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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL NOTE 2333
TRANSIENT AERODYNAMIC BEHAVIOR OF AN AIRFOIL DUE TO
DIFFERENT ARBITRARY MODES OF NONSTATIONARY
MOTIONS IN A SUPERSONIC FLOW
By Chieh-Chien Chang
Johns Hopkins University
'yT. .1'
•\
Washington
April 1951
https://ntrs.nasa.gov/search.jsp?R=19930082938 2018-07-05T09:07:29+00:00Z
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TEC}INICAL NGE 2333
TRANSIENT AERODYNAMIC BEHAVIOR OF AN AIRFOIL DUE TO
DIFFERENT ARBITRARY MODES OF NONSTATIONARY
WYTIONS IN A SUPERSONIC FLOW
By Chieh-Chien Chang
SUMMARY
The present treatment investigates the transient behavior of an airfoil in supersonic' flow due to pitching, flapping, and vertical gusts by means of the Fourier integral method. It is an extension of the earlier work "The Transient Reaction of an Airfoil Due to Change in Angle of Attack at Supersonic Speed" by Chieh-Chien Chang, Journal 'of the Aeronautical Sciences, volume 15, number 11, November 19148, pages 635-655. If a control surface is attached to the airfoil and deflected according to a prescribed time schedule, the effect can be evaluated from the result found for the main wIng. An example is shown.
By using a convolution integral as shown in the above-mentioned article, the aerodynamic effect of any arbitrary motion of the airfoil can be determined, if the aerodynamic response of the airfoil to a step-function disturbance is known. As useful examples, the harmonically oscillating airfoil with different modes or degrees of freedom is analyzed for the complete time history, for the case in which the motion starts abruptly from rest. The results check exactly with those of Garrick and Rubinow in NACA TN 1158, after the transient effect of the abrupt beginning has died out, that is, for times later than (M M
from the beginning of harmonic oscillation where M is the Mach number, U is the free-stream velocity, and C is the airfoil chord in con-sistent units.
In carrying out the investigation, a new integral or function C(13,M) has been introduced. This new function is sufficiently funda-mental to the supersonic flutter problem that the behavior of all the different degrees of freedomcan be expressed in terms of C(13,M) and the usual transcendental function (here 13 is related to the frequency of oscillation). It is believed that this new function plays an equiva-lent part in supersonic flutter to the Theodorsen function in the flutter problems of incompressible flow. Preliminary investigation of both the
2 NACA TN 2333
complete C-function C ( 13 , M ) and the incomplete C-function is made. The incomplete C-function together with the incomplete Bessel function, so designated by Garrick and Rubinow, are important to the transient
region for times between (__M__ and (__M__ • An extensive calcu-\M+]JU, \M-l/U
lat ion of this new function will be very useful in supersonic flutter analysis.
INTRODUCTION
Recently, the transient problem of the nonstationary motion of an airfoil in supersonic flow has been treated by a number of investigators. With the Fourier integral method the present author first presented a paper on the transient reaction of an airfoil due to change in angle of attack at supersonic speed (reference 1) in January 19 )48. As a by-product of the investigation, the transient effect due to a vertically descending airfoil is also shown in reference 1. Concurrently, Biot (reference 2) investigated Schwarz's problem (reference 3) of vertical gust with a different approach. Later Heaslet and Lomax ( reference Ii.) considered the cases of the vertically descending airfoil and anair-foil encountering a vertical gust with the frame of reference at rest with the undisturbed fluid. Miles treated the above gust and flapping case with Green's function (reference 5).
The present paper is an extension of the author's earlier work (reference 1) to all possible modes of nonstationary motion, and it shows that the results can be applied to the transient effect of con-trol surfaces. With the concept of the convolution integrals as developed in reference 1, the transient results can be applied to many types of nonstationary motion, particularly the harmonic oscillation.
There are four types of problems to be treated in this paper.
(a) Investigation of the case of a suddenly descending airfoil with a constant rate - so-called flapping degree of freedom of the wing
(b) Investigation of the case of the airfoil encountering a vertical gust of constant strength
(c) Application of the convolution integral to evaluate the wing loading due to different modes of nonstationary motion, such as harmonic oscillations, and so forth
(d) Extension of the present results to other degrees of freedom, such as aileron deflection, and so forth
NACA TN 2333 3
The investigation of drag-characteristic behavior of the wing in steady flow is omitted. entirely from this paper, because it is not a difficult matter for a designer to calculate the drag characteristics with the information contained in this paper and available solutions of the steady case.
The present investigation was conducted at the Johns Hopkins University under the sponsorship and with the financial assistance of the National Advisory Committee for Aeronautics. The author woula like to express his appreciation to Misses Vivian O'Brien and Patricia Clarken for their assistance in carrying out the project.
TRANSIENT REACTION OF AN AIRFOIL DUE TO UNIT-STEP CHANGE
0 OF FLAPPING AND VERTICAL GUST
As shown in references 1 and 6, the linearized partial differential equation of the irrotational flow of a compressible nonviscous fluid is
(1)
where 0 is the disturbance potential of the flow, the x-axis is positive in the opposite direction of the flight, and the y-axis is positive in the upward direction. The origin is attached to the air-foil leading edge. The details of the notations are given in appendix A. In the supersonic case, the disturbance potential of such an airfoil is equivalent to that due to a source sheet at the y = 0 plane with time-dependent strength and is
- T)
Ø(x,y,t) = 1 2
______________ dT d (2) (M2 - i)h/2 J0 JT1 (T - Tl)(T2 - T)
where v(,+O,t - T) is the vertical velocity component on the wing surface at the point (,+o) at an earlier time t - T • The term
- T) S
is just the source-sheet strength per unit area. In (M2 -
addition,
NACA TN 2333
M(x-) r - T1=a(M2_l)_
(3)
M(x-) r T2=a(M2_l)+•
where
r = M2 - 1(x )2 - (.M2 - i)(y - )2
- yM - 1 . ()
(l°)'
If the point of interest is on the lower side of the wing surface (x,-O,t) .equation (2) reduces to
Px PT
i i I I v(,-O,t - T)
Ø(x,-O,t) = - , I I _______________ dT d (7) / 2 lj2 i
- ) Jo jT \j( T1)(T2 T)
With the above equation the transient case due to. sudden change of angle of attack of the airfoil about any arbitrary point has been treated in reference 1. The important results of that paper are given in table 1 for use in the present development. Two additional interesting cases will be treated as follows.
