def setUp(self):
# FE model for beam - no modes, i.e. rigid
x = linspace(0, self.length, 20)
density = (2 * self.mass / self.length) * (1 - x / self.length)
fe = BeamFE(x, density=density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in Z direction
load = np.zeros((len(x), 3))
load[:, 2] = self.force
self.beam.loading = load
# Offset from hinge axis
self.conn = RigidConnection('offset', [self.offset, 0, 0])
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.conn)
self.conn.add_leaf(self.beam)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
def setUp(self):
# FE model for beam - no modes, i.e. rigid
self.x = x = linspace(0, self.length, 20)
fe = BeamFE(x, density=2, EA=0, EIy=0, EIz=0)
# Build the elements
self.shaft = Hinge('shaft', [1, 0, 0])
self.roots = []
self.blades = []
self.pitch_bearings = []
for ib in range(1):
R = rotations(('x', ib*2*pi/3), ('y', -pi/2))
root_offset = dot(R, [self.root_length, 0, 0])
root = RigidConnection('root%d' % (ib+1), root_offset, R)
bearing = Hinge('pitch%d' % (ib+1), [1, 0, 0])
blade = ModalElementFromFE('blade%d' % (ib+1), fe, 0)
self.shaft.add_leaf(root)
root.add_leaf(bearing)
bearing.add_leaf(blade)
self.roots.append(root)
self.blades.append(blade)
self.pitch_bearings.append(bearing)
# Build system
self.system = System()
self.system.add_leaf(self.shaft)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def setUp(self):
# Rigid body with offset centre of mass
self.body = RigidBody('body', self.mass, Xc=[self.offset, 0, 0])
# Hinge with axis along Z axis
self.hinge = Hinge('hinge', [0, 0, 1])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.body)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def setUp(self):
# FE model for beam - no modes, i.e. rigid
x = linspace(0, self.length, 20)
density = (2 * self.mass / self.length) * (1 - x / self.length)
fe = BeamFE(x, density=density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in Z direction
load = np.zeros((len(x), 3))
load[:, 2] = self.force
self.beam.loading = load
# Offset from hinge axis
self.conn = RigidConnection('offset', [self.offset, 0, 0])
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.conn)
self.conn.add_leaf(self.beam)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
def test_simple_prescribed_integration(self):
s = System()
h = Hinge('hinge', [0, 1, 0])
s.add_leaf(h)
s.setup()
s.prescribe(h)
w = h.vstrain[0] = 0.97 # rad/s
integ = Integrator(s)
t, y = integ.integrate(9.0, 0.1)
# Check time vector and shape of result
assert_array_equal(t, np.arange(0, 9.0, 0.1))
self.assertEqual(len(y), 1)
self.assertEqual(y[0].shape, (len(t), 1))
# Result should be y = wt, but wrapped to [0, 2pi)
assert_aae(y[0][:, 0], (w * t) % (2 * np.pi))
# Check asking for velocity and acceleration works
h.xstrain[0] = s.time = 0.0 # reset
integ = Integrator(s, ('pos', 'vel', 'acc'))
t, y = integ.integrate(1.0, 0.1)
assert_array_equal(t, np.arange(0, 1.0, 0.1))
self.assertEqual(len(y), 3)
for yy in y:
self.assertEqual(yy.shape, (len(t), 1))
assert_aae(y[0][:, 0], w * t)
assert_aae(y[1][:, 0], w)
assert_aae(y[2][:, 0], 0)
def setUp(self):
# Parameters
mass = 11.234
length = 2.54
gravity = 9.81
# Build model
hinge = Hinge('hinge', [0, 1, 0])
link = RigidConnection('link', [length, 0, 0])
body = RigidBody('body', mass)
system = System(gravity=gravity)
system.add_leaf(hinge)
hinge.add_leaf(link)
link.add_leaf(body)
system.setup()
# Custom outputs to calculate correct answer
def force_body_prox_local(s):
theta = s.q[hinge.istrain][0]
thetadot = s.qd[hinge.istrain][0]
