao of the expression AwA.. A.AwWord 2 - the coefficient A, of this expression.
Word 3 - the coefficient At of this expression.
If word 8, card 3, is greater than 1, the program assies the bearing
data will be provided for each value of speed. In this case, a card
is provided with a single speed value and this is followed by the
bearing data as follows: all of the translatory stiffness and dsping
coefficients are provided in the order; two cards for each bearing.
The first card contains the X coordinate translatory cs.fficle
k1(X)wC ,,,1, and (wCa.nd the second card the c coordtiste trsmelatory
coefficients K~.a"1" n pp both cards for benas ons m followed
by two cards for bearing two, etc. , to the total number of bearings
Again, if word 6, card 3 is zero, no rotatiomal parintere are required,
otherwise, they are provided in a similar manner: oae card of values
t.'W, Mk),1Wt j and a second card of M1,wD,Kpeskw for bearingI followed by two cards each for the renainlLs bearings.
The card format in this case is then:
Card: (1PE14.6) Speed.
Card: (1P4E14.6) - Bearing data in the order defined abs wLth theee
values on arch card as follows:
Card I - Word I - Kee,Word 2
Word 3 -
Word 4 - ,
34
Card 2 - Word I - K ,
Word 2
Word 3 -
Word 4 "C3g
35
COKPLTER OUTPUT
The computer output is largely se.f-explanatory and each output item is identified
by a descriptive text. Two samplAcases are shown in Appendix A. *The output
first lists all the inputdata, i.e. the two heading cards, the control words,
Youngs Modulus, the rotor data, the bearing stations, the unbalance data, the
coupling stations, the pedestal data, the speed data and finally the bearing
data. Thereafter follow the results of the :alculdtiona with one set of output
for each specified rotor speed. First, the speed to given in RN which may be
followed by the input bearing data if it is now Jpr every ersed. Next, a
9 column list gives the rotor amplitude and bending moment at each rotoritation.
Both the amplitude and the bending moment require four quantities for their
description. Since each rotor station whirls in an elliptical orbit it is con-
venient to express the four quantities in terms of the dimenaions of the
ellipse. Then the four quantities become:
1. the major axis of the ellipse: a(i.e. the maximam aplilude or the
maximum bending moment during one revolution.
2. the minor axis of the ellipse: b
3. the angle between the x-axis of the overall reference system and the"major axis of the ellipse: f, degrees (in output identified by:ANGLE X-MAJOR)
4. the phase angle with respect to the cosine-component of the unbalance:k , degrees.
The amplitude is given in thousands of an inch (mile) and the bending moment
is given in lbs.in.
The selected method of presentation is illustrated by Fig. 5 and is given in de-
tail in the analysis by Eqs. (37) to (41). However, a general description will
also be given here.
The presentation is based on two reference boordinate sysi•m. The first reference
system is the stationary x-y systan fixed with respect to ground, end which has at
each rotor str on its origin in the center of the statically deflected rotor
(i.e. the defLectLon due to gravity). The x-y-settes o 'cmunicated" to ths
rotor via the specified values of the bearing spring and damping coefficients
36
( K.,kJ,,C s4,etc.) and the corresponding pedestal data. In other words, the
directions of the x-axis and the y-axis are chosen when preparing the computer in-
put and the choice reflects in the input values used for K k etc. Then the
elliptical rotor orbit is centered in the origin of the x-y-syrtem (i.e. the
steady state shaft center), it has a major axis a, a minor axis b, and the
orientation of the ellipse is defined by the angle' 0 between the x-axis and
the major axis, measured in direction of rotor rotation. Note, that both a, b
and 1 vary along the rotor. A negative value for b signifies backward whirl.
Thus aib and 0 specify tie dimensions and the orientation of the elliptical
orbit but one more quantity is needed to specify the position of the moving shaft
center on the ellipse at any given time. The phase angle. at serves this purpose.
Let the major axis be the x1 -axis and the minor axis the y1 -axis (see Fig.5),
i.e. the xl-yl-system is obtained by rotating the x-y-system an angle 0 in
the direction of rotor rotation. Then the instantaneous position of the shaft
center is given by:
Note that the orientation of the xI-y,-syspem changes along the rotor since 0
does with respect to the x-y-system the iuttantasuous, shaft center position is
given by:
112 (aiaqt (~.t)'fill (cut +t1+ talt( ta.q$)
In addition to determining the instantaneous position of the shaft center with
respect to a stationnry coordinate system 4t also may be desired to know the
position with respect to the rotor unbalance. The location of the unbalance in
the rotor is defined by a coordinate system which is fixed in the rotating shaft
and whose axes are called "the cosine axis" and "the sine axis". Hence, each
unbalance consists of two components: a cosine component and a mine component
(in the analysis the symbols Ux and U., are used, respectively, see Eq.s(lO)
and (11). The instantaneous orientation of the cosine-eystom is defined by the
angle ( (at ) between the fixed axis and the cosine-axis. Thue, the instantaneous
phase angle between the amplitude vector (i.e. the radius vector from the center
of the elliptical orbit going through the instantaneous shaft center position)
and the total rotor unbalance vector is:
37
angit by which dmplitude vector lags unbalance vector-
FE U'. aq. 4 vcs
Here, Ar, and ' , indicates the summations of the cosine-components
and the aine-components, respectively, of all unbalances. It is seen that the
lag-angle is not constant as the shaft center movesarouni its orbit. It
attains its maximum and minimum values.when:
Aithough the discussion above is primarily aimed at describing the notion of
Lhe shaft center (i.e. the computer output for the amplitude) the same des-
crLption applies to the output for the bending umant. However, for each rotecr
station the output lists one line for the amplitude biut two lines for the bend-
-.. 6 moment. Whereas the output for the amplitude applies at the rotor station
itbelf the bending moment has one value imediately to the left of the station
and another value immediately to the right of the station. The output gives
the left hand value first (i.e. the output gives ?% sod MO respectively, see
Fig. 2). The tvo values are in general the sme Uiless the particular station is
a bearing station which resists tilting. The last listed value of the bending
moment should always be zero (i.e. corresponding velues of the major and minor
axis should be zero). In general they are not exactly zero but very small. The
amount by which the values differ from zero gives am indication of the accurncy
of the calculation. Note, that for this reason the ls.t values of the angles
and d are meaninglecS.
38
Following the output for the amplitude and for the bending moment come ti
suits for the force transmitted to the bearing housing (equal to the dynav
bearing reaction). If the pedestals are flexible, the force transmitted I
the foundation and the amplitude of the pedestal mass are also given. Eac
the three quantities are presented in two ways: first in terms of the cox
ponding ellipse (i.e. in analogy to the rotor amplitude) and secondly by I
x and y-components. Thus, if the transmitted force is F the output gives
quantities: the major axis, the- minor axis, the orientation angle je the
angle 9( J IFwl J 'Aw, IFJI and w7y where the last four items are defi
force in x-direction: F, IG I cos (ct+ 06)
force in y-direction: F IF,,I('ut+a,)
The transmitted force is given in lbs and the pedestal amplitude in thousar
of an inch (mils).
The next line of output serves as a check on the calculation. It gives the
energy per revolution put into'. the system by the unbalance forces and the e
dissipated per revolution in the bearings and pedestals. Theoretically, th
two values should be equal but numerical inaccuracies in the calculations a
cause discrepancy. Normally they \iffer in the fifth or sixth decimal placi
The energy is given in lbs.inch/revolution.
To convert it into H1 multiply the energy value by the speed in RPM and divJ
by 3.96 10.
If the input does not include any gyroscopic moment effects the calculations
repeated for a new rgtor speed and the output will follow the description gi
above. If the gyroscopic moment is included there are two sets of output fo
each rotor speed, each set having the format as explained above. The first
applies to a rotor without any gyroscopic moments, and the second set gives
the final result for the calculation with the gyroscopic moment included.
39
two sets are separated by a twc column list giving the sequential results of the
iterations needed to perform the gyroscopic moment calculation. The first
column idencifies the iteration number and the second column gives the relativc
cor,.ergence of the iteration procedure as explained in describing the computer
input.
40
COMPUTER PROGRAM: THE STABILITY OF A ROTOR IN FLUID FILM BEARINGS
This section sets forth the basic analysis and the detailed instructions for
using the computer program: PNO017: "The Stability of a Rotor in Fluid Film
Bearings" for the IBM 704 digital computer. The program calculates the speed
at onset of instability (the threshold speed) and the corresponding whirl
frequency.
Each rotor support consists of a fluid film bearing mounted in a pedestal, both
members possessing flexibility and damping. The bearing fluid film is represent.
by 8 dynamic coefficients and it is the value of these coefficients which prim-
arily govern the instability mechanism. For a given application they vary with
the speed of the rotor and they zzu.t be specified in the computer input for each
speed to be tested. The bearing pedestals are represented by 2 spring coefficiet
and 2 damping coefficients (in the vertical and the horizontal direction, re-
spectively) and the corresponding masses may also be given.
The program is to a large extent compatible with the unbalance response program
such that much of the input data used in the latter program also applies to the
stability program.
THEORETICAL ANALYSIS
The analysis is an extension of the methods used in the previous section to deter-
mine the unbalance response of the rotor. Thus, the following discussion assumes
familiarity with the earlier given analysis. The. rotor is again represented
by a finite number of mass stations connected by weightless but stiff shaft
sections which can be brought to approximate the actual rotor to any degree of
accuracy depending on the number of mass s~itions. Each bearing is represented
by.8 dynamic coefficients: lxx, Cxx, Kxy, Cxy, Kyx, Cyx, Kyy and Cyy, which
depend on the operating speed of the rotor: w, radians/sec.
The purpose of the analysis is to establish the onset of instability of the rotor-
bearing system. No external forces act on the rotor (i.e. there are no unbalance
forces), instead the dynamical equilibrium of the steady-state operation of the
41
system is tested for a series of discrete speed values over a speed range. This
is done by disturbing the rotor with an assigned frequency V radians/sec. To
this end the previously established equations are used as sum.arized in Eqs.(32)
by replacing W with V or since W.J is given, the disturbing frequency is speci-
fied as a ratio of the speed:
S(note:CJ
The rotor unbalance components U and U are eliminated. Applying Eqs.(32)xn ynand the brundary conditions Eqs. (33) and (34) yields Eq.(35) where the right
hand side is now equal to zero.
Since the assigned frequency is a pure frequency and does not contain a transient
term the outlined procedure applies to the threshold of insLability. In other
words, the calculation determines the state of neutral stability at which the
effect of any disturbance continues indefinitely without either increasing or
decreasing. Hence, at the point of neutral stability there must be a finite,
although undetermined, solution for the rotor amplitudes, i.e. the 8 end values,
ca CDsl0 ...... , ys cannot all vanish. This implies that the determinant on
the left hand side of Eq.(35) must be zero for the system to be neutrally stable.
On this basis, a calculation procedure can be dev jed. Select a sufficiently
low value of the rotor speed that the system is known to be stable and scan the
entire frequency range to obtain the value of the determinant for each frequcncy.
Repeat the calculations for an increased rotor speed and proceed in this way
until a speed is encountered at which the determinant becomes zero. At that
particular sreed the rotor-bearing system is on tbe threshold of instability
and any further increase in the rotor speed will make the system unstable.
Although the method is quite simple in principld certain difficulties arise in
applying the method. Considering the matrix in Eq.(35) it is seen to be an
8 b. 8 matrix in which all elements are real. Actually, it can be witten as a
4 by 4 matrix with complex elements. Therefore, its determinant will always
be positive and in applying the outlined calculation procedure the nro-point
of the determinant will appear as a minimum. There will be no cross-over
from a positive value to a negative value, or, in other words, at the onset of
42
instability the chosen form of the determinant has a repeated root. Thus, it
is necessary to plot the dcterminant as a function of the rotor speed to be able
to establish when it becomes zero. However, the determinant also depends on
the disturbing frequency and it only vanishes for a particular value of that
frequency. It is, therefore, also necessary to know the frequency-value at
which the determinant should be evaluated at each rotor speed.
For this reason it is chosen to calculate the real and imaginary part of the
4 by 4 complex determinant which is equivalent to the 8 by 8 real determinant.
Denote the real matrix:rb b V
B2n (n = 4 in the present case)b2n, I .. b 1n,2nj
and let the corresponding complex matrix by:
n a
an1 ann/
With the convention followed in the present analysis the two matrices are re-
lated by:
b 1 ,Re ( all b12 -Mjl
b -1 -IMta lai b 2 2 ' Re {alJ
etc.
Let these matrices be the coefficient matrices in a set of linear equations:
(50) B X - U2n
(51) A z Z Wn
where:
S vI Z2 w
43
\M
such that:! "-]I - i j j - ln
w Su - iv j1-,In
Since the two equations dre equivalent they must yield the same solution. There-
fore, first set: wI = w2 - - Wn1-i 0 and wn - i which means: ul, u 2 -- un = 0,
v1 = v 2 a .. v = 0 and v 1l. Solve Eq.(51) for z2n-1 n - n
(52) %A,, ~ (,,~AL
where A is the determinant of , oAna is the determinant of An with the n'th
column and row removed, and:
i.e., &nr and 6 are the real and imaginary parts,respectively of the complex
matrix and it is desired to calculate them.
Next, solve Eq.(50) for xn and yn with vn - -1:
(53) X4 "
where B2n is the determinant of B2n' B 2nI is the determinant of B with the n'th
column and row removed, and Bn; is the determinant of B with the (n-l)'th
column and the n'th row removed. Furthermore, it is known that:
Equate Eqs. (52), (53) and (54) to get:
a.,
or:
(55) O ,LI
44
where B2n-2 is the determinant of B2n with the two last columns and rows deleted
and:
Thus, Eq.(55) gives a recurrence formula where the order of the original deter-
minant is reduced by I. If it is applied repeatedly the final result becomes:
( 5 6 ) a t , -i A l
with the definition:
and:
(57) B2 k-l is the determinant corresponding to the first (2k-I) columns and rows
of the matrix B2n
(58) Bk-kl is the determinant corresponding to the first (2k-l) columns and rows
of the matrix B2n but where the (2k-l)'th column has been interchanged with
the 2k'th column,
(59) B2 k- 2 is the determinant corresponding to the first (2k-2) columns and rows
of the matrix B2n.
Thus a method has been established to calculate the real and imaginary part of an
n by n complex determinant. The method is used to evaluate the determinant of
the 8 by 8 matrix in Eq. (35).
It may be noted that Eq. (35) and Eq. (50)are identical except for the change in
nomenclature. Hence, in Eq. (50) X represents the 8 rotor end coordinates:
0c1' 0s.. ...... ' Ycl' ys, and V represents the moment and shear components at the
other-end of the rotor: Mxcr M .xsr ..... , V V . Eq. (50) is solved in
Eqs. (53) and (54) for the case of v - -I, i.e. for V .- -1 which means that an ysr
force:=oc - s;.vt
has been applied to the rutor end. The corresponding amplitude at the same
rotor end can be computed as described in the rotor response analysis. Let the
y-componevt be: amplitude ycrCOS.Vt + Ysr sin V t
45
The energy input imparted to the rotor motion is:
(60) E""17 Pr ,c:le 2 4oS;nVt&'I',;*tvt ,ro~yt) 2t 1T1,
A positive energy input implies that energy is required to sustain the particular
vibratory motion, i.e. the 'notion is stable. On the other hand, a negative energy
input signifies an unstable motion. When the energy input is zero the motion is
neutrally stable. It must be noted that the motion itself may not be possible
unless the previously discussed determinant is also zero, i.e. the outlied energy
criterion can not be used alone to test the stability of the rotor-bearing system.
However, the criterion can be used to determine the value of the instability
frequency.
A simple example may serve as an illustration. Let a single mass M be supported
on a spring with a coefficient K and a dashpot coefficient C. The amplitude is:
J1USYt +L I S;hVt
Apply a force V sin Vt such that the equations of motion become:
{(k -MVe)VC4C 0
-VC IV.' •
in analogy to Eq. (35). The determinant becomes:
(K-M')'(VC)O (note: ti,e determinant is always positive)
2which vanishes when V C - 0 and V m K/M, i.e. the frequency must be such that
it simultaneously makes the damping VC zero and also equals the natural frequency
of the system. Next, solving for 11 and computing the energy input yields:
Energy input per cycle --71 =, fl, 1V,~ kM,)+,)Applying the energy criterion from above the motion is unstable when VC is negative
and vice versa, which is of course evident in this simple case. However, the
system 1a only n.eutrallv stable if in addition the frequency also equals the
natural frequency: V - L
46
In calculating the energy input from Eq.(60) it is seen that a difficulty arises
when the system determinant is equal to zero in which case the amplitude ycr
cannot be determined. To circumvent this problen', ycr is not computed as its
actual value. In evaluating 9cl, 9sl .....Ysl from Eq. (35) with Vysrn -1, Cramer's
rule is used but such that the system determinant is always set equal to I re-
gardless of its true value. Thus, the resulting calculated energy should actually
be divided by the system deteiminant to obtain the real value of the energy input.
To solve for the onset of instability the procedure is:
a. Select a rotor speed sufficiently small that the system is known to
be stable.
b. Scan the frequency rangc and determine those frequency values at which
the real part and the immginary part of the system determinant equal zero.
c. Repeat the calculation for several values of the rotor speed covering
a sufficiently large speed range.
d. Plot curves of frequency versus rotor speed, obtaining one (or more)
curve corresponding to the real part of the determinant being zero and one
(or more) curve corresponding to the imaginary part being zero. The
intersection of the curves determines the threshold speed.
To assi3t in searching for the instability frequency the single bearing frequency
value is determined for each bearing. This'value is derived by considering a
stiff, symmetric rotor supported in similar bearings. Let the rotor mass per
bearing be M whereby the equations of motion becomes:I(1(wr-My'%+iY~x) (Kv (4iYxj(61) = 0
The real and the imaginary parts of the determinant have to equal zero separately:
(62)c)o
47
Solve the equations to get:
(64) (V WU (Jcl1 7- CL) C., Wcr~
(65) MVz IvW + & g j 'i ,$
Computing Eq. (65) first and substituting into Eq. (64) yields the instabilit';
frequency ratio Y for the bearing. Since for the actual rotor the bearingW
reactions are in general unequal and the 8 bearing coefficients, therefore,
differ in value among the bearings the instability frequency will not have the
same value for all the bearings. However, the instabill.y frequency for the
rotor-bearing system will lie between the minimum and maximum value of the
bearing frequencies.
4
48
COMPUTER INPUT
The input data is prepared according to the instructions given in the following.
Note, that unless stated otherwise no input card may be omitted.
Card I FORMAT (49 cols. Hollerith) Any descriptive text, used to identify
the particular calculation, may bp punched in cols. 2-49.
Card 2 FORMAT (615) Control parameters:
Word I (NS) Number of rotor mass stations. The number of stations is selected
according to the previouc discussion. There must be a mass station at each rotor
end and at each bearing. The mass at a station may be zero. The maximum nu...uer
of stations is 30.
Word 2 (NB) Number of bearings. This integer denotes the total number of bearings
along the rotor. A maximum of 10 bearings is allowed.
Wora 3 (NFR) Number of frequency ratios. This integer specifies the number of
items in the input list for the frequency ratios.
Word 4 (NCAL) Number of speed and bearing data input sets. Each set of data
consists of a speed range and the values of the 8 dynamic coefficients for each
bearing. The speed range is specified by an initial speed, a final speed and
a speed increment. For each speed value in the speed range the frequency range
is scanned and the corresponding values of the system determinant are calculated.
There is no limitation on the value of the input item.
Wocj 5 (NPST) Pedestal flexible/rigid. If this integer is zero the program
assumes the pedestals--to be rigid and no pedestal data can be furnished. If :he
integer is 1 the pedestal has both flexibility and damping and t;e pedestal data
must be given.
Word 6 (INP) Input. If this integer is zero the program will return to read in
a new set of input upon completion of the computation. For the last set of input
the integer should be 1.
49
,j-, 3: 4'nUMAT (IXE'.6)
This -. jrd cnt 1ijs ,i single word. Yourgs modulus E in lbs/in. It is constat
thrroughout the rotor. Since the program never usesE by itself but always in -he
product El (1 = cross-sectional moment of inertia) any actual variation in E
can be absorbed in a corresponding change of I.
R,'tor Data: FORMAT (4(IXE13.6) )
ihe rotor data consist of as many card'. as there are mass stations (card 2,
word 1). Each card has 4 words:
Word 1: the mass at the station in lbs.
Word 2: the, length of the shaft section to the right of the station in inches
Word 3: the cross-sectional moment of inertia of ihe shaft section to the right4
of the station in in
Word 4:the polar mass •oment of inertia minus the transverse mass morment ofinertia, lbs.in'.
For the last mass station the shaft letigt'hand the cross-sectional moment ofinertia has no meaning and may be set equal to zero.
LOCATION OF BEARING SUPPORTS: FORMAT (10(WxI4))
This list of integers provides the number of each mass station at which there
is a bearing. The stations should be listed in sequence, beginning with the
lowest number.
PEDESTAL DATA: FORMAT (6(IXElI.4))
If the pedestals are rigid no pedestal data are required and item 5, card 2,
must be zero. Otherwise, the data for each pedescal must be provided. There
is one card for each pedestal and they should be in the same sequence as the
bearing station numbers in the previous ligt. Each card contains 6 words:
Word 1: the vibratory mass of the pedestal for motion in the x-direction, lbs.
Word 2: the pedestal stiffness in the x-direction, lbs/in
Word 3: the pedestal damping coefficient in the x-direction, lbs.sec/in
Word 4: the vibratory mass of the pedestal for motion in the y-direction, lbs.
Word 5: the pedestal stiffness in the y-direction, lbs/in
Word 6: 'he pedestal damping coefficient in the y-direction, lbs.sec/in
.0
LIST OF FREQUENCY RATIO VALUES: FORMAT (4(1XE13.6))
This input list gives thr2 values of the frequency ratio at which the program
evaluates the system determinant (V = disturbance frequency, radians/sec, w
angular speed of rotor, radians/sec). The values should be given in descendir
order, for instance: - .55, .50, .49, .43, .40 ...... The program automat
inserts in the list the "eigen-instability" freq,'e-icy ratio from each bearing
determined from Eqs. (64) and (65). In most cases these values are equal to a
ximately .5 but may be less for heavily loaded bearings. Unfortunately, the '•
instability" frequency ratio is very sensitive to even smalldeviations in the
dynamic bearing coefficients from their accurate values. I is, therefore, re
commended that the irput values of the bearing coefficients be checked beforeh
by means of Eqs. (64) and (65). If the thus calculated frequency ratio value
differs much from .5 the bearing coefficients should be checked.
