Stiffness Scaling for Solid-Cloth Parachutes

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-29264

AIAA 2001 -2007

SCALING FOR SOLID CLOTH PARACHUTES

H. Johari* and K.J. Desabrais*Mechanical Engineering Department

Worcester Polytechnic InstituteWorcester, MA 01609

Abstract

Dimensional analysis of the parameters affecting the inflation of solid cloth canopies was carried out. The primarydimensionless parameters are the mass ratio, Froude number, permeability, and a measure of stiffness. Comparisonof sub- and full-scale C9 parachutes along with small-scale wind and water tunnel models have revealed that therelative stiffness index ^is able to correlate the filling and opening times of various canopy sizes. The filling andopening times can be expressed as k £"°-064

j where the proportionality constant k is equal to 1.3 for the filling time.The value of k used for correlating opening times is 0.86 for mass ratios in the range of 0.1 to 0.2.

Introduction

From a parachute designer's point of view, thethree most significant performance parameters besidesparachute stability are the peak opening force, opening(or filling) time, and the steady state drag. Theseparameters are typically obtained from full-scale testingof parachute prototypes. Full-scale testing for designpurposes is time consuming and not cost effective. Abetter understanding of the flow field surrounding theparachute canopy may significantly help with thepreliminary design of new parachute systems, and evenaddress questions regarding stability. Assessment ofthe flow field in the immediate vicinity (and near wake)of a full-scale parachute canopy is quite difficult unlessthe parachute is tested in a wind tunnel or simulatedusing sophisticated computational Fluid-StructureInteraction (FSI) algorithms.1'2 However, at the presenttime only the steady state descend of a parachutesystem can be simulated realistically. Owing to thelarge deformations of the canopy during inflation, andthe necessary computational mesh movement, theinflation phase appears to be beyond the capability ofthe FSI computations at the present time. On the otherhand, wind (or water) tunnels provide a controlledenvironment for measuring the flow field of parachutecanopies. Except for very large-scale wind tunnels,blockage limitations typically require the canopy to be

* Associate Professor, senior member AIAA.Graduate student, member AIAA.

Copyright © 2001 by H. Johari and KJ. Desabrais.Published by the American Institute of Aeronauticsand Astronautics, Inc., with permission.

scaled down. Cockrell3 states that a blockage ratio of~ 5% or less is required to faithfully reproduce the flowfield around a parachute canopy in a wind tunnel.

Scaling of parachute performance parameters hasbeen pursued by several investigators, both for roundand parawing type canopies. Fu,4 Lingard,5 Heinrichand Hektner have performed dimensional analysis tocorrelate parachute parameters through dimensionlessforce coefficient and opening time. Although manystudies have attempted to relate full-scale performanceparameters of different parachute systems, few studieshave considered the scaling down of one type ofparachute canopy. For example, Lingard measured theopening forces on a 40-cm solid cloth, flat circularcanopy in air and water under finite and infinite massconditions. More recently, Lee7 has compared theopening time and peak opening force of free-fallingfoll-scale, Vi-scale and V4-scale C9 solid cloth, flatcircular parachutes. Although he found that thesmallest scale canopy generally had the highestnormalized peak opening force and the smallestnormalized opening time, no concrete relationship wasfound to correlate the full-scale and sub-scale results.Changes in the canopy stiffness and permeability (meanflow velocity through the fabric) were thoughtresponsible for the observed variations in the measuredopening parameters. Furthermore, Lee suggested thatlighter fabrics be used in the construction of sub-scalemodels. A difficulty with this suggestion is that lighterfabrics generally have different permeability andstrength characteristics.

Niemi8 analyzed the available literature on thescaling of parachute canopies, paying particularattention to stiffness. He derived a relative stiffnessindex, based on the earlier work of Kaplun9 andKnacke10, that also contained dynamic parameters such

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as the snatch velocity. It appears that the relativestiffness index of Niemi correlated reasonably well thenormalized opening times of Lee's C9 parachutes. Itturned out that a power law relationship exists betweenthe normalized opening time and the relative stiffnessindex of Niemi. In contrast, the peak opening force didnot appear to correlate with this relative stiffness index.

