Addendum to the PDR
At the time of the uploading of our completed document, and to the credit of the Milestone Review Flysheet which drew attention to it, we realized that we had failed to consider what the kinetic energy of our descending sections would be, and what the sizes of the necessary parachutes would have to be in order to have a safe recovery. This addendum to our Preliminary Design Review is to address these topics.
Add 1: Kinetic Energy Calculations
The determination of kinetic energy values for our descending sections of our rocket follows; the kinetic energy for any moving object is given by…
…for non-relativistic speeds.
During the initial descent, when our rocket is descending under drogue, the rocket is relatively integral (in one section) with a spent propellant weight of 40 lbs (mass equivalent to 1.35 slugs). The Kinetic Energy is then…
During the final stage of descent, after main chute and payload deployments, our rocket will consist of three sections; the main part of the rocket consisting of the booster, avionics, and payload section with a weight of 27.5 lbs, (mass equivalency of 0.86 slugs). The second section is the nosecone, carrying the GPSFlight and Walter, having a weight of 142 oz (mass equivalency of 0.28 slugs). The third section consists of the ASTRID (SMD) experiment with an estimated weight of 50 oz (mass equivalency of 0.10 slugs). The payload section will be descending at 18 ft/s while all other units will be descending at 22ft/s. The kinetic energy for each section then follows…
The kinetic energy of the first term appears to exceed the maximum allowed kinetic energy of 75 ft-lbs, this needs to be further analyzed section by section.
As was mentioned before, the first term consists of the booster section (having a weight of 189 oz, or a mass equivalency of 0.37 slugs), the avionics section (having a weight of 113 oz, or a mass equivalency of 0.22 slugs), and the payload section (having a weight of 140 oz – less the weight of the ASTRID payload - or a mass equivalency of 0.27 slugs). Since these three sections will be tethered together, all units descend at the same estimated speed of 18 ft/s. The kinetic energy for each section (after main chute deployment) is:
Even though it appears that we have five descending units, since three of them are tethered together, we have counted them as a single descending unit.
Add 2: Chute Size Determination and Descent Speed
Estimating the chute sizes for the descending sections follows the usual procedure; the drag force acting on an object, having a cross-sectional area A, moving with a velocity of v, through a fluid is given by…
…where it is standard practice to take the dimensionless drag coefficient to be 0.75, and kg per cubic meter is the density of air. For a steady descent rate, we require the drag force to balance the weight of the rocket, . Assuming a circular shape for the parachute , and solving for the radius, yields the following:
The RocSim results gave an initial descent speed of 22 ft/s for the first section, however, we believe this result to be inaccurate because it was determined using an overestimated mass for this section. The original chute size for this descending unit was determined using the following method, and yielded a result of a 130 inch diameter chute. However, we decided to use a chute of 144 inches in diameter.
Reverse calculating, yields a descent rate of…
…or a descent rate ~17 ft/s. We believe, based on actual descent observations made at the Black Rock dessert test flights, that this is closer to 18 ft/s, which is why we used this descent value in the above calculations.
Anyway, estimating the chute diameters for the nosecone section and Astrid follow; we take the mass of the nosecone assembly to be 4.1 kg, the ASTRID payload to be 1.4 kg, and for the requested descent speeds of 22 ft/s (or 6.8 m/s), we get…
Based on these calculations, we expect a nosecone chute diameter of around 60 inches, and an ASTRID chute diameter of about 36 inches.