Vertical Flapping
At time t0, the airfoil suddenly descends with constant velocity i. The unit-step downwash at any point on the wing can be expressed in terms of Dirichlet's integral as -
v(x,-O,t) = (t) = P sin w (t - to) (6) w 2
NAcATN2333 5
wiere U) is the angular velocity per second or a frequency parameter. h(t) Figure 1(a) shows = a(t). Substituting equation (6) into equa-
tion (5), there results,.after manipulation,
_______[x PT2 = - _________________
1 r dJ dTj1(tTt0)&D+
- l) u/2 L o T1 0 - Tl)(T2 - T)
rT2
Hd JT1dT
T1)(T2 - T)j
- 'ix - _____ I &D. - sn w(t - t0 - MX)J0(wX) - - -
- )l/2 J Jo
2(M2 )1/2(7)
or after integration the potential is expressed in three zones according
to the value of.X = v as follows: u(t_t0)
øh(x,_0,t) = --;-
1 (M2 - 1)1/2 sin_i ( - M) - -
M(l - ) -(M - 1 <M + i)
Mv M <=
M
(8)
(M2 i)1/2(
M - 1) M
It fM^i
NACA TN 2333
where
x - X' I u(t - t0) (tt - t0 ') =
M2-1
tu t' M C
______
toU
a(M2_J)to' =__
and J0(wX) is the Bessel function of the first kind of the zero order.
øh C h CLh Table 2 giyes the detailed characteristics of -, f, =- = Fh, 'h
and = as functions of time. Figures 2 to show these charac-
teristics. The detailed methods of calculation are quite similar to those given in reference 1, and are omitted in this paper.
In figures 2 to 14. a few important features may be explained. Within / tI tQ' M •Ch the zone III (0 ) f = -i-- is inversely proportional x M^l,
to Mach number and independent of time and the location of x'. But in
time zone i (_Mt' _o'
= inversely proportional to \M_l x / - 1 which corresponds to the steady case of Ackeret's result. In
zone (_M <- < M ,
varies not with x' or t' - t0' \M+l x' M--l/ ht' - to'
alone, but with the conical coordinate . It is a complicated x,
t t t( function of M and . Thus the present solution is similar to
x
CLh Busemann' s conical flow. So is the behavior of Fh against t' - to'.
NASA TN 2333
7
As far as Gh =M is concerned, it varies both with 1/M and t' - t0'
in zone I. In general, f, Fh, and Gh increase with decreasing Mach t t - to'
number, if
or t' - t0' is kept constant. x,
Vertical Gust
At time t, if the airfoil begins to encounter a gust of velocity , such unit-step downwash at a airfoil can also be expressed in terms of Dirichiet' s
- - v(x,-0,t) = g(t) = gf
LUo
uniform vertical ay point on the integral
(9)
Figure 1(b) shows t) = a g(t) as a function of both t and x.
Substituting equation (9) into equation (5), the disturbance potential Øg due to vertical gust can be written as
pT2 pcw(t_ T -to
øg(xO,t) = 2 (M2 ' 1)h/2[ 'd. dT J
- - 0 o
rT2
I. dd ______
Jo JT1 (T - Ti) ( T2 T)
= (M2 1/2d w(t - to - - MX)JO ) -
2(M2)l/2 (10) -
8 NACA TN 2333
or, after integration, 0g has to be also expressed in three zones,
[ +
( - (M
±
1)
• (0Mi(ll) g
(M2l)h/2 M)
0 (M±l<)
It is interesting to note that Øh and Øg are similar in form. The Fourier integral method of Bessel functions used will take care of the three zones automatically without considering each zone indivIdually. This is one of the main advantages of the present method.
øg Cpg
Table 3 shows the detailed characteristics of c—, = Lg 0g
CL CM = Fg and = Gg as functions of time. Figures 5 to 7 show
fg, Fg, and Gg as functions of time. The detailed calculations are also omitted.
Pitching
For convenience, the essential results,of reference 1 for the pitching case are given in table 1. Figures 8 to 10 show some charac-teristics of this. case.
In the case where x0 ' = 0 or the airfoil rotates about the leading C.
edge at a constant rate (&l - &o) starting at t' - ta', &' = - x(cLl_aO) t t - to' •
varies linearly with but inversely with - M in zone III
I tt - t0t M to ^ I if the angle of attack is small at any instant.
- x' M±1J
NACA TN 2 333 9
/ - ty\ t' - t I In zone I I
M ________ I it also varies linearly with 0 , but xl J
C. inversely with - 1. If M > 1.5, also varies approxi-
x'(al - °o) ti-to t .
mately linearly with and' is continuous in value with zones I
and III although the exact expression as shown in table 1 is quite com-plicated. Of course, as the Mach number is near to 1, this linear approximation becomes worse. It seems better to use the exact solution
C. in the table. Similar statements can be math about F . = a
CM& and G&=,
-
If x0 ' 0 some additional contributions have to be considered as
cpa - = - - Af' (12)
- o)
CLtEL F& =
=F&t -& (13) (al-aO)
G = -. CM&= G&' - (l1.)
(a1-aJ)
f•St LG' a ____ where a = = Fh, and a = Gh are given in table 2 and
X j x0
figures 2 to )4•
10NACA TN 2333
REACTION ON AN AIRFOIL DUE TO MOTION OF AN ARBITRAJy
TIME -DEPENDENT FUNCT ION
In reference 1 it 'has been pointed out that, if the Solution 'of a linear, ordinary or partial, differential equation with a simple boundary condition is known and if the given boundary condition can be obtained by superposition 'Of the simple boundary condition, the general solution o the same differential equation can then be obtained by superimposing the elementary solution corresponding to the simple boundary. In the case of the initial-value problem, the statement is also true. With the transient problem, the important relation of superposition is the con-volution integral. This integral relation is associated with many names in modern mathematical physics. In the transient problem of heat transfer,, it is called Duhamel' s integral in honor of the French mathematician Duhamel (reference 7, pp. 4O3- )4O li-). In England it is commonly known as Borel's theorem (reference 8, pp . 321-328). In this country it is commonlyknown as Carson's integral, particularly in electrical trans-mission. In Germany it is usually called the Faltung theorem (refer-ence 9, pp . 159-167), and this was known to Tricomi. The concept of the convolution integral plays a very important part in applied mathematics.
In aerodynamics, Jones applied this integral to the problem of air-plane dynamics in incompressible flow (reference 10). The results may be considered natural in his problem of ordinary differential equations. Garrick (reference 11) applied this Integral to find, the relation of the Wagner Function to the Theodorsen function in the case of two-dimensional incompressible nonstationary flow.
