The plane problem of the flapping wing

The plane problem of the flapping wing


Material Information

The plane problem of the flapping wing
Series Title:
Physical Description:
38 p. : ill ; 27 cm.
Birnbaum, Walter
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Airplanes -- Wings -- Testing   ( lcsh )
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


A theoretical study, based on vortex theory as applied to the linearized equations of motion, is made of the air forces on wings of infinite aspect ratio in incompressible flow. Expressions for forces and moments associated with steady harmonic oscillations in vertical translation and pitching of wings are derived in the form of power series in terms of a reduced frequency parameter. Use is made of the derived forces first to treat the problem of propulsion due to wing flapping and second to determine theoretical flutter speeds of some simple spring mounted configurations.
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Walter Birnbaum.
General Note:
"Report date January 1954."
General Note:
"Translation of "Das ebene problem des schlagenden flugels" Zeitschrift für augewandte Mathematik und Mechanik, Band 4, 1924."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003780265
oclc - 99996738
sobekcm - AA00006150_00001
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Full Text
Ach-TOl / 3/

79 97 7% 7 o




By Walter Birnbaum

In connection with my report on the lifting vortex sheet2 which
forms the essential basis of the following investigations, I shall show
how the methods developed there are also suitable for dealing with the
air forces for a wing with a circulation variable with time. I shall,
in particular, develop the theory of a propulsive wing flapping up and
down periodically in the manner of a bird's wing. I shall show how the
lift and its moment result as a function of the flapping motion, what
thrust is attainable, and how high is the degree of efficiency of this
flapping propulsion unit if the air friction is disregarded. Finally,
I shall treat an interesting case of dynamic instability for a spring-
suspended wing; this phenomenon was confirmed by experiments at the
Gottingen aerodynamic test laboratory. Professor Prandtl gave me his
guidance concerning the present report, and I want to express here also
my sincere gratitude for the abundant stimulation and energetic assist-
ance he gave to me at all times.

1. General statements.- The calculations refer to the two-dimensional
problem, that is, to the wing of infinite length, or bounded by plane side
walls, or to a wing with so large an aspect ratio that the boundary effect
is negligible. For the rest, the same assumptions as in the first report
are valid concerning smallness of the air forces, slight camber of the
wing, and so forth. The wing is assumed to extend from x = -1/2 to
x = +3/2 so that the point x = 0 becomes the center of pressure of
a plane wing with fixed angle of attack. Let v, in the direction of the
positive X-axis, be the air velocity at a large distance. Let the posi-
tive Y-axis point downward. By 7 = 7(x,t), I denote the density of the
lifting vortices (now a function of the time). At every variation with
time of the density of circulation, free vortices of the density c(x,t)
will separate from the lifting vortices and will drift away with the air
flow. On the basis of the theorem about the conservation of circulation,
or by integration of Euler's differential equation once along the pressure
side and once along the suction side of the wing (compare the unabbreviated

*"Das ebene Problem des schlagenden Flugels" Zeitschrift fur augewandte
Mathematik und Mechanik, Band 4, 1924, pp. 277-292.
1Abstract from the GCttinger dissertation of the same title. Avail-
able in the University library at GCttingen and the state library at
Berlin. Referent: Professor Prandtl.
2This periodical (Z.f.a.M.M.), vol. 3, 1923, pp. 290-297.

NACA TM 1364

report), one finds easily that the vortices must satisfy at every point
and at every time the following continuity equation of the vortex density3:

S+ LE+ +v =
ht 6t dx

Generally one will make suitable assumptions regarding y(x,t) so that
E may be found from the equation. If 8y/dt is designated by y(x,t),
its integral is

E(X,t) = x-vt)

7(t + x\
7,v /---;dS

Where p is an arbitrary function to be determined by boundary condi-
tions. After E has thus been found, there results the induced vertical
velocity in first approximation as principal value of the integral

1 t) 7 + E
w(x,t) d-
2x x [

Similarly to the former case, w now has
moved wing contour. If the motion of the
of the ordinate at the point x, thus by
lation one could also include a moved and
the kinematic boundary condition reads

to be put in relation with the
wing is indicated by the value
y = y(x,t) (with this formu-
simultaneously deforming wing),

y v =y
-y + v -- = w(x,t)
ot ox

3Details on the derivation of this equation may be found in a lecture
of Prandtl "On the Formation of Vortices in the Ideal Fluid" at the hydro-
aerodynamic conference at Innsbruck 1922. (Lectures concerning the field
of hydro and aerodynamics, edited by Th. v. Karman and T. Levi-Civite,
Berlin 1924) and in the original report quoted.

NACA TM 1364

This is the
obtained if
above said

equation from which contour and motion of the wing are
7, and therewith w, is prescribed. In analogy to the

y = 1

w(tt +t d

If one takes into consideration that the air velocity relative to the
supporting vortex line has the components v and w there result
as before the air forces for the unit length of the wing

K = pv y1 dx

K = pv 7 dx

M = pv 7yx dx



6 1)dx

1-W p

a2 2
- p T = pv
f-~ 2

The last term in W represents as in my first report the suction
force at the leading edge of the wing, now of course with a coefficient, a
variable with time of the first fundamental function.

