The Design Space
Our redesigned piston must have the same form factor as a McMaster piston to allow for easy descope. Given the team's Fall 2017 semester experience with tie-rod pistons, we elect to continue using this style.
DTEG Requirements
As of 1/4/2018, the latest edition of the Spaceport America Cup's Design Test and Evaluation Guide has the following requirements for SRAD pressure vessels. These requirements can be read in more detail here (add link).
4.2.2 DESIGNED BURST PRESSURE FOR METALLIC PRESSURE VESSELS
4.2.4.1 PROOF PRESSURE TESTING
4.2.4.2 OPTIONAL BURST PRESSURE TESTING
You can find a complete list of DTEG requirements that affect the Recovery system on the Hermes Recovery System page.
Desired Performance
The piston must be able to supply enough force at its operating pressure to break the shear pins with a 2x safety factor, which is the safety-critical guideline for parachute components presented by NASA (Section 3.3.1.5). As of January 4, 2018, we are designing for 180lbs of shear pins, and thus the piston must supply 360lbs. The same source (Section 3.3.1.6) dictates a design burst pressure factor of 2x the maximum design pressure, which aligns with DTEG requirement 4.2.2. [5] Thus, we expect the piston to burst when it supplies 720 lbs. Here, we make use of thin-walled pressure vessel theory [2], paraphrased below:
Neglecting end effects, the limiting factor will be the hoop stress in the piston bore:
\sigma_{hoop} = \frac{pR}{t}
Given Aluminum 6061-T6 as the material, which typically has a tensile yield strength of approximately 276 MPA (this analysis neglects the internal temperature of the piston due to the gas produced by the combustion of black powder. A transient thermal spike could degrade material properties when the piston is pressurizing, but we assume that the magnitude of energy released is negligible compared to the thermal mass of the aluminum). The tensile yield strength can be used to calculate the design burst pressure. For this preliminary analysis, the wall thickness is chosen to be a 0.25x reduction in that of the previously-qualified piston bore (part 6491K254 on McMaster):
t_{new} = \frac{1}{4}*t_{6491K254} = \frac{1}{4}*0.25in = 0.0625 in
\sigma_{burst} = \sigma_{tensile \ yield} = 276*10^6 Pa \approx 40030 psi
Applying P = F/A where F is 720 lbs at burst and A is the area of bore, we find:
\sigma_{burst} = \frac{F*r}{A*t}
Then, assuming a circular bore, area takes the form A = πr2 and we can solve for the radius of the cylinder.
r = \frac{F}{t*\sigma_{burst}*\pi}
At this stage of the analysis, radius can be constrained further: another requirement of the piston is that it cannot break the shear pins prematurely due to an internal build-up of pressure caused by the altitude difference. Between 4,245 ft (the altitude of Truth or Consequences, NM) and 152,945 ft ASL (a simulated upper bound on performance as of January 4, 2018), the pressure difference is approximately (given by the 1976 Standard Atmospheric Calculator using no temperature offset) -86600 Pa ≈ 12.56 psi.
Applying a 2x safety factor to premature separation (again in accordance with safety-critical recovery components as dictated by NASA), we calculate the maximum allowable radius of the piston [5]:
F_{sep} = P_{diff}*A_{bore}
F_{sep} = P_{diff}*\pi*r_{bore}^2
\sqrt\frac{F_{sep}}{P_{diff}*\pi} = r_{bore}
\sqrt\frac{0.5 * 180lb}{12.56 psi * \pi} = r_{bore}
rbore, max = 1.51 in
Terminology
Resources:
The following resources are useful materials for learning about pressure vessel and piston theory:
[1] Jeff Hanson, Texas Tech: Intro to Thin Walled Pressure Vessels
[2] University of Colorado, Boulder: Thin-Walled Pressure Vessel Theory
[3] NASA Aerospace Pressure Vessel Safety Standard, 1974: NSS/HP-1740.1
Note that this standard was cancelled in July, 2002.
[4] Aerospace Corporation, Operational Guidelines for Spaceflight Pressure Vessels
[5] NASA, Structural Design Requirements and Factors of Safety for Spaceflight Hardware