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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WA RTIIME REPORT ORIGINALLY ISSUED April 1944 as Advance Restricted Report L4D18 STABILITY OF A BODY STABILIZED BY FINS AND SUSPENDED FROM AN AIRPLANE By W. H. Phillips Langley Memorial Aeronautical Laboratory Langley Field, Va. WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results Lo an authorized group requiring them for the war effort. They were pre viously held under a security status but are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. L28 DOCUMENTS DEPARTMENT Lylb 9 i,. Digitized by the Internet Archive in 2011, with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/stabilityinbodys001ang RESTRICTED '"" NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ;.VAlICE RESTRICTED REPORT 0O. L4D18 STABILITY OF a BODY STaBILIZED BY FINS AiD SUSPIEDDED FROM AlN A.IPLAiHE By W. H. Phillips S LPU',IA' Y A theoretical investigation has been made of the oscillations pnerfor'.ed by suspended bodies of the type co.mmonly used for traili,ar airspeed heads and similar towed devices. The ori'mairy p'rpFos of the investiga tion v'as to design an instrLunirit that will remain stable as it is d swnL up to a support u nd rneath an airplane without attention on the part of the nilot. Flight tests of a model airspeed head were .iade to supplement the theoretical study. Unstable oscillations of the body at short cable lengths were predicted by the theory, but the rate of increase o' amplitude of these oscilla tions was very small. In flight tests, more violent types of instability vere believed to te caused by unsteady or nonuniform air flow in the region where the cable was lowered from tlih airplane. iTo practical method was found to provide large dampinE of the oscil lations at short cable lengths, but the degree of stability present in a suitably designed suspended body was shown to be satisfactory if the body was lowered into a uniform air stream. IITF T ODUC TIO if Suspended devices that consist of heavy streamline. bodies stabilized by fins have ben used in the past for various purposes. A frequent application of this type of device is the suspended airspeed head used for the accurate measurement of airplane speed (reference 1). Certain difficulties have been encountered in the use of these instruments because of unstable oscillations of the cable and suspended body. One co:,rnon type of instability has been a tendency of the instrument to swing violently back and forth and from side to side as it was drawn up close to the airplane. Because of this FSTFI'T D I'..CA AFR No. L4D18 tendency, these instruments need considerable attention in handling and usually require the services of a person other than the pilot. Another type of instability has been an oscillation of the body and x.hipping action of the cable when the body was being towed at the full length of the cable. This motion has occurred only when the instrument was lowered from certain airplanes. The present investigation was undertaken in an effort to develop a type of trailing airspeed head that could be lowered from and drawn up to a support under neath the airplane without close attention. This requirement necessitates that unstable oscillations of the instrument be avoided at any cable length. A study of both lateral and longitudinal oscilla tions of the body and cable system was made in refer ence 2. This study was based on the assumption that the damping of the motion due to air forces on the cable could be neglected. The present investigation shows that this assumption leads to :erroneous conclusions with regard to the boundaries of stability. SYMBOLS m mass of towed body y lateral displacement Cy coefficient of side force on vertical tail Yc lateral displacement of cable element I angle of yaw N yawing moment P angle of sideslip V forward velocity I tail length Y lateral force C moment of inertia about vertical axis /dCa sloe of tail lift curve (negative a slope of tail lift curve (negative) ydy \dpl YACA ARR No. L4D18 S verticaltail area p air density Z ':rtical distance between airplane and body z v'er t!ic l distance g icceleration of Cravitv D effeciiv drag, also, operator indicating differ;ntia tion with resooct to time D' nccndirrmensional oDerator (DT) D dcira:c 0 body Dc total cable drag C dra coefficie.nt of bod 7  Db (A3) CD eciui'salent lag croeffi'cient cf .cb!ile CT LN, z J (( CD .effecti ve drag cos'fficient D' + Dt X Ltcrizorital distAnce between airplane and body cd drag coeff'LciLeent of cable ;:er unit vertical c /cabl0 drcae height  \ R",I'wcz / i relativedensit7 coefficient ( 7) R ratio of tail l ntr, to vertical