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110 lines
4.9 KiB
Markdown
110 lines
4.9 KiB
Markdown
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The following describes a Trick-based simulation of a model rocket under the forces of thrust,
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gravity and drag, using the parameters and equations shown below.
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### Thrust Force
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In our model rocket simulation, we use an Estes E9 motor by default. It's thrust profile is shown below.
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| Name | Units | Value |
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|-------------------------------------|------------|------------|
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| rocket.motor.thrust\_sample\_times | s | 0.000, 0.010, 0.020, 0.030, 0.040, 0.050, 0.100, 0.110, 0.120, 0.130,0.140, 0.155, 0.165, 0.180, 0.190, 0.200, 0.215, 0.230, 0.250, 0.275, 0.300, 0.400, 0.500, 2.600, 2.700, 2.725, 2.750, 2.800, 100.0 |
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| rocket.motor.thrust\_sample\_values | N | 0.0, 0.1, 0.3, 0.5, 0.8, 1.5, 10.0, 12.6, 15.2, 17.8, 20.4, 23.0, 24.6, 25.0, 24.8, 24.4, 22.0, 17.0, 13.7, 12.6, 11.9, 10.7, 10.6, 10.5, 10.3, 10.0, 9.7, 0.0, 0.0 |
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| rocket.motor.thrust\_sample\_count | -- | 29 |
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### Drag Force
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The drag on the model rocket will be calculated using the drag equation shown below:
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| Symbol | Name | Units | Value |
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|--------|-------------------------------|------------|------------|
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| ρ| rocket.env.air\_density | kg/m³ | 1.2 |
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| Cd | rocket.coefficient\_of\_drag | (unitless) | 0.75 |
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| A | rocket.cross\_sectional\_area | m² | 0.0015 |
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| v | rocket.velocity | m/s | Initially 0.0, Updated by integration of rocket-acceleration over time.|
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### Gravitation Force
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| Symbol | Name | Units | Value |
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|--------|-----------------------------|-----------|------------|
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| g | rocket.env.gravity | m/s² | -9.81 |
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| m | rocket.total\_mass | kg | Initially [Eq#2]. Updated by integration of rocket-mass-rate. |
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### Mass
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As the rocket motor burns, it expels mass, at high speed. Initially the rocket
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motor contains all of its propellant mass. When it burns out, it contains no propellant mass.
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| Name | Units | Value |
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|-------------------------------|-------|--------|
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| rocket.empty\_mass | kg | 0.1162 |
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| rocket.motor.total\_mass | kg | 0.0567 |
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| rocket.motor.propellant\_mass | kg | 0.0358 |
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## Initialization
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| Name | Units | Value |
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|------------------------------|-------|-------|
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| rocket.motor.exhaust\_speed | m/s | Calculated by [Eq#1] |
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| rocket.motor.total\_impulse | N s | Calculated by integrating motor thrust over time. |
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| rocket.total\_mass | kg | Calculated by [Eq#2] |
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| rocket.velocity | m/s | 0.0 |
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| rocket.position | m | 0.0 |
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| rocket.mission\_time | s | 0.0 |
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[Eq#1] **exhaust-speed** = rocket.motor.total\_impulse / rocket.motor.propellant\_mass
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[Eq#2] **rocket.total\_mass (@t=0)** = rocket.empty\_mass + rocket.motor.total\_mass (@t=0).
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## State Derivatives
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| Name | Units | Value |
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|-----------------------------|-------|--------|
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| rocket.mission\_time\_rate | s | 1.0 |
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| rocket.thrust\_force | N | Interpolation of E9 Engine Thrust function |
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| rocket.mass\_rate | kg/s | Calculated by [Eq#3] |
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| rocket.drag\_force | N | Calculated by [Eq#4] |
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| rocket.gravity\_force | N | Calculated by [Eq#5] |
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| rocket.total\_force | N | Calculated by [Eq#6] |
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| rocket.acceleration | N | Calculated by [Eq#7] |
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[Eq#3] **rocket.mass\_rate** = ‖ rocket.thrust\_force ‖ / rocket.motor.exhaust\_speed
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[Eq#4] **rocket.drag\_force** = - 0.5 × rocket.coefficient\_of\_drag × rocket.env.air\_density × rocket.cross\_sectional\_area × rocket.velocity × ‖ rocket.velocity ‖
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[Eq#5] **rocket.gravity\_force** = rocket.total\_mass × rocket.env.gravity
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[Eq#6] **rocket.total\_force** = rocket.thrust\_force + rocket.drag\_force + rocket.gravity\_force
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[Eq#7] **rocket.acceleration** = rocket.total\_force / rocket.total\_mass
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## State Integration
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| Name | Units | Value |
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|------------------------|-----------|------------------------------------------------------------|
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| rocket.mission\_time | s | Numerical integration of rocket.mission\_time\_rate. |
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| rocket.total\_mass | kg | Numerical integration of rocket.mass\_rate. |
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| rocket.velocity | m/s | Numerical integration of rocket.acceleration.|
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| rocket.position | m/s² | Numerical integration of rocket.velocity. |
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## Results with Default Parameterization
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