Report number E -73.3, published by this Laboratory in June 1957, (Ref. 1) presents certain dynamic coefficients for a model of the Basic Finner Missile (Fig. 1) which had been measured on the angular dynamic balance in the High Speed Water Tunnel at this Laboratory. Several of the desired coefficients, specifically 



Y_r' coefficient of rotary force derivative



Y_v' virtual inertia coefficient (lateral acceleration)



N_r' coefficient of rotary moment derivative



N_v' virtual moment of inertia coefficient (lateral acceleration) 



remained undetermined at that time.



By employing the translational dynamic balance and its associated internal moment balance, it had been hoped that the missing values for these coefficients would be supplied. Only partial success has been achieved, insofar as numerical results are concerned, at contract expiration

time.



The coefficient of static force derivative, Y_v', and the virtual inertia coefficient, Y_v', have been measured as part of this investigation. These coefficients have been designated Z_w' and Z_ẇ' in this report to comply with the new direction of model motion with respect to the tunnel coordinate system. Since the first of these, Z_w', had already been determined in the angular dynamic measurements, only the presentation of a value for

Z_ẇ' is new. This coefficient had appeared in linear combination with the coefficient of rotary force derivative; hence the latter important quantity

also is now uniquely determined.



In addition to the force reactions, the moments arising from transverse velocity and acceleration components were also measured, but under conditions of undetermined deflection of the model-spindle assembly. For

this reason the moment coefficients have not been presented here, nor have the experimental procedures used to obtain them been included. Instead, a detailed discussion of both the apparatus and the experimental

procedures has been planned for reference 3

Kiceniuk, Taras

Caltech Authors - Main

An Experimental Determination of Dynamics Coefficients for the Basic Finner Missile By Means of the Translational Dynamic Balance