In reference 1, the present author constructed an integral relation entirely from the physical concept because in this problem of supersonic transient flow the kernel function is discontinuous in the first deriva-tive at a certain instant of time. The ordinary concept of the convolu-tion integral cannot be applied without detailed examination, although the result can be rewritten in quite a similar form to the ordinary convolution integral. For example, the pressure coefficient Cp(tt,xt),
the lift coefficient CL(t'), and the moment coefficient CM( t ') for
any arbitrary a(t) = ____ ____ U
convolution integral of the angle of attack as follows:
or g(t) can be obtained by means of the U
kernel function. due to unit-step change in
NACA TN 2333
11
Cp(tt,x) f(tt,xt)
f(t' - T' ,x'
CL( t ') = F ( t ') [ai(+O)-(-O)] + • F(t - T')
CM(t t ) G(t') - G(t' -T')
-(15)
where F, G, and f are kernel functions discontinuous in first - derivatives as shown in the following table.
ConditionM
t'M M
ot' M M-1 M+l M-1 M+l
f(t t ,xt) f1(t',x') f2(t',x') f3(tt,xt)
F(t') F1(tt) F2(tt) F3(t')
G(t') G1(t') G2(t') G3(t')
The subscript I should'be equal to 1, 2, or 3 according to the above range of time in the term of the left-hand side. Owing to the discontinuity of these functions, the integral on . the right must be broken up into parts according to the range of time for each function. For example, in the flapping case, the lift is.
(a) For .M - M-1
tt
CL( t')I = [h+o - h(_OFl(tt).+.Io
M-1 h( T t )Fl(t t - Tt) dT' +
nt_JL. rtt I M+1
I h(T')F'2(t' T') di-' + J&(i-')F3(t'- T') di-'
Jt'__L M-1 M+1
(16)
12 NACA TN 2333
(b)For M M M+l M-1 I
t , ------M+1
CL( t ')II = [h( +o - ah(-0F2(t') ah(T')F2(t' - T') dT' +
rt'
Ih(T')F3( t ' - • T') dT' (17)
M+l
(c) For o t' M
CL( t ')III = [a(+O) - %(-o)]F3(t') + I ah(T')F3(t' - T') dT' Jo
(18)
Tables 2 and 3 give the expressions for f's, F's, and G's in the case of flapping and vertical gusts. In the case of pitching, the above relation is the same as equation ( 15), except that CL should be replaced by ci as given in reference 1 except for C which is replaced by
in equation ( 15). Another complication in this pitching case is the
location of the axis of rotation x 0 ', which is the ratio of the axis location from the leading edge divided by the chord. The contribution due to the nonzero value of x0 ' is denoted by if&, LF&, and and should be added respectively to F&, aiid G&. As expected, f . .
= -. = Fh, and = as shown in figures 2, 3, and x0 x0 respectively.
As an application of the above convolution integrals, the cases of the harmonically oscillating airfoil starting abruptly from rest at t = 0 have been investigated as follows:
NPLA TN 2333 13
Pitching oscillation about the leading edge (xrf = o)._ In the case of pitching oscillation about the leading edge,
a,(t') = for t t <0
(19) cL(t') = aeiwt = for t' >0
U) where - is the frequency of oscillation, = -, and. t = -. (See 2it U C table it.) Actually, equation ( 19) is a complex quantity, and the real' part is
Real a.(t') = for t' <0
(so) Real a.(tt) = cos t' for t' > 0
j
The expression a.(t') = E for t' <0 is necessary in order that a( t t) be finite at t' = 0, because only finite a.(t') is allowed for pitching. (For details see reference i.) The imaginary part is
Iaginary a(t') = 0 for t t <0 1 (21), Imaginary a.(tt) = sin t t for t' > 0 J
With the complex a(t'), the analysis is more convenient than with the real or the imaginary part alone. If the axis of rotation is not at the leading edge x0 ' 0, the additional effect can be obtained from the flapping case.
Flapping oscillation. - In the case of flapping oscillation,
a.(t) = 0 for t <0 1(22)
- iWt %(t) = U ahe for t > 0
]Jt NACA TN 2333
The solution is given in table 5 for the three time zones. Some of the - integrals are given in appendix B.
Harmonically oscillating gust.- In the case of a harmonically oscillating gust,
a.g(t)=o t<o
(23) a.g(t) = &geiYt - t > 0 J
The solution is given in table 6. Some of the integrals are given in appendix B. All three cases are essential to the supersonic flutter. The present analysis gives the aerodynamic behavior of the airfoil for the whole time history, if the harmonic oscillation In flapping and
M pitching start abruptly at t = 0. If t > M
the present work
should check with reference 6 exactly. Appendix C shows the comparison.
To show the present analysis graphically;. CL and CM have been calculated with M = 1.5 and 3 = ,T/2 for the three oscillations. Figures 11 and 12 show the CL and CM of the pitching oscillation. Figures 13 and 14 show them for flapping and figures 15 and 16 show them for vertical gust. The corresponding values of a.(tt) are also shown for comparison of phase shifts. The maximum CL and CM are larger in the transient beginning than the steady case which is represented by.
M dotted curves. The transient effect dies out completely when ' > M -
It is interesting to note that the extremes (maximum or minimum) of CL or CM always lead the corresponding extremes of a. But in the case
of flapping oscillation and oscillating gust, the extremes of CL or always lag behind the corresponding extremes of a.. As the supersonic Mach number nears 1, the transient effect becomes more pronounced and lasts longer.
In the time zone iii (oM M ),
the expressions for CL(t')
and CM( t ') are very simple in all three cases of harmonic oscillation
f or pitching, flapping, and vertical gust. (Refer to tables 4, 5, and 6.) In the time zone ] (_M t M , the expressions are rather
M-l/ complicated and cannot be represented in a closed form. Three new functions or integrals have to be defined. Let
NACA TN 2333 15
= dO exp( cos (21i-)
where m = M2 1, > 0, M > 1, 0 01 it, and is usu1ly a
function of time t'. As designated by Garrick and Rubinow,
may be called the incomplete Bessel function of the zero order. it is complex except when , 01 = 0. At 61 = 0, equation (2k. ) reduces to the
ordinary Bessel function of the zero order which is real. Simi-
larly, the incomplete Bessel function of the first order can be defined as
r
IT
(
cos e) (25) - =-- I d6coseexp
Ue1
which is also complex, but reduces to 1(), the ordinary Bessel
function of the first order when 6 = 0. The new integral is now defined as
c( ,M; 01) = dO exp(_)
(26)
When 61 = 0, it reduces to
c(,M) = rd0 exp(_-iM ,)
(27) it110 M-cos6
For convenience, C(B,M) is called the C-function and C(6,M; ei)
is called the incomplete C-function. The incomplete C-function occurs in time zone II for all three cases. In time zone III, it reduces to C(,M). This function, -to the author's knowledge, has never been explored before and is investigated in appendix D. In time zone III, CL( t ') and CM( t ') can be expressed with the known funãtions
k C - c (28)
16
NACA TN 2333
except C(13,M) in all the three cases. It is expected that C(3,M) is as important to supersonic flutter as the Theodorsen function is to flutter in incompressible flow.
