Canning, Thomas N

English

NASA Technical Reports Server

cm
*
N
w
c
c
a
2
NATIONALADVISORYCOMMITTEE
FOR AERONAUTICS
TECHNICAL NOTE 3125
MECHANICAL ANALOGUE
STABILITY OF AIRCRAFT
FOR STUDYING THE DYNAMIC
HAVING NONLINEAR
MOMENT CHARACTERISTICS
By Thomas N. Canning
Ames Aeronautical Laboratory
Moffett Field, Calif.
Washington
February 1954
AFWX
m 2811
https://ntrs.nasa.gov/search.jsp?R=19930083913 2020-06-17T20:09:28+00:00Z
“.Y g~~@~[
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS uobsqa~’
TECHNICAL NOTE 3125
A SIMPLE MECHANICAL ANALOGUZ FOR STUDYING THE DYNAMIC
STABILITY OF AIRCRAFT EAVING NONLINEAR
MOMENT CHARACTERISTICS
By Thomas N. Canning
SUMMARY
The snalogy between a ball rolling over a contoured surface and a
pitching and yawing aircraft is developed. Test results from a model
representing a missile with linear moment characteristics are presented
to verify the analogy. Several examples of the behavior of nonlinear
systems are also given. These examples include results from ballistic-
range firings as”well as analogue tests.
INTRODUCTION
Designers of aircraft are frequently confronted with the problem
of predicting the behavior of aircraft having nonlinear pitching- and
yawing-moment characteristics. The usual approach to these problems
involves tedious calculations or use of elaborate simulators. A simpler
approach suitable for some of the problems encountered, particularly
those of missiles, is the subject of this paper. ‘Theapproach uses the
analogy between a ball rolling in a suitably shaped bowl and a missile
pitching and yawing In flight. This analogue was devised in order to
underst~d a peculiar motion executed by a projectile in the Ames super-
sonic free-flight range.
SYMBOLS
A,B constants defining variation of ztithxandy
a radius of ball, ft
ag radius of gyration ofball, ft
C,D constants defining variation of ~ and ~ with a and ~
% pitching-moment coefficient of missile
2c~
co
F
K
I
K
Ki,2
L
z
21,2
r
w
X,y,z
xcg/2
a
P
5
e
e
u
w
l
yawing-moment
vector sum of
NACATN 3125
coefficient of missile
~ and Cn for axially symmetric missile
force component acting on ball parallel to bowl surface at point
of contact which prevents slippage
acceleration of gravity, ft/se8
moment of inertia of ball about a diameter, slug-ti2
constant, ft/sec2
constants, radians/sec2
constant defining
body length, ft
constant defining
rolling friction of.ball, l/see
damping of missile, l/see
J X2 + y2
weight of ball.,lb
coordinate system for bowl description, x and y horizontal and
z vertical
position
angle of
angle of
angle of
tin-l ~
x
tan-’ f!
a
of center of gravity as fraction of body length
attack, radians
sideslip, radians
inclination of bowl
m
rotational velocity of ball,
differentiation
surface, radians
radians/see
Superscript
with respect to time
.
—
9
*
,,
m
NACATN 3E!5
.
3
M
Consider
surface.
R!lSICDE.RIVATIQN
the force system acting on a ball.rolling on an inclined
Two forces act on the ball to produce translator acceleration: the
ccmponent of the weight force W parallel to the eurface at the point of
contact, and the tangential force F at the point of contact which pre-
vents slippage.
..
Wsin5-F=~~ (1)
g Cos E
Fa = Iti
Since slippage does not occur
.
x= au cos 8
..
x = a~ cos 6
Using this relationship
..
Fa=~~
a Cos 5
.
Substituting in equation (1)
4where
NACA TN 3125
*
l
and ag is the radius of #ration of the ball.
If .Ib[<0.15 radisn,
az
sin&cos5=sin 5=5 =—
b
and it follows that
. .
x =I&
ax
The derivation has thus far been limited to motion in the X,Z plane.
.
In the general case of motion anywhere on the bowl, the component accel-
erations parallel to the x and y axes, respectively, are: c
..
x = K ~ (xjy)
ax
Y =K* ~y (x)y)
Stiilar equations govern the motion of an undamped pitching and
yawing missile
..
$ =K2k (a,~)
Comparison of these equations shows that a and ~ are represented
.
in the analogy by x and y and that ~ and ~ are simulated by the
bowl S1OPSS, bZ~X and hZ/&. The ball rolling about the bowl may be *
visualized as describing the path traversed by the missile tip relative
to the center of gravity.
NACATN 3K!5 5
Inserting a friction force proportional to rolling velocity and
aerodynamic moments proportional to angular velocities results in
for the ball, and
for the missile. In the case of a missile free to plunge, as well as to
pitch, the coefficients contain several terms, but the differential equa-
tion is in the same form as that presented.
This, then, is the analogy between a ball rolling on a surface of
given contour and a corresponding missile pitching and yawing with the
above relations between surface slope and aerodynamic moments.
ExAMmEs
