For the Hermes Pathfinder vehicle, the relevant dimensions are:
| Variable | Length (in inches) | Description |
|---|---|---|
| hmp | 57.4 | Height of the mission package from nose cone tip to the end of the airframe |
| lmp | 14.5 | Depth into the cup |
| lfc | 7.6 | Depth of booster coupling section |
| hfc | 86.23 | Total height of booster including coupling section. |
| RB | 7.5 | Height of the main deployment bag, not including flaps. |
| DM | 114 (9.5 ft) | Diameter of the main parachute |
| RM | 80.6 | Height of the main parachute (found using 0.707 aspect ratio): RM = DM*0.707 |
| SM | 120 | Length of shroud lines on the main parachute (not the vertical length). |
| RD | 31 | Height of the drogue parachute (measured on test article 2/15/2018) |
| SD | 100 | Vertical length (accounts for confluence angle) of the drogue shroud lines. Taken from the DGB parameters spreadsheet. |
| FD,D | 17 | Forebody diameter used for drogue parachute shock cord calculation. Please see the extended note on calculation below. |
| FD,M | 29.5 | Forebody diameter used for main parachute shock cord calculation. Please see the extended note on calculation below. |
| DD | 38 | Disk diameter of the drogue parachute. Taken from the DGB Parameters spreadsheet. |
| HTD | 2.7 | Height of the Tender Descender, including 1.5 quicklinks. (Since two Tender Descenders will be used in the calculation below, this will yield the correct total length) |
| WA,E | 5.4 | There is no webbing between junctions A and E, just two Tender Descenders. Thus, this is calculated by 2*HTD |
| WE,F | 13.5 | This is the total height of the main deployment bag (RB) plus the 6 in for the straps on the top of it.. Placing the junction between the tender descender (TD) and the main chute load paths as far inside the mission package as possible reduces the likelihood that the main parachute will enter the airstream prior to the designated release altitude of 2,000 ft. |
Constraints and Calculations v2.0
These calculations and constraints are only a preliminary estimation, with the following assumptions:
- Neglect drogue deployment bag as of right now, we will take this into account on calculations further down the line. Or alternatively, there will likely be enough wiggle room with all the lengths that we don't need to worry about it.
- Vehicle dimensions are subject to change
- Main parachute shroud line length are subject to change–I also don't think this value accounts for the confluence angle, so it will actually be a little lower than advertised.
| Constraint | Description |
|---|---|
| WC,D > RB | The line connecting the top of the main chute canopy to the inside of the deployment bag must be long enough such that the main parachute is not stuck inside the deployment bag upon full line extension. |
| WA,H > WA,E,F,G + RD + SD + lfc | When lines between the mission package and fin can are fully extended, the drogue parachute must be within the airstream. |
| WA,H > lmp + hmp + lfc | Assuming full line extension, the mission package and booster section should not hit one another during descent. This neglects the effect of drag on these sections and their respective masses. |
| WA,E,F,G - lmp > 8.5*FD,D- SD | This constraint reduces the impact of forebody wake on the drogue parachute during drogue descent. Please see the extended note below on the calculation of forebody diameter. We subtract the vertical height of the shroud lines to reach a total canopy distance of ~9 forebody diameters.1 |
| WA,E,F < WA,B + SM + RM + WC,D | Prior to firing the tender descender, the load should go through the shorter (TD) path. Note that this is automatically satisfied by virtue of the fact that the main shroud lines alone are significantly longer than the deployment bag. There is no need to check this constraint. |
| WA,B > lmp + 6*FD,M - \sqrt{S_{M}^2-(D_{M}/2)^2} | "[T]he parachute should be ejected to a distance equivalent to more than four times–and preferably six times–the forebody diameter" (Knacke, 5.2.2). Please see the extended note below on the calculation of the forebody diameter. We subtract the vertical length of the shroud lines to reach a total canopy distance of ~6 forebody diameters. |
Note on Drogue Forebody Diameter
As noted in Hermes Descent Dynamics, it is difficult to predict the falling orientation of the mission package and booster during drogue descent. For this reason, we evaluate the worst case possibility (supersonic descent is already somewhat of a worst case scenario, so we have multiple methods of redundancy): the booster falls horizontally below the drogue parachute. In this scenario, the actual forebody area (FA,D) is calculated by multiplying the drogue diameter by the booster diameter, yielding approximately 227 in2. The effective forebody diameter is calculated below via the equivalent circle technique presented in Knacke section 5-23 [2]:
F_{D,D} = \sqrt{\frac{4F_{A,D}}{\pi}}
where FA,D = 227 in2
FD,D = 17 in
Similarly, the main parachute forebody area (FA,M) is calculated as 684 in2. The effective forebody diameter is calculated:
F_{D,D} = \sqrt{\frac{4F_{A}}{\pi}}
where FA = 684 in2
FD,D = 29.5 in
| Webbing Length Variable | Length (in) | Factor of Safety | Reasoning |
|---|---|---|---|
| WA,E | 5.4 | 1 | Described above–not a webbing length. |
| WE,F | 13.5 | 1 | Described above–not a webbing length. |
| WF,G | 40.1 | 1 (Try to keep around 1) | |
| WA,B | 85.9 | 1 (Try to keep around 1) | |
| WC,D | 75 | 10 (Suggested around 10) | In subsonic flow, upstream disturbances could disrupt the airflow if there is only minimal length on this webbing. |
| WA,H | 395 | 2 |
Resources
[1] Clark & Tanner, "A Historical Summary of the Design, Development, and Analysis of the Disk-Gap-Band Parachute"
[2] T.W. Knacke, Parachute Recovery Systems: Design Manual