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NASATechnicalMemorandum81636 NASA-TM-8163619810008913

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_: Computer Program Documentationfor the Dynamic Analysis ofa Noncontacting MechanicalFace Seal

Fort _EFEIW2_CE

B. M. Auer and I. Etsion t'_._.r_r-,_-=_._,-,=r_r_:Lewis Research CenterCleveland,Ohio

; t! :,:i.:_ ", ' 'j_NTEt_

t November 1980

N/ A

https://ntrs.nasa.gov/search.jsp?R=19810008913 2018-05-22T09:03:27+00:00Z

COMPUTER PROGRAM DOCUMENTATION FOR THE DYNAMIC ANALYSIS

OF A NONCONTACTING MECHANICAL FACE SEAL

by

B. M. Auer and I. Etsion

National Aeronautics and Space AdministrationLewis Research Center

Cleveland, Ohio 44135

o

ISUMMARY

A computer program is presented which gives a numerical solution to a set

of second order ordinary differential equations which describe the dynamic be-havior of a noncontacting mechanical face seal. The equations were based on an

analytical solution of the Reynolds equation for the fluid film pressure com-

bined with a numerical solution, using time integration techniques, of theequations of motion of the flexibly mounted seal ring.

The face seal consists of two discs, a seal seat attached to a shaft and

rotating with it and a seal ring attached to a stationary mount. The seal ring

is flexibly supported to allow self alignment and tracking of the mating seal

seat. Program input includes variables describing the geometry and the physical

characteristics of the face seal and seal operating conditions. Output from the

program includes velocities and displacements of the seal ring along and aboutthe axes of an inertial reference system.

Five appendixes are included, containing the nomenclature (A), the diction-

ary of Fortran symbols (B), the computer program (C), the program flow chart (D),and a sample problem (E).

INTRODUCTION

A dynamic analysis of a noncontacting, coned, mechanical face seal is pre-sented in Ref. [i]. The seal consists of two discs, a rotor which is attached

to the shaft and rotates with it (see Fig. i) and a stator which is flexibly

supported to allow self alignment and tracking of the rotor. The stator, orseal ring, can move axially and tilt about two orthogonal diameters while its

circumferential rotation is prevented by anti-rotation locks. The motion of

the flexibly mounted stator is controlled by the forces and moments acting upon

it, by its mass inertia, and by the stiffness and damping properties of theseal system.

The dynamic analysis in Ref. Eli is based on a numerical solution of the

equations of motion of the stator. For this purpose, a computer program was

prepared. Input data for the program describes the geometry and the physicalcharacteristics of the face seal, as well as its operating conditions. These

/VY/- 17 3s-

are the inner and outer seal ring radii, the seal ring mass and radius of gyra-

tion, the coning angle, the external moment acting on the seal ring, the spring

constant of the flexible support, the radius of the springs location, the seal

design clearance, the inside and outside pressures, and the shaft speed. The

computed results at any time t, include the seal ring axial and angular dis-

placement and the minimum clearance between the ring and the seat. The first

time derivatives of the ring displacements are also computed.

" The object of this report is to describe the equations of motion of theface seal and the method of solution of these equations, to describe the input

and output of the program, and to document the computer program.

STATEMENT OF THE PROBLEM

The program solves the set of three second order ordinary differential

equations, (i), (2), and (3) for _x, _y, and z as a function of time. Thefull set of equations, (i) through (28), is listed below, and this is followed

by a brief description of their derivation.

ex = (Mx + Me)/I (i)

= (2)

"Z = Fzlm (3)

1I = mR 2 (4)

g

i. +Ri

Rm = 2 (5)

Mx = MlcOS(_ ) - M2sin _ (6)

= Mlsin(_ ) + M2cos _ (7)

27 i

o _R i K Co (8)• MI = pR2cos BdRdB - _ R2p y ro

M2 = pR2sin 0 dRd0 (9)

f2_ /R IFz = PR dR dO - F (i0)c

i

Fc = K- Z+_Rm(I - Ri) o + Pi + (Po - Pi (ii)r 2 + B(I - Ri)o

P = Pd + Ps If (P < 0), then P = 0 (12)4

Pd = i - YRmSin 8 - Z -YRmCOS A (13)(i + Z)3 1

2

Ps Po (Po Pi) Hi(H° + H)= - - A (14)2

2 + 2Y = _x y (15)

= (&xex + &yey)/y (16)

= tan-l(_y/_ x) (17)

= (_yex - _xey )/Y2 (18)

= _I(1+ z) (19)

6 = sl(1+ z) (20)

H = i + cR cos(8) + _(R - Ri) (21)

Hi = 1 + _Ricos(8 ) (22)

Ho = i + s cos 8 + 5(1 - Ri) (23)

Hm = (Hi + HO)/2 (24)

A = (1 - R)/_mH2(I - Ri)_ (25)

4

HMINI = i - ERi (26)

HMIN2 = i - g + 6(1 - Ri) (27)

HMI N = MIN(HMINI, HMIN2) (28)

The theoretical model is shown in Fig. 2. The seal seat is parallel to

- the xy plane of an inertial reference system xyz and is rotating at a

constant angular velocity m about the z axis. The primary seal ring has

three degrees of freedom. It can move axially along the z axis and tilt

about the x and y axes. A rotating coordinate system 123 coincides withthe principal axes of the ring so that axis 3 is perpendicular to the plane

of the ring. The coordinate system 123 rotates in the inertial reference xyz

so that axis i always remains in the plane xy and axis 2 is directed to the

instantaneous point of maximum film thickness. Thus, the orientation of coor-dinate system 123 in the inertial reference xyz can be defined by the nuta-

tion angle y* measured from axis z to axis 3, and the precession anglemeasured from axis x to axis i.

Equations (i), (2), and (3) are the general equations of motion of the

seal ring where _ and &_ are the accelerations of the ring about the xand y axes, respectively, and Z" is the axial displacement acceleration.

Equations (6) and (7) transform the ring moments from the rotating 123 coor-

dinate system to the xyz inertial system. Equations (8) and (9) are used2 Co

to calculate the moments MI and M2 where Rsp Y T_° is the restoring

moment provided by the flexible support. Equation (i0) calculates the axial

force, Fz, and Eq. (ii) calculates the closing force, Fc, contributed by boththe flexible support and the pressure balance of the seal. Equations (12), (13),

and (14) calculate the fluid film pressure distribution, where Ps is the hydro-

static pressure component and Pd is the hydrodynamic and squeeze pressurecomponents. Equations (21), (22), (23), and (24) give the film thickness dis-

tributions H, Hi, Ho, and Hm, and Eqs. (19) and (20) give the tilt and theconing parameters g and 6.

The velocities _., _., and Z and the new displacements ex' e ' and Zare found from the solution of the system (i), (2), and (3) to give t_e trans-

ient dynamics of the seal. From these values, the nutation y and the pre-

cession _ and their derivatives can be found by using Eqs. (15), (16), (17),

and (18). After each time step, the minimum film thickness, HMIN, is calculatedfrom Eqs. (26), (27), and (28), and is tested against a failure criterion.

Since contact can take place on either the inner or the outer radius of the

ring, the minimum film thickness is set equal to the smaller of HMINI orHMIN2.

METHOD OF SOLUTION

The program solves the set of three second order differential equations,

(i), (2), and (3), where initial conditions are given for _x, _y, Z, _x, _y,

5

and Z. These equations were rewritten in terms of the six new variables YI,

Y2, Y3, Y4, Y5, and Y6 as the following set of six first order equations,

with initial conditions given for YI' Y2' Y3' Y4' Y5' and Y6"

YI = ex (29)

Y2 = _'y (30)

Y3 = _ (31)

" Y4 = (Mx + Me)/l (32)

Y5 = My/l (33)

Y6 = Fz/m (34)

The dependent variables are given by Eqs. (35) through (40) as follows:

YI = _x (35)

Y2 = ey (36)

Y3 = Z (37)

Y4 = _'x (38)

Y5 = _'y (39)

Y6 = _ (40)

To find the solution of Eqs. (29) through (34), the program uses the GEAR

differential equation solver software package in the Lewis Research Center's

International Mathematical and Statistical (IMSL) software library of scientific

Univac ii00 Fortran V programs• The Lewis Research Center subscribes to this

software package which is supplied by the International Mathematical and Sta-

tistical Libraries, Inc. software company in Houston, Texas•

The GEAR package finds the numerical solution of initial value problems

• for systems of first order ordinary differential equations. This package has

the option of using two basic methods of solution, either an implicit multistep

variable order Adams predictor corrector method or an implicit multistep back-

ward differentiation formula method, also known as Gear's stiff method [2].

Let YI, _2, Y3' Y4' Y_, and. Y6 be denoted by Y and let YI' Y2, Y3'--_Y4'

95, and Y6 be denoted by Y. Then the call to the GEAR package is CALL DGEAR

(N6, FCN, FCNJ, T, DELH, Y, TEND, TOL, METH, MITER_ INDEX, IWK_ WK, IER), where

the parameters passed in the call statement are defined as follows:

N6 - Input giving the number of first order differential equations tobe solved.