2. Periodical wing motion.- From these general expressions, I now
pass to treating the special problem of the flapping wing. It is to be
expected that 7 itself in this case will be periodical, except for
a constant mean value. Thus, I determine first from the expression

7 = 70(x)eivt = 7eivt = 70e'vt

7 r-

- p -2

NACA TM 1364

the free vortices e; for 7 which alone enters into e is independent
of the constant mean value 7 so that this expression is sufficiently
general. As customary in oscillation theory, one tacitly deals with
the imaginary constituent of 7. v is the circular frequency of the
oscillations, w = V a dimensionless quantity which I call "reduced
frequency" (v has the dimensions of an angular velocity since it equals
the flight velocity, measured with half the wing chord as unit length).
If one designates the distance travelled by the wing after a full period
as the "wave length" k, one has

X = 2A V 2-
v a2
X = _F

The further results are obtained by series developments with respect to
the reduced frequency ca which have the form

,cmo m(log W)n m > n

These series converge satisfactorily only for small w up to about
o = 0.1 corresponding to a wave length of 30 wing chords or more. Thus
I presuppose for the present report slow quasi-steady oscillations of
the wing; this does not imply, of course, that the actual frequency can-
not assume high values, too, if only the wing chord is sufficiently small.
If I calculate, finally, with u' = iw, the formulas obtained, at least
outwardly, real coefficients which offer many advantages for the

I now assume that parallel flow free from vortices prevails ahead
of the wing. Then I obtain with x = x 1/2 for E and w, under the
presupposition that the "switching-on process" of the wing motion has
already run its course (the motion thus has become steady), the fol-
lowing expressions:

E =0 ; X < -

E = = a= e"C'(-) eW' 70()d -1 x 5 1


E = E = CeO'(vt-) e e o0()dS ( 3 > 1

NACA TM 1364

2rr(x,t) f(jt) d
-l x T

S+1 Ei ,t)

As in my first report, I determine
fundamental functions4

70(x) from the three first

70 = ayOa + byOb + cyOc

Where a, b, c now are complex numbers. One has then also

w = wOeu'vt

0 = aw0a + bw0b + cWOc

The integrals (9) and (10) cannot be evaluated by elementary functions.
The last integral (10) in particular leads to an integral logarithm.
The Euler constant C which appears as a consequence always occurs in
combination with log 2 so that I shall introduce as transcendent

Z = log 2 C = 0.11595

especially for the present report. If one takes
sideration one obtains the w in the form

this fact into con-

WOa = CO + al(m' + ago2' log m' + a,3w'2 + aQC 21og ,o' + aC5m'3 + cag'3og co'


One obtains

70a = +

Ob = 1 x2


d + oo


70c = xl 2

NACA TM 1364

and corresponding expressions with Pi and Ti are valid for wOb
and VWc. The coefficients ai, Pi, Ti are polynomials in x the
coefficients of which are linear and rational in Z (regarding their
values compare the original report). Besides, the appearance of log o'
shows that a treatment of the problem with a lifting line instead of
the supporting surface must fail since to the transition to the lifting
line there corresponds a transition to a' = 0 (v increases arbitrarily
for decreasing wing chord). For this reason, the theory of the flapping
wing here presented is the simplest possible one.

Bypassing the integral for y, we are now concerned with determining
the wing motion in an appropriate manner and bringing it into accord with
the expression for y. Since the methods of the lifting vortex sheet are
linear and since a deformation oscillation will not be taken into con-
sideration, shape and mean angle of attack of the wing may be disregarded:
It is then sufficient to calculate the oscillations of a plane wing with
the mean angle of attack 0 and to superimpose these oscillations as
small fluctuations with time of the profile chord linearly on the arbi-
trarily prescribed profile. For the amplitudes which are assumed to be
small, one may disregard the fluctuation of the abscissa of a wing point
as small of higher order, and has then as the most general motion still
possible (compare fig. 1)

y = p(t) + xp(t) (12)

p and cp are to be developed as Fourier series with respect to ov;
I retain only their first term

y = (A + Bx)m'vt (15)

Higher-harmonic oscillations would again have to be superimposed linearly.
Fluctuations in the flight direction (which could be expressed by periodic
fluctuations of v) also will be disregarded.

I am designating the quantities A and B as complex stroke ampli-
tude and amplitude of rotary oscillation, respectively, since equation (15)
is formed by combination of the special translator and rotary oscillations

A = 1, B = 0 (a) and A = O, B = 1 (p)

Thus 70 and all quantities linearly connected with it will be linear
and homogeneous in A and B. For simplification of the calculation,
the cases (a) and (0) may therefore be calculated separately and after-
wards be combined linearly, for instance

a = Aa, + Bap etc.

NACA TM 1364

The complex circulation coefficients are found from equation (4) in the
following manner: a, b, c are expressed in the form of the series

a = a + alw' +

aS 21og2 ,W
a5w' 3log2w'

b = b +

c = CO +

a2w' log *' + a 3'2 + a '2log w' +

+ a6n'3 + a-w'31og w' +

+ a9g'31og3w' + .

