Present state of development in nonsteady motion of a lifting surface


Material Information

Present state of development in nonsteady motion of a lifting surface = Lo stato attuale delle ricerche sul moto instazionario di una superficie portante
Portion of title:
Lo stato attuale delle ricerche sul moto instazionario di una superficie portante
Physical Description:
96 p. : ; 28 cm.
Cicala, P
United States -- National Advisory Committee for Aeronautics
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aeronautics   ( lcsh )
Aerofoils   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Includes bibliographic references (p. 80-84).
Additional Physical Form:
Also available in electronic format.
Statement of Responsibility:
by P. Cicala.
General Note:
"Technical memorandum 1277."
General Note:
"Report date October 1951."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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Full Text
NAIc4 7 r 12177





By P. Cicala

A summary is given of the principal results thus far obtained
from studies of the nonsteady motion of a lifting surface in an
incompressible fluid; the methods followed by various investigators
are indicated.

The aerodynamic problem of the nonsteady motion of an airfoil
has been the subject of numerous investigations, which in 20 years
have brought a degree of development such that an entire branch
(that of two-dimensional motion) may be said to have been completely
solved. The mass of existing publications is very large and among
these naturally many overlap; moreover, the study of the problem has
produced a variety of methods such that the same phenomenon is
endowed with rather diverse aspects that, although of considerable
speculative interest, do not always facilitate the task of those who
wish to learn only the results of the research. A synthetic deduc-
tion of the results thus far obtained is given hereinl in order to
expound the principles of the various methods of investigation and
particularly to collect the latest results in a form that is best
suited to application.

As previously stated, the two-dimensional problem has been
exhaustively studied. In this field, the problems of unsteady motion
reduce to computations that at times are laborious, but which can
always be conducted without uncertainties of the approximations and
which are considerably facilitated by existing tabulations. Only a
few attempts have been made to develop a rigorous computation for
the wing of finite aspect ratio inasmuch as the existing methods
present approximate solutions that contain many inaccuracies.

In part I of this report, the results relative to the wing of
infinite aspect ratio are described, always with the assumption of a
perfect and incompressible fluid with regard to the components of

*"Lo Stato Attuale delle Ricerche sul Moto Instazionario di Una
Superficie Portante." Estratto Da "L'Aerotecnica" Vol. XXI, N. 9-10,
Settembre-Ottobre 1941, XIX.

iThe analytic developments have been shortened and the rigor of
the demonstrations has at times been relaxed.

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of the normal force on the wing plane. In part II, procedures are
indicated for the computation of the wing of finite aspect ratio;
a comparison is made of the various approximations thus far employed.
In part III, the results are given relative to the force component
in the direction of motion (propulsive or drag). In part IV, the
results of experimental investigations are considered. In each part
an evaluation is given of the existing publications, which may serve
as a guide to a more detailed study of any aspect of the problem.

Once a reference system is established with respect to which
the fluid is motionless at infinity, it is assumed that a fixed
plane exists from which the points of the lifting surface are at
distances that may always be considered very small with respect to
the dimensions of the projection of the surface on the plane. The
thickness and the curvature of the profile are therefore considered
infinitesimal, as are the displacements along the normal to the
fixed plane. Almost all the investigations up to the present time
on the nonsteady motion of a wing use simplifications that are
derived from the preceding assumption and thus assume that the sin-
gularities in which the body in motion is schematized are permanently
contained in the fundamental plane. In this plane lie the orthogonal
axes x and y, which are displaced with respect to the fixed refer-
ence and remain parallel to themselves with a velocity V parallel
to the x-axis but opposite in direction; the z-axis is at right angles
to the xy-axes. With respect to these axes, which follow the motion
of the wing, the relative velocities of the points of the wing are
small with respect to V. It is thus assumed that the perturbations
produced by the motion of the wing are sufficiently small. Hence, in
the relations that are used, all terms of higher order than the linear
terms in the velocities induced by the motion of the obstacle in the
surrounding fluid are neglected. This fundamental simplification
gives the problem under consideration the advantages of the linear
theories (the most important of which is the principle of superpo-
sition) from the effects of which by the analysis of particularly
simple motions, solutions can be obtained that through linear combi-
nations make possible the study of motions of a more complicated

The velocity V is assumed to be small compared with the velo-
city of sound although investigations have been conducted that consider
the compressibility of the medium (reference 1).

The variations in time or space must frequently be measured.
The symbols occurring in the derivations herein are therefore defined
in the following paragraph:

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Let q be any magnitude that is a function of a point and that
for every point varies in time. The symbol 8q/6x (or 6q/6y)
denotes the dependence of the law of variation of q as a function
of x (or of y), measured by giving to q, at each point, the
values that correspond to a certain fixed time t. The symbol
6q/6t indicates the rate of variation of q at a fixed point
relative to the x, y, and z axes, which in the absolute refer-
ence system is displaced with the velocity V. In general, with
the assumed linearization, the derivative also represents the rate
of variation of q at a fixed point of the wing because the velocity
of a point of the wing with respect to the moving axes is very small;
if the gradient of q is not too great, the variation of q in time
will be the same whether measured at the fixed point relative to the
moving axes or measured at the point that follows the lifting surface.
In order to express the rate of change in time of a q always
measured at the same point of the stationary reference, the symbol
d'q/dt is used. Because of the assumed linearization, the previous
derivative coincides with the derivative that measures the change in
q with time for the same material point. Inasmuch as the absolute
velocities of the fluid particles are very small, if the gradient of
q is not too large, the change of q with time will be the same
whether measured at the fixed point or following the molecule. It is
preferred in this report to use a distinctive sign in d'q to differ-
entiate the local derivative from the derivative of the quantities
that depend for the problem under consideration only on the time param-
eter; for this problem the notation dq/dt is used.


1. The results of modern research on the two-dimensional problem
will first be described. The simplification introduced by the assump-
tion that the phenomenon develops in the plane of the x- and z-axes
is such that it can be stated that each problem within this range
can be reduced to the computation of integrals that, with graphical
procedures aided by analytical considerations, can be computed with
suitable accuracy (limited to the approximation of the existing
tables), which eliminates the expansion into series. In the three-
dimensional case (the wing of finite span), except for some parti-
ular problems, only approximate solutions exist.

In order to simplify the expressions of the two-dimensional
problem, the semichord of the wing is assumed to be of unit length.
The abscissas of the leading and trailing edge of the wing are assumed
to be given by x = -1 and x = 1, respectively. In order to define

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the points of the profile, the parameter 6 is introduced. The
variation with x is shown by the relation

x Cos 0

The values 0 and x of the parameter therefore correspond to
the leading and trailing edges of the wing, respectively.

The coordinate z, normal to x, is considered positive in the
downward direction (fig. 1). The vertical component v of the velo-
city of the fluid is therefore considered positive if turned in the
direction of positive z. The difference in pressure p between the
two faces of the wing has the positive sign directed upward. The
same sign convention is true for the vertical force P, the lift of
a segment of unit chord. The moment M on the same segment is con-
sidered positive if it is a diving moment.

2. Condition of tangency. The condition that the profile be
impenetrable to the fluid is expressed by making the relative velo-
city of the fluid, with respect to the wing, tangent to the profile;
or, in other words, the absolute velocities Vc of a point on the
contour and Vf of the fluid particle in contact with the contour
have the same projection on the normal n (fig. 2). Inasmuch as
only the linear terms in the coordinate z of the points of the con-
tour and their derivatives, or in the velocities (perturbations)
created in the fluid by the motion of the wing are considered, the
component on n of the velocity Vf may be supposed equal to the
component w of the velocity of the fluid particle, which is parallel
to z. By using the simplification of neglecting the quadratic terms
in the computation of the projection on n of the velocity Vc,
which has the components V and 6z/ot, there is obtained

bz 6z
S= + (1)

3. Circulation and pressure on profile in steady motion. Under
the conditions of steady motion, it is known that the relative velo-
cities of the fluid with respect to the profile, even if the profile
is considered to be of infinitesimal thickness, are, in general,
different on the two surfaces. In the motion under consideration,
if V1 and V2 are the velocities at the corresponding points of
the two surfaces (fig. 1), the pressure rise p between the surfaces
is given by the Bernoulli equation

:IACA T 1277

p = (v2-v21) = P(V -V1) V2 PVu

where it may be assumed, because of the linearization hypothesis, that
the average of the velocities V1 and V2 is V, and u denotes
the difference between them.

If the wing is in the positive aspect, the smallest velocity is
found at the bottom surface. The velocity difference u is con-
sidered positive under these conditions and the positive pressure p
is therefore in the opposite direction to positive z. In order to
represent the field created by the wing, the skeleton of the wing
composed of a vortex film is considered; this system of singularities
is capable of giving the existing velocity increment u between the
two surfaces if the circulation in an element dx is equal to u dx.

4. Circulation and pressure on profile in unsteady motion. -
For the case where the motion is unsteady, the velocity increment may
be represented by a vortex distribution of intensity g u. The only
difference, when compared with the preceding case, is in the fact that
the discontinuity in the velocity field exists not only on the points
of the wing but also in the wake behind the wing. In order to describe
this phenomenon, consider two fluid layers that pass above and below the
profile. Because of the linearization assumption, the distances of the
points of the profile from the x-axis can be neglected in the following
discussion. It can easily be verified that the terms of the second
order will therefore be neglected. The difference between the momen-
tums of the two layers is computed (fig. 3)2 at the time t0 and at
the time t0 + dt and therefore the variation that the difference has
undergone in the interval dt is also computed. This variation,
divided by the interval dt, must be equal to the difference between
the forces along x that are applied to the two layers. Inasmuch as
tangential actions do not exist (perfect fluid), and because no pressure
difference exists on the anterior face (upstream of the profile), the
previously mentioned momentum will be given by (Pl-P2) dy = p dy.
The difference in the momentum of the molecules of the two layers,
which is given by the product of the mass and the difference in the
velocities u, varies in the interval dt because:

2The two layers are indicated in figure 3 by hatched lines in the
two directions; the position of the leading edge of the profile is
Indicated by a semicircle.

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1. New molecules come in contact with the profile3 and there-
fore acquire the velocity increment u = g.

2. The increment for a given point of the profile varies with

If the graphs of g relative to t = t0 and t = t0 + dt on
the same position of the profile (fig. 3) are plotted corresponding
to the two causes, respectively, it is found that the area (which
gives the difference in the momentum when multiplied by P dy) varies
in the interval dt by the amount

gV dt + dt dx

The first term is indicated in figure 3 by the obliquely hatched
area (which, except for infinitesimal of higher order, is equivalent
to a rectangle of base V dt and altitude equal to the value of g
corresponding to the abscissa x of point P); the second term is
indicated by the vertically hatched area. Equating the impulse to
the variation of the momentum and dividing by p dy dt yield4

g + dx (2)
P -i

The first term on the right side measures the pressure at P,
which is obtained under the conditions of steady motion; this pres-
sure depends on the local value of the velocity increment. By the
effect of the other term, the pressure under the conditions of non-
steady motion depends on the variation that g undergoes at the
instant considered in the entire strip ahead of P.

3The difference of the momentum of the two layers is observed to
be zero before arriving at the wing because the velocity increment in
front of the wing is zero if there are no other liTting surfaces.

4This relation is obtained by another method in differential
form in reference 2.

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5. Vorticity of wake. If at the instant t = tO the point P
coincides with the trailing edge of the wing, equation (2) still
holds, but the first term becomes zero because a difference in pres-
sure cannot exist where there is no wing surface. It follows that
the increase in velocity gs existing in the points of the wake,
which is zero in the case in which the distribution of g on the
wing is independent of the time, is given by

SdK (3)
gs = V dt


K = g dx

is the circulation about the wing and the derivative is measured at
the instant at which the trailing edge passes through the point of
the wake under consideration. In other words, in the distance V dt
that the wing moved in the time interval dt a vorticity is distri-
buted equal to the variation that the circulation about the profile
has undergone in the same time. This conclusion can also be derived
from the'principle of the conservation of vorticity.

In figure 4, the diagram of the vorticities on the profile and
in the wake is given for the case in which the wing executes a trans-
latory oscillation with frequency 0 related to L and V by

OL = 2V

The curve a refers to the instant in which the wing crosses
the middle position; b refers to the position at the end. The
scale of the vortices is indicated by assuming the vertical maximum
velocity of the profile to be unity. For comparison, figure 4 shows
the graph of the circulation g' corresponding to the maximum veloc-
ity in the case of steady motion. The increment of velocity in the
wake is considerable. In the theories of a wing in nonsteady motion,
it is generally assumed that this velocity increment remains in the
position in which it originated, or, in other words, that the vortices

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shed by the wing maintain their original position and intensity
unaltered in time. It is evident that the error inherent in this
assumption, from which the real phenomenon certainly deviates,
impairs to some extent the results of the theory.

6. Bound and free vortices. An original physical interpre-
tation of equation (2) was given by Birnbaum (reference 3). By

P (4)

( = -d (5)
o -1

Equation (2) is written as

g = 7+ (6)

With the aid of this relation, the value of the total vorticity
g at a point on the profile at which the velocity increment between
the corresponding points of the top and bottom surfaces is divided
into two parts:

(1) The bound vorticity 7, which is sustained by the aerody-
namic action

(2) The free vorticity C, which trails in the fluid in its
relative motion and which therefore gives no pressure

In order to clarify the relation between the free vortices and the
bound vortices, equation (2) is differentiated with respect to x
after g is expressed in terms of equation (6). Thus,

-yt + V =C d(7)
= +Vx- dt (7)

rIACA TM 1277

This relation indicates that the variation undergone in an inter-
val of time by the vortex C at a point of the fluid is equal (and of
opposite sign) to the variation undergone in the same time by the
bound vortex at the point of the profile in contact with it. In
other words, the bound vortices leave at each instant, in the fluid
with which they come in contact, an effect represented by a vorticity
of intensity equal to that which they have produced at that instant.
According to this representation, each of the inducing elements at
the wing is considered in isolation and therefore produces a pres-
sure rise expressed by equation (4) and has a proper vortex wake.
The free circulation at any point on the wing or downstream of it
is given by the sum of the circulations of the wakes corresponding
to the bound vortices that are upstream of the point considered.

The decomposition of the total circulation in the two vortex
systems previously described is shown in figure 4. It can be seen
that the graph of the free circulations on the chord are joined in a
continuous manner with the system of the wake at the point where the
free circulation is equal to the total circulation.

7. Relations between circulations and normal velocities. -
Ordinarily, the law of variation of w along the chord and with time
is known from equation (1), with the aid of which these quantities
are derived from the characteristics of the motion. The vorticities
on the wing and in the wake must induce at each instant on the points
of the chord the assigned w; that is,

I rg(x, ) dx,
w(x) = X- (8)

The integration is extended from the leading edge of the wing
to the entire part of the wake in which the passage of the wing has
created the vorticity discontinuity.

It Is known that, for the wing in steady motion, the field of
motion would not be determined if the point of separation (trailing
edge of the wing) were not fixed. This condition is also assumed
for the wing in unsteady motion and is translated into the analytical
condition that g be finite at the trailing edge of the wing.

The vorticity in the wake is connected with that at the wing
by equation (3). When equation (8), which is completed by the condi-
tion of separation and by equation (3), is solved the total vorticity

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g is obtained. The pressures are then obtained by making use of
equation (2). As will be shown in more detail in section 17, this
scheme is followed by various procedures of solution by Theodorsen,
who makes use of conformal transformations (reference 4), Schwarz
(reference 5), who makes use of Betz's solution of equation (8), and
S6hngen (reference 6). Wagner and Glauert also refer to the total
vorticity, but determine the total action on the profile and not the
pressure distribution (as do von Kermdn and Sears, reference 7).

When a different method is used, the free circulation on the
chord and in the wake can be expressed as a function of 7 by
making use of the integrated equation (7) and thus transforming
equation (8) so that only the unknown function 7 appears in it.
Then the pressures can be directly obtained by means of equation (4).
This procedure was followed by Birnbaum (reference 3), by Kussner
(reference 8), and by Cicala (reference 9).

8. Acceleration potential. A different interpretation of the
same problem can be made on the basis of the acceleration potential.
For a perfect and incompressible fluid, the equation of Euler

a= grad

expresses the equality between the force of inertia, which corresponds
to the acceleration a of the fluid particle, and the resultant of
the forces that are transmitted to the particle by the medium surround-
ing the particle. The Euler equation also permits stating that the
components of the acceleration can be obtained from the function p/p
through the same operations of differentiation with which the com-
ponents of the velocity are deduced from the corresponding potential.
The generating singularities are arranged on the surface of discontinu-
ity (wing + wake) for the acceleration field whose potential satisfies
analytical properties similar to those of the velocity potential in
the same manner as for the velocity field. The singularities are
arranged where the discontinuity exists in the pressures; that is,
on the lifting surface. The simplification that is introduced by
this concept is not, however, as great as might appear from the fact
that the singularities (which are called the dipoles of the pres- are limited only to the wing. In each case, it is necessary
to pass from the acceleration field to the velocity field and this
passage requires an integration through which the effects of all the
preceding states of motion are felt and which, according to the

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vortical representation, leaves a trace in the system of free vortices.
In other words, the velocity field of the pressure dipoles depends
on the history of the formation of the dipole.

