6.10. Estimation models with dimensional analysis and linear regressions (student version)#
Written by Marc Budinger (INSA Toulouse), Scott Delbecq (ISAE-SUPAERO) and Félix Pollet (ISAE-SUPAERO), Toulouse, France.
6.10.1. Propellers#
The application chosen for this notebook are UAV propellers. Propellers characteristics can be expressed by \(C_T\) and \(C_P\) coefficients. These coefficients are function of dimensions and conditions of use of propellers. Dimensional analysis and linear regression of suppliers data can be used for the generation of \(C_T\) and \(C_P\) prediction models.
Fig. 6.9 APC MR (Multi-Rotor) propellers#
6.10.2. Dimensional analysis and \(\pi\) numbers#
The propeller performances can be expressed with 2 aerodynamic coefficients:
The thrust: \(F = C_{T} \rho_{air} n^2 D^4\)
The power: \(P = C_{P} \rho_{air} n^3 D^5 \)
The dimensional analysis and especially the Buckingham \(\pi\) theorem enable to find this results.
6.10.2.1. Dimensional analysis of the propeller thrust#
The thrust \(F\) of a propeller depends of multiple parameters (geometrical dimensions, air properties, operational points):
\(F=f(\rho_{air},n,D,p,V,\beta_{air})\)
with the parameters express in the following table.
Parameter |
M |
L |
T |
|---|---|---|---|
Thrust \(T\) [N] |
1 |
1 |
-2 |
Mass volumic (Air) \(\rho_{air}\) [kg/m\(^3\)] |
1 |
-3 |
0 |
Rotational speed \(n\) [Hz] |
0 |
0 |
-1 |
Diameter \(D\) [m] |
0 |
1 |
0 |
Pitch \(p\) [m] |
0 |
1 |
0 |
Drone speed \(V\) [m/s] |
0 |
1 |
-1 |
Bulk modulus (Air) \(\beta_{air}\) [Pa] |
1 |
-1 |
-2 |
\(=\pi_0\) |
|||
\(=\pi_1\) |
|||
\(=\pi_2\) |
|||
\(=\pi_3\) |
Remark: The dimension of a parameter \(x\) is function of dimensions L, M and T : \([x]=M^aL^bT^c\). The previous table gives the value of \(a\), \(b\) and \(c\) for each parameter of the problem.
Parameter |
M |
L |
T |
|---|---|---|---|
Thrust \(T\) [N] |
1 |
1 |
-2 |
Mass volumic (Air) \(\rho_{air}\) [kg/m\(^3\)] |
1 |
-3 |
0 |
Rotational speed \(n\) [Hz] |
0 |
0 |
-1 |
Diameter \(D\) [m] |
0 |
1 |
0 |
Pitch \(p\) [m] |
0 |
1 |
0 |
Drone speed \(V\) [m/s] |
0 |
1 |
-1 |
Bulk modulus (Air) \(\beta_{air}\) [Pa] |
1 |
-1 |
-2 |
\(=\pi_0\) |
|||
\(=\pi_1\) |
|||
\(=\pi_2\) |
|||
\(=\pi_3\) |
6.10.2.2. Effect of the rotational speed#
APC suppliers give complete propeller data for all their propellers. From the file APC_STATIC-data-all-props.csv, we find all static data provided by APC:
import pandas as pd
# Read the .csv file with bearing data
path = "https://raw.githubusercontent.com/SizingLab/sizing_course/main/laboratories/Lab-multirotor/assets/data/"
df = pd.read_csv(path + "APC_STATIC-data-all-props.csv", sep=";")
# Print the head (first lines of the file)
df.head()
| LINE | COMP | TYPE | RPM | DIAMETER(IN) | PITCH(IN) | BLADE(nb) | THRUST(LBF) | POWER(HP) | TORQUE(IN.LBF) | Cp | Ct | AREA(m^2) | THRUST(N) | POWER(W) | ANGLE | EFF | N.D | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | NaN | 1000 | 10.5 | 4.5 | 2 | 0.03 | 0.01 | 0.02 | 0.03 | 0.08 | 0.06 | 0.1335 | 7.457 | 0.43 | 60.180222 | 10500.0 |
| 1 | 2 | 1 | NaN | 2000 | 10.5 | 4.5 | 2 | 0.13 | 0.01 | 0.08 | 0.03 | 0.08 | 0.06 | 0.5785 | 7.457 | 0.43 | 60.180222 | 21000.0 |
| 2 | 3 | 1 | NaN | 3000 | 10.5 | 4.5 | 2 | 0.29 | 0.01 | 0.17 | 0.03 | 0.08 | 0.06 | 1.2905 | 7.457 | 0.43 | 60.180222 | 31500.0 |
| 3 | 4 | 1 | NaN | 4000 | 10.5 | 4.5 | 2 | 0.52 | 0.02 | 0.30 | 0.03 | 0.08 | 0.06 | 2.3140 | 14.914 | 0.43 | 60.180222 | 42000.0 |
| 4 | 5 | 1 | NaN | 5000 | 10.5 | 4.5 | 2 | 0.81 | 0.04 | 0.47 | 0.03 | 0.08 | 0.06 | 3.6045 | 29.828 | 0.43 | 60.180222 | 52500.0 |
For next steps, we keep only the Multi-Rotor type propellers (MR).
