Equations of sizing scenarios (INSA)

Written by Marc Budinger, INSA Toulouse, France

The purpose of this notebook is to illustrate on an existing drone example the main equations to be taken into account during the sizing. The selected drone is the MK-Quadro drone of Mikrokopter. The technical informations about the MK-quadro drone can be found in Annex 1 or on the website of MikroKopter company.

MK-Quadro from MikroKopter company

Propellers fundamental equations

Propellers can be described in an ideal manner by the disk momentum theory. The dimensional analysis theory enables to describe with more details the propeller performances thanks to aerodynamic coefficients:

• The thrust: \(T = C_{T,h} \rho \Omega^2 A R^2=C_{T,p} \rho n^2 D^4\) which gives \( = k_{TD} \Omega^2\) for a given propeller and a given diameter

• The power: \(P = C_{P,h} \rho \Omega^3 A R^3=C_{P,p} \rho n^3 D^5 \) which gives the power \(P=k_{PD}\Omega^3\) or the torque \(M = k_{PD} \Omega^2\) for a given propeller and a given diameter

Remark: Be carefull, litterature shows 2 notations :

• the propeller notation with n the propeller speed in [rev/s], D the diameter in [m], \(\rho\) the air density in [\(kg/m^3\)]

• the helicopter notation with \(\Omega\) the propeller speed in [rad/s], R the radius in [m], \(A\) the disk area in [\(m^2\)], \(\rho\) the air density in [\(kg/m^3\)]

Exercice: compare the caracteristics of the following APC propellers to those of the MikroKopter MK-Quadro (GWS).

Characteristics	APC Propeller	GWS Propeller
Diameter [inch]	10	10
Pitch [inch]	4.5	4.5
Thrust coef \(C_t\) [-]	0.1125
Power coef \(C_p\) [-]	0.0441
Thrust coef \(k_{TD}\) [\(\frac{N}{(rad/s)^2}\)]		\(1.799.10^{-5}\)
Torque coefficient \(k_{PD}\) [\(\frac{N}{(rad/s)^2}\)]		\(3.054.10^{-7}\)

with: \(1\) \(inch = 0.0254\) \(m\), \(\rho=1.18\) \(kg/m^3\)

D=10*0.0254 # [m] Propeller diameter
rho=1.18 # [kg/m^3] Air mass volumic (25°C)

# APC coefficients 
# Ct=T/(rho * n**2 * D**4) (thrust coef.)                                                              
Ct=0.1125 # [-] Thrust coefficient, propeller notation
# Cp=P/(rho * n**3 * D**5) (power coef.)    
Cp=0.0441 # [-] Power coefficient, propeller notation

Hover flight scenario

The flight conditions in static hover are no forward speed, no vertical speed. This scenario influences in a very important way the autonomy of the vehicle. The thrust generated by propellers have to compensate the global weight of the drone and the load.

Hover flight (V=0)

Questions: with the given technical informations, calculate the requested thrust for each propeller for a no load hover flight.

Characteristics	MK-MikroKopter drone
Total mass ready to take off	1350 g
Number of propellers	4

from math import sqrt

# Hover scenario
# ------------ 
Mass_total=1.35 # [kg] mass of the drone 

N_prop=4 # [-] propeller number
d_prop=2.54e-2*10 # [m] propeller diameter

# to be completed

Questions: with the technical informations of GWS propellers calculate the propeller torque, speed and the mechanical power.

# Propeller characteristics
# -----------------------------

Kw =1.799e-5 # [N/(rad/s)^2] thrust coef
Mw = 3.054e-7 # [N.m/(rad/s)^3] torque coef

# to be compared to the ones calculated with regression
# GWS 10*4.5

d_prop=10*.0254 #[m] diameter

# Torque and RPM of motors
# -----------------------

# to be completed
#

print("----------------")
print("Propeller speed: %.2f rad/s or %.0f RPM"%(W,W/2/3.14*60))
print("Propeller torque: %.2f N.m "%(M))
print("Propeller power: %.2f W "%(M*W))

----------------

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[3], line 19
     12 # Torque and RPM of motors
     13 # -----------------------
     14 
     15 # to be completed
     16 #
     18 print("----------------")
---> 19 print("Propeller speed: %.2f rad/s or %.0f RPM"%(W,W/2/3.14*60))
     20 print("Propeller torque: %.2f N.m "%(M))
     21 print("Propeller power: %.2f W "%(M*W))

NameError: name 'W' is not defined

Questions: with the following technical informations calculate the motor voltage and current, the autonomy of the drone if we assume an efficiency of 95% for electronic speed controllers (ESC) and a depth of discharge of 80% for the battery.

Component	Characteristic	Value
Motor	Poles pair number	7
Load current	6‐9A (DC)
Max current	10A (DC)
Speed constant Kv	760 tr/min/V
Mechanical power	110 W
Dimensions	28.8 x 29 mm
Max efficiency	76 %
mass	65 g
Resistance	R = 0.26 ohm
Inertia	J= 2.5e-5 kg.m²
Battery	Voltage	LIPO 4S (4*3.7V)
Capacity	3300mAh
mass	329g

# Voltage / current calculation
# -----------------------------

K_mot=(1/760)*60/2/3.14 # [V/(rad/s)] Motor torque coef 
R_mot = 0.26 # [Ohm] Motor resistance

# Voltage / current calculation

# to be completed

print("----------------")
print("Motor current: %.2f A "%I)
print("Motor voltage: %.2f V "%U)

# Autonomy of the battery

# to be completed

print("----------------")
print("Battery current: %.2f A " % I_bat)
print("Battery voltage: %.2f V " % V_bat)
print("Autonomy of the battery : %.1f min" % Aut_bat)

----------------

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[4], line 12
      7 # Voltage / current calculation
      8 
      9 # to be completed
     11 print("----------------")
---> 12 print("Motor current: %.2f A "%I)
     13 print("Motor voltage: %.2f V "%U)
     15 # Autonomy of the battery
     16 
     17 # to be completed

NameError: name 'I' is not defined