thetadotdot = s.qdd[hinge.istrain][0]
Fx = mass * (-gravity * np.sin(theta) - length * thetadot**2)
Fz = mass * (+gravity * np.cos(theta) - length * thetadotdot)
return [Fx, 0, Fz, 0, 0, 0]
def force_hinge_prox(s):
theta = s.q[hinge.istrain][0]
thetadot = s.qd[hinge.istrain][0]
thetadotdot = s.qdd[hinge.istrain][0]
A = np.array([[+np.cos(theta), np.sin(theta)],
[-np.sin(theta), np.cos(theta)]])
Fxz = -mass * length * np.dot(A, [thetadot**2, thetadotdot])
return [Fxz[0], 0, Fxz[1] + gravity * mass, 0, 0, 0]
# Solver
integ = Integrator(system, ('pos', 'vel', 'acc'))
integ.add_output(LoadOutput(hinge.iprox))
integ.add_output(LoadOutput(link.iprox))
integ.add_output(LoadOutput(body.iprox))
integ.add_output(LoadOutput(body.iprox, local=True))
integ.add_output(CustomOutput(force_hinge_prox, "correct ground"))
integ.add_output(
CustomOutput(force_body_prox_local, "correct link distal local"))
self.system = system
self.integ = integ
class TestReactionForcesForCentrifugalForce(unittest.TestCase):
"""
System
------
A rigid body with offset mass, attached to a spinning hinge.
Tests
-----
Check centrifugal force reaction on hinge is in correct direction.
"""
mass = 5.0 # kg
offset = 3.2 # m
def setUp(self):
# Rigid body with offset centre of mass
self.body = RigidBody("body", self.mass, Xc=[self.offset, 0, 0])
# Hinge with axis along Z axis
self.hinge = Hinge("hinge", [0, 0, 1])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.body)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def test_reactions(self):
# Set angular acceleration
w = 5.21 # rad/s
self.hinge.vstrain[0] = w
self.system.update_kinematics() # Update nodal values based on DOFs
self.system.update_matrices()
self.system.solve_reactions() # Solve reactions incl d'Alembert
# Some parameters
L = self.offset
m = self.mass
# Check reactions at beam root
Pg = self.system.joint_reactions["ground"]
P0 = self.system.joint_reactions["node-0"]
Fx_expected = -m * L * w ** 2
assert_aae(P0, [Fx_expected, 0, 0, 0, 0, 0])
assert_aae(Pg, P0)
def setUp(self):
# Parameters
mass = 11.234
length = 2.54
gravity = 9.81
# Build model
hinge = Hinge('hinge', [0, 1, 0])
link = RigidConnection('link', [length, 0, 0])
body = RigidBody('body', mass)
system = System(gravity=gravity)
system.add_leaf(hinge)
hinge.add_leaf(link)
link.add_leaf(body)
system.setup()
# Custom outputs to calculate correct answer
def force_body_prox_local(s):
theta = s.q[hinge.istrain][0]
thetadot = s.qd[hinge.istrain][0]
thetadotdot = s.qdd[hinge.istrain][0]
Fx = mass * (-gravity*np.sin(theta) - length*thetadot**2)
Fz = mass * (+gravity*np.cos(theta) - length*thetadotdot)
return [Fx, 0, Fz, 0, 0, 0]
def force_hinge_prox(s):
theta = s.q[hinge.istrain][0]
thetadot = s.qd[hinge.istrain][0]
thetadotdot = s.qdd[hinge.istrain][0]
A = np.array([[+np.cos(theta), np.sin(theta)],
[-np.sin(theta), np.cos(theta)]])
Fxz = -mass * length * np.dot(A, [thetadot**2, thetadotdot])
return [Fxz[0], 0, Fxz[1] + gravity*mass, 0, 0, 0]
# Solver
integ = Integrator(system, ('pos', 'vel', 'acc'))
integ.add_output(LoadOutput(hinge.iprox))
integ.add_output(LoadOutput(link.iprox))
integ.add_output(LoadOutput(body.iprox))
integ.add_output(LoadOutput(body.iprox, local=True))
integ.add_output(CustomOutput(force_hinge_prox, "correct ground"))
integ.add_output(CustomOutput(force_body_prox_local,
"correct link distal local"))
self.system = system
self.integ = integ
class TestReactionForcesForCentrifugalForce(unittest.TestCase):
"""
System
------
A rigid body with offset mass, attached to a spinning hinge.
