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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 1337
ANALYTICAL STUDY OF SHIMMY OF AIRPLANE WHEELS*
By Christian Bourcier de Carbon
SUMMARY
The problem of shimmy of airplane heels is particularly important
in the case of the tricycle nose wheel.
There is no rational theory of this dangerous phenomenon, although
some tests have been made in the U. S. A. and in Germany.
The present report deals with a rather simple theory that agrees
with the known tests and enables dimensions of the landgear to be com
puted so as to avoid shimmy without resorting to dampers. Tests with a
fullscale landing gear are described.
I. THE PROBLEM OF SHIMTMY
When a wheel fitted with a tire is designed to pivot freely about
a vertical pivot AA' (fig. 1) and this pivot is given a horizontal
forward motion while the wheel is made to roll on the ground, it happens
that it spontaneously assumes a selfsustained oscillating motion about
the pivot AA'. This phenomenon popularly termed shimmy or wiggle is
frequently observed on the tail wheel of a conventional airplane and
on the nose wheel of tricycle landing gears. In the latter case it. can
become extremely violent and conseuently very annoying, and occasionally
induce failure of the landing gear. In fact, it has been called the
"bete noire" (bugbear) of the tricycle landing gear: "the problem of
nose wheel shimmy, long the bugaboo of tricycle landing gear . .
(Aerodigest, March 15, 1944, p. 134).
The Americans, promoters of the tricycle landing gear, have already
made largescale theoretical arid experimental studies of this phenomenon,
but as yet, there seems to be no complete and acceptable theory. Thp
writer proposes to supply this theory in the present report. The sug
gested explanation involves only the elementary mechanical properties
of the tires which are all well dfined and asily measurable.
"Etude Theorique du Shimmy des Roues d'Avion", Office National
d'Etudes et de Recherches Aeronautiques, Publication No. 7, 1948.
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Before any particular hypothesis, it is, in fact, quite evident
that shimmy can have no cause other than the reactions exerted by the
ground on the 'heelreactions transmitted by the tire. These reactions
can only be contact reactions, due to the adherence of the tire; they
are necessarily reduced to a force and to a torque. To analyze these
reactions more correctly it is necessary, first of all, to form the
theory of the elementary mechanical properties of the tire.
II. THE MECHANICS OF THE TIRE
A tire is not merely an elastic element possessing vertical elas
ticity which is essentially its reason for existence, that is, a simple
vertical spring placed between ground and wheel; it is a complex
mechanical unit having a certain number of other elementary mechanical
properties. For the ensuing theory the four following fundamental fac
tors are considered:
1. Lateral Elasticity
2. Torsional Elasticity
3. Drifting
4. Turning
1. Lateral Elasticity
At rest, under the action of a relatively moderate force F, applied
parallel to the wheel, that is, perpendicular to its plane, the wheel is
displaced with respect to the ground, and through the simple effect of
the deformation of the tire, by an amount : proportional and parallel
to the force F. This displacement is due, as shown in figure 2, to the
simple deformation of the tire near the surface of contact with the
ground, without any modification of that contact surface as long as the
force F does not exceed the limit of adherence.
In the formula
A = TF
T is the coefficient of lateral elasticity of the tire.
2. Torsional Elasticity
At rest, under the action of a relatively moderate moment torque M
and vertical axis applied to the wheel, the latter is subjected to a
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rotation of angle a proportional to the moment M. As in the preceding
case, the mechanism of this rotation is easy to understand. It results
from the deformation of the tire near the surface of contact with the
ground, this surface itself remaining unaltered as a result of the
adherence.
In the formula
a = SM
S is called the coefficient of torsional elasticity of the tire.
3. Drifting
Under the action of the side force F a wheel mounted with a tire
is, according to the foregoing, subjected, first, to the lateral dis
placement = TF, that is, its center is shifted from 0 to O' and its
plane from P to P' (fig. 3). In addition, if the wheel is made to roll,
it is seen that its track is no longer contained in the plane of the
wheel, it moves without sliding along a straight line 0'M, forming an
angle 5 proportional to the force F.
In the formula
S = DF
D is termed the drift coefficient of the tire. This phenomenon is in
effect, a drift comparable to the drift of a ship in side wind. It is
particularly evident on a bicycle, or better yet, on a motorcycle with
an insufficiently inflated front wheel. It is this deflection of the
rubber tires that enables mothers to steer with ease baby carriages with
nonswiveling wheels without being forced to make the wheels skid on
the ground or to raise tvo of the wheels in order to turn the carriage.
A remarkable thing, this drift is produced without the least sliding
of the tire over the ground. It, as well as the other properties of
the tire, can be explained by a very simple diagram. Supposing the
tire is equivalent to an infinite number of small coiled springs mounted
radially on the periphery of the wheel and endowed with elasticity in
the radial and the lateral directions (fig. 4). This evidently repre
sents the vertical and lateral elasticity of the tire, as well as the
torsional elasticity. It also explains the drift. In effect:
Suppose that the wheel is at rest; a certain number of springs n
are in contact with the ground, as outlined in figure 5. Now, when a
side force F is applied to the wheel, it first is subject to a lateral
displacement 1 = TF as a result of the simple lateral deformation of
the n springs in contact with the ground without the least sliding
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of their contacts with the ground. As a consequence each of these
springs assumes a certain lateral tension, the sum of these n tensions
balancing the force F. Now the wheel is made to roll. Obviously, on
account of the displacement A the contact of the spring n + 1 with
the ground is no longer in the alignment of the first n's, but dis
placed laterally by an amount equal to S. On the other hand, at the
moment that the spring n + 1 makes contact with the ground, the
spring 1 loses this contact; but this spring has a certain lateral
tension which suddenly stops as soon as the spring leaves the ground,
while at the instant of making contact with the ground the spring n + 1
has as yet no tension. The sum of the reactions of the ground is there
fore diminished by the tension of spring 1 and it no longer balances
the force F. To recover this reduction the wheel is again subjected to
a slight lateral displacement so that the point n + 2 is again dis
placed or forced off laterally with respect to the point n + 1. The
process is repeated every time one spring leaves the ground and another
makes contact. It is seen that the wheel rapidly assumes the state of
a lateral displacement proportional to the path traveled in the direc
tion of rolling of the wheel. The normal state can be considered as
reached as soon as the point n has left the ground, that is, as soon
as the wheel has covered a distance equal to the length I of the
surface of contact. This length being slight, it all happens as if
that state was instantaneous. Since each one of the elementary dis
placements involved in this analysis is proportional to F, the same
applies to their sum, hence the formula
drift = DFx
x = distance traveled in direction of rolling, and consequently
angle of drift 5 = DF
Thus the drift is explained by the natural elasticity of the tire
without involving the least effect of skidding or sliding of the tire
over the ground. This phenomenon is almost the same as the longitudinal
creep of transmission belts; it is fully comparable with pseudoslipping
in which the drive wheels of a vehicle make a number of revolutions
greater than that resulting from the distance traveled by the vehicle
and greater than that of nondriving wheels of the same diameter carrying
the same load. Pseudoslipping which has been the subject of many
experimental studies for locomotives, is explainable in the same manner
without involving real slipping between wheel and rail; it results from
elastic deformations of the steel near the surface of contact and its
simplified theory is the same as that of drift. Pseudoslipping is a
longitudinal drift and the drift a lateral pseudoslipping. Likewise,
braking also produces negative pseudoslipping.
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Within certain limits the drifting of a wheel can be compared to
the deformation of a solid body. The displacement A plays a part
similar to that of the reversible elastic deformation, and drift DFx
similar to that of the irreversible permanent deformation arising from
a viscous creep, the path covered x in the drifting playing the part
of the time in creep.
The torque accompanying the drift. The preceding analysis indi
cates that the drift is necessarily accompanied by a very important
mechanical phenomenon, namely, the displacement of the lateral reaction
of the ground toward the rear of the wheel. Return to figure 4 and
consider the wheel at the moment the spring k + n comes in contact
with the ground. On account of the drift the points k + 1, k + 2, .
k + n, are alined, conformably as in figure 6, along a straight line
forming an angle 8 with the plane OP of the wheel. On the other hand,
at the instant point k + n makes contact with the ground, the plane
of the wheel obviously passes through the point k + n, which is termed
the head point of contact of the wheel with the ground. Since the ten
sion is proportional to the distance between ground contact and wheel
plane the result is a lateral tension of some of the n springs in
contact with the ground proportional to the distance between ground
contact and head point. This tension varies therefore from zero for
head point k + n to a maximum value for the tail point k + 1. The
reaction F' of the ground, the resultant of these different tensions,
therefore passes through the point G located at twothirds of the
line of contact from the head point. The ground reaction F' therefore
does not pass through the center M of the line of contact. While
parallel, equal and of opposite sign of F, the reaction F' is there
fore not directly opposite to the force F which, passing through the
projection 0 of the center of the wheel, passes also through M. The
result is a torque C about the vertical axis of the wheel.
With e denoting the distance MG and noting that the angle of
drift 5 is practically always small, this torque has the value
C = EF
In the case of the wheel shown in figure 4 it is readily apparent
that c = MG = I, where I = length of contact, that is, distance between
head point and tail point. But it is fitting to note at the same time
that, while the scheme is helpful for understanding the drift and the
phenomenon of the accompanying torque, the tire is nevertheless a more
complicated mechanism. The simplified scheme supposes that the small
radial springs, equivalent to a tire, are independent. But owing to the
continuity of the pneumatic tire, these springs must be considered
strongly tied elastically to one another in such a way that the deforma
tion of one of them also involves almost as much deformation of the two
adjacent ones.
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However, it can be shown that between e and Z the important
relation
6 2
exists.
For the present, without stopping to demonstrate this formula, the
following are assumed experimental facts:
(1) Every relatively moderate lateral force F applied to a tire
in motion produces a drift proportional to the angle 5 = DF.
(2) This drift is accompanied by a couple C = EF tending to
orient the wheel in the true direction of displacement, that is to
reduce the angle of drift spontaneously.
The coefficient e for a given load supported by the wheel is a
characteristic constant of the tire which is termed length of displace
ment; and torque Fe the torque accompanying the drift.
4. Turning
Under the action of a couple of relatively moderate torque M
about the vertical axis, applied to the center of the wheel, the latter
is subjected, first, to a static rotation of angle a, as shown in the
foregoing. But, when the wheel is made to roll, it no longer moves
along a straight track, but along a circular trajectory of radius p,
according to figure 7. The arc of circle AB is such that its curva
ture is proportional to the moment M. Hence
1 = R
P
p 
that is, for a trajectory AB of length s, the wheel has turned through
an angle
= RMs
R is termed the coefficient of turning of the tire.
To a certain degree the rotation of the wheel can be compared with
the deformation of a solid body. The angle a then plays a part simi
lar to that of the reversible elastic deformation, and angle 0 similar
to that of the irreversible permanent deformation.
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It is seen that the moment M necessary to make the wheel turn
to a permanent angle 0 is inversely proportional to the traveled
distance s. It is this property of turning that explains the ease of
steering the wheels of an automobile in operation while turning the
wheels of an automobile when standing still is much'more difficult
because it can only be achieved by an entirely different mechanism,
namely, by exceeding the limit of adhesion of the tire with the ground.
If this adhesion were perfect, that is, if no sliding were possible,
the permanent turning of the wheel at rest would be impossible, whereas
the sliding of the wheel plays no part in steering the wheel when
rolling, as will be proved elsewhere in a more rigorous analysis of
this phenomenon. As for the drift, turning can be explained on the
basis of figure 4; the process is exactly the same; simply replace the
lateral displacements by rotations. Turning is to the torsional
elasticity what deviation is to the lateral elasticity. The reality
of turning is assumed as an experimentally established fact. As
regards the proportionality of rotation 0 to moment M (like the
proportionality of drift to side force) from the moment the existence
of such rotation is assumed the proportionality must be assumed proved
from the following very general reasoning: The tire is a complex
elastic system, whatever the effects of the forces, they must be pro
portional, at least as long as the forces do not exceed certain limits.
The properties of turning and drift in question indicate that a
wheel fitted with a pneumatic tire, or more general, any wheel fitted
with any tire has the important property of being able to roll while
making a certain angle with its trajectory. This is the essential
phenomenon which, as will be shown, underlies and basically explains
the shimmy of pneumatic tires.
The five characteristic tire factors produced by the foregoing
analysis are
(1) Coefficient of lateral elasticity T defined by the equation
A = TF
(2) Coefficient of torsional elasticity S defined by equation
a = SM
(3) Coefficient of drift D defined by equation
8 = DF
(4) Coefficient of turn R defined by equation
= = RMs
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(5) Length of displacement E defined by equation
C = EF
III. MATHEMATICAL THEORY OF ELEMENTARY SHIMMY
In order to obtain a clear picture of the mechanism of this
phenomenon, the number of parameters are reduced by starting the study
with what may be called elementary shimmy or shimmy with one degree of
freedom, that is, the shimmy obtained when the spindle axis of the
wheel is assumed to be impelled only by a straight, uniform horizontal
motion. This is equivalent to assuming the pivot absolutely rigid and
the mass of the airplane very great with respect to that of the wheel,
so that the effects of wheel reactions on the trajectory of the caster
axis can be disregarded. The position of the wheel with respect to
the airplane is thus defined by one parameter, the angle e of its
plane with the axis of the airplane (fig. 8).
The study of elementary shimmy is divided in two parts.
In the first part is assumed that the reactions of torsion and
turn are negligible compared with the reactions of lateral elasticity
and drift, that is that the effects of torque M are negligible com
pared with those of force F, which is the same as assuming that the
coefficient R is sufficiently great with respect to coefficients T
and D. Later on it shall be shown that this is practically the case
for ordinary tires. It produces a simplified theory of elementary
shimmy. The second part contains tbh complete theory of elementary
shimmy.
A. Simplified Theory of Elementary Shimmy
It is assumed that the pivot is vertical and the pivoting free,
that is without restoring torque and without brakirg.
Let P denote the track on the ground of the pivoting axis at
instant t, x the distance traveled by this point, 0 the projection
of the wheel center, e the angle of the line PO with the trajectory PX
of the point P, that is with the axis of the airplane, AMGB the axis
of symmetry of the contact surface of the tire with the ground, A being
the head point, B the tail point, M the middle of AB and G the
center of the ground reactions, that is the point so that MC = e, y
and z the distances from PX of points M and 0 and finally
that is the angle of the line of contact AB with PX.
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Considering now the
is the straight line AB,
through points B and
the formula (fig. 8)
place of point M on the ground whose tangent
since the point M passes successively
A due to the rolling of the wheel. Hence
dy
dx
It is readily seen that the angle of drift 5 is equal to 0 + r, and
since 0 z hence
aF
dy z
dx a
Now, if F is the resultant of the ground reactions, the pre
liminary study made on drift makes it possible to write
S= DF
hence the first equation reads
dy z
dy + = DF
dx a
On the other hand, the action of the lateral elasticity yields the
second equation
z y = TF
Lastly, the force F being applied at G, that is a distance a + e
from point P, produces a torque equal to F(a + 6) with respect to
this point. In other words, to the primary torque Fa the accompanying
torque FE must be added. So, if I signifies the inertia of the
whole oscillating system with respect to the pivot P, the third equa
tion reads
d2e
I
= F(a + E)
Noting that in the first equation
dy = dy dt 1 dy
dx dt dx v dt
10 NACA TM 1337
where v = speed of airplane, and that in the third equation
e z
a
the system
1 dy + = DF
v dt a
z y = TF
I dz = F(a + e)
a dt2
is obtained.
The elimination of F from equations (1) and (3) leaves
1 dy z ID d2z
v + a + = ) 20
V dt a a(a +E) dt2
On the other hand, the elimination of F from equations (2) and (3)
leaves
IT d2z
a(a + C) dt2
Substitution of this value of y into the preceding equation finally
gives the differential equation of the third order
IT d3z
v(a + e) at3
ID dz a dz
++ z = 0
a + Edt2 v dt
which governs the motion of the wheel center. The general solution of
this equation is
z = Clet + C2et + C 3t
z = Cle + C2e + C3e
NACA TM 1337 11
where Cl, C2, and C3 are arbitrary parameters and sl, s2, and s3 are
the roots of the characteristic equation
IT s3 + ID s2 + a s + i = O. (7)
v(a + e) a + e v
Now, while the mathematical representation of the wheel center can be
expressed by a unique formula, it is well to bear in mind that the
physical appearances of this motion are entirely different depending
upon the positive or negative, real or imaginary values of the charac
teristic equation. Hence the necessity for discussing the values of
si, S2, and s in terms of the coefficients of the characteristic
equation, and as a consequence, as function of six parameters T, D,
e, a, I, v which characterize the tire, the wheel and the airplane
speed.
It should be noted that the characteristic equation has no posi
tive coefficients, hence can have no positive root; so if this equation
has real roots, they must naturally be negative. On the other hand,
the equation has always at least one real root since its first member
varies continuously from m to +co.
Suppose sl = a is this root.
From the point of view of real roots there can be only two possible
cases:
(1) Three negative real roots.
The general solution is then of the form
at Pt 7t
z = Cle + C2e + C3e
a, C and i being positive numbers. Whatever the initial conditions
that cause the variations of CI, C2, C3, the motion of the wheel is,
in this case, an periodic and convergent oscillation.
(2) One negative real root sl = a and two conjugate imaginary roots.
It is known that, when s2 and s3 are conjugate imaginary, of
s2t 3"t
the form i wi, the two terms C2e and C 3e combine to give
a term of the form
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BeXtsin(Wt + qp)
w being a number essentially positive and X a positive or negative
real number.
