Deprecated Classes (Docstrings)¶

Deprecated since version 1.13: Body and JointsMethod have been deprecated. The functionality of Body is fully captured by RigidBody and Particle and the functionality of JointsMethod is fully captured by System.

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.

Deprecated since version 1.13: The Body class is deprecated. Its functionality is captured by RigidBody and Particle.

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.

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

As Body has been deprecated, the following examples are for illustrative purposes only. The functionality of Body is fully captured by RigidBody and Particle. To ignore the deprecation warning we can use the ignore_warnings context manager.

>>> from sympy.utilities.exceptions import ignore_warnings

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
>>> with ignore_warnings(DeprecationWarning):
...     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)
>>> with ignore_warnings(DeprecationWarning):
...     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')
>>> with ignore_warnings(DeprecationWarning):
...     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.utilities.exceptions import ignore_warnings
>>> from sympy.physics.mechanics import Body, ReferenceFrame
>>> with ignore_warnings(DeprecationWarning):
...     A = Body('A')
>>> N = ReferenceFrame('N')
>>> with ignore_warnings(DeprecationWarning):
...     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

angular_momentum(point, frame)[source]¶

Returns the angular momentum of the rigid body about a point in the given frame.

Parameters:

point : Point

The point about which angular momentum is desired.

frame : ReferenceFrame

The frame in which angular momentum is desired.

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.

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

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

Add force to the body(s).

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 opposite forces on self and other body if both are given on the desired point on the bodies. The force applied on other body is taken opposite of self, i.e, -force.

Example

>>> from sympy.utilities.exceptions import ignore_warnings
>>> from sympy import symbols
>>> from sympy.physics.mechanics import Body, Point, dynamicsymbols
>>> m, g = symbols('m g')
>>> with ignore_warnings(DeprecationWarning):
...     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
>>> with ignore_warnings(DeprecationWarning):
...     N = Body('N') #Newtonion Frame
>>> x = dynamicsymbols('x')
>>> with ignore_warnings(DeprecationWarning):
...     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.

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

Add torque to the body(s).

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 opposite torques 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.utilities.exceptions import ignore_warnings
>>> from sympy import symbols
>>> from sympy.physics.mechanics import Body, dynamicsymbols
>>> t = symbols('t')
>>> with ignore_warnings(DeprecationWarning):
...     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
>>> with ignore_warnings(DeprecationWarning):
...     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.

property central_inertia¶: The body’s central inertia dyadic.

clear_loads()[source]¶

Clears the Body’s loads list.

Example

>>> from sympy.utilities.exceptions import ignore_warnings
>>> from sympy.physics.mechanics import Body
>>> with ignore_warnings(DeprecationWarning):
...     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
[]

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.utilities.exceptions import ignore_warnings
>>> from sympy.physics.mechanics import Body
>>> with ignore_warnings(DeprecationWarning):
...     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)]])

property frame¶: The ReferenceFrame fixed to the body.

property inertia¶: The body’s inertia about a point; stored as (Dyadic, Point).

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.utilities.exceptions import ignore_warnings
>>> 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')
>>> with ignore_warnings(DeprecationWarning):
...     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)
>>> with ignore_warnings(DeprecationWarning):
...     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

linear_momentum(frame)[source]¶

Linear momentum of the rigid body.

Parameters:: frame : ReferenceFrame

The frame in which linear momentum is desired.

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.

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

property mass¶: The body’s mass.

property masscenter¶: The body’s center of mass.

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.utilities.exceptions import ignore_warnings
>>> from sympy.physics.mechanics import Body
>>> with ignore_warnings(DeprecationWarning):
...     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

property name¶: The name of the body.

parallel_axis(point, frame=None)[source]¶

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.utilities.exceptions import ignore_warnings
>>> from sympy.physics.mechanics import Body
>>> with ignore_warnings(DeprecationWarning):
...     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]])

property point¶: The body’s center of mass.

property potential_energy¶

The potential energy of the body.

Examples

>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
>>> P.potential_energy
g*h*m

remove_load(about=None)[source]¶

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.utilities.exceptions import ignore_warnings
>>> from sympy.physics.mechanics import Body, Point
>>> with ignore_warnings(DeprecationWarning):
...     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)]

property x¶: The basis Vector for the Body, in the x direction.

property y¶: The basis Vector for the Body, in the y direction.

property z¶: The basis Vector for the Body, in the z direction.

class sympy.physics.mechanics.jointsmethod.JointsMethod(newtonion, *joints)[source]¶

Method for formulating the equations of motion using a set of interconnected bodies with joints.

