def test_inertia_when_offset_axially(self):
density = 230.4
length = 20.0
offset = 5.0
element = _mock_rigid_uniform_beam(density, length)
conn = RigidConnection('offset', offset=[offset, 0, 0])
joint = FreeJoint('joint')
system = System()
system.add_leaf(joint)
joint.add_leaf(conn)
conn.add_leaf(element)
system.setup()
# Calculate reduced system to get rigid body matrices
rsys = ReducedSystem(system)
# Expected values: rod along x axis
m = density * length
Iy = m * (length**2 / 12 + (length/2 + offset)**2)
expected_mass = m * eye(3)
expected_inertia = diag([0, Iy, Iy])
expected_offdiag = zeros((3, 3))
# Y accel -> positive moment about Z
# Z accel -> negative moment about Y
expected_offdiag[2, 1] = +m * (length/2 + offset)
expected_offdiag[1, 2] = -m * (length/2 + offset)
assert_aae(rsys.M[:3, :3], expected_mass)
assert_aae(rsys.M[3:, 3:], expected_inertia)
assert_aae(rsys.M[3:, :3], expected_offdiag)
assert_aae(rsys.M[:3, 3:], expected_offdiag.T)
def build_structure(rigid_body_configs):
# Free joint represents the rigid-body motion of the platform
free_joint = FreeJoint("base")
for i, conf in enumerate(rigid_body_configs):
name = conf.get("name", "body{}".format(i))
conn = RigidConnection("conn-" + name, offset=conf["CoM"])
body = RigidBody(name, mass=conf["mass"], inertia=np.diag(conf["inertia"]))
free_joint.add_leaf(conn)
conn.add_leaf(body)
system = System(free_joint)
return system
def build_structure(rigid_body_configs):
# Free joint represents the rigid-body motion of the platform
free_joint = FreeJoint('base')
for i, conf in enumerate(rigid_body_configs):
name = conf.get('name', 'body{}'.format(i))
conn = RigidConnection('conn-' + name, offset=conf['CoM'])
body = RigidBody(name,
mass=conf['mass'],
inertia=np.diag(conf['inertia']))
free_joint.add_leaf(conn)
conn.add_leaf(body)
system = System(free_joint)
return system
def test_distal_forces_cause_acceleration(self):
j = FreeJoint('joint')
b = RigidBody('body', mass=3, inertia=np.diag([7, 7, 7]))
s = System()
s.add_leaf(j)
j.add_leaf(b)
s.setup()
# Constant loading
j.distal_forces = np.array([2, 0, 0, 0, 0, 0])
s.update_kinematics()
s.update_matrices()
s.solve_accelerations()
s.update_kinematics()
assert_array_equal(j.ad, [2. / 3, 0, 0, 0, 0, 0])
def test_three_elements_forming_a_disc_about_X_have_correct_inertia(self):
density = 5
length = 20.0
offset = 1.25
m = density * length # mass of one beam
joint = FreeJoint('joint')
# Make 3 elements spaced by 120 deg about z axis
for i in range(3):
# Rotation of -pi/2 about y aligns local x axis of ModalElement
rotmat = rotations(('x', i * 2*pi/3), ('y', -pi/2))
offset_vector = dot(rotmat, [offset, 0, 0]) # offset // local x
conn = RigidConnection('offset%d' % i, offset_vector, rotmat)
element = _mock_rigid_uniform_beam(density, length,
'element%d' % i)
joint.add_leaf(conn)
conn.add_leaf(element)
system = System()
system.add_leaf(joint)
system.setup()
# Calculate reduced system to get rigid body matrices
rsys = ReducedSystem(system)
# Expected values: perp inertia using projected lengths of beams
Iy = m * (length**2 / 12 + (length/2 + offset)**2)
Iperp = Iy + Iy/4 + Iy/4
Iaxial = 3 * Iy
expected_mass = 3 * m * eye(3)
expected_inertia = diag([Iaxial, Iperp, Iperp])
expected_offdiag = zeros((3, 3))
assert_aae(rsys.M[:3, :3], expected_mass)
assert_aae(rsys.M[3:, 3:], expected_inertia)
assert_aae(rsys.M[3:, :3], expected_offdiag)
assert_aae(rsys.M[:3, 3:], expected_offdiag.T)
def test_distal_forces(self):
j = FreeJoint('joint')
# Constant loading
j.distal_forces = np.array([2, 3.1, 2.1, 4.3, 2.5, 1.0])
j.calc_external_loading()
assert_array_equal(j.applied_forces[:6], 0)
assert_array_equal(j.applied_forces[6:], j.distal_forces)
assert_array_equal(j.applied_stress, 0)
# Loading function
j.distal_forces = lambda element, t: np.ones(6)
j.calc_external_loading()
assert_array_equal(j.applied_forces[:6], 0)
assert_array_equal(j.applied_forces[6:], 1)
assert_array_equal(j.applied_stress, 0)
def test_interal_forces(self):
