def test_angular_momentum_and_linear_momentum():
"""A rod with length 2l, centroidal inertia I, and mass M along with a
particle of mass m fixed to the end of the rod rotate with an angular rate
of omega about point O which is fixed to the non-particle end of the rod.
The rod's reference frame is A and the inertial frame is N."""
m, M, l, I = symbols('m, M, l, I')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
a = ReferenceFrame('a')
O = Point('O')
Ac = O.locatenew('Ac', l * N.x)
P = Ac.locatenew('P', l * N.x)
O.set_vel(N, 0 * N.x)
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
A = RigidBody('A', Ac, a, M, (I * outer(N.z, N.z), Ac))
expected = 2 * m * omega * l * N.y + M * l * omega * N.y
assert linear_momentum(N, A, Pa) == expected
raises(TypeError, lambda: angular_momentum(N, N, A, Pa))
raises(TypeError, lambda: angular_momentum(O, O, A, Pa))
raises(TypeError, lambda: angular_momentum(O, N, O, Pa))
expected = (I + M * l**2 + 4 * m * l**2) * omega * N.z
assert angular_momentum(O, N, A, Pa) == expected
def test_linear_momentum():
N = ReferenceFrame("N")
Ac = Point("Ac")
Ac.set_vel(N, 25 * N.y)
I = outer(N.x, N.x)
A = RigidBody("A", Ac, N, 20, (I, Ac))
P = Point("P")
Pa = Particle("Pa", P, 1)
Pa.point.set_vel(N, 10 * N.x)
assert linear_momentum(N, A, Pa) == 10 * N.x + 500 * N.y
def test_linear_momentum():
N = ReferenceFrame('N')
Ac = Point('Ac')
Ac.set_vel(N, 25 * N.y)
I = outer(N.x, N.x)
A = RigidBody('A', Ac, N, 20, (I, Ac))
P = Point('P')
Pa = Particle('Pa', P, 1)
Pa.point.set_vel(N, 10 * N.x)
assert linear_momentum(N, A, Pa) == 10 * N.x + 500 * N.y
def test_linear_momentum():
N = ReferenceFrame('N')
Ac = Point('Ac')
Ac.set_vel(N, 25 * N.y)
I = outer(N.x, N.x)
A = RigidBody('A', Ac, N, 20, (I, Ac))
P = Point('P')
Pa = Particle('Pa', P, 1)
Pa.point.set_vel(N, 10 * N.x)
assert linear_momentum(N, A, Pa) == 10 * N.x + 500 * N.y
def test_linear_momentum():
N = ReferenceFrame("N")
Ac = Point("Ac")
Ac.set_vel(N, 25 * N.y)
I = outer(N.x, N.x)
A = RigidBody("A", Ac, N, 20, (I, Ac))
P = Point("P")
Pa = Particle("Pa", P, 1)
Pa.point.set_vel(N, 10 * N.x)
raises(TypeError, lambda: linear_momentum(A, A, Pa))
raises(TypeError, lambda: linear_momentum(N, N, Pa))
assert linear_momentum(N, A, Pa) == 10 * N.x + 500 * N.y
def test_rigidbody2():
M, v, r, omega = dynamicsymbols('M v r omega')
N = ReferenceFrame('N')
b = ReferenceFrame('b')
b.set_ang_vel(N, omega * b.x)
P = Point('P')
I = outer (b.x, b.x)
Inertia_tuple = (I, P)
B = RigidBody('B', P, b, M, Inertia_tuple)
P.set_vel(N, v * b.x)
assert B.angularmomentum(P, N) == omega * b.x
O = Point('O')
O.set_vel(N, v * b.x)
P.set_pos(O, r * b.y)
assert B.angularmomentum(O, N) == omega * b.x - M*v*r*b.z
def test_gravity():
N = ReferenceFrame("N")
m, M, g = symbols("m M g")
F1, F2 = dynamicsymbols("F1 F2")
po = Point("po")
pa = Particle("pa", po, m)
A = ReferenceFrame("A")
P = Point("P")
I = outer(A.x, A.x)
B = RigidBody("B", P, A, M, (I, P))
forceList = [(po, F1), (P, F2)]
forceList.extend(gravity(g * N.y, pa, B))
l = [(po, F1), (P, F2), (po, g * m * N.y), (P, g * M * N.y)]
for i in range(len(l)):
for j in range(len(l[i])):
assert forceList[i][j] == l[i][j]
def test_angular_momentum_and_linear_momentum():
m, M, l1 = symbols("m M l1")
q1d = dynamicsymbols("q1d")
N = ReferenceFrame("N")
O = Point("O")
O.set_vel(N, 0 * N.x)
Ac = O.locatenew("Ac", l1 * N.x)
P = Ac.locatenew("P", l1 * N.x)
a = ReferenceFrame("a")
a.set_ang_vel(N, q1d * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle("Pa", P, m)
I = outer(N.z, N.z)
A = RigidBody("A", Ac, a, M, (I, Ac))
assert linear_momentum(N, A, Pa) == 2 * m * q1d * l1 * N.y + M * l1 * q1d * N.y
assert angular_momentum(O, N, A, Pa) == 4 * m * q1d * l1 ** 2 * N.z + q1d * N.z
def test_gravity():
N = ReferenceFrame('N')
m, M, g = symbols('m M g')
F1, F2 = dynamicsymbols('F1 F2')
po = Point('po')
pa = Particle('pa', po, m)
A = ReferenceFrame('A')
P = Point('P')
I = outer(A.x, A.x)
B = RigidBody('B', P, A, M, (I, P))
forceList = [(po, F1), (P, F2)]
forceList.extend(gravity(g*N.y, pa, B))
l = [(po, F1), (P, F2), (po, g*m*N.y), (P, g*M*N.y)]
for i in range(len(l)):
for j in range(len(l[i])):
assert forceList[i][j] == l[i][j]
def test_angular_momentum_and_linear_momentum():
m, M, l1 = symbols('m M l1')
q1d = dynamicsymbols('q1d')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, q1d * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
assert linear_momentum(N, A, Pa) == 2 * m * q1d* l1 * N.y + M * l1 * q1d * N.y
assert angular_momentum(O, N, A, Pa) == 4 * m * q1d * l1**2 * N.z + q1d * N.z
def test_kinetic_energy():
m, M, l1 = symbols('m M l1')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
assert 0 == kinetic_energy(N, Pa, A) - (M*l1**2*omega**2/2
+ 2*l1**2*m*omega**2 + omega**2/2)
def test_angular_momentum_and_linear_momentum():
m, M, l1 = symbols('m M l1')
q1d = dynamicsymbols('q1d')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, q1d * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
assert linear_momentum(N, A, Pa) == 2 * m * q1d* l1 * N.y + M * l1 * q1d * N.y
