def test_nonzero_prescribed_acceleration(self):
# Test reduction where a prescribed acceleration is non-zero:
# two sliders in series, with a mass on the end. If the second
# slider's acceleration is prescribed, the first slider's DOF
# sees an inertial force corresponding to the acceleration of
# the mass.
mass = 36.2
s = System()
s1 = PrismaticJoint('slider1', [1, 0, 0])
s2 = PrismaticJoint('slider2', [1, 0, 0])
b = RigidBody('body', mass)
s.add_leaf(s1)
s1.add_leaf(s2)
s2.add_leaf(b)
s.setup()
s.prescribe(s2, acc=0)
# With hinge angle = 0, no generalised inertial force
rsys = ReducedSystem(s)
assert_aae(rsys.M, mass)
assert_aae(rsys.Q, 0)
# With hinge angle = 90deg, do see generalised inertial force
s.prescribe(s2, acc=2.3)
rsys = ReducedSystem(s)
assert_aae(rsys.M, mass)
assert_aae(rsys.Q, -mass * 2.3)
def test_single_rigid_body(self):
mass = 36.2
inertia = np.diag([75.4, 653, 234])
s = System()
j = FreeJoint('joint')
b = RigidBody('body', mass, inertia)
s.add_leaf(j)
j.add_leaf(b)
s.setup()
# Calculate reduced system to get rigid body matrices
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (6, 6))
self.assertEqual(rsys.Q.shape, (6,))
assert_aae(rsys.M[:3, :3], mass * np.eye(3))
assert_aae(rsys.M[3:, 3:], inertia)
assert_aae(rsys.M[3:, :3], 0)
assert_aae(rsys.M[:3, 3:], 0)
assert_aae(rsys.Q, 0)
# Now if some freedoms are prescribed, don't appear in reduced system
s.prescribe(j, part=[1, 2, 3, 4, 5]) # only x-translation is free now
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (1, 1))
self.assertEqual(rsys.Q.shape, (1,))
assert_aae(rsys.M[0, 0], mass)
assert_aae(rsys.Q, 0)
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 test_three_rigid_elements_as_disc_have_ends_in_right_place(self):
length = 20.0
offset = 5.0
# Make 3 elements spaced by 120 deg about z axis
system = System()
elements = []
for i in range(3):
rotmat = rotations(('z', i * 2*pi/3))
offset_vector = dot(rotmat, [offset, 0, 0])
conn = RigidConnection('offset%d' % i, offset_vector, rotmat)
element = RigidConnection('element%d' % i, [length, 0, 0])
elements.append(element)
system.add_leaf(conn)
conn.add_leaf(element)
system.setup()
r = offset
R = offset + length
assert_aae(elements[0].rp, [r, 0, 0])
assert_aae(elements[1].rp, [-r/2, r*sqrt(3)/2, 0])
assert_aae(elements[2].rp, [-r/2, -r*sqrt(3)/2, 0])
assert_aae(elements[0].rd, [R, 0, 0])
assert_aae(elements[1].rd, [-R/2, R*sqrt(3)/2, 0])
assert_aae(elements[2].rd, [-R/2, -R*sqrt(3)/2, 0])
def test_adding_elements(self):
conn = RigidConnection('conn')
body = RigidBody('body', mass=1.235)
s = System()
s.add_leaf(conn)
conn.add_leaf(body)
s.setup()
# Should have dict of elements
self.assertEqual(s.elements, {'conn': conn, 'body': body})
# Number of states:
# 6 ground
# + 6 constraints on conn
# + 6 <node-0> between conn and body
# ---
# 18
self.assertEqual(s.lhs.shape, (18, 18))
for vec in (s.rhs, s.qd, s.qdd):
self.assertEqual(len(vec), 18)
# Check there are no dofs
self.assertEqual(len(s.q.dofs), 0)
self.assertEqual(len(s.qd.dofs), 0)
self.assertEqual(len(s.qdd.dofs), 0)
def test_solve_accelerations(self):
# solve_accelerations() should find:
# (a) response of system to forces (here, gravity)
# (b) include any prescribed accelerations in qdd vector
g = 9.81
s = System(gravity=g)
j = FreeJoint('joint')
b = RigidBody('body', mass=23.54, inertia=52.1 * np.eye(3))
s.add_leaf(j)
j.add_leaf(b)
s.setup()