Deprecated since version 1.13: The JointsMethod class is deprecated. Its functionality has been replaced by the new System class.

Parameters:

newtonion : Body or ReferenceFrame

The newtonion(inertial) frame.

*joints : Joint

The joints in the system

Examples

As Body and JointsMethod have been deprecated, the following examples are for illustrative purposes only. The functionality of Body is fully captured by RigidBody and Particle and the functionality of JointsMethod is fully captured by System. To ignore the deprecation warning we can use the ignore_warnings context manager.

>>> from sympy.utilities.exceptions import ignore_warnings

This is a simple example for a one degree of freedom translational spring-mass-damper.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import Body, JointsMethod, PrismaticJoint
>>> from sympy.physics.vector import dynamicsymbols
>>> c, k = symbols('c k')
>>> x, v = dynamicsymbols('x v')
>>> with ignore_warnings(DeprecationWarning):
...     wall = Body('W')
...     body = Body('B')
>>> J = PrismaticJoint('J', wall, body, coordinates=x, speeds=v)
>>> wall.apply_force(c*v*wall.x, reaction_body=body)
>>> wall.apply_force(k*x*wall.x, reaction_body=body)
>>> with ignore_warnings(DeprecationWarning):
...     method = JointsMethod(wall, J)
>>> method.form_eoms()
Matrix([[-B_mass*Derivative(v(t), t) - c*v(t) - k*x(t)]])
>>> M = method.mass_matrix_full
>>> F = method.forcing_full
>>> rhs = M.LUsolve(F)
>>> rhs
Matrix([
[                     v(t)],
[(-c*v(t) - k*x(t))/B_mass]])

Notes

JointsMethod currently only works with systems that do not have any configuration or motion constraints.

Attributes

q, u	(iterable) Iterable of the generalized coordinates and speeds
bodies	(iterable) Iterable of Body objects in the system.
loads	(iterable) Iterable of (Point, vector) or (ReferenceFrame, vector) tuples describing the forces on the system.
mass_matrix	(Matrix, shape(n, n)) The system’s mass matrix
forcing	(Matrix, shape(n, 1)) The system’s forcing vector
mass_matrix_full	(Matrix, shape(2n, 2n)) The “mass matrix” for the u’s and q’s
forcing_full	(Matrix, shape(2*n, 1)) The “forcing vector” for the u’s and q’s
method	(KanesMethod or Lagrange’s method) Method’s object.
kdes	(iterable) Iterable of kde in they system.

property bodies¶: List of bodies in they system.

property forcing¶: The system’s forcing vector.

property forcing_full¶: The “forcing vector” for the u’s and q’s.

form_eoms(method=<class 'sympy.physics.mechanics.kane.KanesMethod'>)[source]¶

Method to form system’s equation of motions.

Parameters:: method : Class

Class name of method.
Returns:: Matrix

Vector of equations of motions.

Examples

>>> from sympy.utilities.exceptions import ignore_warnings

This is a simple example for a one degree of freedom translational spring-mass-damper.

>>> from sympy import S, symbols
>>> from sympy.physics.mechanics import LagrangesMethod, dynamicsymbols, Body
>>> from sympy.physics.mechanics import PrismaticJoint, JointsMethod
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> m, k, b = symbols('m k b')
>>> with ignore_warnings(DeprecationWarning):
...     wall = Body('W')
...     part = Body('P', mass=m)
>>> part.potential_energy = k * q**2 / S(2)
>>> J = PrismaticJoint('J', wall, part, coordinates=q, speeds=qd)
>>> wall.apply_force(b * qd * wall.x, reaction_body=part)
>>> with ignore_warnings(DeprecationWarning):
...     method = JointsMethod(wall, J)
>>> method.form_eoms(LagrangesMethod)
Matrix([[b*Derivative(q(t), t) + k*q(t) + m*Derivative(q(t), (t, 2))]])

We can also solve for the states using the ‘rhs’ method.

>>> method.rhs()
Matrix([
[                Derivative(q(t), t)],
[(-b*Derivative(q(t), t) - k*q(t))/m]])

property kdes¶: List of the generalized coordinates.

property loads¶: List of loads on the system.

property mass_matrix¶: The system’s mass matrix.

property mass_matrix_full¶: The “mass matrix” for the u’s and q’s.

property method¶: Object of method used to form equations of systems.

property q¶: List of the generalized coordinates.

rhs(inv_method=None)[source]¶

Returns equations that can be solved numerically.

Parameters:: inv_method : str

The specific sympy inverse matrix calculation method to use. For a list of valid methods, see inv()
Returns:: Matrix

Numerically solvable equations.