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American Institute of Aeronautics and Astronautics1Axisymmetric Parachute Shape StudyVladimir S. Drozd1Airborne Systems North America, Santa Ana, CA, 92704, United StatesParachute inflated shape is one of the main parameters that drives parachuteperformance in steady descent. This paper looks at axisymmetric parachute shapeformation, particularly the relationship between parachute constructed geometry, numberof gores, gore shape, pressure distribution and inflated parachute shape. Stresses in aparachute structure are discussed in relation to parachute inflated shape. Analyticalestimates derived from one dimensional parachute model are extended by a finite elementanalyses performed with LS DYNA software.Nomenclature0A - canopy reference area, nominalN - number of canopy gores,mS - length of the canopy radial from an apex to a skirtmSSs /= - dimensionless distance from apex to a current point along the radialmSsYsXyx /)(),(, = - dimensionless yx, coordinate of a radial pointnr- unit vector in plane of the radial and normal to the radialr- unit vector tangent to the radiala2 - distance between the radials on inflated canopyr - fabric strip arch local radiusR - distance from the canopy centerline to the radial along the normal nrto the radial - fabric strip arch half anglel - inflated gore half length ( gore arch half length ) - half angle between the adjacent radial normal vectors - angle between a tangent to the radial and X-axesF - confluence forceq - dynamic pressureqspp /)(= - dimensionless pressure distribution along radialCp - pressure coefficient,Cd - drag coefficientCc - carrying capacity coefficientT - tensile force in a radial,)(sFt - tangential distributed force component along the line,)(sFn - normal distributed force component along the line,fT - distributed tensile force in a fabric strip,L - lines lengthDimensionless parameters are used as follows:1Systems Analyst, Space and Recovery Systems, 3000 Segerstrom Avenue, Santa Ana, CA 92704 USA.AIAA Member20th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar4 - 7 May 2009, Seattle, WashingtonAIAA 2009-2944Copyright 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.American Institute of Aeronautics and Astronautics22mSqTT =mnnSqFF =mttSqFF =mSss =The dashes above the dimensionless values are omitted in a text belowAmerican Institute of Aeronautics and Astronautics3IntroductionDemanding and changing performance requirements to modern parachute systems keep up continual interest tonew parachute designs and improvements of the existing systems. The broad spread of todays requirements varyfrom a single use cargo delivery system where cost is likely a design driving factor to a space recovery systemswhere chute performance per unit of system mass is often a major efficiency criteria.Industry advances in textile materials and new parachute assembly methods enhanced with modernsimulation techniques open additional opportunities in satisfying todays rigorous requirements, often allowingengineers to take another look at already existing concepts and consider a new ones.Despite seemingly being around forever and studied thoroughly circular parachutes are not an exclusion, andon the contrary prove time over time their necessity and superiority for many existing and new applications.Mostly experimental approach of early days to a parachute design provides engineers with valuable factualdata of parachute performance. Today those data can be complimented by advancements in parachute simulationmethods. Achievements in a finite element analyses (FEA) and fluid structure interaction (FSI ) allow forsimulation of the real life structures with acceptable accuracy and more and more realistic outcomes, parachutes notbeing an exclusion. Quickly growing computing power makes simulation results also feasible within a reasonablecomputing time.FSI methods by their definition involve coupling of fluid and structure motions and therefore represent themost advanced and adequately complex approach to a parachute behavior analyses.On the other hand FEA method thoroughly considers structural part of parachute behavior, provideddistributed aerodynamic loads applied to a parachute surface are known. In some cases those loads may beavailable from experiments or estimated separately, which allows structural analyses of parachute to be complete.As universal computational methods FSI and FEA methods are applicable to any type of a parachute structureprovided parachute geometry is known.Being very powerful in their ability to analyze local effects ( like stress concentrations for complicated shapesfor example ) on another hand they do not provide that analyses convenience compared to the cases when systembehavior can be described and analyzed through analytical relationships between major design defining parameters.Fortunately a circular parachute has some specific construction features that allowed researchers to come upwith a relatively simple and meanwhile productive parachute structural model.Circular parachute canopy structure consist of radials forming vertical wire frame, and a fabric segments fillingspace between the radials, fig. 1 ( A horizontal circumferential reinforcing is present quite often as well ). Thisschematic of a circular parachute closely matches the pathway the loads are distributed in a canopy structure andtransferred to the confluence point.Due to a fabric curvature a distributed pressure load )(sp normal to the canopy surface get transformed into atensile loadfT in a fabric. Tensile loads in a fabric in turn get transferred as a distributed loads to the radials.Eventually radials collect distributed load and transfer it to the lines.Since a radial is a linear flexible element the radial shape and a tension T in a radial can be described byflexible line equations:)(/ sFtdsdT = ( 1 )TsFndsd /)(/ = ( 2 )cos(/ =dsdx ( 3 )sin(/ =dsdy ( 4 )WhereT - tensile force in a radial,s - linear coordinate along the radialAmerican Institute of Aeronautics and Astronautics4)(sFt - tangential distributed force component along the radial,)(sFn - normal distributed force component along the radial, - angle between a radial curve tangent and X-axis ,yx, - rectangular coordinates of the radial pointThis approach has been used successfully to simplify the case and produce a model that adequately describes astatic structural loading of the circular parachute in many important cases. The concepts of the model have beenimplemented in CANO and later in CALA codes for a circular parachute structural analyses 1 .Although aimed primarily to determine structural loads in a canopy for known constructed geometry and apressure distribution, the developed parachute model and corresponding equations provide a valuable tool for acircular parachute shape study.Canopy loadingMajor assumptions of the model deal with the question of how the load comes from a canopy fabric to theradials. It is assumed that fabric itself is a one dimensional structure consisting of a set of independentcircumferential elements ( ribbon parachute is a good example ). Actual solid cloth parachute falls into thatcategory when radial strains ( stresses ) in a fabric are negligible, compared to the circumferential ones.Since pressure along the fabric strip element is considered constant, the fabric strip takes a shape of a circle archof angle 2 between two attachment points to the adjacent radials.Due to an assumed absence of structural interference between fabric strips the value of tensile loadfT inthem are statically determined when pressure )(sP and the fabric strip radius of curvature r are known.In general case a radial is not perpendicular to the plane of a fabric strip, therefore a tensile forcefTrappliedfrom a fabric strip to a radial may have both along the radial component )(sFt and a normal to the radialcomponent with projection )(sFn in plane of the radial.These components may be approximated by the following dimensionless expressions :sin)sin()sin(2)(=NxpsFn ( 5 )coscossin)cos()(sin2)(2=NxpsFt ( 6 )Equilibrium equations ( 1-2 ) and geometrical relationships ( 3 - 4 ) form enclosed system of differentialequations for four unknown functions )(),(),(),( ssTsysx and known boundary conditions. With XY referencesystem origin being at canopy apex the boundary conditions are:,0)0( =s and ( 7 )ms = )0.1( ( 8 )wherem is the radial angle at the point of line attachment.When right sides of the equation (1-2) are known the equations can be integrated numerically. Howeverpressure distribution along the radial is not known in advance ( coupling problem, mentioned before ), what makesthe case open.Nevertheless there is still a certain reason in studying solutions of equations ( 1- 4 ). Available experimentaldata of pressure distribution around the canopies ( including various shaped canopies and solid models ) give someidea of pressure distribution along the radial and a range of pressure variation. Given that it is possible to run someparametric studies without knowing the exact pressure distribution.Since forces developed by a parachute depend in part on its inflated shape, and mainly on its projected area, itseems to be important to keep a projected area at its maximum for a given constructed dimensions. For a givenconstructed radial length only a gore width distribution differ one solid cloth canopy design from another ( providedfabric permeability and lines length remains the same for all designs ).American Institute of Aeronautics and Astronautics5Historically canopy shape evolved from a flat circular to a conical, and then to a more complicated quarterspherical and extended skirt designs. That evolution resulted in a canopy gore width decrease compared to a flatcircular gore. Fig.2 shows gore material distribution in each of them for a 30 gore canopy. Since flat circular canopyhas the widest gore does it also produce the maximum possible inflated diameter for a given constructed diameter?Is it possible to increase a canopy inflated diameter by adding more material into the gore? In general, how does amaterial distribution in a gore control a canopy inflated shape?Some observations about the canopy shape can be made based on the structure of the equations describing a radialloading .Free shaped (pressure shaped) parachutesFigure 1 shows canopy schematic and the components )(sFn , )(sFt of the distributed load applied to theradial. Beside a pressure function both of these components (5-6) contains two angles: and . Angle 2 isan angle between the normals of the adjacent radials . It is determined by number of the gores in a canopy and by aposition along the radial. Thus angle describes a radial geometry and can be expressed as)sin()(arcsin(sin =N( 9 )Angle varies along the radial and has its maximum value of N/ = at the maximum inflated canopydiameter ( 2/ = ) For a typical 30 gore canopy it is in a range of060 .On another hand fabric strip arch angle indicate a gore material fullness at a particular point along the radialwhich may be characterized by a ratio of a fabric strip arch length l over the distance a between the radials at thepoint of consideration: sin/ =al ( 10 )Theoretically a range of possible angles may vary from 0 ( no fullness in a canopy ) to a 2/ = whengore has maximum fullness.This particular case is a subject of interest. In this case as it seen from (5-6), 0)( =sFtPhysically it means that tensile force in a fabric strip lies in plane of the radial and is perpendicular to the radial,thus not having a component in a direction along the radial. As a result a tensile force T in a radial remains constantall the way along the radial which formally results from eq.1.It is reasonable to expect a small variation of tensile radial force for the cases when angle still.On the other hand for normal