Tests
-----
Check centrifugal force reaction on hinge is in correct direction.
"""
mass = 5.0 # kg
offset = 3.2 # m
def setUp(self):
# Rigid body with offset centre of mass
self.body = RigidBody('body', self.mass, Xc=[self.offset, 0, 0])
# Hinge with axis along Z axis
self.hinge = Hinge('hinge', [0, 0, 1])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.body)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def test_reactions(self):
# Set angular acceleration
w = 5.21 # rad/s
self.hinge.vstrain[0] = w
self.system.update_kinematics() # Update nodal values based on DOFs
self.system.update_matrices()
self.system.solve_reactions() # Solve reactions incl d'Alembert
# Some parameters
L = self.offset
m = self.mass
# Check reactions at beam root
Pg = self.system.joint_reactions['ground']
P0 = self.system.joint_reactions['node-0']
Fx_expected = -m * L * w**2
assert_aae(P0, [Fx_expected, 0, 0, 0, 0, 0])
assert_aae(Pg, P0)
def test_call(self):
s = System()
c = RigidConnection('conn', [1, 0, 0])
h = Hinge('hinge', [0, 1, 0])
b = RigidBody('body', 1)
s.add_leaf(h)
h.add_leaf(c)
c.add_leaf(b)
s.setup()
# Set hinge angle
h.xstrain[0] = 0.82
h.vstrain[0] = 1.2
h.astrain[0] = -0.3
s.update_kinematics()
s.solve_reactions()
# Test load outputs
out = LoadOutput('node-1')
assert_array_equal(out(s), s.joint_reactions['node-1'])
out = LoadOutput('node-1', local=True)
F = s.joint_reactions['node-1']
assert_array_equal(out(s), np.r_[np.dot(b.Rp.T, F[:3]),
np.dot(b.Rp.T, F[3:])])
def build_system():
# Calculate inertia, stiffness and damping
I2 = mass * length**2
k = (2*np.pi*natfreq)**2 * I2
c = 2 * damping_factor * I2 * (2*np.pi*natfreq)
# Build model
hinge = Hinge('hinge', [0,0,1])
hinge.stiffness = k
hinge.damping = c
link = RigidConnection('link', [length, 0, 0])
body = RigidBody('body', mass)
system = System()
system.add_leaf(hinge)
hinge.add_leaf(link)
link.add_leaf(body)
system.setup()
return system
def setUp(self):
# FE model for beam - no modes, i.e. rigid
self.x = x = linspace(0, self.length, 20)
fe = BeamFE(x, density=2, EA=0, EIy=0, EIz=0)
# Build the elements
self.shaft = Hinge('shaft', [1, 0, 0])
self.roots = []
self.blades = []
self.pitch_bearings = []
for ib in range(1):
R = rotations(('x', ib * 2 * pi / 3), ('y', -pi / 2))
root_offset = dot(R, [self.root_length, 0, 0])
root = RigidConnection('root%d' % (ib + 1), root_offset, R)
bearing = Hinge('pitch%d' % (ib + 1), [1, 0, 0])
blade = ModalElementFromFE('blade%d' % (ib + 1), fe, 0)
self.shaft.add_leaf(root)
root.add_leaf(bearing)
bearing.add_leaf(blade)
self.roots.append(root)
self.blades.append(blade)
self.pitch_bearings.append(bearing)
# Build system
self.system = System()
self.system.add_leaf(self.shaft)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def __init__(self, length, radius, mass, spin):
self.length = length
self.radius = radius
self.mass = mass
self.spin = spin
self.bearing = Hinge('bearing', [0, 0, 1])
self.pivot = Hinge('pivot', [0, 1, 0])
self.axis = Hinge('axis', [1, 0, 0])
self.body = self.build_body()
self.pivot.damping = 200
self.system = System(gravity=9.81)
self.system.add_leaf(self.bearing)
self.bearing.add_leaf(self.pivot)
self.pivot.add_leaf(self.axis)
self.axis.add_leaf(self.body)
self.system.setup()
# Prescribed DOF accelerations: constant rotational speed
self.system.prescribe(self.axis)
def test_applied_force(self):