The general solution can then be written as
z = Aet + Bextsin(at + qp)
the three constants of integration which depend upon the initial condi
tions of motion, are the parameters A, B, (p.
From the point of view of the physical appearances of the motion
this case is split in two:
(a) If X < 0 the oscillations are convergent periodic
(b) If X > 0 the oscillations are divergent periodic
From among the possible forms of the solution z, the only case
in which there is a divergence of oscillation is therefore X > 0.
Now an attempt will be made to calculate the angular frequency w
and the divergence X of the oscillations in terms of the coefficients
of the equation of motion of the wheel, that is in terms of the charac
teristic coefficients of the tire and the wheel. The condition of
divergence of the oscillations, that is the condition of instability
is derived by writing X > 0. In all other cases there is convergence,
that is stability.
Calculation of divergence X and frequency a,
As X and w are independent of A, B, (p, it is assumed that
A = 0, B = 1, p = 0. Hence
z = etsin ot
dz = et r sin (t +
dt
d2z = et [x2 a2)sin wt
dt2
d3z = et (X2 302)sin at +
dt3
W Cos Dt]
+ 2 Xw cos
Wt
m(3X2 wv)cos t
NACA TM 1337
which, written into the equation of motion of the wheel gives an
equation of the form
eXt tg(k,w)sin wt + h(X,cm)cos Wt] = 0
This equation, which must be checked whatever t may be, gives the
system
Sg(X,w) = 0
h(X,a~) = 0
of two equations that define X and 0. Hence the system
 ~2)
IT (3%2
v(a + E)
ID (%2
S+ (X
a+ 6
 2) 2ID + a=
a + e v
which can also be written
2TX(X2 + w) + vD(2 + 2) v (a+ ) = 0
I
T(3X2 2) + 2vDX + a(a + e) = 0
I
The second of these equations gives
2 = 3x2 + 2vD + a(a + e)
T IT
which, entered in equation (8), and a eliminated, gives the equation
rT2IX3 8DTIvX2 + 2[DIv2 + a(a + c)Tx + (aD T)(a + e)v = 0 (11)
IT
v(a + 6)
 2) + + 1 = 0
v
(10)
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and defines the values of X in terms of the velocity v and the
characteristic coefficients of the tire and the wheel. With regard to
the abscissa v and ordinate X, this equation represents a cubic
passing through the origin, symmetrical with respect to the origin and
asymptotic to the axes of the abscisses. It is easy to plot this cubic
and to identify the characteristic elements.
Plotting of curve X(v). The preceding equation is of the second
degree in v and can be written
2D2Xv2 + [DTIX2 + (aD T)(a + ] v + 8T21x3 + 2a(a + E)TX = 0 (12)
Hence the curve of the values of v is easily plotted against X. It
is readily apparent that for each given real value of X there always
are two real values of v, the only condition being that the discriminant
be positive, that is
16DT2IX2 < (a + e)(aD T)2
or
l aD T a + e
4T DI
Maximum or minimum X. When
2 (aD T)2 a + E
16T2 DI
the curve X(v) passes therefore through a maximum or minimum M and
the value vm of the corresponding speed is such that
2 Br2IX2 + 2a(a + e)T
vm =
2D2I
we get, by replacing X by its value
Ivm = aD + T a + e(
S (12a)
2D DI
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Tangent at the origin. This is the straight line of the equation
"2aTX + (aD T)v = 0
hence its slope is
T aD
2aT
Representative curve. In consequence of this value of the slope
at the origin, the representative curve can assume different shapes,
depending upon whether aD > T or aD < T.
First case: aD >T
The curve X(v) has the shape represented in figure 9.
velocity v being necessarily positive, it is seen that X
negative.
The
is always
Second case: aD < T
The curve X(v) has the form represented by figure 10.
The speed v being necessarily positive, X is, in this case,
always positive. The only condition for divergence of oscillations of
the wheel is, as was shown before, that X > 0, and the only condition
of convergence is X < 0. In consequence the condition of divergence
or instability is reduced to
aD
and that of convergence or stability to
aD > T
This result could have been reached more rapidly by application
of Routh's general rules (or the equivalent method by Hurwitz) of which
the particular application to an equation of the third order with posi
tive coefficients of the form
d3z
A
dt3
d2z dz
+B +C Dz = 0
dt2 dt
reduces to the following rule: the necessary and sufficient condition
of convergence is BC > AD.
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In fact, in the case of equation (6) it is seen that this rule
yields immediately aD > T as the condition of convergence. But this
method involves the use of the original works by Routh ("On the Stability
of a Given State of Motion," Adams Prize Essay, 1877) and his treatise
("Advanced Rigid Dynamics") or the works of Hurwitz.
Moreover, it supplies no accurate data about the variations of
X in terms of the airplane speed v.
Resuming the preceding discussion it is seen that when aD < T,
the oscillations are unstable, that is the wheel, while rolling,
assumes a motion of divergent oscillations.
In practice the aforementioned conclusions are not vigorously
checked except for the start of the phenomenon. This is due to the
fact that our equations are applicable only to small deformations and
cease to be rigorous when the oscillations attain sizeable amplitude.
So, in the initial phase there is a phenomenon of unstable oscillations,
exhibited by a divergence which can become very accentuated when the
velocity is near that of the previously defined speed vy. But the
motion is never infinitely divergent. As soon as the oscillations
have reached sufficient amplitude, a state of continuous oscillatory
motion is established.
The process was frequently believed to be a phenomenon of resonance,
arising from a lack of symmetry of the tire or the wheel or similar
causes, and interpreted as such in most theories of front wheel shimmy
in automobiles. This explanation cannot be sustained in the face of
the severe cases of shimmy over the smoothest of grounds with perfectly
balanced wheels. Furthermore, the violence of the phenomena in some
of its manifestations and the extent of the speed range in which it can
manifest itself, are enough to prove that no resonance phenomenon is
involved but rather a phenomenon of vibratory instability. The present
theory shows that shimmy can be explained in the most elementary way
by the inherent mechanical properties of the tire.
The simplicity of the condition of convergence aD > T is sur
prising. In particular, it is extremely unusual to find that neither
the wheel inertia I, nor the offset e or the velocity v figure in
the condition of convergence. This condition gives a rule of extreme
simplicity; it is sufficient to follow this rule in the construction of
T
the undercarriage a > r in order to eliminate elementary shimmy.
Third intermediary case: aD = T
In this case, equation (11) gives immediately X = 0.
NACA TM 1337 17
On the other, equation (10) gives
= a(a + ) a +
IT DI
Therefore the general solution takes the form
z = Aeat + B sin(cot + 9)
The term Ae~t rapidly approaches zero, leaving only the second term
which represents a sinusoidal motion whose frequency o =  is
not dependent on the airplane speed v.
Frequency wn
The quantity X can be eliminated in the same manner as w( from
equations (8) and (9). Multiplying equation (8) by 3 and equation (9)
by 2X and then adding up, gives the equation
vIDX2 + 2 a(a + e) 41ITo + 3v a + e) ID = 0
On the other hand, equation (9) can also be written
3ITX2 + 2vIDX + a(a+ ) IT2 = 0
thus giving two equations of the second degree in X. It is known that
the result of the elimination of x from the two equations
ax2 + bx + c = 0
a'x2 + b'x + c' = 0
is the relation
(ac' ca')2 (ab' ba')(bc' cb') = 0
obtained by equating the result of these two equations to zero.
Applying this formula to the two preceding equations in X, gives the
equation
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v21 D(a + e) IT] 9T a + ) I = h 2ID2 3T a(a + )
4IT {[a(a + e)( + ) IT 4TAv2ID a + e) ITW 3v2Da + E IDu2]J
(13)
which defines the values of wu in terms of v and the characteristic
coefficients of the tire and wheel. This equation of the sixth degree
in v and wo is reduced to one of the third degree by a change of
the coordinates v2 = x and ao2 = y. Unfortunately the direct study
of this equation proved to be rather difficult, and was therefore
abandoned in favor of a more simple method of defining the variations
of w in terms of v. In fact it can be stated that the writer has
studied the variations of X in terms of v, and was thus able to
study the variations of cow in terms of X, and the problem of the
variations of the frequency c with respect to v can be solved the
same way.
The velocity v is easily eliminated from equations (8) and (9);
hence the equation
(x2 + a2)2 + a 3 a) 2 a + a + a(a + ) = o (14)
I \D T) I \ T/ DT
which defines the values of c in terms of X and the characteristic
coefficients of the tire and wheel. This equation of the fourth degree
in X and ao is reduced to one of the second degree by changing the
coordinates 2 = X and of = Y. The result is
2 a + E a a + a a(a + e
(X + Y) + X Y + = 0 (15)
I T I D T 2DT
The equation of the group of asymptotic directions
(X + y)2 = 0
indicates that the conic involved is a parabola whose axis is parallel
to the second bisectrix of the coordinates: Y = X. Only the portion
corresponding to the conditions X > 0, Y >0, that is located in the
first quadrant, will be considered.
NACA TM 1337
This arc of the parabola is easily plotted. For X = 0, that is
X = 0, the two values
S 2 = a(a + E)
IT
YB = a2 a + E
ID
represented by point A
represented by point B
are obtained.
By referring to equation (13) or simply to equation (10) or equa
tions (8) and (9) it is easy to identify that the first of these values
corresponds to zero velocities and the second to infinite velocities.
On the other hand,
YB YA = a (T aD)
IDT
hence, when aD < T, that is in the case of divergence
tions, point B is above point A (fig. 11); while, when
convergence, point B is below point A (fig. 13).
of the oscilla
T < aD, that is
Arranging equation (15) according to the decreasing powers of Y
under the form AY2 + BY + C = 0 and posting the condition B2 4AC > 0
to be fulfilled so that the values of Y are real, results in
S= 2 < (aD T)2 a +
16T2 DI
a value already obtained as maximum of X2 in the study of X in
terms of v.
The corresponding value of Ym = (Un2
sidering equation (15) developed under the
is readily obtained by con
form
AY2 + BY + C = 0, hence Ym = B
2A
NACA TM 1337
which gives
y + E L a m
M = 21 \D T)
hence
Ym I= a2 a + (7T2 + 1OaDT a2D2)
16DIT2
Returning to the study of variations of Y, that is of u2 when
v varies from zero to infinity, it is apparent that the representative
point of the variations of u2 and X2 (figs. 11 and 13) passes from
A to B along the parabolic arc defined by equation (15) and that B
is above A when aD < T and below in the opposite case. In order to
define the shape of the representative curve of the variations of the
frequency cu in terms of velocity v, it is necessary to establish
whether the variations of ow are increasing or decreasing as in fig
ures 11 and 13 or can pass through a minimum L as in figure 15. As
to the possibility of a maximum it is apparent that, geometrically, it
is excluded by reason of the general position of the parabola.
For it to have a minimum L
the slope of the tangents to A
equation (15) is differentiated,
point is
a +
dy
p = d=
dx
a +
I
Since the coordinates X and Y of points
preceding formula gives the slopes pA and
gives
it is necessary and sufficient that
and B have the same sign. So, when
the value of the slope p at any one
i a)+ 2(X + Y)
+ 2(X + Y)
A and B are known, the
PB; forming the product
(aD + 3T)(aD 5T)
PA X PB =
(aD T)
which for pAPB > 0 reduces to aD > 5T.
NACA TM 1337
This condition involves a fortiori aD > T; hence there can be no
minimum except in the case of convergence.
The discussion will be concluded with an examination of the possi
bility of periodic solutions. For this possibility to exist it is
necessary and sufficient that Y = co2 can assume negative values, that
is that the parabolic arc AB intersect axis OX. This geometrical
condition is expressed by the double analytical condition that the
equation in X obtained by making Y = 0 in equation (15) can have
at least one real and positive root.
The two roots of this equation are both real or imaginary; to be
real it is necessary and sufficient that the discriminant be positive,
which gives the condition
9T2 10aDT + a2D2 > 0
This condition requires that
aD < T
or
aD > 9T
a(a + E)2
As, on the other hand, the product of the roots a(a + ) is positive,
I2DT
the roots are either both positive or negative. To be positive, their
3 a
sum must be positive, which gives the condition < 0 that is
D T
aD > 3T.
To satisfy the required double analytical condition it is therefore
sufficient to know the unique relation aD > 9T; this is the condition
necessary and sufficient for the existence of periodic solutions for
suitably chosen v. These velocities of periodic conditions are,
obviously, those comprised between the velocities vl, and v2, positive
solutions of the biquadratic equation obtained by making w = 0 in
equation (13). The corresponding convergences are obtained the same
way by making w = 0 .in equation (14).
Now the curves m(v) representative of the variations of the
frequency w or, which is the same, the curves w2 in terms of veloc
ity v can be accurately plotted. It is accomplished by associating
to each curve a2(v) the corresponding curve w2(X2).
NACA TM 1337
The foregoing discussion distinguished four cases:
(1) aD < T divergent periodic oscillation (figs. 11 and 12)
(2) T < aD < 5T convergent periodic oscillationwith decreasing
frequency (figs. 13 and 14)
(3) 5T < aD < 9T convergent periodic oscillation with maximum
frequency (figs. 15 and 16)
(4) 9T < aD convergent periodic oscillation when v1< v v2
convergent periodic in the other cases. (figs. 17 and 18)
B. Complete Theory of Elementary Shimmy
The equations (1), (2), and (3) which form the basis of the sim
plified elementary shimmy were obtained by assuming the moment torque M
arising from the reactions of torsion and turning, a torque exerted
directly on the tread by the ground to be negligible. To take this
moment M into account, equations (1) and (3) must be suitably modified,
while equation (2) remains the same.
First of all, the plus sign is placed before M when a restoring
torque due to a positive displacement z is involved, just like the
force F was given the plus sign when a restoring torque due to a
positive displacement z was involved.
Next, consider equation (3); its first term represents the angular
acceleration of the wheel about the caster axis. The quantity F(a + c)
was the generating moment of this acceleration. In the complete theory,
the moment M simply added to this moment in such a way that this
equation becomes
I d2,
Sd = F(a + E) M
a 2
a dt
As to equation (1), it expressed the relation between the direction
dy/dx of the contact surface and force F. It had been assumed by
writing this equation that the drift, that is the orientation of the
contact surface immediately followed the variations of force F; it
implicitly implied that the turn was instantaneous, or in other words,
implicitly attributed an infinite value to the turn coefficient R. To
understand the line of reasoning that leads to the equation desired,
we shall start with the analysis of a particularly simple phenomenon:
drift.
NACA TM 1337
Complete Study of Drift
Consider a pneumatic tire at rest (fig. 19). A force F normal
to its plane is applied and it is made to roll while maintaining its
plane parallel to a fixed direction OP. Up to now it had been assumed
that the contact surface shifts along a straight line CM forming an
angle of drift 5 = DF with the plane of the wheel, that is instanta
neous drift was assumed. But experience indicates that the trajectory
of the contact surface or rather its center is actually a curve CAB
tangential to OP in 0 and asymptotic to O'M' parallel to OM.
This contact area can turn only according to the law of turning
dy
0 = d dy = B dx
dx 
that is
d2Y= RM
dx2 
But this couple M, the
ference of two effects: the
couple of the tread equal to
generator of turning, is itself the dif
accompanying couple Fe and the torsional
S dy
S dx
in such a way that
M = FE 1
S
hence
d2y
dx2
d2y
S
dx2
dy
+ R  = RSeF
dx
=R [F Ld
S dx]
NACA TM 1337
This then is the differential equation of the trajectory OAB of the
center of the contact surface in drift. This equation is easily
integrated
S2EF x 
y = SEFx + 1
an equation that closely represents a curve having the form of that
plotted in figure 19. Moreover, this equation makes it possible to
present an unusually interesting result: consider the drift dy/dx;
when x increases indefinitely, dy/dx tends toward SeF. Comparison
with the formula
dy
= DF
dx
it is seen that
Se =D
This relation is retained and applied repeatedly in the subsequent
study. Another significant feature of this relation is that it reduces
the number of characteristic parameters of the ordinary tire by one.
Take for example, T, D, e, R or else T, D, S, R. The equation of
drift then reads
R
y = DFx + D F S 1
R
In the light of this digression on drift we now return to the most
general motion problem. The path curvature of the center of the contact
surface is always
d2Y 
2 
dx2
The generating moment M of turning of the contact surface being still
the difference of two effects, the torsion couple of the tread equal to
a/S, a being the angle between the axis of contact surface and wheel
dv z
plane, that is a = dy + and the accompanying couple F.
dx a
NACA TM 1337
Hence
M 1 dy F
dx Sa
can still be written
four equations
1 d2y
 if
V2 dx2
d2y RM
dt2
v is constant.
Hence the
(16)
z y = TF
I d2z
Sd2 F(a + e) M
a dt2
M = d +z FE
 Sv dt Sa
Entering the values of F amd M in
then in equations (16) and (18) gives
1 d2y
R2 '2
Rv dt2
equations (17) and (19) and
 d + f y + z = 0
Sv dt T \Sa T
(20)
(21)
1 dy a I d2z 1 S0
Sv dt T a dt2 +Sa T
The elimination of y from the two preceding equations then leaves the
differential equation in z defining the motion of the wheel. Thus
I dz I d3z IE + 1 a d2z a + dz a+ e
Rav2 dt4 Say dt3 Ta SRav2 TR dt2 TSv dt TSa
(22)
d2y
But d2y
dx2
system of
(17)
(18)
(19)
NACA TM 1337
Note i. Setting R = o in the preceding equation and bearing in
mind the relation SE = D, yields the equation (6) of the simplified
theory.
