Aerodynamic Theory of the Oscillating Wing-Aileron of Finite Span Citation Boehnlein, Charles Thompson (1944) Aerodynamic Theory of the Oscillating Wing-Aileron of Finite Span. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/KJZ7-FG07. https://resolver.caltech.edu/CaltechTHESIS:04142011-111534450 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Aeronautics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): von Kármán, Theodore Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1943 Record Number: CaltechTHESIS:04142011-111534450 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04142011-111534450 DOI: 10.7907/KJZ7-FG07 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6328 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 15 Apr 2011 17:04 Last Modified: 10 Nov 2023 19:42 Thesis Files Preview PDF - Final Version See Usage Policy. 37MB Repository Staff Only: item control page

AERODYNAMIC THEORY OF THE CSCILLATING AING-AITLERON OF FINITE SPAN Thesis by Charles Thompson Boehnlein In Partial Fulfuliment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena 19435 This thesis consists of two parts as listed belows Part A. Aerdiynamic Theory of the Oscillating Wing Aileron of Finite Span; Pert B. Flutter Report No. 5, (Due to the classification of this report, its in- clusion es Part B of this thesis cannot be allowed until the lifting of such restrictions). Preface Sunmery of Symbols ss < 2 se isd ce] & Chapter I ~- Wing Lift and Wing Voment Ii 1-4 1-5 1-6 TH? I-€ I-9 A Preliminary Remark Purpose Summary Introduction Wing-Aileron Oscillations Replaced by Equivalent Downwash Derivations of 3°, 32, , Bz, Wing Lift and Wing Moment Wing Lift and Ving Moment, Coefficient Form Wing Lift and Wing Moment, Theodorsen Form I-10 Wing Lift and Wing Moment, Lombard Form Chapter II — Aileron Hinge Moment TI-1 TI-2 Ti-3 li-4 Tl+5 II-6 II? Il=8 TI-9 TI-10 TI-11 Tl-12 II-135 The Pressure Expression Afileron Hinge Moment, Fundamentai Form Integreble Forms of £,,£,, E, Derivations Derivations Derivations Derivations Assembly of ¢&.f.42 of FoF", ES fs &. of £. Ec", 22 ae a Aileron Hinge Moment Aileron Hinge Moment, Coefficient Form Aileron Hinge Moment, Theodorsen Form Aileron Hinee Moment, Lombard Form Conclusion 25 30 CONTENTS (continued) References Appendices A B Theodorsen Constants Lombard Constants Aspect Retio Integrals Integrals of Section II-5 iv 81 &2 108 Shortly after arriving at Pasadena in June, 1941, Dr. M. A. Biot presented the anthor with the problem which js given in reference 1. For shertneas of form, reference 1 is simply referred to as Report 5S. Dr. Biot had already dow gare work on this protlem and Led conceived the ides of the vortex aurroxi- metfions which ¢re set forth in section I-] vart II of Revort 5, Under the goidence of Dr. Biot this protlem was sterted, and after it was well along he sugvested thet it be used for the author's thesis. This wag brought to the attention of Dr. Treoders von Kernsn who approved the suggestion crovided turt some part would be solitary work so thet the requirerentsa of a thesis would be satisfied. Thet part of the problem which wes to be coneidered as solitsry work wag not settled until Sentexber, 1542, at which time Report 5 wac nearly completed. It is to be oteerved tht Rerort 5 considers only the simple wing and does not trest the wing-aileron combination. I¢ was then agreed upon by Drs. in Kernan and Biot thet tre treatment of this phase 9f the sutject would satisfy the above mentioned requirevents, end thus the following pagee have been written. It is to be pointed ovt that thie thesis can be considered as Fa sunrlenent or a sequel to Report 5 and in reading it is necessary to mve a copy of tis report at hand. Acknowledgment has been made to the authors concerned within the body of the text except where reference is mede to Report 5, in which @ve credit is given. At this voint, bowever, the author wishes to exoress hie zratitude and thenks for the inforration and help given him by the staff merbere of the California Institute of Tecrmmology. GC. Tt. Boehniein Minneavolis, Minnesota. Noverter, 15940. Summary of Symbols The following grouvs of symbols are not included in this list: the symbols introduced in Report 5, certain other symbols of a temporary nature which are self-understood, and more or less standard symbols such as are used for the Bessel functions, the logarithnic base ©, the imaginary unit ¢, etc. Roman, Upper Case Ag instantaneous value of the total circulation about an oscillating wing= aileron combination introduced in section I-4, Bz, defines an integral concerning the downwash along the chord, introduced in section I-4, corresponds to Sa of Report 5. BS, defines an integral which is a component pert of B2g , introduced in section I-4, derived in section I-6. B39 defines an integral concerning the downwash along the chord, introduced in section 1-4, corresponds to 63 of Report 5. 834 defines an integral which is a component part of £32 » introduced in section 1-4, derived in section I-6. G e function of reduced velocity, defined by Theodorsen in reference 2, introduced in section 1-9, same as / used by Lombard. Fo. bis Ee define integrals of the vorticity along the aileron chord, introduced in section TI-2 as expressions (°.1),(2.2),(2.3). FE. Es, Ea", Ee, Ei, = EL" Ee define integrals which are component parts of £., £,, and FE, introduced in section II-%, es expressions (2.9), (3.10), etce Ke the real part of 5. introduced in II-13. L the wing lift per unit of span, positive upward. Ln, Ly , ha ep coefficients in tha wing lift expression, used only in section I-€. JM the wing moment per unit of span about the mid-point of the chord, positive when stalling. /tq the wing moment per unit of span ebout the axis of rotation, positive when stslling, introduced in section 1-9, Theodorsen's notstion. Me identical to M, introduced in section I-10, Lombard's notation. Mg the moment of the aileron per unit of span about its hinge, positive when stelling, introduced in section II-2 as expression (2.4) M,, »Mg Me, Me coefficients in the wing moment expression, used only im section I-8. Nap imaginary part of Se introduced in section II-1c. P a function of reduced velocity, defined by Lombard in reference 4, introduced in section I-10. C. symbolizes an integral, introduced in section II-5 aa expression (5.4). kK, Fey v0, Fig primarily nondimensional functions of aileron chord, Lombard's notation, see appendix B. Se a function of reduced velocity, aspect ratio, and aileron chord, defined by expression (9.9) section II-9 Solk, 6) , S,(K, G0) ,-+ 7, Se (4s, Ba) | functions of aspect ratio and aileron chord, A = oe or SR? introduced in section I1I~4, Wea ay ***, Te primarily nondimensional functions of aileron chord, Theodorsen's notation, see appendix A. Homan, Lower Cane oY ratio of distance between midpoint of the chord and aileron hinge divided by semichord, Theodorsen's Notation. om , , ’ Ye, functions of aspect ratio, repeated here from Report 5, part I], page 30. In this thesis see section Ii-é. . Qn coefficients of a Fourier series, used in aprendix C only. Cc chard, same as in Report 5. Co, Ey Cx auxiliary notation used to assemble aileron hings monent expression futroduced in section Il-8 ag expressions (€.7), (6), and (£.9). f used here to designate a function of aspect ratio, reduced velocity, and aileron chord, introduced in section II-9, as expression (%.4) , fa % functions of same variables as * above, introduced in section I1-}8 ss expressions ({.2}, (84), and (£.€), component parts of for , tir, far , Tox, 2° % 5 Fox, component parts of Fe hy and fe , introduced jo sections ITI-4, Ils, and IJ-€, A instantaneous vertical displacement of the midnoint of the chord due to wing oscillations, positive downward,same as in Report 5. hy instantaneous vertical displacement of the point on the chord designated eg the axis of rotation due to wing oscillations, positive downward, introduced in section I-9. k uged in appendix © to designate 34 and “gis Pi, mass of an air cylinder per unit of span with the chord as diameter, introduced in section II-ic, S downwash velocity, powgitive downward. velocity of relative downwash from leading edge te aileron hinge, due to wing oscillations, introduced in section I=o. S UW, velocity of relative downwash along allieron chord, due to wing oscillations, introduced in section I-s. OY uged to symbolize the velocity of relative downwash due to wing oscillations, includes both Wo and Wr , introduced in section II-g. t ni afl Wy, Wa", We downwash due to weke treiling, shed, tip and bound vorticity respectivaly for wing-aileron combination, introduced in section IIl-c. x designates distance measured along the chord, origin at midpoint, positive toward trailing edge. Xo distance from midpoint of chord to aileron hinge. Xi distence from midpoint of the chord to the center of Lift, used in section I-° only. Ye inatantaneous vertical displiecement of the point on chord designated as the axis of rotation, due to wing oscillations, positive upward, intro- duced in section I-16, Lombard's notation. Greek, Upper and Lower Cage Fe, angle of aileron deflection relative to its neutr:l position, positive when it increases the Lifte See figure 5.1 section I-5. 7,° circulation due to the oscillations of a wing-sileron: combination, introduced in section I—4. hae i, i" circulation due to the wake trailing, shed, tip and bound vorticity respectively of a wing-alleron combination. a the rat .o of the distance between the forward quarter chord yvoint and the axis of rotation divided by the wing chord. See figure 1C.] section I-1C, Lomberd's notation. used as a variable of integration, related to X by expression (3.2) section IIl-d. | a term which designstes the position of the aileron hinge. See ex- pression (5.6) section I-5. | . used as a veriable of integration, related to x by expression (5.5) section 1-5. This term does not appear in final expressions for wing lift, wicg moment, and aileron hinge moment. See line below. ratio of aileron chord divided by total chord, used only in sections I-10 and appendix B. Appears in final expressions only. See definition of T given above. Chapter I T=1 WING LIFT AND WING MOMENT I-l] A Preliminary Remayk As is stated in the preface this dissertation is a supplement or a sequel to reference 1 which was written by Dr. M. A. Biot and the author. For shortness of form reference 1 is referred to a8 Report 5. The besic theory used herein is described in Report 5 and many formulae are taken vodily from this report. In reading this dissertation it is therefore essential thet a copy of Report 5 is at hand. Tad Purpose A formula for the lift of en oscillating wing and one for the wing moment are derived in Heport 5. The moment formula is taken about a lateral axis through the midpoint of the chord. The formulae of Report 5, however, assume a wing without an aileron, consequently, the effects of a wing with an aileron having oscillatory motion are unknown in so far as Report 5 is concerned, It is the purpose of this thesis to derive the formulae for an oscil lating wing of finite span, which wing has an aileron. These formulae are, the wing lift, the wing moment end the moment of the aileron sbout its hinge, I-3 I~3 Swnmary All the fundamental considerations as set forth in Report 5 elso apply here. fhe summary of Report 5 (pege 2 part I) written by Dr. Biot is hereby thought of as @ part of this summary. This is especially true of the wing lift and wing moment formuiae developed herein, since the introduction of an aileron produces no new functions of the reduced velocity and aspect ratio. The particular functions referred to in Report 5 sre R and Ap. These functions occur unchanged in the formulae for the wing lift and the wing moment for an oscillating wing with an aileron. From this point of view the contents contained herein are pure ly @ sequel to Report 5. The moment of the aileron about its hinge, as derived here, forms a more or less separate derivation, although it is dependent on some of the results of Report 5. The same vortex pattern and the same epvroximations as used in Report 5 are also used here. Further, there ere no new approximations intro- duced but a new function designated as i arises from the derivation, This new function Sa is similar to e end Gin given in Report 5 in thet it is a function of reduced velocity He and aspect ratio, but different in thet —— is also a function of the ratio of aileron chord to wing chord or as given here a nondimensional parameter which — the location of the eileron hinges When the aspect rstio is infinite teen Re = Qe=5,=C where C" is the function described by Theodorsen in reference 2. The numerical values of Se have not as yet been calculated, In order to integrate certsin expressions pertaining to = Fourier series end termwise integration are used. The coefficienta of this series are the modified Bessel functions, the veriable being proportional to the reciprocal of the aspect ratio, while the trignometric terms depend on the above mentioned nondimensional parameter which locates the sileron hinge. Ali other integrals are teken from Report 5 with the exception of one for which no integration was found, Fortunately, this latter integral cancels from the finished formulae. The results are given in four forms. The first form is as derived, assuming thet the vertical displacement of the wing is measured at the midpoint of the chord. In order thet the formulee can be easily compared to the results of Report 5 they are given in a second form which is called here the coefficient form. The third form, called here the Theodorsen form, is given so that direct comparison can be made with Theodorsen's work in reference 2. The fourth form ig given because it perhaps lends itself better to the numerical calculations of wing flutter analysis, it is called here the Lombard form, If a comparison is, made it will be found that this fourth form is identical to thet used by Lémbard in reference 4. I—4 Ied Introduction As mentioned in the summary the fundamental considerations set forth in Report 5 apply here. The same vortex pattern will be asaumed except that in the determination of the circulation the effect of the aileron will be in- cluded. On page 53 of Report 5 the expression for the total circulation is given by means of equation (6.1) which is Aa fer ( +l ere” where ["', f" lr axe the component parts of the circulation about the wing respectively due to the wake trailing vortices, the shed vorticity, and the tip trailing and bound vortices, while T° is due to the wing oscillations. Since the only item of change is the addition of an aileron the only term which changes directly is /°° . ‘This means that /"° must be derived so as to include the effects of the aileron. The terms J7' [7 and [7 to de sure include A as ® common factor and do change but only as a consequence of the change of the factor A . Following Theodorsen's notation given in reference 2, the Greek letter r) is chosen to designate eileron deflection, hence S used as subscript will indicate thet the terms taken from Report 5 must be adjusted so as to Loviaiie the effect of the aileron. Following this scheme expression (6.1) of Report 5 can now be written as Ag = hth +i + where As designates the total circulation about an oscillating wing with aileron. It is evident that ia must be derived. For a wing with aileron equation (6.3) page 53 of Report 5 now takes the following form: ° - BE As = FrG~ a, (4.1) [4 where now the effect of the aileron is included. The terms F, Qo, and Q, have here the seme meaning as in Report 5 and likewise the numerical values of Report 5 can be used here. Before the wing lift and wing moment formulae for the case of a wing with ean aileron can be determined two more terms of Report 5 must be modified, viz., Be end 63. If the sudscript S is properly attached to expression (2.23) which is given on page 70 of Report 5, it becomes for the wing with aileron 287 BigtAeclS+metige tA] — Here By,mast be calculated for a wing with aileron but fF can be taken from Report 5. If the same operation is applied to expression (2.36) given on page 76 of Report 5, the expression becomes / I z Go _ 28 = 2/7 2 ~ <= =. & Se bp where Bag must be derived for a wing with aileron. The values for however can be taken directly from Report 5. The equations of section III-3 of Report 5 can now be put in form g0 that they indicate the inclusion of an aileron. From Report 5 page 7& the lift: expression (3.4) is given as If A is replaced by A, and Bz vy Bag the expression becomes L=pUAsg +£iwcds = piu Bo, (4.4) where thie expression designates symbolically the Lift of an osclll«ting wing of finite span with aileron. Tt is to be pointed out that the subscript 4 is omitted from the lift symbol L . This should not cause any confusion, since if B=B =B=0 expression (4.4) reduces to expression (2.4) paze 78 of Report 5. If expression (2.5) given on page 78 of Report 5 is treated in the same menner as the lift expression it becomes Tah M=-pU Bz - £5 iwAs +4 tw bo, (4.5) which gives the moment of a wing with aileron about a lateral axis through the midpoint of the chord positive when it tends to stall the wing. Here agein the subscript 4 has been omitted from the symbol M for the seme reason as in the case of expression (4.4). A symbol Wg will be employed however, to designate the aileron hinge moment. Before introducing A and Qg in expressions (4.4) and (4.5) it is advisable to eliminate A from expressions (4.2) and (4.3) by means of the relation that W= AU (406) which relstion is found as equation (6.3) on page 39 of Report 5. Performing this operation on expression (4.2) it becomes ° U 4 Bag = Bagt Asc| e+ me + =e * F | (4.7) and likewise expression (4,3) can he written as U 2U U Qe = fe] 2 — es Looe Big = Bz, + Ag E “twe * Goce * Biwe 55 of 2U*QA | Ciwc)* +h (4.8) Substituting (4.1) end (4.7) in expression (4.4) it becomes on algebraic reduction as follows: Cwe Gh * Ser . w = (w Bs U ° L P 28 ~ f Ls i. +Qi- - (4.9) On page 79 of Report 5, i is defined by expression (3.6) which expression is _ Q. + sit Fp = (4.10 RY “Qo Q-F 1-4 Thus expression (4.9) can be written as L= pewBk,, + pUls” Fe (4.11) In like manner expression (4.5) can be bronght into the following form by means of substituting expressions (4.1) (4.7) and (4.8) and reducing algebraically; thus M=-pUB, + 5lwB35z ~£uc Ig » UWF +4F Speer Qo+t Q,-F (4.12) +2Uci, On page 80 of Report 5, a is defined by means of expression (3.8) which en is Loe — Qa + 4F - e U fe Qr = sal Qo +Q, - Fr (4.13) Substituting this in expression (4.12) it becomes a FP. ° r ° £ a Oy: M>=-pUB,, + ziwB,,- 7 Ucie +Z UCi, Ar (4.14) From the appearance of expressions (4.11) and (4.14) it is evident that the wing lift and the wing moment formulae willbe completed as soon as /e , bre and B,’, are derived. The expression for the moment of the aileron about its hinge however, requires considerably more development. Since Report 5 considers the entire wing, 1. @. from leading edge to trailing edge, the results of Report 5 do not lend themselves directly to this phase of the problem. The solution of the -ileron hinge moment must be introduced by considering the air pressure acting on the aileron. Since the development of the aileron hinge moment is rather lengthy no formulae will be introduced here. Therefore, in the following pages the wing lift end the wing moment equations will be developed first. The eileron hinge moment will be taken up last starting with the pressure equation which is given as equation (1.3) page 55 of Report 5. I-5 Wing-Aileron Oscillations Replaced by Equivalent Downwssh In Keport 5 section I-6 pages Of-41 the expression for the relative downwash is derived from the kinematics of the wing's oscillatory motion. Tris is worked out for a wing without en 2ileron. Here it is easential to include tie effect of the sileron. Since the solution of this derivation is linear it is only necessary to derive the effect of the sileron slone, then the expression for the wing-nileron combination is obteined by the process of superposition. Figure 5.1 hes been drawn sinilsr in some respects to figure 6.2 page 39 of Revort 5 but in addition figure 5.1 possesses an eileron. This figure shows the coordinate ayaten and depicts the angle BS 4. @., the angle of aileron deflection. The point F designates the leading edge, Fo the trail- ing edge, and G the aileron hinge. The origin is et the midpoint of the chord and is designeted as point OQ . The X-axis is considered as the lineE£G produced. The TF , abscisea of the eileron hinge is taken as Xo The point P with abscissa Fig. 5.1 ‘ing-Aileron Combination. % 4s an arbitrary point on the aileron. From figure 5.1 it is apoarent thet expression (f.6) given on page 40 of Report 5 is applicable to th=t ort of the wing designated by the line EG hence it can be said thet the downwash is wes ef ste mia & , for ~$€X SXo (5.1) where s designates the vertical velocity of the point O , positive when downward. From the information given on page 40 of Renort 5 it follows that the anguler velocity and meking use of this relation,expression (5.1) becomes wy, = -h -U“K~-ax , for - s<X =x, (5.2) The latter expression being the same as (5.1) is however, in = simpler fors. From figure 5.1 it is apparent thot the angle of aileron deflection © ig measured relative to the line E G produced, which as drawn has an engle of attack & , consequently the angle of attack of the aileron GF is A+B. The angles & and 4 are taken so gntll that with good avvroxime- tion the sin(« 7B) = ax+A end the cos(a@ +1) = /. Since the methods of superposition are to be used the engle & will be set equal to zero. Under the imposed conditions the undisturbed air stream wy ’ has an upward component normal to the aileron of magnitude Us . Since an upward velocity is here considered ag a negative downwash the contribution of this component to the downwach is —U. As in Report 5 the anguler velocity Xx is considered positive when it ig in the direction of increasing ol . In like manner let B designate the angular velocity of the aileron GF , positive with increasing RK es indi- cated in figure 5.1. Since the methods of superposition are used here teke “~=O0 . Under this condition the linear velocity of the point P due to the angular velocity B is (X~ Xe) 3 The angular velocity ye} causes the point F of figure 5.1 to move downward, hence the motion of the undisturbed air relative to point fP is upward. ne contritution of this motion to downwash is evidently — (x ~Xo) y) The total effect vroduced on the downwash by aileron when hs X= X= O is —UB —(*- Xo) AB The downwash equivalent to the aileron GF when hi, co, and XK are not zero is obtained by superposing the above expression on equation (5.2), thet is ‘ x: le We= We ~UB -—(X-xX)8 , fer xX,SXSB (5.2) ye Le? 3 I-5 Wr = -UK ~ KX -UB -(x-x) 8 for xX, XS = (5.4) where “= the downwash velocity equivalent to the aileron. For the convenience of subsequent integration a new variable @ will be introduced, defined by the following relation a ©. Ag eae (5.5) In order to designate the hinge GC (figure 5.1) a constant ©, is so chosen that Ce Xo = BF 605 Oo (5.6) The particular choice of the Greek letters CT and © are consistent with e Report 5 except that no constant © , 1s introduced in that report. Relation (5.5) is introduced on page 44 of Report 5 as equation (1.3) and O decomes is one of the variables of integration on page 58 of Report 5. However, as will be seen the present use of T will in no way confuse its use as designating the ratio of aileron chord to wing chord. Making use of relations (5.5) end (5.€) expressions (5.2) and (5.4) can be put in the following forms: We =-h —-Ua~Hcost for @6,< TST (567) and . " cm _ ca ui = -h -VxK — > cost ~Us= = (cost— cos Gy) for O<t<& (908) IHé. I-6 Derivatione of GB) Bae, Bos From page 43 of Report 5 Munk's integral for the determination of the circulation is given as expression (1.2) and is here reproduced as expression (6.1); thus = Pane| ur(i+ costae (2 where C = the length of the chord. To eveluate this integrel, however, the (6.1) downwash U7 must be a known function of T . The expression for iE" can now be obtained by subetituting relations (5.7) and (5.€) in Munk's integral, From the limite of integration it igs clear however, that w7 must be integrated from O to © , and Wo fron © to TT . Following this scheme Ip becomes 8, ; gr--of [-h ~ Ux - $8 cost -U - SB (cos T- cos @)](i +c05t) At {e} r -ef [-n -U« ~ cost ] (1+ ws t) At 9 (<) This integral can be written as Ir ls = cf [A+ Ok. + cost] (1+ cos T) AT ( Q , tof [ug + F(cos t - cos @)] (i+ ost) Ae and on integration it becomes [p= Tro(h+ Uc +) + UcB(@ + sin6.) +28 Se — @ cos 2, sine, = $sin2e.) (€.2) 6 It is very convenient to express [,'° in terms of the Theodorsen's constants (see reference 2). These constants are given in eprendix A in terms of the notation used here. Introducing these constants expression (6.2) I~6 becomes A= reeth +Uq + S& ) +UceT, + S47, (6.3) To derive Big integral (1.7) page 58 of Report 5 mst be used. This integral is ir B2=& | wsinerdr (6.4) (o) This integral presenta the same problem in regards to the interval of integra— tion as Ig’° it being necessary to integrate from O to ©, and then from 6, to 77 . Substituting expressions (5.7) and (5.8) integral (6.4) becomes 9 -2/" fa- =Va.~ % cos T= ie — L (cos t — cos @)] sina tT dt +$ [-k- Ux — 6s S cos t | sin?t dr & which can be written as Bee = -¢/ [A + UVa + cos t]sinet dt -£f [Ue a SB (cost - cos @) J sin? T dt fe] (2) Integrating and rearranging Dag tekes the following form: c2 : Bee = =| lah +Ux) +UB(e, — 5 @, cos @, ) + ef Sin3@, —O, cos O, + 31n @, cos? ©, ) (6.5) Making use of the Theodorsen constents as given in appendix A expression (6.5) becomes Fo, 2~Slr(h +Ua) -UeTG - ET] 1-6 The last integral of this section is B36 ‘ This is given as integral (1.12) on page 61 of Report 5 as follows: T B=) wrsiner cost at (6.7) O Substituting expressions (5.7) and (5.8) integral (6.7) becomes 34 =] [-f-Us-Seost -UB ~ SF (cost - cos @,\] sin? Teos tT AT © 1 ; +g f [-B-Un-B coe] sine cost de © or 1T 3 ™ : et . Ban => =] [t +Ux+ SP cost] sintt cost AT “+o 6, iE ~ $f [Ver # (cost -cos0,)|sin* tos TAT Oo On integrating, the above expression is brought to the following form: 2 °° 2 . (6.8) Again, making use of Thecdorsen's constants it can be shown that Bs, =-$/£crx * UcA(T,-T,) - SET; +Tcose,) (6.9) The formulee for Bie wie lift and the wing moment can now be put in their final form, The formula first to be assembled is the wing lift, expression (4.11). This expression requires the substitution of Bro expression (6.6), and fg ° expression (6.3). Making these substitutions the formla for wing lift becomes = — piw }~$]mr(F +Ux) ~UBT, ~T,]} + pUR hie(h tUR+ 52%) +UCA Tro + SBT, ¢ or L =Fectiwfh+Ux - YP 7, - SET] Tn | }0, +rpcURlh+U« + + UBT, + 5 (7.1) which is one form of the expression for the wing lift per unit of span positive when upwerd. Expression (7.1) is not however, in a suiteble form for applica- tion. Expression (7.1) can be put in a more suitable form as follows: From page S part I of Report 5 the following relations are obtained: h-heiwt iwi A third expression concerning the angle B can be written as _— where Fes) is the complex emplitude. Ditferentiating the above three expressions gives cwet h=ztw he a=iwaeirvs - =iw~e'v= which by means of the first three expressions can be put in the following form: h =iwh & ALarm (7,2) 6 =(wBe If the seme operation is applied again the above three relations become oe h = lwh=-w*h as pan Hs = (7.3) B=(wWh=-wrg The dines bracketed quantity of (7.1) has (WwW as a factor. If each term of the brackets is multiplied by this factor and relations (7.2) and (7.3) are applied, expression (7.1) becomes L=gpct | h tUc- Sel, — seh | tr pcUlg[ h+Ua+ ce 875 + $27] (704) This form is suitable for application, but for betterment and comparison three others will be introduced. Before introducing these other forms the wing moment equation will be assembled. The wing moment is given by expreasion (4.14). To agsemble expression (4,14), the expressions for /,”° , i and Bg are needed. These are given by expressions (6.3), (6.6), and (6.9). Substituting, M becomes M= -pU}-$ |n(h+Ux) -U8T; ~ 8 r]} 1 EMER AE 8 sesh) SE (ietoved — 4c} rel h 4+UK4£ SY 4 UB Tro + 7. +4 UCT, } me(h+Ua+S #)+U co To + & ce Ta Before applying the operations contained in relations (7.2) and (7.