Note
This page was generated from a Jupyter notebook.
Angle of attack (AoA) vs Calibrated Airspeed (CAS) of a Global 5000 aircraft in wings-level flight
Credits:Sean McLeod (Computer Software Consultant and Professional)Guilherme Araujo Lima da Silva (CEO @ ATS4i Aerothermal Solutions)Agostino De Marco (Prof. Flight Mechanics, Univ. Naples Federico II, Italy)
The Bombardier Global Express is a large, long-range business jet produced by Bombardier Aerospace. Introduced in the late 1990s, it is known for its exceptional range, speed, and luxurious cabin, making it popular among corporate and VIP travelers. The Global Express can fly intercontinental distances without refueling, connecting cities such as New York and Tokyo or Paris and Singapore nonstop. Its advanced avionics, high cruise speed, and spacious, comfortable interior set a benchmark in the business aviation market. The aircraft has also served as the basis for several variants, including the Global 5000 and Global 6000.
See this Wikipedia page https://en.wikipedia.org/wiki/Bombardier_Global_Express to learn more.
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Bombardier Global Express in flight (Photo: Wikipedia). |
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Bombardier Global 5000 (Photo: Wikipedia). |
Plotting trim angles of attack vs flight speed
Consider the following generic flight conditions:
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Aerodynamic axes (From JSBSim reference manual). |
We are interested in finding particular cases of the above condition, where:
wings are not banked, i.e. roll angle \(\phi = 0\) deg,
the flight trajectory is a straight line,
the altitude is constant, i.e. the trajectory is a straight, horizontal line (FPA, Flight Path Angle \(\gamma_0=0\) deg),
the aircraft translates, i.e. the angular rates \((p,q,r)\) are constant and zero,
the True Air Speed \(V\) (TAS) is constant, i.e. the motion is unaccelerated (that is, equilibrated),
the sideslip angle \(\beta\) is zero (that is, symmetrical aerodynamics).
These are the trimmed flight conditions in level flight.
An aircraft in trimmed flight with a non-zero flight path angle \(\gamma\) is shown below:
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Aircraft in straight, wings-level, climbing flight. |
An aircraft in wings-level, constant altitude (zero flight path angle \(\gamma\)), trimmed flight is shown below:
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Aircraft in straight, wings-level, constant altitude flight. |
For a given flight altitude \(h_0\), for a given aircraft weight \(W_0\), there are certainly a minimum speed \(V_\mathrm{min}\) and a maximum \(V_\mathrm{max}\) the aircraft can fly in the above trimmed conditions. Now, for all \(V_0\) within a speed interval \([V_1, V_2] \subset [V_\mathrm{min}, V_\mathrm{max}]\), we want to plot the required flight angle of attack \(\alpha_\mathrm{B}\) (AoA) and the required (normalized) elevator deflection.
See next image introducing aerosurface deflection angles and their standard symbols.
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Aircraft standard aerosurface deflections. |
Setting up a simulation script
[1]:
import sys
if 'google.colab' in sys.modules:
print('Running on Google Colab – installing jsbsim …')
!pip install jsbsim
PATH_TO_JSBSIM_FILES = None
Initialize FDM
[2]:
import jsbsim
# --- JSBSim Initialization ---
# These lines initialize the flight dynamics model.
# Create a flight dynamics model (FDM) instance.
fdm = jsbsim.FGFDMExec(PATH_TO_JSBSIM_FILES)
fdm.set_debug_level(0) # Suppress verbose JSBSim console output
if fdm is not None:
print("FDM created successfully")
fdm.set_debug_level(0) # Suppress verbose JSBSim console output
else:
print("Failed to create FDM")
JSBSim Flight Dynamics Model v1.3.0 Apr 9 2026 10:00:08
[JSBSim-ML v2.0]
JSBSim startup beginning ...
FDM created successfully
Tweak aircraft XML file: remove <output name="..."/> nodes from the officially released files
[3]:
import os
import xml.etree.ElementTree as ET
import matplotlib.pyplot as plt
# Global variables that must be modified to match your particular need
# The aircraft name
# Note - It should match the exact spelling of the model file
AIRCRAFT_NAME="global5000"
ac_xml_file_path = os.path.join(fdm.get_root_dir(), f'aircraft/{AIRCRAFT_NAME}/{AIRCRAFT_NAME}.xml')
print(f"Aircraft original XML file: {ac_xml_file_path}")
print("Parsing XML...")