# Set up a hinge with a mass offset on a rigid body. The
# reduced system should have 1 DOF -- the hinge rotation --
# with the associated mass being the inertia of the mass about
# the hinge, and the associated generalised force being the
# applied moment.
mass = 36.2
zforce = -30
L = 3.2
s = System()
h = Hinge('hinge', [0, 1, 0])
c = RigidConnection('conn', [L, 0, 0])
b = RigidBody('body', mass, nodal_load=[0, 0, zforce])
s.add_leaf(h)
h.add_leaf(c)
c.add_leaf(b)
s.setup()
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (1, 1))
self.assertEqual(rsys.Q.shape, (1, ))
self.assertEqual(rsys.M[0, 0], mass * L**2) # inertial about hinge
self.assertEqual(rsys.Q[0], -zforce * L) # moment about hinge
def test_applied_force(self):
# Set up a hinge with a mass offset on a rigid body. The
# reduced system should have 1 DOF -- the hinge rotation --
# with the associated mass being the inertia of the mass about
# the hinge, and the associated generalised force being the
# applied moment.
mass = 36.2
zforce = -30
L = 3.2
s = System()
h = Hinge('hinge', [0, 1, 0])
c = RigidConnection('conn', [L, 0, 0])
b = RigidBody('body', mass, nodal_load=[0, 0, zforce])
s.add_leaf(h)
h.add_leaf(c)
c.add_leaf(b)
s.setup()
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (1, 1))
self.assertEqual(rsys.Q.shape, (1,))
self.assertEqual(rsys.M[0, 0], mass * L**2) # inertial about hinge
self.assertEqual(rsys.Q[0], -zforce * L) # moment about hinge
def setUp(self):
# Rigid body with offset centre of mass
self.body = RigidBody('body', self.mass, Xc=[self.offset, 0, 0])
# Hinge with axis along Z axis
self.hinge = Hinge('hinge', [0, 0, 1])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.body)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def test_call(self):
s = System()
c = RigidConnection('conn', [1, 0, 0])
h = Hinge('hinge', [0, 1, 0])
b = RigidBody('body', 1)
s.add_leaf(h)
h.add_leaf(c)
c.add_leaf(b)
s.setup()
# Set hinge angle
h.xstrain[0] = 0.82
h.vstrain[0] = 1.2
h.astrain[0] = -0.3
s.update_kinematics()
# Test node outputs
out = StateOutput('node-1')
assert_array_equal(out(s), np.r_[b.rp, b.Rp.flatten()])
out = StateOutput('node-1', deriv=1)
assert_array_equal(out(s), b.vp)
out = StateOutput('node-1', deriv=2)
assert_array_equal(out(s), b.ap)
out = StateOutput('node-1', local=True)
assert_array_equal(out(s), np.r_[np.dot(b.Rp.T, b.rp),
np.eye(3).flatten()])
out = StateOutput('node-1', deriv=1, local=True)
assert_array_equal(out(s), np.r_[np.dot(b.Rp.T, b.vp[:3]),
np.dot(b.Rp.T, b.vp[3:])])
out = StateOutput('node-1', deriv=2, local=True)
assert_array_equal(out(s), np.r_[np.dot(b.Rp.T, b.ap[:3]),
np.dot(b.Rp.T, b.ap[3:])])
# Test strain outputs
out = StateOutput('hinge-strains')
assert_array_equal(out(s), 0.82)
out = StateOutput('hinge-strains', deriv=1)
assert_array_equal(out(s), 1.2)
out = StateOutput('hinge-strains', deriv=2)
assert_array_equal(out(s), -0.3)
# Strains cannot be transformed to local coordinates
with self.assertRaises(RuntimeError):
out = StateOutput('hinge-strains', local=True)
out(s)
class Gyroscope:
def __init__(self, length, radius, mass, spin):
self.length = length
self.radius = radius
self.mass = mass
self.spin = spin
self.bearing = Hinge('bearing', [0, 0, 1])
self.pivot = Hinge('pivot', [0, 1, 0])
self.axis = Hinge('axis', [1, 0, 0])
self.body = self.build_body()
self.pivot.damping = 200
self.system = System(gravity=9.81)
self.system.add_leaf(self.bearing)