TRANSIENT AERODYNAMIC BEHAVIOR OF THE CONTROL SURFACE
Since the principle of superposition holds for the two-dimensional linear problem if the deflection angle of the control surface is measured from the main wing, the control surface itself may beconsidered as an independent airfoil at the corresponding deflection angle ö and with its chord CS. Under- suèh a consideration, the lift and moment of the control surface itself in nonstationary motion can be obtained directly from the result of the airfoil in the early section, if the proper time scale is used. The time required to travel:-a length Cs
is t = , and, if t* is used as the time unit, the nondimensional
time t" = - = is connected with the true time and nondimensional t* CS t Ut . -
time t' = - = - in the relation t C -
- (t" - t0 )t* = t = t'
and -
where t"t is the difference in starting time of the control movement from that of the main wing. Thus for an individual increment in as the step function at time t j ", the lift and moment coefficients of the control surface at later times are
CL( t") = j (t i tt)Fj (t tt - ti") I - (j = 1,2,3) (29)
C.(t") = 1(t1")G.(t" - t1") J where F and G are the same as shown in table 1 except that t' is replaced by t'.
NACA TN 2333 17
The lift coefficient of the control surface alone is
cL(t) = 0(o)F(t") + I - t j" ) dt " +
nt"- ^L M+l (t
j")F2 (t" - t i") at" + Jt"--&-
M-1
rt"
-(t1")3(t" - t i") at" (30)
M+l
Then, the increnient of the total wing lift coefficient due to the control surface is
LCL = kcCL()
kc [O(0)j() + ficc (T' )F(tt -
d Tt]
where
F(_) = F1( ._) 0
-M M
J \kcl 2\kcj kc kc kc
,, M t F(—\=F(--\
- i \k/ 3 \kc/ kc kc
(31)
NACA TN 2333
Similarly, the increment of total moment coefficient due to the control surface can be found.
As an example of the effect of the control surface on the total lift, the following calculation is made.
(1) M = 1.5 a = 1000 ft/sec t1 = 0.02 sec (t1t = C6ft CS2ft(kcl/3) x0 0 x14ft
(2) Operation schedule of the main wing:
t'.<O
Ott tit
ti ? <tt
(3) Operation schedule of the control surface:
5(t') = (t) = 0 t' <0
(t')=2 O^t'
=3 3
= -2& t' t 3
5(tt) = (t') = 0 .t'
The distribution of &(tt) and (tt) against t t and the geometry of the airfoil and control surface are given in figure 17. The contri-bution of the control surface to total lift, that is, LCL(t'), is shown
in the lower curve; cL(t') of the main wing is shown in the dotted curve which has been given in reference 1. Since the ratio of (tt) and &(tt) is given, this curve can be used for any arbitrary &. In
4ât1 this curve CLO = is used instead of &.
(M2 - 1)1/2
NACA TN 2333 19
DISCUSSION
For the airfoil in a flow of constant supersonic speed, the tran-sierit effect due to pitching, flapping, or vertical gust will damp out in a time period C immediately after the change in angle of attack
U-a ceases. Although the result is obtained from the linearized theory, it is expected to be approximately true for the nonlinear theory if the angle of attack is reasonably small. After that time period, the lift, wave drag, and moment become the same as given by Ackeret in steady two-dimensiOnal linearized supersonic flow. The transient effect on C,
CL, and CM becomes more pronounced and lasts longer as the supersonic
Mach number approaches 1. The same is also true of the transient period of the harmonic oscillations in pitching, flapping, or vertical gust which start from rest at t = 0.
C. In the case of pitching with constant rate, ,
Xt(& &cj ) (& - CM
and can be approximated satisfactorily with a straight line (al-O)
in the time zone II, if M > 1 .3. When such an approximation is adopted, the case of pitching harmonic oscillation can be evaluated very quickly. It is also easy to evaluate any arbitrary motion in pitching. In the present analysis, the exact expressions in table 3 were used instead of the above approximation because no simple approximation can be obtained for either flapping or vertical gust.
In solving the transient problems with the convolution integral, one new function c(3,M) is discovered. For the time from the abrupt
start ( > M , cp, CL, and in the cases of harmonic oscil-
\ M-1UJ lations have to be expressed in terms of C(r3 ,M ). It seems of comparable importance to supersonic flutter as the Theodorsen function is to flutter in incompressible flow. In the transient time zone II, where M ^ M , a new function C(c3,M; ei), called the incomplete
M+1U M-1U C-function, occurs which assumes an importance equal to that ofthe incomplete Bessel functions. More complete calculation of C(13,M) seems useful for the analysis of supersonic flutter in order to cover wider ranges of Mach number and frequency than those given in reference 6. Appendix D gives some of the properties of C(3,M), C(3,M; 01), and the incomplete Bessel functions.
20 NASA TN 2333
C. Cli The pressure coefficients , in pitching, in x(a.1_cL0)
flapping, and - in vertical gust are functions of Mach number and
t trtt a conical parameter 'only. They are analogous to the behavior x' -
of Busemann's conical flow. As pointed out in reference 1, these results can be applied to a yawing ininite wing, if the leading edge is ahead of the Mach line.
The effects of additional degrees of freedom such as the cOntrol surface or servoflaps can be evaluated with the result of the pitching and flapping of tie main wing as shown under Transient Aerodynamic-Behavior of the Control Surface. With the present basic approach, a good aeronautical engineer with ingenuity should be able to solve all two-dimensional problems of flutter and any other arbitrary motion. Of course in the case of complicated time-dependent functions of angle of attack some numerical or graphical integration of the convolution integral might be necessary as shown in references 1 and 10.
For the case of the harmonic oscillation with constant maximum amplitude with abrupt start at t = 0 the absolute magnitudes of the extremes in CL and CM for pitching and flapping degrees of freedom
are larger in the transient region N Q < Mthan those
M+lU M-lU • M C in the steady region M 1 - < particularly as the supersonic Mach
number nears 1. In addition, the maximum and minimum of the oscillating angle of attack are not in phase with the maximum and minimum of CL or CM. In the case of ' pitching, the load leads the angle of attack; in the case of flapping, the angle of attack leads the load. Also, the angle of attack leads the load in the case of vertical gust.
As an interesting example of the harmonic oscillation building up to flutter or damping out, the case with complex w should be inves-tigated. With the convolution integral, it seems within the reach of the present analysis. Of course, the imaginary part of w must be determined from the interaction of aerodynamic forces and the elastic behavior of the wing.