Note 2. The elimination of y from equations (2Q) and (21)
requires algebraic equations which can be simplified by the following
considerations; the elimination must result in a linear differential
equation in z. If n is the degree of this equation it will have
n particular solutions of the form z = Ae"t, the coefficients a
being real or imaginary. Conversely, the elimination of z leaves a
unique equation in y. Equation (20) indicates that, if y is of the
form y = Alemt, z is also of the form z = Aeat. Equation (21)
indicates that, when z is of the form z = Aeat, there also is a
solution y of the form Ale"t. Therefore, every solution of the
differential equation in z is also a solution of the differential
equation in y and conversely. Thus these two equations are deduced
by simply changing y for z and vice versa. The elimination of z
is much easier; simply enter its value in equation (20) and then sub
stitute in equation (21). This method makes it possible to write equa
tion (22) very quickly.
Stability of oscillation. The preceding equation is a linear differential
equation of the fourth order of the form
aOzIV + alzIII + a2zI + a3zI + a4z = 0
and all its coefficients are positive.
Referring to Routh's "Advanced Rigid Dynamics" it is seen that
the conditions necessary for such an oscillation to be stable are
ala2 > aoa3
ala2a3 > a0a32 + a12a4
Applied to equation (22) the first of these conditions gives,
after reductions,
IE T E
a Sav2 Rv
a SRav2 Rv2
NACA TM 1337
which, with allowance for the fundamental relation SE = D, can be
written
v2 >aD T
Iv >
IRD
(23)
The second of these conditions yields, after reductions
IE 1 E I
+Ta SRav TR2 
Ta SRav2 T9v2 Sa2
(IRv2 a)(Sea T) > 0
that is with the fundamental relation previously advanced,
(IRv2 a)(aD T) > 0
(23a)
IV. COMPARISON OF THE PRESENT THEORY WITH THE EXPERIMENTAL RESULTS
Let us interrupt the examination of these two stability conditions,
and begin by studying the second. To interpret it, assume that speed u
is defined by the equality
u2 =a
IR
so the discussion can be summed up as follows:
First case: aD > T
The condition (23a) is confirmed if v > u.
Second case: aD < T
The condition (23a) is confirmed if v < u.
It is readily apparent that in both cases equation (23) is confirmed
when the condition (23a) is confirmed. Therefore the conditions of the
stability of oscillation are as follows:
First case: aD > T
Second case: aD < T
with v > u
with v < u
NACA TM 1337
This u is termed the inversion speed of stability. The determina
tion of the order of magnitude of this inversion speed u for the usual
pneumatic tires seems to be essential. But unfortunately no exact
determination of the value R is available so that this value could be
introduced in the formula
u2 = a
IR
Nevertheless, it shall be shown that the apparently very different
experimental results can be used to define the order of magnitude of
the inversion speed for the ordinary pneumatic tires.
Referring to the earlier study of drift and to figure 19, and
assuming that the equation for the curve OAB is given, it is readily
Rx
seen that CD' = e s at any point C of this curve. Now, for the
DD'
usual pneumatic tires the curve OAB rapidly tends toward its asymp
totic O'D'M', and according to the data given by various experimenters
it seems permissible to assume that after a travel of 50 cm
CD' 1
DD' 4
which, with the centimeter as the unit of length, gives
R
.50>
es >4b
hence
50 > 1.40,
S
So, if the
be solved.
But it was
that is R > 0.028S
values of S for the usual tires are known, the problem can
Unfortunately, no such determinations are as yet available.
seen th9t SE = D. Thus the preceding relation becomes
R > 0.028 D
C
NACA TM 1337
On the other hand, it has been indicated that if I denotes the length
of surface contact
6 2
So a length of 3 cm for e is an altogether admissible value and prob
ably very close to the real value in a great number of cases. The
preceding inequality consequently becomes
R > 0.028 D or R > 0.0093D
J
Experiments made on automobile tires have shown that under a side force
of 100 kg the drift 6 = DF reaches a value approaching 30, that is
= 0.0524 radians
180
Using the C.G.S. system of units the coefficient D is
0.0524 radians
D = rad s = 5.35 1010 (C.G.S. system)
100.000 x 981 dynes
hence
R > 5.1012 (C.G.S. system)
Supposing now that the caster length a is 5 cm and the inertia I
is 10 kg at 1 m from the axis, that is 108 (C.G.S. system). In that
case
u2 = J = 5 10
IR 108 x 5 x 1012
u = 100 cm/sec = 1 m/sec or 3.6 km/hr
NACA TM 1337
With a 10 cm caster length it would be
u = 1.4 m/sec = 5.1 km/hr
With a 20 cm caster length
u = 2 m/sec = 7.2 km/hr
The essential part to remember is that the inversion speed u is
small. This is, as shall be shown, an extremely important result and
it can be regarded as incontestable, although R has been derived by
indirect means. The form
u2 = a
IR
of the equation used to define u has, in fact, shown that a given
variation of R involves only a very small variation of u, so that,
for R onefourth as large, for example, the real value of. u would
only be doubled. Furthermore, the results of this analysis, and in
particular the 1 m/sec obtained for the inversion, coincide exactly
with the experimental curves published by John Wylie of the Douglas
Aircraft Company (ref. 1). That writer gives the exact coefficient of
friction K of the hydraulic damper necessary to avoid shimmy on the
landing gear of the OA4A airplane. This curve is reproduced in fig
ure 20. There is no need for a damper at velocities below 1 m/sec.,
but there is for all speeds above this value. The case exhibited by
this airplane was the case a< . Therefore Wylie's curve evidences
D
an inversion speed with stability of oscillations below this speed and
instability above it. Moreover, the inversion speed resulting from
Wylie's curve is exactly coincident with that evaluated in the present
numerical example.
The same writer also gives a similar curve for the nose wheel of
the Douglas DC4. The inversion speed of the stability for this air
plane is around 1.80 m/sec, hence is still close to that evaluated in
the present article.
The Douglas DC4 in 1940 was the largest airplane which had been
equipped with a nose wheel landing gear. It was a fourengine all
metal, low wing monoplane. Each engine developed 1165 hp., its weight
was 29.6 tons and its one nose wheel weighed 300 kg. It was designed
for 47 passengers, had a 382 km/hr top speed and 119 km/hr landing speed.
A similar curve was obtained by E. Maier and M. Renz in Germany
during the war, on the nose wheel of the Douglas DB7 "Boston." The
inversion speed seems to have been near 4 m/sec.
NACA TM 1337
As a result of these experimental data the existence of a low
inversion speed u can be regarded as proved by experiment. Therefore
referring to the stability condition (23) it can be seen that, in order
to eliminate shimmy at all speeds above u, it is sufficient to realize
the condition aD > T. At speeds below u, which are low speeds, it is
evident that shimmy should not be very annoying and can, perhaps pass
completely unnoticed in certain cases, because, as will be shown later,
simple relatively moderate friction in the hinge is enough to damp it
out instantly. The consequence of this analysis, that is, the condition
aD > T
can be practically regarded as the fundamental stability condition of
the wheel.
Can this condition be easily achieved for the permissible values
of caster lengths a? This is the next problem to be treated.
If the exact values of the coefficients T and D for the usual
tires are known, the answer is immediate, but unfortunately there are
no such data. So far it has been possible to evaluate the order of
magnitude of the coefficients D, S, R, and e on the basis of earlier
experiments, but still we know nothing of coefficient T. However,
the difficulty can be overcome by suitable interpretation of seemingly
very dissimilar experiments.
We refer to the report by B. v. Schlippe and R. Dietrich, entitled
The Mechanics of Pneumatic Tires (ref. 2) which is a detailed study of
the behavior of the pneumatic tire. Unfortunately, the authors failed
to give the simple properties that control the mechanics of the tire,
as they did not take up the study of the most complex phenomenon,
shimmy. Nevertheless, their report supplies some significant informa
tion because it contains a certain amount of numerical data. They
made experiments and measurements on a 260 x 85 tail wheel fitted with
a Continental balloon tire with small longitudinal grooves carved
around the circumference. The tire pressure was 2.5 atm and the load
180 kg. The wheel was connected to different dynamometers and kept
stationary, and rolling was accomplished by a rotating drum 90 cm in
diameter, covered with emery paper to assure good adhesion.
The first experiment by these writers to which attention is called,
is the following: when a tire is made to roll by keeping the plane of
the wheel fixed and imposing on it a rectilinear trajectory making a
certain angle 0 with the wheel plane (fig. 22), the center M of the
contact surface describes a curvilinear path MM' having as asymptote
the straight line (D) located a distance m from the trajectory 00' of
the wheel center.
NACA TM 1337
The authors established, by experiment, the proportionality
m = Ke
K being equal to 10 if e is expressed in radians and m in cm.
When considering the side force F necessary to keep the wheel
on the trajectory 00', the angle e is, obviously, the angle of drift,
hence e = DF. On the other hand, m is evidently the displacement of
the lateral elasticity and consequently m = TF, hence
m T
9 D
hence
m = T e
m=T_
D
Their coefficient K is none other than the quotient T/D, and
the experiment in figure 22 gives this quotient directly. The quotient
for this wheel is, thus, equal to 10 cm, and the caster length necessary
to eliminate shimmy is a > 10 cm.
This particular example indicates that the.caster lengths to which
the present theory leads are of a reasonably approximate magnitude, at
least in certain cases. Hence nosewheel shimmy can be probably
eliminated by a simple modification of the position of the pivoting
axis of the wheel, thus making the use of shimmy dampers unnecessary.
The experiments by Schlippe and Dietrich are of further interest
for another reason: they enable the characteristic coefficients T, D,
S, R, and E of the tire to be found indirectly, which, in the absence
of more adequate measurement, helps in defining the approximate magni
tude of these fundamental factors. Their direct measurement of the
lateral elasticity of the wheel indicated that a side force F of
2 x 32.5 kg, or 65 kg, was necessary to produce a 1 cm lateral displace
ment A of the tread with respect to the wheel. Now if A = TF, we
get in C.G.S. units
T= 1
65000oo x 981
NACA TM 1337
T = 157 x 1010 (C.G.S. system)
Since T = 10, we get D = 15.7 X 1010 (C.G.S. system)
D
These authors also indicated that, to maintain an angle 0 = 10
or 0.0175 radians in the experiment described above (fig. 22), a side
force of 11.4 kg and a torque having a moment C of 49.7 cm x kg must
be exerted. So, since e = DF, we should have
0.0175 = D x 11.4 x 981.000
hence
D = 15.6 X 1010 (C.G.S. system)
The agreement between this figure and that obtained independently some
lines back is perfect.
The accompanying torque measured by these authors was C = 49.7 cm. kg.
It is known that
C = EF
49.7
hence e = or E = 4.35 cm.
11.4
This figure is compared with the length
tact surface. The authors indicate that
Hence
I of the tread or ground con
this length was about 9 cm.
9
9. = 2.1
4.35
This result is in close agreement with the formula
2< < 6
given at the beginning of the present article while analyzing the
accompanying torque.
NACA TM 1337
The torsional elasticity was measured by the difference in lateral
displacement Z of the lead point and Z' of the tail point of the
tread contact in such a way that if 2 is the length of the tread, the
torsion angle a is such that
Z Z'
sin a =
therefore, for the small angles
Z Z'
For these conditions the authors set up the formula
M = d(Z Z')
where M is the moment necessary to produce the torsion and d is a
constant equal to 317 kg, where Z and Z' are expressed in cm and
M in cm. kg, that is d = 317 X 981000 (C.G.S.). Now Z Z' = la,
hence the formula giving M becomes
M = dla
However, since a = SM and i = 9 cm,
S = 1 that is, S = 3.57 x 1010 (C.G.S. system)
dZ 9 x 317 x 981000
On the other hand, D = Se and e = 4.35 cm. Therefore,
D = 4.35 x 3.57 x 1010 that is D = 15.5 X 1010 (C.G.S. system)
Two independent methods have already given
D = 15.7 X 1010 and D = 15.6 X 1010
The agreement between these three figures is really remarkable, it even
seems to exceed the probable accuracy of the measurement. At any rate,
these results give a completely satisfactory experimental check of the
present theory.
The one characteristic left to define is the turn coefficient R.
Its determination will be based upon the following experiment: by
causing the tire to roll against or on a circular wheel of 1 m radius,
the authors claimed that the motion produced an axial force F of
21.2 kg, tending to keep the tire away from center of the periphery.
NACA TM 1337
If, as seems likely, according to the conditions of the experiment, it
is assumed that the generating moment of M of the rotation arises
solely from the accompanying torque
Fe = 21.2 x 981000 X 4.35 = 90.5 X 106 (C.G.S. system)
the coefficient R can be deduced by the formula
d2y
= RM
dx2
d2y 1
If p signifies the curvature radius of the track the formula  
dx2 P
must be used, hence
R = that is R = or R = 1.1 1010 (C.G.S. system)
pM 100 x 90.5 x 106
It will be noted that this Value of R is in good agreement with
the inequality R > 5 X 1012 indicated previously. Still, this deter
mination does not offer the same degree of certainty as the preceding
ones: in fact since it was necessary to assume that the wheel was
exactly perpendicular to the radius of the track, any error in this
special circumstance will modify the centrifugal force F by the
superposition of a drift effect. To eliminate this potential source
of error completely it is necessary to assure the equality of the
centrifugal force F by changing the direction of rotation. In the
case of a minor discrepancy, the average should be taken.
U.S. Reports on Shimmy
The problem of nosewheel shimmy has already formed the object of
numerous theoretical and experimental studies, especially in the
United States where the socalled tricycle landing gear was born. It
was studied in great detail by the Douglas Aircraft Company, then a
little later, by the Lockheed Aircraft Corporation. On the occasion
of these studies a mathematical theory of shimmy was suggested by Wylie
(ref. 1) and by Arthur Kantrowitz of the Langley Memorial Aeronautical
Laboratory of the National Advisory Committee for Aeronautics (NACA
Rep. 686).
Because it is the only theory giving a numerical account of the
phenomena of shimmy, it is deemed practical to reproduce a literal
NACA TM 1337
exposition of Kantrowitz's article. This theory is proposed as being
"based on the discovery of a new phenomenon called kinematic shimmy."
Here is the exposition of Kantrowitz.
1. Kinematic Shimmy
Some preliminary experimental results on shimmy were obtained by
the N.A.C.A. with the aid of the beltmachine apparatus shown in fig
ure 23. This machine consists of a continuous fabric belt mounted on
two rotating drums and driven by a variablespeed electric motor.
Provision is made for rolling a castering wheel up to about 6 inches
in diameter on the belt in such a way that it is free to move vertically
but not horizontally.
On this belt machine, the following phenomenon (see fig. 23) was
discovered while pushing the belt very slowly by hand. With the wheel
set at an angle with the belt as in (a) and the belt pushed slowly, the
bottom of the tire would deflect laterally as is shown in (b). When
the belt was pushed farther, the wheel straightened out gradually as
is shown in (c). The bottom of the tire would then still be deflected,
however, and the wheel would continue to turn as in (d). The wheel
would thus finally overshoot, as shown in (e) and (f). The process
would then be repeated in the opposite direction.
Figure 24 is a photostatic record of the track left by the bottom
of the tire on a piece of smoked metal. Two things will be noticed:
First, that the bottom of the tire did not skid; and, second, that the
places where the wheel angle is zero (indicated by zeros on the track)
correspond roughly to the places where the lateral deflection of the
tire is a maximum. Thus the wheel angle lags the tire deflection by
onequarter cycle.
It was noticed that the oscillation could be interrupted at any
point in the cycle by interrupting the motion of the belt without
appreciably altering the phenomenon. From this observation it was
deduced that dynamic forces play no appreciable part in this oscillation.
The distance along the belt required for one cycle was also found
not to vary much with caster angle or caster length. (See fig. 25.)
Caster length was therefore considered not to be of fundamental importance
in this type of oscillation.
It should be pointed out that, in order to observe the kinematic
shimmy, lateral restraint of the spindle is necessary to prevent the
spindle from moving laterally when the bottom of the tire is deflected
and thus neutralizing the tire deflection. This restraint is supplied
by the dynamic reaction of the airplane when the airplane is moving
NACA TM 1337
forward rapidly but is not ordinarily present when the airplane is
moving forward slowly. It has been observed, however, on airplanes
towed slowly with two towropes so arranged as to provide some lateral
restraint.
Figure 23(a) shows that, when the center line of the wheel is at
an angle 9 (see fig. 25) with the direction of motion, the bottom of
the tire deflects. This situation is represented schematically in fig
ure 26. It is seen that a typical point on the peripheral center line
must have a component of motion perpendicular to the wheel center line
if the tire is not to skid. Thus
dX = sin 0 ds
(The minus sign follows from the conventions used as shown in fig. 25.)
Since only small oscillations are to be considered, the approximation
dX = (1)
ds
may be substituted.
The effect of tire deflection on e will now be considered. For
the purposes of rough calculation, it will be assumed that, as illus
trated in figure 27, the projection of the peripheral center line on
the ground is a circular arc intersecting the wheel central plane at
the extremities of the projection of the tire diameter. (See fig. 23(d).)
Thus, in figure 27, r is the tire radius. (It will be assumed for
the time being that the caster length and the caster angle are zero.)
