from sympy import Symbol, S
from sympy.physics.vector import ReferenceFrame, Dyadic, Point, dot
from sympy.physics.mechanics.body_base import BodyBase
from sympy.physics.mechanics.inertia import inertia_of_point_mass, Inertia
from sympy.utilities.exceptions import sympy_deprecation_warning
__all__ = ['RigidBody']
[docs]
class RigidBody(BodyBase):
"""An idealized rigid body.
Explanation
===========
This is essentially a container which holds the various components which
describe a rigid body: a name, mass, center of mass, reference frame, and
inertia.
All of these need to be supplied on creation, but can be changed
afterwards.
Attributes
==========
name : string
The body's name.
masscenter : Point
The point which represents the center of mass of the rigid body.
frame : ReferenceFrame
The ReferenceFrame which the rigid body is fixed in.
mass : Sympifyable
The body's mass.
inertia : (Dyadic, Point)
The body's inertia about a point; stored in a tuple as shown above.
potential_energy : Sympifyable
The potential energy of the RigidBody.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody
>>> from sympy.physics.mechanics import outer
>>> m = Symbol('m')
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer (A.x, A.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, A, m, inertia_tuple)
>>> # Or you could change them afterwards
>>> m2 = Symbol('m2')
>>> B.mass = m2
"""
def __init__(self, name, masscenter=None, frame=None, mass=None,
inertia=None):
super().__init__(name, masscenter, mass)
if frame is None:
frame = ReferenceFrame(f'{name}_frame')
self.frame = frame
if inertia is None:
ixx = Symbol(f'{name}_ixx')
iyy = Symbol(f'{name}_iyy')
izz = Symbol(f'{name}_izz')
izx = Symbol(f'{name}_izx')
ixy = Symbol(f'{name}_ixy')
iyz = Symbol(f'{name}_iyz')
inertia = Inertia.from_inertia_scalars(self.masscenter, self.frame,
ixx, iyy, izz, ixy, iyz, izx)
self.inertia = inertia
def __repr__(self):
return (f'{self.__class__.__name__}({repr(self.name)}, masscenter='
f'{repr(self.masscenter)}, frame={repr(self.frame)}, mass='
f'{repr(self.mass)}, inertia={repr(self.inertia)})')
@property
def frame(self):
"""The ReferenceFrame fixed to the body."""
return self._frame
@frame.setter
def frame(self, F):
if not isinstance(F, ReferenceFrame):
raise TypeError("RigidBody frame must be a ReferenceFrame object.")
self._frame = F
@property
def x(self):
"""The basis Vector for the body, in the x direction. """
return self.frame.x
@property
def y(self):
"""The basis Vector for the body, in the y direction. """
return self.frame.y
@property
def z(self):
"""The basis Vector for the body, in the z direction. """
return self.frame.z
@property
def inertia(self):
"""The body's inertia about a point; stored as (Dyadic, Point)."""
return self._inertia
@inertia.setter
def inertia(self, I):
# check if I is of the form (Dyadic, Point)
if len(I) != 2 or not isinstance(I[0], Dyadic) or not isinstance(I[1], Point):
raise TypeError("RigidBody inertia must be a tuple of the form (Dyadic, Point).")
self._inertia = Inertia(I[0], I[1])
# have I S/O, want I S/S*
# I S/O = I S/S* + I S*/O; I S/S* = I S/O - I S*/O
# I_S/S* = I_S/O - I_S*/O
I_Ss_O = inertia_of_point_mass(self.mass,
self.masscenter.pos_from(I[1]),
self.frame)
self._central_inertia = I[0] - I_Ss_O
@property
def central_inertia(self):
"""The body's central inertia dyadic."""
return self._central_inertia
@central_inertia.setter
def central_inertia(self, I):
if not isinstance(I, Dyadic):
raise TypeError("RigidBody inertia must be a Dyadic object.")
self.inertia = Inertia(I, self.masscenter)
[docs]
def linear_momentum(self, frame):
""" Linear momentum of the rigid body.
Explanation
===========
The linear momentum L, of a rigid body B, with respect to frame N is
given by:
``L = m * v``
where m is the mass of the rigid body, and v is the velocity of the mass
center of B in the frame N.
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v = dynamicsymbols('m v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (N.x, N.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, N, m, Inertia_tuple)
>>> B.linear_momentum(N)
m*v*N.x
"""
return self.mass * self.masscenter.vel(frame)
[docs]
def angular_momentum(self, point, frame):
"""Returns the angular momentum of the rigid body about a point in the
given frame.
Explanation
===========
The angular momentum H of a rigid body B about some point O in a frame N
is given by:
``H = dot(I, w) + cross(r, m * v)``
where I and m are the central inertia dyadic and mass of rigid body B, w
is the angular velocity of body B in the frame N, r is the position
vector from point O to the mass center of B, and v is the velocity of
the mass center in the frame N.
Parameters
==========
point : Point
The point about which angular momentum is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> 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')
>>> P.set_vel(N, 1 * N.x)
>>> I = outer(b.x, b.x)
>>> B = RigidBody('B', P, b, m, (I, P))
>>> B.angular_momentum(P, N)
omega*b.x
"""
I = self.central_inertia
w = self.frame.ang_vel_in(frame)
m = self.mass
r = self.masscenter.pos_from(point)
v = self.masscenter.vel(frame)
return I.dot(w) + r.cross(m * v)
[docs]
def kinetic_energy(self, frame):
"""Kinetic energy of the rigid body.
Explanation
===========
The kinetic energy, T, of a rigid body, B, is given by:
``T = 1/2 * (dot(dot(I, w), w) + dot(m * v, v))``
where I and m are the central inertia dyadic and mass of rigid body B
respectively, w is the body's angular velocity, and v is the velocity of
the body's mass center in the supplied ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The RigidBody's angular velocity and the velocity of it's mass
center are typically defined with respect to an inertial frame but
any relevant frame in which the velocities are known can be
supplied.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody
>>> from sympy import symbols
>>> m, v, r, omega = symbols('m v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (b.x, b.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, m, inertia_tuple)
>>> B.kinetic_energy(N)
m*v**2/2 + omega**2/2
"""
rotational_KE = S.Half * dot(
self.frame.ang_vel_in(frame),
dot(self.central_inertia, self.frame.ang_vel_in(frame)))
translational_KE = S.Half * self.mass * dot(self.masscenter.vel(frame),
self.masscenter.vel(frame))
return rotational_KE + translational_KE
def set_potential_energy(self, scalar):
sympy_deprecation_warning(
"""
The sympy.physics.mechanics.RigidBody.set_potential_energy()
method is deprecated. Instead use
B.potential_energy = scalar
""",
deprecated_since_version="1.5",
active_deprecations_target="deprecated-set-potential-energy",
)
self.potential_energy = scalar
[docs]
def parallel_axis(self, point, frame=None):
"""Returns the inertia dyadic of the body with respect to another point.
Parameters
==========
point : sympy.physics.vector.Point
The point to express the inertia dyadic about.
frame : sympy.physics.vector.ReferenceFrame
The reference frame used to construct the dyadic.
Returns
=======
inertia : sympy.physics.vector.Dyadic
The inertia dyadic of the rigid body expressed about the provided
point.
"""
if frame is None:
frame = self.frame
return self.central_inertia + inertia_of_point_mass(
self.mass, self.masscenter.pos_from(point), frame)