The relation between the vortex representation of the phenomenon
and that based on the pressure dipoles is already implied in the con-
cept of Birnbaum of the vortices shed from all the points of the
airfoil that is the only source of the bound vorticity 7, which is
proportional to the difference in pressure between the two faces of
the profile. A method will be shown, on the basis of the concept of
vortices, for the derivation of some fundamental relations that in
other publications are justified with the procedure of the accelera-
tion potential.

The system of bound vortices induces at a point 0 at the
instant t = T a velocity that is denoted Va; at a succeeding
instant t = T + AT, at a point that occupies, with respect to the
wing, the same position that 0 occupied in the first condition,
there will be a velocity differing from Va by an amount denoted
by AVa. The velocity AVa will depend on the variation in the con-
stitution of the induction system or, with changed sign, will be the
velocity induced by the system of vortices that, at the instant con-
sidered, are freed from the bound system and constitute the trace
that the bound system has left in the fluid with which it has been
in contact. The quantity AV /AT as AT-0. as a limit, represents
the derivative vYa/6t. Hence, if it is desired to express analyti-
cally the property that the actual velocity Vf of the fluid parti-
cles results from the sum of the velocities induced by the actual
configuration of the bound vortices and from those freed at the pre-
ceding instants, then

Yf =Ya (9)

where Va and Vf are computed at tne point and at the instant t,
c/J6t being computed at the point of the fixed space being considered,
and the system of bound vortices is in the corresponding position at
the instant T preceding t. In the case of translational motion
or, more precisely, in the linearization assumed, the terms of the
second order in the velocities of the points of the wing relative to
a system that is displaced with translational motion with velocity V

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are neglected; the term Va/bt is measured by having the point
of induction at the abscissa x-V(t-T), or, with respect to the
point 0 of the abscissa x to which the velocity Yf refers,
in a position upstream, advanced by the quantity V(t-T) if V is
assumed constant.

The following relation can be obtained from equation (9):

d'V (10)
dt ox

The quantity Va is the velocity induced at 0 at the constant
t by the bound-vortex film layer. From the instant t-dt to the
instant t, the vortex system changes only in the number of the bound
vortices leaving a trace and that occupy the position corresponding
to the time t. The variation that the velocity Vf of the fluid
has undergone at the point 0 of the fixed space because of the
effect of this change (local variation d'Vf/dt dt) is that which
would be measured by leaving the vortices 7 in position and moving
the point of induction of the segment dx = Vdt so that the point
of induction occupies the position that corresponds to the instant t;
the variation that is measured in this manner is expressed by
Vya/6x dx and is thus added to equation (10), which equates the two
expressions of the variation.

Relations (9) and (10) are equivalent5 to the relations derived
by Possio, which are based on the concept of the acceleration poten-
tial, if it is assumed that the stationary field C, defined in
reference 10, coincides with the field produced by the system of
bound vortices in a uniform stream V. Inasmuch as the concept of

5Equation (10) is derived directly if the acceleration field is
considered to depend only on the actual values of the pressures on the
profile and, with the pressures equal, has the same configuration as
though the motion were steady. In this case, according to equa-
tion (4), the actual pressures would be obtained by having only the
actual values of 7 on the profile: the velocity of the fluid would
be Va and the acceleration that, by the assumed linearization, is
computed from d'Va/dt (and not from the derivative formed in following
the fluid particle) would be expressed by d'Va/dt = 8Va/6x V (without
8Va/ot because this virtual field is steady). When the acceleration
of this virtual field is equated to the effective acceleration, which
is written d'Vf/dt, equation (10) is obtained, which when integrated,
gives equation (9).

rACA TM 1277

the acceleration potential lends itself less to physical interpre-
tations than the concept founded on vortices although some simpli-
fications can be obtained in the analytical development, it is
preferred, in the following sections, to employ the classical method
of description.

9. Cases of total zero circulation. If the total circulation
about the wing is constant in time and the system has no vortices in
the wake, the induced velocities can be computed on the basis of the
total circulation about the wing at the instant considered in the
same manner as for steady motion. From the analysis relative to this
case (reference 11, p. 185), it is known that if the circulations are
represented by one of the following functions

1g = cot 2 sin 9

gn = 2 sin n&


(n = 2, 3, ...)

the corresponding velocities are respectively given by


wl = 2 + cos

wn = cos n5

Inasmuch as the values of gn represented by equation (11)
satisfy the condition

g dx = 0

the preceding result holds for any motion because no wake exists down-
stream of the body. Thus,

w =2An wn

It -S 11

14 NACA TM 1277

where An are quantities independent of 0 but functions of time.
The total circulations are represented by the corresponding sum

g = ZAn gn (13)

The corresponding bound vortices are represented by Z 7, where,
as derived from equations (5) and (6),

71 = A1 (cot 2 sin ) + A'l(sin 8 + sin 6 cos 9)

= 2 BA sin n + A'l(sin(n+l)t --if(-l)0)(n =2, 3,.

A' = d (15)
n V dt

In general, the values of w can be expressed by expanding in
a Fourier series:

w= A + An, cos n& (16)

The coefficients An, general functions of time, can be obtained by
harmonic analysis by setting

= w ZAn wn

There is easily obtained

S= (AO Al) (17)

It is therefore concluded that, if the vertical velocities on
the profile are developed in Fourier series, the circulations and
the pressures can be directly computed with the aid of equations (13)

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and (14) if AO = Al. In the general case, it will be necessary to
sum the circulations (equation (13)), which correspond to the

w = w(1 cos 6) d- (17')

which is a function of time, but is constant, over the chord.

10. Case of velocity w constant over chord. Applying equa-
tion (10) for the potential on the z-axis, inasmuch as w is the
projection of Vf, yields

d'w Va
V dt = (18)

where Va is the projection of the vector Va on z.

Because at each instant w assumes, for all the points of the
chord, the value w, the following relation is obtained:

d'w d
dt dt

and, therefore, from equation (18), integrating along the chord yields

V w-t x + C
a V dt

where C is a constant with respect to x, but is a function of time.
From the analysis of steady motion, it is known that the distribution
of the vorticity capable of giving velocities satisfying the condition
at the edge is given by

70 = 2 dt sin + 2"C cot (19)

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In order to investigate the dependence of C on the condition
of motion, the elementary case is considered in which w, which has
been zero for an indeterminate time, suddenly acquires, at the
instant t', the value AW and maintains it unchanged. For such
a case, for t > t', C, which must be proportional to Aw, may be
put in the form

C = (1-R) Aw (20)

where R is a function of the space passed through by the wing
since the instant t'. Because, with the passing of time, the phe-
nomenon tends toward the steady conditions for which the circula-
tion tends to assume the distribution

70 = 2 AW cot (21)

the asymptotic value of R must be zero. The law of the variation
of R was studied in one of the first publications on the wing in
unsteady motion (reference 12, which gives a resume of the work of
Pistolesi to whom reference is made). The more general case is
obtained from the elementary case by superposition of the effects.
According to equation (20), the second term of expression (19) is
decomposed into the asymptotic term (equation (21)) and a term that
contains the function R and represents the distribution of the
circulation that would be realized if i possessed, from an indeter-
minate time, the value Aw up to the instant t', and then for
t > t', V = 0. The pressures corresponding to this term, which
represents the effect of the preceding variation and diminishes to
zero with time, can be referred to as "transitory pressure."

In the computation of C in a general case, sunming the effects
of all the increments that V has received from the start of the
motion, when it may be supposed w = 0 (so that LAw = w), there is

S= (2)
C = W R dw (22)

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The integral is taken as the sum of the products of all the vari-
ations that W has undergone in the preceding instants (the procedure
is not modified whether the variations are abrupt or gradual) for the
corresponding values of R. The distance traveled by the wing,
measured from an arbitrary origin, is denoted by s; the value of s
for the position at which the pressures are measured is denoted by
sO and the value of s for the distance s0-s referred to the semi-
chord is denoted by O. The values of the function R(o) are given in
table I. These values have been obtained from the recent tabulations
of Kussner and Schwarz (reference 13).

11. General solution. When the expressions of the two pre-
ceding sections are collected, the resultant pressures for a general
case of motion may be computed. If the values of w are expressed
by the series of equation (16), which is put in the form

w w + An Wn (16a)

the pressures corresponding to the first term, according to equa-
tions (19) and (22), are given by

p = PV 2 sin 5 + 2 ( R d cot
V dt 0 2

whereas, the values of w that constitute the summation of equa-
tion (16a) correspond to the pressures

p = PV Z7

where 7n is expressed by equation (14).

It is convenient to divide the total pressures into a part that
depends on the history of the motion and has been denoted as the
transitory pressure given by

2 cot R dV (23)

NACA TM 1277

and into a part p, which depends only on the actual values of the
parameters that characterize the motion and is denoted by the
"instantaneous" pressure, given by

py = 2 f cot 2 + 2V dt sin& + y7n

When equations (14) and (17) are used, this relation assumes
the form

SA' n+l A'n-l(24)
S= AO cot 2 A 2n- sin n 4 (24)

For the computation of the pressures on the profile, the fol-
lowing operations are therefore required:

(I) From the law of motion there is obtained, with the aid
of equation (1), the expression for w as a function
of time and of the coordinate 4.

(II) By developing w in a Fourier series, the values of A
as functions of time are computed.

(III) From the instantaneous values of A and the derivatives,
the values of p are computed from equation (24).
When w has been determined from equation (17), the
values of T are computed by equation (23) and hence
the resultant pressure p = p + p. The total force
and moment are obtained by the simple integral

P =f p dx

S= p d
M = px dx

IACA TM 1277

The formulas given here are also valid without change if V
varies from one instant to the next. In such a case, it is con-
venient to assume as a reference variable instead of time the dis-
tance s traveled by the wing. Therefore

A'n= -; where ds = V dt.

In the calculation of the transitory pressures, the expres-
sion (23) is usually computed by graphic integration. When the dia-
gram of w as a function of 0 is known and the value of R is
obtained from table I, it is convenient to draw the graph of w(R)
(fig. 5), which is obtained by laying off, for each ordinate W,
the corresponding value of R for the same position. The area
enclosed by this curve, by the vertical axis, and by the two hori-
zontal lines through the ends (one of which is the axis i = 0)
represents the integral of equation (23) (cross-hatched area in
fig. 5). If when tracing around the contour from a to b the
area is on the right, the area is considered positive; in figure 5
the value would be considered negative. Even though the graph of w
presents abrupt variations (as in the case of the figure), no compli-
cations are thereby introduced.

When it is possible to proceed by the analytic method in com-
puting the preceding integral, it is convenient to assume as the
variable of integration the distance s. Equation (23) is there-
fore written as

= 2 cot R(sO-s) ds (23')
PV 2 de

or as

= 2 cot R(0) do (23")
PV 2 dc

12. Example of application. Let it be assumed that the wing
always displaced with constant velocity in magnitude and direction
undergoes a sudden small rotation (of small amplitude) about a point

NACA TM 1277

of the chord. Let there be determined the law of the variation of
lift and of the focal moment at the instant of the rotation.

It is convenient first to assume that the rotation occurs in
a finite interval of time and then passes to the limit to let the
interval approach zero. Let a be the angle of rotation, (c the
final value of a, and. = 40 the coordinate of the axis of rotation.
While the wing travels through the distance from s = 0 to s = A,
the angle a increases continuously. Let

s < 0, w O 0

0 < < A,

s >A,

AO (
-2 f

w = aV + dc (cos 0 cos )

v = cV

S+ L cos 00

A = V-

r 2+ cos r

for s between zero and

A the rotation is V VCP.

In the first

dwi dac
ds ds



and for



NACA TM 1277 21

and in the second phase, dw/ds = 0. If s = so (the position for
which the pressures are measured), let RO, R'O ... be the values
of R and the derivatives for 3 = s0. Equation (23') is then written
by expanding R in a power series in s:

-p = 2pV2

cot =

(RO '0 + ...)( + r d ds



f A


d 2 ds = 0

S1 A ds =
's 2

and moreover, the quantities

sn da ds
| ds

sn+lj d2 ds

for A approaching zero become zero at An.

NACA TM 1277

Thus, from equation (25), for A approaching zero,

-p oPV72 cot (RO r R'O)

From this relation it is concluded that the transitory pressure
decreases according to the function R if r = 0 or if the rota-
tion occurs about the neutral rear point. If r is different from
zero, a term is added in the law of variation of p that decreases
as the derivative of R; this term corresponds to the pressure dis-
tribution that is created on the wing in uniform rectilinear motion
and that executes an instantaneous displacement in the direction
normal to the trajectory and then continues with the initial speed
and direction.

The pressures f after the rotation are given by

S-= 2p2 cot 6

Hence, the lift after the rotation is expressed by

P -= (p + p) dx = itp(LV2 (1 R + rR') (25d)

where L is twice the chord of the profile and R and R' are the
function of table I and its derivative, respectively, both approaching
zero with an increase of the independent variable, which is repre-
sented by the distance of the actual position from that at which the
rotation has occurred.

In take-off of the wing, that is, when the wing starts its
motion from rest,


In the interval 0 < s < A, V passes from zero to its final
value VO. For A--0,

NACA TM 1277

lim R ds w = RO cP

and the lift is therefore still expressed by equation (25a), in which
the factor in parentheses reduces to 1-B (Wagner's case). For the
moments, because the pressures are always proportional to cot t/2,

Mg.E (p + ) x + x 0

In the preceding computation, the impulsive pressures that are
generated at the instant of the rotation and cease when rotation
has occurred were considered. The values are therefore immediately
obtained with the aid of equation (24).

13. Computation in finite terms of instantaneous pressures. -
For the determination of the instantaneous pressures, a convenient
expression is given by Sbhngen, by means of which these expressions
are obtained directly from w without expanding in a Fourier series.
The expression, modified to conform with the notation used herein,
may be given in the following form:

9/P V A0 cot sin co',s) ds' (26)

S 2 cos cos


H(',s) -w(x',s) +x 6dx (27)

and where x = cos 6 is the coordinate of the point to which p is
referred and x' = cos 6' is the variable of integration.

NACA TM 1277

The lower limit of the integral in equation (27) is arbitrary,
so that H is defined except for an additive constant. This arbi-
trariness does not affect the results because

I0 ^d = 0
J cos cos 00

The proof of equation (26) can be given in a manner that is not,
however, entirely satisfactory from the mathematical point of view
by substituting the expression of equation (16) for w in equa-
tions (26) and (27) and verifying that the relation thus obtained
agrees with equation (24).

The practical computation of the integral that occurs in equa-
tion (26) presents difficulties for the singularity of the function
integrated at the point 6 = %0. For cases that are encountered in
practice, however, by dividing the chord into a certain number of
strips it is found that in each strip H can be represented by a
combination of a few terms of the type cos n@. Hence, to simplify
the applications, Sihngen gives the following formula, which in this
case permits conducting the computation in closed form:

sin 6 f d6' cos n0 In [1-cos(-6)] [l-cos(4+4B)]
J cos cos 2 l-cos(+2) [l-cos(4-4)

2 Z 1 sin(n-v) 6 (sin 42 sin ui9) + (2 6) sin (28)


f=0 for 0< <

f = cos n6 for 6, < < 62

f = 0 for 62 < < n

NACA TM 1277

In each case, by means of a few terms of this type, the func-
tion H can be expressed and the pressures can therefore be com-
puted as a sum. The computation of the resulting actions involves
easy integration. In order to simplify this part of the computa-
tion, S6hngen gives the expression

in [1-cos(&- )] sin n& d = cos n- sin no +

2 cos[(n-) +vCl 1 (cos na cos n() In[1-cos(-c)]
n n- V n

The use of these expressions will be clear from the examples
that follow.