# Prescribe horizontal acceleration. Vertical acceleration
# should result from gravity.
s.prescribe(j, 2.3, part=[0]) # x acceleration
# Initially accelerations are zero
assert_aae(j.ap, 0)
assert_aae(j.ad, 0)
assert_aae(j.astrain, 0)
# Solve accelerations & check
s.solve_accelerations()
s.update_kinematics()
assert_aae(j.ap, 0) # ground
assert_aae(j.ad, [2.3, 0, -g, 0, 0, 0])
assert_aae(j.astrain, j.ad) # not always true, but works for FreeJoint
def test_solve_accelerations_coupling(self):
# Further to test above, check that coupling between prescribed
# accelerations and other dofs is correct. For example, if there
# is a rigid body vertically offset from the joint, then a
# prescribed horizontal acceleration should cause an angular
# acceleration as well as the translational acceleration.
s = System()
j = FreeJoint('joint')
c = RigidConnection('conn', [0, 0, 1.7])
b = RigidBody('body', mass=23.54, inertia=74.1 * np.eye(3))
s.add_leaf(j)
j.add_leaf(c)
c.add_leaf(b)
s.setup()
# Prescribe horizontal acceleration, solve other accelerations
s.prescribe(j, 2.3, part=[0]) # x acceleration
s.update_kinematics() # update system to show prescribed acc
s.solve_accelerations() # solve free accelerations
s.update_kinematics() # update system to show solution
# Ground shouldn't move
assert_aae(j.ap, 0)
# Need angular acceleration = (m a_x L) / I0
I0 = 74.1 + (23.54 * 1.7**2)
expected_angular_acc = -(23.54 * 2.3 * 1.7) / I0
assert_aae(j.ad, [2.3, 0, 0, 0, expected_angular_acc, 0])
assert_aae(j.astrain, j.ad) # not always true, but works for FreeJoint
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 test_1dof_nonlinear_system(self):
s = System()
j = PrismaticJoint('joint', [0, 0, 1])
k = 0.45 # quadratic stiffness coefficient
j.internal_force = lambda el, t: -k * el.xstrain[0]**2
b = RigidBody('body', 10)
s.add_leaf(j)
j.add_leaf(b)
s.setup()
# Linearise around z0 = 0: stiffness should be zero
linsys = LinearisedSystem.from_system(s, z0=0)
assert_aae(linsys.M, [[10.0]])
assert_aae(linsys.C, [[0.0]])
assert_aae(linsys.K, [[0.0]])
# Linearise about z0 = 2: stiffness should be 2kx
linsys = LinearisedSystem.from_system(s, z0=[2])
assert_aae(linsys.M, [[10.0]])
assert_aae(linsys.C, [[0.0]])
assert_aae(linsys.K, [[2 * k * 2]])
# Test setting z0 in another way
linsys = LinearisedSystem.from_system(s, z0={'joint': [4.2]})
assert_aae(linsys.M, [[10.0]])
assert_aae(linsys.C, [[0.0]])
assert_aae(linsys.K, [[2 * k * 4.2]])
def test_prescribe_free(self):
s = System()
j = FreeJoint('joint')
s.add_leaf(j)
s.setup()
s.time = 3.54
# Initially all 6 joint motions are free
self.assertEqual(len(s.q.dofs), 6)
# Prescribing joint to be fixed results in no dofs
s.prescribe(j)
self.assertEqual(len(s.q.dofs), 0)
# Freeing joint results in 6 dofs again
s.free(j)
self.assertEqual(len(s.q.dofs), 6)
# Prescribing 2 of joint motions leaves 4 dofs
s.prescribe(j, lambda t: t, [0, 2])
self.assertEqual(len(s.q.dofs), 4)
# Prescribing another joint motions leaves 3 dofs
s.prescribe(j, 2.0, part=3)
self.assertEqual(len(s.q.dofs), 3)
# Check accelerations are applied to qdd
assert_aae(s.qdd[j.istrain], 0)
s.apply_prescribed_accelerations()
assert_aae(s.qdd[j.istrain], [3.54, 0, 3.54, 2.0, 0, 0])
# Freeing joint results in 6 dofs again
s.free(j)
self.assertEqual(len(s.q.dofs), 6)
def test_nonzero_prescribed_acceleration(self):
