/
dynamic-joined-wing.jl
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/
dynamic-joined-wing.jl
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# # [Time-Domain Simulation of a Joined-Wing](@id dynamic-joined-wing)
#
# In this example we use the same joined-wing model as used in the [previous example](@ref
# static-joined-wing), but with the following time varying loads applied at the wingtip:
# - A piecewise-linear load ``F_L`` in the x and y-directions defined as follows:
# ```math
# F_L(t) = \begin{cases}
# t10^6 \text{ N} & 0 \leq t \leq 0.01 \\
# (0.02-t)10^6 & 0.01 \leq t \leq 0.02 \\
# 0 & \text{otherwise}
# \end{cases}
# ```
# - A sinusoidal load ``F_S`` applied in the z-direction defined as follows:
# ```math
# F_S(t) = \begin{cases}
# 0 & t \lt 0 \\
# 5 \times 10^3 (1-\cos(\pi t /0.02)) \text{ N} & 0 \leq t \lt 0.02 \\
# 10^4 \text{ N} & 0.02 \leq t
# \end{cases}
# ```
#
# We will also use the same compliance and mass matrix for all beams, in order to simplify
# the problem definition.
#
# 
#
#-
#md # !!! tip
#md # This example is also available as a Jupyter notebook:
#md # [`dynamic-joined-wing.ipynb`](@__NBVIEWER_ROOT_URL__/examples/dynamic-joined-wing.ipynb).
#-
using GXBeam, LinearAlgebra
## Set endpoints of each beam
p1 = [0, 0, 0]
p2 = [-7.1726, -12, -3.21539]
p3 = [7.1726, -12, 3.21539]
Cab_1 = [
0.5 0.866025 0.0
0.836516 -0.482963 0.258819
0.224144 -0.12941 -0.965926
]
Cab_2 = [
0.5 0.866025 0.0
-0.836516 0.482963 0.258819
0.224144 -0.12941 0.965926
]
## beam 1
L_b1 = norm(p1-p2)
r_b1 = p2
nelem_b1 = 8
lengths_b1, xp_b1, xm_b1, Cab_b1 = discretize_beam(L_b1, r_b1, nelem_b1, frame=Cab_1)
## beam 2
L_b2 = norm(p3-p1)
r_b2 = p1
nelem_b2 = 8
lengths_b2, xp_b2, xm_b2, Cab_b2 = discretize_beam(L_b2, r_b2, nelem_b2, frame=Cab_2)
## combine elements and points into one array
nelem = nelem_b1 + nelem_b2
points = vcat(xp_b1, xp_b2[2:end])
start = 1:nelem
stop = 2:nelem + 1
lengths = vcat(lengths_b1, lengths_b2)
midpoints = vcat(xm_b1, xm_b2)
Cab = vcat(Cab_b1, Cab_b2)
## assign all beams the same compliance and mass matrix
compliance = fill(Diagonal([2.93944738387698e-10, 8.42991725049126e-10,
3.38313996669689e-08, 4.69246721094557e-08, 6.79584100559513e-08,
1.37068861370898e-09]), nelem)
mass = fill(Diagonal([4.86e-2, 4.86e-2, 4.86e-2, 1.0632465e-2, 2.10195e-4,
1.042227e-2]), nelem)
## create assembly
assembly = Assembly(points, start, stop;
compliance = compliance,
mass = mass,
frames = Cab,
lengths = lengths,
midpoints = midpoints)
F_L = (t) -> begin
if 0.0 <= t < 0.01
1e6*t
elseif 0.01 <= t < 0.02
-1e6*(t-0.02)
else
zero(t)
end
end
F_S = (t) -> begin
if t < 0.0
zero(t)
elseif 0.0 <= t < 0.02
5e3*(1-cos(pi*t/0.02))
else
1e4
end
end
## assign boundary conditions and point load
prescribed_conditions = (t) -> begin
Dict(
## fixed endpoint on beam 1
1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0, theta_z=0),
## force applied on point 4
nelem_b1 + 1 => PrescribedConditions(Fx=F_L(t), Fy=F_L(t), Fz=F_S(t)),
## fixed endpoint on last beam
nelem+1 => PrescribedConditions(ux=0, uy=0, uz=0, theta_x=0, theta_y=0, theta_z=0),
)
end
## time
t = range(0, 0.04, length=1001)
system, history, converged = time_domain_analysis(assembly, t;
prescribed_conditions=prescribed_conditions,
structural_damping=false)
