Tables of the Theodorsen circulation function for generalized motion

(1)

-e

ARCHIEF

Stik.ntARy-Important in all stabitity aeroela,,tic, and flutter probleins

tlac Theodorsca CoO fimei ion.

las of

his fun, lion are well

known if the Motion is 'harmonic

Heretofore, tables of the

fbcodorscri function for generalized mot ionlaave_not been avail

in this report the latter tables are given foi a range that

should be sufficient to ',fleet the need of aerodyniiT ifrists P

Jones has given ri disenssion of .thr, generalized Theod:orsen flute-tim

but his result- appear

cOr.- Lt, for, in his. 'analysis, the

Th odorsen function asderived by approaching ha,monic motion

frt u the side of 'stability and from the. side of 'xistability is

lit,

.

If the jomN

-were- true,. the norma ,meness of the

function would imply that the Bes,i I functions are

Alia continuous, which is indeed false, The dirE, nity is

clanin

his paper.

(f.,))

.imeizooncnolic

()

m i

NE OF TIM MOST BASIC FUNCTION,,

,snb

sOnie.

unsteady aerodyna

ics

s. the Theodorsen func

ti-oiel

This function is important in all types of

sta-bility, aeroelastic,. and flutter problems.

In the case

if incompressible flow,. it

is needed to compute the

values of the flutter coefficients.

Inipro3: rig an

iter-teee method, Dietze" uses the function to calcuirtg

flutter coefficients for subsonic flow.

Au alternative

iterative procedure for calc ulating Subsonic

flutter

,coefficients has been suggested by .1\ilde

arid here

tor, the starting point is a knowledge of the Thriodorser

function.,

Another application is found in the acre),

dynamic theory of oscillating wings of finite spar .4

The curt discussion rendered is by no means complete

1)ot is presenttel only to show the signilic mice of the

Theodorsen fin

on.

-(3)

DEVE.0113ENT OP THE TliEdnoRSEN

'CIRCULATIoN

/ON PoR OENERALI: En MoTtote

In the treatment Of semi:drain-nit oscillatory prob

'Itecis, it is assiiikried that the- mutton follows the law

.where

in the time varialki and

411

(ii

us,,i g-ives

c'uquency-o.i mcitiore, w rule p deefint

damping.

4

0, the 'motion is t nyerg;ent

bk.; if

> LI, the .niotiou is di iergerii or unstabli

July . ' I

ieritber3

10.01

The angle,

to thaw t

'teal nate

pi 1.3.olore.s elf tille..e!: iiu staff if the Midure!

,eitthi. cm-April 114.. ..reporti.

1 Tke.sia,,i I.fri t.

r..eseareb K r

-

Lab,

y, Scheepsbotmkunde

Technische Hogeschooi

rfct

s

t lie The( (lorsen

-Ftinction

Gelteralized

UDELL I-. LLKEt A.N'u, MAX A Li ENGLE i

11 dwed A,o, arch In ill

4703.

- Fr?

When p

the motioe is of constant amplitude ,ana

is said to be nentrallystable.

Heretofore, for the -,nost part, the nattirt

Theodoesen function h

_only been considered

a,

= O.

le a paper by Joues,7 the generalization

r etns

function is studied.

his conclusions are rat/it r

dis-turbing, and it seems appropriate at this time tha the

difficulties be clarified.,

To accomplish this efferiii

a derivation of the Theodorsen function C(k) for

eral InetiOnS will be presented.

For u

stalling pi

n"'

consider tht equalitys'

_a7 +

C(k) =

----

dx

_

\/2

where it is supposed that

L oe.:ItYgoi .+

and that w >

It is well known in the there

Bessel fimetions9 that

fK

, (z)

cosh ny dy

which is valid if targ.

< ir/2,. This requires L

Re(z) >0

Let

z. =

Thus, if Re(s) > 0, it follows that p > 0, so Ole th

motion is assumed to he divergent.

Combine

--(2) through (5) with the well-known relation

Ki(z) =

it is re.adil y deduced that

= /4(21,0

,V02)(k)

which is the generalii,ed Theodorsen ;CIO funct

is true th it, in the development of Eq (7), it w

sary to, re:

vire that a > 0.

HoweVer, Eq.

,

need for -.lick a. re, ,eietion, and therefore

method 01 analytie

ijiu,uatin. n e can at

Eq (7' is

ibd for ull" A, a.

In tiiu

Leeching ar4

,4s the freciii, ecy of tt

wasted. c P spoihtzN,

(),

nr.,1

SO lb

t(1,)1I.

ced la:7 its

r

dam k

y tf.y. &e:1 t s,

(01-44

*gate

'.3"

t

A) = it

I t ) W

= 0,

1 I

Delft

(2)

TA

S

()F

')'HI,.

1'11 E01)01(SEN C11.4 CULATION FlN-(24'10N

479

1 'ere the frequency of the motion is negative.

In all

problems in

the Theodorscn function, current

practice is to assume tha I the frequency part of the

motion is positive.

In this manner, many difficulties

can he avoided.

