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NASA TT F-14,576 


W. Roth 

c o P -* 

Translation of: "Eine Theorie zur Berechnung von Flatterschwingungen 
bei Unterschallstroemungen," Acta Meehaniaa, Vol. 6, 
pp. 22-41, 1968. 



NASA TT F-14,576 

W. Roth 

ABSTRACT. Calculation of a critical flutter velocity for a 
membrane in a nonconservative system. It is shown that the 
damping forces have a destabilizing effect and that a dis- 
continuity appears in the stability criterion which depends 
on the damping coefficient and is characteristic of noncon- 
servative systems. By means of this theory, the flutter 
of weather vanes or sails, as well as the related traveling 
transverse waves, can be explained. 

I . Introduction . /22_* 

the calculation of the flutter oscillations of a thin profile or a plate 
without flow may be accomplished using the singularity method by means of a 
fictional vortex layer that changes with time. The calculation is relatively 
complex and vague; it eventually leads to the problem of the solution of an 
integral equation. Kuessner [1], Schwarz. [2] and Soehngen [3] performed 
calculations of flutter oscillations in 1936-1940. The calculation would be 
much simpler if there were a pressure law that would describe the reaction 
of the flowing medium on a plate element as an explicit function of the 
deformation of the elements. Such a pressure law would have the advantage 
that the problem of calculating the flutter oscillations would be separated 
from concepts of aerodynamics and become a pure problem of technical oscilla- 
tion theory. Now such a pressure law is available for supersonic flows, /23 
provided by Ashley and Zartarian [4] according to the "Piston Theory." One 
is therefore led to wonder whether or not such a pressure law could be 
obtained for incompressible flows as well. 

x The present paper was delivered in abbreviated form by the author at the 
GAMM Congress in Zuerich (1967), under the title "A Contribution to the 
Calculation of Flutter Oscillations." 
*Numbers in the margin indicate pagination in the foreign text. 


II. Derivation of the Pressure Law . 

In the following discussion, it will be assumed that we are dealing with 
incompressible flow which is essentially friction-free. The flow around an 
oscillating plate will be compared with the flow inside an oscillating tube 
through which a medium is flowing! Such tubes have recently been studied 
in connection with their stability [5] , and the results of the calculations 
are also in agreement with the phenomona that occur in technology. In order 
to understand the forces that are created at the tube wall by the flow 
velocity v of the medium in the tube and those causing the movement of the 
tube, see Figure 1. A medium element with relative velocity v^ = v is 
moving through the oscillating tube element with a deflection z(x,t). In the 
kinematic sense, the tube element constitutes a vehicular element, toward which 
a medium element moves with relative'velocity v . A familiar principle of 
kinematics [6], however, says in connection with such a relative movement 
toward a vehicle that the absolute acceleration is composed additively of 
three components, so that ', 

* - V +b t + v 

where b f is the vehicular acceleration, b is the relative acceleration, and 

b is the Coriolis acceleration. Since in the present case the motion of the 
c : 

tube elements occurs primarily in the z-direction, b f = z . The relative 

acceleration at constant r consists of the relative centripetal acceleration 

b =v 2 z . The Coriolis acceleration is, calculated as follows /24 

r xx | 

b c * 2 (uj >i v r ), 


so that with an angular velocity of the vehicle to = z we will have 

b = 2 v z . If we let q equal the force exerted by the tube wall on the 
c xt 

flowing medium, based on the unit length of the tube axis, the motion equation 
of the medium element in the direction of the instantaneous perpendicular 
to the trajectory plane will be 

q dx = y p (v2 Zxx ^ 2 v z xt + z tt ) d x, 

from which we obtain as the reaction of the flowing medium against the tube 
wall, the line load q as follows:,,. . 

y F (v 2 

z +2 vz . + z..) 

XX xt tf 

(II. 1) 

Figure 1. Kinetic Values For a Tube 
Element with Flow Through It. 

In this pressure law, y through 
the internal cross-section of the tube 
according to the filamentary flow 
theory applied here is a known mass 
of the flowing medium directly in- 
volved in the transverse oscillations 
of the tube, based on the unit length 
of the tube axis. As far as the tube 
is concerned, there is no objection 
to using (II. 1), at least when a 
very thin tube is employed, for which 
the ratio r/L composed of the tube 
radius j of the internal cross section 

and the tube length 1 is much less than; 1. A theoretical expansion of (II. 1) 
occurs which is valid with respect to technical applications as well, if the 
right-hand side of (II. 1) is increased by a damping factor, so that we have 

M F (V 

Z x* + 2 V Z xt + Z tt> + 6 V (H - 2) 

Such an assumption of damping with damping factor <5 Q is necessary in the sense 

of modern stability theory according to Lyaponov [7], and to a certain degree 

is even necessary, as we shall see later on in the discussion of the results. 

