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1

EXTENDED PARACHUTE OPENING SHOCK ESTIMATION METHOD

Karl-Friedrich Doherr1

Am Sundern 11D-38108 Braunschweig

Germanye-mail: [email protected]

1 Dr.-Ing., M.Sc., Associate Fellow, AIAA

ABSTRACT

Based on Pflanz' earlier work of 1942, Ludtke in 1973 developed an expression for the non-dimensionalopening shock of an inflating parachute in horizontal flight and at medium to large initial velocity at linestretch. In the present paper, this expression is extended to include the cases of arbitrary descent flightpath angle and small initial velocity at line stretch. The results of the Extended Pflanz-Ludtke Method agreewell with the corresponding results from the numerical solutions of the two-degrees-of-freedom equationsof motion.

NOMENCLATURE

A ballistic parameter = Fre / nfALx limit ballistic parameterB parameter = Frs / nf = A (vs /ve)2

CD0 drag coefficientCDS drag area(CDS)0 steady state drag area = CD0*S0CK opening force factor, non-dimensional

opening shock CKL Ludtke's analytical expression of CKCKP Pflanz’ analytical expression of CKCx opening force coefficient = (CDS)max/(CDS)0D0 nominal parachute diameter =

(4/� S0)1/2

Fx opening shock, maximum of the inflationforce

Fre Froude number = ve2/ gD0

Frs Froude number = vs2/ gD0

g acceleration of gravityj drag area exponentnf non-dimensional inflation time =

tf vs / D0nx load factor = Fx / mgmt system massS0 nominal reference areatf inflation timetmax time at which CDS has reached its

maximumtm time at which Fx has reached its maximumv system velocity

ve terminal velocity = steady state velocityvs velocity at line stretchx non-dimensional inflation force� angle of flight path, downward negative� non-dimensional drag area� density of air� mass ratio = mt / (�D0

3)

INTRODUCTION

The calculation of the parachute opening shock,which is the maximum of the inflation force, is ofprime interest for the lay-out of parachute systems.Earlier, fundamental work was done 1942 byPflanz1. Assuming analytical functions for thedevelopment of the parachute drag area with time,he integrated the non-linear differential equation ofmotion and calculated the inflation force for thesimplified case of horizontal flight (� � 0). The Pflanzmethod is based on the also simplifying assumptionthat each parachute type has a characteristicchange of non-dimensional drag area with non-di-mensional time (Fig. 1), independent of size, initialvelocity, density, and canopy loading, see alsoLingard2.

In Fig. 2 the non-dimensional opening forcereduction factor, X1, calculated by the Pflanzmethod, is plotted versus the ballistic parameter, A,

17th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar19-22 May 2003, Monterey, California

AIAA 2003-2173

Copyright © 2003 by Karl-Friedrich Doherr. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2

Fig. 1: Top: Averaged measured non-dimensionaldrag areas of several parachute types vs.non-dimensional inflation time, fromKnacke3, Fig. 5-40, p. 5-47 (with permissionof Para Publishing).Bottom: Typical drag area vs. time increaseof several parachute types, from Knacke,Fig. 5-38, p. 5-45

for three drag area polynomials with exponents ½,1, and 2, respectively. X1 corresponds to theopening force factor, CK, when no over-inflationoccurs (Cx � 1).

In the Pflanz method three essential effects areneglected:

(i) the over-inflation coefficient or opening forcecoefficient, Cx(ii) the initial trajectory angle, �0, at the beginning ofthe inflation (at line stretch)(iii) the initial velocity ratio, vs/ve, at the beginning ofthe inflation (at line stretch)

Cx plays a role only, when the opening shock

Fig. 2: Opening-force reduction factor, X1, vs. bal-listic parameter A, from Knacke3, Fig. 5-51,p. 5-59. Terminology: Read X1 � CK for thecase Cx = 1; n � j; (CD

.S)p � (CDS)0; V1 � vs

occurs in the region of over-inflation. This usuallyhappens when, for example, the parachute is smallin comparison to the payload (reefed parachutes,drogue chutes, pilot chutes) or when the drag areaexponent, j, is large (Circular Flat canopies).

In 1973 Ludtke4 applied Pflanz’ solution for theopening force factor to the analysis of Circular Flatcanopies, taking j = 6. The more general case ofarbitrary j and the effect of over-inflation weretreated by Ludtke5 1986.