distance below airplane (1/1) C1 miomntofirertia factor (k/L)2 NACA APR No. L4D18 k t 4 T a,b,c,d,e Wc K Yb Yc P radius of ;:rL .tion time time unit =I coefficients of quartic cable diameter angle of cable with horizontal fraction of cable side force applied to body side force on body side force on cable period of oscillation number of cycles to damp to onehalf amplitude ( m o3 S1 6Y y. V N 6N y nt 1 6Y N 1 A dot over a symbol indicates the first derivative of the quantity with respect to time t and two dots over a symbol indicates the second derivative. THEORETICAL INVESTIGATION Because of the axial cTin.rtry of a trailing airspeed head, its longitudinal and lateral motions occur NACA ARE No. L4D18 5 independently. These types of motion miay, therefore, be treated separately. The lateral motion of the instrument will be analyzed in considerable detail because this mode of motion is theoretically .nost likely to become unstable. Lateral Oscillations Mathematical treatment is possible only for the case of small oscillations, fo' which the forces acting on the body var, linarly with the displacemmnts and angular velocities. The instrument will swinrr from side to side like a pendulum but, for small amplitudes, its motion may be considered to take place in a hori zontal plare. A restoring fcrce depending on the cable length under consideration '.vill be assumed to act through ;he oivot point. The subsequent analysis indicates that tne drag force on the cablEs Cid todt hts an important influence on the damipinr cf the oscillations. In practice, al.nost all the drag acts si, the cable. For purposes of naalysis, however, an e fectivs drag force due to the cable will be assumed to act on the body at its center of :ravity. The relation between this effective drag force and the characteristics of the cable will be discs.sed later. Th3 nctation used in considering the lateral motion is shown in figure 1. The equations of motion with respect to a fixe d system of axes are as follows: yop 6Y 6Y op y 0 7 o 6* d In order to s'imlify the notation, let D = d and define the stability derivatives 1 6Y 1 CS 6 NTACA ARR No. L4D18 and define the other derivatives similarly. These equations may be solved by the usual procedure of setting the determinant of the coefficients equal to zero. This determinant may be expanded to give the quartic aD4 + bD3 + cD2 + dD + e = 0 (1) where a~l a = 1  b = ..  c = N + T v k+ d = NpY + Y, e = YyNp In order to find the nature of the motion from equation (1), it is necessary to evaluate the stability derivatives in terms of the dimensions and aerodynamic characteristics of the. instrument. Tn setting up a simplified form for the stability equation, it is sufficiently accurate to assume that aerodynamic forces other than drag forces will act only on the vertical fin of the instrument. The derivation of the expression for YP is given as an example: YY = pa V2S 2S where dCv a  Y a S2 Y3 m op aV S n 2 IN.CA AER No. L4D8 ' The other aerodynamic derivatives may be determined in a similar manner. Only the derivatives related to the forces exerted by the cable require special consideration. The Yforce caused by a lateral displacement of the body is found by assumiing that the body and cable system, when _viewed front, the front, deflects as a simple pendiilu. (fig. '). The restoring force c;ue to a small deflection y of the body suspended a vertical dis tance Z below the irlplane is Y6 r.tg: Y = HE 1 6y Y = y m y =7 z The derivative Y is found fraon the drag force acting on the body andd cable. A drag force acting on the body will have a component of side force as shown in figure 3. Thus Y = ,( + W) Db7 V Db r Pv23 = CDb iVS (2) 67 .' The component of side force due to the bodr is 1 Y .Svs Db M Inasmuch as the drag of the cable ordinarily far exceeds the drag of the body, the value of the deriva tive Y:, will be principally determined by the cable jd 8 NACA ARR 1o. L4D18 drag. The method of calculating an equivalent drag coefficient CDc to take into account the effect of the cable is given in the appendix. The derivative Y is then given as follows: \pv b Y DDb c vs C D m The value of the coefficient CDC may be determined as a function of the ratio of horizontal length to height of the cable X/Z from figure 4. All the aerodnriirc derivatives have been evaluated in terms of the dimensions of the body and cable system. In order to reduce the number of variables, it is con venient to express these derivatives in terms of non dimensional ratios of the quantities involved, which are given as follows: Froude number, y2 F =  Relativedensity coefficient, m. Ratio of tail length to vertical distance of body below point of