# Test reduction where a prescribed acceleration is non-zero:
# two sliders in series, with a mass on the end. If the second
# slider's acceleration is prescribed, the first slider's DOF
# sees an inertial force corresponding to the acceleration of
# the mass.
mass = 36.2
s = System()
s1 = PrismaticJoint('slider1', [1, 0, 0])
s2 = PrismaticJoint('slider2', [1, 0, 0])
b = RigidBody('body', mass)
s.add_leaf(s1)
s1.add_leaf(s2)
s2.add_leaf(b)
s.setup()
s.prescribe(s2, acc=0)
# With hinge angle = 0, no generalised inertial force
rsys = ReducedSystem(s)
assert_aae(rsys.M, mass)
assert_aae(rsys.Q, 0)
# With hinge angle = 90deg, do see generalised inertial force
s.prescribe(s2, acc=2.3)
rsys = ReducedSystem(s)
assert_aae(rsys.M, mass)
assert_aae(rsys.Q, -mass * 2.3)
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 test_single_rigid_body(self):
mass = 36.2
inertia = np.diag([75.4, 653, 234])
s = System()
j = FreeJoint('joint')
b = RigidBody('body', mass, inertia)
s.add_leaf(j)
j.add_leaf(b)
s.setup()
# Calculate reduced system to get rigid body matrices
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (6, 6))
self.assertEqual(rsys.Q.shape, (6, ))
assert_aae(rsys.M[:3, :3], mass * np.eye(3))
assert_aae(rsys.M[3:, 3:], inertia)
assert_aae(rsys.M[3:, :3], 0)
assert_aae(rsys.M[:3, 3:], 0)
assert_aae(rsys.Q, 0)
# Now if some freedoms are prescribed, don't appear in reduced system
s.prescribe(j, part=[1, 2, 3, 4, 5]) # only x-translation is free now
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (1, 1))
self.assertEqual(rsys.Q.shape, (1, ))
assert_aae(rsys.M[0, 0], mass)
assert_aae(rsys.Q, 0)
def test_dof_index(self):
s = System()
j1 = FreeJoint('j1')
j2 = FreeJoint('j2')
s.add_leaf(j1)
j1.add_leaf(j2)
s.setup()
# Prescribe some of the strains:
# _____j1____ _____j2____
# Strain: 0 1 2 3 4 5 0 1 2 3 4 5
# Prescr: * * *
# Dofs: 0 1 2 3 4 5 6 7 8
s.prescribe(j1, 0, part=[1, 4])
s.prescribe(j2, 0, part=[3])
self.assertEqual(s.dof_index('j1', 0), 0)
self.assertEqual(s.dof_index('j1', 2), 1)
self.assertEqual(s.dof_index('j1', 5), 3)
self.assertEqual(s.dof_index('j2', 5), 8)
# Strain index out of range should give IndexError
with self.assertRaises(IndexError):
s.dof_index('j1', 6)
# Asking for index of prescribed strain should give ValueError
with self.assertRaises(ValueError):
s.dof_index('j1', 1)
def test_solve_reactions(self):
# Check it calls the Element method in the right order: down
# the tree from leaves to base. It must also reset reactions.
s = System()
c0 = RigidConnection('c0')
c1 = RigidConnection('c1')
c2 = RigidConnection('c2')
b1 = RigidBody('b1', 1)
b2 = RigidBody('b2', 1)
s.add_leaf(c0)
c0.add_leaf(c1)
c0.add_leaf(c2)
c1.add_leaf(b1)
c2.add_leaf(b2)
s.setup()
# Check elements' iter_reactions() are called
def mock_iter_reactions(element):
calls.append(element)
calls = []
import types
for el in s.elements.values():
el.iter_reactions = types.MethodType(mock_iter_reactions, el)
# Test
s.joint_reactions[:] = 3
s.solve_reactions()
self.assertEqual(calls, [b2, c2, b1, c1, c0])
assert_aae(s.joint_reactions, 0)
def test_1dof_nonlinear_system(self):
s = System()
j = PrismaticJoint('joint', [0, 0, 1])
k = 0.45 # quadratic stiffness coefficient
j.internal_force = lambda el, t: -k * el.xstrain[0]**2
b = RigidBody('body', 10)
s.add_leaf(j)
j.add_leaf(b)
s.setup()
# Linearise around z0 = 0: stiffness should be zero
linsys = LinearisedSystem.from_system(s, z0=0)
assert_aae(linsys.M, [[10.0]])
assert_aae(linsys.C, [[0.0]])
assert_aae(linsys.K, [[0.0]])
# Linearise about z0 = 2: stiffness should be 2kx
linsys = LinearisedSystem.from_system(s, z0=[2])
assert_aae(linsys.M, [[10.0]])
assert_aae(linsys.C, [[0.0]])
assert_aae(linsys.K, [[2 * k * 2]])
# Test setting z0 in another way
linsys = LinearisedSystem.from_system(s, z0={'joint': [4.2]})
assert_aae(linsys.M, [[10.0]])
assert_aae(linsys.C, [[0.0]])
assert_aae(linsys.K, [[2 * k * 4.2]])
def test_accounts_for_offset_centre_of_mass_in_applied_force(self):
# check applied force due to gravity is correct
b = RigidBody("body", mass=5.6, Xc=[1.2, 3.4, 5.4])
