Mechanics: Masses, Inertias, Particles and Rigid Bodies

This document will describe how to represent masses and inertias in 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 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. 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 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 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 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

In 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:

\[\begin{split}\mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_x} &= \mathbf{\hat{a}_x} \mathbf{\hat{a}_x}\\ \mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_y} &= \mathbf{\hat{a}_x} \mathbf{\hat{a}_y}\\\end{split}\]

Where \(\mathbf{\hat{a}_x}\mathbf{\hat{a}_x}\) 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:

\[\begin{split}(x \mathbf{v}) \otimes \mathbf{w} &= \mathbf{v} \otimes (x \mathbf{w}) = x (\mathbf{v} \otimes \mathbf{w})\\ \mathbf{v} \otimes (\mathbf{w} + \mathbf{u}) &= \mathbf{v} \otimes \mathbf{w} + \mathbf{v} \otimes \mathbf{u}\\ (\mathbf{v} + \mathbf{w}) \otimes \mathbf{u} &= \mathbf{v} \otimes \mathbf{u} + \mathbf{w} \otimes \mathbf{u}\\\end{split}\]

A vector in a reference frame can be represented as \(\begin{bmatrix}a\\b\\c\end{bmatrix}\) or \(a \mathbf{\hat{i}} + b \mathbf{\hat{j}} + c \mathbf{\hat{k}}\). Similarly, a dyadic can be represented in tensor form:

\[\begin{split}\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\\\end{split}\]

or in dyadic form:

\[\begin{split}a_{11} \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} + a_{12} \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} + a_{13} \mathbf{\hat{a}_x}\mathbf{\hat{a}_z} + a_{21} \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} + a_{22} \mathbf{\hat{a}_y}\mathbf{\hat{a}_y} + a_{23} \mathbf{\hat{a}_y}\mathbf{\hat{a}_z} + a_{31} \mathbf{\hat{a}_z}\mathbf{\hat{a}_x} + a_{32} \mathbf{\hat{a}_z}\mathbf{\hat{a}_y} + a_{33} \mathbf{\hat{a}_z}\mathbf{\hat{a}_z}\\\end{split}\]

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:

\[\mathbf{\hat{a}_x} \otimes \mathbf{\hat{b}_y} = \mathbf{\hat{a}_x} \mathbf{\hat{b}_y}\]

Dyadics can also be crossed and dotted with vectors; again, order matters:

\[\begin{split}\mathbf{\hat{a}_x}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &= \mathbf{\hat{a}_x}\\ \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &= \mathbf{\hat{a}_y}\\ \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} \cdot \mathbf{\hat{a}_x} &= 0\\ \mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &= \mathbf{\hat{a}_x}\\ \mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} &= \mathbf{\hat{a}_y}\\ \mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &= 0\\ \mathbf{\hat{a}_x} \times \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &= \mathbf{\hat{a}_z}\mathbf{\hat{a}_x}\\ \mathbf{\hat{a}_x} \times \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &= 0\\ \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_z} &= - \mathbf{\hat{a}_y}\mathbf{\hat{a}_y}\\\end{split}\]

One can also take the time derivative of dyadics or express them in different frames, just like with vectors.

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