14. Examples of application. Let the expressions of the pre-
ceding section be applied to the determination of the instantaneous
pressures corresponding to the rotation of the elevator; that is,
it is assumed that the forward part of the wing, corresponding to
values of 6 between 0 and 60, remains immovable while the rear
part rotates rigidly about the hinge located at the point of the
abscissa x = cos 60. The angle of the elevator is denoted by B
(positive downward) and the primes denote the derivatives d' = dP/ds,
P" = d2P/ds2. Hence,

z = P(cos o cos 9)

and therefore, from equation (1),

w/V = P + P3 (cos 40 cosB )

H = v+V (0'+B" cos40-W" cos 6) sine d6 =B0+B1 cos 4 + B2 cos 26

NACA TM 1277


BO =V P + 2p' cos50 + 2 + o cos 230

B1 = V (20 + 0" cos 6o)

B2 = V cos 20

For each of
is applied with
i1 = 0, 2 = a.

the three terms of the expression of H, equation (28)
n = 0, n = 1, and n = 2, respectively, and with
Making the substitutions yields

0 H d6'
Ssin cos 1' cos 6

= 1 (BO + BI cos + B2 cos 26) In 1 cos (A + 40) +
2 1 cos ( 60)

( 60) (B1 sin % + B2 sin 26) 2 B2 sin% 0 sin

On the other hand,

+ 'in

2(1[ 0o)
w d = (P +P0 cos s0)

AO = 2
I t

NACA TM 1277

There is thus finally obtained from equation (26)

x p/2 PV2 =- p(-0) + 0['sin 6 + (i-60) cos ] cot +

2p'(i-o0) + "(x-60) cos + sin sin (n-o0) sin 26 +

F0 1 ,22 l-cos(10o)
S+ 23'(cos 60 cos ) + 2 B"(cos 60 cos ) l -cos( -o0)

As a second example, the pressures in the case of the stationary
gust are computed; that is, the wing is assumed displaced with the
velocity, which is constant in magnitude and direction, encountering
air layers that move in a vertical direction perpendicular to V
with velocities that, at each point of the fixed space, are main-
tained constant in time. The graph of w along the wing trajectory
(shape of the gust) is assumed given and P is a general point of
the chord that is indicated by the positions corresponding to t = to
and t = tO + dt in figure 6. For the point considered, the value
of w dx is represented at the first instant by the obliquely
hatched area and in the succeeding instant by the same area increased
by the horizontally hatched strip and decreased by the vertically
hatched part. The intervening variation in the interval considered
will be represented, except for infinitesimal of the higher order,
by the quantity (wl-w) ds. Hence,

6 rx
H = w + w dx =

The quantity H is therefore constant for all points of the
chord. The same conclusions evidently hold for the point P if
the point P has not yet entered the gust. (The only difference
with respect to the preceding case is that in this case w = 0.)
Hence, from equation (26) (the expression for p), the only nonzero
term will be cot 6/2. It is therefore concluded that for the wing
that crosses a stationary gust, whatever the form of the gust, the
pressures are distributed proportionally to cot % /2. The same result

NACA TM 1277

could also be deduced by considering equation (18), for which, in
this case, the first member should be zero from the hypothesis that
w does not vary in time. The result given by Kassner that the
aerodynamic actions on the wing that enters a stationary gust have
a resultant passing through the focal point is thereby obtained.
This result holds for the case where the values of w do not vary
locally; in general, in agitated air the velocities vary rapidly
with time.

By following ths analysis of the problem of the stationary gust,
the case of the elementary gust (the step diagram in fig. 7) is first
considered. During the time in which the front of the gust lies
within the wing,

Ao = 2 w d = 2 w
0 ') i

A1 = 2 w cos d6 = 2 sinO' Aw

S sin 6'
W = Aw

where x' = -cos 6' is the abscissa of the gust front. When the
entire wing is enveloped by the velocity Aw,

A0 = 2 Aw Al = 0

The instantaneous pressures are therefore represented by

7p = 2PV Aw cot in the first case,

and by

= oPV Aw cot in the second.

iACA TM 1277

For the computation of equation (23). only during the phase of
the crossing of the gust front is dw different from zero, and there
it has the value

dw Aw (1-cos ) d' = Aw i1 dx'
'K t l-x'

If s denotes the distance of the midpoint of the wing at the
actual position from the gust front, for which the values of p are
measured, at this instant

-x p/PV Aw cot = 2 (s-x') f dx'

The pressure may therefore be computed as a function of R, in
the case of the gust, by means of a simple integration. The resultant
pressures may therefore be expressed in the form

p + = PV Aw Rl cot (29)

where R1 is a function of the distance sa = s + 1 of the front
of the gust from the leading edge of the wing. The function is
evaluated in table 2; the values are obtained from reference 13.
When s1 is negative, then evidently R1 = 0.

From the solution relative to the elementary case, the solu-
tion for a gust of any shape can be obtained by substituting in equa-

tion (29), in place of RE Aw, the quantity J R dw (taken as

the sum of the products of all the variations that the values of w
undergoes for the corresponding values of R1).

15. Profile in harmonic oscillatory motion. The same relations
permit solving the case of harmonic motion. Assume V constant.
Using the complex variable notation yields

w = We

NACA TM 1277

where 0 is the frequency and W is a function of 6 but not
of t. Equation (26) for this case yields

Ssin 6 i s' C
cos 61 cos x

W(x) dx




Xt Cos 4'

For v,

10t -we (W constant)
S-We =We W constant)

(For the position

8 -0,

;(s-a) v(s)e -e *i e

(Hence, from equation (23"),

iPpweL cot w -R) Y()eWo do (31)

The same problem can be attacked by making use of the general
relations initially given. By procedures that are developed in
numerous publications and that, in part, are herein presented, rela-
tions are arrived at that are equivalent to equations (30) and (31).
In this manner, which is more rapid than direct computation, it is

M /2Veeint

= cot i W d, sin co'W cos

NACA TM 1277 31

found that the quantity that enters the second member of equa-
tion (31), a function of the reduced frequency 9 = OL/2V, is iden-
tified with the quantity denoted by X in references 2 and 9 and is
therein expressed by means of the Hankel function of parameter M:

H= (2) i(2)
HO 1H^

Reference 14 gives a tabulation of this function, which is rather
important in the study of the aerodynamic phenomenon for the oscil-
lating wing and which is reproduced in table 3. Kiissner uses instead
the function T correlated with X by the relation

T = 1 2X

whereas in the paper by Kassner and Fingado (reference 15), the func-
tion P = 1 X is used with argument V = 1/2.

American publications use the function C introduced by Theo-
dorsen, which has the same definition as P.

By means of equations (30) and (31), the pressures are easily
computed for any type of oscillation (for example, translator, rota-
tional of the entire wing, or rotational of the flap). The coeffi-
cients of the aerodynamic actions have been determined in various
publications. A complete tabulation for the case of a wing with a
flap hinged at the forward edge is found in reference 16, the com-
putations for which were developed by the national institute for
theoretical applications on the basis of the formulas of Kussner.
In a recent publication by Kussner (reference 17), the case of the
profile with a flap and with a tab hinged to the flap is treated.
The fact that the hinges of the two movable parts can be retracted
with respect to the corresponding leading edges is taken into account.

16. Analysis of pressures on airfoil in motion in nonperturbed
air. Equations (27) and (28) are considered, with the assumption
that the values of w are due only to the motion of the wing and can
therefore be expressed by equation (1). It is first assumed that V
is constant. Then

= + =V tdx +

NACA TM 1277

Inasmuch as adding a constant to the value of H does not
change the result of equation (26),

T dx = z

and therefore

H = V + 2 + fx 2 z (32)

The first term in the second member together with the corres-
ponding term contained in A0 gives rise to the pressures that are
denoted by pO:

,p0/pv2 = cot Z db sin { z d (33)
0 2 ot x | xi cos ', cos

where 6z/6x is computed at the point of integration x' cos 6'.
These pressures are those that are obtained if the wing in the actual
configuration is under the conditions of steady motion.6

The second term in the second member of equation (32) with the
corresponding term contained in A0 gives the pressures that are
denoted by pi:

S 8z 0 dj' (34)
,Pl/2oV = cot 2 -o d0' 2 sin J cos -co (34)

These actions, which are proportional to the vertical velocities
of the points of the wing, have the characteristic of damping forces
and can, in part, be interpreted by cinematic considerations. Thus,
if the wing is displaced without rotation with vertical velocity v,

6In fact, substituting in this expression the value of u/V given
by equation (37) of reference 11 for 8z/ax and developing the computa-
tion yields the value of y = p/PV expressed by equation (31) pre-
sented herein.

HACA TM 1277

the intuitive result that the wing is in the same condition as if it
were at an angle of attack v/V is obtained from equation (34).
Also, in regard to the effect of a torsional motion, a qualitative
interpretation of pi can be given'. If the wing is, for example,
in diving rotation, it behaves with respect to the fluid as though
it were curved upward. It is seen from such considerations that the
focal moment that arises from this effect is turned in the opposite
sense to the angular velocity (damping action). This consideration
of the dynamic curvature has been put at the basis of the numerous
approximate investigations on the aerodynamic coefficients of the
vibrating wing (reference 9). These considerations would lead in
substance to the computation of the values of pi with the same rela-
tion (equation (33)) that holds for pO, in which az/V 6t is substi-
tuted for 6z/6x. This procedure leads to results that are quantita-
tively in error because it is necessary to halve the second term in
the expression of pl.

The third term of equation (32) gives rise to the pressures p2:

P2/2p = sin cos cos dx (35)
Jo 0 -1

These pressures, which result independently of the vertical
accelerations of the points of the wing, represent an inertia effect
of the mass of the circulating air. As evident from equation (35),
the pressures P2 do not depend directly on the local values of the
accelerations, but on the entire distribution. Thus, in the case of
the oscillation of a flap, although the forward part of the wing
remains fixed, there are pressures over the entire chord. There can
therefore,be no distribution of masses that are apparently capable
of reproducing the inertia effect of the medium.

The result of equation (35) is of interest for the rigid motion
of the airfoil. In the case of translational oscillation, pressures
are obtained that are distributed proportionately to sin 9 and give
rise to the same resultant as though the mass of the cylinder of air
circumscribed about the wing underwent the motion of the wing. It
is easy in such a case to compute also the actions on a part of the
chord. In the case of the rotational oscillation of the profile
about the mean point, the pressures of inertia still correspond to a

IACA TM 1277

system of masses distributed according to sin %, but the total mass
in this case is equal to one-half that of the preceding case. The
rigid wing thus undergoes in its motion an inertia action that can be
thought of as reduced to two masses, each of a value equal to one-
half of the mass of the circumscribed cylinder of air concentrated at
a distance L/4!3T from either side of the mean point (always in
regard to the computation of the resultant actions).

It is observed that the pressures P2 have values that are
independent of the velocity of advance and must therefore be sustained
in air at rest. For such conditions, the results would not rigour-
ously apply because the assumption of the smallness of the perturba-
tions, as compared with V, does not hold. The results nevertheless
agree with those that, for any particular case, have been obtained
without the preceding assumptions and moreover apparently reproduce
sufficiently well the actual phenomenon as it is found from some
measurements by Cicala, in which the periods of the oscillation of
a wing model in rarefied air and at normal pressure were compared.
In the case of the phenomenon of the wing vibrations, the inertia
pressures are not of great importance, whether because the additional
masses represent a small part of the mass of the structure or because
the previously described actions, which exist independently of the
velocity V, can be directly included in the computation if measure-
ments of the mass of the structure are made by dynamic procedures.

Equation (35), which permits computing the inertia pressures
in closed form, is derived in reference 2, and the expression for pi
is contained in the same reference. The expression for Pl presents
a certain difference when compared with equation (34) in that it con-
tains a term that in the preceding scheme is added to the transitory

If the velocity V is not constant in the expression for H,
there is derived from the term I dx, in addition to the
6J t z
quantity expressed by equation (32), the term d dx, which
because H is defined except for a constant is written z dV/dt.
Hence, to the preceding computed pressures there are added the pres-
sures p3, given by

dV z d9'
p3/2 = sin os cos 6'

NACA TM 1277

From this relation, there is obtained, for example, the following
result: If the rectangular wing is displaced with velocity V not
constant and with angle of attack a, which is constant, there is a
force normal to the flight path represented by the distribution

P3 = 2pa d sin

which corresponds to the mass of the cylinder of air circumscribed
abcut the wing subjected to the acceleration dV/dt.

It is emphasized that the pressures p = p0+P1+P2+p3 are still
added to the transitory pressures, depending on the values that the
quantity W has assumed in the preceding instants. If the values
of w are distributed linearly over the chord, w represents the
value of w at the neutral rear point. The instantaneous pressures
have the resultant passing through the focal point.

The decomposition of the total pressures into instantaneous and
transitory pressures has no absolute character in the sense that a
part of the instantaneous pressures can be combined with the transi-
tory (not vice versa, because a term containing the history of the
motion is clearly distinguishable from the terms depending on the
actual values). It nevertheless appears that the definition given
herein is more natural because in the limiting case of steady motion
the actions resulting from the group comprised of the instantaneous
and the transitory pressures become zero.

17. Remarks on treatment of unsteady motion of wing in two-
dimensional case. The first investigator to study the aerodynamic
problem of the oscillating wing was Birnbaum, who made use of the
concept indicated in Section 6 of the splitting of the circulation
about the wing into bound and free components. Equation (7) is
integrated for harmonic motion. By use of the complex variable
notation, in this case,

Y =j (x) eit

E = Z (x) eit

NACA TM 1277

As is easy to verify, equation (7) is obtained if

(x) = e-.J 7 (x') eWx' dx'



W M io/v

Inasmuch as the free circulation is zero in correspondence with
the leading edge of the profile (if no other sources of vortices
occur upstream of the wing), in order that x = -1 and e = 0, the
lower limit of the integral in equation (36) must be equal to -1.
For the points of the wake, the integration is evidently limited to
the chord, because Y is zero outside the wing.

On the basis of these results equation (8) assumes the form

e' dx'
x' -x


eUaX 73(x") dx" -

X' -
=-1 x -x
1- xl-


eWx" 7(x") dx"

where x' and :' are variables of integration and

v = w0et.

Birnbaum takes into account the condition of separation by
expressing Y by means of a combination of functions that satisfy
this condition. The solution is sought in the form of a series expan-
sion in powers of the reduced frequency OL/2V. The series converge
rather slowly so that the results of Birnbaum are applicable only to
rather low values of the reduced frequency.

Wagner (reference 12) considered the problem of unsteady motion
of the win of infinite span. In the first part of reference 12, the
treatment refers to the case in which w is constant over the entire
chord but variable in time, a case of fundamental importance, as has


Wo r(J -o ) dX x
2avo X -X

NACA TM 1277

been stated in Section 10. Making use of the condition of separation
that imposes the circulation of the wing capable of giving rise to
a finite velocity at the trailing edge, Wagner arrived at an integral
equation that defines the circulation in the wake of the wing. With
a procedure based on the moment theorems, Wagner gave an expression
for the computation of the lift and of the moment on the wing on the
basis of the circulation in the wake. In particular, the computa-
tions were developed for the take-off motion of the wing. The case
of rotation of the wing about a point of the chord was also considered.

The problem of the unsteady motion of a wing was also considered
by Glauert. In an initial paper (reference 18), he made use of the
hypothesis that the circulation remains constant, a hypothesis that
considerably limits the importance of the results.7 In a succeeding
paper (reference 20), he took into account the variation of the
circulation. In contrast to Birnbaum, Glauert sought to obtain
directly the total circulation.of the wing, which is divided into a
part that would correspond to the case in which the vortex wake would
be absent, and into a circulation induced by the vortices downstream
of the wing. It is simple to show the equivalence of the relations
assumed by Glauert for computing the lift P and the moment M with
respect to the center point of the wing with the relations that are
obtained on the basis of the bound circulation. From equation (4),

P/PV = (g-e) dx = K d dx


M/PV = (g-e) x dx

Considering that for x = -1, E = 0, and for x = 1, according
to equation (3), VC =- dK/dt, integration by parts yields

Sdx = dt x dx

1 1V d _1 x1

2 V dt 2 W dx
J-1 -1

7Lamb (reference 19) also treated the problem with the same

NACA TM 1277

With the aid of these relations, equations (37) become

P/p + + gx

M/P.V g ax+ -d gx2dx
+2 dt R

considering that, according to equation (5),

and that the sign of the differentiation can be taken outside the
integral because the limits are independent of the time. Equa-
tions (38), which are the equations used by Glauert, serve for the
computation of the resulting aerodynamic actions but do not lend
themselves directly to the determination of the pressure distribution,
a computation that is necessary if it is desired to know the actions
on part of the wing (flap). The equations obtained by Wagner are
also subject to the same limitation.

With the values of g expressed with the aid of the instantaneous
characteristics of the rigid motion of the wing on the basis of the
circulation existing in the wake, Glauert determined the lift and the
moment with the aid of equations (38) and arrived at expressions
agreeing with those of Wagner. For harmonic motion, in which the
circulation is distributed in the wake according to the sinusoidal
law, the solution of the problem reduces to the determination of
certain integrals that Glauert obtained by approximate procedures
that limit the results to values not much higher than the reduced

On the basis of the work of Birnbaum, Kiissnsr (reference 8) again
took up the problem of the oscillating wing, assuming for the bound
circulation the functions that are used for steady motion (refer-
ence 11, p. 184). The corresponding w, expressed as a function of x,
consists of a polynomial and of a trigonometric function multiplied
by a factor depending on c, which Kissner computed by means of a

lIACA TM 1277

power series. After rather laborious computations, Ktssner obtained
the pressure distribution on the wing corresponding to the motion of
translational or rotational oscillation of the wing and also, approx-
imately to the motion of the flap, for which he derived the coeffi-
cients of the aerodynamic actions for the field of variation of 3,
which is of interest for the phenomenon of wing vibration.

