Responsible Engineers:
Max Kwon (2022) maxkwon@mit.edu
Pedro Pavao (2022) ppavao@mit.edu
Purpose
The Staging Cone serves as an attachment interface between the booster and the sustainer stages of Phoenix. After burn of out the booster motor, drag forces should be larger on the booster due to its larger freestream area causing the sustainer should passively separate from its resting position on the Staging Cone. Recovery soft goods are stored in the Staging Cone and are released when the piston separates the Staging Cone from the booster airframe.
Requirements
- Must interface with the Sustainer nozzle
- Must withstand the weight of the Sustainer (during flight)
- Must be able to be machined using tooling in the Deep (MIT Machine Shop)
Design Details
Left: Isometric view of the staging cone. Right: Cross-sectional view of the staging cone
Materials:
Aluminum 6061-T6 Round Stock (Cone)
Aluminum 6061-T6 Tube Stock (Base)
#8-32 Fasteners (18-8 Steel)
Software Used:
SolidWorks 2019 (CAD and FEA)
Design and Analysis
Design Brief:
The Staging Cone is designed to be able to attach by way of shear pins to the booster airframe and to the sustainer by a geometrical fit. The upper cone (henceforth the cone section) of the Staging Cone is to be the exact geometry of the sustainer nozzle (expansion section) so that it sits properly with no tilt and such that the entire wall area inside the nozzle is in contact with the cone. This calls for fine, and currently unspecified, tolerances. If manufactured correctly, the top lip of the base section of the cone should also be in contact with the bottom of the sustainer nozzle, dividing the weight of the sustainer between the base and cone sections of the Staging Cone. The number of bolts being used to attach the cone and base sections was derived assuming worst case tolerances and all of the sustainer's weight is on the cone section.
Hand Calculation Analysis:
The spreadsheet that completed these hand calculations can be found here: https://docs.google.com/spreadsheets/d/1kWIMJ8-9FpM9AI0ZJvyIqJiwlM8lQ-BzgPI3EAgFaO4/edit#gid=0. If you would like edit access, contact Max Kwon (maxkwonkorea@gmail.com or maxkwon@mit.edu)
There were three failure modes that were taken into account when modifying the design of the Staging Cone.
1) Exceeding the shear strength of the bolts
To run calculations on the shear strength of the bolts, it was assumed that all of the sustainer's weight would be on on the cone section and reacting against the bolts. This would yield the most conservative estimate. The forces accounted for on the cone were the weight of the sustainer under a maximum acceleration of 20 G's and drag force at Max-Q (from RASAero). To find the shear stress, this total force is this divided by the minor area of a bolt multiplied by the number of bolts to be used. To find the equivalent or von mises stress, this stress is then multiplied by the square root of 3. This comes from the equation of von mises stress under the condition of shear only.
2) Exceeding the bolt tear out strength of the aluminum
To calculate the tear out stress from the bolts reacting against the aluminum of the base (smaller thickness), that same derived total sustainer force is then divided by the projected area of the bolt onto the walls of the hole. This is A = minor diameter * wall thickness. This area calculation was used knowing that there are more correct and conservative estimation methods but it was felt that this was sufficient and simple. After finding this stress, a stress concentration factor of Kt = 3 was multiplied. Without this, stress calculations will be severe underestimates.
3) Exceeding the maximum moment the cone could impart on the sustainer
To calculate the pure moment that would be imparted by the cone section onto the sustainer in the event of non-zero angle attack, a maximum angle of attack is specified. In the case of my calculations, it was 5 degrees. Flight conditions including velocity and dynamic pressure are specified at their maximum values (these values are coupled). Experimental data on the drag coefficient of a cylinder based on the Reynolds number of the flight was then used with a compressibility factor given that the rocket will be supersonic to find the drag on the projected area of the rocket. This force is then assumed to be concentrated at half the length of the rocket and a moment is calculated from there.
This moment can then be used to find the stress on the cone after calculating the minimum second moment of area of the cone. We use the minimum since a pure moment does not have one reference position and therefore this minimum point is where failure would occur. The equation for a circular ring is posted below. Stress can be found using max_stress = M*r_max/I_min where r_max is the maximum distance away from the center line of the cone, M is the moment and I_min is the minimum second moment of area.
In progress
Changes from Demo II
The main design change from Demo II is that the Staging Cone (Phoenix) will be made in two separate pieces. This was done for the purpose of making machining faster and more practical. It may also have the unintended (but welcome) consequence of making the part cheaper. This is due to the fact that there is less material that has to be cut away since the smaller diameter cone is made from a separate piece of stock from the larger diameter base (booster airframe retention).
Manufacturing
Preliminary Plan:
- Take both stock and face both sides on the lathe
- Turn outer profile (for both)
- Bore our inner profile (for both)
- Use Radial Indexer on the mill to drill holes (in both)
Testing