self.bearing.add_leaf(self.pivot)
self.pivot.add_leaf(self.axis)
self.axis.add_leaf(self.body)
self.system.setup()
# Prescribed DOF accelerations: constant rotational speed
self.system.prescribe(self.axis)
def build_body(self):
Jx = self.radius**2 / 2
Jyz = (3*self.radius**2 + self.length**2) / 12
Jyz_0 = Jyz + (self.length/2)**2 # parallel axis theorem
inertia = self.mass * np.diag([Jx, Jyz_0, Jyz_0])
return RigidBody('body', self.mass, inertia, [self.length/2, 0, 0])
def simulate(self, xpivot=0.0, vpivot=0.0, t1=10, dt=0.05):
# reset
self.system.q[:] = 0.0
self.system.qd[:] = 0.0
# initial conditions
self.system.q[self.pivot.istrain][0] = xpivot # initial elevation
#self.system.qd[self.pivot.istrain][0] = vpivot # initial elevation spd
self.system.qd[self.axis.istrain][0] = self.spin # constant spin speed
# simulate
integ = Integrator(self.system, ('pos', 'vel'))
integ.integrate(t1, dt, nprint=None)
return integ
class Gyroscope:
def __init__(self, length, radius, mass, spin):
self.length = length
self.radius = radius
self.mass = mass
self.spin = spin
self.bearing = Hinge('bearing', [0, 0, 1])
self.pivot = Hinge('pivot', [0, 1, 0])
self.axis = Hinge('axis', [1, 0, 0])
self.body = self.build_body()
self.pivot.damping = 200
self.system = System(gravity=9.81)
self.system.add_leaf(self.bearing)
self.bearing.add_leaf(self.pivot)
self.pivot.add_leaf(self.axis)
self.axis.add_leaf(self.body)
self.system.setup()
# Prescribed DOF accelerations: constant rotational speed
self.system.prescribe(self.axis)
def build_body(self):
Jx = self.radius**2 / 2
Jyz = (3 * self.radius**2 + self.length**2) / 12
Jyz_0 = Jyz + (self.length / 2)**2 # parallel axis theorem
inertia = self.mass * np.diag([Jx, Jyz_0, Jyz_0])
return RigidBody('body', self.mass, inertia, [self.length / 2, 0, 0])
def simulate(self, xpivot=0.0, vpivot=0.0, t1=10, dt=0.05):
# reset
self.system.q[:] = 0.0
self.system.qd[:] = 0.0
# initial conditions
self.system.q[self.pivot.istrain][0] = xpivot # initial elevation
#self.system.qd[self.pivot.istrain][0] = vpivot # initial elevation spd
self.system.qd[self.axis.istrain][0] = self.spin # constant spin speed
# simulate
integ = Integrator(self.system, ('pos', 'vel'))
integ.integrate(t1, dt, nprint=None)
return integ
def build_system():
# Calculate inertia, stiffness and damping
I2 = mass * length**2
k = (2 * np.pi * natfreq)**2 * I2
c = 2 * damping_factor * I2 * (2 * np.pi * natfreq)
# Build model
hinge = Hinge('hinge', [0, 0, 1])
hinge.stiffness = k
hinge.damping = c
link = RigidConnection('link', [length, 0, 0])
body = RigidBody('body', mass)
system = System()
system.add_leaf(hinge)
hinge.add_leaf(link)
link.add_leaf(body)
system.setup()
return system
def __init__(self, length, radius, mass, spin):
self.length = length
self.radius = radius
self.mass = mass
self.spin = spin
self.bearing = Hinge('bearing', [0, 0, 1])
self.pivot = Hinge('pivot', [0, 1, 0])
self.axis = Hinge('axis', [1, 0, 0])
self.body = self.build_body()
self.pivot.damping = 200
self.system = System(gravity=9.81)
self.system.add_leaf(self.bearing)
self.bearing.add_leaf(self.pivot)
self.pivot.add_leaf(self.axis)
self.axis.add_leaf(self.body)
self.system.setup()
# Prescribed DOF accelerations: constant rotational speed
self.system.prescribe(self.axis)
class TestReactionForcesOnModalElementFromFE(unittest.TestCase):
"""
System
------
A triangular rigid beam, offset by a rigid link from a hinge.