Now if the tire is deflected in the form of a circular arc, then the
condition that the torque about the spindle axis be zero is that the
strain be symmetrical about the projection of the wheel axle on the
ground. Clearly, if the wheel is displaced, it will be turned about
the spindle axis by the asymmetrical elastic forces until, if it is
allowed time to reach equilibrium, the symmetrical strain condition is
reached. Thus, if the tire is deflected an amount X as in figure 27(a)
and if the wheel rolls forward a distance ds to the condition shown
in figure 27(b), in order for the strain to remain symmetrical the wheel
must turn about the spindle axis an amount dO. From figure 27,
Rd0 = ds. The value of R may be readily obtained from geometry in
terms of r and X. Thus R2 = r2 + (R X)2, from which, if X2 < r2,
it is seen that R = r2/2X. Then substituting for R,
dO 2 (2)
ds r
NACA TM 1337
If the caster length is finite, the strain will not be symmetrical
about the axle, as was assumed here, but will be symmetrical with respect
to some line parallel to the axle but a certain fixed distance ahead.
Hence, the essential elements of the geometry are unchanged and all the
reasoning that led to equation (2) is still valid for this case.
Since the phenomena represented by equations (1) and (2) occur
simultaneously, they must be combined to get the total effect. Thus
d2e 2 (3)
ds2 r2
This differential equation corresponds to a free simple harmonic
oscillation occurring every time the wheel moves a distance
S =21T = r
72/r2
Measurements of the space interval S of kinematic shimmy have
been made for three similar tires of the type illustrated in figure 23.
These tires all had radii of approximately 2 inches so that the theo
retical interval was about 0.74 foot. Their experimental intervals
were 0.65 foot, 0.74 foot, and 0.79 foot. This agreement is closer
than might have been anticipated in view of the roughness of the assump
tion. It will be seen from equation (1) that X is onefourth cycle
out of phase with 8, which is in agreement with the information
obtained from figure 24.
In order to take account of tires for which the assumption made
concerning the projection of the peripheral center line is not quantita
tively valid, an empirical constant K will be used in place of 2/r2
in equations (2) and (3), thus obtaining
dG = (4)
ds
and
d2)
S= K (5)
ds2
The constant K can be measured by observing the space interval of
kinematic shimmy. Where experimental values of K are available, they
will be used rather than the rough theoretical value 2/r2.
NACA TM 1337 39
2. Dynamic Shimmy
In the foregoing derivation for the oscillation called kinematic
shimmy, it was assumed that the strain of the tire was always symmetrical,
that is, the wheel was moving so slowly that any torque arising from
dynamic effects involved in the oscillation would be negligible. For
this case, from equation (4)
dx=0
K ds
If, now, the wheel is assumed to be moving at a velocity such that
the effect of the moment of inertia about the spindle axis is signifi
cant, then the strain can no longer be symmetrical and, for small
asymmetries, the torque exerted by the tire on the spindle will be
proportional to the amount of the asymmetry. Thus, the value in paren
theses will no longer be zero but it can be assumed that it will be
proportional to the dynamic torque; hence
A = (. 1 de
dt2 Kds
where C1 is an appropriate constant of proportionality and includes
the moment of inertia about the spindle axis I. If the forward veloc
ity V of the wheel is constant
d2 = Vd2
2 2
dt2 ds2
and
2 d2 C 1 dG(6)
ds2 K dK
The constant C1 may be determined by deflecting the bottom of
the tire a known amount, moving the wheel forward, and balancing the
torque M exerted by the tire on the spindle so that e stays constant.
The method of deflecting the bottom of the tire a known amount will be
described later. In this case (6) becomes
NACA TM 1337
M
= Cl (6a)
1
from which C1 may be found. It has been observed that C1 increases
with increasing caster angle. Thus, for a tire like the one in fig
ure 23, Cl was 71,000 radians per second2 per foot for a caster
length L of 0.17 inch (caster angle, 50; no fork offset) and was
104,000 radians per second2 per foot for caster length 0.68 inch
(caster angle, 200).
In the study of kinematic shimmy, it was also seen that the only
change in X was due to the fact that a component of the forward motion
was perpendicular to the central plane of the wheel. This circumstance
is expressed by equation (1). It is, however, found that, if the spindle
is clamped at e = 00 and the bottom of the tire is deflected, the
deflection will gradually neutralize itself; that is, the bottom of the
tire will roll under the wheel. Thus the asymmetrical ) strain
ds
that exists in this case contributes to dX/ds. The case of 0 and
de/ds zero is illustrated in figure 28. If the effect is again supposed
to be proportional to the cause, there is obtained
S= C2 (7)
ds
In this equation, the constant C2 is a geometrical constant of
the tire that can be obtained from static measurements. The order of
magnitude of C2 can be obtained by assuming that the periphery of the
tire intersects the extremity of the extended central plane of the wheel.
In that case C2 = .
r
If e is not zero, there will be a component of the forward motion
contributing to dX/ds. As in equation (1), this component will be 0.
Adding this component to the part of dX/ds due to asymmetry, then
(de/ds still assumed zero)
dX
= e C2 (8)
ds
This equation expresses that, for d = 0, the contribution of the
ds
asymmetrical strain to dX/ds was C2X. Also for the symmetrical
strain, in which case X dB/ds 0 (kinematic shimmy), there was,
K
of course, no contribution due to asymmetry. Assume now a linear inter
polation between these two limiting cases. Thus, finally,
NACA TM 1337
d = 0 C2 d) (9)
ds K ds
A method of determining C2 is provided by equation (7) which,
when integrated, gives
loge = C2(s so)
If, with the spindle clamped at e = 00, the tire is deflected a
known amount X0 and rolled ahead a known distance and the new X
measured, C2 may be computed. It was seen earlier that C2 was of
the order of magnitude of l/r; that is, it would be of the order of 6
for a 2inchradius tire. The constant C2 was determined for two
model tires under different loads and found to be 6.2 and 3.4. Con
siderable variations of this constant with tire pressure and load have
been found.
A method of obtaining the constant known X necessary for the
measurement of C1 is provided through equation (8). Here it is seen
that, if the wheel is pushed along at a constant angle 0, X will
dX
increase (negatively) until = C2X, in which case = 0 and
equilibrium is reached.
When the wheel is moving ahead at a finite constant velocity, the
phenomena represented by equations (6) and (9) occur simultaneously.
Therefore, to get the total effect, combine the two equations, thus
obtaining
V2 d3 + + 2d2 + = 0 (10)
C1 ds3 K C /ds2
The solution for the natural modes of motion represented by equa
tion (10) is
aLs Cm2s a_ (}S
e = Ae1 + Be2 + Ce3 (11)
where the a's are the three solutions of the socalled auxiliary
equation
V 3 + + 2 V 2 + 1, = 0 (12)
Cl K CI
NACA TM 1337
One of these a's is real and negative and corresponds to a non
oscillatory convergence. The other roots are conjugate complex numbers
and correspond to the shimmy under consideration. The roots will be of
the form a oi. If the divergence a is positive, the oscillation
will steadily increase in amplitude (while its amplitude is not large
enough for skidding to occur); and, if a is negative, the oscillation
will steadily decrease in amplitude and eventually disappear. The
meaning of the quantity "divergence" may be illustrated by saying that
it is approximately equal to the natural logarithm of the ratio of
successive maximum amplitudes to the distance between them. The
quantity w is equal to 2r times the number of oscillations per foot.
The frequency therefore is WV/2I. The phase angle is obtained by sub
stituting for e in equation (6) the value obtained from the foregoing
procedure and solving for X.
The divergence, the frequency, and the phase relations thus derived
for typical model tire constants are plotted in figure 29. For small
velocities (0 to 6 ft per sec) the frequency corresponds to kinematic
shimmy; it is proportional to velocity. The divergence increases
rapidly, however, because the spindle angle lags on account of the
moment of inertia about the spindle axis, thus allowing more lateral
deflection than would occur in a kinematic shimmy. On the next half
cycle, a larger spindle angle is reached and the process repeats. As
the velocity is further increased, the lag, and hence the asymmetry of
the strain, further increase until the strain becomes almost entirely
asymmetrical. For this condition, de << X. Then the restoring torque
ds K
on the spindle is approximately proportional to X. (See equation (6).)
The tire deflection X (measured negatively) will, however, still lag
somewhat behind 0 because, after the wheel is turned through a given
angle, a certain forward distance is required for equilibrium tire
deflection to be reached. Thus, the restoring force will again lag
the displacement. As the velocity increases in the highvelocity range,
the frequency stays nearly constant (see fig. 29) and the distance
corresponding to a single oscillation increases. Hence this constant
lag becomes a smaller part of the cycle and the divergence decreases
at high velocities.
It will be appreciated that the foregoing theory considers only
the fundamental phenomena taking place in shimmy. Other phenomena
occurring simultaneously have been neglected. Some of the more important
of the neglected phenomena are:
1. Miscellaneous strains (other than lateral tire deflection)
occurring in the tire. A rubber tire being an elastic body will distort
in many complicated ways while shimmying. In particular, there will
be a twist in the tire due to the transmission of torque from the
ground to the wheel.
NACA TM 1337 43
2. Two effects will cause the stiffness constants of the tire to
change with speed. First, centrifugal force on the rubber will make
the tire effectively stiffer at high speeds. Second, much of the energy
used to deflect the tire will go into compressing the air. The com
pressibility of the air will change with the speed of compression owing
to the different amounts of heat being transferred from it.
3. There will be a gyrostatic torque about the spindle axis caused
by the interaction of the rotation of the wheel on the axle and the
effective rotation of part of the tire about a longitudinal axis on
account of the lateral tire deflection. This torque will later be
shown to have a noticeable effect on the results.
The inclusion of items 1 and 2 in the theory would obviously be
very difficult. It is therefore necessary to resort to experiment to
determine whether the present theory gives an adequate description of
the phenomena. If so, the omission of these and any other items will
be justified.
An experimental check on the theory was obtained by measuring the
divergence and the frequency of the shimmy on the belt machine at two
caster angles and at a series of velocities. These measurements were
made by placing a lighted flashlight bulb on a 6inch sting ahead of a
model castering wheel with a ballbearing spindle and then taking high
speed moving pictures of the flashlight bulb with the wheel free. The
photographs were made with time recordings on the film, and the belt
carried an object that interrupted light from a fixed flashlight bulb
and thus recorded the belt speed on the film.
The divergence and the frequency of the shimmy were obtained by
measuring the displacements and the times corresponding to successive
maximum amplitudes (while the amplitude was still small enough to make
all the assumptions valid). The results are plotted in figure 30.
In order to compare these results with the theory, the constants C1,
C2, and K were determined on the same tire at the two caster angles by
the previously described methods. It was found that C2 = 6.2 feetl;
that K = 62.5 feet2; and that, for o5 caster angle, Cl = 71,100 feetl
second2 and, for 200 caster angle, C1 = 104,000 feetlsecond2. The
roots of equation (12) were then found and the divergence and the fre
quency of the shimmy were computed for a series of velocities and at
caster angles of 50 and 20. These results are plotted in figure 31 and,
for purposes of comparison, the experimental curves are also reproduced.
The agreement between theory and experiment is considered satis
factory as regards qualitative results. It will be noticed, however,
that the theoretical values of the divergence are decidedly too large
at high velocities, say 25 feet per second.
NACA TM 1337
Kantrowitz's theory and his experimental results will now be
compared with the new theory proposed in the present article:
First of all, the equation of motion (22) of the wheel has for the
general solution the function
z = Cleslt + C2est + C3eS3t + C4es4t
where C1, C2, C3, C4 indicate four arbitrary parameters and sl, s2,
s3, s4 the four roots of the characteristic equation
I s4 + l + a + + s E a+ E
Rav2 Sav Ta SRav2 TRv TSv TSa
The coefficients of this equation being always positive, the four
roots are real, negative or conjugate negative in groups of two. The
negative roots, if existing,, correspond to convergent terms and there
fore cannot be generators of shimmy; as a result shimmy must arise
from the imaginary roots. Assuming now that sl and s2 are two con
jugate complex numbers of the form X + mi. The corresponding expo
nentials are then imaginary and it is advisable to modify the expres
sion. Transformed in trigonometrical terms by means of the Euler
formulas, these two exponentials are combined to produce a term of the
form Aextsin(wt + (p), that is an exponential sinusoidal oscillation,
that is stable when X is negative and unstable when X is positive.
The theoretical study of shimmy can now be successfully completed
with the calculation of the variations of X and co in terms of the
coefficients of equation (22). The same direct method used in the
simplified theory for defining the curves of X and w in the terms
of velocity v could be applied, but the task would be drawn out and
difficult. In fact it will be shown that it is not absolutely necessary
to interpret the phenomenon of shimmy quantitatively and qualitatively
in its minute details. Use will be made of Kantrowitz's theory and
experimental data.
We shall first examine what the kinematic shimmy can be. The
record of the track of the tire on the ground, the place of the center
of the control surface, of elongation y, was expressed by an equation
where y had the same form as equation (22). Replacing z by y in
equation (22), and taking the path distance x instead of time t as
the independent variable, yields
NACA TM 1337
dy dy dx dy
dt dx dt dx
d3y
dt3
3 d3y
dx3
and equation (22) becomes
d3y (Ic 2
dx3 \Ta SRa
a d2y a + e dy a + e
T + + y=O
TR/ dx2 TS dx TSa
When the velocity v is very small and approaches zero, all the terms
containing v disappear and the preceding equation becomes equivalent
to
(T + Sa2)d2 + Ra(a + )dy + R(a + e)y = 0
dx2 dx
Such an equation represents a damped oscillating motion.
track will therefore be a damped sinusoid of equation
(25)
The tire
y = AeXxsin(wx + cp)
with
S4(T + Sa2)R(a + e) R2a2(a + e)2
ur+ =Sa
4(T + Sa2)2
(26)
Ra(a + E)
2(T + Sa2)
d = 2
dt2
d2y
dx2
v4 dx
dx4
d4y
dt
Iv2 d y
Ra dx
Iv2
Sa
(24)
and
(27)
46 NACA TM 1337
The wave length on the ground, that is the period W, is given by
2a
At zero caster length a
w2 R W = 2 t X = 0
T VRe
So, if a = 0, the damping is zero, that is the tire track on the
ground is a sinusoid of wave length W = 2t . Thus the first point
made by Kantrowitz and the phenomenon of kinematic shimmy is explained.
An attempt will now be made to compare the computed wave length
with that obtained experimentally by the American writer. To this end
the characteristics obtained for the tire studied by Schlippe and
Dietrich, that is
T = 157 x 1010 C.G.S. system
S = 3.57 x 1010 C.G.S. system
R = 1.1 x 1010 C.G.S. system
E = 4.35 cm
are used.
The result is
W = 2rc 157 = 36 cm
S1.1 x 4.35
This wave length is greater than the 19.8, 22.5 and '4.1 cm
obtained by Kantrowitz, but that is iuite natural; the tire of the
German writers was 26 cm in diameter, as against 10.2 cm for the
U. S. tire. It will now be attempted to find the effect of the caster
length a on this wave length from equation (26). We get
NACA TM 1337
for a = 0 W = 36 cm
a = 2 cm W = 34 cm
a = 5 cm W = 36 cm
a = 6 cm W = 39 cm
a = 10 cm W = 74 cm
a = 13 cm W = 0
a< 13 cm W is imaginary
It is seen that the wave length is practically constant for caster
lengths up to about 6 cm, because this wave length passes through a
minimum near a = 2 cm. This explains Kantrowitz's second remark:
"the distance along the belt required for one cycle was also found
not to vary much with caster angle or caster length."
Unfortunately the author believed he could conclude immediately
that the caster length a could be regarded as being devoid of impor
tance, which was a little hasty and not entirely logical. The essential
conclusion of the theory presented here is exactly to the contrary.
The effect of the caster length a on the phenomena of shimmy is of
primary importance.
Furthermore, the preceding tabulation indicates that wave length W
is far from being constant as Kantrowitz's theory assumes; it passes
through a minimum only at a small value of caster length a, and as
soon as a exceeds 6 cm, W is seen to increase in such a way that
W = 74 cm for a = 10 cm. This increase is speeded up more and more
and W approaches infinity when a reaches 13 cm. The wave length for
a values above 13 cm is imaginary, that is the track of the tire on
the ground is an periodic curve. The phenomenon of kinematic shimmy
has then completely disappeared.
This fact of the aperiodicity of kinematic shimmy when a exceeds
a certain value is absolutely sure. It can be easily checked with a
caster fitted with an elastic tire.
Kantrowitz's theory can therefore be only a rather rough theory
since its starting point is based on the assumption of constant wave
length W, which definitely becomes infinite when a reaches values
of the order of these (a > 10 cm). As shown previously, this should
eliminate the instability of the oscillations, that is shimmy. In any
case, the American author was absolutely unable to reach our significant
conclusions regarding the effect of caster length, because his theory
starts by refusing a priori to take this parameter into consideration.
To sum up the theory proposed in this report, fits into a theory
on kinematic shimmy represented by equation (25) in place of the equation
NACA TM 1337
S+ KB = 0
ds2
obtained by making v = 0 in Kantrowitz's equation (lQ), and equa
tion (25) seems to correspond more to reality, because it makes allow
ance for the variations of wave length W in terms of caster length a.
Lastly, it should be added that equations (25) and (26) indicate
that kinematic shimmy is usually a damped oscillation, except when
a = O, in which case the damping is zero. The equation
d20
+ Ke = 0
ds2
on the other hand, cannot allow for a damping, whose presence is easy
to confirm by experiment.
It will next be attempted to compare certain conclusions of
Kantrowitz's theory with the corresponding conclusions of the suggested
theory. For this we shall try to express the coefficients K, Cl, C2,
beginning with the characteristic coefficients T, D, S, R, E.