If t he motion is nonoseilla tory but divergent so that

0 and /./ > 0, them

(AA) = 114(A)i"I1\71(P) -F

IQA)i

(9)

If thc motion is nonoscillalorv and convergent, then

Eq. (9) must be replaced bv

Ki(12) -4- 7rir4)

=- -

KI(A) Ku(p.)

riLlt(A)

-I- UM)]

(10)

where i is now treated as a positive 111,1111Tc.

rev?, ; ,/\

-

F0,0')

=

,

p,

=

0

-pou -0 i 1 0 0 0 01 0.07919 12 0.01396 94 0.98212 15 0 01505 21 0 08549 00 0 141717 15 0 95571 13 0.04851 14 0 02 0 21 01407107 21 01496372 52 0.077...0 79 0.90531 08 0 07539 10 0 9-.121 65 0.05130 30 0 (13 44.'14 ,911 12 0.09:29S 13 0 91701 10 0 07.'71 35 0 ¶474555 0 10209 30 0.07037 13 0 10729 SO

0 0:

0.92127 37 0.11067 :36 (14(24)7)) 18

0 lion) 13

0 93204 31 (4191,7 HI P 93144) W. 0.12539 71 0 07 (4;0359 65 0.12308 :ill 0.00900 90 0.139G: 11 0 9,543 33 0.13521 25 0,02162 01 0.11516 91

in; ;),.,.;,;65 1).1:;:;!t5 Solo:; 7 o 112719 o 5!?,7197 19 11 151:12 08 0. ¶1r1I05 45 9.15)109 51

0 S,J70 17 0 112s1 0.57751 27 0 17248 17 0 s573 (41 0 li;2!:2 0 .7,,2 21 0.17200 30 08 25 0 15004 St 0 501;13 18 0.169!0 2i 0 ,G 70 65 0.17102 35 0 ;,-:109 73 0.18192 70 00

Is;) i9

0.15793 75 0051579 0.16000 15 0 55G10 83 0.17532 92 0.55618 78

0.19000 50

9 In

0.82790 32 0.10071 fir 0.53192 41 0.17230 22 0.53637 59 0 18420 01; 0.81132 73 0 19074 41 9 11 0.51727 81 34 01156 70 0.51577 79 Cl 170,1 95 0 84170 87 0.15911 24 (1.52712 77 0 20218 13 3). 129 92 0 0:7G.; ;;; s116:.:2 7:3 Cl 1s4, 27 0 N,J73 89 I) I I I 51::7ti tft) 0.20655 96 ; 13 0,7910s 1 I",997 10 (4 791,3 72 iI I ;-12 4).191;,9 00

n 80009 20

0.21003 19 I) 0.75128 51 0.17100 17 0 118337 15 014515H 01 0.78573 52 0 (91)7.771 0.758:2 G9 0.21272 77

III.

0.77110 71

0.17325 20

0 77279 91

H18:;5 56

0.771(11 93 0.20025 57 9.776,91 61 21475 60 0.10 0.701 1,9 01

0.17125 15

0.76277 PI 0.18770 59 0.70112 76 n 0 76705 61 0.::1021 22 0.17 0.7.,251 93 0.17tss 111 0 753.6 911 0.15s25 G5 0 75111 20 0 D,f;.-, 70 0 77717 91 0.21717 45 0 15 0.; 1::92 07 0.17720 51 0.71127 70 0 18567 27 H ,4106 -17 I I ₁₄ 0 71515 89 0.21771 21 44 1)'

0.7.1,0 30

0.17525 13 73770 31 0 15577 !s 73500 50

((.21:114(

0.73705 74

0.21758 42

H 20 0.72591 H2 0.17513 50 0 72757 99 0 18862 42 0 72711 59

0.2079 72

0 72002 89 0.21774 11 0021 0.72,16 07 0.17150 22 0 710,4; H8 H 1,520 17 0.71899 12 0.209,1 10 0.71504 09 0.21732 70 9.2" 0 71360 36 0.17440 4

0.712:2 II

H 15772 32 0 71120 59 0 2;4151 90 0,709,5 75 0.21065 02 0 23 0.70700 45 (417367 44 I 70553 74 0 I 57i2 55 0 70391 00

0.2107 03

0 V0212 02 0 171553 31 0.24 0.70000 32 0.17202 25 0 69588 0 4S0I9 94 0 60002 11 0 n1 0.69171 11 0 2!181 47 (427 0.00461 90 0.17200 915 0.00277 20 0 1552; su

0 ;915 86

0 0,95 05

0.08771 04

0.21304 90

0 20 0.08585 39 0.171I3 02 0 Gs1;51 27 0 18.120 13 0.05390 96 0. 04792 10 0.68101 42 0.21235 97 127 0.08:i:37 11 0.17011 70 0.0`,07r, 02 H ,7 70 0 07755 59 0 19,416 75 0.07101 39 0.21090 37

0.25

029 0.678,0 IX

0.07:,97 Ofi

(4.1)90i i0

0.16710 35

0 07521 91 0.G0900 42 14.15185 07

0 45062 !6

0.67,48 HT,0,6611-6 71

0.1,,rn

1ti n 19401 5:: 0 0,;.:71 3:;

0 9,047 51

2791 (.:11) 0.66820 17 11.1007 1.. 1.) (11;1)17

ii

0179:;; II 11.1;01710 II) ((.141)50) 0) (1 81 ;,2,5129 471 0.32 0.079,5 14) 0 16129 17 0 147776 86 H 17,;59 I 0. 4 11 0,61WT, 76 0.20290 27

11.11 70 (4.16;75II H I 4Ii;(1 IN (I (7;;;' ,11 0.1,1.2H 2; o. 186.28 I (I I' 11 98 1). 1,,I;)17 58

0 30 0 (14347 04 0.15914 49 0 03002 Ho 0.17055 77 0 0:1117 ))3

0.15191 78

0 G2577 00 0.19570 92 (1415 ((9114:141)0 0 17071 31 0.04171 79 o 10792 11 0 971.77(1! 0017975 00 0.62983 4S

0.19212 55

0.10 (04)2:103.442 0.15357 11 0.62197 03 9. 16194; 10 0 61956 55 0.17070 60 0.01351 07 0.18517 72 0 42 0.02:14100 0.17123 ;0 0.01873 111 0 012 45 57 0.61309 89 017:141- 01469082 32 0018484 91 0.11 0.01521 22 (4.11502 02 0 61295 75 0.15915 43 0 60711 77 0.17092 15 0.0101,..; 52