The pressure law (II. 2) can be applied to a plate around which a medium is 

flowing if q and y are based on a unit area of the plate and weight per unit 

area u r is assumed known. The pressure law given by the "Piston Theory" for 

supersonic flows, in comparison with (II. 2), is 

6 1 v z x + 6 V 

(II. 3) 

where 6, <=6 n = <p /c represents a coefficient determined by the medium; 
1 °° °° 

here < is the polytropic exponent, p ro and c w are the pressure and speed of 

sound in an undisturbed medium. It is advantageous for further calculations, 
however, to ensure a discussion of 4 the .results that is as^omplete as possible, 
so that 6 f 6.. In (II. 3) the deflection of the plate element is 'contained 1 
in only one term with the derivative zr, the second term in this instance also; 
representis damping. • i 

III. Calculation of the Critical Velocity for a Membrane with Rectangular 
Boundaries . 

We shall now use pressure laws (II - 2) and (II. 3), for comparison of the 
results they yeild, in the case of a membrane (Figure 2) with rectangular 
boundaries subjected to flow in the x-c.irection at velocity v. The membrane 
is assumed to be stressed in the x- anc y-directions by forces S. and S 
based on the unit length, and these forces are further assumed to be positive, 
pressure forces. Since the edges of the membrane are assumed fixed, the 
desired solution z (x, y, t} for the mqvements of the membrane must satisfy 
the boundary conditions 

z (x, 0, t)_-.p ...: 
z (x, d, t) « 

In addition, in the sense of the stabi! 

... z.ttCy. t) = o__ _ 

V ^ n (III.l) 

z ;(c, y, t) =0. 


ity study at time t = 0, the initial 

z (x, y, 0) = £ (x, y) | z (x, y, 0) - * (x,y) (III. 2) 

may be given for the initial deflectiori and the initial velocity, where the 

functions <j> (x, y) and $ (x, y) are rar.dom within rather broad limits. If 

the pressure law (II. 3) is initially ajplied to the membrane, we will obtain 

the partial differential equation 

S. Z + S Z + 
1 xx 2 yy 

W Z tt + & 1 v z x + 6 Z t 

(III. 3) 

as a motion equation, in which y is 1;he weight per unit area of the membrane;. 
It is not necessary to assume an additional external damping in this equation j 
which is proportional to the deflectiori velocity z ; instead, this can be j 


thought of as included in the damping factor 6 Q since 6j t 6 Q and the two 
numbers may therefore be thought of as independent of each other. With the 

7rmy rm _i9-z ^ (III. 4) 

u(x, t) sin —r- (m = !> l > 6 > ■••> y 


z = uix, tj sin -5 
fill 3] is changed to the partial differential equation 

S l U xx + MP U tt + 6 1 V U x + 6 U t " S 2 ~^~ U = °" 


This equation can then be solved by means of the separation expressi. 

ixx igt (III. 5) 

u = e 

where, with maintenance of the boundary conditions in x according to 
(III.l) and summation over all existing particular integrals the following 
solution can be obtained: 

,V' r 


<» i » > (III. 6) 

Here c and c, are freely selectable constants, and the characteristic 

' Imn 2mn 
exponents of the time are therefore 

,> i \v V* * s '. I • T '"l s ; ' s '' f •""' | 2 (II 1. 7) 

These exponents describe the behavior of equation (III. 6) with time. In order 
to develop the solution z (x, y, t) to satisfy random initial disturbances 
(III. 2) according to (III. 6), all particular integrals are required, and 
hence the exponents (III. 7) must be discussed for all combinations of whole- 
-number values of m and n. It is advantageous in conjunction with further 
calculation to consider separately the two different cases 6 Q = and 5 Q > 
with arbitrary 6 > 0. If we initially assume that 5 Q = 0, i.e., if we 
calculate without damping, it becomes evident that instability is avoided 
either because no oscillations developed that become apparent with time or 
(what amounts to the same thing) exponents iS 12mn exhibit no positive real 

