def __init__(self, Ns):
super(Failures, self).__init__()
# inputs
self.add('flags', VarTree(Flags(), iotype='in'))
self.add('yN', Array(np.zeros(Ns + 1), iotype='in', desc=''))
self.add('Finternal', Array(np.zeros((6, Ns + 1)),
iotype='in',
desc=''))
self.add('strain', VarTree(Strain(Ns), iotype='in'))
self.add('d', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('theta', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('nTube', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('nCap', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('yWire', Array([0], iotype='in', desc=''))
self.add('zWire', Float(0., iotype='in', desc=''))
self.add('EIxJ', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('EIzJ', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('lBiscuit', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('dQuad', Float(0., iotype='in', desc=''))
self.add('thetaQuad', Float(0., iotype='in', desc=''))
self.add('nTubeQuad', Float(0., iotype='in', desc=''))
self.add('lBiscuitQuad', Float(0., iotype='in', desc=''))
self.add('RQuad', Float(0., iotype='in', desc=''))
self.add('hQuad', Float(0., iotype='in', desc=''))
self.add('EIQuad', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('GJQuad', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('tWire', Float(0., iotype='in', desc=''))
self.add('TWire', Array([0], iotype='in', desc=''))
self.add('TEtension', Float(0., iotype='in', desc=''))
# all this to get TQuad... maybe should be split out
self.add('b', Int(0, iotype='in', desc='number of blades'))
self.add('fblade', VarTree(Fblade(Ns), iotype='in'))
self.add('mSpar', Array(np.zeros(Ns),
iotype='in',
desc='mass of spars'))
self.add('mChord',
Array(np.zeros(Ns), iotype='in', desc='mass of chords'))
self.add('mElseRotor', Float(0., iotype='in', desc=''))
# outputs
self.add('fail', VarTree(Failure(Ns), iotype='out'))
class Structures(Assembly):
"""
structural computation, first computes the mass of the helicopter based on
the structural description of the spars and chord lengths. It then
computes the deformation of the spars, the strains, and the resulting
factor of safety for each of the failure modes.
"""
# flags
flags = VarTree(Flags(), iotype='in')
# inputs for spars
yN = Array(iotype='in',
desc='node locations for each element along the span')
d = Array(iotype='in', desc='spar diameter')
theta = Array(iotype='in', desc='wrap angle')
nTube = Array(iotype='in', desc='number of tube layers')
nCap = Array(iotype='in', desc='number of cap strips')
lBiscuit = Array(iotype='in', desc='unsupported biscuit length')
# joint properties
Jprop = VarTree(JointProperties(), iotype='in')
# inputs for chord
b = Int(iotype='in', desc='number of blades')
cE = Array(iotype='in', desc='chord of each element')
xEA = Array(iotype='in', desc='')
xtU = Array(iotype='in', desc='')
# inputs for quad
dQuad = Float(iotype='in', desc='diameter of quad rotor struts')
thetaQuad = Float(iotype='in', desc='wrap angle of quad rotor struts')
nTubeQuad = Int(iotype='in',
desc='number of CFRP layers in quad rotor struts')
lBiscuitQuad = Float(iotype='in', desc='')
RQuad = Float(
iotype='in',
desc='distance from centre of helicopter to centre of quad rotors')
hQuad = Float(iotype='in', desc='height of quad-rotor truss')
# inputs for cover
ycmax = Float(iotype='in', desc='')
# inputs for wire
yWire = Array(iotype='in', desc='location of wire attachment along span')
zWire = Float(iotype='in', desc='depth of wire attachement')
tWire = Float(iotype='in', desc='thickness of wire')
TWire = Array(iotype='in', desc='')
TEtension = Float(iotype='in', desc='')
# inputs for 'other stuff'
mElseRotor = Float(iotype='in', desc='')
mElseCentre = Float(iotype='in', desc='')
mElseR = Float(iotype='in', desc='')
R = Float(iotype='in', desc='rotor radius')
mPilot = Float(iotype='in', desc='mass of pilot')
# inputs for FEM
fblade = VarTree(Fblade(), iotype='in')
presLoad = VarTree(PrescribedLoad(), iotype='in')
def configure(self):
self.add('spar', SparProperties())
self.connect('yN', 'spar.yN')
self.connect('d', 'spar.d')
self.connect('theta', 'spar.theta')
self.connect('nTube', 'spar.nTube')
self.connect('nCap', 'spar.nCap')
self.connect('lBiscuit', 'spar.lBiscuit')
self.connect('flags.CFRPType', 'spar.CFRPType')
self.add('joint', JointSparProperties())
self.connect('flags.CFRPType', 'joint.CFRPType')
self.connect('Jprop', 'joint.Jprop')
self.add('chord', ChordProperties())
self.connect('yN', 'chord.yN')
self.connect('cE', 'chord.c')
self.connect('d', 'chord.d')
self.connect('flags.GWing', 'chord.GWing')
self.connect('xtU', 'chord.xtU')
self.add('quad', QuadSparProperties())
self.connect('dQuad', 'quad.dQuad')
self.connect('thetaQuad', 'quad.thetaQuad')