assert angular_momentum(O, N, A, Pa) == 4 * m * q1d * l1**2 * N.z + q1d * N.z
def test_kinetic_energy():
m, M, l1 = symbols('m M l1')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
assert 0 == kinetic_energy(N, Pa, A) - (M*l1**2*omega**2/2
+ 2*l1**2*m*omega**2 + omega**2/2)
def test_rigidbody2():
M, v, r, omega, g, h = dynamicsymbols('M v r omega g h')
N = ReferenceFrame('N')
b = ReferenceFrame('b')
b.set_ang_vel(N, omega * b.x)
P = Point('P')
I = outer(b.x, b.x)
Inertia_tuple = (I, P)
B = RigidBody('B', P, b, M, Inertia_tuple)
P.set_vel(N, v * b.x)
assert B.angular_momentum(P, N) == omega * b.x
O = Point('O')
O.set_vel(N, v * b.x)
P.set_pos(O, r * b.y)
assert B.angular_momentum(O, N) == omega * b.x - M*v*r*b.z
B.set_potential_energy(M * g * h)
assert B.potential_energy == M * g * h
assert B.kinetic_energy(N) == (omega**2 + M * v**2) / 2
def test_kinetic_energy():
m, M, l1 = symbols("m M l1")
omega = dynamicsymbols("omega")
N = ReferenceFrame("N")
O = Point("O")
O.set_vel(N, 0 * N.x)
Ac = O.locatenew("Ac", l1 * N.x)
P = Ac.locatenew("P", l1 * N.x)
a = ReferenceFrame("a")
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle("Pa", P, m)
I = outer(N.z, N.z)
A = RigidBody("A", Ac, a, M, (I, Ac))
assert 0 == kinetic_energy(N, Pa, A) - (
M * l1 ** 2 * omega ** 2 / 2 + 2 * l1 ** 2 * m * omega ** 2 + omega ** 2 / 2
)
def test_rigidbody2():
M, v, r, omega, g, h = dynamicsymbols('M v r omega g h')
N = ReferenceFrame('N')
b = ReferenceFrame('b')
b.set_ang_vel(N, omega * b.x)
P = Point('P')
I = outer(b.x, b.x)
Inertia_tuple = (I, P)
B = RigidBody('B', P, b, M, Inertia_tuple)
P.set_vel(N, v * b.x)
assert B.angular_momentum(P, N) == omega * b.x
O = Point('O')
O.set_vel(N, v * b.x)
P.set_pos(O, r * b.y)
assert B.angular_momentum(O, N) == omega * b.x - M*v*r*b.z
B.potential_energy = M * g * h
assert B.potential_energy == M * g * h
assert expand(2 * B.kinetic_energy(N)) == omega**2 + M * v**2
def test_potential_energy():
m, M, l1, g, h, H = symbols("m M l1 g h H")
omega = dynamicsymbols("omega")
N = ReferenceFrame("N")
O = Point("O")
O.set_vel(N, 0 * N.x)
Ac = O.locatenew("Ac", l1 * N.x)
P = Ac.locatenew("P", l1 * N.x)
a = ReferenceFrame("a")
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle("Pa", P, m)
I = outer(N.z, N.z)
A = RigidBody("A", Ac, a, M, (I, Ac))
Pa.potential_energy = m * g * h
A.potential_energy = M * g * H
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_Lagrangian():
M, m, g, h = symbols("M m g h")
N = ReferenceFrame("N")
O = Point("O")
O.set_vel(N, 0 * N.x)
P = O.locatenew("P", 1 * N.x)
P.set_vel(N, 10 * N.x)
Pa = Particle("Pa", P, 1)
Ac = O.locatenew("Ac", 2 * N.y)
Ac.set_vel(N, 5 * N.y)
a = ReferenceFrame("a")
a.set_ang_vel(N, 10 * N.z)
I = outer(N.z, N.z)
A = RigidBody("A", Ac, a, 20, (I, Ac))
Pa.potential_energy = m * g * h
A.potential_energy = M * g * h
raises(TypeError, lambda: Lagrangian(A, A, Pa))
raises(TypeError, lambda: Lagrangian(N, N, Pa))
def test_potential_energy():
m, M, l1, g, h, H = symbols('m M l1 g h H')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
Pa.set_potential_energy(m * g * h)
A.set_potential_energy(M * g * H)
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_potential_energy():
m, M, l1, g, h, H = symbols('m M l1 g h H')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
Pa.set_potential_energy(m * g * h)
A.set_potential_energy(M * g * H)
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_Lagrangian():
M, m, g, h = symbols('M m g h')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
P = O.locatenew('P', 1 * N.x)
P.set_vel(N, 10 * N.x)
Pa = Particle('Pa', P, 1)
Ac = O.locatenew('Ac', 2 * N.y)
Ac.set_vel(N, 5 * N.y)
a = ReferenceFrame('a')
a.set_ang_vel(N, 10 * N.z)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, 20, (I, Ac))
Pa.potential_energy = m * g * h
A.potential_energy = M * g * h
raises(TypeError, lambda: Lagrangian(A, A, Pa))
raises(TypeError, lambda: Lagrangian(N, N, Pa))
def test_potential_energy():
m, M, l1, g, h, H = symbols("m M l1 g h H")
omega = dynamicsymbols("omega")
N = ReferenceFrame("N")
O = Point("O")
O.set_vel(N, 0 * N.x)
Ac = O.locatenew("Ac", l1 * N.x)
P = Ac.locatenew("P", l1 * N.x)
a = ReferenceFrame("a")
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle("Pa", P, m)
I = outer(N.z, N.z)
A = RigidBody("A", Ac, a, M, (I, Ac))
Pa.potential_energy = m * g * h
A.potential_energy = M * g * H
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_rigidbody3():
q1, q2, q3, q4 = dynamicsymbols('q1:5')
p1, p2, p3 = symbols('p1:4')
m = symbols('m')
A = ReferenceFrame('A')
B = A.orientnew('B', 'axis', [q1, A.x])
O = Point('O')
O.set_vel(A, q2 * A.x + q3 * A.y + q4 * A.z)
P = O.locatenew('P', p1 * B.x + p2 * B.y + p3 * B.z)
I = outer(B.x, B.x)
rb1 = RigidBody('rb1', P, B, m, (I, P))
# I_S/O = I_S/S* + I_S*/O
rb2 = RigidBody('rb2', P, B, m,
(I + inertia_of_point_mass(m, P.pos_from(O), B), O))
assert rb1.central_inertia == rb2.central_inertia
assert rb1.angular_momentum(O, A) == rb2.angular_momentum(O, A)
def test_center_of_mass():
a = ReferenceFrame('a')
m = symbols('m', real=True)
p1 = Particle('p1', Point('p1_pt'), S(1))