support, Momentofinertia factor, Cl 2 m Time is expressed in terms of the time unit T = VS 2 When the derivatives are expressed in terms of these variables, the stability quartic becomes aD'4 + bD'3 + cD'2 + dD' + e = 0 (.) NACA ARR No. L4D18 where D' = DT and a = 1 a b i a + CD a L.R aD ae = aI d Cl C l e The stability of the towed airspeed head may be determined by substituting numerical values in the formulas for the coefficients and factoring the quartic. The two quadratic factors dettermirne the period and damping of two modes of oscillation. One quadratic factor yields values of the rer'od and damping very close to those obtained with a simple pendulum having a length equal to the vertical distance of the towed body below the airplane and damping equal to that supplied by the drag force. The other quadratic factor gives an oscillation that has values of period and damping very close to those of the body rotating as a weather vane about a verticall axis through its center of gravity. The weatherrane oscillation generally has a short period and is always rapidly damped. The damping of the pendulum oscillation is, however, very slow at short cable lengths, because the cable drag is small. The coupling between the two modes of oscillation introduces the possibility of instability of the pendulum oscillation. In order to find the conditions for instability, the coefficients of the quartic may be substituted in Routh's discriminant, which states that the motion will be stable if the coefficients satisfy the relation (bc ad)d b2e >0 (4) 10 NACA ARR No. L4D18 When the values of the coefficients (equation (3)) are substituted in formula (4), Routh's discriminant be c homes 1 CDiR2 aRCD acDR DRC2 ap2R  R + "  F2 a / C2 C1 C1 C1 212CDR 2 2 \ a+CD aCD2 apCD  i CD>R + aC + +. D 2 Cl C12 C12 Cl aCD2 CD3 C Ci C1 1 The expression is given in this form merely for the sake of completeness. In practice, a great simplification may be made, with negligible loss of accuracy, by neg aCD electing the small term C in coefficient c, for C1 mula (3). The simplified form of the discriminant is 2CD a a C C CI a D a R V L 2aCD 1 a The minus sign before the expression CD a gives one condition for stability < 1 (5) a 1+ and the plus sign before the same expression gives another condition for stability F C1(CD a) ( R > C (6) RT aCD Boundaries of stability are plotted in figure 5. It is seen that below a certain small value of the param eter F/RP, given by formula (5), the motion is stable for all values of the drag coefficient. As F/RL is NACA aRE No. L4D18 increased above this value, the motion is unstable until the boundary of stabilit'r given by formula (6) is reached. The motion then becomes stable again at all higher values of F/RV. Examples have been worked out from the general boundaries of stability (fig. 5) to show the variation of stability of an actual airspeed head as the cable length is changed. The characteristics of the airspeed head and cable used in the calculations are as follows: m, slug ......................................... 0.466 b, foot .................................. ......... 0.45 S, square foot .................................. 0125 k, foot .............. .......................... 0.416 Aspect ratio .................................... 2.25 a, per radian .................................. 2.10 Cl .................. ........................... 0.855 Wc, inch ................................... ... 0.375 Cable weight, pound per foot ................... 0.05 p, slug per cubic foot ......................... 0.00233 The cable length is plotted against the effective drag coefficient CD = C + CDc in figure 6. The method for determining this curve is given in the apoendix. The boundaries of stability for this particular case are plotted in the same figure in order that the region of instability may be found. As the body is lowered from the airplane, it will be stable for a very short distance and will then become unstable until the upper boundary of stability is reached. The upper boundary of stability occurs when the body is 2.8 feet below the point of support at an airspeed of 200 feet per second, or 6.3 feet below at 100 feet per second. For all greater cable lengths, the body will be stable. When thu body is drawn up to the airplane, it will again pass through the unstable region. The period and degree of damping of the oscillation at various cable lengths for the airspeed head having the characteristics previously given have been calculated by substituting numerical values in formula (3) and are given in the following table for an airspeed of 100 feet per second: NACA ARR No. L4D18 Distance below Pendulum W. eathervane airplane, oscillation oscillation Z (ft) P P N (sec) N (sec) l4/2. 