Oil Film Thickness Predictions on Roller Bearings
Dipped in Oil Baths
Jeff Bode
MEAE 4960
April 10, 2001
1
Table of Contents:
List of Symbols Used: ................................................................................................................................... 3
Abstract: ........................................................................................................................................................ 4
Introduction: ................................................................................................................................................. 4
Problem Description and Mathematical Formulation: ............................................................................. 4
Numerical Approaches ................................................................................................................................. 7
Results and Discussion ................................................................................................................................. 8
Error Analysis............................................................................................................................................... 9
Discussion ...................................................................................................................................................... 9
Conclusion ....................................................................................................................................................10
Bibliography: ...............................................................................................................................................11
Appendix I: Program .................................................................................................................................12
Appendix II: Samples of Output ...............................................................................................................15
2
List of Symbols Used:

c
p
Ca


F

g

R
U
x
y
Constant of integration
Capillary Length
Change in pressure
Capillary Number
Surface Tension
Oil thickness as measured from bearing surface to air
Adhering oil film thickness
Ratio of oil film thickness to adhering oil film thickness
Acceleration due to gravity
Viscosity
Radius of curvature of oil film between oil bath and bearing
surface
Withdrawal Speed
vertical distance from bearing surface to oil bath
horizontal distance from bearing surface (not limited by oil/air
line)
3
Abstract:
Roller bearings are used in many applications in many different industries. During the
overhauling process of many machines, it becomes necessary to remove roller bearings
for preservation, as bearing material is highly susceptible to corrosion. In order to
preserve the bearings, they are dipped in an oil bath and then sealed in a plastic or other
non-permeable bag or container. It is the intent of this paper to analyze whether the
amount of oil film accumulated on an aerospace bearing may be correlated to the
equations for the static meniscus or the entrained fluid.
Introduction:
There is a continual drive within industry to decrease quality escapes and eliminate
waste. When a bearing is left unpreserved for more than a few hours, atmospheric
corrosion is likely to occur, rendering the bearing unusable. The financial impact of this
can be anything from a three dollar bearing replacement in a common household tool to a
ten thousand dollar aerospace bearing replacement with the loss of critical time in
locating spare assets for replacement. This can hold up a machine build that can be billed
at $5000 per hour of delay when on a scheduled contract.
This paper will assess the effects of removal of roller bearings from oil baths to
determine the thickness of the oil film on the bearing material. The objective is to
determine whether there is a correlation between the oil film thickness and the static
meniscus, and whether this correlation could have any impact to oil film thickness on
bearings. This correlation could be used to develop practices to help prevent bearing
material corrosion damage.
This analysis was conducted using FORTRAN numerical analysis. There were several
numerical integrations that needed to occur in order to arrive at the predicted oil film
coating thickness. These integrations were performed using the Runge-Kutta Method for
Systems of Differential Equations. The goal is to solve a third order differential equation
using three separate linear differential equations. Because this will still require an initial
guess for the film thickness, this becomes an iterative problem.
Problem Description and Mathematical Formulation:
The roller bearings were modeled as an infinite plate being withdrawn from an oil bath.
The geometries involved with every detail involved with the roller bearings are such that
it would be impractical to model the intricacies of these complex parts. Also, because
this is a generic exercise to possibly apply this analysis to various bearings, a specific
bearing geometry is not required to complete the analysis. The final formulation for flat
plate geometry is also known, so that the analysis may be numerically validated.
4
The velocity profile is represented by the equation:

 d 3  y 2

u U 
 y 
3 
 dx  2

Using this formula, the oil flow rate per unit width of plate (or as we assume here –
circumference of the roller bearing) is shown as:
 x
~
 d 3  3
Q   udy  U 
 dx 3 3
0
As x approaches infinity, the oil film flow rate may be shown as
~
Q  U f
~
The substitution of these 2 equations for Q yields:
d 3  3U 
 3U 


  
 f
3
dx
  
  
The goal of this derivation is to determine the initial conditions required for integrating
the value of  with certain initial conditions.
3
The value of delta at x = 0 is numerically impossible to achieve. This is because the
static meniscus equation does not apply here. The best that we can do from a numerical
integration method is to use as small a value of x as practical. But let us understand what
it is that must be solved.
For this derivation, constant temperature is assumed. This gives us the formulation for
the variable (x) shown in the above equation.
5
U is being approximated as 0.2 m/s. The quantity
3U

may then be deemed a constant.
The material properties of the oil were taken at a temperature of 300 K. The following
values were used in the analysis:
 = 0.486 N*s/m2
 = 30.23 N/m
 = 884.1 kg/m3
Also, at or very near x = 0,    F ,
3
d 3  3U 

  0
dx 3   
Consequently,
2
d 3  3U 

0
dx 3   
Because
   at x = 0,
 3U 
 3U 

 

d 
 
 




0
dx 3
2
 2
3
This would mean that
d 3
 dx 3  C1
Probstein argues that the application of static equilibrium makes
p  gx  constant and therefore,
 ' '
 gx  0
3
2 2
1   ' 
Rearranging gives:
gx 1   ' 2 2
 '' 