p2 = Particle('p2', Point('p2_pt'), S(2))
p3 = Particle('p3', Point('p3_pt'), S(3))
p4 = Particle('p4', Point('p4_pt'), m)
b_f = ReferenceFrame('b_f')
b_cm = Point('b_cm')
mb = symbols('mb')
b = RigidBody('b', b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
p2.point.set_pos(p1.point, a.x)
p3.point.set_pos(p1.point, a.x + a.y)
p4.point.set_pos(p1.point, a.y)
b.masscenter.set_pos(p1.point, a.y + a.z)
point_o=Point('o')
point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
expr = 5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
assert point_o.pos_from(p1.point)-expr == 0
def test_center_of_mass():
a = ReferenceFrame('a')
m = symbols('m', real=True)
p1 = Particle('p1', Point('p1_pt'), S.One)
p2 = Particle('p2', Point('p2_pt'), S(2))
p3 = Particle('p3', Point('p3_pt'), S(3))
p4 = Particle('p4', Point('p4_pt'), m)
b_f = ReferenceFrame('b_f')
b_cm = Point('b_cm')
mb = symbols('mb')
b = RigidBody('b', b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
p2.point.set_pos(p1.point, a.x)
p3.point.set_pos(p1.point, a.x + a.y)
p4.point.set_pos(p1.point, a.y)
b.masscenter.set_pos(p1.point, a.y + a.z)
point_o=Point('o')
point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
expr = 5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
assert point_o.pos_from(p1.point)-expr == 0
def test_rigidbody3():
q1, q2, q3, q4 = dynamicsymbols('q1:5')
p1, p2, p3 = symbols('p1:4')
m = symbols('m')
A = ReferenceFrame('A')
B = A.orientnew('B', 'axis', [q1, A.x])
O = Point('O')
O.set_vel(A, q2*A.x + q3*A.y + q4*A.z)
P = O.locatenew('P', p1*B.x + p2*B.y + p3*B.z)
I = outer(B.x, B.x)
rb1 = RigidBody('rb1', P, B, m, (I, P))
# I_S/O = I_S/S* + I_S*/O
rb2 = RigidBody('rb2', P, B, m,
(I + inertia_of_point_mass(m, P.pos_from(O), B), O))
assert rb1.central_inertia == rb2.central_inertia
assert rb1.angular_momentum(O, A) == rb2.angular_momentum(O, A)
def test_center_of_mass():
a = ReferenceFrame("a")
m = symbols("m", real=True)
p1 = Particle("p1", Point("p1_pt"), S.One)
p2 = Particle("p2", Point("p2_pt"), S(2))
p3 = Particle("p3", Point("p3_pt"), S(3))
p4 = Particle("p4", Point("p4_pt"), m)
b_f = ReferenceFrame("b_f")
b_cm = Point("b_cm")
mb = symbols("mb")
b = RigidBody("b", b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
p2.point.set_pos(p1.point, a.x)
p3.point.set_pos(p1.point, a.x + a.y)
p4.point.set_pos(p1.point, a.y)
b.masscenter.set_pos(p1.point, a.y + a.z)
point_o = Point("o")
point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
expr = (5 / (m + mb + 6) * a.x + (m + mb + 3) / (m + mb + 6) * a.y + mb /
(m + mb + 6) * a.z)
assert point_o.pos_from(p1.point) - expr == 0
def test_rigidbody3():
q1, q2, q3, q4 = dynamicsymbols("q1:5")
p1, p2, p3 = symbols("p1:4")
m = symbols("m")
A = ReferenceFrame("A")
B = A.orientnew("B", "axis", [q1, A.x])
O = Point("O")
O.set_vel(A, q2 * A.x + q3 * A.y + q4 * A.z)
P = O.locatenew("P", p1 * B.x + p2 * B.y + p3 * B.z)
P.v2pt_theory(O, A, B)
I = outer(B.x, B.x)
rb1 = RigidBody("rb1", P, B, m, (I, P))
# I_S/O = I_S/S* + I_S*/O
rb2 = RigidBody("rb2", P, B, m,
(I + inertia_of_point_mass(m, P.pos_from(O), B), O))
assert rb1.central_inertia == rb2.central_inertia
assert rb1.angular_momentum(O, A) == rb2.angular_momentum(O, A)
def test_kinetic_energy():
m, M, l1 = symbols("m M l1")
omega = dynamicsymbols("omega")
N = ReferenceFrame("N")
O = Point("O")
O.set_vel(N, 0 * N.x)
Ac = O.locatenew("Ac", l1 * N.x)
P = Ac.locatenew("P", l1 * N.x)
a = ReferenceFrame("a")
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle("Pa", P, m)
I = outer(N.z, N.z)
A = RigidBody("A", Ac, a, M, (I, Ac))
raises(TypeError, lambda: kinetic_energy(Pa, Pa, A))
raises(TypeError, lambda: kinetic_energy(N, N, A))
assert (0 == (kinetic_energy(N, Pa, A) -
(M * l1**2 * omega**2 / 2 + 2 * l1**2 * m * omega**2 +
omega**2 / 2)).expand())
def test_angular_momentum_and_linear_momentum():
"""A rod with length 2l, centroidal inertia I, and mass M along with a
particle of mass m fixed to the end of the rod rotate with an angular rate
of omega about point O which is fixed to the non-particle end of the rod.
The rod's reference frame is A and the inertial frame is N."""
m, M, l, I = symbols("m, M, l, I")
omega = dynamicsymbols("omega")
N = ReferenceFrame("N")
a = ReferenceFrame("a")
O = Point("O")
Ac = O.locatenew("Ac", l * N.x)
P = Ac.locatenew("P", l * N.x)
O.set_vel(N, 0 * N.x)
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle("Pa", P, m)
A = RigidBody("A", Ac, a, M, (I * outer(N.z, N.z), Ac))
expected = 2 * m * omega * l * N.y + M * l * omega * N.y
assert (linear_momentum(N, A, Pa) - expected) == Vector(0)
expected = (I + M * l ** 2 + 4 * m * l ** 2) * omega * N.z
assert (angular_momentum(O, N, A, Pa) - expected).simplify() == Vector(0)
def test_angular_momentum_and_linear_momentum():
"""A rod with length 2l, centroidal inertia I, and mass M along with a
particle of mass m fixed to the end of the rod rotate with an angular rate
of omega about point O which is fixed to the non-particle end of the rod.
The rod's reference frame is A and the inertial frame is N."""