self.connect('nTubeQuad', 'quad.nTubeQuad')
self.connect('lBiscuitQuad', 'quad.lBiscuitQuad')
self.connect('flags.CFRPType', 'quad.CFRPType')
self.connect('RQuad', 'quad.RQuad')
self.connect('hQuad', 'quad.hQuad')
self.add('mass', MassProperties())
self.connect('flags', 'mass.flags')
self.connect('b', 'mass.b')
self.connect('spar.mSpar', 'mass.mSpar')
self.connect('chord.mChord', 'mass.mChord')
self.connect('chord.xCGChord', 'mass.xCGChord')
self.connect('quad.mQuad', 'mass.mQuad')
self.connect('xEA', 'mass.xEA')
self.connect('ycmax', 'mass.ycmax')
self.connect('zWire', 'mass.zWire')
self.connect('yWire', 'mass.yWire')
self.connect('tWire', 'mass.tWire')
self.connect('mElseRotor', 'mass.mElseRotor')
self.connect('mElseCentre', 'mass.mElseCentre')
self.connect('mElseR', 'mass.mElseR')
self.connect('R', 'mass.R')
self.connect('mPilot', 'mass.mPilot')
self.add('fem', FEM())
self.connect('flags', 'fem.flags')
self.connect('yN', 'fem.yN')
self.connect('spar.EIx', 'fem.EIx')
self.connect('spar.EIz', 'fem.EIz')
self.connect('spar.EA', 'fem.EA')
self.connect('spar.GJ', 'fem.GJ')
self.connect('cE', 'fem.cE')
self.connect('xEA', 'fem.xEA')
self.connect('spar.mSpar', 'fem.mSpar')
self.connect('chord.mChord', 'fem.mChord')
self.connect('mass.xCG', 'fem.xCG')
self.connect('zWire', 'fem.zWire')
self.connect('yWire', 'fem.yWire')
self.connect('TWire', 'fem.TWire')
self.connect('fblade', 'fem.fblade')
self.connect('presLoad', 'fem.presLoad')
self.add('strains', Strains())
self.connect('yN', 'strains.yN')
self.connect('d', 'strains.d')
self.connect('fem.k', 'strains.k')
self.connect('fem.F', 'strains.F')
self.connect('fem.q', 'strains.q')
self.add('failure', Failures())
self.connect('flags', 'failure.flags')
self.connect('yN', 'failure.yN')
self.connect('strains.Finternal', 'failure.Finternal')
self.connect('strains.strain', 'failure.strain')
self.connect('d', 'failure.d')
self.connect('theta', 'failure.theta')
self.connect('nTube', 'failure.nTube')
self.connect('nCap', 'failure.nCap')
self.connect('yWire', 'failure.yWire')
self.connect('zWire', 'failure.zWire')
self.connect('joint.EIx', 'failure.EIxJ')
self.connect('joint.EIz', 'failure.EIzJ')
self.connect('lBiscuit', 'failure.lBiscuit')
self.connect('dQuad', 'failure.dQuad')
self.connect('thetaQuad', 'failure.thetaQuad')
self.connect('nTubeQuad', 'failure.nTubeQuad')
self.connect('lBiscuitQuad', 'failure.lBiscuitQuad')
self.connect('RQuad', 'failure.RQuad')
self.connect('hQuad', 'failure.hQuad')
self.connect('quad.EIx', 'failure.EIQuad')
self.connect('quad.GJ', 'failure.GJQuad')
self.connect('tWire', 'failure.tWire')
self.connect('TWire', 'failure.TWire')
self.connect('TEtension', 'failure.TEtension')
self.connect('b', 'failure.b')
self.connect('fblade', 'failure.fblade')
self.connect('spar.mSpar', 'failure.mSpar')
self.connect('chord.mChord', 'failure.mChord')
self.connect('mElseRotor', 'failure.mElseRotor')
# link up the outputs
self.create_passthrough('mass.Mtot')
self.create_passthrough('fem.q')
self.create_passthrough('strains.Finternal')
self.create_passthrough('strains.strain')
self.create_passthrough('failure.fail')
self.driver.workflow.add('chord')
self.driver.workflow.add('fem')
self.driver.workflow.add('joint')
self.driver.workflow.add('mass')
self.driver.workflow.add('quad')
self.driver.workflow.add('spar')
self.driver.workflow.add('strains')
self.driver.workflow.add('failure')
class Failures(Component):
"""
Computes the factor of safety for each of the failure modes of the spar.
"""
# inputs
flags = VarTree(Flags(), iotype='in')
yN = Array(iotype='in', desc='')
Finternal = Array(iotype='in', desc='')
strain = VarTree(Strain(), iotype='in')
d = Array(iotype='in', desc='')
theta = Array(iotype='in', desc='')
nTube = Array(iotype='in', desc='')
nCap = Array(iotype='in', desc='')
yWire = Array(iotype='in', desc='')
zWire = Float(iotype='in', desc='')
EIxJ = Array(iotype='in', desc='')
EIzJ = Array(iotype='in', desc='')
lBiscuit = Array(iotype='in', desc='')
dQuad = Float(iotype='in', desc='')
thetaQuad = Float(iotype='in', desc='')
nTubeQuad = Float(iotype='in', desc='')
lBiscuitQuad = Float(iotype='in', desc='')
RQuad = Float(iotype='in', desc='')
hQuad = Float(iotype='in', desc='')
EIQuad = Array(iotype='in', desc='')
GJQuad = Array(iotype='in', desc='')
tWire = Float(iotype='in', desc='')
TWire = Array(iotype='in', desc='')
TEtension = Float(iotype='in', desc='')
# all this to get TQuad... maybe should be split out
b = Int(iotype='in', desc='number of blades')
fblade = VarTree(Fblade(), iotype='in')
mSpar = Array(iotype='in', desc='mass of spars')
mChord = Array(iotype='in', desc='mass of chords')
mElseRotor = Float(iotype='in', desc='')
# outputs
fail = VarTree(Failure(), iotype='out')
def execute(self):
