Objectives
Here we will show you how to perform an aerodynamic stability analysis of a conventional aircraft. Here, we'll attempt to replicate the design of a Boeing 777. It won't be a realistic replication, but the overall geometry will match to a certain extent.
Source: boeingboeing2, DeviantArt
Refer to the Aircraft Aerodynamic Analysis tutorial before studying this tutorial.
Recipe
- Define the geometries of a fuselage, wing, horizontal tail and vertical tail.
- Perform an aerodynamic analysis of this aircraft configuration at given freestream conditions and reference values.
- Evaluate the derivatives of the aerodynamic coefficients with respect to the freestream conditions.
- Evaluate the quantities of interest for aerodynamic stability.
Let's import the relevant packages.
using AeroFuse # Main package
using Plots # Plotting library
gr( # Plotting backend
size = (800,600), # Size
dpi = 300, # Resolution
palette = :Dark2_8 # Color scheme
)
using LaTeXStrings # For LaTeX printing in plotsAircraft Geometry
First, let's define the fuselage. Here we'll define it by combining a cylindrical definition for the cabin with hyperelliptical curves for the nose and rear.
# Fuselage definition
fuse = HyperEllipseFuselage(
radius = 3.04, # Radius, m
length = 63.7, # Length, m
x_a = 0.15, # Start of cabin, ratio of length
x_b = 0.7, # End of cabin, ratio of length
c_nose = 2.0, # Curvature of nose
c_rear = 1.2, # Curvature of rear
d_nose = -0.5, # "Droop" or "rise" of nose, m
d_rear = 1.0, # "Droop" or "rise" of rear, m
position = [0.,0.,0.] # Set nose at origin, m
)
# Compute geometric properties
ts = 0:0.1:1 # Distribution of sections for nose, cabin and rear
S_f = wetted_area(fuse, ts) # Surface area, m²
V_f = volume(fuse, ts) # Volume, m³
# Get coordinates of rear end
fuse_end = fuse.affine.translation + [ fuse.length, 0., 0. ]3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
63.7
0.0
0.0Now, let's define the lifting surfaces. We'll download a supercritical airfoil for the wing section; note that this is not the same one as used in the Boeing 777-200LR. We'll also define a two-section wing.
# Define one airfoil
foil_w = read_foil(download("http://airfoiltools.com/airfoil/seligdatfile?airfoil=rae2822-il"))
# Define vector of airfoils
foils = [ foil_w, foil_w, naca4((0,0,1,2)) ]
# Wing
wing = Wing(
foils = foils, # Airfoils
chords = [14.0, 9.73, 1.43561], # Chord lengths
spans = [14.0, 46.9] / 2, # Span lengths
dihedrals = fill(6, 2), # Dihedral angles (deg)
sweeps = fill(35.6, 2), # Sweep angles (deg )
w_sweep = 0., # Leading-edge sweep
position = [19.51, 0., -2.5], # Position
symmetry = true # Symmetry
)
b_w = span(wing)
S_w = projected_area(wing)
c_w = mean_aerodynamic_chord(wing)
x_w, y_w, z_w = mac_w = mean_aerodynamic_center(wing)3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
30.683074339989055
0.0
-2.5For reference, let's plot what we have so far.
p1 = plot(
xaxis = L"x", yaxis = L"y", zaxis = L"z",
aspect_ratio = 1,
zlim = (-0.5, 0.5) .* span(wing),
camera = (30,30)
)
plot!(fuse, label = "Fuselage", alpha = 0.6)
plot!(wing, label = "Wing")Stabilizers
Now, let's add the stabilizers. First, the horizontal tail.
htail = WingSection(
area = 101, # Area (m²)
aspect = 4.2, # Aspect ratio
taper = 0.4, # Taper ratio
dihedral = 7., # Dihedral angle (deg)
sweep = 35., # Sweep angle (deg)
w_sweep = 0., # Leading-edge sweep
root_foil = naca4(0,0,1,2),
symmetry = true,
# Orientation
angle = -2, # Incidence angle (deg)
axis = [0., 1., 0.], # Axis of rotation, y-axis
position = [ fuse_end.x - 8., 0., 0.],
)
b_h = span(htail)
S_h = projected_area(htail)
c_h = mean_aerodynamic_chord(htail)
x_h, y_h, z_h = mac_h = mean_aerodynamic_center(htail)3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
60.08867631466764
0.0
0.1532559539584015Now the vertical tail.
vtail = WingSection(
area = 56.1, # Area (m²)
aspect = 1.5, # Aspect ratio
taper = 0.4, # Taper ratio
sweep = 44.4, # Sweep angle (deg)
w_sweep = 0., # Leading-edge sweep
root_foil = naca4(0,0,0,9),
# Orientation
angle = 90., # To make it vertical
axis = [1, 0, 0], # Axis of rotation, x-axis
position = htail.affine.translation - [2.,0.,0.]