m, M, l, I = symbols('m, M, l, I')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
a = ReferenceFrame('a')
O = Point('O')
Ac = O.locatenew('Ac', l * N.x)
P = Ac.locatenew('P', l * N.x)
O.set_vel(N, 0 * N.x)
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
A = RigidBody('A', Ac, a, M, (I * outer(N.z, N.z), Ac))
expected = 2 * m * omega * l * N.y + M * l * omega * N.y
assert linear_momentum(N, A, Pa) == expected
expected = (I + M * l**2 + 4 * m * l**2) * omega * N.z
assert angular_momentum(O, N, A, Pa) == expected
import numpy as np
g, lb, w, h = sm.symbols("g lb w h", real=True)
theta, phi, omega, alpha = me.dynamicsymbols("theta phi omega alpha")
thetad, phid, omegad, alphad = me.dynamicsymbols("theta phi omega alpha", 1)
thetad2, phid2 = me.dynamicsymbols("theta phi", 2)
frame_n = me.ReferenceFrame("n")
body_a_cm = me.Point("a_cm")
body_a_cm.set_vel(frame_n, 0)
body_a_f = me.ReferenceFrame("a_f")
body_a = me.RigidBody(
"a",
body_a_cm,
body_a_f,
sm.symbols("m"),
(me.outer(body_a_f.x, body_a_f.x), body_a_cm),
)
body_b_cm = me.Point("b_cm")
body_b_cm.set_vel(frame_n, 0)
body_b_f = me.ReferenceFrame("b_f")
body_b = me.RigidBody(
"b",
body_b_cm,
body_b_f,
sm.symbols("m"),
(me.outer(body_b_f.x, body_b_f.x), body_b_cm),
)
body_a_f.orient(frame_n, "Axis", [theta, frame_n.y])
body_b_f.orient(body_a_f, "Axis", [phi, body_a_f.z])
point_o = me.Point("o")
la = (lb - h / 2) / 2
import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
frame_a = _me.ReferenceFrame('a')
c1, c2, c3 = _sm.symbols('c1 c2 c3', real=True)
a = _me.inertia(frame_a, 1, 1, 1)
particle_p1 = _me.Particle('p1', _me.Point('p1_pt'), _sm.Symbol('m'))
particle_p2 = _me.Particle('p2', _me.Point('p2_pt'), _sm.Symbol('m'))
body_r_cm = _me.Point('r_cm')
body_r_f = _me.ReferenceFrame('r_f')
body_r = _me.RigidBody('r', body_r_cm, body_r_f, _sm.symbols('m'), (_me.outer(body_r_f.x,body_r_f.x),body_r_cm))
frame_a.orient(body_r_f, 'DCM', _sm.Matrix([1,1,1,1,1,0,0,0,1]).reshape(3, 3))
point_o = _me.Point('o')
m1 = _sm.symbols('m1')
particle_p1.mass = m1
m2 = _sm.symbols('m2')
particle_p2.mass = m2
mr = _sm.symbols('mr')
body_r.mass = mr
i1 = _sm.symbols('i1')
i2 = _sm.symbols('i2')
i3 = _sm.symbols('i3')
body_r.inertia = (_me.inertia(body_r_f, i1, i2, i3, 0, 0, 0), body_r_cm)
point_o.set_pos(particle_p1.point, c1*frame_a.x)
point_o.set_pos(particle_p2.point, c2*frame_a.y)
point_o.set_pos(body_r_cm, c3*frame_a.z)
a = _me.inertia_of_point_mass(particle_p1.mass, particle_p1.point.pos_from(point_o), frame_a)
a = _me.inertia_of_point_mass(particle_p2.mass, particle_p2.point.pos_from(point_o), frame_a)
a = body_r.inertia[0] + _me.inertia_of_point_mass(body_r.mass, body_r.masscenter.pos_from(point_o), frame_a)
d1 = (e1).collect(x).coeff(x, 0)
d2 = (e1).collect(x).coeff(x, 1)
fm = sm.Matrix([i.collect(x) for i in sm.Matrix([e1, e2]).reshape(1, 2)]).reshape(
(sm.Matrix([e1, e2]).reshape(1, 2)).shape[0],
(sm.Matrix([e1, e2]).reshape(1, 2)).shape[1],
)
f = (e1).collect(y)
g = (e1).subs({x: 2 * x})
gm = sm.Matrix([i.subs({x: 3}) for i in sm.Matrix([e1, e2]).reshape(2, 1)]).reshape(
(sm.Matrix([e1, e2]).reshape(2, 1)).shape[0],
(sm.Matrix([e1, e2]).reshape(2, 1)).shape[1],
)
frame_a = me.ReferenceFrame("a")
frame_b = me.ReferenceFrame("b")
theta = me.dynamicsymbols("theta")
frame_b.orient(frame_a, "Axis", [theta, frame_a.z])
v1 = 2 * frame_a.x - 3 * frame_a.y + frame_a.z
v2 = frame_b.x + frame_b.y + frame_b.z
a = me.dot(v1, v2)
bm = sm.Matrix([me.dot(v1, v2), me.dot(v1, 2 * v2)]).reshape(2, 1)
c = me.cross(v1, v2)
d = 2 * v1.magnitude() + 3 * v1.magnitude()
dyadic = (
me.outer(3 * frame_a.x, frame_a.x)
+ me.outer(frame_a.y, frame_a.y)
+ me.outer(2 * frame_a.z, frame_a.z)
)
am = (dyadic).to_matrix(frame_b)
m = sm.Matrix([1, 2, 3]).reshape(3, 1)
v = m[0] * frame_a.x + m[1] * frame_a.y + m[2] * frame_a.z
def test_operator_match():
"""Test that the output of dot, cross, outer functions match
operator behavior.
"""
A = ReferenceFrame('A')
v = A.x + A.y
d = v | v
zerov = Vector(0)
zerod = Dyadic(0)
# dot products
assert d & d == dot(d, d)
assert d & zerod == dot(d, zerod)
assert zerod & d == dot(zerod, d)
assert d & v == dot(d, v)
assert v & d == dot(v, d)
assert d & zerov == dot(d, zerov)
assert zerov & d == dot(zerov, d)
raises(TypeError, lambda: dot(d, S(0)))
raises(TypeError, lambda: dot(S(0), d))
raises(TypeError, lambda: dot(d, 0))
raises(TypeError, lambda: dot(0, d))
assert v & v == dot(v, v)
assert v & zerov == dot(v, zerov)
assert zerov & v == dot(zerov, v)
raises(TypeError, lambda: dot(v, S(0)))
raises(TypeError, lambda: dot(S(0), v))
raises(TypeError, lambda: dot(v, 0))
raises(TypeError, lambda: dot(0, v))
# cross products
raises(TypeError, lambda: cross(d, d))
raises(TypeError, lambda: cross(d, zerod))
raises(TypeError, lambda: cross(zerod, d))
assert d ^ v == cross(d, v)
assert v ^ d == cross(v, d)
assert d ^ zerov == cross(d, zerov)
assert zerov ^ d == cross(zerov, d)
assert zerov ^ d == cross(zerov, d)
raises(TypeError, lambda: cross(d, S(0)))