The Johns Hopkins University Baltimore, Md., November 4, 1949
NACA TN 2333
APPENDIX A
SYMBOLS
a velocity of sound
C wing chord
CL transient lift coefficient
CLO two-dimensional lift coefficient in steady case
CM transient momeit coefficient about axis of rotation
C pressure coefficient (2( -
C3 chord of control surface
c( 3,M) = dO exp I 1 N / -ii3M) - cos =
dO exp tM - cos U ltJo
M I rli(
(a,M) =i I dO
/
) exp
it! M Uo '-cosO
it C(3,M; °i) = I dO exp (M e) - cos Jo1
F( ) kernel function concerning lift coefficient
f( ) kernel function concerning pressure coefficient
G( ) kernel function concerning moment coefficient
velocity of vertical gust
.maxinium velocity of uniform vertical gust
21
22
NPtCA TN 2333
descending velocity
ii maximuxnvelocity of vertical flapping wing
j0 ( ) Bessel function of zero orderri
j1 ( ) Bessel function of first order
k == cC
M Mach number
m parameter of Mach number (M2M__i)
t time
t' nondimensional time (ti)
t*=CS/U
nondimensional time (control surface) (t/t*)
U free-stream velocity
v y-component of velocity
x axis along chord direction
x' percent of chord (x/C)
axis of rotation of entire airfoil
y vertical axis
a. angle of attack
constant angle of attack
a.g /U
hhh/U
NACA TI 2333
23
frequency parameter for oscillating airfoils
deflection angle of control surface
source location along y-axis
0 running variable
lower limit of 9
= x/a(M2 -
v - x/U(t - t0)= x'/(t' -, t0')
source location along X-axis
T time interval
T t nondiniensional time interval (r/E)
0 velocity potential
frequency parameter (angular velocity per second)
Subscripts: -
g vertical gust
h flapping
& pitching (changing angle of attack)
control surface
2 NACA TN 2333
APPENDIX B
SUMMARY OF INTEGRALS
Important Integrals in the Case of Steady Harmonic Oscillation
The following Integrals are used in evaluating CL and CM for the
time zone 'M
(as shown in tables , 5, and 6)
IljM1e1TI + 'sin-1 [(t' - T') - e') + 2e'
12j elTt(t - T') + s1n1 [m(t' - r') - M]} e1(t'_ + i) +
e1(t')[+ -
I f -
- r )2 - - T - 1)2 dT' = Mi e1(t' - t'_ 1M2_1
I r' - eT'(t Tt) (t' - - M2(t' - - 1)2 dT = Mi e1(t'_ 0() +
iM2 -1 M-1
( -
15 rI_1 e1' cos' Hie_1(M0S9)]
NPCA TN 2333
- 25
16j' M1 e 133T ' (t - T') cosl Mt - T - =-
M t .1 133 M+1 t , --M-1
M -i33 1fil +
-e Jol.in)j 133
15
/
M 17 M+1 e33T'(t1 - r) 2 cos 1 Mt. - T - 1 di- = -
Jt M t -T i33 L\M+hi M-1
id43TO(rn) + (M2 - i)3/2 - (M2 - 1)3/2 e 1()] +
18 I et33T'(tI - T') 3 cos 1 M - T 1 - 1 dT' = tJ t ,• ._!L t' - T' 113 L\M i- i/ -
M-1
(2/2+ 1)J() (2M1 + + '7
M
191' - e1'(t' - T )2 (t' - T )2 - (t - Tt - 1)2 di- = M3 e133(t' -
tT_T S (M2 -
1 1 (M2 +1) + (M2 - + 6(-)2 - (2M + 113
26
NACA TN 2333
Important Integrals in the Case of Transient
Harmonic Oscillation
The folloving integrals are used in calculating CL and CM for
the time zone M <, < M(as shown In tables 4-, 7, and 6). M+l M-1
Ii
feP' + sin 1 [m(t' - T') - M]} dT' = -+ cos1 (mt - M) +
2 e1(t' - ) r etBm ãØ iJco1 (mt'-M)
Ct? M+11
cos1 (mt' - M) -2 I eT'(t - T') + sin (t? - T') - M] dT' = - + ()2J
I + 0
2e(t'
1 (nt M)+
+ m m j m
13 ' Lt? - eT' (t' - T' )2 - M2(t' - - 1)2 dT' = 1M
4i - (mt' - M)2
±(t'_) re oDsØ Me m
cos 0 e \M2 - 1 1 J06-1 (mt'-M)
MACA T 2333
27
e'(t' - T')(t' - T') 2 - M2(t' -T' - 1)2 M Er- - ( j)2]1 - (mt' - M) 2 -
M e(t'_
L:s 1 nt' M) '
M cos 0 2 cos cosØ
____ +)e dO
M2_1 i - m
(e 1 ') cos M - T t - 1 dT' = et - - eMos 0 e -
5. -Jot' - 1 i
&Icos_l M(t::1)
M(t' - 1) t'
I
16' I M+1 etT'(t - T') cos1 Mt - T' -1 dT' = !_ e1t _) - cos1 M(t' - 1) - - T' ]. M + 1 i3 t'
M e1(t' _) rcosø
1
____ I e m dØ+—I'
- 1 llcos_1(nitt_M)
17'MM+l eT'(t? - T') 2 cos 1 M ' - T - 1 dT' = (2ei(tt_ ) - cos1 M(t' - I) -
J
. t - j\M +1)
0
M2 e(t'_) r5
3/2 i J(M + cos Ø)e m dØ + 16'
(M2 - i) cos1 (mt'-M)
NACA TN 2333
18' rM+1 eT'(t - r') 3 cos' Mt__-
T' - 1 , n / M 113(t'_ ) M(t' -1) - - cos
- tt -T' 113 t' - Jo
M3 e113(t' _)It
cosø m
1130)e113ãø + + - cos
(M2 - )5/2 113(M2 + 1 + 2M cos 0
(mt' -M)
-- 1 - (mt' - M) 2 + 17' 113 (M2 -. 1)3/2
19' Ml ei13T'(t1 - T t ) 2 (t - T1)2_(t - Tt _1) 2 a7 , = - M3 1 - (mt' -
- Jo(M2 - )5/? 113
IC
(M + t') + 6(\21 - M3
e113(t_m) ios' (mt'-M) +
+ 0 [2 M + 1+—+ 1(3 113 113 \'1 ' (M2 - )/2 113
6(]IcosØ
- dØ
NACA TN 2333
29
APPENDIX C
CHECK WITH NACA TN 1158 -
To check the present results through convolution integrals with Garrick and Rubinow's work, a case of the steady harmonic oscillation of the angle of attack about the leading ed€e is investigated. With their notation, CL due to the harmonic oscillation of the angle of
attack alone (a = a0e t) may be verified from their equation (26) as
2 iwt = k e cx.0 (L3 + iL)
At = 0,
L3 = L3 t L3 = L3 t - 2x0L1
- L4 = 4' 4 = 4' - 2x0L2
At M = 2, for = 1.667, from their table II,
L3 ' = 1.50219 4' = 0.69968
Hence,
= a0(2.16315 + i.0075#i)e1°
(Cl)
30
NACA TN 2333
NGTATION
NACA TN 1158 Present paper
t t'=LL C
CD
b C=2b
v U=aN=v
V
=2k
exp (it') = exp (iwt)
In the present case a exp (it'). From table l. for t' M M-l'
_____ M2 )]
[2 + 2 (2 - i) i 1 CL(t') = i exp [(tt M - 1 - [M2 - - 1 +
( r 2 1(1 +
+ exp (it') + + _____ 1 1
m • iii 2 LTMITM -1]
2 lIj c( M3(i)2jL
whence, at M = 2 and. = 1.2, CL = exp (itt )(2.l6315 + 1.007531). The expressions are identical, within computational limits.
