Source code for sympy.physics.mechanics.rigidbody

__all__ = ['RigidBody']

from sympy import sympify
from sympy.physics.mechanics.point import Point
from sympy.physics.mechanics.essential import ReferenceFrame, Dyadic


[docs]class RigidBody(object): """An idealized rigid body. 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. 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, frame, mass, inertia): if not isinstance(name, str): raise TypeError('Supply a valid name.') self._name = name self.set_masscenter(masscenter) self.set_mass(mass) self.set_frame(frame) self.set_inertia(inertia) self._pe = sympify(0) def __str__(self): return self._name __repr__ = __str__ def get_frame(self): return self._frame def set_frame(self, F): if not isinstance(F, ReferenceFrame): raise TypeError("RigdBody frame must be a ReferenceFrame object.") self._frame = F frame = property(get_frame, set_frame) def get_masscenter(self): return self._masscenter def set_masscenter(self, p): if not isinstance(p, Point): raise TypeError("RigidBody center of mass must be a Point object.") self._masscenter = p masscenter = property(get_masscenter, set_masscenter) def get_mass(self): return self._mass def set_mass(self, m): self._mass = sympify(m) mass = property(get_mass, set_mass) def get_inertia(self): return (self._inertia, self._inertia_point) def set_inertia(self, I): if not isinstance(I[0], Dyadic): raise TypeError("RigidBody inertia must be a Dyadic object.") if not isinstance(I[1], Point): raise TypeError("RigidBody inertia must be about a Point.") self._inertia = I[0] self._inertia_point = 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 from sympy.physics.mechanics.functions import inertia_of_point_mass 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 inertia = property(get_inertia, set_inertia) @property
[docs] def central_inertia(self): """The body's central inertia dyadic.""" return self._central_inertia
[docs] def linear_momentum(self, frame): """ Linear momentum of the rigid body. 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 >>> 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): """ Angular momentum of the rigid body. The angular momentum H, about some point O, of a rigid body B, in a frame N is given by H = I* . omega + r* x (M * v) where I* is the central inertia dyadic of B, omega 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 point O 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 >>> 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) >>> Inertia_tuple = (I, P) >>> B = RigidBody('B', P, b, M, Inertia_tuple) >>> B.angular_momentum(P, N) omega*b.x """ return ((self.central_inertia & self.frame.ang_vel_in(frame)) + (point.vel(frame) ^ -self.masscenter.pos_from(point)) * self.mass)
[docs] def kinetic_energy(self, frame): """Kinetic energy of the rigid body The kinetic energy, T, of a rigid body, B, is given by 'T = 1/2 (I omega^2 + m v^2)' where I and m are the central inertia dyadic and mass of rigid body B, respectively, omega 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 = (self.frame.ang_vel_in(frame) & (self.central_inertia & self.frame.ang_vel_in(frame)) / sympify(2)) translational_KE = (self.mass * (self.masscenter.vel(frame) & self.masscenter.vel(frame)) / sympify(2)) return rotational_KE + translational_KE
[docs] def set_potential_energy(self, scalar): """Used to set the potential energy of this RigidBody. Parameters ========== scalar: Sympifyable The potential energy (a scalar) of the RigidBody. Examples ======== >>> from sympy.physics.mechanics import Particle, Point, outer >>> from sympy.physics.mechanics import RigidBody, ReferenceFrame >>> from sympy import symbols >>> b = ReferenceFrame('b') >>> M, g, h = symbols('M g h') >>> P = Point('P') >>> I = outer (b.x, b.x) >>> Inertia_tuple = (I, P) >>> B = RigidBody('B', P, b, M, Inertia_tuple) >>> B.set_potential_energy(M * g * h) """ self._pe = sympify(scalar)
@property
[docs] def potential_energy(self): """The potential energy of the RigidBody. Examples ======== >>> from sympy.physics.mechanics import RigidBody, Point, outer, ReferenceFrame >>> from sympy import symbols >>> M, g, h = symbols('M g h') >>> b = ReferenceFrame('b') >>> P = Point('P') >>> I = outer (b.x, b.x) >>> Inertia_tuple = (I, P) >>> B = RigidBody('B', P, b, M, Inertia_tuple) >>> B.set_potential_energy(M * g * h) >>> B.potential_energy M*g*h """ return self._pe