raises(TypeError, lambda: cross(S(0), d))
raises(TypeError, lambda: cross(d, 0))
raises(TypeError, lambda: cross(0, d))
assert v ^ v == cross(v, v)
assert v ^ zerov == cross(v, zerov)
assert zerov ^ v == cross(zerov, v)
raises(TypeError, lambda: cross(v, S(0)))
raises(TypeError, lambda: cross(S(0), v))
raises(TypeError, lambda: cross(v, 0))
raises(TypeError, lambda: cross(0, v))
# outer products
raises(TypeError, lambda: outer(d, d))
raises(TypeError, lambda: outer(d, zerod))
raises(TypeError, lambda: outer(zerod, d))
raises(TypeError, lambda: outer(d, v))
raises(TypeError, lambda: outer(v, d))
raises(TypeError, lambda: outer(d, zerov))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(d, S(0)))
raises(TypeError, lambda: outer(S(0), d))
raises(TypeError, lambda: outer(d, 0))
raises(TypeError, lambda: outer(0, d))
assert v | v == outer(v, v)
assert v | zerov == outer(v, zerov)
assert zerov | v == outer(zerov, v)
raises(TypeError, lambda: outer(v, S(0)))
raises(TypeError, lambda: outer(S(0), v))
raises(TypeError, lambda: outer(v, 0))
raises(TypeError, lambda: outer(0, v))
point_p = me.Point('p')
point_o.set_pos(point_p, c1 * frame_a.x)
v = (v).express(frame_n)
point_o.set_pos(point_p, (point_o.pos_from(point_p)).express(frame_n))
frame_a.set_ang_vel(frame_n, c3 * frame_a.z)
print(frame_n.ang_vel_in(frame_a))
point_p.v2pt_theory(point_o, frame_n, frame_a)
particle_p1 = me.Particle('p1', me.Point('p1_pt'), sm.Symbol('m'))
particle_p2 = me.Particle('p2', me.Point('p2_pt'), sm.Symbol('m'))
particle_p2.point.v2pt_theory(particle_p1.point, frame_n, frame_a)
point_p.a2pt_theory(particle_p1.point, frame_n, frame_a)
body_b1_cm = me.Point('b1_cm')
body_b1_cm.set_vel(frame_n, 0)
body_b1_f = me.ReferenceFrame('b1_f')
body_b1 = me.RigidBody('b1', body_b1_cm, body_b1_f, sm.symbols('m'),
(me.outer(body_b1_f.x, body_b1_f.x), body_b1_cm))
body_b2_cm = me.Point('b2_cm')
body_b2_cm.set_vel(frame_n, 0)
body_b2_f = me.ReferenceFrame('b2_f')
body_b2 = me.RigidBody('b2', body_b2_cm, body_b2_f, sm.symbols('m'),
(me.outer(body_b2_f.x, body_b2_f.x), body_b2_cm))
g = sm.symbols('g', real=True)
force_p1 = particle_p1.mass * (g * frame_n.x)
force_p2 = particle_p2.mass * (g * frame_n.x)
force_b1 = body_b1.mass * (g * frame_n.x)
force_b2 = body_b2.mass * (g * frame_n.x)
z = me.dynamicsymbols('z')
v = x * frame_a.x + y * frame_a.z
point_o.set_pos(point_p, x * frame_a.x + y * frame_a.y)
v = (v).subs({x: 2 * z, y: z})
point_o.set_pos(point_p, (point_o.pos_from(point_p)).subs({x: 2 * z, y: z}))
import sympy.physics.mechanics as me
import sympy as sm
import math as m
import numpy as np
g, lb, w, h = sm.symbols('g lb w h', real=True)
theta, phi, omega, alpha = me.dynamicsymbols('theta phi omega alpha')
thetad, phid, omegad, alphad = me.dynamicsymbols('theta phi omega alpha', 1)
thetad2, phid2 = me.dynamicsymbols('theta phi', 2)
frame_n = me.ReferenceFrame('n')
body_a_cm = me.Point('a_cm')
body_a_cm.set_vel(frame_n, 0)
body_a_f = me.ReferenceFrame('a_f')
body_a = me.RigidBody('a', body_a_cm, body_a_f, sm.symbols('m'), (me.outer(body_a_f.x,body_a_f.x),body_a_cm))
body_b_cm = me.Point('b_cm')
body_b_cm.set_vel(frame_n, 0)
body_b_f = me.ReferenceFrame('b_f')
body_b = me.RigidBody('b', body_b_cm, body_b_f, sm.symbols('m'), (me.outer(body_b_f.x,body_b_f.x),body_b_cm))
body_a_f.orient(frame_n, 'Axis', [theta, frame_n.y])
body_b_f.orient(body_a_f, 'Axis', [phi, body_a_f.z])
point_o = me.Point('o')
la = (lb-h/2)/2
body_a_cm.set_pos(point_o, la*body_a_f.z)
body_b_cm.set_pos(point_o, lb*body_a_f.z)
body_a_f.set_ang_vel(frame_n, omega*frame_n.y)
body_b_f.set_ang_vel(body_a_f, alpha*body_a_f.z)
point_o.set_vel(frame_n, 0)
body_a_cm.v2pt_theory(point_o,frame_n,body_a_f)
body_b_cm.v2pt_theory(point_o,frame_n,body_a_f)
ma = sm.symbols('ma')
body_a.mass = ma
point_o = me.Point('o')
point_p = me.Point('p')
point_o.set_pos(point_p, c1*frame_a.x)
v = (v).express(frame_n)
point_o.set_pos(point_p, (point_o.pos_from(point_p)).express(frame_n))
frame_a.set_ang_vel(frame_n, c3*frame_a.z)
print(frame_n.ang_vel_in(frame_a))
point_p.v2pt_theory(point_o,frame_n,frame_a)
particle_p1 = me.Particle('p1', me.Point('p1_pt'), sm.Symbol('m'))
particle_p2 = me.Particle('p2', me.Point('p2_pt'), sm.Symbol('m'))
particle_p2.point.v2pt_theory(particle_p1.point,frame_n,frame_a)
point_p.a2pt_theory(particle_p1.point,frame_n,frame_a)
body_b1_cm = me.Point('b1_cm')
body_b1_cm.set_vel(frame_n, 0)
body_b1_f = me.ReferenceFrame('b1_f')
body_b1 = me.RigidBody('b1', body_b1_cm, body_b1_f, sm.symbols('m'), (me.outer(body_b1_f.x,body_b1_f.x),body_b1_cm))
body_b2_cm = me.Point('b2_cm')
body_b2_cm.set_vel(frame_n, 0)
body_b2_f = me.ReferenceFrame('b2_f')
body_b2 = me.RigidBody('b2', body_b2_cm, body_b2_f, sm.symbols('m'), (me.outer(body_b2_f.x,body_b2_f.x),body_b2_cm))
g = sm.symbols('g', real=True)
force_p1 = particle_p1.mass*(g*frame_n.x)
force_p2 = particle_p2.mass*(g*frame_n.x)
force_b1 = body_b1.mass*(g*frame_n.x)
force_b2 = body_b2.mass*(g*frame_n.x)
z = me.dynamicsymbols('z')
v=x*frame_a.x+y*frame_a.z