Equation (4) has in our case the form

w(x,t) = (AC' + BR'x + B)ve'ovt

= (awa + bwb + wOc) eW'Vt

This is to be valid identically in x and t. If both sides are
interpreted as series in 3'm(log u')n, their coefficients must be
identical. a, b, c therein are unknowns. Each coefficient yields
a relation in x which at first cannot be expected to be identically
fulfillable by only three free values. It was intended to fulfill the
condition at least at three suitably selected locations xl, x2, and
x3. Surprisingly, these conditions are now shown to be precisely of the
second and not of a higher degree in x. At least this is the case up
to the terms of third order, and there is no doubt that this will remain
so for the higher terms as well. The coefficients of second degree
in x determine, therefore, with their three subcoefficients each one
triplet of values, each of the a, b, c. In every new comparative
coefficient there appears a new value triplet of this kind so that the
thirty unknowns for which a formulation has been set up may be found


NACA TM 1364

uniquely from linear relations by successive evaluation.
results in the following values:
a0 = 2Bv

a, = 2v[A + 1 B(l 2Z)

a2 = 2Bv

a3 = 2v AZ B(Z Z2-

a4 = 2vLA + B(l 2Z4

a5 = 2Bv

ag = 2v AZ2 B + Z

a- = 2v 2AZ + B Z+

a8 = 2v A+ B 3z)

ag = 2Bv

bO = 0

bl = 4Bv

b2 = 0

b = 2v(A + B)

c = 0

C1 = 0

c2 = 0

c3 = Bv

The calculation

- Z2 + Z (See footnote 5.)


i 4

5The dimensions of all quantities are affected by the selection
of the wing chord as unit length. A treatise of the author in the
Zeitschrift fur Motorluftschiffahrt on the same subject has been arranged
so that no objections are possible from the viewpoint of similarity

bi = ci = 0,

NACA TM 1364

I want to point out here briefly that approximated values for the
circulation coefficients are obtained also, if the effect of the free
vortices is disregarded and the elementary calculation made in such a
manner as if the momentary apparent angle of attack of each wing element
were decisive for its circulation, namely the angle formed by the air
velocity v relative to the moved wing element and the direction of the
latter. Since in case of rotary oscillations (B j 0) every wing element
has another vertical velocity, the apparent angles of attack of the
elements are all different, that is, the wing assumed to be plane behaves
as if it had an apparent (dynamic) curvature which is periodical. The
simple calculation (compare the original report, third part, beginning
of section II) yields accordingly circulation contributions of the two
first fundamental functions, that is, only the following terms:

a0 = 2Bv b0 = 0

a = 2v(A + B bl = 2Bv

Thus except for higher terms and with consideration of the order of magni-
tude of B (see below) a good approximation is obtained.

All the rest follows readily from the circulation coefficients. I
had introduced the quantity w' in order to enable an easier calculation
and consideration also of complex w', that is, damped or excited oscil-
lations. It is true that for damped oscillations the integrals (10) lose
their meaning, since the oscillation had been assumed to have been going
on for an infinitely long time so that here infinitely Large amplitudes
would have had to precede. However, the formulas obtained may be retained
as approximations, provided a correction is made for the starting process.
For the following consideration of the steady oscillations, I continue the
calculation for reasons of physical clarity with the real frequency a).

Lift and moment are purely periodical quantities, that is, their
temporal mean value is 0. Nevertheless, their amplitude and their phase
angle are of interest. According to formula (6), there is

K = pv2(Ak, + Bkp)eiwvt = pv2t(Ak + Ck)eiwvt

= pv2er(At- + C )evt

M = pv2g(Amz + Bm)eiwt = pv2t(Am, + Cm)eiWVt

= pv2W(.Ai + Cm)einvt

NACA TM 1364

There in

B = Cw; k = ako; my = mnp; km = u4ka;

mia = ua ko = ki; mp = m.

I have introduced here the quantity C as new amplitude of rotary oscil-
lation. This was done because in case of ordinary flapping motions
B is of the order of magnitude uA as a simple consideration shows.
The value of w is assumed to be small; thus A and C will be of
the same order of magnitude so that calculation with C instead of B
will be more convenient in practice. The third form of the air forces
permits, for cases of equal stroke velocity Aieiw t, comparison of
these forces in a simple manner; the roughest approximation theory would
yield constant coefficients ki = 2i, ky = 2. The coefficients k and
m are complex numbers the constituents of which are given by the fol-
lowing series developments. It is noteworthy that the series for m
are finite.

kc = k' + ika"; ma = min' + ml"; a = c,P,7

adk' = ka' = -(1 2Z)u2 2m21og w 2nZu3 + 2nvulog w + .

OuK" = k" = 2m2 22 +Z)w3 a g (16)

2a31og2w + .

NACA TM 1364

k =k

2(1 2Z)w21og w 2w21og2+A + Z 3rZ23 +

n(l 6Z)w3log w + 3=u31og2m +

kB" = =


= (3 2Z)w + 2w log w i(l 2Z)w2

2rra21og c +

( 2 -2
^n -

/ + + Z 2Z -
/n + + Z -
44 2 2

1 + 2Z 6Z2)walog w (1

2+ 2Z3 J3 +

- 6Z)3log2w -

2w31og3w + .

= 1 =9
mm = rig = 2 1

m '-= 5 =^ y' =- a-)2

mp" =-m" 1
g rn U W

It suggests itself to represent k. and m% by the initial posi-
tion of "time vectors," visualized as rotating, in the complex number
plane (as customary in alternating-current techniques) and to combine
from them with the parameters A and C (of which A may be assumed
real without impairing the generality) linearily, in the known manner,
the amplitude coefficients k = Ak, + Ck7 (and correspondingly m)
according to magnitude and phase. The diagram (figs. 2 and 3) shows the




= = 2 + (- + 2Z 2Z2 2

NACA TM 1364

curves of the end points of the vectors k and m as functions of the
parameter w. In the representation of the curves for k and B which
would yield the most accurate values for the graphical evaluation, the
curves of the various approximations have been plotted side by side for
comparison of the convergence of the series. Of particular interest is
the case where the wing without being affected by significant air
forces glides over an undulating streamline course, clinging to it
as much as possible. To this corresponds the parameter C = -iA, which
in fact yields small air forces of second order in m, namely
1 k' = + k"= ZZ +. .

