612 lines
19 KiB
Python
612 lines
19 KiB
Python
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from sympy.core.backend import Symbol
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from sympy.physics.vector import Point, Vector, ReferenceFrame, Dyadic
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from sympy.physics.mechanics import RigidBody, Particle, inertia
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__all__ = ['Body']
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# XXX: We use type:ignore because the classes RigidBody and Particle have
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# inconsistent parallel axis methods that take different numbers of arguments.
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class Body(RigidBody, Particle): # type: ignore
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"""
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Body is a common representation of either a RigidBody or a Particle SymPy
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object depending on what is passed in during initialization. If a mass is
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passed in and central_inertia is left as None, the Particle object is
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created. Otherwise a RigidBody object will be created.
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Explanation
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===========
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The attributes that Body possesses will be the same as a Particle instance
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or a Rigid Body instance depending on which was created. Additional
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attributes are listed below.
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Attributes
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==========
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name : string
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The body's name
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masscenter : Point
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The point which represents the center of mass of the rigid body
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frame : ReferenceFrame
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The reference frame which the body is fixed in
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mass : Sympifyable
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The body's mass
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inertia : (Dyadic, Point)
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The body's inertia around its center of mass. This attribute is specific
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to the rigid body form of Body and is left undefined for the Particle
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form
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loads : iterable
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This list contains information on the different loads acting on the
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Body. Forces are listed as a (point, vector) tuple and torques are
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listed as (reference frame, vector) tuples.
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Parameters
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==========
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name : String
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Defines the name of the body. It is used as the base for defining
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body specific properties.
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masscenter : Point, optional
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A point that represents the center of mass of the body or particle.
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If no point is given, a point is generated.
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mass : Sympifyable, optional
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A Sympifyable object which represents the mass of the body. If no
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mass is passed, one is generated.
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frame : ReferenceFrame, optional
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The ReferenceFrame that represents the reference frame of the body.
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If no frame is given, a frame is generated.
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central_inertia : Dyadic, optional
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Central inertia dyadic of the body. If none is passed while creating
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RigidBody, a default inertia is generated.
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Examples
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========
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Default behaviour. This results in the creation of a RigidBody object for
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which the mass, mass center, frame and inertia attributes are given default
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values. ::
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>>> from sympy.physics.mechanics import Body
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>>> body = Body('name_of_body')
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This next example demonstrates the code required to specify all of the
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values of the Body object. Note this will also create a RigidBody version of
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the Body object. ::
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>>> from sympy import Symbol
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>>> from sympy.physics.mechanics import ReferenceFrame, Point, inertia
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>>> from sympy.physics.mechanics import Body
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>>> mass = Symbol('mass')
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>>> masscenter = Point('masscenter')
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>>> frame = ReferenceFrame('frame')
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>>> ixx = Symbol('ixx')
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>>> body_inertia = inertia(frame, ixx, 0, 0)
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>>> body = Body('name_of_body', masscenter, mass, frame, body_inertia)
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The minimal code required to create a Particle version of the Body object
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involves simply passing in a name and a mass. ::
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>>> from sympy import Symbol
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>>> from sympy.physics.mechanics import Body
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>>> mass = Symbol('mass')
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>>> body = Body('name_of_body', mass=mass)
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The Particle version of the Body object can also receive a masscenter point
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and a reference frame, just not an inertia.
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"""
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def __init__(self, name, masscenter=None, mass=None, frame=None,
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central_inertia=None):
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self.name = name
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self._loads = []
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if frame is None:
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frame = ReferenceFrame(name + '_frame')
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if masscenter is None:
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masscenter = Point(name + '_masscenter')
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if central_inertia is None and mass is None:
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ixx = Symbol(name + '_ixx')
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iyy = Symbol(name + '_iyy')
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izz = Symbol(name + '_izz')
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izx = Symbol(name + '_izx')
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ixy = Symbol(name + '_ixy')
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iyz = Symbol(name + '_iyz')
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_inertia = (inertia(frame, ixx, iyy, izz, ixy, iyz, izx),
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masscenter)
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else:
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_inertia = (central_inertia, masscenter)
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if mass is None:
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_mass = Symbol(name + '_mass')
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else:
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_mass = mass
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masscenter.set_vel(frame, 0)
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# If user passes masscenter and mass then a particle is created
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# otherwise a rigidbody. As a result a body may or may not have inertia.