0.15125 95

11.40 0 61300 22 44,13005 15 0 607:-.8 79 0.15029 10 0 0015,' 53 0 514)0:; 0.791,9 01 0.17772 35

0 48

0.60814 19 0.11371 42 0 00259 21 (4173)7 10 0.50013 07 (41637! 3o 0.58959 22

0.17125 06

0.50 0.09360 816 0.1 1102 21 9,71(7930) 0417070 95 ((51(17407 (019114)4;'? 0.58407 30 0,17084 84 0.75 0.T1,9319 14 0 13501 33 0.78759 0 1 (107 31 0 58196 19 n i.,327 49 0 5;353 7fi 0.10268 29 h0 0.5s157 35 0.12931 36

0 57550 0;

13';;-5 52 0 77211 97 0.146;1 82 0 56171 49 0.15502 89 0 67 0.57740 81 0.12101 91 0 77126 71 0 13191 90 0 5G;50 In 0 13987 06 0.75795 02 0.14758 87 0 70 0,57094 35 0.11903 21 0.56175 95 0 12011 80 (451700 00 1)114>5252 1057.11591 0014124 89 0 77 0.70529 10 9.114116 GI 0,55900 09 0 12129 07 0.57230 52 ((.114)971

(07)0779

0.13500 Go 9)

H.11000 II

0 77011 66 0.11050 21 0.51735 56 0.12295 10

0 5399s 00

0.12932 20 0.55 0.55502 01 0.10791 73 0.51978 06 0 11203 HI 0.71305 II 0.11800 02 0.51576 00 0.1230A 02 (1 90 0.5.9200 74 0.10209 30 0.51799 50 0.10784 00 9 73920 45 0 11319 72 0.53.200 GD 0.11900 61 9.97 005471 09 0.09850 87 0.54219 83 0.10303 77 (5:171(18? (4.10923 53 0.52852 39 0 11430 )14 1 00 0.71537 20 0.09514 49 0 51913 15 0.10027 29 0.53297 69 0 10521 90 0 52590 57 0.11004 17 I 50 (47:.1006 27 0.07045 z7 1472101 32 0 07376 AI 0 51559 (1 II 0761 70 0 70959 31 0.07900 sS

.

00 0.71721 41 0,0.-701 26 0 71297 45 0.07769 13 0 79813 14 0 05951 IS 0.70161 G2 0.90104 35

.

50 (1(4 0.71:138 79 0 50944 74 0.40580 111 0 03888 50 0 50574. 40 9 :9625 HO 0.01720 09

090000 39

0.50100 3110.50290 19 0.04871 71,(401089 20 0.r01,,,7 TO,

0.49950 3)

0.01.950 590004153 13 70 0.50751 23 (4.(91:174 95 0.50171 92 1).11149'! 23 0.70180 75 0 93515 31 0.49578 59 0.08771 63 I 444 0.5. Ci0 45 0.020T9 ;7 0.50307 99 0.03010 GI 0 70108 18 0 01100 27 0.49519 91 0.03130 07 >1111 0,h9114 4.0

0.0,111

.11 0.70230 72

0.02159 5,

0.50028 47 0 02491 GI 0.4081; ;.,i1

0.02505 53

G 00

0.:q11. 78

0 02023 91 0.591.08 PI 0.92059 57 (14)990 47 0.02080 51

0 ip.,0x 84

0.02086 915

7)4

0 5.4271 71 (1.0)712 98 0.50124 70 0.01770 59 0 49971 11 0.01787 40 0.49811 141 0.4)17871? 8 00 0079.25 70 0,01510 21 0 70095 014 0.01552 20 I, .1,4900 ¶19 11.3(1T614 c,14 0.49,421 10 0 01562 PI 9 1111 H.50194 22

((Hi II 53

H 50076 09 0,0135i GI 14 19977 7s 0 01389 59

0.49531 05

0.01387 17 10 (u 0.70105 45 0.01229 Pi 0.50001 78 9.01211 07 19953 29 0.01270 09

0 40513 75

0.01247 29

l'A/H.E

TAI,), of

IIII. r11(odor,-;en Ciryof,(,,,,, Function for 0,twrilizi,r1 :`,fotion

C

-0 1.0 44 83 48 79 75 0.65098 0.56031

(3)

( ()

JoNEs's Ith,;(LussioN

OF TIM TIlEon()R;FN

CIRCULATION FUNCTION FOR C,FNERALIZED MoTION

The essential starting point for Jones's discussion

if

the Theodorsen fnnetion is Eq. (1) I.

I

rephces /,

iw and employs the form of Lq. Oit.

These

equa-tions arc then eninbined with

the following formula

given by Ve'atiti:"'

772ETkak,-.

480

_JOURNAL OFT111,, AFIt ()NAUTICAL SCIENCES--.11' LV,

₁₉₅₁

TANI F 2

I.LIJIcs of Ow Tli,odorsc itCin..ttlation rum:lion for Gencriliz.ol 5101.16,;

sill (.1

/4/1071',

II'"(ze""")

-11`"(z)

-stn

/Hr c- 11 n 1Sill

Mnir

7,1

(11)

Sill nr

If U > 0, ji )11 es takes in

_{= I and derives the form. of}

Eq. (7); but, for p <

_{= -I in}

_ned,

And the result is ditierent flow He].