) # Not a symmetric surface
b_v = span(vtail)
S_v = projected_area(vtail)
c_v = mean_aerodynamic_chord(vtail)
x_v, y_v, z_v = mac_v = mean_aerodynamic_center(vtail)3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
59.17243218476657
2.407305072320327e-16
3.9314275332224553Let's mesh and plot the lifting surfaces.
wing_mesh = WingMesh(wing, [8,16], 10,
span_spacing = fill(Uniform(), 4) # Number of spacings = number of spanwise stations
# (including symmetry)
)
htail_mesh = WingMesh(htail, [10], 8)
vtail_mesh = WingMesh(vtail, [8], 6)
# Plot meshes
plt = plot(
xaxis = L"x", yaxis = L"y", zaxis = L"z",
aspect_ratio = 1,
zlim = (-0.5, 0.5) .* span(wing_mesh),
camera = (30, 45),
)
plot!(fuse, label = "Fuselage", alpha = 0.6)
plot!(plt, wing_mesh, label = "Wing")
plot!(plt, htail_mesh, label = "Horizontal Tail")
plot!(plt, vtail_mesh, label = "Vertical Tail")Aerodynamic Analysis
Now, let's generate the horseshoe system for the aircraft.
aircraft = ComponentVector(
wing = make_horseshoes(wing_mesh),
htail = make_horseshoes(htail_mesh),
vtail = make_horseshoes(vtail_mesh)
)ComponentVector{Horseshoe{Float64}}(wing = Horseshoe{Float64}[Horseshoe{Float64}([41.31884181869616, -30.45, 0.7004239638398483], [39.226616003049074, -27.51875, 0.39233717421733394], [40.29663791737372, -28.984375, 0.5463805690285913], [0.0005456281159104126, 0.029861367098509173, 0.28033310122543664], 0.0) Horseshoe{Float64}([39.226616003049074, -27.51875, 0.39233717421733394], [37.13439018740198, -24.5875, 0.08425038459481993], [38.217098237408095, -26.053125, 0.23829377940607704], [0.0016368843477312378, 0.04627635308596634, 0.42907804209054934], 0.0) … Horseshoe{Float64}([37.13439018740198, 24.5875, 0.08425038459481993], [39.226616003049074, 27.51875, 0.39233717421733394], [38.217098237408095, 26.053125, 0.23829377940607704], [0.0016368843477312378, -0.04627635308596634, 0.42907804209054934], 0.0) Horseshoe{Float64}([39.226616003049074, 27.51875, 0.39233717421733394], [41.31884181869616, 30.45, 0.7004239638398483], [40.29663791737372, 28.984375, 0.5463805690285913], [0.0005456281159104126, -0.029861367098509173, 0.28033310122543664], 0.0); Horseshoe{Float64}([41.37067989644471, -30.45, 0.7004239638398485], [39.31589158416699, -27.51875, 0.39233717421733416], [40.412672379669004, -28.984375, 0.5463805690285911], [0.00019510184898996474, 0.08570486747568318, 0.8135583465329406], 0.0) Horseshoe{Float64}([39.31589158416699, -27.51875, 0.39233717421733416], [37.261103271889276, -24.5875, 0.08425038459481993], [38.39470066708566, -26.053125, 0.23829377940607693], [0.0005853055469684509, 0.13137909281537086, 1.2452329779493736], 0.0) … Horseshoe{Float64}([37.261103271889276, 24.5875, 0.08425038459481993], [39.31589158416699, 27.51875, 0.39233717421733416], [38.39470066708566, 26.053125, 0.23829377940607693], [0.0005853055469684509, -0.13137909281537086, 1.2452329779493736], 0.0) Horseshoe{Float64}([39.31589158416699, 27.51875, 0.39233717421733416], [41.37067989644471, 30.45, 0.7004239638398485], [40.412672379669004, 28.984375, 0.5463805690285911], [0.00019510184898996474, -0.08570486747568318, 0.8135583465329406], 0.0); … ; Horseshoe{Float64}([42.63406946277126, -30.45, 0.7004239638398482], [41.491702158378985, -27.51875, 0.39233717421733383], [42.13227244993827, -28.984375, 0.546380569028591], [0.02071501380863916, 0.09146069787357614, 0.8135583465329401], 0.0) Horseshoe{Float64}([41.491702158378985, -27.51875, 0.39233717421733383], [40.349334853986704, -24.5875, 0.08425038459481993], [41.02672174524036, -26.053125, 