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Mechanics: Masses, Inertias, Particles and Rigid Bodies
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This document will describe how to represent masses and inertias in
:mod:`mechanics` and use of the ``RigidBody`` and ``Particle`` classes.
It is assumed that the reader is familiar with the basics of these topics, such
as finding the center of mass for a system of particles, how to manipulate an
inertia tensor, and the definition of a particle and rigid body. Any advanced
dynamics text can provide a reference for these details.
Mass
====
The only requirement for a mass is that it needs to be a ``sympify``-able
expression. Keep in mind that masses can be time varying.
Particle
========
Particles are created with the class ``Particle`` in :mod:`mechanics`.
A ``Particle`` object has an associated point and an associated mass which are
the only two attributes of the object.::
>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import Symbol
>>> m = Symbol('m')
>>> po = Point('po')
>>> # create a particle container
>>> pa = Particle('pa', po, m)
The associated point contains the position, velocity and acceleration of the
particle. :mod:`mechanics` allows one to perform kinematic analysis of points
separate from their association with masses.
Inertia
=======
Dyadics are used to define the inertia of bodies within :mod:`mechanics`.
Inertia dyadics can be defined explicitly but the ``inertia`` function is
typically much more convenient for the user::
>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')
>>> # supply a reference frame and the moments of inertia if the object is symmetrical
>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)
>>> # supply a reference frame along with the products and moments of inertia for a general object
>>> inertia(N, 1, 2, 3, 4, 5, 6)
(N.x|N.x) + 4*(N.x|N.y) + 6*(N.x|N.z) + 4*(N.y|N.x) + 2*(N.y|N.y) + 5*(N.y|N.z) + 6*(N.z|N.x) + 5*(N.z|N.y) + 3*(N.z|N.z)
Notice that the ``inertia`` function returns a dyadic with each component
represented as two unit vectors separated by a ``|``. Refer to the
:ref:`Dyadic` section for more information about dyadics.
Rigid Body
==========
Rigid bodies are created in a similar fashion as particles. The ``RigidBody``
class generates objects with four attributes: mass, center of mass, a reference
frame, and an inertia tuple::
>>> 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)
>>> # create a rigid body
>>> B = RigidBody('B', P, A, m, (I, P))
The mass is specified exactly as is in a particle. Similar to the
``Particle``'s ``.point``, the ``RigidBody``'s center of mass, ``.masscenter``
must be specified. The reference frame is stored in an analogous fashion and
holds information about the body's orientation and angular velocity. Finally,
the inertia for a rigid body needs to be specified about a point. In
:mod:`mechanics`, you are allowed to specify any point for this. The most
common is the center of mass, as shown in the above code. If a point is selected
which is not the center of mass, ensure that the position between the point and
the center of mass has been defined. The inertia is specified as a tuple of length
two with the first entry being a ``Dyadic`` and the second entry being a
``Point`` of which the inertia dyadic is defined about.
.. _Dyadic:
Dyadic
======
In :mod:`mechanics`, dyadics are used to represent inertia ([Kane1985]_,
[WikiDyadics]_, [WikiDyadicProducts]_). A dyadic is a linear polynomial of
component unit dyadics, similar to a vector being a linear polynomial of
component unit vectors. A dyadic is the outer product between two vectors which
returns a new quantity representing the juxtaposition of these two vectors. For
example:
.. math::
\mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_x} &= \mathbf{\hat{a}_x}
\mathbf{\hat{a}_x}\\
\mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_y} &= \mathbf{\hat{a}_x}
\mathbf{\hat{a}_y}\\
Where :math:`\mathbf{\hat{a}_x}\mathbf{\hat{a}_x}` and
`\mathbf{\hat{a}_x}\mathbf{\hat{a}_y}` are the outer products obtained by
multiplying the left side as a column vector by the right side as a row vector.
Note that the order is significant.
Some additional properties of a dyadic are:
.. math::
(x \mathbf{v}) \otimes \mathbf{w} &= \mathbf{v} \otimes (x \mathbf{w}) = x
(\mathbf{v} \otimes \mathbf{w})\\
\mathbf{v} \otimes (\mathbf{w} + \mathbf{u}) &= \mathbf{v} \otimes \mathbf{w}
+ \mathbf{v} \otimes \mathbf{u}\\
(\mathbf{v} + \mathbf{w}) \otimes \mathbf{u} &= \mathbf{v} \otimes \mathbf{u}
+ \mathbf{w} \otimes \mathbf{u}\\
A vector in a reference frame can be represented as
:math:`\begin{bmatrix}a\\b\\c\end{bmatrix}` or :math:`a \mathbf{\hat{i}} + b
\mathbf{\hat{j}} + c \mathbf{\hat{k}}`. Similarly, a dyadic can be represented
in tensor form:
.. math::
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}\\
or in dyadic form:
.. math::
a_{11} \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} +
a_{12} \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} +
a_{13} \mathbf{\hat{a}_x}\mathbf{\hat{a}_z} +
a_{21} \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} +
a_{22} \mathbf{\hat{a}_y}\mathbf{\hat{a}_y} +
a_{23} \mathbf{\hat{a}_y}\mathbf{\hat{a}_z} +
a_{31} \mathbf{\hat{a}_z}\mathbf{\hat{a}_x} +
a_{32} \mathbf{\hat{a}_z}\mathbf{\hat{a}_y} +
a_{33} \mathbf{\hat{a}_z}\mathbf{\hat{a}_z}\\
Just as with vectors, the later representation makes it possible to keep track
of which frames the dyadic is defined with respect to. Also, the two
components of each term in the dyadic need not be in the same frame. The
following is valid:
.. math::
\mathbf{\hat{a}_x} \otimes \mathbf{\hat{b}_y} = \mathbf{\hat{a}_x}
\mathbf{\hat{b}_y}
Dyadics can also be crossed and dotted with vectors; again, order matters:
.. math::
\mathbf{\hat{a}_x}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &=
\mathbf{\hat{a}_x}\\
\mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &=
\mathbf{\hat{a}_y}\\
\mathbf{\hat{a}_x}\mathbf{\hat{a}_y} \cdot \mathbf{\hat{a}_x} &= 0\\
\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &=
\mathbf{\hat{a}_x}\\
\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} &=
\mathbf{\hat{a}_y}\\
\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &= 0\\
\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &=
\mathbf{\hat{a}_z}\mathbf{\hat{a}_x}\\
\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &= 0\\
\mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_z} &=
- \mathbf{\hat{a}_y}\mathbf{\hat{a}_y}\\
One can also take the time derivative of dyadics or express them in different
frames, just like with vectors.