def test_partial_velocity():
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3')
u4, u5 = dynamicsymbols('u4, u5')
r = symbols('r')
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
C = Point('C')
C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x))
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
vel_list = [Dmc.vel(N), C.vel(N), R.ang_vel_in(N)]
u_list = [u1, u2, u3, u4, u5]
assert (partial_velocity(vel_list,
u_list) == [[-r * L.y, 0, L.x], [r * L.x, 0, L.y],
[0, 0, L.z], [L.x, L.x, 0],
[
cos(q2) * L.y - sin(q2) * L.z,
cos(q2) * L.y - sin(q2) * L.z, 0
]])
def test_partial_velocity():
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3')
u4, u5 = dynamicsymbols('u4, u5')
r = symbols('r')
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
C = Point('C')
C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x))
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
vel_list = [Dmc.vel(N), C.vel(N), R.ang_vel_in(N)]
u_list = [u1, u2, u3, u4, u5]
assert (partial_velocity(vel_list, u_list) == [[- r*L.y, 0, L.x],
[r*L.x, 0, L.y], [0, 0, L.z], [L.x, L.x, 0],
[cos(q2)*L.y - sin(q2)*L.z, cos(q2)*L.y - sin(q2)*L.z, 0]])
u_s1 = [dot(pP1.vel(A), A.x), dot(pP1.vel(A), A.y), q3d]
u_s2 = [dot(pP1.vel(A), E.x), dot(pP1.vel(A), E.y), q3d]
u_s3 = [q1d, q2d, q3d]
# f1, f2 are forces the panes of glass exert on P1, P2 respectively
R1 = f1 * B.z + C * E.x - m1 * g * B.y
R2 = f2 * B.z - C * E.x - m2 * g * B.y
ulist = [u1, u2, u3]
for uset in [u_s1, u_s2, u_s3]:
# solve for u1, u2, u3 in terms of q1d, q2d, q3d and substitute
kinematic_eqs = [u_i - u_expr for u_i, u_expr in zip(ulist, uset)]
soln = solve(kinematic_eqs, [q1d, q2d, q3d])
vlist = subs([pP1.vel(A), pP2.vel(A)], soln)
v_r_Pi = partial_velocity(vlist, ulist, A)
F1, F2, F3 = [
simplify(
factor(
sum(dot(v_Pi[r], R_i) for v_Pi, R_i in zip(v_r_Pi, [R1, R2]))))
for r in range(3)
]
print("\nFor generalized speeds [u1, u2, u3] = {0}".format(msprint(uset)))
print("v_P1_A = {0}".format(vlist[0]))
print("v_P2_A = {0}".format(vlist[1]))
print("v_r_Pi = {0}".format(v_r_Pi))
print("F1 = {0}".format(msprint(F1)))
print("F2 = {0}".format(msprint(F2)))
print("F3 = {0}".format(msprint(F3)))
pP1.vel(A), pP2.vel(A)))
# three sets of generalized speeds
u_s1 = [dot(pP1.vel(A), A.x), dot(pP1.vel(A), A.y), q3d]
u_s2 = [dot(pP1.vel(A), E.x), dot(pP1.vel(A), E.y), q3d]
u_s3 = [q1d, q2d, q3d]
# f1, f2 are forces the panes of glass exert on P1, P2 respectively
R1 = f1*B.z + C*E.x - m1*g*B.y
R2 = f2*B.z - C*E.x - m2*g*B.y
ulist = [u1, u2, u3]
for uset in [u_s1, u_s2, u_s3]:
# solve for u1, u2, u3 in terms of q1d, q2d, q3d and substitute
kinematic_eqs = [u_i - u_expr for u_i, u_expr in zip(ulist, uset)]
soln = solve(kinematic_eqs, [q1d, q2d, q3d])
vlist = subs([pP1.vel(A), pP2.vel(A)], soln)
v_r_Pi = partial_velocity(vlist, ulist, A)
F1, F2, F3 = [simplify(factor(
sum(dot(v_Pi[r], R_i) for v_Pi, R_i in zip(v_r_Pi, [R1, R2]))))
for r in range(3)]
print("\nFor generalized speeds [u1, u2, u3] = {0}".format(msprint(uset)))
print("v_P1_A = {0}".format(vlist[0]))
print("v_P2_A = {0}".format(vlist[1]))
print("v_r_Pi = {0}".format(v_r_Pi))
print("F1 = {0}".format(msprint(F1)))
print("F2 = {0}".format(msprint(F2)))
print("F3 = {0}".format(msprint(F3)))
def test_partial_velocity():
q1, q2, q3, u1, u2, u3 = dynamicsymbols("q1 q2 q3 u1 u2 u3")
u4, u5 = dynamicsymbols("u4, u5")
r = symbols("r")
N = ReferenceFrame("N")
Y = N.orientnew("Y", "Axis", [q1, N.z])
L = Y.orientnew("L", "Axis", [q2, Y.x])
R = L.orientnew("R", "Axis", [q3, L.y])
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
C = Point("C")
C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x))
Dmc = C.locatenew("Dmc", r * L.z)
Dmc.v2pt_theory(C, N, R)
vel_list = [Dmc.vel(N), C.vel(N), R.ang_vel_in(N)]
u_list = [u1, u2, u3, u4, u5]
assert partial_velocity(vel_list, u_list) == [
[-r * L.y, 0, L.x],
[r * L.x, 0, L.y],
[0, 0, L.z],
[L.x, L.x, 0],
[cos(q2) * L.y - sin(q2) * L.z, cos(q2) * L.y - sin(q2) * L.z, 0],
]