0.23829377940607688], [0.06214504142591892, 0.15166120081756396, 1.245232977949374], 0.0) … Horseshoe{Float64}([40.349334853986704, 24.5875, 0.08425038459481993], [41.491702158378985, 27.51875, 0.39233717421733383], [41.02672174524036, 26.053125, 0.23829377940607688], [0.06214504142591892, -0.15166120081756396, 1.245232977949374], 0.0) Horseshoe{Float64}([41.491702158378985, 27.51875, 0.39233717421733383], [42.63406946277126, 30.45, 0.7004239638398482], [42.13227244993827, 28.984375, 0.546380569028591], [0.02071501380863916, -0.09146069787357614, 0.8135583465329401], 0.0); Horseshoe{Float64}([42.7193199413754, -30.45, 0.7004239638398482], [41.6385206043654, -27.51875, 0.39233717421733383], [42.202829279371514, -28.984375, 0.546380569028591], [0.010666838939015766, 0.03320013990913556, 0.2803331012254997], 0.0) Horseshoe{Float64}([41.6385206043654, -27.51875, 0.39233717421733383], [40.5577212673554, -24.5875, 0.08425038459481993], [41.13471607804297, -26.053125, 0.23829377940607688], [0.03200051681704619, 0.05656144082582942, 0.4290780420905902], 0.0) … Horseshoe{Float64}([40.5577212673554, 24.5875, 0.08425038459481993], [41.6385206043654, 27.51875, 0.39233717421733383], [41.13471607804297, 26.053125, 0.23829377940607688], [0.03200051681704619, -0.05656144082582942, 0.4290780420905902], 0.0) Horseshoe{Float64}([41.6385206043654, 27.51875, 0.39233717421733383], [42.7193199413754, 30.45, 0.7004239638398482], [42.202829279371514, 28.984375, 0.546380569028591], [0.010666838939015766, -0.03320013990913556, 0.2803331012254997], 0.0)], htail = Horseshoe{Float64}[Horseshoe{Float64}([62.88890365446351, -10.298058069364341, 1.5162553193557646], [62.54031413176851, -9.794035232054846, 1.44215835982815], [62.76985874207378, -10.046046650709593, 1.4811362068318652], [-0.003889780601881568, 0.013685154467470186, 0.11138874253045755], 0.0) Horseshoe{Float64}([62.54031413176851, -9.794035232054846, 1.44215835982815], [61.528667934930915, -8.331303987175813, 1.2271206079077435], [62.09737450683494, -9.06266960961533, 1.3368354231503647], [-0.012848276001000303, 0.0452032285906627, 0.36792648581346676], 0.0) … Horseshoe{Float64}([61.528667934930915, 8.331303987175813, 1.2271206079077435], [62.54031413176851, 9.794035232054846, 1.44215835982815], [62.09737450683494, 9.06266960961533, 1.3368354231503647], [-0.012848276001000303, -0.0452032285906627, 0.36792648581346676], 0.0) Horseshoe{Float64}([62.54031413176851, 9.794035232054846, 1.44215835982815], [62.88890365446351, 10.298058069364341, 1.5162553193557646], [62.76985874207378, 10.046046650709593, 1.4811362068318652], [-0.003889780601881568, -0.013685154467470186, 0.11138874253045755], 0.0); Horseshoe{Float64}([63.04472763081975, -10.298058069364341, 1.5216968125153882], [62.707577960496174, -9.794035232054846, 1.4479993414354557], [63.03349105386857, -10.046046650709593, 1.4903424500226277], [-0.011077157969957813, 0.038972022690971254, 0.317208301282571], 0.0) Horseshoe{Float64}([62.707577960496174, -9.794035232054846, 1.4479993414354557], [61.7291315083166, -8.331303987175813, 1.2341209501454369], [62.39743171606405, -9.06266960961533, 1.3473136517881916], [-0.03658879445176172, 0.12872790398734174, 1.0477659852371668], 0.0) … Horseshoe{Float64}([61.7291315083166, 8.331303987175813, 1.2341209501454369], [62.707577960496174, 9.794035232054846, 1.4479993414354557], [62.39743171606405, 9.06266960961533, 1.3473136517881916], [-0.03658879445176172, -0.12872790398734174, 1.0477659852371668], 