P arts > if rTTT jn 

Sj < and S 2 < ( TII - 8 J 

t , > n However, if we assume that S < and thereby 
for all values of 6 > 0. However, .„ sta bility imposes the 

assume damping, the retirement for avordance ^ t b 1 P ^ 
condition tm.S). ^though stabiUty » now ssured and^ ^ 

solution (HI. 6, exists which decreases wrth Um C - ^ 

reoul res validity for all combinations numbers^ ^ » ^ ^ ^ 

„ avoid instability. Prom the physical Stan porn t ^ ^ 
really not very convincing that crrterton (III.8, do s n 

vel ociUes V, and that there is consequent y ^ > ^° J J different ^ 

nutter oscillations of the ^ ra c h ^ ^ ^^ 

movement conditions are stable or unstab ^ ^ 

of the magnrtude of the flow veloc. y v ™ - _ ^^ 

Bolotin [8]. When bending resrstance » lne ^ [4] , [9] . 

t he velocity v of the medium can influence the stabrlrty 

, ftt ?l to the membrane leads to a solution 

The application of pressure ia. (1.2) ^^ 

for the z (x,y,t) which is different from (III. 6 , and 

motion equation for the membrane which is to 

differential equation 

(III. 9) 

(III. 10) 
of the flow medium 

,->•? 1-hp tlOW mcuxum 

„ - u + y is the sum of the weight per unit area y p of the 
Id the weigher »« area , p of the membrane „* the assumpt^" ., 
which sattsfres the boundary cond-ons o . ^ . 
(III.l), we obtain from (III. 9) the parx 

71 II). \ 1 „ 

u = 0, 

which can be solved by means of the separation expression (III. 5) or directly 
by means of the expression 

h (x, t) - F (a .»■ + bt) sin --"-'- (» = 1, 2, 3. . . .)• 

Here, F (ax + bt) constitutes a transverse wave moving through the membrane 
in the x-direction at a velocity 

For the function F, we can find the conventional second-order differential 
equation ' ^ ,, 

-;r tt .+-;/ w - ;w;x**' + f '/«''" 'V Hi '' J i ' H J. 

with the argument u = ax + bt and 

•V* - 'Vf "" 'V • (II 1. 12) 

The exponential expression F = exp iru gives the characteristic exponents 
for the time 

s*\ \ . 1/ V , *■•• i*m» . i:''** 'Vl-- 1 ,'"-! 2 1. (HI. 13) 

t r i 

and the solution of (III .9) then has the form 

- , _•'?■-<,-„„ . --"''--"-(*-*.,, (IH-14) 


<*2 w « ^ 

HI - .-.111 I I ■ I /I* M 

1 » 1 

= (•»•..'/.') =--- £ 2> u r • s " 1 -/ ' (III. 15) 

in - i ii i 

<•,»,» *'l »>„("■'• i l>l) l^nn, F 2m A"-*' 


with constants c^ and c^ which are again random. These constants can be 
established so that the initial disturbances (III. 2) are satisfied. The 

«= b f.y + btl and F (ax + bt) alone do not satisfy the 
transverse waves F lmn (ax + Dtj ana r 2mn icia > 



*~~ , rv v t~\ since although the factor sin 
boundary conditions in X for z (, since i 

appears in the solutions of (111.14) and (111.15). these waves are distorted 
uuasilocally or locally and the result is that the edges of *• -*'«™ 
regain at rest and do not undergo deflection. Solutions (III.6) and (III 41 
as well as (III. 15), however, display exponential dependence on the wave oims 
in x Such wave forms were also given by Sparenberg [10] for an infinitely 
long membrane subjected to flow. Different results were obtained here, 
however, with respect to behavior with time. In order to determine the 
behavior of the solutions with time, the exponents („!..« must be discus by 
analogy with the above. It turns out that instability with vanishing dating 

5 =0 can be avoided if 

' S t < and .Sj* < » 

or N, = and S,** < (III. 16) 

or S t > o and V* < and A\* > « 

or \ r r. s .y~ <v ^ p..'. 

