# Structural evaluation of spherical parachute.

Introduction

Many researches have been done regarding to simulation and investigation of parachute flying or stress analyzing among that of surface area. For example some researches in the field of parachute analysis are referred. Aerodynamic equations and structural dynamics equations were developed for describing parachute opening process, and an iterative coupling solving strategy incorporating the above equations was proposed for a small-scale, flexible and flat-circular parachute . With the rise of computational fluid dynamics and structural finite element method, it is possible to accurately describe the dynamic characteristics of a parachute. In order to simulate the flow field around the parachute during terminal descent, a finite volume method and Spalart-Allmaras turbulence model are used to solve the incompressible Navier-Stokes equations or analyzing by numerical method [2 and 3]. By means of establishing parachute's flying physical model, four methods (i.e. tiny segment analysis method, inflating distance method, moment method and simulating canopy shape method) are adopted to analyze and calculate the variations of canopy shape and the parameters of parachute inflation. The results are simply analyzed . Parachute aerodynamics involves an interaction between the flexible, elastic, porous parachute canopy and the high speed airflow (relative to the parachute) through which the parachute falls. Computer simulation of parachute dynamics typically simplifies the problem in various ways, e.g., by considering the parachute as a rigid bluff body . Most of those researches have been focused in dynamic subjects about parachute and there is no analysing of surface parachute in textile aspect.

Since longtime ago, different kind of textiles were used in industrial applications especially in composite field. However, because of the structure of woven fabrics and kind of interactions between fabric elements, we could evaluate woven fabrics as a orthotropic material . The structure of spherical parachute is a part of sphere during falling that for air exhaust a hole is positioned in upper of it. In this paper, first, the equations of orthotropic shell presented and then we simplify them for membrane shell theory. Finally, the obtained equations were used for analyzing the parachute.

The evaluation of air distribution into parachute is an aerodynamic phenomenon. We suppose that distribution of air pressure into parachute is constant and we consider the following conditions as boundary conditions: the bottom edge of parachute is fixing when falling and the upper edge of parachute is fixing when falling.

The equations of orthotropic shell

To get the equations of orthotropic shell, the two dimensional coordinates ([[alpha].sub.1], [[alpha].sub.2]) was selected to conform curvatures of shell and unit vector ([e.sub.n]) was selected vertically on surface. The force equilibrium equations for shell are as follows 

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

By using Kirchhoff hypothesis, we could eliminate shear strains [[gamma].sub.23], [[gamma].sub.13] and normal stress [[delta].sub.33]. So, the stress -strain relationships for orthotropic shell will be as follows [8 and 9]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[[sigma].sub.12] = [G.sub.12] [[omega].sup.(z)] (6)

In mentioned equations [[gamma].sub.12], [[gamma].sub.21] and [G.sub.12] are mechanical properties of material in [[alpha].sub.1], [[alpha].sub.2] direction of coordinates and [[epsilon].sub.1], [[epsilon].sub.2] and [omega] present strains.

In the equations 4, 5, 6, 'z' is the amount of strain in layer with z distance from middle layer. For simplifying the 4, 5 and 6 equations, we substitute the following coefficients:

[E.sub.10] = [E.sub.1]/1-[v.sub.12][v.sub.21] (7)

[E.sub.20] = [E.sub.2]/1-[v.sub.12][v.sub.21] (8)

Hence, we get the new equations

[[sigma].sub.11] = [E.sub.10][[epsilon].sub.1.sup.(Z)] + [v.sub.21][[epsilon].sub.2.sup.(z)]] (9)

[[sigma].sub.22] = [E.sub.20][v.sub.21] [[epsilon].sub.1].sup.(z)] + [[epsilon].sub.2.sup.(z)]] (10)

[[sigma].sub.12] = [G.sub.12] [[omega].sup.(z)] (11)

By using the geometry of shell, the stress-strain relationships are 

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The virtual work method was used to obtain the strains and forces. The strain energy equation for a shell is (equation 15) as follow 

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

In the equation 15 the integration on volume was applied. We get the amount of strains in layer with (z) distance from middle layer as equations 16, 17 and 18 [7, 10]

[[epsilon].sub.1.sup.(z)] = 1/1 + z/[R.sub.1] ([[epsilon].sub.1] + z[k.sub.1]) (16)

[[epsilon].sub.2.sup.(z)] = 1/1 + z/[R.sub.2] ([[epsilon].sub.2] + z[k.sub.2]) (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[k.sub.1] And [k.sub.2] are the variations of curvature of parallel surface in relation to middle surface in [[alpha].sub.1] and [[alpha].sub.2] directions respectively and [tau] is torsion of element belongs to middle surface. The equations 9, 10 and 11 and 16, 17 and 18 can be substituted in equation 15.

