Form with equations for small aerial vehicles (like Drones, remote control, small rockets). Not for designing an Airbus 380 (obviously)
Design of Small Aircraft
Basic Aerodynamics
Flight is governed by four fundamental forces
- Weight: gravitational force downward.
- Lift: counteracts the weight of the aircraft.
- Thrust: produced by the engine, moves the plane forward.
- Drag: opposes the forward motion.
Lift Equation
Lift is generated by the wing as it moves through the air.
Where:
is the lift force (N). is the air density (kg/m³). is the airspeed over the wing (m/s). is the wing area (m²). is the lift coefficient (dimensionless).
Drag Equation
Drag is the force opposing the motion.
Where:
is the drag (N). is the drag coefficient.
Lift Coefficient and Angle of Attack
The lift coefficient depends on the angle of attack
Where:
is the lift coefficient at . is the slope of the lift curve.
Stability and Control
Center of Gravity (CG)
The CG is the point where all the weight of the aircraft is concentrated. It should be near the center of lift to maintain balance.
Pitch Moment
The pitch moment generates longitudinal stability and is calculated as:
Where:
is the pitch moment (Nm). is the dynamic pressure . is the wing area (m²). is the wing chord (m). is the moment coefficient.
Propulsion and Power
Required Thrust
The thrust needed to counteract drag is:
Required Power
Power is the product of thrust and speed:
Propeller Efficiency
The efficiency of the propeller (
Design of Small Helicopters
Forces on a Rotor
Rotor Lift
The main rotor generates lift similar to a rotating wing:
Where:
is the angular velocity of the rotor (rad/s). is the swept area of the rotor ( ). is the rotor radius.
Drag Torque
The torque generated by the rotor’s drag is:
Where:
is the torque coefficient.
Helicopter Stability
Center of Lift and CG
The CG of a helicopter should be just below the center of lift of the main rotor to ensure stability.
Flight Control
- Cyclic: Controls the tilt of the rotor, affecting the flight direction.
- Collective: Controls the angle of attack of all rotor blades, affecting the altitude.
Design of Drones
Stability and Control
Quadcopters
Quadcopters use four propellers to generate lift and control direction. To maintain stability, two of the propellers spin clockwise and the other two counterclockwise.
Total Torque on a Drone
The torque generated by the propellers affects the drone’s rotation.
Where:
is the torque of each propeller.
Actuator Control
PID Control
Drones often use PID controllers to adjust motor inputs:
Where:
are the proportional, integral, and derivative constants. is the error at time .
Design of Small Rockets
Tsiolkovsky Rocket Equation
Describes the change in velocity of a rocket based on mass expulsion (momentum conservation principle):
Where:
is the change in velocity (m/s). is the exhaust velocity of the gases (m/s). is the initial mass of the rocket (kg). is the final mass after fuel consumption (kg).
Specific Impulse (Isp)
Where:
= gravitational acceleration on Earth ( )
Terminal Velocity of a Rocket
When the rocket reaches its maximum height, the velocity can be estimated with the following equation, considering aerodynamic drag and gravity:
Stability and Center of Pressure
Center of Pressure (CP)
For a rocket to be stable, the center of pressure must be behind the center of mass. This prevents the rocket from spinning uncontrollably.
Stability Coefficient
Stability can be evaluated with the coefficient
Where:
is the normal coefficient. and are the positions of the CG and CP. is the diameter of the rocket.
Thrust of a Rocket Engine
The thrust produced by a rocket engine is:
Where:
is the mass flow rate (kg/s). is the exhaust velocity of the gases (m/s). is the pressure of the gases at the exit. is the ambient pressure. is the exit area of the nozzle.
Maximum Height of a Rocket
The maximum height reached by a rocket can be estimated as:
Where:
is the initial velocity at launch (m/s). is the gravitational acceleration (9.81 m/s²).
Total Flight Time
The total flight time (ascent and descent) is:
Thrust-to-Weight Ratio
It is important to ensure that the thrust-to-weight ratio is greater than 1 for takeoff.
Where:
is the total thrust. is the weight of the aircraft or rocket.
Design of Vehicles
Total Aerodynamic Force (R)
Where:
= lift = drag
Center of Pressure
- The position of the center of pressure is calculated to analyze the moment equilibrium of a vehicle:
Where:
= pressure on the surface of the vehicle = coordinate along the longitudinal axis
Aerodynamic Moment (M)
Where:
= aerodynamic moment coefficient = reference area
Longitudinal and Lateral Stability
Pitch Moment Coefficient (C_M)
Where:
= moment coefficient for = angle of attack = angle of attack in equilibrium
Lateral Stability (Yaw Moment)
Where:
= yaw moment = wingspan
Longitudinal Static Stability Condition
A vehicle is stable when the moment coefficient
Sideslip Angle (
This angle is used in lateral stability:
Where:
= lateral velocity = total velocity
Calculation of Total Aerodynamic Drag
Parasitic Drag
Where:
= friction coefficient = wet area (surface in contact with airflow)
Induced Drag
Where:
= aspect ratio (wingspan squared over wing area) = elliptical efficiency of the wing
Power Required to Maintain Level Flight (P)
Where:
= total drag = speed
Thrust Coefficient (C_T)
Where:
= thrust of the engine
Flight Trajectories
Motion Equations for Atmospheric Flight
Ascent Trajectory
The vertical and horizontal motion in atmospheric flight can be modeled by the ascent equations:
Where:
= ascent angle = thrust = drag = lift = vehicle speed
Rate of Ascent
Where:
= rate of ascent = vehicle speed = ascent angle
Orbital Trajectories
Orbital Velocity
Where:
= universal gravitational constant = mass of the central body (e.g., Earth) = distance from the center of mass of the central body
Specific Orbital Energy (E)
Where:
= specific energy (energy per unit mass)
Kepler’s Orbit Equation (Body Orbit Equation)
Where:
= semi-major axis = eccentricity of the orbit = true anomaly angle
Escape Velocity
The minimum velocity for a vehicle to exit the gravitational influence of a body:
Orbital Transfer Equations
Hohmann Transfer
Used to change from one circular orbit to another with minimum energy:
Velocity in the first orbit (perigee)
Velocity in the second orbit (apogee)
Where:
= radius of the first orbit = radius of the second orbit
Orbital Plane Change
The change in inclination of an orbit requires a velocity change perpendicular to the direction of motion:
Where:
= orbital velocity = change in inclination