On the other hand, if we assume doping with 6 Q > 0, in order to avoid in- 
stability or implementation of stability we will have the required 

(III. 17) 

We can enjoy only a slight degree of conviction in accepting crlt n C K J 
in contrast to (111.17) because of its inhomogeneous structure which includes 
numerous possibilities and because it also partly contains the *"» *1 
which cannot be explained physically. Hence, fro, the physical and t «h ni» 
standpoint, it is also unacceptable because it was developed without onsidera 
tion of damping and damping forces are always present in reality Th. 
stability theory of Lyapunov offers the clue that the criterion (II 
obtained with consideration of dating may be considered correct. Criterion 
rill 16) can only guarantee avoidance of instability but cannot guarantee 

• u „c a - n was assumed in establishing 
establishment of stability, inasmuch as 6 Q - was assum 

this criterion. However, according to Lyapunov, this is precisely the 
critical case, in which the theory predicts that the smallest nonlineanties 


a A in the present linearized calculation, can lead to 
which were disregarded in the piesen uncer tain which motion 

,•■.-*. ti- i<; therefore, in fact, unceitaxi 

i::;i:rr;:i ;u — « ^-^^cizr 

the motive power produced by the stream pressure « , ^ ^ 

s „ess on the membrane in the x-direct.on «"'"£'" ' md then allo ws 

calculation of the critical flow velocity v R 

,,...-, , - Sl . (III. 18) 

Th e doping factor «„ U no longer contained in th, criterion,^ criterion 
[IU .16, is not always obtained for S „ C J my 

th ere is a discontinuity wrth respect to am -^ ^ ^ration 
criteria. Since criterron (I II. 17) C indicating that the 

f damping, the present result ^ * ^ , destabilil ing 
damping that exists when pressure law (II. 2) » oscil lations 
eff ect on the oscillating membrane and ensures that flutter 
develop for all flow velocities V > V 

instability phenomena, caused by critical --ciUes^can also develop 
ln the case of rudders or single-bladed »,« * < 
mad e by „eidenha«r [11]. - such a system, « * °" iMl 

the critical velocity depends on the darning factor « 
velocity given by criterion (III. IS) is -depen en ^ ^^ 

with respect to the dam p f r J^ ^ ^ ^ ^.^ 

degrees of freedom. Ziegler L i J reS ultant unsteadiness in 

the stability criterion for a aouoie v 

IV. Determination of the Weight Per Unit Area or Pressure Layer . 

The calculation, as performed thus far, is based on a known weight per 

unit area y of the flowing medium. The critical flow velocity v k according 

to (V.10) is dependent on y and the value of y p is likewise dependent on 

the form of oscillation of the membrane. In the following, we shall describe 

a calculation of y„ which is based on the singularity method. According to 


Birnbaum [13] and Glauert [14], the flow around a thin profile (Figure 3) 
can be obtained through interaction with a vortex layer of intensity k(x). 
The induced velocity v (x) at point x, caused by the totality of interactions 
k(x J ) at points x 1 , is given by the Biot-Savart law 

l '»W=^J -(7' -'*)-■ 

x' « o 



Figure 2. Membrane Plate With Rec- 
tangular Boundaries Subject to Flow. 

The expression 

k (.»•) = 2 v\A ctg J + £ A n sin n &\ 

n - 1 

(IV. 1) 

for the vortex application k(x), with 
substitutions x = j (1 - cos©) and 
X 1 = J (1 - COS<f>) (0 < < tt, < <j> < *) 
with consideration of integral values 


-t ■ ,, In =- 0, 1, 2, . . .) 

C eon u (p dip 
J {eon ip - cos (-)) 



1 r„ M 



and the flow condition, becomes 



A + ]£ A H conn 


(IV. 2) 

Figure 3. Flow Around a Thin Profile 
With Application of a Vortex Layer. 


v.njs and wt obtain 


_, .s ■ ,,^-r.--\ : ^\.* ip. •' •■' - 

1 /' ''" Ai-I A -'{''," cos i' ( ' f ,,< "'- 

r, J ,U ( '°- A " » J "' 

<-■> u 

(i it 

The lift, based on the unit length of the profile depth, is calculated according 
to the Kutta-Joukowski lift fo»»la with consideration of the corre.pondxng 
sign determination which applies here and the density p of the medium at d P/dx 
q = - vpk(x). If we equate this lift to the static pressure components 

, , f TT ?1 hv the relative centripetal acceleration 
developed in the pressure law (II. 2) by the reiauv 

(Figure 1), and use (IV. 1), we willhaye 

and thus obtain for the weight per unit .area__ ... 