Department of the Navy 
Bureau of Ordnance 
Contract NOrd-16200 
To sk 1 
AN EXPERIMENTAL DETERMINATION 
OF DYNAMIC COEFFICIENTS 
FOR THE BASIC FINNER MISSILE 
BY MEANS OF THE 
TRANSLATIONAL DYNAMIC BALANCE 
Taras Kiceniuk 
Hydrodynamics Laboratory 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
Pasadena, Co lifornia 
Report No. E-73.9 
May 1958 
Approved by 
Haskell Shapiro 
Department of the Navy 
Bureau of Ordnance 
Contract NOrd-16200 
Task l 
AN EXPERIMENTAL DETERMINATION OF DYNAMIC COEFFICIENTS 
FOR THE BASIC FINNER MISSILE BY MEANS OF THE 
TRANSLATIONAL DYNAMIC BALANCE 
Taras Kiceniuk 
Reproduction in whole or in part is permitted for any purpose 
of the United States Government 
Hydrodynamics Laboratory 
California Institute of Technology 
Pasadena, California 
Report No. E-73. 9 
May 1958 
Approved by 
Haskell Shapiro 
INTR OD UC TION 
Report number E -73. 3, published by this Laboratory in June 1957, 
(Ref. 1) presents certain dynamic coefficients for a model of the Basic 
Finner Missile (Fig. 1) which had been measured on the angular dynamic 
balance in the High Speed Water Tunnel at this Laboratory. Several of 
the desired coefficients, specifically 
Y r' coefficient of rotary force derivative 
Yv' virtual inertia coefficient (lateral acceleration) 
Nr' coefficient of rotary moment derivative 
N.' virtual moment of inertia coefficient (lateral acceleration) 
v 
remained undetermined at that time. 
By employing the translational dynamic balance and its associated 
internal moment balance, it had been hoped that the missing values for 
these coefficients would be supplied. Only partial success has been 
achieved, insofar as numerical results are concerned, at contract expira-
tion time. 
The coefficient of static force derivative, Y 1 , and the virtual inertia 
v 
coefficient, Y · ', have been measured as part of this investigation. These 
v 
coefficients have been designated Z 1 and Z . 1 in this report to comply 
w w 
with the new direction of model motion with respect to the tunnel coordi-
nate system. Since the first of these, Z 1 , had already been determined 
w 
in the angular dynamic measurements, only the presentation of a value for 
Z. 1 is new. This coefficient had appeared in linear combination with the 
w 
coefficient of rotary force derivative; hence the latter important quantity 
also is now uniquely determined. 
In addition to the force reactions, the moments arising from trans-
verse velocity and acceleration components were also measured, but under 
conditions of undetermined deflection of the model-spindle assembly. For 
this reason the moment coefficients have not been presented here, nor 
have the experimental procedures used to obtain them been included. In-
stead, a detailed discussion of both the apparatus and the experimental 
procedures has been planned for reference 3. 
2 
APPARATUS 
The hydrodynamic measurements were made using the translational 
dynamic balance {Figs. 2, 3} mounted in the working section of the High 
Speed Water Tunnel. This balance is shown schematically in Figure 4. 
From the diagram it can be seen that the balance assembly consists of 
two major components, the driving platform and the model assembly, 
the two being coupled by a linear spring system of known and controllable 
stiffness. A method is provided for measuring the amplitudes of both the 
driving platform and the model assembly, as well as for determining the 
phase angle relating the motion of one to that of the other. The effective 
mass and damping of parts of the system, as revealed by the observed 
steady-state motion, can then be related to the desired hydrodynamic 
coefficients. 
In actual practice, the motion of the driving platform is imparte d 
to it by means of a motor-driven cam. A large, geared flywheel as sure s 
freedom from short period speed variations, while long period variations 
are controlled by varying the input voltage to the driving motor. This 
voltage is supplied by a saturable reactor controlled by the difference 
between the output voltage of a d-e tachometer gen e rator mounted on the 
flywheel shaft and that of an adjustable reference source. At one end of 
the shaft is mounted a 2 -pole a -c tachometer genera tor providing a sinu-
soidal output voltage accurately proportional to the angular velocity of the 
shaft. The phase of the output voltage can be controlled by rotating the 
outer case of the generator to any desired position. This latter asse11!bly 
was originally installed to provide a controllable voltage against which to 
compare the output of a strain gage moment balance located within the model, 
but was subsequently used as a supplementary method of analyzing the mo-
tions of the driving and model platforms. 
At the other end of the main earn shaft is located the contactor as-