_(T).

_{By letting'}

- -V = 15°-

_{0 -}

-p I 1 n --(700.d) 0 FTpeOl I 0 F(Peig) 1.0 -G(pcio) F(Pe) 1.0 ()__(;(p,O) 0 HI 0 99!!96 01 0 '41965 52 0 09553 36 0.05059 20 0.99912 17

0.0510 2n

1.00251 29 0.05177 19 It 02 0 07542 52 0 05109 55 0 05390 25 0.05,,57 01 0.95999 56 0.08574 85 0.(.05T0 96 0 0610;0 23

o A

0 09. 18 22

n.m90 5,

0.06911 31 0 11555 83 0.0;663 69 0.11974 01 0.08100 49 0.12330 07 9,91 1491594 13 U 3111 0.953 1126 15 (P0615415 9 14590 42 0.97025 r111 (1 15129 23 0 05

0.51

0 91696 02 9.169%5 25 0.141521 21 0.16511 41 (61(51.2227 0.17533 21 n n6 0.91167 51) u 19901 21 0 '(05651 _{0.17709 25} _{0.92823 74} _{0.15697 28} _{0.91,75 96} _{0 19501 84} 0 07 O.:0175 31 0 15215 25 0.902t5 35 0.192-15 11 9.91009 37 0.20294 77 0.92013 42 0 21356 23 OHN 0657520 12 0 19312 30 0 55562 f,0 0 20011 04 0.59379 0.21642 35 0.992S9 20 9 22853 18 0.09 0.86230 50 0 20219 53 0.56914 99 0.21114 15 0.57650 79 0 2277' 76 0.855;5 95 0 24116 00 II) 0.84659 ht ,0.,97 30 0.85315 52 0722112 47 0.89019 09 0.23713 84 0 59511. 75 0 95175 35 0 11

0.8210 IS

0 21575 56 0.51770 94 0.23000 71 9.54104 34 0.24490 23 0 55121 36 o 26053 04 0 12 o.51793 37 0 8207, 61

0.5,215 30

0 21559 25 0.5251.; 28 0625123 11 0.81477 75 0.29771 92 0 13 0 80139 79

0 ,21G5 88

080560 73 h 24005 68

0 81346 17

0 256:11 05 0 5155777 _{0.27351 55} 0 II 0.79145 73 0 22;7, 49 0 79.317 94 _{0.24355 04} _{0.70507 14} _{0.29030 3i} _{0.89355 90} _{0.27809 34} 0.15 0.771418 84 0 23004 12 0.75199 GI 0.24920 41 0.75516 10 0 2615;L-, 16 0.75554 79

_{0.28160 58}

0.16 0.76748 11 0.21171 38 0.76955 67 0.24811 10 0 77196 11 9.26557 87 0.77-45 65

0.28418 88

0.17

0.75631 62 0 23382 99 0 75773 55 0.21942 92 0.75039 65 9''I;711641 0.76128 74 0 25.-06 0.18 0.74575 73 0 21046 61 0 71615 31 0 25015 39 0.71739 37 0.26799 04 0.74843 35 0 28703 21 0.49

0.73569 07

It n398 8,

0.73577 714 0.25016 S2 0 7159.; 49 0 26535 27 0.71618 27 0 28749 01 0.2(1 0.72612 11 92:1:15531 (1.725.5964 0.25034'72 0,7259690 9 26.,n5 25 0,72151 86 0257-12 19 0.21 0.71702 36 9 23311 13 0 71501 54 0.21957 6A 0.71171 65 0 26-iT5 31 0.71312 18 0 286,0 35 0 22 0.70517 38 0 21:-L-10 46 0 70671 08 0.24910 16 0 70488 26 0626691 02 0.70257 07 0.28597 24

0.23

0.70011 78 0 23117 08 0.99795 59 21507 12 0 69553 ul 0.24577 0; 69251 27 0.28171 22

0.21

0.(4),.232 28 u 23931 23 0.6,o91 69 0 21652 20 0 6591 77 0.264737 95 0.95131 16 0.25016 H; 0.25 0.w;157 68

0.22104 75

016.:172 14 0.24515 06 0.67520 07 0 26277 IT 0,67426 30 0.25106 32 6.26 0.67778 73 0.2276! 02 0.1,7419 16 0.24377 79 0 97017 05 0.26097 97 0.6651:646 0.27935 43 0.27 0.971113 96 0 22695 12 0 61.702 99 _{0.21201 86} (.(1123; 55 0.259n3 21 0.65719 67 0_27719 74

9.38

0 66140 96

0 2:39 60

0.h9920 G3 0 21018 13 0 65527 47 0.25695 16 0.64071 71 0.27-153 11 0.29 0.655;5 13 0 2",65 30 0365171 22 0.20523 38 0 6IS36 52 0.25476 58 0 61216 46 0.27237 04

0.30

0.05261 79 0 22053 91 0.91752 62 0 23620 35 0 64179 97 0 25218 77 0.63535 85 0.26950 72 0 32 0.61174 :35 0 21704 Ss 0.9.1601 22 0 23195 93 0 62055 71 0 2177" 17 0.62216 79 0.26144 75 0.31_:11 (63151 18 n 21319 71 0.62551 h 22751 65 0 91551 21 0 24277 37 0661961 91 0.25887 79 9.35 (462273 970.61143 76 0 29997 8520:A0 95 9.61599 o,'", 9.69729 25 9 22301 82 6 21515 92 9 6981.5 9; 9 599211 SI 9 2:12C,2 (IS 11.59995 1,6 (1.51.)043 46 25:119 43 6 21716