Thus the value obtained'^ this' fashion for „ F can also be used for the 
terms in motion equation (III. 9) in the sense of an approximation theory 
for describing the flutter oscillations. Since Vf can also be interpreted 
as the weight per unit mass of a flowing medium for a tube with curvature 
,U „ ith flow through it and having a rectangular cross section of width 
equal to 1, division by the density p of the medium gives the 
t he cross section, so that Vp - ph. This height h can be represented 
thickness or the height of a pressure layer, in which the pressure buildup 
on the surface of the membrane subjected to flow is created by the flowing of 
the medium at the velocity v, governed by the centrifugal forces of the 
medium elements in the vicinity of the bent membrane. The flow around the 
Lbrane can therefore be expressed by the totality of an infinite num ber 
"ubes having cross sectional height h and infinitesimal cross sectional width 
a ranged close together and parallel to the X-axis. Hence «»•=«« 
Less of the oscillating membrane with flow over it is attributed primary 
to the interaction between the membrane and the pressure layer or the weight 
per unit area Vf by analogy with the tube with flow through it. The flow 


around the .emhrane plate corresponds to flow through a tuhe with 
resistance E I - 0, or flow throu 8 h a hose [IS], The weight per unit area 

fro, (IV.3J or the thickness h'of the pressure layer will generally vary 
wrth the profile coordinate x pr e. Hence, it is appropriate withtn the 
of an approximate method, such as is developed here to introduce an average 
value for „ p which is still .ore exact instead of Vf and likewise introduce 
an average alue for the pressure layer thickness h as weli. n order t 
ohtain initially an estimate of the order of magnitude of ,.„ and to ascerta.n 
the influence of constant curvature of the profile with a bulge f on the 
weight per unit area, we will assume for the circuLr profile accordtng 

Figure 4 / ,, .> /i\ 

(IV. 4) 

to be the result of the calculation. We 


which has the curvature z 11 = p-> 

„f ja -nr-n-7^4...). Hence, 
will have A Q = 0, A 1 = 4 - and A n - (n - 2,3, 4,... J 

[IV. 5) 

,,,,. :r.= ,., r sin (-), 

as the average weight per unit .area y p we will have 

. /f „„ = ; (,„,/,•=-■: 'jr. ■ 0,785 e r (IV. 6) 

as the average value over the up^rdly convex profile with profile depth c. 

The average thickness of the pressure layer is therefore 

/, = I r : ■: 0,7S5 c. 

(IV. 7) 

qn(1 h is accomplished with particular simplicity in this 
Determination of y and h is accompli*" .,.,„,. for the 

case, since in (iv's) the curvature that appears in the denominator for 
profile (IV. 4) is constant over the entire depth of the profile. 

Fieure 4. Circular Profile With Flow 
Around It as the Lowest Possible Form 
of Oscillation of the Membrane. 



The lowest form of oscillation of the membrane plate can be approximated 
by the circular pattern of the deflections according to (IV. 4). Higher 
oscillation forms can be approximated by lining up such segments of a circle 
with alternating signs, so that a periodic wave "train results, which correspond 
to a sinusoidal pattern for the wave profile at higher wave numbers. This /33 
approximation would have the advantage that in (IV. 3) the curvature z" would 
be constant for each segment and therefore taking the average over y p would 
also be simple from the calculating standpoint. In the meantime, however, 
on the assumption of such a wave profile, the curvature over the base c of 
the profile would behave in an unsteady fashion, and the calculation would 
prove to be quite tedious as far as determining the deflections z according to 
Fourier from the curve of the curvatures^ is concerned. Hence, for the profile 

^/HiupA-D-V-, (A.=-l,2,"a,...r (IV. 8) 

which describes a sinusoidal half-arc over the base c with A = 1 and which 
can then be compared with the circular profile (IV. 4), after which the weight 
per unit area y p can be determined. For values X > 1 (IV. 8) describes a 
profile with 2\ - 1 half-waves. If we ;recall that the relationship 

cos (.«• sin y) = /„ (.<0 4 2 £ / 2 „ (x) coh 2 w y ( IV . 9) 

exists, where the I 2n (x) are the Bessei functions of the first type and the 
second order, we will then obtain for the profile according to (IV. 8) as a 
function of the coordinate using (IV. 9) 

,= -/(_ 1)4/. (A*- -jM + 2£ /,.(**- -J) <-1>»cu,2»6>]. 

