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 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_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 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)
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)
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 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 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 __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 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_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)
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 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
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 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
class TestReactionForcesWithRotatedBeam(unittest.TestCase): """Intended to check the transformation from blade loading to rotor loading in a wind turbine rotor: the loads are applied to the beam in the local rotated coordinate system, check they work through to the ground reactions correctly. """ force = 24.1 length = 4.3 root_length = 0.0 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_reactions(self): # Some parameters L = self.length F = self.length * self.force # Set loading - in local z direction load = np.zeros((len(self.x), 3)) load[:, 2] = self.force self.blades[0].loading = load self.system.update_kinematics() self.system.update_matrices() self.system.solve_reactions() # Check reactions at ground (0, 0, 0) P = -self.system.joint_reactions['ground'] F_expected = [-F, 0, 0] M_expected = [0, -F * (L + self.root_length) / 2, 0] assert_aae(P, np.r_[F_expected, M_expected]) # Reactions on other side of hinge P2 = -self.system.joint_reactions['node-0'] assert_aae(P, P2) # Now set pitch angle to 45deg # NB: hinge rotation is opposite to wind turbine pitch convention self.pitch_bearings[0].xstrain[0] = -pi / 4 self.system.update_kinematics() self.system.update_matrices() self.system.solve_reactions() # Check reactions at ground (0, 0, 0) P = -self.system.joint_reactions['ground'] F_expected = [-F / sqrt(2), F / sqrt(2), 0] M_expected = [-F / sqrt(2) * L / 2, -F / sqrt(2) * L / 2, 0] assert_aae(P, np.r_[F_expected, M_expected]) # Reactions on other side of hinge P2 = -self.system.joint_reactions['node-0'] assert_aae(P, P2)
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])
class TestReactionForcesWithRotatedBeam(unittest.TestCase): """Intended to check the transformation from blade loading to rotor loading in a wind turbine rotor: the loads are applied to the beam in the local rotated coordinate system, check they work through to the ground reactions correctly. """ force = 24.1 length = 4.3 root_length = 0.0 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_reactions(self): # Some parameters L = self.length F = self.length * self.force # Set loading - in local z direction load = np.zeros((len(self.x), 3)) load[:, 2] = self.force self.blades[0].loading = load self.system.update_kinematics() self.system.update_matrices() self.system.solve_reactions() # Check reactions at ground (0, 0, 0) P = -self.system.joint_reactions['ground'] F_expected = [-F, 0, 0] M_expected = [0, -F*(L+self.root_length)/2, 0] assert_aae(P, np.r_[F_expected, M_expected]) # Reactions on other side of hinge P2 = -self.system.joint_reactions['node-0'] assert_aae(P, P2) # Now set pitch angle to 45deg # NB: hinge rotation is opposite to wind turbine pitch convention self.pitch_bearings[0].xstrain[0] = -pi / 4 self.system.update_kinematics() self.system.update_matrices() self.system.solve_reactions() # Check reactions at ground (0, 0, 0) P = -self.system.joint_reactions['ground'] F_expected = [-F/sqrt(2), F/sqrt(2), 0] M_expected = [-F/sqrt(2)*L/2, -F/sqrt(2)*L/2, 0] assert_aae(P, np.r_[F_expected, M_expected]) # Reactions on other side of hinge P2 = -self.system.joint_reactions['node-0'] assert_aae(P, P2)