# Compute factor of safety for each failure mode
# ----------------------------------------------
# computes material failure, euler buckling failure and torsional
# buckling failure. All failure modes are computed at the nodes.
#
# For material failure it looks at a sample laminate on the top,
# bottom, back and front of the spar. For each side of the spar
# it computes the material failure for the cap, the plus angle
# plys and the minus angle plys. For each ply it computes the
# fracture of failure in the fibre direction, matrix direction
# and shear. Thus there are 4x3x3=36 (side x lamina x direction)
# possible failure modes in the tube.
# ex. fail.top.cap = 3x(Ns+1) vector
# fail.top.plus = 3x(Ns+1) vector
# fail.top.minus = 3x(Ns+1) vector
#
# Stresses and strain are given as a 3x(Ns+1) vector
# ex. sigma_11, sigma_22, sigma_12
# Positive sign denotes tensile stresses, negative denotes compressive
# factor of safety for each failure mode
# short aliases
flags = self.flags
yN = self.yN
Finternal = self.Finternal
strain = self.strain
d = self.d
theta = self.theta
nTube = self.nTube
nCap = self.nCap
yWire = self.yWire
zWire = self.zWire
EIxJ = self.EIxJ
EIzJ = self.EIzJ
lBiscuit = self.lBiscuit
dQuad = self.dQuad
thetaQuad = self.thetaQuad
nTubeQuad = self.nTubeQuad
lBiscuitQuad = self.lBiscuitQuad
RQuad = self.RQuad
hQuad = self.hQuad
EIQuad = self.EIQuad[0] # convert single element array to scalar
GJQuad = self.GJQuad[0] # convert single element array to scalar
tWire = self.tWire
TWire = self.TWire
TEtension = self.TEtension
b = self.b
fblade = self.fblade
mSpar = self.mSpar
mChord = self.mChord
mElseRotor = self.mElseRotor
TQuad = np.sum(fblade.Fz) * b - (np.sum(mSpar + mChord) * b +
mElseRotor / 4) * 9.81
Ns = max(yN.shape) - 1 # number of elements
fail = Failure()
# Material failure
fail.top = self.material_failure(Ns, strain.top, theta, nCap, flags)
fail.bottom = self.material_failure(Ns, strain.bottom, theta, nCap,
flags)
fail.back = self.material_failure(Ns, strain.back, theta, [], flags)
fail.front = self.material_failure(Ns, strain.front, theta, [], flags)
# Euler Buckling failure in main spar from wire force
k = 0.7 # pinned-pinned = 1, fixed-pinned = 0.7 with correction factor
#kk = 1.42 # correction factor
kk = 1 # correction factor was in error, never saw buckling failure
thetaWire = atan2(zWire, yWire)
L = yWire # wire attachment provides pinned end
F = TWire * cos(thetaWire) + TEtension
fail.buckling.x = np.zeros(Ns + 1)
fail.buckling.z = np.zeros(Ns + 1)
for s in range(Ns):
if yN[s] <= yWire:
critical_load_x = pi**2 * EIxJ / (k * L)**2
critical_load_z = pi**2 * EIzJ / (k * L)**2
fail.buckling.x[s] = kk * F / critical_load_x
fail.buckling.z[s] = kk * F / critical_load_z
else:
fail.buckling.x[s] = 0
fail.buckling.z[s] = 0
# no buckling at tip
fail.buckling.x[Ns] = 0
fail.buckling.z[Ns] = 0
# Torsional Buckling failure
fail.buckling.torsion = self.torsional_buckling_failure(
Ns, Finternal, d, theta, nTube, nCap, lBiscuit, flags)
# Quad Buckling failure
if EIQuad != 0:
k = 1
L = sqrt(RQuad**2 + hQuad**2)
alpha = atan2(hQuad, RQuad)
P = TQuad / sin(alpha)
critical_load = pi**2 * EIQuad / (k * L)**2
fail.quad_buckling = P / critical_load
else:
fail.quad_buckling = 0
# Quad bending moment failure (does not include torsion since they don't occur at the same time)
RotorMoment = 1400
if EIQuad != 0:
TbottomWire = TQuad / tan(alpha)
BM = TbottomWire * zWire + RotorMoment
strainQuad = -np.array([BM * (dQuad / 2) / EIQuad, 0, 0]).reshape(
1, -1).T # strain on compression side
mf = self.material_failure(1, strainQuad, [thetaQuad], [], flags)
fail.quad_bend = abs(
mf.plus[0,
0]) # use only compressive failure in fibre direction
else:
fail.quad_bend = 0
# Quad torsional material failure
if GJQuad != 0:
strainQuad = np.array([0, 0, dQuad / 2 * RotorMoment / GJQuad
]).reshape(1, -1).T
mf = self.material_failure(1, strainQuad, [thetaQuad], [], flags)
fail.quad_torsion = abs(mf.plus[0, 0])
else:
fail.quad_torsion = 0
# Quad torsional buckling failure
FRotorMoment = np.array([0, 0, 0, 0, RotorMoment]).reshape(1, -1).T
tbf = self.torsional_buckling_failure(1, FRotorMoment, [dQuad],
[thetaQuad], [nTubeQuad], [0],
[lBiscuitQuad], flags)
fail.quad_torbuck = tbf[0]
# Wire tensile failure
wire_props = wireProperties[flags.WireType]
fail.wire = np.zeros(len(yWire))
for i in range(len(yWire)):
stress_wire = TWire[i] / (pi * (tWire / 2)**2)
fail.wire[i] = stress_wire / wire_props['ULTIMATE']
self.fail = fail
def material_failure(self, Ns, strain, theta, nCap, flags):
failure = MaterialFailure()
failure.cap = np.zeros((3, Ns + 1))
failure.plus = np.zeros((3, Ns + 1))
failure.minus = np.zeros((3, Ns + 1))
# Material Properties
tube_props = prepregProperties[flags.CFRPType]
# Cap Prepreg Properties (MTM28-M46J 140 37 %RW 12")
cap_props = prepregProperties[flags.CFRPType]
# Populate Q matrix for tube
Q_TUBE = np.zeros((3, 3))
Q_TUBE[0, 0] = tube_props['E_11']
Q_TUBE[1, 1] = tube_props['E_22']
Q_TUBE[0, 1] = tube_props['E_22'] * tube_props['V_12']
Q_TUBE[1, 0] = Q_TUBE[0, 1]
Q_TUBE[2, 2] = tube_props['G_12']
# Populate Q matrix for caps
Q_CAP = np.zeros((3, 3))
Q_CAP[0, 0] = cap_props['E_11']
Q_CAP[1, 1] = cap_props['E_22']
Q_CAP[0, 1] = cap_props['E_22'] * cap_props['V_12']
Q_CAP[1, 0] = Q_CAP[0, 1]
Q_CAP[2, 2] = cap_props['G_12']
stress = MaterialFailure()
stress.cap = np.zeros((3, Ns))
stress.plus = np.zeros((3, Ns))
stress.minus = np.zeros((3, Ns))
for s in range(Ns):