The results are shown graphically below:
3
Delta Double Prime Initial Condition
Delta Double
Prime
8.E+08
6.E+08
4.E+08
2.E+08
0.E+00
0
0.001
0.002
X (Meters Up Plate)
0.003
It is important to understand the physical
significance of  ' ' . This is the change in the
rate of change of the thickness of the oil film
on the plate. Its integral,  ' , is the change in
the oil film thickness as a function of x. At x =
0, this value is negative infinity. A value of
infinity is not useful for a numerical
integration, so we apply a constant of –1 when
6
we integrate  ' ' .
'
1   ' 
2

gx 2
1
2
Algebraic manipulation then yields:
 2 g 2 x 4 gx 2

1
4 2

 ' 
  2 g 2 x 4 gx 2

1  

 1
2

 4

Delta Prime Initial Condition as a
Function of X
0
0.001
0.002
0.003
0
Delta Prime
It is also noted that the derivation to get to the
above expression required the use of the
negative, instead of positive  ' . We shall
only use the negative value of the above
expression for the numerical calculations.
The reason being that the change in delta as x
increases is negative. The graphical
representation of this is shown to the right.
-2000
-4000
-6000
-8000
X (Meters Up Plate)
Numerical Approaches
The computation of the initial value of  then becomes quite complex, and not very
useful. Therefore, this initial value will need to be approximated.
The x value used for computation was 0.0001m. The following values were obtained
from the above formulas:
 ' ' 0.0001  5904022 =3
 ' 0.0001  590.389 =2
 0.0001  .064684 =1
Though these are the approximations for the initial conditions, the full equations will be
used to determine the 3 governing equations in the numerical integration. The program
will also require an inputted guess for F. When the initial guess yields an answer that
supports the guess, the solution is obtained.
u1  
u2   '
u3   ' '
u1 '   '
u2 '   ' '
u3 '   ' ' '
7
The 3 equations input into the Runge-Kutta Method for Differential Equations are:
f 1 t , x1 , x 2 , x 3   x 2
f 2 t , x1 , x 2 , x 3   x 3
f 3 t , x1 , x 2 , x 3   
.00964604697321 x1   F 
x1 3
The Runge-Kutta program Alg057.for supplied with the class textbook was modified to
show the initial values put into the program, as well as adjust the order of the overall
problem from a second order differential equation to a third order differential equation.
Results and Discussion
Using initial conditions for alpha previously listed, it was shown that the value of  ' ' did
not change during the iterations,  ' increased steadily, and  decreased until it
numerically crossed the plate at a value of x = .0002m with an initial guess of .0001 for
F .
t(i)
1E-04
0.00012
0.00014
0.00016
0.00018
0.0002
0.00022
0.00024
0.00026
0.00028
0.0003
w1(i)
w2(i)
w3(i)
0.064684004
-590.38898 5904022
0.054057028
-472.30853 5904022
0.045791663
-354.22809 5904022
0.039887905
-236.14764 5904022
0.036345758
-118.0672 5904022
0.035165217 0.013244629 5904022
0.036346287
118.09369 5904022
0.039888963
236.17413 5904022
0.04579325
354.25458 5904022
0.054059148
472.33502 5904022
0.064686656
590.41547 5904022
This shows a dominance of the w3 term (  ' ' ) on the overall iteration process. It
becomes clear that an error exists in the initial approximation of  ' ' . An increase in the
precision of the initial condition to x=0.00001 only amplifies the impact of the initial
approximation  ' ' .
1  0.12800545   0.00001
 2  -5903.8311   ' 0.00001
 3  590383230   ' ' 0.00001
8
t(i)
1E-05
3.9E-05
6.8E-05
9.7E-05
0.000126
0.000155
0.000184
0.000213
0.000242
0.000271
0.0003
w1(i)
w2(i)
w3(i)
0.128005 -5903.831 590383230
0.205051 11217.28 590383230
0.778608
28338.4 590383230
1.848678 45459.52 590383230
3.41526 62580.63 590383230
5.478355 79701.75 590383230
8.037962 96822.87 590383230
11.09408
113944 590383230
14.64671 131065.1 590383230
18.69586 148186.2 590383230
23.24152 165307.3 590383230
The value of delta reaches its minimum between the first and second iteration.
Error Analysis
Probstein uses the following equation to derive the exact value of  F :
f
c
 0.946Ca 2 3
Where
Ca 
U

