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Dynamic Stability Analysis of a Tethered AerostatAshok Rajani, Rajkumar S. Pant,and K. SudhakarIndian Institute of Technology, Bombay, Mumbai 400 076, IndiaDOI: 10.2514/1.47010This paper describes a model that has been developed to study the stability characteristics of aerostats. This modelincorporates the concepts of apparent mass, dynamic tether and allows 6 degrees of freedom for the motion of theaerostat. Estimation of aerodynamic coefcients is based on empirical relations and curves available in literature.Weight and buoyancy are calculated based on geometry of the aerostat. Appropriate values for operational altitudeand desired angle of attack of the aerostat are assumed. Moment balance about conuence point gives the optimallocation of the conuence point. Equations of motion for the aerostat and dynamic tether are simulated andappropriate boundary conditions are applied. Force balance gives the tether tension force and its orientation at theconuence point. Based on the tether tension and its orientation at the conuence point, the tether prole is estimatedby breaking upthe tether into several elastic segments, eachinequilibrium. Once equilibriumis established, the windis perturbed and the response of the aerostat is simulated. The paper reports the results of simulation carried out forthe TCOM 365Y aerostat and the aerostat response to various ambient velocity proles.I. IntroductionTETHERED aerostats fall under the category of lighter-than-airsystems. Agas having lower density compared with ambient air(usually hydrogen or helium) is enclosed in an envelope and thedifference in their densities gives rise to buoyancy. In an aerostat,buoyancy is the major source of lift, whereas in heavier-than-airsystems (e.g., xed-wing aircraft or rotorcraft), aerodynamic liftproduced due to relative motion between the ambient air and thevehicle is the major source of lift. The various components of atypical aerostat system are outlined in Fig. 1.The hull or envelope is a bag containing the lifting gas. Fins areattached at the rear end of the hull and provide stability to the aerostat;they are usually in the form inated structures, lled with lifting gasor air. The payload, which is usually a surveillance camera or a radar,is mounted one the envelope. A series of ropes called conuencelines connect the hull to a single point called conuence point, towhich the main tether is attached.Aerostats can remain stationary for long duration in reasonableweather, which makes them a very good choice for surveillance,advertising, and raising antennae for wireless communication, toname a few. In real life, aerostats have to operate in highly varyingweather conditions and winds. Aerostat failures have occurredbecause of abrupt changes in the wind, which result in shock loads.Estimation of these shock loads is an important requirement inaerostat design, and it can be accomplished by modelling thedynamics of an aerostat and predicting its response to sharplyuctuating winds.This paper starts with a brief history on the development ofmodeling and simulation of aerostats. The next section deals withequilibrium analysis of tethered aerostats, in which the angle ofattack at which the aerostat is in equilibrium as a function ofambient wind speed Uis determined. Aerodynamic coefcients needto be estimated for equilibrium analysis, which can be done usingempirical relations based on the aerostat geometry, as explained inthe next section. The next section gives details of the simulationmodel that was developed for simulating the response of aerostat.The results obtained by running the analysis for the TCOM 365Yaerostat are presented next. The aerodynamic and geometricparameters are compared with those fromliterature. The conclusionsdrawn fromthe results and also scope for future work are presented inthe last section.II. Historical Development of Modelingand Simulation of AerostatsInformation related to stability analysis of aerostats in openliterature is quite sparse, as the work has mostly been done by privateorganizations, e.g., TCOM. First-order stability analysis of anaerodynamically shaped tethered balloon was reported by DeLaurier[1], which was extended by him to predict RMS lateral response,using transfer functions [2]. Based on wind-tunnel data of ve scaledmodels of aerostats, Jones and DeLaurier [3] developed empiricaltechniques for approximate the aerodynamics parameters in 1981.Using these formulae, Jones and Krausman [4] developed a 6-DOFnonlinear dynamic simulation model (NDLS) for obtaining theresponse of a tethered aerostat to turbulence and other disturbances,and