# dtk Package¶

## `inertia` Module¶

`dtk.inertia.``compound_pendulum_inertia`(m, g, l, T)

Returns the moment of inertia for an object hung as a compound pendulum.

Parameters: m : float Mass of the pendulum. g : float Acceration due to gravity. l : float Length of the pendulum. T : float The period of oscillation. I : float Moment of interia of the pendulum.
`dtk.inertia.``cylinder_inertia`(l, m, ro, ri)

Calculate the moment of inertia for a hollow cylinder (or solid cylinder) where the x axis is aligned with the cylinder’s axis.

Parameters: l : float The length of the cylinder. m : float The mass of the cylinder. ro : float The outer radius of the cylinder. ri : float The inner radius of the cylinder. Set this to zero for a solid cylinder. Ix : float Moment of inertia about cylinder axis. Iy, Iz : float Moment of inertia about cylinder axis.
`dtk.inertia.``euler_123`(angles)

Returns the direction cosine matrix as a function of the Euler 123 angles.

Parameters: angles : numpy.array or list or tuple, shape(3,) Three angles (in units of radians) that specify the orientation of a new reference frame with respect to a fixed reference frame. The first angle, phi, is a rotation about the fixed frame’s x-axis. The second angle, theta, is a rotation about the new y-axis (which is realized after the phi rotation). The third angle, psi, is a rotation about the new z-axis (which is realized after the theta rotation). Thus, all three angles are “relative” rotations with respect to the new frame. Note: if the rotations are viewed as occuring in the opposite direction (z, then y, then x), all three rotations are with respect to the initial fixed frame rather than “relative”. R : numpy.matrix, shape(3,3) Three dimensional rotation matrix about three different orthogonal axes.
`dtk.inertia.``euler_rotation`(angles, order)

Returns a rotation matrix for a reference frame, B, in another reference frame, A, where the B frame is rotated relative to the A frame via body fixed rotations (Euler angles).

Parameters: angles : array_like An array of three angles in radians that are in order of rotation. order : tuple A three tuple containing a combination of `1`, `2`, and `3` where `1` is about the x axis of the first reference frame, `2` is about the y axis of the this new frame and `3` is about the z axis. Note that (1, 1, 1) is a valid entry and will give you correct results, but combinations like this are not necessarily useful for describing a general configuration. R : numpy.matrix, shape(3,3) A rotation matrix.

Notes

The rotation matrix is defined such that a R times a vector v equals the vector expressed in the rotated reference frame.

v’ = R * v

Where v is the vector expressed in the original reference frame and v’ is the same vector expressed in the rotated reference frame.

Examples

```>>> import numpy as np
>>> from dtk.inertia import euler_rotation
>>> angles = [np.pi, np.pi / 2., -np.pi / 4.]
>>> rotMat = euler_rotation(angles, (3, 1, 3))
>>> rotMat
matrix([[ -7.07106781e-01,   1.29893408e-16,  -7.07106781e-01],
[ -7.07106781e-01,   4.32978028e-17,   7.07106781e-01],
[  1.22464680e-16,   1.00000000e+00,   6.12323400e-17]])
>>> v = np.matrix([[1.], [0.], [0.]])
>>> vp = rotMat * v
>>> vp
matrix([[ -7.07106781e-01],
[ -7.07106781e-01],
[  1.22464680e-16]])
```
`dtk.inertia.``inertia_components`(jay, beta)

Returns the 2D orthogonal inertia tensor.

When at least three moments of inertia and their axes orientations are known relative to a common inertial frame of a planar object, the orthoganal moments of inertia relative the frame are computed.

Parameters: jay : ndarray, shape(n,) An array of at least three moments of inertia. (n >= 3) beta : ndarray, shape(n,) An array of orientation angles corresponding to the moments of inertia in jay. eye : ndarray, shape(3,) Ixx, Ixz, Izz
`dtk.inertia.``parallel_axis`(Ic, m, d)

Returns the moment of inertia of a body about a different point.

Parameters: Ic : ndarray, shape(3,3) The moment of inertia about the center of mass of the body with respect to an orthogonal coordinate system. m : float The mass of the body. d : ndarray, shape(3,) The distances along the three ordinates that located the new point relative to the center of mass of the body. I : ndarray, shape(3,3) The moment of inertia of a body about a point located by the distances in d.
`dtk.inertia.``principal_axes`(I)

Returns the principal moments of inertia and the orientation.