In the following, an approximate extension of thePflanz-Ludtke method is presented, that includesthe effects of initial trajectory angle and initialvelocity ratio.

PFLANZ-LUDTKE METHOD

The peak inflation force, or opening shock, of theparachute is calculated from

)SC( v C = F 0D2s2Kx

� (1)

The opening force factor, CK, is the maximum of thenon-dimensional instant inflation force

3

)SC(SC )

vv( =

)SC( v

SC v = ) (t x0D

D2

s0D2s2

D2

2�

�

(2)

The velocity ratio v/vs can, in general, bedetermined from a solution of the non-dimensionalequation of motion with two degrees of freedom:

tt

tt 0 ; = (0) ; 1 =

vv(0) : 0 =

tt

v / v1

vt g

- = t / t d

d

)vv(

) SC(SC

m 2v t ) SC( -

vt g

- = t / t dv / v d

ff0

sf

ss

f

f

2

s0D

D

t

sf0D

s

f

f

s

max

cos)(

sin)()(

��

��

��

(3)

tf is the inflation time from suspension line stretchuntil the canopy has reached its steady state dragarea, (CDS)0, for the first time, and tmax is the time atwhich the drag area has reached its maximum.

We assume a polynomial change of drag area withtime (Fig. 3):

C = : C t

t =

tt ; 1 = : 1 =

tt ; 0 = : 0 =

tt

)tt( =

) SC(SC = ) (t

xj / 1

xffff

j

f0D

D

��

�

�max

(4)

0 0.5 1 1.5t /tf 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

CKCKxx

v/vsv/vs

� = (t/tf)2� = (t/tf)2

Cx

Non-dimensional inflation variables

tmax/tfNONPAPER | 6.3.2003

Fig. 3: Example of the approximation of � by apolynomial of time together with calculatedv, x, and CK from a 2DoF simulation with j =2, nf = 12, vs/ve = 14.6, A = 0.0236, �0 = 0

The drag area can, for example, be measured in awind tunnel.

We introduce the following four parameters:(i) the non-dimensional inflation time

Dv t = n

0

sff (5)

(ii) the terminal velocity

) SC( g m 2 = v

0D

te

�

(6)

(iii) the ballistic parameter

nF =

n D gv =

n D g )SC( g m 2 =

n D )SC(

m 2 = v t ) SC(

m 2 = A

f

re

f0

2e

f00D

t

f00D

t

sf0D

t

�

�� (7)

and (iv) a combined parameter

)vv( A =

nF =

t gv = B 2

e

s

f

r s

f

s (8)

Then, Eq. (3) becomes

C tt 0 ; = 0) ( ; 1 =

v0) ( v : 0 =

tt

v / v1

B1 - =

t / t d d

)vv( )

tt(

A1 -

B1 - =

t / t dv / v d

j / 1 x

f0

sf

sf

2

s

j

ff

s

��

��

�

cos)(

sin)()(

(9)

In general, Eq. (9) can only be solved numerically.The resulting velocity ratio, v(t)/vs, and the dragarea function, �(t), can be used to calculate thenon-dimensional drag force, x(t), and its maximum,CK, from Eq. (2), see Fig. 3.

Eq. (9) has five independent parameters: A, vs/ve(included in B), Cx, j, and 18) �0. In A are contained

4

the canopy loading, mt / (CDS)0, the mass ratio, � =mt / �D0

3, the Froude number, Fre = ve2 / gD0, based

on ve, and the non-dimensional inflation time, nf.The Froude number, Fre, is the important modelnumber of the steady part of the motion, while thenon-dimensional inflation time, nf, is an importantparameter of the inflation process. The parameter Bcan be interpreted as a Froude number Frs,calculated with the initial speed, divided by the non-dimensional inflation time.