So far, it appears that Niemi's relative stiffnessindex has only been used to correlate finite massinflation of scaled models. The 3-foot (91-cm) canopyof Heinrich and Noreen11 and 80-cm canopy ofLingard5 in a wind tunnel (plus a 40-cm canopy in awater tank) appear to be the smallest scale solid cloth,flat circular canopies that have been inflated undercontrolled conditions. Heinrich and Noreen measuredthe projected area of the canopy during the inflation aswell as the force time-history. From thesemeasurements, Heinrich12 concluded that the projecteddiameter time-history during finite and infinite massdeployments have minimal differences. Lingardmeasured the projected diameter and drag of inflatingsmall-scale canopies under finite and infinite massconditions. The small scale of the canopies chosen inhis studies was dictated by the need to keep theblockage ratio to a minimum.

We are investigating the flow field in the nearwake of flat circular canopies in a low-speed watertunnel during inflation and steady state. We areemploying 15- and 30-cm solid cloth, flat circularcanopies in a 60-cm square cross-section water tunnel.The blockage ratios for these 1728th and 1756th scalemodels during steady state are approximately 10% and2.5% respectively. The low-speed water tunnel waschosen because it produces longer opening times thanin a wind tunnel at comparable Reynolds numbers withthe same scale model, and the Digital Particle ImageVelocimetry (DPIV) technique can be implemented in awater tunnel with relative ease. This technique allowsthe measurement of the velocity field in a plane at asufficient spatial resolution to enable the calculation ofthe vorticity in the near wake of the canopy.

In this paper, we first present a detaileddimensional analysis of parameters which are relevantsolely to the aerodynamics of an inflating parachutecanopy, and then those which are relevant toperformance parameters of an entire parachute system.It is important to point out that the majority of paststudies have only considered the entire system and notnecessarily the specific issues of the canopyaerodynamics. The relative importance of matchingvarious parameters will also be discussed. Followingthe dimensional analysis, a comparison of theparameters for full, 1728th, and 1756th scale solid cloth,flat circular canopy in air and water will be presentedalong with the correlation of the filling time for the verysmall-scale models and full-scale parachute data.

Dimensional Analysis

The parameters affecting the aerodynamics of theparachute canopy is considered first in the analysisbelow, and then the entire parachute system includingthe effects of suspension lines and payload is examined.As far as the canopy aerodynamics is concerned, theindependent parameters to be considered are the fluidforce exerted on the canopy F9 and the mean projectedcanopy diameter Dp. From the time history of thesetwo parameters, the peak fluid force F*, the openingtime t0 (the time from the lines taut instant to the timeof peak opening force), and the filling time tf (the timewhen the projected canopy diameter first reaches themean steady state diameter) can be found. Theindependent parameters to be considered are the fluiddensity p, viscosity /^ and sound speed a, snatchvelocity Vs (or the constant freestream velocity Vx ininfinite mass cases), initial angle of attack a (withrespect to the symmetry axis), canopy mass m,constructed diameter D0, canopy fabric thickness 8, anequivalent modulus of elasticity E, and Poisson's ratiov. For permeable canopy fabrics, permeability ascharacterized by a mean velocity c across the canopyfabric has to be accounted for. This velocity dependson the fabric pore size and the pressure differentialacross the fabric. Although these parameters can beincluded in the analysis, the inclusion of the integralmeasure c at the appropriate pressure differential wouldsuffice. Dimensional analysis results in the followingrelationships.

CF.

•̂ • = /2 r, Re, A/,a,-p—,

If\ r e> / • ! » • » •» f C O

£ = /4 Re, M,a>—,-—,

772—, V

__D

Here, time has been rendered non-dimensional byDJVS9 S0 is the constructed area, and q denotes thedynamic pressure YipVs - If the flow profile upstreamof the canopy is not uniform, then another parameterq I q characterizing the ratio of oncoming dynamicpressure profile (averaged over the canopy area) to the

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assumed dynamic pressure has to be added to the abovefunctions.