FCN - User supplied subroutine for evaluating the function Y.

FCNJ - User supplied subroutine to c_.omputethe N6xN6 Jacobian Matrix ofthe partial derivatives of Y with respect to Y. FCNJ is needed

only when miter = I.

T - The independent variable• T is input on the first call only, and

on the first call passes the initial value to the program. T is

output on subsequent calls and passes to the user the current value

of the independent variable at which integration has been completed.

DELH - On the first call only, DELH is input and contains the next step

size in T to be used. On subsequent calls, DELH is output and con-

tains the step size actually used.

Y - The_dependent variable, Y. On input, Y supplies the initial valuesof Y. On output, Y contains the value of Y computed at TEND.

TEND - Input giving the value of T at which a solution for Y is desired.

TOL - Input giving the relative error bound•

METH - Input indicating whether the Adams Method or the Backward Differen-tiation Formulae Method is to be used.

MITER - An input indicating whether the chord method or functional itera-

tion is to be used and whether a finite difference or analyticJacobian is to be used with the chord method.

INDEX - Both an input and an output parameter. Index tells whether this

is the first call to DGEAR and whether the integration wassuccessful.

IWK - An integer work vector.

WK - A real work vector.

IER - An output error parameter telling the user whether the integrationwas halted due to an error. The value of IER indicates the error

encountered, as described on the following pages.

The GEAR package is called from the MAIN program and before the first call,the parameters N6, DELH, TOL, _TH, MITER, and INDEX are assigned values. On

the first call, T is set equal to TZRO, TEND is set equal to TZRO + DTEND, and

the Y array is assigned the initial values for the dependent variables. The

initial value of the step size, DELH, also must be chosen. This value should

be considerably smaller than the average value expected for the problem since

DGEAR must begin with a not so highly efficient first-order method. For the

first step, as for every step, DGEAR tests for the possibility that the step

size was too large to pass the error test (based on TOL), and so adjusts the

step size down automatically. This downward adjustment, if any, is noted by

IER having the values of 66 or 67. The relative error bound, TOL, should be

assigned a value. TOL is an estimate of the local truncation error, as dis-tinguished from the global truncation error. The local truncation error is

the error resulting from taking a single step, starting with data which is

regarded as exact. TOL should be at least an order of magnitude larger thanthe unit roundoff but generally not larger than 0.001.

DGEAR solves the differential equations from T = TZRO to T = TZRO + DTEND

with the given initial conditions. The value of IER, which is returned to the

MAIN program from DGEAR, is tested and the program stops if IER is between 132and 136. The output error parameter, IER, informs the user of errors encount-

ered by the GEAR package, if any. If the program failed to pass the error test

after reducing DELH by a factor of 1.0EIO from its initial value, then IER is

set equal to 132. If the program failed to achieve corrector convergence afterreducing DELH by a factor of 1.0El0 from its initial value, then IER is set

equal to 133. Incorrect input to the GEAR package is flagged by IER = 135 or

136. If IER is less than 132 or greater than 136, the desired output is

printed, TEND is reset to TEND = TEND + DTEND, and DGEAR is called again. This

process is continued until there is seal failure, TEND = TSTOP, or IER is be-

tween 132 and 136. To solve the problem, the GEAR package used the implicit

Adams Method and used the following values for its parameters; N6 = 6,DELH = 0.001, TOL = 0.0001, METH = I, MITER = 0, and INDEX = i.

To find the moments M1 and M2 and the force Fz, the double integralsof Eqs. (8), (9), and (i0) have to be evaluated. This was done by the applica-

tion of the Guassian quadrature integration formula, one of a family of powerful,

accurate, numerical integration techniques [3]. To evaluate these integrals,the Fortran program SQUAD1 was used. This program was written at the Lewis

Research Center by Charles Goldstein, and it evaluates integrals by repeated

application of Gauss' formula [A].

The call to obtain the inner integrals, which give integration with respectto R, is CALL SQUAD1 (MODE, NR, RI, DDI, XR, YRI, ANSI), where the parametersare defined as follows.

MODE - =i, then obtain the arguments XR(NR)

=2, then use the evaluations of the integrand YRI_R(NR_ toobtain ANSI

NR - The number of points of integration; N = 3(1)16

RI - The lower limit of integration

DDI - The upper limit of integration

SR - An array containing the variable of integration

YRI - An array containing values of the integrand

ANSI - The result of the integration; i.e., the value of the integral

The call to obtain the outer integrals, which gives integration with re-

spect to e, is CALL SQUAD1 (MODE, NTH, THI, TH2, XTH, YTHI, ANSI), where theparameters are defined as follows.

MODE - =i, then obtain the arguments XTH(NTH)

=2, then use the evaluations of the integrand YTHI_TH(NTH_ toobtain ANSI

NTH - The number of points of integration per quadrant; NTH = 3(1)16

THI - The lower limit of integration

TH2 - The upper limit of integration

XTH - An array containing the variable of integration

YTHI - An array containing values of the integrand

ANSI - The result of the integration; i.e., the value of the integral

The program used 5 points of integration for the inner integral (NR = 5), and

8 points of integration for each quadrant of the outer integral (NTH = 8).

Larger values were tried for NR and NTH (e.g., NR = ii and NTH = 16) and the

values of the resulting integrals were compared. It was found that using

NR = 5 and NTH = 8 gave sufficient accuracy for the integrals. The variables

NR and NTH can take on values between 3 and 16. To compute the outer integrals,

whose limits of integration are 0° and 360 °, SQUAD1 was used four times over the

quadrants 0° to 90°, 90° to 180°, 180 ° to 270 °, and 270° to 360°. This was done

because NTH can have a maximum value of only 16, and applying SQUAD1 over four

intervals allowed more points to be used over the total interval of integration0° to 360 °. It was also done so that the program would avoid computing the

integrand at the points 0° and 180 °. The routine SQUAD1 does not compute the

integrand at the end points of the interval over which SQUAD1 is being used, so

that computation at these points was avoided. Singularities can only occur at

0° and 180 °, but because of the complexity of the integrand, it could not easily

be determined whether singularities do occur. As a precaution, computation ofthe integrand was thus avoided at 8 = 0° and 180°.

, FORTRAN PROGR_

General Description

The foregoing analysis has resulted in a computer program for the analysisof a noncontacting mechanical face seal. The MAIN executive program handles

most of the program flow and logic. It assigns values to program parameters,

reads and writes program input, initializes the GEAR package and repeatedly calls

the GEAR package to move ahead in time by DTEND at each call. After each call,

9

it writes the output and tests for seal failureor a failureof the GEAR package.The program stops when a failure is detected or when TSTOP is reached. The sub-

routine FQUAD does most of the computing for the functions which are used in the

DGEAR package.

A brief explanation of the function of each program module follows.

DGEAR the general GEAR package for solving a system of first orderordinary differential equations

ERMIN a subroutine called by ERTST to determine when H varies byINless than HSTAB during several revolutions. I_ this condition

is met, then ERMIN sends a flag through ERTST to MAIN

ERTST a subroutine called by the MAIN program to compute HMI N anddetermine if there is seal failure or GEAR package failure

FCN a subroutine which computes the Y array

FCNJ a dummy subroutine passed to the GEAR package

FQUAD a subroutine used to compute the integrand of the double integralsappearing in Eqs. (8) through (i0)

MAIN the Main executive program

SQUAD1 a subroutine called by FQUAD to perform a single Gaussianintegration

WRT a subroutine called by the MAIN program which writes the outputee •

Z, y, 4, Z, y, 4, _x, _y, HMIN, T, _x, _x, _y, _y, _, DEEM,INDEX, IER, and FC after each call to the GEAR package

Using the Program

The program always starts by reading program input through the two name-

lists, INPUT1 and INPUT2. The input in these namelists are described below (allsymbols are repeated in appendix C).

Namelist INPUT1

DELH a parameter in the GEAR package containing the first step size tobe tried

DTEND for each call to the GEAR package, solve the differential equa-tions from TEND to TEND + DTEND

HMSTOP the program is stopped when HMIN becomes less than HMSTOP

HSTAB the program is stopped if HMIN vairies by less than HSTAB forseveral revolutions

i0

INDEX a parameter in the GEAR package indicating the type of call beingmade to the package

METH a parameter in the GEAR package call indicating the method ofsolution to be used

MITER a parameter in the GEAR package call indicating the iterationmethod to be used

NPLT if punch is true, then write every NPLT-TH array point into data-set 15 to be saved to make a movie

NR a parameter in the SQUAD1 call indicating the number of points of

integration used to obtain the inner integrals

NTH a parameter in the SQUAD1 call indicating the number of points of

integration used to obtain the outer integrals

PUNCH a logical variable. If PUNCH = TRUE, then write and save outputin dataset 15

READ1 a logical variable. If READ1 = TRUE, then read in and processanother case

TOL a parameter in the GEAR package call giving the relative errorbound

TSTOP the differential equations are solved to TEND = TSTOP

TZRO the initial value of time, the independent variable

YIN an input array containing the initial values for ex' ey, Z, _x,e_, and Z

Namelist INPUT2

BETA normalized coning angle,

FKROC the dimensionless parameter, K --Co which is the spring constantr O '

multiplied by the equilibrium center-line clearance divided by ro

, FM dimensionless mass, m

FME dimensionless external moment, Me

PI dimensionless pressure at the inner radius of the ring, Pi

PO dimensionless pressure at the outer radius of the ring, Po

RG dimensionless ring radius of gyration, Rg

ii

RI dimensionless inner radius, Ri

RSP dimensionless radius of the location of the springs, Rsp

If the logical variable READ1 is TRUE, then another seal case is read into

the MAIN program through the Namelists INPUT1 and INPUT2 and this case is pro-

cessed. If READ1 is read in as FALSE, then the program stops.