ROTOR SPEED AND BEARING DATA
The following input data should be repeated as many times as specified by word
card 2. First comes a card giving the speed range for the Lalculatio.s. The
speed range is defined by an initial speed value, a final speed value and a sp
increment, all in RPM. Thus, if the initial speed is given as 3000 RPM, the fi
speed as 9000 RPM, and the speed increment as 1000 RPM calculations are perfor
for 3000, 4000, 5000, 6000, 7000, 8000 and 9000 RPM.
Next follows the 8 dynamic coefficients for each bearing. There are 4 values
per card, hence, there are 2 cards per bearing. The first card gives Kxxv (Cx
K and awC , and the second card gives Kyy • aCyy K and ayC . All the co-xy xy yy yyx yx
efficients are measured in lbs/inch. The cards should be given In the some se-
quence as the bearing station numbers in the previous input list.
COMPUTER OUTPUT
An example of the output is included in Appendix B.
The first page of the output lists the input values for checking and control pu
poses. Next, follows the results of the calculations with the results for each
rotor speed given separately. The first line specifies the particular speed
51
and then foil w- the "eigen" - instability frequency lti RPM ard the "eigen" - mass
in lbs for each bearing as determined from Fls. (64) and (65). They are labeled:
"INST.FREQ, RPM" and "INST.WEIGHT", respectively. Thereafter the results of the
calculations for each frequency are listed in 5 columns. Thefrst column, labeled
"FREQ.RAT.", gives the frequency ratio -. The second cjlumii, labeled "DETERMINANT",
gives the square root of the system determinant (i.e. of the matrix in Eq. (35) ).
The third and the fourth col':mns, .--heled "RE(DET)" AND "IM(DET)" gives the real
and the imaginary pa, t of the system determinant, respectively (i.e. tr and A .
from Eq. (56)). The last colur:n, labeled "ENERGY", is proportional to the energy
input given by Eq. (60).
It should be noted that in order to determine if the rotor is stable or unstable
it is necessary to find at which speed it becomes unstable. Hence, results must
be available over a range of speeds. To determine that speed, at which instability
sets ir, the results must be plotted. One method is as follows: for each speed,
plot the real and imaginary part of the system determinant against the frequency
ratio. Find those frequency ratio values at which the two functions become zero,
(there are usually several values). Next, plot the "zero-point" frequency ratios
against the rotor speed, obtaining a curve corresponding to the imaginary part of
the determinant. Where the two curves intersect is the threshold speed. Thus,
for the output example given in Appendix B the threshold speed is found to be
8,350 RPM at a frequency ratio of .4937.
52
ACKNOWLEDGMENT
The analyses and the computer program described in the present report are
developed from two basic computer programs which resulted from an internal
research program by Mechanical Technology Inc.
53
REFERENCES
I. M.A. Prohl, "A Gerteral Method for Calculating Critical Speeds of
Flexible Rotors," Journal of Applied Mechanics, Vol. 12, 1945,pp. 142-t-.08
2. N. 0. Myklestad, Journal of Aeronautical Sciences, Vol. 11, 1944, p.153.
3. S. Timoshenko, 'Vibration Problems in Engineering," D. Van NosZrand
Company, 3rd Edition, 1955.
4. J. W. Lund and B. Sternlicht, "Rotor- Bearing Dynamic& with Emphasis
on Attenuation," Journal of Basic Engineering, TRANS. ASNE, Vol. 84,
Series D, December 1962, pp. 491-502.
5. J. W. Lund, "The Stability of an Elastic Rotor in Journal Bearings vith
Flexible, Damped Supports," To be published in the TRANS.ASHE
6. "Design Data for Analyzing the Dynamical Performance of Rotor-Bearing
Sys tems" J. W. Lund (Editor). Report issued for the Air Force Aero
Propulsion Laboratory, Wright Patterson Air Force Base, Ohio under
Contiact AF33(615)-1895 by Mechanical Technology Inc.
I5
X •
FIG.I SHAFT SECTION BETWEEN TWO MASS STATIONS
x" Xn+l
in LET n ] I MŽA i Ttl
V,,n-I V1t,*l- V xtf V V1, Vi V, n÷ VtnI
FIG. 2 CONVENTION AND NOMENCLATURE FOR ROTOR CALCULATION
K KI
- cy, .
KARING
HOUSING Nosl
FIG. 3 BEARING AND PEDESTAL SYSTEM FOR TRANSLATORY MOTION
55
FIG. 4 GYROSCOPIC 11OM0ENT CALCULATION
S.0
- - -a--le
MPENDIX A
SAMPLE CALCULATIONS AND INPUtr FORMS FOR THE COMPUTER PROGRAM"UNBALANCE RESPONSE OF A ROTOR IN FLUID FILM BEARINGS"
57
0.'.%ooo 6S-34I*FRfEMANvSESCDSI"~C'3 vA',i HCS.mAP
c ME:$4NICAL 1f:Ch4N0LOGyo1NC0 J0$RGEN We LUND)C 1JN1JALANCL. RFSOPý,'ON OF qJtUq IN NO-NJNIF3O(!A uLAMIN.GS PNW I1
11V~cNsf0'4 d~Er(10. do$ ft1M)-d10. boloBVICi10. dlo).MYS1109 dU)qVXC j~2210,411 *b4i(10.41 VfC 110.41) .JYSIIO.41 2.LAI.I1O,'.O2DA.,IO9'.J2L,Lyc 1i "J 4 v
4 D vxAA( 40.0 9DVx 3 (.*0) 9L (~. 440 1 9 )VXD f40) 9DVYA (40) 9DV Yb 140 1 DVYL.(4U' 1 *05DVYD(402.1Rv(%f))9RL14O) .!4S1a 461 kIP (40JoH 2I T 40 901A140) 9DJS(40 J oIC( IUZ)eC6403 1 or) 1040 ) A44 4lU 1 1%( u s4 (4.0 1UX (40 1 UVI4O ;*LU(401 .LBf 25 1dKEXX UO09139252 .dCAX( 3.?52 .dAr I 3 *25 1*6(XY 3,25 1 ,B(KYY( 3s25 1(.YY ( 3#25 ) 03YXg V1U0
98 S%4Y*V 3.2oi2 51s8r)4Y Y( 3 54 DY 39 C!20)
2C I 94 90( Sol I oF (4*4o2O) of-F01 oal 9ENTI 101 .4N1H51bo,19wUX(so, t 1503 UVUY'8 0 90UM'1Y 1300 1 blA ;32 1 sSHO 13 1 * HC 13 ) 9StL) (3 1AUX (2Zs4,93, 1 .-46048JXI25,'..3)eLm(M1I10O),CLNRIO) U170Cu~qu0N A * 23 0 C * D * CFM * EN 200COMMO0N HS , DUNq 1 CF9 , MAT s KFN * CLNNW 0210.COMMON 3 RN3 0220COMMON NSeN8.k#0,fdC9NPST.N.%40N.NGYk.NCAL.NDIA(,ivNPUIYM,35CFRM.RLo u233
2PIX.PS14X.P!)'*0PIv.PSMV .PDMY.SPST.±,PFN.5PINC~edKXx,23CXX,8KXYRBCXY% 25IqYY98CYY.134X.2<CYX.NR0MXtR--)'XobStAXZV.ifMXYtti4yY.9DMYY.5SMYX*
6ANSP. SPCAL .OVXA.V)vxd .VXCIDVXAo.VYAeOVYB8.DVYCDVVO.DfD4XA.DmEI30MXC, 02d
CU4MON 6MEXC.23u.*9YC*OMYb.JXC.VXS9VYC$VYS.DXC.DXSspyC~oDYbXC*Xs.
COMMON JCAL#Ak'5.NITC20l 1 IIsi2?Pt CALL SUdR 0310
C ROTOR CALCULATION 30004fln CFls0O.O01
CF~wu0 ;02CFmwO.O 303vCF4uao. 3040CF580.0 3050CF6x,0.0 3060CC7uOv0 3070C0880#3 JocONtTCa1 30i0IST~.1 3100
1 r4V931104M1 .Jjrui 3120
NCu 1 3130I#rf%C: 403.403,4.02 314.0
4()2 JFNLCIj2 3150GO TO 4.04 3160
40~3 JFN&NS JAI4m6~ no 40!S .ja1.NS 3100
OVUXI JIN0.0 Woo04O0a DVUYIJ)406O 3J 1w
00 422 [1 19IF 3210
IFIJST-j2 AI11411*'f'16 !230'.06 KFNsJST~jct 32,*0
Pj1XCI 00~41mQSos 3220OVEXSI 1 .(P41 0.0 3260
ýXsl 'J"r 1 -0.133UU
-VC (I19 ý,J T ) - .11 3310
',k)T I. (4i,'4. '4099410*ld4ld.' 4ld9'~4l8o4169416)o,(ST 3330
-,o Tfl 418~ 33:)04^8 D X; 2 9J'ýT I x1.. 3 3ea
ý,:) TO 41d 3310
,j,) TO 418 33iwO410 DYSt'.JST ):1.o 3400
r-4', TO 418 3410
411 4x~cI 2.1 ) --. 0 34206 MX % 1.01 lzooo343
d IV I( 1. 9 11 00 34ý0Vxc(I.1 ý0901 3460V(S1 1l:0.00 347()Vyc'Ilol .ol 3400Vy ( I 1l 0.1)00 3490
X5 I I s1I 10.0 35100
'rc H9l :0.0 3520YS( 1,1 :.00 3530DXC ( 1 1) Z0.0 .3540
DXS( 191120.0 3ýotlOVyc I 1 9 10.0 3560t)y( 1; 1 ) 0.0 3510GO TO l4O7,4,84fl9,4l0,412,4l3,4l4,4l5,416,416I1pKST 3580
412 XC(%,l:1.*C 3590GO TO 41d *3600
411 XStSq,.n.i0 3610G0 TrO 41d 3620
414 YC17*11:1.0 3630GJ TO 410 3640
GJ TO 418d 36*0416 JO, 417 Jjal*N 3670
KST-LUIJI 3680DVJXlK ST ixUX~ Ij I ANSP2 36100
41' DVUvlKST)u'JY(J:.AN.SP2 3700418 ')0 '42 2j:aJý To 'F 3710
KST-J.J 3720'.FN2<St-I 3710KMD:KST~l 374*0IF(JST-11 4d0941~09.'i1 3750
41', IFI-1JS1') 421,421,420 3760420 bM -! i'ACI19KFNI+%A J (x D~ E S(Ie)+MLIJ 3770
IDYC(1,Jl'OMXIf)UI.LYSgIj)..ug..Il J7U0BxS( I 9K S T ) ý%0% I 9 4 x%41i I 90XC I J) +'Jle4X*s( D~ JI .j-D4XL J) * 3790
1vYC( f J)' +-vX CI j ty519J : 3IJ 36001-3 Y IIOK 6 t Ia IMYCF )+;.'vI A I9 +x(. JI ,X41sJ) SY1I I* 3660
I I OVXC)(iJI ty',( I It.S ) *v~ jY 1,1 30600
V slI J l sx (I jjIý;xtfji*A ItjI iVX IJ *SIIejI L~x f : YLlI j J 61
IlrVYSIIJ*1,uYsIJ)*0VUYUJ)*CIJDVI)XIIO~(2#(, 391C"
II .')VYA j I YS (I J' .!)VUX'JK j 39270lF(JF~i-Jl 422,4229421 3930
MMSI ,4I.8N)OBIXSE1,KST)*RI~JIeVXS)IJ41I 3950bM'YCEI ,e24DIei3MYCI I K5T 2RLiJI.VYCI I J.1I 3960RMS *'DOMSIKt*LJOY~e)l 3970
OXC(I *JlI mUXC (I 9JI +AN IJ I 9tMXC I I 9ST I bN IJ) VXC I I J*12 39u0DXSII 9J,1I alxSlI 9JI +AN I J,*otMXSI I 9KSTI I+N( J I VX5 I I j*U 3990I)YC(Ii .J*I ) mYC I I 9J i AN(JI Ja.VY( I I*ST I *dh J I VYC IIJ.1J 4000
DYS(I jI O,.YS ( I 9JI +AN (J I *MS ( I ST )*Bt4(J I VYS I.J*1I )' 010XC I 19 J+ I IXC I I 9J) RL J I DXL I sJ I+ON I J IMXC II vKST I UNIJ I VXc II 9J1 4020
11 4030
11 4050VC(IJ*1,)UYCII,9J I RL I J IDYCI Il.J I#ON I J I084YC I I tST I DNI JI*VYC II J.J* 4060
1) 4070
YS I I 9J,13uI bfI 9J I e I J) #DV (1 911 +N I iI8MYSI I vST ! *0NIJ tVYS(Il#*J 40d0
11 4090
422 CONTINUE 4100IFINDIAGI 423,424#421 4110
C DIAGNOSTIC 2 4120421 WRITE 1691361 4130
WRITE I6.1111IIt.IFP4.JSTJFNKSTOKFN.KM0.NCC 4140
WRI1TE 16.144)IDIXC(I.Jl,0XSII.JI'aDYCII.JI.0YS(I*JI. XC(Iojltx51 4150IOJ).YCI1,JIVSfI.jI.IuI,010IJa1NSI 4160WRITE 16.1451(D1AIJI .DIR(JI.OIC~iJIOIDfiJDVUXIJI, OVUYIJI ,Ja1 4170
1*.4S I 4110KSYT'4S*1 4190W I ITE 16# 104114 VXCI IoJ I*VXS II #J I VYC I I 0A VYSI #J 19 I18,10I.ju1 42001 .<STI 4210KiTaNP4S.S 4220W41TE 16.I0411(dMXLII.J),SI4XbIIJleIIYCII..JI.UNVS(IIJI.1U1910I9Ju1 4230
Iv.~ST I 4Z40
424 IiINS-JFNI 446944404Z5 42!00
C ROTOR hr4S COUPLING STATIONS 4260
42S KSTaJFN*.IFN 4270IFfIST-101 42694389416 4280
426 D0 427 J4194 4290FI 1.J*NCCImdMYCIJoKSTI 4300Fg29J9MCC IadMXS(JsKStl 4310
F( 3*J.PCC ,maM'rCIJoKST 4320F14,I.4CC udP0VS#J#KSTI 4330
KFNaj+4 4340
S(I*Jlv-94XCIKFN*KST) 4350dl2vJ)8-dMXSIKFNoKST 4360OI 3,jI.-OWYCIKFN9KbTI' 4310b14oJIU-8MYSIKFNSS!Tl 6 3.0DO 427 1.1.4 4390All ,J)mF( I .j9CCI 4400
427 CIIOJIOPII$Jo*gCCl 4410CF9mSCF 4420I#ATel 4430IFINOIAGI 429.4309429 4440
C DIAGNOSTIC 3 4450429 WRITE 16*1371 446J
wRiTE 16*10411 ICII8.ipoll 41 eJs*41 ofIISI IJI.IUI4IoJsl,41.II Al 19J 447011l.1a1.41,J.1,4IoiIFIlJ.IECCI,1a1,4IJuls4I 4460
410 KFNaKFN 4490CALL EQS 4500GO TO 11.l.11KN4510
4311fI.I-WC9(T 4520
( ~T I '.J io1, 14. ! %I M~**YC I 9K~ St 4,P40
I I.' -- 4-Y, I KýT 4!) 00
C, 4560
C 4LL F 4li
6< It' I4d'f.,1F 1'4600
DO 4.34 jJSJTojrN 46JO
KST s .4 4650
VC(KST.J+1)zVXC(KSTj*1)+VX*Cfi.j*12*A(I.KI 4610VXS(KST.J$1It*VXS(~b(,J.1).VXS(I419JIe*AE!KI 46800VYC(K5tJ+1)sVYC(K,'T.J.1I4VYLIl.J.1)*leld. 4.6Y0VYS(hKSt.J+1IsVYcI(Kz7,J*13.VYSEj.J*1)OA(II.K 47U0XC(I(STJ,-XCIKST,J',XC(l,J1OAUI,K. 4710XS(KST,.JI2Ab,(KSIJ)+XS(1,,fl*A(I#KCI 4720YC(K'r,..daYCI%:)rJI+YC(I.,JJ.AI oI ,~ 4730YSIK5T.J)YSI(5,TJ;4YS(Ij.J*A(IKI 4740OXIKEISrj)*rX(<jT.J)+DXCIJJ)*A(IKI* 4 7.500)XS(KStJlaDXfI(KSr.J3.fXS(J.JI'A(1.K.I 4760DYC(cStJI=DYCIKST.j)+DYCIIJ).AeIKI 4770
4.34 DYS(gCSTJ IDY(,(KSTJ)+OYS( I J)'AI 1.3( 47b0KFN*JFN+JFN 4790KmosJsr+JSr 4000DO 436 JxK~D.3(FN 4810
KS~aK+44030DO 436 1xl,4 4840BMC(KSTJ)s8imXCIK.btJI,8MXC(1.J)eA(1.KI 4 b!P08MXS(3(it.JlxBMXSI(KTJ.+8P4XS(IJI*A(I.3() 4860RMYCIKT.J3u@YCI3(STJI8MYC(,oJJ*A(I,3(I 48J70
436 VWS(<STJ)zFtYi(3(bTeJ1.8MY5(1,JI*A( J.%. 48a0GO tC 4%'2 4890
438 D(l1.13 -dMXCI 10#KST) 4900D(2#1 )*-81XSI 1OoKST 49 1100(3v1 Is-8MYCfI 0,3(T) 4970D(4*113u-SYSIl0.KJt) 4930IY) 428 Jxl#4 49400) 428 Jul.'. 4VDU
428 C119J)*F(I*JoNCCI. 4960C '9uSCF 4910MAT&I 4960Ke'Nal(FN4 4990CALL COS b 000GO TO 143995IO.5l1)oKFN 5010
4.39 D0 440 JwJSYjFN 5020D0 440 Jul.' 5030
VYC(100J*I)ZVYC(100J*11J4VVCII.J*1IOcIlI. 5040