Tests
-----
Set the angular acceleration of the hinge. Check the reaction
forces at the centre and at the root of the beam.
"""
mass = 5.0 # kg
length = 20.0 # m
offset = 3.2 # m
force = -34.2 # N / m
def setUp(self):
# FE model for beam - no modes, i.e. rigid
x = linspace(0, self.length, 20)
density = (2 * self.mass / self.length) * (1 - x / self.length)
fe = BeamFE(x, density=density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in Z direction
load = np.zeros((len(x), 3))
load[:, 2] = self.force
self.beam.loading = load
# Offset from hinge axis
self.conn = RigidConnection('offset', [self.offset, 0, 0])
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.conn)
self.conn.add_leaf(self.beam)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
def test_reactions(self):
# Set angular acceleration
alpha = 1.235 # rad/s2
self.hinge.astrain[0] = alpha
self.system.update_kinematics() # Update nodal values based on DOFs
self.system.solve_reactions() # Solve reactions incl d'Alembert
# Some parameters
L = self.length
m = self.mass
Ro = self.offset
Rg = L / 3 # distance to CoM along beam
IG = m * L ** 2 / 18
assert_aae(m, self.beam.mass_vv[0, 0])
# Check reactions at beam root
P = self.system.joint_reactions['node-1']
Fz_expected = (-m * (Ro + Rg) * alpha -
self.force * L)
My_expected = ((IG + m * Rg * (Ro + Rg)) * alpha +
self.force * L ** 2 / 2)
assert_aae(P, [0, 0, Fz_expected, 0, My_expected, 0])
def test_update_kinematics_results(self):
# Test system: (all rigid connections of length 1)
#
# [hinge]
# (gnd)---c1---(0)
# (1)---c2---(2)
# |
# y c3
# | |
# |--> x (3)
#
s = System()
c1 = RigidConnection('c1', [1, 0, 0])
c2 = RigidConnection('c2', [1, 0, 0])
c3 = RigidConnection('c3', [0, -1, 0])
hinge = Hinge('hinge', [0, 0, 1])
s.add_leaf(c1)
c1.add_leaf(hinge)
hinge.add_leaf(c2)
c2.add_leaf(c3)
s.setup()
# All velocities and accelerations should be zero
for el in [c1, c2, c3, hinge]:
assert_aae(el.vp, 0)
assert_aae(el.vd, 0)
assert_aae(el.ap, 0)
assert_aae(el.ad, 0)
# (gnd)
assert_aae(c1.rp, 0)
assert_aae(c1.Rp, np.eye(3))
# (0)
assert_aae(c1.rd, [1, 0, 0])
assert_aae(c1.Rd, np.eye(3))
# (1)
assert_aae(c2.rp, [1, 0, 0])
assert_aae(c2.Rp, np.eye(3))
# (2)
assert_aae(c3.rp, [2, 0, 0])
assert_aae(c3.Rp, np.eye(3))
# (3)
assert_aae(c3.rd, [2, -1, 0])
assert_aae(c3.Rd, np.eye(3))
##### now set angular velocity of hinge #####
hinge.vstrain[0] = 1.0
s.update_kinematics()
# (gnd)
assert_aae(c1.vp, 0)
assert_aae(c1.ap, 0)
# (0)
assert_aae(c1.vd, 0)
assert_aae(c1.ad, 0)
# (1)
assert_aae(c2.vp, [0, 0, 0, 0, 0, 1.0])
assert_aae(c2.ap, 0)
# (2)
assert_aae(c3.vp, [0, 1, 0, 0, 0, 1.0])
assert_aae(c3.ap, [-1, 0, 0, 0, 0, 0]) # centripetal acceleration
# (3)
assert_aae(c3.vd, [1, 1, 0, 0, 0, 1.0])
assert_aae(c3.ad, [-1, 1, 0, 0, 0, 0]) # centripetal acceleration
class TestReactionForcesWithRotatedBeam(unittest.TestCase):
"""Intended to check the transformation from blade loading to rotor
loading in a wind turbine rotor: the loads are applied to the beam
in the local rotated coordinate system, check they work through to
the ground reactions correctly.