c 

g
and
 f  0.00121668201825m
The use of this value for the initial guess of  f does not significantly affect the output of
the program. The value of  ' ' still dominates the output. The program’s predicted value
of x to this value is still closest to ~0.0002m which actually only reaches a minimum
value that correlates to a value of  f equal to 0.035165m. This gives an error of 2790%.
This result then causes the reevaluation of the initial conditions. The initial conditions
for  ' ' and  ' are numerically determined. However, the initial condition for  is the
only input that is guesstimated. This input is therefore the condition under suspicion of
detrimentally affecting the analysis.
(0.0001)= 0.31m
Discussion
0.002
Delta (Meters)
The initial results cause scrutiny of
the initial conditions. The initial
conditions for  ' ' and  ' are
numerically determined and have a
basis for their estimates. However,
0.0015
0.001
0.0005
0
0.0189
0.019
0.0191
0.0192
0.0193
X (meters up plate)
0.0194
9
0.0195
 had no basis for its estimate. This input was originally estimated as ~6.4cm.
Subsequent iterations with differing values of  at x = 0.0001m produced a more
acceptable input in around 31cm. However, this still could not produce the results
desired.
The output of the series of first order differential equations was greatly affected by the
value of  ' ' . The magnitude of the initial condition value for  swayed the rest of the
output. So that the  ' became positive at values greater than x=0.0002 m. This should
never have happened.  ' should have remained negative for any value of x.
Delta Double Prime
(0.0001)=0.31m
5904900
5904800
5904700
5904600
5904500
5904400
5904300
5904200
5904100
5904000
5903900
0.0189
0.019
0.0191
0.0192
0.0193
0.0194
0.0195
Probstein reports that the
value where  ' goes to
zero is the transition from
the static meniscus region
to the lubrication /
entrained fluid region.  '
was not able to be
calculated to a value of
zero anywhere near the
correlating value of
   f . Values for
 ' ' were mostly
unchanging until the
region extremely close to the plate were analyzed, and this graph shows a curve
beginning near value of  = 0.
X (Meters up plate)
Conclusion
The goal of obtaining a correlation between the oil film thickness on a bearing surface
after withdrawal from an oil bath was unsuccessful. The correlation for a flat plate being
withdrawn from an oil bath was the first step towards attaining that goal. It was clear that
the inputted initial conditions had a huge effect on the accuracy of the analysis. Though
initially, the large value of  ' ' was the initial suspect for the large errors being obtained,
it was noted that the initial value of  was just as important in obtaining a correct
solution. The use of the Runge-Kutta Method for Systems of Differential Equations was
able to be applied to the system of equations derived for this analysis, but the validity of
the initial conditions prevented accurate results from being obtained. Physical and
chemical analysis of these boundary conditions would be needed to obtain some accurate
predictions on oil film thickness.
10
Bibliography:
Burden and Faires. Numerical Analysis, California, Brooks/Cole, 2001, pp. 313-322.
Levich, Veniamin G. Physicochemical Hydrodynamics, Englewood Cliffs NJ, 1962, pp.
675-683.
Probstein, Ronald F. Physicochemical Hydrodynamics, Boston, Butterworths, pp. 280285.
11
Appendix I: Program
C**********************************************************************
*
C
*
C
RUNGE-KUTTA FOR SYSTEMS OF DIFFERENTIAL EQUATIONS ALGORITHM 5.7
*
C
*
C**********************************************************************