VXS(10.J~aE)(10JI.ES(Ij+1eCIIlojl)*Cll stisVYC(IIO.J+uYC(10J3.YC(Ij,.*C(IZ.1, c l 56100YV1.YS( S(1eI)VYJIO.Y+g1.J,.CI.Rl*lll 5070
DYCIIOJIuOYCIIO.43,XflYCi I*CfgI Jell640
W.F 4.;FNJF 45160K -) IJS T +JS 5110Ell) 441 J4(MO.KFN 518000 441 1.1.4 5 191:
520013mvryoC a~v( 1O.9I: My 1O4?fAYclI JiocfIq.) 5.120
".41 RMYS( 1O9j.*%Vv4ý 1I,)9j av~SI I.JIC(Iql) 52304.42 .$rTsicN 5240
NC CBJCC * 15e,50IFrNCC-NC, '44.9444.443 5260
443 JFNaNS 5270GO Tl 44' 9d
444 JOaCNC 52904.45 G11 T) 404 5300
C END CONDITIONS 5310446 KST*43.NS 5320
K ̂ NaNS*l 5330D) 44.7 jalski 5340C0F0(1 .4IdMxC(J.9STi 5350C..M(?.J)we04X~gJ*KST, 53C0CFM( 3.41 ut).iyC (J*STI 5370CFMf4*.-JI .d4vsJ*KSr) 53a0CFA4(io.,aVXC (J.'(FNI 5390CFM(6*.4) VXSIj.<FNI 5400CFM( 7.JlOV'CIJ*<FNI 5410
447 CFMI8.JIOVYS(JoKF4) 5420RHS(II .&I-8BEC(9*KST) 5430RHS(2 .1Ilw-dMj(;9.KSTI 5440RI4513. lw-~3mYC(9*K5T) 5450RHS(4.1 lu-f3MYc(9.KST) 5460RHSIS*1 Iu-VXC(9*KF4) 5470RH5(6vlI1*-VX519.'(PNj 5480ORN.S(791 '*-VYCg99KFNI 5490RHSI8.1 )u-VV5(9*KFN) 5500IFItST-10) 4499448,448 5510
48RP4S(5.1)%RmSI%,1)-6MXCI10v4SN, 5560
RHSf6qI)uRHS(6*I)-VX5l1O.KFNj 5570RHSI7.11.RHSg7,1I-VYCg10.KFA4, 55S0RH(olR!iI-Y(0K~ 5590
449 CF9=SCF 5600MAT.,4 5610IFINDIAGI 450.4S1.450 5620
cDIAGNOSTIC 4 56304S0 WRITE (6.1381 5640
WRITE (6.104hg(C1:.J,,Iul,4,.Ju1.4,.g AhIJIIol.4,,J.1.5,IICFMII.o 5650lJI.iu1.81.Jalmlog.MHSII.1,Iu1.,i,,)CF9 5660WRITE (6.14411(IIXCI I.JI.0N6 IJI.I)YCII..i.L#YbllI.J) XCII.Jlqxs( 5670IloI.Jlyc(I.4loys(I.j),Iu1.l0)jul,~,5; 5680
451 4e'uKN 5690CLL E05 5700
452 EIT(9161.0 5720I' fN!)IAGI 4969497#4)6 5730
C DIAGNOSTIC S 5A740496 WAITF 16.1191 5 750
WRITE f69I44u1(CFM( .IIujIsj**j~jsjq, 5760497 IF(I ST-10) 4%los00.500 s77u453 IF(NCAL-li 4?4,471.474 5700
c 4 A1 TE :A-L' t 566
*,, ITE 1 ;:FM I 1 1)6O0
f) 51) 1*1.4 59304'? ENT(!3=0.0 5940
ý50 TO 461 5950
461 00 7 15 J -I1.'.U 5970
rVl.!(VI N3 ~'1 ,2 p99,0
7i 1 )VUY(vY):lJNJl*AN.;P2 6000"~1 ý52000 6010
ENGV'0.O 60,10'4N13'1 6viJ0V9R:1 6040
L'IR'iLB( ii 6020
L44'LxLu( I) 6060
')470 J21.'JS 6070I 7 (J-KFNI 46494A294'41460
'462 V) 463 1=194 6.JVO
4#63 ENTtI )aCF4( loll 61U0
464 KiTvJ4J 6110C<C=KSiT-I 6120DO3 465, <'1.' 61 3UDD 465, 1*1.' bil&0
465 H119K)20*0 ti170
.1(211 3( 21 )x~. I ,~rN' )6160-1(3.1Ixf3( 3.1Il.YC( I9Jl'rNT( Il 6190ý'(4.1 )sq('..1 +Y'-( I JI'ENTIl II0-(.41 2! W3 (1.02) +:i%iXC I I 9KC ) rhc? I II 6d 108(2.2)s.J(2.2,+6,dXS(I.(C,.CNTEI; 6210:) (3.2) 'Ii3.? ).oYYC( I .CJ *E~ (lI Idj
of(1. 1) xti ( 1 3 1 .sJMXC( I IVKST I-Y I l J
*d(393lv:)(393)I..'YC( Z.(bTlI.(%TdI I d IV
466 8 (4 93 ) al 14 93 1 E1Y -3 1 9CjT *'II 1 61a01F(J-LORG3 70n.oi.7C? 62 P0
C TRANS14ITT'k0 FPRo PE:O.'TAL V.';TICN 6300?f13 CF1A*Rv(j)*ANIZP7 6310
SHA~llmICF1A*M(l.1 !.%VUX(J) 6320
SHOW 111 CF IA*d 14 * )+011UXJ x I 1D0 704 IxI*IFN 1%J60Se4A(1luS,4A(1)4(VXC(I.J)-Vx~llI.jtl)IOLN1(I 6370
1 s %I4t I) I v-4C ( II* I4(vYS I 1911 .YLI : .4.1II. b1 'ý T.Y I I I6400
IF (NPSTI I05.706v?70%
ON "(:aI143AAP'A" 6430
-P' I PX ( wl. I A%4S*AN.,P64 L
Ci I ý)sI0C Y V,4 41 A 4SOANSP 64 1vj
iHAE12,I -AlIpo' 1A-S..JgI)LrlCI/CFIý 6490
5441(3 1 I ,oA( I ICFIý.4Sm1of I I&e14 )/(.FIE 65ou
CvlftxCFI loCFJR.CFIL)sCF In 6210lýHCfll¶ 2' 4CE Ii.CF1I:-sIHCl I *CF 11) /CF IL 6520
5.i~ ~: l( l 1 CF1,).SH~lE I I *7F I d2C I rpI 6530SHAI' 2 '5HAf 1),CFIK*SI4AjI31 6540
rMm(2lxSH1M1*2CF1I.5NH9M 65-30
SPEC (221 zS'4C 1 I +CF I"*SIC (32 b~bu
SH 2 ) a HIM ! lCF1M0SID(' 65 7UCF 1Aa8I 1,1 2SA( 32 65o0CFlt~vH(2,1 2-SHd(3l 65¶v0CFIsfH 3,1 I-SHC(l) 6600CF 1%4-8(4, 9 '10 32r)11 6610
GO V) 707 6620706 CF1ASB(1.l2 6630
CF 1i3':.(2, 2 1 6640CFIK*Fif1.12 665 0CFIVwug(4, 12 66605HA(21uSH~A( If 6670sP4'(22 'S'4(:U 66d0
SHC 12 9514CI112 6690SNO(22T55b41I( 6700SP4AMU2 0.0 6710SHBI3)O.0 6720SHC( 32 0.0 6730SHDE 32'0*0 671-0CF1Cwo.0 6720CFIDSO.0 6760
7C7 CFAIXIsmlOXI94IAS~CX3VRONP 6770CF2:i.2CXY f I WO I qX Y (2e 204i I*%SP+Oe.CXY ( 1 042A 1 ASPZ 6 78CCcCvC 0A)+4Y -WI*NýPV X1 wsl0N0 6790Cl2xCYIkPIdy#9A)A'~CY14k*NP dl
CF~SYI*4R~KX2vqlAS*4X3Yg*NP 6820DISSaOIS,,.w3.141',V?70(CF2A*(LF lA*C1.o*1FJ01d2U*CF21)*lCF1'CF1I* 6830
1C,;1NeCFIM)IlCFS.#Crd.:2.(tF A*CF1k.*Ci liiCf Id 2*(C.FZM-LF2N)O9(CFlo060ZCFIC~F1AC~ v2,.~1.(S.9A~b.5N~32,HbI32*N5(IJ*~1U~tC3J'6810
MIC 1C3 1 +SH0( 1 OP01,O31 21 6860IF 144041 7119716.I11 6800
717 CFIAaOv0 6880
CF1eauo. 68 ?0CFlKU0.0 6900CFIVN0.0 6910D0 740 l181IF~4 6920CFZA.CFZA+()NCII.J2*LATIIl 6930CF18uCv1-$'OX5aI 1J20EtTI 1 6940CF1(sCF!K+DYCflJJ*EliVIJ 691U
740 CF IPOCF1 ~~,I*A!4ST s lrtI11 6960CF2AR(Ill-8j92-VI~.'l6970
CF2091Z.291-flZ.Z2D1r14j) 69fo0CF2Cw~f3o12-:J~l321-f)RC .f 6990
7000IF I'N0STi 7419742.741 7010
741 CF41SXY1)P~44tl6Cl*P~*144 ?020CF2qa(PSVY(I"t-P1yIVRI/Ie6.O69.4NSP2)/AMT ?030CF IC sPllX f wa tS'0IAVSA4$P 7040
CF1 =P1M(Mdj/AA0S0AN.S0U5CF L -CF'2N4CF 2kCF IC*CF IC %C60CF3A-(CF2A*CFZM-(.F28WCFC1C/CFIE 7U70CF3I1lZ(U'2A*CF1C.(LF2-iCF2MP/CFIE 7000CF 1F:CF2N#CF2%J.CF1D*CFlD 7090CF3Cx ICF2COCF2'.-CF2D*CF1DI /CF1E 1100CF'flw(cr2C.CF10+(C?2D.'CF2N,/CFIE 7110CF1AsCFIA-CF3A 7120CFIL3.Cc18-CF34 7130CF1lk.CF1K-CF3C* 7140C'7M*CFIM-CF31l 7150C9mCr4~F 4(.*(CFIA*(.F3A.CF3S*CF 38I.CF1D*ICF3CICF3C*CF3D*CF30) 7160
7q*2 C'ýZM0.0 7180743 DISS=DISS+3.1415927*(CF2A*CFlb-CF2U'CFIA+CF2C*CFIM-f.FZD*CF1%.*CF2MI 7190716 0) 708 1u1,3 7200
CFIAzS$A( 1) 7210Ct:18*-SI4 (I 1 7220CF1CzSHC I I 7230CF1D*SHD(I I 7240AUX IPOBR* 1 #1 lx0RT(CF1AOCF1AGCFlbOCF18J 7250ALJX(MOIR92,l )ANG(CF1A9CF1It3 7260AUXP48R,3 .1 JSSQRT(CFlC*CFICCF100CFID) 7270AUX1M8Rv491)=ANG(CF1D9CFIC) 72b0CF2AzCFIA*CF IA 7290CF28=CF18*CF18 7300CF2C*CF1C*CFIC 7310CF2DECFID*CFID 7320CF1Es(CF2A.CF28*C.-F2C+CF2CI/2.0 73JOCF1IKs(CF2A.CF28-CF2C-CF2D)/200 7340CFIMuCF1AOCFIC-CF18*CFID 7350CF1NsCFIA*CF1B-CFIC*CF ID 7360CF2AwICF2A-CF28.CF2C-CF2DI /2.0 7370CF2BuSQRT(CFIeC*CFIKCF1M*CFP4,13UCF38wCF1A*CF1D+CFlo4CFIC 7390CF38=CF38/A8S(CF391 74008UX(M8R*1I*I.SQRT(CFIE+CF28) 7410BUXWSR,2,I JUCF35*SQRTICFIE-CFZ8I 7420BUX(M8R,3.I IUANGICF1<i,CF P91/2.0 7430
708 8UX(M8R94,l I.ANG(CF2A9CF1N,/2.0 7440IFIMBR-NO) 702*7019701 7450
let LORG=NS.?. 7460GO TO 709 7470
702? M89.RakfR 7480LBRG=L8INBR I 7490
709 IF IJ-LNdL) 71497109714 7500.710 IF IMP4S-NIJI 712.711,711 7510711 Lt4BLONS+2 7520
Go TO 713 7530712 10489MNB*1 7540
LI4aLuLUIN8I 7550713 E'4GYuENGY+3* 141592 1*(DVLX IJ101612*1 1-013,11 I+OVUY (kIO5IW1 1011 1560
15)NERT ESLT T FLLIPSIS 7580.z714 0) 469 IwI*3 7590
CVIA9(Iq1*9jqI)7600Cr1Bw@I2*I I@5(2*I) ?410CFlC0911*IJ*8(3.Il 7620CF1OaBI49Iffdf',I, 7630CFIEaICF1A.CFIB+CFIC+CFl1D,?.0 ?640CPI'. ICF1A+CFI8-(.F C-CFIPN/2.0 74"0
I~j ;_-,r I .?On7410
d -I to I . ,1?* 1 1 ,J I 1*s' 1* !: 1o~ 4. 1 /6o'C F I C S.7 ( 'IF Ilc CF I9.*C IV *CI 14 11( F 1H *P ( *It IM ) OK .I 4 11-~4 12 *116 ( ll *I 1711'CC 31--CF3 IVA 3-11H 1 711i'Ii I I; S ,R T (CF IE-' IC) 7.,(2,!1)Cý1df.S'QT(CFIr-CFjr.I I7JU
4 IS 34*IJz)AG4CýIA.CFid)/2*0 /7t4.69 C ON T I~ *' 17bu
1210.1 401 77d3
CF IA AM a A4 41 AUS I ;C!C I Sus ISSiCF IA "18 13F'4GYsFjGY/q%4Sh2WRITF 460130) 7oi0nO3 72S Isle? 7b40IF 11-21 12397-'09722 75
720 IF (%-PST 1 721,726,121 7b66121 %qITr (697311 7a10
S1 TO 723 7tsoO122 WRITE 16* 1321 1690121 do'1T 1691331 1u
-C 724. Ja1 .4d 7910
724 w4ITF d61.I.I..,,'xJ2t.JI.,, UX(Jo49II.AU 79301ElJol.iI..JXIJ.2.Il..%UX(j.1.II.AIJXIJ*i..II 7940
725 C AT 1-UE 7950726 W'UTr 169?11V.I4GY*JISS 796t'-
GJ TO I5I3.48o,5SIA.(vD 7970C MAKE READY FOR GYI'0$CCPIC' MOYENT CALCJLATIUh 1900
4.80 ISTslO 19,00IFP4UjO dvJuU
C CALCULATE 5YQISCOPIC 'IOMET O2481i no 482 ImIed o0iu462 ENTIII*CVY(I.I) 8LU43
IF14CI 48143',.4d3 6050463 <F%4LCINC1 6060
DO 4A4. !ule. 01074.84 E'4T111m0.0 8080
GO T! 4.96 009048S <F~aS.1 a 100486 no 493 JalNS 6110
IFIJ-CF'4) 4.o99487*499 81 2(407 DO 48d IwI.4 el's'488 E'IT(IlsCFVII11 ll489 CF1A*0*0 6170
CF1B0.*O o16UCF1C90.O 0170CFIDwO*0 6180
CF1AmCFlA*')XCl ,jlO9-'TlIIj d200CF15uC6Ij.DXSqtjIojj)g!IIj 021CCFIC.CFlC.IWyCI ,Jj u,j1I oic
490 C~~~I).ýyfqjF~~ 6 2i0CF1Eesv1A*CP 1 dd.4uCF1KnCFIIS-CFIC 642vCVI1MmCP1ACFlnI.CFl~uOCF IC % 260CFI4*'rIsEQCz1e!CFlcuCP 1K IU1cIFICF141 4129491.492#d(
491 CIAIJI.0s0 bilv0PC!!IIJ18090 3J00
~~r~jI @310
32 Tf~~ 43203 C
CF i -.'> , ýl'iC- IK(C-Il a3 IjCf IPZ 1TI JI A4AN'-,2 o
z-() ~CFIv.rr2N*(:FIC c410d420
491 C1VJT1NJr '3430IFIN'DIAG) 49494?5#494 o440
C )ICAGJiI' *4ý0494 W31Tý (6#140) d460
1F1CCF1.~CFU,*Cr 1<,CFlV.CIFlNotfF1 CFZtfF3.CF4, CF59CF69CFI 04oU
'495 CiO TC 401 6500c jYROSCOPIC ~'AOENT 1114TIoh ! 1C
500 CFAAiCF"11,F)AI( 2,11-Cf:23A,'iS(CHý'(3,1ICF3)+ Au.S d 521(CFf-(4911 F )%i(F (o lCF )A SC Y6 1-F ) At3'JICF b530PM(79I1)CF7),AS(CFM(691)-CFdId:4CF2t3:0.0 b5jODO 501 1:1,8 d560
$M1 CF2B=CF234,A i(CF4(1qI) d! 87 0IF(CF21) 50295019502 *!)do
50? CF2A=CF2A/CF20 2vJC3 4 R I E 16 91!,e ) . 1T C 9.F 2A o6uj
IF(CF2A-D)GYR) 506#5069504 06105 04 NITC2%ITC,1 0640
IFfITC-NIr) S505,0',.506 d630505 CFizcr,4tioiP *640
C'q:C~v(lq,1 d 663C-4zCF:'(49I I ab 10C'5SCFV(591. o6o0C-6zfF*A(S1 I 06-YOC-;7-CFV( 7.11 0700CF8=CFVfd,1 I 710GJ T') 481 8720
'506 4RITF (69IS4) @730<MD= 3 d740'ENTI 10 * 1 *0 o720
GO TO 456 6760C AtrVICE SP[FP b 770
512 JCALsiCAL41 619i0IF('NCAL-JCAL) $18.991,99ni
901 1I~GO TO 200
501 SOCALzSPCAL.%PI'\L db810ANSPe0.134719760W~fDCAL 6820
902 111:3l00 tr. .100
c 'MJROVAd Er d@1
ST OPc XSI'0C:W )IArphOTIC
GO T'j 41) ooou
GO Ti 513 0910100 F0Cl.AT 172-40 8920
I d930101 FQr'qATI 7- 89401 8950
IM2 FIUN8ALNCE 0HJ 8960U~dA ANCE RES ONSE OF OT~k WIT d910
I-' N-0-UNIdFO-i~m IIEAFINC, Suppu'it: PNoDI RIPNE-~O~i * 010' FIRYAYE123H 1 00IA. TFfHN)L0GjYINC. MEHAI 9000104 FJR-ATlIP4E15.7I 90100105 FORN'ATII151, 9010U
106 FJY~IsIEl6 90.euIn? FOR.VAtI lO4oyCtiNGS ACIWtLtJ SCALE FACTOR 9040100 FORMAltli,,.; '7-:% :;3.9J~UAL* '4O*COUPLe PfD.oF 9050ILEx. F3R~a*.WvENT CVR3*'-O4* N09CASES DIAGNOSTIC INPuT 1 96109 F0Rm-ATeiP5F1I.,71 j 9070110 FORY.AT(1,,91121 9070III FflRYA?(36H* ROTOR DATA 1. 9090112 FORMATIl '3E14961 9109I1I rOR,9eAr1P5F1 4 ,,6 1 9110114 FORVATII?1P3E~20,71 9110115 F0R'MAygI?,1n5F2o.,, 9130116 F0Rm~A'7I72rf STAT!0N No, A~LNGNC~S 9130
1 SECT.1lNETIA IjY ~ EN ~ RS 9140117 FORMAT112QH :iTATION No, MAS~ L~.jC0 9160
15 SLCreIN4ERTIA POLAt M'M,,INkRTjA T6RAISV9hILJMOINLPTIA 1 97118 FORVeAyg14iI - 9170119 F0PMATg136H~~rRAT, IT"DATOCONVERGoLIMT 919012Ml F0 vAvtjgwoqFAR!Rl tyA1JmpýiS 1 1
121 VRMAT(Ii~o9200121 STQAT(13N o,12 HEARING 9210lAY TATU NO* 12192201Z22 FORM4AT1120H Y l(xx v Cxx v X y CXY 9230
123 FvDATgl~j5*j~ 42 Y V Y 9240124x FmwM?32. A4Y OXY 92601 mvv V 0YDW 1 9712S FIR4.iT(IP6EI2,4l
Y I 979280127 FORP'471102H bR,ýSSATIoA, V4;* l (1 CX 93001 4ASS*Y-nljftK Cv p 31128 FORMAyg78~40 'EkTL3IA 9320
ITIANSLATIRy ~iv-yo ELSA A 9330129 FOQ'4ATIKHO PEETLDT, 9330
IR'ITATI?(NAL vo'IO;.j PEETLDT9 9340130 VIRMATg10oi,. dRGoSTATI(Oj 14ENTI',X oex DX 9360I 14RTIASY MY 5 37131 FV)W4AT4I,,)PjFj5*7 2 9370132 F)RsVAT1IITP2jE21 .7) 9390133 FAA14ONFA~t. STS N-UNOD.LANCL V-UWMALANCE I V400114 FO3RMAT I t.cOCOLPL I% bfATICN.%p
9410315 F~JRWAT4Ii,0DIAaN0STIC I 1 9420'36 OrMQ.ATl1S.J~nIAGmOSTtc 7 1 9430117 FORMV61P4~0DIAGN05tIC ? 9440136 FORMATIIIHOOI', 00 5 9:C 4 9420139 F0RPOA1fieg.oooIAGNOSrIC a 9460140 FOUAM44..0DgAWlSTIC 6 1 I7141 FORMAT142..40INITIAL SPkLI) FINAL SDEED '"gap INC**#"04144 FORMATIIPVEIS*71 94yu145 FORMATIItdEIS91) i~0so146 FORMAVI11140NOTOR S0([email protected]~l 9510