ac_xml_tree = ET.parse(ac_xml_file_path)
# Save a copy of the original XML file for backup
backup_xml_file_path = ac_xml_file_path.replace('.xml', '_backup.xml')
print(f"Saving backup XML file: {backup_xml_file_path}")
ac_xml_tree.write(backup_xml_file_path)
# Traverse the XML tree and remove all <output ... /> elements with name attribute
ac_xml_root = ac_xml_tree.getroot()
for x in ac_xml_root.findall('output'):
has_port = x.get('name') is not None
if has_port:
print(f"\tRemoving <output ... /> with name: {x.get('name')}")
ac_xml_root.remove(x)
print("Saving modified XML...")
ac_xml_tree.write(ac_xml_file_path)
print("No CSV file will be generated since all <output ... /> elements with name attribute have been removed from the XML file.")
Aircraft original XML file: /home/vscode/.local/lib/python3.11/site-packages/jsbsim/aircraft/global5000/global5000.xml
Parsing XML...
Saving backup XML file: /home/vscode/.local/lib/python3.11/site-packages/jsbsim/aircraft/global5000/global5000_backup.xml
Saving modified XML...
No CSV file will be generated since all <output ... /> elements with name attribute have been removed from the XML file.
Run a simulation for a single altitude, single weight, single CoG position
[4]:
%matplotlib inline
# Fuel max for Global5000, lbm
fuelmax = 8097.63
# Instantiate the FDMExec object and load the aircraft
fdm = jsbsim.FGFDMExec(PATH_TO_JSBSIM_FILES)
fdm.set_debug_level(0) # Suppress verbose JSBSim console output
fdm.load_model(AIRCRAFT_NAME)
fdm.disable_output() # Prevent JSBSim from trying to open/write model CSV output files
ac_xml_tree = ET.parse(os.path.join(fdm.get_root_dir(), f'aircraft/{AIRCRAFT_NAME}/{AIRCRAFT_NAME}.xml'))
ac_xml_root = ac_xml_tree.getroot()
# Get the empty weight from aircraft xml [assume lbs]
for x in ac_xml_root.findall('mass_balance'):
w = x.find('emptywt').text
empty_weight = float(w)
# Get the original CG from aircraft xml, assum inches from Construction axes origin
for loc in ac_xml_root.findall('mass_balance/location'):
x_cg_ = loc.find('x').text
x_cg_0 = float(x_cg_)
# Assume a payload, midweight, lb
payload_0 = 15172/2
# Assume the mass of fuel, half tanks, lb
fuel_per_tank = fuelmax/2
# Assume a flight altitude
h_ft_0 = 15000
weight_0 = empty_weight + payload_0 + fuel_per_tank*3
# Assume a zero flight path angle, gamma_0
gamma_0 = 0
print("-----------------------------------------")
print("Altitude {} ft, Weight {} lb, CoG-x {} in".format(h_ft_0, weight_0, x_cg_0))
print("-----------------------------------------")
# Run the simulation varying speed
speed_cas_1 = 90
speed_cas_2 = 550
print("Running simulation in the CAS range {} to {} kts".format(speed_cas_1, speed_cas_2))
# Set engines running
fdm['propulsion/set-running'] = -1
results = []
results_fcs = []
# Prepare two stacked subplots (AoA and elevator deflection)
fig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(10, 6), sharex=True)
# Run for different speeds
for speed in range(speed_cas_1, speed_cas_2, 10):
fdm['ic/h-sl-ft'] = h_ft_0
fdm['ic/vc-kts'] = speed
fdm['ic/gamma-deg'] = gamma_0
fdm['propulsion/tank[0]/contents-lbs'] = fuel_per_tank
fdm['propulsion/tank[1]/contents-lbs'] = fuel_per_tank
fdm['propulsion/tank[2]/contents-lbs'] = fuel_per_tank
fdm['inertia/pointmass-weight-lbs[0]'] = payload_0
# Initialize the aircraft with initial conditions
fdm.run_ic()
# Run fdm model
fdm.run()
# Trim
try:
fdm['simulation/do_simple_trim'] = 1
results.append((fdm['velocities/vc-kts'], fdm['aero/alpha-deg']))
results_fcs.append((fdm['velocities/vc-kts'], fdm['fcs/elevator-pos-rad'], fdm['fcs/elevator-pos-norm']))
except jsbsim.TrimFailureError:
print("\n\nTrim failed, continuing with other configurations...\n\n")
pass # Ignore trim failure exceptions