point_o.set_pos(point_p, x*frame_a.x+y*frame_a.y)
v = (v).subs({x:2*z, y:z})
point_o.set_pos(point_p, (point_o.pos_from(point_p)).subs({x:2*z, y:z}))
force_o = -1*(x*y*frame_a.x)
d = me.inertia(frame_a, 1, 1, 1)
point_po1 = me.Point("po1")
point_po2 = me.Point("po2")
particle_p1 = me.Particle("p1", me.Point("p1_pt"), sm.Symbol("m"))
particle_p2 = me.Particle("p2", me.Point("p2_pt"), sm.Symbol("m"))
c1, c2, c3 = me.dynamicsymbols("c1 c2 c3")
c1d, c2d, c3d = me.dynamicsymbols("c1 c2 c3", 1)
body_r_cm = me.Point("r_cm")
body_r_cm.set_vel(frame_n, 0)
body_r_f = me.ReferenceFrame("r_f")
body_r = me.RigidBody(
"r",
body_r_cm,
body_r_f,
sm.symbols("m"),
(me.outer(body_r_f.x, body_r_f.x), body_r_cm),
)
point_po2.set_pos(particle_p1.point, c1 * frame_a.x)
v = 2 * point_po2.pos_from(particle_p1.point) + c2 * frame_a.y
frame_a.set_ang_vel(frame_n, c3 * frame_a.z)
v = 2 * frame_a.ang_vel_in(frame_n) + c2 * frame_a.y
body_r_f.set_ang_vel(frame_n, c3 * frame_a.z)
v = 2 * body_r_f.ang_vel_in(frame_n) + c2 * frame_a.y
frame_a.set_ang_acc(frame_n, (frame_a.ang_vel_in(frame_n)).dt(frame_a))
v = 2 * frame_a.ang_acc_in(frame_n) + c2 * frame_a.y
particle_p1.point.set_vel(frame_a, c1 * frame_a.x + c3 * frame_a.y)
body_r_cm.set_acc(frame_n, c2 * frame_a.y)
v_a = me.cross(body_r_cm.acc(frame_n), particle_p1.point.vel(frame_a))
x_b_c = v_a
x_b_d = 2 * x_b_c
a_b_c_d_e = x_b_d * 2
e2.expand().coeff(x),
e2.expand().coeff(y)
]).reshape(2, 2)
b = (e1).expand().coeff(x)
c = (e2).expand().coeff(y)
d1 = (e1).collect(x).coeff(x, 0)
d2 = (e1).collect(x).coeff(x, 1)
fm = sm.Matrix([i.collect(x) for i in sm.Matrix([e1, e2]).reshape(1, 2)
]).reshape((sm.Matrix([e1, e2]).reshape(1, 2)).shape[0],
(sm.Matrix([e1, e2]).reshape(1, 2)).shape[1])
f = (e1).collect(y)
g = (e1).subs({x: 2 * x})
gm = sm.Matrix([i.subs({x: 3}) for i in sm.Matrix([e1, e2]).reshape(2, 1)
]).reshape((sm.Matrix([e1, e2]).reshape(2, 1)).shape[0],
(sm.Matrix([e1, e2]).reshape(2, 1)).shape[1])
frame_a = me.ReferenceFrame('a')
frame_b = me.ReferenceFrame('b')
theta = me.dynamicsymbols('theta')
frame_b.orient(frame_a, 'Axis', [theta, frame_a.z])
v1 = 2 * frame_a.x - 3 * frame_a.y + frame_a.z
v2 = frame_b.x + frame_b.y + frame_b.z
a = me.dot(v1, v2)
bm = sm.Matrix([me.dot(v1, v2), me.dot(v1, 2 * v2)]).reshape(2, 1)
c = me.cross(v1, v2)
d = 2 * v1.magnitude() + 3 * v1.magnitude()
dyadic = me.outer(3 * frame_a.x, frame_a.x) + me.outer(
frame_a.y, frame_a.y) + me.outer(2 * frame_a.z, frame_a.z)
am = (dyadic).to_matrix(frame_b)
m = sm.Matrix([1, 2, 3]).reshape(3, 1)
v = m[0] * frame_a.x + m[1] * frame_a.y + m[2] * frame_a.z
import sympy.physics.mechanics as me
import sympy as sm
import math as m
import numpy as np
frame_a=me.ReferenceFrame('a')
c1, c2, c3=sm.symbols('c1 c2 c3', real=True)
a=me.inertia(frame_a, 1, 1, 1)
particle_p1=me.Particle('p1', me.Point('p1_pt'), sm.Symbol('m'))
particle_p2=me.Particle('p2', me.Point('p2_pt'), sm.Symbol('m'))
body_r_cm=me.Point('r_cm')
body_r_f=me.ReferenceFrame('r_f')
body_r=me.RigidBody('r', body_r_cm, body_r_f, sm.symbols('m'), (me.outer(body_r_f.x,body_r_f.x),body_r_cm))
r_a = sm.Matrix([1,1,1,1,1,0,0,0,1]).reshape(3, 3)
point_o=me.Point('o')
m1=sm.symbols('m1')
particle_p1.mass = m1
m2=sm.symbols('m2')
particle_p2.mass = m2
mr=sm.symbols('mr')
body_r.mass = mr
i1 = sm.symbols('i1')
i2 = sm.symbols('i2')
i3 = sm.symbols('i3')
body_r.inertia = (me.inertia(body_r_f, i1, i2, i3, 0, 0, 0), body_r_cm)
point_o.set_pos(particle_p1.point, c1*frame_a.x)
point_o.set_pos(particle_p2.point, c2*frame_a.y)
point_o.set_pos(body_r_cm, c3*frame_a.z)
a=me.inertia_of_point_mass(particle_p1.mass, particle_p1.point.pos_from(point_o), frame_a)
a=me.inertia_of_point_mass(particle_p2.mass, particle_p2.point.pos_from(point_o), frame_a)
a=body_r.inertia[0] + me.inertia_of_point_mass(body_r.mass, body_r.masscenter.pos_from(point_o), frame_a)
import sympy.physics.mechanics as me
import sympy as sm
import math as m
import numpy as np
frame_a = me.ReferenceFrame("a")
frame_b = me.ReferenceFrame("b")
q1, q2, q3 = me.dynamicsymbols("q1 q2 q3")
frame_b.orient(frame_a, "Axis", [q3, frame_a.x])
dcm = frame_a.dcm(frame_b)
m = dcm * 3 - frame_a.dcm(frame_b)
r = me.dynamicsymbols("r")
circle_area = sm.pi * r**2
u, a = me.dynamicsymbols("u a")
x, y = me.dynamicsymbols("x y")
s = u * me.dynamicsymbols._t - 1 / 2 * a * me.dynamicsymbols._t**2
expr1 = 2 * a * 0.5 - 1.25 + 0.25
expr2 = -1 * x**2 + y**2 + 0.25 * (x + y)**2
expr3 = 0.5 * 10**(-10)
dyadic = (me.outer(frame_a.x, frame_a.x) + me.outer(frame_a.y, frame_a.y) +
me.outer(frame_a.z, frame_a.z))
import sympy.physics.mechanics as me
import sympy as sm
import math as m
import numpy as np
frame_a=me.ReferenceFrame('a')
frame_b=me.ReferenceFrame('b')
q1, q2, q3 = me.dynamicsymbols('q1 q2 q3')
frame_b.orient(frame_a, 'Axis', [q3, frame_a.x])
dcm = frame_a.dcm(frame_b)
m = dcm*3-frame_a.dcm(frame_b)
r = me.dynamicsymbols('r')
circle_area = sm.pi*r**2
u, a = me.dynamicsymbols('u a')
x, y = me.dynamicsymbols('x y')