¢), an intermediate form of thie expression will be written ag follows: sm ca UP cB M=— Ape jul at (Ty + To) + $2(T +27) tiv] s54+ Sear, - %) - SE(G +1 c0s8)] ~UG,[h+U« +S + ¥ZeT, + sen] (7.5) Applying the operations expressed by relations (7.2) and (7.3) the above expression becomes _ Ucg, Cx , UB +ZER(G Pe ees fr pf) { Fe€) I-7 which gives the wing moment per unit of span, positive when stalling and about a lateral axis through the midpoint of the chord. Tat Ning Lift. and Wing Moment, Coefficient Form Expressions (..7) and (2,°) given in section III-2 of Report 5 are in the so-called coefficient form. This form is exclained in section 3 of Report 5, in particuler, page 6 part I, These forms es shown on page 6 of the above reference are L=Lyh + Lack M=Mph + MaX& If the coefficient of 4 is added the above will appear es L=Lyh + i oe Leff M=Myh + Max + MeB The eynbol Ma is here used to designate a coefficient, and in Chapter II i is used to designete the aileron hinge moment. No confusion should arise for the use of Mo as a coefficient is limited to this section only. In Hevort 5 however, the above two expressions are slightly nodified ss shown below; L=Lgnr + Lae + Lap MaMph + Mar + Mp8 where, by aprlying the first of the relations (7.7) it is seen thet Li= qo hh and My = Applying reletion (7.2) with the excention of the first, to expression a % (7.1) it heomces = Bporiw[h+Us— TET — SEAT | tirpcUR[h+Uu + 2s a+ LET, + weg, | A Cn collecting terms the expressiom becomes in coefficient form the followings I-8 L=mpcul $88 +B] h +1pcU* [age +( + Ys) B Jo + peu" [a (GPT Fy +(To* zET) RIA (8.1) If this is compared to expression (3.7) page 79 of Report 5 it will be observed that the coefficients of h and & are identical. Applying the seme procedure to expression (7.5) as is done to the wing Lift expression, the wing moment can be changed into the coefficient form. This involves the substitution of fond wk uses and p = twpP " Substituting in expression (7.5) it becomes M=-Eper} UL goat YET +To) + A(T + 2%) + iw[ ex + Ul (G-7) ~ wa +Ts8,)) —UG_[ H+ Un + wears oe a 7] On rearranging terms the expression can be written as M =F pc2UGgh +Fpeu| Ss, ~ Sas +(1 + 8s) Q, |x -£2 u*| SET, + Tcos6,) + WET +4 “tnd ae 5: +h +% IB (&,2) Comparing this with expression (3.9) page &0 of Report 5 it is observed that the first two coefficients coincide. IH9 WW . W Moment, Theodorsen Fo As the above subtitle implies the —— forms as they are called herein are the forms of the wing lift end the wing moment expressions as _ Theodorsen presented them in the N, A. —. A., T. R. No. 496, see reference 2, For this purpose figure 2 of Theodorsen's report has been reproduced as figure 9.1 consistent with the notation used here. In figure 9.1 the point A is the axis of rotation. It is located a distance ¢< aft of the midpoint O of the chord. The quentity a is here used in the same sense as it is by Theodorsen and is to be sure a pure number but may be either” positive or negative. The only conflict between Theodorsen's notation and the notation as used here is the lower case c . Here the lower case c is used to designate the length of the chord, while Theodorsen uses it as a nondimensional quantity to designate the distance OG shown on figure 9.1. See also apvendix A. The balance of the notation is the same ag shown in figure 5.1. Here 4 is assumed to be the vertical component of the velocity (positive downward) of the point O, figure 9.1. In order to ac E | f- - distinguish between the [ f vertical velocity of point O and that of pointA let A, designate the vertical velocity of point Fig. 9.1 - Point A , the Axis of Rotation. A which, according to Theodorsen, is considered positive when downward. Due to the oscillatory motion of the wing and the ee of the axis of rotation it follows that the point A is moving on @ ‘ straight vertical line. This is not true for the point O becauce its ° a . motion is the vector sum of h, and “a KX ‘ The path of point © due to 19 =a co is on a circular arc described about point A as a center, and therefore, the resultant path of point O is not a vertical etraight line. Since here the amplitude of « is assumed to be small, it follows with good approximation that the point O can be considered ag moving on a straight vertical line; hence 2 (9.1) and from the above argument it also follows that h a Fiz ~ ZB (9.2) The lift, expreesion (7.4) can now be put in Theodorsen's form by sub- stituting expressions (9.1) and (9.2) as shown below; L= fpc [ha ~% eee) +mpcuR | ha- Se +Ux +S + UE Te + + S27, | which can be written as +7pcu Bf ha +Ux+S(g—a)% + YET + 27] P (9.3) If the notation of{this expression is completely translated into that of Theo— dorsen, it will be found that this expression is identical to expression (XVIII) of N. A. C. A., T. R. No. 496, except that B&B revlaces Theodorsen's C , Attention is also called to the fact that Theodorsen takes the lift ae positive downward. Expression (7.6) gives the moment about the midpoint of the chord which I-9 is considered positive if it is a stalling moment. Theodorsen takes the moment sbout the point A of figure 9.1 but in like manner he considers it elso positive when stalling. Let Ma designate the moment about the point A Figure 9.2 shows the lift vector 4 , at a distance X , from the midpoint O of the chord U AL EF . The point E is ac rE "2 ; Ps the leading edge. The axis 9 a of rotation is shown as point <—— X,—_—_ > A on this figure. From Fig. 9.2 - Showing Lift Vector 1. figure 9.2 it follows that M= -xX,L and — on eee Ma=-(%,-S)L aC gee Lt a. ; from which it follows that M,=M +EL (9,4) Before making use of expression (9.4), relation (9.1) will be substituted in expression (7.6) as shown below: Maa otf GEG + Sk + Lat, +) = "4A PO) 7 352 ae + Ue 8(% +£%) - ae A(T; +1, cos @,) -UQ,| ha + Ux + $(g-a)x I. + $6 T.]} (9.8) Substituting expressions (9.3) and (9.5) in expression (9.4) the wing moment I-9 about the axis of rotation is obtained as follows ce Tw WG. « C& es > Ma=~Fpce{ Bex + 556% + 2O(% +T) Z Uc « 2 + 5HA(B +3) -— S(T +059) UG, [ he +UK + $(g-a) d+ acS7 i* UA 6B +32] Fpc+[hatve-Sa-Belq, - $47] trpcu R[ h,tUx+S(k-a)c US & Fer, + aT] The above expression can be put in the following form; + Ma =-Epct}- Sh, + E(s-a)& +$(grayja +2 (7+) 4+ SE; + 4G +47.) 6 ~ S/F +(<os0,~a)T]A +ZpcrU (2aB+Q,g)] h,+Ux + $(£- ate) 5 nay to + | sig i-9 The above coefficient of A can be altered through an identity which exists between three of the Theodorsen constants. This identity is as follows; Tg = Ti— Tg — cos 6, (9,9) This identity can be quickly proven by the substitution of the constants as given in sppendix A. If this identity is substituted ag written above for Tg in expression (9.8) the coefficient of Q becomes il UE (Te +2 +37): Bi [T ~ Te ~(c0s 6-2) Ty +4 Tu] If the coefficient of B is changed as indicated and expression (9.8) is translated —— notation, it wlll be found identical with ex- pression (XX) of reference 2 provided the factor (2a Ret Qe) ; in expression (9-8) is replaced by 2(a+4)C. Attention is called to the fect that RB =Q,.=C when AR = OG (see Report 5 page 2 pert I) from which it follows thet 2akR + a. = 2(4+3)C when ARQ = OO ‘ I-10 I-10 Wing Lift end Wing Moment, Lombard Form The wing Lift and the wing moment exvression will now be put in what is called here the Lombard form. To do this it is necessary to introduce some of the Lombard notation. The complete explanation of Lombard's notation is found in appendix II of reference 4. The notation of Report 5 will also be retained in this section, however, it is necessary to adopt some new notation end this will be taken from Lombard's work. Here, as well as in Report 5, the symbol T has been used ag one of the varinbles of integration. It will now be used to designate the ratio of aileron chord divided by wing chord. This should cause no confusion gince the aforesaid variable of integration does not occur in the final formulae, In section I-5, XX, is defined as the abscissa of the aileron hinge, see figure 5.1. From this it follows that This relation can be put in terms of cos 6, if expression (5.6) is em ployed, thus j T=sa(l-— cose 2 ( :) (10.1) or wo= | ~ er (10.2) The next veriable to be introduced is € > this is similser to Theo- dorsen's CQ in that it locates the axis of rotation. The relation between a and € can best be derived by means of a figure, and for this purpose figure 10.1 is shown. In this figure FF represents the wing chord where 1-10 the point E is the leading edge «nd F the trailing edge. As in figure 9.1 the point A represents =: +——-$ -—> | a the axis of rotation. From be |A F figure 10.1 it is apparent that Fig. 10.1 - Lombard's € , - ac c in relation to Theodorsen's Cl . eae 8 and from this it follows that e=4(arty (10.3) or (10.4) In order to make the results appear more nearly like those of Lombard, the following additional notation will te used. This notation is defined below, thus Me = Mea _ IT 2 Hh. = Pe and —he } Ye It is to be pointed out that the barred notation is not used here as it is used by Lombard, see reference 4, ¢. g. Lombard writes Yeo . By substitution of the above notation and the R’s given in appendix B, expression (9.3) divided by 177, apvears as follows; —— | j- Ux #c(e-4)& -R,UA-A,C 8 | -4¥ R | g.-Ux -c(f-e)a- Fue - cA] | TAL ch , aon? I-10 The above expression is in reality the Theodorsen form in Lombeard's notation. Lombard expresses the formula for wing lift in the coefficient form. In order to put expression (10.5) in coefficient form, relations (7.2) and (7.3) _ be applied. Avplying these relations and collecting the terms in such a manner as to form the coefficients of Ye, OC, and Be expression (10.5) becomes —[(g-8)* 4% B— 4iw wPi(L-e) nit iw® Jac — 3 j —_ —| Ry 02 —R, Ee —iwS(FeB+Rs)| ac (10.6) The minus signs before the bracketed quantities in the above expression have been introduced in order to make expression (10.6) directly comparable to Lomberd's work. In the arrangenent of the dynamical equations as given by Lombard the aerodynamic coefficients appear with opposite signs, which is the usual thing in vroblems of hsrmonic motion. If the terms due to the elastic constants of the wing and those of the wing mass are omitted in expressions (5.04), (5.05), and (5.06) page 119 of Lombard's Thesis (reference 4), then with a slight change of notation, these expressions coincide precisely with the three bracketed quantities of — — formula (10.6) except that /E replaces the FP’ in Lombard's expressions,* * Largan weellLiin the eo E to R, Kassner and a. Pisgsts, see via 56 of Lombard's Thesis, reference 4. It is identical to the C of Theodorsen. I~10 To put the wing moment formula into the Lombard form, the notations of this section end thet of anpendix B is substituted in expression (9.8), section I-9, Attention is called to the fact that expression (9.2) is the wing moment about the axis of rotation positive when stalling; it is designated as Me in this section. Performing the above mentioned substitutions the expression for the wing moment divided by Cm, can te written as 26 . & ay Ee Me = [(e- a) ie + UlE-9)% 4 C(e*- 5 +H) & | aul U(eB + Set i Ge -Ux —c(4-e)&% Ri Cc 5 - Bue - $7.4] (10.7) which again might be called the Theodorsen form in Lombard's notation, The next step in the procedure is to epply relations (7.2) and (7.3) of section I-7. Applying these relations expression (10.7) becomes I-10 + siwl (eR + Sa )(e-1)- 4 (eBs Ba B) | orc -|—(Ry- Rye)? + oT Rs-R, (eit SoH I +(w 2/R,-R3€ - Fa (age ca5 8 )}t ac (10.8) The above expression is in 8 form so thet it can be compared with expressions (5.07), (5.08), and (5.09) page 119 of Lombard's Thesis (reference 4). Here it will be noticed that ce Sno fe replaces PE€ in the expressions as given by Lombard. On page 2 part I of Report 5, it is pointed out thet for a wing of infinite aspect ratio P. =Qy = C . The symbol C’ used by Theodorsen is identical to the P used by Lombard, hence it can be said that for a wing of infinite aspect ratio fe =Ap =P from which it followa that ER + Ga = Pe when the aspect ratio AR = GO . Under this condition the two expressions become identical. Chepter II II-1 AILERON HINGE MOMENT 1i~l The Pressure Expression As pointed out in the introduction, section I-4, the derivation of the moment expression for the aileron about its hinge starts with the pressure expression given as (1.2) page 55 of Report 5. It will be noticed thet this expression is given on the second page of chapter III of Report 5 and that this chapter starte with Euler's equetion for fluid motion, consequently, it can be rightfully said thst the aileron hinge moment starts with first princi- ples. Attention is called to the fact thst Euler's equstion in Report 5 was linearized, hence this approximation is inherent with the above mentioned pressure expression. The pressure expression referred to is as follows: x Pix) - ped =-pU[ uu, 0) - ux)] -p2 J [ uz (x) - u(x >/dx ~€-§ (1.1) This expression gives the difference in pressure between upper and lower sur- faces of the airfoil at the point X as shown on fizure 1.1. In this figure ‘EF represents the chord where the | * P| point & designates the leading edge. ar seal r Ae the figure indicates the origin O | is at the midpoint of the chord. Fig. 1.1 - Flet Plate Wing The dimension € establishes a point slightly upstream from the leading edge, the reason for which is given on page 55 of Report 5. Expression (1.1) ig written in functional notetion to emphasize the terns dependent on x . As explained on page 56 of Report 5, the strength of the circulation Y , ver unit of length can be written in functional notstion as ¥=-- (| Us(x) — L(x) (1.2) where (4,(*) and (A(x) ere the perturbation velocities at the point (figure 1.1) on the upper ond lower surfaces respectively. Subatituting expreaeton (1.7) in (1.1) and atropping the fonetienal notation It bacomes ‘ "i é At thie point the & In the lower Limit can be set — to sere, It in true thet J bende to infinity as the leading edge of a plane alrfoll is approached, however, the integral existe. Aa a matter of fact the integral fe voy a determiners the circulation about a wing of Infinite epan,. Following the praocedure mentioned above the preasire equation becomea M i- p= pud t re \idy (1.3) II-2 II-2 Ajleron Hinge Moment, Fundamental Form Before starting the derivation of the aileron hinge moment, three terms symbolized by £,, &, and 2, will be introduced. These represent integrals which will appear in the subsequent development and are defined as follows: $ E = | tds (2.1) Xe £ f, =| yxdx (2.2) Ao : = Xo In the above, Y is the strength of the circulation per unit of length, X, (2.3) | & is the point at the trailing edge of the airfoil, and the integration is taken along the aileron chord GF es shown is the coordinate of the hinge, in figure 2el. In figure 2.1 the wing-aileron comtination is schematically represented by the streight line EGF . The point & represents the leading edge, the point G the aileron hinge, and the pee———— x point F the trailing edge, hence the - J ite line segment GF is the aileron chord, aeererer 3 oe Xe If ax is an element of sileron chord, Fig. 2.1 - Wing-Aileron. the lift ver unit of span is (2 -~)dx and the moment of this element about the point G is (R-fe)(x~%o) olx. From this it follows that the sileron hinge moment per unit of span is @. Map =~-} (P-P)(X~%e) dx bee is where the -inus sign is introduced such that stalling moments are positive. II-2 Substituting expression (1.3) in (2.4) it becomes He =-] [pur+ pS Ydx |(x-x) dx Ko “2 which can be arranged as & 2 rs me=- pu} vxdx + SE rdx x ° Xo agit {JL git Jeet ~xof* [ [tex Jaf Use can now be made of integrais (2.1) and (2,2); thus Mg becomes Ma = ap E, ~ Meg ) BL [Pf sda Jude nef [fede] ao (2.5) The two latter integrals of (2.5) can be brought into a better form by means of an integration by parts. Before carrying out this work a function F(x) 4s defined as follows; thus let roa) = f. YAx The firet of the two latter integrals can now be written as ih If. ¥ cb | xelx =" Foo ta ade i In applying the parts formula fad = uv-f[vdu letu= Fox) and dv = xdx (2.6) 2 then du- F'(xjdx and v= 3 . From (2.6) it follows that du= Ydx hence [Lo FF oo) - [Eres £ 2 i e ‘ 2. % a Xo II~2 The terms F(S) and F(Xo) mst be interpreted, the latter term of the above being FE, as defined by expression (2.3). The term F(S&) by expression (2.6) is FD =| “¥dx which is the total circulation about the wing, he re symbolised by Ag , gee section I-~4, hence The term F(x.) in like manner is Xo FOG) = Ydx 2 With reference to the coordinate system shown in figure 2.1 it is evident that F(x.) =| “¥dx ~ | Ors -& Me. i and by the above discussion and expression (2.1) this can be written as F(x.) = As - Eo The first of the two latter integrals from expression (2.5) now becomes - Cc [LS bax] xox = £4, - 38 (Ag-E,) - 4 Ee ” -: oO In e similar way the last integral of expression (2.5) can be put in the following form 5 x | [ | td |dx = SAg-xo(As—Eo) ~E, Xe ~3 The above procedure is similar to that given on pages 57 and 58 of Report 5, Substituting the last two results of the ebove paragranh expression (2.5) becomes II-2 ce ty / ~ PEL 8 Ae Fo) “BAe — Xo | Ae Xo(Me~ Ee) — EJ} which expression can be arranged nr follows: Ma = ~pu(E, = ¥ Es] Making use of the operational methods as set forth in section I-7 the above expression can be written aa Mp = PU( XE, EB, + E bw } ( > ~ xo) Ag —( XP E,- 2%, + bf Substituting relation (5.6) of section I-5S the above expression becomes Me = pu(s cos@, Eo ~E,) ~ 5 (uw ct (I ~ 60-06) Ay 28 Expression (°.