9;

III)

0.90682 90

u 20994 04 0 59903 29 0.2139! 16 0 59095 GS 0 22753 01 0.58157 62 0 21.175 91

l4;

0.59983 47 o 19959 G2 0 59204 14 0 2,912 71 0 55311 35 0 22218 214 0 57161 26 2.1610 37 0.A1 0 59310 19

0 192,9 02

o 55535 70 0 20497 05 0 574:N 4G 9 21750 5! 0.56936 12 0 23053 41

0.16

0.58747 28 0 18599 19 0 57921 36 0.2005s 54

0 5,001 31

0 21291 71 _{0.55971 85} _{0 2'2507 25} 0645 0.55190 88 0 15510 51 0.57356 1111

" 14629 I!

9 56117 10 0 2075 32 0.55371 06 9 21970 45

0.50

0657993 n 15I32 79 0.56534 ¶0 9 10200 69 55580 G4 0 20319 32 0.54818 94

0.21453 19

. 0.55 0.56551 42 0.172_27 10 0,55699 55 0.!5206 19 0 54719 12 o 19202 44 0.51612 72 0 20215

3i-0 93i-0

0,55690 58 0 16382 60 0.51792 14 '.1725)59 0 53770 h1 111)(;7 66 0.52673 87

0 19069 55

0.95 0.5,1,. 1 81 0 15591 97 0 .;77981 92 0P 16103 24 0.1,295';" ti2 0.17210 81 0.51593 34 0.15013 59 0670 0651225 50 It 44 "

8'

9 15699 IS 9.52339 I5 0.1612; 95 0.51254 02 0 17043 22 0.75 0.51669 48 11.11156 05 0.52775 57 0p 14557 51 0.51797 76 0.45514 01 0.50727 51

_{0 16151 21}

0.80

0.53190 41 9.11558 65 0 52307 35 0 14170 59 0 51140 15 0 14763 32 0.50291 85 0 15331 19 0.85 0.52177 53 P115(7563 0.51907 53 0 13534 76 0.50960 37 0 149;0 42 _{0.49930 03} 0.14576 60

0.90

0.52119 89 0.12134 HI 0.51564 68 0 12945 57 0 509:;5 08 0.13130 07 0.4962S 50 0 13881 27 0.95 0.52108 64 0.11930 21 0.51269 40 0 12305 S5

95:94026

1.12,14:7742 _{0.49479 60} _{0 13239} ₅₆ 1.00 0.51'779 410 0 11611 00 0 51014 nS 0 11890 78 0.50125 14 0.12285 04 0.49165 77 0.12646 29 4.50 0.5u:i92 55

0.05126 95

0.19705 70 0 05319 52

9.43009 40

0.05494 10 0.45273 53

0.05565 25

2.00

0.10890 35 0,06225 41 0 403:41 Hi 0.09310 75 0.45775 00 0.09359 An 0.48202 08

00:455 14

2.50 0.49996 87 0.05017 90 0 19220 71 _{0.05051 97} 0.48777 71 0 05050 72

(P1511290

(P0.501(74

3.0n 0.19591 814

0 01159 57

9.49222 20 0 04197 32 _{0.45811 22} _{0.01173 21} _{0.45457 11}

_{0.04115 20}

3.50

0.49597 73 0.04,90 17 _{0.49248 57} 0 01582 86 0.45024 21

_{0.01517 35}

_{(.48595 91} _{0 03152 23} (r) 0 40596 71 0.01)37 62 0.11(2572! h1121 56 0.49001 15 0.01080 57 _{0.48719 63} _{0(1:1)11:4 29} 5 00

0 19,51

'21 0.00501 15 0 19367 99 0 02477 70 0,4911.; 83 0.02133 65 0.45,920 54 0 02398 73

6.00

0.49624 77 0.02077 15 0.49439 59 0 02051 41 0.49254 72 0602005 56 0.49071 73 0.01945 44

7.00

0 ,110,-.7 33 0.01775 17 0649199 15 0 01749 14 0.49:712 39 0.01708 72 0.49157 70 0 n1653 67 s 10

0.I93 44

_{0.01549 20} _{0.4951s 90} _{0 01523 97} _{0.49412 63} _{0 01489 23} 4449275 82

0 01115 55

9 n"

u.1'611514

((.0112310

0.19590 11 0.01149 85 0.491(0 81 0.61314 73 0.49352 02 04)121:5 47 10.00 0 40711 97 0 01214 22 0.49624 SI 0.01211 33 0.49517 17 0.0117s 56 0,49411 98 0 10135 90 I

=

Eq.

is G(pro).

-

25" 22 1) 33 25 32 0,20771 79 '

(4)

where

w ILL =

P > 0,

101 < 71" 2 (

Thus, if the motion is convergent, 0 < 0 K ir 2;

if

(:(=) ==

114(2)(z)

iijo(2)(r1

p

0 from both the positive and negative sides, two

different limits of the Throdorsen function for real

argument are obtained.

Actually, no limiting process

is required, since the Flankel functions

1.11-C.

regular in

the complex plane except at zero and infinity.

A prima

,drie examination shows that his results are incorrect,

if not, one then concludes that the Bessel

func-tions are not continuous, which is false.

It can be

shown that the correct branch of the Hankel function

is taken when m =

I.

