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