12

APPENDIX A

NOMENCLATURE

A (i - R)/HmH2(l - Ri)

C seal center-line clearance

Co equilibrium center-line clearance

Fc closing force

* 2Fc dimensionless closing force, Fc/Sm o

Fz axial force

* 2Fz dimensionless axial force, Fz/Sr o

H dimensionless film thickness, h/C

Hmi n dimensionless minimum film thickness

h film thickness

I* ring mass moment of inertia about a diameter

I dimensionless moment of inertia, l*_2Co/Sr_

K* spring constant

K dimensionless spring constant, K*/Sr o

M* moment

Me external moment

Me dimensionless external moment

M dimensionless moment, M*/Sr 3o

m ring mass

• m dimensionless mass, m*_2Co/Sr _

n number of seal revolutions, t/2_

P dimensionless pressure, p/S

p pressure

R dimensionless radius, r/r o

13

r radial coordinate

rg ring radius of gyration

Rg dimensionless ring radius of gyration

S seal parameter, 6_(ro/Co)2(i - Ri)2

t* time

t dimensionless time, _t*

Z* axial displacement

Z dimensionless displacement, Z*/C°

tilt angle

normalized tilt, _ ro/Co

B* coning angle

B normalized coning, B*ro/C o

¥ nutation

normalized nutation, _*ro/C o

coning parameter, _*ro/C

tilt parameter, Y*ro/C

0 angular coordinate

viscosity

precession

m shaft angular velocity

Subscripts :

1,2,3 axes i, 2, or 3, respectively

d hydrodynamic

i inner radius

m mid radius

o outer radius

14

s hydrostatic

sp springs

x,y,z axes, x, y, or z, respectively

15

APPENDIX B

FORTRAN SYMBOLS

ANSI a parameter in the SQUADI call containing the value of the integral

BETA normalized coning angle, B

CAX normalized tilt, ex' wrt. the x axis

CAY normalized tilt, ey, wrt. the y axis

CAXD the first derivative of ex

CAYD the first derivative of ey

CZ the axial dimensionless displacement of the ring, Z

CZD the first derivative of Z

DD1 the upper limit of integration in the SQUAD1 call

DEL the coning parameter,

DELH a parameter in the GEAR package containing the step size to be used on

the first call to DGEAR and the step size actually used on subse-

quent calls

DTEND for each call to the GEAR package, solve the differential equationfrom TEND to TEND + DTEND

EPS the tilt parameter,

FC the dimensionless closing force, Fc

±FCN a subroutine used by the GEAR package for evaluating the function Y

FCNJ a DUMY subroutine used by the GEAR package

FI the dimensionless moment of inertia, I

CFKROC the parameter K _--0-°,which is the dimensionless spring constant multi-

, r Oplied by the equllibrium center-line clearance divided by ro

FM dimensionless mass, m

FMI dimensionless moment, MI, wrt. axis I

FM2 dimensionless moment, M2, wrt. axis 2

FME dimensionless external moment, Me

16

FMX dimensionlessmoment,Mx, wrt. axis x

_@f dimensionless moment, _, wrt. axis y

FZ dimensionless axial force, Fz

GAM normalized nutation, y

GAMD the first derivative of the normalized nutation, y

H dimensionless film thickness, H

HI dimensionless film thickness at the inner radius, Hi

HO dimensionless film thickness at the outer radius, Ho

HM the average of HI and HO

HMIN dimensionless minimum film thickness, HMIN

HMINI dimensionless minimum film thickness at the inner radius, HMINI

HMIN2 dimensionless minimum film thickness at the outer radius, HMIN2

HMSTOP the program is stopped when HMIN becomes less than HMSTOP

HSTAB the program is stopped if HMIN varies by less than HSTAB for severalrevolutions

IER an output from the GEAR package indicating any errors encountered

while using the GEAR package

INDEX an output parameter from the GEAR package call indicating whether the

integration was successful, and an input to the GEAR package indi-

cating the type of call being made to the package

IWK a parameter in the GEAR package call which is used as an integer workvector

IWRT a parameter indicating whether debug output is wanted

• LOGER a logical variable returned from subroutine ERTST. LOGER = FALSE means

that the program should be stopped

METH a parameter in the GEAR package call indicating whether the Adamsmethod or the backward differentiation formulae method is to be used

MITER an input in the GEAR package call indicating whether the chord methodor functional iteration is to be used and whether a finite difference

or analytic Jacobian is to be used with the chord method

17

N6 a parameter in the GEAR package call indicating the number of firstorder differential equations to be solved

NPLT a program input indicating how much output to save for moviegeneration

NR a parameter in the SQUAD1 call indicating the number of points of

integration used to obtain the inner integrals

NTH a parameter in the SQUAD1 call indicating the number of points of

integration used to obtain the outer integrals

P dimensionless pressure

PD the dimensionless hydrodynamic and squeeze pressure components, Pd

PI the dimensionless pressure at the inner radius of the ring, Pi

PO the dimensionless pressure at the outer radius of the ring, Po

PS the dimensionless hydrostatic pressure component, Ps

PSI the precession,

PSID the first derivative of the precession,

PUNCH a logical variable. If punch = true, then write output into dataset 15to be saved to make movies

RAX,RAY,

RTANG,

RZ arrays in which to save output for movie generation

READ1 a logical variable. If READ1 = TRUE, then read in and process anothercase

RG the dimensionless ring radius of gyration

RI the dimensionless inner radius, Ri

RM the dimensionless mid radius, Rm

RSP the dimensionless radius of the location of the springs, Rap

T dimensionless time, the independent variable

TEND a parameter to the GEAR package giving the value of T at which a solu-tion for Y is desired

THI a parameter in the SQUAD1 call giving the lower limit of integration ofthe outer integral

18

TH2 a parameter in the SQUAD1 call giving the upper limit of integration

of the outer integral

TOL a parameter in the GEAR package call giving the relative error bound

TOUCH a logical variable. TOUCH = T means that the seal ring touches theshaft

TSTOP the differential equations are solved to TEND = TSTOP

TZRO the initial value of time, the independent variable

WK a parameter in the GEAR package call containing a real work vector

XR a parameter in the SQUAD1 call containing the variable of integration

of the inner integrals

XTH a parameter in the SQUAD1 call containing the variable of integration

of the outer integrals

Y a parameter in the GEAR package call containing the dependent variable,Y

YIN an input array containing the initial values for ex, ey' Z, _x, _y'and

YPRI_ a parameter in the FCN call containing the first derivative of the Y

array

YRI a parameter in the SQUAD1 call containing the values of the integrand

of the inner integrals

YTHI a parameter in the SQUAD1 call containing the values of the integrand

of the outer integrals

19 ,

APPENDIX C

FORTRAN PROGRAMS

1:C MAIN PROGRAM2:C

3:C THIS PROGRAM DESCRIBES THE MOTION OF A SEAL RING4:C BY SOLVING SIX FIRST ORDER DIFFERENTIAL EQUAS. (WHICH5:8 WERE DERIVED FROM THREE 2NO ORI,ER ORDINARY DIFFERENTIAL6 :C EQUATIONS )7:C8:C9:C THE SYSTEM OF DIFFERENTIAL EOUAS, TO BE SOLVED IS AS FOLLOWS

10:8 YPRIME(I> = AX-DER111:8 YF'RIME<2) = AY-DERI12:C YF'RIME(3) = Z-DERI13:C YF'RIME(4) = AX-DER214:C YPRIME(5) = AY-DER215:C YPRIME(6) = Z-DER216:C17:8 WHERE THE Y ARRAY IS DEFINED AS FOLLOWS18:C Y(1)=AX19:8 Y(2)=AY20:C Y(3)=Z21:C Y(4)= DER-AX_:C Y(5)= DER-AY23:C Y(6)= DER-Z24:C

25:8 BTEND INCREMENT TEND BY DTENB26:C27:C HSTAD STOF' THE PROGRAM IF HMIN VARIES LESS THAN HSTAB28:C DURING FIVE REVOLUTIONS