I . :2)9? '% pvITHO' 1"OU T G RcC :iCZ I C '. T I eU
I~ - ' r Q 'HTAT 1.1 VAJ(:R AXIS WINUR AXIS ANGLE: X-MAJOR P H A S 1,0 5.1 AN~L~ AJO X!S YJ~.1:4 AXIS AN~GLE XAC PHASt-~.L' ,c
p )P.1 F~'T 171 AMPLITUDE 95109ýE%)IN4G '-CMLNTI
1. *-' 9 Q' T1 I?4? 1~' TFRAT.?Itjr
I 14 FORv'AT(24PI 41T GfROSCOPIC VD"LNT ý61(11'-1 F DqP6 00VR/'ff4LWI XblIt'E-,F AT III VI6dQ156~ c VýTf1,0qTI IS 5INGULA4 IN X~i~~,'F AT 11l )6 JJ15? F3RKMAT(16)4E14s63 9640)110 FORvAT(36tlQ F^R'CE T1RANS,41TTLJ TO) 6zRN H1JUSI4G) 96:)111 rORMA~T(~33HQ F^< TqANSVITTV' TU ) r-xNrATION) 9660712 F09#A T (18HO P~rDFCTAL %11T!1ýN 1 9670
FORm.AT(1204. ri9G#.N') VAJC^R AX':S MINOR -CXIS ANGLE A-MAJOR Pr4ASE 96o0I I NGL X-AYOLITJDE X-PHASL AN'G Y-AYPLITUDkT Y-PHASt *1NGI 96,YO
114 F 0 VA T 115. -0tJE1Vy INPUT=9lPI14*7921H. ENERGY DISSIPATEC=9lPc.14*1 ý00C~I13 97110
E~rý -97.lu$IrTC' ;tJeRS V,94vXR79LIý;T9N0ECK
CJIIRC,.,TINE SURR
2104)*y 1)#09C1C.4OI#Xt11)401.YCI10,40IYS(1O,4.0I~i)MXA(4O ujoso
4PNVXA(40),o)VXP(40II)VXC(40I.,VXrhG.0),CVYAI40),0VYR(4OIDOVYCf401, 00705DVYD(4OI.Rw(4r)),RLC40I R,R(40I,PRIPg40IIkT(40liIAI40OIDlB(4O1i.j'IC( Lp0806401 .DID(40) ,AN(40),iNI 40II)NI4CI 'UXI4C1.4.YI40I.LU14OI.L4(25I .dKXX( U090
73259CX.*2I1Bx(95sCV?2)4KY321bY(9SoKX % 'JiI
83925VuY(8OID 5W9iýYX(3OU,,1,HA 3I,(.bHCI-2iefaMY3I,:,I18DMX.aux[2,,4.,904YU(Y4,IDW(395CdýMYi.CLNR(8).Y(9594)Y( 5 01 iC
C(WOVON A C . 0 C a 0 q C s tNT 0200COMYON IHS *), rkJu 1 (r9 , 004 9 N v CLNR 0210COVVON PRN3 U2.eoCON'VON NS NJ,.'J.NC9, "ST, NYOW. Vi~j NCAL .NI1IAGe INPUT *YM.bCI..IRM91L, 0 230IR5,NlI tOGYýqRIP.R: TL'3,L'ý,UX9t)Y.LC.PMXPK(XoPCAv,PMYP~PC~y KScyL.240
38A(YY.'4CYY*eS'YE.'CYX,8SYX~DVAA X~a4Y954~oVY98MX
6ANSPSPCAL ,1VXA,')VX1.rVXCoVEDD'VYA ,f~lVYFo:)VyC.IWVyflMXAOMX8DMXC9 028
C0NlX0N IMYAC .-vAS, YC,8IVYSoA~v,nNX;VC.v! ,CAYCVSXI,~
C1WOMMO JCAL.AA.5wITC160 FORMAT(72H0 . I0
1 894.0101 F0RYAT(72.4 0 950
1 4960102 VF3RtAT412O,41 UNHALA-14%..~ qSOUJ4Sk OF aqT0, 'WIT ov 1%)
IH NON-U'NIF04-Y. tSEAING SUPP~)NTS Pt400ll 1 096161 FIROPAT(120.4 ~ECbHANIC a
IA'. TECH~41L0GY9144o I1 9000o
IMS F')RlATfIOI', 90200106 F)RMATlI5ID,1?3e6, 9030
107 FORM4AT130MOY0INGS 0ODULUS SCALE FACTOR! 9040108 FORMAT1120O*4 STATIONS N~dRS 1O,,UNUALo NO*COUPL. PEO.F 9050
ILEX* diGeMOMENT GY~4O.00t' N~oCASEý DIAGNOSTIC INPuT 1 9060109 FORMAM(PSE15*7l 9070110 FORPMAT117991171 9uboIII FORMAT(16*', ROTOR DATA 1 9090112 F0RPMAT1123'314*61 9100113 FOR&OAT( IPSE14,61 9110114 FORMAT1I9I?.3S2Oo7I 9120115 FORWAT(IITPSF20*71 9130116 FORMAT(72H STATION 40. MASS LENGTH CROSS 9140
1 SECT*14ERTIA I 9150117 FORVW.TI120H STATIO'uNO 409 MAS LENGTH CROS 9160
1S SECTo1NERT:A POLAR Mok'.I.kLRFIA TRANSVeMOMoINERTIA 1 9110118'FORMAT1lA 15, 9180I119 FORP'A1436HOITERATe I TERAToCONVERGeL IV IT 1 9150O120 FORMATII8HOSEARING STATIONS 1 10200121 FORMATM~HO BEARI S 2?10
IAY STATION N~O* 121 9220122 FORMAT1120H4 KXX Cxx KX CAe 9230
1 KYY CYY KUX CYX I Q240123 FORMATI1PSE15.61 9220i24: FORMAT112ON "XX DXX MEY DXY 9260
1 Myy OVY MYX DYX 1 9270125 FORMATIIP6EI2e41 9280126 FORMATIit.1P60116.41 9290127 FORMATI1102 bRGoSTATID#4 PASS.X-DIRo KX Cx 9300
1 MASSOV-0140 KY CV 1 9310128 FORY-AT178HO PEDESTAL DATA. 93207
ITRANSLATORY M4f)TION 1 9330129 FORM4AY178HO PEDESTAL DATA* 9340
1R2)TATICNAL A0TIC% 1 9350130 F')RMAT11O2H ORGoSTATION INEI4TIAvX Mx Dx 9360
1 INERTIAoV MYDV 9370131 F')RMAYIS91P2r15.7J WSW1112 F)RA0AT(I?91P2r23,?1 9390133 F)RIOATl54HOUNRALAtN.L ST9 X-UNSALANCE Y-IJNBALAhCE 1 9400134 F'ORMATl18HOC0'JPLING STATbONS) 9410135 FO3RMAT1SNODIAGN0STIC 1 1 94d0141 FORMATWHZOINITIAL SPEED FINAL SPEED SPEED INCRe I 9dbbO144 FORMATI1~bE15*71 9#*9u145 FORMATI116E15.71 9500146 FORMAT(1M1OROTOR SPEEDws1PEI4.7.0$RPM1 9510156 FORMAT134'104ATRIX IS SINGULAR IN XSimEQF AT Ill 9630
157 FORMAT 11P4E1~o61__0G TO 19009227*300).111900 READ 15.lCOI
READ Is.1011 0400READ I s 0 105 4SoftR sU#NC NP5T *NMUOMNGYX *NCAL 00 IA6,INPUT 0410READ I5.1041YW.SCF 0420WRITE 16.10O21 0430WRITE 1601031 0440WRITE 16o,100 0450WRITE f6#1011 0440WRITE 16.1001 "' T0WRITE 16o. 110 1NSs4B.NU.NC APST s,4MOMoNGYk 9NCAL .NQ AG# INPUT 0460WRITE 16*1071 0490
_____ WRITE 166.E4Iym.SCF 0500IFINGYRI 202.301.202 -- -ilj 201 READ 65.112) IRWIJI.RLI.JI.RSlJI eJuelNSI 0520WRITE (6el11I 0530WRITE 16.1161 V540WRITE 16,11AIIJ.QWI1JI.RLIJI.RSIJI.JsI.'$%I 0550
2 ý 4 LA; (6 1? u!6 T'ý0
21' REA I So113 LC(R J) .JL(lNCI k[($PI(ljxoS 7J61
z !RI r (69 1 1 1 (L()J N)U6Z0?r7 Tý16 1 , IP'T J2a)d,0 9 1JIoR ;SIJ)9 RITJI92I9N 07Y0
4'1T F (6%,119) 0:7 J I#2IU60
,vR I2 (9 '1.N11 0110
WR ITFtA(6J3) , )9jf ý' J ~ N I 106P
24IFtNC3) 215,227,215 U700
20WRIITE 169 114) u 20:.RITE (6w1I301 ()J2.C u73O
? 0 P 271 T1 20872904 74
v<RTLr- ('6917) 060
211 wRITE 169126)K T,9lý(JI.PKXJ),D0X(JI 1PYJ'DY(J)(JY4 ), Pr4Y(J) 0870
210 ARTEA (69121)~PTFPN 0Ad9.'RITE (6sI31) v.900
n0 2'11 .Juj,\-N4 V50
I(ST*LB( J) 0916021W!RITr (69!26)v'(T#XJ.P-"'i-ormjlpyjosyl0!ýYI U970
WRITE (691471 0 11900
24~REM *.I?')r'(AWUX('J).9'=vvI,,2),i<Xy(I.JI.HCs y(JJ, !oXY(I IJI,f 0920
) 0 1 1C03) 2 ( 01;9216.05 In11010KY J z 3 t(YIIol911319ft jj205 I9) 2 1AO 11 ) 1:.' X (j 9.J .1 9IN1 9(4Vx 3 ) ,s1,3)(100-41 0940
~2064 J819%-3 10950STRL0I(J) 1960
WRITE (691211l(T 1070
20WRITF (69!1218 ~0900
C.0 tNM 216 92690 1010
227 REýAD ( 5 91%12~ ).1130XXII )91T19 1tIAMXII9J p121 9(!MYIslzIREAD1x 31o I 6e15)8AEX.II * A 191 9 t1 0J)1,s( I "J 1 al 93C).' 1 PJDMY(I 1 8 Y1 a 1 1100
WRITE (69121)KST 1070b
221 223 J1,w 1170
WRITE 16*121' KSy
1 0221 WR CYIT 621)PKXXI CYXI9 1 Oj;u 1210
22* 4,7* 2240 ,,,? 1230
D0 226 Ju1.N8 1260KSTBLuhJi 1280WRITE 1691Z1IKST I'M7
25WRITE 16.9123~4Eg3 1290
1)RPY(*fRP,~l~eD4XIJ 1310000 228 13?.) 1310
HKXXgI9.1I1m0*0 13308BXXI'I JI 0.0 1340"'MIO 1.4;0.0 1350j),:x(I~jlt~s01360
sCyltt1j)aQ00 1370B C Y Y I O J I 0 * 01 3 v 0BZYE(I.OJISO.0 I3VOBSIXEII*Jlao* 0 1*008s)MXX; I J)O.40 14108SMEV 11.116066 1420asONEVI I .41.0 1430
so mv Y l ~ fi l s 1 4 40'OSMYYI 1,4100.. 1450
228 CONTINUE 1490218 40 TO 290 15400CCONVERT Ilput UNIZTS 1590250 AM~ss100*0. 1510CF103669004DAMS 130CF2mCFI/2%0 150RSINS,.RS I. 155DO 251 JOIONS 1560
1580RITIJIURI TIJOICrI
1590stBUat4#*I45.qfqJ 1800
St fJlsRLI J)i2 OG9Af#J) 1810251 OftfJ;mRLq(JI/3* b.ftg j) 162000 2 2 JsleftU 1640UJI(JIGUXIJI,167703 1840252 UYIJIDUYI,,,,8 1 1 7 01 66SPCALOSPS 1
1880AftSPmOqlO4?I9780SPCAL 1680MITC aI 1 49030 SPEED DjPFhnkNT PARAMETUaS 1090300 ASP~sMSP#1700M
0,11 101 Julefts 1710
0'IIBIJIGOOO 1730O4xCI.jlw0.o 1740o 'OIJ)01ms 1Moo
04V91Aa.~OoO..04VOIJIMOOO
0dTo0~014v 1.0.04 111001600
ý;VXAtJ~xStF IdDVYAf1jI's tF 1830O)VXC ( J s.0 1840DVXU)(JlsO.0 1.6u
OVYC .J~a* lb.
DIA(Jil0*0 1900OId(Jlm0.0 1910DIC(JI'0.0 1920
Wfl f0 311 JaloNS 1940KST&LBI J) 1950i(FMSC 1960CF1'(wBKXX( i.J).dPKAI2.J)ANSP+dKXX(3.JI.ANSP2 1970CF1Cs cdCXX( 1.j).9CXXE2,JIeANSP*oCXX(3,J)mAtlSP2 1960CF1DuBKXYI 1'JI,8KXY(2,.!)*ANSP+t3XY(39J).ANSPZ 19090CF1Ea dCEY(I I j)bCXY(2.JIOANSP*OCXY1Y3,J)*AN4S6Z 2000CF2Kx8KYY(1,JIpiBgrYYI2,J)*AN!,f.dp&YYE3,JI*.ANSP2 2010CFZC* dCif~l jI~dCYY(2,jieAN4iP*oC~fYE3,J)eANSP2 2020CF20x9YX(19JI~flKYXI29J)*A;*SP*BP.YXI39J)*AN~SP2 2030CF2Em 8CY1,JIojdCYXI2,J)oAA.SP~sCYXg3,JI.ANbPZ 2040IF(NPSTI 30393359,303 2050
303 CF1'~!PKXIJ3-PMX(J,/3S6.069*ANSP2. 2060CF1N=PCXI JI*ANSP 2070CFlAzCFIKi+CFlmq 2060CF1BuCFIC+CFIN 2090C92M*PKYIJI-PVYIJ)/3869069*ANSPd.
-z1GDCý2NaPCYI iI*ANSP 2110C*:2Aa'CF2KCF2m 2120CV2b*CF2C.CIý2N 9Li0Gi) TO 30? 2140
304 'K-Nsl 2150CF1'(uI5MxXI 1 jI*65SMxEt2,j)*At1SP,8SMXXI3,JI.AN5P2 2160Cr 1CS BDMAXI11,J).9()MXXI2.j)eANSPt00.4XXI3,J~oAhSP2 2170CF~gMYIj+5X(9)AS~S4Y3JON0 2100CFIE* 3DX(9)qMV2J*NS*0%X49)A4P 2190CF2Ku8SmYY(I1Jp~dbYYI2,j)mA.NP~dSMYYI3,Jl.A~bi4 J200*CF2Cm 8DMYY~liJ BgUMYY(2,J)oA?.iP~twMYYI3,J,.ANSP2 2210CF2038iMYXIl1.JI*ISS4YX(2,jI*A.*4PIU5MYXI3.JI*ANSP2 2220CF2E* eDf4YX(1,jlNbDMYX(2.J,.ANSP.80MYXf3,J,.ANSP2 2230IFINPSTI 30593069305 id40
305 CF1MuPSMXIJ)-PIXIJI,3866069.AI.S12 2250CFlNaPDMX IJI 'ANSP 2260CF1AsCFIKCFlm 2270CFda.CFIC+CFIN 22.0CF2'4uQSM.YIJI-PIY IJ)/1866069eAN5P2 2290CF2t4u0OMY U I*ANSD 2300CF2AuCF 2K.CF2i' 2310CF2UaCF2C*CF2m di2000 TO 307 2330
306 CF~uCFIK 2340CF2=CFIC dib0CF3*Cf 10 2160CF*SCFIE a370CF~uCF2O 2300CF6uCF2E 2199CF~wCF2K 2400CFSuCF2C 241060OTC 308 2420
307 CF4mCF2A*CF2A*CP2,8.CF2R 2430CFIutCFZAfOC#tCF2cs*CF2EI/CF4 2440
L(bCFI: ýI~I IA- -hC r 1.' OC, 1E 24.0
c-ewCF-d-CI2CP1U-~c r IO'A 1 2500
C-;8O-Cý4*CF 1D-CF1t1 3ois l 5
Cv24mCFO*CF',,CF6*Ck6 2520Cr2AaICPO*CI 1M.CF6*CFlN)/ýi2.,* 2530
CFJtjsfCFOCýI: -CF69CP1MI/Cilh 2540ivuECI OCP I.C1 0C60*- I lc II2s 2
I w 48KIICFOU8CF SeLp 7 1 CF2tt 2560
CFI:Aa-(CF ICF2A+CF dO(F~t1 2570
CF I u aCf I *F~ai.(LF 20(.F A .06.0
CI: V*CF 3-(.0' 1OCF24#%.r 2*LP2N 2590
CF INuF4-CF1*CF2N4.CF20CF2.'4 d600
CF~uCFIKOC9-2A-CF1COCV20*CF1LJ'(.tiA.+CI~lrCF1O c6lu
Cf2aCF1KUCF2UCFlCOCF2A-CFIJ0ý- 1j*CF1E'CF1A 2620
CF~aCF1K*CF2PO-CF1C0CFNCFh)*'Li.4C1fr.CF1I 2630
CFaCFIKO.'F2N.CF1COC~i:M*CF IJ*Crl1I*CFI1ECF1M4 2640
CICF2L.CidA-CF2E.CF2?3.criK(.F1A.CF2l*CFlb 2650
C'Fý6a5:'.2*CI~d,.CF2E*CF2A-CF2K*CPI *CI:2LACFIA 2660
Cf IuCF2DU 2~m-cr2LEE 2N+CI:2~0LC I: -CF2U*CF 14 267%)
(FdmCF20SC~tghCF2L*4CF2m*CF2<(4CI:N9PCI:2LCf m 266V
JOS l If(FN1 51003094J10 2690
309 0VXAfKStJ.OVXAI1KSD-CFI:/AMS 2700
UVX8~ K1 3 aF2jAm5 2710
OvxCl(K5 puCFi/AMS 21.20DVXD1KSTIuCF41AWS 2730
D)VYAI(K5TI 0VYfAIST I-CF flAM5 2740
DVYBI ST UCFd/AMS 2750
I)VYCIKST gmCFS/AmS 2760
')VYOI K5T ) CF6/AM5 2770
IF(NMOMI 104.111.304 2760
310 UMXA(K5TI.CF~IA14b 27il0
DMXS#KSTI* CFI/A'4S 2600
DMfXC1KST~aCF),AMiS ab10
W~0IMX(KT1&CIE.AMS IM2uD'IYA I 4ST IaCF I JAMS Z6su
0MYI(AST I CFV/AMA zus004 fSTaCF5/APS 8i
04MYD1 KST ICF6/AP0S 2v60
Ill C)NTI'4UE 2700
IIINDIAGI 312.313*112 28v0
C DIAG'4!)STIC 1 2.90
312 w4ITE 1691351 2900Wh17E 16.1091CIC7*LOIC.oCPfu.CCI-I:ZI;[email protected],*F2t* CF2ACLFZd*C 2910
IFJJ4.CF2%.LF1ACF1ro.CF1M.*Cj.1N(.CI:CF2,Ll-3eCF4@oCFi~eCF6* CF7.CFO*STF 29202 .ANSPZ .ANP4S$.PC4L 2930WRITE 16.110)KST*K.I-f 2940
WRITE 14.916441iVAAIJI~uV~tijlesJvXCI.iehiVX0(JIqiJVYAlJI O VY161.01OVY 2950
1CIJ) *DVYDIJI ejO1j %.Na 9301)AIl A *4Mbio stMECIJ1*WXD(J)* 0MVA6J1.0MY 29602BE4J I Wyc I J I or.OMY01I je I 9*4i 1 2970WRIT( 16,1451 gRRII JI.RIPlIbd1J~k~i.AhlloD.dmE$Jls0Wlli .jeaI.SI 2940
313 RETUR!4END
SteriC ANCY a946MRlC ARCTA~4 ROUTINE iU020
FUNCTION A,4GjAFC~qAFSNJ 0010ACSmAFCS 030jAsheArs.4 0040
600 IFIA5S4I 11049Se10104%12Sol IFIACS) 602.801.60i U000
002 AG:8oo.0GU o rdl,2d~ ANGa*0.0
VU7lGO to 811206
8d04 IF(ACSI dQ9.8i)508od 009013 f%5 IF#ASN) 8Oh,do13,bo7 00806 ANiGB..O.O 0110GO To 61 CJ120
107 ANG-90.O U 130
GO To $1 014008ANGSO.O lpGI To ,810 616Ud09 ANGR-1ao.o
0170810 ASNaASrN/AC.S O
A':SBABSIASN) 01904 :S-ATAN( ACSp 02200A4G*ANGi+ACS#5,2V57SU 02101UtASN) 811.812.l12 ('220
811 AIGA-ANG U2130
512 RETURN(04
lI8FyC COSS M9494XRF 04160SUB3ROUTINE Eg oz 0270(DICUMIflOOJ .CLNRIBI
U020COoN ,~.CF ,ENT 0060COMMON RHiS v Du.'1 . CF9 .MAT * Fft 9 CLAIR007COMIkOh PIRN3
00oa
KFN~rl0120GO T 12492692602601MATU130260 00 Z54 J21#4
0140PRNRaI.0 0150
PR~43*1.0 u1l/oPRN~m1.o u1ooKR480. 0190DO 248 1=194 Qu10PRNI*AIIg 0210IFIPR41g) 242.,u41#2 43 0230241 PRP4ZSPRN2,l.0 0230GO TO 244 0240242 PRP4I*..PRNI 05243 PRNR*PRNR*IPRAkIP..
