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_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 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):
# 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_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_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_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 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):
# 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
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])
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)
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 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 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 test_adding_elements_with_strains(self):
slider = PrismaticJoint('slider', [1, 0, 0])
conn = RigidConnection('conn')
body = RigidBody('body', mass=1.235)
s = System()
s.add_leaf(slider)
slider.add_leaf(conn)
conn.add_leaf(body)
s.setup()
# Should have dict of elements
self.assertEqual(s.elements,
{'slider': slider, 'conn': conn, 'body': body})
# Number of states:
# 6 ground
# + 6 constraints on slider
# + 1 strain in slider
# + 6 <node-0> between slider and conn
# + 6 constraints on conn
# + 6 <node-1> between conn and body
# ---
# 31
self.assertEqual(s.lhs.shape, (31, 31))
for vec in (s.rhs, s.qd, s.qdd):
self.assertEqual(len(vec), 31)
# Check there is the one slider dof
self.assertEqual(len(s.q.dofs), 1)
self.assertEqual(len(s.qd.dofs), 1)
self.assertEqual(len(s.qdd.dofs), 1)
# After prescribing the slider, there should be no dofs
s.prescribe(slider)
self.assertEqual(len(s.q.dofs), 0)
self.assertEqual(len(s.qd.dofs), 0)
self.assertEqual(len(s.qdd.dofs), 0)
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 setUp(self):
# Parameters
mass = 11.234
length = 2.54
gravity = 9.81
# Build model
slider = PrismaticJoint('slider', [1, 0, 0])
link = RigidConnection('link', [0, 0, length])
body = RigidBody('body', mass)
system = System(gravity=gravity)
system.add_leaf(slider)
slider.add_leaf(link)
link.add_leaf(body)
system.setup()
# Prescribe motion -- sinusoidal acceleration
motion_frequency = 1 # Hz
motion_amplitude = 2.3 # m
# x = motion_amplitude * np.cos(w*t)
# v = -motion_amplitude * np.sin(w*t) * w
# a = -motion_amplitude * np.cos(w*t) * w**2
def prescribed_acceleration(t):
w = 2 * np.pi * motion_frequency
return -w**2 * motion_amplitude * np.cos(w * t)
system.prescribe(slider, prescribed_acceleration)
# Set the correct initial condition
system.q[slider.istrain][0] = motion_amplitude
system.qd[slider.istrain][0] = 0.0
# Custom outputs to calculate correct answer
def force_body_prox(s):
a = prescribed_acceleration(s.time)
Fx = mass * a
Fz = mass * gravity
return [Fx, 0, Fz, 0, 0, 0]
def force_link_prox(s):
a = prescribed_acceleration(s.time)
Fx = mass * a
Fz = mass * gravity
My = length * Fx
return [Fx, 0, Fz, 0, My, 0]
def force_slider_prox(s):
a = prescribed_acceleration(s.time)
x = -a / (2 * np.pi * motion_frequency)**2
Fx = mass * a
Fz = mass * gravity
My = length * Fx - x * Fz
return [Fx, 0, Fz, 0, My, 0]
# Solver
integ = Integrator(system, ('pos', 'vel', 'acc'))
integ.add_output(LoadOutput(slider.iprox))
integ.add_output(LoadOutput(link.iprox))
integ.add_output(LoadOutput(body.iprox))
integ.add_output(CustomOutput(force_slider_prox, "correct ground"))
integ.add_output(CustomOutput(force_link_prox, "correct slider dist"))
integ.add_output(CustomOutput(force_body_prox, "correct link dist"))
self.system = system
self.integ = integ
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 setUp(self):
# Parameters
mass = 11.234
length = 2.54
gravity = 9.81
# Build model
slider = PrismaticJoint('slider', [1, 0, 0])
link = RigidConnection('link', [0, 0, length])
body = RigidBody('body', mass)
system = System(gravity=gravity)
system.add_leaf(slider)
slider.add_leaf(link)
link.add_leaf(body)
system.setup()
# Prescribe motion -- sinusoidal acceleration
motion_frequency = 1 # Hz
motion_amplitude = 2.3 # m
# x = motion_amplitude * np.cos(w*t)
# v = -motion_amplitude * np.sin(w*t) * w
# a = -motion_amplitude * np.cos(w*t) * w**2
def prescribed_acceleration(t):
w = 2*np.pi*motion_frequency
return -w**2 * motion_amplitude * np.cos(w*t)
system.prescribe(slider, prescribed_acceleration)
# Set the correct initial condition
system.q[slider.istrain][0] = motion_amplitude
system.qd[slider.istrain][0] = 0.0
# Custom outputs to calculate correct answer
def force_body_prox(s):
a = prescribed_acceleration(s.time)
Fx = mass * a
Fz = mass * gravity
return [Fx, 0, Fz, 0, 0, 0]
def force_link_prox(s):
a = prescribed_acceleration(s.time)
Fx = mass * a
Fz = mass * gravity
My = length * Fx
return [Fx, 0, Fz, 0, My, 0]
def force_slider_prox(s):
a = prescribed_acceleration(s.time)
x = -a / (2*np.pi*motion_frequency)**2
Fx = mass * a
Fz = mass * gravity
My = length*Fx - x*Fz
return [Fx, 0, Fz, 0, My, 0]
# Solver
integ = Integrator(system, ('pos', 'vel', 'acc'))
integ.add_output(LoadOutput(slider.iprox))
integ.add_output(LoadOutput(link.iprox))
integ.add_output(LoadOutput(body.iprox))
integ.add_output(CustomOutput(force_slider_prox, "correct ground"))
integ.add_output(CustomOutput(force_link_prox, "correct slider dist"))
integ.add_output(CustomOutput(force_body_prox, "correct link dist"))
self.system = system
self.integ = integ
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_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