.9.C30:C HHSTOF' - STOP THE PROGRAM IF HMIN BECOMES LESS31:C THAN HMSTOF'32:C33:C IWRT : 0 DO NOT WRITE DEBUG OUTF'UT34:C = 1 WRITE COMPLETE DEBUG OUTPUT35:C36:C NF'LT - PUNCH EVERY NF'LT F'T. FOR THE MOIVE37:C PLOTTING PROGRAM3B:C39:C NR = NO. OF INTEGRATION INTERVALS IN THE40:C INTERVAL 1 TO RI41:C42:C NTH = NO. OF INTEGRATION INTERVALS43:C IN EACH QUADRANT OF THE INTERVAL 0 TO 2_PIE44:C45:C PUNCH - PUNCH=.T., THEN PUNCH OUTPUT FOR THE MOVIE46:C PLOTTING PROGRAM47:C PUNCH=.F., THEN IlO NOT PUNCH THIS OUTPUT48:C FOR THE MOVIE PLOTTING PROGRAM49:C50:C RAX,RAY,RZ,RTANG - INFORMATION F'UNCHED FROM THIS F'ROG.51:C TO BE READ INTO & USED IN THE MOVIE F'LOTTING52:C PROGRAM53:C54:C TEND FOR EACH CALL TO DGEAR, FINEr SOLUTION55:C AT TEND56:C57:C TOUCH=°T. MEANS THAT THE SEAL RING TOUCHES THE SHAFT,58:C

59:C TSTOF' SOLVE DIFF. EOUASo TO T=TSTOP60:C61:8 TZRO INITIAL T

2O

62:C63:C64:C65:C66: EXTERNAL FCN,FCNJ67:C68: LOGICAL LOGER,TOUCN,READ1,PUNCH69:C70: DIMENSION YIN(6)71: DIMENSION Y(6),WK(144),IWK(6)72: DIMENSION RAX(6200),RAY(6200),RZ(6200),RTANG(6200)73:C74: COMMON/COMI/ PIE,PIE2,N4F'IE,NR,NTH,IWRT75:C76: COMMON/COM2/ FKROC,RO,BETA,RI,CO,77: I RM,PO,PI,FI,FME,FM,RSP78:C79: COMMON/COM3/ HMIN,HMIN1,HMIN280:C81:C COMMON/GEAR/ DUMMY(48),SDUMMY(4),IDUMMY(38)82:C83: NAMELIST/INPUT1/ READ1,METH,MITER,TOL,INDEX,DELH,84: 1 TZRO,YIN,TSTOF',85: 2 DTEND,NR,NTH,HMSTOF',HSTAD,86: 3 NF'LT,F'UNCH87:C88: NAMELIST/INF'UT2/ RI,FKROC,BETA,PI,F'O,8?: 1 FME,FM,RG,RSP90:C91: DATA READ1/.TRUE./92:C93: DATA TZRO,TSTOP,DTEND,YIN /0.,13.,.1,0.,.2,4_0./94tC95:C96: DATA HMSTOF'/.1E-20/97:C98:C99: NR=5

100: N4F'IE=4I01: NTH=8102: N6=6103: IWRT=O104: F'UNCH=.FALSE.105: NPLT=I106! PIE=3.14159265359107: PIE2=2._F'IE108: HSTAB=.O01109:C110: GO TO 14111: 20 IF(.NOT.PUNCH) GO TO 14112: WRITE(15,122) RI_FKROC,BETA,PI,F'O,FME,FM113: WRITE(15,124) RG,RSP,TZRO,TOLPDELH,NR,NTH,METH,MITER114: WRITE(15,122) YIN115: WRITE(15,112) JT,NPLT,TOUCH116: WRITE(15,114) (RAX(J),J=I,JT,NPLT),RAX(JT)117: WRITE(15,114) (RAY(J),J=I,JT,NPLT),RAY(JT)118: WRITE(15,114) (RZ(J),J=I,JT,NPLT),RZ(JT)119: WRITE(15,114) (RTANG(J),J=I,JT,NF'LT),RTANG(JT)120:C121:C122: 14 METH=I123: MITER=O124; INDEX=I125: TOL=.O001126: DELH=.01127: IF(.NOT.READ1) GO TO 9912G_ READ(5,INPUT1)

21

129: WRITE(6,101)130: WRITE(6,1NPUTI)131! READ(5,1NFUT2)132: WRITE(6,INPUT2)133:C134:C135: FI=.5_FM_RG*RG136: RM=(I..RI)/2.137:C138:C

137:C INITIALIZE THE INTEGRATION140:C141: 30 LOGER=.TRUE.

142: TOUCH=.FALSE.143: JT=I144:C145: T=TZRO146: TEND=TZRO.DTEND147:C148: DO 32 J32=I,6149: Y(J32)=YIN(J32)150: 32 CONTINUE151:C152:C153:C WRITE OUTF'UT HEADING154: WRITE(6,104)155: WRITE(7,106)156: WRITE(7,102>157:C158: CALL ERTST(LOGER,TOUCH,Y,IER,T,I:ELH,DTEND,HHSTOP,HSTAB)159:C

160: CALL WRT(N6,JT,Y,T,DELH,TEND,INDEX,IER)161:C162: IF(.NOT.PUNCH) GO TO 40163: RAX(JT)=Y(1)164! RAY(JT)=Y(2)165: RZ(JT)=Y(3)166: RTANG(JT)=TZRO167:C

168:C USE SUB DVOGER TO MOVE ONE STEP IN TIME169:C170: 40 JT=JT+I171: CALL DGEAR(N6,FCN,FCNJ,T,DELH,Y,TEND,TOL,172: 1 METN,MITER,INDEX,IWK,WK,IER)173:C174: IF(oNOT.PUNCH) 80 TO 54175: RAX(JT)=Y(1)176: RAY(JT)=Y(2)177: RZ(JT)=Y(3)178: RTANG(JT)=TEND179:C

180_ 54 rwC_LLERTST(LOGER,TOUCH,Y,IER,T,£1ELH,DTENI_,HMSTOP,HSTAB)181:C182: IF(.NOT.LOGER) GO TO 201B3:C184: CALL WRT(N6,JT,Y,T,DELH,TEN[I,INDEX,IER)185:C186:C187:C18B: TEND=TEND+DTEND1B?:C190: IF(TEND.GE.TSTOP) GO TO 20191: GO TO 40172:C173:C194:C195: 101 FORMAT(1H1)

22

196: 104 FORHAT(IHI,6X,'CZ',gX,'GAM',SX,'PSI',SX,'CZD',BX'197: I "GAMD',7X,'PSID',7X,'CAX',BX,'CAY',BX,'HMIN',7X,19B: 2 'TWRT')199:C200: 102 FORHAT(1HI,'INDEX',2X,'IER',5X,'AXD',BX,'AYD',gx,201: 1 'ZD',9X,'T'_IOX,'H'r9X,'AXDD',7X,'AYDD'tTX,202: 2 'ZDD',9X,'FC')203:C204: 106 FORMAT(1HO,/////)205: 112 FORMAT(214,L4)206: 114 FORMAT(BEIO.4)207: 122 FORMAT(7E11o4)208: 124 FORHAT(SE11o4,4IS)209:C210: 99 STOP211: END

23

I_ SUBROUTINE ERTST(LOGER,TOUCH,Y,IER,T,DELH,DTEND,HMSTOP,HSTAB)2:C3:C4:C THIS SUBROUTINE TESTS THE VARIABLE IER5:C FROM THE DGEAR CALL AND PRINTS6:C THE CORRECT ERROR MESSAGE7:C8:C THE SUBROUTINE ALSO COMPUTES HMIN9:C

10:C11:C

' 12: DIMENSION Y(6)13:C14: LOGICAL LOGER,TOUCH15:C16: COMMON/COM1/ F'IE,F'IE2,N4PIE,NR,NTH,IWRT17:C18: COMMON/COM2/ FKROC,RO,BETA,RI,CO,19: I RM,PO,PI,FI,FME,FM,RSP20;C21: COMMON/COM3/ HMIN,HMIN1,HMIN222:C

23:C TO COMPUTE AND TEST HMIN24:C25: CAX=Y(1)26: CAY=Y(2)27: SOXY=CAX_CAX+CAYICAY28: GAM= SORT(SOXY)29:C30: CAXX= ABS(CAX)31: CAYY= ABS(CAY)32: PSI= ATAN2(CAYY,CAXX)33: IF(CAX.LT.O.AND.CAY.GT.O) PSI=-F'SI34: IF(CAX.GE.O.AND.CAY.LT.O) PSI=-PSI35:C36: CPSI=CAX/GAM37:C38: CZI=I.+Y(3)39: EPS=GAM/CZ140: DEL=BETA/CZI41:C42: HMINI=I.-EPS_RI43: HMIN2=I.-EPS.DEL_(I.-RI)44: HMIN=HMIN1

45: IF(HMIN2.LT.HMINI) HMIN=HMIN246: IF(HMIN.GT.HMSTOP) GO TO 2047: LOGER=.FALSE.48: TOUCH=.TRUE.49: WRITE(6,102) HMIN,HMIN1,HMIN250: GO TO 9951:C52: 20 HI=HMIN

53: CALL ERMIN(DTEND,H1,LOGER,HSTAB)54_C55: 22 IF(IER.NE.33) GO TO 2456: LOGER=.FALSE.57_ WRITE(6,104) IER58: GO TO 9959:C60: 24 IF(IER.LT.132.0R.IER.GT.136) GO TO 9961: LOGER=.FALSE.62: WRITE(6,104) IER63:C64:C65:C66: 101 FORMAT(1H1)67: 102 FORMAT(" FATAL ERROR',SX,'HMIN='E12.5,SX,

24

68_ I 'HMIN1='E12.5,SX,'HMIN2='EI2.5)69_ 104 FORMAT(' FATAL ERROR',SX,'IER='I5)70:C

71_ 99 RETURN72t ENb

25

1: SUBROUTINE ERMIN(DTENDtH1,LOGER,HSTAB)2:C3:C4:C THIS SUBROUTINE TESTS HMIN. IF HMIN DOES NOT VARY BY HSTAB5:C IN FIVE REVOLUTIONS, THEN SET LOGER TO .FALSE.6:C7:C IN CALLING PROG, ERTST, HI=HMINB:C9:C10:C11: DATA IDO,HS /0,1.E20/12:C13: LOGICAL LOGER14:C15: IF(IDO.EQ.1) GO TO 4016:C17: IF(ABS(H1-HS).GT.HSTAB) GO TO 5018:C19: HS=H120: TS=O.