25 j V,?60244 PRNI*m8(ijtj 0.047
'IFIPRNII 246o245#247 uo245 PRN4*PRN4,1.0
09K4mK4+I 0290G0 TO 248---~ U 310246 PRN4a.OqDRI
32247 PRt43.PRN3*(PR~j**o*Z5 1 0340248 CON4TINUE
04JIN1PRIN2 250*25G.149 0340o249 DlN~w4*o/i,I-apN
2 j* 0340P MePJR 01442(o
250 CLNRIJIMP~ANR101IPRA44I 2529252*2S1
039025? PVN4*4.0/g4*0...~h4 )
4Z52 OUM4I(J~mPRN3
043000 253 Ju)*4 0"aoAfl* J)vAjq sjiiP4N*i #1450
254 CONT:AIUE J)410CALL 1AT1NiAv4a'..*'.CF,, U480DO 256 .ijn., 0490PtR~lOI-U,4lIj) 0500DO 255 161e4 U510
25S A'1*JJS8II.JItCLNAI1J*PRhj$J256 CONT:,NUE 03GO to 291 u!)'260 PRN3nT~oV57
PRN400,o 0360K.4u) ('51000 .!7j Jul.'4 0360PRNRu 100 0590PRP42auOo 060000 264 191e4 (1610PRN1SCI I,J)%02
IF(PRp41, 262,261.263 63261 PRN2*pIRN2#,l064
GO TO 264'.,>262 PRNlv-oR41
06601263 PtAMR8PRNR.,PR~)..0,2 S 0670264 CONTINUE 41680IFIPR421 2660266,265 0690265 PAN2-.'../14.0.-PO 2 1 0700
PINRPR~NR *PR42 0710266 C'.NRI.5OPmpq 0720U0) 267 I19'b 73
267 ~j~jv~t~jjjRW074010IPAN1, 26902680a10 U760266 0q%443PRN4'* 100 0770KVS4*1
0700GO To 271 0790269 PRIORNla. I~ 0'.g270 PRN38PRN3*4pq4I** 0 * 2 S1 0slo271 CONTIN4UE 0820IF;PRN##l 273,2?l.272 0640272 IFi K.-41 2716273.273 0040
PAN38PRN3#*PR44' 0600273 00 274 Jal.'. Lbgt0174 D~~jDjljqj L0690CALL MAtIj4v(C,4,0..CF,, 04900279 00 276 1.1.. 19100
G76 toa 0920200 1Rl0 093720PW44-u109 0940
K4wO) 095000 291 jai,# (0970
PRN24.0~ 099009 204 lued 10900
PR"1CF~j9Jj1000IF IOR1IO 2#2,.01,263 10202St ORft20Pmtq2,* 0 102060 To 28'. 1040282 O8NIS-Oiq,4315
263 PRNR*Pqm*P&'1 ~ 1 ,* 1 1 10402S4 CONTINUE
M07IF (PR%213 2&6,26402#b 3100205 2a6 O~t6 g~p~~ j1090
286 C11(J*RNluU
D) 28? 1819d 1110287 C`%q( I ,Ji ) FM( 1 1120
It: IPRNlI 289.288,,290 1140288 PRN4uP~k'4+.l( 1160
<4*K41 116GO TO 291 11/0
289 PRN~m-PqR41 1160290 PRN3uPRN3*lPeHN1..oej25j 1190291 cONTINuf 1200
IF (PRN41 293,293,292 1210292 IF(K4-8) 2'id,293,d,3 122029d PRN428.O/(8,0o.PRN4) 1,43
PRN3UPRN3**PRN4 U240293 DO 294 ~J0198 2l294 RMSlJ,1IluRH5(J.1 I/PRN3 1260
CALL. MATINV(CFP4,8,.RMSo1CF9, 1270295 no 296 1.I98 19se96 CFMU tI 2URHSg I *I )CLNR I I )PRN3 1290297 R~ETURN
1310END 1310SIBFTC MATRIX M94oxR7 12C MATRIX INVERSION wITH ACCOMPANYING SOLUTION OF LINEAR EQUATIONS vuu2
C 0030SUBROUTINE MA.TINViA*N*B*M*DETERM) u010C U040DIMENSION IPIv0Tld It A18* 8 It blb.4 1, IhDfXt8.2 I# PIVQTIS 1 %005
-C EQUIVALENCE (IROW*JROWI* (1C0LUM*JCOLUMI* IAI4AX# To SWAP) 0070C INITIALIZATION' 0090,C
U 10010 DETERM=1.0 ullo15 DO 20 Jul oN 012020 IPIVOTIJ)&O u 130
C30 DO 550 Is1,N U140C S'ARCH FOR PIVOT ELEM1ENT 0150
40 AMAX*0*0 U) 8045 D) 105 Jsl#N 0190SO to (IPIVOTIJ)...) .6L,* 105, 60 uU2060 On 100 K*19N 01?0 IF (IPIVOT(I-114 b0. 100, 740 .022080 IF (A9S(AMAX)..AISfA(J9K),) 85* 100s 100 023085 IROW..j #b90. ICOLUMv;K
2095 AMA)CuA(J*Kl U16100CONINU U9020100 CONTINUE
110 IPIVOT(ICOLUMIUIPIVOT(ICOLUw,,..l a0f
C NTfLRCMAGk Rows TO POT PIVOT ELEMENT 014 DIAG3NAL U310c130 If (IROW-IC0L,1i) 14M,. 260, 140 0320140 DETERMm-DEyERM,
U340150 DO 200 Leloft 0350160 SWAPsA(IRO*.L,
0370200 AIICOLUM9L)8SwAP
UiGo205 IFIfdI 260. 260. 210 U390210 00 250 LO!. 's 0400
220 U'A~I~~L 410230 Eti II O.L I ,IdIICOLUM*L 1 0420250 Bf(ICLUMvLIUSwAP 0430
260 tNDEX(foIleIR'nW 0440270 INDEXII,2lwIC^L''4 0450
310 PI3lv*TI.AfCCLJM9ICOLUMl U460320 VETER~4*DETERM*PIVOT III u470
C DIVIDE PIVOT ROW By PIVUT ELEMENT U490C USQo
330 A(ICOLUM*IC0LljMIm1.0 V510340 0) 350 Ls1.N utl20350 A(ICOLUPOL)uAIIC0LUM4.LI/PIVOTI!I (530355 IF(MI 360, 350, J60 0540
160 0) 370 L*1,M 0550170 R(ICOLUM,*Llu5(IC0LUP.,LI/PIV0TiII 0560
C 0 570C RiDUCE NON-PIVOT ROWS obo0C V590
380 DO 550 LIwIfN 0600390 IFILI-ICOLUM) 400v 550. 400 U610400 TOA( 19 ICOLU1MI 0620420 A(I91ICOLuMIGO*0 0630430 DO 450 LuI*N 0640450 A(LI9L~wAILIgLI-AIICOLUb6oLI0T U650455 IFIMI 550. 5509 460 U660460 DO 500 1.819M 0670S00 81L1,Liud(LIL)-6fICOLUM9L)*T 0680SS0 CON4TINUE 0690
C -0700
C INTERCHANGE COLUMNS u710C U720600 DO 710 Iu1.N 0730610 LN*N.-I 0740620 IF (INDEEELelf-INDEXIL921I 630. 7109 630 V150630 JR0kaINDEXIL~iI 0760640 JCOLUMmI'4OEXIL*2) 0770650 DO 70S KuhsN 0 idO660 SWAPmAIK9JROWI 0790670 AiKoJR0wl.AIKoJCOLU~l 0000760 AIKoJCOLUMIOSWAP 0810765 CONTINUE 0020710 CONTINUE 0J830740 RETURN 06407S0 END
- END OF FILE
.AMP L. CAL OkLUIrUN ')i IR.tCR I qlý;l . )4 PtDFSTAL!b..'40 vY0-LPI :vUNL\%t
27 4 1 1 0 0 9 1 0 03.13O000001+Ol 1.nO')0OOenE+007.600060().01 3.75(00",E+00 5.2!fOOOE+O I
3.OOOOOOE+01 5.IlrOOOOE+0Of 4.bt.,ao~OE.o1l.250000E*02 6,.620000L+30 1*440000E+021.240000E+02 2.25~0000E+10 3.02000Oct0Oe3.SZOOOOE+02 5.410000E+00 7. lVdO00E.O2Is/ 60000E+02 9. IdCe)0'th.0 2.:idoooot0~3?. 710000E+O2 7*19OOOCL..00 do3dOOOOE+025oOOO0OOE+02 b*5?OOOOE.00 3*Y50O00E+0d
4o530000E+02 7*1900O0.0k+ 3*.5OOOL0l+ ~2o35O0Ot0E+02 5*OtMOODOL+00 3.110U0i00t01.OQOOOOE+O? 6*62O000'i+00 6.130000E+023.130000ET+02 4.8i71000E+00 1.930OQOE+Oid.2O00nnE+01 2*113000E0On 6.800000E+013.20flOflfl+.1 3.19O0101h+00 4.1bOO0OE+013.6~72000)E+01 1*4?Qoonlk+0 2.0OOOOOOE,0i2o621f)OOE+O1 d*65O0O0E+OO !,.4t6OOOE+OOP.04.3000E+01 lo237SOO'L+O1 3.120OO~OE-OI4.2911O0.0E0 1.231COL+0O 3.120000E-014*297100E+0O 1.2l7500E+o0 3.120000E-011*85ZIOOE+OI 4.I'iO00t~+00 2.000000t.O13.'.23')0OE+01 5*32000OiE+00 1.092000L+02.9.82fn)OOE+31 6.l90000r,00 a.540000OE.O22.431,)OOE*O2 6.5100O00L+10 1*602OO0et.O33*300100fF#02 6.2110O00E.0O 8. 360000OE+022.8430nE+02 4..310010E+00 I.SS40OOE*026*1700o0E01 OsOMOOOOE+O0 0.Of00i000.oo
3 16 22 2?10 1 tl00000')E~oj O*0aoOO0or)E+00
1.000000E.03 S*IEV)OOCE+13 2.OOOOOOE.031.~'.2000E.O6 O.0#000000k0 O*OOOOOOE.OO3.362000E+06 0.000300t+00 0.OOOOOOE+00
-d.onOOOOE+O5 0.oOOno.oon 1o no.oooout+ooS* 302000E+06 O.OnlOOt)L+10 fl.OOnOOOOE+fl1*54'OO.OE+0 0*orooooEo3 0*OOOQOOE*OOle04OOOOE+16 O.0O0000E+00 O.O0OOOOOE+OO9.72000E+05 O.OnOO~OOE.00 O.OOOOOOE+,oO1.102O0nnF+6 O.nIOOCOE,'no fl.OOCOOOE+nOi .542OflfE.06 O.0')mOnOI'E.O n*OonlOOE.OO3. 362000E+06 O.O000000t+r)0 6.0000O00E+00
-8.O10000E*05 O.Or0ooonf.Q0 nononnlOOf+01. IC2000E+06 0O*flOOOOE.OO 0*0.OOOOOE.OO1.'S42OOOE+05 O*O~OO.O0OOOOOOE~o ~ooo+oo1.O'.OOOOE+06 0*001OOOE+OO O.OOOQOOEO*op9*720OOOE.oS Oeoooo0ot~aoO ln.foodooooo1.302000E+06 O.OflOCOOE.00 M.OOOOOEOOE01.46300OE*O6 O.OOOOOOL*~0O 1r.OOflOt+OO3*296000E+06 O.onOOOOE.OO 0.000000E+00-1*045000E*06 O~o#nooon+Io . n o.oOOOOOE.oo1 .7810O'n~O6 O.oO1Oq0lk~o 0.00',rb0Lt.O
-1*197DOOE405 OoOnOnOOOE~n f~O*OfOOO&OO*I. 75'.ooE*06 OonflA~onE ,., n.COrO~t'nE+M1.1 75OOnE*06 O*COO)00ý+0O nlonoooofnOE.
1.d.83000E*O6 O.OOOOOOE.o0 fl.GOOOOL*O3#196*OOE*06 O*OfOOO.OOEo noOoO 00O
1*781')Ot+0.6 O*OMCCQIE00. I1.OOOOODE*OO-10197lon'E.os O.O'o000OL+0O 1000OOOOtOO.O
I *1 lr~Eo~ OsOMOQOOOL+)o oool.oo~oEoo1. 18 10OOE406 OO"OOOOEo(.o O.OoonvooE.oo
0
w0 0
z ~.0
a 0
10 0
4 0 0
I.I.
20 A0 @00fftt4;4 0 ftAtf
'". . 0 0
04 0
.3 0 o A 090 93
0. 0190. 0
14 O.040 .04000
W ~~ 00000000
0 0
" I so0 0000000000IO 000 0000lu~~~~~E. It..0 A , 0
0 ~ 00U w
A i- ft 6U #0.0 0 f -0 .0 0 ft Aft a O joE*i .
Z4 0s. z. -0.4SM M MM~SS MMMS MM~ 4f0 fts0 w 0
-- 0 000 #A f
-W 14.40.4000 .o000 4 2,C4 4 0: A 3.A v : ft :07" S2 0000000 0 Q600
0090.00 0 Ot0t a0O 0 o. 00000 vil .M .1ii Cg~g t~g~~ ggS~sftp
3 wo@*oocs ooeeo6 ~6og66 T Z".. IS POM.U# 90.4M@SSMM MS~r MM
A SM04SMPSMSM~i46MM*S~rdtP~M.4MSM"0 0 .
MO aC 0
.010 10 N N N . N d O * WN 0 z:~ 10 1* W~ V 4%: '*
a0 .40 aS Of 0 00aeemuuesPes,,a 41702" M O'r4
CO a 0 O @0 Jr 0J J -r 0 J 0J J 0J J0 J0J00P000q00
W 1"; 4W
noIm0 MC 0ONNNNN.000@eS0 000 A m~4.00
.. P a 2- ..-- ; ,c f= 0
* ::2 ??? ooo eei oa 1o0coe oo0oooo???V ??? a?00 4.0 0 WW4w w 44444.W4 .W .4 . W4 U .W .MW4 W I W W
ida ft ;. I*:I; * & 0"
00 -00
N ~ ~ b W4 WW W . NO WWWWWWW WWWWWdJ U.aU wU WuW WW W '0 0- 0 -4 4 b0. ja Q . N a o0 . N O .. 0aO 4 04 Pv-
0 a NNNNNNt ft N@OfNNNN*0op
-. 000 J
446a Cc) VO N a 0- .4 .4 9 a 9 - V4 9
0 a
dA W W U W IM.4 UPN O4WD V% f-N . . .0 ft w 0 6 .: NO 0
W act wk 0 4 0 V 0S0 0 0 0 0 0 0
S-0 .4 W0 0, as 4 . 4 4 0 . .40 a .4 .0 -49 a 0 0"0 0 0 0 0 06 0 0 0
I* 0p0 Is-W41 IW It4 W M J. J1.4 00 P.M P. P.
.4. O N NNo4 N 4 a N 0 4*43. N N N E .440 N N 0 a
Mo t ft f tA Of0 ft At4 ft Of At At t a AN AN m ORN?0 T T * 6 ? 6 ? 7 7 T ? 7 SAt a % N N' Nt N 40 w '41 N N0 N NallNN 4 0
I 0I &4 .0 Jo NUO 40 ANO j mo :so ~ "0 0 ~ 4
0~0 so 00. .4 Vs .0 N
--- - -r WWI on Of" -4 .4 Oft- - - - -- -- - -- - -- - -
19 gf ... 4 1 'f 0 V900 .00 0 .. n *ff4* .% 99 f t t0 0 .0 .. 94!4 ...**., 7 30 ~-7 N4 &o f--7ftt00444.4'' 4 . 4*-..ft""" .... t'ettttfftttpm
* e , , 8 , I 1 1 11 6 6 1 # S S * U S I 1 S~ I6 * 6I
.0 .0 4 -e .4 P 4
**ý W d4 ..0 .e .f~t 4''009449. .. . . 9009944 oft4'4 0f9P.0"mu O4 f" K P .ft a0 S-W 00 4 I ' NN14 f00 ft 4 f aomf P. . - ftp. tf444'44 '.0 n " &4..a%4f0%P.F.4 4'
00 000490 Igg-00 N ? 00 4'''444494.-9tt444ff4fv4eff00
004 ...... ft I * * *.0004499 in#499 ftt.f PZ044ft4ff .F4ftC4 004 4 4 vft P . m m 0 v a.ftInS.in
04 Or in gseg gee gu ft
oocooe O 90000 *' ooooooooooocoooo%a I u %I Il w I 6ww 4w w-- II
fttt-tftO Weaaft4' i * 0F .4 4tt~tt9OtfO ff944 0 .0 cct~tfftt4f0 f440I- f. f-'oftaN e W Sttff0 4000ffttf"A"# see t'PPt.444#
W%0in0e v F F 4 P-.f4 04 *04.f.. ftpf t94 4ttt49ftwd4'440 .0.
.......... i I. .. ! - * . . * . * .. .. *.... * .Ot*. *4i4*i~ ii. l aa0000 l e'
uj-- 000 500 4'000 000100000"000000000000000
99 fft I .0or-Oft4 f" 00 % 00 9044; tf4'4 a 049tttttpffcffft W i f af"a I P iff "4..4 4 It 7 0 # P4. 4ftfteo 4'4f 0.4
. 0 ff.4f"9 -t.4. -K 0 A 4 -fin 0. 9f4 44'4'4'44Nftf.'a n004 4 ff
4
-t - P. -4.Z a4 WFt- -. a a P. 0 01 4 4 .4 - 4 - - - .
000c 04 -T0a0 0 ~ a 00 0@ .0 0 000 00 00 @000
Ii 0
.I t.40 3..44. a Nw- S4 Oft S,ft 0 ' 0OfttO ' a. 10, Vt f 0 P. f at V, po ft4
ft P. in v'.f~t P . N~ V- 4 i It 19 4: 1 . 4 4: 1 1: 1t 1
40& AftO " ft ft400 0 ' ft4 ft09 0 ft 0t ft *t oft40 4 f"
0- - I I 0.0t f
W% f. -Gl0' 0 000. 0a. 0 0 in 0'e o a. Iftp 4g 14a . 4 0 0 f t f t4 4 ~ f t4* ~Z O .' on - t.0 4 O ft0 f 'f * 0 4 c 4 a 0 ft
ft ~ ~ ~ i P. : 0 'S44f W% 4'9 4 4 f t t f 4 4 4 ' 4 S 030ft0"ft ' at I .4f oo -4 P '.7f 44. 9. 'ao A. 0.4 ' p
4 ' . 4 f
.4 4. .4 .NJ W .~W 1. I P.
46 0 t f 40 4 .0 .0 09. ft 9
: 2 : is :: : ,0f 44 a 0. 94 ft 0.$. '0 a .4 4' f O t t 0-I ftOft.&.OEO Mf 04ZtfO 4 ~. 44 'f t f
. ft p -. 4 f tt0 P. ft -4ft t- I-f ft ft . 0 4 t- t 0 9 t 0 ft p
.4 0
00 t0 -4904 - ft 4' .4 ii 4' i. it 0 - P: 4: P:t4
ft . t [email protected] At ft ft ft ft 44' p.9 9 p. ft ft.09 0 t tf
IAD@ 0
ft ft ft ft M4 "ft"tf MB 0fw4 f 4 . 4 . 4 . 4 f
MA doe, wmwo- Wi A&6ft O0VC4p 0OR3CC ItO4O
-P a 0 -a ii -. I- Cý ft 0 in w~ Nn. d% 0 . . 9 N 0% 00 b99 pb P.
&01-0 W0 0 o
4.O9WS909 ;44999; *o 09 09 9 0 -0 9 9 -.. 9 --.- N @ O 4
04 4 7N9~ 4 4J ?0. 4 .wt .1 & C . . . . .A . 0 : P ft * M * * * * * * * *. I. ; A . . S * *44444 4~N £4 #N 'r ftN ft or w99 N P NN N N N 9 99NNNNNNNNN
v a in
0OOC000C10 0000 IA cc0 a0 0O 020 00O0000 coo
ML.,A4J WM UW VWA.W W w w b u.1 0 w w ~'u~w w W WuW& MW WW W WW aWANW W ww ow ww wa-~~~-4I 4p9A AU 9N4o4 .#.tft '.N9A N o mzo= _^9A..N44 9ft
999w%'A44e~~~~~~ea00~ N45P 0 0 N0 U NN N 9 N 9 99 N W 9
WO 440C4 4 5V N4 0"0 00 U40 0O 09.49 V C" 9. 4 U
a #on aMN 00 vSNN a~44 99 9N
00000000000000 vOftO 0Z0a
P409 44 0 40ft 4" 4L F. f a 44 4 9 4 -r .04 4 4 Wk 400 4
* . .. . .. ... . .. . a . .. . ~~4. 9 * a * * . ... * * .. . .
I. 4 ., 1
ft 4.5 W, 40 wa -' 49 4 Q W.0 e 0 N 0 - N - 4 *-~~~~~~~ 0 U 0 9 9 0- N P 0 N0 0
- .- - - - -~ .P - - - - P. a N P
,A0I of 1 U
- - -- 00 9- - ~ - -0 -b - - - -0 00 0r 00t 0. 0 0 4 a0 00 00 0 0 a00 0
W. P 1 45# N. -9 v N a N gJ4 A g0 : 4t #- 4 A N 0 0
* 4 - - 9 4 ftAPP o- t 0 *N N a 0
00 0 O 00 0 0000 0 .094 0
.. 0 0 O0 01~'A44t4 Ma.A
P. P.V. P.0 4P- .a a P. ft A P.P*P4
l. 01 1 01 P01P.P 0OU@Q .4.04~O ~ t 4 P. ft W, Nft LONA
;! .! . a*. * * .* 0 a * 4. * **
9~~ ~ 1 1: S-I I . . I
-- Ma f-t N -
OaftAft
-rO00100 %b1 % N f JI4 0P . &fP
a. OA to0p e,- 4 .6tf 4a .Pfr . a. * a * . a& * " a *a a S **
-4.!00.0400144p1010117" *1
P0 -1 1%- ---- 14- LIf ' %f"0 ft ft 7 -c
O@C00000000 coo00000 c 000
aSJWWWWWWW~aWWWWWWW Maa.aO
: 0 v .. 00 4 3P P0114 a 00,010
Affff NA 0-A A0ft fff~~~ " w P.
sa. sItahwteaWt I Is
O~~ftftftft~ aft.AA ftoP011 40 %0
*. 1 0 0 0 0 A 0 1 P * a f f 0 Oft o oO04--A*4P.Pt~ 4t%00100
A# . a w w &M w w .l al * a a * . . a S . *
- % - 0 - -t -0 - 0 - - - - M
LI 01
WI ~ ~ ~ ~ ~ Z fm 01004 0 aA 0 00.
ft U 0 01 4 0 4 P-; 044 M
ft P. 0 m"WtM"4P - w0 4P. . P 1 0 P UPO.
ft 0 014 A 4 A 4f A. 4.4*P*~ .a a a . .: ia a
00
of - - - - 0 t- -f 0Ima P
SAMPLE CALCULATION NO.?ROTOR IN FLEXIdLE OEDESTALS AND GYROSCOPIC MOMENT
27 4 3 0 1 1 1 1 0 13. 1300000E#0 7 1 .OOOOOOOE.OO7 1*0000OOE-03
7.6O0n)ooE+01 3.75O0000E+0 5.25O00OOE4O1 le2#2000E+O4 7*11OOOOE*032.l40000E+01 4.090000E.O0 4.d8OOOOE.01 O.O0OO0OEoo0 0.0OOoOE*003.OOOIOnE*O1 5.310CO0E+0O 4*~080DCOE+Oi 0.000000E+00 0.oOOOOOE0OO1.2'5O'00'.O2 6.620000E+00 1*440000E*02 O*OOOOOOE*00 0o.OOOOOE*OO1*24O)OOE*02 2*2S0000E+30 3.020O000E+02 O.0OO3000k*O O*OOOOOOEO003*820')OOE+O2 5*910000E+00 7.1800O00E+02 O.OOOOOOEdoo 0*OOOOOOE.o08.760OCQE*02 9.780000F#10 2.58SOO0OE+03 O.OOOOOOE+00 O*0.0000E+00T. 7701O0E+02 7.190000L+33 S.360000DE+02 O.OOOOOOE+OO 09000000E+005.7OO0flOE+O2 3.57Cn)00E,0fl 1.95000OE*02 O*.0000 +0 9000000E+O004.51n000fE.02 7.190000E+00 3*85n000E+02 0*OOOOOOE+00 0*OOOOOOE4002.35O0000E+2 5.Of00000E+00 3.210000E+02 O.OOOOOOE+0O 0eOOOOOOE+OO3.000001)E*O? 6*620000E.O0 6*130000E.02 0.OOOOOOE.00 09OOOOOOE*O03*130000E*02 4o8700OOE0EO 1.930000E+OZ02 O00E0 O.OOOOOOE*0 e0O0+O0i.?00Of1OE+t,1 2.13OOOOE+00 6*00ooOOE.01 09OOOOOOE.00 0.OOOOOOEO003.200000E*01 3.190000E*O0 4.160000E+01 0.OOOOOOE+00 O*OOOOOOE*OO3.672000E+01 7*42OOOOE,00 Z.OOOOOOE+01 0.OOOOOOE+0O O.000000L+002.821000E+01 8.650000E+00 59476000E+00 19020000E+0Z 9o70000OLE019*043O00!.O1 1*2175O0E+O1 3s120000E-Ol 1*O5O0O0E*03 Se450OOOE+024*297000E+O0 1*23?500E,0O 3*1200OOE-01 0*OOOOOOE+O0 09000000E+004.297000E+00 1.217500C#00 3*1Zf0O0OE-O1 O*OOOOOOE+00 0.OOOOOOE*0O?*SS20OOE*01 4.750000E+00 2.OOOOOOE*O1 1*65OO00E*03 8*UOOOOOE+O23*423000E+01 5#320000E+00 1*092000C*0Z O.OOOOOOE*00 O.OOOOOOE+OO9.SZOOOOE*01 6*190000E+00 8o54OOO0E+02 0*OOOOOOE+00 0.OOOOOOE*0OZ.431000E*02 6.510000E+00 1.602000E+03 O.OOOOOOE+0O 0*0000006+003*300000E+0Z 6*21OOOOE+0O 89360000E.OZ 09OOOOOOE.O0 0*000000E+OOZ.843000E*OZ 4.310000E+00 1.Sb40OOE+02 O.O00000E+00 0OOO0OO0E+00S. 3?0000E+01 09000000E+00 O.OOOOOOE+O0 4e0OO0Q0k*0O 0*OOOO0tk*00
3 16 22 271 1*OOOOOOOE+01 O*0O0OOOOEO00
10 1.00OO000E+01 19OOOOOOOE.0118 -1.O000000OE~tfl O*OnOOOOOOE+00
5*ZOOOE.01 I.2O0nmE*0 2*2000E*01 992000E.01 bo3000f+O5, 2*2000E*O15*2000E+01 S.Z00fE.05 2*2000E+01 5.ZOO0E*01 3*3000E+05 2*2OOOE+0l494000E+01 19300flE.06 3.4000OE+01 494000E+01 990000E+05 3*4000E+O14.4000E*O1 1*3000E.06 3*4000E+01 4s4000E+01 9.0O00E+0~ 394000E*0l9.*O0OE*01 1.7OO'E*06 1*80oOE*01 9*5O00E+01 193000E*04 198OOOE+019*5O00E,01 1-7000E+06 190OOOE*01 VeS000E+O1 193000E+06 1*IO00E+O18.500OE-01 le9000EO6 2*TOOOE+Cl *.S000E+01 1*2000E+04 2.7000E+0189500OE.01 1*9000E+O6 20000OE+01 i.500E+01 195aooE*06 2*7OOOE+011 .000000E*03 S. lnOOOOE+03 29000000E+03195~42000E+06 0.OOOOOOE*00 0.O00000E.O03*362')OOE+06 0o.OOOOE+00 0.OOOOOOEO00
-S.OllOOS40!50 0.0n0OODE400 fl.000000E+0O10302100OE+06 0OeflOOOOE,')f fl.OOOOOOE*00loS42100E.O5 o.fO*OOFO.OE0 0.OOOOOOE+001 .O40)OOE+06 0O.OnOOfF.IO090~00f0OOE4009o7ZO300E+O5 0*0O000O0L*30 09000000E+001 .IO2000E+06 0sofoOOO0E*O0 0.OOOOOOE+00
- 1.42,00E,406 0*OoOOOOE+0O 09OOOOOOL*003*362000E*06 O.000000L+00 0o.OOOOOE..0O
-8*010000E+05 0*OflOOOE+00 O.OOOOOOE*00193OZOOOE*06 0.OflOOOOE*O0 09O00000E*001.*52000E.05 090000OOE400 0.OOOOOOE*0019.040000E+06 0.0'n0000E.00 0000OO0OE+009*720000E+05 0.OflOOOOE*00 fi.OOOOOOE+00lo302OOOE#06 0*0f0000*E.*0 0*000000E*001*4630OO006 O 00OflOOOE*00 ft~noO000Eo003*29S000E+06 0*000000E*00 fl.O00000E+00
I. to IOC()kCo O.o()OOOCE.3O 0'.OOQOOOL.OO
-I.IflOOOE+0 OeOnOOOOL.OO O.OOQOOOOE.OOI * 7l'OOOE4O6 O.OI00OQL+10 ').OO'OOOOE.OO1.81 00OOE+Ob 0.010000E+00 Q.00900E01.01.7iilOOOE*05 0.tn(nOE,00+O n.0O'OOOE.r)3.2'7300~E+36 a9"nronoooE.,v ,f.030rOO3E.00
-I n4,30Nt )E.6 3.'1O0OO1E*')0 rI.00)OOE+)OE01.7681000E+06 O.00OO'(.E.9O n.OOOOOOE.O)0
-1.*197300ý+05 0OIOnO00E,00 0.001000E*001.754000~E+06 0.O*OO0GOO+0 0l.000000tA+001.175000E+06 0.0iO000OE+00 O.O0000E.O001.781O00EOE. 0.flflnn)OE,00 C*.OOO E0000~OI.O700t~nE+Ob O.CCV3lOE.O0 (n.OOOOOOE,0O2.240000E+06 0.OOODOOCE*00 0.000000E+00
-5.?OOOOOE+05 0-0603)01E+00 fl.OOOOOE*0O9. l'000CE+05 0.0 33OE4OC 0'.003000E+00i.oi0o0nE*o5 0.0flfl'nfEi00 O.000000OE006.800100E+35 0.OflOCOOE+00 0OO00000OE006.32011Mý+O, Oo'0.7M3flfl+0 O*OO.OOL,')09914'0100E*05 O*OIOO90OE.O0 0.000C.000+001.0?010OE.06 O.0#)OOOE+00.f f.OOOOOOEO002.240),flCE.06 0.at~onelnFlF.1 fl.000()00tE.OC
-5. ?00,)OOE+05 0.tn0OC0CL+30 fl.00000OE.009*140000E+05 0Of)O0n.+OOL0 0-0000O0t.001 .07OOOOE.O5 0.0fOO00E+30 00O00300t+006*O00000E+05 0.0tM030OE.OO fl.OOOOOOE.0O6. 32oo0oS.Os osonooooet.oo oeoooooca..oo9. 1400OOE+0t O.OAfl00L0.0 O.O00000L+O09e6OO0O0E+05 O.')VnO03E,')( eO.Or000E.O2ol60000E+06 0.0tOOO9L0~+0 0.0300OOE*00
7. oO,,nP*'9.9 l)IOOOOE,9C 6.0'VIOOE.0OL *31I0C~fnL*Db 0.0fn0C00L O n,.OOOO0OL.0O
-d*?OOcO.O3E0 O*()ooor0(,v~c~ n.O9 OOt..Oo1* 200000E.O6 0O*flOOOOL.aO( fl.O0fl()OE.O1.bOOO0nflOtfl l.On0OOOOE~fO n.O00OO00t*O1*33000DE.06 O.OOOOOOE*OO 0*OOOOOOL.0O9.800000[+').j )..)O000OE.90 (1.003030L.,002. 160000E*Ob 0.I'nOOOOE.Oo 0*OOOOOOt*OO
-1.ooaornnk.o5 n.onaooom.o 'C.OOOOaOE.OO1.33001n'!+06 3060000tk.00 'C.0O0nOnt.O
-8. 7OOnPF.'n6 O.Ofl0OO'Ot+iO n.OOOOOOck.OO
- ~ r Or' tP iL1.
