dtk Package¶
bicycle
Module¶
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.
Returns: 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.
Returns: 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 xaxis. The second angle, theta, is a rotation about the new yaxis (which is realized after the phi rotation). The third angle, psi, is a rotation about the new zaxis (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”.
Returns: 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
, and3
where1
is about the x axis of the first reference frame,2
is about the y axis of the this new frame and3
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.Returns: 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 * vWhere 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([[ 7.07106781e01, 1.29893408e16, 7.07106781e01], [ 7.07106781e01, 4.32978028e17, 7.07106781e01], [ 1.22464680e16, 1.00000000e+00, 6.12323400e17]]) >>> v = np.matrix([[1.], [0.], [0.]]) >>> vp = rotMat * v >>> vp matrix([[ 7.07106781e01], [ 7.07106781e01], [ 1.22464680e16]])

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.
Returns: 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.
Returns: 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.
Returns: 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 threedimensional 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 xaxis, the second about the yaxis, and the third about the zaxis. All rotations are with respect to the initial fixed frame, and they occur in the order x, then y, then z.
Returns: 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)
Threedimensional 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)
Threedimensional cartesian inertia tensor describing the inertia of a mass in a rotated coordinate frame.
Returns: 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.
Returns: 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.
Returns: 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.
Returns: 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.
Returns: 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.
Returns: 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.