Masses, Inertias, Particles and Rigid Bodies in Physics/Mechanics¶

This document will describe how to represent masses and inertias in sympy.physics.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 sympy.physics.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. sympy.physics.mechanics allows one to perform kinematic analysis of points separate from their association with masses.

Inertia¶

See the Inertia (Dyadics) section in ‘Advanced Topics’ part of sympy.physics.vector docs.

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 sympy.physics.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 sympy.physics.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.

Linear Momentum¶

The linear momentum of a particle P is defined as:

\[L_P = m\mathbf{v}\]

where \(m\) is the mass of the particle P and \(\mathbf{v}\) is the velocity of the particle in the inertial frame.[Likins1973]_.

Similarly the linear momentum of a rigid body is defined as:

\[L_B = m\mathbf{v^*}\]

where \(m\) is the mass of the rigid body, B, and \(\mathbf{v^*}\) is the velocity of the mass center of B in the inertial frame.

Angular Momentum¶

The angular momentum of a particle P about an arbitrary point O in an inertial frame N is defined as:

\[^N \mathbf{H} ^ {P/O} = \mathbf{r} \times m\mathbf{v}\]

where \(\mathbf{r}\) is a position vector from point O to the particle of mass \(m\) and \(\mathbf{v}\) is the velocity of the particle in the inertial frame.

Similarly the angular momentum of a rigid body B about a point O in an inertial frame N is defined as:

\[^N \mathbf{H} ^ {B/O} = ^N \mathbf{H} ^ {B/B^*} + ^N \mathbf{H} ^ {B^*/O}\]

where the angular momentum of the body about it’s mass center is:

\[^N \mathbf{H} ^ {B/B^*} = \mathbf{I^*} \cdot \omega\]

and the angular momentum of the mass center about O is:

\[^N \mathbf{H} ^ {B^*/O} = \mathbf{r^*} \times m \mathbf{v^*}\]

where \(\mathbf{I^*}\) is the central inertia dyadic of rigid body B, \(\omega\) is the inertial angular velocity of B, \(\mathbf{r^*}\) is a position vector from point O to the mass center of B, \(m\) is the mass of B and \(\mathbf{v^*}\) is the velocity of the mass center in the inertial frame.

Using momenta functions in Mechanics¶

The following example shows how to use the momenta functions in sympy.physics.mechanics.

One begins by creating the requisite symbols to describe the system. Then the reference frame is created and the kinematics are done.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, Particle, Point, outer
>>> from sympy.physics.mechanics import linear_momentum, angular_momentum
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, M, l1 = symbols('m M l1')
>>> q1d = dynamicsymbols('q1d')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> Ac = O.locatenew('Ac', l1 * N.x)
>>> P = Ac.locatenew('P', l1 * N.x)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, q1d * N.z)
>>> Ac.v2pt_theory(O, N, a)
l1*q1d*N.y
>>> P.v2pt_theory(O, N, a)
2*l1*q1d*N.y

Finally, the bodies that make up the system are created. In this case the system consists of a particle Pa and a RigidBody A.

>>> Pa = Particle('Pa', P, m)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, M, (I, Ac))

Then one can either choose to evaluate the momenta of individual components of the system or of the entire system itself.

>>> linear_momentum(N,A)
M*l1*q1d*N.y
>>> angular_momentum(O, N, Pa)
4*l1**2*m*q1d*N.z
>>> linear_momentum(N, A, Pa)
(M*l1*q1d + 2*l1*m*q1d)*N.y
>>> angular_momentum(O, N, A, Pa)
(M*l1**2*q1d + 4*l1**2*m*q1d + q1d)*N.z

It should be noted that the user can determine either momenta in any frame in sympy.physics.mechanics as the user is allowed to specify the reference frame when calling the function. In other words the user is not limited to determining just inertial linear and angular momenta. Please refer to the docstrings on each function to learn more about how each function works precisely.

Kinetic Energy¶

The kinetic energy of a particle P is defined as

\[T_P = \frac{1}{2} m \mathbf{v^2}\]

where \(m\) is the mass of the particle P and \(\mathbf{v}\) is the velocity of the particle in the inertial frame.

Similarly the kinetic energy of a rigid body B is defined as

\[T_B = T_t + T_r\]

where the translational kinetic energy is given by:

\[T_t = \frac{1}{2} m \mathbf{v^*} \cdot \mathbf{v^*}\]

and the rotational kinetic energy is given by:

\[T_r = \frac{1}{2} \omega \cdot \mathbf{I^*} \cdot \omega\]

where \(m\) is the mass of the rigid body, \(\mathbf{v^*}\) is the velocity of the mass center in the inertial frame, \(\omega\) is the inertial angular velocity of the body and \(\mathbf{I^*}\) is the central inertia dyadic.

Potential Energy¶

Potential energy is defined as the energy possessed by a body or system by virtue of its position or arrangement.

Since there are a variety of definitions for potential energy, this is not discussed further here. One can learn more about this in any elementary text book on dynamics.

Lagrangian¶

The Lagrangian of a body or a system of bodies is defined as:

\[\mathcal{L} = T - V\]

where \(T\) and \(V\) are the kinetic and potential energies respectively.

Using energy functions in Mechanics¶

The following example shows how to use the energy functions in sympy.physics.mechanics.

As was discussed above in the momenta functions, one first creates the system by going through an identical procedure.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, Particle
>>> from sympy.physics.mechanics import kinetic_energy, potential_energy, Point
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, M, l1, g, h, H = symbols('m M l1 g h H')
>>> omega = dynamicsymbols('omega')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> Ac = O.locatenew('Ac', l1 * N.x)
>>> P = Ac.locatenew('P', l1 * N.x)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, omega * N.z)
>>> Ac.v2pt_theory(O, N, a)
l1*omega*N.y
>>> P.v2pt_theory(O, N, a)
2*l1*omega*N.y
>>> Pa = Particle('Pa', P, m)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, M, (I, Ac))

The user can then determine the kinetic energy of any number of entities of the system:

>>> kinetic_energy(N, Pa)
2*l1**2*m*omega**2
>>> kinetic_energy(N, Pa, A)
M*l1**2*omega**2/2 + 2*l1**2*m*omega**2 + omega**2/2

It should be noted that the user can determine either kinetic energy relative to any frame in sympy.physics.mechanics as the user is allowed to specify the reference frame when calling the function. In other words the user is not limited to determining just inertial kinetic energy.

For potential energies, the user must first specify the potential energy of every entity of the system using the sympy.physics.mechanics.rigidbody.RigidBody.potential_energy property. The potential energy of any number of entities comprising the system can then be determined:

>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * H
>>> potential_energy(A, Pa)
H*M*g + g*h*m

One can also determine the Lagrangian for this system:

>>> from sympy.physics.mechanics import Lagrangian
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> Lagrangian(N, Pa, A)
-H*M*g + M*l1**2*omega**2/2 - g*h*m + 2*l1**2*m*omega**2 + omega**2/2

Please refer to the docstrings to learn more about each function.

Classical Mechanics

Kane’s Method in Physics/Mechanics