Connected with the investigations of Wagner and Glauert is the
theory developed by Theodorsen (reference 4), in which there are
separately determined by use of the methods of conformal transfor-
mation the potential function on the wing for simple fundamental
motions of the wing with flap. From the potential function, Theodorsen
determined the pressures with the aid of the Bernoulli's equation
generalized for nonsteady motion (reference 11, p. 38):

P, + +2 = (39)
p 2 TE

(where T is the potential function).

In the computation of the difference in the pressure between the
lower and upper surfaces of the wing, it is necessary to consider,
according to equation (39), the quantity derived from the variation
with time of the difference in potential existing between the two sur-
faces of the wing. This difference is measured from the circulation
of the velocity, which is determined by following a path (fig. 3) that
joins the two points situated on the opposite surfaces and passes,
always in the proximity of the wing, through the forward edge of the

wing, thus obtaining the quantity i g dx; it is thus seen that
the second term in the second member of equation (2) represents the
corresponding term in 89p/t in equation (39).

Theodorsen's treatment of the problem represents a marked advance
with respect to the preceding work because it determines the distri-
bution of the pressures on the wing and hence, in contrast to the work
of Glauert and Wagner, permits the computation of all the coefficients
of the aerodynamic actions for the wing with a flap. Also, because
the integral with which the effect of the vortical system of the wake
is computed are solved by means of Hankel functions, all restrictions
on the value of the reduced frequency are thus eliminated by use of
the existing tabulations..

NACA TM 1277

Independently of Theodorsen, and almost simultaneously, Cicala
arrived at the solution of the problem of oscillatory motion of a
deformable wing with any law by the same method followed by Birnbaum
and Kissner. In reference 9, it is proven that a class of functions
exists, depending on the reduced frequency, that represents the dis-
tribution of the bound vortices, which correspond to the velocity w
distributed over the wing by a particularly simple law. The first
of these functions, corresponding to constant w on the chord, con-
tains the Hankel functions of the reduced frequency; the others are
essentially represented by the 7n of equation (14) and give rise to
the values of wn in equation (12). In this manner, relations were
obtained by means of which the coefficients of the Fourier series of
the bound circulation on the wing can be computed in closed form as
a function of the coefficients of the series for w. The coefficients
of the aerodynamic actions for the wing with flap were thus computed.
In a succeeding report (reference 2), it was also shown how the pres-
sures depending on the second power of the reduced frequency (inertia
pressures) and those proportional to the first power (pressures pl)
could be computed in closed form without developing them into Fourier

At the same time, Kissner arrived at.the solution of reference 9
by a procedure described in reference 21, some results of which were
anticipated in reference 22. The general case of nonsteady motion
was also treated in reference 21, where the discontinuous motion was
studied on the basis of the solution for the harmonic motion with the
use of the Fourier integral. The case of a stationary gust was studied
and tests were conducted (reference 23) confirming the result obtained
that the focal moment remains zero during passage through the gust.
The solution of Kgssner was used by Dietze (reference 24) to compute
the resultant of the actions on the flap (in the preceding papers
only the hinge moments were computed); it was also used by Krall
(reference 16) to elaborate, with the aid of the National Institute
for Applied Computations, the tables of the aerodynamic coefficients
for the oscillating wing, and was used by Dietze again (reference 25)
for the computation of the coefficients for the wing with a flap and
a tab hinged to the flap.

Kassner and Fingado (reference 15) also succeeded in computing
the actions corresponding to the oscillatory rigid motion of the wing,
making use of the Wagner's expressions by which, in the case of har-
monic motion, the evaluation of the integrals relative to the effects
of the vortices in the wake was investigated with the aid of Hankel
functions by Borbely (reference 26). With the aid of the Wagner's

NACA TM 1277

expressions, Ellenberger computed the resultant actions on the wing for
a flap rotating according to a general function of time (reference 27).

In a summarizing note, Jaeckel (reference 28) established a coor-
dination between the procedures of Glauert, of Lamb, and of Birnbaum-
KICssner for the solution of unsteady motion of wings and considered
also the case of the wing with a variable chord. Jaeckel also pub-
lished a systematic derivation (reference 29) of the results that were
given with rather synthetic justification by Kissner. A clear review
of the theories on the wing in unsteady motion was given by Lyon
(reference 30).

The results of the preceding studies, with some further develop-
ment, are treated by von Karmdn and Sears (reference 7); a derivation
procedure is developed that presents in an intuitive form the mathemati-
cal fundamentals of the investigation. The computation is restricted to
the determination of the lift and the aerodynamic moment, which are
computed with the aid of the expressions, respectively,

P/P =d ri xi (40')

M/P = rX2 (40")

The second member of the first expression represents the rate of
variation of the moment of circulation of the system measured with
respect to a fixed point; in an interval of time dt, this moment
should vary as the bound vortices are displaced by the amount V dt,
while the position of the free vortices and therefore the moment with
respect to the fixed axis have not changed. The total variation will
therefore be given by the product of the total bound circulation and
V dt and therefore, when multiplied by the density and divided by dt,
will give the lift. By analogous reasoning, the second of equa-
tions (40) is verified. Thus, the square of the distance from the
fixed point varies by the amount 2VX dt for the bound vortices while
it remains constant for the free vortices. Hence, the variations of
the second member of equation (40") will be represented by the moment
of the bound circulation or the aerodynamic moment except for the
factor p.

From these relations, von Karmdn and Sears derived expressions for
the lift and the aerodynamic moment, which present a better generali-
zation than those of Glauert inasmuch as they can also be applied to

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the deformable profile; but the relations still do not permit compu-
tations on parts of the wing. The authors divided the total circula-
tion into a part that would be obtained in the absence of the vortex
wake (so-called quasi-stationary system) and into the induction of the
vortices of the wake. The case of the general motion of the profile
and of a stationary gust is also considered.

Garrick (references 31 and 32) brought out the relation (Laplace
transform) that exists between the Wagner function, which gives the
circulation by the elementary discontinuity, and the function that
holds for the harmonic motion and proposed the use of approximate
expressions to represent the function of Wagner, which would be use-
ful in the analytic solution of various problems of unsteady motion.

Possio, making use of the acceleration potential, considered the
problem of the discontinuous motion of a wing (reference 10) and the
case of the stationary gust.

In a recent publication (reference 17), Kgssner and Schwarz indi-
cate the relations with which the pressures on the profile can be
computed without making use of Fourier series but using integration.
These equations were applied to the determination of the aerodynamic
coefficients for the wing with flap and hinged tab on the flap, when
the case is considered in which the hinges are set back relative to
the leading edge of the moving parts. The same relation for the com-
putation of the pressures on the oscillating wing was derived in a
different manner in a report by Schwarz (reference 5), in which a
clear and rigorous derivation of the known solution of the aerodynamic
problem of the oscillating wing is developed.

In reference 13, Kissner gives the general solution of unsteady
motion in the two-dimensional case and the functions of Wagner and
those relative to the aerodynamic actions produced by a gust are com-
puted with greater precision.

The general case of unsteady motion has also been treated in a
report by Sghngen (reference 6). The solution is put in substantially
the form indicated in Section 15.

NACA TM 1277


The relations that connect the velocities induced by the vor-
ticity of the inductor system of a wing of finite aspect ratio are
derived herein and a form for these expressions is sought that lends
itself to a future refinement of the investigation that, up to the
present, has been conducted with approximate procedures. An evalua-
tion is then given of the various approximations that have been used
in the theory of the finite wing in unsteady motion.

The consideration is restricted to essentially rectilinear
motion of the lifting surface; that is, it is assumed that the velo-
city of the points of the wing give small deviations relative to a
mean value V, which maintains its direction unchanged with respect
to the motionless fluid at infinity. It is assumed that V is small
compared with the velocity of sound. The origin of the orthogonal
axes x, y, and z is located at the point at which the induced velo-
city is measured (point of induction). The x-axis is taken parallel
to V and in opposite direction; the y-axis is normal to the x-axis
and is contained in the fundamental plane that, during the motion, is
at a very small distance from the points of the lifting wing, which
is assumed to be of infinitesimal thickness and curvature; the z-axis
is perpendicular to the x- and y-axes and directed downwards. The
vortices having an axis parallel to the x-axis (longitudinal vortices)
and to the y-axis (transverse vortices) are considered positive if
turned in the sense that carries the positive directions of x and y
on z. In the middle section of the wing is located the origin of the
t- and T-axes parallel to x and y. The semispan of the wing is
denoted by b, so that for the points -b <_ i b, to and T0 denote
the coordinates of the point of induction, hence

x = -t

y = 1 nO

Let tn(I) be the equation of the leading edge; tp(q) the equa-
tion of the trailing edge; pp-tn = L the chord; 2 the pulsation;

NAGA TM 1277

o the imaginary factor 10/V; @ the reduced frequency OL/27;

L the integral taken over the chord from the leading to the trailing
edge. The other symbols are defined in the text or in Part I.

18. Inductor elements in tridimensional case. Let P (fig. 8)
be a point of the wing that moves relative to the fluid with vector
velocity v of magnitude v and having any direction; the linear ele-
ment dl through P normal to v support the bound vorticity r;
that is, supports the aerodynamic action pvr dl. In the interval
dt that precedes the actual instant, the point P starting from P'
is displaced by the segment do = v dt. The total inductor is then
changed in that the bound vorticity element has come to occupy the
position 1-2 from the position 3-4, leaving behind it free vortices
and simultaneously creating the two longitudinal elements 2-3 and 1-4
of equal circulation F. There is thus added to the preexisting
inductor system a closed vortex element 1-2-3-4; on the side 3-4, which
at the instant t there exists the vorticity element of intensity
-dF liberated from the bound vortex, it may be assumed that there
simultaneously exist the vorticity r- do, which existed at the
time t-dt and the element of intensity F with oppositely directed
sign constituting the fourth side of the circuit 1-2-3-4.

With the aid of the formula of Biot, it is found by simple com-
putation that the closed vortex element induces, at a point 0 at
distance r from P, a velocity that, except for infinitesimal of
the higher order, may be written as8

d2w = r dl da/4xr3

Whatever happened in the interval dt occurred in all the pre-
ceding time starting from the instant at which a force on the lifting
element has arisen; that is, since an element of the bound vorticity
was created. In the problems that ordinarily present themselves, it
is assumed that v always has the same direction parallel to the
x-axis, or more precisely, the deviations with respect to this direc-
tion are considered sufficiently small. Let do and dl therefore

8The symbol d2 is used to indicate a quantity that, divided by
the product Al AO, has a finite limit when these elements approach zero.
With the assumed signs, w is directed upward and is therefore

NACA TM 1277

be parallel to x and y, respectively. The symbol a denotes the
distance measured from the point P of coordinates x and y,
parallel to x, assuming that the circulation r on the element of
the lifting surface considered is expressed as a function of a.
Summing the effects of the closed vortex elements created in the
preceding time, the velocity at 0 is obtained:

dv = dy -do 1'do (41)
4) rZ 4a @(x+0)2 4 y23
41J r3 4 0J

The velocity induced by the bound-vortex element and by the
system generated by it for r 0 is identical to that which is
obtained on the basis of the concept of the acceleration poten-
tial and which is defined as the field of the pressure dipoles
(reference 33).

The particular case of steady motion ( constant) is con-
sidered. Inasmuch as

do (42)

fo 4(x+y)2 + y2 y22 + y 2

a quantity that is denoted by f(x+0o,y), then from equation (41),

4x dw rf(x,y) dy (43)

where dw denotes the velocity induced by the elementary horseshoe
vortex of frontal side dy and circulation F. The velocity (fig. 9b)
induced by the semivortex 2 is and therefore,

summing the velocity corresponding to the other semivortex, which
has the opposite sign to the first, there is obtained, except for
infinitesimal of higher order, the quantity

dya[1 X)] d,[- dy1-)+r

NACA TM 1277

Adding to this quantity the velocity corresponding to the
frontal segment yields the value of dw given in equation (43).
The equivalence, which in this case holds between the inductor
system of the pressure dipole and the elementary horseshoe vortex,
can also be seen by considering that the sides parallel to y of
the continuous vortex circuits (indicated by the small circles in
fig. 9(a)) are canceled if r has the same value for all. This
inductor element is called a bivortex and a section of it closed
in front and behind is called a segment of a bivortex.

Making use of equation (42) and integrating by parts yields,
from equation (41),

4r dw =r f(x,y) dy + dfy o f(x+oy) d (44)

The operation that leads from equation (41) to equation (44)
transforms the inductor system constituted by the closed circuits
into a system of bivortices: a generating bivortex originating
on the wing, denoted as a bound bivortex, and a row of free vortices.
In a distance do of the wake of the bound bivortex, free bivortices
originate the circulation of which is represented by the variation
that the intensity of the bound bivortex has undergone in passing
through the distance do; that is, do.

For harmonic motion,

where 7 is constant and 0 is the frequency of the motion. In
this case, V is always assumed constant. The circulation of
element 12 at the time in which the point P of the wing was set
back with respect to the actual position by the amount 0 is repre-
sented by

r (o) Fe-10

that is, with respect to the actual circulation I, a lagging phase
shift represented by ij/V = *o. Making use of this result and
denoting by FO dy the velocity induced by the bound bivortex and
by the system of the wake in the case of harmonic motion yield

NACA TM 1277

- 4Ai -=


e- do

,V(,x,.)2 + y2

By setting

x- a x/V y = y/V

equation (45) can be written in the form

4w U? 0(

where 4 is the function of two variables

I and y defined by the


S(0) e"-i do

, Y)2 2] 3/2

For x = 0, this function has been computed by
ence 34, p. 36). For F / 0,

0 (,.y) = e

- 7

MUller (refer-

e-iu du
(u2+ 2)3/2

With the aid of this relation, which is easily obtained from
equation (46a), the computation is reduced to the tabulated function
and to an integration between finite limits that can be carried out
by graphical or numerical methods. The computation of 4 was carried
out for y = 2. When

S= Q' + 1"t

there are plotted as abscissas and ordinates in figure 10 the real
and imaginary parts $' and 0" of the function and the curve with



NACA TM 1277

the values of 3 drawn. This curve serves for the computation of
the velocity induced by the pressure dipole at the points of a sec-
tion at distance y = 2V/0 from the inductor elements. The point
! = 0 is at the origin of the dipole. The points that are found
upstream correspond to the positive values of 3. For this segment,
the values of Q decrease continuously but all have slightly dif-
ferent phases. Proceeding toward the downstream region, the points
shifted back with respect to the front of the bivortex generator
(7 < 0) encounter velocities that vary little in magnitude but rela-
tively more in phase. It is evident that at a great distance from
the origin the velocities vary on the parallel considered by a sinu-
soidal law and therefore the representative point of figure 10 for
negative values of x increasing in absolute value will approach a
certain limiting circle with center at the origin.

19. Correction of divergent expressions. The expressions of
the preceding section are sufficiently well adopted for the computa-
tion of the velocities corresponding to the inductor system of a wing.
Special methods are required, however, for the points of infinity that
are presented by the functions under the integral sign present.

The case of steady motion is first considered. A bound vortex
filament AB (fig. 11(a)) of circulation P varying from point to
point is considered and with this filament is associated the system
of longitudinal vortices that is shed from the points of AB according
to the condition of Helmholtz. The velocity that these elements
induced at a point 0, which is outside the vortex wake of the fila-
ment AB, can be computed by means of the relation

4w r f(x,y) dy (47)

where x and y are the coordinates of a point of the filament AB
at which P is the circulation. This expression reduces the inductor
elements to a system of bivortices that originate on the filament AB
(scheme of fig. 1l(b) for A--0). The demonstration of this equa-
tion is given in reference 35. An intuitive justification is obtained
from figure 11 (b). The segments of the dotted vortices in the limit
reproduce the effect of the generator filament AB, whereas the other
longitudinal elements constitute the system of the marginal vortices;
each pair of contigous elements function as a single inductor ele-
ment of circulation equal to the difference of the circulations of
the two elements that compose it.

NACA TM 1277

This expression is applied to an indefinite vortex of unit cir-
culation at a distance x0 from the point of induction. If x0 is
positive, f always remains finite. In fact,

1 ( 1
lim 1 -
y-0oyI o ) + Y ) 2x 0

If x0 is negative, f increases indefinitely for y -O.
Hence, in a singular integral, it is necessary in computing equa-
tion (47) to exclude from the integration the small segment from
y = 8 to y = +5 and then make 8 approach zero. The expression
that is thus obtained, however, has no limit, as is intuitively evi-
dent from an examination of figure 12. The bivortices (fig. 11(a)),
into which equation (47) transforms the inductor system when the ele-
ments contained in strip 28 (which includes the point 0) are
excluded, are equivalent to two angular vortices (fig. 12(b)) (because
the semivortices joined by the small circles are canceled) and these
elements, on approaching 0, induce a velocity that increases without
limit. For this reason, the integral equation (47) diverges if the
abscissa x0 from the point at which the filament AB cuts the x-axis
is negative, as in the simplified case of figure 12; in the computa-
tion of the principal value of the singular integral, an expression
is obtained that increases indefinitely and that represents the velo-
city induced by the two semivortices, which are indefinitely removed
from the point 0. (See fig. 12.)

