Research on the influence of length-width ratio on cruciform parachute airdropping performance

Mathematical model

In this work, the structural domain and flow field are described by Finite Element Method (FEM). Since the entire calculation process does not involve heat transfer issues, and the FEM based on Lagrangian description can naturally describe the change of the material boundary, the entire calculation process only needs to solve the momentum equation. Then, momentum conservation equations of the flow field and structural domain are show as following.

ρ f v i , t + ρ f v i , j c j = σ j i , j + ρ f b i

(1)

ρ s v i , t = σ j i , j + ρ s b i

(2)

The canopy is in a folded state at the initial stage, and a large number of random folds will be formed during the canopy inflation, which makes it difficult to use body-fitted grids to discretize the space inside the parachute. Therefore, the structural domain grids are directly inserted into the flow field to avoid the establishment of body-fitted flow field grids. Based on the Lagrangian description, the problem of information coupling between the two domains can be transformed into a contact problem. The penalty function method commonly used in the explicit finite element method to realize the coupling.

First at all, the node of flow field element is defined as slave node n S and the node of structural domain element node is defined as master node m S . Then, the master node m S and element S i closest to the slave node n S are searched, and the contact point C is obtained by equation (3) (Figure 1).

{ ∂ x c ∂ ξ ( ξ c , η c ) ⋅ [ x n S − x c ( ξ c , η c ) ] = 0 ∂ x c ∂ η ( ξ c , η c ) ⋅ [ x n S − x c ( ξ c , η c ) ] = 0

(3)

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Figure 1.

Determine the location of the touch element node.

The dot product l ( l = n i ⋅ [ x n S − x c ( ξ c , η c ) ] ) is obtained by the vector [ x n S − x c ( ξ c , η c ) ] and the external normal unit vector n i of point C . If the vector dot product l is negative, the penetration occurs. Then a normal contact force f c ( f c = − l k i n i , where k i is the stiffness factor of the triangular structural element) taken as coupling force between the flow field and the structure is applied. The coupling force f c of each node of the structural domain element is obtained by interpolation of the shape function N I :

f c = − ∑ I = 1 3 f c I = ∑ I = 1 3 N I ( ξ c ) f c

(4)

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Figure 2.

Schematic diagram of flow field reconstruction and information update.

In addition, in order to ensure that the flow field always wraps the structural domain and moves with it, the transformation matrix T of Computer Graphics is used to realize the flow field movement equation (5). Here, T i j ( i , j = 1,2,3 ) mainly implements grid scaling, rotation, symmetry and shear transformation, [ T 14 T 24 T 34 ] T implements projection transformation, [ T 41 T 42 T 43 ] implements displacement transformation on X, Y, and Z axes, and T 44 implements overall expansion and contraction transformation. Because the parachute is connected with the projectile through the anti-rotation joint, the random rotation of the projectile in the air has no effect on the parachute system. Instead, the parachute system can control the trajectory change of the projectile through the joint. At the same time, in order to reduce the amount of calculation, this work only defines that the projectile has displacement degrees of freedom on the Y-axis and Z-axis related to the trajectory and trajectory angle. This simplification has no effect on the displacement and rotation of the parachute system in three-dimensional space, nor does it affect the deceleration and stability analysis of the parachute system. After the above reasonable simplification, the movement of the flow field will not involve transformations such as scaling, rotation, and shear. Therefore, the final transformation matrix T is mainly determined by the displacement of the projectile’s gravity center.

T = [ T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44 ] = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 y t 0 + Δ t − y t 0 z t 0 + Δ t − z t 0 1 ]

(5)

Case study

Generally, the design manuals recommend the value of LWR to be 3.0–3.8.^1,2 But, in actual projects, the value of LWR is almost large. As described in the design manual, the values of LWR in the two actual cases are 3.9 and 4.6, respectively. ² The main purpose of this work is to study the influence of the design parameter LWR on deceleration performance and terminal trajectory stability. Therefore, the value range of LWR is expanded and two limit values are added in this work. Here, five different LWR cruciform parachutes are used for research (Figure 3), and the LWRs are 2.28, 3.06, 3.78, 4.5, and 5.22, respectively. The remaining design parameters and material parameters of the above five models are exactly the same. The total area of all cruciform parachutes is 1.28 m², the fabric elastic modulus is 0.42 GPa, the thickness is 0.4 mm, and the air permeability of the fabric is 0.327 m/s under the pressure difference of 49 Pa (Russian standard). The parachute cord length is 1.4 m (total 12 cords), and the elastic modulus of cord is 0.6 GPa.OPEN IN VIEWER

