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_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_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_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_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_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_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_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_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 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):
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):
# 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_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 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 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_find_equilibrium(self):
g = 9.81
m = 23.1
k = 45.2
s = System(gravity=g)
slider = PrismaticJoint('slider', [0, 0, 1])
slider.stiffness = k
body = RigidBody('body', mass=m)
s.add_leaf(slider)
slider.add_leaf(body)
s.setup()
# Initially position should be zero and acceleration nonzero
s.solve_accelerations()
assert_aae(slider.xstrain, 0)
assert_aae(slider.astrain, -g)
# At equilibrium, position should be nozero and force on body zero
s.find_equilibrium()
s.update_matrices() # recalculate stiffness force
s.solve_accelerations()
assert_aae(slider.xstrain, -m * g / k)
assert_aae(slider.astrain, 0)
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)
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 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