€) appeers to be in the most convenient form for the work as presented here. II-~3 Integrable Forms of Fo,4,, EF. The terms £,,£,, and £, are anslogons to the terms /" , B,, and Bs of Report 5. It is now necessary to evaluate these terms. In order to evaluate the terms &,,f,,and E, they mst be put in more definite form. From page 100 of Report 5 expression (1D) gives J as iT hx oo ur sin*T a 2. ~ 77 3in® cos O~ cost ITC sin® (3.1) é) In this expression “4, the downwash is considered as a function of T , and T is related to xX through relation (5.5) of section I-5. In the same way the variable © of expression (5.1) is related to the X of the terms &,,£,, and Fo » see expressions (2.1), (2.2) and (2.3). This relation is a es + cos 0 (3.2) Substituting (3.1) and making use of relation (2.2), integrals (2.1), (2.2) and (2.3) can be put in the following form @, ;IT =e wr sin®T T' 66 (3.3) =f COSO — COST drde + TT Oo “oO afr" jn=T rc = € Ursin c . Ey =: nal | éSs@ = cost ©°> edtde sf 21r Sun 8, (3.4) Oo “oO 8, cit c3 °) wr sin?tT . Es cr Lh 4 (2.5) In the above expressions one integration has been performed, viz., that part which embraces the term sts of expreseion (2.1). To complete the evalua— tion the function of T which determines the downwash w must be given. In Report 5 the downwash uw is determined by a piecemeal process where the resultant downwash in the notation of that report is given as Ht oy ds wr = Ur! pur" + ur!" + Ww a see page 44 of Report 5. The formulae for w"'. ur" and ware the same as given in Report 5 except that the A in these formulae is here replaced by Ag , see expression (4.1) section 1-4. The We must be modified for that pert of the wing which is aileron, otherwise it is the sane. Let the symbol wg be defined such that Us= WG for Ox TSO, and We = We for ©,<T<T7T Both we and Wr are given here in section 1-5 as expressions (5.7) and (5.8) respectively. The resultant downwash aes used here can now be written as bars ig" a ale eal (3.6) where the subscript S has been attached to the terms w', uw" and «4 to indicate that Ae is used in their respective formulae in place of the A of Report 5. Attention is called to the fact that the 7” present in expressions (300), (3.4) and (3.5) is the total circulation about the wing and is here equal to Ag - From this and expression (3.6), E, can be written as follows; o (T ipa cal | (ost uae 05) sur dtde of As ®o (6) cos 9 COST TT O ° (3.7) In section 1-4 the term Ag is given as Substituting the above relation in expression (3.7) it becomes Ber Cc i . , sin*T a We' + ug" + wel +W3)——_—— dtde a [ic a a ENG a) cos @— a he o Jo + (B'+1g"+ B+ (3°) & i] 7 (3.8) Let four symbols E3,E',,Eci and Eo be defined as follows; =s Wwe sin2t set dca 4 ie, BR cosO-cose TT (3.9) a seer la © ca acc Ores + A (3.10) 7 > mh Eo == UR qa = acane dt de + a . aii (3.11) seer = of I,"° , ——- tde + B Cos@ —tosT VT a4 “I e (3.12) Comparing these four expressions with (3.8) it follows that E,= Eo t+ ES + Eo + Ee — and following similsr logic with &, and Ez it can be shown that (= E/ +E +E} +9 . Coote ' (3.14) and that E,= EL+EL +E;" + £2 a za where expressions similar to (3.9), (3.10), (3.11) and (3.12) exist defining the component parts of (3.14) and (3.15). These eight — en will not be given here as in the case of FE, but will be tazen up as each one of these parts is evaluated. As in Report 5, it can be said that FS is thet part of E, due to the trailing vortices, that &,' is that part of fF, due to the shed vorticity, etc. II~4 Ii-4 Derivation of Eo, E., Fe In the evaluation of the twelve terms given in expressions (3.13), (3.14) and (3.15) it is, from the nature of the integrals, more bagicedl to derive these terms: in sequence, £,,£F,,F,, F,&,\E,etc. The first term to be considered is E,. The formula for wy which represents the downwash induced on the wing by the trailing vortices, can be written from the expression given at the top of page 45 Report 5 as S Age * A+LALR, S,cost “B= 21TH, (2+ta,) | 2 FLAN, +o (enor™ _ CEST Ons )| (4.1) 1 It is here to be noticed that As has replaced the A in the aforesaid refer~- ence, otherwise the two expressions are identical. alee edly (4.1) in expression (3.9) it becomes . Cc $f | _ Ase * Sete tn IG Quest. am | aah camy Atria 4+ S; ( enor _ enosTty, ost) ase on. dtd +2 cose—cost Tr which can be arranged as follows. wc , tke LAT ens ce 2 E.= ie lb aes ( sint Tv aderde 277°C, (a+ tan) 20m * cos@—Cost @ (4.2) $608 T cost gin* tT atde + 2% 8, “ C05 OQ —COSsT “TT In the above there are ‘two | en integrals which are symbolized as Sol F > ®,) and S,(f>6) thus S04) = ae] jon" 25 sine dacde (4.3) cos @— cost and S (€ 1@) = . at i@ Scost cost sim? tT drde (4,4) cos@-— cost II-4 These symbols are written with functional notation to emphasize the fact that both 5,(§:6.) and 5, (£)8.) are functions of 5 and 6,. With this new notation expression (4.2) can now be written as -£ 1 cAse”™ A+tah, ¢ . E ~ ATT 4(2+6h%) ; 2+tirA, ¥ § |5o( 6c) i 4.5 ~GSiGeohs Es “9 The integrals which So( #2) and S,(%.,) represent are worked out in appendix C. The values of these two expressions are given below: Saki) = Eee a(S +B.) (4.6) and S\(% 8) =] I,(%) sine,— 2( # - cos &)O( $e) | (4,7) where i, {e) is the modified Bessel function of zero order. The symbol O ($9) represents the following Fourier series; Co _ a 0($>6,) = 2.1, (G) sin 16, (4.8) This series is the result of the integration described in Appendix C, and as shown, is also a function of - and ©,.- fo finish the above formulae substitute for /Z, the expression, C AR 5 SU = — (4.9) which relation is expression (2.2) part II — page 135 of Report 5. Thus ex- vression (4,5) _ peg ase | A 4 Eo = WAR ( 1 + ide er) Neeson + Rr] So(& » 8d) I 'e@. (4.10) -~ 4 5.(free) { + . and expressions (4.6) (4.7) and (4.2) become hea _ AS . FR *) = ce TCR 9) (4.12) 5.(e 8.) = | 1 (4) sine, -2(#- case) + ( #1.) (4412) O- (6) = > Ma) sin ne, (4013) Following the sane scheme as exemplified in expression (2.6) page 46 of Report 5a symbol Tor is chosen such that ne _ e { 4 A for = RUT EKER) i+ tae Ri Rt 1 | 5.(4 x) “8. 4 4 (4.14) — RH ( ro) b The subscript O of the symbol Ky is chosen so es to design-te it as part of the £, group and the T associates it with the trailing vortices. By means of this &, can be written as Aa f, a) Ey = a + a (4,15) It is to be pointed out thet Ar is a function of AR , @ and AC, which means that it is a function of aspect rution, aileron chord, and the reduced velocity. As outlined at the beginning of t~is section, the &5 would be evaluated in the order &,' , £, » E> , etc. . The next term in this sequence ie F/’ . This term is defined by substituting ug’ (exoression (4.1)) foruw and ae for in *aryensicn (S04); thus ' ge artsy |S past | Sma (era) A+iAN, eS iB! Foes = £ cos T ) sin a (e" e cost cose—lost ses (4. 16) This expression can be arranged as follows, liI—, = 7 is As ©! 7 pane ++ lA2, " feost sin? T 60S6 1, 477 7/0,(2 + iAN,) 2+ikR, +5] tas @ ~cos & ah € cost 20 cla’ (4017) _—— ew sin cose Cla _: ok” i} fe a Mapenea en a 280 deol «$ qr 5n Ge 0 4o As was done for expression (4.2) here also the two integrals will be symbolized by S,($)6.) and alr, ‘ thus e%* sin*T cos © a (Fe @2) = t[[e COS@— mab ese 2 8 (4.1&) ‘ $cosT 2 A= a Cf Cu” cos SLE SEE drde Oo “Oo Cose-Ccost (4.19) Subatituting these two expressions Ey becomes og . _ czAge™ 4+lhh, , Cc c E,= FTTH (24644) . | 24 tA + a) SK @o) (4.20) ~Jo! -£5,(§.0)$ + Seesin Expressions (4.18) and (4.19) as worked out in appendix © are given below in terms of aspect ratio AC , rather than ; thus 5, (#8) = 4 | WR) @. + I(#) sine ~2 (B-.cos G0) o (# 160)| (4.21) 5, (410) = $1 + Be Acosa, + c0s 265] 0(#1 2) + La(4) & + B[-(£ +1), (4) + 2Rb(4) +31,($) | 22 + [CDG (E+ NLA) + RL BD -RI,(4) =I (4)) sin: 2, (4.22) Substituting relation (4.9) in expression (4.20) it becomes Ij—4 (4.23) Let fir designate the following expression: _ e ® od 4 rs TRI + EXE 72) it Ke RR + OR +1 |S, UR») A «x (4.24) a a “)} Here the subscript / of fr designates its relation to A, while the T as before, associates it with the trailing vortices, The term E;' can now be written as AgChr + c Ie" ; EY = er ao 8, (4.25) It is to be noted that 7, differs fron fy in that S,(Q»6.) and 5, (4,6) replace So( % 10) and S,\ (#702) respectively. The term £' is defined by substituting expression (4.1) for wr in expression (2.4) and 4’ for I” 3; thus @, 17 -$ " £2 Age” At LAM pcos ait 4 RTA, (24irkR,) LAt LAN, £cost &¢osT 2 +e i — en cost) lf Saree oes dtde CO56 — Cos T EE Be + asin 2 8.) (4.26) which can be written as “3 2 . —cosT Els CAs C wees Ari +¢ i fet , Sin*T OS*® 1 oe a 8172, (24 lan, 2+irn, cos 8—COST ae ee 2 2 Te ary ae 2 al “fe EOE ogs am are dtcle ( oO “0 ct 9.5 / : es - +a le tas Z@o) (4.27) This expression involves two more integrals which are defined as . cos T sin®T cos*@ Sa CF,» Ge) " =a] “fet cos @— COST At de Ze (4.26) c / 9% 77 Scost sin*T cos?#@ ded Tf oe 1G : en fo (5) Oo) = 7 sl COST cose — cost ° (4.29) oO “0 ; Substituting the above two symbols in expression (4.27) it becomes = _ c3 Ag Ct A+taAk, S| & ea.” 817 4A,(2 4+ ¢427,) rene + 7, an (4.30) -55 p(gre)t +S a. & + z5in2@,) The integrals (4.28) and (4.29) are worked out in Apvendix C and are shown below in their Ar forn; 5, (4 18.) =Fli+Z — Ros 6, + cos 20, | (4,0) + Le, + [-(F-)l(® +2R1,(4) +3 (4) | sire. +e | (R*+DI(R) -R( + 0L(4#) + LR) RL (¥) ~I;(%)] sin 2, (4.31) If expression (4.22) is subtracted from expression (4.31) it will be found that Sal& »O6) can be written as follows; S4(% > Co) — (4 Og) + # I (&)sin G5 (4.3 32) In like manner it can also be shown that AR 52(4&) = 5, (#> 6.) ap a 1a 65 (4.33) The above two relations can also be obtained if the two integrands defining these expressions are first subtracted and then integrated, see appendix C. The last expression of this group is Ss (%>%e) which is given below: 11-4 + 3R(1+®)cose, a ay O (RG) 5 [le +282 28) 1g) 2B +R) —3(I- R15 (4 ) ~1,(4)| sin ®, 3 : , +g AZ Ue) (kta PSL) @| ~ 44 3H )1,(4) +74 4A =I, faj= “Ts (4) -; 51,(4)| sinze, + gl($ 2 D4) B04 2) EG) +R" (142) 1(S)-B (1+ 2B) LR) + 9A T,(4) -~-#1,(4)-3f 4)| S138, (4,34) Expression (4.30) in terms of aspect ratio is as follows; 4 > R E/= c# Ag C rie +441) 5,(#.8) 477 R(1+ iAS RR) I+ EAE mR (4,35) ~ R25 £0) t+ Se oF (& +47 5in28,) A symbol fer © can now be defined as shown below: - A fre arp wg {leew + #7] 5.14) _4 (4036) Hol Hd The term Fy. can now be written as & 2A ole &. / H 53 Sa er Zw t+ qZ5!7 26.) (4.37) 11-5 Derivation of Fo , E. , Ee The second group of terms to ve derived is FE’, EF, , and &;'. This group arises from the shed vorticity. To obtain the downwash for this group take the expression given at the top of page 47 of Report 5 and replace A by As 3; thus “ds 1.1356 LAAs gieg i-eeet) Wg = case 7T(1+6iAA,) As in appendix C page 95 of Report 5 the variable § will be changed to the nondimensional variable S through the relation that Cc S=S (s- Making this substitution, Wg" becomes “Z ait €— cost TUFELAN) (5.1) , Substituting the above in expression (3.10), EJ’ takes on the following form; gee GT Ac «_ € CAAa oS c= f | J . | Sarees 6 “oO _ 11358 thhg oa ~ 8D) 2 sin® dede T(1+6ikN) cose@- wt ra a Rearranging the above expression it becomes 2 > zert 2 " ~ £AC, sin* Tv o> errs e* |" y [os (Sa ae Toe 8- cost) ds de cde ~~ 772(1+60AN,) COS @- COST + (Se?) The first integral of expression (5.2) is worked out in appendix D. The value of this integral is II~5 8, iT; © ~ X~ _ 6 * sin*T dedtde (§ -ws T)(cos@-cost) ~ oe ee oO fe) / LAG CRC emp 2 27 "= ; = FAG = (7r - 60) ~ Ae e * Q23in O, (5.3) where QA, is defined in appendix D as cAc cxe 7 © “a wl =€* E As (5.4) Ga | (P= cos euro? No integration was found for the above integral, however, it fortunately cancels from the aileron hinge moment equation. The second integral of expression (5.2) is seen to be the same as inte- . Cc ; gral (4.3) if = is replaced by j37,. Following the scheme of integral (4.3) let B77 C P el Tee Ge) ~ # if € COS@~—COST dtde (5.5) Oo “0 and from appendix C it follows thet =7€ _ BA Cc P=) Sal ems @o) i O (727 , Dy) (5.6) where in ‘this case ; (a2) Or (iE% >) = 2 Tn (zen) si 18, In order to write the above expressions in terms of aspect ratio substitute relation (4.9); thus So( sR Co) = ER T( sR O) (5.7) and = } Or (she 8) = a Ty (sm) sin ne, (5.8) where [,(zm@) is the modified Bessel function of order 77. II-5 Substituting expressions (5.