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if central_inertia is None and mass is not None:
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self.frame = frame
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self.masscenter = masscenter
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Particle.__init__(self, name, masscenter, _mass)
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self._central_inertia = Dyadic(0)
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else:
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RigidBody.__init__(self, name, masscenter, frame, _mass, _inertia)
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@property
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def loads(self):
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return self._loads
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@property
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def x(self):
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"""The basis Vector for the Body, in the x direction."""
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return self.frame.x
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@property
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def y(self):
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"""The basis Vector for the Body, in the y direction."""
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return self.frame.y
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@property
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def z(self):
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"""The basis Vector for the Body, in the z direction."""
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return self.frame.z
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@property
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def inertia(self):
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"""The body's inertia about a point; stored as (Dyadic, Point)."""
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if self.is_rigidbody:
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return RigidBody.inertia.fget(self)
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return (self.central_inertia, self.masscenter)
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@inertia.setter
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def inertia(self, I):
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RigidBody.inertia.fset(self, I)
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@property
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def is_rigidbody(self):
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if hasattr(self, '_inertia'):
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return True
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return False
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def kinetic_energy(self, frame):
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"""Kinetic energy of the body.
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Parameters
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==========
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frame : ReferenceFrame or Body
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The Body's angular velocity and the velocity of it's mass
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center are typically defined with respect to an inertial frame but
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any relevant frame in which the velocities are known can be supplied.
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Examples
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========
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>>> from sympy.physics.mechanics import Body, ReferenceFrame, Point
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>>> from sympy import symbols
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>>> m, v, r, omega = symbols('m v r omega')
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>>> N = ReferenceFrame('N')
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>>> O = Point('O')
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>>> P = Body('P', masscenter=O, mass=m)
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>>> P.masscenter.set_vel(N, v * N.y)
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>>> P.kinetic_energy(N)
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m*v**2/2
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>>> N = ReferenceFrame('N')
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>>> b = ReferenceFrame('b')
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>>> b.set_ang_vel(N, omega * b.x)
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>>> P = Point('P')
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>>> P.set_vel(N, v * N.x)
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>>> B = Body('B', masscenter=P, frame=b)
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>>> B.kinetic_energy(N)
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B_ixx*omega**2/2 + B_mass*v**2/2
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See Also
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========
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sympy.physics.mechanics : Particle, RigidBody
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"""
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if isinstance(frame, Body):
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frame = Body.frame
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if self.is_rigidbody:
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return RigidBody(self.name, self.masscenter, self.frame, self.mass,
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(self.central_inertia, self.masscenter)).kinetic_energy(frame)
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return Particle(self.name, self.masscenter, self.mass).kinetic_energy(frame)
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def apply_force(self, force, point=None, reaction_body=None, reaction_point=None):
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"""Add force to the body(s).
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Explanation
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===========
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Applies the force on self or equal and oppposite forces on
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self and other body if both are given on the desried point on the bodies.
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The force applied on other body is taken opposite of self, i.e, -force.
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Parameters
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==========
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force: Vector
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The force to be applied.
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point: Point, optional
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The point on self on which force is applied.
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By default self's masscenter.
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reaction_body: Body, optional
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Second body on which equal and opposite force
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is to be applied.
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reaction_point : Point, optional
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The point on other body on which equal and opposite
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force is applied. By default masscenter of other body.
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Example
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=======
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>>> from sympy import symbols
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>>> from sympy.physics.mechanics import Body, Point, dynamicsymbols
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>>> m, g = symbols('m g')
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>>> B = Body('B')
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>>> force1 = m*g*B.z
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>>> B.apply_force(force1) #Applying force on B's masscenter
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>>> B.loads
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[(B_masscenter, g*m*B_frame.z)]
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We can also remove some part of force from any point on the body by
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adding the opposite force to the body on that point.