A case representing a missile with linear moment characteristics was
tested to verify the analogue, snd three nonlinear cases were tested to
demonstrate the application. All four cases were chosen axially symmetric
to economize on model construction. The feature of axial symmetry sug-
gests use of cylindrical coordinates to define the moment characteristics
of the missile and the surface of the bowl.
The moment tending to realign the missile axis tith the flight path
for the cases tested is given by
C(-J= Cu + Das
NACA TN 3125
r.
4
6
and cd is related to ~ and Cn by
% = co Cose
& = Ca sine
The bowl-surface slope is given by
dz
— = 2Ar -t4Br3
dr
-—
and the equation of the bowl surface is then
z = zo+Ar=+Ikr4
Converting the buwl-surface
z = 20
The slopes of the bowl are then
dz
—=2Ay
dy
equation to Cartesian coordinates gives
.
+ A(x2+~) + B(x2+~)2
.
+ 4Bx3 + 4BXY2
+ 4~s + 4B39c2
.-
Using the analogy between the bowl and the missile moment characteristics
gives
—
q.pca+r@+Ik@2
Cn = C13+ DD3 + D@2
.
These equations illustrate that Cm, for instance, is a function of both
a and ~, even for this simple nonlinear system. .
NACA TN 3X25
l
The simplification attained
apply to conventional electronic
s rolling-ball analogue.1
7
by use of cylindrical coordinates may
analogue computers as well as to the
The four cases were obtained by assigning various positive and nega-
tive values to the coefficients C and D. The moment equations and the
resulting bowl profiles are presented in figure 1. ‘I’heterms “stable”
and “unstable” in this figure refer to conditions
The cubic case has zero
Several techniques
s~ability near the center
CONSTRUCTION OF’MODELS
near the bowl center.
of the bowl.
of construction were tried. The most accurate
bowl, also the least expensive, was cast in hard plaster over a die made
of plaster. The first step in making a model was to make the die on a
flat work surface by sweeping a steel template about.a vertical axis
through the center of the die. While this die was hardening, a round
frame was made using a piece of plywood for the base and a strip of sheet
metal for the side. The hardened male die was greased lightly and the
frame, with access holes cut in the base, was placed upside dowm over it.
Plaster was then poured in through the access holes. The final contoured
. surface was then sanded and sprayed with flat, black lacquer. About eight
man-hours of work were required per bowl for the axially symmetric cases
built. These bowls were 2 feet in dismeter.
TESTING
The bowls were placed directly below a motion-picture csmera. Lag
screws in the plywood base permitted accurate leveling of the bowls. The
balls were all solid and were made of steel, brass, and aluminum and
vsried in size from 0.25 inch to 1.00 inch in diameter. No elaborate
technique was required for stating the balls rolling, but it was found
%e equations of motion of the ball and missile maybe converted to
cylindrical coordinates. The equations for the ball are, assuming
axial symmetry
The
the
2$<+rg=-
term 21?; is the Coriolis force.
missile sre
l2
G-se = k%
266+a~=-
Lr:
The equivalent equations for
- 26
2@”
8 NACA TN 3125
convenient to roll the balls down a chute rather than roll them by hsmd
a
because good repeatability of initial conditions was desired. The motion
pictures, which were taken at either 8- or 16-fremes per second, were
projected frame by frame onto a screen for measurement. The camera fram-
*
ing rate was used.as the time standard. Oil layers of various thick-
nesses on the bowls were found to provide good control of the damping.
Using balls of variouB densities also provided some control over damping
without oiling the surface. Actually, the damping was measured rather
than predetermined.
RESULTS
The results obtained with the linear-case bowl are easily compared
with anal@ic results and serve to check the validity of the snalogue.
In figure 2 are plotted the data for three runs corresponding to simple
pitching motion with progressively heavier damping. This large variation
in damping was attained by using a l-inch-diameterbrass ball.on the bowl
with no oil for the light damping (fig. 2(a)) and a l/4-inch-diameter
aluminum ball with a thin coat of oil for the case of heaviest dsmping
(fig. 2(c)). Superposed oneach set of data is a damped sine wave fitted
by the method of least squares (ref. 1). That the experherttal data fit
the theory quite well is clear. The unbed natural period, as predicted -
on the basis of bowl profile and the inertia characteristics of the solid
balls used, is indicated in figure 2.
.
Several plots of angle of attack versug time for pitching missiles
with nonlinear moment curves, obtained with the bowls, are presented in