0.0) Horseshoe{Float64}([62.707577960496174, 9.794035232054846, 1.4479993414354557], [63.04472763081975, 10.298058069364341, 1.5216968125153882], [63.03349105386857, 10.046046650709593, 1.4903424500226277], [-0.011077157969957813, -0.038972022690971254, 0.317208301282571], 0.0); … ; Horseshoe{Float64}([65.32850451577563, -10.298058069364341, 1.6014480586855144], [65.15901884058357, -9.794035232054846, 1.533605543331635], [65.40109993639021, -10.046046650709593, 1.5730211740557802], [-0.011077157969957705, 0.03897202269097152, 0.31720830128257105], 0.0) Horseshoe{Float64}([65.15901884058357, -9.794035232054846, 1.533605543331635], [64.66715225377132, -8.331303987175813, 1.336718895359434], [65.0921625288351, -9.06266960961533, 1.4414157253432798], [-0.036588794451761386, 0.12872790398734185, 1.0477659852371666], 0.0) … Horseshoe{Float64}([64.66715225377132, 8.331303987175813, 1.336718895359434], [65.15901884058357, 9.794035232054846, 1.533605543331635], [65.0921625288351, 9.06266960961533, 1.4414157253432798], [-0.036588794451761386, -0.12872790398734185, 1.0477659852371666], 0.0) Horseshoe{Float64}([65.15901884058357, 9.794035232054846, 1.533605543331635], [65.32850451577563, 10.298058069364341, 1.6014480586855144], [65.40109993639021, 10.046046650709593, 1.5730211740557802], [-0.011077157969957705, -0.03897202269097152, 0.31720830128257105], 0.0); Horseshoe{Float64}([65.58280216782836, -10.298058069364341, 1.6103283283751404], [65.43198581212043, -9.794035232054846, 1.543137760023534], [65.56264383893216, -10.046046650709593, 1.5786624114392451], [-0.003889780601881672, 0.013685154467470526, 0.11138874253046113], 0.0) Horseshoe{Float64}([65.43198581212043, -9.794035232054846, 1.543137760023534], [64.99429970069268, -8.331303987175813, 1.3481431359431886], [65.27602622989178, -9.06266960961533, 1.4478363872657793], [-0.01284827600100058, 0.045203228590662406, 0.3679264858134668], 0.0) … Horseshoe{Float64}([64.99429970069268, 8.331303987175813, 1.3481431359431886], [65.43198581212043, 9.794035232054846, 1.543137760023534], [65.27602622989178, 9.06266960961533, 1.4478363872657793], [-0.01284827600100058, -0.045203228590662406, 0.3679264858134668], 0.0) Horseshoe{Float64}([65.43198581212043, 9.794035232054846, 1.543137760023534], [65.58280216782836, 10.298058069364341, 1.6103283283751404], [65.56264383893216, 10.046046650709593, 1.5786624114392451], [-0.003889780601881672, -0.013685154467470526, 0.11138874253046113], 0.0)], vtail = Horseshoe{Float64}[Horseshoe{Float64}([53.84630872675395, 0.0, 0.0], [54.18486988563193, 2.137860520751363e-17, 0.34913911868137404], [54.304865633097386, 1.0689302603756815e-17, 0.17456955934068702], [0.0, -0.40399072732320807, 2.47372975550789e-17], 0.0) Horseshoe{Float64}([54.18486988563193, 2.137860520751363e-17, 0.34913911868137404], [55.14901049489124, 8.225972198474937e-17, 1.3434032088602492], [54.9433606669902, 5.1819163596131497e-17, 0.8462711637708116], [0.0, -1.0993398152055094, 6.731514929333348e-17], 0.0) … Horseshoe{Float64}([61.439011047620205, 4.794447948899935e-16, 7.829927701992145], [62.40315165687952, 5.403259116672293e-16, 8.824191792171021], [62.05432531043232, 5.098853532786114e-16, 8.327059747081583], [0.0, -0.5299187314164414, 3.244816391186853e-17], 0.0) Horseshoe{Float64}([62.40315165687952, 5.403259116672293e-16, 8.824191792171021], [62.741712815757495, 5.617045168747429e-16, 9.173330910852394], [62.692820344325135, 5.510152142709861e-16, 8.998761351511707], [0.0, -0.16812879171659315, 1.0294919331011913e-17], 