I3y utili7ing-, the principle of

analytic continuation, the resulting form of the

Theo-rsen function is valid for all p's.

( / DISCUSS IN OF 1-111: TAIILFS OF TUE FITEODORSEN

C 1RCULATION Furs:cm IN FOR (;ENF:RAI:um) MI rrioNt

In the sequel, the Theodorsen function is defined as

ii,(2)(z)

=

0) + IGO), 0)

(la)

the motion is divergent, ir'2 < 0 < 0.

However, if

the motion is harmonic, 0 = 0 and, therefore, p = k.

f n this case, value.; 2 if C(k) are well known."

However,

if it is desired to study an aerodynamic system for

non-harmonic motion,

it

is then necessary to know the

value of C(z) for genera/ f.

Heretofore, tables of C(r..)

have not been available.

Many studies in stability

have been hypothesized on a pseudo-expansion of

C(k) about. k = 0.

That this can lead to serious

dis-crepancies is the subject of a recent report by Goland. I2

In other studies, the necessity of tables of (7(z) hits

been obviated by using curve fits of C(k) for k real and

then substituting

pew for k in the curve lit

expres-sions.

Au example of this approach is to be found in a

paper by Goland and Luke."

l-iince the tendency ill aerodynamics is always toward

greater accuracy and refinement, it has bec,inte apparent

that a table of C(z) would prove most useful for future

research.

In 'fables I and 2, C(..:;) is tabulated for the

following range of p at

0:

p:

0(11.111)0.3(0.02)(1.5(0.05)1.00.5110.0

0: 5° (.;')1:1(1'

For 0 = 0, the range of values is a bit more extensive

than that given above.

the tables \V&'7 1" C(mtputed

using values of the Ilessel functions tabulated in refer.

P(k) G kk)

Table of the Theotlot,II

CirculAtion

Function for 11,11

'I,

Mot 990

kt.) ) 11) - ftk, 11,002 1 0 0.99,70 96 01257 97 0.072 0 071 0 87269 76 050)14' II 0 15111 63 0 15:.7s 41 0 310 0 350 0 616,4 AO 0.111290 011 0 17,75 8. 0 1721 37 0 910 1150 0 31315 0. 51219 31 01.1A1, 0 1019 06 77 0 PM 0.99325 79 0 02225 66 0 076 0 56630 19 0 15738 is 0.360 0.61.02 00 0.170(5 75 0,0 O. 5115- 1,4 0 1(4215 .311 0 106 0.95970 52 0 03077 52 0 075 0 56315 12 0 15892 36 0 61171 79 II )6792 II 0 1,0 0 ..54061 117 . 10171 05 II 008 0 010 0 9,605 97 0.98212 15 0 II 02821 015115 S0 21 0 080 81101,i 18 0 (s3741 35 10010 21 0 .1111,2 0.4110 0 62197 0.61871 111 92 0 16/98 40 0 162.5 37 1 000 I 100 0.5:101'3 11 :12 121 49 17 o 1.027 0 09,60 2, W.: 0..12 0.97S71 74 02230 15 '1)) 1/ 551454 02 0.11121, 01 04 40 0 01295 0 15915 43 1.2011 0.52',1/5 611 0 05770 S. 0 ,41 0.97i95 50 1) 02951 45 0 056 11.8.'533 99 0.101 PI (I. 0.61022 37 0.15771 73 ..:;1117 115211)2 117 0 08214, 1: 0 016 01, 0 02. 0 02'2 0 97124 18 0.96745 36 0 96372 52 0,95,96 57 0

.

0 n 1,1110 1,6995 1,7520 0.419 43 15 79 99 o O's 0 090 )9)., 0 WI 1) 5I5111 4:1 1179 92 S121111 II 0 141012 97 0.111575 11 0 I110,11 15 0 1092:1 0 180 0 520 0.40758 0.60259 0.597,3 0.59358 79 21 61 96 0 1562, 10 0 15217 40 0.15070 05 0 115,0 10 1 4,1 1 21,1 ow,0 1 700 Ii22519 0 .:)2101 0 51889 4.51705 57 32 92 45 0 07777 0,07350 0.06070 0 061,31 5, 29 71 0 1134 0,,7921 II 0 111,1111 91 II 096 537:15 .111 0 170211 0.340 0.38932 55 0.14535 41 ,500 0.51551 50 0 06318 17 0 026 0 028 0 030 0 952I6 51 0.94573 OS 14,1501 10 0 0 0,917 00:;79 09791 12 10 35 09, 0 100 110 ii 83103 II. 04,192 41 S1S77 79 0.17132 10 ii.172,0 22 0.17001 I12 0.550 0 560 II 580 0.58759 0.55572 0.58215 22 22 0.11105 33 0.11276 82 0.1,024 50 1 (110 2 011, 2 51111 51415 5)2(42 50874 02 4S 40 0 06031 0 057,,I 0 01729 72 13 69 0 0.9)!31 7,, 0 9371.2 35

.