# Compute stresses in structural axes for each lamina angle
# Q is the matrix of elastic constants in the material axis
# Q_bar is the matrix of elastic constants in the structural axes for a
# lamina at a ply angle theta.
# Failure is computed at each node, but using the ply angle of the
# element (this shouldn't cause a large discrepency). Failure at tip is
# zero, since stresses/strains at tip are zero.
# Transform the elastic constants for the plus angle ply
x = theta[s] # Composite angle (in radians)
T_PLUS = np.array([[cos(x)**2,
sin(x)**2, 2 * sin(x) * cos(x)],
[sin(x)**2,
cos(x)**2, -2 * sin(x) * cos(x)],
[
-sin(x) * cos(x),
sin(x) * cos(x), (cos(x)**2) - (sin(x)**2)
]])
# MATLAB version: Qbar_TUBE_PLUS = (T_PLUS\Q_TUBE)/T_PLUS'
tmp = np.linalg.solve(T_PLUS, Q_TUBE)
Qbar_TUBE_PLUS = np.linalg.solve(T_PLUS, tmp.T)
# Transform the elastic constants for the minus angle ply
x = -theta[s] # Composite angle (in radians)
T_MINUS = np.array([[cos(x)**2,
sin(x)**2, 2 * sin(x) * cos(x)],
[sin(x)**2,
cos(x)**2, -2 * sin(x) * cos(x)],
[
-sin(x) * cos(x),
sin(x) * cos(x), (cos(x)**2) - (sin(x)**2)
]])
# MATLAB version: Qbar_TUBE_MINUS = (T_MINUS\Q_TUBE)/T_MINUS'
tmp = np.linalg.solve(T_MINUS, Q_TUBE)
Qbar_TUBE_MINUS = np.linalg.solve(T_MINUS, tmp.T)
# Compute stresses in structural coordinates
stress.cap[:, s] = np.dot(Q_CAP, strain[:,
s]) # using Q for the cap
stress.plus[:, s] = np.dot(
Qbar_TUBE_PLUS, strain[:,
s]) # using Q_bar for the + angle plys
stress.minus[:, s] = np.dot(
Qbar_TUBE_MINUS, strain[:,
s]) # using Q_bar for the - angle plys
# Transform stresses to material axes for each lamina angle
stress.plus[:, s] = np.dot(T_PLUS, stress.plus[:, s])
stress.minus[:, s] = np.dot(T_MINUS, stress.minus[:, s])
# Determine fraction of failure for each lamina angle
# ULTIMATE_11_TENS and ULTIMATE_11_COMP are both positive values
# indicating the maximum tensile and compressive stress before failure.
# failure.cap[0,s] will be positive for tensile failures and negative for
# compressive failures.
# Cap failure
if len(nCap) == 0 or (nCap == 0).all():
failure.cap[0, s] = 0
failure.cap[1, s] = 0
failure.cap[2, s] = 0
else:
if stress.cap[0, s] > 0: # tensile stress in fibre
failure.cap[
0,
s] = stress.cap[0, s] / cap_props['ULTIMATE_11_TENS']
else:
failure.cap[
0,
s] = stress.cap[0, s] / cap_props['ULTIMATE_11_COMP']
if stress.cap[1, s] > 0: # tensile stress in matrix
failure.cap[
1,
s] = stress.cap[1, s] / cap_props['ULTIMATE_22_TENS']
else:
failure.cap[
1,
s] = stress.cap[1, s] / cap_props['ULTIMATE_22_COMP']
failure.cap[2, s] = stress.cap[2, s] / cap_props['ULTIMATE_12']
# Plus angle ply failure
if stress.plus[0, s] > 0: # tensile stress in fibre
failure.plus[
0, s] = stress.plus[0, s] / tube_props['ULTIMATE_11_TENS']
else:
failure.plus[
0, s] = stress.plus[0, s] / tube_props['ULTIMATE_11_COMP']
if stress.plus[1, s] > 0: # tensile stress in matrix
failure.plus[
1, s] = stress.plus[1, s] / tube_props['ULTIMATE_22_TENS']
else:
failure.plus[
1, s] = stress.plus[1, s] / tube_props['ULTIMATE_22_COMP']
failure.plus[2, s] = stress.plus[2, s] / tube_props['ULTIMATE_12']
# Minus angle ply failure
if stress.minus[0, s] > 0: # tensile stress in fibre
failure.minus[
0, s] = stress.minus[0, s] / tube_props['ULTIMATE_11_TENS']
else:
failure.minus[
0, s] = stress.minus[0, s] / tube_props['ULTIMATE_11_COMP']
if stress.minus[1, s] > 0: # tensile stress in matrix
failure.minus[
1, s] = stress.minus[1, s] / tube_props['ULTIMATE_22_TENS']
else:
failure.minus[
1, s] = stress.minus[1, s] / tube_props['ULTIMATE_22_COMP']
failure.minus[2,
s] = stress.minus[2, s] / tube_props['ULTIMATE_12']
return failure
def torsional_buckling_failure(self, Ns, Finternal, d, theta, nTube, nCap,
lBiscuit, flags):
# Material Properties
tube_props = prepregProperties[flags.CFRPType]
V_21_TUBE = tube_props['V_12'] * (tube_props['E_22'] /
tube_props['E_11'])
# Coordinate system: x is axial direction, theta is circumferential direction
mu_prime_x = tube_props['V_12']
mu_prime_theta = V_21_TUBE
Q = np.zeros((3, 3)) # Preallocate Q-matrix
# Matrix of elastic constants (AER1401, Composite Lamina, slide 8)
Q[0, 0] = tube_props['E_11'] / (1 - tube_props['V_12'] * V_21_TUBE)
Q[1, 1] = tube_props['E_22'] / (1 - tube_props['V_12'] * V_21_TUBE)
Q[0, 1] = tube_props['E_22'] * tube_props['V_12'] / (
1 - tube_props['V_12'] * V_21_TUBE)
Q[1, 0] = Q[0, 1]
Q[2, 2] = tube_props['G_12']
failure = np.zeros(Ns + 1)
for s in range(Ns):
if nCap[s] != 0:
AF_torsional_buckling = 1.25 # See "Validation - Torsional Buckling.xlsx"
else:
AF_torsional_buckling = 1
# Determine elastic properties of rotated tube laminate
x = theta[s] # Composite angle (in radians)
# Transformation matrix
T = np.array(
[[cos(x)**2, sin(x)**2, 2 * sin(x) * cos(x)],
[sin(x)**2, cos(x)**2, -2 * sin(x) * cos(x)],
[-sin(x) * cos(x),
sin(x) * cos(x),
cos(x)**2 - sin(x)**2]])
