¶ Recovery system design theory
¶ Chalmers Aerospace Society for Advanced Rocketry
The parachute system is a critical part of a rocket that, ironically, often fails. This section will therefore cover the basic mathematics behind the design of a pressure-actuated recovery system and explain how to determine the appropriate sizes for the drogue chute and main parachute.
¶ Parachute calculations
The formula used to calculate the parachute area is based on the drag equation. This equation provides the most accurate results when pressure drag dominates over skin friction drag — as is typically the case for parachutes or objects moving at high speeds. The drag equation is expressed as follows:
Where F is the drag force, the drag coefficient, the cross sectional area, the density of the air at the deployment altitude (can be found in a table), and the velocity of the moving body, relative to the air. For a freely falling body, the drag force can be set equal to the gravitational force, . From this relationship, the required parachute area can then be determined as follows:
Calculation example:
Nota that is shit for a parachute, but it is a way to create a margin of error.
The force will be lower in reality because the speed will have decreased du to higher air dencity at the lower altitude.
¶ Parachute ejection calculations
The ejection system is pressure-based, where the parachute chamber is sealed and gas pressure is used to separate the rocket sections. Therefore, the ideal gas law, along with adiabatic process equations, will be used to model and design the system:
Area of a cylinder:
The pressure at apogee can be looked up in a table, or calculated with the folowing formula:
The diffrence in pressure between the outside of the rocket and inside will then be:
Where is the atmospheric pressure at ground level, and is the pressure at apogee. The ejection system will be designed to increase the internal pressure of the rocket by . Consequently, the shear bolts can be designed to fail at a pressure difference of , resulting in a total pressure difference of under nominal conditions. This total pressure difference will be used to calculate the shear force acting on the bolts as:
And the total chamber pressure when the system activates is:
The first step of calculating the required canister volume is to approximate the volume with only the ideal gas law:
Where is the carbon dioxide cartridge pressure, and is the pressure for parachute ejection.
Where is the parachute chamber temperature before ejection and the temperature of the carbon dioxide if it expanded in vacum after the ejection. A better estimate can be made by taking a weighted average (this works because the relation between energy and temperature is linear + concervation of energy) of the temperature in the chamber and the previus calculated one:
This temperature will be an okay colder estimate of the final volume that will be calculated with the ideal gas law:
These calculations should provide a reasonable estimate. If you are not happy with it, just repeat to do the last 3 equations with the new volume ratio and it will converge. But the pressure will likely be higher in reality after the first step because the actual volume is smaller, which increases the temperature. Heat from the surroundings will also raise the temperature, further increasing the pressure. This should compensate for any variations in the initial temperature inside the parachute compartment.
Calculation example:
(air, carbon dioxide has but it gets higher with lower temperature so air should be fine)
¶ Cartridge puncture device
The task of puncturing a CO₂ cartridge is challenging because it requires a significant amount of force. A servo motor, combined with a cam mechanism that drives a needle, offers a practical solution by providing the necessary torque and controlled motion. The equations for designing such a system are presented below.
T is the torque, F is the force, r is the radius and is the angle along the cam. This equation gives a relationship between the force the cam can push and the torque that is required by the motor to push that force.
l is the distance the needle needs to move in order to properly puncture the CO2 cartridge membrane. This will give the minimum angle that the motor needs to move in order to push the nedle. Remember that a smaller angle gives a faster ejection time of the parachute. This formula does not account for friction, so some additional angle will be needed due to lost torque from the friction.
¶ Fuselage pressure calculation
The pressure that a cylindrical vessel can handle, given that the wall thickness is much smaller than the diameter of the cylinder:
where r is the radius of the cylinder and t is the wall thickness, can be calculated with the hoop stress formula:
where is the yield strength and SF is the safety factor. Remember that temperature is an important factor for the yield strength, so make sure that the expected operating temperatures do not affect the yield strength too much.
¶ shear bolt calculation
A simple formula for the diameter of a shear bolt, d, that shears under a force F, assuming the bolt is homogeneous, solid, and fails in single shear, is the following:
where is the yield strength.