"""
force = 24.1
length = 4.3
root_length = 0.0
def setUp(self):
# FE model for beam - no modes, i.e. rigid
self.x = x = linspace(0, self.length, 20)
fe = BeamFE(x, density=2, EA=0, EIy=0, EIz=0)
# Build the elements
self.shaft = Hinge('shaft', [1, 0, 0])
self.roots = []
self.blades = []
self.pitch_bearings = []
for ib in range(1):
R = rotations(('x', ib * 2 * pi / 3), ('y', -pi / 2))
root_offset = dot(R, [self.root_length, 0, 0])
root = RigidConnection('root%d' % (ib + 1), root_offset, R)
bearing = Hinge('pitch%d' % (ib + 1), [1, 0, 0])
blade = ModalElementFromFE('blade%d' % (ib + 1), fe, 0)
self.shaft.add_leaf(root)
root.add_leaf(bearing)
bearing.add_leaf(blade)
self.roots.append(root)
self.blades.append(blade)
self.pitch_bearings.append(bearing)
# Build system
self.system = System()
self.system.add_leaf(self.shaft)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def test_reactions(self):
# Some parameters
L = self.length
F = self.length * self.force
# Set loading - in local z direction
load = np.zeros((len(self.x), 3))
load[:, 2] = self.force
self.blades[0].loading = load
self.system.update_kinematics()
self.system.update_matrices()
self.system.solve_reactions()
# Check reactions at ground (0, 0, 0)
P = -self.system.joint_reactions['ground']
F_expected = [-F, 0, 0]
M_expected = [0, -F * (L + self.root_length) / 2, 0]
assert_aae(P, np.r_[F_expected, M_expected])
# Reactions on other side of hinge
P2 = -self.system.joint_reactions['node-0']
assert_aae(P, P2)
# Now set pitch angle to 45deg
# NB: hinge rotation is opposite to wind turbine pitch convention
self.pitch_bearings[0].xstrain[0] = -pi / 4
self.system.update_kinematics()
self.system.update_matrices()
self.system.solve_reactions()
# Check reactions at ground (0, 0, 0)
P = -self.system.joint_reactions['ground']
F_expected = [-F / sqrt(2), F / sqrt(2), 0]
M_expected = [-F / sqrt(2) * L / 2, -F / sqrt(2) * L / 2, 0]
assert_aae(P, np.r_[F_expected, M_expected])
# Reactions on other side of hinge
P2 = -self.system.joint_reactions['node-0']
assert_aae(P, P2)
class TestReactionForcesOnModalElementFromFE(unittest.TestCase):
"""
System
------
A triangular rigid beam, offset by a rigid link from a hinge.
Tests
-----
Set the angular acceleration of the hinge. Check the reaction
forces at the centre and at the root of the beam.
