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Vibrations of thin plane wings in tandem in plane incompressible flow
VIBRATIONS
OF THIN
PLANE
WINGS
IN TANDEM
IN PLANE INCOMPRESSIBLEFLaOW (K~LKBANIIA
~0~~000
POLIPLANA
TANDEW
v PLOSKOH
NESZlIlAElOl POTOKE) PMM Vol.22,
No.4, M.
(Received
1958,
pp.465-472
D. KHASKIND (Odessa) 29
November
1957)
This paper deals with small vibrations of thin plane wings arranged in tandem in plane incompressible flow. The solution is obtained by breaking the problem down into two simpler ones [ 1 1. One of these problems is nonhomogeneous and represents noncirculatory flow past a system of wings, whilst the second is a homogeneous problem which can be solved by means of functional combinations containing several constants. Thin wing theory is used to solve these problems [2 I. the whole investigation consisting essentially of finding the constants by means of linear equations. Closer attention is given to the problem of vibrations in a tandem biplane system in which one of the wings is fixed. Approximate expressions are given for the hydrodynamic forces and the energy characteristics of the system, regarded as a moving group.
1. Kinematic relations for vibrating planes in tandem. We consider
a system of thin plane wings of infinitespan in tandemwhich is underwing small harmonicvibrationsof frequency(Iin a stream of incompressiblefluidmoving with constantvelocityvo. We introducethe coordinatesystemOzy (FiR.1)which moves with the undisturbedstream at velocityuo; then if we assume the disturbedfluid flow to be irrotational, the velocitypotentialcP(x,y, t) of the total motion satisfiesthe linearizedflow conditions a@ (k=l,. . , ,n; j=1’z) - = Vnk(z)ejof at akbk (I.11
8Y
Fig.
649
1.
650
M.D. Khaskind
Roth here, (with
and in what follows,
respect
involve
the exponential
restricting average bodies
to the imaginary
we shall
number j)
time term. Moreover,
the validity
of expression of incidence is zero. If
angle
only deal with the real
of complex expressions
the complex amplitude
for brevity,
part
which
but without
(1.1) it is assumed that the the thin winKs vibrate as solid
of the nonal
velocity
r,&(x)
can be deter-
mined from the expression Vnk (4
=
Ck +
(PO = -2)
(k + x)6%
where uk and aj are the complex amplitudes . velocities of the plate akbk. Assuming stationary
the disturbed
vibrational
The harmonic function In fact,
for
from s-try
$
vortex
pressure
p. = pvo 3 (
considerations
= -9(4
$!!-&09=0 It
to be quasi-
= rnk (z)
on ok bk
several
additional
we have the linearized and density -
(1.3) condi-
expression
in the undisturbed
jpo~) ejaf
(1.4)
and the condition
of continuity
of
we obtain ‘p (x7 -Y)
lines
motion of the fluid
the pressure
p Thus
and angular
$ ( n, y) must satisfy
the fluid
(pO and p are respectively fluid)
pressure
the vertical
we find @ (5, 21,t) = ‘p(5, y) &‘,
tions.
of
(1.2)
follows
Y),
‘P(X9 O)=O
for y=O
from these conditions
of discontinuity
that,
in horizontal
sheet oriKinatinK
trying to find should trailing edges ak.
at these
satisfy
outside
trailina
(1.5)
akbk (z
behind the wing-trailing
velocity
the condition
for z>bl
exist,
edws. of
representing
‘lhe solution
finite
velocity
ed=s, a
we are at the
To solve the problem we introduce the function w = 6 + i $ of the complex variable z = x + iy, where the imaginary number i = d-1 is not interchangeable with the imaginary number j. Later on we shall split the above problem into the two simpler problems; w(z) = we(t) + w1(L) where the functions w. = q5,,+ i l/lo and 70~= J,, + i @iI satisfy conditions lm 2
m-z-
v,,k (x)
Im zuI (5) = ilk
on ~66k,
on u,( h,,
Rc w,, (2) = 0 outside uk b,i (1.6) Rcwl(z)=O
for
r>b,
Vibrations
Re2(
jpow1
of
thin plane
) -0
(for y=O
651
in tandtn
ring*
outside akbk (Z
(I-7)
'Ihefunctionx0(z) representsthe complexpotentialof fluid flow without circulatiou,while the function~~(2) is a solutionto the homoReneousproblem and is linear in terms of the constantsA,. It is obvious that the functionw(z), determinedfrom conditions(1.6) and (1.7), satisfies(1.3)and (1.5).'lheconstantsA, in this functionmust be found from the conditionof firrite velocityat the trailingedges ok. UsinR thin wing theory[2.3 1 we find imnediatelvthat dwo
dz-
1 Zxig(r)
_-(r,
k=O
g(z) =
“k
n
n-2
&Z
+2kzr'"';z(')
(fitz-
a,)
tz-
b,)
&)
(I.81
j”
s=1
n
g,
(z) =
((bk
- z) @-
ak)
n (z -
‘I*
a,) (z -
b.))‘-
(1*9)
r+k
?he constants Bk(k = o#l, . n - 2), which are real with respectto i, are foundby equatingto zero the circulationround n - 1 sections akbk ..)