sembly. It consists of a secondary cam which closes an electrical circuit 
at any de sired angular position of the main cam. This angle is then dis-
played on a mechanical counter face for the observer to record. The 
contactor assembly is part of the trigger circuit for a repeating electronic 
flash lamp which illuminates a pair of targets, one carried by the driving 
platform and the other by the model platform. A white line on the target 
3 
card is lined up with the cross-hairs of a cathetometer by the observer 
making the run. The contactor position is recorded, as are the cathetom-
eter readings showing the instantaneous positions of both the driver and 
model. Readings of this type are taken at 24 equal angular intervals to 
permit a Fourier analysis to be made of the motions of both elements. 
An alternate and somewhat faster method employs variable linear 
differential transformers (Schaevitz gages} to measure the amplitudes and 
phase angles of the two platforms. To do this, the output of the a-c tachom-
eter generator is made equal and opposite to the a -c output of the Schaevitz 
gage on the platform being measured. The resulting error signal is then 
displayed along the vertical axis on the face of an oscilloscope tube, while 
the a -c generator signal alone is fed to the horizontal sweep. The re suit-
ing Lissajous figure can be used to obtain phase and amplitude null balance. 
Control of the output voltage is achieved by means of precision potentiom-
eters, and control of the phase angle is effected by rotating the outer case 
of the a-c generator. A more complete description of the electronic equip-
ment will be presented in reference 3. 
The entire assembly is mounted on the High Speed Water Tunnel with 
the support spindle projecting into the tunnel water. By controlling the 
clearance space between the spindle and the hole through which it passes, 
it was possible to limit the leakage of the water from the tunnel. In addi-
tion, provision for air-pressurizing an annular space around the shaft 
reduced the leakage to about one gallon per hour when the pressure within 
the working section was held near the working value of one atmosphere, 
absolute. 
THEORETICAL ANALYSIS 
The hydrodynamic forces and moments actin g on a submerged body 
are assumed to be equal to the sum of the separate reactions arising from 
displacement, velocity, and acceleration in each degree of freedom. For 
small, effective instantaneous angles of attack, a body executing longi-
tudinal and transverse motion (along the z, or vertical body axis} is acted 
upon by total hydrodynamic force and moment reactions which can be written: 
Z = Z w+Z· w+Z u+Z. u 
w w u u 
( 1) 
M = M w+M· w+M u+M. u 
w w u u 
( 2) 
where the subscripts indicate partial derivatives in accordance with the 
procedure outlined in reference 2., and where 
Z normal component of hydrodynamic force (in the positive 
direction of the z fixed body axis) 
M hydrodynamic pitching moment about the y-axis (positive 
in accordance with the right-hand screw rule) 
w = z velocity of origin of body axis in the direction of the z 
(vertical) body axis, positive downward. 
w = z acceleration of origin of body axis in the direction of the 
z (vertical) body axis, positive downward. 
u = x velocity of origin of body axis in the direction of the x 
(longitudinal) body axis, positive forward. 
u = x acceleration of origin of body axis in the direction of the x 
(longitudinal) body axis, positive forward. 
4 
It is to be noted that equations (1) and (2.) contain no angular velocity terms, 
since the model was constrained to permit translation only. 
Because the body possesses quasi rotational symmetry, longitudinal ve-
locity and acceleration cannot contribute transverse reactions; therefo re 
Z=Z w+Z.w 
w w 
(3} 
M=M w+M.w 
w w 
(4) 
Only force reactions will be considered here. Rewriting equation (3) 1n 
terms of the displacement z, 
Z=Z z+Z.z 
w w 
( 5) 
Newton's law, written for the missile alone and accounting for Z(s) 
the force exerted on the n1issile by the spindle, can be written as 
( 6) 
1n the absence of spring or damping forces acting on the mass of the model, 
mb. If it can be assumed that the hydrodynamic contribution to the model 
can be considered equivalent to the addition of a spring, dashpot, and mass 
to the existing model, then the following equation of motion can be formulated: 
5 
( 7) 
where 
force exerted by the spindle on the model 
= mass of model 
= apparent mass of model due to hydrodynamic reactions 
total effective mass of model assembly 
b 
Kd 
= 
= 
damping coefficient due to hydrodynamic reactions 
rate of coupling spring = 85 7. 04 1 b/ft 
k = effective spring coefficient due to hydrodynamic re-
actions. 
That the latter quantity k equals zero is easily verified by disconnecting 
the coupling spring (Fig. 4) and manually displacing the model spindle. 
