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