# Transform the elastic constants using the transformation matrix to obtain the
# elastic constants at the composite angle.
# MATLAB version: Qbar = (T\Q)/T'
tmp = np.linalg.solve(T, Q)
Qbar = np.linalg.solve(T, tmp.T)
# Breakout tube elastic constants at the transformed angle
E_x = Qbar[0, 0]
E_theta = Qbar[1, 1]
# Calculate tube geometric properties
t_tube = nTube[s] * tube_props[
'T_PLY'] # Shell thickness, CR-912 p.xxxvi
R = (
d[s] + t_tube
) / 2 # Radius from axis of rotation to centroidal surface of cylinder wall, CR-912 p.xxxiv
L = lBiscuit[s] # Unsupported length of cylinder, CR-912 p.xxx
# Calculate tube elastic properties (CR-912 p.576)
D_x = E_x * (1. / 12) * (t_tube**3)
D_theta = E_theta * (1. / 12) * (t_tube**3)
B_x = E_x * t_tube
B_theta = E_theta * t_tube
# Calculate Gamma
rho = ((D_x * D_theta) / (B_x * B_theta))**(1. / 4)
Gamma = 3.6125e-07 * ((R / (rho * 1000)) ** 6) + \
-1.9724e-05 * ((R / (rho * 1000)) ** 5) + \
0.0004283 * ((R / (rho * 1000)) ** 4) + \
-0.0048315 * ((R / (rho * 1000)) ** 3) + \
0.031801 * ((R / (rho * 1000)) ** 2) + \
-0.12975 * (R / (rho * 1000)) + \
0.88309
# Calculate factors required in critical torque equation
Z = ((B_theta * (1 - mu_prime_x * mu_prime_theta) * (L**4)) /
(12 * D_x * (R**2)))**(1. / 2)
Z_s = ((D_theta / D_x)**(5. / 6)) * ((B_x / B_theta)**(1. / 2)) * Z
K_s = 0.89 * (Z_s**(3. / 4))
N_x_theta = (Gamma * K_s * (pi**2) * D_x) / (L**2)
# Calculate critical torque
critical_torque = AF_torsional_buckling * N_x_theta * 2 * pi * (R**
2)
failure[s] = abs(Finternal[4, s] / critical_torque)
failure[Ns] = 0 # no torsion at tip
return failure
class FEM(Component):
"""
Computes the deformation of the spar
"""
# inputs
flags = VarTree(Flags(), iotype='in')
yN = Array(iotype='in', desc='')
EIx = Array(iotype='in', desc='')
EIz = Array(iotype='in', desc='')
EA = Array(iotype='in', desc='')
GJ = Array(iotype='in', desc='')
cE = Array(iotype='in', desc='chord of each element')
xEA = Array(iotype='in', desc='')
fblade = VarTree(Fblade(), iotype='in')
mSpar = Array(iotype='in', desc='mass of spars')
mChord = Array(iotype='in', desc='mass of chords')
xCG = Array(iotype='in', desc='')
yWire = Array(iotype='in', desc='location of wire attachment along span')
zWire = Float(iotype='in', desc='depth of wire attachement')
TWire = Array(iotype='in', desc='')
presLoad = VarTree(PrescribedLoad(), iotype='in')
# outputs
k = Array(iotype='out', desc='local elastic stiffness matrix')
K = Array(iotype='out', desc='global stiffness matrix')
F = Array(iotype='out', desc='global force vector')
q = Array(iotype='out', desc='deformation')
def execute(self):
# short aliases
yN = self.yN
EIx = self.EIx
EIz = self.EIz
EA = self.EA
GJ = self.GJ
xEA = self.xEA
cE = self.cE
mSpar = self.mSpar
mChord = self.mChord
xCG = self.xCG
yWire = self.yWire
zWire = self.zWire
TWire = self.TWire
fblade = self.fblade
presLoad = self.presLoad
Ns = len(yN) - 1 # number of elements
dy = np.zeros(Ns)
for s in range(Ns + 1):
dy[s - 1] = yN[s] - yN[s - 1] # length of each element
# FEM computation for structural deformations
# -------------------------------------------
# Initialize global stiffness matrix
K = np.zeros(((Ns + 1) * 6, (Ns + 1) * 6)) # global stiffness
F = np.zeros(((Ns + 1) * 6, 1)) # global force vector
# Create global stiffness maxtrix and force vector
k = np.zeros((12, 12, Ns))
for s in range(Ns):
# Local elastic stiffness matrix
k[0, 0, s] = 12 * EIx[s] / (dy[s] * dy[s] * dy[s])
k[0, 5, s] = -6 * EIx[s] / (dy[s] * dy[s])
k[5, 0, s] = k[0, 5, s]
k[0, 6, s] = -12 * EIx[s] / (dy[s] * dy[s] * dy[s])
k[6, 0, s] = k[0, 6, s]
k[0, 11, s] = -6 * EIx[s] / (dy[s] * dy[s])
k[11, 0, s] = k[0, 11, s]
k[1, 1, s] = EA[s] / dy[s]
k[1, 7, s] = -EA[s] / dy[s]
k[7, 1, s] = k[1, 7, s]
k[2, 2, s] = 12 * EIz[s] / (dy[s] * dy[s] * dy[s])
k[2, 3, s] = 6 * EIz[s] / (dy[s] * dy[s])
k[3, 2, s] = k[2, 3, s]
k[2, 8, s] = -12 * EIz[s] / (dy[s] * dy[s] * dy[s])
k[8, 2, s] = k[2, 8, s]
k[2, 9, s] = 6 * EIz[s] / (dy[s] * dy[s])
k[9, 2, s] = k[2, 9, s]
k[3, 3, s] = 4 * EIz[s] / dy[s]
k[3, 8, s] = -6 * EIz[s] / (dy[s] * dy[s])
k[8, 3, s] = k[3, 8, s]
k[3, 9, s] = 2 * EIz[s] / dy[s]
k[9, 3, s] = k[3, 9, s]
k[4, 4, s] = GJ[s] / dy[s]
k[4, 10, s] = -GJ[s] / dy[s]
k[10, 4, s] = k[4, 10, s]
k[5, 5, s] = 4 * EIx[s] / dy[s]
k[5, 6, s] = 6 * EIx[s] / (dy[s] * dy[s])
k[6, 5, s] = k[5, 6, s]