4.0 2.22 23.4 to double amplitude 1.08 3.87 6.3 2.86 '(neutrally.stable) 1.07 4.10 10.0 3.51 20.0 toonehalf amplitude 1.06 4.28 20.0 4.95 4.95 to onehalf amplitude 1.07 4.34 From these calculations it is seen that the damping or rate of divergence of the pendulum oscillation is very small for cable lengths some distance on either side of the stability boundary. For the longest cable length, how ever, the oscillation damps to onehalf amplitude fairly rapidly. The boundaries of stability determined theoretically are in good qualitative agreement with the observed behavior of the NACA trailing airspeed head. Actually, there is no sharply defined boundary of stability because the oscillation is only slightly damped after the body has been lowered some distance into the stable region. As will be explained later, disturbing influences not taken into account in the theory may cause an unstable oscillation of the body when it would theoretically have a slightly damped oscillation. The boundaries of stability shown in figure 5 indicate that, when the drag coefficient is zero, the body will be unstable at all values of cable length greater than that corresponding to the lower stability boundary. For very small values of the drag coefficient, such as would be obtained by neglecting the cable drag, the theory indicates that the body will be unstable over a large range of values of the cable length. The results of reference 2, in which the damping effect of air forces on the cable is neglected, are therefore believed to be in error. Investigation of Modifications to Improve Stability In order to investigate the changes that might be made to improve the stability of a conventional type of airspeed head, it is convenient to express the condition for stability (formula (6)) in the following form, where ;TACA aRR No. L4D18 the nondimensional expressions have been removed by substituting the dimensional quantities that they replace 1 k2/g2 k2 ,/,2 1 > k2/2 + !(7) mg aV2S D Z 2 where k is the radius of gyration about an axis through the pivot point and D is the effective drag obtained by multiplying C = CD + CD by V2S. D D Db 2VS. The curves of figure 5 show that the region of instability for a conventional type of towed body can never be entirely eliminated. The following changes would tend to restrict the unstable region to a region closer to the airplane (a) Decrease in weight mg. (b) Increase in drag (c) Increase in area and aspect ratio of fin (d) Decrease in ratio of radius of gyration to tail length k/l The first two changes are impractical because they interfere with the usefulness of the instrument as an airspeed measuring device. The second two chances, however, provide practical methods of impro"ement. For example, the greatest distance below the airplane at which unstable oscillations occur in the example previously given could be decreased from 6.Z feet to 4.2 feet by doubling the tail length withoutt increasing the radius of gyracion. This chane could be accomplished by mounting a light set of fins on a boom behind the instrument. Formula (7) indicates that increasing the speed will restrict the unstable region to shorter cable lengths. Once the oscillation becomes unstable, however, it will probably increase in amplitude faster at higher airspeeds. It may be advantageous, therefore, to raise and lower the body at low flying speeds. The use of special devices to improve the stability will now be considered. It has been found by the writer that the two modes of oscillation given by a quartic 14 NACA ARR No. L4D18 will damp to onehalf amnplitude in the same time if the coefficients satisfy the relation b3 bc + d = 0 +d=O S 2 Such a condition will give the optimum use of damping in the system. Through examination of the coefficients of the quartic, formula (1), it is found that this relation may be satisfied by greatly increasing the damping in yaw N or by reducing the directional stability t almost to zero. Physically, a condition is thus reached at which the body remains approximately parallel to the average direction of flight as it swings from side to side instead of turning into the relative wind. Forces are thereby brought into play to damp out the pe ."luia oscillation. AIe foregoing method of obtaining stability may also be explained in terms of the stability boundaries plotted in figure 5. In the small stable region below the Icwver boundary of stability, the pendulum oscil lation is damped out by the mechanism just described. By reiuci". the directional stability and increasing the daring in yaw, the lower boundary of stability is raised to higher