To determinew1 we introducea furtherfunction
f (2) = r+is-_
- ho WI
(1.11)
From (1.7)we have the followingconditionsfor the functionf(z): Im f (5) = -
jplJ &
Ref (2) = 0 outsideakbk (1.12)
on Okb,,
It followsfrom these conditionsthat the expansionof the function f(z) in the neighborhoodof a point at infinitytakes the followingform (a, iS real with respectto i)
f(z)=$+!gf..
.
(1.13)
The functionf(z) is thus determined,and so is dtq,/dz. In fact,using the expansion(l-13),we shall have (C, is real with respectto i) n-1
fcZ)=&
(I:CkZ'+ k=O
2jp0 i /&,. YE*(i)&) k=o
(1.14)
ak
To determinethe constantsC, and A, we make use of the conditionof
652
M.D. Khoskind
finite fluid velocity at the trailing edges akP which can be written (Ll,...,)L) liln(z- al)'Iz($+f)-0 z-1 Applying this condition we arrive at the system of linear equations
n
n-2 +
bk v,k
~BkQk+2xI;I\ k=o
-
(t) gk 4__al
(t) di=O
(1.15)
(l=Ir...,d
“k
in which all the constants Ck are given linearly in terms of A,. To determine the latter we must satisfy the first of the conditions (1.7). To do this let us regard (1.11) as a differential equation in wk. Bearing in mind that at large distances in front of the planes the fluid is undisturbed we find $Q(z, 0) = ejp+i e-jpaxs(x,0) dx (1.16) ac Making use of this expression and satisfying the first of the conditions (1.7) we have the following system of linear equations ~~ =ejpobl [[ e-jr"axs(x,
0) dx
+
Q)
2
lis(e-jfi@~ _
e--jp&)
+
8'1
z-lb+1 +
zl
\
e--jpox~(x,
0) dx]
(I=l,...,n)
(1.17)
as In this way, the final determination of the function w(z) consists of finding the constants A,, B, and C, by means of the linear equations (l.lO), (1.15) and (1.17). However,an analysis of transient motion of multiplanes in tandem [3,4 1 in the general case, with arbitrary variation of normal velocities with time, leads us into great difficulty because some unwieldy integral equations have to be solved.
2. Vibrations of one wing in a tandem biplane. We now discuss a tandem biplane, the edges of which are determined thus: - a1 = - a, b, = a, a2 = - c - b and b, = - c + b, where c is the abscissa of the center of gravity of the second wing, and 2b is the width of this wing. We shall now assume the second wing to be at rest (u,,(x) = O)*, then formulas (1.8), (l.o), and (1.14) for this case take the form
l
In all at the change
the stationary wing is our calculations we assume c > 0, i.e. If the stationary wing is in front(c < 0) the following rear. (($4)1/2 should be made in the formulas, (c’ - &1/2=_
Vibrations
of
plane
thin
dwo
‘dz =&(Bo+2
f(z) =
&
g (2) = ((2” -
rings
To determine
(2.1)
-rf;
di + 2jpo Aa -T
di ) (2.2)
-c-b
b21)“% g, (4 = ((a” - x2) I@ + 4” - b21P
d) [(z + c)” -
(2) = -- (I x2 -
the constants
which in particular
tandem
f vn~~:(E)d) -a
(c” + CIZ + 2jpt, A1 j e -a
g,
in
(2.3)
(5 + C)2])“*
u2 ) [b2 -
we have equations
(1.101,
(1.15)
and (1.171,
give
=v
a
&I _-.