k[5, 11, s] = 2 * EIx[s] / dy[s]
k[11, 5, s] = k[5, 11, s]
k[6, 6, s] = 12 * EIx[s] / (dy[s] * dy[s] * dy[s])
k[6, 11, s] = 6 * EIx[s] / (dy[s] * dy[s])
k[11, 6, s] = k[6, 11, s]
k[7, 7, s] = EA[s] / dy[s]
k[8, 8, s] = 12 * EIz[s] / (dy[s] * dy[s] * dy[s])
k[8, 9, s] = -6 * EIz[s] / (dy[s] * dy[s])
k[9, 8, s] = k[8, 9, s]
k[9, 9, s] = 4 * EIz[s] / dy[s]
k[10, 10, s] = GJ[s] / dy[s]
k[11, 11, s] = 4 * EIx[s] / dy[s]
# Perform dihedral and sweep rotations here if needed
# Assemble global stiffness matrix
K[(6*s):(6*s + 12), (6*s):(6*s + 12)] = \
K[(6*s):(6*s + 12), (6*s):(6*s + 12)] + k[:, :, s]
Faero = np.zeros((6, 1))
if self.flags.Load == 0: # include aero forces
# aerodynamic forces
xAC = 0.25
Faero[0] = fblade.Fx[s] / 2
Faero[1] = 0
Faero[2] = fblade.Fz[s] / 2
Faero[3] = fblade.Fz[s] * dy[s] / 12
Faero[4] = fblade.My[s] / 2 + (xEA[s] -
xAC) * cE[s] * fblade.Fz[s] / 2
Faero[5] = -fblade.Fx[s] * dy[s] / 12
Fg = np.zeros((6, 1))
Fwire = np.zeros((12, 1))
if (self.flags.Load == 0) or (self.flags.Load == 1):
# gravitational forces
g = 9.81
Fg[0] = 0
Fg[1] = 0
Fg[2] = -(mSpar[s] + mChord[s]) * g / 2
Fg[3] = -(mSpar[s] + mChord[s]) * g * dy[s] / 12
Fg[4] = (xCG[s] - xEA[s]) * cE[s] * (mSpar[s] +
mChord[s]) * g / 2
Fg[5] = 0
# Wire forces (using consistent force vector)
for w in range(len(yWire)):
if (yWire[w] >= yN[s]) and (yWire[w] < yN[s + 1]):
thetaWire = atan2(zWire, yWire[w])
a = yWire[w] - yN[s]
L = dy[s]
FxWire = -cos(thetaWire) * TWire[w]
FzWire = -sin(thetaWire) * TWire[w]
Fwire[0] = 0
Fwire[1] = FxWire * (1 - a / L)
Fwire[2] = FzWire * (2 * (a / L)**3 - 3 *
(a / L)**2 + 1)
Fwire[3] = FzWire * a * ((a / L)**2 - 2 * (a / L) + 1)
Fwire[4] = 0
Fwire[5] = 0
Fwire[6] = 0
Fwire[7] = FxWire * (a / L)
Fwire[8] = FzWire * (-2 * (a / L)**3 + 3 * (a / L)**2)
Fwire[9] = FzWire * a * ((a / L)**2 - (a / L))
Fwire[10] = 0
Fwire[11] = 0
else:
Fwire = np.zeros((12, 1))
Fpres = np.zeros((12, 1))
if self.flags.Load == 2:
# Prescribed point load (using consistent force vector)
if (presLoad.y >= yN[s]) and (presLoad.y < yN[s + 1]):
a = presLoad.y - yN[s]
L = dy[s]
Fpres[0] = 0
Fpres[1] = 0
Fpres[2] = presLoad.pointZ * (2 * (a / L)**3 - 3 *
(a / L)**2 + 1)
Fpres[3] = presLoad.pointZ * a * ((a / L)**2 - 2 *
(a / L) + 1)
Fpres[4] = presLoad.pointM * (1 - a / L)
Fpres[5] = 0
Fpres[6] = 0
Fpres[7] = 0
Fpres[8] = presLoad.pointZ * (-2 * (a / L)**3 + 3 *
(a / L)**2)
Fpres[9] = presLoad.pointZ * a * ((a / L)**2 - (a / L))
Fpres[10] = presLoad.pointM * (a / L)
Fpres[11] = 0
# Prescribed distributed load
Fpres[0] = Fpres[0] + presLoad.distributedX * dy[s] / 2
Fpres[1] = Fpres[1] + 0
Fpres[2] = Fpres[2] + presLoad.distributedZ * dy[s] / 2
Fpres[
3] = Fpres[3] + presLoad.distributedZ * dy[s] * dy[s] / 12
Fpres[4] = Fpres[4] + presLoad.distributedM * dy[s] / 2
Fpres[
5] = Fpres[5] - presLoad.distributedX * dy[s] * dy[s] / 12
Fpres[6] = Fpres[6] + presLoad.distributedX * dy[s] / 2
Fpres[7] = Fpres[7] + 0
Fpres[8] = Fpres[8] + presLoad.distributedZ * dy[s] / 2
Fpres[
9] = Fpres[9] - presLoad.distributedZ * dy[s] * dy[s] / 12
Fpres[10] = Fpres[10] + presLoad.distributedM * dy[s] / 2
Fpres[11] = Fpres[
11] + presLoad.distributedX * dy[s] * dy[s] / 12
# Assemble global force vector
F[(s * 6 +
0)] = F[(s * 6 +
0)] + Fpres[0] + Fwire[0] + Fg[0] + Faero[0] # x force
F[(s * 6 +
1)] = F[(s * 6 +
1)] + Fpres[1] + Fwire[1] + Fg[1] + Faero[1] # y force
F[(s * 6 +
2)] = F[(s * 6 +
2)] + Fpres[2] + Fwire[2] + Fg[2] + Faero[2] # z force
F[(s * 6 +
3)] = F[(s * 6 + 3
)] + Fpres[3] + Fwire[3] + Fg[3] + Faero[3] # x moment
F[(s * 6 +
4)] = F[(s * 6 + 4
)] + Fpres[4] + Fwire[4] + Fg[4] + Faero[4] # y moment
F[(s * 6 +
5)] = F[(s * 6 + 5
)] + Fpres[5] + Fwire[5] + Fg[5] + Faero[5] # z moment
F[(s * 6 +
6)] = F[(s * 6 +
6)] + Fpres[6] + Fwire[6] + Fg[0] + Faero[0] # x force
F[(s * 6 +
7)] = F[(s * 6 +
7)] + Fpres[7] + Fwire[7] + Fg[1] + Faero[1] # y force
F[(s * 6 +
8)] = F[(s * 6 +
8)] + Fpres[8] + Fwire[8] + Fg[2] + Faero[2] # z force
F[(s * 6 +
9)] = F[(s * 6 + 9
)] + Fpres[9] + Fwire[9] - Fg[3] - Faero[3] # x moment
F[(s * 6 +
10)] = F[(s * 6 + 10)] + Fpres[10] + Fwire[10] + Fg[4] + Faero[
4] # y moment
F[(s * 6 +
11)] = F[(s * 6 + 11)] + Fpres[11] + Fwire[11] - Fg[5] - Faero[
5] # z moment
# Add constraints to all 6 dof at root
if self.flags.wingWarp > 0: # Also add wingWarping constraint
raise Exception(