Coefficient K
Kantrowitz's differential equation (5) assumes 0 = A sin s K. The
wave length of the kinematic shimmy is then
W =B
T
But the present theory indicates that W = 2A /i. Therefore
K = R (28)
T
Coefficient Cl
M
Kantrowitz's equation (6a) gives CI = , that is with our notations
M 1IA
Cl = and since
IA
M = F(a + e) and A = TF
NACA TM 1337
we get
a+E
Cl = a e
IT
Coefficient C2
Kantrowitz's equation (7) expressed in our notation reads
(29)
d _C2 so that = = D = D
dx dx T
hence
D
c2 = D
This coefficient C2 is essentially tied to the drift which the
American author implicitly assumes to be instantaneous.
Numerical results
(30)
Kantrowitz obtained W = 24.1 cm, or K = 22, hence K = 0.068
W2
(C.G.S. system). Formula (29) indicates that Cl must be dependent
in a large measure on caster length a. Kantrowitz in fact noted it
and found
for a = 0.43 cm
a = 1.73 cm
C1 = 2.330 (C.G.S. system)
C1 = 3.410 (C.G.S. system)
Moreover, formula (29) makes it possible to write
0.43 +
1.73 + E
2.330 0o.43
3.410 i1.73
This equation permits the computation of the length e by assuming
I0.43 = 11.73, which is approximately true on account of the smallness
of a.
Hence
E = 2.4 cm
NACA TM 1337
For C2 Kantrowitz obtained C2 = 6.2 feet, or C2 = 0.2
(C.G.S. system).
Calculation of T, D, S, R, and E
Coefficient e has already been defined. The other characteristic
coefficients of the tire studied by Kantrowitz can be defined by the
formulas (28), (29), and (30), in
Kantrowitz gives I = 1.06 X 104
mass unit of 14.6 kg and 1 foot =
which the inertia I
in slugs and feet.
30.48 cm, so that
is involved.
The slug is a
I = 1.440 C.G.S. units
Formula (29) gives then
a a+ e
TC TI
1
0.43 + 2.4
1.440 x 2330
T = 84 x 108 in C.G.S. units
Formula (28) gives
R KT 0.068 x 84 x 108
E 2.4
R = 2.38 x 108 in C.G.S. units
likewise formula (30) gives
D = C2T = 0.2 X 84 x 108
Lastly,
D = 16.8 X 108 in C.G.S. units
S = 7 x 108 in C.G.S. units
by reason of the formula Se = D.
NACA TM 1337
The cinematic shimmy of Kantrowitz
By the use of the derived characteristic coefficients together
with formula (26) the determination of the wave lengths W of kinematic
shimmy made on the SchlippeDietrich tire can be applied to Kantrowitz's
tire.
For a = 0
a = 2 cm
a = 4 cm
a = 6 cm
a = 11 cm
a > 11 cm
W = 24 cm
W = 21 cm
W = 27 cm
W = 38 cm
W = 0
W is imaginary
This tabulation proves that the conclusions valid for the German tire
are even more valid for the American tire.
Inversion velocity u
Having defined the characteristic coefficients the inversion
velocity u defined previously by the equation
2 a
IR
can now be computed.
Since a high caster
make the calculation for
was 200 but only for the
to be regarded as zero.
angle is likely
the test series
series where it
Then
to modify this u, we shall not
for which this high caster angle
was 50, which is small enough
u2 =
hence
that is
0.43
1.440 x 2.38 x 108
u = 112 cm/sec
u = 3.70 ft/sec
52 NACA TM 1337
But on considering figure 30 and the curve A of the convergence
(for a caster angle of 50 and a = 0.43 cm) it is plain that the exten
sion of the experimental curve does not pass through the origin. This
circumstance is also relevant for the curves of figures 20 and 21
established by Wylie. The extension of the experimental curve A
plotted by Kantrowitz exactly intersects the axis of the velocities in
the axis of the abscissa v = 3.7 ft/sec; as stipulated by the theory.
In contrast to Kantrowitz's theory, this is an additional accom
plishment of the present theory. In fact, it will be shown that by
Kantrowitz's equation (10) the curve X(v) passes through the origin
which does not correspond at all to the experimental results.
In order to make the comparison between the two theories more
accurate, consider Kantrowitz's equation (10) and compare it with equa
tions (6) and (22) into which the suggested theory fits. First of all,
equation (10) is transformed by taking the time t as independent
variable in place of the distance s, and then the amplitude z com
puted in place of the angular elongation 0. On the one hand
z de 1 dz d2e 1 d2z d38 1 d3z
a ds a ds ds2 a ds2 ds3 a s3
on the other
dz 1 dz d2z d2z d3z 1 d3z
s = vt
ds v dt ds2 v2 dt2 ds3 v3 dt3
so that equation (10) becomes
1 d3z 1 C2 d2z
+ + . +  + z = 0
C21 dt3 \Kv2 Cjdt2
that is
IT d3z / ID T _d2z
+  +  + z = 0 (31)
(a+ E)v dt3 \a+ e Rev/dt2
by expressing the constants K, Cl, C2 by equations (28), (29), and
(30). We repeat equation (6) of the simplified theory
NACA TM 1337
IT d3z ID d2z a dz (6)
+ +a + z =0(6)
(a + )v dt3 a + dt2 v dt
TSa
and equation (22) of the complete theory after multiplying by 
a + E
taking into account the relationship Se = D
ITS dz IT d3z ID T + Sa2 d2z a dz'
+ + + + + z = 0
2dt 4 '(a I3a+ E v dt
(22)
Now equation (31) can be compared with equations (6) and (22). Inci
dentally, the turn coefficient R (consequently the phenomenon of
torsion) was implicitly introduced in Kantrowitz's theory,' although
its author claims to have ignored the torsion of the tire. Kinematic
shimmy can only be explained by the effect of this torsion and it is,
in fact, easy to prove that equation (6), which would be a complete
theory if this phenomenon of torsion could be disregarded, cannot give
an account of kinematic shimmy. A second preliminary remark is needed.
When the linear differential equation of the third order with positive
coefficients
Az'' + Bz" + Cz' + D = 0
is considered, Routh's rule is summed up as follows: for the oscilla
tion to be convergent it is sufficient and necessary that BC > AD. In
equation (31) C = 0; therefore the oscillation can never be convergent.
As a result Kantrowitz could not be led to suspect the possibility of
combating shimmy by a simple judicious combination of parameters. This
was unfortunate since it led to the design of landing gears having as
short a caster length as possible. Lastly, in comparing equation (6)
of the simplified theory with equation (22) of the complete theory it
is seen that the only difference is the presence in equation (22) of
two supplementary terms:
ITS d4z T + Sa2 d2z
 and
R(a + e)v2 dt4 R(a + E.)v2 dt2
Each one of these terms carries the factor v2 in the denominator. So,
when the velocity increases infinitely, the motion represented by
equation (22) tends asymptotically towards the motion represented by
NACA TM 1337
equation (6). In other words, when the velocity is large enough the
two equations (6) and (22) can be regarded as practically equivalent.
So, for a comparison of equations (22) and (31) at high velocities,
it is sufficient to compare equations (6) and (31).
We shall now define the relationship existing between velocity
and divergence X in the case of equation (31) (that is, in the
Kantrowitz theory), as we did before for equation (6).
Employing the same method as for equation (11), the system
IT X(X2 2)
(a + E)v
+ + ID (2 D2) + 1 = 0
a + E e+
IT (32 _2) + (D + 2 = 0
(a + e)v a e RT
is obtained. To obtain the relationship looked for between v and X,
simply eliminate ca from the two preceding equations. Thus
a + ID
(a + e
T h2 +
N 2
R Er2
2(a+ E)V ID+ T*
I +T = l
IT \a + E Rev2
This equation assumes that time t is the independent variable. Now,
in his calculations as well as in the curves of figures 30 and 31,
Kantrowitz has used the distance x = vt as the independent variable.
The fundamental oscillation of shimmy
z = etsin ot
is then
z = e Xsin a x
v
by putting
X
To obtain the equation giving the new expression of the divergence,
is simply replaced by vpi in equation (32). We then get, after
8IT 3
(a + E)v
(32)
NACA TM 1337 55
arrangement of the terms of this equation
2T D 8 4 42 + 4D V2 +2(a + c)
2i i + v + ( 2 + = + a=0
a + TRTRc IR2C2
With respect to velocity v this is an equation of the fourth power,
whose two primary roots, if they are real, have the same sign. In
order that these roots are real, the discriminant must be positive.
Therefore after some simplifications we obtain
l6T12 + 8D4 Re .< 0
which is true when i has a value between i1 and i2 of this
trinomial. These roots being of opposite sign, it is sufficient that
p is less than the positive root, hence
2 + TRg D
4T
It is readily apparent that, when this condition exists, the roots
are, of necessity, positive, because the coefficient of v2 is then
negative. In fact, this coefficient is a trinomial of the second degree
in 1 allowing two roots of opposite sign, hence the trinomial is
negative when n is less than the positive root. But this root is
2 + 2TRE D
1 = T
4T
and it is obviously greater than the positive root of the discriminant.
Thus the first condition is sufficient to assure the second, i < 1.
The result is that the divergence passes through a maximum
VD2 + TRE
M= 4T
for a given velocity vm given by the equation
2 a+ e
IRe(2 +
\ T)
NACA TM 1337
Thus, for the tire studied by Kantrovitz (curve A in fig. 36)
Pm = 0.0322 C.G.S. units
v = 364 cm/sec
or in English units of feet (30.5 cm)
4n = 0.98 English units
vm = 11.9 ft/sec
Shape of curve X(v) at high velocities
The variation of X(v) when the velocity increases infinitely,
K
will be examined by putting X = and trying to define K in terms
v
of the characteristic coefficients. Entering this value in the preceding
equation while ignoring the infinitely small terms gives the relationship
2(a + e)v ID 2 K
IT + e v
so that
K T(a + E)
2D21
The same calculation with equation (11) gives
K' = (T aD)(a + e)
2D2I
Thus it is seen that the present theory produces lower values for X
at high velocities than Kantrowitz's theory and seems to explain to
some extent the remark made by him that the values deduced by his
formula are clearly too high for high velocities.
To complete the comparison of Kantrowitz's theory, of the preceding
theory and of the experimental results, the calculations of X and 1
at various velocities were made
(1) By Kantrowitz's theory (equations (32) and (32a))
NACA TM 1337
(2) By the present theory in its approximate form (equation (12))
(3) By the present theory in its complete form (equation (22))
Present theory
Kantrowitz's theory Approximate
Approximate
r Complete form
form
mA, r. rEnglish English
cm/sec C C.G. S. Engih C.G.S. C..S. C..S English
unit unit
S0 0
100 .87 0.0087 0.265
112 0 0 0
200 4.77 .0238 .730 10.5 .0525 1.6
294 km = 24.7
300 9.36 .0312 .95 19.5 .0650 1.98
364 11.7 .0322 .98
400 12.7 Im.0318 m.97 20.6 .0515 1.57
500 14.8 .0296 .90 20.8 .0415 1.27
600 16.0 .0267 .815 20.8 .0346 1.05
700 16.6 .0238 .725
800 16.71 .0209 .635 19.6 .024 .73
900 16.65 .0185 .555
1000 16.4 .0164 .50 17.6 .0176 .54
1500 13.62 .0091 .28 15.3 .0102 .31
2000 11.48 .00575 .175 11.3 .00565 .17
2500 10 .004 .12 10 .004 .12
The table is graphically represented in figure 32 with p plotted
against the velocity v for
(1) Experimental data (curve E)
(2) Data from Kantrowitz theory (curve K)
(3) Data from present theory in its complete form (curve C)
The velocity is expressed in cm/sec and i in English units used by
Kantrowitz. To obtain curve E, plot the 15 experimental points given
by Kantrowitz, then draw the most likely mean curve defined by those
15 points.
NACA TM 1337
The plotting of curve K involves the resolution of a linear
differential equation of the third order which presents no special
difficulties, as the necessary calculations can be made by any of
several classical methods; the plotting of curve C reduces to
resolving equation (22) which is a linear differential equation of the
fourth order. It is a more difficult problem and involves considerable
paper work, so a graphical method of producing quick and excellent
results is desired.
Graphical method for solving linear and homogeneous linear equations
The equation (22) that is to be solved, is of the form
AzIV + Bz 1 + CzII + Dz + Ez = 0
which, putting z = Keext, gives the characteristic equation
f(x) = Ax4 + Bx3 + Cx2 + Dx + E = 0
Sought are the real numbers or, complex x, solutions of this equation.
Consider, therefore, the complex plane 0, X, Y, (fig. 33). To every
point m of the. prefix x the preceding equation makes correspond a
point d' with prefix f(x). This point M is easily obtained from
point m by simple graphical operations with rule and dividers.
Point m corresponds to a root x when M reaches 0. Then consider
a point m' near m to which corresponds a point M' near M, and
which gives by definition
MM'
== f'(x)
mm'
However, the value of f'(x) is not dependent on the position of m';
but solely on that of m. Therefore, if two vectors Pm" and PM"'
equipollent to mm', and MM' are involved, the triangle Pm'M"'
remains similar regardless of the position of point m' near m. Con
sequently, in the only condition where the prefix x of m is not one
of the roots of the equation f'(x) = 0 (which would be an exceptional
case since it is impossible to have more than three points in the entire
plane satisfying f'(x)) a point m' near m is, as a rule, easily
found, so that its transformed M' is closer to 0 than point M.
After a sufficient number of operations, the point M can be gradually
brought as near to point 0 as desired. This is the principle of a
graphical solution of algebraic equations by successive approximations.
NACA TM 1337
It should be noted in passing that the remark contains implicitly the
rigorous proof of d'Alembert's theorem which states that any algebraic
equation with real or imaginary coefficients has always at least one
real or imaginary root. In many cases the foregoing operations can be
made to converge very quickly. The method consists in taking as paint m
the second point mi so that
Md
f'(x) =
mml mm'
M and M' being constructed on the line of m and m', only m'
being chosen near m on the radius OM. The graphical method of
MM '
defining the point mi by utilizing the ratio F is much more simple
mm'
than trying to determine ml directly from the derivative f'(x).
After obtaining mi deduce M1, then begin again with torque mM1,
the operations to be made on torque mM. The result is a torque m2M2.
As a rule, the successive points M, Ml, M2, . come closer and
closer to 0 and the points m, ml, m2, .. tend to a limiting point
whose prefix X + iw is the root of the equation f(x) = 0. The
function
z = e sin wt
is then the corresponding solution of the differential equation that
was to be solved.
The unusual feature of this method, whose principle is apparent
from Newton's method for finding real roots of algebraic equations is
the small number of operations necessary to obtain convergence as soon
as a root is approached. This graphical method is absolutely general
and can be applied to algebraic equations of any degree.
After this digression the study of the three curves in figure 32
is resumed.
The comparison of the experimental curve E with Kantrowitz's
theoretical curve K has shown thst the maximum of the latter curve
is not clearly enough indicated. Curve K is much too flat and drops
too slowly when the velocity increases. Specifically curve K gives
slightly too small divergences near the maximum and by way of contrast,
definitely too great divergences at high velocities, as the author
himself admits.
NACA TM 1337
The comparison of the experimental curve E with curve C given
by the present theory indicates that the latter gives value sensibly
too great, but that the general shape of the curve C (existence of
the same inversion velocity u, acuteness of the maximum, rapidity of
decrease at increasing velocity) approaches curve E' more closely
than the flat shape of curve K. The correspondence between curve C
and curve E is almost a simple expansion of the ordinates in a ratio
approaching 1 as the velocity increases. So in spite of the similarity
of forms between E and C there still is a certain discrepancy
between the experimental and theoretical results of the present theory.
It is the first discrepancy of this kind that has been found. A careful
examination to see whether this can be explained within the framework
of the present theory, was therefore indicated.
It seems preferable to have the theoretical discrepancies on the
high side rather than the low (as in the case of curve K in the
neighborhood of the maximum). The divergence of the oscillations
corresponds, in fact, to a storage of mechanical energy in the castering
wheel and the aim of the theory of shimmy is precisely to discover the
source and the laws of this energy. Shimmy is accompanied by various
energy dissipating phenomena for which the theory makes no allowance,
such as friction in the castering axis, the slight slippage of the tire
close to the periphery of the contact surface and particularly of the
tail point, the reactive effects at the inside of the tire, etc. The
last cause is far from being negligible: according to the general
scheme of oscillatory motions shimmy is accompanied by a periodic and
continuous transformation of kinetic energy of the wheel into potential
stress energy of the tire and vice versa. It is known that a stressed
tire recovers only part of the stress energy. Drop tests on tire indi
cate, for example, the height of rebound is always from 15 to 30 percent
less than the height of drop, the loss being due largely to the reaction
effects inside the tire. It is therefore to be expected that in a
correct theory of shimmy the values of divergence are always greater
than the experimentally observed values. Still, this explanation
hardly justifies a difference exceeding 10 to 20 percent. The difference
between the maximum point of the theoretical curve C and the experi
mental point amounts to 2 = 40 percent. There must be some other
20
cause for explaining such a marked discrepancy. There is only one
possibility, namely, the numerical values used for T, D, S, R, and E
are probably not rigorously exact. Unfortunately these factors had
been defined indirectly from Kantrowitz's factors K, C1, and C2, hence
there is a possibility that some of the defined factors, T, D, S, R,
and E were only approximate.
This matter will be discussed in a little greater detail: For
mulas (28), (29), and (30) were taken as a starting point. Formula (29)
NACA I4 1337
resulted in
0.43 + e 2330 10.43
0.73 + E 3410 I1.73
which made it possible to compute e = 2.4.
Obviously, this value for e is only approximate, and for two
reasons: first, it was assumed that
i0.43 = '1.73
although the second of these values is certainly greater than the first.