As you know [integral]zdz = 0 and another side because of thin shell the terms of [z.sup.2] and upper power is negligible; so the final equations on the basis of coefficient of (z) and some constant parameters are as follow (equation 19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

In the latter equation, if we separate the parts with constant coefficient and (z) coefficient and then present them with [Q.sub.0] and [Q.sub.z] respectively, we get (equations 20 and 21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Now, if we negligible the z/R term (thickness to curvature radius) in compare with one value and then substitute [delta]/2 term instead of (z) finally we obtain the strain in [delta]/2 distance (equations, 22, 23, and 24) as follows

[k.sub.1] = 2/[delta] ([[epsilon].sub.1.sup.(z)] - [[epsilon].sub.1]) (22)

[tau] = ([[omega].sup.(z)] - [omega])/[delta] (23)

k/2 = 2/[delta] ([[epsilon].sub.2.sup.(z) - [[epsilon].sub.2]) (24)

By substituting the equations 22, 23 and 24 in equation 21 and by eliminating the terms of [delta]/R and [[delta].sup.2]/R z, the strain energy will be as follow (equation 25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

We have to apply a virtual variation on shell for using virtual work principle i.e. derivative of strain energy with respect to strain (equation 26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Another side, we could obtain the equation 27, by using forces and moments obtained from geometry analytic of a shell.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Where (S) and (H) are

S = [T.sub.12] - [M.sub.21]/[R.sub.2] = [T.sub.21] - [M.sub.12]/[R.sub.1] (28)

H = [M.sub.12] + [M.sub.21]/2 (29)

With taking variation of strain energy from equation (25) and equalize the same terms with equation (27), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

S = [G.sub.12] [delta][omega] (34)

H = [[delta].sup.3][G.sub.12]/6 (35)

By using equations 30, 31 and 34, we could calculate strain in related to stress (equations 36, 37 and 38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[omega] = S/[delta][G.sub.12] (38)

Membrane theory

All of the momentum are assumed zero in membrane theory. We can accept this hypothesis, if shell doesn't have any resistance to bending or variation of curvature and torsion of middle layer is negligible. Also eliminating of moments is possible, if the bending rigidity to be very low.

In respect to fully flexible shells such as a shell made with thin fabric, it seems that resistance to pressure forces are negligible and the stability of shell is destroyed by exerting smallest pressure force. This shell can only tolerate tension forces. Furthermore, we can conclude:

S = [T.sub.12] = [T.sub.21] (39)

[N.sub.1] = [N.sub.2] = 0 (40)

Therefore, equilibrium equations 1, 2 and 3 will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

[T.sub.1]/[R.sub.1] + [T.sub.2]/[R.sub.2] = [q.sub.n] (43)

Also, by calculating the strains from equations 12, 13, 14 and equations from 36 to 38, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

Solving the existing equations and analysis of parachute

To find the equations of parachute under the weight of parachutist in air, we consider the following hypothesis:

1. Parachute is assumed as a membrane.

2. The shape of parachute under the exerted force is as a part of sphere and there is a hole on top of it [11 and 12].

3. Normal force exerted on surface of membrane is distributed uniformly on the entire surface of parachute.

4. Boundary conditions were assumed as follows

[T.sub.1]([[theta].sub.1]) = 0 (47)

[[theta].sub.1] is assumed as the angle of air outlet hole.

u([[theta].sub.2]) = 0 (48)

Hence, by considering the geometry of parachute as Figure 1, we get

[[alpha].sub.1] = [theta] (49)

[[alpha].sub.2] = [theta] (50)

[R.sub.1] = [R.sub.2] = R (51)

[A.sub.1] = R (52)

[A.sub.2] = R sin [theta] (53)

By replacing mentioned parameters on membrane equations (41, 24, and 43), the equations 49, 50 and 51 will be obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

Since, derivative of T, S and R respect to [phi] is zero and the values of [q.sub.1] and [q.sub.2] is zero; finally the above equations are as follows

1/R d[T.sub.1]/d[theta] + 2/R [T.sub.1] cot g[theta] = [q.sub.n] cot g[theta] (57)

1/R dS/d[theta] + 2S/R cot g[theta] = 0 (58)

[T.sub.2] = R[q.sub.n] - [T.sub.1] (59)

To solve the differential equation (57), if we use P = [T.sub.1 [Sin.sub.2][theta] as a change of variable, the mentioned equation is converted to a perfect differential equation and after integrating, the equation (60) was obtained:

P = 1/2 [q.sub.n][R.sup.2] [sin.sup.2] [theta] + C (60)

By exerting the boundary conditions, as figure 1, [theta] = [[theta].sub.1] and T = 0, finally, we get