*
C
C
C
C
TO APPROXIMATE THE SOLUTION OF THE MTH-ORDER SYSTEM OF FIRSTC
ORDER INITIAL-VALUE PROBLEMS
C
UJ' = FJ(T,U1,U2,...,UM), J=1,2,...,M
C
A <= T <= B, UJ(A)=ALPHAJ, J=1,2,...,M
C
AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].
C
C
INPUT ENDPOINTS A,B; NUMBER OF EQUATIONS M; INTIAL
C
CONDITIONS ALPHA1,...,ALPHAM; INTEGER N.
C
C
OUTPUT APPROXIMATIONS WJ TO UJ(T) AT THE (N+1) VALUES OF T.
C
CHARACTER NAME1*30,AA*1
INTEGER OUP,FLAG
LOGICAL OK
c
WRITE(6,*) 'ENTER GUESS FOR DELTA SUB F'
c
READ(5,*) C
C
CHANGE FUNCTIONF F1, F2 AND F3 FOR A NEW PROBLEM
F1(T,X1,X2,X3)=X2
F2(T,X1,X2,X3)=X3
F3(T,X1,X2,X3)=-0.00964604697321*(X1-.0012)/(X1*X1*X1)
C
DEFINE FUNCTIONS F1,...,FM
WRITE(6,*) 'This is the Runge-Kutta Method for systems with m=3.'
WRITE(6,*) 'Have the functions F1, F2, and F3 been defined?'
WRITE(6,*) 'Enter Y or N '
WRITE(6,*) ' '
READ(5,*) AA
IF(( AA .EQ. 'Y' ) .OR. ( AA .EQ. 'y' )) THEN
OK = .FALSE.
10
IF (OK) GOTO 11
WRITE(6,*) 'Input left and right endpoints separated by'
WRITE(6,*) 'blank'
WRITE(6,*) ' '
READ(5,*) A, B
IF (A.GE.B) THEN
WRITE(6,*) 'Left endpoint must be less'
WRITE(6,*) 'than right endpoint'
ELSE
OK = .TRUE.
ENDIF
GOTO 10
11
OK = .FALSE.
12
12
13
6
C
C
WRITE(6,*) 'Input the three initial conditions.'
WRITE(6,*) ' '
READ(5,*) ALPHA1, ALPHA2, ALPHA3
IF (OK) GOTO 13
WRITE(6,*) 'Input a positive integer for the number'
WRITE(6,*) 'of subintervals '
WRITE(6,*) ' '
READ(5,*) N
IF ( N .LE. 0 ) THEN
WRITE(6,*) 'Must be positive integer '
ELSE
OK = .TRUE.
ENDIF
GOTO 12
CONTINUE
ELSE
WRITE(6,*) 'The program will end so that the functions'
WRITE(6,*) 'F1, F2 and F3 can be created '
OK = .FALSE.
ENDIF
IF(.NOT.OK) GOTO 400
WRITE(6,*) 'Select output destination: '
WRITE(6,*) '1. Screen '
WRITE(6,*) '2. Text file '
WRITE(6,*) 'Enter 1 or 2 '
WRITE(6,*) ' '
READ(5,*) FLAG
IF ( FLAG .EQ. 2 ) THEN
WRITE(6,*) 'Input the file name in the form - '
WRITE(6,*) 'drive:name.ext'
WRITE(6,*) 'with the name contained within quotes'
WRITE(6,*) 'as example:
''A:OUTPUT.DTA'' '
WRITE(6,*) ' '
READ(5,*) NAME1
OUP = 3
OPEN(UNIT=OUP,FILE=NAME1,STATUS='NEW')
ELSE
OUP = 6
ENDIF
WRITE(OUP,*) 'RUNGE-KUTTA METHOD FOR SYSTEMS'
WRITE(OUP,*) 'FUNCTIONS USED ARE:'
WRITE(OUP,*) 'F1(T,X1,X2,X3)=X2'
WRITE(OUP,*) 'F2(T,X1,X2,X3)=X3'
WRITE(OUP,*) 'F3(T,X1,X2,X3)=-0.00964604697321*(X1-.0012)/(X1*X
+1*X1)'
WRITE(OUP,*) 'ALPHA1=',ALPHA1
WRITE(OUP,*) 'ALPHA2=',ALPHA2
WRITE(OUP,*) 'ALPHA3=',ALPHA3
WRITE(OUP,*) 'LIMITS ARE ',A,' TO ',B
WRITE(OUP,*) 'WITH ',N,' SUBINTERVALS'
WRITE(OUP,6)
FORMAT(12X,'t(i)',11X,'w1(i)',11X,'w2(i)',11X,'w3(i)')
STEP 1
H=(B-A)/N
T=A
STEP 2
W1=ALPHA1
13
C
C
C
C
C
C
C
C
C
110
C
400
1
W2=ALPHA2
W3=ALPHA3
STEP 3
WRITE(OUP,1) T,W1,W2,W3
STEP 4
DO 110 I=1,N
STEP 5
X11=H*F1(T,W1,W2,W3)
X12=H*F2(T,W1,W2,W3)
X13=H*F3(T,W1,W2,W3)
STEP 6
X21=H*F1(T+H/2,W1+X11/2,W2+X12/2,W3+X13/2)
X22=H*F2(T+H/2,W1+X11/2,W2+X12/2,W3+X13/2)
X23=H*F3(T+H/2,W1+X11/2,W2+X12/2,W3+X13/2)
STEP 7
X31=H*F1(T+H/2,W1+X21/2,W2+X22/2,W3+X23/2)
X32=H*F2(T+H/2,W1+X21/2,W2+X22/2,W3+X23/2)
X33=H*F3(T+H/2,W1+X21/2,W2+X22/2,W3+x23/2)
STEP 8
X41=H*F1(T+H,W1+X31,W2+X32,W3+X33)
X42=H*F2(T+H,W1+X31,W2+X32,W3+X33)
X43=H*F3(T+H,W1+X31,W2+X32,W3+X33)
STEP 9
W1=W1+(X11+2*X21+2*X31+X41)/6
W2=W2+(X12+2*X22+2*X32+X42)/6
W3=W3+(X13+2*X23+2*X33+X43)/6
STEP 10
T=A+I*H
STEP 11
WRITE(OUP,1) T,W1,W2,W3
CONTINUE
STEP 12
CLOSE(UNIT=5)
CLOSE(UNIT=OUP)
IF(OUP.NE.6) CLOSE(UNIT=6)
STOP
FORMAT(3(1X,E15.8))
END
14
Appendix II: Samples of Output
RUNGE-KUTTA METHOD FOR SYSTEMS
FUNCTIONS USED ARE:
F1(T,X1,X2,X3)=X2
F2(T,X1,X2,X3)=X3
F3(T,X1,X2,X3)=-0.00964604697321*(X1-.0350)/(X1*X
1*X1)
ALPHA1=
6.46840E-02
ALPHA2=
-590.389
ALPHA3=
590402.
LIMITS ARE
