Time temperature characteristics of thin skinned models as affected by thermocouple variables - Laplace transfor

Larson, M. B.

English

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--.-- 
-MI-ANNUAL REPORT __I__.- 
U S A  Grant No. NGR-35-002-003 
b M Z i E R A T u R E  C H i  AmER isTics OF TH i N-sKiNNED 
MODELS AS AFFECTED BY THERMOCOUPLE VARIAELES;;~ 
f^ 
by 
MILTON B . X O N -  
University of North nilkata 
Grand Forks, North Dakota 
October, 1965 
L 
I' 
ff 653 July 65 
E N G I N E E R I N G  E X P E R I M E N T  S T A T I O N  
U N I V E R S I T Y  OF NORTH D A K O T A  
GRAND FORKS N O R T H  DAKOTA 
https://ntrs.nasa.gov/search.jsp?R=19660003719 2020-03-16T22:11:18+00:00Z
subject to the boundary csnditlans 
Tp%,8) = z;-(6/@) 
FIRST SEMI-ANNUAL REPORT 
NASA GRANT NQ NGR-35-002-003 
of the work on 
"Time Temperature Characteris tics of Thin -Skinned Models as Affected 
by Thermocouple Variablee" 
Milton B. Larson 
University of North Dakota 
Grand ForksI North Dakota 
October, 1965 
by 
The initial report has been delayed approximately six months 
because of dif€iculty in obtaining graduate students to work on the 
project and because of the prees of other academic activities, Request 
for extension has already been made, 
As outlined in the proposal, the first part of the problem consists 
of solving the family of equglticm 
and 
I 
where  
6 2  
#- LF< e 
a t j m s  have been 
01, -- 
e, Jp dx‘ 
-3-  
with boundary conditions 
zh 4) = W&!! 
and initial condition 
Using the Laplace Transform, the following set of equations is obtained: 
+ d ‘t, 
de‘ 
and 
( 5 )  
- 4- - 
subjec t  to the  conditions 
(.u-p”/Jdj = 0 
and t2 are the  t r a n s f o r m s  of and . 
A solut ion of (5) sa t i s fy ing  (m) is 
A solution of (6) sa t i s fy ing  (n) is 
U s i n g  condition (k) 
I. 5- 
Therefore, 
Using equations (7) and (8) in condition (1) provides 
where 
When inversion of this transform is complete the problem is 
If in condition (b) the t e r m  
essentially solved. 
Q,7TmZ is  omitted (this represents 
the energy input to the small  area directly behind the w i r e  on the skin), 
the equation for zfz ( ' 9 ~ )  becomes 
Using the approach that 
we obtain 
and 
I A -6- 
If the denominator of the integral has no zeros inside the contour 
shown in the sketch below, the integral above can be replaced by an 
integral around the contour A-B-C-D-E-F, 
W e  have not yet been able to show that there are no zeros in the contour 
in question, but we are continuing to w o r k  on this. 
The contour integral f rom A to B and E to F wi l l  have zero contribution 
in the l imit  as the radius tends to infinity. 
For the small  circle about the origin, A may be expressed as ,LI =pl" LP 
while var ies  f rom -n to 7r. 
I 
-7 -  
For the integral B-C, $Ld -- -ar 
For the integral over D-E  ( = aa) we have the negative conjugate of 
the above. When this is added in 
-8 
Therefore 
An inversion of equation (9) has not been attempted at  this point. 
Howeverp solutions of equation (10) suitable for short  time have been 
worked on by use of asymptotic solutions a s  suggested by Carslaw and 
Jaeger in Operational Methods in Applied Mathematics pp. 276-270. 
The result obtained is 
Carslaw and Jaeger provide justcficatisn for this approach for small  
enough time. Physically it would appear that the period of time of interest 
in the problem would be small enough so that this would be valid. That is, 
the interest  is in t h e  period of time during which the temperature is jus t  
start ing to rise. However, the magnitude of 9 calculated for typical 
values of the parameters and variables is about 1,000. This obviously 
will prevent the use  of equdtion (12) for any solution. At this time no 
resolution of this present problem has been reached. At the present time, 
it would seem that some numerical integration of (11) will provide a 
satisfactory solution. 
Future efforts will be expended in being certain o€ the validity of 
previous assumptions and in gaining a perspective as to the type of 
solution which should be obtained. 


Time-temperature characteristics of thin- skinned models as affected by thermocouple variables First semiannual report

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