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>>> f1, f2 = dynamicsymbols('f1 f2')
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>>> P = Point('P') #Considering point P on body B
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>>> B.apply_force(f1*B.x + f2*B.y, P)
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>>> B.loads
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[(B_masscenter, g*m*B_frame.z), (P, f1(t)*B_frame.x + f2(t)*B_frame.y)]
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Let's remove f1 from point P on body B.
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>>> B.apply_force(-f1*B.x, P)
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>>> B.loads
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[(B_masscenter, g*m*B_frame.z), (P, f2(t)*B_frame.y)]
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To further demonstrate the use of ``apply_force`` attribute,
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consider two bodies connected through a spring.
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>>> from sympy.physics.mechanics import Body, dynamicsymbols
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>>> N = Body('N') #Newtonion Frame
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>>> x = dynamicsymbols('x')
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>>> B1 = Body('B1')
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>>> B2 = Body('B2')
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>>> spring_force = x*N.x
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Now let's apply equal and opposite spring force to the bodies.
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>>> P1 = Point('P1')
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>>> P2 = Point('P2')
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>>> B1.apply_force(spring_force, point=P1, reaction_body=B2, reaction_point=P2)
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We can check the loads(forces) applied to bodies now.
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>>> B1.loads
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[(P1, x(t)*N_frame.x)]
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>>> B2.loads
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[(P2, - x(t)*N_frame.x)]
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Notes
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=====
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If a new force is applied to a body on a point which already has some
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force applied on it, then the new force is added to the already applied
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force on that point.
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"""
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if not isinstance(point, Point):
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if point is None:
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point = self.masscenter # masscenter
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else:
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raise TypeError("Force must be applied to a point on the body.")
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if not isinstance(force, Vector):
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raise TypeError("Force must be a vector.")
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if reaction_body is not None:
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reaction_body.apply_force(-force, point=reaction_point)
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for load in self._loads:
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if point in load:
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force += load[1]
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self._loads.remove(load)
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break
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self._loads.append((point, force))
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def apply_torque(self, torque, reaction_body=None):
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"""Add torque to the body(s).
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Explanation
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===========
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Applies the torque on self or equal and oppposite torquess on
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self and other body if both are given.
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The torque applied on other body is taken opposite of self,
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i.e, -torque.
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Parameters
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==========
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torque: Vector
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The torque to be applied.
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reaction_body: Body, optional
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Second body on which equal and opposite torque
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is to be applied.
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Example
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=======
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>>> from sympy import symbols
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>>> from sympy.physics.mechanics import Body, dynamicsymbols
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>>> t = symbols('t')
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>>> B = Body('B')
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>>> torque1 = t*B.z
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>>> B.apply_torque(torque1)
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>>> B.loads
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[(B_frame, t*B_frame.z)]
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We can also remove some part of torque from the body by
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adding the opposite torque to the body.
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>>> t1, t2 = dynamicsymbols('t1 t2')
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>>> B.apply_torque(t1*B.x + t2*B.y)
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>>> B.loads
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[(B_frame, t1(t)*B_frame.x + t2(t)*B_frame.y + t*B_frame.z)]
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Let's remove t1 from Body B.
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>>> B.apply_torque(-t1*B.x)
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>>> B.loads
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[(B_frame, t2(t)*B_frame.y + t*B_frame.z)]
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To further demonstrate the use, let us consider two bodies such that
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a torque `T` is acting on one body, and `-T` on the other.
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>>> from sympy.physics.mechanics import Body, dynamicsymbols
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>>> N = Body('N') #Newtonion frame
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>>> B1 = Body('B1')
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>>> B2 = Body('B2')
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>>> v = dynamicsymbols('v')
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>>> T = v*N.y #Torque
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Now let's apply equal and opposite torque to the bodies.