Subsequently, this work establishes finite element models of the folded parachute system based on the above geometric parameters, in which the canopies are discretized by triangular elements with same size. And the number of canopy elements of different designs is between 7900 and 8500. The flow field is discretized by 930,600 hexahedral elements. The flow field and the structure are assembled by interpenetrating with each other (Figure 4). The projectile mass is 7.5 kg, the initial trajectory angle is 60°, the initial speed is 300 m/s, and the flight time of the projectile in the air is 3.5 s. In addition, this work also considers the impact of downwind (+Y direction) and upwind (-Y direction) with 8 m/s on deceleration performance and terminal trajectory stability.OPEN IN VIEWER

Analysis of the terminal trajectory of the projectile

Figure 5 shows the comparison of the projectile’s terminal trajectory and resultant displacement. The host graph is the comparison of the projectile’s terminal trajectory, and the upper left sub-graph is the comparison of the projectile’s resultant displacement.OPEN IN VIEWER

From Figure 5(f) (upper left sub figure), it can be found that the resultant displacement of the model with an LWR of 5.22 (hereinafter referred to as Model 5.22, other models are named similarly) is 107.48 m, while the resultant displacement of Model 2.28 is 82.41 m. The maximum difference between those two models can be up to 25.07 m. And the former is almost 1.3 times than that of the latter, while the resultant displacements of the other models are between those two models. And the resultant displacement increases with the increase of the design parameter LWR. This indicates that under the condition that the canopy area, material and other designs are exactly the same, the smaller the LWR, the stronger the deceleration ability of the cruciform parachute. From the perspective of the influence of the external wind field on the resultant displacement of the projectile (Figure 5(a)–€), the resultant displacements are in order of downwind conditions, no wind conditions and upwind conditions. The maximum difference of the resultant displacements of all models does not exceed 15m. Therefore, the influence of the external wind field on the resultant displacement is less than the design parameter LWR.

From the terminal trajectory comparison shown in the host graph of Figure 5, it is found that LWR has a more obvious impact on trajectory stability. Since the space coordinates of different models are different at 3.5 s, the maximum horizontal displacement difference at this time cannot be used as the basis for trajectory stability analysis. Therefore, this work uses 70 m as the base height for comparison (Table 1). From the analysis of the maximum horizontal displacement difference, it can be found that the influence of the external wind field on the trajectory stability decreases with the growth of LWR, and the time required to reach the height of 70 m also proves that the deceleration characteristic of the cruciform parachutes decreases with the increase of LWR. On the other hand, the influence of the external wind field on the terminal trajectory of Model 2.28 began to appear after 0.06 s, while the influence of that on the Model 5.22 only began to appear after 0.18 s. This effect becomes more and more obvious over time. It is foreseeable that when Model 2.28 finally landed, its horizontal displacement difference will further expand.

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Table 1.

Comparison of the time required to reach 70 m height and the maximum horizontal displacement difference.

	Model 2.28	Model 3.06	Model 3.78	Model 4.5	Model 5.22
Time required to reach 70 m (s)	2.98–3	2.72–2.9	2.26–2.28	1.94–1.98	1.46–1.56
Maximum horizontal displacement difference (m)	29.08	21.01	20.23	14.46	13.49

In addition, it is obvious from the trajectory comparison shown in Figure 5. As the LWR increases, the terminal trajectory curve changes more smoothly. The Model 2.28 trajectory curve (Figure 5(a)) has slight mutations locally, which leads to irregularity in the trajectory comparison of all models under windless conditions (Figure 5(f)). This is mainly due to the poor stability of Model 2.28 itself.

Analysis of the deceleration characteristics of the projectile.