¢) and (5.5) in expression (5.2), it becomes 7) tArE a ajlrT, 2 eal ~2° 2 By are As Ee (1r-06, ~The Qz sin 6. | $J878 LAE 7K. ' -— [ome Ta" Ge T* (1+ Bh") ileal [ Do (mre) + 7 Making use of relation (4.9) again the above expression can be brought to the following form; Ee, = al T-@. — G2 sin 6, | ! AC > aR : : I./358 ¢ Ap @ 3& S, (sz ' 6.) +4 Oe _ i T (1 + 2B8R) (5.9) As in section II-4 let &, be defined as follows . i 1.AG58 AC ~ sR ! hee SSE © OP* 5, (se &) (5210) [+ 3eac ) a Ae The letter S , in the subscript indicates its association with the shed vorticity. The term £g can now be written as EM Ag | ‘ Aste I,” = = | 77 — Go ~ Qa sin © | + i a 2a. (5,33) The next term of the group is &," . This term is defined when wz, '' (expression (5.1)) is substituted in expression (3.4) of section II-3. Per~ forming these operations, a becomes on? pw LE LA ™ LL TS aT © “fs S = cos'r AG ~ TF SIAN) cos @~cosT os cls" + — STF sin, which can be put in the following form: II-5 ep «» sre* = 7 5in*T cos @ E, = ape? EPL: An BN eee. ded de 1A35B CAC* > SH —— Ea) COST cin*T cose si ro Ag Ce e ee 21T*(1+6LAN) 5 cos @- cost ce. 4- = sift @, 2 (5.12) Like expression (5.2), expression (5.12) involves two integrals, the first of which is treated in anvendix D, In this appendix the following result is given, a [ly BO *T 6088 edrde (era cosT)(cos @ — COST) A Vv eS 4. — Fe (a | sin2e + Tie sing | Qe athe a 4 2 - ARC * sineQ, +Aaee * @,/1+ Q,| (5.13) The terms Q, and Qi: which avpear in the above integrel are defined on page 49 of Report 5 as expressions (2.3) and (3.4) which are here repeated as follows:. (Ac Qo= Gt ‘a er CHS) (5014) éac _Z; FT yRpac Q, =#iace*® (4s) (5.15) where nr) and Hs) are the so-called Henkel functions, The term Q2 is defined by expression (5.4). The second integral of (5.12) can be made the same as the integral (4.16) if £ is replaced by St >» trom this it follows that cost 2 Oren sine’ T cose i 52 (2% e.) =a) ‘fe cosT — cos ® (5.16) es Cc 2 * 2 af ¢ aon, 2 4 2 This integral is worked out in apoendix C and Sa (7a? &) is given below in terms of aspect ratic, thus Ii-5 5,( sag? Go) = 3a | L.(s5R) & + Io(5_m) 57 © = Z@ (3R- 605 6.) T (se oa » QB) (5,17) Substituting expression (5.13) and replacing the second integral by iT 3, (sar Go) expression (5.12) becomes «_. LAC* “2 (2m ao Bots £,"= Ae4,e* | 2h e T- sin, + 72-(Tr~6,)| - He Vsinze, + fe sine,|Q irc tre a - (Ac gac ATT » 2 + GRE sine, Qo + Bee “e[i+ alt _, LAF oLAE* BR Coe 2TT*(7+ Ela) Ag | 7 S2(se >) cm oF - siti 6, Substituting for = BE , the above expression reduces to me a) em 2SING@, 20, Ey = Sap |7 sin + He + AEP tH HG —(ssin 29, +7 ae sitt oy) Qa | eerie C Ig" + Say SING, — (5.18) Like expression (5.10) a f, is defined such that fs ‘ _ _. At 35B tke , aR ! lis = — —Saep © 22( 3 %) (5.19) Blac 4+ oo aati With this, expression (5.18) can be written as follows; vu CAg . 2 Si11 Bs 2e, ie Bie | 7(1 + tee) + sie, + SEE Qe ire A —(sin@, cose, + 77 3iN&e) Qa | cAg = €r , eb oar fis + “aap aut B, on (5.20 The last term in this group is £2 which is derived in the esme manner as Ey and &," To define this term substitute expression (5.1) in exoression (2.5) of section II-3; tius iAc 6, Be © sa S- cos T oO (2, / _ Air 8 iAAs e = geal wee SIN*T cos?® TT (1+66AN,) dvde cos 6 ~—cCoSsTt + Sed sin 205) iriting the above in the same form as was done in the preceeding work it be- comes éAc n_ £ac3 = e 3 * ieee este as 8 IT 2 Ae € “| yy ( § — cos T}(cos @~cosT) at de _ 41/358 LACRA “46 Oe 72%, 72 69ST sin2T cosze aire(i+etan,) © * | [en anes Cos @ - Cost - Lr" gQ + ——( 2% +7 :5IN26 arr (2 +2 :) (5.21) The first integral of the above is treated in apvendix D and the second in appendix C. The result for the first integral is - iAu 205-7 90 3" 7, Sin?T cos Lf | }. } j (¢- cos T)( cos 6— oT adtdtde r os ; 42 ~» 2 o Bede =Fhe © fir 3 sit 26, + AR (i+ <=) a _— 4sir1 8, — sing, cos*e, + a (cos ©, oe a) | at where Q,» Q, » and @, are defined by expressions (5.14), (5.22) (5.15) and (5.4) respectively. If in expression (4, 28) a is replaced by a it becomes TEqeost sine T cos* 6 eal 2 5 a (ram O) = =) a e aes Ode (5023) where on comparison it is seen that the above integral is the same as the second integral of expression (5.21). As worked out in appendix C expression (5.23) in-terms of aspect ratio is as follows; 5Sa(gR Co) =3R [i +36R*-/2 AR cos, + cos 20" (slar > 8) +3 I, (sz) e, + = (72 R-II (s'@) + 24 RE (sk) +3 R(x |5in 6, +f] (1442+ 1, (agg) — 12 (72 +1) L, a) + 144 R*T(sR) 12 RI, (sz) -1, (sia) | Sin 2@, (5024) Substituting expressions (5.22), (5.23) and (4.9) in expression (5.21) it reduces to II-5 aw ¢24 ts = 42 Sir- S-Lsin2e+ So (14+ & Bf 451 , ; + rac | Z5in 28, +The «Ot e.| Qc A 285 ale + sine, | Q, +sine,|cos*e, + aaah eos a, 77 a | Q [ ! 1Lt35@ tAC3 ee ! Ag @ saat Bay Go) eed ETECE) c*# I," / e., I. +33 (4 +e 280) (5.25) A symbol f,, will now be defined for the aspect ratio term of £,' as follows: ! 1I3BS58 LAC pp FR ! — 2s ~~ p+ StAc aR c D4 (sm Qy) (5426) + a ee Making use of expression (5.26) ES can be put in its final form; a fA, & 1. Ar 2 Afi... 4 sine, + Te [ goin 2e, + S578 + @| Qs + elie A + sine, | Q, — sing , | cos? Q+ = at (cos, + Az ] @a Saar Pe (5.27) " # B I. i ATT ing t ge = +35in2@,) TI-6 II-6 Derivations of 4. , 4" , Fi" The group of terms &,'", E," , and Ez" arise from the bound and tip trailing vortices discussed on pages 7-10 part II of Report 5. The effects of the tip trailing and the bound vortices on the downwash are summed in a single expression given on page 32 part II of Report 5. It is expression (5.5). A table of constants to be used in connection with this expression is given on page 35 part II of Report 5. This same table can ve used here. The only item which is changed in the above mentioned expression is to replace A by Ag - Expression (4.1) given on page 51 of Report 5 is more convenient than the above mentioned expression and is here introduced with the subscript ‘3 3; thus wo As a U2 ' 4 4 " Az 2 W3 = Ala, + a+ > - Bo (4, +42) cos T + Peas Tt (6.1) where Q, , @% , and GQ, are given in Table I page 35 part II of Report 5. The first term £,'" of this group is defined by expression (3.11) of section II-3, which on substitution of expression (6.1) becomes fo’= rey, a 1 Qe. 8 } sin*t A ped [3 '@, + ——Cos*T — AtTde + 4. ers TE This expression is now put in the following form, ®p um As Qa oe _sin? T a, “ger Hao +#)/ [: ee ae — £(aray | J cost sin’. drds Cos cos O- cos T 452 2 We ep l cos ST dede | si a8 COSO— COST (602) Expression (6.2) involves a double integration, the first of which can be intesrated by means of the following formula, Pa: eto tT . siInNS sa ee HS BS ee HS j bes T-éo80 . Tv sine (6.3) ao} where /(}= 0 »/,25353°°* : This integral is not discussed here but is well known in the subject of thin airfoil theory and its solution represents a Cauchy principal value. The derivation of this formula ig given by von Karman and Burgers on pages 173 and 174 of Aerodynamic Theory, volume II, reference 3. The second part of this integration being elementary in nature will not be discussed end the final value of &,"' will be given at once; thus m _ A ; fo" = ten [ (480, + 244, +154,)5in®, - 6(a,+ a.) sinz2e6, Ise + A, 5iIn36 »| + + ———. * IT (6.4) A symbol f[, will be defined such that he rye 56 [(484, +24q, +/S5Q,) 5in 8, ~ 6(4+4,) sin26, to + Qo Sin ze | + . _ (6.5) from which it follows that 248 A [,'""6 ie sal tr ibs + “TT 2 (607) The second term of this group is &,"" . It is defined by substituting expression (€,1) and replacing /” by &'’ in expression (3.4) of section 11-4; thus | a / sae én | yi 144 lat — 3 (44+ 4) cost in?T cos @ + Boos? SIT SE 2. gterdls 4. SE gin @, cos O~ COST ar (608) The technique of the integration of expression er being the same as that T1=6 used on expression (6.2), the intervening steps will be omitted and the result written down directly; mn cA ; £. = sade | (48% +244, +15 4,) 6, -12(A, +42) Sin 6, + (24&+124,+94,) 5in20, —4(a,+ G2) sin3G +Za,sinae,) + cb sing, 27T (6.9) A symbol f,, will now be established as follows; I ee fe = 735 4840+ 244, +152) -12 (a+ 42) =i, +(244,+124,+90,)sin 20, ~4(4, +42) 3in 3, +3aqa sine, | 4? ° (6.10) Using this expression, £,"’ can be put in the following fora; EL = 35r he + ay 3 tn &. (6.11) The last term of this group is &, end defining it by substitution ex- pression (6.1) for uw and JQ" for [” in expression (3.5) it becomes @, wT Esa! | {Belardes P- Fla +ag)eost [o) ° , Gz ncd SIn*T cos*#@ 7, + A O° al cos @-—cosT dt de oe rs CY le (Be44s5; + Se ( Se + 45/7 26.) (6.12) Here egain, owing to the similarity of the integration, the outline of this procedure will be omitted and the result given directly as shown below: a C* ” f, = S367 [Uta ae+ 724, +4802) 5in@, + (/64@4+BA+7a2) sin 38, + §$A,5/N 50, -3(4,+,)(46,+ 45in2@, + sinae,)| c2 ‘Mies 6 i. of Me gaedin ch —_s<: (3 a sin2 @) (6.1%) A term fe is defined as I , Fes sea [ (144, + 72a, + 4843) sin@, + (IEG+84, +72) SINZe, 3 +F Q, sitt FO, — 3(4,+ 42) (46,+4sin2e, + sin 46,) | (6.14) | vee re The term 4, can now be written as wa C*As ce I," 9g, Ios z Zz 47t ie a sar (2 + ZF sit726,) (6.15) lin? Derivation of 624 Fy The last group of terms are FS , FY , and £° . This group arises from the motion of the wing-aileron combination, The downwash equivalent to this motion is given in section I-5. The expressions are Ww; = -—h-Ua« - SF cost-UB = SE (cost ~ cos 6.) (Tel) for OF TX<O,» and =—-f Gq — S cost (762) for © ST STs Substituting these two expressiong in expression (3.12) and integrating in their respective intervals as defined above £,° becomes a “Af Uf [-# -Ua-Seosr - uf a £4 (cas t — cos 6,)) Se _sintt dr cos@8- cost - ‘ 2 © +f [-h-Ua- ss cost) SIE fy de + 2& i", TT cos 6@- COST oO The above ie can be arranged as follows; ff [A+ Ua + $F cost | az dtde +f [ [us + F (cost cose) ] Set dete} fo) oO Is’ 8o + TY (Fea) The first integretion of the first integral of expression (7.2) is r 1 co sin? T | | h +Ua + F cost] eet fo} vr which can be handled by the integral formula (€.2) section II-6. After soply= ing thie formals the second integration becomes elementary. 4 5 =; ‘5 ep 5 2 tb aw - ee pe fer oy 4 It will he neticed that tie sécond integral‘ of expression (7.2), i.e. ae @ Go . J] | 9+ S(cost-cos6,)) = Jak ee eT i ia symmetrical, which means that T and © can be interchanged without changing the value of the integral. Bo evaluate this integral the same technique can be applied here as was applied by von Karman and Burgers in Volume II of “Aerodynamic Theory" section 11 pages 53-56, (reference B)o The integral can also be integrated in the ordinary way, however, as it turns out it is a Couchy principal value. If the above mentioned process of inte- eration is carried out the value of F, becomes sin 6, Ep =-c| (H+Ua) sine, + sinze, + Wee 68 Kh . — Bir (GsinZe, + cos 26, :| + - The definition of £,° involves expressions (7.1) end (7.2) as well as expression (3.4) of section II-3, however, /” must be replaced by [@° Following this procedure E,’ becomes fo af ron = Ua- Seost- UB - CB (cost - cose) | 28 sin CES 8 ae cos @ — cost cosg -cos T = . os ee tr sO ° . a of~ Ve - esr |S ote tele + SE sing, a 2Tr oO which can be arranged as r) o_ c? ° CH sin®t cos 8 E, = 2m t | i ‘[h+Uas x= | cos@- cost dtde ff [ua+ ‘8 (wst- cos0,)| sorte? deo (7.5) The outline Of the integration is tre same here as in the case of expression li-? (7.3) hence the result will be giver directly; thus c ry } Cle . F Ek: a [(A+ 00) (a+ 4sin RE.) + a ee @, + $SiN36,) TD ++ 2? (e2 + sin® 2») a £ (a, airra, ~6, *cos@)| + $2 sing, (7.6) The last term of this group is Ee « This ig defined by substituting expressions (7.1) and (7.2) for wr and replacing [ by [Z° in expression (2.5); ’ thus s) 8, oO fret. “) of (fel ox So epery 2-47 =e ee fo) 46 ca 2 26 ~U8-~ 3 (cost — cos a)] gure E atr COS @~COST ae +f [-h-Uax~ feos r | EE cost ze Ate ° c*hs’, Gp 5 bes + aT ge + Zsin 26) which can be written as C3 a alr c& sin “T cos*@ a4] | [hsv 8 cosT| —p woes at de Oo Zo ‘By £ Pa , pees 2 +] J [ue + BL (cost ~ cos a,)| sarees? de dol o ~o cO> Q@— LOST hl a a d c Here also, the integration procedure being the same as in the preceeding, the result will be given directly, thus Eos -£[(A +Ua)(2sine,- §$sin3@,) _ a ; (a, + 614 2G, — 4 sin 4&,p ) T <= 6, sin a “ cg é 62 38... aS 57 (sin 6) : a = 11-8 II-€ Assembly of Fo, Fi, Fa. Expression (2.13) section II-~3 indicates that Ao is the sum of expres-— sions (4.15), (5.11), (6.7), and (7.4). In assembling £, the last term &° will be written first; thus Fe =-c[(A+Ua)sing, + SF sin 26. '* ae >in @, la Sin ZO, + cos 26,-~ 1) | + a Aa for I5''8, + “Gt + A [ ai As a Ig’ @o + 2-6, Geng, + “2 + 4S Ag foc le 6, + TT iu TT which can be written as Q i E,=-Cc [(h + Ux) sin &, + SS sin26, a Se 9, sInQ, 7 (@,5in 26, + cos 28, - -1)] mines Gy — Qa SING + Tor + hos + fe | In section I-4 under the discussion of circulation it was pointed out that Ag=6 °+ + Is" +h" where Ay is the total. circulation about the wing. If this is substituted in the expression for f, it will be noticed that the last term will cancel the ©, term in the second bracketed quantity and that E, becomes II-8 Cg = =i [CA + Ud)siti a + “Be 3128, + a 0, 31NG, Cg ; ~ Br ( 9,sin AO, + Cos 26.— i)| Ag - + 42| rr @, sin@, + f,| (Bel) where fo = fer + ths + foe (2.2) From expression (3.14) it is anpsrent that £, ia the sum of expressions (4.25), (5.20), (6.11) and (7.6). The procedure for determining E, being similar to E., the result will be written as follows: 2 . . Ca ; £2 — = \(h + Ux) (6,4 £si0 2 Qo) + (sine + ¥ 917 38,) + FE (a2 Be 311?6,) + a (@ sin & — @,* cos &.)| CAs Rstn®, + 29 | ar (1+ Be) + “EAC. Gs+ FS ar — sine, (cos @,+ 752) Qe + fi] (8.3) where , (8.4) Expression (3.15) designetes £, as a sum of expression (4.37), (5.27), (6.14) and (7.8). This result is also given without comment as follows; II~8 a. etre E,>~ g |(h + Ua)(2 sing, - & sin%e,) Co : to: aug = (+ sin2@, + 3 sin 4e,) + a7 % SING, foo) C8) cine OF _ 3% 5; L sind + Se &+ F a SINLG, + 7E sin 2@,) | cA» AIT 2 tod ee Bo he Be) a | if 451 @ q + xe B57 ze, + Sa eh On ie + <.| = + sin @,| Q, As = | Fag (C05 O, + =) Oa+h t (8.5) where (8.6) Owing to the fact that the expressions which make up the aileron hinge moment are bulky, the following symbols will be defined as a temporary expedient; thus let = ° In? Bo ‘ Co = Fe ~ (6.7) = cf,'°.. @= £,- a SING, (8.8) (€.9) From this end expressions (@.1), (8.3), and (85) it follows that Eo= co + fl tr-sine, Qs + f | i, = &, + Ae | ar (1 +52) + rp a Qo + S* “: Q, ~ sin (cos @,+ 2 ya, o t | E,= e+ Slr {i+ ee + #-)| ric 6 LAC _2& [.: ASIN ®, +a - cos Q@,+ s% he + 8. | Qo 2746, _ + x50| Fe +25in6,| Q, = aif 29 +4 *)] sin @, | cos 6, + Ae (42> @ + oe Q.+ _Also from expressions (7.4), (7.6) and (7.@) it follows that e,= el (h + Ux) sine, + S sinze, a Be'O.3in &,, = — $|(h+Ua)/ (2, +r 4sin26,)+ EX (sine, + 3 5iN36,) Ue 3 r F @,==— = (A a Ue) (2 sin é,- € sin?6,) + < (a, +s5in 265+ +sin<e,) + 22? 0, 5in@, cA a =. ‘ 1. af + £& (sin* o,* oa +O, sin 28, + sin@2@,)| II-8 (8.1) @.11) (8.12) (8.13) (8.14) (8.15) II-9 II-9 Aileron Hinge Moment Before assembling the aileron hinge moment forumla it will be found con- venient to replace w in expression (2.