06
-C 0. 00 0
70 a0 At C.03 -~ 00 0
"4 00 uC A
44 3 00004;J
.4.Ux
00000c -00 0000 -Aa
IO 'A0 00 a 0 00a3 CA C0 fm £ WO .
0 %0 CO 0 000
2O o c 00 20rA 02 0. r.;C Iu b 0 C5 00 0
2 pp
IL 0 Ia 0
20. 10 Ca OrWW .4* ~ ~~~~~~~~~~~~ A, . o.-- - - 4a4f .00
3z *7 T 40 0Z00 Q48 ;.QIU~~~~~ ~~~~~~ 0~ .. 1M ~ J'A.. * WW .A N 4 7
PA9 vC CV0 ~ 0 0 03 20 9 4 ****ax~~~~~~ .0 9 0 C 00 u 0 P 0'C U 0 9 C
a Ir 4A -T-j 1
NP 00 4 L C
A0 o
if~ ~ -P 4 o0U e 0V C 0 3.C O Al V.2 .f . .. of.
4L 0 ** , 4 0 0
.4r.-~i --
a0 0 0 0 0 a 0
0 0 0 a aa 04
.NN 00 00 a 00 100.60ft t uW; c 0 a co 'a .4
-00 C0 .00 a C0 ;00 0100 .40o -00
04 44 4 -%4
0 0 0 0O 0 0 0 0 .4
O 0 t 0 0o m 0 o 0 0 w0D0 .O a. 3-0 V. 0 --o P.3 P.O 0. 3-C6."
o S000 w N Now 3N N Nt a 0 0a NO
&00N COO W -00 -00 4000 400 NOO 0 30
* 0
4 a %A 44-0 0 0 0 0 01 0 0 0 x
Z 4.4 . U U.41 di
-0000 0 0 0 a 0 0 a 0 34
O30C )- 3-0 4 40 03-00000 Q0 0coC C
* *. 00 J40 a.4 Z 0 00a 400 -00 .00
'A 4 44 0 4 c.
0 0 0 0 a z
30 30 310 30 0- 30 30 310
0000 -f F. NP. 0 00000 x 4 . .4. - C. a0 P. X PN.0300 0 . 0 . 0.. 0 0.0 . .0 .. C'
4444 ZOO0- 0 M0 0 - .400 - -00 - moo a joo
on 44 4 4
C0000. 0 0 %2N~~$ z l NNZ
4 i. 4 'd . ' 4 d 4 A .4 us w 441 4w 0 G 00op-C 3. c 1 P. 0 OP.0 JP.-0 030-0 433-0 03.0 C U 6
55000 24x r- 074 P40 x4 7MO Zsc0 z I0 240 ' 4"UN ;U 41 ft -0 -00 -00 -00 441"aN .Af 0 0 x W 4 4 4 .
0000 4 " -C 4 m~ 49 N. 4w - *00CC to *S* # * ** N ** * N ** N ** N * N .0000 Zo a40 6 ; .40 cc Zia Zi -00 4; 4;0 p@ 0 .0@ C-0 0 -@0
0 0 0 C . 0 0 0 0 x w
%4 3- 3. J. in. P. N 3-9 3.
£0 C0C0 0 0 0 0P
000 - o M>. 40 =-I 3 0 0C 0 coo .45 - "cc 00 0 00 0 004
GiO £ 0 0 . C 0 0 a a 0
*1 It S SS--W~W W WA A. "
.0 0 0 42 104 3 M6 0u 0 0 w 0 30 0 0 Aw40.4 0 m 0 w 0 w 0
=cc 3.Oo z4 01 a .44 00 . 0 a0 00 0£k 3ol4COCO C ~ . 4. -0
.40~~ ~ 0 r4NN .
000m -% a 0000 a.4 0 &1 4ofAAa 0 *.Jw 4 IL
oea w - 0 0 .4
Nm -0 ft F- to- "s a4W Oft 04 C, It 0C 0 NP 4
0 43 43 M 0 C-400 -0N "Go "004 3 £
fttc7c wlOO @C C OOc" eo@fOOO**9O0 00 00 001000 00 00 Z, ec
0 00
-N^4 a.... -- 050,00 4 41,4
~~ .1NN11141441P C0. NNNNoff$@$4A.e5
~~~~'444IZN IVI N 1 . 4 3 . . . . 14 N t V 44N-gN 11N n441.44 . . 4 4 . 4 . . . 4 3 3 1 rA4
weels .O W: ee s&s, .aasWi, ' ' O'A , A 6 WW *sega IgeeAes a AN 0.WW:tm N"L'WU. Ujwc4
v .. Iam:;PNN..sontr...ft..N% f.NN5- - - - - -- -S .o 0- P. .. %0I3 e 0 c ~ 0 C C o 0 2 0 ~ 2 e 0 vQ O O 0 0 2 O ~ 3C~
NA000- & N
'0 -1 CO I ft, - --- P 44 i44I .1N 9. - 0- - - -'t--4 .1P 1P ft - , .0 s U
of 3
v C27i fCOC4NP1. 4 44.d..N 54
.N.4 r 44.s3m40
V.. 44 aftNP~ IN* 4 4P1 NN NN W1 .iN 4 00 C'S.. 4.4 99 w 0 , lo
- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ t - W-- . S.4
P. 4 - N 40 -2N 4 3 . C C' N 0 -0 0 4 .
OP.#
49.5 L .5 5. 5 9 9 4 - & . - . .9 I. I. 5 5. - u 5. IS ** .5 a~too 5
-~ ~~ ~~~~~~~~ C 1 45 p. - S -. N14 4 . 4404'
4P MA4- o tfawaw~" ,W,:C 00f aa aS N P.WWI W W WW W I PWI W W W W W W. MW S
* S 4*** S * * * ** * * * * . a *... . . . . . . . .** * * * 4 * * *
9L 0411 mlI4 00 0I ~ ~ ~ . PWa W ~ a a ~ a a w a a a a a a, 1. : I: I
NN3d 004~4 .01 4 0 00.0 0-0 . . 0 00 Q0 o 4N.0. .ý.0 0 *. 000
~o ~ r 0~.30 0.400.4 N * . . 4 N O 4* . . O e O o I co0
*~0 .m m n #,M A" *.M~ * * *0wf at,.... Ir**** *, 4* ** ** 4* ** ** * 4 **
0.4 4016 I6 1 6 6 11 l S I 11 1
70000 ~ ~ ~ ~ c 7000 I' 000000000000 0000000000000 000 cc0
4' m -- m' a I41 -F , 4I 'M44O.- ~ ~ ~ ~ ~ im in# in 0 0 Z4 0 .4 . . . 4 0 43 2 . 4 4 4 40 . 0# 0
.40.1C 4.040 *44IA00000 W000,0 N040 II
* I
14.4~~I, -400l illiN1vi . Vila..NN N NN N V NNNN. . . . .2000 00000 ft0 on000 0 00 0 0 W r6 %- fI t "tfe ~t. 0 Mot00 00 0 00 0
- Oils, - --
-u L. ýj4 W ip ja w maS.SS Lus Lw w Ww ulw-A& wtu ww L u aUAwwwwu im I"
W%0%4 . 0% &OfN f . F . a.44 .40 0..0 a, f. 00.4 440 4 N 00,oo 10.0,.-.tn 4 04 9 n4014 I.40 A00 004.00..0 A40 000"'4 9 0r.. 'U O N 4pNm4N CAP.1& *0' Q 0-.N I A C IA00m
,A f- * 'a4 *4 4 4 . . . 4 4 . * * * * *4 4 4 4 4 4 4 4
-c 0- 4fa q& w.a 7'a jaa , N w1 W.U w u M 541 "i a bi a 4. 14*U s5 0. as A as kS a
P04 4 .414 0 - 40 14n 04o ON 0 1 0 % ft In ad M 0 N*a14NN W,00 a- eS 6 4 - N 0 m, - AA . 4 0 w.
riN O a CP 4 2009 %33 ZXJ 0 0 4 4 4 N 4 4
44 N 400 *0 4A Nt 0o -o ^04 Ch 0A 4a N 0 aA C 0 0 0L a44 I. 4 144 L. 4u 41 4. t4 0. w V . .4 %6 . Li .4 .m &
5N,4400w m. 4 in4 IA A 4 4 U S N IA . 4 Np 0 -01 0
Z' 14 'A If4 t L % a* % .
e4 4 a. 4
1414ha55145 IWWW~1 4 , & 1 17 a, 1I I 4 4. u 4 .5 14 a a a a SW4J'.4N f4.44 V% .14 0 cc w4 ft 1 .' N N
CN . 0. . 4 4 C N 0 4 4 ~ 0 0 4 - I114140 a.0 0 .J .. ) 4 N 0 1' N N £ 0 0 0 0r4
ID41044 01N ,2 4 4 4 4 4 4 4 4 A IA Z ao
A"W --
"4.I00 NfNO aw4 4 0 0
0000 00 00 000 00 0 0 0 0 0 0 0 00 0 00 00
N 4 0 ,oJSNO 0 .0 IN P4 a a 4 -AA N a, 0 N .4 .f4N4 Na
04 4 P- 4444 on tvU WN WN ow fft CNA
-~~ 0 CI - - - - - - - - -A -'~ - -. - - -OA A A -A -4
0 000 0 000 00000 "t 00O0ooo0ooooo0oeo
0as.. CL*4 V P..a a- 4 - o Ile -ki ua- 4
f't 4 f't' N 444P- m% 10 1-5- .N m. 4A W% rd W
A .M w .j Iu *. w *j *l A. o a %,% A . .44.1~~ W44 4 4 A .s A -- A N4N - - w0 4A4z.z.
04 Sa' pS ,P -- AO I'. S4p
-. 00 4D~.N w 4)"'' .0. 075 4Df -41 P0 4 4 -
44 000- F IL 0 i% a . 5-t- .5 4 a 2
-~~NC.C.~.N4403OSo.' m. N v.O 4 . 4. N w O ~ N ~ S A ft N a- -
415! P' 0 S1 4pS .A A. . . . e I5. -*. 0 w s O .. 's s .4N A A 4 4 4 - " 4.
-, -* '. S a or ~ .4 .1.. .4 . .' 5 S ' .. . .u .s . . J W . . .L
~~~~ c, 4N4 k0N4 wO-~ 0 04E N N N
2' T' 44g~ A 4 ID 0 a 04-4S * * * . S .lot . .. . . . . . . . . . . S
z c
- - - 5 -- - .5 -- - *5- .~ SS- - - - - - - - - -t. i- C 'L. 0 A V AO.U CC.(5( U0.U 0 1. 0 0 0 £5 0 C. 0!,5 !. ..5 , 5. . -...~ .s '. : 1 1. 1 J N I* ad Is I~ I. Is I I IO ~ U 4 0 C 4 4c s~ ~ eN N 4F .C .4 N. 10 - - L, #1 N. 6'ft S 4 4 Wt. O U U S 'C 4 P .' P. 4N -6 w- v C
41 C ~ P... ... ... A3 wN 4'~ E 4 A N N-- V N z -~34% AN ' 4 A 4 N 6' U Ut N
P. 4 5a ,.-C4 k.-- - - - - -0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ b A 3 0 U 9 0 c C C 3 U 3 0 0 0
u 4r u-*~ ~~ ~~ ~~~~~~~~~~~~~ 3' . 5 . = g .. S a . 5 *.~A . . 5 s 59 u 5 a uN ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ f A . 4 S S * 515' C A 8 1. * s N 64 55 - ~ N 1 5. O ... .. 4 N 5' .s N 6' 4 ( A - N 6 2,
-t o~dm w -------------- - - - - - - - - - % -on0-. S-ft0 -0 ft004- 4.3 -f f4400 0 0 0 0 00 0 0 0 0 0000400 co0m on00.MaZ 08 I, Z 0 ~0 a4,0 V 0000
rt O '4000044fft 0W14M-a 44F. 4'O44444..4' W4'4 ft4- a~. T~ 4O WOftf. .4qv% on P.44
.4: it 4 It I I ii i
--- --- ---- -- -- -- ------- N N - 5 "N -- wO ft ftoo o~oO oO~o eO OO OO OO OO OO O 000010, 00000 00000,
4,ftftW4. 'aam444aN * v P. 0600" N0 A -'0. a. t0a44tft440004''44 N 0 44NN4'0In06 0 in-.- .J0 4 J00 4 0
-. 4IP 03000 m44 N O N '4a 44 0 ZN-44 XNOM4'0 I 0 .0,0i . . .lif . . . . lif *1:1i *: 0: ft * 34 *: ..
ft o44. ne On 4'4.ft If b, i -n44 4J4 '0 .In43f N4.Nf 3. f" aft
4'44' ftffttft~.'fft~tfftfftt~ffttfftfftt--.4 LSNN" ONftmt ý3ftmft".000co000c000000ccoccoo0 000000000000 20000 Z0000 Zoooo
-4 40 -W~4f f co~ft0'4'0f.d004.-40t4fN0 5'4 N' 444'C-4.44 N 40 ff44044'.4 0404 oo4ze 40-ft
00000 0000000000003000000000000000 co 0000 00000 3000o
f"C0Z m 400.044' ft04f.44N N4 ft m''0 4' .ft m'N a.I40v 4 .0 ftft O..00.' I.a. I 440 44..4 0l 4 0 4t x44'' 0,04.. ft fft
ft 0 0 0 4400fN O44SN ''lV4 &444P' ft0.f .0 0 0U4#- Off
* *** * ** . . t . . * 4 * * 4 * .* * *. * . . a . * .P. Id, -f ftft.4ft & a Ifft ft 01 ItIt . P wW% n4'ft ft-444 I"rn IVa xV P 39. -,a4
Z- 0 - 0- a - c L 0 0 a 0 10 0 L,0 .3(ee u100 Jo oc
on m m w' f- o- 4' f-a 0 N 19 .do~ 4~ 4v P4 f" v f. E ' 4 - .4 2,f* 0. n'N 4 N 4 0 -. -0 4' C. f" in .4 4. 4.ftO 0 &"- - P - f 0 ft P_ 0A N% #- P. #0 : 0ý 4 4N 4 w04'oft m'
4 on N. 0 4r 4 4'f ' N N, 0r a LA Q. N 0. a &OF-4 am 'A t A4 ' N
4' 4 4 4)N " 4 9 r4 4 4' 4 f N o. .1 00%op d4',0N Tvame Z*~F G. 4 M a, . S * * * * a be r. * ft It **.
A 4' 4' - P. 4 F. 4' -Z ft4ft 4 t . f4' & f4 g '44
O~~ - 0 @b -f - -l .. p -t -0 in4 - -4 . 400d
a 0 0 0 0 a 00 0 0 3 0 0 0 0 0 X00000 90000 4400004I I J, I 1 W A 0 I :* !UM 2 1 1 .A 4" J 6.1AJ
4 40 .0 ..0* f # 0. 4 . ft Z4'I 0 4 ft 4N 49 0 " ft P. in'O ton. 3 0 0 0Z. ft #- 4 in ' N .. 4 4 0 W 't 4 M '0 4 0 t
p- 4' 40 4 4 0 0 N I ft -a ' 0 o M'N..t 4 2f4 g 0.p. N m Zt 4' - 4X4 i a r 4m 0 - * 4 3 4 WI.ft4 U fe~t
* ~ ~ ~ 2 pt a a. * .0 *4 2** . 5
P.4
C~~ 00 000o CI 0 0 00 0 0 000 04 0 0 .00a a a a a * a a a a, I I T- -ella0 4 Nj N -4 a N ." 4 J, ft 0l 44 up ft 244 0f .f 4N0VA 0-D w Na' C 4' Nr Dflt efc 0'w.. a 0 . 0 0 fO f# N 40 &^ N1 a a' f 0 % oft 4 'r't4f o.40 40004
a ft 4' Na 0 4c 0 0 f6 0 0ý 4' a,4' 4 c--.. O f4.2t 0044 5 I -
-7 -P P a u* in a a a v*d * i-
on A f .4 Isft dot I & f 1 " .f
0 4 4 4 4 4 - 4 44 - 4tN ftCP - ffk 0 .4 U
U~~~~~~~P P.4 igg@ 4''' *'4444441- 0~4 ''0% 0 .4 ''4 c '400 w' 4.% op* '4404''I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~P .11",e44'Pg4p~.44 44444~00 %''40 '0
*4 A** * **4* * * ** .... 4 * so Who*** 4**9 . .Aw***
MI I ~ -AIL#44 '.AM ' 4 % p W.-4 WUW4A4IWJUM O, W40I 0% '4 '40 '4400%6 IM#WNJWJ Lý 4L W40 WWW MII0
0 .4 t" aC W` IA%44'''. U 6.4 4J0 4 *4444''0 4 * P.A.9. .0..IV Ot V 2 ft*S * ft ft ft** * *0 *0 V* C 0% In44 4 k" 0 *&X044 %4. .4'I.,%4EN0 E44illp4*4:% U.0*.e.
P~ ON ii d-i iI IS% -6558-oo#mo *--
a ~ .0= 'PSIP SUP M0%4'4..Z4'%.0 # 49 P-0: If 4 9 .4a b,. 4 V51 ~ ~ ~ ~ ~ ~ . P.''0 ''4 % % 4444..0 9 P.4440f44d0 00% 44 4%4'4@ %4P. 4 00P.
o 0 u444 4 P'44444494..04 w 0 @ 0 *%%44''0%4''0.-4
'A 0 p
00 00Oo *a... 144 400'r40 % 000.4 9 A%0 0%0%00644004'4;L% . f, 6 , u 0, wP44 'So ... 44t w 4 wt L. 4% 4 w I4'44'a4u..- v4* ** aA A* A A A .a .. * *
a41 A a . 0 0 0 U 0 0
do o
co a% 2.0 L 9- 4% v44 ft0 4 ft f-.
4 AO 0 0 a4 a% a 4% Q 19 0% ' 0 4% 0 4 .2 - - -4 -% -4 -4 a-0 a4 Q% a 0 '4 o 7 c, 4a%4 0 4 4 ' 4 0 4 a 04 0 4 4
* . A 4 * A * * 4 * 4 * * A * * * ow ow * AIS ~ 1*'4 0% 4 fts 4 % % .0 'a A% 4% 4 % % - - 4 0 - 0 % '
4.1~~ a v aFC9 op - fm 0% P. 0 4 0 r ft 0. 'n OR t4- - -- -t to-
,0~~~~ 0k 0 '0n00 0 0 0 0 0 t
w 4 P
70300 1000 p008 cc 00;00000000000000000eo" A4 4fi 14.