sA+2 s g1(4
(ngl ID
j +g_ *F;--z -a
dE
=
()
(2.4)
a
vn “:: i,” A’ g, (E)di + 2jpoA2-i+b -f$
B~+Co-Cla-+2
-a
B,+Co-Cl(c+b)+
2 [
SC
(2.5)
-c-b
(Vn(i)+j~oAl)((u2-E2):=~T~)111dE-
-c+b
--&,A,
& = 0
IE2-a2j
b -(e
+ c) ‘II
b+E+e)
di=O
(2.6)
-c-b
la
Al = &a
s
e-h= s (z, 0) dx
(2.7)
cn --e f-b
A 2 z A, & (a+b-c) + & (b-c)
s
e--jpax
0) dx
s (x,
(2.8)
-a Lst us suppose that.the vibrating
stationary
one so that c >> b. In this
us to find explicit these equations, if
wing is much shorter
than the
case equations
-
approximate expressions ior we represent the stationary
(2.41
(2.8)
sinmlarity. section L
In fact, it follows from (2.1) and (2.41 that outside c - 6, - c + b) the following expressions are valid to of the order of (b/c)*.
accuracy
no = _
2
5p,
(;) (~2 ._
--n
;-?)‘h&,
“d”p= 3
xi (;’
’
allow
the constants enter& winK as a concentrated
?
\
- nzP -a
p’
.”
(C)
(n?
’ 4
_
--:
the
5”)“’
df
(2.9)
654
Y.D. Khaskind
Expressions (2.9) represent fluid flow without circulation round an isolated thin vibrating winE. It therefore follows that the effect of the stationary wing on the noncirculatory part of the fluid flow close to the vibrating win,q,which gives rise to the supplementary effect of connected masses, is only manifest in dipole, quadripole and higher approximations. Equations (2.5) and (2.6) appear in the following simplified form with this degree of accuracy C,-&a Gc
CO-
+ 2"jPoU2(+-;SjA,=(c-a)D, ++,c2Ar
= b(O2
+
G+W~oc4
(2.10)
+
4jp0 (3 - 2'!z)(c2 - a2)'1, AZ)
+
(2.11)
If follows from (2.2) that outside the section (- c - b, - c + b), to the following expression prevails an accuracy of terms of order (b/c)*,
while it follows from (2.11), that the second factor in (2.13) is of order b/c. Thus the approximate expressions (2.9) and (2.13) correspond to representation of the effect of the stationary wing on the fluid flow close to the vibrating wing, by that of a concentrated vortex of given strength. Using expressions (2.7) and (2.13) we can obtain the following equations to determine the constant A,:
G HlJ2) (I*) - 2xpA, Hi(f)(p)= -ii-li,&,- C,c + x’jpnAl) E,
s
c1
p+
,-jv
dz
a”)(z2_l)‘lr
d11,7
9
-;icc--ia”&
-=-pJ%)
(p x- potI) (2.14)
(
a,, = “a)
(2.15)
Here H,(*) = I, - jlv, is a Hankel function, and from relation (2.15) we obtain F _ In (cztl- (a2 -- I)“‘) & + + ,jv H,(2) (+ ” , .,) - jHl,(“) ( ;; , v)] (v 7:W) A0 (au2 -
1fi2
”
[
(2.16) where the functions H (*I and H (*I have been tabulated [5 1 pressed in terms of H&kel functions by means of relations
and are ex-
Vibrations
Equations
Ci
and
(2.10),
of
(2.11)
thin
plane
(b/c)(‘).
Ci = Cio + bCir, Then from these equations,
ili =
(2.18)
Aio + bAi,
we obtain
2who = (1- C(PM4 COO = a [(a0 C(1-4) - $ (1 - C(f-43 4
C ol=
II(2a0--1)C(~f+(PL0-~)(l--C(II))--
C 11=
C
2xjpdll Here C(p),
b0+
-1 L
-1)EoT(~~]
C (EL)$ a
HJ2) (I*)
wP) (IL) ' G = D, + 2;E.j vdlo -t_ Cl0 +
A,,.
determination
From equation
of all
(2.8)
$@_j
(a0 -
f
1) -&J (P)]
the fol~~in~
HI(“) (EL)t
For the final
(2.19)
(a0 -w0w]~
-$(a0
C(P)-
=
$)
and G represent
T(p)
C(P) =
the constants
we put
c(El)Dl t
-4.
of
To do this,
c*
10 =
value
655
in tandem
a 11ow us to determine
and (2.14)
Ai to an accuracy of order
wings
T(P)
a (a*G_ 1)
expressions 1
=
at2)(II) + iffoC2) (I.4(2.20)
4&At0 (3 -
2,~) (ao2 -
the constants
and expressions
e
1)1”
must find
(2.13)
the
and (2.19)
we
find
A2o =
I-&
(1 E I=
C (p)) (1-
e-jr E,) -
=0 e-_jp.r s1 ++-
Note that the functions
&
e--iv E,
F -.dE’l ‘2 dp
Q/z’ dr,
E, and E2 can be evaluated
functions Ik(p). Actually, if we make the substitution the expression for E, takes the following form
El=
(2.21)
C (p)] ,!I%
-Icxp[j>(~+-A-j]%. (U,=a,-
(2.22)
in terms of Bessel U = x -
(x2 -
(ao2--l~*~~)
Now, making use of the expansion cn
cxp
[$- (u + -&j] = IO(PI +
&z; jk (Uk +
U-k) !k h-4
1)1’2
656
M.D.