"""
mass = 5.0 # kg
length = 20.0 # m
offset = 3.2 # m
force = -34.2 # N / m
def setUp(self):
# FE model for beam - no modes, i.e. rigid
x = linspace(0, self.length, 20)
density = (2 * self.mass / self.length) * (1 - x / self.length)
fe = BeamFE(x, density=density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in Z direction
load = np.zeros((len(x), 3))
load[:, 2] = self.force
self.beam.loading = load
# Offset from hinge axis
self.conn = RigidConnection('offset', [self.offset, 0, 0])
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.conn)
self.conn.add_leaf(self.beam)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
def test_reactions(self):
# Set angular acceleration
alpha = 1.235 # rad/s2
self.hinge.astrain[0] = alpha
self.system.update_kinematics() # Update nodal values based on DOFs
self.system.solve_reactions() # Solve reactions incl d'Alembert
# Some parameters
L = self.length
m = self.mass
Ro = self.offset
Rg = L / 3 # distance to CoM along beam
IG = m * L**2 / 18
assert_aae(m, self.beam.mass_vv[0, 0])
# Check reactions at beam root
P = self.system.joint_reactions['node-1']
Fz_expected = (-m * (Ro + Rg) * alpha - self.force * L)
My_expected = ((IG + m * Rg * (Ro + Rg)) * alpha +
self.force * L**2 / 2)
assert_aae(P, [0, 0, Fz_expected, 0, My_expected, 0])
class TestReactionForcesWithRotatedBeam(unittest.TestCase):
"""Intended to check the transformation from blade loading to rotor
loading in a wind turbine rotor: the loads are applied to the beam
in the local rotated coordinate system, check they work through to
the ground reactions correctly.
"""
force = 24.1
length = 4.3
root_length = 0.0
def setUp(self):
# FE model for beam - no modes, i.e. rigid
self.x = x = linspace(0, self.length, 20)
fe = BeamFE(x, density=2, EA=0, EIy=0, EIz=0)
# Build the elements
self.shaft = Hinge('shaft', [1, 0, 0])
self.roots = []
self.blades = []
self.pitch_bearings = []
for ib in range(1):
R = rotations(('x', ib*2*pi/3), ('y', -pi/2))
root_offset = dot(R, [self.root_length, 0, 0])
root = RigidConnection('root%d' % (ib+1), root_offset, R)
bearing = Hinge('pitch%d' % (ib+1), [1, 0, 0])
blade = ModalElementFromFE('blade%d' % (ib+1), fe, 0)
self.shaft.add_leaf(root)
root.add_leaf(bearing)
bearing.add_leaf(blade)
self.roots.append(root)
self.blades.append(blade)
self.pitch_bearings.append(bearing)
# Build system
self.system = System()
self.system.add_leaf(self.shaft)
self.system.setup()
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
def test_reactions(self):
# Some parameters
L = self.length
F = self.length * self.force
# Set loading - in local z direction
load = np.zeros((len(self.x), 3))
load[:, 2] = self.force
self.blades[0].loading = load
self.system.update_kinematics()
self.system.update_matrices()
self.system.solve_reactions()
# Check reactions at ground (0, 0, 0)
P = -self.system.joint_reactions['ground']
F_expected = [-F, 0, 0]
M_expected = [0, -F*(L+self.root_length)/2, 0]
assert_aae(P, np.r_[F_expected, M_expected])
# Reactions on other side of hinge
P2 = -self.system.joint_reactions['node-0']
assert_aae(P, P2)
# Now set pitch angle to 45deg
# NB: hinge rotation is opposite to wind turbine pitch convention
self.pitch_bearings[0].xstrain[0] = -pi / 4
self.system.update_kinematics()
self.system.update_matrices()
self.system.solve_reactions()
# Check reactions at ground (0, 0, 0)
P = -self.system.joint_reactions['ground']
F_expected = [-F/sqrt(2), F/sqrt(2), 0]
M_expected = [-F/sqrt(2)*L/2, -F/sqrt(2)*L/2, 0]
assert_aae(P, np.r_[F_expected, M_expected])
# Reactions on other side of hinge
P2 = -self.system.joint_reactions['node-0']
assert_aae(P, P2)
