How is Takeoff Rotation Speed Calculated? A Simple Python Model

 Why does the same aircraft need a longer runway at Leh than Delhi? To understand this, let's break down what actually happens in an aircraft's takeoff roll.

As an aircraft accelerates down a runway, there is a precise moment when the pilot pulls back on the controls and the aircraft takes flight. This speed is not arbitrary but rather is calculated based on factors including runway length, wind speed and direction, and aircraft weight.

V stands for 'vitesse', which is the French word for speed. 

V1 (Decision Speed): V1 refers to the maximum decision speed during takeoff. In the event of an emergency, the pilot can only stop the aircraft safely, within the runway, before V1. If the aircraft is past V1, the pilot must continue takeoff even with an emergency such as an engine failure.

VR (Rotation Speed): VR is the speed at which the pilot begins to rotate the aircraft. At this point, the nose lifts, increasing the wings angle of attack, and thus allowing the aircraft to generate lift.

V2 (Takeoff Safety Speed): V2 is the speed an aircraft must achieve after liftoff to maintain a safe climb. It provides a margin over stall speed and ensures the aircraft continues climbing safely, even with engines failures.

While all three speeds are critical, this article focuses on VR- Rotation Speed.

To understand how VR is determined, we must first understand how an aircraft generates lift.

The Lift Equation

L = ½ ρ V² S Cₗ, where:

• ρ → air density
• V → speed
• S → wing area
• Cₗ → lift coefficient

(1) Air Density (ρ):

The density of air which affects how much lift is produced. Higher density (cold weather / low altitude) leads to more lift.

(2) Speed (V):

The speed of the aircraft relative to the surrounding air.

(3) Wing Area (S):

The total surface area of the wings. Larger wings can generate more lift. This is why the Boeing 777 has large wings.

(4) Lift Coefficient (Cₗ):

A value that represents how efficiently the wing generates lift, depending on the angle of attack and flap configuration.

Derivation

L = ½ ρ V² S Cₗ

L = W

∴ W = ½ ρ V² S Cₗ

⇒ 2W = ρ V² S Cₗ

⇒ V² = 2W / (ρ S Cₗ)

⇒ V = √ (2W / (ρ S Cₗ))

Explanation

This equation provides the minimum speed for the aircraft to generate enough lift to overcome its weight.

In practice, aircraft rotate at a slightly higher speed, known as VR, to provide a margin, thus preventing a stall.

Simulation Overview

• This simulation calculates the VR (rotation speed) of an aircraft based on several factors including mass, wing area, wind speed, etc.
•  It models key forces including thrust, lift and drag. 
• It also includes environmental factors such as wind speed (headwind/tailwind), altitude (affecting air density), and runway length. 
• At each step, forces are calculated, net acceleration is determined, and velocity and the position of the aircraft are updated. 
• The loop continues until VR speed is reached.
•  The key concept is that takeoff isn't governed by a single equation, but rather the interaction of different factors such as lift, drag and thrust.
•  Inputs: Aircraft type (either predefined figures or custom inputs), wind speed, altitude and runway length. 
• Outputs: VR speed, stall speed, distance covered, graphs of motion (velocity vs time, distance vs time), and time to takeoff.

Key Physics

1. Lift Force Equation: The lift generated by an aircraft depends on air density, velocity, wing area and the lift coefficient.
2. Drag: Drag opposes the motion of the aircraft. It depends on air density, wing area, and the drag coefficient. As velocity increases, drag becomes more significant and reduces acceleration.
3. Thrust vs Drag: When thrust is greater than drag, the aircraft accelerates. As speed increases, drag also increases, thus reducing net acceleration. This balance is imperative in determining VR.
4. Kinematics: The aircraft's motion is updated step-by-step through small time intervals. v = v + a x dt --> Speed increases based on acceleration.  x = x + v x dt --> Position changes based on current speed.

The Simulation

Here is the GitHub link for my python code: https://github.com/aryan-441/aircraft-vr-simulation-python

1. Initialisation

• Before the simulation begins, the program sets or calculates parameters such as weight and air density.
• Velocity, distance and time are all set to 0 and dt (time step) to 0.1.
• Data including aircraft mass, wing area, max headwind, max tailwind, etc. are either taken as input or extracted from a pre-defined dictionary.

2. Iterative Loop

• The simulation runs in a loop until the aircraft reaches rotation speed (VR).
• At each step of 0.1 seconds,
• drag and rolling resistance are calculated
• thrust is adjusted based on speed
• velocity and position are updated
• acceleration is calculated
• ground speed is calculated
• velocities, distances and times are all added to their own respective lists. This data is later used to create graphs.

3. Screenshots

Loop

Pre-defined aircraft specifications 

4. Flap Settings

• Flap combinations are pre-defined, both in aircraft selection and manual data input.
• The loop runs with every flap setting and compares the results to determine the best configuration, that is, the one which provides the shortest distance to VR.

5. Examples

• A320 at DEL/VIDP (Indira Gandhi International Airport).
• Altitude: 237m
• RWY Length: 4430m (RWY 11R/29L)
• Wind: +5 kts (headwind)

•    B777 at DEL/VIDP (Indira Gandhi International Airport).
• Altitude: 237m
• RWY Length: 4430m (RWY 11R/29L)
• Wind: +5 kts (headwind)

• A320 at Leh Airport (IXL/VILH).
• Altitude: 3526m
• RWY Length: 2750m (RWY 07/25)
• Wind: +5kts (headwind)

 Conclusion

• This simulation demonstrates how modelling aircraft takeoff performance is not limited to a single equation, but the complex interaction of distinct factors.
• Initially, the model was based purely on stall speed, leading to unrealistic results, where different aircraft produced nearly identical VR speeds.
• By refining the model, several important insights have emerged. Firstly, aircraft behaviour cannot be derived from lift alone. Thrust varies with speed and altitude, and factors like rolling resistance and flap configurations further affect acceleration.
• Environmental conditions like wind speeds and altitude further complicate the picture. At high-altitude airports like Leh, we observe that reduced air density affects lift and takeoff performance. While there is little variation in takeoff speeds, the required runway length increases substantially.
• As seen in the example, an A320 at Delhi requires 995m of runway while it requires 1183m at Leh.
• In conclusion, this simulation highlights how a reliable model is not based solely on mathematics and equations, but also on how different physical factors interact, as well as safety margins, regulatory limitations, and certification constraints.

Bibliography

https://skyvector.com

https://www.aircraftperformance.org

https://www.skybrary.aero

https://www.engineeringtoolbox.com/standard-atmosphere-d_604.html

https://github.com/aryan-441/aircraft-vr-simulation-python --> Full source code available here