NACA TN 2333
31
APPENDIX D
INVESTIGATION OF TEE FEW IMPORTANT INTEGRALS RELATED TO
SUPERSONIC FLUTTER AND TRANSIENT PROBLEMS
C-Function
The problem is investigation of the C-function or the integral
______ ( >0) c(,M) = dO e (M -iM
(M > 1) It , I - cos Jo
If the above integral is differentiated with respect to J3 and M,
(Dl) - 13 r deEexp (-i cos 6
-M - cos - cos 0)2
C -i _________ _________ =
dO(_- i3M )1 M (D2)
- cos 0_JM - cos e
= -. 1' dO[exP (
M(D3)
J0 LM - cos eij (M - cos e)2
With the above three relations, it is found that C (13 ,M ) = CR + iC1
satisfies the differential equation
(D14)
32 NACA TN 2333
If the real and. imaginary parts of C(3,M) are taken separately,
CR CIMCR 2 f34
(D5)
CICRMCI 2 3M
which are two simultaneous differential equations. The required six boundary conditions can be obtained from C(,M) as follows.
For =O,
CR(O,M) = 1 C1(o,M) = 0
-M
For M--*oo,
CRCOS C1=-sin
flit ii 1 The integral -
d exp - M \ can be written as (M - cos
(JO
(a,M) = 1' exp ( ltd M-cose
where a = 3M. Under the new definition, the function C(a,M) can be shown to satisfy the differential equation
(D8)
(D6)
(D7)
NPCA TN 2333
33
or
a CR - 2 M $
(D9)
a =
It is easy to show that the real and imaginary parts of C(a,M) must satisfy the equations
2 / 2'-\ 2-a—k 1= O^a^oo
a2 \ 2kr2 / M2
and
(Dio) -
2 / 2\
___I l^M^o
a2\ a2/ M2
The corresponding boundary conditions are:
- - r 'I -
rur U%J Ni,
CR =l -I=o
-1 a
2CR M ___
____ - (M2 - i)31"2 a2 -
For M—oo,
CR=l
—=0
CR —=0 —.=o
(Dli)
(D12)
3 NACA TN 2333
The preceding function is so important to the supersonic flutter problem that thorough investigation of the function and the other solutions of the corresponding differential equation seems advisable. To show the general behavior of this new function, a case with M = 1.3 and 0 J3 3 has been calculated numerically. See table 7 and figure 18.
Incomplete C-functionLI
The incomplete C-function is
flTC
c(3,M; 01) -
tJ01
exp (M
dO - cos (D13)
It can be shown that this incomplete C-function also satisfies the differential equations (D5) except the boundary conditions are obtained from equation (D13).
For J3 = 0,
C1=O -
(D114.)
= 0 = -M - a tan- AI M + 1 tan
3 4M2-1L IC \IM-1
For M—)co,
CR (ei\ = - -J cos C1 = _(i -
sin (Dl7)
To show the nature of the incomplete C-function, a case with M = 1.7, = and. 0 Oi it has been calculated by means-of numerical inte-
gration. See table 8 and figure 19. The above data are used in the tran-sient behavior as shown in figures 11 to 16.
NACA TN 2333 35
Incomplete Bessel Functions
The incomplete Bessel functions are
nit
J0 (z,Oi) = I exp (+iz cos 6) d6 (D16)
J e'
J1(z,01) =
cos 6 exp (+iz cos 6) de (D17)
These two functions beconie J 0(z) and J1(z), respectively, when
= 0. They have been called the incomplete Bessel functions by
Garrick and Rubinow. The case M = 1 .5, 13 = and 0 61 < it has
been calculated by numerical methods. It is interesting that each has a real and an imaginary part. At 61 = 0, both become real and Ordinary Bessel functions. Both play as important rOles as the incomplete C-function in harmonic oscillations at the transient time zone II. See table 8 and figures 20 and 21.
36 . NACA TN 2333
REFERENCES
1. Chang, Chieh-Chien: The Transient Reaction of an Airfoil Due to Change in Angle of Attack at Supersonic Speed. Jour. Aero. Sd., vol. 17, no. 11, Nov. 19148, pp. 637-657.
2. Blot, M. A.: Loads on a Supersonic Wing Striking a Sharp-Edged Gust. Rep. No. SA-247-S-7, Cornell Aero. Lab., Cornell Res. Foundation, Inc., 19)4.8.
3. Schwarz, L.: Plane Nonstationary Theory of the Wing at Supersonic Speed; an Abstract of a Report with the Same Title. Translation No. F-TS-93 14--RE (ATI 225611. ), Air Materiel Command, Army Air Forces, March 19 )4.7. -
. Heaslet, Max. A., and Lomax, Harvard: Two-Dimensional Unsteady Lift Problems in Supersonic Flight. NACA Rep. 914.5, l91t9. (Formerly NACA TN 1621.)
5. Miles, Joim W.: Transient Loading of Airfoils at Supersonic Speeds. Jour. Aero. Sd., vol. 15, no. 10, Oct. 19 )48, pp. 592-598.
6. Garrick, I. E., and Rubinow, S. I.: Flutter and Oscillating Air-Force Calculations for an Airfoil ma Two-Dimensional Supersonic Flow. NACA TN 1158, 1914.6.
7. V. Krmn, Theodore, and Blot, Maurice A.: Mathematical Methods in Engineering. McGraw-Hill Book Co., Inc., 19)4.0.
8. McLachlan, N. W.: Complex Variable and Operational Calculus with Technical Applications. The MacMillan Co., 19)4.6.
9. Doetsch, G.: Laplace Transformation. Dover Publications, 1914.3.
10. Jones, Robert T.: Calculation of the Motion of an Airplane under the Influence of Irregular Disturbances. Jou'r. Aero. Sc!., vol. 3, no. 12, Oct. 1936, pp. 41914.25; A Simplified Application of the Method of Operators to the Calculation of Disturbed Motions of an Airplane. NACA Rep. 560, 1936.