Figure 6 shows the comparison of the resultant acceleration and acceleration components of the projectile. The host graph is the comparison of the resultant acceleration of the projectile, and the upper left sub-graph is the comparison of the acceleration components in the Z direction and the Y direction of the projectile.OPEN IN VIEWER

It can be found that the acceleration change trend of the same model under windless, downwind and upwind conditions is basically the same (Figure 6(€(e)). Due to the additional drag generated by the wind field, the upwind condition causes the projectile’s overload to be slightly higher than the windless condition and the downwind condition. Therefore, the inherent deceleration characteristics are extremely limited by the external wind field. Meanwhile, it can be found that the canopy area of all models is exactly the same, and the projectile reaches its maximum acceleration almost at the same time in Figure 6(f). And the maximum resultant acceleration values (arranged from small to large according to LWR) are 3026.4 m/s², 2744 m/s², 2495.6 m/s², 225.3 m/s², and 2046.9 m/s², respectively. The smaller the LWR value, the greater the maximum overload that the projectile has, and the faster the overload decreases. However, as the parachute system enters the steady dropping stage after 0.6 s, the acceleration curve of each model does not change much. This shows that the smaller the LWR, the stronger the deceleration ability of the cruciform parachute, and the higher the deceleration efficiency.

According to the flow field results at the time of maximum overload (0.02s) of each model projectile reached, it can be found that Model 2.28 is the first to complete the canopy inflation. The canopy arm of Model 2.28 is the shortest (only 1.366 m), but the canopy can form an efficient aerodynamic deceleration surface. That the outflow of air inside the canopy is blocked, forming a high-pressure area inside, and the low-pressure area formed by the external turbulence causes the Model 2.28 to form a relatively large aerodynamic drag in a short time (Figure 7(a)). As the LWR increases, more and more air inside the canopy can flow out from the gaps between canopy arms. Therefore, it is difficult to form a high-pressure zone again (Figure 7(a)–(e)). At the same time, the increase in LWR means that the longer the parachute arm with the same area (Figure 3). The long canopy arms will further hinder the surrounding air from entering the inside of the canopy at the bottom. Even a local high-pressure area is formed outside the canopy, which further extends the inflation time of the cruciform parachute. The smaller the LWR, the shorter the inflation time of the cruciform parachute, and the higher the deceleration efficiency. However, the smaller LWR design would lead to greater overload and stress, which will put forward higher requirements for the strength design of the whole system.OPEN IN VIEWER

Figure 8 shows the comparison of the resultant velocity and velocity components of the projectile. The host graph is the comparison of the resultant velocity of the projectile, and the upper left sub-graph is the comparison of the velocity components on the Z direction and the Y direction. In the initial stage of deceleration, the projectile speed of the same model in the downwind conditions is slightly higher, and the speed in the upwind conditions is slightly lower (Figure 8(a)–(e)). Meanwhile in the early stage of deceleration, the smaller the LWR, the higher the deceleration efficiency of the cruciform parachute, and the lower the speed of the projectile (Figure 8(f)). However, after 0.6s, as the deceleration process enters the steady dropping stage, the above-mentioned speed curve changes gradually stabilize, and the difference gradually becomes smaller. At about 3.5 s, the steady dropping speed of all models remains at 12–13.8 m/s. In the end, the speed curves of all models show that the external wind field has a limited effect on deceleration, but the steady dropping speed is positively correlated with LWR.OPEN IN VIEWER

Analysis of cruciform parachute stability

Figure 9 shows the comparison of the oscillation angle and trajectory angle of the cruciform parachute. The host graph is the comparison of the oscillation angle and the sub-graph in the upper left is the comparison of the trajectory angle of the cruciform parachute.OPEN IN VIEWER

From the comparison of the oscillation angle, it can be found that the oscillation angle change frequency of all models is higher in the early stage of deceleration. However, as the cruciform parachute enters the steady dropping stage, the oscillate frequency decreases, and the external wind field has little effect on the change of the cruciform parachute oscillation angle (Figure 9(a)–(e)). The design parameter LWR has a greater influence on the oscillation angle amplitude. The larger the LWR, the smaller the oscillation amplitude of the canopy during the steady dropping stage (Figure 9(f)). This shows that the stability of the canopy itself is positively correlated with the design parameter LWR.