0.0); Horseshoe{Float64}([54.6849577820983, 0.0, 0.0], [55.00436743353075, 2.137860520751363e-17, 0.34913911868137404], [55.63498023034437, 1.0689302603756815e-17, 0.17456955934068702], [0.0, -1.1037231928337101, 6.7583553762425e-17], 0.0) Horseshoe{Float64}([55.00436743353075, 2.137860520751363e-17, 0.34913911868137404], [55.91396916385321, 8.225972198474937e-17, 1.3434032088602492], [56.21436308536693, 5.1819163596131497e-17, 0.8462711637708116], [0.0, -3.003452229924818, 1.8390840798847047e-16], 0.0) … Horseshoe{Float64}([61.84816105614035, 4.794447948899935e-16, 7.829927701992145], [62.75776278646279, 5.403259116672293e-16, 8.824191792171021], [62.66699118485762, 5.098853532786114e-16, 8.327059747081583], [0.0, -1.4477648981121707, 8.865003241954816e-17], 0.0) Horseshoe{Float64}([62.75776278646279, 5.403259116672293e-16, 8.824191792171021], [63.07717243789525, 5.617045168747429e-16, 9.173330910852394], [63.24637403988018, 5.510152142709861e-16, 8.998761351511707], [0.0, -0.45933640118489183, 2.8126242672147173e-17], 0.0); … ; Horseshoe{Float64}([60.65210209711992, 0.0, 0.0], [60.835245204102755, 2.137860520751363e-17, 0.34913911868137404], [61.53399127314118, 1.0689302603756815e-17, 0.17456955934068702], [0.0, -1.1037231928337075, 6.758355376242484e-17], 0.0) Horseshoe{Float64}([60.835245204102755, 2.137860520751363e-17, 0.34913911868137404], [61.3567926472095, 8.225972198474937e-17, 1.3434032088602492], [61.85121371233109, 5.1819163596131497e-17, 0.8462711637708116], [0.0, -3.0034522299248185, 1.8390840798847047e-16], 0.0) … Horseshoe{Float64}([64.75933961381432, 4.794447948899935e-16, 7.829927701992145], [65.28088705692106, 5.403259116672293e-16, 8.824191792171021], [65.38414259892375, 5.098853532786114e-16, 8.327059747081583], [0.0, -1.4477648981121634, 8.865003241954773e-17], 0.0) Horseshoe{Float64}([65.28088705692106, 5.403259116672293e-16, 8.824191792171021], [65.46403016390389, 5.617045168747429e-16, 9.173330910852394], [65.70136503811365, 5.510152142709861e-16, 8.998761351511707], [0.0, -0.4593364011848968, 2.812624267214748e-17], 0.0); Horseshoe{Float64}([61.99757944912139, 0.0, 0.0], [62.14999704659526, 2.137860520751363e-17, 0.34913911868137404], [62.36306457476276, 1.0689302603756815e-17, 0.17456955934068702], [0.0, -0.40399072732321056, 2.4737297555079053e-17], 0.0) Horseshoe{Float64}([62.14999704659526, 2.137860520751363e-17, 0.34913911868137404], [62.58404564147047, 8.225972198474937e-17, 1.3434032088602492], [62.64344182076147, 5.1819163596131497e-17, 0.8462711637708116], [0.0, -1.0993398152055023, 6.731514929333304e-17], 0.0) … Horseshoe{Float64}([65.4157549123554, 4.794447948899935e-16, 7.829927701992145], [65.84980350723059, 5.403259116672293e-16, 8.824191792171021], [65.76602316797545, 5.098853532786114e-16, 8.327059747081583], [0.0, -0.5299187314164555, 3.24481639118694e-17], 0.0) Horseshoe{Float64}([65.84980350723059, 5.403259116672293e-16, 8.824191792171021], [66.00222110470447, 5.617045168747429e-16, 9.173330910852394], [66.04640041397415, 5.510152142709861e-16, 8.998761351511707], [0.0, -0.16812879171659068, 1.0294919331011762e-17], 0.0)])Note that the fuselage is not included in the analysis for now, and is only included for plotting. Support for its aerodynamic analysis will be added soon.
Now, let's define the freestream conditions.
fs = Freestream(
alpha = 3.0, # degrees
beta = 0.0, # degrees
omega = [0., 0., 0.]
);Similarly, define the reference values. Here, the reference flight condition will be set to Mach number $M = 0.84$.
