Electrical analogy technique for determining rotational apparent masses of body in two- dimensional fluid flo

Andrew, G. M.

Cahn, M. S.

Garcia, J. R.

English

NASA Technical Reports Server

ROTATIONAL APPARENT MASS BY 
ELECTRICAL ANALOGY 
NOR 66-364 
December 8, 1966 
Prepared Under NASA Ames  Contract No. NAS 2-2671 
bY 
NORTHROP CORPORATION 
Nora i r  Division 
3901 West Broadway 
Hawthorne, California 90250 
for 
National Aeronautics and Space Administration 
Ames  Research Center  
Moffett Field, California 
https://ntrs.nasa.gov/search.jsp?R=19670015217 2020-03-16T18:27:07+00:00Z
NORTHROP NORAIR 
FOREWORD 
This  report  was prepared by the Norair  Division of Northrop Corporation, 
Hawthorne, California, on NASA Ames Research Center  Contract NAS 2-2671, 
Revision 1, "Rotational Apparent Mass  by Electrical  Analogy. The  study pre- 
sented represents  a n  effor t  by members of the Research and Technology Section 
of Northrop Norair. 
PRINCIPAL CONTRIBUTORS 
Maurice S. Cahn 
J. R. Garcia 
G. M. Andrew 
T h e  Technical Monitor f o r  the NASA Ames Research Center  was Mr. E .  R. Keener. 
-3 
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NORTHROP NORAIR 
Title -
CONTENTS 
Page_ 
SUMMARY ...................................... ii 
SYMBOLS ...................................... i i i  
INTRODUCTION .................................. 1 
THEORY ....................................... 2 
TESTS ........................................ 5 
CONCLUSIONS ................................... 8 
REFERENCES .................................. 9 
ILLUSTRATIONS 
Figure  
2 la. W=Z FLOWPATTERN ............................... 10 
l b  . CALIBRATION STRIP ................................ 10 
2. CIRCUIT SCHEMATIC..  .............................. 11 
5 EQWALENTNETWORK ............................. 12 
4. CORRELATION BETWEEN EXPERIMENTAL AND CALCULATED 
VIRTUALMASS .................................... 13 
5, VIRTUAL MASS VS SUMMATION OF RESISTANCES . . . . . . . . . . .  14 
ii 
NORTHROP NORAIR 
SUMMARY 
An investigation was conducted to study methods for the determination of 
rotational apparent masses by electrical analogy techniques. The change in resistance 
2 in a W = Z flow field due to 
class of shapes. A method of 
between four points was found 
the presence of a body was found to work for a certain 
measuring the summation of the resistance changes 
to work for a larger class of shapes. 
iii 
NORTHROP NORAIR 
SYMBOLS 
m33 
P 
V 
+ 
W 
Z 
R 
e 
Q 
apparent mass 
density 
velocity 
potential function 
complex potential function + + i \k 
physical plane x + i y 
distance from origin i n  physical plane 
angle between x axis and R 
stream function 
e1ectriczi.l resistacce 
RAB, CD resistance between AB shorted together and CD shorted together. 
moment of inertia of test section 
moment of inertia of body 
IT 
IB 
v33 virtual mass + m33 
iv 
NORTHROP NORAIR 
INTRODUCTION 
The estimation of aerodynamic stability derivatives for a body can be made by 
the measurement of the added apparent masses due to the fluid flow about the body. 
These apparent masses are derived from the kinetic energy imparted to the fluid by 
the body as it moves through the fluid. It has been shown (Reference 1) that the 
apparent mass due to a fluid flow can be easily measured using the electrical analogy. 
An electrical flow of the type desired is created in a medium, such as  Teledeltos 
paper, and electrical resistances recorded. The body shape is then cut out of the 
medium and again the electrical resistances recorded. The increase in electrical 
resistances caused by cutting out the body shape will give the apparent mass. This 
two- dimensional shapes. Aerodynamic stability derivatives were then calculated 
I from these measured apparent masses, using the equations from Reference 2. 
In order to determine certain rotary derivatives, the apparent mass due to the 
I solid body rotation of a two-dimensional shape is needed. The purpose of this report 
is to study methods by which the apparent mass due to rigid body rotation can be 
determined by electrical analogy in two-dimensional flow. 
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NORTHROP NORAIR 
THEORY 
Consider a body moving with unit velocity in a fluid with a frame of reference 
fixed to the body. The apparent mass for this body is the summation of the kinetic 
energy in the fluid with respect to moving axes fixed in  the body: 
m = $ p s  V2 dt 
This volume integral is shown in Reference 2 to be equal to the surface integral: 
The flow of an electricai current is analogous io thst of a pteEtial fluid flow. 
The velocity potential q5 becomes the voltage V. At the boundary of the electrical field, 
which is sdficiently far from the body, the voltage can be measured on all conducting 
edges and since is zero on all cut edges then Equation (2) can be evaluated. For 
a unit current flow it is the measured electrical resistance of the flow. This gives the 
total kinetic energy of the flow in the entire field. When a body is cut out of the field 