1m, = 1 2 1m" = 13
A 2 A 8

3. The induced drag.- The induced drag is no longer linear in the
circulation so that the complex method could not be retained without
new stipulations. It offers no longer any simplifications, and it is
advisable to continue from here on the calculation with the imaginary
constituent of all quantities in real form. If the calculation is
carried out according to equation (7), W assumes the form

W = WO + W1 sin(wvt + qPI) + W2 sin(2cnt + q2)

Here W1 is different from zero only when the oscillation is superimposed
on a constant angle of attack different from zero. W is in a small,
of second order. The purely periodical terms are, therefore, hardly sig-
nificant; however, the temporal mean value WO, which is different from
zero is important. I calculate only this value and write for it, for
reasons of simplicity, again W. I equate

A = A' + LA", B = B' + iB", C = C' + iC"

and may assume, without restricting the generality, A" = 0. Then W
becomes a quadratic form in A' = A, B', B" (or A, C', C"): a simple
deliberation shows that the coefficients of B'2 and B"2 (or C'2
and C"2) are equal and that the coefficient of B'B" (or C'C") is
zero. Thus W becomes

W = pvA2 A2wa + 2AB'w,' + 2AB"wp" + (B'2 + B"2)wpp

= pv2 A + 2AC'w',' + 2AC"w'7" + (C'2 + C"2)W1

NACA TM 1364

The Wik are again series in aw(m log w)m and have up to higher terms
the values

'w. = _-2 + (3 7 2 Z2) 4 2Zlog + w4log2w+ .

,p b1 1 (3 2Z)w2 121og w +

S(3 2Z)w3 + a31og w +
4 2

=1 = + 3 2 -5 n2 Z2
af aa02 4 "=\8 2

1 (1 2Z)(u31og w = 1 uylog2c +
2 2

1 = = 2 + 2 (32 +
S-0w0 7 = 7 W (n + l)j + + 3 +
8w -

+ z)w3 +

2 4Z2 4Z))u3 +

S( + 2Z)w31og a 1 w31og2w + .
2 2

W may be positive or negative, according to the selection of parameters.
Depending on the type of motion, one has, therefore, to expect drag or
thrust. I postpone detailed discussion until after calculation of the
power requirement and the efficiency. The case of gliding over an
undulating flow mentioned above, corresponding to C' = 0, C" = -A,

W = A2pv2n 0log Wr ( > 0

Thus the selection of parameters made does not yet correspond exactly to
the case W = 0; a small correction would have to be provided for this


NACA TM 1364

4. Work done at the wing.- In the free vortices behind the wing,
energy is contained which must be produced by mechanical work on the
airplane. This may be done in two ways. The flapping motion may result,
as mentioned before, in positive or negative thrust. In the case of neg-
ative thrust a propeller which overcomes this and all other resistances
to flight is required for maintenance of equilibrium of motion. In the
case of positive thrust a propeller is needed only until the thrust due
to flapping exceeds the resistances to flight, whether the flight be
uniform or accelerated. As to the work performed at the wing itself, the
wing motion consumes, of couse, energy if thrust exists; the ratio of
thrust power -Lw = -Wv and the total mechanical power Lf to be applied
to the wing may then be denoted as aerodynamic efficiency of the flapping
wing. In case of drag, two more possibilities exist. First, the flapping
motion may require additional work. The efficiency defined above then
becomes negative and arbitrarily large when the wing power Lf decreases
more and more. Second, the case may occur that Lf becomes negative,
that is, the wing then is supplied with energy from the air (indirectly
by the propeller) and may use that energy for surmounting the resistances
in the oscillation mechanism, or may store it in the oscillation itself,
that is, increase its amplitudes. Aside from this "incremental power"
the propeller must, of course, in this case yield additionally the energy
of the free vortices so that one may define the quotient of -Lf and
the total propeller power as efficiency referred to the power absorption
of the wing. This is then exactly the reciprocal value of the efficiency
defined above which in this case, as the quotient of two negative numbers
becomes positive but larger than one, thus loses its physical meaning.
Lf is divided into two parts. One has

S= p + px = iwn(A + Bx)eirvt

Therewith Lt = Kp becomes the "flapping power" and Lr = Mp the power
opposed to the rotary oscillation; thus the wing power is Lf = Lt + Lr.
The power opposed to the drag is denoted by Lw = Wv. In the air there
then remains in all L = Lf + Lw = Lt + Lr + Lw. I indicate of all Li
again only the temporal mean values for which I obtain quadratic forms
of the same type as for W

Li = pv33{A2liaa + 2AB'impa + 2AB"ic0,, + (B'2 + B"2) Zip

= 3 ia + 2, + 2(22)
= pv3 A2 + 2AC 11 M, + 2AC",, 1 + (C 2 + C"2) l Y) I

NACA TM 1364

Lr has finite series and is, in general, small compared to
coefficients are individually

Zwik = Wik

St = : 2 + 3 +


z2 4

Z 1 = y = (3 2Z)w2
taa i w tay 4

+ 2Zi log

S- wlog2w .