'FEM is untested and surely broken for wingWarp > 0')
ii = np.array([])
for ss in range((Ns + 1) * 6 - 1):
if (ss > 5) and (ss != self.flags.wingWarp * 6 + 5):
ii = np.array([ii, ss]).reshape(1, -1)
Fc = F[(ii - 1)]
Kc = K[(ii - 1), (ii - 1)]
else:
Fc = F[6:]
Kc = K[6:, 6:]
# Solve constrained system
qc, _, _, _ = np.linalg.lstsq(Kc, Fc)
if self.flags.wingWarp > 0:
self.q[ii, 1] = qc
else:
# self.q = np.array([0, 0, 0, 0, 0, 0, qc]).reshape(1, -1)
self.q = np.append(
np.array([0, 0, 0, 0, 0, 0]).reshape(-1, 1),
qc).reshape(-1, 1)
# output the stiffness and force arrays as well (for use in failure analysis)
self.k = k
self.K = K
self.F = F
def __init__(self, Ns):
super(FEM, self).__init__()
# inputs
self.add('flags', VarTree(Flags(), iotype='in'))
self.add('yN', Array(np.zeros(Ns + 1), iotype='in', desc=''))
self.add('EIx', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('EIz', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('EA', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('GJ', Array(np.zeros(Ns), iotype='in', desc=''))
self.add(
'cE', Array(np.zeros(Ns),
iotype='in',
desc='chord of each element'))
self.add('xEA', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('fblade', VarTree(Fblade(Ns), iotype='in'))
self.add('mSpar', Array(np.zeros(Ns),
iotype='in',
desc='mass of spars'))
self.add('mChord',
Array(np.zeros(Ns), iotype='in', desc='mass of chords'))
self.add('xCG', Array(np.zeros(Ns), iotype='in', desc=''))
self.add(
'yWire',
Array([0],
iotype='in',
desc='location of wire attachment along span'))
self.add('zWire',
Float(0., iotype='in', desc='depth of wire attachment'))
self.add('TWire', Array([0], iotype='in', desc=''))
self.add('presLoad', VarTree(PrescribedLoad(), iotype='in'))
# outputs
self.add(
'k',
Array(np.zeros((Ns + 2, Ns + 2, Ns)),
iotype='out',
desc='local elastic stiffness matrix'))
self.add(
'K',
Array(np.zeros((Ns + 2, Ns + 2, Ns)),
iotype='out',
desc='global stiffness matrix'))
self.add(
'F',
Array(np.zeros((6 * (Ns + 1), 1)),
iotype='out',
desc='global force vector'))
self.add(
'q',
Array(np.zeros((6 * (Ns + 1), 1)),
iotype='out',
desc='deformation'))
def __init__(self, Ns):
super(Structures, self).__init__()
# flags
self.add('flags', VarTree(Flags(), iotype='in'))
# inputs for spars
self.add(
'yN',
Array(np.zeros(Ns + 1),
iotype='in',
desc='node locations for each element along the span'))
self.add('d', Array(np.zeros(Ns), iotype='in', desc='spar diameter'))
self.add('theta', Array(np.zeros(Ns), iotype='in', desc='wrap angle'))
self.add(
'nTube',
Array(np.zeros(Ns), iotype='in', desc='number of tube layers'))
self.add('nCap',
Array(np.zeros(Ns), iotype='in', desc='number of cap strips'))
self.add(
'lBiscuit',
Array(np.zeros(Ns), iotype='in',
desc='unsupported biscuit length'))
# joint properties
self.add('Jprop', VarTree(JointProperties(), iotype='in'))
# inputs for chord
self.add('b', Int(0, iotype='in', desc='number of blades'))
self.add(
'cE', Array(np.zeros(Ns),
iotype='in',
desc='chord of each element'))
self.add('xEA', Array(np.zeros(Ns), iotype='in', desc=''))
self.add('xtU', Array(np.zeros(Ns), iotype='in', desc=''))
# inputs for quad
self.add('dQuad',
Float(0., iotype='in', desc='diameter of quad rotor struts'))
self.add(
'thetaQuad',
Float(0., iotype='in', desc='wrap angle of quad rotor struts'))
self.add(
'nTubeQuad',
Int(0,
iotype='in',
desc='number of CFRP layers in quad rotor struts'))
self.add('lBiscuitQuad', Float(0., iotype='in', desc=''))
self.add(
'RQuad',
Float(
0.,
iotype='in',
desc=
'distance from centre of helicopter to centre of quad rotors'))
self.add('hQuad',
Float(0., iotype='in', desc='height of quad-rotor truss'))
# inputs for cover
self.add('ycmax', Float(0., iotype='in', desc=''))
# inputs for wire
self.add(
'yWire',
Array([0],
iotype='in',
desc='location of wire attachment along span'))
self.add('zWire',
Float(0., iotype='in', desc='depth of wire attachement'))
self.add('tWire', Float(0., iotype='in', desc='thickness of wire'))
self.add('TWire', Array([0], iotype='in', desc=''))
self.add('TEtension', Float(0., iotype='in', desc=''))
# inputs for 'other stuff'
self.add('mElseRotor', Float(0., iotype='in', desc=''))
self.add('mElseCentre', Float(0., iotype='in', desc=''))
self.add('mElseR', Float(0., iotype='in', desc=''))
self.add('R', Float(0., iotype='in', desc='rotor radius'))