On the other hand, formula (29) makes no allowance for inclination of
caster. Rigorously written it should have been
M_ F(a + e) + pe
Z =A IA
with i = TF and 0 = DF, that is
a + E pD
C 1 + 
IT IT
the parameter p being defined later. However, T was computed by
formula (29) from the derived value for e, which is the reason that
the value obtained for this parameter cannot be entirely rigorous.
Then R was computed by formula (28). This formula, utilizing the
length of kinematic shimmy, seems to be rigorous. Unfortunately, it
contains q and T which limits the accuracy to be expected from R.
Lastly, D Jas computed by formula (30) and S by the relation SE = D.
Our formula (30) is simply Kantrowitz's formula (7), which is equivalent
to assuming that drift is instantaneous. And this assumption is, as
already shown in section IIIB, only a simplifying assumption frequently
permitted, but which, in this instance, is likely to taint the calcula
tion of C, with an error.
A rigorous determination of the equation of the curve studied by
Kantrowitz in figure 28 will now be attempted. This curve was used to
determine the coefficient C2.
2'
62 NACA TM 1337
Let y represent the distance of the center M of the contact
tread with respect to the undeflected plane of the wheel. With our
usual notations (fig. 34)
d2y 
= RBM
ax2
M =  Fe
S
dy
dx
F = (z y) = 
T
So the final equation after
ceding equations reads
eliminating M, a and F
from the pre
1 d2y 1 dy e
 + + y = 0
R dx2 S dx T
The most general solution of this differential equation is of the
following form
y = Ae sx + Bes2x
A and B are two coefficients depending on the initial conditions
and sl and s2 the two roots of the equation of the second degree
s2 +s + E 0
 + + = o
R S T
Therefore, the equation of curve L, which represents the path of the
center M of the contact tread (fig. 34) is
y = Ae + Bes2x and not y cx Ce
y = Ae + Be and not y = Ce = Ce L
NACA TM 1337
as assumed by the American author without adequate reason. As a con
sequence the parameter D computed by wrongfully assuming that it is
the equation of curve L, may very well be tainted with an appreciable
error, which.by itself might be sufficient to explain the discrepancy
between curve E and C in figure 32.
V. EFFECT OF RESTORING TORQUE
(Inclination of Castering Axis)
EFFECT OF FLUID AND SOLID FRICTION
(Damping of Shimmy)
A. Effect of Restoring Torque
In the foregoing it has been assumed that the castering axis was
vertical. The next logical question is: what is the effect of an
inclination of the castering axis on the stability of the vibrations?
It is readily seen that, provided it is not too great, such an inclina
tion should be equivalent to the existence of a restoring torque propor
tional to the angle 0, this torque being moreover, negative in the
sense of figure 25, that is tending to move the wheel from its mean
position = 0.
Now the general problem of the effect of a restoring torque Cr = p9
will be analyzed. To express the problem in equation form, the same
line of reasoning used for the system of four equations (16), (17), (18),
and (19) is followed. It is apparent that, in these conditions, equa
tions (16), (17), and (19) are not changed by the existence of a
restoring torque. Equation (18) can be replaced by
d2
I = F(a + E) M pe
dt
or
I d2z 9
F(a + E) M  z
a dt2 a
since = z. To the moments F(a + E) and M produced by the
a
reactions of the ground on the contact tread, must be added the moment
pO of the restoring torque, p being positive when a true restoring
torque is involved and negative when a torque of opposite effect is
NACA TM 1337
involved (as for example in the case of a
illustrated in fig. 25).
The result is the equation system
disposition of the caster as
1 d2y =
v2 dt2
z y = TF
I d2z
 d2= F(a + ) M z
a dt2 a
M = 1 dy + FE
Sv dt Sa
(16)
(17)
(18a)
(19)
The values of F and M of equations (17) and
tions (16) and (18a) give the two equations
1 d2y 1 dy E
+ + y +
Rv2 d2 Sv dt T
1 sdy a I d2zT
Sv dt T a dt2 a
(19) entered in equa
 z =0
+  + Z = 0
Sa T
The values of z from equation (20) posted in equation (21a) give the
differential equation
I d4y
Rav2 dt
I dy + i + 1 a d___
+ + T+ +p+
Say at3 Ta SRav2 TRv2 Rav2Idt2
(+ e p \dy+ a + E 0
+ + avdt Ta
TSv Sav dt TSa aT/
(33)
It is easily seen that the second remark made below equation (22) is
still applicable here. Therefore, to obtain the differential equation
(20)
(21a)
NACA TM 1337 6!
of motion of the wheel y is simply replaced by z in the preceding
equation. The result is, then, a linear differential equation of the
fourth order of the form
aOzV + alzII + a2z I+ a3z + a4z = 0
In connection with equation (22) it was shown that Routh's conditions
for stable oscillation are as follows
ala2 > aoa3
aa2a3 > aa 32 + al 2a
Applied to equation (33) the first of these
reductions
Ic T
a SRav2
SRHav
conditions gives, after
>
Rv2
or, if the fundamental relation Sc = D is taken into account
2 aD T
v > i
IRD
The second of the preceding equations gives, after reductions
IE 1
+S
Ta 2
SRav2
PT
SRa2v2(a + e)
Rv2 a  (Sa T) > 0
~Za+
(34)
or
E
>T
I
S+
Sa2
Ep
Rav2(a + e)
66 NACA TM 1337
that is if the fundamental relationship is taken into consideration
(IRv2 a pT (aD T) > 0 (34a)
\ a + E/
To complete the discussion of these stability conditions, two cases are
distinguished according to the sign of the factor p.
(a) When a true restoring torque is involved, that is when p > 0,
the discussion presented following equations (23) and (23a) applies
here also. Consider the velocity u defined by the equality
IRu2 = a + pT
a + E
First case: aD > T. The condition (34a) is proved when v > u.
Second case: aD < T. The condition (34a) is proved when v < u.
It is plain that in either case, when equation (34a) is proved
equation (34) is also confirmed. Therefore, the conditions of stability
of oscillation are
First case: aD > T with v > u
Second case: aD < T with v < u
The problem therefore is like that discussed following equation (22)
and it is clear that the effect of a true restoring torque on the sta
bility conditions is simply an increase in the inversion velocity u.
(b) When a negative torque is involved, that is when p is nega
tive, which is reached at an inclination of the caster as shown in
figure 25, it can no longer be asserted that the confirmation of equa
tion (34a) automatically entails the confirmation of equation (34).
It is seen that, if aD < T this condition (34) is always proved,
but if aD > T, the condition (34) is proved only when v > w, w
denoting the velocity defined by the equality
IRw2 = a T
D
The stability conditions of the oscillation are therefore
NACA TM 1337
First case: aD > T with v > u and simultaneously v > w
Second case: aD < T with v < u
On trying to compare u and w it is readily apparent that in
a + E
order that u > w it is necessary and sufficient that p > 
From the previously defined values for E and D and the value of p
that will be computed subsequently (formula 35) it follows that, except
for large inclination of the axis and substantial wheel loads, u > w,
regardless of the caster length a.
The discussion is again similar to that relevant to equation (22)
to the effect that a negative restoring force on the stability condi
tions is simply a decrease of the inversion velocity u.
Let us now refer to figure 30 and compare the experimental curves A
and B given by Kantrowitz. By extending these curves up to their
intersection point with the velocity axis the corresponding inversion
velocities are obtained. Obviously the inversion velocity relative to
curve B is substantially lower than that with respect to curve A.
But curve B is exactly that which has been obtained for a 20 degree
inclination of the caster axis, corresponding to a negative restoring
torque. This experimental check although purely qualitative (because
the author failed to mention the load applied to the wheel, and which
is needed for defining the factor p) is an added proof of the present
theory.
The decrease in u following the inclination of the caster axis
is much more evident in this example since the experiments relative to
the curve B offigure 30 had been made .'ith a caster length (a = 1.73 m)
greater than that relative to curve A (a = 0.43 m). So without incli
nation the inversion velocity would have been greater, and it actually
would have resulted in
u= 1. 0 = 225 cm/sec
u =1i. 440 x 2. 38
for curve B, instead of u = 112 cm/sec for curve A.
Calculation of coefficient p for an inclination rp of the spindle
axis.
For the case of an inclination p of the spindle axis, the coef
ficient P of the negative restoring torque can be defined in terms
of the load P of the wheel on the ground, the caster length a, and
NACA TM 1337
the angle (p. Consider figure 35 which represents a tire touching the
ground at M and assume that AP is the spindle axis. When the wheel
turns about AP it can be assumed for first approximation that M
describes a circle centered on AP, that is a circle.of center H.
Axis AP is assumed to be stationary while the wheel turns through a
small angle e around this axis. Now it is readily seen that the
point M is raised up to a height
 e2 a2
h = MH  sin P = a sin p cos p
2 2
The work performed by the force F is therefore
e2
Ph = Pa  sin .7 cos C
2
with Cr = pe representing the negative restoring
that the foregoing work is equal to that performed
that is
0
Cr d = J
0
torque it is evident
by the torque Cr
Sd = P 2
Comparison of this expression with the preceding one, gives the sought
for expression of the coefficient p
p = Pa sin Cp cos !p
(35)
So, if the American author had not forgotten to give us the pres
sure P of the wheel on the ground, we would have been able to use his
experimental results for a numerical check of the present theory. In
the absence of these data the foregoing considerations can, however,
still be utilized for a numerical calculation by adopting an inverse
method of looking for the evaluation of P from an examination of the
curve B.
From an examination of the curve
the inversion velocity u is reduced
the inclination of the spindle axis.
seen that
B in figure 30 it is
to a value approaching
Hence 1.e assume u = 0
found that
zero, by
It was
u2 = a pT
IR (a + c)IR
NACA TM 1337
hence
a Pa sin (p cos qpT
IR (a + E)IR
consequently
a + E 1.73 + 2.4
T sin (p cos qp 84 x 108 x 0.342 x 0.94
that is
= 153 X 105 dynes
P # 15.3 kg weight
So, for
we get
the first series of experiments (fig. 30, curves A and A')
p = 107 x 0.43 x 0.087 x 0.996 # 3.5 x 105 C.G.S.
and for the second (curves B and B')
p = 107 x 1.73 x 0.342 0 x 0.94 # 55 x 105 C.G.S.
Comparative study of the curves of figure 30.
Curve B presents a slight deviation with respect to curve A.
It is logical to hunt for an explanation for this different shape with
the aid of the present theory. To start from the complete theory,
that is from equation (33) would involve considerable paper work.
Recourse is therefore had to the simplified theory which obviously
produces the equation
I d3y
Say dt3
dt'
IE dcy a + E p dy a + E ep
+ + + + y = O
Ta dt2 TSv Say dt TSa aT
obtained from equation (33) by disregarding all the terms having R
in the denominator. The same method that produced equation (11) gives,
therefore, when starting from equation (35), the system
(35)
NACA TM 1337
2TX(X2 + a)) + vD(X2 + aw) ( + E + Dp) = 0
I
T(3X2 n2) + 2vDX + a(a + ) + Tp = 0
I
The second of these equations gives
S= 3X2 + 2 D + a(a +
T IT I
(36)
which, substituted in the first equation, gives
[ n 2 vDx a(a + E) v(a + E + Dp)
(2TX + vD) 2 + 2 + + + = 0
T IT I
that is after transformations
2D2Iv2 + 8DTI + (aD T)(a + ev + 2TX ITX2 + a(a + e) + Tp] = 0
(37)
Compared to equation (12) this equation differs only by the p:
of coefficient p in the constant term. It is seen that for a rer
value of X there always are real values of v, provided only tha
determinant is positive, that is that
16DT2I(X2 + Dp) < (a + E)(aD T)
The maximum value Xm is then
2 (a + e)(aD T)2
16DT21
So, for the curve A (a = 0.43 cm, p = 3.5 x 105 C.G.S. system)
presence
al
t the
(38)
Xm = 21.47
NACA TM 1337
and for the curve B (a = 1.73, P = 55 X 105 C.G.S. system)
Xm = 21.38
Thus we obtain two practically equal values, which is in good agreement
with the shape of curves A and B.
We shall now turn to the calculation of the corresponding veloc
ities vm. Since the discriminant of equation (37) is zero ini this
case,
v2 = TX2 + a(a + e) + Tp
which gives
vm = 255 cm/sec = 8.4 ft/sec for curve A
vm = 314 cm/sec = 10.3 ft/sec for curve B
These values are almost identical with the experimental values in
figure 30. But above all this calculation has proved that the present
theory allows for the deviation of curve B relative to curve A,
which consists essentially of a displacement of the maximum and the
adjacent parts toward the higher values of the velocity.
Equation (36) enables the values of the angular frequency w
corresponding to the maximum Xm to he computed and consequently also
the values of the frequency n = . Thus
2Tr
n = 10.5 cycles/sec for curve A'
n = 12.4 cycles/sec for curve B'
These values very closely approximate those of the experimental curves A'
and B' of figure 30. It should be remembered above everything else
that the present theory justifies a higher frequency at corresponding
velocities for curve B' which is plainly indicated by curves A' and
B'.
The considerations are completed by the calculation of the limit
of the frequency N when the velocity increases infinitely. Concerning
equation (33), it is readily apparent that, when v increases infinitely,
the majority of its terms approach zero, so that at the limit this
72
equation is reduced to
IE d2y a +
Ta dt2 TSa
This equation represents a sinusoidal
hence
W2)
IT l D
NACA TM 1337
+ y = 0
function of the frequency
function of the frequency a
a + E +
ID I
IDI
Since N = 
2v
N = 17 cycles/sec for curve A'
N = 18.3 cycles/sec for curve B'
These figures are
and B'.
in perfect agreement with the experimental curves A'
In conclusion it should be borne in mind that the inclination of
the spindle axis, that is the angle of play, has only a small effect
on the stability of the oscillations. Therefore this angle might just
as well be determined from other considerations such as structural,
geometrical, or operational characteristics.
B. Effect of Viscous Friction
Viscous friction is called a resistance proportional to the veloc
ity of motion. Now suppose that the castering of the wheel is braked
by a torque equal to
de
f = , that is
dt
f dz
a at
The same line of reasoning employed in the foregoing for treating
the case of restoring torque will produce the system
Sd2 = RM (16)
v2 dt2
NACA TM 1337
z y = TF
I d2z = F(a + ) M f dz
a dt2 a dt
M = dy z F
Sv dt Sa
The values of F and M from equations (17) and
equations (16) and (18) yield the two equations
(19) entered in
1d2y 1 dy E 1 =
+ + + y z = 0
Rv2 dt2 Sv dt T Sa T
1 dy a
Sv dt T
I d2z f
a dt2 a
The values of z from equation (20)
the differential equation
d 4y
I dth
Ray2 dt4
+ 
~Sav
f dty
 d
Ravydt]
entered in equation (21b) give
+ 
\Ta
1
SRav2
SRav2
a
+ +v2
TRv2
f \ d2y
Sv) dt2
+ + 
\ TSv a Tdt TSa
It is readily seen that the second remark made following equa
tion (22) is applicable here too. To obtain the differential equation
of motion of the wheel, simply replace y by z in the preceding
equation. The result is a linear differential equation of the 4th order
of the form
aOzI + alz" + a2zI + a3z + a4z = 0
(17)
(18b)
(19)
(20)
(21b)
(39)
dz /1 a\
 + + z = 0
dt Sa T
NACA TM 1337
In connection with equation (22), it was seen that Routh's condi
tions for such a stable oscillation are
ala2 > a0a3
ala2a3 > a0a32 + a12a4
The direct study of these equations being too laborious, will be
abandoned in favor of the simplified theory. It is evident that this
theory produces the equation
I d3y lE f \d2y a +E fe dy a+ E
S+ + + + y = 0 (40)
Sav dt3 Ta Sa dt2 TSv aT dt TSa
obtained by disregarding all terms having R in the denominator of
equation (39).
Incidentally, this is the same equation obtained by disregarding
all the terms of higher degree in l/v, in equation (39), that is the
terms with 1/v2.
The motion defined by the simplified theory can, therefore, still
be regarded as the asymptotic limit of motion defined by the general
equation when the velocity increases indefinitely. This remark applies
also to equation (35) obtained in the theory of the restoring torque
from equation (33), and to equation (6) with respect to equation (22).
Equation (40) is of the form
aoyII + alyI + a2y + a3y = 0
and all its coefficients are, of necessity, positive. The unique con
dition of Routh's stability is then
ala2 a0a3
that is, by taking the fundamental relationship SE = D into account,
NACA TM 1337
(ID (a + E fD I(a + E)
T v v a av
which can also be written
ID2fv2 + [DTf2 + I(a + e)(aD T)v + Ta(a + e)f > 0 (41)
In relation to the velocity v the left hand side represents a
trinomial of the 2nd order admitting of two real roots of the same sign,
vl and v2, or else imaginary. Thus the condition of stability is
realized in the following two cases:
(1) If the discriminant A of the trinomial (41) is negative
(2) If the discriminant A is positive and if, in addition,
v < v1 or v2 < v
This argument assumes that v can take any algebraic value. But v
is necessarily positive, which also restricts the scope of the con
clusions, as will be seen. The discriminant A is written
A = D2T2f4 2IDT(a + c)(aD + T)f2 + 12(a + E)(aD T)2
With respect to f2, A is a trinomial of the second degree having
two positive roots
(a + E)
,I(a + E)(42)
f2 + aD (42)
DT
Before proceeding to the discussion, three preliminary remarks are noted:
(1) If aD > T, the left side of the inequality (41) is a sum of
positive terms and the stability condition is confirmed.
NACA TM 1337
(2) If DTf2 + I(a + E)(aD T) > 0, that is if f > p by
substituting
=I(a +c) T aD
DT
the stability condition is proved again for the same reason.