+ 2 log w -

S(1 2Z)w3 a w3log

1 1
Icl 2Z)IIlog

( ( Z)w3log

A 2 + (+-Z-

w w3log2w +

1tpp = -2 zt77 = 0

2 =0

B r7' =y

1 1 3
2 r c r 7" =

S = 1
r3~ =7

Ir77 =

1 z2






-8 -

NACA TM 1364

w w41og2w + .

zfp, =- Ifay' = (3

- 2Z)32 +1 2log a -

S(1 2Z)w3 1 w3og
4 2

11 = i 2
2fa.n 2 14.bU)

(1 2Z)w31og


1 1 2
=fpp = Ify7 =

uD3 2
Caa 2 2

1 3
Z4 = I,, = a +...
43' =- a =- 2

'a3 = 1- la7,o
a co

S =-I =-i -
) U2 77 2

= 2 2 3 +
2 2

(3 3

n2 2

+ Z2
4 2 2

(1 + 2Z)m31og w3 og2w + .
2 2

Ifan = (2

- c03 + f^
2 \

+ 24og
+ 2Zjn log

2 =

Z2) )3 -



a3log2) +


- 2)W4


NACA TM 1364

L, as the energy of the vortex trail, can of course never become nega-
tive and must therefore be a positive quadratic form definitive in the
amplitudes. The fact that L in the form here noted is capable also of
small negative values is caused by the neglect of higher terms in the
series development. What is obtained in this case, is therefore only the
error, accidentally negative, of the almost vanishing vortex power.
Since, when C is used, all coefficients contain the factor u2, the
latter has been cancelled in the graphical plotting which is in agree-
ment with the presentation of the power for constant flapping velocity
(fig. 4). The curves show that the essential terms always stem from the
stroke amplitude A, possibly in combination of the latter with the
amplitude of rotary oscillation, and that the corresponding coefficients
increase somewhat more slowly than w2. By rotary oscillation I meant
above an oscillation about the z-axis. More generally, every oscillation
where the ratio of A and C is real is a rotary oscillation about the
fixed axis with the abscissa a where A then is A = -aCc. It is
shown that for not too large values of aw, that is, for axes which do
not lie at too great a distance, W and L are always positive; that
is, it is not possible to obtain thrust or power absorption by rotary
oscillations about fixed axes. Production of thrust always requires a
stroke amplitude different from zero, power absorption requires addi-
tionally a rotary oscillation lagging by about 900. Pure stroke oscil-
lation without rotation also produces thrust which in roughest approxi-
mation results as -W = A2pv2Ma2, similar to the so-called Knoller-Betz
effect (if one calculates with the y-axis as the "polar" in the plane

One obtains good insight into the variation of drag and power if
one varies, for fixed absolute value of the amplitude ratio C/A = c,
only the phase angle cp between the two oscillation components where
one then has to put

C' = Ac' = AlIc cos q, C" = Ac" = AIcI sin (

W is shown to become a minimum, the thrust thus a maximum, when the
rotation leads by somewhat more than 900, namely by ( = arc tan--.

It is plausible physically, too, that the thrust will assume large values
precisely then when the phase is shifted by about 1800 compared to the
phase which is present for gliding free from air forces over the wave
course. For every Icl there exists an w and vice versa for which
the thrust maximum is absolute. One then has

18 NACA TM 1364

w Y '

w rY


Wmin = A2pv2 Wmin

"min = w on -

7,2 + wy,,2

- ---2 +
2r 2n2 2,3

3 2). .

L becomes a minimum and disappears for suitable w except for terms
of the fifth order when the rotation is lagging by somewhat more than 900.
Finally, the wing power Lf becomes a minimum thus, the absorbed power
a maximum when the rotation lags by somewhat less than 900, namely for
fct "
p = are tan This maximum too becomes absolute when between c

and a) the following condition is satisfied

c'(W) = f'y

c"(W) = -
SA 77

Lfmin = A2 P "fmin

fay'2 + 2

Ifmin = 1faa.

-1 + i +

Altogether, L becomes negative only when Ic| > 1.

NACA TM 1364

The efficiency

1 Lw L 1i is valid for Lf > 0 and]
1 = =- = 1 where (28)
=2 Lf Lf 12 for Lf < 0. J

is the quotient of two quadratic forms and capable of a great many values.
Generally, it becomes negative for small or completely vanishing stroke
amplitudes, and arbitrarily large with w--O. The same is valid for
rotations about a fixed not too remote axis. If, however, A 6 0 and
also lim A A 0, n approaches for w --~0 to the limit 1.

By a rotary component (c" = 0) of equal or opposite phase q is
always deteriorated compared to c = 0; the same is true for c' = 0,
c" > 0. In contrast, the efficiency is improved for c' = 0,
-1 $ c" $ O, as can be seen from the diagram (fig. 5). For c' = 0,
c" = -1, T becomes identically 1, and for c' = 0, c" < -1 there
results power absorption. The representation for fixed Icl in depend-
ence on C as a discontinuous single-wave-harmonic function is very
graphical for the efficiency as well. Figure 6 shows clearly at what
phase angles the transition from power absorption to power production
takes place. The most important ones among the coefficients found from
the series developments have been compiled in the numerical table
(table I).

5. Application to the flutter of elastically supported wings.- The
derived laws could be practically applied in the investigation of a
phenomenon our pilots observed in the last war. In the so-called sesqui-
planes, the lower wing was fastened to one single spar only, thus was
only slightly elastic against small deflections and rotations. In case
of increased flight velocity, for instance in steep dives, there occurred
sometimes vigorous flutter of the lower wing tips which underwent obviously
unstable oscillations in the increased air flow. Of course, such unstable
oscillations are possible only if energy is supplied to the oscillating
system, and this occurs, according to my investigations, only when the
vector of the amplitude of rotation lags by about 900 and when the ampli-
tude of rotation itself is sufficiently large (Ic must be > IAI). Let
us visualize again the gliding almost free from air forces of the wing
over an undulating flow course where the airspeed (relative to the wing
elements) has no vertical component. If the amplitude of rotation is
smaller than corresponds to this case, the motion is damped by the
counteracting air force. If the amplitude of rotation is, on the con-
trary, larger than in the case above and the wing therefore scoops more
deeply into the air, the air force always acts in the direction of the
motion, and the motion is excited.