No detectable spring-like restoring force exists at any velocity; therefore 
Rewriting equations ( 6) and ( 8) 
z(s) = mb z - z 
z(s)=(mb + mf) z + b z 
Combining equations (9) and ( 1 0) 
Z = - mf z - b z 
Equating coefficients of like terms in equations (5) and (11) yields 
z = - b . 
w 
These can be expressed in din'lensionless form (see ref. 2} 
z. = z . I (1/2 PAd) 
w w 
( 8) 
(9) 
( l 0) 
( 11) 
( 12) 
( 13) 
( 14) 
( 15) 
where 
p = rna s s density of the fluid 
A = cross- sectional area (i2 is used in ref. 2) 
d = diameter of missile (1 is used in ref. 2) 
To obtain the steady-state solution, the equation of motion of the 
entire system is written in terms of the effective physical parameters 
introduced by the hydrodynamic reactions as well as those belonging to 
the mechanical system 
mz +bz +Kd(z -z2 ) = 0 (16) 0 0 0 
where Kd is the spring constant of the coupling spring. Assume 
z = A sin (.l)t 
0 0 
( 1 7) 
z2 = A2 sin ((.l)t + ~ ) ( 18) 
6 
6 being the phase angle between the driver and the model platform motions 
and A
0 
and A 2 the amplitudes of the model and driving platforms, re-
spectively. Substituting 
z = A (.l) cos (.l)t ( 19} 0 0 
.. A (.l)2 sin (.l)t {20) z = 0 0 
into the equations of motion and equating coefficients of like terl:(ls 
m A (.l) 2 + KdA = A2 cos 6 ( 21) 0 0 
bA (.l) = 0 KdA 2 sin 6 {22) 
or 
Kd ~ A2 a] ( 23) m = cos (.l)2 Ao 
b 
Kd [~: sin~ {24} = (.l) 
DATA SUMMARY 
The following tables show the data used to compute the de sired 
coefficients. 
I. Amplitude Ratio 
Frequency - Velocity - feet per second 
cycles per 
second In Air 0 5 10 15 
3.0 .928 .912 . 896 .908 .911 
3.5 . 912 . 875 . 872 . 870 . 892 
4.0 . 876 . 834 . 827 . 832 .84 7 
4.5 .848 .775 .786 . 783 .900 
5 . 0 .791 . 713 .722 .729 .757 
6.0 .704 .589 . 595 . 605 . 630 
6.5 . 644 .538 .524 .541 .564 
7.0 .594 .503 .475 .488 .503 
II. Phase A ngle 6 = a2 - a0 , in degrees 
Frequency - Velocity - feet per second 
cycles per 
second In Air 0 5 10 15 
3.0 0.4 0.8 2.2 4.0 6.3 
3.5 0.4 0.8 2.8 4 . 7 7.4 
4.0 0.6 0.9 3.2 5.7 9.9 
4.5 0.5 1.0 3.7 7.0 ll. 7 
5.0 0.5 1.4 4.3 7.0 11. 7 
6.0 0.6 2.3 5.6 10. 1 16. 5 
6.5 0.7 3.4 6.6 12.6 19.6 
7.0 0.8 5.0 8. l 15.9 23.7 
7 
8 
SUMMARY CF RESULTS 
The quantities listed below are the arithmetic mean values of the 
coefficients for frequencies ranging from 3 to 7 ~ycle s per second, and 
velocities from 0 to 15 feet per second. Because of the limited range 
of velocities tested (due to structural limitatio n s of the model support 
system} no con s istent dependence of the coefficients on either velocity 
or frequency could be established: 
z 
(1) z I w 13.5 = = -w 1/2. PA U 
z . 
z I \V 17.0 = = -w 1/?. pDA 
(2) 
Further experim ental work would be requi red to detect the effect 
of scale on the values o: t h ese coefficients. It should be noted here that 
the sign of the c oefficient 
reference 1. 
z I 
w 
was incorrectly reported as positive in 
REFERENCES 
l. Kiceniuk, Taras, "An Experimental Determination of Dynamic 
Coefficients for the Basic Finner Missile by Means of the Angu-
lar Dynamic Balance", California Institute of Technology, Hydro-
dynamics Laboratory Report No. E-73. 3, June 1957. 
2. "Nomenclature for Treating the Motion of a Submerged Body 
through a Fluid", The Society of Naval Architects and Marine 
Engineers, New York, Technical and Research Bulletin No. l-5. 
3. Kiceniuk, Taras, and Shapiro, H. , "An Experimental Method for 
Determining the Dynamic Coefficients of Submerged Bodies", 
California Institute of Technology, Hydrodynamics Laboratory 
Report No. E -35. 5. {To be published.) 
9 
~ t 
---------------e~~~JJ. g g 
I I l ~ <D 
_l_ 
SUPPORT SPINDLE 
11.96 -------------------18~.00~0===~------------------_j 20.000-----------------~ -1 
Fig. l. Important geometrical dimensions of the Basic Finner Missile. 
Fig. 2. Translational dynamic balance on calibrating stand. 
r 
-
-----
• 9 
\.. •'- "' 0 
9 + -
• •• 
0 
'" • 
~ 
Fig. 3. Translational dynamic balance in place at the working 
section of the High Speed Water Tunnel. 
K 
mb 
mf 
b 
r-------
I 
I 
I ( I 
I 
I I I I I I I I I I I L-----------' 
DRIVING PLATFORM 
DRIVING SPRING RATE 
MASS OF MODEL 
VIRTUAL MASS 
EFFECTIVE DAMPING RATE DUE 
HYDRODYNAMIC REACTIONS 
TO 
----------., 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I I 
~-------------.J 
MODEL 
z 2 , z0 DISPLACEMENT OF PLATFORM AND MODEL 
A 2 , A 0 MAXIMUM AMPLITUDE OF PLATFORM AND MODEL 
8 2 , 80 PHASE CONSTANT OF PLATFORM AND MODEL 
z2= A 2 SIN (wt ... 8 2 ) 
z0 =A 0 SIN (Wt+8 0 ) 
Fig. 4. Schematic diagram of the translational dynamic balance. 


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An Experimental Determination of Dynamics Coefficients for the Basic Finner Missile By Means of the Translational Dynamic Balance

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