# NB minus sign because convention for applied_stress is that
# stiffness loads are positive.
j = FreeJoint('joint')
# Constant loading
j.internal_forces = np.array([2, 3.1, 2.1, 4.3, 2.5, 1.0])
j.calc_external_loading()
assert_array_equal(j.applied_forces, 0)
assert_array_equal(j.applied_stress, -j.internal_forces)
# Loading function
j.internal_forces = lambda element, t: np.ones(6)
j.calc_external_loading()
assert_array_equal(j.applied_forces, 0)
assert_array_equal(j.applied_stress, -1)
def test_velocity_transforms_depend_on_joint_orientation(self):
j = FreeJoint('joint')
j.rp = array([0, 0, 8.6])
# x-velocity: shouldn't depend on speed
j.vstrain[:] = [2.3, 0, 0, 0, 0, 0]
j.calc_kinematics()
assert_array_equal(j.F_vp, eye(6)) # movement of p node moves d node
assert_array_equal(j.F_ve, eye(6)) # with zero rotation, direct
assert_array_equal(j.F_v2, 0)
# now apply a 90 deg yaw angle (about z axis): now 'pitch'
# causes a -ve global x-rotation, and 'roll' causes a +ve
# global y-rotation
j.xstrain[5] = pi / 2
j.calc_kinematics()
assert_array_equal(j.F_vp, eye(6)) # movement of p node moves d node
assert_array_equal(j.F_v2, 0)
F_ve = eye(6)
F_ve[3:, 3] = [0, 1, 0]
F_ve[3:, 4] = [-1, 0, 0]
assert_array_almost_equal(j.F_ve, F_ve)
def test_interal_forces(self):
# NB minus sign because convention for applied_stress is that
# stiffness loads are positive.
j = FreeJoint('joint')
# Constant loading
j.internal_forces = np.array([2, 3.1, 2.1, 4.3, 2.5, 1.0])
j.calc_external_loading()
assert_array_equal(j.applied_forces, 0)
assert_array_equal(j.applied_stress, -j.internal_forces)
# Loading function
j.internal_forces = lambda element, t: np.ones(6)
j.calc_external_loading()
assert_array_equal(j.applied_forces, 0)
assert_array_equal(j.applied_stress, -1)
def test_velocity_transforms_depend_on_joint_orientation(self):
j = FreeJoint('joint')
j.rp = array([0, 0, 8.6])
# x-velocity: shouldn't depend on speed
j.vstrain[:] = [2.3, 0, 0, 0, 0, 0]
j.calc_kinematics()
assert_array_equal(j.F_vp, eye(6)) # movement of p node moves d node
assert_array_equal(j.F_ve, eye(6)) # with zero rotation, direct
assert_array_equal(j.F_v2, 0)
# now apply a 90 deg yaw angle (about z axis): now 'pitch'
# causes a -ve global x-rotation, and 'roll' causes a +ve
# global y-rotation
j.xstrain[5] = pi / 2
j.calc_kinematics()
assert_array_equal(j.F_vp, eye(6)) # movement of p node moves d node
assert_array_equal(j.F_v2, 0)
F_ve = eye(6)
F_ve[3:, 3] = [0, 1, 0]
F_ve[3:, 4] = [-1, 0, 0]
assert_array_almost_equal(j.F_ve, F_ve)
def __init__(self, structure_config):
s = structure_config
#### Load details of flexible elements ####
if "definition" in s["tower"]:
if "mass" in s["tower"]:
raise ValueError("Both tower definition and explicit mass!")