NACA TM 1364

I consider a wing supported on spars, elastic with respect to trans-
lation in the y-direction and with respect to twist. In order to be able
to go on from my formulas used so far, I introduce the directional forces
per unit length in z-direction; I calculate therefore as if these forces
were distributed continuously over the length of the wing. This assump-
tion does not lead to any contradictions if the wing in itself, aside
from its support, is sufficiently stiff. Since I disregard deflections
in the flight direction, I can show that a wing supported on spars always
has only three essential elasticity parameters, corresponding to the
three constants of the work of deformation quadratic in p and cp.
(Compare fig. 1 and the original report.) With respect to its elastic
properties, this wing may therefore always be replaced by a wing which
is supported only on one spar (the "elastic axis") with the abscissa a,
elastic with respect to translation by means of the directional force c,
and with respect to rotation by means of the directional moment 7, as
schematically indicated in figure 1. The resultant of the elastic forces
and its moment at the origin then are

K = -c(p + ar)

M = -cap (ca2 + y)r

If s is the abscissa of the center-of-gravity axis (point S,
fig. 1), the momentum and its moment at the origin have the values (with
the mass m per unit length and the corresponding moment of inertia
9 = mr2 for the center-of-gravity axis)

C = m(p + sp)

U = msp + (ms2 + d)q = ms + m(s2 + r2)

The equations of motion of the wing then read

S= K U =M

For K and M the air forces have to be added. If I put as before

p = Aem'vt p = Bew'vt

NACA TM 1364 21

and take into consideration that the air forces have on the wing the
opposite effect from the one the forces of the wing calculated above
have on the air, K and M become

K = -e'vt jcA + caB + pv2n(Ak. + Bkp

M = -e'vt caA + (ca2 + 7)B + pv2t(Amm + Bmp)

For abbreviation, I introduce the following designations

Po c 2 7 2
mT = o2 =

Therewith there result finally as the equations of oscillation

A(cu'2 + aO2 + poka) + B(sw'2 + amO2 + pokp) = 0

A(sw'2 + a +b2 + P0m) + B((r2 + s2)w'2 + (a2 + q2)(+2 + Pnmp) = 0 (29)

Equating the determinant to zero yields the equation for the fre-
quencies w = -io'. If I first disregard the forces (for this purpose
I put pO = 0), there result two main oscillations as a consequence of
the coupling of the oscillation of the center of gravity with the fre-
quency o0 and the oscillation about the center of gravity with the
frequency With b = (s a)+2 + q2 there is

2 12 2 b 2 + 4q2r2)
l.2 1.2 2r2

A s( + t 82 4q2r2) 2ar2 (30)
B 2 4q 2r2 2r2 -al.2

yk = Bk(x ak)eiwk't; k = 1.2

NACA TM 1364

The main oscillations therefore are rotary oscillations about fixed
axes at the distances al,a2. The quantity s a forms a measure for
the coupling. In general, there develop beats from both main oscilla-
tions. If I write those in the form

y = i-aiBleiVt a2B2e-iC^ + x [leit + B2e-tcvt] ei

01m + m2 = 2O

cmL ag2 = 2E <<

The motion may be interpreted as an ordinary oscillation with an ampli-
tude ratio slowly variable as to magnitude and phase. Therein there
appears of course, periodically recurrent, the phase angle which corre-
sponds in the air flow to the power absorption. If the air forces are
to counteract the change of this phase angle, corresponding to con-
tinuous supply of energy, it can be shown that the case of slight coupling
(s a small) for balanced or almost balanced frequencies of the uncoupled
system is the best presupposition for this phenomenon. I am anticipating
from the results of the following calculation with consideration of the
air forces that without coupling (s = a) no unstable oscillation at all
would be possible.

The air-force coefficients occurring in the oscillation determinant
are themselves functions of the frequency w' which cannot be indicated
by simple analytical expressions so that the roots of the oscillation
determinant cannot be obtained in a simple manner. However, since their
existence is secured by the physical meaning of the problem, it is per-
missible to introduce into the equation instead of the coefficients k
and m, the first terms of their series developments, all the more so
since the occurring factor pO generally is a small number; for it is
sensible to break off the series for cos x in order to find from the
polynomial obtained for instance approximately the first zero of this
function whereas the same method is meaningless for ex, since no zeros
of this function exist in the finite domain.

The oscillation determinant obtains the following form

r2 t4 + +O2(O2 + q24 + '2 2 + psO2 2 2)k0 skg smn + m]I +

pW 2a + 2) k ak -am, + ] + P2(kmp kp _) =0 (31)

NACA TM 1364 23

The roughest approximation is the result of the calculation which
bases the determination of the lift and moment coefficients in the
manner of the theory of the Knoller-Betz effect on the momentary
apparent angle of attack and the apparent (dynamic) curvature of the
wing. Even with this procedure, there result in certain cases complex
roots u' with positive real parts which result in an "increment" of
the oscillations and thus correspond to dynamic instability. I shall
not here discuss this approximation more closely. Also, I shall only
briefly mention the case of the greatest instability since this case
is trivial. It occurs if the elastic axis lies so far to the rear that
at the slightest displacement of the wing from equilibrium position the
air flow simply causes the apparatus to tip over periodically toward
the rear. This always occurs as soon as a is positive and the air
2 o2
velocity sufficiently large, namely for a > -- or v2 >
2p0 2pna
for a > 0.