# Plot results
speed, alpha = zip(*results)
speed, dElev_rad, dElev_norm = zip(*results_fcs)
dElev_deg = [d * 180 / 3.14159 for d in dElev_rad]
# Plot AoA vs KCAS
ax1.plot(speed, alpha,
label="Weight {0:.0f} lb, Altitude {1:.0f} kft, CoG-x {2:.2f} in".format(weight_0, h_ft_0/1000, x_cg_0),
marker='o', linestyle='-', color='blue')
ax1.set_title("Trimmed flight conditions (FPA = {0:.0f} deg)".format(gamma_0))
ax1.set_ylabel('Angle of Attack (deg)')
ax1.set_ylim(-5, 15)
ax1.legend(loc='upper right')
# Plot elevator deflection vs KCAS
ax2.plot(speed, dElev_deg,
label="Weight {0:.0f} lb, Altitude {1:.0f} kft, CoG-x {2:.2f} in".format(weight_0, h_ft_0/1000, x_cg_0),
marker='o', linestyle='-', color='red')
#ax2.set_title("Trimmed flight conditions (FPA = {0:.0f} deg) - Elevator Deflection vs KCAS".format(gamma_0))
ax2.set_ylabel("Elevator Deflection (deg)")
ax2.set_ylim(-15, 10)
ax2.legend(loc="upper right")
# Plot normalized elevator deflection vs KCAS
ax3.plot(speed, dElev_norm,
label="Weight {0:.0f} lb, Altitude {1:.0f} kft, CoG-x {2:.2f} in".format(weight_0, h_ft_0/1000, x_cg_0),
marker='o', linestyle='-', color='green')
#ax3.set_title("Trimmed flight conditions (FPA = {0:.0f} deg) - Normalized elevator deflection vs KCAS".format(gamma_0))
ax3.legend(frameon=False)
ax3.set_ylabel("Normalized elevator\n deflection (-)")
ax3.set_xlabel("KCAS (kt)")
plt.tight_layout()
plt.show()
-----------------------------------------
Altitude 15000 ft, Weight 67967.445 lb, CoG-x 790.82 in
-----------------------------------------
Running simulation in the CAS range 90 to 550 kts
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Different aircraft weights, CoG positions and altitudes
Plots the variation in AoA versus CAS for level trim for different aircraft weights, cg and altitude.
Calculation required for aircraft icing engineering.
[5]:
# Control whether to print results to console in addition to generating a plot
print_output = False
# Function to change CG in aircraft xml. Change the directory to the aircraft to be studied.
# Note - Moments of inertia are not updated.
# Input x_cg is a float. Returns a float.
def changeCG(fdm, x_cg, readOnly):
tree = ET.parse(os.path.join(fdm.get_root_dir(), f'aircraft/{AIRCRAFT_NAME}/{AIRCRAFT_NAME}.xml'))
root = tree.getroot()
for loc in root.findall('mass_balance/location'):
x = loc.find('x').text
if not readOnly:
loc.find('x').text = str(x_cg) # replace with the new value
# Save the modified XML back to the file
tree.write(os.path.join(fdm.get_root_dir(), f'aircraft/{AIRCRAFT_NAME}/{AIRCRAFT_NAME}.xml'))
# return the same value
x = str(x_cg)
return float(x)
# Fuel max for Global5000
fuelmax = 8097.63
# Assume a zero flight path angle, gamma_0
gamma_0 = 0
# Prepare subplots to overlay plots
# Prepare two stacked subplots (AoA and elevator deflection)
fig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(10, 12), sharex=True)
# Define here the payloads to be studied
payload = [1500, 15172/2, 15172]
# Define here the mass of fuel (per tank)
fuel = [1000, fuelmax/2, fuelmax]
# Three cases for weight
weight = ["light", "mid", "heavy"]
# Marker settings for each weight case: (marker, fillstyle)
# i=0 (light): hollow square, i=1 (mid): solid circle, i=2 (heavy): triangle up
marker_settings = (
('s', 'none'), # light: hollow square
('o', 'full'), # mid: solid circle
('^', 'full'), # heavy: triangle up
)
fdm = jsbsim.FGFDMExec(PATH_TO_JSBSIM_FILES)
fdm.load_model(f'{AIRCRAFT_NAME}')
fdm.disable_output() # Prevent JSBSim from trying to open/write model CSV output files
# Get the original CG from aircraft xml
x_cg_Orig = changeCG(fdm, 0, True)
# Vary CG in the study
xs_cg = [x_cg_Orig*0.95, x_cg_Orig*1.05]
# Vary altitude
h_ft = [8000, 30000]