Other canopy properties such as the mean fabricstrain can also be written as a function of the aboveparameters. If the strain is expressed as the tension inthe fabric (r Ap) over (££), where r is the local radiusof curvature and Ap is the pressure differential acrossthe canopy, then strain is represented by (q/E)(D</8).These terms are clearly present in the above set ofparameters.

Although stiffness of the canopy is not directlypresent in the above set of parameters, it is a function ofthe last four dimensionless parameters, i.e. the canopythickness ratio SD0, the canopy mass ratioMRc = ml(pDl), normalized modulus of elasticity

E/q, and fabric Poisson's ratio. Alternatively, thestiffness index 77 proposed by Heinrich and Hektner6

can be replaced in the above expressions. This index isdefined as « = Anax " ; where D^^ is the maximum

Da wS0

canopy width when hung upside down, W and w are thecanopy weight and fabric areal density, respectively.Lee7 reports that a standard C9 has an 77 of 0.15, withAnax = 0.077 D0. However, it should be stated that 77appears to be applicable only to canopies with similarconstruction and from the same fabric.

If the entire parachute system including thesuspension lines and the payload is considered, thefollowing parameters have to be added to the previousones: payload mass mh gravitational constant g,suspension line length 4, diameter <%, spring constant k,and damping dS9 as well as number of suspension lines nand their drag coefficient c</. Once these parameters arenon-dimensionalized, the force F\ experienced by thepayload will have a form as follows:

Fr =gD0'pD%'D0'D0'qD0'pVsS0'

n*cd

In addition to geometric similarity, all of the abovedimensionless parameters have to be similar between amodel and full-scale parachute to achieve completesimilarity. Since it would be quite difficult to match allof these parameters, the significance of each isdiscussed.

Reynolds number, which denotes the ratio ofinertia to viscous forces, is on the order of 106 -107 forfull-scale parachute canopies. It is thought thatReynolds number does not play a significant role duringthe inflation and vertical descend of round canopies.5As long as the flow separates at the skirt of model

canopies (and Re > 104), no Reynolds number effect isexpected. Although Cockrell3 provides data on thesteady state drag coefficient of full-scale canopies atlower speeds that indicate a dependence on Reynoldsnumber, these variations are thought to be due to thevariation of the mean projected diameter with Re. Oncethe drag is normalized with the projected area (asopposed to the constructed area), the Reynolds numbereffect should be minimal.

Mach number is an indicator of the onset ofcompressibility effects. Macha13 reports that Machnumber effects appear gradually for 0.4 < M < 0.9. Forthe low-speed applications and models tested in low-speed tunnels, Mach number has little effect on theparachute performance parameters.

The initial angle of attack a of the parachutecanopy is determined from the flight trajectory, and isan important parameter in determining the drag if itchanges significantly during the inflation phase. Forround canopy models held in a wind tunnel, smallchanges in a do not have an appreciable effect on theopening time and drag coefficient.

Permeability of the canopy fabric as characterizedby c/Vs is an important similarity parameter because itchanges the flow in the near wake of the canopy.Permeability of the canopy fabric is thought to affectlarge-scale vortical motions behind bluff bodies.Increase of drag coefficient and decrease of openingtime is expected with a reduction in permeability.Permeability appears to be even more important underinfinite mass conditions. Wolf44 presents data fromflight and wind tunnel tests which reveal that porosity isa dominant factor in the peak opening force ofparachutes at infinite mass.