validated it against experimental results. In this study a frozen-eld turbulence model was used, in which a continuous turbulencespectrum is simulated by combining a number of discretewavelengths in random phase, where the amplitudes are calculatedthrough integration of the Dryden power density spectrum. TheNDLS model was further improved and validated against full-scaleght tests of an instrumented TCOM 71Mtethered aerostat by Jonesand Schroeder [5].Lambert and Nahon [6] have also presented a nonlinear model forinvestigating the dynamics of a tethered aerostat. However, they havelinearized the system using a nite difference approach. They havepresented results for several longitudinal and lateral modes, and havealso considered the cross coupling between the tether and aerostat.Stanney and Rahn [7] have used an improved atmospheric model,and validated the experimental results listed cited by Jones andSchroeder [5], using a nonlinear 2-D aerostat model, which includesa rigid body with aerodynamic loading attached to a continuoustether. They have also generated gust inputs using real hurricane datato predict the aerostat response to severe turbulences.Surviving harsh weather conditions, especially in the presence ofstrong winds, is a major challenge in aerostat operation, particularlyfor those intented for long duration operations. Rawat [8] has carriedout a simulation of tethered aerostats using nonlinear differentialequations derived using the Lagrangian theory, on a lumped param-eter model, without linearizing them. He has investigated threetypical scenarios of aerostat-weather interactions, viz., effect of aReceived 11 September 2009; revision received 22 January 2010; acceptedfor publication 15 June 2010. Copyright 2010 by Ashok Rajani, RajkumarS. Pant, and K. Sudhakar. Published by the American Institute of Aeronauticsand Astronautics, Inc., with permission. Copies of this paper may be made forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 0021-8669/10 and $10.00 in correspondencewith the CCC.Undergraduate Student, Department of Aerospace Engineering.Associate Professor, Department of Aerospace Engineering. MemberAIAA.Professor, Department of Aerospace Engineering. Member AIAA.JOURNAL OF AIRCRAFTVol. 47, No. 5, SeptemberOctober 20101531passing downdraft on an aerostat, a gust during aerostats recovery,and a lateral gust on a moored aerostat.III. Equilibrium and Stability Analysisof Tethered AerostatsStability is dened for an equilibrium state. An equilibriumanalysis for the aerostat needs to be carried out and then thatequilibriumstate needs to be analyzed for stability. The methodologyfor equilibrium analysis of tethered aerostats has been provided byPanda and Krishnamurthy [9], which is described in the followingsection.A. Force and Moment BalanceThe following forces and moments act on an aerostat: 1) weightacting about the center of mass, 2) buoyancy acting about the centerof buoyancy, 3) aerodynamic forces acting about the center ofpressure, that may be represented as forces and a constant momentacting at aerodynamic center, and 4) tether tension acting about theconuence point.The coordinate system used for the equilibrium analysis is thebody xed system (BFS) with origin at the nose of the body, X axispointing to the rear of the body along its axis of symmetry and the Zaxis upward. The aerostat experiences steady horizontal winds at anangle of attack as shown in Fig. 2.Aerodynamic forces and moments act at xa, buoyancy force acts atxb, and the weight at xg; all these points are assumed to be on the axisof symmetry. The conuence point is located at (xc; zc). Twodistances cx = xg xc and cz = zg zc are dened.Balancing forces in the X-Z plane, we get the following equations:Fax Bf sin W sin Th cos Tv sin = 0 (1)Faz Bf cos W cos Th sin Tv cos = 0 (2)Balancing moments about the conuence point gives:Ma Faz(xa xg cx) (Fax Bf sin W sin )cz Bf cos (xb xg cx) cxW cos = 0 (3)whereFax = 12U2A|CL sin CD cos | andFaz = 12U2A|CL cos CD sin | (4)Ma = 12U2AlCM0 (5)CL and CD in turn are given byCL =av sin CD =CD0 K2(6)Let us dene cx, cz, xa, xg, and xb as distances nondimensionalizedwith the envelope length l.Simplications after substitution yields Eq. (7)CM = CM0 av2 sin 2 CD sin ( xa xg cx) |av sin 2CD cos | cz| Bf12U2A|cos ( xb xg cx) cz sin | W12U2A| cx cos cz sin | (7)For a given speed U, the angle of attack adjusts itself to satisfyCM = 0 and results in Eq. (8)CM0 (av sin cos (CD0 K2) sin )( xa xg cx) (av sin 2 (CD0 K2) cos ) cz= 112U2A|Bf(cos ( xb xg cx) cz sin ) W( cx cos cz sin )| (8)Using Eq. (8), it can be shown that the equilibrium angle of attack ismaximum at U = 0; and as U increa