self.add('mPilot', Float(0., iotype='in', desc='mass of pilot'))
# inputs for FEM
self.add('fblade', VarTree(Fblade(Ns), iotype='in'))
self.add('presLoad', VarTree(PrescribedLoad(), iotype='in'))
# configure
self.add('spar', SparProperties(Ns))
self.connect('yN', 'spar.yN')
self.connect('d', 'spar.d')
self.connect('theta', 'spar.theta')
self.connect('nTube', 'spar.nTube')
self.connect('nCap', 'spar.nCap')
self.connect('lBiscuit', 'spar.lBiscuit')
self.connect('flags.CFRPType', 'spar.CFRPType')
self.add('joint', JointSparProperties(Ns))
self.connect('flags.CFRPType', 'joint.CFRPType')
self.connect('Jprop', 'joint.Jprop')
self.add('chord', ChordProperties(Ns))
self.connect('yN', 'chord.yN')
self.connect('cE', 'chord.c')
self.connect('d', 'chord.d')
self.connect('flags.GWing', 'chord.GWing')
self.connect('xtU', 'chord.xtU')
self.add('quad', QuadSparProperties(Ns))
self.connect('dQuad', 'quad.dQuad')
self.connect('thetaQuad', 'quad.thetaQuad')
self.connect('nTubeQuad', 'quad.nTubeQuad')
self.connect('lBiscuitQuad', 'quad.lBiscuitQuad')
self.connect('flags.CFRPType', 'quad.CFRPType')
self.connect('RQuad', 'quad.RQuad')
self.connect('hQuad', 'quad.hQuad')
self.add('mass', MassProperties(Ns))
self.connect('flags', 'mass.flags')
self.connect('b', 'mass.b')
self.connect('spar.mSpar', 'mass.mSpar')
self.connect('chord.mChord', 'mass.mChord')
self.connect('chord.xCGChord', 'mass.xCGChord')
self.connect('quad.mQuad', 'mass.mQuad')
self.connect('xEA', 'mass.xEA')
self.connect('ycmax', 'mass.ycmax')
self.connect('zWire', 'mass.zWire')
self.connect('yWire', 'mass.yWire')
self.connect('tWire', 'mass.tWire')
self.connect('mElseRotor', 'mass.mElseRotor')
self.connect('mElseCentre', 'mass.mElseCentre')
self.connect('mElseR', 'mass.mElseR')
self.connect('R', 'mass.R')
self.connect('mPilot', 'mass.mPilot')
self.add('fem', FEM(Ns))
self.connect('flags', 'fem.flags')
self.connect('yN', 'fem.yN')
self.connect('spar.EIx', 'fem.EIx')
self.connect('spar.EIz', 'fem.EIz')
self.connect('spar.EA', 'fem.EA')
self.connect('spar.GJ', 'fem.GJ')
self.connect('cE', 'fem.cE')
self.connect('xEA', 'fem.xEA')
self.connect('spar.mSpar', 'fem.mSpar')
self.connect('chord.mChord', 'fem.mChord')
self.connect('mass.xCG', 'fem.xCG')
self.connect('zWire', 'fem.zWire')
self.connect('yWire', 'fem.yWire')
self.connect('TWire', 'fem.TWire')
self.connect('fblade', 'fem.fblade')
self.connect('presLoad', 'fem.presLoad')
self.add('strains', Strains(Ns))
self.connect('yN', 'strains.yN')
self.connect('d', 'strains.d')
self.connect('fem.k', 'strains.k')
self.connect('fem.F', 'strains.F')
self.connect('fem.q', 'strains.q')
self.add('failure', Failures(Ns))
self.connect('flags', 'failure.flags')
self.connect('yN', 'failure.yN')
self.connect('strains.Finternal', 'failure.Finternal')
self.connect('strains.strain', 'failure.strain')
self.connect('d', 'failure.d')
self.connect('theta', 'failure.theta')
self.connect('nTube', 'failure.nTube')
self.connect('nCap', 'failure.nCap')
self.connect('yWire', 'failure.yWire')
self.connect('zWire', 'failure.zWire')
self.connect('joint.EIx', 'failure.EIxJ')
self.connect('joint.EIz', 'failure.EIzJ')
self.connect('lBiscuit', 'failure.lBiscuit')
self.connect('dQuad', 'failure.dQuad')
self.connect('thetaQuad', 'failure.thetaQuad')
self.connect('nTubeQuad', 'failure.nTubeQuad')
self.connect('lBiscuitQuad', 'failure.lBiscuitQuad')
self.connect('RQuad', 'failure.RQuad')
self.connect('hQuad', 'failure.hQuad')
self.connect('quad.EIx', 'failure.EIQuad')
self.connect('quad.GJ', 'failure.GJQuad')
self.connect('tWire', 'failure.tWire')
self.connect('TWire', 'failure.TWire')
self.connect('TEtension', 'failure.TEtension')
self.connect('b', 'failure.b')
self.connect('fblade', 'failure.fblade')
self.connect('spar.mSpar', 'failure.mSpar')
self.connect('chord.mChord', 'failure.mChord')
self.connect('mElseRotor', 'failure.mElseRotor')
# link up the outputs
self.create_passthrough('mass.Mtot')
self.create_passthrough('fem.q')
self.create_passthrough('strains.Finternal')
self.create_passthrough('strains.strain')
self.create_passthrough('failure.fail')
self.driver.workflow.add('chord')
self.driver.workflow.add('fem')
self.driver.workflow.add('joint')
self.driver.workflow.add('mass')
self.driver.workflow.add('quad')
self.driver.workflow.add('spar')
self.driver.workflow.add('strains')
self.driver.workflow.add('failure')