If I retain of k and m all terms up the second order in o',
the period-equation of the wing becomes

f(w') = au'4 + ao'm' log U' + a O'11Wlog2' + b0w'3 + bo'w'3lpg u' +

CU',2 + O'mw'21og w' + Co"w' 21g2 + dOU' +

do't' log u' + e0 = 0 (32)

With the coefficients

a0 = r2 + P + P2 Z2) > 0

bo= POl + P02 2> 0

C = m02 + P) 2p0s + p02

dO = o02PO > 0

NACA TM 1364

eo = 02 (q22 2pa)

aO' = po[2(r2 + s2)

- 2s(l 2Z) -


a0" = -P0(2s + PO)

bo = -Po(2s pO)

Co, = 2po2[2 + q2 a(l

ll = do' =

-2p0oao 2

= (s a)2 + r2 + q2 > 0

i = 1 + 2(r2 + s2)

X = I + 2(a2 + q2)

- s(3 2Z) > 0

- a(3 2Z) > 0

e = (r2 + s2)(1 2Z) s(l 2Z + 2Z2) +

v = (q2 + a2)(1 2Z) -

a(1 2Z + 2Z2) +

Z = 0.11595 .

- 2Z)



NACA TM 1364

6. Numerical evaluation.- The equation can be solved only approxi-
mately, of course. For this purpose, I first omitted the logarithmic
terms and determined the roots e' of the algebraic equation

g(-') = ao'4 + bOw'3 + cO'2 + do-' + e0 = 0


I regard this equation as an approximation, equate w' = m' + X, and
can now develop the logarithmic terms retaining the linear terms in X.
Thus I obtain

X = -;' log aW'


R = aoc3' + a "m'l3og + b'' + + do'(1 + 0' log i')

S = 4ac '3+ ao'a'53(l + 4 log W') + 2ao"W'31og W'(l + 2 log i') +

3bo0'2 + b0o''2(l + 3 log 5) + 2c0l' + Co'i'(1 + 2 log M') +

do + do'(1 + 23' log 3')(1 + log ')

The procedure may be continued and yields the further approximation

R(-w')w' log w' + g(w')

Finally, there follows the complex amplitude ratio B:A = b from
one of the equations (29).

NACA TM 1364

Finally one now has to learn the conditions for which the equation
for w' has complex roots with positive real part, corresponding to
excited oscillations or critical support of the wing. For the limiting
case of dynamic indifference, I formulate the roots of the equation (32)
as purely imaginary and obtain, by setting the real and the imaginary
constituent of the equation equal to zero, the conditions

(a0 aO'") + ao'w log w + ao",4log2w + b' 3 -

0c CO' )2 co'a2log w dogm + w log2c + e = 0


ao' 3 + ao"0o l og a b&2 bo'w2log -

CO' 0 (J + do' log w(l nv) + dO = 0

From them w would have to be eliminated in every special case
whereby a conditional equation for any of the parameters s, a, r2,
q2, w0, p0 or for the flight velocity v results. As an approxi-
mation, it is sufficient to investigate the equation (33). Since the
approximation (34) shows in the case of instability, a small additional
damping, the criteria for instability derived from the following equa-
tions are necessary but not always sufficient; they are therefore valid
only with the reservation of checking with the approximation (34). When
they are satisfied, the wing may at any rate be regarded as critically
supported. Before I set down the most general criterion for instability,
I shall mention one which is very simple but suffices only for cases of
great instability. This is the condition that the coefficient co becomes

> c(5 + POv)6)
p 2 > s o
pi(2s pO)

NACA TM 1364

Written as a function of s, the same condition reads

s2U2 2s (P + a2) + 02 + u2 (a2 + r2 + q+ < 0

Further limits are yielded by the Routh condition

Co < a0 + e0
0 0 d


102 (B + + P02 2p~s < [r + Po +

/\1 PO 2 2 + P 0(j z)
p02 + Z Z+ q 02- +2p ) -
8 4 + POO1 z

In order to satisfy this equation, it will above all be required
that v be sufficiently large, that is, u0 sufficiently small. Fur-
thermore, s must be positive; therewith s a is positive, too,
because, as mentioned before, a > 0 would for sufficiently large v
always result in great instability. One can readily understand that
this must be so. The centroidal axis of the wing as axis of inertia
generally lags behind the elastic axis as line of application of the
directional force. If the centroidal axis therefore lies, in flight
direction, behind the elastic axis (s a > 0), the wing has in its
upward motion, on the average, a positive angle of attack; the opposite
is true for the downward motion. Thus the air forces always take effect
in the direction of the motion and amplify the oscillation; whereas in
the case s a <0 the opposite, that is, damping occurs.

It can easily be confirmed that both degrees of freedom must act
together for achievement of the oscillation. The calculation always
yields roots with negative real parts if one of the degrees of freedom
is suppressed (corresponding to aO = m or q2 = =). This fact is
confirmed by the failure of tests undertaken formerly in Gittinger with
a wing with only one degree of freedom.

NACA TM 1364

The power L produced or, respectively, absorbed by
according to previous formulas

the wing is

Lf = A2p7v 3f

2f = Ifa= + 2b'lfap +

If I put temporarily w' = 3 + iw, the mean
wing energy for constant amplitude A is


value of the entire

1 2 2
E = I mvy A Q

Q = j2 + w)2 + 2b' (amjL2 + s2) +

(b'2 b"2) [(a2 + q2)u2 (r2 + s2)W2]

A = Agei3t, and hence, in a different form

-Lf = d =1 v2AAQ = mv3A2Q
dt 2 2

if = 2 O
f Q


If the oscillation calculation has been carried out, the agreement
of the values If found by different methods offers therefore a control
for the calculation.