#!jl nothing #hide
# We can visualize tip displacements and the resultant forces accessing the post-processed
# results for each time step contained in the variable `history`. Note that the fore-root
# and rear-root resultant forces for this case are equal to the external forces/moments,
# but with opposite sign.
#md using Suppressor #hide
#md @suppress_err begin #hide
using Plots
pyplot()
point = vcat(fill(nelem_b1+1, 6), fill(1, 6))
field = [:u, :u, :u, :theta, :theta, :theta, :F, :F, :F, :M, :M, :M]
direction = [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3]
ylabel = ["\$u_x\$ (\$m\$)", "\$u_y\$ (\$m\$)", "\$u_z\$ (\$m\$)",
"Rodriguez Parameter \$\\theta_x\$", "Rodriguez Parameter \$\\theta_y\$",
"Rodriguez Parameter \$\\theta_z\$", "\$F_x\$ at the forewing root (\$N\$)",
"\$F_y\$ at the forewing root (\$N\$)", "\$F_z\$ at the forewing root (\$N\$)",
"\$M_x\$ at the forewing root (\$Nm\$)", "\$M_y\$ at the forewing root (\$Nm\$)",
"\$M_z\$ at the forewing root (\$N\$)"]
#nb ph = Vector{Any}(undef, 12)
for i = 1:12
#nb ph[i] = plot(
plot( #!nb
xlim = (0, 0.04),
xticks = 0:0.01:0.04,
xlabel = "Time (s)",
ylabel = ylabel[i],
grid = false,
overwrite_figure=false
)
y = [getproperty(state.points[point[i]], field[i])[direction[i]] for state in history]
if field[i] == :theta
## convert to angle
@. y = 4*atan(y/4)
end
if field[i] == :F || field[i] == :M
y = -y
end
plot!(t, y, label="")
plot!(show=true) #!nb
#md savefig("../assets/dynamic-joined-wing-"*string(field[i])*string(direction[i])*".svg"); #hide
#md closeall() #hide
end
#nb ph[1]
#nb #-
#nb ph[2]
#nb #-
#nb ph[3]
#nb #-
#nb ph[4]
#nb #-
#nb ph[5]
#nb #-
#nb ph[6]
#nb #-
#nb ph[7]
#nb #-
#nb ph[8]
#nb #-
#nb ph[9]
#nb #-
#nb ph[10]
#nb #-
#nb ph[11]
#nb #-
#nb ph[12]
#nb #-
#md end #hide
#md nothing #hide
#md # 
#md # 
#md # 
#md # 
#md # 
#md # 
#md # 
#md # 
#md # 
#md # 
#md # 
#md # 
#-
# These graphs are identical to those presented in "GEBT: A general-purpose nonlinear analysis tool for composite beams" by Wenbin Yu and Maxwell Blair.
#
# We can also visualize the time history of the system using ParaView. In order to view the small deflections we'll scale all the deflections up by a couple orders of magnitude. We'll also set the color gradient to match the magnitude of the deflections at each point.
airfoil = [ #FX 60-100 airfoil
0.0000000 0.0000000;
0.0010700 0.0057400;
0.0042800 0.0114400;
0.0096100 0.0177500;
0.0170400 0.0236800;
0.0265300 0.0294800;
0.0380600 0.0352300;
0.0515600 0.0405600;
0.0669900 0.0460900;
0.0842700 0.0508600;
0.1033200 0.0556900;
0.1240800 0.0598900;
0.1464500 0.0640400;
0.1703300 0.0675400;
0.1956200 0.0708100;
0.2222100 0.0733900;
0.2500000 0.0756500;
0.2788600 0.0772000;
0.3086600 0.0783800;
0.3392800 0.0788800;
0.3705900 0.0789800;
0.4024500 0.0784500;
0.4347400 0.0775000;
0.4673000 0.0759600;
0.5000000 0.0740900;
0.5327000 0.0717400;
0.5652600 0.0691100;
0.5975500 0.0660800;
0.6294100 0.0627500;
0.6607200 0.0590500;
0.6913400 0.0551100;
0.7211400 0.0508900;
0.7500000 0.0465200;
0.7777900 0.0420000;
0.8043801 0.0374700;
0.8296700 0.0329800;
0.8535500 0.0286400;
0.8759201 0.0244700;
0.8966800 0.0205300;
0.9157300 0.0168100;
0.9330100 0.0134200;
0.9484400 0.0103500;
0.9619400 0.0076600;
0.9734700 0.0053400;
0.9829600 0.0034100;
0.9903900 0.0019300;
0.9957200 0.0008600;
0.9989300 0.0002300;
1.0000000 0.0000000;
0.9989300 0.0001500;
0.9957200 0.0007000;
0.9903900 0.0015100;
0.9829600 0.00251;
0.9734700 0.00377;
0.9619400 0.00515;
0.9484400 0.00659;
0.9330100 0.00802;
0.9157300 0.00941;
0.8966800 0.01072;
0.8759201 0.01186;
0.8535500 0.0128;
0.8296700 0.01347;
0.8043801 0.01381;
0.7777900 0.01373;
0.7500000 0.01329;
0.7211400 0.01241;
0.6913400 0.01118;
0.6607200 0.00951;
0.6294100 0.00748;
0.5975500 0.00496;
0.5652600 0.00217;
0.532700 -0.00092;
0.500000 -0.00405;
0.467300 -0.00731;
0.434740 -0.01045;
0.402450 -0.01357;
0.370590 -0.01637;
0.339280 -0.01895;
0.308660 -0.021;
0.278860 -0.02275;
0.250000 -0.02389;
0.222210 -0.02475;
0.195620 -0.025;
0.170330 -0.02503;
0.146450 -0.02447;
0.124080 -0.02377;
0.103320 -0.02246;
0.084270 -0.0211;
0.066990 -0.01913;
0.051560 -0.0173;
0.038060 -0.01481;
0.026530 -0.01247;
0.017040 -0.0097;
0.009610 -0.00691;
0.004280 -0.00436;
0.001070 -0.002;
0.0 0.0;
]
section = zeros(3, size(airfoil, 1))
for ic = 1:size(airfoil, 1)
section[1,ic] = airfoil[ic,1] - 0.5
section[2,ic] = 0
section[3,ic] = airfoil[ic,2]
end
mkpath("dynamic-joined-wing-simulation")
write_vtk("dynamic-joined-wing-simulation/dynamic-joined-wing-simulation", assembly, history, t, scaling=1e2;
sections = section)
#md rm("dynamic-joined-wing-simulation"; recursive=true) #hide
# 