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>>> B1.apply_torque(T, B2)
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||
|
|
|
||
|
|
We can check the loads (torques) applied to bodies now.
|
||
|
|
|
||
|
|
>>> B1.loads
|
||
|
|
[(B1_frame, v(t)*N_frame.y)]
|
||
|
|
>>> B2.loads
|
||
|
|
[(B2_frame, - v(t)*N_frame.y)]
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
If a new torque is applied on body which already has some torque applied on it,
|
||
|
|
then the new torque is added to the previous torque about the body's frame.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if not isinstance(torque, Vector):
|
||
|
|
raise TypeError("A Vector must be supplied to add torque.")
|
||
|
|
|
||
|
|
if reaction_body is not None:
|
||
|
|
reaction_body.apply_torque(-torque)
|
||
|
|
|
||
|
|
for load in self._loads:
|
||
|
|
if self.frame in load:
|
||
|
|
torque += load[1]
|
||
|
|
self._loads.remove(load)
|
||
|
|
break
|
||
|
|
self._loads.append((self.frame, torque))
|
||
|
|
|
||
|
|
def clear_loads(self):
|
||
|
|
"""
|
||
|
|
Clears the Body's loads list.
|
||
|
|
|
||
|
|
Example
|
||
|
|
=======
|
||
|
|
|
||
|
|
>>> from sympy.physics.mechanics import Body
|
||
|
|
>>> B = Body('B')
|
||
|
|
>>> force = B.x + B.y
|
||
|
|
>>> B.apply_force(force)
|
||
|
|
>>> B.loads
|
||
|
|
[(B_masscenter, B_frame.x + B_frame.y)]
|
||
|
|
>>> B.clear_loads()
|
||
|
|
>>> B.loads
|
||
|
|
[]
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
self._loads = []
|
||
|
|
|
||
|
|
def remove_load(self, about=None):
|
||
|
|
"""
|
||
|
|
Remove load about a point or frame.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
about : Point or ReferenceFrame, optional
|
||
|
|
The point about which force is applied,
|
||
|
|
and is to be removed.
|
||
|
|
If about is None, then the torque about
|
||
|
|
self's frame is removed.
|
||
|
|
|
||
|
|
Example
|
||
|
|
=======
|
||
|
|
|
||
|
|
>>> from sympy.physics.mechanics import Body, Point
|
||
|
|
>>> B = Body('B')
|
||
|
|
>>> P = Point('P')
|
||
|
|
>>> f1 = B.x
|
||
|
|
>>> f2 = B.y
|
||
|
|
>>> B.apply_force(f1)
|
||
|
|
>>> B.apply_force(f2, P)
|
||
|
|
>>> B.loads
|
||
|
|
[(B_masscenter, B_frame.x), (P, B_frame.y)]
|
||
|
|
|
||
|
|
>>> B.remove_load(P)
|
||
|
|
>>> B.loads
|
||
|
|
[(B_masscenter, B_frame.x)]
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if about is not None:
|
||
|
|
if not isinstance(about, Point):
|
||
|
|
raise TypeError('Load is applied about Point or ReferenceFrame.')
|
||
|
|
else:
|
||
|
|
about = self.frame
|
||
|
|
|
||
|
|
for load in self._loads:
|
||
|
|
if about in load:
|
||
|
|
self._loads.remove(load)
|
||
|
|
break
|
||
|
|
|
||
|
|
def masscenter_vel(self, body):
|
||
|
|
"""
|
||
|
|
Returns the velocity of the mass center with respect to the provided
|
||
|
|
rigid body or reference frame.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
body: Body or ReferenceFrame
|
||
|
|
The rigid body or reference frame to calculate the velocity in.