The changing law of the trajectory angle of the canopy is also the same as the changing law of the oscillation angle. The trajectory angle of all models changes from the initial 60°–90°. In this process, the influence of the additional aerodynamic drag caused by the external wind field on the deceleration characteristics also becomes weaker. This also explains the reason that the velocity or acceleration difference caused by external wind gradually becomes smaller in steady dropping stage (Figures 6 and 8). However, the difference in the terminal trajectory of the projectile does not decrease (Figure 5), and the influence of the external wind field on the terminal trajectory can be divided into indirect and direct influence. In the early stage of parachute dropping, due to the existence of an initial trajectory angle other than 90°, the external wind field will form additional aerodynamic drag on the tangent direction of the trajectory, which directly affects the deceleration characteristics and indirectly affects the trajectory. In the steady dropping stage, the additional aerodynamic drag gradually disappeared and the external wind field hit the canopy vertically, which directly affects the trajectory.

According to the above analysis, it can be found that the design parameter LWR has a greater influence on the stability of the cruciform parachute. The canopy stability will further affect the trajectory stability of the projectile. This also explains the reason that the curve changes of Model 2.28 shown in Figure 5(a), Figure 6(a) and Figure 8(a) are not smooth. However, this conclusion only applies to models with smaller parameter of LWR.

According to the results of Model 3.78 in upwind conditions (Figure 9(c)), it can be found that the oscillation angle and trajectory angle of the canopy change drastically at 2.6s. This change has almost no effect on the trajectory stability and deceleration characteristics of the projectile. The reason for this dramatic change is that there was a partial twist at the bottom of the canopy at that moment. Meanwhile, Model 3.78 has a larger gap between the canopy arms, and the high-pressure area at the top of the parachute cannot expand downward to the bottom (Figure 10). The slack bottom cannot be restored to its original shape in a short time, and then the asymmetric flow at the bottom aggravates the instability of the cruciform parachute. The Model 4.5 and Model 5.22 also have bottom slack or local twist in different working conditions. However, their arms are relatively longer and the windward area at the bottom is smaller, and the slack bottom of the canopy has limited influence on its own stability.OPEN IN VIEWER

Therefore, the smaller the design parameter of LWR, the worse the stability of the cruciform parachute. While the influence of the parachute stability on the trajectory stability and deceleration decreases with the increase of LWR.

Quantitative analysis of deceleration characteristics of cruciform parachute

In order to verify the accuracy of the numerical method, Model 4.5 is used for airdropping experiment. The experiment conditions are the same with the calculation conditions shown in Analysis of cruciform parachute stability section. But the direction and speed of external wind field are uncontrollable, and the wind speed near the ground is about 5 m/s in airdropping experiment. In addition, due to the limitations of the experiment conditions and consideration of safety factors, only the steady dropping speed is obtained based on the high-speed camera records. The experimental value of 13.5 m/s is relatively close to the calculation value of Model 4.5 (12.6–13.5 m/s).

In order to further analyze the influence of other factors on the deceleration characteristics, this work continues to take Model 4.5 as the research object. The deceleration processes of the projectile with different initial trajectory angles and different external wind field are calculated. From the calculation results shown in Figure 11, it can be found that the initial trajectory angle and wind speed have almost no effect on the deceleration of the projectile. The additional drag caused by the wind field only has an impact on the early stages of deceleration. When the parachute system enters the steady dropping stage, the speed difference is further reduced. The final speed difference is even smaller than that caused by parameter of LWR (Figure 8(f)).OPEN IN VIEWER

Based on the results shown in Figure 6, Figure 8, and Figure 11, it can be found that the external wind field has a limited impact on the deceleration ability of the cruciform parachute. LWR is another design parameter that has a greater impact on the deceleration performance of the cruciform parachute after the canopy area. Finally, the correction equation (6) of drag coefficient C d related to LWR is fitted according to the calculation results of the steady dropping stage in windless conditions (Figure 12).

C d = 2.75 − 1.4 L W R + 0.34 L W R 2 − 0.03 L W R 3 ( 2.28 ≤ L W R ≤ 5.22 )

(6)

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Figure 12.

Fitting of correction equation of drag coefficient.