Ciphering the algorithms in the making: Figures are important!
Seeing the above figure might have given you a slight impression about the amount of mathematical calculations, physics laws and algorithms, as well as vector and positional analysis, is required for a successful run of a drone! Don’t worry, I am gonna make this all easy for all of you by briefing out what are the basic mathematics which you will be required for making a drone work without any unwanted hindrance!
From the previous blogs we have already come to know about various technicalities, essentials and materials required to build a drone. But these essentials will only become handy by the application of physics and maths combined and the complete knowledge about the calculations involved.
1. Motor Assumptions:-
The standard motors used in drones are brush-less DC Motors. For these types of motors, the motor shaft is rigidly attached to the outer shell of the motor, and the motor axle is rigidly attached to the base of the motor, which is fixed on the drone frame. When the motor is turned on, the axle of the motor is fixed relative to the drone frame, and the outer shell is spinning around the axle.
The rotation of the motor will induce a torque on the frame of the drone. It can be mathematically stated as : —
‖𝜏𝑖‖ = 𝑘1𝜑̇ 𝑖 2 (𝜏𝑖=torque induced, k1= proportionality constant, 𝜑̇ 𝑖=Rotational Speed)
The direction of this torque is always opposite to the direction of the motor shell in which it’s spinning.
Further Analysis :
The above-induced torque is generated in all the four motors in a drone. This will be more clear when you look at the following diagram: -
Here, the magnitude of the forces generated by rotors are proportional to each rotor’s rotational speed squared: ‖𝐹𝑖‖ = 𝑘2𝜑̇ 𝑖 2. 𝜑̇ 𝑖 is the rotational speed of each motor, and 𝑘2 is a proportionality constant.
2. Flight Time : -
Here since we are talking about the mathematical analysis of a drone, one of the most important measures of a drone is its Flight Time.
Flight Time for an Unmanned Aerial Vehicle can be described at the total time taken during the flight. For an electric fixed-wing aircraft this is directly related to the capacity of the battery and the amount of current the motor produces to keep the aircraft in the air. It may depend on various other factors as well but for simplicity, we narrow the formula down to:-
FLIGHT TIME= (BATTERY CAPACITY (Ah)/ AVG.CURRENT DRAW(Amp)
One must also note here that as the payload of a drone increases , the flight time of the drone decreases.
It also sometimes depends on Aircraft Size, weight and Payload weight. The above formula gives us a rough idea about the flight time of a drone.
3. Range:-
After calculating the total time in a flight, one must also be able to roughly calculate how far a drone can fly. Calculating the range helps us do that.
The range of a drone depends on various characteristics such as the amount of current the drone is producing, flight time, flight speed and the aerodynamic performance.
Using maths we can easily determine the range of a drone. It can be expressed through the following formula : -
RANGE = FLIGHT TIME(hr)* AVG SPEED(km/hr)
Accurately using wing area, weight and coefficient of the lift of the airfoil range can be calculated.
The signal strength of the radio controller also has a greater impact on the Range of the drone. The greater the signal strength, the more the range.
4. Further Mathematical Calculations and Analysis : -
After knowing about the basic physics behind the working of a drone there are still some mathematical calculations required which are essential for the proper working of the drone. The calculations depend largely on the type of setup we do while making the drone. There are two types of setups which can be done while making a drone which are: -
- Newtonian Setup
- Lagrangian Setup
Understanding the calculations and the mathematical derivations for the above two setups is too complex to understand in the present context.
In Newtonian physics you stand at a point and watch something move in relation to the stationary observation point based on forces applied.
In Lagrangian physics you are the something moving that experiences the forces.
The above two things are the most important concepts which you need to know for now.
Conclusion
Through this blog, my main focus was to explain about the various forces and torques required in the construction of the drone. We saw that all the motors work on the same principle just in different directions. Also, we came to know about the flight time and range of a drone and how to calculate such mathematical measures. My main focus here was on the dynamics of the modelling of the drone, I still did not add the control theory and the mechanism of the working of the drone which is beyond the scope of the blog and is too complex to understand.
What Next?
We have come a long way through a series of blogs on the basic understanding and development of drones. We have already learnt the basics of drones, its motors, various configurations, designing it and of course the calculations, laws and algorithms required in the making!
So now we are moving towards the most important step in the making, The final arrangement of the parts of the drone and our drone will be ready to fly!
So stay tuned as we move towards the assembling of the parts!