0 10,5 10362 )1 35 i '0 II I;11 0 50332 73 0 79153 72 0 15007 27 0 1525, 33 0 Goo 0.620 0.57880 0.57565 10 12 0 13778 ,2 0.13525 53 000 3 500 50,23 50471 10,) 142 0 0100. 0 01462 39 0:3,; U 038 0 010 0 93396 03 92.31 92 0.92670 IS 0 0 0 10923 11208 11600 23 1,5 12 11 1111 1:,1) 1) 1110 o 75137 15 0 77270 46 . 76277 19 0 I5is9 04 II _1,64, 0 15756 0 640 0.65n 6G0 0 57268 0.57126 0,36955 53 71 9S n 13305 44 0.131,1 06 0.13075 22 010 .1 5011 5 000 0 203(17 0 20293 11 302:19 0, 41 72 1 03019 0.02723 0 02459Sr II 042 0 014 14,2310 91 0.91951 30 0 0 11917 12222 72 33 0 170 11 II 753:6 0 7112- 70 0.48, 7 27 . 680 0.700 0.56725 0.56475 15 95 0.12557 OS 0.12611 89 5 74W, G 000 0 2019, 0 501118 11 11) 0 02212 0 02059 JO 0 046 0 .1440 36 0 12511 61 1,0 0 7357. 21 15577 Is _{0.56210 26} _{0.12432 52} _0.500 0 5.414 14 0.01901 21 0 048 0 91219 21 0 12795 13 Cl 21'0 0 72757 19 0.1581,2 12 _O 74(2 0.36.17 12 1222S 82 7 000 0-7,4121 70 0.0177050 I) 030 0 90900 90 0 13061 11 If)

.

71956 05 n 15522 47 O 771) 0.55909 99 0 1212, 05 7 500 0 50108 0.01654 23

.

.52 0 94555 50 0 1.3322 01 220 0 71252 II 0 1,772 22 0 55505 67 0.12020 65 8.000 0 50995 96 0.01552 20 0 054 0 90213 07 0 13571 11 2:9) . 70553 73 0 15702 55 742) 0.33603 ,Y. 0.11537 SI S 300 0.50085 17 0 01101 96

.

056 0.59873 65 0 135)0 00 II 210 6,985 79 0 11,,,i9 III _{0 500} _{Ill 61)} _{0.11650 21} 9.041)1 0 50070 09 0.01381 61

0 0,5 0 59537 27 0 11039 21 0 69255 26 0 Is321 so . 520 1)552113 (19 11.4117 70 9.500 0 50068 28 0.01309 55 II'.)) 0 80203 97 0 112:1111 0 2,0 0.05651 25 0 18120 4; 0 81. 5211,1 2,/0./1290 05 10.000 U 50061 78 0.01244 67 0 062 0.98873 77 0 14471 06 0.270 0 65075 02 0 18107 7G 0 850 () 5197S 0.11203 01 20.000 0 50015 58 0.00621 32 0 064 0,58.1)46 GS 0 111,7I 4:I

.

280 0 67521 91 0 1515, .7 11 0 515,7 71 0.11117 11 30 000 0 50006 94 1).001111 411 0 060 0 ,.8222 74 0 14809 )411 0 290 II 9,01, 12 0 1,3102 10 ,..S1,11 0 0 51711 GG 0 10918 81 40 0110 0.50003 90 0.00312 42 0.068 0.57901 93 0 151157104 0 300 0 661,7 11 17931 91 n 900 0.5-1 101 sti 0 10781 96 50 000 0 50002 50 0.00249 96 0 070 0.87384 27 0 23235 17 0 320 0 655)6 86 0 17659 29 0 "20 0 54150 s, 0 11.25 38 100 000 0 50000 0 50000 62 00 0.00124 0.00000 99 00

'r A It LES

T II

I)()1<ti

CIRC

I.A T1ONFrNCTION

481

-

F(p,

(13)

All

8.340 420 75; 0.460 11.400 88, 05 0 04 0.18s28 55 11.720 0

(5)

vrx.Txr:"(47;77 '77472"'".-"".' --,17-"7'

-

" ';.>.+4C4..threiy.o..1112.,...,.4.,Agwai

.f 01-1,Z NAI. Oar

Al3R0,1sTA1.'r-PCAL SC! E.NC.F.S.JULY,

.3 PAW=4IUU

nreswausksame

idommiuslam

rt-sokiwatabbobt

11111111TrrV1Mft

w

111111

apiloitimam

,e)

1.0 -9 ..3 .7 .6 r.C.717-1","''''.7;77,"107#1707577.. . -PrEr7

FIG .. 1 - PLOT OF THE THE0DOR:1E2f FUNCTION FOR GINERALI ZED Marl ON F (

_{,) vs. e FOR DU'i MEN'.}

.

t . .3 '4' 41%0 .5 . 6 :1i .7 .8

FIG.. 2 - Pair OF THE THEODOR:, FUNC:ION FOR G121 .NALIZED MOTION. G ( e 09.1 VS. (0 FOR Di} FM -e-.

.9 30 3 0 -5 .6 .7 . :9

1O(

, 4 : 1 143

.1

1

I

1 1 , 1 1

III

0 5

r

__

,

__15

. il 1

En

27°5 .,

-Fv4.---.---30, 1

111-

....%ft,,,memummowirAmw,

_,-,....-...

leglessi

A

ME

, .3 5 p 9₅ -/ 0 .1 .8

(6)

1 A B L 1.

0 l

I' 11

T II E 0 I) 0 SEN CI 1?. C IT I. AT ION I, (1 NCT1ON

ewes 11, 1,5,

0i, and 17.

Seven decimals were carried,

and it is estimated that the error is < 3

X 01 .7.

11

believed that the present range is sufficiently large to

cover current needs.

The range of 0 can easily be

ex-tended using the last two references.

Plots of F and c;

versus p for different 0 are shown in Figs. 1 and 2.

For a number of applications, linear single or

double

interpolation, as the case may be, will afford sufficiently

reliable results.

To achieve greater accuracy, one can

employ the series expansions of the Hankel

functions

in the neighborhood of the origin.