s = u*me.dynamicsymbols._t-1/2*a*me.dynamicsymbols._t**2
expr1 = 2*a*0.5-1.25+0.25
expr2 = -1*x**2+y**2+0.25*(x+y)**2
expr3 = 0.5*10**(-10)
dyadic=me.outer(frame_a.x, frame_a.x)+me.outer(frame_a.y, frame_a.y)+me.outer(frame_a.z, frame_a.z)
frame_n=me.ReferenceFrame('n')
x1, x2, x3 = me.dynamicsymbols('x1 x2 x3')
l=sm.symbols('l', real=True)
v1=x1*frame_a.x+x2*frame_a.y+x3*frame_a.z
v2=x1*frame_b.x+x2*frame_b.y+x3*frame_b.z
v3=x1*frame_n.x+x2*frame_n.y+x3*frame_n.z
v=v1+v2+v3
point_c=me.Point('c')
point_d=me.Point('d')
point_po1=me.Point('po1')
point_po2=me.Point('po2')
point_po3=me.Point('po3')
particle_l=me.Particle('l', me.Point('l_pt'), sm.Symbol('m'))
particle_p1=me.Particle('p1', me.Point('p1_pt'), sm.Symbol('m'))
particle_p2=me.Particle('p2', me.Point('p2_pt'), sm.Symbol('m'))
particle_p3=me.Particle('p3', me.Point('p3_pt'), sm.Symbol('m'))
body_s_cm=me.Point('s_cm')
body_s_cm.set_vel(frame_n, 0)
body_s_f=me.ReferenceFrame('s_f')
body_s=me.RigidBody('s', body_s_cm, body_s_f, sm.symbols('m'), (me.outer(body_s_f.x,body_s_f.x),body_s_cm))
body_r1_cm=me.Point('r1_cm')
body_r1_cm.set_vel(frame_n, 0)
body_r1_f=me.ReferenceFrame('r1_f')
body_r1=me.RigidBody('r1', body_r1_cm, body_r1_f, sm.symbols('m'), (me.outer(body_r1_f.x,body_r1_f.x),body_r1_cm))
body_r2_cm=me.Point('r2_cm')
body_r2_cm.set_vel(frame_n, 0)
body_r2_f=me.ReferenceFrame('r2_f')
body_r2=me.RigidBody('r2', body_r2_cm, body_r2_f, sm.symbols('m'), (me.outer(body_r2_f.x,body_r2_f.x),body_r2_cm))
v4=x1*body_s_f.x+x2*body_s_f.y+x3*body_s_f.z
body_s_cm.set_pos(point_c, l*frame_n.x)
v1 = x1 * frame_a.x + x2 * frame_a.y + x3 * frame_a.z
v2 = x1 * frame_b.x + x2 * frame_b.y + x3 * frame_b.z
v3 = x1 * frame_n.x + x2 * frame_n.y + x3 * frame_n.z
v = v1 + v2 + v3
point_c = me.Point('c')
point_d = me.Point('d')
point_po1 = me.Point('po1')
point_po2 = me.Point('po2')
point_po3 = me.Point('po3')
particle_l = me.Particle('l', me.Point('l_pt'), sm.Symbol('m'))
particle_p1 = me.Particle('p1', me.Point('p1_pt'), sm.Symbol('m'))
particle_p2 = me.Particle('p2', me.Point('p2_pt'), sm.Symbol('m'))
particle_p3 = me.Particle('p3', me.Point('p3_pt'), sm.Symbol('m'))
body_s_cm = me.Point('s_cm')
body_s_cm.set_vel(frame_n, 0)
body_s_f = me.ReferenceFrame('s_f')
body_s = me.RigidBody('s', body_s_cm, body_s_f, sm.symbols('m'),
(me.outer(body_s_f.x, body_s_f.x), body_s_cm))
body_r1_cm = me.Point('r1_cm')
body_r1_cm.set_vel(frame_n, 0)
body_r1_f = me.ReferenceFrame('r1_f')
body_r1 = me.RigidBody('r1', body_r1_cm, body_r1_f, sm.symbols('m'),
(me.outer(body_r1_f.x, body_r1_f.x), body_r1_cm))
body_r2_cm = me.Point('r2_cm')
body_r2_cm.set_vel(frame_n, 0)
body_r2_f = me.ReferenceFrame('r2_f')
body_r2 = me.RigidBody('r2', body_r2_cm, body_r2_f, sm.symbols('m'),
(me.outer(body_r2_f.x, body_r2_f.x), body_r2_cm))
v4 = x1 * body_s_f.x + x2 * body_s_f.y + x3 * body_s_f.z
body_s_cm.set_pos(point_c, l * frame_n.x)
def test_operator_match():
"""Test that the output of dot, cross, outer functions match
operator behavior.
"""
A = ReferenceFrame("A")
v = A.x + A.y
d = v | v
zerov = Vector(0)
zerod = Dyadic(0)
# dot products
assert d & d == dot(d, d)
assert d & zerod == dot(d, zerod)
assert zerod & d == dot(zerod, d)
assert d & v == dot(d, v)
assert v & d == dot(v, d)
assert d & zerov == dot(d, zerov)
assert zerov & d == dot(zerov, d)
raises(TypeError, lambda: dot(d, S(0)))
raises(TypeError, lambda: dot(S(0), d))
raises(TypeError, lambda: dot(d, 0))
raises(TypeError, lambda: dot(0, d))
assert v & v == dot(v, v)
assert v & zerov == dot(v, zerov)
assert zerov & v == dot(zerov, v)
raises(TypeError, lambda: dot(v, S(0)))
raises(TypeError, lambda: dot(S(0), v))
raises(TypeError, lambda: dot(v, 0))
raises(TypeError, lambda: dot(0, v))
# cross products
raises(TypeError, lambda: cross(d, d))
raises(TypeError, lambda: cross(d, zerod))
raises(TypeError, lambda: cross(zerod, d))
assert d ^ v == cross(d, v)
assert v ^ d == cross(v, d)
assert d ^ zerov == cross(d, zerov)
assert zerov ^ d == cross(zerov, d)
assert zerov ^ d == cross(zerov, d)
raises(TypeError, lambda: cross(d, S(0)))
raises(TypeError, lambda: cross(S(0), d))
raises(TypeError, lambda: cross(d, 0))
raises(TypeError, lambda: cross(0, d))
assert v ^ v == cross(v, v)
assert v ^ zerov == cross(v, zerov)
assert zerov ^ v == cross(zerov, v)
raises(TypeError, lambda: cross(v, S(0)))
raises(TypeError, lambda: cross(S(0), v))
raises(TypeError, lambda: cross(v, 0))
raises(TypeError, lambda: cross(0, v))
# outer products
raises(TypeError, lambda: outer(d, d))
raises(TypeError, lambda: outer(d, zerod))
raises(TypeError, lambda: outer(zerod, d))
raises(TypeError, lambda: outer(d, v))
raises(TypeError, lambda: outer(v, d))
raises(TypeError, lambda: outer(d, zerov))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(d, S(0)))
raises(TypeError, lambda: outer(S(0), d))
raises(TypeError, lambda: outer(d, 0))
raises(TypeError, lambda: outer(0, d))
assert v | v == outer(v, v)
assert v | zerov == outer(v, zerov)
assert zerov | v == outer(zerov, v)
raises(TypeError, lambda: outer(v, S(0)))