we arrive
at the following
Khosk ind
expression
for E,,
which is convenient
for
calculation,
E, = -II,(p)lnU,-
(2.23) k=l
3. Hydrodynamic forces which act on the tandem biplane. We now work out the hydrodynamic forces which act on the vibrating wing in the tandem biplane. If we use expression (1.4) and represent the function w in terms of wO snd lift
f, we arrive
at the following
relations
for
and moment:
(3.1)
where K is
the contour which bounds the section
(-
a,a)
and is
anticlock-
wise. From formulas
(2.91,
(2.13)
and (2.191
and the theorem of residues
we
find Y = Y, + $.Y1,
M=M,+;M,
(3.2)
where YO and M,, are the lift and moment respectively vibration of an isolated wing:
il * (I-
corresponding
-$)“‘I
to the
v,, (x, t) dx (3.3)
and Ylb/a action
and M,b/a
are the supplementary
of the stationary
Y, = poGeJ ‘a,
( c (t*)1 a”-
M, = pvo uG;ejol (%s To calculate general
the suction
formula 12
1
force
wing as a concentrated -
f
vortex.
E, 1’ (p) + (ao2- l)-llz) (3.4)
+ z ,zo’2 1j forces
and moment caused by the
4 E, T (p) -
at the leading
a0 (aoz -
edges b,,
I)-‘1%) we use the
Vibrations
of
thin
plane
ringr
in
tandem
557
(3.5) Using (2.9), (2.13) and (2.19), we Ret an expression for the mean value of the suction at the leadinn eh z = a of the vibrating wing (34 where
Xl0 l is the averam suction on an isolated
x1;=
$I-
vibrating
wing
i vn(x)d: +&(c(~)-l)D,~a
(3.7)
_-D(a’ - a+) ‘2
and XIz*b/a is the supplementary component in the suction expression caused by the stationary winR acting as a concentrated vortex. X,,‘=&Re
G [( u
F=-
s
2c,~)_~1
&T(p)
+ -!a0
v, (4 dx
la8 _
-f
xa)‘la +
& (C(I4 -
F +
1>
1
(3.3)
1) a
To work out the s&\&m forces ac’ting on the stationary wit, we must start with the accurate formulas (2.1), (2.2) and (2.5) and after this we must transform the boundary. The result of this fives
(3.9) The cmlete expression for the prqjection of the hydrodynamic forces on the x axis takes the form (B(t) is the angle of incidence of the vibrating wina) :
T=X1+X,-Yp
(
p (t) = -
& &at)
(3.10)
For several modes of vibration 19 > 0 and in this case a tensile force appears, i.e. the vibrating wing in the biplane tandem system can be reerded as a moving group. In this case the avera- value of useful and wasted power is Riven by the
F=T*q,,
IV'=-(YV+MSZ)'
(V= v&t,Q = wjaf) (3.11)
The relations obtained above allow one to work out these energy characteristics and the thrust efficiency q = P/P. BIBLIOGRAPHY 1.
Khaskind, II. D. , Kolebaniia kryla v dozvukovom potoke gaze (Vibrations of a wing in subsonic gas flow). PMMVol. 11, No. 1. 1947.
2.
Sedov,
L. I.,
Problem
Ploskic tadachi in Hydrodynaaicr
gidrodinariki i aerodinaniki (Plane and Aerodynamics). Gostekhlzdat, 1950.
658
M.D.
Kharkind
3.
tandem i o glissiroSedov. L. I., K zadacham o tonkikh polipilanakh vanii na neskol’nikh redanakh (Problems of thin tandem multiplanes and sliding over rough steps). Trud. Tscut. Acre-gidrodin. In8t. Yosk. No. 325, 1937.
4.
Nickel, K. iiber Tragfltigelsysteme in ebener Strbmung belliebigen Bd. 23. No. 3, 1955. instationiren Benegungen. Ingen. -Arch.
5.
einiger Schwarz, L., Untersuchung Ordnung verrandter Funktionen.
mit den Zylinderfunktion Bd. 20,
Loftfahrtforrch.