self.tower_definition = Tower(s["tower"]["definition"])
assert np.all(self.tower_definition.stn_pos[:, :2] == 0) # vert.
z_tower = self.tower_definition.stn_pos[:, 2]
self.tower_modes = ModesFromScratch(
z_tower - z_tower[0],
self.tower_definition.density,
1,
self.tower_definition.EIy,
self.tower_definition.EIz,
)
else:
self.tower_definition = None
self.tower_modes = None
if "blade" in s:
self.blade_modes = load_modes_from_Bladed(s["blade"]["definition"])
else:
self.blade_modes = None
#### Create the elements ####
# Free joint represents the rigid-body motion of the platform
free_joint = FreeJoint("base")
# This is the rigid-body mass of the platform structure
conn_platform = RigidConnection("conn-platform", offset=s["platform"]["CoM"])
platform = RigidBody("platform", mass=s["platform"]["mass"], inertia=np.diag(s["platform"]["inertia"]))
free_joint.add_leaf(conn_platform)
conn_platform.add_leaf(platform)
# Make a rigid body to represent the added mass
# (approximate to zero frequency)
# XXX this is skipping the coupling matrix
# A = whales_model.A(0)
# added_mass = RigidBody('added-mass', mass=np.diag(A[:3, :3]),
# inertia=A[3:, 3:])
# Flexible tower or equivalent rigid body
if self.tower_modes:
# move base of tower 10m up, and rotate so tower x-axis is vertical
conn_tower = RigidConnection("conn-tower", offset=[0, 0, z_tower[0]], rotation=rotmat_y(-pi / 2))
tower = DistalModalElementFromScratch("tower", self.tower_modes, s["tower"]["number of normal modes"])
else:
# move tower to COG
conn_tower = RigidConnection("conn-tower", offset=s["tower"]["CoM"])
tower = RigidBody("tower", s["tower"]["mass"], np.diag(s["tower"]["inertia"]))
free_joint.add_leaf(conn_tower)
conn_tower.add_leaf(tower)
# The nacelle -- rigid body
# rotate back so nacelle inertia is aligned with global coordinates
if self.tower_modes:
nacoff = s["nacelle"]["offset from tower top"]
conn_nacelle = RigidConnection(
"conn-nacelle", offset=dot(rotmat_y(pi / 2), nacoff), rotation=rotmat_y(pi / 2)
)
tower.add_leaf(conn_nacelle)
else:
conn_nacelle = RigidConnection("conn-nacelle", offset=np.array([0, 0, s["nacelle"]["height"]]))
free_joint.add_leaf(conn_nacelle)
nacelle = RigidBody(
"nacelle", mass=s["nacelle"]["mass"], inertia=np.diag(s["nacelle"].get("inertia", np.zeros(3)))
)
conn_nacelle.add_leaf(nacelle)
# The rotor hub -- currently just connections (no mass)
# rotate so rotor centre is aligned with global coordinates
if self.tower_modes:
rotoff = s["rotor"]["offset from tower top"]
conn_rotor = RigidConnection("conn-rotor", offset=dot(rotmat_y(pi / 2), rotoff), rotation=rotmat_y(pi / 2))
tower.add_leaf(conn_rotor)
else:
conn_rotor = RigidConnection("conn-rotor", offset=np.array([0, 0, s["nacelle"]["height"]]))
free_joint.add_leaf(conn_rotor)
# The drive shaft rotation (rotation about x)
shaft = Hinge("shaft", [1, 0, 0])
conn_rotor.add_leaf(shaft)