Canopy mass ratio MI^ = m/(pD%) is anindicator of the inertia of the canopy fabric and hasbeen recognized in the past by Kaplun9 and Niemi8 asan important factor. During infinite mass inflations, thecanopy mass ratio is a dominant factor since the overallmass ratio becomes an inappropriate parameter.Moreover, this parameter shows up in Niemi's relativestiffness factor. For full-scale canopies, the canopymass ratio is typically less than 0.01, indicating that theinflation rate is minimally affected by inertia of thecanopy. Thus, for sub-scale testing, care should beexercised so that the canopy mass ratio is kept as smallas possible.

Payload mass ratio MR = mi/(pD%),representing the ratio of the payload mass to the addedmass, is a primary factor in determining the peakopening force of finite mass parachute systems. Forsub-scale testing purposes, this parameter has to beequal to that of the full-scale system. Theoreticalmodels of Fu4 and Lingard15 have both shown that thepeak opening force increases at first with increasing

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payload mass ratio, and then decreases following apeak.

Froude number Fr = V?/gD0 has long beenrecognized as a principal parameter governing the peakopening load of parachute systems. Theoretical modelsand experimental evidence reveal that the shock factor(the peak opening force over the payload weight) is alinear function of Froude number.15 Of course, forinfinite mass deployments in a wind tunnel, Froudenumber does not play a role.

Non-dimensional parameters associated withsuspension lines are the fineness ratio A/k ds/D0, kjqD0, d/p V5 S0, n, and c^ The first two parameters alongwith the number of suspension lines n are purelygeometrical parameters and are simple to duplicate insub-scale models. The cable drag coefficient is afunction of the Reynolds number of the cable and canbe a non-negligible portion of the total fluid dynamicdrag at slow speeds. The remaining two parameters arethe effective parachute spring and damping constants.If the latter is re-written as (n ds DJmi Vs\ Fu hasshown that this damping related parameter has aminimal effect on the inflation rate. The effectiveparachute spring constant can also be written as (n ksDJmi Vs), which is also expected to have a minimaleffect on the parachute performance parameters as longthe latter has a value greater than 2.4 For infinite massconditions, the effective spring constant k/(q D0) isthought to be an important parameter, and sub-scalemodels should have values similar to that of full-scaleprototypes.

The remaining two parameters E/q and v are thecanopy fabric parameters that contribute to the stiffnessof the canopy. As mentioned earlier, E, vand 5 can bereplaced with the empirical stiffness index of Heinrichand Hektner6. However, we suggest that the relativestiffness index of Niemi be used in place TJ since theformer can be determined from the fabric properties andthe drop conditions (without requiring the measurementof the manufactured parachute). Moreover, Niemi'sindex combines several of the other dimensionlessparameters, thus reducing the total number of theparameters. The relative stiffness as first proposed byKaplun9 has the following form E l/q Z)0

4, where / is thesecond moment of area of the canopy. Utilizing platedeflection theory, Niemi8 writes the relative stiffnessindex as /*- ( \> This index can be

rewritten as &J

if the snatch

velocity is replaced by the Froude number. The termsin the square bracket are only a function of the canopyfabric and scale, whereas the ratio in the parenthesis

depends on the drop conditions. The relative stiffnessindex can be applied to both finite and infinite massdeployments.

Based upon the above discussion, the time historyof the force on the payload in low-speed applicationsbecomes a function of the dimensionless parametersstated below for a given parachute geometry.

The dynamic pressure q in the above expression is interms of the snatch velocity. Therefore, a sub-scaleparachute model is expected to accurately reproduce theforce time history if the Froude number, payload massratio, relative stiffness index, angle of attack, fabricpermeability, the fineness ratio, and the parachutespring constant are matched between the sub- and full-scale systems. The peak opening force and opening (orfilling) time can be written in a similar manner asfollows:

Tn = A, --,-D0 qD

Filling time data of Berndt and De Weese16 for full-scale C9 parachute suggest that the filling time Tf isonly a function of the mass ratio, and the formerbecomes nearly constant for MR greater than -0.1. Fu4

also shows that the theoretically predicted filling time isvery nearly independent of the Froude number andspring constant. Moreover, for standard rip-stop nylonat typical airdrop snatch velocities, the permeability isquite small, of the order of a few percent. Of course, ifthe permeability (or geometric porosity) increases, weexpect the filling time to increase as well. Thus, for therange of parameters in airdrop applications, it isexpected that the filling (or opening) time be a functionof the relative stiffness index £ the initial angle ofattack a, and only a very weak function of MR, exceptat very low mass ratios. Data from Lee's (1989)vertical drop tests of full-, V£-, and !4-scale C9s at amass ratio of 0.13 were fitted reasonably well with apower law dependence on the relative stiffness index.8The expression for the opening time wasr0 = 0.86£"~°'064. We will examine this power lawdependence on £ for our very small-scale (relativelystiff) canopies. It is expected that the proportionalityconstant is a function of the canopy geometry andpermeability. Filling time is also expected to have asimilar form but with different constants.

As far as the peak opening force is concerned,Lingard15 has demonstrated that the shock factor, i.e.

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(F})peak/mjg, is a linear function of the Froudenumber. Rewriting the load factor in terms of thenormalized peak opening force multiplied by Fr/MRreveals that C/ is only a function of the mass ratio,relative stiffness index, permeability, and a. Lingard'ssemi-empirical theory for canopy inflation furtherassumes that Cp is a unique function of time and massratio. It is worth mentioning that Lee's load factor datafor l/4-scale C9 follow a weaker than linear dependenceon/r.

For our small-scale solid cloth, flat circularcanopies inflating horizontally in a water tunnel underinfinite mass conditions, the peak opening force Cpshould only depend on the relative stiffness index andthe parachute spring constant. The permeability ofstandard nylon fabrics in water is quite small andeffectively equivalent to impermeable Fill fabric. Thefilling time Tf is expected to be dependent on therelative stiffness index alone. In the next section,various dimensionless parameters are compared amongfull- and sub-scale C9 inflating in air and small-scalemodels in water.

Experimental Data

Five different solid cloth, flat circular parachutecanopies are considered here. They are a full-scale C9,Heinrich's 3' (91 cm) wind tunnel model, our 12" (30cm) water tunnel model, and 6" (15 cm) model tested ina wind and water tunnel. The very small scale of themodels considered in this study required that specialconstruction techniques be employed so that thestiffness could be reduced as much as possible. Thesecanopies were made from one solid piece of standard1.1 oz/yd2 rip-stop nylon fabric instead of individualgores being sewn together (see Fig. 1). The canopieswere cut by a soldering iron, thereby searing the edgeand preventing fraying. The suspension lines, whichwere 100 jam nylon thread, were attached to the canopyby melting them to the edge of the parachute canopy.All the models had a fineness ratio of unity (/5 = D0)and 24 suspension lines. The suspension lines wereattached to a stationary streamlined forebody in thetunnel, as shown in Fig. 2. The forebody diameter wasabout 4% of the smallest model canopy diameter,resulting in minimal interference with the inflation.The model canopies were inflated under constantfreestream velocity mimicking infinite mass conditions.Further details of the canopy construction and theexperimental setup can be found in Desabrais andJohari17.

The drag force on the canopy was measured by aload cell located inside the forebody and attacheddirectly to the suspension lines. The load cell output

