Understanding the physics of drones

         In order to be able to plan for drone integration and subsequent design and fabrication, the physics behind drone working needs to be well understood.The theory behind lift required to be generated by a Unmanned Aerial Vehicle (UAV) or commonly called drone is determined primarily by the weight of the drone. The weight of the drone is balanced by the lift produced by its motors. The normal force produced during the forward motion of the aircraft, at a given velocity, is called lift force. The required lift can be produced either by an increase of the forward velocity or by increase in the Angle Of Attack .

            Angle of attack is the angle between the reference line of the aircraft (which runs along the longitudinal axis of the aircraft) and the relative wind. In the case of multi-copter drones, this angle is zero. The rotors must be able to create enough lift for the drone to take of at the required velocity of climb termed as Voc.

            The Drag is the opposing force to the motion of drone produced by virtue of the viscosity of air. This drag force is overcome by the thrust force produced by the propeller. The total drag force has two components in it viz., – the zero lift drag and the induced drag. The zero lift drag is the force that is produced because of the friction caused by the air molecules while it flows over the aircraft surface and hence it is directly related to the smoothness of the surface and the area exposed to the airflow itself. The induced drag is caused by the downwash, which is influenced by aspect ratio of the rotors. Down wash is simply the downward velocity component induced beneath the rotor surface and hence altering the relative velocity itself. The zero lift drag coefficient (𝐶_𝐷).

            The required thrust for drone to take off must overcome the force due to gravity and the drag force. This thrust required for drone to vertically take off = Weight of drone + Drag Force experienced during take-off

T_VTO        =  (M*g) + (0.5*ρ_SSL*V_ROC²*S_W*C_D-TO)

The required thrust for drone to land must overcome the opposing force due to gravity and now this phase of the mission is aided by drag which in mathematical terms is defines as below:

Thrust required for drone to vertically land = Weight of drone – Drag Force experienced during take-off

T_VLD  =  (M*g) – (0.5*ρSSL*V_ROD²*S_W*C_D-LD)

We know that thrust required during take-off is much more than thrust required for landing, hence we will calculate thrust per motor from thrust required for vertical take off which is given as,

Thrust per motor = (Thrust during take-off (T_VTO) / No of motors on the drone)     

This value must always be <= 0.5 times the maximum thrust of the motor to be utilized as per its data sheet. Also, 𝐶_{𝐷−𝑇𝑂}  and C_D-LD are the coefficient of drag. I have taken value of 2 based on the results for drag of a flat plate, facing the flow at 90°. The thrust for hover is thrust required to maintain weight

T_H = (M*g)

            In terms of velocity we have two concepts, the induced velocity of hover denoted as V_ih and induced velocity for vertical take off V_iVTO.  This induced velocity is generally in a downward direction also called downwash. Its important consequences is that it modifies the flow of air around the frame and impacts its aerodynamic characteristics. Induced velocity of hover is given by:

V_ih =  √ T_H/(2 * ρ_SSL * A_PROPS)

Where area of Propellers (A_PROP) = 2*π*(D_P/2)² and total area under all propellars (A_PROPS)= A_PROP * No of propellers

Similariliy, the induced velocity for vertical take off (V_iVTO)= -(V_ROC/2) + √((V_ROC/2)² + V_ih) and induced velocity of landing (V_iLD) = -(V_ROD/2) – √((V_ROD/2)² – V_ih)         

            Now power required during vertical take-off denoted by P_VTO = (T_VTO * V_i) / η₀ where    η₀ is the overall efficicncy of the power plant.It is the product of efficiency of propellars, motors and ESCs (η_{0 =} η_P * η_M * η_ESC) . The typical values are 0.66*0.85*0.98 = 0.55 for η₀

Similarly the power during vertical landing denoted by P_VLD = (T_VLD * V_i) / η_0.  The power for hover denoted P_H and power to cruise denoted by P_C is related to take-off and landing by a simple intuitive relation.

P_VTO > P_C > P_H > P_VLD

Thus, if we know what altitude we wish to achieve during vertical take-off based on rate of climb V_ROC, we can calculate energy E_VTO given by power * time. Similarly time for hover can give energy required for hover denoted by E_H while time for cruise can give energy denoted by E_C which is required to cruise from one point to another . The rate of descent V_ROD and the altitide from where descent begins can give total time for descent which can help us calculate the energy required for descent denoted by E_VLD. Thus, we see

Total energy required denoted by E_TR= E_VTO + E_H + E_C + E_VLDwhere E_TR <= 0.85 * Battery Energy Capacity

I have done sample calculations for the X-Type Quadcopter, V-Type Quadcopter and Hexacopter and given in figure 1.11 and figure 1.12. We can see that the thrust requirement per motor in each of the frame is <50% maximum trust capacity of motots selected.