s = System(gravity=9.81) # need a System to define gravity
s.add_leaf(b)
s.setup()
b.calc_mass()
b.calc_external_loading()
assert_array_equal(b.applied_forces, b.mass * 9.81 * array([0, 0, -1, -3.4, 1.2, 0]))
def test_rigid_body_with_no_dofs(self):
s = System()
b = RigidBody('body', 23.7)
s.add_leaf(b)
s.setup()
# Calculate reduced system to get rigid body matrices
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (0, 0))
self.assertEqual(rsys.Q.shape, (0, ))
def test_rigid_body_with_no_dofs(self):
s = System()
b = RigidBody('body', 23.7)
s.add_leaf(b)
s.setup()
# Calculate reduced system to get rigid body matrices
rsys = ReducedSystem(s)
self.assertEqual(rsys.M.shape, (0, 0))
self.assertEqual(rsys.Q.shape, (0,))
def test_print_functions(self):
# Not very good tests, but at least check they run without errors
joint = FreeJoint('joint')
body = RigidBody('body', mass=1.235)
s = System()
s.add_leaf(joint)
joint.add_leaf(body)
s.setup()
s.print_states()
s.print_info()
def test_accounts_for_offset_centre_of_mass_in_applied_force(self):
# check applied force due to gravity is correct
b = RigidBody('body', mass=5.6, Xc=[1.2, 3.4, 5.4])
s = System(gravity=9.81) # need a System to define gravity
s.add_leaf(b)
s.setup()
b.calc_mass()
b.calc_external_loading()
assert_array_equal(b.applied_forces,
b.mass * 9.81 * array([0, 0, -1, -3.4, 1.2, 0]))
def rigid_body_mass_matrix(element):
joint = FreeJoint('joint')
system = System()
system.add_leaf(joint)
joint.add_leaf(element)
system.setup()
for el in joint.iter_leaves():
system.prescribe(el, 0, 0)
system.update_kinematics()
rsys = ReducedSystem(system)
return rsys.M
def rigid_body_mass_matrix(element):
joint = FreeJoint('joint')
system = System()
system.add_leaf(joint)
joint.add_leaf(element)
system.setup()
for el in joint.iter_leaves():
system.prescribe(el, 0, 0)
system.update_kinematics()
rsys = ReducedSystem(system)
return rsys.M
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)
def test_dofs_subset(self):
s = System()
j = FreeJoint('joint')
s.add_leaf(j)
s.setup()
# 2 nodes, 6 constraints, 6 dofs
self.assertEqual(len(s.q), 2 * 12 + 6 + 6)
self.assertEqual(len(s.qd), 2 * 6 + 6 + 6)
self.assertEqual(len(s.qdd), 2 * 6 + 6 + 6)
self.assertEqual(len(s.q.dofs), 6)
self.assertEqual(len(s.qd.dofs), 6)
self.assertEqual(len(s.qdd.dofs), 6)
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', 'C')
fe.set_dofs([False, False, True, False, True, False])
beam = DistalModalElementFromFE('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 setUp(self):
joint = PrismaticJoint('joint', [0, 0, 1])
joint.stiffness = self.K
joint.damping = 2 * self.damping_coeff * (self.K * self.M) ** 0.5
body = RigidBody('body', self.M)
system = System()
system.add_leaf(joint)
joint.add_leaf(body)
system.setup()
system.update_kinematics()
self.joint, self.body, self.system = joint, body, system
def setUp(self):
joint = PrismaticJoint('joint', [0, 0, 1])
joint.stiffness = self.K
joint.damping = 2 * self.damping_coeff * (self.K * self.M)**0.5
body = RigidBody('body', self.M)
system = System()
system.add_leaf(joint)
joint.add_leaf(body)
system.setup()
system.update_kinematics()
self.joint, self.body, self.system = joint, body, 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_1dof_linear_system(self):
s = System()
j = PrismaticJoint('joint', [0, 0, 1])
j.stiffness = 5.0
j.damping = 2.3
b = RigidBody('body', 10)
s.add_leaf(j)
j.add_leaf(b)
s.setup()
linsys = LinearisedSystem.from_system(s)
assert_array_equal(linsys.M, [[10.0]])
assert_array_equal(linsys.C, [[2.3]])
assert_array_equal(linsys.K, [[5.0]])
def test_1dof_linear_system(self):
s = System()
j = PrismaticJoint('joint', [0, 0, 1])
j.stiffness = 5.0
j.damping = 2.3
b = RigidBody('body', 10)
s.add_leaf(j)
j.add_leaf(b)
s.setup()
linsys = LinearisedSystem.from_system(s)
assert_array_equal(linsys.M, [[10.0]])
assert_array_equal(linsys.C, [[2.3]])
assert_array_equal(linsys.K, [[5.0]])
def test_get_set_state(self):
s = System()
j = FreeJoint('joint')
s.add_leaf(j)
s.setup()