In order to eliminate the previously discussed divergence, the
following artifice may be applied: There are added to the elements
of the filament AB those of the indefinite vortex I (fig. 11)
passing through the point Q at which the x-axis through the point 0
intersects the filament AB and having the circulation -rO, where
rF is the value of F at Q. The system thus obtained gives a velo-
city at 0 that is denoted by Dw and, according to equation (47),
is represented by

4 Dw = [{ of(xo,y) rf(x,y)]dy (48)

The first term of this integral represents the effects of the
bivortices having their origin on vortex I; the second term refers
to the bivortices having origins on filament AB (hence the integra-
tion is extended from y of the point A to that of B). The two

NACA TM 1277

terms are divergent when taken separately, but give a correct expres-
sion when added. The intuitive reasoning for this lies in the fact
that the two semivortices that, according to the scheme of fig-
ure 12, include the point 0 and arise from the vortex I find
compensating terms in the elements arising from the filament AB.
The velocity induced by the vortex I is then added to Dw.

In the case of a system Z of inductor elements analogous to
the filament AB with corresponding marginal vortices, the following
procedure may be used:

I) The transverse vortices cut out from the section by the
induced points are prolonged indefinitely, thus obtaining a system
.L, the induced velocity of which w' is computed by the formulas
of the two-dimensional motion.

II) The quantity Dw, which constitutes the velocity induced
by the system Z2 = E Zl obtained by superposing on the inductor
system 2 the system 1 with reversed sign, is computed with the
aid of equation (47).and added to w'.

The advantage of the preceding procedure lies in the fact that
both w' and Dw have a homogeneous inductor system (indefinite
vortices for l1, bivortices for Z2) and therefore free or bound
longitudinal and transverse vortices need not be separately considered.
The advantage is reflected in the simplicity of the formulation of w
and in the rapidity with which the results of the existing theories
for the approximate solution of the problem under consideration are
derived from this scheme.

The same procedure can be applied if the motion is unsteady.
The bound-vortex system impresses on the fluid an imprint represented
by a similar vortex configuration corresponding to the variation of
the intensity of the generating system. These vortex systems carried
by the stream may be treated as shown for the bound system. The
plane system 1 including the free vortices can be analyzed by the
formulas of part I. The system Z2 is made up of bound vortices
with the corresponding wake of the row of free elements (that is,
pressure dipoles) and can therefore be analyzed with the aid of
equation (48).

20. Vortex system of wing. The velocities induced by the vortex
system of a wing can thus be computed by adding to w', induced by

NACA TM 1277

the plane system that would obtain if the transverse vortices cut
out from the section through the point of induction were indefinitely
extended, the velocities Dw that, as a function of the bound vor-
tices, are written as

Dw = V d dy -S 70oQ dx dy (49)

If harmonic motion is considered, the functions w can be
derived from equations (45) and (46).9 If the intensity of the gen-
erator bivortex varies according to a different law, it is necessary
to make use of equation (41), for which the function r(a) must
be known. From this relation the function 0 = dw/dy F(0) is
obtained, which, in general, varies with time.

The first of the two integral of equation (49) is extended to
the surface S of the wing, 7 being the bound vorticity, the
second integral is extended to the strip S1 included between the
lines 11 and t1 parallel to y passing through the points at
which the chord through the point 0 cuts the leading edge I and
trailing edge t; the vorticity 70 on this strip is that which is
on the wing at the section through 0. On each element of the sur-
faces S and SI there originates a bound bivortex of circulation
7 dx dy connected with the pressure, which acts on the element by
the relation

p = PVY

In the computation of Dw, as in the computation of w', the
total circulation g along y can be referred to instead of 7,
the two quantities being connected by the relations of part I. When
the effects of the elements of a section y constant are considered,
a system of bivortices will be obtained the origins of which are dis-
tributed either on the chord or in the wake, the intensity of the dis-
tribution being represented by g. The expressions are restricted to
harmonic motion and are written as

9 A
As the reduced frequency w increases, the modulus of w
decreases and therefore also Dw, which represent the correction
velocities due to the finite span.

g e(|.,T) e

-"I g ax ( An) e

In the wake at a distance 0 from the rear edge in a segment doa
bivortices originate that have the circulation

d K(o) do

equal to the
the interval
at the time
actual value

change that the total circulation K has undergone in
in which the rear edge has passed through the distance dO
t-O/V in which K(O) was displaced with respect to the
by Qo/V:

NACA TM 1277

K(o) = K e-,

The velocity induced by the elements of the section considered
is therefore proportional to

Sgf dx + d(Ke-W)f fdo = Lgf dx -WK
L do JL J2o

The symbol xp denotes the abscissa
chord (that is, xp tp t0), therefore

e- Wf do


of the rear edge of the

f'(x,y) f(x,y) f(xp,y)
It is easily found by using equations (44), that equation (46)
can be written in the form

- w2 4 = f(x,y) w IO

f(x+o,y) e-W" do

NACA TM 1277

Hence, equation (50) can be transformed to


gf' dx 2QK (Ep,Y)

The expression for the velocities Dw at the
therefore obtained:

- 4c Dw = dr 1

-W L( TI)
f^ f 6(1,,

J-OO,, Jl(Tlo

point 0, Tr0

g(E,i) f'(x,y) d -

,TIO) f'(xy) dt -

W2 (,) K(Tj) dT + (2 (io.y)
-b -B

* K(o0) dTj

where JL(
and with x0

indicates the integral taken from tn(TI)
the quantity [t p(9) 0 Q/V

and tp(),

The first of the integrals in the second member of equation (51)
represents the induction of a system of segments of bivortices having
their origin on the points of the wing and ending on the rear edge of
the chord, and which are ot intensity equal to the total vorticity at
the point of origin. The second integral represents the effects of
the analogous inductor system existing in the strip S1. The third
integral expresses the induction of the wake system constituted by a
system of pressure dipoles with origin on the trailing edge of the
wing. The fourth integral refers to the elements of the wake having
their origin on the line tl.


G(t,T) = fn



g(Q',U) dt'

NACA TM 1277

the first two terms in the second member of equation (51), which
corresponds to the velocity denoted by w2, can be transformed by
integration by parts so as to assume the form

- 41w2 = : d ()
-b y L(i)

G(r3) d

Iy 2)

G(Q,t0) dT


It is then observed that, in regard to the effects of the ele-
ments of the wake, the difficulties arising from the divergence of
equation (47) cannot appear because all the points of the wing are
located upstream of the line from which the system of the wake
originates. Hence, for these elements the decomposition into the
systems El and Z2 can be avoided and the total w = w' + Dw can
be directly computed. The velocity denoted by w3 is thus obtained
and is expressed by

4 "b 2 K(0)
4v3 y) K(TI) d) K( )0).0
3 -b

If Wl
is obtained
the section

denotes the velocities induced by the plane system that
by indefinitely prolonging the total vorticities cut by
containing the point O; that is, if

2 1, -i(IO)

g(t ,o) dt

the resultant velocities are expressed by the sum w = w1 + w2 + w3.
In general, wl must represent the preponderant part of w, which
facilitates development of the methods of iteration for the computa-
tion of the function g for an assigned w. The velocities w2,
which represent the part whose computation presents the greatest
analytical complexities, are independent of the reduced frequency and

NACA TM 1277

are therefore determined only once for any frequency of oscillation.
The values of w3, which depend on the frequency of the motion, are
expressed by simple integrals and are, moreover, zero for all the
distributions of the circulation (equation (11)) for which the
integral K is zero.

21. Approximate theories. A first approximate treatment of
the problem of unsteady motion of the finite wing was developed by
Cicala (reference 36). The principle of approximation there assumed
finds simple formulation if reference is made to the expressions of
the preceding section. Although use is made of the exact solution
regarding w', in the computation of Dw the segments of the bivor-
tices of Z2, to which the first two terms of equation (51) correspond,
are neglected and, moreover, it is assumed that on every chord the
velocity induced by the system of the wake is constant and equal to
that which would obtain if the system started at the point of induc-
tion. Hence, referring to equation (51), in addition to neglecting
the first two terms of the second member, in the computation of the
other two by 4, a function of the abscissa and of the ordinate of
the point of t with respect to 0, the value corresponding to
x = 0 is substituted. As a consequence of the first approximation,
Dw = 0 if K = 0. Hence, if g is represented by a linear combi-
nation of gn of equation (11), the corresponding velocities are the
wn of equation (12) and the pressures are obtained from the 7n
defined by equation (14). Reciprocally, if the vertical velocities
on the various chords can be represented by a combination of wn,
the corresponding circulations and pressures can be computed on the
various chords, section by section, as though the motion were two
dimensional. This fundamental simplification permits reducing the
tridimensional problem to a single case; for example, that of ver-
tical velocities constant on the different chords: any distribution
of w that is developed in the series of equation (16) (in this
case, An is a function of the coordinate measured along the
span) requires particular examination only for the circulation
corresponding to the W defined by equation (17) (this simplifica-
tion is general, variable from one section to the next).

The bound circulation corresponding to the velocity
w = w = constant over the entire chord is obtained on the basis of
equation (19) in which, if harmonic motion is considered, according
to equation (22) and the results of Section 15, the following rela-
tion must be substituted:

C = W(l-X)

56 NACA TM 1277

From 7, C is then computed with the aid of equation (36)
and then the total circulation yields

K (y7+) dx

When the computations are made, it is found that between W
and K or between the total amplitudes of the quantities I and K
there exists the relation

w (H(2) iH(2)1) ei k =

where H is the Hankel function of parameter w

The velocity W is that induced by the system 1l; to this
quantity is added the velocity due to the system Z2, which, for the
assumed simplifications, is also constant on every chord. With the
computations developed in reference 35, equation (51) is transformed
into the relation

4s Dwr f N d" (53)


q coordinate measured from middle section of wing parallel
to y (which is measured from point 0)

N=- 7 d

F function of variable jy = ly/ defined by

F = -i 1 1 1 du
u y u2 2

NACA TM 1277

This function is computed by this relation for y > 0. For
y < 0, F(y) = -F(-y). The function F is tabulated in reference 37;
in a recent report by Kfssner (reference 33), the function N is

Adding the velocities induced by the systems E1 and Z2 yields

.- ((2)o)- iH(2)1) ei K 1 b dIN(
2L Ti d-

This equation, on the basis of known values of W, defines the
distribution of the total circulations along the span. For 3 = 0,
this reduces to the integrodifferential equation of Prandtl. Inasmuch
as, for the simplifications assumed, the distribution of 7 over the
chords is similar to that of the two-dimensional motion, on the basis
of the values of the total circulations, the problem is completely
solved. Given the series of equation (16) on w, the circulations
and the pressures corresponding to the part that is developable in
the series of wn is computed as if the motion were two-dimensional,
whereas for the remainder w, the circulations are distributed on the
chord as if the motion were two-dimensional and along the span of the
basis of the solution of equation (54).

In a succeeding note (reference 37), the procedure was applied
to the determination of the aerodynamic coefficients for the oscil-
lating wing.

Independently of reference 36, Borbbly proposed a type of approxi-
mation for the computation of w for the finite wing (reference 38).
Reference 35 shows that the expressions that Borbely elaborated for
the computation of Dw in the particular case of the elliptical
distribution of K along the span agree with the results that, for
this case, were derived on the basis of equation (53).

Possio, concerned with the problem of the stability of small
oscillations of the wing considered as a rigid body, also analyzed
the problem of the oscillating wing of finite span. Making use of
the concept of acceleration potential, he derived equations (9) and
(10), which, however, as has been shown in Section 8, are also justi-
fiable on the basis of the concepts of vortices. The solution is
expressed by Possio in the form of a power series of the parameter
Cb/V. The value limited by this parameter and the smallness of the
ratio L/b (large aspect ratios) in the series containing the powers
of QL/2V Justify, for the computation of the w' corresponding to

NACA TM 1277

the system 2, the introduction of simplifications that were not
adopted in references 36 and 37. The assumed simplification in the
computation of Dw, when it is reduced to the scheme represented by
equation (49), can be thus defined: The function t is computed
by means of equation (46), substituting for 4 the value that,
according to equation (46a), corresponds to x = 0 and that is there-
fore constant for the elements of each chord. Reference 39 contains
the principles of the procedure. Some of the results are described
in reference 40 and in greater detail in reference 41. More general
cases of the motion are considered in references 42 to 44. Refer-
ence 44 analyses the law of variation of the lift on a rigid wing of
elliptical plan form during the start of the motion (the same problem
that was studied by Wagner in the two-dimensional field). In the
computation of the velocities induced by the transverse vortices,
there was assumed (in the simplification of equation (14) of refer-
ence 44) an approximation different from that used in reference 43;
the approximation gives for the case considered a greater precision
of the results.

Sears (reference 45) also studied the problem of the oscillating
wing of rectangular plan form with approximate procedures. Reducing
the computations to the scheme of the preceding section, the simpli-
fication assumed consists of the suppression of w2, while a rigorous
computation is proposed for w3, and making use of the results that
are obtained for the infinite wing with sinusoidal distribution of the
circulations along the span. The computation, which is intended to
eliminate the errors inherent in the approximate theory of refer-
ences 36 and 37 criticized by Sears, does not give results more accu-
rate than that theory. As an example,_the effects of the inductor
elements of a chord L, for the case w = 1 0, are considered with
the point of induction at a distance y = L. Apart from factors that
need not be considered in a comparison, the induction can be expressed
by the equation (50'), which can be written as

y2 Jl d

where 1 is the nondimensional factor

10In the cases of wing vibrations encountered in practice, the
reduced frequency turns about this value.

NACA TM 1277

l(xy) = y2 [f (x,y) W2 0 -Y 0 z' 1 + 1

x being the abscissa of the point, with the vorticity g and xp
the abscissa of the rear end of the chord measured with respect to
the position of the point of induction. The quantity 41 was com-
puted for the elements of the circulation g at the leading edge,
at the middle, and at the rear edge of the chord considered, and in
the three cases, on modification of the position of the induction
point, there were obtained (see fig. 13, in which are drawn as the
abscissas and ordinates the real and imaginary parts of .1,
respectively) the curves I, II, and III in the figure with the values
of I = xp/L. If the approximation that was made in reference 36
is assumed, in the three cases, for any position of the induction
point, the end of the representative vector is the point indicated
by the double circle; with the aid of the approximation of Sears
for all three cases, the representative points are those of curve III.
The error with either approximation is large. If the greater com-
plexity of the computations required by the solution of Sears is
considered, the advantage of his approximation is questionable.

In a recent publication (reference 33), Kissner, making use
of the acceleration potential, developed a new approximate theory
of the oscillating wing of finite span. The approximation assumed
is easily related to the expressions of the preceding section;
referring to equation (49), the value of w given by equation (45),
setting u = x + o, can be written in the form

4o* ee' du (45')
/(u2 + y2)3
Xu X

Kussner's solution can be obtained by setting the lower limit
of this integral equal to zero. The Dw, on the basis of this
assumption, are proportional to e-vw (where t is the coordinate
measured parallel to x from a fixed origin of the wing in an arbi-
trary position); that is, the values of Dw are distributed by the
sinusoidal law over each chord. If the distribution of the vorticity
is considered, which on the basis of the solution of the two-dimensional
motion corresponds to the velocities distributed according to this
law, and if for this case the values of w of the system El are
summed, velocities are obtained that can be represented by the

INAA TM 1277

w = t- e -vt) (55)

where W is a function of q.

By obtaining, on the basis of the solution of the two-dimensional
problem, the velocities induced by the system Ij and adding the
values of Dw computed with the aid of equation (49), simplified
according to the preceding assumption with respect to the limit of
the integral in equation (45) and transformed by operations analogous
to those that led to equation (54) (see reference 35), the final equa-
tion is obtained

Z(H(2)0 iH(2)1)K' 1 -b d t
2 l)- IN dT (56)
2L(J0-iJ1) -4x d N

in which H and J are cylinder functions of the parameter W and


This quantity has the same modulus as the total circulation K,
but has a certain phase displacement with respect to it.

By solving the integrodifferential equation (56), which has the
same kernel as equation (54), the law of distribution of the vorticity
on the wing corresponding to the values of w given by equation (55)
can be obtained. If it is assumed that t-Vt is the abscissa
measured in a system of reference fixed with respect to the fluid, it
is concluded that equation (55) represents a distribution of velocity
having local values constant in time, as would be the case of dis-
turbed air that presents, along the trajectory of the wing, vertical
currents of constant velocities in time (stationary gust). The solu-
tion indicated refers to the case of the stationary gust of sinusoidal
form. The more general case of motion can be studied with the aid of
the preceding solution and the solution of the two-dimensional problem
when it is considered that, according to the approximation of Kissner
(as with the procedure of references 36 and 37), the values of Dw
are zero for the distribution of the g that gives rise to zero
values of the integral K and therefore of K', which has the same
modulus as K. Hence, in this case, to the values of w represented
by a combination of wn expressed by equation (12) there correspond

NACA TM 1277

the values of g expressed by equation (11), as if the motion were
two-dimensional. On this basis, reference 35 indicates the extension
to general cases of the solution based on equation (56). Kissner
makes the generalization by a different principle, which leads,
however, to expressions that, in the limiting case of the infinite
wing, give the exact solution already known.