&) by its equivalent AU, and write expression (2.8) as follows; Mig =p U(S cos, E. —E,) + LVihe| ¢ cos? @,E, — cos QE, + £2 - Ae/1- 003.4)? | (9.1) Tre fect that w= AU is set forth on pages 28 end 39 of Report 5 end is here given as expression (4.6) of section I-4, From the above it appears thet there ere two quantities to construct with the F's . The firet is given in the parenthesis and the second in the brackets. The quantity first to be treated is the parenthesis end substituting expressions (8.10) and (8.11) it becomes £ cos 6, £, — E,=§ cos i c z 2 sine tar = [Te (c0s @, =i fie) ~ “Te? Qe 28, 2siné. - We Q,+ “tXC 2 Q, + f, cos 6, - f | (9,2) The second expression of (9.1) requires the substitution of expressions (8.10), (€.11) and (8.12); performing this substitution it reduces to cA © ¢os* 6, E.= cod 6, Eb, + Ee = ee cos @.)° a = 2 gta, & = ue + & + SF tie fer[ i+ ; ric ~ £0 6| +| t352 +0, - sing cose,| Q +2[sin@, + 5 we & 0€05 Go| Q,~ t2n® Q, +AS[f, cos*6,~ 27 cos 8, +n)h (9.3) Ly~9 Substituting exsressions (9.7) end (9.5) in expression (9.1), M. becomes —_ C > Ms = pu 2 C08 O & ae e, + tae & cos* O, €, - cos O@,€, + alt puc + PE Ae} [e,— sine, cos] Q,+ 2| sine, — e, cos @,| Q, This can be pit in a more condensed form by making use of the Theodorsen con- atants and by the introduction of a new symbol -_ From appendix A it anvears that | 7 = —(@ ~ 3in@, cos @,) and Te-G = 2 (sin, ~ @, cos B,) For the symbol Ff let i? , cos@,—-f + nis f, cos*@,-RT, cos @, + fh , 2 (9.4) From this it follows that the above expression for Me, can be written as cos @, €, — c086,€, + S) + OEE ae [~Ta Qo t (Te- TQ 4+ F (945) Before substituting expressions (€.15), (£14), and (£.15) for @, , G , and @, respectively, it is advisable to change the trigonometric functions in these expressions to functions of a single angle, in order that the Theodorsen constants of apvendix A may be quickly applied. If these changed expressions ere substituted and the factor (w in expression (9.5) treated by expressions (7,2) and (7.2%) of section I-7, expression (9.5) becomes Ii-9 Ms = a | af (9, C05 4% -3In@® +5 217? @) +U(h +U&)(6, -sin@, cos 4,) (2, ane — $5in 70, -sin@, cos? @, ) Lice +e + (gine, 5 @, + 73 sin?0, «020, - $e) ™ DS ae? + 5(n*6, ~24, sin @, cos 6.) oe 2 2(s cos*@, —% sin*@, éps*a, -z ~ $0 — 9,* cos? 6, + ne , ‘ ; . +*3 sinO, cos?@, + + 8, Sin yg cose.) | + EF As[- QT, +9. (Te - Ty) +7] (9.6) The above expression lends itself to the use of the Theodorsen constants (see appendix A). In some cases, however, the constants must be transformed by means of the trigonometric identities. Making use of these constants ex— pression (9.6) becomes Ms = =| CAT -U(h+Ua) TG + UAE 4 we C28 aris + oF B| +22 A[-Q.% + A(t2.-%) +f | (9.7) The next step in this development is to insert the value of Ag . In order to minimize the work the latter part of expression (9.7) will be treated slone. This part will be combined later with the first pert thus completing the formula for the aileron hinge moment. Substituting expression (4.1) of section I={4 the above mentioned latter part of expression (9.7) becomes As| - Qu Ty + Q(Te~ Ty) + | = eof [en + a(t %) +4] which can be put in the following form; As|-@Ty + @(Ta~ tm) + F | 0) j Q, -E& + a = /y Tim 12 12 , 4 12 Qo t Q -F (9,8) At this point let a quantity 5S be defined such that AR Q, af Q, —F 9,9) provided OC6,<S77T . The reason for the interval being open at the lower endnoint is due to the fact that 4 co as G&—O. The quantity F is independent of 6, and therefore remains constant, A rovgh investiga- tion was made of the term £ which indicated that this too tends to infinity ns @o— O. For 6.=7 however, 7,=1 , 7,=-7m and f is finite, hence no difficulty is to be expected at the upper end point. In the expression for Mg , it is seen that 3, is multiplied by Tre and from the rough investi- gation it appears that te Se —+O *8 @& —->0O. The reason for defining * Bigs as shown in expression (9.9) is to make the hinge moment forma es it appears here comparable to the existing formulee for infinite aspect ratio. if Sin ig written in terms of Lombard's notation (see appendix B) it becomes FR; £ = _ Q, “Fea + 7T Feg = Q, + Q, -F (9.10) provided O<TX<Ii. Here T designates the ratio of aileron chord divided by wing chord, see expression (10.2) section I-10. Since Si, is complex, eo) TI~9 notation is here given for its reel and imaginary parts; thus Se = Ae + tNg (9.11) Substituting expression (9.9) and expression (6.3) of section I-6 for f’° , in expression (9.8) it becomes Ag [-Q,.% - Q.(Ta~ a) * f | = | rtc(h+u« +o) + UCP T+ G2 Tt — Tae | (9.12) Placing (9.10) in expression (9.7) and combining terms, it becomes 2t- . i Me = oe E Ay + UES (T+ 2%) + SE (T+ T cose) 4 [2 ry* U’cB a +224 (7, To ~ Ty + 41r %T, +4 4 7T B , 7, ¢% « OS cB ~The Se (+ +4 id Ar ad (9,13) which is the complete formula for the aileron hinge moment. TI-10 TI-10 Aileron Hinge Moment, Coefficient Form Although expression (9.13) is the finished formula, it might be desirable to put this expression in coefficient form as is done with the wing lift and wing moment formulae in section I-8. In order to put formla (9.13) in co— efficient form it is necessary only to apply identities (7.2) and (7.2) of section I” and then collect the terms which form the coefficients of }; , ~ and & (see section 1-8). Applying identities (7.2) and (7.3) to expression (9.13) it becomes Ma= FE | Sent + (2 € 17% (Ty +21) ~ WC” OX (T, +7 <058,) 29 P wrc* ee (Whe -% + ows US Ty ~ aa A Is + -UF 5, (A+ + 2S x + 28 vr, + (we 8T,,)| ao) The terms of the above expression will now be collected and arranged as co- efficients of h ,A& , end A ; thus rs = C e 2 fees a0 rae oo Sa & +2 | 7, Te = Te ‘oe Wt a — (The + ee Ti) 5 )5, 7.4 + (10.2) The above result is now in the form used in Report 5, however, it ia obvious that no comparison can be made with the formulae of Report 5. Ii-11 II~l1 Aileron Hinge Moment, Theodorsen Form The Theodorsen form of the aileron hinge moment equation is practically the same as expression (9,13) except that and h must de replaced by h, and 4, by means of relations (9,1) and (9.2) of section I-9, If this is done expression (9.13) becomes Ma= EVE (K-48) 7+ UE8 (G+ 2%) + a (Ts +7,c0s@,) + LP(7,7,-%) which on collecting terms takes the following form, 2 ee ° Ma = 5S 1S hc + USS (7; +275) 2 + O68 7; + (cos 6, -a)7,| + SE (7,-7%) ,Uc& c?fé tai + Far Ts ~UT25,| hat Ux +CH(s— - : e + S47} (Kila) If the above formula is compared with expression (XIX) of Theodorsen's work (reference 2) it will be found that the coefficient of Coe does not appear the same as the corresponding term in Theodorsen's expression. This coefficient as given in expression (11.1) can be brought into the form as given in Theodorsen's work if use is made of the following identity existing between the Theodorsen constants. This identity is B=2% + -at, (11.2) From this it follows that Tgt+ 21g = Ty + 2|2T x; - 4% | ™ | - | =2|2% +7 T4(4 5)| (1103) Also from avpendix A it follows that “2 Te = 1 + (cosrG,—a)T (11,4) which is the coefficient of [°%, substituting expressions (11.3) end (11.4) a in formula (11.1) it becomes c2 se ° F C*o = U am Ai. + UE (TyTe-G) + WAT + GAG - UT..Fq| hat Vex +8 (L-a)+ = Bey io + S27, robe Kang (11.5) which, aside from a difference of notation and a slightly different arrangement, is identical to formula (XIX) of reference 2 if Be is replaced by Theodorsen's Os II~12 II-12 Aileron Hinge Moment, Lombard Form To put the eileron hinge moment formula in the Lombard form, the terms defined in section I~10 and the /’’s given in appendix B are substituted in expression (11.1). Performing these substitutions the aileron hinge moment divided by CI, becomes Ma CM 2 e ee ~ Ryo 2 6 -Ry UB - Rye CP — Re ¥ 5 n|-Ye + Ua + c(5-~e)& +B Ue + Bed | (12.1) Attention is also called to the fact that Y.-- h, and that 14, is the moment per unit of span. | To put expression (12.1) in its final form, apply relations (7.2) snd (7.3) of section I+? and arrange in coefficient form as shown below; Mg cm, , — 2 — na. ED 2 The above expression is in a form which is comparable with Lombard's work, see expressions (5:10), (5:11), and (5:12), page 119, reference 4. TI-13 Ii-15 Conclusion | The principal theme of this thesis is to develop serodynamic formulae for an oscillating wingealileron combination of finite span. This ig done in the preceeding pages. Hence, the principal theme of this conclusion is the effect of span or, tetter eay, the effect of aspect ratio on the aerodynamic formulae of an oscillating wing-aileron combination, From the nature of this problem it eapvears adviaable to eplit the con- clusion into two parts, one concerning the wing lift and wing moment, the other concerning the aileron hinge moment. The derivations of the formulae for wing lift snd wing moment are in reality an extension of the work which is given in Report 5, while the derivation of the aileron hinge moment is a problem which starts from first principles. Part. I In chapter I, the wing lift and wing moment formulae as developed in section I-8 are comparsble to the corresponding formulae of Herort 5. On comparing expressions (8,1) and (62) of section I-@ it will be observed that they are respectively identical with expressions (2.7) pave 79 and (3.9) page £0 of Report 5, except that the two former expressions have terms due to the presence of the aileron, i. e. the coefficients of A. Further, if the co- efficients of & in expressions (§.1) and (&.2) of section I-€ sre examined it will be observed that the only terns effected by aspect ratio are A and Q,. ‘From section I-9 it follows that expression (XVIII) for the wing lift as given by Theodorsen in reference 2 can be made to apnoly to a wing of finite soan if Theodorsen's C is replaced by the B of Report 5. In like manner it also follows from section I-~9 that the wing moment expression, 1. e. ex- pression XX of reference 2, can be made to apply to a wing of finite span if the factor 2(A+4S)C in Theodorsen's expression is replaced by the factor (2d fe + Qe) where B and Qn are taken from Heport 5. A similar TI-13 comparison is pointed out in section I-10 concerning Lombard's work in reference 4o From the above paragraph it can be concluded that there are no basic differences between the formulee for finite aspect ratio and those for infinite espect ratio. Hence, to study the effect of aspect ratio it is necessary only to study its effect on FR and Q, . This has been done in Report 5 and the effect of the aspect ratio on R and em is shown in figure 3 part I of that report. It can therefore be said that the conclusions given in section 7 part I of Report 5, which were written by Dr. M. A. Biot, can be taken over as a unit insofar as the wing Lift and wing moment formulae of chapter I are con- cerned, Part I] The aileron hinge moment formula as developed here has been compared with Theodorsen'’s formula in section II-11 and with Lombard 's formula in section II-e1°. From these comparisons it can be concluded that there are no basic differences. For example, it is necessary only to replace the C’ in formula XIX of reference 2 by Se in order to obtain the aileron hinge moment formula for 8 wing of finite aspect ratio. It was also pointed out that Sp 7 C=P es AR —» go , in this case the formulae become identical. Hence, to study the effect of aspect ratio on the aileron hinge moment means the study of a in comparison with C or PF. Attention is called to the fact that Ss. is a function of reduced velocity > aspect ratio, and aileron chord or, more precisely, T whereas C” or I Ele fF ere functions of reduced velocity only. In addition, the quantity Sx like C or P is complex. Consequently, since the numerical work is somewhat lengthy, the author was excused from evaluating ee insofar as this thesis is concerned. It is therefore deemed as inadvisable to attempt any concrete con= clusion about the variations 3a with aspect ratio, without the numerical values, Lo [es] 4 References Biot, M. A. and Boehnlein, C. T.; Aerodynemic Theory of the Oscillating Wing of Finite Span; Repert No. 5; Approved by Th. von Karman; California Institute of Technology, Pesadena 1942. Herein referred to as Report 5. Theodorsen, Theodore, General Theory of Aerodynamic Instability end the Mechanism of Flutter; T. R. No. 496, N. A. C. A. 1940. Herein referred to as N.: aA. C. Res i T. Re ae 496. von Kerman, Theodore and Burgers, J. M.; . Aerodynamic Theory, Volume II; W. F. Durand, Editor-in-chief. Lombard, Albert E. Jr.; An Investigation of the Conditions for the Occurrence of Flutter in Aircraft and the Development of Criteria for the Prediction and Elimination of such Flutter, (a doctor's thesis); Celifornia Institute of Technology, Pasadena; 1939. Hardy, G. H.; A Course of Pure Mathematics; seventh edition; Cambridge at the University Press, 1938; in particular, Chapter VIII, section 203. TR-496, reference 2, however, the lower care C Appendix A Theodorsen Constants The definitions of the following symbols are taken from N. A. C. A. » used by Thecdorsen has been replaced by cos6, likewise the Vvi-c? has been replaced by sin @, » and cos 'C by Oo. 7, = Wom i l {{ il jl I t{ I 1) ot i} 4 pp ( KI ~ w Po 4 ~ sin 6,(2 + C03? Go) + 6,¢cos 9, = 00s @, (1 — 60570,) — @, sin 6, (1 + cos? ©.) + cos O, ~(g +€0s*6,) 8 + $6,cos @ sin 0, (7+ 2 cos*@) — J (1-¢os* ©)(Sc0s*6, + 4) -~ @ + cos@, siNnG, -(i~ 603*6,)~ Of + 2G, cos 655178, Tp ~(% + ws? @) @ + g cos 6, sin@, (7 +2c05* G) _ 5 5in®, (2 cos*6, + ») + O, C05 0, 5 | $sin?e,+ a7) = 5(-p + aTy) where pas sin7@, sin@, + 8, | @o(i- 2003@,) + SIN@® (2 — 60S, ) = $in@,( 2+ 0s.) ~ &(2c0s 6+ 1) i 2 ca / rs |-7 - (cos @,—- 4) | - + C05 0, Identities: ~ \\ 2Tg +17, -aATy see section I-9 see section II-~-1l Appendix B Lombard Constants Relow is given a table of the Lombard-A's in terms of the Theodorsen 7 5, see appendix A. This table is taken from the nomenclature, sppendix II of Lombard's Thesis, reference 4. % i _ Ay vil (Ty + hho) =, df 3 . Jz 8” 77 l TT 2 Ten Tee Ay Ate ie i Ts "a> ~ arr R Tio Ti2 — ie 13 TT? _ 4 — Tite — Rete \4Q~ 47T® ~— 4 ote: Ry and RR, were taken from a report on wing flutter written by Dro Mo A. Biot. The second equivalents for AR, and As are not given in Lombard 's Thesis. Appendix Aspect Ratio Integrals The integrals of this appendix are called aapect ratio because they are the results of the aspect ratio terms in the functions F, , E£, , ond £, as given in chapter II, however, these integrels are also functions of the aileron chord. For the purpose of working ont these irtegrals a symbol A will be defined such that k= (c 1) or (a k = 53% (C 2) Py this means integrate (4.3) and (5.5) of chapter IJ arpear the same, and one derivation suffices for both. Integral (4,3) now becomes keos € 56(k@ a sin? T el ) = a je cos OQ — Cost at ae (c 3) \\ cost In order to integrate, the first step im to write @ in a Fourier half rang@ cosine series, thus ay Nh COS e cost _ Ao + 4, cost tf Q9t0s BUF + A,cos ht af isa (ce 4) Multiply the above expression by AT and integrate from O to 7 as shown below: Tv Ww Tv Ww [eter ae = | a Ut + [aosear + J dscosat de bere ral) o o . OC in +] qcosnt de tere Oo All of the above integrals of the right member reduce to tero except the first, and from it the following reault is obtained: 7 0 (Cc 5) Inte,ral (B 1) «iven on page 8&5 of Report 5 becomes after substitution of (C i) the following; iv ] ot casnt de = 1 1,(k) 0 (c 6) where [,(k) is the modified Bessel function. If 47 takes on the value zero the above becomes Tr : | gre? dese TFL i) ° (Cc 7) Substituting this result in (C 5) the value of Go becomes = I,(k) (c 8) To obtain the balence of the coefficients, multiply expression (C 4) by cosrit dt and integrate from O to 77. Following this procedure qd, becomes a, = 21,(K) (c 9) for H= /,2,3,:+, Making use of (C 8) and (C 9) the series (C 4) can be written as Qe ekest—1(k) +2 2 1,(k) cosnt iad (c 10) Before substituting series (C 10) in (C 3) the integral will be slightly modified in form; thus (yc ee: ak Po. __sin*@ S, (4,4) = 4) Je } cost + cos om ate ore} dedo 0 “o With the exception of the fraction the above can be integrated by means of (C 6) and becomes 5.