figure 3. The curves for the cubic case (fig. 3(a)) were obtained at
different times during the same test and illustrate the extreme depend-
ence of frequency on aplitude of oscillation, a characteristic attribu-
table to the nonlinear moment curve. The large-smplitude curve has the
sharp peaks typical of cubic systems. The stable cubic case (fig. 3(b))
exhibits these same characteristics to a lesser extent. The unstable
cubic case (fig. 3(c)) exhibits 6 points of inflection per cycle instead
of the 2 per cycle showr.by all the other cases.
Two plots of motions obtained with the unstable cubic case are shown
in figuxes 4 and ~. The motion of figure 4 gradually degenerates and
takes on a form much like that of figure 5. These two motions were not
obtained on the same test run.
In the course of experimental firings of a shuple body of revolution
in the Ames supersonic free-flight range, the pitch-yaw records shown in
figure 6 were obtained. In the absence of-large gyroscopic forces (the .
models were fired.from a smooth-bore gun), it was difficult to explain
this behavior. In attempting to do this, the analogy between the ball
and the missile was discovered sad the improved Intuitive feel for the
*
:Y
NACATN 31.25 9
l
problem made the explanation clear. The moment curve for this body has
not been determined, but the motions observed indicate that the unstable
l cubic case is a fair representation. The data were obtained using bal-
listic “yaw csmls.” The technique consists of deducing the model orien-
tation at my instant from the shape of the hole it punches in a piece
of paper. Therefore, the curves are not exactly defined, but the general
nature of the motion is illustrated.
LIMITATIONS
The most serious limitation on the use of the analogue is that only
pitching snd yawing motion may be simulated. The missile which is being
simulated must be assumed to be roll stabilized to zero rate of roll.
The dsmping coefficient must be the same for pitch and yaw, which also
limits the value of the analogue for airplane-like cases.
It was noted during the test with the linear example that when the
ball was executing elliptical motion, the axes of the ellipse precessed
slowly. This distortion of the motion is believed to result from second-
order momentum effects (arising from the fact that the bowl surface had
finite slope) not considered in the derivation. The salient features of
. the motion are not masked by this effect.
l CONCLUDING REMARKS
!l%euse of a ball rolling on a suitably shaped surface to represent
the pitching and yawing motion of a missile has been described. Intuitive
understanding of some complicated motions is simplified through the use
of this device. The use of the analogue for axially symmetric cases sug-
gests that the use of cylindrical coordinates for describing nonlinear
moment characteristics may prove helpful in solving problems on conven-
tional analogue computers.
AriesAeronautical Laboratory
National Advisory Committee for Aeronautics
Moffett Field, Calif., Dec. 10, 1953
REFERENCE
w
1. Shinbrot, Marvin: A Least-Squares Curve-Fitting Method With Appli-
cations to the Calculation of Stability Coefficients From Transient-
. Response Data. NACA ~pskl, 1951.
r a , 1
NACATN 3125 IL
.
.
l
.
Cg I
Figure 1- Mment
21
Z=A r2+Zo
L ineor
)Iillli Ilflllllllflll r
Cubic
z
z=/fr2+ Br4 + 20
I Stub/e cubic
z
Z= -Ar2+Br4 +Zo
I
r
Unstuble cubic
~
curves und surfuce profiles for onologue
surfaces f ested.
NACA TN 31.25
(o.) Light dumping.
(b) Moderute dumjxhg.
0 fiperj~~t
—ftttect damped
\
sinusoid
T Theomticol undamped
noturul period
x
{w
I I I I ‘qS=’
77me, t
(c) Heavy damping.
.
.
Figure 2.- Angl+of-othmk history for linear case.
,.-
.
, P
Time , #
(u) Pure cubic.
17gum 3.- Ang/iw&@tffck his@ries of nonlineor cases, /?s O,
!3
‘G’
‘%
Q
/
)
Time , f
(b) Stable cubic.
Figute 3.- GOfiti~Uc?d
.
.
. . f >
\s I
I
Time, f
!2
(c) U@oM9 CUbiC.
Figure 3- cO/W’Udt?d
16
Time, t
(“) Pitch history.
(c)
Figure 4.-
NACA TN 3125
—
.
.
.
..
.
Time , t
(b) Yaw history.
.
Combined pitch und yaw histories.
Pitch and yuw histories for unstable
cubic cose of large umplitude.
.
.
NACA TN 3123 17
.
.
.
.
Time, t
(0) Pitch history.
77me, t
(b) Yaw history.
I
Angle of yuw, ~
(c} Combined pitch am’ yaw histories.
Figure 5. - Pitch und yuw histories for unstable cubic
cuse of smull amplitude.
NACATN 312518
Figure 6.- Experimental @#-yuw Mtories for u body
of revoh’lon in free f4#. Mmh Nnl&er = 4.5.
NACA-Lwwley - 2-8-E4 -1000
.“
.


A simple mechanical analogue for studying the dynamic stability of aircraft having nonlinear moment characteristics

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server