Parameters: I : ndarray, shape(3,3) An inertia tensor. Ip : ndarray, shape(3,) The principal moments of inertia. This is sorted smallest to largest. C : ndarray, shape(3,3) The rotation matrix.
`dtk.inertia.``rotate3`(angles)

Produces a three-dimensional rotation matrix as rotations around the three cartesian axes.

Parameters: angles : numpy.array or list or tuple, shape(3,) Three angles (in units of radians) that specify the orientation of a new reference frame with respect to a fixed reference frame. The first angle is a pure rotation about the x-axis, the second about the y-axis, and the third about the z-axis. All rotations are with respect to the initial fixed frame, and they occur in the order x, then y, then z. R : numpy.matrix, shape(3,3) Three dimensional rotation matrix about three different orthogonal axes.
`dtk.inertia.``rotate3_inertia`(RotMat, relInertia)

Rotates an inertia tensor. A derivation of the formula in this function can be found in Crandall 1968, Dynamics of mechanical and electromechanical systems. This function only transforms an inertia tensor for rotations with respect to a fixed point. To translate an inertia tensor, one must use the parallel axis analogue for tensors. An inertia tensor contains both moments of inertia and products of inertia for a mass in a cartesian (xyz) frame.

Parameters: RotMat : numpy.matrix, shape(3,3) Three-dimensional rotation matrix specifying the coordinate frame that the input inertia tensor is in, with respect to a fixed coordinate system in which one desires to express the inertia tensor. relInertia : numpy.matrix, shape(3,3) Three-dimensional cartesian inertia tensor describing the inertia of a mass in a rotated coordinate frame. Inertia : numpy.matrix, shape(3,3) Inertia tensor with respect to a fixed coordinate system (“unrotated”).
`dtk.inertia.``rotate_inertia_about_y`(I, angle)

Returns inertia tensor rotated through angle about the Y axis.

Parameters: I : ndarray, shape(3,) An inertia tensor. angle : float Angle in radians about the positive Y axis of which to rotate the inertia tensor.
`dtk.inertia.``torsional_pendulum_inertia`(k, T)

Calculate the moment of inertia for an ideal torsional pendulum.

Parameters: k : float Torsional stiffness. T : float Period of oscillation. I : float Moment of inertia.
`dtk.inertia.``total_com`(coordinates, masses)

Returns the center of mass of a group of objects if the indivdual centers of mass and mass is provided.

coordinates : ndarray, shape(3,n)
The rows are the x, y and z coordinates, respectively and the columns are for each object.
masses : ndarray, shape(3,)
An array of the masses of multiple objects, the order should correspond to the columns of coordinates.
Returns: mT : float Total mass of the objects. cT : ndarray, shape(3,) The x, y, and z coordinates of the total center of mass.
`dtk.inertia.``tube_inertia`(l, m, ro, ri)

Calculate the moment of inertia for a tube (or rod) where the x axis is aligned with the tube’s axis.

Parameters: l : float The length of the tube. m : float The mass of the tube. ro : float The outer radius of the tube. ri : float The inner radius of the tube. Set this to zero if it is a rod instead of a tube. Ix : float Moment of inertia about tube axis. Iy, Iz : float Moment of inertia about normal axis.
`dtk.inertia.``x_rot`(angle)

Returns the rotation matrix for a reference frame rotated through an angle about the x axis.

Parameters: angle : float The angle in radians. Rx : np.matrix, shape(3,3) The rotation matrix.

Notes

v’ = Rx * v where v is the vector expressed the reference in the original reference frame and v’ is the vector expressed in the new rotated reference frame.

`dtk.inertia.``y_rot`(angle)

Returns the rotation matrix for a reference frame rotated through an angle about the y axis.

Parameters: angle : float The angle in radians. Rx : np.matrix, shape(3,3) The rotation matrix.

Notes

v’ = Rx * v where v is the vector expressed the reference in the original reference frame and v’ is the vector expressed in the new rotated reference frame.

`dtk.inertia.``z_rot`(angle)

Returns the rotation matrix for a reference frame rotated through an angle about the z axis.

Parameters: angle : float The angle in radians. Rx : np.matrix, shape(3,3) The rotation matrix.

Notes

v’ = Rx * v where v is the vector expressed the reference in the original reference frame and v’ is the vector expressed in the new rotated reference frame.