values of F/Ry. It is theortically possible; by using s".cial devices that arbitrarily increase the damping in yaw or reduce the directional stability, to raise this stability boundary so that the unstable region is eliminated. It will be noted that ? ,bisi method of improving stability is different in principle from the one described following for mula (7)o The method based on formula (7) consisted in extending the stable region by lowering the upper boundary of stability The method now being considered consists in raising the lower stability boundary. It has been found impossible, in practice, to reduce the directional stability of a conventional towed body to the extremely small value required. The bo. of the instrument is generally unstable ind some fin area is required to give neutral directional stability. Any small chaige in the characteristics of the body due to Reynolds number or due to small changes in shape would be sufficient to mako it either directionally unstable or too stable to obtain damping by vi:u.te of its low directional stability. The alternative, greatly increasirnj the damping in yaw, NACA ARR No. L4D18 miirht be accomplished by operating the rudder of the instrument by means of a gyroscopic element to cause the rudee' to deflect an amount proportional to the yawing velocity. The complication introduced by such a mechanism would probably make the method impracticable. Another method of increasing the damping in yaw and at the same time reducing the directional stability is to use two fins, one at the front and one at the rear of the body. Calculations show that the directional stability must be reduced to a very small value (approxi mately 4 percent of the stability contributed by the rear fin) in order to avoid the unstable oscillations. A moderate decrease in directional stability, even when combined with a dampinr in yaw of 20 times that for a conventional body, will not avoid the unstable oscilla tions. If the required small directional stability could be obtained, any slight misalinement of the front and rear fins would cause the body to trim at a high lift coefficient. This condition would cause the body to fly out to one side and would also make it undesirable as an airspeed measuring device. Longitudinal Oscillations The longitudinal notion of a towed body has been treated theoretically in reference 2. This analysis neglected the damping of the motion contributed by air forces on the cable. The boundaries of stability calculated in reference 2 are therefore believed to be unconservative. In practice, the foreandaft pendulum motion of the body has never: been observed to become unstable at long cables lengths. It is noted that the effect of the cable could be taken into account as an equivalent drag coefficient, as it was for the lateral oscillations. If a value of drag coefficient of the correct order of magnitude is substituted in the relations presented in reference 2, the pendulum oscillations may be shown to be well damped at long cable lengths. At short cable lengths and moderate speeds, the body hangs approximately vertically below the point of support; therefore, very little coupling exists between foreandaft movement of the body and pitching motion. The oscillation is simply a pendulum motion 16 NCA ARR No. L4D18 with damping supplied by the cable drag. Inasmuch as this cable drag is small at short cable lengths, the oscillation, though theoretically stable, is slowly damped and may become unstable if disturbing influences are present. Ti: analysis of reference 2 shows that other modes of longitudinal oscillation involving bowing of the cable and pitching of the body are theoretically possible, but such oscillations have never Le.n observed in practice. It is believed that the dra on the cable rrvents these oscillations from becoming unstable. F' .7 T'.I'TAL INVESTIGATION Flight tests were made of an approximately 1scale model about : yrnamicallly~ sl:, 1 i1. tote.CA trailing airspeed head suspended from a Stinson SRb& airplane. A drawing of the model is shown in figure 7. In order to simulate pulling the head up to a su i:ort under the airplane, the cord was run through an eyelet on the cabin steps. The instrument was stable when towed on the end of a 75foot cable at speeds between 0O and 150 miles per hour. Lateral and foreandaft oscillations da:'ed out in a small number of cycles. It should be noted that the corresponding cable lengths on a fullscale towed airspeed head, twice the size of the one tested, would be twice as great. The corresoc ,din speeds would be \/2 times as great in order to maintain the same value of the Froude number F = V2/ L. When the model was drawn