For the special case of horizontal flight, � = 0, thesimplified equation of motion

)vv( )

tt(

A1 - =

t / t dv / v d 2

s

j

ff

s

)()(

(10)

can be integrated:

C tt 0

)

tt (

) 1 +(j A1 + 1

1 = vv

j 1/x

f

1 +j

f

s

��

(11)

Substituting Eqs. (11) and (4) into Eq. (2) anddifferentiating x with respect to time, provides acondition for the time, t = tm, at which the maximumof x occurs

]2 +j

A 1) +(j j [ = tt

1 +j 1

f

m (12)

For small A the opening shock occurs early duringthe inflation and for large A at the end of theinflation, when �(tm/tf) = Cx, tm = tmax, tm/tf = Cx

1/j and

C ) 1 +j (j

2 +j = A A j / ) 1 +j ( xLx� (13)

For A < ALX the opening shock occurs at tm < tmax.Then, taking v(tm) and �(tm), with tm from Eq. (12),Eq. (2) yields Ludtke’s solution for the openingshock factor

] 2 +j

A ) 1+j (j [] ) 1 +j ( 2

2 +j [ =

)tx = C) 1 +j ( /j 2

mKL �( (14)

For A ≥ ALx, tm = tmax, CKL is limited by

x 2jj

x

x2

sKL

C]CjA

[

C ) vtv ( = C

��

�

��)1(

max

)1(11

)(

(15)

At infinitely large mass (in the wind tunnel) thevelocity stays constant. Then CKL(tmax) = Cx.

Ludtke5 also considered the case that some initialdrag area, �0, exists at the beginning of theinflation, � = �0 + (1 – �0)*(t/tf) j, but found nocorresponding analytical solution for CK, whereasPflanz1 found a solution for the case � = [a + b*(t/tf)]j, with a = �0

1/j and b = 1 - a: (16)

}]a - A ) 1 +j ( b [ 2 +j

j {

4

j b A = C

) 1 +j ( )/ 2 +j ( - 1 +j

222

KP

*

*

Eq. (14) can be derived from Eq. (16), setting a = 0and b = 1 and rearranging terms. Thus, CKL is thespecial case of CKP for �0 = 0.

Fig. (4) shows a plot of CKL over a large range of Afor Cx = 1 to 1.8 and j = 6 (stars). Added are thecorresponding results of two-degrees-of-freedom(2DoF) computer simulations (solid lines) forhorizontal flight (�0 = 0�) and large initial velocityratio, vs/ve = 12.5. The agreement between theanalytical and the numerical results is excellent.

But, in the case of low initial velocity ratio, forexample vs/ve = 2.5, (Fig. 5), there are largedeviations between the analytical and the numericalresults in the range of small to medium A < 1. Thedeviations become even larger in vertical flight, �0 =-90�, and at small vs/ve (Fig. 6).

5

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK

Opening Force FactorCx 1.8 1.6

1.4

1.2 1.0

2DoF simulations ***** Pflanz-Ludtke method

j = 6vs/ve = 12.5

�0 = 0°

Fig. 4: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 6, �0 =0°, vs/ve = 12.5.

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK

Opening Force FactorCx 1.8 1.6 1.4

1.2 1.0


j = 6 vs/ve = 2.5

��0 = 0°

Fig. 5: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 6, �0 =0°, vs/ve = 2.5.

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


1.4

1.2 1.0


j = 6

�0 = -90°

vs/ve = 2.5

Fig. 6: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 6, �0 =-90°, vs/ve = 2.5.

EXTENDED APPROXIMATION

At small initial velocity gravity causes the system toaccelerate first and gain velocity before enoughdrag from the inflating canopy has developed to

decelerate the system. This causes a higherinflation force than estimated by the Pflanz-Ludtkemethod. This effect is only pronounced at smallvelocity ratios, see Fig. 7, where the load factor, nx= Fx / mg, calculated numerically for an arbitraryexample of A = 0.05, j = 6, and �0 = 0, is plottedversus the square of the initial velocity ratio, (vs/ve)2,(solid line).

-50 0 50 100 150 200-5

0

5

10

15

20

25

(vs/ve)2

n x

Load Factor nx = Fx / mg

�0-90°-67.5°-45°-22.5° 0°

------ 2DoF simulations***** Pflanz-Ludtke method A = 0.05 j = 6

(1-e-B) j0.5

e-B j0.5

�0-90°-67.5°-45°-22.5° 0°


(1-e-B) j0.5

e-B j0.5

�0-90°-67.5°-45°-22.5° 0°


(1-e-B) j0.5

e-B j0.5

�0-90°-67.5°-45°-22.5° 0°


(1-e-B) j0.5

e-B j0.5

�0-90°-67.5°-45°-22.5° 0°


(1-e-B) j0.5

e-B j0.5

CKL

j0.5

Fig. 7: Load factor, nx = Fx / mg, vs. the square ofthe initial velocity ratio (vs/ve)2 for j = 6, dif-ferent �0 and small ballistic parameter A =0.05.