was sampled at 150 Hz with a 12-bit A/D card. Thecanopy was imaged by a high-resolution progressive-scan CCD camera at a 30 Hz framing rate. The imageswere analyzed by an automated image processingprogram which computed the maximum projectedcanopy diameter to within 1%. A typical force andmaximum projected diameter trace for the 12" modelinflated in the water tunnel at a speed of 0.2 m/s isshown in Fig. 3. As expected, the drag force risesslowly at first while the canopy is unfolding. Theprojected diameter trace has an inflection point atapproximately Q.5D0, beyond which both the projecteddiameter and drag increase rapidly. The forcecoefficient reaches its maximum value of 3.5 at a timeof 2.33 s and then drops rapidly. There are severallarge-amplitude oscillations in the drag force before itsettles into its steady state value. The maximumprojected diameter reaches a maximum value of 0.88D0, at the over-inflated state prior to settling into itssteady state breathing mode. The solid line in Fig. 3shows the diameter during steady state. The point thatthe diameter curve crosses the solid line is the fillingtime, as indicated in Fig. 3. The opening and maximumdiameter times are also shown in the Figure. Table 1lists the mean measured filling and opening times,along with the maximum force coefficient for the 12"and 6" models inflating in the water tunnel at constantvelocity. For our small-scale models the filling time isgenerally within 5% of the opening time. This isconsistent with the commonly-accepted notion thatinfinite mass inflations have the same opening andfilling times.10 The maximum force coefficients in ourexperiments is also consistent with those measured byLingard5 for similarly scaled water tunnel models.

Table 1. Dimensionless inflation characteristics ofsmall-scale water tunnel models

A,(m)0.30 (12")0.15(6")0.15 (6")

Vs (m/s)0.200.390.20

Tf1.912.072.10

TO1.942.182.07

Cf3.612.913.48

Major dimensionless parameters of Reynoldsnumber, permeability, thickness ratio, canopy massratio, and the two stiffness indices are listed in Table 2for full- and small-scale C9s. The snatch velocity forthe two water tunnel models were chosen to producethe same Reynolds number of 6.7x104; the wind tunnelmodel had a snatch velocity resulting in a Reynoldsnumber comparable to the water tunnel models. Thelast column in the Table shows the range of payloadmass ratio used in the larger scale finite massexperiments.

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Table 2. Dimensionless parameters for C9 parachutes with different scalesCanopyC9 (28')

3'12"6"

6" (air)

Re~107

1.3xl06

6.7x1 04

6.7x10"6.3 xlO4

c/V,<0.020.0170.0120.0210.06

SfD08.3X1Q-6

7.8xlO'52.3x10-"4.7x10-*4.7x10-"

MRC5.4x1 0'3

0.081.1x10-"2.1x10""

0.17

C10-" -10*4.0xlO"7

1.5x10-"S.lxlO"4

l.lxlO"3

770.150.60.430.50.5

MR0.15-0.4

0.2400

00

00

ZW,(s)0.1-0.5

0.41.50.4

0.025

Since our small scale canopies were made of asolid piece of fabric, the Heinrich and Hektner stiffnessindex becomes 77 = I>max I DO - Measurement of Dmax

for our models revealed that 77 was 0.5 and 0.43 for the6" and 12" models, respectively. These values are to becompared against a value of 0.15 for the full-scale C9as reported by Lee7, and 0.6 for the Heinrich 3' model.The large value of TJ for the latter case is due to the full-scale construction technique employed on that model.

The permeability for the different canopy sizes inTable 2 was calculated based on the standard airpermeability at 0.5" water pressure differential. Thepermeability of the water models is based on themeasurements carried out by Lingard5.

There are several interesting observationsregarding the dimensionless number in Table 2. Thepermeability parameter is comparable among all casesconsidered except for the smallest 6" air model whichhas a value approximately 3 times higher than theothers. Thus, we would expect a longer inflationprocess and filling time for this case when comparedwith the others. The canopy mass ratio is smallest forthe two water tunnel models, even smaller than thatassociate with the full-scale canopy. This is due to thelarge density difference between air and water. Thus,the inertia of the canopy itself is negligible in the watertunnel models, as is the case for the full-scale. On theother hand, the 3' and 6" air models have substantiallylarger canopy mass ratios. As far as the relativestiffness index is concerned, it decreases with scale andsnatch velocity, as expected. The range ^ covers 8orders of magnitude among the cases considered. Evenwith such a large range for £ the stiffness index 77 ofHeinrich varies over a relatively narrow range. This isdue to the fact that both the snatch velocity and canopyscale determine the relative index.