up to about 3 feet from the airplane, it was sufficiently stable at 80 miles per hour. Unstable oscillations did not start while the body was left in this osition for about a inute. This behavior does not necessarily indicate "at the oscilla tions would have damped out once they had started. The theory shows that a large number of oscillations is required to double amplitude; the body might, therefore, have to be towed for a considerable length of time before oscillations would become notice'cl9e. Unfortunately, no means were available to start an oscillation. As the s:ped was increased, the motion became less stable, until at 95 miles per hour increasing oscillations occurred. As predicted by the theory, both the NACA ARR :Jo. L4D18 foreandaft and lateral oscillations had periods close to the period of a simple pendulum. The lateral and foreandaft oscillations inevitably combine.s to cause the instrument to travel in an elliptical orbit. The direction of rotation was such that the instrument swung back as it came closest to the fuselage. Proba bly the increased velocity near the fu.selage fed energy into the motion with each oscillation and caused a greater rate of increase in the amplitude than would have been predicted by the theory. Several modifications of the model were tried in an effort to improve the stability. Two modifications appeared to improve the stability of the pendulun oscil lation at short cable lengths. One of these changes consisted in shifting the pivot point rearward 1/2 inch, and the other consisted in equioping the model with a hinged rudder with weight behind the hinge line and viscous damping. These changes prevented the oscilla tion from appearing spontaneously as the speed was gradually increased from 80 to 140 miles per hour. A theoretical study indicates that these changes should have only secondary effects o. stability. These tests are not considered to be a conclusive der.monrtration of the stability: of the body because it is not known whether oscillations would have damerd ou.t once they were started. Various other modifications that were tried resulted in unstable shortperiod oscillations of the body. These tests were made at a speed of 80 miles per hour. A forward shift of the pivot ooint caused a pitching oscil lation. This motion was believed to be the result of elasticity of the cable and mass unbalance of the body and was similar in nature Lo flutter. A freely hinged rudder with weight behind the hinge line caused a short period yawing oscillation. The use of an asymmetrical vertical fin, extending only below the body, caused a shortperiod rocking motion of the body. Another type of instability has been encountered on a few occasions when the fullsize l.aCa airspeed head was lowered at the full length of the cable (approxi mately 200 ft). In one case in which this motion was observed, the airspeed hed ed was lowered from the door of a twinengine lowwing cabin monoplane. The head was steady at speeds below 150 miles per hour, but at this speed oscillations of about Zfoot wave length in NACA ARR No. L4D18 the cable originated at the airpla ne and traveled down to the body. As the speed was increased to 165 miles per hour, the oscillations became very violent and caused a pitching motion of the body. The whipping action at the lower end of the cable eventually caused it to break. A metal sphere was later towed from the same airplane and the oscillations occurred as before. The oscillation was therefore not related to the aero dynamic characteristics of the body. It was believed to be caused by the action of unsteady air flow from the wingfuselage juncture on the tow cable. The same airspeed head has been used without difficulty at much higher speeds on other airplanes. Several relatively light, largesize towed bodies have been tested in flight. The pendulum oscillation of these bodies has never been known to become unstable, even when the body was raised or lowered from the air plane quite slowly. This behavior is in agreement with the theoretical prediction. These bodies have a small value of u compared with that of the towed airsps.1 head; the unstable region at normal flying s'eds is therefore very small. DISCUSSION OF RESULTS The theoretical and experimental investigations have shown that the pendulum motion of a towed body may become unstable when the body is drawn up close to the airplane. The theory shows that the instability is not serious because the amplitude of the oscillations increases very slowly. The maximum cable length at which unstable oscillations can occur may be reduced by reduction of the ratio of radius of :yration to tail length of the body and by increase