Elsewhere, Taylor

series expansions of C(z )

are valid, for, if z is small,

neglecting terms of second order and higher,

C(.s

iz [log (.7; 21 + y + i(77,.2)1

(14 )

Wick;

i[log (z '2)

-y

Or '2)]

(IT)

where 7 is Euler's constant.

'Pius, rIC 'ilz does not exist

at the origin.

It follows from Eq. (14) that

if 0 = 0, then the latter limit becomes 7r/2.

It can

easily be shown that the higher derivatives of F with

respect to p do not exist at the origin.

Differentiating

Eq. (12), one easily derives that

z(dC,/(10 = C(C

1) + lz(2C

(1

and, therefore, calculation of all higher derivatives is

simple.

If 7, is real,

kVG/dfr) = (2F

(G + k)

(19)

.a

nd

k(dF/dk) = F(F

1) G2 2Gk

(20)

It can be shown that

irk

JuJI)

F(F

1) +

-2G

1

(21)

and, if Eq. (21) is placed in Eq. (20), then dF'dk has

the form as given by Rauseher and Heilenday."

This

reference contains a typographical error in that

the

coefficient of id, should be positive.

Values of C(.r.,-) for

large ezin be culenlated rapidly

by employing the asymptotic expansions of the Frankel

functions.

fly employing these expansions, it is readily

deduced that lirn Ct.zi

112.

REFERENCES

' Theodorse 11, Th., General Theory of Aerodynamic

In,tability

and the Mechanism of Flutter, N.A.C.A. T.R. No. 490, 19111.

2 Dietze, F., Die 1.uftkraq7e des Harmonised; sehwingenden

Fends im hompressiblen .1Iedium bei

1:il(,'rsehallgesch,indiffit (Plane Problems).

Part 1. llethod crf Computation, MB Fit

January, 191:1. Translation by A.A.F. No, -TS-500-R 0, November, 1910.

Miles, j. W., A Not,' on a Solution to Possio's integral Equa-tion for an Oscillatinr:f in Suhsonie Flow, Quart. of Appl. Math., Vol. 7, pp. 21:3-216, July, 1949.

(tint, M. A., and Poehnlein, C. T., Aerodynamic Theory er

the Oscillating TVimt (11 Finite Span, Report No. 5 for submittal

o the Army Air Forces, Scptember, 19-12.

'

Fl- our, E.

Etr, (-1 of Fill Spit 0 on the Airload

Distribu-p. 75.

" Kassner, R., and Finga do, II., 'Ike Two-DimensionalProblem II in Vibrations, jmr. Royal Act o. Soc., Vol. 41, 19:37, Sec

also Luke, V. L., Tables of Coellicients for Compressibk Duller

Orientations, NI idNvest. Re,eareh institute, Kansas City, Mo.,

Report No. R111E-105, August, 1949 (Prepared for U.S.A.F.).

12 Goland, M., 71re Ouaci-.Ctrady Air Forces for Use in

Loam-Frequency Stabriitv Calculations, journal of the Aeronautical

Sciences, Vol. 17, No. II, p. 601, October, 1959.

Goland, M., and Luke, V. L., .4 .Study of the Bending-Torsion

Aermlostic .1117deS r A ireraft IVings, journal of the

AL-rattan-tival Sciences, Vol. lft, No. 7, p. 11S9, July, 11149

Bc.ssel Functions, Part 1, Functions. of Order Zero and Unity; British Association for the Advancement of Science, Ourdaridgc University Press, 111:17.

Tables of the Becs.el Functions

0 < a < 1;

Nat ional Bureau of Standards, Appl. Math. Series I, Washington, Felarmcry, It 148.

Tableofthe Functions ..10(rf: and J,(:) for Complex

..1r,o-mcnts, 2nd Ed.; t\1 at

Tables Project, National Bureau

of Standards, Columlia University Press, 11)17.

17 Table of the Besse! Functions Ir,;(z) and Ft(t0 for Complex A rguments;

Mathematical Tables Project, National Bureau of

Standards, Columbia University Press, 1951)

Rauscher, M., and II-Acrid:1y, F., Jr., Theoretical and Experi-mental Methods of' Flutter Analysis, Vol. VI. p. 57; Massachusetts Institute of Technology, June 15, 1949.

p, 1 P

[(log -1)

_ 7r

sin 0 +

(

- + 0

cos 0

o_{_} ( I( il

tion, for Oscillating

Part I--Aerodynamtc Theory of

Wings ti Finite Span, N.A.C.A. TAN., No, 1191,

March, 1917.

Reissner, E., and Stevens, j. E., E(Tert of Finite Span rm the .1 Irload l)istributions for 0,CI1/a II int.:,

Part 11Methods of

G(p, 0)

Oriculation and Example, of .1 pplication,

T.N. No.

P

[(log

cos 0

f

0)

sin

01 (17

1195, October, 1947.

roues, \V. l'., Aerodynamic 1,ces on II

)IS in ,Von_Umforpi NI. No. 2117, Augt,t, 1945.

Thus,

C(z) = I.

Note further that lim eX,;

--,

Reference 1 of this ,ection, p. S.

Watson, N., .1 Treatise on th, Theory of Bessel Functions,

co for all 0 and that him

'OF 0/31

r7, for 0 / 0.

But,

21h1 191., p. 1."11 ; Cambridge Unhersity Press, 1945.

+

+ +

0)

+

p p 0 1) .1713, Airfoil Oscillating. Notion, & G. Ibid., of , 14

K.i(x

Besse!