11. Garrick, I. E.: On Some Reciprocal Relations in the Theory of Non-stationary Flows. NACA Rep. 629, 1938.
NACA TN 2333
37
TABLE 1
C008ACT001000CS OP AN AI000IL WITH CHANGING ANGLE OP ATOACK
[on' 0; are flgo. 8, 9, ao,t 10. For o' 0 refer tO table 2 nod fIgo. 2, 3, nod
ZOOe I loGe II Zone III , t -to
^ N ' O' 8T 3 <H
(.)2 f t ' if 1ikt' -tO')- *
(2 - 1)u/2l,,[ °' 2(02 . l)j° 1 L o'
-O,t') -)2 r (02 - 1) 1/2 -
1 ...—.coo o__)o (M2 l)l/2Tj] N
(t'-t&)2( 0
n(t' -tO'))- M2(1 - __)2])
2((O+ ![+ 010 [o(t -to) - * (02
C(',-O,t') 2( ... t' (w)
coo-i (o' * (02
(202 _ i)(t' - tO') _Mh(1 20(021)hi2 ' t' .0'jj
-20000
-to'
20 I 00 I
--t' -to' !L1Zone III
ot' o_1Lr
•-tO)l4]}c
(02l)V2([2lj1[2
- tO') I[, - 0,• (02 - 1)V2[, - t'
0- t')° 11 -1 (' - to' - l\ 'c°° M
(t' -to') 2 (t' - ')c
- 8) (o2 - i)i/OL 2(02 - i)J , Mo [ 102 j - to
-•
___________ (M2.i)i/(1, - tO') M2(t - tO'- 1)2)
FO 13to)2
0lo [o(.' -ta') (o2 1) i/2 ([ 2 3(M2l)]
(02 - i) i/2 r(t . t')3 t' - to' ii -i ft - - A * * -j coo
M()
c (t' - to') 3 2(t' - to') ta') I It ' - to' 02 - 2 1Ma [ 02 - 0
(&j6)010 (M2l)l/2[2 3(M2l)j
- 1)02[02 - , ,2 - o,, - ) ° I - -
:.:_]j(t - to , ) 2 - M2(t to, - 1)2)
tFrOm reference 1. -
0 0
iL
0 A H en
en H
H 0
0 en
0 k-I
en
H
H C)
en
cc
c.
C, C,
C,)
UJ
38
NACA TN 2333
H HC)
H H Vii 0U
C. H Vu - 0J
:9 ,Z' .9 :;.,
C
C, C.
C
-IZU
H
Vii
CUIX
Vi,
ten
0 0c')1
H
-8-
,9 +
+
- 0 - OH
•1
.9 - .9 .9,•1
Z I' -
0.9
-Z ri .8 -
H C') .8.-
r?
- _.
c•:
4± H H
-H I.
,C, T
?Jcc (kJjCC -
-..- - H
_
IC C') ccii z
C.
:9L...
c Hk
H
:9
__
c9 c I:9
Q Vii C') H
C,
C')Vi'
-H - H H X C') -4 - cci
z I I
c II C -
.9
Ii U H
' i -t
H
d'
C)
0 H
C) -
0 H
V
NACA TN 2333
39
C-
C, Cu 1-4 .
I.t
H Cu C'
In 0 0 Cu 0
Cu Cu - Iii
Cu Cu Cu
H H E-4 0
0. 0 01 Cu 0
C-I Cu
C'I.Cuu
L
'-1
InCu
Cu Cu I-I Vii 1-i H -
Cu en
H i-i 1-4
VII
- — -
II
Cu
o 01 +0 0 -
14
o 4, ii
o
0) 0
0 4'
0 4
-0 4.
Vii Vu -OlCu
o 0CCIX
Cu CU
0.
-
I
- 41 0
-
0 0
4' I
I
4' I
I - 41
- Cu
Cu
Cu
'-I I 0 14
H 4? H
_Cu
-
CI)-Q
H H H H HO
o 4'
-- 0 44 CI
- Vu
-14
C') -__- H -
14 I I
CU
•Cu H
4, II
H __._ H
CU
1-4
0
-0 4'
I -
4' VI
-4 0. II
-
HH
H C, U
HH
Cl)
CuL:
- -
44 -' 0
Cu
II
Cu C,
4) - -
o Cu I Id
I-
- III
-
0 4' 0
4'0
4'
Cu 0
- 44
Cu - Pd
I Cu -
Cu 4,
Cu 0
Cu Cu
0
V
- -
NACA TN 2333
TABLE
CEAJOGING ANILE OP ATTACK
rm .Ae12t t >0 I 10 the C000 0. See P1gm. 11 emS 12. Lma 0>1,
2000 i TAme 13 ZC iii
< N I0+1
i0eiEt' C * 2 r - MMII
2 CEO 4 IleIlt 203)10)2] -
211 1E( H
Lo-lotM - j)1/20
['' CL(')
e( (G*)Jø(
-
(t + (M2 )1/2 -
0
(;iPt
2Ole ), -
] - o(M2 - j)1/2
(2 )1/2
-
- (mt - H)2 - T(t + 2} 1) o-i M() + t 412 -
1101Et'1.. 2 2 E P0
Ii . I ';we "
L34 + 201 C 03(11)3]
LJcom1N(1)
J -
11010t 2(312-2) 2]t0E --
e(12 - i)1/2J(mt-N)
- j10 1Et 2M3UE)3]1 -
)0
o1l012t' *
lE) - CN(t' -
- /2j(Oh-M)
+ COO
CN(t') 0 21Bo ,
34250- 1]) -
[1(M2 - 4)
1 r 2 2t 3 II 2)12 * 2] +
1E1 (' - )2
L34b0( - 1)1/2 Seo(- )3/2 -
2111(t__)C 2
M(M2 - 1)h/2A(-3) 1
- GE.2 0 0 2412 C com- M(!') - - 1)h/2Ad (12)2]
Oim(' - M C2
- -1)' (M2 - 1)1/2
mC-fu000tloro C(),M). - 00 lmComplete C-f000ctioC C(B,M; O j); see appondlo i.
tlnCCmplete Bessel P0000CtI005J see mpp000dl+ 11.
NACA TN 2333
+
HH
-
H1HII -
Vii Hci H
-+
cli Ix
+
1 II ci +
+ '1E I
I
'lb.
__
1'°
0I "ci H 'I'
ci
H- I
L'lb ii H
I 'H I ci ii) cii
L U H- I
Vu
H.