1.00000E-04 TO
3.00000E-04
WITH
100 SUBINTERVALS
t(i)
w1(i)
w2(i)
w3(i)
0.99999997E-04 0.64684004E-01 -0.59038898E+03 0.59040219E+06
0.10200000E-03 0.63504405E-01 -0.58920819E+03 0.59040219E+06
0.10400000E-03 0.62327169E-01 -0.58802740E+03 0.59040219E+06
0.10600000E-03 0.61152294E-01 -0.58684662E+03 0.59040219E+06
0.10800000E-03 0.59979782E-01 -0.58566583E+03 0.59040219E+06
0.10999999E-03 0.58809631E-01 -0.58448505E+03 0.59040219E+06
0.11199999E-03 0.57641841E-01 -0.58330426E+03 0.59040219E+06
0.11399999E-03 0.56476414E-01 -0.58212347E+03 0.59040219E+06
0.11600000E-03 0.55313349E-01 -0.58094269E+03 0.59040219E+06
0.11800000E-03 0.54152645E-01 -0.57976190E+03 0.59040219E+06
0.12000000E-03 0.52994303E-01 -0.57858112E+03 0.59040219E+06
0.12200000E-03 0.51838323E-01 -0.57740033E+03 0.59040219E+06
0.12400000E-03 0.50684705E-01 -0.57621954E+03 0.59040219E+06
0.12600000E-03 0.49533445E-01 -0.57503876E+03 0.59040219E+06
0.12800000E-03 0.48384547E-01 -0.57385797E+03 0.59040219E+06
0.13000000E-03 0.47238011E-01 -0.57267719E+03 0.59040219E+06
0.13200000E-03 0.46093836E-01 -0.57149640E+03 0.59040219E+06
0.13400000E-03 0.44952024E-01 -0.57031561E+03 0.59040219E+06
0.13600000E-03 0.43812573E-01 -0.56913483E+03 0.59040219E+06
0.13800000E-03 0.42675484E-01 -0.56795404E+03 0.59040219E+06
0.13999999E-03 0.41540757E-01 -0.56677325E+03 0.59040219E+06
0.14200000E-03 0.40408392E-01 -0.56559247E+03 0.59040219E+06
0.14399999E-03 0.39278388E-01 -0.56441168E+03 0.59040219E+06
0.14600001E-03 0.38150746E-01 -0.56323090E+03 0.59040219E+06
0.14799999E-03 0.37025467E-01 -0.56205011E+03 0.59040219E+06
0.14999999E-03 0.35902549E-01 -0.56086932E+03 0.59040219E+06
0.15199999E-03 0.34781992E-01 -0.55968854E+03 0.59040219E+06
0.15399999E-03 0.33663794E-01 -0.55850775E+03 0.59040219E+06
0.15599999E-03 0.32547958E-01 -0.55732697E+03 0.59040219E+06
0.15800000E-03 0.31434484E-01 -0.55614618E+03 0.59040219E+06
0.16000000E-03 0.30323371E-01 -0.55496539E+03 0.59040219E+06
0.16200000E-03 0.29214621E-01 -0.55378461E+03 0.59040219E+06
0.16400000E-03 0.28108232E-01 -0.55260382E+03 0.59040219E+06
0.16600000E-03 0.27004205E-01 -0.55142303E+03 0.59040219E+06
0.16800000E-03 0.25902539E-01 -0.55024225E+03 0.59040219E+06
0.17000000E-03 0.24803236E-01 -0.54906146E+03 0.59040219E+06
0.17200000E-03 0.23706295E-01 -0.54788068E+03 0.59040219E+06
0.17400000E-03 0.22611715E-01 -0.54669989E+03 0.59040219E+06
0.17600000E-03 0.21519495E-01 -0.54551910E+03 0.59040219E+06
0.17799999E-03 0.20429637E-01 -0.54433832E+03 0.59040219E+06
0.18000000E-03 0.19342141E-01 -0.54315753E+03 0.59040219E+06
0.18199999E-03 0.18257007E-01 -0.54197675E+03 0.59040219E+06
15
0.18400000E-03
0.18599999E-03
0.18800001E-03
0.18999999E-03
0.19200001E-03
0.19399999E-03
0.19600001E-03
0.19799999E-03
0.19999999E-03
0.20200000E-03
0.20400000E-03
0.20600000E-03
0.20800000E-03
0.21000000E-03
0.21200000E-03
0.21400000E-03
0.21600000E-03
0.21800000E-03
0.21999999E-03
0.22200000E-03
0.22399999E-03
0.22600000E-03
0.22799999E-03
0.23000001E-03
0.23199999E-03
0.23400001E-03
0.23599999E-03
0.23800001E-03
0.23999999E-03
0.24200001E-03
0.24400000E-03
0.24600001E-03
0.24800000E-03
0.24999998E-03
0.25200000E-03
0.25399998E-03
0.25600000E-03
0.25799999E-03
0.26000000E-03
0.26199999E-03
0.26400000E-03
0.26599999E-03
0.26800000E-03
0.26999999E-03
0.27200000E-03
0.27399999E-03
0.27600001E-03
0.27799999E-03
0.28000001E-03
0.28199999E-03
0.28400001E-03
0.28599999E-03
0.28800001E-03
0.29000000E-03
0.29200001E-03
0.29400000E-03
0.29599998E-03
0.17174235E-01
0.16093824E-01
0.15015774E-01
0.13940087E-01
0.12866760E-01
0.11795795E-01
0.10727191E-01
0.96609499E-02
0.85970694E-02
0.75355507E-02
0.64763939E-02