(3) When aD < T, then fl < p < f2.
The second of this double inequality is evident. As to the first,
it is sufficient to show that
aD + T 2DTi < T aD
that is
2aD < 2VaIf
or else
aD < T
which is true by assumption.
So, if aD
whatever the velocity may be, because
(a) either fl < f < f2 and A < 0
(b) or f > p
When f < fl, then A > 0 and the oscillation is stable only when
v
To sum up: stability exists no matter what the velocity may be
(1) if aD >T
(2) if aD fl
But, when aD < T and f < fl, the oscillations are unstable for any
velocities between vI and v2.
NACA TM 1337
According to these conclusions when a tire shimmies, the phenomenon
is, as a rule, observed only in a certain speed range (vl, v2), which
can become quite extended.
To overcome shimmy, if aD is not >T, it is sufficient to use a
hydraulic damper having a coefficient of viscous friction f greater
than the coefficient of friction fl defined by equation (42). This
solution has been frequently used, especially by the Americans.
Unfortunately, this solution is not without drawbacks, because it is
not simple, it increases the cost, requires a certain amount of main
tenance, and increases the weight of the vehicle, which is particularly
annoying when an airplane is involved. The best solution is very
likely the simple realization of the relation aD > T.
The coefficient of minimum viscous friction to overcome shimmy is
therefore fl. For a value of f slightly below fl, shimmy still
exists for velocities near to v0 given by the equation
2 Ta(a + E) (43)
vo (43)
ID2
obtained by substituting f = fl in equation (41). This velocity v0
can therefore be regarded as the most dangerous velocity for shimmy.
With respect to the velocity vm defined by equation (12a), that is
relative to the maximum inversion velocity in the absence of the damper,
we get therefore
vo W (44)
Vm aD + T
It follows, as is readily seen, that, as a rule
v0 Vm
m v
C. Effect of Solid Friction
Solid friction or, as it is also called, constant friction, is a
constant resistance independent of the speed of motion. Suppose that
the castering of the wheel is braked by a constant torque C. The case
of solid friction is more difficult to treat analytically than that of
NACA TM 1337
viscous friction, but if it is only a question of defining the torque
of solid friction C sufficient to prevent shimmy, the following
expedient can he resorted to: the solid or the viscous friction damper,
both act through the dissipation of energy transmitted to the wheel by
the reactions of the ground. It therefore seems logical to assume that
the solid friction damper will be sufficient to eliminate shimmy when
it absorbs at each oscillation the same amount of energy as the viscous
friction damper, that is with fl (formula (42)) for coefficient of
viscous friction.
The energy dissipated during a quarter cycle of amplitude em by
solid friction damper is W = COm. On the other hand, the energy dis
sipated by a viscous friction damper under the same conditions is
W = f de
de
oJ dt
ihen this damper is exactly able to make this oscillation represent
the boundary between stability and instability, a sinusoidal oscillation
0 = 8, sin wt and f = fl
is produced.
The preceding integral gives
W = f m2
Equating the two values of W produces
C= flaem (45)
In this formula all parameters, save the angular frequency m, are
known. It was shown that m is an increasing function with the veloc
ity but varying little with this velocity. So, when v increases
indefinitely the limiting value of w should be taken. This value is,
as seen,
NACA TM 1337 79
and formula (45) becomes thus
c a + e (46)
4 D
Hence the expression of the solid friction torque is sufficient and
even a little more than sufficient to stifle shimmy. This torque is
proportional to the maximum amplitude 8m while the viscous friction
coefficient fl necessary to achieve the same purpose is independent
of 0m. This remark is extremely important, because it explains the
fact pointed out frequently that, when solid friction is involved,
shimmy is not produced in general in the absence of an initial dis
placement of sufficient amplitude. In other words, when solid friction
is involved and there always is more or less friction in the spindle
axis, even in the absence of a special damper shimmy, in order to be
produced, has to be initiated or stimulated.
To avoid such initiation at landing, it might be useful to employ
a wheel castering mechanism to keep the angle small at the moment of
contact with the ground. So in formula (46) for the practical calcu
lation of torque C, one should use for Om the maximum amplitude
likely to be accidently produced. The determination of this angle 0m
raises some difficulties. It has already been proposed to take for ,@
the angle at which the skidding of the tire on the ground occurs. This
is justified if one assumes that for the greater angles sufficient
braking action is assured by the skidding of the tire on the ground.
Under these conditions formula (46) can be given a different form.
Consider a sinusoidal oscillation
J = m9 sin ut
and assume, as in the approximate theory, that the oscillation of the
wheel is largely due to the lateral force F exerted at a distance e
behind the point of contact. In this case
z = aem sin wt d2 = am2sin oat
dt2
I d2z = F(a + e) hence F(a + e) = IemaI2sin ut
a dt2
NACA TM 1337
and consequently
e= Fm a +
9m = Fm
Im2
Fm being the lateral force which produces the skidding. On assuming
that ac2 is still approximately equal to
a + E
ID
formula (46) becomes
C = i(a + e) Fm (47)
4 T
The limitation of tW amplitude of shimmy as a result of the skidding
of the tire when the lateral force F reaches the limit of adhesion Fm,
explains in particular why the terrain most favorable to intense shimmy
is macadam and dry and rough concrete, and why on the other hand, gravel
and light soil have a certain damping effect on the oscillations of
shimmy and thus limit the amplitude. It also explains why the load is
a factor having considerable effect on the intensity of shimmy. All
these facts have been checked many times on automobile wheels as will
be shown later.
Many designers have adopted the solid friction damper as a means
to stop shimmy. This solution has the advantage of being simpler and
cheaper than the hydraulic damper, but it has the drawback of "stiffening"
the castering of the wheel and so making it harder to maneuver the air
plane on the ground. This drawback makes this solution to the problem
extremely annoying in the case of airplanes of large weight, and prac
tically limits its use to light aircraft. Furthermore, it is very
difficult to ensure a well defined and perfectly constant value for
solid friction. There always is the risk of too much or not enough
tightness. All these drawbacks show that solid friction is not the
ideal remedy against shimmy.
Tests made during the war in Germany by the Motor Institute, a
branch of the Institute of Technology at Stuttgart, and by the
Consolidated Vultee Aircraft Corporation in the U. S. A. (See Aero
Digest of March 15, 1944, pp. 134 to 137; Conquest of Nose Wheel Shimmy
by C. B. Livers and J. B. Hurt) have shown that nose wheel shimmy of
the tricycle landing gear could be eliminated by the use of two wheels
side by side on one axle. This remedy is similar to the preceding one,
NACA TM 1337
because the two wheels are clamped on the same axle with the possibility
of differential rotation. The castering of the unit is, of necessity,
accompanied by a certain skidding of the wheels on the ground, skidding
equivalent to a solid friction. This solution present the same draw
backs as the foregoing, it makes handling on the ground difficult,
especially when a heavy airplane is involved.
Considerations of practical hydraulic dampers.
In the aforementioned study of viscous friction it had been
assumed that the resistance was rigorously proportional to the rate
of motion. In practice this law of damping is never realized for
actual hydraulic dampers, because the true law of resistance in terms
of velocity is the resultant of three effects:
(1) An effect of the viscosity of the liquid corresponding some
what to a resistance proportional to the velocity.
(2) A kinetic effect or socalled hydraulic friction or turbulence,
corresponding to a resistance proportional to the square of the velocity.
This is a resistance of purely kinetic origin that is produced in every
device forcing a liquid through a narrow orifice, even if a liquid
without appreciable viscosity is involved.
(3) A valve effect, the preceding orifices being often duplicates
of spring valves that remain closed as long as the pressure remains
below a certain value, depending on the force of the spring. On opening
they increase the area of the passage and prevent the torque from
increasing with the square of the speed. All sorts of dampers with
varied characteristics are obtained by appropriate control of the two
compensating effects.
As a rule, a damper is characterized by the curve representing
the resistance it offers to motion in terms of the rate of the motion.
The ordinary dampers have generally characteristic curves similar to
that shown in figure 36. The dashed lines represent the characteristic
curve of a solid friction damper and the theoretical curves of pure
viscous and pure hydraulic friction. The viscosity of a liquid decreases
when the temperature increases, so that the typical curve of an actual
damper is always deformed through diminution of all its ordinates.
With such a graphical representation it can be immediately determined
whether or not shimmy disappears if one assumes that two dampers are
equivalent from the point of view of oscillation stability when they
absorb the same amount of energy during one period. In that case, the
viscous damping indicated by the theory (formula(42)) is simply trans
ferred on to the characteristic diagram of the damper which gives a
straight line passing through the origin. Considering the shape of
the curve given for an actual damper it is seen that shimmy is likely
NACA TM 1337
to occur at low angular velocities and hence at low amplitudes. If
the curve of the,viscous damping required is tangent to the damper
characteristic at zero velocity, the damper is exactly strong enough
to prevent divergent oscillations. If the slope is less than that of
the damper characteristic at zero velocity, it is more' than sufficient
to eliminate shimmy. Conversely, if the slope is greater, the torque
at low velocities should be greater than that that can be given by the
damper, and shimmy results. Lastly, if curve of the damping necessary
intersects the curve of the damper at a comparatively high velocity
above which the damper gives a more than ample torque, shimmy amplitude
is limited to that at which the energy received by the wheel at each
cycle is equal to the energy dissipated by the damper. If the curve of
the resistance is definitely above the characteristic curve of the
damper, the latter is powerless to limit the amplitude of shimmy. As
the smallest amplitudes possible are already negligible, the only solu
tion is to stifle shimmy completely. On the other hand, the theory of
shimmy being applicable only to low amplitudes, if the damping provided
is insufficient for stifling shimmy completely, the amplitude attained
may be high enough so that the equations are no longer applicable, and
the motion continues then to increase in intensity instead of remaining
in a certain state of equilibrium. This is the reason why, in certain
cases, it was deemed preferable to adopt a greater damping than strictly
necessary in order to assure convergence; any oscillatory effects due
to deviations from linearity that could occur are thus damped out.
Equation (47) presents an unusual feature. Taking the coefficients
which we computed for the SchlippeDietrich tire, that is
e = 4 cm T = 157 X 1010 (C.G.S. D = 15.7 x 1010 (C.G.S.
system) system)
and computing by this equation the variations of the friction coeffi
cient C necessary to stifle shimmy for various caster lengths a,
the following results are obtained:
a = 0 cm C = 3.14 Fm C.G.S. system a'= 5 C = 2.09 Fm C.G.S. system
0.5 2.75 6 1.79
1 2.69 7 1.44
2 2.62 8 1.04
3 2.5 9 .56
4 2.32 10 0
The data are represented graphically in figure 37. Although this curve
decreases consistently when a varies from 0 to = 10, it is seen
D
NACA TM 1337 83
that C still is comparatively great for a values of the order of
magnitude of 6 cm, the decrease of C being especially rapid when a
approaches T/D. But since C measures to some extent, the eventual
intensity or the danger of shimmy, it follows that the usual caster
lengths are displaced precisely in the zone particularly dangerous for
shimmy. The matter becomes plainer if one notes that on certain tires
the curve C passes through a maximum for a value of a varying
between 0 and T/D. In fact if it is assumed that e = 2 instead
of c = 4 for the preceding tire, a figure still likely for a tire of
that size, equation (47) then permits the calculation of the following
values:
a = 0 cm C = 1.57 Fm C.G.S. system a = 5 cm C = 1.63 Fm C.G.S. system
0.5 1.52 6 1.44
1 1.61 7 1.18
2 1.74 8 .865
1.77 9 .475
4 1.74 10 0
The results are represented by the curve of figure 38. The curve C
passes through a maximum near a = 3. This remark can be illustrated
by an analytical calculation: computing the derivative of C from
equation (47) gives
dC r fD
a= 2fa + ,)
when a is infinitely small the foregoing quantity assumes a negative
infinite value which makes it possible to define the shape of the
T
curve C for very small values of a. When > 3E, the bracketed
term (a trinomial of the second degree with respect to J/) cancels
out for the two roots
^T i3E
3
It is readily apparent that the smallest of these values of a
corresponds to a minimum of C and the largest to a maximum. In the
present case this calculation gives
NACA TM 1337
a1 = 0.15 cm Cmin = 1.48 Fm C.G.S. system
a2 = 2.96 cm Cmax = 1.77 Fm C.G.S. system
The condition necessary and sufficient for curve C to assume a
maximum is
T
> 3c
This condition is realized on certain tires. The usual caster lengths
are then precisely the most dangerous for shimmy. This remark stresses
the importance of increasing the caster length to combat shimmy at its
very source. Equation (12) makes it possible to establish that, when
the velocity v varies, the maximum divergence Xm is given by the
equation
16DT2IXm2 = (a + e)(aD T)2
T
But a calculation readily proves that when T > 2e (a condition realized
for a large number of tires) and a is made to vary, this divergence Xm
passes through a maximum
A T + DE T + DE
6DT 31
for a caster length
T 2DE
a =
3D
The coefficient C seems better than the divergence X for
appraising the danger of shimmy.
Doctor Langguth's Experiments (1941).
It is timely to compare those theoretical conclusions with the
experimental results obtained in 1941 by Doctor Langguth and reported
by M. P. Mercier. He dealt with experiments on 260 X 85 tires, mounted
on a rolling mat, at a speed of 50 km/h. The author recorded the
maximum angular swivel or deflections of the wheel in terms of caster
lengths for castering angles of 30, 00, and +30. Figure 39 summarizes
the results. The four essential conclusions of this work are the
following:
IACA TM 1337
(1) The caster angle is of little importance
(2) The angular deflections pass through a maximum at a = 7 cm
approximately
(3) They cancel out at around a = 12 cm
(4) Increasing the load increases the angular deflections prac
tically in the same proportions
Before interpreting these results, it is advisable to understand
the meaning of angular deflection (swivel) studied by the German author,
since this quantity does not figure directly in the present theory.
Nevertheless the theory, considered superficially, seems to indicate
that, every time there is an oscillatory divergence, the angular deflec
tion should increase indefinitely. This apparent contradiction is due
to the fact that, in order to treat the problem, the data must be
linearized, which is, perfectly legitimate so long as the angular
deflections attain no unduly great values. Furthermore, the present
theory assumes the adhesion of the tire to the ground to be infinite,
while in reality this adherence is limited. The limit of adherence
explains in particular why the maximum angular deflections were approxi
mately proportional to the load on the tire.
Undoubtedly the present theory does not permit the ordinates of
the curves of Langguth to be tied quantitatively to the characteristic
parameters of the tire, but it does seem justified to assume that,
everything else being equal, the maximum angle must vary as C, and the
existence of the maximum discovered by Langguth seems to be in good
agreement with the foregoing theoretical conclusions. But from the
present writer's point of view the major importance of these experiments
is that they establish the existence of a caster length of about 12 cm
below which (for a < 12) the oscillations are unstable and above which
(for a > 12) the oscillations remain stable. In other words, Langguth's
experiments confirm quite well the essential conclusion of the present
theory, that is the stability of the oscillations with a sufficiently
large caster length. And this conclusion is quantitative as well as
qualitative: Langguth's 260 X 85 tire was of the same size as that
used by Schlippe and Dietrich. Therefore, it is legitimate to assume
that these tires have practically the same characteristics T and D.
T
But, it has been shown that the caster length a = of the Schlippe 
Dietrich tire, necessary for stable oscillations was about 10 cm. The
quasiagreement of this figure with Langguth's experimental value is
extremely remarkable, in as much as the slight residual discrepancy
between the two values can be readily attributed to a difference in
tire inflation in the two test series.
NACA TM 1337
VI. MATHEMATICAL THEORY OF COMPLEX SHIMMY
The term complex shimmy describes shimmy with two degrees of
freedom, the second degree of freedom resulting from a certain lateral
elasticity of the pivoting axis.
As in the study of elementary shimmy, we shall start with the
simplified theory, disregarding the torsional moment of the bottom
part of the tire, that is assume that the drift directly accompanies
the lateral force F which, in more exact terms, amounts to assuming
the coefficient of turn R of the tire to be infinite.
To put the problem into mathematical form we return to figure 8
and to the line of reasoning that produced the system of equations (1),
(2), and (3). But, first, in order to account for the lateral elasticity
of the pivoting axis, it is advisable to replace figure 8 by figure 40:
With I as the lateral displacement of the pivot under the action
of the lateral force FO,
2 = TOFO
where TO is called the coefficient of the lateral elasticity of the
pivot.
Equation (1) always gives the relation
= e + = DF
but now
e = z
a
so that equation (1) must be replaced by
1 dy z 1
St a a DF
v dt a a
Equation (2) is obviously not modified.
NACA TM 1337
As to equation (3), it is transformed as follows: If IO is the
inertia of the wheel about the vertical axis passing through the center
of gravity of the wheel, then I = IO + ma2, where m = mass of wheel.