.40.-CE4 .. #. *MA:.0-. 4"f04-0~ &:^ NP 4 0 O 00 N 4 0 04 4 d 0 0*4.
P. I.4M ft 3.44 0 &4 *4 4 0N104 ft00ftN0044memoN00.4aNoo40
zo C30 ole
ad 14 o- Id c 4N40 t 444 0 0 0 . N 0 N 4 0 0 N 4* a . a . . I * ** *.8 . *: W 3 .0 . .,!. . . .,! . . . a*"".A.P. - 1X00.- - - - 0.4 M4M4M4M4. % 140w444 I44 V044 N4 .. ft- A.f
o04 1111111gI 9 41Iae 1111141x
fty r;; 1 1"4.. AJN . M4." 0% N N ft ft ft4M.M.M0.4.4.M4.4.4.M.M .4I"4^ 4.4.4^ A 42000 200 Z000O 000000000000000000000000000000 0
4 41 4P
.00CU0 0.0.40 90.00.0 b1 440000000 a 00000a0 0 0 400 Coco
ala~ ~ ga m s s-a
0 NaW -f p1 0 0- SC "-
*~ ~ 0r N J-4. 0 L) 4.0 0000 0' 0 0 0 0 ud Na0440 0~e.-..4 w. 0.0.. Wc c U, eoW0w W lN N O O NNw0000u.-4W...4,.0W0
Z00.t 10n08.8."M4 1 440ca-0 &09- 004 N 0 0 0 0 N 4 4 N N O 8 4 0 N
4L 9L4. Ud.14. -V 54 I --n A 4 a a - - -. do - A -r -e - -n
a~,U0 aOClJ .JOi0 0 .-%c 0 U 0 0 0 0 0 U 00 "0
# L.VoN I 08 O"Mr0 0A cr -r 4 li 4 0 4a 9 M 01 F. g.i8 0 0 # NN0 04 4 N - 4o 04 0 -4 N P. s 0 0 4
4000 4000d W040 0 4 N 0 . 0 - 1, -4 0 0 .1 -1 - 40.. . ... a* I. . . . . a In ft 4 a c
w We t . . .a : a . I . 0 ! ai * & * i
w 4w
::000 2!0000 !20000 C"00 00 0 0 0 0 0
& P 4 f4 -9 W% av P. toa0 a t
.n 8p 04 gQP8 aO-r P. (P pa0 0 N I 0f 0 0 0 0n N ft a 0 N Px C004 I" zxv0-f0a4 atN 4 0 4 A4 9 IV 0 'a N w N & 0 ft 0 4p 0 e
00188 40 4 .. 10 0 do NA 0t 0" 0 0% 'D vd . 4 4 Pd P'0 .. P.a w0 4 5 :O ' i v aa A. r - (P 0n *d 0 4 .
ZNN4 ZN 4 .4 0 80 02.- .4 -1N 1: . 4 - 4 .4 N N 4 N 58
'j -
Q00 t!00 0000 4?5. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
- I-- 'Aaa i- a j a aW k a A a a WfuN ft P-O (PO8 V. 0 1 p =8 ft 0 8 0. 4 & a4 .
,N N !04 *4 0 C8 ft A ; 0, 0- - 0 : " 0 0. N 0n 0 0(300P. I00.C ft80 t ON 00 di0 4 4 4 N N P 4 0 0 0
-- o Ca. I
493 0O0% 4 rd. ~0 A F. f" 00. M P. a oz 0 0a v on 0t 0 1. - 0 0 0 0 0 0 0 0t 0
INPUT FOR PNOOIA:
UNBALANCE RESPONSE OF A FLEXIBLE ROTOR IN FLEXIBLE, DAMPED BEARINGS
Card I Text Col. 2-72
Card 2 Text Col. 2-72
Card 3 (1015)
I. NS. Number o. rotor mass stations (!S 80)
2. NB. Number of bearings (:5 25)
3. NU. Number of unbalance stations (5 80)
4. NC. Number of coupling stations (L 20)
5. NPST. 0: Rigid Pedestal I: Flexible Pedestal
6. NMGO4. 0: No bearing resistance to moment 1: Moment resistance t
7. NGYR. 0: No gyroscopic moment 1: Gyroscopic moment calculation
8. NCAL. 1: 1st type of bearing data input > 2: 2nd type of bearlidata input.
9. 0: no diagnostic I: diagnostic given
10. 0: More input follows 1: last set of input
Card 4 (iP4Ei5.7)
21. E, Youngs modulus, lbs/in
2. Scale factor in simultaneous equation solutli
IF NGX - l
Card (I5, 1PE23.6)
1. NIT. Number of iterations in gyroscopic moe.
2. Convergence limit for gyroscopic moment calm
Op97
ROTOR DATA
If NGYR - 0, use only first 3 columns, FORMAT (1P3E14.6)
If NGYR - 1, use all 5 columns, FORMAT (IPSE14.6)
Give one card for each rotor station, in total NS cards
Rotor Station Mass Length of Cross sec Polar Mass Transverse MassStation shaft sec- tional Moment Moment of Moment of Inertia(don't tion of Inertia Inertia2punch) lbs. ipch in4 lbs. in lbs.in2
I
23456789101112131415
Rotor Stations with Bearinm Supnort
(1415)
Give NB items
Unbalance Data(I5, 1P2E15.7)
Give one card for each rotor station with unbalance, in total NU cards
.1. Rotor station number
.2. X-component of unbalance, os.inch
.3. Y-component of unbalance, oz.inch
98
Rotor Stations with Couplina
(1415)
Applies only if NC 0 0. Give NC items.
Pedestal Data for Translatorv Motion
([P6K12.4)
Applies only when NPST ;- I. Give one card for each bearing, in total NB card
Pedes.Mass Pedes.Stiffn Pedes.Damping Pedes.Mase a edes.Stiffn. Pedes.Dx-direction x-direction x-direction y-directic y-direction y-direc
lbs. lbs/in lbs.sec/in lbs. lbs/in lbs.se
Pedestal Data for Tilting
(MP6EI2.4)
Applies only when NPST - I and NMGI - 1. Give on card for each bearing,
in total NB cards.
Mass Moma. Angular Angular Mass Mom. Angular Angularof Inert. Stiffn. Damping of Inert. Stiffn. Dampingx-directi n x-direction x-direction y-directioq2 n y-direction y-directi,
ibs.in2 Ibs.in/rad 1ba.in sec/rad lbe.in- lbs.in/rad Ib.ain.se
9,
Tvye I Bearina Data. NCAL- I
Speed Data (1P3E14.6)
11. Initial speed, RPM"2. Final speed, RPM
3. Speed increment, RPM
Bearing Coefficients for Tranalatory Motion
(1P3E14.6)
Give 8 cards per bearing, in total 8eNB cards. Eqch card gives one coefficient
in the form: Kxx- xx,o +Kxx,l+ 2 C +C w + C Cx2w , etc.xx x ,0 x~lw+ Kxx, uc x xXo xx,l xx,2
Kxx
xxK •
xy
xyK
yy
yyKyx
yx
Kxx
xx
KYK
yy
yy
Yxac
yx
100
IL
Bearinit Coefficients for Tiltinit
(lP3E 14.6)
Applies only when NNMCI a 1. Give 8 cards per bearing, in total 8.NB cards.
Each card gives one co&efficitnt in the form:
M uxM xxo+ m t~lW + M 2 w 2 ,wDu'D + D wa)+ D xw 2, e tc
xx____ ___ ____ ___ __ ____ ___ ___ ____ __ ___ ____ ___ ___wD
xx___ ___ ___ ___ ___ __ ___ ___ __ ___ ___ __ ___ ___ ___ ___ ___ _
xy____ ___ __ _ ___ ___ _ __ ___ ___ wD
xy
____ ___ ____ ___ ___ ___ ____ ___ ____ __ ____ ___ ____ ___ __
yy
yy
uDyx
xx
xx
u xy
yy____ ___ __ _ ___ ___ _ __ ___ ___ wD
Yy
yx
101
Type 2 Eearing Data, NCAL > 2.
Repeat the following input as many times as given by FCA'.
Stpeed Data (1P4EI4.5)
___ Speed, RPM
Bearing Coefficients for Translatorv Motion
(MP4E14.6)
gK ,.wC ,K , C___ ___ __ _ _ __ ___ __ __ __ ___ __ __ __ ___ __K ,.•C ,K ,a•Cyy yy yx yx
K , c ,K ,uCxx xx xy xy
_K ,uC ,K ,wC- yy yy yx yx
Bearing Coefficients for Tiltina
(lP4El4.6)
Applies only when 101(M-l. Give 2 cards per bearing, in total 2*NB cards
M xxaix ,aZ ,WDxyMyy UDyy 'Myx Q yx
Mxx aD xx M.xy, wDxy
_ yy__C_ yy,__ _yx_,wDyx
102
S• ., _ _ I I I I i
SAMPLE CALCULATION AND INPUT FORMS FOR THE COMPUTER PROGRAM"TýE STABILITY OF A ROTOR IN FLUID FILM BWARINGS"
103
-' • III I i ij
IJOB095.5000 65-3419FREEMANtSESCDSEXECUTE IBjosSIBJ:)g ~fVRAN M~APSItIFTC ROTOR2 4~94#XR344'6 DECK
COMMON X~t.30).XS(9.30),YC19,30),YS(9.30).DMXA(30) .DMYA430).DV u010IXA13019DVXd(30),DVXU(30).DVYA(30).DVrb(30;.DVYC(301.OVYD(30).OVXCI vi02u230).4m(30),RL( 30).RS(30I .RIP(30) .AN(30),8N(30).ON(30). 033 LB( 10),dgXX(10),BCXX(10).8KXY(1O),8CXY(10),8IKYY(101)WCyYY( U0,4043O).8KYX( 1O).RCYX(10).PRMS(1O)'BR~FR(I0),FRQ1(1001.FREU(1f0i.l PMX( 0050510) .PMvflQ) ,PKXU10).oPCX( 10) .PKY(10) .PCY(IO). (.FM(8*8) .ENtl9) U0606,CLNR(8).932N1(b.8).t32NP(8,~l8)t2N2(8.8),CF9,FRW.DT.DR.DE.ENGY1.d2l. U070702P98~22 0080
260 READ (5,100) 0120READ 159105)NS*NB*NFR*NCAL#NPST*INP U130READ 159103)YM 0140WRITE (6.1001 u1:ý0WRITE 16910d) 0160WRI TE (6. 110)NSsNd9NFR9NCAL .NPST9INP ii 70WRITE (69104)YM 4dWRITE (69111) 0190WRITE (69116) 0200DO 201 JzI*NS 0210READ 15,103)RM(J)9RL(J) .RS(J)9RIP(J) 0220
201 WRITE (6. 114)JRM(J) .RLIJ) .RS(JI .RIP(JI 0230READ (5.118) (L8(J) .J'1N8) 0240WRITE 169120) 0250WRITE (6*1lb) (L8(J) ,J~lN8) (J260IF(NPST) 20892289208 0270
208 WRITE 16.1281 0260WRITE (6,127) 0290DO 209 J1.9N8 0300READ (5.125)PMX(.J),PKX(J).PCX(J).PMYIJl.PK.YIJ).PCY(JI 031aKST=L8(J) 0320
209 WRITE 16.126)KST.PV.X(J).PKX(J) ,PCX(J).PMY(J).PKY(J),PCYIJ) 0330228 RSAD (5v103)lFRQ1(J)9J=l.NFR) 0340
W'lITE (6.101) 0350WlITF (6.103) (FRQ1(J) .Jx1,NFRI 0360
C CONVERT INPUT UNITS 0370250 A4S81000.0 0360
C-1a386*069*AMS 0390RS(NS)*RSIL) 0400DO 251 JU1.NS 0410RMCJ)xRM(JI/CFI 0420RIP(J)*RIPIJ)/CFI 0430STFaYM/AMS*RS( J) 0440AN(J)aRL(J)/STF 0450SN(J) aRLI J)/2.0*AN( J) 0460
251 ON(J)zRL(J) /3.O*8N(J) 0470229 READ (591031SPST95PFN.SPINC 0480
WRITE 16.141) 0490WRITE 169103)SPSToSPFN*SPINC 0500DO 204 Jz1.Na U5 10KST-LB(J) 0520READ (5.103)RKXX(J).BCXX(J).RKXY(J).eCXY(JI u530READ (S.103)BKYY(J).I3CYY(J) ,BKYX(J).8CYX(JI 0540WRITE 1691211KST 0550WRITE (6,1221 0560WRITE 16.103)nKXX(J) .OCXX(J) .3KYIJ).8CXY(JI 0570WRITE (6.123) 0500WRITE (69103)RKYY(J).BCYY(J) .4KYX(JhtRCYY(JI 0590CIuRCXE(J'I+BCYY.tJ) 0600Cl. (fKXX( J)*iBCYY (J) +BKYY(J) 'HCXX(J)-8KXY(J)*SCYX (J)-8KYX(J)*8CXY(J 06101) )/C1 0620
( *C? EI.) I tC vyIj -dcE y j ~ ( X i U63U
IFIC21 1060105,205 J65 0205j C2-5S2TgC21 ,b
ClC2U6 10GO TO 207 068V0206 Cla-I.0 06900207 '41ITE 16o1241 0700011TE (6*1031C29C.C4~g U720
OJXA(J~aC]
0730204 B-IFRqjauC3
0740MdR8o
0770214 KST*O Q7MBOmS*esq,
1
0190FRWwFR01 I ('FRj 0 0800DO 211 JvI#N8 0810CIaOVXAIJi 0810IFCIC! 213%211,210 0830210 IFICI-FRWI 211,212.211 0830211 FRW*Cl 84WFRa*q.. 1
0860212 KSTaJ
U870213 CON4TIN4UE uIFIKCST) 2169216,215 ou215 OVkA(I(sTIs..*o 0990
216 FREQIMgRIUFaw 0900I~FMFQI P17.21a.218 0910217 mFrpe 02218 MlFR84Fq~j
0930IFIMFR;P4FRI 214*214.219 0950
219 C2a-oo
0'950KSTsO
096000 222 J- A).t - -0980Cl ,DvxAtji 0990IFICI) 2229222*22 10900221~ IF I C-c,, 222,222,221 10100221 C2uC1
1010KSTvj
1020222 CONTIN~UE
10-TO1FfKSTI P749PP4,221 1050221 MBRaM9R.l
1060FRrof~M, laC2
1070OVXAfKST3U...1*
1 0?001~ TO 219
1090224 t4'RlueqB
11090S'CAL .SPST 11
22S WAITE 169146IPCAL 1120A lSPRuo. lO4?j9?6*SwC&L 1130
WQITE (6.1481 1140DO) 226 J-1.NS 1160
KSTO RIJI11 60CloPR~sigj,,c2
1150* ClG8RFRfJ)#SP(.AL
1190226 WRITE 16%11lKISV.C3.C, 12900WRITE i6swi? 12100KDC*( 121
C ýFROI1220FREQUENWCY DEPENDENIT IDARAMETERb 1 230300 FRWaFREoq(epR,
1240
'~-1250
1^ .Jz*19N, ld /0'TP.4a~j~vN<P212a0")'-'XA ( JI - ' T F-12 9 C
fPYYA fjl .sr 1300
":VXA ( J I ., TF 1 3.eo:VYA( Jj S rr 1 33Vj-'VX,3(J)*O.O 1 340r'vxcLjI.O.0 1 3207'VX- I Jlao.Q 1360O~VY~f J o.Q 1310EDvYC jl0.0 1 Jd0
'(,I )VYD(J).O~.O 1 3iv302 ýýO ill j~l,-4t 1400
K.STmLQ (J) 1410CFI<:=.rxxIj) 1420CF1C=3Cxx (J I'FR'A 1430Cc'I :ýX(J I 1440CCIE."Cxyfj*~ J 4ý0C`2K*R<YY J) 146UC02CzRCYy I I~ 1470C '21)-3KYXIJl 1400C-:2Ea'f3Cyy f j) *r,:j~ 14VOIF(NPST, 303*106,301 1500
303 Ci IM-PKX ( J) -0.X JI 3206*.'5CF19IzPCXIJI.ANSCo 1520CFIA*Cr1IC.CFlv 15430CFII3:=CF C+CF1N4 1540CF2M=-PKY( JIpMy(J2'386.069gA.'iSP2
1560CF2II3PCYIJ)*AN-S0 127UCF2ASCF2K.CFi~v
CF~tdzCF2C+CF20q~vGO TO 307
10306 CF'-CFI< 16100CF2=CFIC 1620CFI.Crll) 1630CF48CF IF 1640
CF6-CF2E 1600CF7*CF?( 66CFBSCF2C 1670GO TO 308 1 6o0
3V7 CF4=CF2A*CF2A+CF2.t*CFi- 17900'F':I(CF2A*C6'20,C;2~.jCFL )/C.F4
1710CF2 I CF2A*CIF2F-F2I0.CF2DI ,CF4
1720CF 3 8(IC2A*(VF2m+Cr~aCF2u4 I/CP4 1130CF4aICF2AECF2M..-'F2I.CF2%IIC.F
4 1 7411CF~uCFIA-Cplv(F1')-4Cr?*CFIE 7,CF6sCF113ý7F2*CF1ýF11..CrII. r 1760
CF72-Cp.F3*C~lnCF4*CFIE 1770CF--F4CF)-F-*C* 1770CF2N*4CF5*CFS+CFJ*CF 6 1 700CF2AsfCF5oCf1M+CF6.C~'l.C,,C
2 - 1auy,C ý2 *I C F 5OC F I 04-CF 6 *,s 1*4hIC F2, NuCF2vwICF5,.CV 1CF 5OC.Iit)/CF2N 01
C C IA a CF I OCF 2 A~ +C1, C ;2. lasuC c1 Cr I *1CF 2 tj C F 2&Cr 2 A kiboC'1vsCF 3-CF I4eF2v,.~F2*Cý PN 1850
18 70C'lC4CfF2-F2C2 ~ Ilac
CF2aCF14eCF~o.CP1COCF2A-CFIDOCF1I3*CFIE*CFIACF3.CF1'(OCF2m-CFlC@CF2N*CF1O*CF1I4CF1E*CFIN 19'03CF4UCFIKOCIZN.CPACSC-F2NU1FIOCFIN*CF1E*CFIM 1910CF~,uCF200CF2A-CF2EOCF2d*CF2KUCF1A*CF2('CF15 192V
CF6.CF2DOCF2j.CF2EeCF2A-CF2'(*F1o+CF2(*CF1ACF ?-tF2DDCF 2,4-CFZE*CFN+CF2KOCF I o-CF2COCF IN 1q4J)
CF-FDC2#FELZ+C2#FNC2*FM15308 DVXA(KStloDVXAIKSTI-CFI/AMS 1960
oVXe(KST I&CF2/AP'S 197clC'VXC('(ST P uU3/AMS 1960DVXD('(ST I CF4/AMS - 19DVYA(K'ST I DVYA('(ST l-CF?/AMS 20100flVYS(KSTI* CFS/AMS 01
DVYCI'(STIRCFS/AMS 2020
DVYDE KSrI UCF6/AMS 2030
311 CONTINUE 2u'#Cc ROTOR CALCULATION 2050S
DO 428 1.1.8 2060'(STmI 2070BMXC*O.O 2ab0
IBMESB0.0 2390
B NYCOO * *0 co
SMYSOO.O 7110VXCBOO0 2120VXSw0.O -2130VYCwoO. 2140VYSM0.O 2150XCII .1 .0.0 2160
XSI #I eO.0.-2 r7CaYCI 1.1)mO#O 2160YSlIq911.0. 2190D'ICUOSO 2200
f)ISWOOO2210D'cwo.o 2220DfSaO*O 773T0G) TO l40?o408o4O9o4t0s4lZ,4l3o414#4lbI PKST 2240
4fl7 DqC@1*O 2250GO TO 418 2260
408 DASsl.O 2270GO TO 418 2280
409 DYCw1.Q 7770GO TO 418 2300
410 Oyso1.O 2310GO to 418 2320
412 XCIS91I*1.0 23306O TO 418 2340
411 XSI69l1ul.0 2S33O0O TO 416 2360
414 YCf701I18.0 23706O TO 418 2360
415 YSIO.)laoo 239041$6 00 424 JaelNS -2400
8INXC.8'IEC.DNx*Ilo .941C iTBMXSU88IES.OMXAIJI*DNS 2420SMYC.UP4YC.DMYAIJl*OYC 2430*NYSuw9VS*OI'4V4fIJ DYS 2440Cl.DVXAilJ*XCgI.JI-UVXe3IJIeESI1.JI-DVXCEJI*VCII.J)-0VXDfjI*YSII.ji 2450
_ C2.DVXSIJIOECII.Jp.OVXAEJIOXSIIJI.OVNOIJIOYCII.JI-OVXCEJI@YS(IJI 2460C3.-ovycijIOxcfhjI-o3v,..EjIoNsiI.jI.OVVA(j)'ycI I*JP-DVYfsfjlOysf.J *747011 2480C4o~vYD(jP.XC(i.Ji-0VYCIJI.XSII.JI.OVY5iJ1OYCII.JI.OVYA6JIOYS(I.JI 2490VXC-vxc.ct 2500vseuvxsC*2 Z310
1 VCzv V C C3 25,eu
422 l7(~45S-J) 424,424,9421 2540
421 XC119J*.1l"C(19JI*RL(JD'DXC .t3N(J)*BMXC *DNIJ)*VXC 2 550XS(lJ+1IESX(!,JI4RL(J['0XS +81% 1J) *fxs +ON (J )*V XS .2560
Y'(I.J.1I)2C(IoJI+-4L(Ji'eDYC *BN(JI@BMYC *DN (J I VYC 2570Y'f,(IJ+1lavS(IJ)+RLCJ))*CrYS +4N(,jI*BMYS *DN(J)*VYS 25d0
~'XCx0XC.AN( Jl t3MXC+8.4( J)*VXC 2590rNSrX+NJ*BX*NJ*X 260001CaDYC*AN(.J)*tiMYCA+,,4IJI*VYC .2610
'YS-DYS+AN(J)*B~MYS~tJNIJI*VYS 2620
El *XCzf!YXC+RLC Ji*VXC 2630
BMX5S,8YXSRL (J)*VXS 264063MYC*8MYC4RL (JI 'VYC 2~
B'lvSI3mYS .RLfIJ)*VY5 2660424 CONTINUE 2670
CF4 (1 t* :BMXC 26o0CFM(?.I )zBMXS 2690CFfMI3,I IBMYC 2700
CFY(491 )z93vf5 2710CFM(5,I I2VXC 2120CFY(6,I )%VE' 2730CFM(7.I (=VYC 2740
CrM4(99IzIVYS 2750
428 CONTINUE 2760CALL EQS 2770
506 %IFR~mFl41 2 7d0
IF(MFR-NFR1) 300910G.507 2190C. ADVANCE SPEED 2800
507 S;PCAL=SDCAL+SOINC 2810IF(SPFN-SPCAL) 51192259225 2820
C PROGRAM END 2a30511 KDCs(')C+1 2b40
IF(NCAL-I(DC) %1095109229 2850
510 :F(INP) 509*2009509 2860Smq~ STOP. 2d7bICO FORIYAT149HO 28to0101 FQRMAT(1IfHOFRFll.ikkNCY RATIOS) 2d90103 FORMAT14(IXE13*6)) 2900104 VORMAT(I6H0Yu',iNG5 A00ULUSus E£14.7 1 29101rn5 FORMAT(7151 2920108 F')RFAT(Sam0 T ATI IONS NOed3RGS. N0.F(4LO NO..SPtED PED.FLEX I le930
INPI 2940110 F')RMAT(4XI496(6XI4.) .2950III F')RMAT(14HO ROTORI DATA) 2960114 F)RMAT(4X14,7XE13.6.IXE'I3.6,1XEi3.6,1lXk13.6) 2910
116 F)RMAT(67Hl STATION NO. MASS LENGTHi CHO.StCT*INER 2960IT IIP-IT)l 29v0