# The blades
if self.blade_modes:
rtlen = s["rotor"]["root length"]
Ryx = dot(rotmat_y(-pi / 2), rotmat_x(-pi / 2)) # align blade modes
for i in range(3):
R = rotmat_x(i * 2 * pi / 3)
root = RigidConnection("root%d" % (i + 1), offset=dot(R, [0, 0, rtlen]), rotation=dot(R, Ryx))
blade = ModalElement("blade%d" % (i + 1), self.blade_modes)
shaft.add_leaf(root)
root.add_leaf(blade)
else:
rotor = RigidBody("rotor", s["rotor"]["mass"], np.diag(s["rotor"]["inertia"]))
shaft.add_leaf(rotor)
# Build system
self.system = System(free_joint)
# Constrain missing DOFs -- tower torsion & extension not complete
if self.tower_modes:
self.system.prescribe(tower, vel=0, part=[0, 3])
def test_distal_node_is_transformed_by_joint_freedoms(self):
j = FreeJoint('joint')
j.rp = array([3.5, 9.21, 8.6])
j.Rp = eye(3)
# Test distal transform -- translation
j.xstrain[:3] = [2, 3, 4]
j.calc_distal_pos()
assert_array_equal(j.rd, j.rp + [2, 3, 4])
assert_array_equal(j.Rd, j.Rp)
# Now add a rotation of 60 deg about x axis
j.xstrain[3] = pi / 3
j.calc_distal_pos()
assert_array_equal(j.rd, j.rp + [2, 3, 4])
# new unit vectors after rotation
assert_array_almost_equal(j.Rd, c_[[1, 0, 0],
[0, cos(pi/3), sin(pi/3)],
[0, -sin(pi/3), cos(pi/3)]])
# Check combination of Euler angles: 90deg yaw and pitch
j.xstrain[3:] = [0, pi/2, pi/2]
j.calc_distal_pos()
assert_array_equal(j.rd, j.rp + [2, 3, 4])
# new unit vectors after rotation.
# 1) 90deg yaw -> y, -x, z
# 2) 90deg pitch -> -z, -x, y
assert_array_almost_equal(j.Rd, c_[[0, 0, -1], [-1, 0, 0], [0, 1, 0]])
def test_distal_node_is_transformed_by_joint_freedoms(self):
j = FreeJoint('joint')
j.rp = array([3.5, 9.21, 8.6])
j.Rp = eye(3)
# Test distal transform -- translation
j.xstrain[:3] = [2, 3, 4]
j.calc_distal_pos()
assert_array_equal(j.rd, j.rp + [2, 3, 4])
assert_array_equal(j.Rd, j.Rp)
# Now add a rotation of 60 deg about x axis
j.xstrain[3] = pi / 3
j.calc_distal_pos()
assert_array_equal(j.rd, j.rp + [2, 3, 4])
# new unit vectors after rotation
assert_array_almost_equal(j.Rd, c_[[1, 0, 0],
[0, cos(pi/3), sin(pi/3)],
[0, -sin(pi/3), cos(pi/3)]])
# Check combination of Euler angles: 90deg yaw and pitch
j.xstrain[3:] = [0, pi/2, pi/2]
j.calc_distal_pos()
assert_array_equal(j.rd, j.rp + [2, 3, 4])
# new unit vectors after rotation.
# 1) 90deg yaw -> y, -x, z
# 2) 90deg pitch -> -z, -x, y
assert_array_almost_equal(j.Rd, c_[[0, 0, -1], [-1, 0, 0], [0, 1, 0]])
def _make_random_joint(rdm):
# Make free joint with random axis and transform
return FreeJoint('joint', post_transform=random_rotation_matrix(rdm))
def __init__(self, structure_config):
s = structure_config
#### Load details of flexible elements ####
if 'definition' in s['tower']:
if 'mass' in s['tower']:
raise ValueError("Both tower definition and explicit mass!")
self.tower_definition = Tower(s['tower']['definition'])
assert np.all(self.tower_definition.stn_pos[:, :2] == 0) # vert.