Translated
nullter 1943.
by
V.H.B.
OF THIN
PLANE
WINGS
IN TANDEM
IN PLANE INCOMPRESSIBLEFLaOW (K~LKBANIIA
~0~~000
POLIPLANA
TANDEW
v PLOSKOH
NESZlIlAElOl POTOKE) PMM Vol.22,
No.4, M.
(Received
1958,
pp.465-472
D. KHASKIND (Odessa) 29
November
1957)
This paper deals with small vibrations of thin plane wings arranged in tandem in plane incompressible flow. The solution is obtained by breaking the problem down into two simpler ones [ 1 1. One of these problems is nonhomogeneous and represents noncirculatory flow past a system of wings, whilst the second is a homogeneous problem which can be solved by means of functional combinations containing several constants. Thin wing theory is used to solve these problems [2 I. the whole investigation consisting essentially of finding the constants by means of linear equations. Closer attention is given to the problem of vibrations in a tandem biplane system in which one of the wings is fixed. Approximate expressions are given for the hydrodynamic forces and the energy characteristics of the system, regarded as a moving group.
1. Kinematic relations for vibrating planes in tandem. We consider
a system of thin plane wings of infinitespan in tandemwhich is underwing small harmonicvibrationsof frequency(Iin a stream of incompressiblefluidmoving with constantvelocityvo. We introducethe coordinatesystemOzy (FiR.1)which moves with the undisturbedstream at velocityuo; then if we assume the disturbedfluid flow to be irrotational, the velocitypotentialcP(x,y, t) of the total motion satisfiesthe linearizedflow conditions a@ (k=l,. . , ,n; j=1’z) - = Vnk(z)ejof at akbk (I.11
8Y
Fig.
649
1.
650
M.D. Khaskind
Roth here, (with
and in what follows,
respect
involve
the exponential
restricting average bodies
to the imaginary
we shall
number j)
time term. Moreover,
the validity
of expression of incidence is zero. If
angle
only deal with the real
of complex expressions
the complex amplitude
for brevity,
part
which
but without
(1.1) it is assumed that the the thin winKs vibrate as solid
of the nonal
velocity
r,&(x)
can be deter-
mined from the expression Vnk (4
=
Ck +
(PO = -2)
(k + x)6%
where uk and aj are the complex amplitudes . velocities of the plate akbk. Assuming stationary
the disturbed
vibrational
The harmonic function In fact,
for
from s-try
$
vortex
pressure
p. = pvo 3 (
considerations
= -9(4
$!!-&09=0 It
to be quasi-
= rnk (z)
on ok bk
several
additional
we have the linearized and density -
(1.3) condi-
expression
in the undisturbed
jpo~) ejaf
(1.4)
and the condition
of continuity
of
we obtain ‘p (x7 -Y)
lines
motion of the fluid
the pressure
p Thus
and angular
$ ( n, y) must satisfy
the fluid
(pO and p are respectively fluid)
pressure
the vertical
we find @ (5, 21,t) = ‘p(5, y) &‘,
tions.
of
(1.2)
follows
Y),
‘P(X9 O)=O
for y=O
from these conditions
of discontinuity
that,
in horizontal
sheet oriKinatinK
trying to find should trailing edges ak.
at these
satisfy
outside
trailina
(1.5)
akbk (z
behind the wing-trailing
velocity
the condition
for z>bl
exist,
edws. of
representing
‘lhe solution
finite
velocity
ed=s, a
we are at the
To solve the problem we introduce the function w = 6 + i $ of the complex variable z = x + iy, where the imaginary number i = d-1 is not interchangeable with the imaginary number j. Later on we shall split the above problem into the two simpler problems; w(z) = we(t) + w1(L) where the functions w. = q5,,+ i l/lo and 70~= J,, + i @iI satisfy conditions lm 2
m-z-
v,,k (x)
Im zuI (5) = ilk
on ~66k,
on u,( h,,
Rc w,, (2) = 0 outside uk b,i (1.6) Rcwl(z)=O
for
r>b,
Vibrations
Re2(
jpow1
of
thin plane
) -0
(for y=O
651
in tandtn
ring*
outside akbk (Z
(I-7)
'Ihefunctionx0(z) representsthe complexpotentialof fluid flow without circulatiou,while the function~~(2) is a solutionto the homoReneousproblem and is linear in terms of the constantsA,. It is obvious that the functionw(z), determinedfrom conditions(1.6) and (1.7), satisfies(1.3)and (1.5).'lheconstantsA, in this functionmust be found from the conditionof firrite velocityat the trailingedges ok. UsinR thin wing theory[2.3 1 we find imnediatelvthat dwo
dz-
1 Zxig(r)
_-(r,
k=O
g(z) =
“k
n
n-2
&Z
+2kzr'"';z(')
(fitz-
a,)
tz-
b,)
&)
(I.81
j”
s=1
n
g,
(z) =
((bk
- z) @-
ak)
n (z -
‘I*
a,) (z -
b.))‘-
(1*9)
r+k
?he constants Bk(k = o#l, . n - 2), which are real with respectto i, are foundby equatingto zero the circulationround n - 1 sections akbk ..)