(4,&) = { [400 + I,(k) cos 8 | de oO ¢ -#f [etree sin?6 arda cos T— Cos O (¢ 11) | Substituting i 10) and (C 11) it becomes sti) = | | Lik) + L.(k) cova} de “af | “f 104) +2 Ss Int) os it} === deco C Assuming that the above satisfies the requirements such that it can be written as s,(K.a)={ [L0H +14) cose] de ~ L004) | i] Se dede (>) ° o cos T—cose Sop . -22 100 [ f sin 8 60S HIT elo o “oO cost~-—cos@ (C 12) It is apparent that in expression (C 12) the following integration is involved; thus Wr i cos NT dt = 7 sin tie ° cosStT—COSG sine (C 13) where /7=0,/,2,-+, This is integral (2 D) given on page 101 of Report 5 except that the symbols have been changed to suit thie problem. The result is a Cauchy principal value, see also pages 173 and 174 of reference 3. Using (C 13) expression (C 12) can be written as & 20 8, —2 2,L(%) | sine sinne de te 3a) From the trigonometric identities the integrand in the above integral can be transformed as follows; sine sinns = 3 [cos (n-i)e “ cos(7+1) 0} (c 15) Substituting (C 15) in (C 14) and integrating it becomes S0(k,®) = I.(k) @ + I,(k) sine, —_ [oo] a =f 80 " T(t) f | cos(n-1) ~ cos(n+ie| de 40 In order to avoid the difficulty apparent in the remaining integral when += / the series will be separated and the above expression written as CG S,(4,8&) = L(k)e + 1,(4 sing, - 1k) | ‘de wo So ° — 22k) | cos(n-1)ede oO a e, F > In(t) | cos(n+yN ede ) which becomes on integration So(K, >) = Pe (kK) sin®@, _ 2. In () sid inf Bo ; [o) Sin(m+)D6 sg 2_ I,(x) 2in ttt) Ge Before changing the form of (C 16) the limit 5,(k,©,)as the aspect ratio tende to infinity will be evaluated. From expression (C 1) and (C 2), and from the fact that fey which ig given in Report 5, it is evident that 4 . Kk =e (6 17) or / a 8 A a (Cc 18) hence as A—~oo it follows for either of the above two equivalents that A—~O From the properties of the Bessel functions it follows that limit L,(«) = 0 Jimit L,(x) for vis lg2@saa°" and: for =o int L. = Lint L.(«) / hence if G6, #O it follows from expression (C 16) that limit S,( 4,8.) = Sik — (c 19) The procedure of rearranging expression (C 16) will now be given in detail. In the first series in this expression let m<nmh-I! and in the second let m=Htl then So(H,8.) = Io) sin ©, — 2 Teves (K) See e ‘in 1, + 2D) (4) es By including the first term of the right member in the second series the above can be written as S,(K,0.) = is | Zn (A) 2 Deny (x)| ae (Cc 20) From the properties of the Bessel functions the following recurrence formula exists; thus en Zn-(k) — Tn (K) = Fo Tr(a) (c 21). Substituting this in expression (C 20) end at the same time writing m for 177 it becomes 2 (2) 5,(k,6,) = & > I, (k) sin ne K meal ° « (c 22) At thia point a symbol 0(k,@) will be defined such that fo) T (k,0,) = 2 1,(k)sin ne, (¢ aa Expression (C 22) can now be written in its final form as S,(k,@,) = Es 7 (K,®,) , K ‘ (GC 24) This expression is given in section II-4 as expression (4.6) where Aa and in section I1-5 es expression (5.6) where kK = eK . Next in order of derivation is 5, (%> 8.) which is first defined by the integral (4.4) given in section II~4. Making use of expression (C 1) thie integral can be written as ee % akeos tf 5b sine 3,(K,® =+} | en — a At do 0 ~O ° (c 25) To bring this into a form easier to handle, the fraction in the integrand will be changed as follows, cos T sin? Tt cos? T-cost cos 8 = cos T COSO = COST sin*@ cos @ a I cost — sin?e — cos + cos 6cos 9 eos Tr Cose (C 26) \t Substituting this in (¢ 25) it becomes Oo 7 S,(K,@,) 7 7 ] ere) coset + cos@ cosT Oo Vv 42 ~ antg « 27S me f deeds cost ~ cose (¢: 27) The first three terms of the integrand can be integrated by means of (C €) as shown below, k 6.) =[ {4 [a0 + I,(«)| + I,(k)cose - h(n) int @ ble — tt ekeosT 2 tf sin*@ cosO ?_drde cos T— CC5@ (Cc 28) Substituting expression (C 10) and rearranging it becomes sine) =| SETAC) + I(k)cose + $1,(k) cos 20| de = | | $L (x) +221, (k) cosn6 Us ee ee drde . = st - 0056 (c 29) which as in the case of S,(k,@,) can be written as ® S,(k,8,) =| | £1(k) + I(k)cos@ + $1,(k)cos2o| de Oo S10 [s —~ k) ] si de 22_I,,( | inn@ sin @ cose TO where use of integral (C 13) has been made at this point. By means of the trigonometric identities SING sin® cos @ = 4 | <os(n-2)@ ae cos(n+2)6| substituting this end integrating the first three terms (C 50) becomes 3,(k,@.) = & 1,(k)@ + L(k)sine, + 4 f.(k)sin 20, fore) Go - sh 1,(x) | [ cos(n-2) 8- cos(n+2)6] de (9) In this case the term which gives trouble is obtained for =2 , hence the above is written as S,(k,8) = £1,(4) 6, + [(k)sine, + 4L(k)sin Ze, _ £40) ss -$1,(K) ‘de 80 -4 >_I,(k) | cs(n-2)6 de Ls nt + £2 (00 | cos(n+2)6 de - O which on integration becomes S,(k,a) = sl (k)sine, +41,(k) sin 20, -41, (ass +4D1, (K ipa Sirt(11+2) 0, nena (C 31) From expression (C 31) it is also quite evident thet . . ” / . - (rast S,(k,@,) = 7 sin 20, ; (c 32) The procedure to be used in transforming (C 31) into a form more suitable for numerical calculation is the same as thet used in the case of S.(x,@,) but here the work is somewhat more involved. To begin, let M7=mM-1' in the first series of (C 31) and let m=2n+2 in the second, then 35,(k,6,) = $L(x)sine, + L(x) sinze, sin Me ~ 2 Retna) c-,8} in re +4 2, Lpaa(k) eae ~ which can be written as 5,(k,0,) = $1,(k)sine, - 34k) sin @, +420 | Tpia(k) —Linve() | Ee 0 In the recurrence formula (C 21) first let m='-1 and then let M-/7+1 , which given the following two formulae: Ima (#) - Im(a) = 2°82 7, Cg and Lin (k) Lm egl W) ra ee 1 ai( A) Adding the above two expressions gives as 20h ‘ Lipa ®) _ Leg 42(K) ad kh | Lins (K) ¥ Leet («)| Applying the recurrence formula (C 21) to the latter brackets, the expression can be written as digwgl®) =~ Forget) BF | In-iCk) + Dina (Kh) ~ = Tn (Kk) | (C 34) Substituting the above in (C Sc) and separating the series it can be brought to the following form; S,(4.8,) = $1,(k) sine, — E(k) sine. co oo l ! Ty 2 Imi (K) INE, T “e 2 Lmyi(k) Sin mG, co 2 ois ii mee pa ae, (C 25) Now consider each of the above series seperately. In the first of the above series let N=m-i , then 2_Im.,(K) sin me, = 2 In (k) sin(7+1) 6, n-2 co = > I,(ix)[3in 116, COS @,, + cos Ne, sin e.| In the second series let H=am+!/ , then co > Im, (K) SiN 118, = > In| (k)sin(17-1) 6, 722 = 20 k}|sin he, cos 6, — cos Ne » 3n@,| —I,(k) sin Adding the above two expressions gives oO co 2 I,,-, (K) Sin me, + B Sots (‘)sinme, I (oe) 2cos6,2_ 1,(k)sin ne, — I1.(k)sine, cae | 2 cos Oy O'(k,6,) —1,(k) = I,(k) sine, i where expression (C 23) has been applied. Combining the first two terms of {C 25) by means of the recurrence formula and substituting the above expression (C 35) becomes 5,(4,@) = = [,(k) sine, + tlacose, 7(k,@,) — I, (k) sin @| which can be written as i. = = 5,(k,6,) = 71.(K) sine, + —O°(k,6,) cos8, = - a | > [,(k) sin me, ~ L(x) sine, | Again applying expression (C 23) to the above expression end eliminating L, (4) by means of the recurrence formula the above expression becomes S,(%,8,) = rae) sine, — 2 ( i ~ cos 6, ) o(k,0,) | (C 36) which is identical to expression (4.21) of section IIl—4 if K ie set equel to J FR For the next integral substitute A= € in integral (4.18) section II-4 as shown below; as ot | ts 4%]TCcose ° (7) coo8 —-cos T (C 37) 6 Comparing the above intezrand with the integrand of (C 2) it is seen that the two are the same except that (C 37) has an edditional factor cos6, in the nunerator. Hence, multiplying the integrands of (C 14) by cosO, the function Sa(kK,@.). can be written at once as 5,(k,@,) = - [(k) cose + I,(k) cos? @| de ey | co Be = 22 I,(a) | sine cos6 sinne dea To bring the second integrand into form substitute the following trigonovetric identity; sin@cces@ sinne= 4|cos(n-a)e - cos(n+2)o| (c 39) Substituting (C 39) and integrating the first expression (C 3&) becomes 5,(k,8.) =f (k) 9in@, + 1,(«)( $ + Zs Ze) v7et es Ly Iy()] [eosln-2)e ~ cos(r+2) 6] de (Cc 40) Before integrating the series, they will be seprrated and the one containing the cos(n-2)6 terms will be expanded for+72! and n=2 ; thus integrating (C 49) becones S2(k,0,) = T(k) sine, + 1,(4)(¥ + 4sin26,) ~ I (x)sine, - 3 1,(4)e, ao ! sin (172) 8. ~ 42, Ln) STS ce | sin(1t+ 2) 6. * 22 Inlh) “at (c 41) At this voint the Limit S2(k,6) can be seen at once to be Ko limitS,(K,@) = 2 + +sin26 (C 42) k~7o 2’? a «+ i —— {The procedure used in reducing (C 41) to its final form is much the same cg thet used in reducing (C £1) and will not be given in detail. If, however, the procedure is carried out, it will be found thet the result becomes 53(k,0,) = (L(x) e, +1,(k) sine, — 2{ *z ~ os Go) o-(k,8.) | (C 43) Before leaving S,(K,©,) it might be well to show the direct derivation of formula (4.28) of section II-4 which in A notation can be written ss / S.(k,6) = 5, (-,) + KL(A)e, To derive this directly subtract integral (¢ 25) from (C 37); the resulting expression is 4°77 in? tT Ccos®@ COST SINnzT S,(k,0,) — 5:(k,8,) = 7 i | eres | Sn ee ~ ossin | dtde o “oO cose@-cosT cos@ -cosT This result reduces to 8 oi. 5, (K, 0.) — S,(k,6,) = al | e kot oper dt de (4) Oo which integrates at once as fs) S,(k,8,) — 3, (k,@,) =(L.(W) ~I,(k)\ & The above can be reduced by means of the recurrence fornula to : @)-s K,6,)=7l(ne (C 44) S2(k, ° vf A.) = 3% (AK ° . From section II-4 expression (4.19) becomes on substitution of k-£ the following ; a ial in? Tcos@ ot Kk cost sin COS 53 (KG) = +f | e Cost cosezcost ATO oO “0 This integrel is similar to (C 25) for if the integrand of (C 25) be multivlied by COSG the integral becomes (C 45), Multiplying the integrends of (C 30) by COS@ the first integral of (C 45) is obtained; thus Qo _ 5,;(k,6,] -/ | SL,(k) cose + 1(k)cos?@ + $1,(1k) cos20 coso | de o [ce] °, , am 2 Inlay [ sinhne sinacos20de 7 fo) Since the above can be used in the derivetion of 5,(k,60) it is well to rewrite this exrnresrion as 5; (k,6.) -[ | eke (k) cose + $1,(k)(/ + cos 2) +28 (k)(cos@ + cos 3@)| de co 5 - > z.(00| sinnesin26 cose de a! O (C 46) The trigonometric identity which must be substituted for the second integrand in (C 46) is shown below; SINH®6 SiIn26 cose = = |cos(n -)@ + cos(H -3)6 ~cos(N+3)@ — ccs (n+io| (c 47) From this expression it is apparent that after substitution, each term of the resulting series will be made up of four terms, From the procedure which ‘ie it ig advisable to write the result with four sevarate series. As in the preceeding cases it is necessary before integrating to separate from the series containing the cos(n-1)6 , the term for which#-=! and in addition the series containing the CoS(4-3)0 , the term for which 7=3 . In the latter series however, the terms for whicht7=/ and H=2 will also be separated. Performing this substitution ond integrating (C 46) becomes 5,(4,4) = 4-1,(k) sir 8, + Lil (4)(6,+ 3510 2Q) +4 Liksine, + £sin3@) ~ +L (n)@, +> 1x) se ~0, nee - gAlk)sin2e - al, (k) sine, ~F4I,(kKe, ~42 1 aie. co oO a j Bin (74+3)@ 0 7 sin(i +4, +4 4 2 Inlh) A+B +42, 1(h) N+] (c 46) which can be written 2-8 a 53(k,@) = - +1, (kK) (sin@, + 5 717 38,) + 1(k)(@, + $3in Ze) I(A)sine, — 13(k)e, <- oO sin (+1) 0, _ sin(H7-1)@, a 2 Lak kK ane rer In(h) ti-f ) siin(n +3), in(M-3)e diy _ 2 a +2. (Kaa 2, T4005 A-3 (G49) From the above exrregmaion it te aopsrent that lirt S 33(k,a,) = 46ine, +45inse) (G 50) Following procedure eimiler te that vaed in S51 (Kk ,8,) » the series of (GC 4°) ean be written tn terne of (7(K,Q,) % The mantpulation of the terme tea quite Teng and anly the final reanlt will he elven here, which ie 4, (4A) rack aa i, COS, + Ces 26, a] (Ks, @,) - + £1(K)@, {. & r a . + { - sal, 4 1) 1) 1 (2 1a(4) + oa (1) | SIN a. Pe | 17 8 7) \ |( is + )LOO = (pe + Oda & Ga Ls) chk i ly / ' 5, Ly(n) ~ a 1) sin 20, [ (GO #1) The shove @apreaston ta nef me yeh in tte beat form for numericeal celeulationrm, ainee the Reseal funetiona sheuld be rediused te ardere of pero and one, Thhe form, however, tea much better for numerlesl caloulation than (UC 42) sinee here it te anty nevyesnary fo compute one eeries 1, @., AK B)yand thie muet be come muted for S06,@.); Rapreacton (4.07) eection 11.4 defines on (Qa) which tn terme of the ie negation begcamees e,, 3 1 Apiseesal - - Sy) He) [ e aut cesta r ‘ at f sti 6€0? { i” ) oe eo Ce 27 aes @ &. Piste litegral sat be obfetued ty mit iolytug the titegs ais of Ur & } by (Iso C The first integral is therefore derived from expression (C Z&) by introducing the factor Cc56 as follows; 5, (k, &) -| [ L(A) cos* + L.(k)cos2e | de co Qo ~2, 10 sin @ cos%e sinnede °o which can be written as 8, S,,(K,8) -f [EL (a(i+ cos 26) +41,(k)(3cos 6 + cos 36) ] dé ° fea) ©, =y In(k) | Sinfie sin2e@ cos e@ de (0 2 = Ow " 6) If the above expression is compared with (C 46) it will be noticed that the two integrals involving the seriss are the same. With the aid of (C 48) the above expression can be integrated at once; thus Sy(4,) = $1(k)(6, + 3 sin 20) + 3 L(k)(3sine, + $sin 36,) ~3MK)e, - 4 2, Bae ~ 3 1(W) sin 20, - Lh (k)sine,~ SL (We, ~ AF-S, (x) res ne co i y oer sin(tt +1) @, +42 14) et L thay Oe which can be written as S, (K,Q,) = Z [Z(k)(3sine, + sin 3@) + I(k)(@, +S sinze,) — I, (k)sine, — 1,(4) 6, 4. >I (x) sin(nt!) e, _ AG, y sin(n-)e% z h +i fis sin(n +3)@, Sin (tt 3) 6, + Rodney Serres ~ Fay Soe From the above it can be seen that the Minit 3i(k,@.) = 4(33ine, + + 5iN3@,) (c 55) Manipulating the series of (C 54) by methods used heretofor and by using the recurrence formula it can be brought to the following form, S,(4,@,) = ral 4 ~ # cos 8, + cos20,|T(K,0) + qa (k) 2, +|—she (He LW) + fz I, (k) + = I,(k)]sin 6, + (A+d) lw -s6(8+ 0b + 2h | l ‘ = ap Ly{h) —_ £1,(k)|5In 26, (C 56) The derivation of expression (4.32) section II-4 will now be shown which in the k notetion can be written as S4(k,6,) ~ 53(k,8) = ¢1,(x) sine, (c 57) Making use of the expressions (C 45) and (C 52) the above difference can be written es TT - 6, 2 rt f A cost sit?