The approximate theory of KXssner therefore cannot be derived on
the basis of the vortices concept developed in Section 18, as is true
for the solution proposed by Cicala, which is criticized in the pre-
ceding note by Kissner as presenting arbitrary assumptions; the disa-
greement between equations (54) and (56) finds its justification in
the different principle of approximation rather than in a fault of the
derivation method based on the vortex concepts, which, according to
Kissner, would lead to erroneous results. The approximations thus far
assumed all lead to a somewhat inexact value of the induced velocities,
as is shown in reference 35; the theories are all, except that of
Sears, constructed so as to converge in the case of steady motion to
the theory of the vortex filament of Prandtl, whose approximation has
thus far been proven sufficient. On analytically examining some local
values of the errors committed,11 all are shown to be of little pre-
cision, from the simplest to that of Sears, which consists of the most
laborious application, or that of Kussner, which is based on the
concept of pressure dipoles. Only the fact that in a limiting case

11If the solution is expressed in the form of.a decreasing power
series of the aspect ratio X, it is found that the error in the
existing theories starts from the term in log /X2. In reference 35,
rather than analyzing the order of magnitude of the error, it is pre-
ferred to carry out the computation for concrete cases so as to be
able to compare the various approximations.

The comparison is particularly evident by making reference to the
concept of pressure dipoles. According to the principle followed in
reference 39, the dipoles of the system Z 2 are transported parallel
to the direction of the x-axis up to the induction point; according
to the principle adopted in reference 33, these dipoles are given the
same displacment and, in addition, the same phase displacment
0 x/V (x is the abscissa of the dipole with respect to the induc-
tion point); according to references 36 and 37, these elements are
given the same displacement and phase shift 0(x-Xp)/V. It is shown
in reference 35 that the three approximations alter somewhat the
value of the velocity produced by the pressure dipole. The affinity
of the three principles is evident.

62 NACA TM 1277

the approximations lead to a theory that has shown itself satis-
factory in applications indicates that a compensation of the errors
will, in a certain measure, be found in the values of the resultants
of the actions. A solution of greater rigor would, however, be
greatly desirable.

NACA TM 1277


The component parallel to the velocity V of the aerodynamic
action of the wing of infinite aspect ratio in nonsteady motion can
be readily computed on the basis of the solution of the problem of
two-dimensional motion, as given in part I. Under the assumptions
made, for any law of motion of the infinite wing, the theory per-
mits computing the drag or propulsive force. For the hypothesis of
a perfect fluid, the profile drag is not considered, nor are the
variations of this drag due to the unsteady motion computable with
the aid of this analysis. These actions are therefore added to
those that are here computed.

For the wing of finite aspect ratio, in the problem under
consideration, the uncertainties mentioned in part II are also
encountered. The analysis will therefore be limited to the results
obtained for the two-dimensional motion.13

Drag and thrust in unsteady two-dimensional motion. The
symbol Ri is the instantaneous value of the force that arises in
the direction of V on a segment of unit chord of the wing in a
uniform flow of velocity V. With the notation of the preceding
parts and in the same range of validity of the theory there given,
Ri, considered positive if it has the sign of a resistance and neg-
ative if it corresponds to a propulsive force, can be computed with
the aid of the expression given by Birnbaum (reference 3).14

Ri = v i 7 dx pa2L/4 (57)


2a = lim --0 7 sin (58)

12The numbers of the figures, the equations, and the paragraphs
follow from the preceding part.

13The treatment of Schmeidler (reference 46) examines the aero-
dynamic action corresponding to assigned vorticity of the wing. The
method cannot, however, be generalized.

14The sign JL indicates the integral taken over the chord of the
wing from the leading to the trailing edge.

NACA TM 1277

The aerodynamic action on a segment dx of the chord is repre-
sented by the force PV7 dx normal to the line of the axis and is
therefore inclined to the z-axis normal to V by the angle 3z/ax.
The integral in the second member of equation (57) therefore repre-
sents the action along x that is exerted on the points of the
chord. The negative term in the expression corresponds in every
case to a propulsive force and arises from the suction that is
exerted at the leading edge by the surrounding fluid and produces
a lowering in pressure that becomes infinite for the wing of infini-
tesimal thickness. If 7 is expressed by means of the customary
series of functions cot 6/2, sin 6, ..., sin n6, only the first
term can give rise to a suction at the leading edge because the
other terms represent circulations that vanish. The quantity a
defined by equation (58) gives the coefficient of the first term as
found when considering that for 6--*,

lim sin cot 2 = 2

The steady motion that is considered corresponds to a chordwise
distribution of vorticity represented by the same 7 as for the
steady motion. Inasmuch as the suction at the leading edge depends
on the instantaneous value of 7, the value of 7 must be the
same for nonsteady motion as for steady motion. In addition, when
it is considered that for steady motion the resultant force in the
wing direction must be zero,

O = PV J( )0 7 di pa2L/4 (59)

where (3z/bx)o is the slope of the axis of the wing on which, under
the considerations of steady motion, the vorticity distribution 7

If w7 denotes the velocity induced by these vortices, then

( x -)

NACA TM 1277

From this relation and from equations (57) ani (59),

Ri = P v -w) y dx (60)

which is a second form of equation (57) given by Jaeckel (refer-
ences 28 and 29) and is entirely equivalent to the equation given
in the development of the computations.

Denoting by wl the velocity induced by the free vortices and
taking account of equation (1) yield

w= wz z
w = v7 + Wv V a +-

Equation (60), on the basis of this relation, becomes

Ri = P w7 dx -P 7 dx (61)

This third form of equation (57) was used by Schmeidler (refer-
ence 46). It lends itself to interesting interpretations: If the
forces in the z-direction are distributed with density PVy and
the corresponding velocities of the points of application are 3z/6t,
the instantaneous power Ni absorbed by the motion of the points of
the profile in the direction normal to V is expressed by the

Nj/V = P 7 dz (62)

It is noted that the values of z are assumed positive down-
ward, whereas the pressures PV7 are positive upward. Hence Ni.
according to equation (62), is positive if work is done in overcoming
the aerodynamic action.

NACA TM 1277

When equation (62) is considered, it is observed that the
second term in the second member of equation (61) represents the pro-
pulsive force (or thrust)15 that would arise if the phenomenon occurred
without dissipation of energy; that is, without increase in the kinetic
energy of the fluid surrounding the wing. The first term represents
the drag RE that must be overcome by this phenomenon, or in other
words, by the creation of the vortex wake.

The computation of the instantaneous values Ri and Ni and,
successively, of the mean values Rs and Nm of the same magnitude,
in the case of harmonic motion, is immediately obtained on the basis
of the expressions given in part I. Poggi (reference 47), on the
basis of the investigations of Glauert (reference 20), computed the
propulsion and power corresponding to a rotary oscillation and, as a
limiting case, to the translator oscillation of the wing. Kissner
(reference 21) determined these values for the wing in translational
and rotary oscillation of the wing or flap. The same computations
were made by Garrick (reference 48), who made use of equation (61)
and of the energy interpretation of the term

Rm = L -V2 L J + (63a)
2 ^ Hl(2) + 10

m LV3 (63b)


15This is the quantity called "Vortrieb" by Schmeidler; the
first term is called "Widerstand."

NACA TM 1277

SH(2) + (2)oV

J + 21iw (1 cos 0) d6

ItJl = Z d + J2

J2 21 Z cos & d6
L fL

and z = Zet is the ordinate of a point of the wing according to
the complex notation with f )(C), CI, and C the real part, the
modulus, and the conjugate complex, respectively, of the complex
quantity C, with H the Hankel function of reduced frequency
S= L/2V.

The first addend that appears in the parentheses of the expression
of Rm corresponds to the resistance Rs. If E denotes the ampli-
tude of the sinusold that represents the distribution of the free vor-
ticity in the wake, it is found that, according to equation (63), the
following relation holds:

Rs = pLE2/16ao

This relation can also be derived by considering that the work
done by the resistance R, during the displacement 27V/Q corres-
ponding to an entire oscillation of the wing must be equal to the
increment that the kinetic energy of the fluid has received in the
same time, and therefore to the kinetic energy of the fluid (considered
stationary at infinity) in a strip included between two parallels to
the z-axis at 2irV/ distance from each other and located in the wake
at a great distance downstream of the wing.

68 NACA TM 1277

Applications to particular cases of oscillatory motion. In the
case of a nondeformable wing (a wing in motion of translation and

Z Zm Z0 co8 B

in which Zm represents the complex amplitude of the oscillation of
the middle point of the wing and Z0 the amplitude of the incident
oscillation. Equation (63) yields, by simple computations,

Rm = Og 2 0[ JZo) h J 2

-Nm =- pLV3[ (jOK) k J1 2]




J = ZO + I Zm + 1 iwzo

E 1(2) 1
K ='2 (1-iw) + iw
I(2) (2) 2

SH(2) 2
b(= (2) 7_727
El + iH0


k l 0

NACA TM 1277

The quantities K, h, and k are functions of the reduced
frequency. The variation of h and k as functions of w are
shown in figure 14. In figure 15, the real part is plotted on the
abscissa of the quantities K/h and K/k and the imaginary part
on the ordinate assuming the segment OA as unity and the positive
sign of the imaginary axis in the downward direction. The points
on the curve give values of the reduced frequency.

In the particular case of translator oscillation (Z0=0), from
equation (64), denoting by v = n(Zm the maximum velocity corres-
ponding to this oscillation,

-Rm = j pLv2h (65a)

Nm= pLv2kV (65b)

The force that arises in this type of motion is a propulsive one.
The efficiency of the wing considered as a means of propulsion is
h/k, which is equal to 1 for w=0 and decreases continuously toward
0.5 as w increases.

If .a wing of velocity vl normal to the wing velocity V is
considered under conditions of steady motion, the lift P, the coeffi-
cient of which is equal to nvl/V, under the usual assumptions gives
a component in the direction V represented by

Pvl/V = rpLv21

for a segment of unit chord. This component is directed forward
whether vl is directed downward or upward. On varying v1 harmo-
nically, if it were valid to apply at each instant the expressions of
the steady motion, vl would be the mean value of the propulsive force
given by equation (65a), in which h = 1. It is therefore concluded
that the exact analysis corrects this approximate consideration by
reducing the propulsive force by a factor depending on the reduced
frequency. This factor approaches 1 when the reduced frequency is
decreased and approaches 1/4 when w is decreased. The power absorbed
is reduced according to a factor that varies from 1 to 0.5 with an
increase in w.

NACA TM 1277

It is of interest to examine how, by combining a torsional motion
with translational and oscillatory motion, the propulsive force can be
increased. By simple computations, it is found., on the basis of equa-
tion (64a), that for a given amplitude Zp of oscillation of the rear
neutral point of the wing, for every reduced oscillation frequency a
certain value of the amplitude and of the phase of the torsional motion
exist for which the propulsive force is a maximum. If K' and K"
denote the real and imaginary parts of the quantity K/h, respectively,
the component f of the rotation in phase with the translator motion
is given by

Zp K"
f T Kt- (66a)
L K'-l

The component in quadrature is expressed by

q = P 2-K' w (66b)
L K'-l

The propulsive force under these conditions is given by

n pLv2h K'2 + K"2
z 2h 4(K'-1) (67)

With respect to the propulsive force of the purely translator
oscillation expressed by equation (65a), the effect of the rotation
introduces the factor dependent on the reduced frequency
(K'2+K"2)/4(K'-l). This quantity assumes decreasing values with
increases in the reduced frequency, tending asymptotically to the
value 1.125. For w = 0.5, this value is equal to 1.445; for
reduced frequencies not too small, the adding of the torsional motion
does not greatly modify the value of the maximum obtainable propulsive
force, the base value of which is always that of the purely flectional
motion. For sufficiently small reduced frequencies, there are con-
siderable increases. By neglecting the higher powers of the reduced
frequency, the expression for R~n can be assumed

K a zpVV

LACA TM 1277

On decreasing W, the torsional motion that must be combined with
the flectional motion to obtain the maximum propulsive force tends to
assume a phase displacement of 90 ahead with respect to the trans-
latory oscillation (that is, in the phase in which the translator
velocity is a maximum upward, the wing is nearest its maximum negative
incidence angle).

The analysis of the variation of the propulsive force as a func-
tion of the flectional motion for a given amplitude of the torsional
motion was given in reference 49 by means of a graph that permits
computing directly from the propulsive force (or drag) and the power
absorbed (or emitted) in the oscillation. All the values of the
efficiency from 1 to 0 can be obtained by suitably varying the ampli-
tude and the phase of the flectional motion. The region of maximum
efficiencies is nearest the point J = 0, which corresponds to the
motion without drag and without absorbed power, with zero vorticity
in the wake. Under these conditions, for a small w, the torsional
motion is displaced by about a 900 lag with respect to the flec-
tional motion; that is, in the phase in which the translator velo-
cityl6 upward is a maximum, the wing is nearest the maximum positive
incidence. This result, in relation to that of the analysis of the
maximum propulsive force previously indicated, leads to the conclu-
sion that the conditions of maximum efficiency are not compatible
with those of maximum efficiency of the wing considered as a propul-
sive means, which can be obtained with a certain loss in efficiency.

On the basis of equation (63), it may also be determined whether,
by combining with the flectional motion a deformation that alters the
curvature, any advantage in the value of the thrust can be obtained.


Z = Zm + Z2 cos 26


J = 2Z2 + iwZm

RE = 2_qJZ2 B1(2- -h J|2 I pLV2
Rm = H(2) iH(2) pL2
HMo +i oi o

16More precisely, the velocity of the rear neutral point.

NACA TM 1277

In this case, results are obtained that are entirely analogous
to those of the motion of translation and rotation. When

2Z2 = (f+lq) Zi

there are obtained for f and q the same expressions of equa-
tion (66) in which K' and K" are substituted for the real and
imaginary parts of Hl(2)/h(Hl(2)+iO(2)). With this modification,
the factor of increase in the maximum thrust has the same expres-
sion as for the preceding case. This factor, which can be expressed
by 1/4(k-h), has the value for w--i0 and the value 1.148 for
S= 0.5. In this case also, for a not very small w, no great
increases are obtained in the thrust as compared with the purely
translator motion. As in the preceding case, for a given frequency
of oscillation on increasing the wing velocity, then maximum thrust
for a given translator amplitude first increases rather slowly;
only when sufficiently low values of the ratio QL/2V are obtained
does the maximum thrust tend to increase linearly with the velocity.

NACA TM 1277


The experimental investigations that have thus far been conducted
on the aerodynamic actions on a wing in unsteady motion are not as
numerous as would be required by the complexity and importance of the
problem. It is from the measurement of the forces on the oscillating
wing that conclusive data are expected that would permit a reliable
computation of the critical velocities of the wings and tail surfaces,
Various problems relative to the stresses of the wing structures
during flight in agitated air also require experimental clarification.
The experimental investigation should furnish the necessary control
for the fundamental hypotheses of the theory of wings of infinite
aspect ratio and for the finite wing, the actual theory that makes
use of approximations that have not yet been completely checked should
be integrated. The research presents, in addition to the difficulties
common to all problems for which forces variable in time are to be
measured, serious obstacles for the requirement of absolute regularity
of the stream in which the experiment is conducted. Small fluctuations
in the velocity and in the direction of the wing, which do not have
any great effect in the measurements of a steady flow, can render the
measurements of the forces on the oscillating wing entirely unreliable.

In this part, the results obtained up to the present by various
experimenters will be discussed, and the results compared with theory.