21: IDD=I22: GO TO 9923:C

24: 50 IDO=O25: HS=H126: GO TO 9927:C

.B. 40 IF(ABS(HS-H1).LT.HSTAB) GO TO 5529: IDO=O30: HS=H131: GO TO 9932:C33: 55 TS=TS.DTEND34: IF(TS.GT.31.416) LOGER=.FALSE.35:C36: 99 RETURN37: END

26

1_ SUBROUTINE FCN(N,T,Y,YF'RIME)2:C

3_C THIS SUB IS A PARAMETER IN THE CALL4;C TO DGEAR5:C6_C THIS SUB COMPUTES YPRIME7:CB; DIMENSION Y(N),YPRIME(N)9_C

10_ COMMON/COM1/ F'IE,PIE2,N4PIE,NR,NTH,IWRT11_C

121 COMMON/COM2/ FKROC,RO,BETA,RI,CO,13_ 1 RM,PO,PI,FI,FME,FM,RSP14:C

15: COMMON/COM4/ CAX,CAY,CZ,CAXD,CAYD,CZD161 COMMON/COM5/ FMX,FMY,FZ,GAM,PSI,GAMD,PSID,FC,17: 1 FM1,FMIC,FZ1,FM2,EPS,DEL18:C19_C

20_C COMF'UTE THE RIGHT HAND SIDE OF THE 6 FIRST ORDER2i_C DIFFERENTIAL EOUATIONS22_C23: 20 CAX=Y(1)24: CAY=Y(2)25: CZ=Y(3)26_ CAXD=Y(4)27; CAYD=Y(5)28_ CZD=Y(6)29_C30t CALL FOUAD31:C32: YF'RIME(1)=Y(4)33: YPRIME(2)=Y(5)34: YPRIME(3)=Y(6)35: .YPRIME(4)=(FMX.FME)/FI36; YF'RIME(5)=FMY/FI37: YPRIME(6)=FZ/FM38_C39:C401 I0 RETURN41: END

27

1: SUBROUTINE FCNJ(N,T,Y,PD)2:C TO COMPUTE THE NSN JACOBIAN MATRIX OF PARTIAL3:C DERIVATES FOR THE DGEAR IHSL ROUTINE4_C THIS IS A DUMY ROUTINE FOR DOEAR5:C6: DIHENSION Y(N),PD(N,N)7:C DOUBLE PRECISION T,Y,PDB:C9: DUMY=DUMY

10: RETURN11_ END

28

11 SUBROUTINE FQUAD2:C3:C4:C THIS SUBROUTINE PERFORMES DOUBLE INTEGRATION5:C USING THE GOLDSTEIN QUADRATURE ROUTINES6:C TO FIND M1,M2,AND FZ7:CSIC INTEGRATE THE INNER INTEGRAL WRT R9:C INTEGRATE THE OUTER INTEGRAL WRT THETA

10:C111C NR = THE NO. OF PTS. IN THE R INTEGRAL12:C NTH = THE NO. OF PTS IN EACH QUADRANT OF THE THETA13:C INTEGRAL. WE INTEGRATE FROM 0 TO 21PIE,14:C I USE NTH PTS. IN EACH QUADRANT15:C16:C IWRT = 0 DO NOT WRITE DEBUG OUTPUT17:C =1 WRITE DEBUG OUTPUT1SIC19:C20: DIMENSION XTH(16),XR(16),YTH1(16),YTH2(16),21: I YTH3(16),YR1(16),YR2(16),YR3(16)22:C23: COMMON/COM1/ PIE,PIE2,N4PIE,NR,NTH,IWRT24:C25: COMMON/COM2/ FKROC,RO,BETA,RI,CO,26: 1 RM,PO,PI,FI,FME,FM,RSP27:C28: COMMON/COM4/ CAX,CAY,CZ,CAXD,CAYD,CZD29:C30: COMMON/COMS/ FMX,FMYyFZ,GAM,PSI,GAMD,PSID,FC,311 1 FM1,FMIC,FZI,FM2,EPS,DEL32:C33:C341 FN4PIE=N4PIE35: F'IEN4=2._PIE/FN4PIE36:C37: DOt=l.3B: FMI=O.39: FM2=O.40: FZI=O.41:C42: SQXY=CAX_CAX . CAY_CAY431 GAM= SQRT(SQXY)44:C45: CAXX= ABS(CAX)46: CAYY= ABS(CAY)47: PSI= ATAN2(CAYY,CAXX)48: IF(CAX.LT.O.AND.CAY.GT.O) PSI=-PSI49: IF(CAX.GE.O.AND.CAY.LT.O) PSI=-PSI50:C51: CPSI=CAX/GAM521 SPSI=CAY/GAM53:C541 GAMD=CAXDICF'SI . CAYDISPSI55: PSID=(CAYDICPSI - CAXD_SPSI)/GAM56: IF(IWRT.GE.1) WRITE(11,116) CAX,CAYPCZ,CAXD,CAYD,CZD,GAM,GAMD57:C5B: RIMI=I.-RI59: ZI=I..CZ60: EPS=GAM/Z161: DEL=BETA/Z162: EPSRI=EPSIRI63: DELRI=DEL_RIMI64: Z3R= I./Z1/Z1/ZI/RIM165: PDI=(.5-PSID)IGAM_RM&6: F'D2=GAMD_RM67: PSI=.5_(PO-PI)

29

68:C69:C70: CALL SOUADI(I,NR,RI,D[II,XRrYRIgANSI)71:C72: THI=O.73: TH2:0.74:C75: IF(IWRT.EO.1) WRITE(II,102) NR,RI,ANSI,PSI,CPSI,SPSI,76: 1 GAM,GAMD,PSID,Z1,EPS,DEL,EPSRI,DELRI,Z3R,PD1,PD2,77: 2 PS1,XR,YR178:C79: DO 40 J4=I,N4PIE80: THI=TH281: TH2=TH2.PIEN482: CALL SOUADI(I,NTH,TH1,TH2,XTN,YTH1,ANS1)83:C84: IF(IWRT.EO.I) WRITE(II,I04) J4,NTH,THI,TH2,ANSI,XTH,YTHI85: DO 30 J3=I,NTH86: THA=XTH(J3)87: STHA= SIN(THA)88: CTHA= COS(THA)89:C90: DO 20 J2=I,NR91: R=XR(J2)92: R2=R_R93: ECTHA=EPS_CTHA94: H=I.+R_ECTHA+DEL_(R-RI)95: HI=I.+EPSRI_CTHA96: HO=I,+ECTHA+DELRI97: HM=.5_(HI.HO)98: A=(1.-R)/(HM_H_H_RIM1)99:C

100: PD=Z3R*(PDI_STHA-CZD-PD2_CTHA)_At(R-RI)101: PS=PO-PSI*HI_HII(HO.H)IA

302: F'=F'D+PS103: IF(P,LE.O) P=O.104: YRI(J2)=F'IR2_CTHA105: YR2(J2)=P*R21STHA106: YR3(J2)=F'IR107: IF(IWRT.EO.1) WRITE(11,106) R,R2,ECTHA,H,HI,108: 1 HO,NM,A,PD,PS,P109: 20 CONTINUE110:C111: CALL SQUADI(2,NR,RI,DD1,XR,YR1,ANS1)112: CALL SOUADI(2,NR,RI,DD1,XR,YR2,ANS2)113: CALL SQUADI(2,NR,RI,DD1,XR,YR3,ANS3)114:C115: YTNI(J3)=ANS1116: YTH2(J3)=ANS2117: YTH3(J3)=ANS3118:C119: IF(IWRT.E_.I) WRITE(11,108) NTH,THA,STHA,CTHA,ANS1,120: 1 ANS2,ANS3,YR1,YR2,YR3121: 30 CONTINUE122:C123: CALL SQUADI(2,NTH,TH1,TH2,XTH,YTH1,ANS1)124: CALL SQUADI(2,NTH,TH1,TH2,XTH,YTH2,ANS2)125: CALL SOUADI(2,NTH,TH1,TH2,XTN,YTH3,ANS3)126: FMI=FMI+ANS1127: FH2=FH2+ANS2128: FZI=FZI+ANS3129: IF(IWRT.EO.1) WRITE(11,110) TH1,TH2,ANS1,130: I ANS2,ANS3,YTH1,YTH2,YTH3131: 40 CONTINUE132:C133: FMIC=.5_FKROC_RSP_RSP_GAM134: FMI=FM1-FMIC

