# Body (Docstrings)#

## Body#

class sympy.physics.mechanics.body.Body(name, masscenter=None, mass=None, frame=None, central_inertia=None)[source]#

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.

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 of the body. If none is passed while creating RigidBody, a default inertia is generated.

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.

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.

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.
ang_vel_in(body)[source]#

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

apply_force(force, point=None, reaction_body=None, reaction_point=None)[source]#

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.

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.

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_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_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_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)]
[(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.

apply_torque(torque, reaction_body=None)[source]#

Parameters:

torque: Vector

The torque to be applied.

reaction_body: Body, optional

Second body on which equal and opposite torque is to be applied.

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.

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_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_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_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_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.

Example

>>> from sympy.physics.mechanics import Body
>>> B = Body('B')
>>> force = B.x + B.y
>>> B.apply_force(force)
[(B_masscenter, B_frame.x + B_frame.y)]
[]

dcm(body)[source]#

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)]])

kinetic_energy(frame)[source]#

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


sympy.physics.mechanics

Particle, RigidBody

masscenter_vel(body)[source]#

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


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_masscenter, B_frame.x), (P, B_frame.y)]

>>> B.remove_load(P)