To determinew1 we introducea furtherfunction
f (2) = r+is-_
- ho WI
(1.11)
From (1.7)we have the followingconditionsfor the functionf(z): Im f (5) = -
jplJ &
Ref (2) = 0 outsideakbk (1.12)
on Okb,,
It followsfrom these conditionsthat the expansionof the function f(z) in the neighborhoodof a point at infinitytakes the followingform (a, iS real with respectto i)
f(z)=$+!gf..
.
(1.13)
The functionf(z) is thus determined,and so is dtq,/dz. In fact,using the expansion(l-13),we shall have (C, is real with respectto i) n-1
fcZ)=&
(I:CkZ'+ k=O
2jp0 i /&,. YE*(i)&) k=o
(1.14)
ak
To determinethe constantsC, and A, we make use of the conditionof
652
M.D. Khoskind
finite fluid velocity at the trailing edges akP which can be written (Ll,...,)L) liln(z- al)'Iz($+f)-0 z-1 Applying this condition we arrive at the system of linear equations
n
n-2 +
bk v,k
~BkQk+2xI;I\ k=o
-
(t) gk 4__al
(t) di=O
(1.15)
(l=Ir...,d
“k
in which all the constants Ck are given linearly in terms of A,. To determine the latter we must satisfy the first of the conditions (1.7). To do this let us regard (1.11) as a differential equation in wk. Bearing in mind that at large distances in front of the planes the fluid is undisturbed we find $Q(z, 0) = ejp+i e-jpaxs(x,0) dx (1.16) ac Making use of this expression and satisfying the first of the conditions (1.7) we have the following system of linear equations ~~ =ejpobl [[ e-jr"axs(x,
0) dx
+
Q)
2
lis(e-jfi@~ _
e--jp&)
+
8'1
z-lb+1 +
zl
\
e--jpox~(x,
0) dx]
(I=l,...,n)
(1.17)
as In this way, the final determination of the function w(z) consists of finding the constants A,, B, and C, by means of the linear equations (l.lO), (1.15) and (1.17). However,an analysis of transient motion of multiplanes in tandem [3,4 1 in the general case, with arbitrary variation of normal velocities with time, leads us into great difficulty because some unwieldy integral equations have to be solved.
2. Vibrations of one wing in a tandem biplane. We now discuss a tandem biplane, the edges of which are determined thus: - a1 = - a, b, = a, a2 = - c - b and b, = - c + b, where c is the abscissa of the center of gravity of the second wing, and 2b is the width of this wing. We shall now assume the second wing to be at rest (u,,(x) = O)*, then formulas (1.8), (l.o), and (1.14) for this case take the form
l
In all at the change
the stationary wing is our calculations we assume c > 0, i.e. If the stationary wing is in front(c < 0) the following rear. (($4)1/2 should be made in the formulas, (c’ - &1/2=_
Vibrations
of
plane
thin
dwo
‘dz =&(Bo+2
f(z) =
&
g (2) = ((2” -
rings
To determine
(2.1)
-rf;
di + 2jpo Aa -T
di ) (2.2)
-c-b
b21)“% g, (4 = ((a” - x2) I@ + 4” - b21P
d) [(z + c)” -
(2) = -- (I x2 -
the constants
which in particular
tandem
f vn~~:(E)d) -a
(c” + CIZ + 2jpt, A1 j e -a
g,
in
(2.3)
(5 + C)2])“*
u2 ) [b2 -
we have equations
(1.101,
(1.15)
and (1.171,
give
=v
a
&I _-.