z_tower = self.tower_definition.stn_pos[:, 2]
self.tower_modes = ModesFromScratch(
z_tower - z_tower[0],
self.tower_definition.density, 1,
self.tower_definition.EIy, self.tower_definition.EIz)
else:
self.tower_definition = None
self.tower_modes = None
if 'blade' in s:
self.blade_modes = load_modes_from_Bladed(s['blade']['definition'])
else:
self.blade_modes = None
#### Create the elements ####
# Free joint represents the rigid-body motion of the platform
free_joint = FreeJoint('base')
# This is the rigid-body mass of the platform structure
conn_platform = RigidConnection('conn-platform',
offset=s['platform']['CoM'])
platform = RigidBody('platform',
mass=s['platform']['mass'],
inertia=np.diag(s['platform']['inertia']))
free_joint.add_leaf(conn_platform)
conn_platform.add_leaf(platform)
# Make a rigid body to represent the added mass
# (approximate to zero frequency)
# XXX this is skipping the coupling matrix
#A = whales_model.A(0)
# added_mass = RigidBody('added-mass', mass=np.diag(A[:3, :3]),
# inertia=A[3:, 3:])
# Flexible tower or equivalent rigid body
if self.tower_modes:
# move base of tower 10m up, and rotate so tower x-axis is vertical
conn_tower = RigidConnection(
'conn-tower', offset=[0, 0, z_tower[0]],
rotation=rotmat_y(-pi/2))
tower = DistalModalElementFromScratch(
'tower', self.tower_modes,
s['tower']['number of normal modes'])
else:
# move tower to COG
conn_tower = RigidConnection(
'conn-tower', offset=s['tower']['CoM'])
tower = RigidBody('tower', s['tower']['mass'],
np.diag(s['tower']['inertia']))
free_joint.add_leaf(conn_tower)
conn_tower.add_leaf(tower)
# The nacelle -- rigid body
# rotate back so nacelle inertia is aligned with global coordinates
if self.tower_modes:
nacoff = s['nacelle']['offset from tower top']
conn_nacelle = RigidConnection('conn-nacelle',
offset=dot(rotmat_y(pi/2), nacoff),
rotation=rotmat_y(pi/2))
tower.add_leaf(conn_nacelle)
else:
conn_nacelle = RigidConnection(
'conn-nacelle',
offset=np.array([0, 0, s['nacelle']['height']]))
free_joint.add_leaf(conn_nacelle)
nacelle = RigidBody(
'nacelle',
mass=s['nacelle']['mass'],
inertia=np.diag(s['nacelle'].get('inertia', np.zeros(3))))
conn_nacelle.add_leaf(nacelle)
# The rotor hub -- currently just connections (no mass)
# rotate so rotor centre is aligned with global coordinates
if self.tower_modes:
rotoff = s['rotor']['offset from tower top']
conn_rotor = RigidConnection('conn-rotor',
offset=dot(rotmat_y(pi/2), rotoff),
rotation=rotmat_y(pi/2))
tower.add_leaf(conn_rotor)
else:
conn_rotor = RigidConnection(
'conn-rotor',
offset=np.array([0, 0, s['nacelle']['height']]))
free_joint.add_leaf(conn_rotor)
# The drive shaft rotation (rotation about x)
shaft = Hinge('shaft', [1, 0, 0])
conn_rotor.add_leaf(shaft)
# The blades
if self.blade_modes:
rtlen = s['rotor']['root length']
Ryx = dot(rotmat_y(-pi/2), rotmat_x(-pi/2)) # align blade modes
for i in range(3):
R = rotmat_x(i*2*pi/3)
root = RigidConnection('root%d' % (i+1),
offset=dot(R, [0, 0, rtlen]),
rotation=dot(R, Ryx))
blade = ModalElement('blade%d' % (i+1), self.blade_modes)
shaft.add_leaf(root)
root.add_leaf(blade)
else:
rotor = RigidBody('rotor', s['rotor']['mass'],
np.diag(s['rotor']['inertia']))
shaft.add_leaf(rotor)
# Build system
self.system = System(free_joint)
# Constrain missing DOFs -- tower torsion & extension not complete
if self.tower_modes:
self.system.prescribe(tower, vel=0, part=[0, 3])