The column designated by DJVS in Table 2 hasdimensions of time and is included to provide a timescale for the inflation process. The largest time scale of1.5 s belongs to the 12" water tunnel model; thesmallest time scale of 25 ms belongs to the 6" windtunnel model. The rest have time scales of order a fewtenths of a second. These data reveal the extremefrequency response required of the instrumentation usedfor testing small-scale canopies in a wind tunnel.

Relatively long inflation times can be had in a water atreasonable Reynolds numbers.

Correlation of Data

Since the peak opening force occurs at differenttimes with respect to the canopy inflation for differentmass ratios, it would be difficult to correlate our infinitemass opening times with the widely available openingtimes for mass ratios in the 0.1-0.2 range. For low massratios, the opening time is typically about one third ofthe filling time. The opening time is comparable tofilling time for high mass ratio inflations. Therefore,filling time was chosen as the parameter to becorrelated for small models and the full-scale C9.Unfortunately, there are no filling time data availablefor !/2- or tt-scale C9s.

The filling times of full-scale C9 as reported byBerndt and DeWeese,16 the 3! model of Heinrich andNoreen,11 along with our small scale models are plottedin Fig. 4 against the effective stiffness index £ Thebest-fit power law to the data is 1.3 £"°'064, indicatingthe efficacy of this stiffness index in correlating fillingtimes for different scale canopies over 8 orders ofmagnitude. The exponent of-0.064 in this expressionis the same as that found by Niemi when correlatingLee's7 opening times of !/£-, 1A-, and full-scale C9s at amass ratio of 0.13.

Based on the above correlation, we suggest thatthe power-law first proposed by Niemi is a valuabletool for scaling the filling and opening time of solidcloth, flat circular canopies. The above expression isdeemed applicable to both infinite and finite massinflations. It is thought that the exponent is 'universal'for all solid cloth, flat circular type canopies, and itwould depend only on the canopy geometry (not onscale). On the other hand, the proportionality constantdepends whether filling or opening time is considered.Furthermore, this constant should be independent of themass ratio (based on the payload or system mass) aslong as the mass ratio is greater than about 0.1. All paststudies have indicated a rapid increase of filling timewith MR as it gets smaller than -0.1. Lastly, theproportionality constant is also expected to depend onthe permeability of the fabric, as tested under the

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pertinent pressure differential across the fabric. As thepermeability increases, the proportionality constantshould be increased, perhaps with a weak lineardependence. Lastly, care should be taken when datafrom drop tests are considered in evaluating therobustness of any correlation because of the variabilityof the canopy angle of attack at the start of the inflationprocess.

Conclusions

A detailed dimensional analysis of the parametersaffecting the inflation of solid cloth canopies wascarried out. The two cases of the canopy aerodynamicsalone and the entire parachute system were consideredseparately. Comparison of sub- and full-scale C9parachutes along with small-scale wind and watertunnel models have revealed that a single parameter,namely the relative stiffness index, is able to correlatethe filling and opening times of various canopy sizes.The filling time, normalized by the constructeddiameter and snatch velocity, is

L064

The opening time for mass ratios in the range of 0.1 to0.2 can also be expressed by ^as follows:

The proportionality constant in the last expressiondepends on the mass ratio, and can increase by up to60% for infinite mass cases. Fabrics with largepermeability are expected to produce proportionalityconstants that are different than those in the aboveexpressions. The utility of the effective stiffness indexin correlating the peak opening force needs to beinvestigated for a range of canopy sizes.

Acknowledgement

This material is based upon work supported by theU.S. Army Research Office under grant numberDAAG55-98-1-0171.

References

1. Stein, K.R., "Simulation and Modeling Techniquesfor Parachute Fluid-Structure Interactions," Ph.D.dissertation, Department of Aerospace Engineering,University of Minnesota, December 1999.

2. Stein, K., Benney, R., Kalro, V., Tezduyar, T.E.,Leonard, J., and Accorsi, M., "Parachute Fluid-Structure Interactions: 3-D Computation," ComputerMethods in Applied Mechanics and Engineering, Vol.190, 2000, pp. 373-386.