*t cos?O 4 (4,85) ~ Salk, Go) = | ad apes ° ° Sin*T cos@ — COS T——_—__-_—— vos @-cost ] at de which reduces to = a Sy(K,O.) — 53(4,8) = « | { eT sinzt cosadt de o *0 (C 58) The first integration of (C 5&) is made by use of (C 6) which becomes By S 4(k,@) ~ 53,8.) = 4{ 1,0 ~~ I.(k) | it cose de (a) Since I,(k) — [,(k) = 21 (kK) expression (C 57) follows at once. The last expreesion to be derived is S,(k,@,) . This is defined by integral (4.29) which in A notation becomes QQ 8, 7 ! kK cost Sin*T cos?e@ ‘S.5 == “CO s (K,@) 2 | / € cog se tos rT dt de (c 59) Oo 6 This integral can be obtained if the integrand of (C 45) is multiplied by cos® and from this it follows that the first integration can be obtained by multiplying the integrand of (C 46) by cos® . This is done es shown below; @, se s ss(K.9,)= [ | 31, (k) cos?@ +5 1,(k) (cos @ + cos 26 cos @) 0 renee + cos30 cose) | de oa i on} ele ne sin 26 cos? 6 de 1 (Cc 60) which can be arrenged as &, _ a l | + Ss (4,6,) -3/ | BL(k) (1+ 2cos 20 + cos 46) fe] + 1)(k)(3cose + ¢0S36) + 1,(k)(1 +05 2e)|de eo - = 4 >_ I,(k) sinne sin26 (1+ cos 20) de al - (¢ 61) Integrating the first integral of (C 61) and rearranging the second, it becomes S,(4,&) = 4(1 o(k) (@. + sin 20, + 4 sin 46.) + I(k)(3sine,+ $sin 36,) + I,(4)(@,+4sin 26,)| 9, + S 1,(69 f sintia sin 2ede 7 oO fe) o, - 42, Inte) sinne sin 4ede ‘ (c 62) The trigonometric identities to be epplied in this case «re sinne sin2e= 3 | cos (n- -2)6 - cos(n+z2)e| and Sin ne Sin 46 = $[<0os(n-4)e _ cos(n+a)e | 21 Apnlying these trigonometric identities, and sepsrating the proper terms, expression (C 62) can be written as S5 (4,6) = 54 [I (k) - I ix) a,+| aL (n)-1 3(k)] sine, + lla) + 3 ae 24)) 5 in 2@, + + L(k)sin 30, + rE (x) sina} = in(H-2 sin(?+2)6, ~ 420 1,4) Be 4 4 +5 1 > Ip, (k) ee _— 7 sin(n-A)9, sin( N+) 6. x 2 1,(K) ne + 4 7 43 (k) = +a (C 63) The value of Ss5(k,@,.) for A=0 will be taken from (C 63); thus limit $5(k,6,) = 3 (@,+ Sinze, + x sin 48) (c 64) The function will now be given in its final form as shown below; S.(x,a)= 51H) —L (H)] 6, +44 cos 30,~ § cos 20, +( + 33 )eos 6, ~(& + 24 ox) +e (a+ 34 =o + ja) te) -(# + $$ )Io(k) -(3 - 8)L(W) -1,(1)| sine, + g[- Ls (S458 + HL -(2 + 48) ZW) + 12 1,(K) ~21,(0 ~ $1. (4) |sinze, +o [(g+Byiin~ (2 +88 )L(0+ (18 + 28)E —(e fe) I, (k)+ & I,(k) — Rle(K) - 5 L,(4)\sin3e, (Cc 65) Although the higher order Bessel functions in (C 65) can be reduced, they must be computed for 7(H,6) . It wes therefore thought best to leave expression (C 65) as is, for it may be necessary to compute the Bessel functions up to and including /] 7(h) in order to obtain a satisfactory value for GO (Kk,6o) Using tre results of the preceeding paragrephs the limit of the function © f will now be shown to be zero es the aspect ratio tends to infinity. From section II-9 expression (9.4) is given as = 2(£c0se,-f,) + LXE ( f cos 8, - 2ficos@ +f) (c 66) It can now be shown that linit T = 0 | (c 67) From section IImW8 expressions (€.2), (&.4), and (8.6) are respectively Ip St Be Toe # lee (c 62) t= f, v is t+ fh, (C €9) (c 70) f= hr + het fe It can be shown for the nine terms given in the right hand members of (C 68), (C 69), and (C 70), that limit fy = limit bs zee = limit £. =9 (c 71) R- + 00 Ro AR —- oO from which it follows that Hirt = tanith = tirmtt Te = © (¢ 72) and hence the limit (C 67). To start this proof take the function hr which is given as expression (4,14) section II-4. Since the limit of a product is equal to the product of the limits it tethers thet the limit of (4.14) is lint i bet oe {[ 7" aR += 4 «Mimi S.(# 19) - Z limit SCZ 09 From (C 19) end (C 32) it follows that the ahove can be written as lirnit fF. = hint # _ 6% _ [ er R20 97 Reco AI + LAC 2) [+ AS gg + #41 ]sine, - Z(4 sinze) R SH RNA ° from which it followe that lint for = (C 74) AR + CO C The function fs is given as expression (5.10) section II-5. The limit of this expression can be obteined as shown below; os det tos = rit |- 75st CO | Menit Sosa) (6-78) Substituting (C 19) this expression becomes Pit f= tpeit| = EBS ere | ing, x Ane Reel 1+ ade R Ss (o 76) The lest function of fi is ie and it is given as expression (€.5) sec~ tion II-6. The limit equation is irr be = 56 Limit | (48 a,+ 24a, +154,)s5in @, ~ 6(a,+ @) 5iINn2@ + a, sin 26, | (c 2%) COW From expressions (5.6), (5.7) end (5.8) of section I+S part II of Report 5 it follows that lint Ay = limit @, = lint es - foe es ee = ee (c 78) It can now be said that hinit &. = O (c 79) Rreo From exoressions (C 74), (C 76) ond (C 79) it follows that limit lh = O (c 80) Av 0 In order to obtain the limits for f and &R expressions (4.24), (4.26), (5.19), (5.26), (6.10) and (6.14) from sections II-4, II-5, and II-6 are needed; from appendix C the results given by (C 42), (C 50), (C 55) and (C 64) are elso needed. The procedure is the same as thet for K and the details will not be given here, but if this work is carried out it will be found that expression (C 72) is true and hence the validity of (C 67) follows. Appendix D Integrals of Section II-5 In section II-5 the integral (5.0) is given es @, ;1T 2 tee [TI Eo ae dsdtde 0 Yo 4 CS Gos' (608 ~E057) (D 1) Reversing the order of integration the above expression becomes (Ac a i e 2 sine IL) ees aay ea fo} (D 2) The integrand will be separated in partial fractions as shown below; ll a int | (G6 —cost)(cOS@ — COs T) €-cos@ §~cost sin*?Q \ me | €-cos@ LOST-—COSe Substituting this in (D 2) it becomes ($ ~ cosT)(coS @ — COS T) (D 3) @ , 6, -1T ~Ase , o 2 ae. IT] e€ ani"? dtdedc ce) f) ®o ~ ise — . l sin®@ | -| | e Ti ee cos. G=cost S$-—cose , Cos T—005S 8 ~ i jar dede | Within the brackets of (D 3) are three quantities which form three sepa— rate integrals with respec’ to C which will be taken up one at a time. The first integral is the type given by B. ©. Peirce as No. 300 which is a+ bcoosx ~ V a2— p = a+ bcosx f .~( b+acosx lax aX | sin’ | ox) (D 4) where the constant of integration hes been omitted. Arplying (D 4) to the first term of integral (D 3) and omitting the fractional coefficient it becomes [Ss = ah siri'[ SPS ER) _ ease ei =a a Fed (D 5) The second quantity in the brackets is the tyne given by (C 13) and therefore - COS T ~ COS @ ‘ (o) D The third quantity is the -/ which on integration yields —7T. Substituting these three results the integrel (D 2) takes the following form; (AC 2S sine at } Ty _e cos (cass —cosT) ig ay enf e = ate see if de de (D 7) The integration with respect to O in the above integral is given by Atdede expression (D 4) and applying this formula (D 7) becomes 00 7 O& - Ace ine []l sit't _ dedods 4 “0 “0 ~ COS T)(COS 6 -COS T) “OO tre ~-3= 5 =~! 6COSO,— TT =-7r]| e@ 2 | sir ee os, = Se + 0,\d5 (D 8) é For the convergence of this integrel ag well ee those which follow see G. H. Hardy, section 203, reference 5. To bring (D &) into its final form the parts forma Judy = ur ~/ vd is used. In avvlying the parts formula let — ai rt Scosa, — ae U = sin v-cs6, — 2 1% and oat ade = dt From this it follows that si] @, ds du = (3 — 0056) fezar ard > = VAC c == oe © Substituting the above in the parts forma, integrel (DB) becomes a Ac fey e 2 Ssin? Tt [J le Ste ogg At deds Cc “O CAC = = Ale (sii Gees! - g+0,)] OAC 5 = cos9, 5 ! ao fg c a a SLE OB: Goa sae Pome ae de . iAc ], (G — cos @) 7 e321 a (Dp 2) The hracketed guoentity in (D 9) tende to gern ae * tente to infinity, however, lower lttmft, i. e@. eT | $t henome -T!I Oe... Ne lution wee found for the remaining integral of (D 9) so a symbol (Q, was defined as shown below; a This is given as expression (5.4) section II-5. Integral (D 9) can now be written in its final form as cAc (ry CPA he dade Oo ’Oa (€—cosT)(cos 6 — COST) } err te er, |. = face = Or-a) - Gee * Qsine * (D 11) The above is given as integral (5.5) section II-5, The next integral of this appendix is given as (5.13) section II-5, On compesring this integral with (D 1) it will be observed that if the integrand of (D 1) is multiplied by cose it becomes integral (5.15) of section II-5, From this it follows that a first integral cen be obtained by mitiplying the integrand of (D 7) by cos@®. Performing this multiplication (D 7) becomes 0 ,6.;,iT «SAS 7 2 intt c Ee sin*T cos @ — , & & ($ — cos T)(cos 6 ~ cosT) 00 / &, irc, . > =r] j e Bs | USF cose ~ cose} do ds (o) o G- cose (D 12) which can be written as 09 7, IT ~ AG — € Sin?T 6058 ardeds dy dp ($—cosT)(cos 6 ~cosT) “(*iMefeyl er ae - 2 ae ee me [J e eo yS?=I cos 8} de ds (p 13) / [o) Within the brackets of (D 13) are three expressions, insofar es @ is concerned, The first forms an integral of type (D 4) and the remaining two integrals are very simple,hence (D 13) becomes ~/% r™ le ‘ i > sin TCOne a } = COST)(Cos 6 ~ COST) ar aes i 1 %0 (“pis pore S cos 6 =n! CF? | te - twee i eC | a 75s —T ! star sine,| os (D 14) The parts formula fu dyr= uv — frrdu must now be applied, ond for this let ITS aa cos O, > : a = Gee SG, ~ GoVS*—! ~ S17, and ° wiht av = S de o From which it follows that _ (tr. § SiIN®> _— ap 2 60s8~1 _ OF Au =(¥ (S- cos®,) Vez ne S — cos 6, fern 4s end (Ac es er = tdc Substituting the above in (D 14) it becomes + sin@t cose [J |: (o> Caras Sainte eee he co cs acs ME as (ms of Scos@.-! _ x3 ; =~ ae le" (4 -Ss ft ae ay | = sina,)| ! 2] @ ther It S sing, 1 Scos Oo-I d +>] e€ * [4 -— - in -— — | D 15) eg) 2 (S$-cos@,)f52_/ mie S-cs@ — 5 ( In the above expression the lower limit of the Uv-term offers no difficulty, the upper limit, however, must be evoluated by means of 1l'Hospitel*s rule considering the following expression; thus wr a7! S905 9-1 h- de — 21086 2 aan S054, ~ & From which it follows thet the limit of the above expression is zero. The remeiniog integral of (D 15) can also be seperated into two convergent integrals thus (DB 15) cen be written as ~Xe = * sin “TCODO [fle (S- cos T)/(COSE ~ COST) kt de ds ~ Ac (7 ~ sine.) ® « Hite ane _ S€*" ue ihe (¢ -cos®)/s?-6 fab - 6:5 Mm a sin Beret / — ca G Cperating once with long division on the integrand of the first integral of the right hand member of (D 17) it can be separated into two convergent integrals. If © is adied end subtracted in the integrand of tte second, it also can be separated into two convergent integrals. By these means (D 17) can be written in the following form; LAC "EF sin?T cose [IT] i (S$ — cosT)(cos 6 ~ casey AT eds ,% thee _ = on Co 2 rr sin 6.) - Rena] a= as _ ihe = ara sind 05 @ — AG 7 am le ree -B+ 8, ) dg (D 18) Considering the integrals of the right hand member of the above expression, the first is given on page 96 of Report 5 and is tAc pees = BL (2g) Vor! (D 19) where AL (AS) is the Hankel function. The second integral is ez@Q, , see expression (D 10), and the third is integral (D 8). The fourth integrel is viven on pe,e 9 of Report 5 which is 2 Ae Y akc aa — 2 _ MyM 2c [é (77 i)A5= nae BH) where by 28) is the Hankel function of order one. (D 20) Before substituting it is advisable to write (D 19) end (D 20) in terms of Qo and Q, respectively. This is done by making use of expressions (5.14) and (5.15) of section II-5 which converts (D 19) and (D 20) into the forms shown below; ric pe oes fac c 2 2 5 dS=-7—-e€ Q | vs oe ° (D 21) or - ire ~ = ] ae Substituting the above results in (D 1€) it becomes a & a oihee — EF sin'Tcos® dr deds (S$ — cosT)(cos@ ~CosT) 1 40 %o - ame Bi ng + Zr- Se” ~ [r sin, + ett 2.) | ok =a pe * [sina o+ 6+ resin. Qa ike ATT3in @ os ATTO, 57 "3° + Tine)? Qo + + tot? [1 * a, | (D 23) which is the form as presented in section II-5, see integral (5,13). The last integral of this appendix is integral (5.22) of section II+5. On examination of this integral it is seen that the first integration is obtained if the integrands of expression (D 13) are multiplied by cose ; thus 3 tAc C78, -3- 5 se rl 2 sin Bae dedede ds (€—cosT)(cos 9 ~COsT) / 02 -e iC ~ Stn ; =r | | e = cos@ — /€*-|' cose ~cos*6 | dedc a) (D 24) GS - cose which can be written as o;,e@, 1 ors ; ; sin®T cos? @ , It (¢ — cost)(cose- cost) aArdeds le ee . nf fe *[vSi (oS - cose - cS) Pe) ~4£-4 dead e 2 cos26 Cas (D 25) Integral (D 4) applies to the first term in the above parenthesis, end the balance of the inieaiiaden is evident; hence (D 25) becomes ° a 2 II] e a ae at de ds 1 =O (€= cosT)(cos @~ Cost) fs .-1 GCosG, ~ = -nf ee [ s*siri ee, — By + ¥s*41 (sine, + 5) = Be J In the above integral the part in the brackets can he shown by 1'Hospital's rule to tend to zero as a limit when CG tends to infinity, hence, according to G@. H. Herdy, section 203, reference 5, the integral converges. To continue the integration the parts formula fudv = uv —fuv cu ig used, where = 6* sin 1 Sc0sG.—! _ om iM Oo 2 | ‘ S = COS Go 25 3 a reosyets" + Ni & + =SINZO6, A~ and cAc Au 3 ee? ode The expressions for duo and Y will not be given here, however, after inteysrating by parts, (D 26) takes the following form, ar oi ES cinet 0s* m jist : > sin co a a il 2 I. ; \ r>) = +e 21T- A - -s sin 2@e [f] fe. - COST (cos @ ~ COST) e 5S * iXe . 0 2 > ERE é a -1 S cos Oot S*sin Be ae eC *[agsin S- cose, + (S- ET mS ine, ones an + 9, (ee jas (D 2” D The deteils of the following integration will not be given here, however, the procedure will be outlined below. The first step is to taxe the second term in the brackets of the integral and separate it by means of long division so that it appears es shown below; S*sine, = Ssine | SIN 8, COS 8. 51n 6, cos? 8, ($-cos6,)fe?-r Ver" Very (S-cos @,)~er_ P (D 28) If (D 2&) ia resubstituted in (D 27) the second fraction can be integrated by (D 21) and the third fraction can te handled by (D 10) which leaves the first fraction unintegrated. Assuming that the above operations have been performed, add to the integrand of the remaining integral the following two zeros; 2efe*-" - 2efst-r 25in Go — 2351InNe, The remaining integrel can now be separeted into three convergent integrals, one of which, say the first, can be recognized through (D 14) to be integral (D 23), except for the common factor 77. The second reduces algebraically to integral (D 21), and the third is integral (D 22). If this method is followed, integral (D 27) becomes eee eC iG sin?t cos* — j (G— cost)(cosS @ — cost) ei 4g % - = eee anes + aT + ===) +4 78, anal - sing| cos*6, + Fa (cose, ¢ re) at (D 29) This is integral (5.22) of section II-5.