# Data Filtering
dfMR =
dfMR.head()
Cell In[2], line 2
dfMR =
^
SyntaxError: invalid syntax
We plot the \(C_p\) and \(C_t\) for the a 10x4.5 propeller (COMP n° 8 in the previous table). We can notice that these coefficients are constant up to a certain value of speed of rotation. The manufacturer recommends using these propellers for a product speed of rotation x diameter less than a limit (depending on the type of propeller technology) and given here:
Maximum speed(RPM) x prop diameter (inches) = 105,000
for MR type which gives a blade tip speed of 135 m/s.
Question: Explain the origin of this operating limit comes from and the \(\pi\) number that can express it.
# Keep only the component n°8
df8 =
# Extract forbidden ND product
df8ND =
import numpy as np
import matplotlib.pyplot as plt
# plot the data
plt.plot(
df8["RPM"],
df8["Cp"],
"bo",
df8["RPM"],
df8["Ct"],
"ro",
)
plt.plot(
df8ND["RPM"],
df8ND["Cp"],
"ko",
df8ND["RPM"],
df8ND["Ct"],
"ko",
)
plt.xlabel("Rotational Speed [RPM]")
plt.ylabel("Cp (blue) and Ct (red)")
plt.grid()
plt.show()
Cell In[3], line 2
df8 =
^
SyntaxError: invalid syntax
6.10.3. Linear regression#
For next calculations, we keep only data with following criteria:
Type ‘MR’ (Multi-Rotor)
Maximum RPM < 105,000/prop diameter (inches)
# Keep only operating points with ND<105000
dfMRND =
Cell In[4], line 2
dfMRND =
^
SyntaxError: invalid syntax
The APC static data correspond to the hover operational point where the speed V=0. The aerodynamic coefficients are thus only a function of \(p/D\) (called ‘ANGLE’ in the .csv file) dimensionless number.
\(C_t=\frac{F}{\rho_{air} n^2 D^4}=f(\frac{p}{D})\)
\(C_p=\frac{P}{\rho_{air} n^3 D^5}=g(\frac{p}{D})\)
The following code uses the Scikit-learn package in order to set up a \(C_t\) estimator for the static case (\(V=0\) or \(J=0\)).
# Import packages
from sklearn import linear_model
from sklearn.metrics import r2_score
import matplotlib.pyplot as plt
# Data
x =
y_Ct =
# Matrix X and Y
X =
Y_Ct =
# Create a new object for the linear regression
reg_Ct =
# Y vector prediction
Ct_est =
# Ct Parameters
# ----
coef =
intercept =
r2 =
# Plot the data
plt.plot(x, Y_Ct, "o", label="Reference data")
plt.plot(x, Ct_est, "-g", label="Data prediction")
plt.xlabel("Pitch/Diameter ratio")
plt.ylabel("Ct")
plt.title("Comparison of reference data and regression")
plt.legend()
plt.grid()
plt.show()
print(f"Ct estimation model : Ct = {intercept:.2f} + {coef:.2f} * p/D with R2={r2:.3f}")
Cell In[5], line 8
x =
^
SyntaxError: invalid syntax
y_Cp =
reg_Cp =
# Cp Parameters
# -----
coef = float(reg_Ct.coef_)
intercept = float(reg_Ct.intercept_)
r2 = r2_score(Y_Ct, Ct_est)
# Plot the data
plt.plot(x, Y_Cp, 'o', label='Reference data')
plt.plot(x, Cp_est, '-g', label='Data prediction')
plt.xlabel('Pitch/Diameter ratio')
plt.ylabel('Cp')
plt.title('Comparison of reference data and regression')
plt.legend()
plt.grid()
plt.show()
print(f"Cp estimation model : Cp = {intercept:.2f} + {coef:.2f} * p/D with R2={r2:.3f}")
Cell In[6], line 1
y_Cp =
^
SyntaxError: invalid syntax