[T1] = 1/2 R[q.sub.n] (1 - [sin.sup.2][[theta].sub.1]/[sin.sup.2][theta] (61)

And by substituting [T.sub.1] in equation 59, we get [T.sub.2] as follow

[T2] = 1/2 R[q.sub.n] (1 + [sin.sup.2][[theta].sub.1]/[sin.sup.2][theta] (62)

[FIGURE 1 OMITTED]

And by exerting boundary condition, the value of (s) in membrane will be zero. Also, by replacing the geometric parameters of material in equations 44 and 45 and by finding the w/[R.sub.2] value from equation 44 and replacing it in equation (45), the following equation are obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63)

If we use [xi] = u/Sin[theta] as a change of variable and by exerting boundary condition, the equation 63 is changed to equation 64 and we get 'u' as follow

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (64)

That:

[L.sub.1] = R[q.sub.n](1-[C.sub.1])/2[B.sub.1] (65)

[L.sub.2] = R[q.sub.n](1-[C.sub.2])/2[B.sub.2] (66)

[H.sub.1] = R[q.sub.n]([sin.sup.2] [[theta].sub.1] + [C.sub.1] [sin.sup.2] [[theta].sub.1])/2[B.sub.1] (67)

[H.sub.2] = R[q.sub.n]([sin.sup.2] [[theta].sub.1] + [C.sub.2] [sin.sup.2] [[theta].sub.1])/2[B.sub.2] (68)

Where [C.sub.1], [C.sub.2], [C.sub.3] and [B.sub.2] are as follows

[C.sub.1] = [v.sub.12]/2 + [E.sub.10] [v.sub.21]/2[E.sub.20] (69)

[C.sub.2] = [v.sub.21]/2 + [E.sub.20] [v.sub.12]/2[E.sub.10] (70)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (71)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (72)

Also from equation 45 'w' is as follow (equation, 73):

w = R ([T.sub.2] - [T.sub.1]([C.sub.2])/[B.sub.2] - cot g[theta]/R u) (73)

In figure 2, the variation of [T.sub.1]/[q.sub.n] and [T.sub.2]/[q.sub.n] in respect to [theta] for a parachute with radius of 320 cm and [[theta].sub.1] 10[degrees] is shown. The figures clearly show that by increasing the [theta] in the meridian direction the [T.sub.1]/[q.sub.n] increases and [T.sub.2]/[q.sub.n] decreases. As mentioned, the distribution of [q.sub.n] was assumed uniformly, so we could conclude that [T.sub.1] i.e. the axial force in meridian direction increases and [T.sub.2] i.e. the axial force in orbital direction decreases.

[FIGURE 2 OMITTED]

Conclusion

In this paper, method of force distribution [q.sub.n] on the surface of parachute is assumed uniformly. We could solve the equilibrium equations if an aerodynamic solution would be presented for method of force distribution [q.sub.n] in parachute.

In this study, the existing equations of an orthotropic shell were obtained, by using the hypothesis of membrane theory. Then, the equations were solved for a sphere shell as a parachute and it appears that the values of force in main axels are always positive and we can conclude that all of the forces into the parachute are tension force.

Also, the above equations are confirmed when the parachute has been reached maximum speed during falling and this is the quasi-static situation.
```Symbols

[E.sub.1], [E.sub.2]                   Axial Elasticity Modulus
[G.sub.12]                             Shear Modulus
[K.sub.1], [K.sub.2]                   Change of Curvature
[M.sub.1], [M.sub.2]                   Bending Momentum
[M.sub.12], [M.sub.21]                 Torsion Momentum
[N.sub.1], [N.sub.2]                   Shear Force
[q.sub.1], [q.sub.2]                   Component of External Force
z                                      Distance from Middle Layer
[theta]                                Angle in Meridian Direction
u, v, w                                Displacement in Axial Direction
[[alpha].sub.1], [[alpha].sub.2]       Coincident Vertical Vector on
Shell Surface
[q.sub.n]                              Component of Force in Unit
Vector Direction
[[epsilon].sub.1], [[epsilon].sub.2]   Axial Strain
[delta]                                Thickness of Shell
[v.sub.12], [v.sub.21]                 Poisson's Ratio
[[sigma].sub.11], [[sigma].sub.22],    Stress
[[sigma].sub.12]
[tau]                                  Torsion
[omega]                                Shear Strain
[e.sub.n]                              Unit Vector
V                                      Strain Energy
[T.sub.21], [T.sub.12]                 Shear Force
[phi]                                  Angle in Orbital Direction
[T.sub.1]                              Axial Force in Meridian
Direction
[T.sub.2]                              Axial Force in Orbital
Direction
```

Reference

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