# State is [q_dofs, qd_dofs].
# Here we have 6 dofs:
s.set_state([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
self.assertEqual(list(s.q.dofs), [1, 2, 3, 4, 5, 6])
self.assertEqual(list(s.qd.dofs), [7, 8, 9, 10, 11, 12])
s.q.dofs[2] = 100
s.qd.dofs[0] = -1
self.assertEqual(list(s.get_state()),
[1, 2, 100, 4, 5, 6, -1, 8, 9, 10, 11, 12])
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', 'C')
fe.set_dofs([False, False, True, False, True, False])
beam = DistalModalElementFromFE('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 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
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)
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)
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 test_iter_elements(self):
# /-- c2
# c1-|
# \-- c3 --- c4
s = System()
c1 = RigidConnection('c1')
c2 = RigidConnection('c2')
c3 = RigidConnection('c3')
c4 = RigidConnection('c4')
s.add_leaf(c1)
c1.add_leaf(c2)
c1.add_leaf(c3)
c3.add_leaf(c4)
s.setup()
# Should iter elements depth-first
self.assertEqual(list(s.iter_elements()),
[c1, c2, c3, c4])
class hinged_beam_tests(unittest.TestCase):
density = 5.0
length = 20.0
force = 34.2 # N/m
hinge_torque = 0.0
free_beam = False
def setUp(self):
# FE model for beam
x = linspace(0, self.length, 20)
fe = BeamFE(x, density=self.density, EA=0, EIy=1, EIz=0)
fe.set_boundary_conditions('C', 'F')
self.beam = ModalElementFromFE('beam', fe, 0)
# Set loading - in negative Z direction
load = np.zeros((len(x), 3))
load[:, 2] = -self.force
self.beam.loading = load
# Hinge with axis along Y axis
self.hinge = Hinge('hinge', [0, 1, 0])
self.hinge.internal_torque = self.hinge_torque
# Build system
self.system = System()
self.system.add_leaf(self.hinge)
self.hinge.add_leaf(self.beam)
self.system.setup()
if not self.free_beam:
# Prescribe hinge to be fixed
self.system.prescribe(self.hinge)
# Initial calculations
self.recalc()
def recalc(self):
self.system.update_kinematics() # Set up nodal values initially
self.system.update_matrices()
self.system.solve_accelerations() # Calculate accelerations of DOFs
self.system.update_kinematics() # Update nodal values based on DOFs
self.system.update_matrices()
self.system.solve_reactions() # Solve reactions incl d'Alembert
def test_assemble(self):
# Test system:
#
# /-- m2
# m1 -----conn----|
# \-- m3
s = System()
m1 = RigidBody('m1', 1.3)
m2 = RigidBody('m2', 3.4)
m3 = RigidBody('m3', 7.5)
conn = RigidConnection('conn')
s.add_leaf(conn)
s.add_leaf(m1)
conn.add_leaf(m2)
conn.add_leaf(m3)
s.setup()
# Check starting mass matrices of elements are as expected
assert_aae(np.diag(m1.mass_vv[:3, :3]), 1.3)
assert_aae(np.diag(m2.mass_vv[:3, :3]), 3.4)
assert_aae(np.diag(m3.mass_vv[:3, :3]), 7.5)
# Initially make system matrix empty for testing
s.lhs[:, :] = 0
assert_aae(s.lhs, 0)
# After assembly, the mass matrices are put in the correct places:
# 0:6 -> m1 node
# 6:12 -> conn constraints
# 12:18 -> m2 node
# 12:18 -> m3 node
s.assemble()
M = s.lhs.copy()
# Subtract expected mass
M[0:6, 0:6] -= m1.mass_vv
M[12:18, 12:18] -= m2.mass_vv + m3.mass_vv
# Subtract expected constraints
M[0:6, 6:12] -= conn.F_vp
M[6:12, 0:6] -= conn.F_vp
M[12:18, 6:12] -= conn.F_vd
M[6:12, 12:18] -= conn.F_vd
assert_aae(M, 0)