a new resistance value can be measured. The difference between the resistance values 
before and after cutting the body out is then the kinetic energy imparted to the fluid by 
the body. This is the method used to determine apparent masses for translational flows 
in Reference 1. 
a n  
I 
l 
I 
In order to obtain the apparent mass due to rigid body rotation the change in 
resistance due to cutting a body out of a rotating electrical field could be used. It is 
difficult to induce in a simple manner a rigidly rotating electrical field. The simple 
resistance flaw of electricity in a medium is an irrotational flow obeying Laplace's 
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NORTHROP NORAIR 
equation, but a rigidly rotating field is rotational and does not obey Laplace's 
equation. However, to determine the apparent mass of a body in this type of flow 
it is only necessary to measure the kinetic energy induced by the body and not the 
total kinetic energy. This induced flow is irrotational and does obey Laplace's 
equation. Therefore it seems possible to induce a flow field or  a combination of flow 
fields which will cause the body to impart the same kinetic energy as that due to rigid 
body rotation. 
Now consider the flow 
2 w1 = z 
which can be written 
w1 = R~ cos 2 e  + i R~ sin 2e 
w1 = <pl + i Ql Since 
then the potential function is 
2 
= R COS 2e 'p1 
and the stream function is 
2 
= R sin 28 % 
Lf this flow field is rotated 45 degrees the resulting flow function is: 
= R~ sin 28 + i R~ cos 2e 
w2 
= R~ sin 2e 
'p2 
q2 = R~ cos 2e 
and 
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The sum of the absolute values of the stream function and potential function of the 
two flows are 
According to Reference 3 this is the value of the stream function needed to 
simulate the flow induced by rigid body rotation. This analysis suggests that by the 
proper addition of the apparent masses of a body in a W = Z flow and a W = Z 
flow rotated by 45 degrees, the rotational apparent mass can be determined. 
2 2 
4 
NORTHROP NORAIR I 
TESTS 
2 Following the suggestion produced by the above analysis,a W = Z electrical 
flow was designed. This was made by cutting a 12-inch diameter octagon out of 
Teledeltos paper as shown in Figure la .  Alternate edges of the octagon were painted 
with silver paint to make them infinitely conducting compared to the general field. A 
calibration strip was prepared. This consisted of a rectangular sheet with silver paint 
at each edge (Figure lb). A Wheatstone bridge was hooked up using the two Teledeltos 
2 sheets as arms (Figure 2). With this arrangement the resistance ratio of the W = Z 
field to the calibration strip could be measured. 
The first measurements were made using a circle as the rotating shape. The 
2 W = Z resistance of t i e  octagonal sheet was mezsisured with aad withxt 2 circle 
cut out. The percentage change in resistance thus obtained was multiplied by the 
moment of inertia of the octagonal sheet. The resulting number for various size 
circles was equal to the moment of inertia of the circles. The next figures measured 
were right angle crosses. These were measured in one position then rotated forty-five 
degrees and measured again. By adding the answers resulting from the two positions 
the theoretical rotational apparent mass was obtained. This method therefore seemed 
to be a promising approach to the problem. It w a s  further pursued and it was found 
that i t  gave good answers for  combination circle and fin bodies. The equation used to 
get the rotational apparent mass is: 
I 
I 
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NORTHROP NORAIR 
The first term on the right side of this equation is the virtual mass of the body. 
When the actual rotational mass or  moment of inertia is subtracted out the result is 
the added apparent mass. 
Next ellipses and single straight lines were measured. Here the method seemed 
to break down. The answers obtained were much smaller than the theoretical values. 
After further testing, using many techniques to try and obtain the correct answer, it 
became apparent that these asymetrical shapes distorted the W = Z field. In other 
words it prevented an equal current flow in each of the four branches of the octagonal 
sheet. To bypass this difficulty the octagonal sheet with its four contacts was studied 
as a simple resistance network. 
connected by six resistances as shown in Figure 3. The problem now was to determine 
the values of the equivalent resistances. This was accomplished by taking measurements 
with various sides shorted together. The following equations were used to determine 
the equivalent resistances. 
2 
The equivalent network consists of four points 
+ 1 
BCD RB, ACD 
J. 
R ~ ~ ,  CD 
1 - 1 4- - 1 - 2 - 
RBC RB, ACD R ~ ,  ABD RBC, AD 
d - 1 ' +  1 
RC, ABD RD, ABC 
1 
RCD, AB 
1 - 1 + - 1 - 2 - 
R~~ RD, ABC RA, BCD R ~ ~ ,  BC 
The commas in the subscript in the above equations indicate how the points in the 
