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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

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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

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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

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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

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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

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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

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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

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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

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C DIAGNOSTIC S 5A740496 WAITF 16.1191 5 750

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c 4 A1 TE :A-L' t 566

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461 00 7 15 J -I1.'.U 5970

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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

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717 CFIAaOv0 6880

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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

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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

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725 C AT 1-UE 7950726 W'UTr 169?11V.I4GY*JISS 796t'-

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4.80 ISTslO 19,00IFP4UjO dvJuU

C CALCULATE 5YQISCOPIC 'IOMET O2481i no 482 ImIed o0iu462 ENTIII*CVY(I.I) 8LU43

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DO 4A4. !ule. 01074.84 E'4T111m0.0 8080

GO T! 4.96 009048S <F~aS.1 a 100486 no 493 JalNS 6110

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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(f~pjPk14o7@Jlll.P~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;29.CF2C@C.F2U,*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

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-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

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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

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3 16 22 271 1*OOOOOOOE+01 O*0O0OOOOEO00

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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

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- 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

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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

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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

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860 F0RMATtISI1EE13.611 v.44vEND 045 0

SIBfrTC SMATI'4 *4949Xq7SV9ROUTINS MATI94V gAtNedeme0ETER1 uolo

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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,<

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U57C 0500C REDUCE NON-Pivot .RowS tj600C

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8(L1xd1IJ-(ICOu)*wAXU610t50 CONTINUE 6*600 RETURN

ii700

- ENn OF FILE

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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