|
||
|
|
|
||
|
|
Example
|
||
|
|
=======
|
||
|
|
|
||
|
|
>>> from sympy.physics.mechanics import Body
|
||
|
|
>>> A = Body('A')
|
||
|
|
>>> B = Body('B')
|
||
|
|
>>> A.masscenter.set_vel(B.frame, 5*B.frame.x)
|
||
|
|
>>> A.masscenter_vel(B)
|
||
|
|
5*B_frame.x
|
||
|
|
>>> A.masscenter_vel(B.frame)
|
||
|
|
5*B_frame.x
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if isinstance(body, ReferenceFrame):
|
||
|
|
frame=body
|
||
|
|
elif isinstance(body, Body):
|
||
|
|
frame = body.frame
|
||
|
|
return self.masscenter.vel(frame)
|
||
|
|
|
||
|
|
def ang_vel_in(self, body):
|
||
|
|
"""
|
||
|
|
Returns this body's angular velocity with respect to the provided
|
||
|
|
rigid body or reference frame.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
body: Body or ReferenceFrame
|
||
|
|
The rigid body or reference frame to calculate the angular velocity in.
|
||
|
|
|
||
|
|
Example
|
||
|
|
=======
|
||
|
|
|
||
|
|
>>> from sympy.physics.mechanics import Body, ReferenceFrame
|
||
|
|
>>> A = Body('A')
|
||
|
|
>>> N = ReferenceFrame('N')
|
||
|
|
>>> B = Body('B', frame=N)
|
||
|
|
>>> A.frame.set_ang_vel(N, 5*N.x)
|
||
|
|
>>> A.ang_vel_in(B)
|
||
|
|
5*N.x
|
||
|
|
>>> A.ang_vel_in(N)
|
||
|
|
5*N.x
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if isinstance(body, ReferenceFrame):
|
||
|
|
frame=body
|
||
|
|
elif isinstance(body, Body):
|
||
|
|
frame = body.frame
|
||
|
|
return self.frame.ang_vel_in(frame)
|
||
|
|
|
||
|
|
def dcm(self, body):
|
||
|
|
"""
|
||
|
|
Returns the direction cosine matrix of this body relative to the
|
||
|
|
provided rigid body or reference frame.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
body: Body or ReferenceFrame
|
||
|
|
The rigid body or reference frame to calculate the dcm.
|
||
|
|
|
||
|
|
Example
|
||
|
|
=======
|
||
|
|
|
||
|
|
>>> from sympy.physics.mechanics import Body
|
||
|
|
>>> A = Body('A')
|
||
|
|
>>> B = Body('B')
|
||
|
|
>>> A.frame.orient_axis(B.frame, B.frame.x, 5)
|
||
|
|
>>> A.dcm(B)
|
||
|
|
Matrix([
|
||
|
|
[1, 0, 0],
|
||
|
|
[0, cos(5), sin(5)],
|
||
|
|
[0, -sin(5), cos(5)]])
|
||
|
|
>>> A.dcm(B.frame)
|
||
|
|
Matrix([
|
||
|
|
[1, 0, 0],
|
||
|
|
[0, cos(5), sin(5)],
|
||
|
|
[0, -sin(5), cos(5)]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if isinstance(body, ReferenceFrame):
|
||
|
|
frame=body
|
||
|
|
elif isinstance(body, Body):
|
||
|
|
frame = body.frame
|
||
|
|
return self.frame.dcm(frame)
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
Example
|
||
|
|
=======
|
||
|
|
|
||
|
|
>>> from sympy.physics.mechanics import Body
|
||
|
|
>>> A = Body('A')
|
||
|
|
>>> P = A.masscenter.locatenew('point', 3 * A.x + 5 * A.y)
|
||
|
|
>>> A.parallel_axis(P).to_matrix(A.frame)
|
||
|
|
Matrix([
|
||
|
|
[A_ixx + 25*A_mass, A_ixy - 15*A_mass, A_izx],
|
||
|
|
[A_ixy - 15*A_mass, A_iyy + 9*A_mass, A_iyz],
|
||
|
|
[ A_izx, A_iyz, A_izz + 34*A_mass]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
if self.is_rigidbody:
|
||
|
|
return RigidBody.parallel_axis(self, point, frame)
|
||
|
|
return Particle.parallel_axis(self, point, frame)
|