I ciI
U _
L__,,H 'I-v
H - H ci ciH
x + H ci 0 ci H U 'D H
________
HIH 0 HIci ZIE H H + +
HHI
I_ i '
'H I + ' -
x "H H T ci ci ci,
+ *_
+lb
U ZIo
(cc, Z ci cOlE H o H ci, I,
LH
,
+
H Vii (H
lb
ci+.
cOlE HIci
Ix0
I H ____ H H H HI ci +
+
__ H
I
HH H
-x1E1>
I ZlE ZIEH
Z(HI H
-lb
Z
Iii Ilf
______ 01 _____ ______________________________
"I H 'ci ci ci in H ci
if\ q-i cci -ii H ii ci) H
ii 'ci ci
V
0
1<
'ci
0,
ci H 'ci "ci "ci 'up' ci' H ii) lb ii
lb ci 0 H 4.' C) ci 0 H 4-i
U H ZOci
10 lb 4-, ci '—Ic,
00 ci"', 0+14.' H ci ii, +1 H H
00 100 o H -I *
NACA TN 2333
H z+ II + x
VH
4-'U '4 '
I- I
* * _________
+ U r_ I 01 C)
I
_
"-I F 0
I U
U
Xl "4
X Vii
U "4U 0 U
U
1+ z H + r1
U(U
-' H iU h 0 H
H I X I CO
0 U
I
HH I H ____
l I I
[ H
_____
ii + "4 "4 --4
-H
I XH
+
rzzi
FH -
I-II '4 U U U 0
(U HIUHIU
II H
-
14-' (U
X '4 "44"4
- X
-' 0 C)
p..,-' 14/
V
'4 0• p'
0)0 0)0)
HO) U
U)- 0 '-0 0 '4 4-, C) 0 - 0 '4 -.4-' oH 04 U I U 0
00)0) 0+' 4.' H U U +'H-4
2 0 0 c-IOU '00 o I-I H * *+-*
hi i C) H E4
T H U
'4
U 0) C/)
4-' U)-"40)
Iii
LJ
NACA TN 2333
43
TABLE 7
VALUES OF C-FUNCTION c(,M) FOR M = 1.3
Real . Imaginary
0 1.0. 0
.3 .84153 -. 41796i
.5 .62101 -.567971
.8 / .32000 -.569621
1.0 .21844 -.498441
1.2 .19147 -.
1.5 .17883 -.477121
1.7 .12058 -.534231
:2.0 . -.4808i . -.562791
2.2 .i6005 -.511541
2.4 -.22697 -.42491i
2.5 -.23982 -.38123i
2.93 . -.22265 -.293551
w
141;. NACA TN 2333
TABLE 8
VALUES OF INCOMPLETE C-FUNCTION AND INCOMPLETE BESSEL FUNCTIONS
[3 =/2, M = 1.5]
c(,M; °i)
O -0.07339 .O.14.71)481 0.29057 0 0.581148 o
.2 -.04952 -.534931 .30946 .06078± .52110 .018801
4 -.02526 -.79317i .32378 .122781 .46198 .O3249i
.6 .02500 -.630791 '.32891 .i86i4± .4o647 .037071
.8 .08770 -.634931 .32078 .249101 .37837 .030881
1.0 .14434 -.607201 .29582 .307701 .32204 .017731
1.2 .1811.35 -.75819± .2514.17 .357361 .30020 -.00289±
.20575 -.4981481 .19871 .385801 .29172 -.015721
1.6 .21084 -.435171 .13598 .3942Oi .29074 -.021901
1.8 .20310 -.372061 .074511. .379001 .288i1 -.O14O9i
2.0 .18769 -.310871 .02257 .3112811 .27614.3 .0021181
2.2 .161211- -.252121 -.01429 .291251 .25027 .020821
2.4 .13173 -.195731 - . 0 3 407 .230961 .21010 .033771
2.6 .09867 -.lIi-1331 -.03808 .167561 .15938 .036831
2.8 .06326 -.o8844± -.02980 .10450± .102148. .029291
3.0 .02644 -.036501 -.01367 .042951 .04279 .O1379i
3.1 .00778 -.010711 -.00-io8 .012591 .01259 .004111
0 0 0 0. 0 0
F'
t
(a) - ah().
x
0 -
NACA TN 2333 15
(b) = ag(t).
Figure 1.- Angle of attack as a function of time t for two cases: (a) Flapping wing and (b) wing meeting a vertical gust.
NACA TN 2333
('J - 0
\
3
\
\
--- -------
- - IIIIErTiIJIiI
0 •e
pI 0
•-. 0
-4-,
0
o r o
-1--4 -4-, 0
Cl) 4-4
--
•d 0 czlb
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NACA TN 2333 1
Fh
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3 t,—to,
Figure 3.- Plot of Fh or
numbers. Fh =
a as a function of t' - t ' at various Mach xo I
C C. -; AF', contributionto La
a h a1 -a0
NACA TN 2333 'Nil
i.5
Gh or
xo,1.c
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I AA
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0 1 2 3 t - to!
Figure 4.- Plot of Gh or a as a function of t - at various Mach
numbers. Gh = CMh; AG a' contribution to . a1-a0
NACA TN 2333
iiiii iiiiiii. \ - - - \.
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50 NACA TN 2333
riD
Old
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NACA TN 2333 51
---
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NACA TN 2333
^9 N•IX 4-iiI
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NACA TN 2333 53
C!) a)
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511. NACA TN 2333
Cl]
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NACA TN 2333 75
T ('A b
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NACA TN 2333
tiiizi
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NACA TN 2333
57
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58 NACA TN 2333
2
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NACA TN 2333
79
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60
- NAA TN 2333
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6
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I'L&CA TN 2333
61
0 2 4 6 8
Figure 17.- Distribution of a (t') and (t') against tt and geometry of airfoil and control surface. Contribution of control surface to total lift shown by curve at bottom of figure. CL(t') shown by dotted curve (from reference 1). t 1 ' = 5.
62 NACA TN 2333
- 3
IIIEfIIIIIIII,±
: ---
Imagina y rt --/
-
-
/ /
/ / N
-_I 3
Figure 18.- Complete C-function C(p,M) against p for M = 1.3. M
C(p,M) = eM - °S e de; = nJ0
2
11 C (, M)
_2o
NACA TN 2333 63
OID3S IJDUIbDWI
0 - N IL)
N
.7N
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/
/
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c.
0
64. NACA TN 2333
4
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Jo(1Oi)2
0
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0 I 2
Figure 20.- Incomplete Bessel function of zero order 0 (P... e 1 ) for M = 1.5 and = n/2.
NPCA TN 23-33 65
.51 I\ I
IiI\IIIIIIIIII H
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02
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0 I 2 31T
Figure 21.- Incomplete Bessel function of first order i (I , e ) for M=l.b and =n/2.
NACA-Langley - 4-4-51 - 1050