3O

135_ FMX=FMI_CPSI-FM2_SPSI136_ FMY=FMI_SPSI.FM2_CPSI137_C138_ FCI=FKROC_CZ139_ FC2=BETA_(PO-PI)IRIM1/(2.+BETA_RIM1)140: FC3=PO+PI.FC2141_C142_ FC=FCI+F'IE_RM_RIMI_FC31431 FZ=FZI-FC144;C

145_ IF(IWRT.GE.1) WRITE(It,112) FMX,FMY,FZ,FC,FM1,146_ 1 FM2,PSI,PSID,FZ1,ANS1,ANS2,ANS3147_C148_C149_C150_ 102 FORMAT(1HO,' FQD2 NR'I2,' RI='E12.5/151_ 1 4E14.5/(4E14.5))152_ 104 FORMAT(1HO,'FQD4 J4'12," NTH'I2,3E15.5/' X'SF8.4/' Y'5FB.4)153:C

154_ 108 FORMAT(1HO,'FQD8 NTH',I3,' THA='E12.5," STHA=',E12.5,155_ 1 ' CTHA=',EI2.5,' ANSI=',EI2.5,' ANS2=',E12.5,15&_ 2 ' ANS3='E12.5,/,(4E15.5))157:C

158! 110 FORMAT(1HO,'F010',' THl='E12.5,' TH2='E12.5,159: i " ANSI='E12.5,' ANS2='EI2.5,' ANS3='E12.5,/(SE15.5))160_C161_ 112 FORMAT(1HO, 'FMX'PE12.5_ 'FMY',EI2.5,' FZ',E12.5,162: i ' FC'E12.5," FM1',E12.5,' FM2',E12.5,' F'SI',EI2.5,163_ 2 " PSID',EI2.5,/,' FZI'EI2.5,' ANSI'El2.5,164_ 3 ' ANS2'EI2°5,' ANS3'E12.5)165_C

166_ 106 FORMAT(' FQD6'," R='EI2.5,' R2=',E12.5," ECTHA='E12oS,167_ 1 ' H='EI2.5,' HI='E12.5/' HO='EI2.5," NM='EI2°5,168_ 2 ' A='EI2.5," PII='EI2.5,' PS='E12.5,' P='E12.5)169_C

170_ 116 FORMAT(1H1 "CAX',E12.5, "CAY'E12.5, 'CZ',E12.5,171_ 1 "CAXD',E12.5," CAYD',E12.5," CZD',E12.5,172; 2 " GAM',E11.4,' GAMD',EI1.4)173_C174_C175_ RETURN176; END

31

1: SUBROUTINE SQUABI(MODE,N,XI,XO,X,Y,ANSWR)2:C SOUO00203:C NUMERICAL INTEGRATION BY GAUSS FORMULA SgUO00304:C SOUO00405: DIMENSION A(70),H(70),X(N),Y(N) SOUO00506= DOUBLE PRECISION A,H

71 DATA (A(I),H(1),I=I,28)/ SOUO00608: I 7.74596669241483D-01, 5.55555555555556D-01,-0. D-OO,SOUO00709: 1 8.88888888888889D-01, 8o61136311594053D-01, 3.47854845137454D-O1,SQUO0080

10: 1 3.39981043584856D-01, 6.52145154862546D-01, 9.06179845938664D-OI,SQO0009011= 1 2.36926885056189D-01, 5,38469310105683D-01, 4.78628670499366D-O1,SQUO010012: 1-0. D-O0, 5.68888888898889D-01, 9.32469514203152D-O1,SOUO011013: 1 1.7132449237917011-01 6.61209386466265D-01, 3.60761573048139D-O1,SQUO012014: 1 2.38619186083197D-01 4.67913934572691D-01, 9.49107912342759b-OI,SQUO013015: I 1.29484966168870D-01 7.41531185599394D-01, 2.79705391489277b-O1,SQUO014016: 1 4.05845151377397D-01 3.81830050505119D-01,-0. D-OO,BOUO015017: 1 4.17959183673469B-01 9.60289856497536D-01, 1.01228536290376b-O1,SQUO016018_ 1 7.96666477413627D-01 2.22381034453374D-01, 5.25532409916329D-O1,SOUO017019! 1 3.13706645877887D-01 1,83434642495650D-01, 3.62683783378362b-01,S000018020: 1 9.68160239507626D-01 8.12743883615740D-02, 8.36031107326636D-OI,SOUO019021: 1 1.80648160694857D-01 6.13371432700590D-0], 2.60610696402935D-OI,S_UO0200221 1 3.24253423403809D-01 3°12347077040003D-01,-0. D-OO,SOUO021023_ 1 3.30239355001260D-01 9.73906528517172D-01, 6.66713443086880b-O2,SOUO0220241 1 8.65063366688985D-01 1.49451349150581D-01, 6.79409568299024D-O1,SQUO023025: 1 2.19086362515982D-01 4.33395394129247D-01, 2.69266719309996D-O1,SOUO024026: 1 1.48874338981631D-01 2.95524224714753D-01/ SOU00250271 DATA (A(1),H(1),I=29,56)/ SQUO026028: I 9.78228658146057D-01, 5.56685671161740D-02, 8.87062599768095D-O1,SQUO027029: 1 1.25580369464905D-01, 7.30152005574049B-01 1.86290210927734D-O1,SOUO028030: 1 5.19096129206812D-01, 2.33193764591990D-01 2.69543155952345D-O1,SOUO029031: 1 2.62804544510247D-01,-0. D-O0 2.72925086777901D-O1,SOUO030032: 1 9.81560634246719D-01, 4.71753363865120D-02 9.04117256370475B-OI,SOUO031033: I 1.06939325995318D-01, 7.69902674194305D-01 1.60078328543346D-O1,SOUO032034: 1 5.87317954286617D-01, 2.03167426723066D-01 3.67831498998180D-O1,SOUO033035: 12.33492536538355D-01, 1.25233408511469D-01 2.49147045813403D-OI,SQUO034036: 1 9.84183054718588D-01, 4.04840047653160D-02 9.17598399222978D-O1,SOUO035037: 1 9.21214998377280D-02, 8.01578090733310D-01 1.38873510219787D-O1,SOUO036038: i 6.42349339440340[=-01, 1.78145980761946D-01 4.48492751036447D-O1,SOUO037039: 1 2.07816047536889D-01, 2.30458315955135D-01 2.26283180262897D-O1,SQUO038040: 1-0. D-O0, 2.32551553230874D-01 9.86283808696812D-O1,SOUO039041: 1 3.51194603317520D-02, 9.28434883663574D-01 8.01580871597600B-O2,SQUO040042_ 1 8.27201315069765D-01, 1.21518570687903D-01 6.87292904811685D-OI,SOUO041043: 1 1.57203167158194D-01, 5.15248636358154D-01, 1.85538397477938D-O1,SQUO042044: 1 3.19112368927890D-01, 2.05198463721296D-01, 1.08054948707344D-OI,SQUO043045_ 1 2.15263853463158D-01, 9.87992518020485D-01, 3.07532419961170D-O2,SQUO0440461 1 9.37273392400706D-01, 7o03660474881080D-02/ SOU0045047= DATA (A(I),H(1),I=57,70)/ SQUO046048_ 1 8.48206583410427D-01, 1.07159220467172D-01, 7.24417731360170D-O1,SOUO047049: 1 1.39570677926154D-01, 5.70972172608539D-01, 1.66269205816994B-O1,SgUO048050: 1 _,94151347077563D-01, 1.86161000015562D-01, 2.01194093997435D-O1,SOUO049051: 1 1.98431485327112D-01,-0. D-O0, 2.02578241925561D-O1,SQUO050052: 1 9.89400934991650D-01, 2.71524594117540D-02, 9.44575023073233D-O1,SOUO051053: 1 6.22535239386480D-02, 8.65631202387832D-01, 9.51585116824930D-O2,SQUO052054: 1 7.55404408355003D-01, 1.24628971255534D-01, 6.17876244402644D-OI,SOUO053055: 1 1.49595988816577D-01, 4.58016777657227D-01, 1.69156519395003D-O1,SQUO0540561 1 2.81603550779259D-01, 1.82603415044924D-01, 9.50125098376370b-O2,SQUO055057_ 1 1.89450610455068D-01/ SQUO056058:C59_C60_ NN=N