117 F )RAAT(4X 14s7XEE13et,1XEI3*6) .30U00
118 FO)RWATE I0EIXIAI 3 3010
120 F..ýRMAT(18BHOfEARI'4ý STATIONS 33020121 FOfO,-AT(24,ium t'ARIJN At STATION NO&9I31 3030
122 FOR14AT152-0 KXX CEX KY'ix CYJ 3040121 FORMAY152Z40 (V YY (cy YX WCYX) 3050124. F 0qNA T ( 43 b4 I4ST.*tRE3RT %4ASS*(F4F0l**2 WFIGHT*(W)*411 3060125 FORMAT16( lxEll.43 I 3070
127 FORMAT(76-0 d*G*!To 'ASS*9-')IRs EX CY MASS*V-DIR 3U90le KYcv, 3100
- 126 r0RMAT(I5HO P-DESTAL ')ATA) 3110
141 FOR-AT(42H01N!T!AL SPEED FINAL SPL&.P SPEED INCReJ 3120I140.~ C3RAT(IlIR 1ft(. SDEEI~ug E13.693"R3PMI 313U
147 rOR.uAt(60,40 F4E'.*RAT* DLETEMINANT REWLLT) IPiIDET) 3140
148 FCRVAT142H0 diqG*STATION ItjsT.IIRE^9RP" I %SToE I GHTI 31b(E Nr 31 7C
SIAFTC.ýEOQS V194XR7SUAR(C.IT 11E O1 944004 xC (9 o I XSt 9 30 1 9 YC f9 v30 ) 9 ) 9301 %DMXA (301 *[,MYA 30) 9OV U3ýc
!XA13019.3VX i I3el) 9DVE0( I0 J I9rVYA 130) orVVB 430) 9OVYCf 311 ,VYO(f3uJl,0VXC 4~6.,230)91'41301 RL f301 *4 %t 30 ! *Q I P 110 1AN I 3DOIN( 301 *ON( 30 1 vk .I,I Ld I 10 1 98KX X10I 19rXX i I U I 99 9Y I lUI ,BCX1 ( l),I YY ( 1) .tiCYYI f .sjk
510o(PmyI1OI.PVXf10I.PCXE11O.')KY(IOIPCYIIOI9 LFM(8*819ENT191 .aILo6 9CLNR (8 1 o 2NI13981 9 6l32P 18 *a I 62d (8 ed ICF9 9FRWo #!jot 9E oEttjYl 9021 # .0 11-792P82 .132?PRN3.1 .0 013C
FNT1III 0o0 0150r") eSO Jules 016C
Dr-S';RTIqRJ30iT1 U I0.Y), 0200D)ACFMI 91, 0210tnEuCF'4I I * 0220
ds1 nj dsl I161#10 0230
92NI1 J1. CFW( I .J) 0250821IP(IojUCFWII.Jl 0260
aS2 8?N2(I*J~oCFMjIIJI 0270853 02SPqI.v')IwCFV(I.MD+1 0280
CALL %4ATINVI,?NI1P.V)*E%4T#9 r.,2II 0290CALL WATINVI82NP9MW)vENT#IUZD, 0300IFimn-?j 685.a04*85 0310
854 ENGY*1*1141S927002P 032045 '400M0-1 0330
CALL WATINV~ti2N2oMJ9ENTI.9*221 0340
F321*42'I1322 0360
C1.92f32 RO-12P.DE 0370nF.82q100F+82p.)R 03tsODRECI 0390UD.40.. I 0400IF100r)-31 b6o8SI98Sl 0410
I56 WRITE 16,UIOIFRW*OT .DR DE9LN(V1l 04.20RE TURN' 0430
860 F0RMATtISI1EE13.611 v.44vEND 045 0
SIBfrTC SMATI'4 *4949Xq7SV9ROUTINS MATI94V gAtNedeme0ETER1 uolo
C MATRIX INVERS10'% WITH ACCOVPANYING SOLUTION Of LIN~EAR EQUATIONS U020DIMENSION A(864 1961(9 )oTPIV3(9 19PIVOT19 1 1430DETER *1*0 =00~DO 17 JujP4 0060KC u0 0070K R. coodoDO 14 IsI.N U090
11K':G1 011012,If1AfJ9I)I 13,14.13 (42013 KqeI Qulo14 C)NTINUF (1140
f 14 DcTERP0.0 017c
17 CONTINUE
0DO 20 jai,%~ 019020 IPIVOij).O U190
D O 5 5 0 1 2 1 .N 0 2 00
W220C SEARCH FOR PIVOT ELEMENT
%J230
DOAX105 . j u2b0DO 1~Jai.M0260IF UfPIVO(Ja-1) 60.105960 027060 00 100 KA1.N 00290IF (IPIVO(KI -1) 80. 100. 800 203080 IF 1A8SIANAXI..A8SEA(J9K1I, 85.100,100
030085 IR0WSJ 0310ICOLU RK 0 330AM~AXRAJ*K
U30340100 CONTINUE 0340105 CONTINUE 05IP!VO( ICOLUI*JPJVOI ICOLUI.! 0370c 0370C INTLRCHANGE ROW~S TO PUT PIVOT ELEMENT ON DIAGONAL u3 90C
IF IIRO(V-ICOLUI 140. 260. 140 00140 DETER -~DETER 0410DO 200 L-19N 0420A%4AXwAf J10W#L1 13AC IROW*LCSA( ICOLUsL) 44200 A(I1COLU#L*AMA,<
0450AMAX*(.IIROWj 0470bfI ROW~mdg ICOLUI J47008f JC0LUJ.AMAx L460260 PIVOTU;ZsA(IC1LU9ICMLU) U4900DE.TER xDETEROPIVOT, ~i 0510C 01C DIVIDE PIVOT ROW 'dy PIVOT ELEMENT 0520C 03A( ICOLUICOLU)41.0. 0540'ýD 350 Lxl#N U.1.7350 A(ICOLUvLIUAI ICDLU.L)/PIVOTiL) 5661 ICOe.Ulxd( ICOLUI/PIVOT()
U57C 0500C REDUCE NON-Pivot .RowS tj600C
06100383 DO 550 LI-I#N 06160IFI1.ICOLU) 4.009 550* 400 06204M.O AMA~e2ALI.1C0L(,; 0630A(L1,1COLJ) unto 0640DO 4.50 LxI$N 06604.50 A(LloL)sAILI.LI..A(ICOLU*)AýX 0660
8(L1xd1IJ-(ICOu)*wAXU610t50 CONTINUE 6*600 RETURN
ii700
- ENn OF FILE
TLST .At1. JN3YMNI4l-4IAL QCTJIR9 1 4 1 0 1
fM. 3O0000,) +08ri,1e20ýEoj 0.370")00t.31 (l.biuDfOE*01 0.114200E+030q!J10rnE.OZ 00775000L#31 fl.:o'1oot+oi O.de.O40c*O3Me4'2O000E+02 0*dlO0Oc.'310 0qd3~')C0Lt1 0*000fLOOO..Q3.1-elOOD0L.2 0*6l1OOOL.O1 0*bV2Q03L+3l O.OOOOOOL+OoO.391000E*02J CeIM4100E+32 C*I0iI00E*0l O.15d.000E.O3
0 3t/dOrO~j.02 0.*7?'O0Ot.01 0~9692000E01 09000GOO&L-000*IdlO00L.01 Os425000k,01 Oo6VQ000I0 0O*OOOOOOE.OOn*212.IOCE.OJ ooo.vo0oot.O 09030O0.00E0 Colb66ojot~u2 a
050.lO'n6E.oo 0o~qbGo3E.0o n*%'vlnOfrO O.'.dOk00to0*b3VSV1PF.04 0#850000L+04 1*200000t#020*12k )40E+06 0.1I *035oEOL*0: o.1A710t#oe O.118a30E*ObOe.108 161)E.06 0*181920E*O6 -0.aa4d6OE+Ob 0*1102304+06Oe f.78 #20t+06 0.?113d 30E**06 0. 3 e9'0E0b 0q1~d120L'Ci60.12417OE+06 Oo92470OGL*J, O.1~vJOQL#35 (i#57o2QL+O6
E-4 OF FILE-E,4rn OF FILE.
00
0-0
000 0 0
00 00SNO 00
* ~~~ 40000 00
06 0 onfi x 0O . 0 0 -0 . 00 uF f
a-v-04 3bN
*~~~0
so-C00 0wft N u
m~n N N0. LA UN F' w 00000000 0 *.1 0 0 0 000 W W -A . -A
I :z -1 0 9 39 ty ac
NIV 0 w00
on IO X4MCNO bW O~~Z 3' M ;3fm~~~ ~~~ owo yNA t0 0f
I' .00 ZNU.0: . 0 b- Up00 0 0 0 Sl "O0 3.0
aM O O oooOR2 4 0.00*
we, *m# ). 4L0 0 0 ; 0 v 04p *A 0 0
N C;
*03 C; 1
-to 0U 0 0 9r
0 AO~ 0 0 ML . 0 two"O fF
0 -I 0 d0 Cor ~0 0 0 42 IF' UP 3'' 2 . N 51
Oti * o 0 42F... 0 N A 40 20 b..P
2 6 A e a '
Afe~~' 04zIa.~4 A 4 4 30 q 3-0
sUl kI wU Iu usr N"I N 40
I, - C N 0 so
emI-N -4 E .* * C* ro
A.- A--. - -
o ~ 4a a% 4 ry
-0000
I A-O o In 0-
.N 4 OA 0 0 ,
0 * * 0*
C C eNow
U, Mi ii JM
r- N 10 In N0.i
.im-h '0' P4Z i .i 0' i
Mi N a N -, c
0To00
et V.ff 0 Mi
04 N4 a4NJ
-a000040
* O
49 ,* *u P a, 0
0 4 0 W0 Ul.It
- 4
'A
,. w* - �flN '4 '4 -g N '4
�m,,.m, mu mu m�. 13 4 4 3
0 -� j*�e� - -. .* -
A *' - � � - �74 '2 4 .CuiN � - - 4 -
* . . . . .00(.00c''S
U. .13* - .13 4- .- - - �. -3
-. awwu .J .3- m� � - .i. e -m'.1gm - � .3 .4 "C � .�'- - p. - £ - N� ?¶ .3�'A.?
eec 3�1'*IS.
TOO �
� -u LJJ .J'.*
�*J3'
� 9 .2 9O'J. . . . . .
& '
0 � -- *.1 * 4
.�e --- y03 h
3 '4
-� �W �P 'N 3.'
� �e*� p.NC ... t -
* . . . . * . . 3'& �-O -. -
C
7- elmS
-. - 4 .- - - -. .3 � C. 9
� 4
4 3'.J u..ISU.444
- if . . . 3' . 3'0 �'V
W LU WA w w I
WNf a N = -0i
*w em f
-NO&
! NE 0 O
U0 m *. P . 9:
000000
W14 - - W W WuW "
IL 0 a 04
00m O 0 00 000
o, W-6O. 0 rAgo ASASC SA
U £00 7-4~.....
* cocoais WJ 7W ~ w
INPUT FOR PNO017;
STABILITY OF A FLEXIBLE ROTOR IN FLUID FILK BEARINGS
Card I Text Col. 2-49
Card 2 (615)
1. NS Number of rotor mass stations (< 30)
2. NB Number of bearings (< 10)
3. NFR Number of frequency ratios (5 100)
__ 4. NCAL Number of rotor speeds
5. NPST 0 Rigid Pedestal I: Flexible pedestal
6. 0: more input follows I: last set of input
Card 3 (lXE13.6)
1. E, Youngs modulus, lbs/in2
ROTOR DATA
FORMAT (4(lXE13.6))
Give one card for each rotor station, in total NS cards
Rotor Station Mass Length of Shaft Sec, Cross sectional Mcent Polar-TranaveiStation of Inertia, in 4 Mass Moment ot(don't Inertia 2punch) lbs. inch lbs.in123456789101112131415
117
Rotor Stations with Bearingt Supp~ort
(lO(1X14))
Give NB it~ems
Pedestal Data
(6(IXEl1.4))
Applies only when NPST - 1. Give one card for each bearing, in total NB cards.
Pedes.Mass Pedes.Stiffn. Pedes.Damping PeJes.Mass. Pedes.Stiffn. Pedes.Dampingx-direction x-direction x-direction y-direction y-direction y-direction
lbs. lb/i lbs.sec/in lbs. lbs./in ibsesec/in
FREUEFCY RATIO VALUES
(4(IXE13.6))
Applies only when NFR > 1. List as many values as given by Nfl, 4 values per card.
118
BEARING DATA
Repeat the following input as many times as giver. by NCAL
Speed Data (4(IXE13.6))
1. Initial Speed, RPM
2. Final Speed, RPM
3. Speed Increment, RPM
Bearing Coefficients
(4(IXE13.6))
Give 2 cards per bearing with 4 coefficients per card, in total 2 NB Cards
_________ ________ _________K ,wC ,K uC
xx xx xy x)
'__K ,WC ,K ,C'-y O Y,•yy, yx Y
Kxx ,'C ,K ,y y>
K , aC yy ,Kyx ,4.C y),
119- t. ... aI.. .5.. t--- - - ' l t A A
DOCUMENT CONTROL DATA- R&D
t 'OIA I , f *1 xe>i- •,,!.r N,V° I. . Cr ,f.r n-... .,,t•r, ;. .nf r~oy nfl.,. .on •r. ¢*f!, c.. , a l$'-.
;2...SEC.,- 1 C&AtC';'.•):n.ci; i ,",hn I ,•fy i n -',rp~rated 1 CL :;2 .... f[jI Abx...iv. . " ,Z bC ~r?
Lat , Nei York 1)i1jU
r-1 'PORT TITlt
lICTOR- 9'-..AR I NG D YNAVTC"' Dt,.'I:IGN TECHNOLOGY
P'•rt V. (Gornu.t.r Prodgram Manual for Rotor fte3p.onse and Stability
4 DESCRIPTIvE NO0ES (Ty"' .o .OpEo od Ochm" d•ee.)
Final Report for Period I April 196h - I April 19655 AUTMOR(S) (Loot atmn.. fI,.t Se..e. Ori.Wo!
Lund, Jorgpn d.
* REPORT DATE 7. ,or,,, No. or PAggi F b OS Oa RE, Alp
r4y 1965 6SCONTRACT OR GRANT NO AF33(615)-1895 10.. OR.,.GNA, ., tORo'str ,u,,wn
6 P-ojcT NO 30hL M'I-65TRl5
'Task No. 304.402 . g].m.•,.PoRl No(, N A.' .,.Any S *A' ,m,,.,y be ..t.e-e,
d AFAPL-TR-65-4s, Part IIn AV A•IAILITY/LOITA-TOOKNOTICES Qualified requesters may obtain copies of the reportfrom DDC. Foreign announcement and dissemination _f this rercrt by DDC is notauthorized. Release to for, n nationals is not authorized. DDC release to theClearinghouse for Federal 6cientific and rechnical Information is not authorized.
I I SUPPLE MENTA111Y NOTES IS. SPONSORING MILITARY ACTIVITY
USAFRTD, Air Force Aero Propulsion LaboratoryWright-Patterson AI. Ohio h5k33
ig AUSTRACT
This report is a manual for using the two computer programs:
1. "Unbalance Response of a Rotor in Fluid Film Bearings."
2. "The Stability of a Rotor in Fluid Film earings."
The report gives the analysis on Sihich the programs are based, and theinstructions for preparing the computer input and for interprefing the computeroutput.
DD I;O!,:... 1473Seadty Claseificadew
a LINK A LINK 8 LIN4K C
jon ca tiona..! Filmi ruiyn i mi c
[1i'or-fl~arina DJynamdc3i'.-iblitit
Turbuie~nt film
INSI UICTONIII.ORIGINATING ACTIVITY: 3mw erthe name and adrs inposed by meeauty classslticatloo. uing atmadmed statememeaofthe contractor. subcontrtactor. grantee, Department of Do. suich so:
tons* tictivity or other organization (corporate aufh~tJ iselilaS (1) "'Qualified requmestes my fiiaini copies of thisthe report. report hrom DDC.112s. REPORT SECURITY CLAWSFICATION: &me the swap (2) "Foreigna smovirscmou sad aissanmlualos of thisall security classilfication of the report. Ini~cate wbelhse reen by DDIC Is wet authaetrd."""Restricted Data" is included, Marking is to be in occardonce with approprsote secuiwty tr*g~siatons. (3) 1U. L. Caviiinassd agencies my ohieala copies of2h. GROUP: Automalic downgrading is specified is D* Di. this spot9 directly hasu DDC. Otba qWlleid DDICrective S2%(. 10 and Armed Forces Industrial U11atiaL. Eate users Pholl ire1031 threatolb. garoup number. Also. when applicable, show that optiontalmarkings have been used for G'oup 3 aend ;roup 4 as autho.- (4) 11U. L milwitar agencies my sbtain copies of this&Red. a"of directly ban DDC. Other ilalifled siesr
3. REPORT TITLE: Eunst the complete repot title is all aOal request throutocapital letters. Title* i" all cases should be unclaetsifted.It a meaningful this caniot be selected without clagalfice'lion, show title classifffraloa in all capitals in parenthesis (3) "All diatributie. of thise irepI is Converlled, qoalImmediately following lb.e Usl. dieid DDC ko5695 shall reioqua Iheguh4. DWSCRIPTIVE NOTE& It appropriate, enter the type of-report. e.g.. Ilmenm. progress. summary. anfi'sal. or tinad. If the repeat has boen fiwoaihad to the Office of TechnicalGive lb. inclusive dates when a Specific roportlag period Io Sanwicu. Dopartatmoo of Cameece.r fisr mil to lb. pulic. indi-coveted. cate this fact and mew the price it heiaawS. AUTHOR(SX Emirtheb nam*s of eulhat(s) as sdow on I L SUMOLZMENTART 111TIM Use for sddthiaeal e*Woo&or n tif- report. Enter lost name. first name. middle IitiaIL. two wieIt rmilitaa'y. show raiu -. I be..........# Tb. a" oth, principal . -hat to ask absolute minImuma requrmesaL it ir. -~iAL..~A~' - .rt o'~m f
we iteamota Proete alliceof0 loeabs w speovseef (per-6. REPORT DATE. Ernt. lb. dafte of lb. reor as dayt. lud f-or the researchs aad dotalopenm laclude addrena.month. year. or weigh, yeas9 It mowe thae ami daftsappearson the report. use date of publicatios I3. ABSTRtACT: 3318, as oshabact giving a brief mid factual
OP PAER Tb tota pa cmin misar of the document isidicative of lb. " r seats evnhemeb7s. TOTAL NUMBR O PAE Thtoapls m it may alIe appear elsewber, is die hady of lb. tochiatm vo-should follow nomal pagination pnoced-mea. La.* ~io the lt speIarqld.actamlssetshlnumiber of pages Contaalinig loformatlea. ha~l atta t PSOI@ched. Otitti bm &A
7b. NUSbalt or RapaRaNcza Matter the total mokrO Itls hioly desirable that the ahaoc of claesifted repabeereferences Cited in the report. ha unclassified. Each patiolpap ad the abdatimactW shalod withSe. CONTRACT ON GRANT NUMBER If appreprisse. sdam OR iadicatim of dide milliaiy security classification of lb. is-the applicable nmiber of lb. coatract or arainvoider whilch faution, in the parayap. Kepaseorned as rs). (a). f c). .. (u)the report was Ulilla.t Twose isi ndime ba asa~ en lea sof0 ahsesect. MEW.lb. or. 6 04. PROJECT NUMBER latinerb th ep e e the 1111141eobted koak is bar 19 to 3,2Sw~
SWNI11.0I~dir ytrtaimes o ibo et 14. ? 3~ ES Us-ly ased anos. aheleelly nmaaio tar9s. ORIGINATOR'S RZPORWT WMSinm(S)fat lb. the- lade aee for I'm le C itheo respor. SeW madeSO m eatciall report atmiber by which tb. dacumalt wUil ha ideninfied mInilec"a adott as aecrity clmasacatuam is iqaisqbdl. ldine-sod contolled by the orihinating activity. Tha s n mbe nw - -ie, suc mi sqp model desaigtmr. fkadl sams. SLIM"be maique to this irepast. P-ta~c code-0 go 0191 -lce ms.n my be wood me bow96. .)rzitE REPORT NJMB3ME): If lb.s rap ee a bgasi a~d bot will be fo~llowd by m asilseadrn ad tocisieI cosm-assigned any other repcit numbers (either by the arnteiilose UM 7M31 sbs omw 011 Imek. Muise. and goe Is "glow.er by the openaw0. &I" miatet this ninihmerlo10. AVAIL ABLIT Y/LIMITATION NOTCES Zaoe ay Unit Itfalso"as father diassemeatias of the reoprt. other dm Z
INCLASS!713