English tests. The first series of tests was conducted by
Duncan at the National Physical Laboratory and published in 1928
(reference 50). The object of the tests was to check the mechanical
theory of the wing oscillations. From these measurements Duncan
obtained, for a particular wing model, the values of the aerodynamic
coefficients to be introduced in the expression of the velocity in
order to compare the calculated value with the experimental velocity
obtained. The greater part of the tests was conducted on a model
that was deformed during the oscillation, according to an incompletely
defined law. The tests therefore do not lend themselves to a check
of the aerodynamic theory, a check with which the experimentor was
not concerned, as he did not then have the results of the theory.
A series of tests were, however, conducted by Duncan on a model that,
during the oscillation, rotated rigidly about an axis parallel to the
span. The wing was rectangular, with RAF 15 profile, 152-millimeter
chord, and 686-millimeter span. The axis of rotation was at 1/10 of
the chord from the leading edge. The damping of the oscillations was
measured in the presence of a wind and in still air for various angles

NACA TM 1277

of attack and frequencies of oscillation. Inasmuch as the oscilla-
tions did not have very rapid damping, the results can be compared
with those of the theory on harmonic motion. According to the theory,
the moment of the aerodynamic force due to the rotational oscillation
possesses a component in phase with the motion and a component Mq
in quadrature and therefore in phase with the angular velocity q;
the moment Mq, which has its sign opposite to q, therefore consti-
tutes a damping action and may be put in the form

Mq = npbq L2VS

where S is the wing area. For a segment of an infinite wing, the
coefficient b depends on the reduced frequency and on the position
of the axis of oscillation. On increasing the reduced frequency, b
tends to the valuel7

b =2 (68)

where t is the distance of the axis of rotation from the focus con-
sidered positive if the axis is in the rear. For the finite wing, b
depends also on the plan form. Its values for not-too-small reduced
frequencies are not, however, considerably removed from that given by
equation (68). For the position of the axis of rotation of the tests
of Duncan, this gives b = 0.211. The values of b obtained on the
basis of the damping moments measured by Duncan are given in figure 16.
For the computation of b, the value of the aerodynamic damping
moment is considered to be the difference between the measured values
in the presence of wind and in still air. The values are all below
that given by equation (68). For equal velocities, these values indi-
cate a decrease with a decreasing w, as would also be given by theory
for positions of the axis of rotation ahead of the focus. For equal
frequency, on decreasing the velocity (hence on increasing w), the
experimental values in general, indicate a decrease that can be
ascribed to the effects of the Reynolds number. There is also a
decrease on decreasing the angle of attack (at least in the region
investigated). For angles of attack from -4 to -5, Duncan found

17Which can be derived on the basis of the expressions given in
references 2 and 21.

NACA TM 1277

a vanishing of the aerodynamic damping. This phenomenon, called by
Studer "oscillations of separation," cannot be studied by the theory
of the preceding parts.

The measurements conducted by Duncan of the damping of the
oscillations of the flap also lend themselves to a comparison with
theory. In these tests, the damping due to the friction of the
suspension is rather large. By assuming for this case also that the
aerodynamic damping can be obtained from the difference between that
measured with wind and that measured in still air, it is found that
the experimental value of this damping is equal to about one-'alf the
theoretical. This disagreement should not be surprising, because the
derivatives relative to the flap are always markedly less than the
theoretical values. In the tests conducted by Duncan, the value of
the derivative of the hinge with respect to the angle of the elevator
under conditions of steady motion was equal to 0.6 of the theoretical

Tests conducted at the Laboratorio di Aeronautica di Torino. -
The aerodynamic actions on the oscillating wing were measured by Cicala
in the free-jet wind tunnel of 600-millimeter diameter at the
Laboratoria di Aeronautica di Torino. The chord of the models on
which the tests were conducted was about 13 centimeters and the span
about 50 centimeters. Because of the relatively small dimensions of
the jet, which was free in the region in which the model was located,
the wing operated with a rather low effective aspect ratio; the
value of 3Cp0/& under steady conditions (referred to pV2) was equal
to about 0.5g because the wing projected from a plane that was placed
tangent to the jet in order to mask the suspension and measuring
apparatus. This rather low value of the aspect ratio is one dis-
advantage of these tests, which are described in references 14, 51,
and 52.

In a first series of tests (reference 14), the aerodynamic
damping of the flectional oscillations (rotation of the model about
an end chord) and the damping of the oscillations about the axis con-
taining the foci of the various sections (also a rigid rotation) were
measured. The measurements relative to the flectional motion were of
little importance because of imperfections in the construction of the
model and of the measuring apparatus. These tests were later repeated
(reference 52). The measurements relative to the torsional oscilla-
tions gave for b a value of about 0.11 (against 0.125 given by equa-
tion (68) for t = 0), which was almost constant when the reduced
frequency was varied as required by theory. This value was confirmed,
at least for the range of not very large angles of attack, by succeeding

NACA TM 1277

tests on a model different from the one described in reference 14,
this model also being of symmetric profile but of greater rigidity.
The principle of the measuring apparatus for these tests and for
those of series III can be briefly described as follows: The oscil-
lation of the model was controlled through an intermediary element
that possessed two simultaneous motions, a rotation depending on the
displacement imposed on the model and a rotation about an axis per-
pendicular to the first rotation and depending on the magnitude of
the force transmitted. This element carried a mirror that reflected
on sensitive paper a luminous point that, by describing the motion
of the intermediary, gave the force-displacement diagram (and there-
fore the moment-rotation diagram). The test conducted for equal fre-
quency in still air and in the presence of wind permitted isolating
the aerodynamic action. In figure 17 are given some of the oscillo-
grams thus obtained that give the simultaneous values of the angular
position of the wing and of the moment transmitted. The field of
the coordinates can be retained as Cartesian, so that the diagrams
are approximately ellipses. The enclosed area measures the work
absorbed in the oscillation. In the figure are given the scale of
motions and also the lines a = constant corresponding to the extreme
positions for one of the oscillograms obtained for a wing velocity of
9.4 meters per second and for a number of oscillations equal to 570.
For each measurement, the oscillogram was obtained by permitting the
luminous point to run through two or three cycles. The paper was
successively advanced, thus intercepting the light point. In order
to obtain a reference point of the angles of attack for each oscillo-
gram, a point in a fixed position was marked (points F in the figure).

In the third series of tests (reference 52), the damping measure-
ments of the translator flectional oscillation (in the sense pre-
viously defined) were repeated. The component of the lift in phase
with the displacement of the wing can be expressed by the derivative

1 cp
S(v = a2

where v is the translational velocity normal to V and v/V is
the corresponding variation of the angle of attack. According to the
theory of the infinite wing, the factor a2 varies from -1 to -0.5
on increasing the reduced frequency (always negative because the vari-
ation of the lift has a sign opposite to the vertical velocity).
According to the approximate theory of the finite oscillating wing,
the range of variation of this factor is reduced, starting from

NACA TM 1277

w = 0, from the corresponding value of the steady motion computed
with the aid of the vortex-filament theory of Prandtl. The tests
gave for a2 a value nearly constant and equal to 0.43 for not
too small a velocity within the range of reduced frequency in which
the tests were conducted (0.2 < w <0.6). At small velocities, a
decrease of a2 was found that was ascribed to the effect of the
low Reynolds number.

The lift component in phase with the flectional motion and the
focal moment in phase with the torsional motion were rather small,
as required by theory, the aerodynamic inertia effect being included
with the measurements of the mass of the model on the basis of the
oscillation data in still air.

The variations of lift that arise from an oscillation of the
wing about the focus were also measured (reference 51). By expressing
the lift component in phase with the rotation by means of the derivative

1 Cp
-;--= a3

values of about 0.52 were obtained for this coefficient for all the
reduced frequencies at which the tests were conducted (0.1 < w < 0.7).

The component Pq of the lift, proportional to the angular velo-
city q and therefore in quadrature with the motion, can be expressed
by means of the relation

Pq = Rpa4qSLV

The coefficient a4, which, according to the theory of the wing
of infinite aspect ratio, has values increasing with o and approaching
to 0.5, was found from the tests to be almost always equal to 0.38.

The principle of the apparatus for the measurement of the lift
due to the torsional moment was the following: The wing was put in
forced torsional oscillation by guiding, according to the harmonic
law, a point of an end section while a point of the same section was
attached by means of a steel wire. Under these conditions, in addi-
tion to the torsional motion, a flexional motion of rotation arose

NACA TM 1277

about the chord of this section. In another section, a force having a
component in phase and one in quadrature with the excited motion, was
introduced, the amplitudes of which could be varied during the test.
In the presence of wind, the amplitudes of the two components were
controlled so as to eliminate the flectional motion and to balance the
aerodynamic action and the inertia force. From the tests conducted at
equal frequency in still air and with wind, the aerodynamic forces were
obtained. By the same principle, the focal moment due to the flectional
motion was measured. In agreement with theory, the value of the
flectional motion was so small that it could not be measured.

For all the aerodynamic derivatives expressed in the preceding
form, almost constant values were thus obtained in these tests by
varying the reduced frequency. The theory of the wing of finite aspect
ratio developed in reference 37 justifies this result for the range
of not-too-small reduced frequencies and low aspect ratios at which
the tests were conducted. In figure 16, the dotted line gives the
value of the coefficient b that would be obtained by these tests,
a value that does not diverge much from the tests of Duncan.

American tests. Tests have recently been conducted in the
United States for the measurement of the aerodynamic forces on the
oscillating wing to check the theory of the infinite wing. An
interesting series of tests was conducted by Reid and Vincenti at
the Guggenheim Laboratory (reference 53). The model used had a chord
of 38 centimeters and therefore permitted the attainment of suffi-
ciently high Reynolds numbers. The span was not large (about 91 cm).
Nevertheless, a large aspect ratio was obtained because the model was
placed between two walls normal to the plane of the wing. The wing
of NACA 0015 profile was put in oscillation about an axis at a dis-
tance of 4/10 chord from the leading edge. At the opposite edge to
that at which the motion was excited, the aerodynamic action was
measured. The wing support, consisting of a ball bearing, was sus-
tained by a rigid spring the inflections of which were recorded by
means of mechanical and optical amplification on a strip of sensi-
tive paper with uniform forward motion. On the same strip were marked
the instants at which the wing occupied the extreme and middle posi-
tion. With the aid of a harmonic analysis of a graph of the forces,
which were necessarily irregular, the amplitude and the phase of the
fundamental harmonic with respect to the motion of the wing were
derived and thus the ratio r of the amplitude of the lift under
conditions of oscillatory motion and the ratio corresponding to the
steady motion for equal rotation and phase angle (leading) 6
between the lift and the rotational motion were measured. The
results are plotted as a function of the reduced frequency in
figures 18 and 19 and compared with the theory of wings of infinite

NACA TM 1277

aspect ratiol8 (continuous curve) and with those that were obtained
in the Torino tests. The Torino tests must necessarily present a con-
siderable divergence from the American tests because of the difference
in aspect ratio. The phase displacements predicted by theory are
somewhat greater than the experimental values. A considerable devi-
ation is presented by the theoretical and experimental curves of the
ratio r.

A similar series of tests was conducted by Silverstein and
Joyner (reference 54). The model had a chord of 13 centimeters,
considerably smaller than that used in the tests by Reid. In this
case also, a large aspect ratio was attained by using end walls.
The oscillation axis passed through the forward quarter chord and
the aerodynamic lift force was measured by means of an apparatus
based on the same principle as previously described. In these tests,
only the phase displacement between the lift and the rotation was
measured and the values shown in figure 20 were obtained; the con-
tinuous lines give the values of the theory of the infinite wing and
the dotted line give the values obtained from the Torino tests. The
scatter of the test points is large for the high values of w; that
is, for the tests conducted at low velocity.

Because of the small number of the results that are available,
no deductions of a conclusive character can be given. The different
conditions under which the tests were conducted also does not provide
a good basis for comparing the different results.

The tests in which the conditions for a check of the theory of
the oscillating wing of infinite aspect ratio were best realized are
those of the Guggenheim laboratory. The comparison is not, however,
completely satisfying. The probable cause of the divergence encountered
seems to lie in the agglomeration and dissipation of the wake vortices,
the mutual positions and intensities of which the theory assumes to
be maintained indefinitely. The problem should be investigated more

18The theoretical curves of the graphs of reference 53 do not
coincide with those given in figures 18 and 19 because the aerodynamic
inertia effect represented by the terms in w2 in the expressions of
the derivatives is not considered. In fact, this action, which remains
unchanged with and without wind, is already compensated in the pre-
liminary operation of putting the center of gravity on the axis of
rotation, a compensation that, it seems, was effected under dynamic
conditions. The correction is small and makes the test points
approach the theory more closely.

NACA TM 1277

thoroughly, especially in an experimental manner. The verification
of the theory of the infinite wing is less urgent, however, than the
investigation of the finite wing, particularly for the phenomenon of
wing vibration, in which, because the motion is more pronounced toward
the tip of the wing, the conditions are considerably removed from two-
dimensional motion. This limiting case is also difficult to obtain
experimentally because of the considerable importance assumed by the
wake over a large distance behind the wing. The Torino tests make
use, however, of low aspect ratios for which the approximations of
the theory of reference 31 are less justified for giving an account
of the results of such tests. There would therefore be required:
First, a perfecting of the theory of the oscillating wing of finite
aspect ratio; and second, the extension of tests to wings of greater
aspect ratio. The range of angle of attack within which the coeffi-
cients can be held constant must be defined and the field of coeffi-
cients relative to the oscillating wing with flap must be investigated.

Translated by S. Reiss
National Advisory Committee
for Aeronautics.


1. Possio, Camillo: L'azione aerodinamica sul profile oscillante in
un fluido compressible a velocity iposonora. L'Aerotecnica,
vol. XVIII, fasc. 4, Aprile 1938, P. 441-458.

2. Cicala, P.: Le azioni aerodinamiche sul profile oscillante.
L'Aerotecnica, vol. XVI, fasc. 8-0, Agosto-Sett. 1936, P. 652-655.

3. Birnbaum, Walter: Das ebene Problem des schlagenden Flugels.
Z.f.a..M., Bd. 4, Heft 4, Aug. 1924, S. 277-292.

4. Theodorsen, Theodore: General Theory of Aerodynamic Instability
and the Mechanism of Flutter. NACA Rep. 496, 1935.

5. Schwarz, L.: Berechnung der Druckverteilung einer harmonisch
sich verformenden Tragflache in ebener Stromaung. Luftfahrt-
forschung, Bd. 17, Lfg, 11/12, Dez. 10, 1940, S. 379-386.

6. S6hngen, Heinz: Bestimmung der Auftriebsverteilung fir beliebige
instationrre Bewegungen (Ebenes Problem). Luftfahrtforschung,
Bd. 17, Lfg. 11/12, Dez. 10, 1940, S. 401-420.

NACA TM 1277

7. von Karmin, Th., and Sears, W. R.: Airfoil Theory for Non-Uniform
Motion. Jour. Aero. Sci., vol. 5, no. 10, Aug. 1938, pp. 379-390.

8. Kissner, Hans Georg: Schwingungen von Flugzeugflugeln.
Luftfahrtforschung, Bd. 4, Heft 2, Juni 10, 1929, S. 41-62.

9. Cicala, P.: Le azioni aerodianamiche sui profile di ala oscillanti
ecc. Memorie d. Reale Ace. della Sci. di Torino, 1935.

10. Possio, C.: Sul problema del moto discontinue di un'ala. Nota 1.
L'Aerotecnica, vol. XX, fasc. 9, Sett. 1940, P. 655-681.

11. Pistolesi, E.: Aerodinamica.

12. Wagner, Herbert: Uber die Entstehung des dynamischen Auftriebes
von Tragflugeln. Z.f.a.M.M., Bd. 5, Heft 1, Feb. 1925, S. 17-35.

13. Kussner, H. G.: Das zweidimensionale Problem der beliebig bewegten
Tragflache hunter Berucksichtigung von Partialbewegungen der
Flissigkeit. Luftfahrtforschung, Ed. 17, Lfg. 11/12, Dez. 10,
1940, S. 355-361.

14. Cicala, P.: Ricerche sperimentali sulle azioni aerodynamiche sopra
1'ala oscillante. L'Aerotecnica, vol. XVII, fasc. 5, Maggio 1937,
P. 405-414.

15. Kassner, R., and Fingado, H.: The Two-Dimensional Problem of
Wing Vibration. R.A.S. Jour., vol. XLI, Oct. 1937, pp. 921-944.

16. Krall, G.: Problemi non stazionari dell'idrodinamica. P Ibbl.
dell'Inst. Naz. per appl. del Calcolo, n. 26, 1938.

17. Kussner, H. G., and Schwarz, I.: The Oscillating Wing with
Aerodynamically Balanced Elevator. NACA TM 991, 1941.

18. Glauert, H.: The Accelerated Motion of a Cylindrical Body through
a Fluid. R. & M. No. 1215, Jan. 1929.

19. Lamb, H.: The Hydrodynamic Forces on a Cylinder Moving in Two
Dimensions. R. & M. No. 1218, Feb. 1929.

20. Glauert, H.: The Force and Moment on an Oscillating Aerofoil.
R. & M. No. 1242, March 1929.

21. KIssner, H.G.: Zusammenfassender Bericht uber den instationiren
Auftrieb von Fligeln. Luftfahrtforschung, Bd. 13, Nr. 12,
Dez. 20, 1936, S. 410-424.

NACA TM 1277

22. iissner, H. G.: Status of Wing Flutter. NACA TM 782, 1936.

23. Kissner, H. G.: Untersuchung der Bewegung einer Platte beim
Eintritt in eine Strahlgrenze. Lufatfrtforschung, Bd. 13,
Nr. 12, Dez. 20, 1936, S. 425-429.

24. Dietze, F.: Zur Berechnung der Auftriebskraft am schwingenden
Ruder. Luftfahrtforschung, Bd. 14, Lfg. 7, Juli 20, 1937,
S. 361-362.