M = 0.84 # Mach number
refs = References(
sound_speed = 330.,
speed = M * 330.,
density = 1.225,
span = b_w,
area = S_w,
chord = c_w,
location = mac_w
);Let's run the aerodynamic analysis first.
system = solve_case(
aircraft, fs, refs;
compressible = true, # Compressibility option
# print = true, # Prints the results for only the aircraft
# print_components = true, # Prints the results for all components
)VortexLatticeSystem -
368 Horseshoe{Float64} Elements
Freestream:
alpha = 0.05235987755982989
beta = 0.0
omega = [0.0, 0.0, 0.0]
References:
Mach = 0.84
Reynolds = 1.9693498716934252e8
speed = 277.2
density = 1.225
viscosity = 1.5e-5
sound_speed = 330.0
area = 427.9435545
span = 60.9
chord = 8.699310326413222
location = [30.683074339989055, 0.0, -2.5]
You may receive a warning that the results are incorrect; this is due to the limitation of the vortex lattice method being able to primarily analyze only subsonic flows under the physical assumptions.
The freestream derivatives can be obtained by passing the resultant system as follows:
dvs = freestream_derivatives(
system, # VortexLatticeSystem
axes = Wind(), # Specify axis system for nearfield forces (wind by default)
# print = true, # Prints the results for only the aircraft
# print_components = true, # Prints the results for all components
# farfield = true, # Print farfield derivatives
);┌ Warning: Results in transonic to sonic flow conditions (0.7 < M < 1) are most likely incorrect!
└ @ AeroFuse.VortexLattice ~/work/AeroFuse.jl/AeroFuse.jl/src/Aerodynamics/VortexLattice/system.jl:112You can access the derivatives of each lifting surface based on the keys defined in the ComponentVector.
ac_dvs = dvs.aircraft9×7 LabelledArrays.SLArray{Tuple{9, 7}, Float64, 2, 63, (:CX, :CY, :CZ, :Cl, :Cm, :Cn, :CDi, :CY, :CL, :CX_Ma, :CY_Ma, :CZ_Ma, :Cl_Ma, :Cm_Ma, :Cn_Ma, :CDi_Ma, :CY_Ma, :CL_Ma, :CX_al, :CY_al, :CZ_al, :Cl_al, :Cm_al, :Cn_al, :CDi_al, :CY_al, :CL_al, :CX_be, :CY_be, :CZ_be, :Cl_be, :Cm_be, :Cn_be, :CDi_be, :CY_be, :CL_be, :CX_pb, :CY_pb, :CZ_pb, :Cl_pb, :Cm_pb, :Cn_pb, :CDi_pb, :CY_pb, :CL_pb, :CX_qb, :CY_qb, :CZ_qb, :Cl_qb, :Cm_qb, :Cn_qb, :CDi_qb, :CY_qb, :CL_qb, :CX_rb, :CY_rb, :CZ_rb, :Cl_rb, :Cm_rb, :Cn_rb, :CDi_rb, :CY_rb, :CL_rb)} with indices SOneTo(9)×SOneTo(7):
:CX => 0.0022641332528671202 … :CX_rb => -1.5827149500136481e-12
:CY => -3.1961713265252737e-16 :CY_rb => -3.157862528537969
:CZ => 0.488779300272069 :CZ_rb => 7.974628825637478e-13
:Cl => -3.250612744503939e-17 :Cl_rb => 0.33851067064045093
:Cm => 0.01236035958459586 :Cm_rb => -3.074932469430325e-12
:Cn => 1.6465191416848013e-16 … :Cn_rb => 1.7043644190221912
:CDi => 0.009363051346220326 :CDi_rb => 1.7542638011755205e-13
:CY => -3.196582110622816e-16 :CY_rb => -3.153938658278606
:CL => 0.4887854408983533 :CL_rb => 6.327189013599491e-15These quantities are the force and moment coefficients $(C_X, C_Y, C_Z, C_l, C_m, C_n, C_{D_{i,ff}}, C_{Y_{ff}} C_{L_{ff}})$ generated from the nearfield and farfield analyses, and their derivatives respect to the Mach number $M$, freestream angles of attack and sideslip $(\alpha, \beta)$, and the non-dimensional angular velocity rates in stability axes $(\bar{p}, \bar{q}, \bar{r})$. The keys corresponding to the freestream derivatives should be evident:
keys(dvs.aircraft)(:CX, :CY, :CZ, :Cl, :Cm, :Cn, :CDi, :CY, :CL, :CX_Ma, :CY_Ma, :CZ_Ma, :Cl_Ma, :Cm_Ma, :Cn_Ma, :CDi_Ma, :CY_Ma, :CL_Ma, :CX_al, :CY_al, :CZ_al, :Cl_al, :Cm_al, :Cn_al, :CDi_al, :CY_al, :CL_al, :CX_be, :CY_be, :CZ_be, :Cl_be, :Cm_be, :Cn_be, :CDi_be, :CY_be, :CL_be, :CX_pb, :CY_pb, :CZ_pb, :Cl_pb, :Cm_pb, :Cn_pb, :CDi_pb, :CY_pb, :CL_pb, :CX_qb, :CY_qb, :CZ_qb, :Cl_qb, :Cm_qb, :Cn_qb, :CDi_qb, :CY_qb, :CL_qb, :CX_rb, :CY_rb, :CZ_rb, :Cl_rb, :Cm_rb, :Cn_rb, :CDi_rb, :CY_rb, :CL_rb)These can be accessed either like a dictionary, or by 'dot' syntax.