sA+2 s g1(4
(ngl ID
j +g_ *F;--z -a
dE
=
()
(2.4)
a
vn “:: i,” A’ g, (E)di + 2jpoA2-i+b -f$
B~+Co-Cla-+2
-a
B,+Co-Cl(c+b)+
2 [
SC
(2.5)
-c-b
(Vn(i)+j~oAl)((u2-E2):=~T~)111dE-
-c+b
--&,A,
& = 0
IE2-a2j
b -(e
+ c) ‘II
b+E+e)
di=O
(2.6)
-c-b
la
Al = &a
s
e-h= s (z, 0) dx
(2.7)
cn --e f-b
A 2 z A, & (a+b-c) + & (b-c)
s
e--jpax
0) dx
s (x,
(2.8)
-a Lst us suppose that.the vibrating
stationary
one so that c >> b. In this
us to find explicit these equations, if
wing is much shorter
than the
case equations
-
approximate expressions ior we represent the stationary
(2.41
(2.8)
sinmlarity. section L
In fact, it follows from (2.1) and (2.41 that outside c - 6, - c + b) the following expressions are valid to of the order of (b/c)*.
accuracy
no = _
2
5p,
(;) (~2 ._
--n
;-?)‘h&,
“d”p= 3
xi (;’
’
allow
the constants enter& winK as a concentrated
?
\
- nzP -a
p’
.”
(C)
(n?
’ 4
_
--:
the
5”)“’
df
(2.9)
654
Y.D. Khaskind
Expressions (2.9) represent fluid flow without circulation round an isolated thin vibrating winE. It therefore follows that the effect of the stationary wing on the noncirculatory part of the fluid flow close to the vibrating win,q,which gives rise to the supplementary effect of connected masses, is only manifest in dipole, quadripole and higher approximations. Equations (2.5) and (2.6) appear in the following simplified form with this degree of accuracy C,-&a Gc
CO-
+ 2"jPoU2(+-;SjA,=(c-a)D, ++,c2Ar
= b(O2
+
G+W~oc4
(2.10)
+
4jp0 (3 - 2'!z)(c2 - a2)'1, AZ)
+
(2.11)
If follows from (2.2) that outside the section (- c - b, - c + b), to the following expression prevails an accuracy of terms of order (b/c)*,
while it follows from (2.11), that the second factor in (2.13) is of order b/c. Thus the approximate expressions (2.9) and (2.13) correspond to representation of the effect of the stationary wing on the fluid flow close to the vibrating wing, by that of a concentrated vortex of given strength. Using expressions (2.7) and (2.13) we can obtain the following equations to determine the constant A,:
G HlJ2) (I*) - 2xpA, Hi(f)(p)= -ii-li,&,- C,c + x’jpnAl) E,
s
c1
p+
,-jv
dz
a”)(z2_l)‘lr
d11,7
9
-;icc--ia”&
-=-pJ%)
(p x- potI) (2.14)
(
a,, = “a)
(2.15)
Here H,(*) = I, - jlv, is a Hankel function, and from relation (2.15) we obtain F _ In (cztl- (a2 -- I)“‘) & + + ,jv H,(2) (+ ” , .,) - jHl,(“) ( ;; , v)] (v 7:W) A0 (au2 -
1fi2
”
[
(2.16) where the functions H (*I and H (*I have been tabulated [5 1 pressed in terms of H&kel functions by means of relations
and are ex-
Vibrations
Equations
Ci
and
(2.10),
of
(2.11)
thin
plane
(b/c)(‘).
Ci = Cio + bCir, Then from these equations,
ili =
(2.18)
Aio + bAi,
we obtain
2who = (1- C(PM4 COO = a [(a0 C(1-4) - $ (1 - C(f-43 4
C ol=
II(2a0--1)C(~f+(PL0-~)(l--C(II))--
C 11=
C
2xjpdll Here C(p),
b0+
-1 L
-1)EoT(~~]
C (EL)$ a
HJ2) (I*)
wP) (IL) ' G = D, + 2;E.j vdlo -t_ Cl0 +
A,,.
determination
From equation
of all
(2.8)
$@_j
(a0 -
f
1) -&J (P)]
the fol~~in~
HI(“) (EL)t
For the final
(2.19)
(a0 -w0w]~
-$(a0
C(P)-
=
$)
and G represent
T(p)
C(P) =
the constants
we put
c(El)Dl t
-4.
of
To do this,
c*
10 =
value
655
in tandem
a 11ow us to determine
and (2.14)
Ai to an accuracy of order
wings
T(P)
a (a*G_ 1)
expressions 1
=
at2)(II) + iffoC2) (I.4(2.20)
4&At0 (3 -
2,~) (ao2 -
the constants
and expressions
e
1)1”
must find
(2.13)
the
and (2.19)
we
find
A2o =
I-&
(1 E I=
C (p)) (1-
e-jr E,) -
=0 e-_jp.r s1 ++-
Note that the functions
&
e--iv E,
F -.dE’l ‘2 dp
Q/z’ dr,
E, and E2 can be evaluated
functions Ik(p). Actually, if we make the substitution the expression for E, takes the following form
El=
(2.21)
C (p)] ,!I%
-Icxp[j>(~+-A-j]%. (U,=a,-
(2.22)
in terms of Bessel U = x -
(x2 -
(ao2--l~*~~)
Now, making use of the expansion cn
cxp
[$- (u + -&j] = IO(PI +
&z; jk (Uk +
U-k) !k h-4
1)1’2
656
M.D.