The flow condition (IV. 2) then becomes 

, . i / i ■* \ Hin 2 n (-) 

A- 1 
If we use the relationship 

^l?_!i®. = 2 V cos (2 i - 1) 6>, (n = 1, 2, 3, . . .) 



we will then have 

v, (-r) clz 

v dx 

00 00 

= (- 1)' 16 I £ £ **P [ X n - t) (~ l ) p P cos ( 2 m-\)G 

m — 1 p — m 

= (- 1)* 16 -{- £ «2m-i cos (2 m - 1) <9 

w- 1 



"2W.-1 = ^ 7 2 p (A 7T - J I (_ ]) P/J . 


If we compare this result with the flow condition (IV. 2), we will obtain the 
coefficients of the vortex interaction with A = 0, A = (n = 2,4,6,...) and 

" m = Vr^,) / z " < /j: = (~ 1) A 2 n -/, (2 A - 1)» 

For calculating the average weight per unit area p_ we must now define an 


average value 

!» = (- 1)» 16 A a 2m ^ = (_ l)M6 { £ /„ [A * - -J-] (- i), p . 


^ = (- 1)» 16 A fl^.j = (_ i)M6 A 2 7 2p (A * - J] (- i), p . 

P -„, l j (IV. 1( 

(n= 1, 3, 5, ...) 

for (IV. 3), in which the integration extends over the central portion of the 
average half-wave of the wave profile (IV. 8) from the point x = C ^ X " ^ 
to the point x - . UA . .. . Here j. 

^ = -^v.-^r~- 

represents the arithmetic mean over the same region of the profile (IV. 8) 
It is attained in such a fashion that the curvature of the profile (IV. 3) 
behaves constantly approximately like a circular wave profile. Further 
calculation then gives 

1 Q C 

l' F 2 (/,-«,)? 

A x (2e + sin 2 f) -f 

(IV. 11) 

* - z "J 


for the profile (IV. 8) or 

!<r = " n (ik'-i) 

j«, (2f + «in 2t) - 

V i_-J)".^» ii ((2 m - 1) Bin (2 /« - 1) e co» £ - cos (2 m - 1) >' sin f ) 

2, "i {M - 1) u 

(IV. 12) 

The angle Tis the complementary angle of the angles e, and e, associated 


with boundaries Xj^ and x 2> 

sins = - ( 2T-"I)" 

between which the relationships &1 = Ji" E ' °2 " 2 ' * ^ for x = 1 
and x -_£(l- cosGO hold. The numerical evaluatxon of (IV.12) a 
Teads'to e = f and then to the disappearance of the entire sum expressxon x„ 

(IV. 11) and (IV.12), so that ~ f:= ___ 4cffli 

remains, for this lowest half-wave number, we obtain the value of coefficients 

/,",.= 0.800 e r (IV. 13) 

a to a = - 0.223, so that 

and for the thickness of the pressure layer we will have the value 

A = 0,890 r (IV. 14) 

The differences between the numerical values (IV.6) and (IV.7) for the 
icu rofile and the values (1V.13) and (IV.14) for the s nuso.dal of.le 
relatively sHg- according to the method used for ^- ^ ~ 

which represents a certain confirmation of the accuracy of thx yp ° 

approximation, figure S shows the manner in which*, » ^^ 

k«^c of the profile. In this diagram, the curve or y p /P 

h-iaher half-wave numbers of tne pruiJ-xc. 

according to (IV.!,) and therefore that of h/c over the higher half-wave 

numbers X is shown. 

Figure 5. Dimensionless Weight Per "nit Area 
I /pc or Dimensionless Pressure Layer Thxckness 
h/c as the Function of the Wave Factor X of the 
Oscillating Membrane. 


■„,. definition used for calculation of the average -eight pen una volume 
„ „ pressure .oyer thtckness h fp W d in C«V.10, contains a certaan degr 
f arbitrariness As indicated by a comparison of numerical values wath he 

r. t— — «- -— - v the r - n r t/:: : . " , 

average (IV.10) does make sense from a mechanical standpoant. If » ».. y 
t al po nt of profile 3 - \ on the middle half-wave for eva uata g 
ow pro ess around the wave profile due to the pressure distrabutto 
and perform the calculations for this point, for which the curvature of the 

profile z „ = ( _ iy . 1*'- (2 A - 1)» 