# Linestyle settings for each altitude case:
# j=0 (8000 ft): densely dashed (5 pts on, 1 pt off), j=1 (30000 ft): solid
linestyle_settings = ((0, (5, 1)), 'solid')
# Run the simulation varying CG, altitude, speed and total weight
fdm = []
# Run for different CG's
for j in range(2):
fdm = jsbsim.FGFDMExec(PATH_TO_JSBSIM_FILES)
fdm.load_model(f'{AIRCRAFT_NAME}')
fdm.disable_output() # Prevent JSBSim from trying to open/write model CSV output files
# Set engines running
fdm['propulsion/set-running'] = -1
# Change CoG-x in the aircraft xml
xcg_ = changeCG(fdm, xs_cg[j], False)
# Run for different weights
for i in range(3):
results = []
results_fcs = []
# Run for different speeds
for speed in range(90, 550, 10):
fdm['ic/h-sl-ft'] = h_ft[j]
fdm['ic/vc-kts'] = speed
fdm['ic/gamma-deg'] = gamma_0
fdm['propulsion/tank[0]/contents-lbs'] = fuel[i]
fdm['propulsion/tank[1]/contents-lbs'] = fuel[i]
fdm['propulsion/tank[2]/contents-lbs'] = fuel[i]
fdm['inertia/pointmass-weight-lbs[0]'] = payload[i]
# Initialize the aircraft with initial conditions
fdm.run_ic()
# Run fdm model
fdm.run()
# Trim
try:
fdm['simulation/do_simple_trim'] = 1
results.append((fdm['velocities/vc-kts'], fdm['aero/alpha-deg']))
results_fcs.append((fdm['velocities/vc-kts'], fdm['fcs/elevator-pos-rad'], fdm['fcs/elevator-pos-norm']))
except jsbsim.TrimFailureError:
pass # Ignore trim failure exceptions
if print_output:
print("-----------------------------------------")
print("Altitude {} - Weight {} - CG {}".format(h_ft[j], weight[i], xs_cg[j]))
print("-----------------------------------------")
print("KCAS, AoA (deg)")
for result in results:
print(result[0], result[1])
print("---------")
print("KCAS, Elevator deflection (rad), Normalized elevator deflection (-1 to 1)")
for result_fcs in results_fcs:
print(result_fcs[0], result_fcs[1], result_fcs[2])
speed, alpha = zip(*results)
ax1.plot(speed, alpha,
label="{0} weight, {1:.0f} kft, {2:.2f} % CoG".format(weight[i], h_ft[j]/1000, (float(xs_cg[j])/float(x_cg_Orig)-1)*100),
marker=marker_settings[i][0], fillstyle=marker_settings[i][1], linestyle=linestyle_settings[j])
speed, dElev_rad, dElev_norm = zip(*results_fcs)
# Convert elevator deflection from radians to degrees
dElev_deg = [d * 180 / 3.14159 for d in dElev_rad]
ax2.plot(speed, dElev_deg,
label="{0} weight, {1:.0f} kft, {2:.2f} % cg".format(weight[i], h_ft[j]/1000, (float(xs_cg[j])/float(x_cg_Orig)-1)*100),
marker=marker_settings[i][0], fillstyle=marker_settings[i][1], linestyle=linestyle_settings[j])
ax3.plot(speed, dElev_norm,
label="{0} weight, {1:.0f} kft, {2:.2f} % cg".format(weight[i], h_ft[j]/1000, (float(xs_cg[j])/float(x_cg_Orig)-1)*100),
marker=marker_settings[i][0], fillstyle=marker_settings[i][1], linestyle=linestyle_settings[j])
# Restore original CG for the aircraft xml
x_cg__ = changeCG(fdm," {:.2f} ".format(x_cg_Orig), False)
print("-----------------------------------------")
print("Restored CoG-x {} in".format(x_cg__))
print("-----------------------------------------")
# Plot final results
ax1.set_title("Trimmed flight conditions (FPA = {0:.0f} deg)".format(gamma_0))
ax1.set_ylabel("Angle of Attack (deg)")
ax1.legend(frameon=False)
ax1.legend(loc='upper right')
ax1.set_ylim(0, 15)
ax2.set_ylabel('Elevator Deflection (deg)')
ax2.legend(frameon=False)
ax2.legend(loc='lower right')
ax2.set_ylim(-20, 0)
ax3.set_ylabel("Normalized elevator deflection (-)")
ax3.set_xlabel("KCAS (kt)")
ax3.legend(frameon=False)
ax3.legend(loc='lower right')
ax3.set_ylim(-1.0, 0)
plt.show()
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Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
Sorry, udot doesn't appear to be trimmable
-----------------------------------------
Restored CoG-x 790.82 in
-----------------------------------------