The Pflanz-Ludtke method (stars) shows nx to beproportional to the square of the velocity ratio withconstant gradient CKL:

)vv( C =

g m)SC( v C =

g mF = n 2

e

sKL

0D2s2

KLx

x

�

(17)

Added to Fig. 7 are the simulation results for �0 = -22.5�, -45�, -67.5�, and -90� (solid lines). The loadfactor increases with �0 approximately equal to j0.5

*sin(-�0). This observation stimulated theintroduction of two correction terms, containing fourof the above mentioned five independentparameters:

e ) vv ( j = C B-

s

e1

2 (18)

e ) (- ) e - 1 () vv ( j = C j

6A -

0 B- 2

s

e2

)( 25.0sin � (19)

C1 takes care of the initial velocity effect, whichdiminishes with increasing vs/ve, while C2 takes careof the influence of the initial flight path angle �0. Fora better match with the numerical solutions of the

6

2DoF equations of motion the factor e ) j 6A - ( 0.25

wasincluded in C2. Then, the opening force factorfollows from

C + C + C = C 21KLK (20)

CKL is calculated either by Eq. (14), when A < ALX,or by Eq. (15), when A ≥ ALX, with ALX from Eq.(13).

The agreement between the 2DoF simulationresults (solid lines) and the Extended Pflanz-LudtkeMethod (stars) is satisfying as can be seen in Figs.8 to 12.

In Figs. 8 to 12 the exponent j assumes the values j= 1/2; 1; 2; 3; and 6. Realistic Cx - values arechosen. Reefed parachutes, for example,experience little over-inflation, Cx = 1.0 to 1.2,

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK

Opening Force Factor

2DoF simulations ***** Extended Planz-Ludtke method

1.2

Cx

1.0

j = 1/2 vs/ve = 12.5

��0 = 0°

Fig. 8a: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 1/2 (typical for reefedcanopies), �0 = 0°, vs/ve = 12.5.

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


Cx

1.2 1.0

2DoF simulations ***** Extended Pflanz-Ludtke method

j = 1 vs/ve = 12.5

��0 = 0°

Fig. 9a: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 1 (typical for Ribbon andRing Slot canopies), �0 = 0°, vs/ve = 12.5

whereas Circular Flat parachutes experience largeover-inflation, Cx = 1.6 to 1.8. According to Knacke3

reefed canopies have a drag area exponent of j=0.5, Ribbon and Ring Slot canopies about j = 1,Extended Skirt canopies about j = 2 to 3, andCircular Flat canopies about j = 6. The author hasfound j to be about 3 for Cross canopies.

In Figs. 8a to 12a the cases of horizontal flight andlarge initial velocity ratios and in Figs. 8b to 12b thecases of vertical flight and small initial velocity ratiosare shown.

The Extended Pflanz-Ludtke Method, Eq. (20),handles the cases of arbitrary descent flight pathangle and arbitrary initial velocity within practicallimits.

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


Cx

1.2 1.0

2DoF simulations

***** Extended Pflanz-Ludtke method

j=1/2 vs/ve = 2.5

��0 = -90°

Fig. 8b: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 1/2 (typical for reefedcanopies), �0 = -90°, vs/ve = 2.5

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


Cx

1.2

1.0


j = 1 vs/ve = 2.5

��0 = -90°

Fig. 9b: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 1 (typical for Ribbon andRing Slot canopies), �0 = -90°, vs/ve = 2.5

7

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


Cx

1.4

1.2 1.0

*****

2DoF simulations Extended Planz-Ludtke method j = 2 vs/ve = 12.5 ��0 = 0°

Fig. 10a: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 2 (typical for ExtendedSkirt canopy), �0 = 0°, vs/ve = 12.5

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


Cx 1.6 1.4

1.2


j = 3 vs/ve = 12.5

��0 = 0°

Fig. 11a: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 3 (typical for Cross andExtended Skirt canopies), �0 = 0°, vs/ve =12.5

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK



j = 6 vs/ve = 12.5��0 = 0°

Fig. 12a: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 6 (typical for Circular Flatcanopy), �0 = 0°, vs/ve = 12.5

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


Cx

1.4

1.2 1.0


j = 2 vs/ve = 2.5

��0 = -90°

Fig. 10b: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 2 (typical for ExtendedSkirt canopy), �0 = -90°, vs/ve = 2.5

