def test_complete_simple_double_pendulum():
q1, q2 = dynamicsymbols('q1 q2')
u1, u2 = dynamicsymbols('u1 u2')
m, l, g = symbols('m l g')
C = Body('C') # ceiling
PartP = Body('P', mass=m)
PartR = Body('R', mass=m)
J1 = PinJoint('J1', C, PartP, speeds=u1, coordinates=q1,
child_joint_pos=-l*PartP.x, parent_axis=C.z,
child_axis=PartP.z)
J2 = PinJoint('J2', PartP, PartR, speeds=u2, coordinates=q2,
child_joint_pos=-l*PartR.x, parent_axis=PartP.z,
child_axis=PartR.z)
PartP.apply_force(m*g*C.x)
PartR.apply_force(m*g*C.x)
method = JointsMethod(C, J1, J2)
method.form_eoms()
assert expand(method.mass_matrix_full) == Matrix([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 2*l**2*m*cos(q2) + 3*l**2*m, l**2*m*cos(q2) + l**2*m],
[0, 0, l**2*m*cos(q2) + l**2*m, l**2*m]])
assert trigsimp(method.forcing_full) == trigsimp(Matrix([[u1], [u2], [-g*l*m*(sin(q1 + q2) + sin(q1)) -
g*l*m*sin(q1) + l**2*m*(2*u1 + u2)*u2*sin(q2)],
[-g*l*m*sin(q1 + q2) - l**2*m*u1**2*sin(q2)]]))
def test_chaos_pendulum():
#https://www.pydy.org/examples/chaos_pendulum.html
mA, mB, lA, lB, IAxx, IBxx, IByy, IBzz, g = symbols('mA, mB, lA, lB, IAxx, IBxx, IByy, IBzz, g')
theta, phi, omega, alpha = dynamicsymbols('theta phi omega alpha')
A = ReferenceFrame('A')
B = ReferenceFrame('B')
rod = Body('rod', mass=mA, frame=A, central_inertia=inertia(A, IAxx, IAxx, 0))
plate = Body('plate', mass=mB, frame=B, central_inertia=inertia(B, IBxx, IByy, IBzz))
C = Body('C')
J1 = PinJoint('J1', C, rod, coordinates=theta, speeds=omega,
child_joint_pos=-lA*rod.z, parent_axis=C.y, child_axis=rod.y)
J2 = PinJoint('J2', rod, plate, coordinates=phi, speeds=alpha,
parent_joint_pos=(lB-lA)*rod.z, parent_axis=rod.z, child_axis=plate.z)
rod.apply_force(mA*g*C.z)
plate.apply_force(mB*g*C.z)
method = JointsMethod(C, J1, J2)
method.form_eoms()
MM = method.mass_matrix
forcing = method.forcing
rhs = MM.LUsolve(forcing)
xd = (-2 * IBxx * alpha * omega * sin(phi) * cos(phi) + 2 * IByy * alpha * omega * sin(phi) *
cos(phi) - g * lA * mA * sin(theta) - g * lB * mB * sin(theta)) / (IAxx + IBxx *
sin(phi)**2 + IByy * cos(phi)**2 + lA**2 * mA + lB**2 * mB)
assert (rhs[0] - xd).simplify() == 0
xd = (IBxx - IByy) * omega**2 * sin(phi) * cos(phi) / IBzz
assert (rhs[1] - xd).simplify() == 0
def test_simple_pedulum():
l, m, g = symbols('l m g')
C = Body('C')
b = Body('b', mass=m)
q = dynamicsymbols('q')
P = PinJoint('P', C, b, speeds=q.diff(t), coordinates=q, child_joint_pos = -l*b.x,
parent_axis=C.z, child_axis=b.z)
b.potential_energy = - m * g * l * cos(q)
method = JointsMethod(C, P)
method.form_eoms(LagrangesMethod)
rhs = method.rhs()
assert rhs[1] == -g*sin(q)/l
def test_jointmethod_duplicate_coordinates_speeds():
P = Body('P')
C = Body('C')
T = Body('T')
q, u = dynamicsymbols('q u')
P1 = PinJoint('P1', P, C, q)
P2 = PrismaticJoint('P2', C, T, q)
raises(ValueError, lambda: JointsMethod(P, P1, P2))
P1 = PinJoint('P1', P, C, speeds=u)
P2 = PrismaticJoint('P2', C, T, speeds=u)
raises(ValueError, lambda: JointsMethod(P, P1, P2))
P1 = PinJoint('P1', P, C, q, u)
P2 = PrismaticJoint('P2', C, T, q, u)
raises(ValueError, lambda: JointsMethod(P, P1, P2))
def test_jointsmethod():
P = Body('P')
C = Body('C')
Pin = PinJoint('P1', P, C)
C_ixx, g = symbols('C_ixx g')
theta, omega = dynamicsymbols('theta_P1, omega_P1')
P.apply_force(g * P.y)
method = JointsMethod(P, Pin)
assert method.frame == P.frame
assert method.bodies == [C, P]
assert method.loads == [(P.masscenter, g * P.frame.y)]
assert method.q == [theta]
assert method.u == [omega]
assert method.kdes == [omega - theta.diff()]
soln = method.form_eoms()
assert soln == Matrix([[-C_ixx * omega.diff()]])
assert method.forcing_full == Matrix([[omega], [0]])
assert method.mass_matrix_full == Matrix([[1, 0], [0, C_ixx]])
assert isinstance(method.method, KanesMethod)
def test_two_dof_joints():
q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2')
W = Body('W')
B1 = Body('B1', mass=m)
B2 = Body('B2', mass=m)
J1 = PrismaticJoint('J1', W, B1, coordinates=q1, speeds=u1)
J2 = PrismaticJoint('J2', B1, B2, coordinates=q2, speeds=u2)
W.apply_force(k1*q1*W.x, reaction_body=B1)
W.apply_force(c1*u1*W.x, reaction_body=B1)
B1.apply_force(k2*q2*W.x, reaction_body=B2)
B1.apply_force(c2*u2*W.x, reaction_body=B2)
method = JointsMethod(W, J1, J2)
method.form_eoms()
MM = method.mass_matrix
forcing = method.forcing
rhs = MM.LUsolve(forcing)
assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m)
assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 *
c2 * u2) / m)