trick/trick_sims/SIM_rocket

The following describes a Trick-based simulation of a model rocket under the forces of thrust, gravity and drag, using the parameters and equations shown below.

Thrust Force

In our model rocket simulation, we use an Estes E9 motor by default. It's thrust profile is shown below.

Name	Units	Value
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
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
rocket.motor.thrust_sample_count	--	29

Drag Force

The drag on the model rocket will be calculated using the drag equation shown below:

Symbol	Name	Units	Value
ρ	rocket.env.air_density	kg/m³	1.2
Cd	rocket.coefficient_of_drag	(unitless)	0.75
A	rocket.cross_sectional_area	m²	0.0015
v	rocket.velocity	m/s	Initially 0.0, Updated by integration of rocket-acceleration over time.

Gravitation Force

Symbol	Name	Units	Value
g	rocket.env.gravity	m/s²	-9.81
m	rocket.total_mass	kg	Initially [Eq#2]. Updated by integration of rocket-mass-rate.

Mass

As the rocket motor burns, it expels mass, at high speed. Initially the rocket motor contains all of its propellant mass. When it burns out, it contains no propellant mass.

Name	Units	Value
rocket.empty_mass	kg	0.1162
rocket.motor.total_mass	kg	0.0567
rocket.motor.propellant_mass	kg	0.0358

Initialization

Name	Units	Value
rocket.motor.exhaust_speed	m/s	Calculated by [Eq#1]
rocket.motor.total_impulse	N s	Calculated by integrating motor thrust over time.
rocket.total_mass	kg	Calculated by [Eq#2]
rocket.velocity	m/s	0.0
rocket.position	m	0.0
rocket.mission_time	s	0.0

[Eq#1] exhaust-speed = rocket.motor.total_impulse / rocket.motor.propellant_mass

[Eq#2] rocket.total_mass (@t=0) = rocket.empty_mass + rocket.motor.total_mass (@t=0).

State Derivatives

Name	Units	Value
rocket.mission_time_rate	s	1.0
rocket.thrust_force	N	Interpolation of E9 Engine Thrust function
rocket.mass_rate	kg/s	Calculated by [Eq#3]
rocket.drag_force	N	Calculated by [Eq#4]
rocket.gravity_force	N	Calculated by [Eq#5]
rocket.total_force	N	Calculated by [Eq#6]
rocket.acceleration	N	Calculated by [Eq#7]

[Eq#3] rocket.mass_rate = ‖ rocket.thrust_force ‖ / rocket.motor.exhaust_speed

[Eq#4] rocket.drag_force = - 0.5 × rocket.coefficient_of_drag × rocket.env.air_density × rocket.cross_sectional_area × rocket.velocity × ‖ rocket.velocity ‖

[Eq#5] rocket.gravity_force = rocket.total_mass × rocket.env.gravity

[Eq#6] rocket.total_force = rocket.thrust_force + rocket.drag_force + rocket.gravity_force

[Eq#7] rocket.acceleration = rocket.total_force / rocket.total_mass

State Integration

Name	Units	Value
rocket.mission_time	s	Numerical integration of rocket.mission_time_rate.
rocket.total_mass	kg	Numerical integration of rocket.mass_rate.
rocket.velocity	m/s	Numerical integration of rocket.acceleration.
rocket.position	m/s²	Numerical integration of rocket.velocity.

Results with Default Parameterization