ac_dvs[:CZ_al], ac_dvs.CZ_al, ac_dvs.CL_al # Lift coefficient derivative wrt. alpha(6.8101016101310154, 6.8101016101310154, 6.801125646406997)Note that the nearfield forces and moments $(C_X, C_Y, C_Z, C_l, C_m, C_n)$ depend on the axis system used ($C_Z$ is not lift if body axes are used!). You can also pretty-print the derivatives for each surface.
print_derivatives(dvs.aircraft, "Aircraft", farfield = true)
print_derivatives(dvs.wing, "Wing", farfield = true)
print_derivatives(dvs.htail, "Horizontal Tail", farfield = true)
print_derivatives(dvs.vtail, "Vertical Tail", farfield = true)─────────────────────────────────────────────────────────────────────────────────────────────
Aircraft Values Freestream Derivatives
∂/∂M ∂/∂α, 1/rad ∂/∂β, 1/rad ∂/∂p̄ ∂/∂q̄ ∂/∂r̄
─────────────────────────────────────────────────────────────────────────────────────────────
CX 0.00226413 0.00627624 0.0870775 0.0 -0.0 -1.69412 -0.0
CY -0.0 -0.0 0.0 2.61907 -0.556502 0.0 -3.15786
CZ 0.488779 0.637111 6.8101 -0.0 0.0 113.021 0.0
Cℓ -0.0 0.0 0.0 -0.0662498 -0.524179 0.0 0.338511
Cm 0.0123604 -0.0528664 -2.82087 0.0 -0.0 -92.7847 -0.0
Cn 0.0 0.0 -0.0 -1.38315 0.224446 -0.0 1.70436
CDi 0.00936305 0.0246163 0.195791 -0.0 0.0 2.55536 0.0
CY -0.0 -0.0 0.0 2.60569 -0.658042 0.0 -3.15394
CL 0.488785 0.636387 6.80113 0.0 0.0 112.523 0.0
─────────────────────────────────────────────────────────────────────────────────────────────
────────────────────────────────────────────────────────────────────────────────────────────
Wing Values Freestream Derivatives
∂/∂M ∂/∂α, 1/rad ∂/∂β, 1/rad ∂/∂p̄ ∂/∂q̄ ∂/∂r̄
────────────────────────────────────────────────────────────────────────────────────────────
CX 0.00334069 0.00925357 0.0763912 -0.0 -0.0 -2.68788 0.0
CY -0.0 -0.0 0.0 -0.0402579 -0.0387053 0.0 0.113127
CZ 0.52069 0.706475 6.06875 0.0 0.0 88.5232 -0.0
Cℓ -0.0 0.0 -0.0 -0.132842 -0.50224 -0.0 0.387917
Cm -0.0960197 -0.290382 -0.311637 -0.0 0.0 -9.07936 -0.0
Cn -0.0 -0.0 -0.0 -0.0021049 -0.0452094 -0.0 -0.00205007
CDi 0.00909333 0.023449 0.208213 0.0 0.0 2.96871 -0.0
CY -0.0 0.0 -0.0 -0.0341629 -0.145646 0.0 0.0852901
CL 0.52054 0.705389 6.06549 0.0 0.0 88.1339 -0.0
────────────────────────────────────────────────────────────────────────────────────────────
────────────────────────────────────────────────────────────────────────────────────────────────────────
Horizontal Tail Values Freestream Derivatives
∂/∂M ∂/∂α, 1/rad ∂/∂β, 1/rad ∂/∂p̄ ∂/∂q̄ ∂/∂r̄
────────────────────────────────────────────────────────────────────────────────────────────────────────
CX -0.00107656 -0.00297734 0.0106863 0.0 -0.0 0.993763 -0.0
CY 0.0 0.0 -0.0 -0.0862034 0.0109909 -0.0 0.130153
CZ -0.0319106 -0.0693637 0.741355 0.0 -0.0 24.4973 0.0
Cℓ 0.0 0.0 -0.0 -0.0547872 0.00257755 -0.0 0.0569628
Cm 0.10838 0.237515 -2.50923 -0.0 -0.0 -83.7053 -0.0
Cn -0.0 -0.0 0.0 0.0454955 -0.00568786 0.0 -0.068279