is y according to (IV. 3), we will have 

It is, however 

;t2 _ _ _e_ c _.._ r fll - «. + «,-, -«,+ •• •]• 

K _ „ 3 + «.-«,+ ...] = -! '«*« [A» - 2 J ( l + 2 f ) 
and therefore the formula 

/'#■ = -„• (2 A- 11" 2rf 2,4< l 2 J 

( - n 

wbich is much simplier in contrast to (1V.11, and (IV.12,. i. «^-»^» 
calculation of the weight per unit area or pressure layer thacaness. T, 

alculated for „ in this fashion can also he used as representataves 

:;: r « *.*.»" «- » lu .. *« -now from (I v. 15 , ^ 

I d ewise those for pressure layer thickness with h- and carry ou the 
aLulation for the odd numerical values X - 1,3,5,7 and >*'™££^ 
unit area and pressure layer thickness according hoth to (W.12) C™. 

deviation hetween the numerical valuea occurs in the case of X - 1^ **« . 
since the factor of ,-, or h* calculated according to t e samp = , e f . 

c n f rxsrilation of the membrane plate is located 

nv 151 for these lowest forms of osciiation vx 

„ :„ the factor ( 1V,, and t .V.7, for the eircula rly curve d eontour and 
th e factor CIV.l, and C.V.14, for the sinusoidal a f-wave the valu a 
calculated in simplified fashaon according to ( V 5). ( 

smaller than those from (IV.12), are also acceptable. T*e 


i « lie slightly below the curve 
*„ the case of even X-values, lie siignw 
in those ways, xn the o ^ ^ ^ ^ in the case 

plotted for odd ™= r * t the point x . c relative 

of ov en x-values their downwash of the prof ^ ^ ^ ^ ^ 

„ the bulge of the profile at porn X - P^ ^^ ^ ^ . 
odd values of X. Numerically, these responding 

sUght ly s nusoldal pattern of the curv en r ^ ^ 

— "** ^""^^rhryl «1 undlteoted. nnall, these 
Figure 5). are so — 1 1 tl»« they c representati ons of the average 
discrepancies are a conscience ° ^ ™ b ? combining several half- 
values, which can, however, he •^»^ lly ^ . sm00thing of the 

— of the profile in the avera ging^ ^J ^ f ^ „ _ 
curves results. For practical requir m ^ ^ 
suf£ice; since it is better to Keep , numer U, ^^ ^^ 
small, there is a safety factor provided xn calcula 

hlc; frlQC | _°^i 3 I.„ 





V Prospects. 

' t rt . .ritical flow velocities v of the membrane 

The emulation of the cr t * ^ ^ ^ 

according to («".» » - ° "^ „, velocity Vfc is obtained for 
which we have determined. The lowest half . wa ve as a deflection 

th. lowest form of oscillation of the membrane with 
for which (according to IV.13) „ F - 0.890 PC so that 

irTi24-' s ''- 

i,o« I 


■ „n higher wave numbers for the membrane. 
High er critical velocities are « ^^ ^ ^ ^^ ^ 

z^7T£~^*» •■;» «~ ai r° tiv - i2) cTable5, 

so that Bt := j,r_"o,173 ^ = *.**\ -V (V.2) 


pe £r T Fi r: 5 , r a" — : - - -» — - - — 

the radical in (V.1J ana iv.zj s theorV with the aid 

appropriate critical velocities is — >^ , "^ "tafcen. In the 
of Figure 5, from which the factors of the Vf /pc values c 
case of even half-wave numhers, the calcuiation was not performed. U ^ 

corresponding to (IV. 8). Instead,™, v crit erion (IV. 1) 

give the factors in front of the radical sign m - a. «, « ^ 
or aiso (1V,> -ith suffic ent accu - • ^; ^ ^ half „„ aves 

l U; Vecause one can calculate fro, it the lowest .deed 
critical velocity v, and achieve .utility of the membrane for all 
velocities V below v,. according (V.l). 

Recent lv, Thwaites [16]. Melsen [17], and, and Zierep [18] 

rrr. rr:"r:::"';,::.;::'-r= : .— , 

fashion, i.e., witnou have critlcal 

tion that resting critical wave shapes of the sal resu ^ 

- — -;• F Tv ™ tc:rrr:-ti. ., ». ^. 