network were  grouped by shorting them together. By taking the resistance measurements 
indicated by these equations the reciprocal of the four equivalent resistances represent- 
ing the four branches of the w = Z2 flow was calculated. The values of these four 
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NORTHROP NORAIR 
resistances were added to get a value with no shape cut out. The same values were 
obtained with the shape cut out and the percent change of these values was multiplied 
by the moment of inertia of the octagonal sheet. This method now seemed able to give 
answers for shapes without a quadrilateral symmetry such as a horizontal line. The 
results using this method for  various configurations a re  shown in Figure 4. 
With the method thus far described there was still some difficulty in obtaining 
correct answers for shapes with no symmetry (Figure 4). One problem seems to be 
involved with how the actual moment of inertia of the body is determined. Since the 
right answer can be obtained for some shapes it is important that further work be devoted 
to make the method work for all shapes. 
One more method was studied and the results indicated the possibility of 
success. This method was to get the percent change in resistance in each of the four 
branches measured one at a time. The summation of the change in these resistances 
was empirically correlated with the virtal mass of var ims  bndies. A good 
correlation was obtained as shown in Figure 5. 
It was decided that the apparent mass of both the M-2 and HL-10 cross sections 
be measured by this technique. A cross section of the M-2 was taken a t  fuselage 
station 230 which includes the vertical fin and a cross section of the HL-10 a t  the base 
of, and including the vertical fin. The changes in resistance for the M-2 and HL-10 
I 
I were measured and a virtual mass determined for both (Figure 5). By subtracting 
the moment of inertia of the cross sections of the M-2 and HL-10 from the virtual 
mass,  apparent mass ( - md3)were obtained. They a r e  22.4 and 15.4  for the M-2 and 
the HL-10, respectively. The model scales are 1/20 and 1/30 for the M-2 and HL-10, 
respectively. 
There was not enough time on this program for  complete investigation of other 
techniques that might have been fruitful. A method which has been used is to measure 
7 
NORTHROP NORAIR 
the voltage around a body and express this voltage with a Fourier series. The coeffi- 
cients of the Fourier series can be related to the transformation function required to 
get all the apparent masses. This method has been applied to arbitrary shapes and 
its feasibility proven. However it does not seem capable of being used for arbitary 
three-dimensional shapes. It is believed that a method based on this same concept but 
which measures the voltage on a unit circle surrounding the body can be used in a 
similar manner to get all of the coefficients of the complete flow field. This method 
may be useful for three-dimensional shapes. 
Another method would be to measure the body's resistance when placed in the 
3 flow field due to w = z ,  w = z2, w = z and so forth to the highest power required for 
accuracy. This could be used to determine coefficients of a Laurents series expansion 
for the transformation of the body into a circle. It seems that this method would also 
apply to 3-dimensional bodies by assuming them to be psuedo two-dimensional bodies. 
Further investigation of these concepts should prove useful to the overall method 
for determining aerodynamic forces for arbitrary three-dimensional bodies. 
8 
NORTHROP NORAIR 
CONCLUSIONS 
A study of methods to determine the rotational apparent mass for arbitrary 
two-dimensional shapes by electric analogy has resulted in the following: 
1. The rotational apparent mass for shapes with quadrilateral symmetry 
can be obtained using a W = Z electric field. 
2 
2. The change in summation of the resistances between the four sides may 
correlate a larger class of shapes for rotational flow. 
9 
NORTHROP NORAIR 
REFERENCES 
1. Cahn, M. S., "Rapid Estimates of Stability Derivatives for Reentry Vehicles 
by Rheoelectric Analogy, AIAA Paper  No. 66-58. 
Neilson, J. N. , "Missile Aerodynamics," McGraw-Hill, New York, 1960. 
Milne-Thomson, L. M. , "Theoretical Hydrodynamics, '' Fourth Edition, The 
MacMillan Company, New York, 1963. 
2. 
3. 
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NORTHROP NORAIR 
A 
D 
C 
FIGURE la. w=z2 FLOW PATTERN 
B 
FIGURE lb .  CALIBRATION STRIP 
11 
NORTHROP NORAIR 
FIGURE 2 .  CIRCUIT SCHEMATIC 
12 
NORTHROP NORAIR 
D 
A 
C 
FIGURE 3. EQUIVALENT NETWORK 
13 
NORTHROP NORAIR 
100 
m 
I 
M 
v33 
Q 
60 
40 
2 0  
0 
NOTE: 
Shape of bymbolwindiouFte: 
shape of body 
0 40 60 80 roo 
CALCULATED . _  VIRTUAL MASS , \133 
CAL 
FIGURE 4. CORRELATION BETWEEN EXPERIMENTAL 
A I D  CALCULATED VIRTUAL MASS. 
14 
I NORTHROP NORAIR 
R -$  
3 
0 * 
0 
15 


Rotational apparent mass by electrical analogy

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