61: XIN= XI62: ANS=O.63: IF(XIN .EO.XO) GO TO (400,500),MODE64: IF (NN.LE. 16> O0 TO 202651 WRITE (6,200) NN66_ 200 FORMAT(47HO_IXINUMBER OF POINTS FOR SOUAB1 INTEGRATION =I4,22H IS67: A OUT OF RANGE._IIII)

32

68: RETURN69; 202 INDKT = MOD(NN,2).I70_COMMENT--INDKT=I, N IS EVEN71;C =2, N IS ODD72: GO TO (204t210),INDKT73; 204 MIN= (NN_NN)/4 - 174: MAX = (NN_(NN.2))/4 - 275: GO TO 21576: 210 MIN= (NN%NN-?)/4 + 177: MAX = (NN_(NN.2)-11)/47B: 215 MINM1 = MIN-179: HDELX= 0.5%(XO-XIN)80: M= NN/281; DO 220 I=I,M82: IM= I.M83: J= MINMI.IB4: GO TO (216,217),MODE85: 216 X(I)= XIN . HDELX_(1.-A(J))86: X(IM)= XIN . HDELX_(1..A(J))87: GO TO 22088: 217 ANS= ANS + (Y(I)+Y(IM))%H(J)89: 220 CONTINUE90; IF(INDKT .EQ. 2) GO TO (230,240),MODE9 : GO TO (501,500),MODE92: 230 X(NN)= XIN.HDELX93; RETURN94: 240 ANS = ANS + Y(NN)_H(MAX.I)95: 500 ANSWR= HDELX_ANS96; 501 RETURN97; 400 DO 402 I=I,NN98: 402 X(I)= XIN99: 999 RETURN

100: END

33

1: SUBROUTINE WRT(NG,JT,Y,T,DELH,TEND,INDEX,IER)2:C3:C4:C THIS SUB WRITES OUTPUT FOR ETSION'S PROGRAM5:C6:C

7: DIMENSION Y(N6)8:C9: COMMON/COMI/ PIE,PIE2_N4PIE_NRtNTH_IWRT

10:C11: COMMON/COM2/ FKROC,RO,BETA,RI,CO,12: 1 RM,PO,PI,FI,FME,FM,RSP13:C14: COMMON/COM3/ HMIN,HMIN1,HMIN215: COMMON/COM4/ CAX_CAY_CZ,CAXD_CAYD,CZDl&: COMMON/COM5/ FMX,FMY,FZ,GAM,PSI,GAMD,PSIB,FC,17: 1 FM1,FMIC,FZ1,FM2,EPS,DEL18:C19:C20: TWRT=TEND21: IF(JT.LE.I) TWRT=O.22:C23: CAX=Y(1)24: CAY=Y(2)25: CZ=Y(3)26: CAXD=Y(4)27: CAYD=Y(5)2B: CZD=Y(6)29: CALL FQUAD30: YPRSVI=(FMX+FME)/FI31_ YPRSV2=FMY/FI32: YF'RSV3=FZ/FM33:C34: WRITE(7,102) INDEX,IER,Y(4),Y(5),Y(6),T,DELH,YPRSV1,YPRSV2,YPRSV3,35: 1" FC36:C37:C38:C39: S_XY=CAX_CAX + CAY_CAY40: GAM= SQRT(SQXY)41:C42: CAXX= ABS(CAX)43: CAYY= ABS(CAY)44: PSI= ATAN2(CAYY,CAXX)45: IF(CAX.LT.O.AND.CAY.GT.O) PSI=-PSI46: IF(CAX.GE.O.ANDoCAY.LT.O) PSI=-PSI47:C48: CPSI=CAX/GAM49: SPSI=CAY/OAM50:C51: GAMD=CAXD_CPSI+CAYB_SF'SI52: PSID=(CAYD_CF'SI-CAXD_SPSI)/GAM53:C54:C55: TWRTP=TWRT/PIE256_ WRITE(6,104) CZ,GAM,PSI,CZD,GAMD,PSID,CAX,57: I CAYFHMIN_TWRTP58:C59:C60: 104 FORMAT(1H ,10Ell.4)61: 102 FORMAT(2IS,gE11.4)62:C63: 90 RETURN64: END

34

APPENDIX D - FLOW CHARTS

MAIN EXECUTIVE PROGRAM

Initializeconstants&variables

Read&writenamelistinputs

tAssigninitialvaluesforgear package

tWriteinitialvalues

" ' Store

initialvalues inarrays

No

().

35

Call DEGARmoveaheadone timestep

Storeout-

put inarrays

,, No

Call ERTST Itestfor sealor gearpack- | -agefailure J

Yes

Write out- I Write out-

put at cur- put intorent time file 15

1IncrementtendbyDTEND

Yes

ComputeY. i/J. E. {)

ComputeHMIN.HMINl. andHMIN2

CallERMIN

36

No Set loger = FalseTouch = True

No Setlager =

false

37

tComputeintegralfunctionsandconstants

1Calculate thedoubleinte-grals forMI,M2,& Fz

1ComputeMx,My, Fc, &Fz

38

Is

Set HMIN- IDOEQ.IDO 0 savedHMIN 1HS=HI LT. HSTAB

Yes No

I Set HMIN- Set

TS=TS+ DTEND savedHMIN IDO=0GT.HSTAB HS =HI

SetNo HS =HI

TS=0IDO=1

SetIoger=false

39

APPENDIX E

SAMPLE PROBLEM

A computer output listing is shown as an example of a representative run.

The computer run generated output for twenty revolutions but only output from

the first, second, nineteenth, and the twentieth revolutions is shown. Input0

data of the two namelists, INPUT1 and INPUT2 is shown preceeding the listedoutput and this input is defined in the report section, USING THE PROGRAM.

In this case, the Adams predictor corrector method was used (METH = i), aninitial step size of 0.001 was tried (DELH = 0.001), functional iteration was

used (MITER = 0), and the input relative error bound, TOL, was set to 0.0001.

The initial values T = 0 (TZRO = 0) and _x = 0, ey = 0.i, Z = 0, _x = 0,_'v = 0, and Z = 0 (YIN = 0, 0.i,0,0,0,0) were set. TSTOP was set equal to

125.66 radians, or twenty revolutions of the rotor• Also, the inputs Ri = 0.9,Co

K _o = 0.0025, B = i0, Pi = 0, Po = 0.25, Me = 0, m = 0.08, rg = i, and RSP = 1were used.

The execution of this case required two minutes of computer time on theUnivac 1100/42.

$]NPUT!READ1 - FMETH : .1MILER : _'0TOL : .I0000000C-03INDEX : 41DELH : • IOOOOOOOE-O2TZRO : .OOOO0000E*O0YIN : "DOl'O00OO[eO0t • lOOOO000E't00, • OOOOO000E"OO, oO000OO00E*O0,

.Oooooo00E*OOt • O0000001_E*O0TSTOP : • 12566 37i E.D3DTENO : .1 25663T1E400NR : +5NTH : +B

HHSTOP : .II)O00000E-ZOHSTAB : • | ODODODOE -02NPLT : +1PUNCH : F

SEND$] NPUT2RI : .goooonooE .cOOFKROC : .25000000E-02BETA : . lO000000E 402PI : .OOOOOOO0E',,O0PO : .2SOODOOOE+ODFHE : .OOQOOOQOE400fM : .80OO00OOE-OlRG : • lOOOOOOOE40IRSP : •IDOOOOOOE*Ol

SEND

CZ 5AH PSI CZO GAPO P$10 CAX CAY HArN TWRT

• 0000 .lO00oO0 .1571401 .0000 .0000 .0000 .0000 .1000,00 .9130"00 .0000.1153-03 .9985-01 -.1569401 .1881-02 -.?279-02 .2177-01 -.1335-03 .9985-01 .Q101*00 .2000-01

• 4605-03 .99qq-01 -.1565401 ,3654-02 -.4422-02 .4282-01 -.5332-03 .9944-01 .9105400 .qOoo-Ol.1021-02 .9876-01 -.1559401 o5318-02 -.6421-02 .6338-01 -.1183-02 .9876-01 °9112400 .6000-01.1784-02 .9784-01 -.1550401 o6873-02 -.8268-02 .8367-C1 -.2072-02 .9782-01 .9121400 .8000-01