raises(TypeError, lambda: outer(S(0), v))
raises(TypeError, lambda: outer(v, 0))
raises(TypeError, lambda: outer(0, v))
x, y = _me.dynamicsymbols('x y')
x_d, y_d = _me.dynamicsymbols('x_ y_', 1)
e = _sm.cos(x)+_sm.sin(x)+_sm.tan(x)+_sm.cosh(x)+_sm.sinh(x)+_sm.tanh(x)+_sm.acos(x)+_sm.asin(x)+_sm.atan(x)+_sm.log(x)+_sm.exp(x)+_sm.sqrt(x)+_sm.factorial(x)+_sm.ceiling(x)+_sm.floor(x)+_sm.sign(x)
e = (x)**2+_sm.log(x, 10)
a = _sm.Abs(-1*1)+int(1.5)+round(1.9)
e1 = 2*x+3*y
e2 = x+y
am = _sm.Matrix([e1.expand().coeff(x), e1.expand().coeff(y), e2.expand().coeff(x), e2.expand().coeff(y)]).reshape(2, 2)
b = (e1).expand().coeff(x)
c = (e2).expand().coeff(y)
d1 = (e1).collect(x).coeff(x,0)
d2 = (e1).collect(x).coeff(x,1)
fm = _sm.Matrix([i.collect(x)for i in _sm.Matrix([e1,e2]).reshape(1, 2)]).reshape((_sm.Matrix([e1,e2]).reshape(1, 2)).shape[0], (_sm.Matrix([e1,e2]).reshape(1, 2)).shape[1])
f = (e1).collect(y)
g = (e1).subs({x:2*x})
gm = _sm.Matrix([i.subs({x:3}) for i in _sm.Matrix([e1,e2]).reshape(2, 1)]).reshape((_sm.Matrix([e1,e2]).reshape(2, 1)).shape[0], (_sm.Matrix([e1,e2]).reshape(2, 1)).shape[1])
frame_a = _me.ReferenceFrame('a')
frame_b = _me.ReferenceFrame('b')
theta = _me.dynamicsymbols('theta')
frame_b.orient(frame_a, 'Axis', [theta, frame_a.z])
v1 = 2*frame_a.x-3*frame_a.y+frame_a.z
v2 = frame_b.x+frame_b.y+frame_b.z
a = _me.dot(v1, v2)
bm = _sm.Matrix([_me.dot(v1, v2),_me.dot(v1, 2*v2)]).reshape(2, 1)
c = _me.cross(v1, v2)
d = 2*v1.magnitude()+3*v1.magnitude()
dyadic = _me.outer(3*frame_a.x, frame_a.x)+_me.outer(frame_a.y, frame_a.y)+_me.outer(2*frame_a.z, frame_a.z)
am = (dyadic).to_matrix(frame_b)
m = _sm.Matrix([1,2,3]).reshape(3, 1)
v = m[0]*frame_a.x +m[1]*frame_a.y +m[2]*frame_a.z
import sympy.physics.mechanics as me
import sympy as sm
import math as m
import numpy as np
g, lb, w, h=sm.symbols('g lb w h', real=True)
theta, phi, omega, alpha = me.dynamicsymbols('theta phi omega alpha')
thetad, phid, omegad, alphad = me.dynamicsymbols('theta phi omega alpha', 1)
thetad2, phid2 = me.dynamicsymbols('theta phi', 2)
frame_n=me.ReferenceFrame('n')
body_a_cm=me.Point('a_cm')
body_a_cm.set_vel(frame_n, 0)
body_a_f=me.ReferenceFrame('a_f')
body_a=me.RigidBody('a', body_a_cm, body_a_f, sm.symbols('m'), (me.outer(body_a_f.x,body_a_f.x),body_a_cm))
body_b_cm=me.Point('b_cm')
body_b_cm.set_vel(frame_n, 0)
body_b_f=me.ReferenceFrame('b_f')
body_b=me.RigidBody('b', body_b_cm, body_b_f, sm.symbols('m'), (me.outer(body_b_f.x,body_b_f.x),body_b_cm))
body_a_f.orient(frame_n, 'Axis', [theta, frame_n.y])
body_b_f.orient(body_a_f, 'Axis', [phi, body_a_f.z])
point_o=me.Point('o')
la = (lb-h/2)/2
body_a_cm.set_pos(point_o, la*body_a_f.z)
body_b_cm.set_pos(point_o, lb*body_a_f.z)
body_a_f.set_ang_vel(frame_n, omega*frame_n.y)
body_b_f.set_ang_vel(body_a_f, alpha*body_a_f.z)
point_o.set_vel(frame_n, 0)
body_a_cm.v2pt_theory(point_o,frame_n,body_a_f)
body_b_cm.v2pt_theory(point_o,frame_n,body_a_f)
ma=sm.symbols('ma')
body_a.mass = ma
frame_n = me.ReferenceFrame('n')
x1, x2, x3 = me.dynamicsymbols('x1 x2 x3')
l = sm.symbols('l', real=True)
v1=x1*frame_a.x+x2*frame_a.y+x3*frame_a.z
v2=x1*frame_b.x+x2*frame_b.y+x3*frame_b.z
v3=x1*frame_n.x+x2*frame_n.y+x3*frame_n.z
v=v1+v2+v3
point_c = me.Point('c')
point_d = me.Point('d')
point_po1 = me.Point('po1')
point_po2 = me.Point('po2')
point_po3 = me.Point('po3')
particle_l = me.Particle('l', me.Point('l_pt'), sm.Symbol('m'))
particle_p1 = me.Particle('p1', me.Point('p1_pt'), sm.Symbol('m'))
particle_p2 = me.Particle('p2', me.Point('p2_pt'), sm.Symbol('m'))
particle_p3 = me.Particle('p3', me.Point('p3_pt'), sm.Symbol('m'))
body_s_cm = me.Point('s_cm')
body_s_cm.set_vel(frame_n, 0)
body_s_f = me.ReferenceFrame('s_f')
body_s = me.RigidBody('s', body_s_cm, body_s_f, sm.symbols('m'), (me.outer(body_s_f.x,body_s_f.x),body_s_cm))
body_r1_cm = me.Point('r1_cm')
body_r1_cm.set_vel(frame_n, 0)
body_r1_f = me.ReferenceFrame('r1_f')
body_r1 = me.RigidBody('r1', body_r1_cm, body_r1_f, sm.symbols('m'), (me.outer(body_r1_f.x,body_r1_f.x),body_r1_cm))
body_r2_cm = me.Point('r2_cm')
body_r2_cm.set_vel(frame_n, 0)
body_r2_f = me.ReferenceFrame('r2_f')
body_r2 = me.RigidBody('r2', body_r2_cm, body_r2_f, sm.symbols('m'), (me.outer(body_r2_f.x,body_r2_f.x),body_r2_cm))
v4=x1*body_s_f.x+x2*body_s_f.y+x3*body_s_f.z
body_s_cm.set_pos(point_c, l*frame_n.x)
import sympy.physics.mechanics as me
import sympy as sm
import math as m
import numpy as np
frame_a = me.ReferenceFrame('a')
frame_b = me.ReferenceFrame('b')
q1, q2, q3 = me.dynamicsymbols('q1 q2 q3')
frame_b.orient(frame_a, 'Axis', [q3, frame_a.x])
dcm = frame_a.dcm(frame_b)
m = dcm*3-frame_a.dcm(frame_b)
r = me.dynamicsymbols('r')
circle_area = sm.pi*r**2
u, a = me.dynamicsymbols('u a')
x, y = me.dynamicsymbols('x y')
s = u*me.dynamicsymbols._t-1/2*a*me.dynamicsymbols._t**2
expr1 = 2*a*0.5-1.25+0.25
expr2 = -1*x**2+y**2+0.25*(x+y)**2
expr3 = 0.5*10**(-10)
dyadic=me.outer(frame_a.x, frame_a.x)+me.outer(frame_a.y, frame_a.y)+me.outer(frame_a.z, frame_a.z)