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from sympy import sin, cos, tan, pi, symbols, Matrix, S, Function
from sympy.physics.mechanics import (Particle, Point, ReferenceFrame,
RigidBody)
from sympy.physics.mechanics import (angular_momentum, dynamicsymbols,
kinetic_energy, linear_momentum,
outer, potential_energy, msubs,
find_dynamicsymbols, Lagrangian)
from sympy.physics.mechanics.functions import (
center_of_mass, _validate_coordinates, _parse_linear_solver)
from sympy.testing.pytest import raises, warns_deprecated_sympy
q1, q2, q3, q4, q5 = symbols('q1 q2 q3 q4 q5')
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q1, N.z])
B = A.orientnew('B', 'Axis', [q2, A.x])
C = B.orientnew('C', 'Axis', [q3, B.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_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_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_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_msubs():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
# Test simple substitution
expr = Matrix([[a*x + b, x*y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
sol = Matrix([[a + b, y],
[x.diff().diff(), 1]])
sd = {x: 1, z: 1, z.diff(): 0, y.diff(): 0}
assert msubs(expr, sd) == sol
# Test smart substitution
expr = cos(x + y)*tan(x + y) + b*x.diff()
sd = {x: 0, y: pi/2, x.diff(): 1}
assert msubs(expr, sd, smart=True) == b + 1
N = ReferenceFrame('N')
v = x*N.x + y*N.y
d = x*(N.x|N.x) + y*(N.y|N.y)
v_sol = 1*N.y
d_sol = 1*(N.y|N.y)
sd = {x: 0, y: 1}
assert msubs(v, sd) == v_sol
assert msubs(d, sd) == d_sol
def test_find_dynamicsymbols():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
expr = Matrix([[a*x + b, x*y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
# Test finding all dynamicsymbols
sol = {x, y.diff(), y, x.diff().diff(), z, z.diff()}
assert find_dynamicsymbols(expr) == sol
# Test finding all but those in sym_list
exclude_list = [x, y, z]
sol = {y.diff(), x.diff().diff(), z.diff()}
assert find_dynamicsymbols(expr, exclude=exclude_list) == sol
# Test finding all dynamicsymbols in a vector with a given reference frame
d, e, f = dynamicsymbols('d, e, f')
A = ReferenceFrame('A')
v = d * A.x + e * A.y + f * A.z
sol = {d, e, f}
assert find_dynamicsymbols(v, reference_frame=A) == sol
# Test if a ValueError is raised on supplying only a vector as input
raises(ValueError, lambda: find_dynamicsymbols(v))
# This function tests the center_of_mass() function
# that was added in PR #14758 to compute the center of
# mass of a system of bodies.
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_validate_coordinates():
q1, q2, q3, u1, u2, u3, ua1, ua2, ua3 = dynamicsymbols('q1:4 u1:4 ua1:4')
s1, s2, s3 = symbols('s1:4')
# Test normal
_validate_coordinates([q1, q2, q3], [u1, u2, u3],
u_auxiliary=[ua1, ua2, ua3])
# Test not equal number of coordinates and speeds
_validate_coordinates([q1, q2])
_validate_coordinates([q1, q2], [u1])
_validate_coordinates(speeds=[u1, u2])
# Test duplicate
_validate_coordinates([q1, q2, q2], [u1, u2, u3], check_duplicates=False)
raises(ValueError, lambda: _validate_coordinates(
[q1, q2, q2], [u1, u2, u3]))
_validate_coordinates([q1, q2, q3], [u1, u2, u2], check_duplicates=False)
raises(ValueError, lambda: _validate_coordinates(
[q1, q2, q3], [u1, u2, u2], check_duplicates=True))
raises(ValueError, lambda: _validate_coordinates(
[q1, q2, q3], [q1, u2, u3], check_duplicates=True))
_validate_coordinates([q1, q2, q3], [u1, u2, u3], check_duplicates=False,
u_auxiliary=[u1, ua2, ua2])
raises(ValueError, lambda: _validate_coordinates(
[q1, q2, q3], [u1, u2, u3], u_auxiliary=[u1, ua2, ua3]))
raises(ValueError, lambda: _validate_coordinates(
[q1, q2, q3], [u1, u2, u3], u_auxiliary=[q1, ua2, ua3]))
raises(ValueError, lambda: _validate_coordinates(
[q1, q2, q3], [u1, u2, u3], u_auxiliary=[ua1, ua2, ua2]))
# Test is_dynamicsymbols
_validate_coordinates([q1 + q2, q3], is_dynamicsymbols=False)
raises(ValueError, lambda: _validate_coordinates([q1 + q2, q3]))
_validate_coordinates([s1, q1, q2], [0, u1, u2], is_dynamicsymbols=False)
raises(ValueError, lambda: _validate_coordinates(
[s1, q1, q2], [0, u1, u2], is_dynamicsymbols=True))
_validate_coordinates([s1 + s2 + s3, q1], [0, u1], is_dynamicsymbols=False)
raises(ValueError, lambda: _validate_coordinates(
[s1 + s2 + s3, q1], [0, u1], is_dynamicsymbols=True))
_validate_coordinates(u_auxiliary=[s1, ua1], is_dynamicsymbols=False)
raises(ValueError, lambda: _validate_coordinates(u_auxiliary=[s1, ua1]))
# Test normal function
t = dynamicsymbols._t
a = symbols('a')
f1, f2 = symbols('f1:3', cls=Function)
_validate_coordinates([f1(a), f2(a)], is_dynamicsymbols=False)
raises(ValueError, lambda: _validate_coordinates([f1(a), f2(a)]))
raises(ValueError, lambda: _validate_coordinates(speeds=[f1(a), f2(a)]))
dynamicsymbols._t = a
_validate_coordinates([f1(a), f2(a)])
raises(ValueError, lambda: _validate_coordinates([f1(t), f2(t)]))
dynamicsymbols._t = t
def test_parse_linear_solver():
A, b = Matrix(3, 3, symbols('a:9')), Matrix(3, 2, symbols('b:6'))
assert _parse_linear_solver(Matrix.LUsolve) == Matrix.LUsolve # Test callable
assert _parse_linear_solver('LU')(A, b) == Matrix.LUsolve(A, b)
def test_deprecated_moved_functions():
from sympy.physics.mechanics.functions import (
inertia, inertia_of_point_mass, gravity)
N = ReferenceFrame('N')
with warns_deprecated_sympy():
assert inertia(N, 0, 1, 0, 1) == (N.x | N.y) + (N.y | N.x) + (N.y | N.y)
with warns_deprecated_sympy():
assert inertia_of_point_mass(1, N.x + N.y, N) == (
(N.x | N.x) + (N.y | N.y) + 2 * (N.z | N.z) -
(N.x | N.y) - (N.y | N.x))
p = Particle('P')
with warns_deprecated_sympy():
assert gravity(-2 * N.z, p) == [(p.masscenter, -2 * p.mass * N.z)]
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