3. Cockrell, D.J., "The Aerodynamics of Parachutes,"AGARDograph No. 295, Neuilly sur Seine, France,July 1987.

4. Fu, K-H., "Theoretical Study of the Filling Processof a Flexible Parachute Payload System," German Airand Space Research Institute, Braunschweig, Germany,August 1975, Research Report DLR-FB 75-76.

5. Lingard, J.S., "The Aerodynamics of ParachutesDuring the Inflation Process," Ph.D. thesis, Departmentof Aeronautical Engineering, University of Bristol,England, October 1978.

6. Heinrich, H.G. and Hektner, T.R., "Flexibility as aModel Parachute Performance Parameter," Journal ofAircraft, Vol. 8, No. 5, 1971, p. 704-709.

7. Lee, C.K., "Modeling of Parachute Opening: AnExperimental Investigation," Journal of Aircraft, Vol.26, No. 5, 1989, pp. 444-451.

8. Niemi, E.E., "An Improved Scaling Law forDetermining Stiffness of Flat, Circular Canopies," U.S.Army Natick Research, Development and EngineeringCenter, Natick, MA, March 1992, Technical ReportTR-92/012.

9. Kaplun, S., "Dimensional Analysis of the InflationProcess of Parachute Canopies," Aeronautical Engineerthesis, California Institute of Technology, Pasadena,CA, 1951.

10. Knacke, T.W., Parachute Recovery Systemsdesign Manual Para Publishing, 1992.

11. Heinrich, H.G. and Noreen, R.A., "Analysis ofParachute Opening Dynamics with Supporting WindTunnel Experiments," Journal of Aircraft, Vol. 7, No.4, 1970, pp. 341-347.

12. Heinrich, H.G., "Opening Time of ParachutesUnder Infinite-Mass Conditions," Journal of Aircraft,Vol. 6, No. 3, 1969, pp. 268-272.

13. Macha, J.M., "An Introduction to TestingParachutes in Wind Tunnels," Proceedings of the AIAA12th Aerodynamic Decelerator Systems TechnologyConference, 1991, San Diego, CA; AIAA paper 91-0858.

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14. Wolf, D., "Parachute Opening Shock,"Proceedings of the CEAS/AIAA 15th AerodynamicDecelerator Systems Technical Conference, Toulouse,France, June 1999, p. 253; AIAA paper 99-1702.

15. Lingard, J.S., "A Semi-Empirical Theory ToPredict The Load-Time History of An InflatingParachute," Proceedings of 8th AerodynamicDecelerator and Balloon Technology Conference,Hyannis, MA, April 1984, pp. 117-185; AIAA paper84-0814.

16. Berndt, R.J. and DeWeese, J.H., "Filling TimePrediction Approach for Solid Cloth Type ParachuteCanopies," Proceedings of AIAA AerodynamicDeceleration Systems Conference, Houston, TX,September 1966.

17. Desabrais, K.J. and Johari, H., "Reynolds numberand Scale Effects on Inflating Parachute Canopies,"Proceedings of the CEAS/AIAA 15th AerodynamicDecelerator Systems Technical Conference, Toulouse,France, June 1999, pp. 258-268; AIAA paper 99-1736.

Figure 1. The 6-inch canopy model.

Figure 2. The forebody support arrangement in the water tunnel.

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4.0

o.o1.0 2.0 3.0 4.0 5.0

t(s)Figure 3. Time history offeree coefficient and projected diameter for a 12-inch canopy inflated at a constant

freestream velocity of 0.2 m/s. The horizontal line in the plot represents the steady state diameter of the canopy.

10

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• full-scale C-9o water tunnel modelsT wind tunnel (6")v Heinrich (31)

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Relative Stiffness Index (QFigure 4. Dependence of filling time on the relative stiffness index.

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Stiffness Scaling for Solid-Cloth Parachutes