of the fin area and aspect ratio. More violent instability of the pendulum oscilla tion than would be predicted by the theory, as well as other types of instability, may be introduced by unsteadiness or lack of uniformity of the air flow in the region where the body is lowered from the air plane. Inasmuch as no practical method has been found to provide large damping of the pendulum oscillations when the body is close to the airplane, it is desirable to lower the body from a point where it is not subjected TThCA ARR No. L4D18 to these disturbing infllunces. IA suitable location would probably be on the plan of syrmetry of a t.jin sn.rine airplane, or on the wing of a singlepngine airplane at a point outside the slipstream. It also appears desirable to lower and raise the body at low flying speed, because the unstable oscillations then increase in amplitue t very slowly. If these precautions are taken, it should bo possible to lower a towed body without attention on the part of the pilot. The only possibility for unstable oscillations to develop would be if th3 body wer'i left fori long periods suspened only a few feet below the airplane. Oscillations of the system at large cable 1ineths ars racpidly damped because of the cable drag. COITCLUSIOirS 1. A theoretical study of the motion of a suspended body stabilized Ly fins showed that it had two modes of lateral oscillation with the following characteristics: (a) Weathervane oscillation The weathervane mode of oscillation was rapidly damped and had a period about equal to that of the instrument oscillatinrr as a weather vane about a vertical axis through its center of gravity. (b) Pendulum oscillation The period of the pendulum mode of oscillation was about the sime as that of a simple pendulumn of length equal to the vertical distance of the body below the airplane. The oscillation was damped by the cable dra~ at large cable lengths but was unscable at short cable lengths. The rate of increase of amplitude in the unstable region was very small. It was found that the unstable region could be restricted. to short cable len'tbs at normal airplane seeds by keening the radius of gyration of the body small and increasing the fin area, aspect ratio, and tail length. 2. In flight tests, more violent instability of the pendulum motion was enco',i.ntered than would have been expected from the theory and other tyoes of 20 NACA ARR No. L4D18 instability occasionally occurred. These conditions were attributed to the action of unsteady air flow on the cable. It is believed that unsatisfactory behavior of a towed suspended body can be avoided by lowering and raising the body at low flying speeds from a point on the airplane where the air flow is uniform. Langley Memorial Aeronautical Laboratory, National Advisory Committee for Aeronautics, Langley Field, Va. NACA ARR No. L4D18 APPENDIX DET;F1TJIIHTIOH OF EUTV.ALJITI DRAG COEPFICICHT OF THE CABLE The drag of each cable element of height dz is dDc = Cd ;'c EV2dz c  where cde is the drag coefficient of the cable per unit vertical height. The variation of this drag coefficient with inclination of the caole has been obtained from the data of reference 3 and is presented in figure R. If the assumption is made that the cable remains straight when viewed from the front, each cable element has a lateral velocity proportional to its distance below the airplane. The side force acting on each cable element is 2c d70 = dD c ScW, z dz dc 2 2 z The total side force is z z c= w V Cdc This side force has been determined by graphical integration for cables with various values of X/Z, the ratio of horizontal length to height. The cable form was assumed to be that of onequarter of a sine wave, as shown in figure 9(a). Although the shape of the actual cable may deviate somewhat from a sine curve, the error in the calculated side force will be small. The location of the resultant side force may also be determined graphically as the center of gravity of the area representing the sideforce distribution. IT the inertia of the cable is neglected, the lateral force NACA ARR No. L4D18 will be balanced by reactions on the body and on the point of support; the magnitude of the reactions will depend on the position of the resultant side force on the cable. As shown by figure 9, most of the side force on the cable is transmitted to the body. Let the fraction of the total side force that is applied to the body be K. The side force applied to the body is then Yb = KY f1 11 dYb K Zwc/V Cdc d VTS d7 S 2' By comparison with formula (2), the quantity in brackets may be substituted as an equivalent drag coefficient in the formula for Y The relations may be summarized