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 test_static_deflection(self):
x = array([0.0, 4.0, 10.0])
EI = 144.0
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
fe = BeamFE(x, density=10, EA=0, EIy=EI, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe)
# Distributed load, linearly interpolated
load = np.zeros((3, 3))
load[-1, 2] = -100 # Load in z direction at tip
element.apply_distributed_loading(load)
defl = -element.applied_stress / np.diag(element.K)
# Check against directly calculating static deflection from FE
Q = fe.distribute_load(interleave(load, 6))
defl_fe, reactions_fe = fe.static_deflection(Q)
assert_aae(dot(element.shapes, defl), defl_fe, decimal=2)
def test_static_deflection(self):
x = array([0.0, 4.0, 10.0])
EI = 144.0
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
fe = BeamFE(x, density=10, EA=0, EIy=EI, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe)
# Distributed load, linearly interpolated
load = np.zeros((3, 3))
load[-1, 2] = -100 # Load in z direction at tip
element.apply_distributed_loading(load)
defl = -element.applied_stress / np.diag(element.K)
# Check against directly calculating static deflection from FE
Q = fe.distribute_load(interleave(load, 6))
defl_fe, reactions_fe = fe.static_deflection(Q)
assert_aae(dot(element.shapes, defl), defl_fe, decimal=2)
def setUp(self):
x = linspace(0, self.L, 15)
fe = BeamFE(x, density=self.m, EA=0, EIy=self.EI, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
beam = ModalElementFromFE('beam',
fe,
num_modes=1,
damping=self.damping_coeff)
system = System()
system.add_leaf(beam)
system.setup()
system.update_kinematics()
self.beam, self.system = beam, system
def test_first_mode_frequency(self):
# From Reddy1993, p. 160
x = np.linspace(0, 1, 16)
# Using the z axis as the transverse direction gives the same
# sign convention as Reddy uses in 2D, namely that rotations
# are positive clockwise.
fe = BeamFE(x, density=1, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe)
Mmodal = element.mass_ee
Kmodal = element.K
w = np.sqrt(np.diag(Kmodal) / np.diag(Mmodal))
assert_aae(w[0], 3.5160, decimal=4)
def test_rigid_element_reference_loading(self):
# Make an element with no modes (rigid)
fe = BeamFE(np.linspace(0, 1, 11), density=1, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe, 0)
# Distributed load, linearly interpolated
load = np.zeros((11, 3))
load[:, 2] = -100 # Uniform load in z direction
element.apply_distributed_loading(load)
assert_aae(element.applied_stress, [])
assert_aae(element.applied_forces[0:6], [0, 0, -100 * 1, 0, 50, 0])
assert_aae(element.applied_forces[6:12], 0)
# Distributed load, triangle at tip
load = np.zeros((11, 3))
load[-1, 2] = -100 # tapered load in z direction
element.applied_forces[:] = 0 # reset
element.apply_distributed_loading(load)
F = -100 * 0.1 / 2
assert_aae(element.applied_forces[0:6],
[0, 0, F, 0, -F * (0.9 + 0.1 * 2 / 3), 0])
assert_aae(element.applied_forces[6:12], 0)
def test_rigid_element_reference_loading(self):
# Make an element with no modes (rigid)
fe = BeamFE(np.linspace(0, 1, 11), density=1, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
fe.set_dofs([False, False, True, False, True, False])
element = ModalElementFromFE('elem', fe, 0)
# Distributed load, linearly interpolated
load = np.zeros((11, 3))
load[:, 2] = -100 # Uniform load in z direction
element.apply_distributed_loading(load)
assert_aae(element.applied_stress, [])
assert_aae(element.applied_forces[0:6], [0, 0, -100 * 1, 0, 50, 0])
assert_aae(element.applied_forces[6:12], 0)
# Distributed load, triangle at tip
load = np.zeros((11, 3))
load[-1, 2] = -100 # tapered load in z direction
element.applied_forces[:] = 0 # reset
element.apply_distributed_loading(load)
F = -100 * 0.1 / 2
assert_aae(element.applied_forces[0:6],
[0, 0, F, 0, -F * (0.9 + 0.1 * 2 / 3), 0])
assert_aae(element.applied_forces[6:12], 0)