0.54195984E-02
0.43651648E-02
0.33130925E-02
0.22633818E-02
0.12160328E-02
0.17104519E-03
-0.87158068E-03
-0.19118447E-02
-0.29497470E-02
-0.39852876E-02
-0.50184666E-02
-0.60492838E-02
-0.70777396E-02
-0.81038335E-02
-0.91275657E-02
-0.10148936E-01
-0.11167944E-01
-0.12184591E-01
-0.13198877E-01
-0.14210800E-01
-0.15220362E-01
-0.16227562E-01
-0.17232399E-01
-0.18234875E-01
-0.19234989E-01
-0.20232741E-01
-0.21228133E-01
-0.22221163E-01
-0.23211831E-01
-0.24200138E-01
-0.25186082E-01
-0.26169665E-01
-0.27150886E-01
-0.28129745E-01
-0.29106243E-01
-0.30080378E-01
-0.31052152E-01
-0.32021564E-01
-0.32988615E-01
-0.33953305E-01
-0.34915633E-01
-0.35875600E-01
-0.36833204E-01
-0.37788447E-01
-0.38741328E-01
-0.39691847E-01
-0.54079596E+03
-0.53961517E+03
-0.53843439E+03
-0.53725360E+03
-0.53607281E+03
-0.53489203E+03
-0.53371124E+03
-0.53253046E+03
-0.53134967E+03
-0.53016888E+03
-0.52898810E+03
-0.52780731E+03
-0.52662653E+03
-0.52544574E+03
-0.52426495E+03
-0.52308417E+03
-0.52190338E+03
-0.52072247E+03
-0.51954163E+03
-0.51836078E+03
-0.51717993E+03
-0.51599908E+03
-0.51481824E+03
-0.51363739E+03
-0.51245654E+03
-0.51127567E+03
-0.51009479E+03
-0.50891391E+03
-0.50773303E+03
-0.50655215E+03
-0.50537128E+03
-0.50419040E+03
-0.50300952E+03
-0.50182864E+03
-0.50064777E+03
-0.49946689E+03
-0.49828601E+03
-0.49710513E+03
-0.49592426E+03
-0.49474338E+03
-0.49356250E+03
-0.49238162E+03
-0.49120074E+03
-0.49001987E+03
-0.48883899E+03
-0.48765811E+03
-0.48647723E+03
-0.48529636E+03
-0.48411548E+03
-0.48293460E+03
-0.48175372E+03
-0.48057285E+03
-0.47939197E+03
-0.47821109E+03
-0.47703021E+03
-0.47584933E+03
-0.47466846E+03
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040219E+06
0.59040231E+06
0.59042606E+06
0.59043769E+06
0.59043731E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
0.59043725E+06
16
0.29800000E-03 -0.40640004E-01 -0.47348758E+03
0.29999999E-03 -0.41585799E-01 -0.47230670E+03
0.59043725E+06
0.59043725E+06
RUNGE-KUTTA METHOD FOR SYSTEMS
FUNCTIONS USED ARE:
F1(T,X1,X2,X3)=((20578.06384*T**4-286.901124711*T**2124711*t**220578.06384*T**4))**.5
F2(T,X1,X2,X3)=286.901*T*((1+X2**2)**1.5)
F3(T,X1,X2,X3)=-0.00964604697321*(X1-.0012)/(X1*X
1*X1)
ALPHA1=
0.310000
ALPHA2=
-590.000
ALPHA3=
5.90402E+06
LIMITS ARE
1.00000E-04 TO
2.40000E-02
WITH
1000 SUBINTERVALS
t(i)
w1(i)
w2(i)
w3(i)
0.99999997E-04 0.31000000E+00 -0.59000000E+03 0.59040220E+07
0.12390000E-03 0.29736018E+00 -0.47633777E+03 0.59040220E+07
0.14779999E-03 0.28695172E+00 -0.39936487E+03 0.59040220E+07
0.17170000E-03 0.27810499E+00 -0.34379791E+03 0.59040220E+07
0.19560001E-03 0.27041230E+00 -0.30180133E+03 0.59040220E+07
0.21950000E-03 0.26360717E+00 -0.26894626E+03 0.59040220E+07
0.24339999E-03 0.25750589E+00 -0.24254143E+03 0.59040220E+07
0.26730000E-03 0.25197643E+00 -0.22085735E+03 0.59040220E+07
0.29120001E-03 0.24692076E+00 -0.20273203E+03 0.59040220E+07
0.31510001E-03 0.24226408E+00 -0.18735587E+03 0.59040220E+07
0.33899999E-03 0.23794799E+00 -0.17414752E+03 0.59040220E+07
0.36290000E-03 0.23392604E+00 -0.16267876E+03 0.59040220E+07
0.38680001E-03 0.23016070E+00 -0.15262715E+03 0.59040220E+07
0.41070001E-03 0.22662118E+00 -0.14374533E+03 0.59040220E+07
0.43459999E-03 0.22328192E+00 -0.13584030E+03 0.59040220E+07
0.45850000E-03 0.22012147E+00 -0.12875934E+03 0.59040220E+07
0.48240001E-03 0.21712165E+00 -0.12237997E+03 0.59040220E+07
0.50630001E-03 0.21426691E+00 -0.11660284E+03 0.59040220E+07
0.53020002E-03 0.21154389E+00 -0.11134652E+03 0.59040220E+07
0.55409997E-03 0.20894095E+00 -0.10654361E+03 0.59040220E+07
0.57799998E-03 0.20644796E+00 -0.10213787E+03 0.59040220E+07
0.60189998E-03 0.20405599E+00 -0.98081993E+02 0.59040220E+07