we arrive
at the following
Khosk ind
expression
for E,,
which is convenient
for
calculation,
E, = -II,(p)lnU,-
(2.23) k=l
3. Hydrodynamic forces which act on the tandem biplane. We now work out the hydrodynamic forces which act on the vibrating wing in the tandem biplane. If we use expression (1.4) and represent the function w in terms of wO snd lift
f, we arrive
at the following
relations
for
and moment:
(3.1)
where K is
the contour which bounds the section
(-
a,a)
and is
anticlock-
wise. From formulas
(2.91,
(2.13)
and (2.191
and the theorem of residues
we
find Y = Y, + $.Y1,
M=M,+;M,
(3.2)
where YO and M,, are the lift and moment respectively vibration of an isolated wing:
il * (I-
corresponding
-$)“‘I
to the
v,, (x, t) dx (3.3)
and Ylb/a action
and M,b/a
are the supplementary
of the stationary
Y, = poGeJ ‘a,
( c (t*)1 a”-
M, = pvo uG;ejol (%s To calculate general
the suction
formula 12
1
force
wing as a concentrated -
f
vortex.
E, 1’ (p) + (ao2- l)-llz) (3.4)
+ z ,zo’2 1j forces
and moment caused by the
4 E, T (p) -
at the leading
a0 (aoz -
edges b,,
I)-‘1%) we use the
Vibrations
of
thin
plane
ringr
in
tandem
557
(3.5) Using (2.9), (2.13) and (2.19), we Ret an expression for the mean value of the suction at the leadinn eh z = a of the vibrating wing (34 where
Xl0 l is the averam suction on an isolated
x1;=
$I-
vibrating
wing
i vn(x)d: +&(c(~)-l)D,~a
(3.7)
_-D(a’ - a+) ‘2
and XIz*b/a is the supplementary component in the suction expression caused by the stationary winR acting as a concentrated vortex. X,,‘=&Re
G [( u
F=-
s
2c,~)_~1
&T(p)
+ -!a0
v, (4 dx
la8 _
-f
xa)‘la +
& (C(I4 -
F +
1>
1
(3.3)
1) a
To work out the s&\&m forces ac’ting on the stationary wit, we must start with the accurate formulas (2.1), (2.2) and (2.5) and after this we must transform the boundary. The result of this fives
(3.9) The cmlete expression for the prqjection of the hydrodynamic forces on the x axis takes the form (B(t) is the angle of incidence of the vibrating wina) :
T=X1+X,-Yp
(
p (t) = -
& &at)
(3.10)
For several modes of vibration 19 > 0 and in this case a tensile force appears, i.e. the vibrating wing in the biplane tandem system can be reerded as a moving group. In this case the avera- value of useful and wasted power is Riven by the
F=T*q,,
IV'=-(YV+MSZ)'
(V= v&t,Q = wjaf) (3.11)
The relations obtained above allow one to work out these energy characteristics and the thrust efficiency q = P/P. BIBLIOGRAPHY 1.
Khaskind, II. D. , Kolebaniia kryla v dozvukovom potoke gaze (Vibrations of a wing in subsonic gas flow). PMMVol. 11, No. 1. 1947.
2.
Sedov,
L. I.,
Problem
Ploskic tadachi in Hydrodynaaicr
gidrodinariki i aerodinaniki (Plane and Aerodynamics). Gostekhlzdat, 1950.
658
M.D.
Kharkind
3.
tandem i o glissiroSedov. L. I., K zadacham o tonkikh polipilanakh vanii na neskol’nikh redanakh (Problems of thin tandem multiplanes and sliding over rough steps). Trud. Tscut. Acre-gidrodin. In8t. Yosk. No. 325, 1937.
4.
Nickel, K. iiber Tragfltigelsysteme in ebener Strbmung belliebigen Bd. 23. No. 3, 1955. instationiren Benegungen. Ingen. -Arch.
5.
einiger Schwarz, L., Untersuchung Ordnung verrandter Funktionen.
mit den Zylinderfunktion Bd. 20,
Loftfahrtforrch.
Translated
nullter 1943.
by
V.H.B.