25. Dietze, F.: Die Luftkrifte der harmonisch schwingenden, in sich
verformbaren Platte (Ebenes Problem). Luftfahrtforschung,
Bd. 16, Lfg. 2, Feb. 20, 1939, S. 84-96.

26. v. Borb6ly: Mathematlscher Beitrag zur Theorie der
Fligelschwinnggen. Z.f.a.M.M., Bd. 16, Heft 1, Feb. 1936,
S. 1-4.

27. Ellenberger, G.: Luftkrgfte bei beliebig instationirer Bewegung
eines Tragflugels mit Querruder und bel Vorhandensein von Bsen.
Z.f.aM.M.., Bd. 18, Heft 3, Juni 1938, S. 173-176.

28. Jaeckel, K.: Uber die Krafte auf beschleunigt bewegte,
ver~nderliche Tragflugelprofile. Ing.-Archiv, Bd. IX, 1938,
S. 371-395.

29. Jaeckel, Karl: Uber die Bestimmung der Zirkulationsverteilung
fur den zweidimensionalen Tragflugel bei beliebigen periodischen
Bewegungen. Luftfahrtforschung, Bd. 16, Lfg. 3, Marz 20, 1939,
S. 135-138.

30. Lyon, H. M.: A Review of Theoretical Investigations of the Aero-
dynamical Forces on a Wing in Non-Uniform Motion. R. & M.
No. 1786, British A.R.C., April 1937.

31. Garrick, I. E.: On Some Reciprocal Relations in the Theory of
Nonstationary Flows. NACA Rep. 629, 1938.

32. Garrick, I. E..: On Some Fourier Transforms in the Theory of Non-
Stationary Flows. Proc. Fifth Int. Cong. Appl. Mech.
(Cambridge, Mass.), Sept. 12-16, 1938, pp. 590-593.

33. Kussner, H. G.: General Airfoil Theory. NACA TM 979, 1941.

NACA TM 1277

34. Miller, Reinhard: Uber die zahlenmassige Beherrschung und Anwendung
einiger den Besselschen verwandten Funktionen nebst Bermerkungen
zum Gebiet der Besselfunktionen. Z.f.a.M.M., Bd. 19, Nr. 1,
Feb. 1939, S. 36-54.

35. Cicala, P.: Le teorie approssimate dell'ala oscillante di
allungamento finito. Atti d. Reale Acc. della Sci. di Torino,

36. Cicala, P.: Sul moto non stazionario di un'ala di allungamento
finito. Rend. Reale Ace. Naz. d. Linoae (Classe Sci. fis.,
mat. e naturali, vol. XXVI, 1937, P. 97-102.

37. Cicala, P.: Comparison of Theory with Experiment in the Phenomenon
of Wing Flutter. NACA TM 887, 1939.

38. v. Borbely: Uber einen Grenzfall der instationaren raumlichen
Tragfligelstromung. Z.f.a.M.M., Bd. 18, Heft 6, Dez. 1938,
S. 319-342.

39. Possio, C.: Sulla determinazione dei coefficienti aerodinamici
che interessano la stability del velivolo. Comm. Ace.
Pontificia d. Sci., 1939.

40. Possio, C.: Aerodynamic Forces on a Lifting Surface in Oscillatory
Motion. Air Ministry Trans. No. 987, Oscillation Sub-Comm.,
British A.R.C., Dec. 6, 1939.

41. Possio, Camillo: Determinazione dell'azione aerodinamica
corrispondente alle piccole oscillazione del velivolo.
L'Aerotecnica, vol. XVIII, fasc. 12, Dic. 1938, P. 1323-1351.

42. Possio, C.: Sul moto non stazionario di una superficie portante.
Atti d. Peale Ace. della Sci. di Torino, 1939.

43. Possio, C.: L'azione aerodinamica su di una superficie portante
in moto vario. Atti d. Reale Acc. della Sci. di Torino, 1939.

44. Possio, C.: Sul problema del moto discontinue di un'ala. Nota 2.
L'Aerotecnica, vol. XXI, fasc. 3, Marzo 1941, P. 205-230.

45. Sears, William R.: A Contribution to the Airfoil Theory for Non-
Uniform Motion. Proc. Fifth Int. Cong. Appl. Mech (Cambridge,
Mass.), Sept. 12-16, 1"' -n. 483-487.

NACA TM 1277

46. Schmeidler, Werner: Vortrieb und Widerstand. Z.f.a.M.M., Bd. 19,
Heft 2, April 1939, S. 65-86.

47. Poggi, L.: Azione aerodinamiche parallel al movimento su di
un'ala animata da moto traslatorio uniform e da moto oscillatorio.
L'Aerotecnica, vol. XI, fasc. 6-7, Guigno-Luglio, 1931, P. 767-779.

48. Garrick, I. E.: Propulsion of a Flapping and Oscillating Airfoil.
NACA Rep. 567, 1936.

49. Cicala, P.: Il problema aerodinamico del volo ad ala battente.
L'Aerotecnica, vol. XVII, fasc. 11, Nov. 1937, P. 955-960.

50. Frazer, R. A., and Duncan, W. J.: The Flutter of Aeroplane Wings.
R. & M. No. 1155, British A.R.C., Aug. 1928.

51. Cicala,

52. Cicala,

P.: Ricerche sperimentali sulle azione aerodinamiche
l'ala oscillante (Ser. II). L'Aerotecnica, vol. XVII,
12, Dic. 1937, P. 1043-1046.

P.: Ricerche sperimentall sulle azione aerodinamiche
l'ala oscillante (Ser. III). L'Aerotecnica, vol. XXI,
1, Gennalo 1941, P. 46-53.

53. Reid, Elliott G., and Vincenti, Walter: An Experimental Deter-
mination of the Lift of an Oscillating Airfoil. Jour Aero.
Sci., vol. 8, no. 1, Nov. 1940, pp. 1-6.

54. Silverstein, Abe, and Joyner, Upshur T.: Experimental Verifica-
tion of the Theory of Oscillating Airfoils. NACA Rep. 673, 1939.

NACA TM 1277


a J a R a

0 0.5000 1.7 0.3490 7.5 0.1588
0.1 0. 488 1.8 0.3427 8.0 0.1509
0.2 0.4762 1.9 0.3366 8.5 0.1436
0.3 0.4651 2.0 0.3307 9.0 0.1368
0.4 0.4545 2.1 0.3250 9.5 0.1307
0.5 0.4443 2.2 0.3195 10 0.1250
0.6 0.4346 3.3 0. 3141 11 0. 1147
0.7 0.4253 2.4 0.3088 12 0.1058
0.8 0.4163 2.5 0.3038 15 0.0852
0.9 0.4077 3.0 0.2805 20 0.0634
1.0 0.3994 3.5 0,2600 26 0.0499
1.1 0.3914 4.0 0.2420 30 0.0408
1.2 0.3837 4.5 0.2261 40 0.0298
1.3 0.3763 5.0 0.2118 50 0.0232
1.4 0.3691 5.5 0.1990 100 0.0109
1,5 0.3622 6.0 0.1875 500 0.0020
1.6 0.3555 6, 5 0.1770 1000 0.0010
7.0 0.1675 co 0.0000

NACA TM 1277






1. 6


' RA s, 1, s, P,

NACA TM 1277


i-A A I -A' I A"

0 1,0000 0 0.62 0,5757 0.1354
0.002 0.9967 0.0126 0.64 0.5727 0. 1330
0.01 0,9826 0.0456 0.66 0,5699 0.1308
0.02 0.9637 0.0752 0.68 0,5673 0.1286
0.04 0,9267 0.1160 0.70 0.5648 0.1264
0.06 0.8920 0.1426 0.72 0.5624 0.1243
0.08 0,8604 0.1604 0.74 0.5602 0,1223
0.10 0.8319 0.1723 0.76 0,5581 0.1203
0.12 0.8063 0.1801 0.78 0.5561 0.1184
0.14 0.7834 0.1849 0.80 0.5541 0.1165
0.16 0.7628 0.1876 0.82 0.5523 0.1147
0.18 0.7443 0.1887 0.86 0.5490 0.1112
0.20 0.7276 0.1886 0,90 0.5459 0.1078
0.22 0. 7125 0,1877 0.94 0.5432 0.1047
0.24 0.6989 0.1862 0,98 0.5406 0.1017
0.26 0.6865 0.1842 1.00 0.5394 0,1003
0.28 0.6752 0.1819 1.1 0,5342 0.0936
0.30 0.6650 0.1793 1.2 0.5300 0.0877
0.32 0,6556 0.1766 1.3 0.5265 0.0825
0.34 0.6469 0.1738 1.4 0.5235 0.0778
0,36 0.6309 0.1709 1.5 0.5210 0.0735
0,38 0.6317 0.1679 1.6 0,5189 0.0697
0,40 0,6250 0.1650 1.7 0.5171 0.0663
0.42 0.6187 0.1621 1.8 0.5155 0.0632
0.44 0.6130 0.1592 1.9 0.5142 0.0603
0.46 0.6076 0.1563 2.0 0,5130 0.0577
0.48 0,6026 0.1535 2.5 0,5087 0.0473
0.50 0.5979 0.1507 3.0 0.5063 0.0400
0.52 0.5936 0.1480 3.5 0.5047 0.0346
0.54 0.5895 0.1454 4.0 0.5037 0.0305
0,56 0.6857 0.1428 4.5 0.5029 0.0273
0.58 0.5822 0.1402 5,0 0.5024 0.0246
0.60 0.5788 0,1378 10.0 0.5006 0,0124
> 10 0.5000 + 1/8
+ 1/16;'

NACA TM 1277



Fig. 1.
Fig. 1.

Fig. 2.

x~ ~

Fig. 3.

NACA TM 1277 a 89



// \

*~ l -J -



NACA TM 1277

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 7.


NACA TM 1277

."- 1 % .
do -,,4

r-4 d

Fig. 8.

dy {p IP : .
do r

Y r
dy P(--yY

Fig. 9.

NACA TM 1277


Fig. 10.

y 8 ,' .

AI -- -- --

F. 1
0 g.

Fig. 11.

0.05 0.1
' i7"1


NACA TM 1277




_. -A b
6 0 '

6 OQ
... --

I "- '~ OiiiiSE- -

+b 0 0

Fig. 12.

-rI I C I Q I I I I I I I
-0=2 0 02 0.4 0.6 0.8 I


Fig. 13.


94 NACA TM 1277

Fig. 14.

Fig. 15.

NACA TM 1277


Fig. 16.

S -. rn, -. "

.. ..

. ,n ,| | 2 i '

S- ..J ...2' ; -- .' 1..
Fig. 17

NACA TM 1277

V-L V-- --V V-

9 9 o.
\ o /

\- i. 1.--

Fig. 18.

Fig. 19.

Fig. 20.

NACA-Langley 10-2-51 1000



ni nr nR r~ril

n a

wi E-' .

t -l il U !IN
0 0
u u w i rDa. E- ,
Q) bl a LOn


4 3< i 1 C 4 S
.4 C4 4 o- e

E S a.q S 3
to Cu Vu .
-= O- cyo<

or.lm %910
c.~ 0 d 0 C C*4 j0 .
r- m to.. uC
ba m C nCu o"

ca N w T ED w o W;ri5
0 0C bb Q Cd D E rut
.2 w aR q E
5:~u C lir ~ rnl0 mL-
CuCuO. 4 *L6 U

:8 a". 4-. Cu' 4 sf 1'. -M t .U
00t.k d P )wQ 0 cud .4 n4.k C
Z ~ ~ CJ _uC tW-
o0 USC C0 =0j ; 4
Cu 0 q t.C u Cu 0 C
Z FA **C mk z u 7 4
z ka t .. .: 'm0k

N ~ 00 C. Cu0~~ .
0-, 0 0 rt- m ca

Cu~E 0. W rCu0 ~
*Cu ;- Wo C0 .4.U 02 W w a) 0
ZCu .0 C e Z -- _

auu~~u)u w. N'.'C Cu wCu q
cl c 0 k 0 C4 CC d a
op,, E 10 cd k 1:4 E S .-

w- n ZZ2Cu- u0' n M
r M ) r
w 0 N w Q) ) I U

18 a m rn E ) Lf k

-0 0 k W*P
r" .
kSa j k n t 0) :s -4 '0c Q
Z >.q W UO CM 5 k 0 M 0)o
cd W 0 E
R42 C.) Lm k ca 0
CL -.4 m 0: Cd c CL k : 3 *
Cd -4 d 9 co 0Z :3 0 m E-4 0 to a B
-C -g -SC, u-Q a 5 ; 1
04 m~O~c R 0. z z W to 0) 4 CL 4

Pi eq 'D

C a- -0 Of
~jE 12 rccV Cd i

0 0
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ba 0bc ) It Mb
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o C rt .o 5S S4 -L
51, 0 =In
r\q Z~t -. f- p m ^

ei CM dl n



'I b
L Cu



g C

r 06
^| <

^1 o
'la. Cu
itCu. s
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>C Cu 0 '


(D ) E 0) t
o -4 t-4

I ",I ci I vl
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2 ol I z.^
-e '* UD "u -T( ^ tE
Cu .ku E Q
C o Ciu iC Ca*uo Cu (0 D t-
b sa 0 a) .0 Cu
cu' ~C E-4cu

gI~bCD kb $Xq
> (M 04

c i c 'i .
-4 eq eor. U

o 0


0 E- -C

0 ilZ

> Ul

u 00 C4 r- C
4 0 iJ-


slgwt 5 < %

C4 -44
w F1..l eq _u

41q0 v~* C~

0~~ ~ 00a U
z .s as Cu
ed3 g u 1. a. *M

D~b~u., f 4) al
, t li il W-
U0bOO (L)c bD -- X 3j o ir

E-4 ;> .1^ UK to.'~-

i4 i c l; L; .4 m i 0M

0 o 02

) .0 C4

0,,4 fz COde-
C)00 Cd0

0Cu o
Ci ) C3 o
0 PRpk 2 6

0 0 1 (1 EC <

0 B o < -<
3 h C d C54
z > Z 1 o a 0 m

;za ;z ;z -4
ci C4u1 0-

EO .0 L
Cu U eq COL
*1 =~~ Ae~ k
E9C: 0 cd = ul Cu Cd). Co
0 0. Cu r. c 0.
0 Cc ~ k 04) -
i ~ ~ b M- in L- ig a-i"i

rg is aa) r. 8 ( ko o

C4 a,2r2

:8 -4a

0 0 a

i 7 'E
"Z .s

al r 4 o -
rA 00 2 ; ,

0 g.

.4~- I
&, Cu 0 0 ....0

r 6O h o o LW
>> C a Q

rzzl (A30 -'4 C
Cuxa M Cd.

-4 e4 C4 eq-
4 o I a-4

eq eq eq
Cu C 0 a .0 j -4
I I OC M 0 S-<>l
u C) Im C,
- s- - 2 s' Eo o C 2 u *C
oi 0~ '0 0. -0 S
a Lo -u"
bi 0 bo j)S Mo u 0
q. a) m.R 0)f LO aI

C4 Pi 'r; u


0 0C,0

= "
0 -'a 0S

E~ UN~

o9rzWuw ,c c
> a) 00 'U 6L-
3. C4 5 4C) ..4t

ZZ 4rj (D -I)~ Cd.L
0 .2 0 t

E2 0 g ^g
EL 1 (3) m > 0,M

>o we u Wt*S o
z 0 k k *2 -

51 -^S tL2"

s< alE-. ca-t ,.
g_ z >U3 -M m C04 C

C1 4 -4 .4 e
I- "" I:: ;? I??
eq C,. I. CSIu rt
.Cu L( S3
0 o -4-.w
ru C~u'-4 k Cu C.) eq
Cu Cu ^ Cu Cu Cu 0
03. 0.3. 04
0 0~C .0s V u Cu2
0 M '0 ;. u DO'

ku~u~~ 0 ~ CuW, vc
4 vc noa 4l v .2 -4 Li
&^ -4> : msq u zf4a',

iCM V; 4I Ln _; Hi d

0 0 I g
z .
0u kCu

z t
o E 0 -4
~40 *q

O000" C
*0o "o 3 *

0 IZI Cu o0)< 0.
g -"5U. a '

c. i

g. 0 o P4Q c L
u| 0 0 g
ZZD0= ti 3s<

".. 4 ,
IC I: C u$. 'i~0',=
0 0. 0 1

.0 0 ,

1.. i r. > ,

Cu,. 0 i~'
,:: B =>a g* a' o

0 ,. 0 Q .. L
0 0 0 o 0.
,n 0u *o Cu Cu ('4 gu CuC
cis 0 000 C
00C CD~ >
rj Cu ow a)
S0 U Cu W 0

C4 clu CC
k.. C C 0 0, Cu M

Fu0Cu 0.
0 Cu w

bD ~ ~ 0. 0baU
l W a ..Cu a 2. 0
Ell all s 0.

S oJ k *w w- 2 2 .g


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