as follows: 2VS S= (CDb + CDc) m where KWcZ z CDc ZI cd c dZ/ The value of CDc may be determined from figure 4 for cables of various values of the ratio X/Z. In order to determine the variation of CDc with cable length as the body is drawn up to the airplane, it is necessary to know how the ratio X/Z changes with cable length. Typical examples have been worked out for two airspeeds for the airspeed head and cable previously described. The shape of the cable was NACA ARR No. L4D18 determined from consideration of the forces acting on the cable elements, obtained from reference 3. The cable shape for each speed is shown in figure 10. Any point on the cable may be considered as a point of suspension. The variation of tha ratio X/Z as the cable is drawn in or let out may therefore be determined graphically from this figure. REFERENCES 1. Thompson, F. L.: The Neasurement of Kir Speed of Airplanes. TN No. 616, NACA, 1937. 2. Glauert, H.: The Stability of a Body Towed by a Light Wire. I. & hI. No. 1312, British A.R.C., 1930. 3. McLeod, A. R.: On the Action of Wind on Flexible Cables, with Applications to Cables Towed blow Aeroplanes, and Balloon Cables. R. & .i. No. 554, British A.C.a., 1918.. NACA ARR No. L4D18 Figs. 1,72 I I d re 0 .4 5 w4 0 41 *4 JJ 4 0 t 4 t s5. LU rl e .he .a o "O r .$^ P ^ or , u D rt IZ, I. Fig. 3 NACA ARR No. L4DI8 Figure 3. Top view of trailing airspeed head showing component of side force due to drag. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS NACA ARR No. L4D18 ^*t "P" ! ia 'i *(+t '*''1 "g.tct W : ^ SiLtL ^ :ri~~~tjt~ ^: ^::: ' f *'^ ^ 411 4  .r .4 .r .&.. ... ...r . ;ilj;^^;j;;^?^^^#;: llll7llljlllillillS4li VW~:thtt~tflT1Zr wtrmt41rwQnltztlmm+~~..trtrn4 ttt~tA.~Iict WT. ..a ~ :::.? .it::: : . ^. .H t Tt 5'L~r~ T tirTtofpE F::!^:!? i+: t; Ia: l"+: ;: It^dr: ~  4I 41a +ri i .4; . .; . ......  4 IE4^ 4f^ ^ 4 1 glit IN :1 ......tj.4 4ll il :7~"tF~FF~~Fi~. .F'~i~ ..... at . .; il s ^ t' t;t^ ;i ^ . Fillli ililii llllllll llllill+lii 2l l7l 44+i Fig. 4 0 Q j z z I t t ++ __ at I!ACA ARR No. L4D18 ~f I,; i. :  ] I  . ] .. t "_ I I 4d  14 V I2L t 1. .4: 1 ifrit 2~ 7t~i~ S: i i I : *:i 4 t  . 4 F :  43 A T T 1 7 g  i.. i L f .. I \ ... .. . "     t :^ iA,::ij S_ _1 1* SII j A. 41 Li 44 ]i I  Value of a is 2.10 per radian. :  I I .l t: 1 4 . 4 ....J i ,1 I. ,r II I . ig re B I , a towed boy. Value o i is p ~ ~ ,' / .1 Iz_ L... _t. _' .i .. 'L "~l Fig. 5 "Ii ' !_ i.. .t ' I kTfle NACA ARR No. L4D18 .... .... .. .. r . 44 .Th.... ...... (a) Airspeed, 100 feet per second. F .;F +eTe o ve..a .istanes t ad boCy tt ive i ili  irsee head an I abi le se i ns l ii Id I 4, A I f p s and body that give instability for airspeed head and cable used in calculations. used in calculations. Figs. 6a, b Fig. 7 NACA ARR No. L4D18 1. ^ ^ cc 4 z1 I__ Ikoe Sa a 4 4. toj0 m00 5 U ( o S. ^ ^.^ , NACA ARR No. L4D18 iI *. :; LNATIONAL ADVISORY 1" fCOMMITTEE FOR AERONAUTICS. SHJ 44::: ^::;:s^ i tI::^fit:gj5.,^:^I;^lI):t!ttfitj *i1 .'  .44 4: 4= ~ ~ ~ .. .... H ..4 4 . ~ c~f~~*'+ r:~cP L 4  IAIL, .... ... 1TL; fr L4uSrj Fr , 447 4 7 ttj'i. rrrm dr,t IiK~~~ 4 4 : .1 . s ~~ ~ I .Ii4 I  # I T .~ . f .;_ _! f i K , F PGtX' L4Ptij:3 Figure 8. Variation of cable drag coefficient per unit vertical height with inclination. (Data taken from reference 3.) Fig. 8 NACA ARR No. L4D18 Ttt 't I4 I. ... ^  4 L14 w 4 T^ 4 tL ti 7r;+ti 7 HThT t 77# F. T.2trT ttt 4 4iji^ ^y ^S  1^^^I^^II^^^ ^ ^I l  ^^1 itr f;^F I': I ^^^^^^w^^p^^^^47 7:i g~gl^^lS~iTiS IglS to .0 0 C w S * II o L So o * 0 0 o4 * .f 0 8 ^ " ? ..a *. a Bl 0 I t ' 'd a' %, fl" A U :4 ^a C Fig. 9 NACA ARR No. L4D18 Fig. 10 J: IL f ', :4 I 'T n.L:aItIilS~tf^^^^i' 1^^^^2!^ 14 I *4 r"4 T if.ItV+ iT. r ~ ..i jn ,' "' LI t I t t t ,+ ... ... Z4. t4 i ..L i I I' iii+: :lli .'" i :, i '.. ^ ' ++ +.: ,+ ,+ If~ ~":I Y : .i ( A d re AION ir4 'i t AS .. . .I .V+ ;;+ .^ ^ ''E .E , U_ 7 .q + (U) Alrspeed, 100 feet per second, NATIONAL ADVISORY :OMNITTEE FOR AERONAUTICS. T 1 .. K 1' ...  ..lI. i^ +  U + 1 l  ii .i _ I: l IT 1l iB 4 4. T T If 4LI:~YY ~a LJ1l~i~~1T~~~f T r :lr~l 1* :tr,: R4E~i~rtTIprttT4v44r ;iN4 RUF iV 44 ..... ,m+' i +"+ (b) Airspeed, 200 feet per second. Figure 10. Cable shape for two values of airspeedor a body and cable that have the characteristics used in the calculations. Y~LY~YL~ J FrJ~t#~t~t~:~,~###aw$4FFH"~'17~~"''"" ''""i '''';:l+'~ir..H nl ~tlr!tit~tl+c~ltrm~~,t'all;~' i!MIti ,P# mtt '' "' ~""" ""~"I""''' i"' ""'"~' ~#ilE~i~Ht:if~~Btli~f~ ~ li~ Ahi 'r?t Ha 'i *. r 1( i I C 1 UNIVERSITY OF FLORIDA iIIllll111 Il IIII 3 1262 08104 987 5 IUN '"RSITY OF FLORIDA I: C: .'.1ENTS DEPARPTME' JT 120 r.1.FSTOrI SC;Er CE LIBRARY P.O. SOX 117011 GC NESVILLE, FL 326117011 USA 