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from sympy.core.backend import (cos, sin, Matrix, symbols)from sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame, Point, KanesMethod, Particle)
def test_replace_qdots_in_force(): # Test PR 16700 "Replaces qdots with us in force-list in kanes.py" # The new functionality allows one to specify forces in qdots which will # automatically be replaced with u:s which are defined by the kde supplied # to KanesMethod. The test case is the double pendulum with interacting # forces in the example of chapter 4.7 "CONTRIBUTING INTERACTION FORCES" # in Ref. [1]. Reference list at end test function.
q1, q2 = dynamicsymbols('q1, q2') qd1, qd2 = dynamicsymbols('q1, q2', level=1) u1, u2 = dynamicsymbols('u1, u2')
l, m = symbols('l, m')
N = ReferenceFrame('N') # Inertial frame A = N.orientnew('A', 'Axis', (q1, N.z)) # Rod A frame B = A.orientnew('B', 'Axis', (q2, N.z)) # Rod B frame
O = Point('O') # Origo O.set_vel(N, 0)
P = O.locatenew('P', ( l * A.x )) # Point @ end of rod A P.v2pt_theory(O, N, A)
Q = P.locatenew('Q', ( l * B.x )) # Point @ end of rod B Q.v2pt_theory(P, N, B)
Ap = Particle('Ap', P, m) Bp = Particle('Bp', Q, m)
# The forces are specified below. sigma is the torsional spring stiffness # and delta is the viscous damping coefficient acting between the two # bodies. Here, we specify the viscous damper as function of qdots prior # forming the kde. In more complex systems it not might be obvious which # kde is most efficient, why it is convenient to specify viscous forces in # qdots independently of the kde. sig, delta = symbols('sigma, delta') Ta = (sig * q2 + delta * qd2) * N.z forces = [(A, Ta), (B, -Ta)]
# Try different kdes. kde1 = [u1 - qd1, u2 - qd2] kde2 = [u1 - qd1, u2 - (qd1 + qd2)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1) fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2) fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
# Check EOM for KM2: # Mass and force matrix from p.6 in Ref. [2] with added forces from # example of chapter 4.7 in [1] and without gravity. forcing_matrix_expected = Matrix( [ [ m * l**2 * sin(q2) * u2**2 + sig * q2 + delta * (u2 - u1)], [ m * l**2 * sin(q2) * -u1**2 - sig * q2 - delta * (u2 - u1)] ] ) mass_matrix_expected = Matrix( [ [ 2 * m * l**2, m * l**2 * cos(q2) ], [ m * l**2 * cos(q2), m * l**2 ] ] )
assert (KM2.mass_matrix.expand() == mass_matrix_expected.expand()) assert (KM2.forcing.expand() == forcing_matrix_expected.expand())
# Check fr1 with reference fr_expected from [1] with u:s instead of qdots. fr1_expected = Matrix([ 0, -(sig*q2 + delta * u2) ]) assert fr1.expand() == fr1_expected.expand()
# Check fr2 fr2_expected = Matrix([sig * q2 + delta * (u2 - u1), - sig * q2 - delta * (u2 - u1)]) assert fr2.expand() == fr2_expected.expand()
# Specifying forces in u:s should stay the same: Ta = (sig * q2 + delta * u2) * N.z forces = [(A, Ta), (B, -Ta)] KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1) fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
assert fr1.expand() == fr1_expected.expand()
Ta = (sig * q2 + delta * (u2-u1)) * N.z forces = [(A, Ta), (B, -Ta)] KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2) fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
assert fr2.expand() == fr2_expected.expand()
# Test if we have a qubic qdot force: Ta = (sig * q2 + delta * qd2**3) * N.z forces = [(A, Ta), (B, -Ta)]
KM1 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde1) fr1, fstar1 = KM1.kanes_equations([Ap, Bp], forces)
fr1_cubic_expected = Matrix([ 0, -(sig*q2 + delta * u2**3) ])
assert fr1.expand() == fr1_cubic_expected.expand()
KM2 = KanesMethod(N, [q1, q2], [u1, u2], kd_eqs=kde2) fr2, fstar2 = KM2.kanes_equations([Ap, Bp], forces)
fr2_cubic_expected = Matrix([sig * q2 + delta * (u2 - u1)**3, - sig * q2 - delta * (u2 - u1)**3])
assert fr2.expand() == fr2_cubic_expected.expand()
# References: # [1] T.R. Kane, D. a Levinson, Dynamics Theory and Applications, 2005. # [2] Arun K Banerjee, Flexible Multibody Dynamics:Efficient Formulations # and Applications, John Wiley and Sons, Ltd, 2016. # doi:http://dx.doi.org/10.1002/9781119015635.
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