CDi 0.00026972 0.00116723 -0.012422 -0.0 0.0 -0.413348 0.0
CY 0.0 0.0 -0.0 -0.0691799 0.00944901 -0.0 0.11129
CL -0.0317545 -0.0690021 0.735637 0.0 -0.0 24.3893 0.0
────────────────────────────────────────────────────────────────────────────────────────────────────────
─────────────────────────────────────────────────────────────────────────────────────
Vertical Tail Values Freestream Derivatives
∂/∂M ∂/∂α, 1/rad ∂/∂β, 1/rad ∂/∂p̄ ∂/∂q̄ ∂/∂r̄
─────────────────────────────────────────────────────────────────────────────────────
CX -0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0
CY -0.0 -0.0 0.0 2.74554 -0.528787 0.0 -3.40114
CZ 0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0
Cℓ -0.0 -0.0 0.0 0.12138 -0.0245174 0.0 -0.106369
Cm -0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0
Cn 0.0 0.0 -0.0 -1.42654 0.275343 -0.0 1.77469
CDi 0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0
CY -0.0 -0.0 0.0 2.70903 -0.521845 0.0 -3.35052
CL 0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0
─────────────────────────────────────────────────────────────────────────────────────Static Stability Analysis
You can evaluate the static stability of the aircraft using the various quantities computed in this process, following standard computations and definitions in aircraft design and stability analysis.
l_h = x_h - x_w # Horizontal tail moment arm
V_h = S_h / S_w * l_h / c_w # Horizontal tail volume coefficient
l_v = x_v - x_w # Vertical tail moment arm
V_v = S_v / S_w * l_v / b_w # Vertical tail volume coefficient
x_cp = -refs.chord * ac_dvs.Cm / ac_dvs.CZ # Center of pressure
x_np_lon = -refs.chord * ac_dvs.Cm_al / ac_dvs.CZ_al # Neutral point
# Position vectors
x_np, y_np, z_np = r_np = refs.location + [ x_np_lon; zeros(2) ] # Neutral point
x_cp, y_cp, z_cp = r_cp = refs.location + [ x_cp; zeros(2) ] # Center of pressure
@info "Horizontal TVC V_h:" V_h
@info "Vertical TVC V_v:" V_v
@info "Wing Aerodynamic Center x_ac (m):" x_w
@info "Neutral Point x_np (m):" x_np
@info "Center of Pressure x_cp (m):" x_cp┌ Info: Horizontal TVC V_h:
└ V_h = 0.7977744718731725
┌ Info: Vertical TVC V_v:
└ V_v = 0.06132559058467538
┌ Info: Wing Aerodynamic Center x_ac (m):
└ x_w = 30.683074339989055
┌ Info: Neutral Point x_np (m):
└ x_np = 34.286491892544305
┌ Info: Center of Pressure x_cp (m):
└ x_cp = 30.463084246888542Let's plot the relevant quantities.
stab_plt = plot(
xaxis = L"x", yaxis = L"y", zaxis = L"z",
aspect_ratio = 1,
zlim = (-0.5, 0.5) .* span(wing_mesh),
camera = (30,60),
)
plot!(fuse, label = "Fuselage", alpha = 0.6)
plot!(stab_plt, wing_mesh, label = "Wing", mac = false)
plot!(stab_plt, htail_mesh, label = "Horizontal Tail", mac = false)
plot!(stab_plt, vtail_mesh, label = "Vertical Tail", mac = false)
scatter!(Tuple(r_np), color = :orange, label = "Neutral Point")
scatter!(Tuple(r_cp), color = :brown, label = "Center of Pressure")This page was generated using Literate.jl.