0.62579999E-03 0.20175718E+00 -0.94335892E+02 0.59040220E+07
0.64970000E-03 0.19954453E+00 -0.90865387E+02 0.59040220E+07
0.67360001E-03 0.19741185E+00 -0.87641136E+02 0.59040220E+07
0.69750001E-03 0.19535354E+00 -0.84637833E+02 0.59040220E+07
0.72140002E-03 0.19336458E+00 -0.81833511E+02 0.59040220E+07
0.74530003E-03 0.19144046E+00 -0.79209030E+02 0.59040220E+07
0.76919998E-03 0.18957709E+00 -0.76747635E+02 0.59040220E+07
0.79309999E-03 0.18777074E+00 -0.74434570E+02 0.59040220E+07
0.81699999E-03 0.18601803E+00 -0.72256828E+02 0.59040220E+07
0.84090000E-03 0.18431588E+00 -0.70202866E+02 0.59040220E+07
0.86480001E-03 0.18266143E+00 -0.68262421E+02 0.59040220E+07
0.88870001E-03 0.18105210E+00 -0.66426338E+02 0.59040220E+07
0.91260002E-03 0.17948548E+00 -0.64686417E+02 0.59040220E+07
0.93650003E-03 0.17795938E+00 -0.63035294E+02 0.59040220E+07
0.96040004E-03 0.17647174E+00 -0.61466343E+02 0.59040220E+07
0.98429993E-03 0.17502068E+00 -0.59973576E+02 0.59040220E+07
0.10082000E-02 0.17360444E+00 -0.58551575E+02 0.59040220E+07
0.10320999E-02 0.17222139E+00 -0.57195423E+02 0.59040220E+07
0.10560000E-02 0.17087001E+00 -0.55900650E+02 0.59040220E+07
17
0.10799000E-02
0.11037999E-02
0.11277000E-02
0.11515999E-02
0.11755000E-02
0.11993999E-02
0.12233000E-02
0.12472000E-02
0.12710999E-02
0.12950000E-02
0.13188999E-02
0.13428000E-02
0.13666999E-02
0.13906000E-02
0.14144999E-02
0.14383999E-02
0.14623000E-02
0.14861999E-02
0.15101000E-02
0.15339999E-02
0.15579000E-02
0.15817999E-02
0.16057000E-02
0.16296000E-02
0.16534999E-02
0.16774000E-02
0.17012999E-02
0.17252000E-02
0.17490999E-02
0.17730000E-02
0.17968999E-02
0.18208000E-02
0.18447000E-02
0.18685999E-02
0.18925000E-02
0.19163999E-02
0.19403000E-02
0.19642001E-02
0.19881001E-02
0.20120000E-02
0.20359000E-02
0.20598001E-02
0.20837002E-02
0.21076000E-02
0.21315000E-02
0.21554001E-02
0.21793002E-02
0.22032000E-02
0.22271001E-02
0.22510001E-02
0.22749000E-02
0.22988000E-02
0.23227001E-02
0.23466002E-02
0.23705000E-02
0.23944001E-02
0.24183001E-02
0.16954888E+00
0.16825669E+00
0.16699217E+00
0.16575418E+00
0.16454163E+00
0.16335350E+00
0.16218883E+00
0.16104668E+00
0.15992624E+00
0.15882666E+00
0.15774721E+00
0.15668714E+00
0.15564579E+00
0.15462250E+00
0.15361665E+00
0.15262766E+00
0.15165499E+00
0.15069808E+00
0.14975645E+00
0.14882961E+00
0.14791711E+00
0.14701851E+00
0.14613339E+00
0.14526136E+00
0.14440203E+00
0.14355505E+00
0.14272004E+00
0.14189669E+00
0.14108469E+00
0.14028370E+00
0.13949345E+00
0.13871366E+00
0.13794404E+00
0.13718434E+00
0.13643429E+00
0.13569367E+00
0.13496223E+00
0.13423975E+00
0.13352601E+00
0.13282081E+00
0.13212396E+00
0.13143525E+00
0.13075449E+00
0.13008149E+00
0.12941609E+00
0.12875812E+00
0.12810741E+00
0.12746380E+00
0.12682715E+00
0.12619729E+00
0.12557410E+00
0.12495743E+00
0.12434714E+00
0.12374311E+00
0.12314521E+00
0.12255331E+00
0.12196730E+00
-0.54663181E+02
-0.53479294E+02
-0.52345581E+02
-0.51258919E+02
-0.50216438E+02
-0.49215496E+02
-0.48253662E+02
-0.47328682E+02
-0.46438480E+02
-0.45581131E+02
-0.44754848E+02
-0.43957973E+02
-0.43188965E+02
-0.42446384E+02
-0.41728893E+02
-0.41035240E+02
-0.40364258E+02
-0.39714848E+02
-0.39085991E+02
-0.38476723E+02
-0.37886147E+02
-0.37313412E+02
-0.36757721E+02
-0.36218327E+02
-0.35694519E+02
-0.35185635E+02
-0.34691044E+02
-0.34210152E+02
-0.33742397E+02
-0.33287251E+02
-0.32844208E+02
-0.32412792E+02
-0.31992550E+02
-0.31583054E+02
-0.31183897E+02
-0.30794691E+02
-0.30415070E+02
-0.30044683E+02
-0.29683199E+02
-0.29330299E+02
-0.28985682E+02
-0.28649057E+02
-0.28320152E+02
-0.27998703E+02
-0.27684458E+02
-0.27377178E+02
-0.27076635E+02