The angular acceleration of the wheel about this axis .is a function of
the sum of the moments of the forces F and FO, hence
IO  = Fa FE
dt2
now
S= z and
a
F0 =
0 = T
Therefore
a0 dt2
a dt2
d2 1 al F
at2 TO
The introduction of a new variable Z calls for a fourth equation,
which is obtained by noting that the acceleration of the center of
gravity 0 is due to the sum of the forces F and FO. Thus
dz 
m  = F
dt2
 FO = F 
To
Hence the following system
1dy+ z_ DF
v dt a a
z y = TF
I d2z
a dt2
2 a2
S= F
dt2 TO
(48)
(49)
(50)
88 NACA TM 1337
d2z z
m  = F (51)
dt2 TO
To obtain the differential equation in z defining the motion of
the wheel, simply eliminate F, 1, and y from these four equations,
as follows:
The four equations are linear and homogeneous. So, if
y = Aest
is a particular solution of the system, three particular solutions
for z, 1, and F of the form
z = Best 2 = Cest F = Dest
are obtained. Thus the equation in s of the foregoing system or the
characteristic equation of the system can be obtained directly by
replacing the unknowns y, z, 1, and F by the values above and then
eliminating A, B, C, and D from the four equations of the system by
Cramer's method of determinants. Hence we obtain an equation in s
in the form
(y) (z) (z) (F)
1 1 1 D
s D
v a a
1 1 0 T
= 0
0 LO0 s2 10 s2 a
a a TO
0 ms2 1 1
To
NACA TM 1337
The development of the determinant gives the characteristic equation
mlO s5 mD 1 10 + ma2 s3 1D IO + ma2) s2 +
v T V\T To T T
a(a + g) s + a + = 0
TTov TTO
The differential equation sought is thus
mlITTO d5z
 dt5
v dt5
d4z ITOT0 + IT d3z d2,
mCP 4 T d + (ID + mTO) +
dt4 dt3 dt2
a(a + F) dz
a(a ) (a + E)z = 0 (52)
V dt
It should be noted that, when TO = 0, that is when the pivot has no
lateral elasticity, the foregoing equation reverts to equation (6).
Two particular cases with important conclusions will be analyzed.
1. Wheel Absolutely Rigid
Assume that simultaneously
T= 0
D= 0
e =0
that is a wheel without any lateral elasticity, hence without drift.
Equation (52) is simplified and becomes
IOTO d3z a(a + e) dz
 + + (a + E)Z = 0
v dt3 v dt
It is readily seen that the condition of Routh's convergence can
never be realized with such an equation. In such a case, if TO is
not zero, that is if the pivot has a certain lateral elasticity, there
always will be a divergent oscillation, that is shimmy.
NACA TM 1337
The lateral elasticity of the pivot can be the sole generator of
shimmy without it being necessary for the mechanical properties of the
tire themselves being involved. For certain particular combinations
of tire factors and speed, the lateral elasticity of the pivot can
therefore become the factor facilitating shimmy.
2. Pivot With Great Lateral Freedom
Centering attention on parameter TO it is assumed that TO = m,
that is that the pivot is absolutely free to shift laterally under the
action of a force no matter how small. Equation (52) then becomes
mIoT d5z d4z 10 d3z d2z
 + mI0D + + m 0e 0
Sdt5 dt4 d3 dt2
d2z
Putting = u, the preceding equation, which is independent
dt2
of a is transformed into a differential equation of the 3rd order
for which Routh's stability condition becomes, after simplifications,
I0 >m T E
It is readily apparent that this condition is realized generally
with ordinary tires. The ratio T/D is a length generally very close
to the radius r of the tire; and considering that
1 mr2
K is a coefficient generally near 2. The preceding inequality thus
becomes r > 2e, which is amply confirmed for all customary tires.
Since the pivot is completely free to shift laterally, the oscil
lations are, naturally, convergent, that is there can be no shimmy.
Hence the possibility of a new remedy for removing shimmy by giving
the pivot complete lateral freedom.
Based upon more elementary reasoning of qualitative rather than
quantitative nature, various American technicians had been led to
assume that the possibility of a wheel to shift laterally without
pivoting should be capable of suppressing shimmy. To realize this aim,
two types of mechanical arrangements were recommended and tested, either
on the airplane or on the beltmachine apparatus.
NACA TM 1337
The first method consists in letting the free wheel slide on its
axis, by allowing the necessary play at the end of the hub. The axis
was slightly curved so that the wheel had a tendency to remain in the *
center. This method was tested by the NACA on the Weick Wl airplane
and revealed itself as being effective as long as the pivoting remained
below 130 in one direction or the other. But other tests have given
no satisfaction. In the light of the present theory it is easy to
understand the reason for this failure. Whatever the perfection of
lubrication, the friction of the wheel on its axle, in the motion of
the lateral displacement, cannot be considered as negligible. This
friction reestablishes, in a certain measure, a lateral restraint. It
follows that, if the stability condition
IO > m e
is not greatly exceeded, the phenomenon of shimmy may reappear.
The second method consists in utilizing two pivots coupled as
illustrated in figure 41. The NACA tested this system in the laboratory
and found that it eliminated shimmy. It was also studied by E. R. Warner
(ref. 4). The operation is obvious. This arrangement realizes complete
lateral freedom for the wheel pivot. It seems quite superior to the
first, although structural complications are involved and landing gear
retraction is made more difficult. The American authors, E. S. Jenkins
and A. F. Donovan (ref. 5) believe, however, that it could be made as
light as a damped system and that the elimination of the detrimental
effects of damping friction on ground handling, as well as the elimina
tion of the upkeep of a damper, would have appreciable advantages. The
additional complications introduced by this system of double pivot are
compensated for in part by dispensing with the scissors which transmit
the shimmy torque to the damper in the ordinary arrangements.
Clearly, all the advantages presented by this arrangement are
found in that advocated in the present report (simple lengthening of
caster length) and which has the added advantage of greater simplicity.
The two foregoing examples prove that, in the fight against shimmy,
the existence of a certain amount of lateral elasticity may sometimes
turn out beneficial, sometimes harmful, depending on the magnitude of
this elasticity and the values of the characteristic coefficients and
the velocity. A clarification of this point calls for a complete study
of the stability condition of equation (52). Unfortunately, too much
paper work is involved to be undertaken by us. Furthermore there
does not seem to be much practical interest in such a discussion; in
fact, it has been shon that it is comparatively easy to stop shimmy
in the case of a rigid pivot; and the realization of a lateral elasticity,
92 NACA TM 1337
properly speaking, in the pivot will present difficulties of construc
tion and drawbacks of practical nature.
Effect of a restoring torque
Assume a restoring torque of the form Cr = P9. Obviously, equa
tions (48), (49), and (51) remain unaltered. Equation (50) is replaced
by
IO 2z ad2 a p
 z (50a)
a t2 dt2 T a
the same way as equation (18) was replaced by equation (18a) to obtain
equation (33).
The result is a system of four equations which is resolved by
means of the method used for equations (48), (49), (50), and (51).
Consequently
mIoTT0 d5z d4z IT + 10TO + mTTOP d3z
 + mIoDTo  +  + (ID + mcT0
v dt t v dt3
d2z a(a + e) + (T + TO)P dz
DTop) +  + (a + E + Dp)z = 0
dt2 v dt
If p = O, equation (52) is obtained again.
Complete Theory of Complex Shimmy
If the torsional elasticity and the turning effect of the tire
are involved, the foregoing line of reasoning results in a system of
five equations.
S= Rd
v2 dt2
z y = TF
NACA TM 1337 93
I0 2z d2\ a2
I aF +  M
a t2 dt2/ T 
1 dy z l
M = + FE
sv dt Sa
d22 z
m  F + 
dt2 TO
The differential equation sought is obtained by eliminating F, M,
1, and y from the five equations. This system can also be studied and
resolved more rapidly by the method employed for equations (48), (49),
(50), and (51). Even then the calculations are long and therefore not
attempted.
VII. SHIMMY OF AUTOMOBILE WHEELS
Ever since the first automobiles started to appear in public
certain cars revealed the existence of a very peculiar oscillatory
phenomenon of the front wheels consisting of a periodic lateral
wavering. This phenomenon generally occurred when the car reached or
exceeded certain critical speeds. At first, it was called waddling or
tacking. In the beginning, it seemed to present no particularly serious
problem and car designers paid little attention to it. Moreover, at
the slow speeds of cars in that era, it was successfully overcome by
all sorts of empirical means, among which might be mentioned at random:
Toein of front wheels
Increasing the angle of camber
Use of tight shock absorbers
Stiff front springs
Stiff and irreversible steering
Higher tire pressure, etc.
NACA T' 1337
All these attempts were no more than palliatives, the simple effect
of which was to put the critical speed for the appearance of shimmy
above the speed of utilization of the car. Shifting the critical speed
was comparatively easy, except for racing cars where more energetic
remedies were called for. The only truly effective remedy for this
type of car was the use of very stiff suspensions and hard shock
absorbers which meant the complete loss of comfort.
But this state of affairs became suddenly worse when the Americans
tried to introduce the balloon tire. The much greater elasticity of
this tire, so pleasant for greater comfort, actually, lowered the
critical speed of oscillation of the front wheels considerably and.this
was aggravated as the suspension became more flexible and front wheel
brakes came into wider use. On some cars, these phenomena were so
violent that it became impossible to speed up even moderately. The
oscillation was transmitted integrally to the axle and, because of
gyroscopic interactions, was not just limited to a simple motion in a
horizontal plane but also in the vertical plane, which accentuated and
compounded the manifestations to an unusual degree. In some cases the
axle seemed to do a fervent acrobatic dance, and this is why the
Americans got to designate the whole of these phenomena by the name
of an exotic dance then in vogue in the U. S. A. and the name shimmy
has remained classical ever since.
For some years shimmy was regarded as a particularly grave ailment
of balloon tires and even as a redhibitory vice to their commerical
distribution. Nevertheless, the greater riding comfort on balloon
tires tempted the engineers of every country to all sorts of ways to
combat this parasitic and annoying phenomenon. But the technicians
themselves did not agree on the origin and nature of shimmy, and even
less on a rational method of avoiding its effects. The question
reached such importance that a whole session of the U. C. Congress of
Automobile Engineers in 1925, was devoted to it. Everybody stuck
loyally to the results of his own observations and to his personal
opinion. But the diversity of opinion of assumptions and remedies
suggested, the utter lack of a general viewpoint on the nature of the
subject, far from bringing any enlightenment, actually left the impres
sion of profound inconsistency. It is only necessary to consult "La
technique automobile et aerienne," vol. 16, no. 130, 3rd quarter 1925)
which published the principal reports read at that session. It is
particularly surprising to find that not one referred to the true origin
of shimmy. The underlying cause of shimmy lies in the peculiar mechani
cal properties of the tire. It is a complex shimmy, the sizeable
elasticity of the longitudinal suspension springs plays the part of the
lateral elasticity of the pivot.
However, the complete mathematical theory of this shimmy of auto
mobile wheels is more complicated than that of complex shimmy described
NACA TM 1337
in the foregoing and its external manifestations themselves are a little
more dissimilar by reason of the following two factors:
(1) The reactions of the linkage or of the steering gear
(2) The gyroscopic reactions
The first idea of seeking the cause of shimmy in the mechanical pro
perties of the tire is much earlier than the recent reports of.American
experimenters on airplane tires. On May 22, 1925 a French engineer,
known for his many publications on the dynamics of the automobile and
for his frequently original views, A. Broulhiet, gave a very interesting
speech before the Socie4t des Ingenieus Civile de France (see the
report of this society (1925, pp. 540554)) in which he stated that
the drift of the tire was the true cause of shimmy. Perhaps Broulhiet's
idea was not convincing enough, perhaps it was not accompanied by a
sufficiently correct mathematical theory nor developed enough to have
attention called to it as the only correct explanation of shimmy. In
any case, it was not taken up by other theorists and did not receive
the publicity it should have had. Incidentally, the development of
automobile engineering has a rather unusual history and which proves
that even a wrong theory can sometimes be very practical.
At this meeting of the Society of American Automobile Engineers
in 1925, A. Sealy, of the Dunlop Tire Company, presented a report in
which he charged the gyroscopic effect of the wheels as being the
cause of shimmy. The idea was taken up in France by an engineer of
Russian extraction, Dimitri Sensaud De Lavaud, a specialist in automobile
dynamics, who undertook to publish a complete mathematical theory of
this gyroscopic effect in two installments refss. 6 and 7).
It is true that such a study is of technical interest, because
the gyroscopic reactions have an undesirable influence on the shimmying
motions if the two front wheels are mounted on the same axle. But
they can in no case be the true cause of shimmy because they are con
servers of energy. Consequently the gyroscopic effect absolutely can
not explain the fundamental characteristic of shimmy which is the
divergence or instability of oscillations resulting from a continuous
transformation of kinetic energy of translation of the car into oscil
lating energy of the wheel.
This matter did not escape De Lavaud's attention, who explicitly
remarked on it. To explain the divergence of the oscillations he
attempted to bring in the rocking motion which naturally accompanies
shimmying. To be sure, there is a real possibility of divergence but
this scarcely perceptible rocking motion is so minute that its amount
of energy per cycle is very small and surely too small to.explain the
intensity of shimmy especially when considering the many sources of
damping which restrain the turning motion of the wheel.
NACA TM 1337
In De Lavaud's theory this imperceptible rocking motion of the
car is the sole possible cause of the divergence of oscillation. But
it is plainly inadequate. The author was therefore compelled to regard
shimmy not as a phenomenon of instability, that is divergence of oscil
lations, but as a simple resonance phenomenon arising from defective
balancing or centering of the wheels. Today it is known as an absolute
fact that shimmy is not a resonance phenomenon but due solely to the
divergence of oscillations, which is entirely different. Even if it
is true that the unbalance of the front wheels may facilitate and, in
certain cases, actually cause artificial shimmy, experience proves
that defective balancing of wheels or tires, whether automobile or
airplane, is not the true cause of shimmy. No resonance phenomenon
could ever explain the intensity so often observed. Furthermore,
shimmy has been observed on perfectly balanced wheels rolling on level
ground. In addition the phenomenon designated by the Americans with
the name of kinematic shimmy, indicates clearly, that it can have an
oscillatory motion without the least effect of inertia, which, however,
is necessary in order to have resonance.
To sum up, the gyroscopic actions which accompany automobile wheel
shimmy and which complicate the phenomenon can never be entirely the
cause. This is readily apparent when considering airplane wheel shimmy
studied under the name of elementary shimmy. In this case the gyro
scopic actions are actually without any effect on the motion of the
wheel, since their moment is then perpendicular to the pivoting axis
which itself is absolutely rigid.
In short, De Lavaud mathematically defined the concept of the
gyroscopic coupling of two oscillations, very important, it is true,
but which did not evolve the true source of energy brought into play
by shimmy. As a logical consequence of his theory, De Lavaud was
forced to advocate the elimination of the gyroscopic effect by an
appropriate mode of linkage of the front wheels to the chassis per
mitting the wheels only a vertical displacement in their own plane.
Through this De Lavaud became the champion and forerunner of this type
of suspension, which under the name of independent front wheel suspen
sion achieved a certain amount of success for two reasons:
First, the independent front wheel eliminated, as foreseen, the
gyroscopic reactions and their vitiating effects, not only in the case
of shimmy but also for the passage of the wheel over any obstacle, and
this advantage alone was enough to justify its use. But, and this is
a curious historical fact, the independent front wheels also completely
eliminated shimmy. In the face of these advantages one after the other
automobile builder adopted this mode of suspension. So, with the pro
blem of shimmy practically solved, the automobile engineers dropped its
study.
NACA TM 1337
Without any other justification it was accepted as an accomplished
fact that the gyroscopic effects were the true cause of this phenomena,
and up to this day the theory of De Lavaud is regarded as complete
explanation. But now it has been proved that the gyr.oscopic actions
are simply a phenomenon accompanying but not a primary cause of shimmy.
How then can the fact that the independent wheels eliminate shimmy be
explained?
Actually, all independent front wheel suspension systems have in
common this peculiarity of totally suppressing the lateral elasticity
of the front wheels, a lateral elasticity which was formerly consider
able with the longitudinal springs generally used for suspending the
chassis from the axle common to the two wheels. Owing to this fact,
automobile wheels can develop only elementary shimmy with a restoring
torque due to the elasticity of the steering gear. The law of motion
then follows equation (33). To establish the stability conditions,
reference is made to the discussion following this equation. In auto
mobile wheels with a given caster length
aD < T
It follows that the unique condition of stability is v < u, that is
a pT
v2 < a PT
IR (a + E)IR
To find the conditions of stability at any speed the car may reach
simply consider a direction in which the steering is rigid or in other
words, large enough p. This is precisely the combination realized
on all modern cars.
VIII. CONCLUSIONS
The present report, instead of starting with long series of experi
ments and involving costly materials, begins with the discussion and
exploitation of all previous experiments in enough detail to be prac
ticable. Life is short and it serves no useful purpose to lose time
in repeating the experiments already made with care and of which the
results have been published. This method of work certainly saves time
and money, because, taking into consideration that the comparison of
the present theory with the many results obtained earlier by other
experimenters, could leave no doubt about the physical value of this
theory and it is this physical value which is the principal achievement
of this work.
NACA TM 1337
However, some experiments were carried out by the O.N.E.R.A. at
Toulouse in order to support this study which involved an experimental
check of the fundamental stability equation (23a).
(IRv2 a)(aD T) > 0
This formula can be checked by ascertaining that the conditions of
stability and those of shimmy in terms of caster length a and velo
city v are those represented in figure 42. The experiments themselves
merely involve checking the pressure or absence of shimmy, hence com
paratively easy and inexpensive measurements, which are, in any case,
less difficult and shorter than Kantrowitz's divergence measurements.
In all cases, the practical use of this theory requires the
preliminary and systematic measurement of the characteristic factors T,
D, S, R, and E of the different tires used in aeronautics. It also is
desirable to study the variations of these factors in terms of tire
inflation and load. Such a study would make it possible to decide the
form, size, and inflation pressure required to obtain the minimum for
the quotient T/D, so as to be able to combat shimmy without having to
resort to long caster lengths.
To conclude, it is a theory which seems to have already been
supported by many experimental proofs and which is herewith presented
to tire technicians and experimenters; I now let them speak.
Translated by J. Vanier
National Advisory Committee
for Aeronautics