• 2735-02 .9670-01 -.1538401 .8320-02 -.9957-02 .1038"00 -.3180-02 .9665-01 o9132500 .1000"00.3871-02 .9535-01 -.1524401 .9658-02 -.1148-01 .1241500 -.4508-02 .952_-01 .91454G0 .1200500.5150-02 .9383-01 -.1507401 .1089-01 -.1284-01 .I445*00 -.5999-02 .9353-01 .9160400 .1400500.6590-02 o9214-01 -.1487401 o1202-01 -.1403-01 .1653400 -.7676-02 o9182-01 09176400 .1600400.8171-02 .9030-01 -.1465401 .1305-01 -.1504°0! .1866*Q0 -.9516-02 .8980-01 .9194,00 .1800400°9878-02 .8835-01 -.1440*01 .1399-01 -.1587-01 .2085400 -.1149-0| .8750-01 .921]+00 .2000500.1169-01 .8631-01 -.1413"01 .1482-01 -.1653-01 .2310400 -.1359-01 .8523-01 .9232+00 .2200*00.1361-01 .8420-01 -.1382401 .1556-01 -.1701-01 .2541*00 -.1579-01 .8271-01 .9252+Q0 .2400400.1560-01 .8204-01 -.1349*01 .1622-01 -.1732-01 .2780-00 -.1807-Q1 .8003-01 .9273"00 .2600500.176B-01 .1985-01 -.1312"01 .1678-01 -.1745-01 .3027400 -.2042-01 .7720-01 .9294400 °2800400• 198_-01 o7766-01 -.1273401 .1727-01 -.1739-01 .3282.00 -.2282-01 .7q23-0| .9315"00 .3000500o220_-01 .7549-01 -.1230,01 .1767-01 -.1716"01 .3543*00 -.2525-01 .7114-01 .9335400 .3200400.2426-01 .7336-01 -.1183401 .1800-01 -.1675-01 .3810"00 -.2]71-01 ,6792-01 .9355400 .3400*00• 2654-01 .7129-01 -.]134401 .1825-01 -.1616-01 .4080400 -.3017-0! .6459-01 .9375"00 .3600400.2885-01 .6930-01 -°1081401 .1843-01 -.1541-01 .4351,00 -.3261-01 .6115-01 .9194400 o3800400o3117-01 .6742-01 -.1024+01 .1855-01 -.1449-01 .4619+00 -.3503-0! .5761-01 .9412*00 .4000"00.3351-01 .6567-01 -o9048*00 .1860-01 -.1341-01 o4880"00 -.3741-01 .5398-01 .9428400 .4200400o3585-/31 o6406-01 -0901940 C o1860-01 -01219-01 05128400 -.3973-01 .5025-01 .94q3*00 04400*00.3819-01 .6261-01 -.8359*0Q o1853-01 -.1084°01 °5358400 -o4198-01 o4545-01 .9457400 04600400.4051-01 .6133-01 -.7673"00 .1841-01 -.9382-02 .5554400 -.4415-01 .4257-01 .9470500 .4800500• 4281-01 .6024-01 -°6962400 .1824-01 -.7845-02 .5739,00 -.4623-01 .3863-01 .9480400 .5000500.4509-01 .5936-01 °.6231500 .1802-01 -.6254-02 .5878"00 -.4820-01 .3464-01 e9489+00 .5200500o4734-01 o5867-01 -.5485,00 o1776-01 -o4641-02 05978500 -.5006-01 o3059-01 09496400 0540040004956-01 .5819-01 -.4730400 .1745-01 -.3037-02 .6037400 -.5180-01 .2651-01 o9501400 05600400 (_• 5173-01 .5791-01 -.3969_00 o1710-01 -.1473-02 .6053,00 -.5341-01 02239-_1 .9504400 eSBO0*O0o5386-01 .5783-01 -01209400 o1671-01 .2301-04 06029500 -.5487-01 01824-01 .9506400 0600040005594-01 05793-01 -02455400 .1629-01 01424-02 05969500 -05619-02 .1q08-01 .9506400 06200500o5797-01 .5820-01 -.1710400 .1584-01 .2710°02 .5879,00 -.5735-01 .9904-02 .9505500 .6400*00.5993-01 .58_2-01 -.9784-01 o1536-01 .3866-02 .5763"_0 -.5834-01 .5726-02 .9502400 .6600"00.6183-01 .5917-01 -.2627-01 .1485-01 .4882-02 .5629400 -.5915-01 .1555-02 *9498400 *6800500o6367-01 e5985-01 .4350-01 .1432-01 .5755-02 .5482400 -.5979-01 -.2603-02 .9494400 .7000"00.6543-01 .6063-01 .111440[ .1376-01 .6485-02 .5327400 -.6026-01 -.6737-02 °9488400 .7200400o6713-01 .6150-01 .1772+00 .1319-01 .7074-02 .5170"00 -.6053-01 -.1084-01 .9481-00 .7400500o6816-01 06242°01 .2411400 .1260-01 .7530-02 .5015400 -.6062-01 -.1491-01 .9474400 07600400o7030-01 .6339-01 .3031,0C .1199-01 07863-02 .4865*00 -o6050-01 -o1892-01 °9467+00 .7800.0007177-01 .6440-01 .3633,00 o1137-01 .8080-02 .4723,00 -.6020-01 -.2289-01 .9459400 08000400• 7317-01 06543-01 04217400 .107_-01 08192-02 .4589400 -o5970-01 -.2678-01 09451400 08200400o7448-01 06647-01 .4785-0 0 o1011-01 08209-02 04465500 -05901-Q1 -.3061-01 .9443+00 .8400400o7572-01 o6751-01 05338,00 .9467-02 .8144-02 .4352,00 -.5812-0! -o3435-01 .9435.00 °8600500o7687-01 o6853-01 .587740£ .8822-02 .8006-02 04251400 -.5703-01 -.3800-01 .9427500 .8800*00°7794-01 .6953-01 06405400 .8176-02 °7806-02 .4160.00 -.5575-01 -.4155-01 .9420"00 .9000,00o7893-01 .7050-0! .6922400 °7530-02 .7553-02 .4080400 -.5427-01 -04499-01 .9412400 .9200"00°798_-01 07143-01 .7429,00 06855-02 07254-02 04010"fl0 -.5261-01 -.4832-01 .9405+Q0 .9400500.8067-01 .7233-01 .7928400 o6245-02 .6918-02 .3950"00 -.5076-01 -.5152-_1 .9398+00 .9600"00.8142-01 .7318-01 °8420400 .5610-02 .6553-02 .3899490 -04874-01 -.5460-01 09391500 .9800"00°8208-01 .7359-01 .8907400 ._983-02 .6166-02 .3857-00 -.4653-01 _.5753-0| .9385_00 .1000501.8267-01 .7474-01 .9389500 04365-02 .5764-02 .3823*00 -.4415-01 -.6031-01 .9379400 o1020"01• 8318-01 .7544-01 .9867400 .3757-02 05351-02 03797500 -.4160-_| -.6293-01 09373500 01040401o8362-01 .7609-01 .1034501 .3161-02 .4934-02 .3778400 -.3_90-_I -.6540-01 .9368+00 .1060501o8399-01 .7669-01 .I082401 .2578-02 .4516-02 .3766400 -o3604-01 -o6770-01 .9363,00 o1080+01

• 8_28-_1 07724-01 .1129+01 02010-02 04103-02 .3759400 -.3305-0! -o6982-D! .9159500 °1100501

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44

REFERENCES

i. Etsion, I: Dynamic Analysis of Noncontacting Face Seals. NASA TM-79294,1980.

2. Gear, C. William: Numerical Initial Value Problems in Ordinary Differential

Equations. Pretice-Hall, Inc., 1971.

' 3. Scarborough, J. B.: Numerical Mathematical Analysis. Third ed. JohnHopkins Press, 1955, pp. 145-150.

4. Goldstein, Charles M.: Numerical Integration by Gaussian Quadrature. NASA

o TM X-52088, 1965, pp. 9-10.

I

_-Spring ,"Primarysealring;

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__ Shaft#

Figure1. - Schematicof a radialfaceseal.

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_SEALSEATJ

x

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Figure2.-Sealmodelandcoordinatessystems.

CS-BO-3199

3 z

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x y

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Yl

Figure3. * Orientationofrotatingcooridinatesystem]23 in inertialreferencexyz.

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1. Report No. 2. Government Accession No. 3. Recipient's Catalog No

NASA TM-81636

4. Title and Subtitle 5. Report Date

COMPUTER PROGRAM DOCUMENTATION FOR THE DYNAMIC November 19806. Performing Organization Code

ANALYSIS OF A NONCONTACTING MECHANICAL FACE SEAL

7. Author(s) 8. Performing Organization Report No.

B. M. Auer and I. Etsion E-650 s10. Work Unit No.

9. PerformingOrganizationNameand Address

NationalAeronauticsand Space Administration11. Contract or Grant No. A

Lewis Research Center

Cleveland,Ohio 4413513. Type of Report and PeriodCovered

12. Sponsoring Agency Name and Address TechnicalMemorandumNationalAeronauticsand Space Administration

14. Sponsoring Agency CodeWashington,D.C. 20546

15. Supplementary Notes

16. Abstract

A computer program is presented which achieves a numerical solution for the equations of

motion of a noncontacting mechanical face seal. The flexibly-mounted primary seal ring motion

was expressed by a set of second order differential equations for three degrees of freedom.

These equations were reduced to a set of first order equations and the GEAR software package

was used to solve the set of first order equations. Program input includes seal design parame-

ters and seal operating conditions. Output from the program includes velocities and displace-

ments of the seal ring about the axis of an inertial reference system. One example problem isdescribed.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Computer program Unclassified-unlimited =

Software STAR Category 37

Sealdynamics

Mechanicalfaceseals

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price*

Unclassified Unclassified

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