to the radial force component )(sFn does not change a lot as well. Thatmeans a radial shape is not going to change substantially with angle variation provided angle is high. Sinceangle is a measure of a gore fullness, it is expected that canopies with a lot of gore fullness all have a similarradial shapes.Based on the features of available fullness it is convenient to call these class of canopies a free shapedparachutes. For these parachutes their shape is controlled by a pressure distribution ( or rather a presence ofpressure difference at the canopy surface, since shape itself is almost independent of a pressure distribution itself )Shaped parachutesFree shape parachutes are those where at any point along the radial. Opposite to this case is the casewhen = . That makes,0=Fn and 0/ =dsd ( 11 )In this case angle remains constant and the following condition must be met:)sin()sin()sin( =N( 12 )Apparently, canopies with constant along a certain part of the radial resemble the conical surface shape atthat area. The shape of the conical part of the canopy does not depend on a pressure distribution along the conicalpart of the canopy, but rather is determined by a geometry of the gore.For a small angle is also small and may be approximated asAmerican Institute of Aeronautics and Astronautics6N = ( 13)Thus knowing ( or assigning desirable ) angle , a variety of conical canopies with different cone angles can begenerated. Knowing angle , and the number of gores N, all other canopy parameters are determined for the areawhere const=)sin(Nxa= (14 )sin()sin(=Nxl (15 )Angle can not remain constant all the way along a radial since at the line attachment point it must be higherthan 90 degrees to satisfy boundary conditions. Consequently a transition area from a conical to different shape isrequired. Since in this area angle must change substantially, angle deviates from angle to produce arequired Fn .Theoretically the conically shaped parts of the canopy may exist at any , even when is greater than 90degrees ( at skirt area ) provided there is a positive pressure difference across the canopy shell in this area. In thiscase a general equation ( 12 ) must be used to obtain angle gamma for this part of the canopy.When “ conical “ load conditions are met at the cylindrical part of the canopy it experiences hoop stresses only.Cylindrical part of the canopy has another interesting feature. The angle 2/ = for a cylindrical part of thecanopy ( maximum diameter of the inflated canopy area ). Therefore in accordance with eg. (1,6) tensile force in aradial must remain unchanged along the radial at that areas of the canopy regardless of gore fullness.Thus there are two ultimate cases of canopy shapes free shaped, or pressure shaped parachutes, and a shapedparachutes, where canopy shape in certain areas is determined purely by a gore geometry. In between thoseultimate cases a general expression ( 5-6 ) must be used when integrating system of equation ( 1-4 ) for angle .Tension in a canopy fabric.Tension in a fabric strip is given by the expression)sin()sin()sin(/NpxpaprT = ( 16 )It become evident again the importance of angle ( or fullness in a canopy gore ). The more fullness in a goreis, the less is the tension in a canopy fabric. In a conical canopies circumferential stresses therefore are higher thanstresses in a free shaped canopies.Pressure distribution approximationSince governing equations are not going to be solved for the exact pressure distributions, some reasonablepressure distribution approximations are used. Linear pressure distribution along the radial is perhaps the simplestone to considerqCpSksmqSsqkmqsPmm=+=+= )/(/)( ( 17 )The first constant term “m” is used to characterize say “average” pressure acting on a canopy surface, or pressureat some characteristic point along the radial, say at canopy apex or close to it. The second term “k” describespressure variation along the radial.The following range of pressure parameters was considered:m = 0.5 - 1.4 , k = 0.0 0.5 ( 18 )Gore geometry parametersAs it was discussed earlier a pressure distribution is not the only parameter that controls the solution ofgoverning equations describing canopy shape. Expressions (5-6) for distributed forces contain an angle - aparameter that characterizes a gore fullness distribution along the radial. It is convenient to use this parameterAmerican Institute of Aeronautics and Astronautics7instead of a gore width. When then canopy has a lot of gore fullness. When 0 then canopy doesnot have any fullness in a gore. These special cases were touched above giving an idea of canopy shape for them.The intermediate cases require solution of the governing equations.As a first step a system of equations was integrated for a range of angles and linear pressure distribution. Arelative lines length ratio 2/ =mSL was used as a typical value. Fig.3 shows the respective computed radialshapes for a range of angles , while gore width distribution along the radial for various angles is shown in fig.4. As it is seen from fig.3 a maximum gore fullness provides maximum canopy inflated diameter. A substantialdecrease in a gore fullness ( decrease in angle from 90 degrees to 30 degrees ) does not lead to a drasticdecrease in an inflated diameter, but substantially decreases a gore width.Fig.5 and 6 show tension distribution in the radial itself and in a fabric material respectively. It may be seen thatthe radials tension remains the same for a wide gore, but starts changing when gore fullness drops. The radialtension change is not pronounced at the skirt of the canopy area, where canopy has close to a cylindrical shape, as itwas discussed earlier.Similarly to a parachute drag coefficient Cd a canopy efficiency can be estimated by a carrying capacitycoefficient Cc defined as AqFCc = / . Carrying capacity coefficient is not a truly drag
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