2 T he kinetic stability criteria for such sails wit g^ r for branes 
free are, however, the same as ^ criteria ci ^ ^ ^ solutlon 
with laterally fastened edges This follows ^ ^.^ £unction 
(III 4) for the motion equation (lli.yj 
instead of the sine function in y. 

th. first line and factors taken from [16] and [17] on the second line 

ft that even at five naif-widths the difference between the numerical 
u f 2. 7 3 and 2 .4S is still ve ry sli g ht constitutes a con -• ^ 

•w.a in this naoer In mechanics, it amounts to the agree 
the theory described an this paper. ^ 

nent of the static with the kinetac stability criteria 
7 the present case, the situation is somewhat more complex, .inc. th 
In the present da ™r,i nE in the motion equation 

inclusion of the supporting member and the damping 
„„ 9) gives a correct result only in conjunction with the proper pre 

( I] The calculated results indicate that the pressure law (1.2) 
i P i ly correct. It is not without interest from the mathematical 
a a oint that even with a more general pressure la. which is com, 

. * r tt ?■) and fll 3) the calculation again yields criterion 
additively from (II.^J ana U 1 -^ . 

aaai ' . trt ; c witerion is concerned, it is 

. As far as the derivation of this criterion 

Zrl « have inclusion of the inertial term in the motion equation of 

~a kv t-hP foriolis acceleration 2 v z t - 
the membrane caused by the torioi^ a xt 

The flutter oscillations can be seen particularly well in ^ <"*°J 
•bhln or flags Figure 6 shows a photograph of a ribbon in a wind tunne 
ribbons or flags, ngu n.ttern which changes with 

nne can clearly see the instantaneous sinusoidal pattern, 

Une can cxcm./ j^~ ri~, ■ TVii <; wave 

:;;:: r;;^ ""::: velocity. „*. * k according „ ,,,, 

n cttt ill the simple relationship 
we will obtain from (III. 11 J tne simpi 

■ r H * (V.3) 

V -- 


„hich expresses the wave velocity « as a function of the flow velocity^^ 

-: rMii::::;; r;::::i:::-5 ~ — - • - — r 

v < v k (subcritical veloci ty v through the membrane and 

" ' \T:l X:^2» o run llll state) the transverse waves travel 

3The author would like to fxpresa his |^* c ^/Sf ^rt 1 had t beS°c^pleted, 
S^rLaMinglhlfcoSari-retween-umrerical values to be made subsequently. 

■ vTv.j ' w» 'g .i>wii». W;-jij .-up. i u i- ' jy 1 '"."-wm ma- nn,i-t.i| i iMH a 

(P|ii;uwi H 'I 

Figure 6. Photograph of a Fluttering Cloth in a Wind Tunnel 

In the first case, nothing can be seen jof the flutter, since the slightest 
disturbances are immediately damped. Ijn the second case, the travel of the 
transverse waves can be seen clearly. However, in the fluttering membrane 
the same types of phenomona canbe^eeT^ as in an oscillating drive belt, 
which served in [15] as a special case of a tube with flow through it. In 
particular, the test in the wind tunnel also revealed that as the flow velocity 
v increases the wave velocity w likewise increases and according to (V.3) vw+v. 
Thus, in addition to the good agreement of the aboye described numerical values, 
it is a phenomenological confirmation of the theory given above. In addition, 
the sails of sailboats moving with the wind show transverse oscillations of 
the type under discussion. These flutter oscillations are not caused by 
some disturbing objects in the airstream, such as the mast supporting the 
sail of a sailboat, which could create vortices and thus stimulate the 
flutter oscillations, but they arise solely from the supply of energy in 
the airstream. An energy balance based on the fluttering system composed 
of the membrane is not possible, since any amount of energy can be drawn from 
a medium flowing with a constant flow velocity v. This is a typical feature 
of nonconservative systems, to which the membrane plate with flow over it 
must be assigned and which leads to the described lack of steadiness in 
the derivation of the stability criterion. 




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Mechanics)," Second Edition, pp. 207, Berlin, 1930. .. 
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[9] Bolotin, V.V., "Nonconservative Problems of the Theory of Elastic « 
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::r i4 
J 15 


J 17 



25 i. 

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Coveh Pa< 

-Translated for the National Aeronautics 
Contract No. NASw-2037 by Techtran Corp< 
Maryland 21061; translator, Williap J 




i- source 

and Space 
d, rat ion, F 
Grimes , N . I . L 

Administration- ur^der - 
.0. Box 729, Glen Bumie, 


HtV, ! 

i 22