7. Comparison with a test result.- For confirmation of the theory,
tests with a light wooden wing of 60 cm length and 10 cm chord were
performed in the small wind tunnel of the GWttinger aerodynamic test

NACA TM 1364

This wing was suspended on an axis between plane side walls by
means of springs. The springs acted on cross-shaped metal sheets
which carried moreover sliding weights for variation of the distance
from the center of gravity and of the moment of inertia. The suspen-
sion springs or, respectively, their initial compression transferred
the directional force or, respectively, the directional moment corre-
sponding to my formulation to the suspension axis. I shall not discuss
here further details of the apparatus; I refer instead to the drawing
figure 7. The purpose of the tests was the determination of the incre-
ments or decrements of the oscillations; for this it was sufficient to
plot the oscillation of a point, for instance the suspension axis.
This was done with the aid of a registering drum kindly put at our
disposal by the physiological Institute of the University Gottingen.
A few of the diagrams of damped and increasing oscillations thus
obtained are reproduced in figure 8. From these diagrams the ratios

a = e of amplitudes succeeding one another were determined and were
compared with the values found by calculation with consideration of the
frictional damping produced by Oot and aor of the translator or
rotary oscillations. Within the considerable limits of error which
could have been cut down only by a major expenditure of time and means,
the agreement may be called satisfactory.

The variation of the numbers was in excellent agreement with the
theory. An increase or damping of the small oscillations resulted
according to whether the wing was overweight toward the rear or the
front; the ratio of amplitudes succeeding one another increased when
the overweight was increased or when the air was blowing more strongly
against the wing. A precession of the leading edge of the wing was
clearly evident when the wing was supported so as to be unstable. This
corresponds in the usual notation to a lag of the rotary oscillation
as must be the case for power absorption.

Finally, it is noteworthy that for small air velocities of about
5 m per second the influence of the viscosity was shown by the fact that
the lift was not yet fully developed at small angles of attack. For the
smaller characteristic values of about 500 m/sec. mm a symmetrical wing
has a small range "dead angle of attack" where the lift remains near
zero (as shown also by other tests in Gottingen). The small oscillations
were therefore still damped when oscillations with a larger initial
amplitude were already increasing. The limiting amplitude lying between
these two cases decreased of course more and more with increasing v.

Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics

NACA TM 1364



W = 0.02 0.04 0.06 0.08 0.10 0.12

k' = 0.00262 0.00773 0.01351 0.01892 0.02317 0.02572
ka" = 0.03852 0.07387 0.10605 0.13532 0.16182 0.18601
k' = 0.03860 0.07449 0.10806 0.13982 0.17029 0.19989
k" = -0.00184 -0.00468 -0.00694 -0.00788 -0.00710 -0.00452
mro' = -0.0002 -0.0008 -0.0018 -0.0032 -0.0050 -0.0072
mn" = 0 0 0 0 0 0
my' = -0.000003 -0.000024 -0.000081 -0.000192 -0.000375 -0.000648
my" = 0.0004 0.0016 0.0036 0.0064 0.0100 0.0144

a-2waa = -0.9336 -0.8684 -0.8073 -0.7514 -0.7014 -0.6576
m-2w = 0.02369 0.03209 0.03480 0.03419 0.03150 0.02752
-2wby", = -0.4529 -0.4094 -0.3707 -0.3371 -0.3089 -0.2862
L-2y77 = 0.02924 0.05414 0.07468 0.09088 0.10273 0.11024

Lu-2a = 0.02944 0.05494 0.07648 0.09408 0.10773 0.11744
or-22,1 = 0.00063 0.00251 0.00566 0.01005 0.01571 0.02262

w-21 0.02944 0.05494 0.07648 0.09408 0.10773 0.11744

W-21fa = 0.9631 0.9233 0.8837 0.8455 0.8091 0.7750
a-2. f, = -0.02306 -0.02958 -0.02914 -0.02414 -0.01579 -0.00490
W-21fc 7 = 0.48235 0.4643 0.4471 0.4312 0.4166 0.4036
-2Zfyy = 0.0002 0.0008 0.0018 0.0032 0.0050 0.0072

NACA TM 1564

Figure 1.- Elastic suspension of a Joukowsky profile with plane "spine."


1 1.5

32 NACA TM 1364

Figure 2.- Vectorial lift and moment coefficients for equal beat amplitudes.

NACA TM 1364

Figure 3.- Vectorial lift and moment coefficients for equal beat velocity.

NACA TM 1564

Figure 4.- Power and drag coefficients for equal beat velocity.
(1) w-2ffa; (2) w-2,fac-'; (3) fifa'"; (4) ~lf,
(5) a-2Waa; (6) '-2wo-,'; f7) ,,~wa"; (8) ("W?-7.

NACA TM 1364

Figure 5.- Degrees of efficiency as a function of w for different amplitude
ratios c.

NACA TM 1364

Figure 6.- Degrees of efficiency as a function of the phase (p.

NACA TM 1564

Sliding weights

;ross-shaped metal sheet

Spring joint

0o 0I

Figure 7.- Test arrangement.

NACA TM 1364

2)V\JV\AAM ^ -



20)Figure .- 21) sci

Figure 8.- Osci.


llation diagrams.

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