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


Cx

1.6

1.4

1.2


j = 3 vs/ve = 2.5

��0 = -90°

Fig. 11b: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 3 (typical for Cross andExtended Skirt canopies), �0 = -90°, vs/ve =2.5

10-2 10-1 100 101 1020

0.5

1

1.5

2

A

CK


2DoF simulations Extended Pflanz-Ludtke method *****

j = 6 vs/ve = 2.5 ��0 = -90°

Fig. 12b: Opening force factor, CK, vs. ballistic pa-rameter, A, for j = 6 (typical for Circular Flatcanopy), �0 = -90°, vs/ve = 2.5

8

The validity of the estimated opening force factorsfrom Eq. (20) depends, of course, very much on thevalidity of the mathematical models of the drag areaand of the non-dimensional inflation time as well ason the accuracy of the other parameters.

An analysis by the author of Ludtke’s 6 wind tunneldata on Cross parachute models with different clothpermeability showed a decrease of the exponent jwith increasing cloth permeability, leading to adecrease in CK. This implies that the drag areapolynomial is not an invariant for a selectedparachute type, but varies with the specific designdata.

Wolf 7 reported 1999 a decrease of CK of Ribbonparachutes with increasing geometric porosity.Certainly both j and nf vary with porosity.

Fu 8 reported 1975 in his theoretical PhD-thesis onthe inflation of Circular Flat canopies that nfdecreases with increasing suspension line ratio,ls/D0, and also decreases with increasing massratio, � = mt / D0

3.

Based on Fu's work, in 1977 Saliaris 9 analyzedflight test data of Circular Flat canopies withconstant suspension line ratio and found thefollowing approximation:

�

0.364 + 3.350 = n f (21)

Only for sufficiently large mass parameter (and thusfor large ballistic parameter) the non-dimensionalinflation time of Circular Flat canopies approachesconstant nf. As a consequence, the widely madeconvenient assumption that nf is a constant for agiven type of parachute must be uestioned.

SUMMARY

For the calculation of the opening shock factor ofparachutes an extension of the Pflanz-LudtkeMethod was developed that includes the influenceof the trajectory angle and of the initial velocity ratioat line stretch. This method can serve as a tool for afirst layout of parachute systems or in parametricparachute system studies.

REFERENCES

1 E. Pflanz, "Zur Bestimmung derVerzögerungskräfte bei Entfaltung vonLastenfallschirmen", Forschungsanstalt GrafZeppelin, Stuttgart-Ruit, Bericht 231, 1942,published as ZWB FB 1706 (1942)

2 J.S. Lingard, "The Effect of Added Mass onParachute Inflation Force Coefficients", AIAA 95-1561, 13th Aerodynamic Decelerator SystemsTechnology Conference, 15-19 May 1995,Clearwater Beach, Florida, USA

3 T.W. Knacke, "Parachute Recovery SystemsDesign Manual", NWC TP 6575, Para Publishing,Santa Barbara, California, 1992

4 W.P. Ludtke, "A Technique for the Calculation ofthe Opening-Shock Forces for several Types ofSolid Cloth Parachutes", AIAA Paper No. 73-477,Palm Springs, California, May 21-23, 1973

5 W.P. Ludtke, "Notes on a Generic ParachuteOpening Force Analysis", NSWC TR 86-142, 1March 1986, Naval Surface Weapons Center,Virginia, Silver Spring, Maryland

6 W.P.Ludtke, "Effects of Canopy Geometry on theInfinite Mass Opening-Shock Factor of a CrossParachute with a W/L Ratio of 0.264", NOLTR 73-157, 31 July 1973, Naval Ordnance Laboratory,White Oak, Silver Spring, Maryland

7 D. Wolf, "Parachute Opening Shock", AIAA-99-1709, 15th CEA/AIAA Aerodynamic DeceleratorSystems Technology Conference, 8-11 June 1999,Toulouse, France

8 K.-H. Fu, "Theoretische Untersuchung zumFüllungsvorgang eines flexiblen Fallschirm-Last-Systems", PhD. Thesis, Technische UniversitätBraunschweig, 21 June 1975.

9 C. Saliaris, "Beitrag zur experimentellen Ermittlungder Massstabseinflüsse auf die Leistungsdaten beiEbenen Rundkappen-fallschirmen", IB 154-77/49,DFVLR Institut für Flugmechanik, Braunschweig, 15October 1977

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