-0.26782610E+02
-0.26494892E+02
-0.26213282E+02
-0.25937584E+02
-0.25667616E+02
-0.25403200E+02
-0.25144169E+02
-0.24890356E+02
-0.24641607E+02
-0.24397774E+02
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
0.59040220E+07
18
0.24422000E-02
0.24661000E-02
0.24900001E-02
0.25139002E-02
0.25378000E-02
0.25617001E-02
0.25856001E-02
0.26095000E-02
0.26334000E-02
0.26573001E-02
0.26812002E-02
0.27051000E-02
0.27290001E-02
0.27529001E-02
0.27768000E-02
0.28007000E-02
0.28246001E-02
0.28485002E-02
0.28724000E-02
0.28963001E-02
0.29202001E-02
0.29441002E-02
0.29680000E-02
0.29919001E-02
0.30158001E-02
0.30397000E-02
0.30636000E-02
0.30875001E-02
0.31114002E-02
0.31353000E-02
0.31592001E-02
0.31831001E-02
0.32070000E-02
0.32309000E-02
0.32548001E-02
0.32787002E-02
0.33026000E-02
0.33265001E-02
0.33504001E-02
0.33743000E-02
0.33982000E-02
0.34221001E-02
0.34460002E-02
0.34699000E-02
0.34938001E-02
0.35177001E-02
0.35416002E-02
0.35655000E-02
0.35894001E-02
0.36133002E-02
0.36372000E-02
0.36611001E-02
0.36850001E-02
0.37089002E-02
0.37328000E-02
0.37567001E-02
0.37806001E-02
0.12138706E+00
0.12081248E+00
0.12024345E+00
0.11967985E+00
0.11912160E+00
0.11856859E+00
0.11802072E+00
0.11747790E+00
0.11694004E+00
0.11640704E+00
0.11587882E+00
0.11535530E+00
0.11483639E+00
0.11432201E+00
0.11381208E+00
0.11330654E+00
0.11280529E+00
0.11230828E+00
0.11181542E+00
0.11132666E+00
0.11084193E+00
0.11036115E+00
0.10988427E+00
0.10941122E+00
0.10894194E+00
0.10847636E+00
0.10801445E+00
0.10755613E+00
0.10710135E+00
0.10665006E+00
0.10620221E+00
0.10575774E+00
0.10531659E+00
0.10487873E+00
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33
0.22876700E-01
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0.59579315E+07
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0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
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0.59579315E+07
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0.59579315E+07
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0.59579315E+07
0.59579315E+07
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0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
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0.59579315E+07
0.59579315E+07
0.59579315E+07
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0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
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0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
0.59579315E+07
This is the output when I tried alternate formulas for F1 and F2 (the equations are shown
below).
RUNGE-KUTTA METHOD FOR SYSTEMS
FUNCTIONS USED ARE:
F1(T,X1,X2,X3)=((20578.06384*T**4-286.901124711*T**2124711*t**220578.06384*T**4))**.5
34
F2(T,X1,X2,X3)=286.901*T*((1+X2**2)**1.5)
F3(T,X1,X2,X3)=-0.00964604697321*(X1-.0012)/(X1*X
1*X1)
ALPHA1=
6.46840E-02
ALPHA2=
-590.389
ALPHA3=
5.90402E+06
LIMITS ARE
1.00000E-04 TO
1.00000E-03
WITH
10 SUBINTERVALS
t(i)
w1(i)
w2(i)
w3(i)
0.99999997E-04 0.64684004E-01 -0.59038898E+03 0.59040220E+07
0.18999999E-03 0.29629294E-01 -0.27709256E+03 0.59040220E+07
0.28000001E-03 0.85913576E-02 -0.19948599E+03 0.59040220E+07
0.37000002E-03 -0.71573816E-02 -0.15449405E+03 0.59040140E+07
0.46000001E-03 -0.19672878E-01 -0.12566929E+03 0.59040140E+07
0.55000000E-03 -0.30034926E-01 -0.10576426E+03 0.59040140E+07
0.64000004E-03 -0.38867485E-01 -0.91239487E+02 0.59040140E+07
0.73000003E-03 -0.46560280E-01 -0.80191368E+02 0.59040140E+07
0.82000002E-03 -0.53371966E-01 -0.71513130E+02 0.59040140E+07
0.91000000E-03 -0.59482604E-01 -0.64520065E+02 0.59040140E+07
0.99999993E-03 -0.65022424E-01 -0.58766853E+02 0.59040140E+07
35