Compute inertia matrix for simple solids: cube, sphere and cylinder

inertia_matrix.py

#/usr/bin/env python

# Based on:
# http://mathworld.wolfram.com/MomentofInertia.html


def get_cube_inertia_matrix(mass, x, y, z):
    """Given mass and dimensions of a cube return intertia matrix.
    :return: ixx, ixy, ixz, ixy, iyy, iyz, ixz, iyz, izz
    From https://www.wolframalpha.com/input/?i=moment+of+inertia+cube"""
    ixx = (1.0 / 12.0) * (y**2 + z**2) * mass
    iyy = (1.0 / 12.0) * (x**2 + z**2) * mass
    izz = (1.0 / 12.0) * (x**2 + y**2) * mass
    ixy = 0.0
    ixz = 0.0
    iyz = 0.0
    return [[ixx, ixy, ixz], [ixy, iyy, iyz], [ixz, iyz, izz]]


def get_sphere_inertia_matrix(mass, radius):
    """Given mass and radius of a sphere return inertia matrix.
    :return: ixx, ixy, ixz, ixy, iyy, iyz, ixz, iyz, izz
    From https://www.wolframalpha.com/input/?i=inertia+matrix+sphere
     """
    ixx = iyy = izz = (2.0 / 5.0) * radius**2 * mass
    ixy = 0.0
    ixz = 0.0
    iyz = 0.0
    return [[ixx, ixy, ixz], [ixy, iyy, iyz], [ixz, iyz, izz]]


def get_cylinder_inertia_matrix(mass, radius, height):
    """Given mass and radius and height of a cylinder return inertia matrix.
    :return: ixx, ixy, ixz, ixy, iyy, iyz, ixz, iyz, izz
    https://www.wolframalpha.com/input/?i=inertia+matrix+cylinder&rawformassumption=%7B%22C%22,+%22cylinder%22%7D+-%3E+%7B%22Solid%22%7D
     """
    ixx = (1.0 / 12.0) * (3.0 * radius**2 + height**2) * mass
    iyy = (1.0 / 12.0) * (3.0 * radius**2 + height**2) * mass
    izz = (1.0 / 2.0) * radius**2 * mass
    ixy = 0.0
    ixz = 0.0
    iyz = 0.0
    return [[ixx, ixy, ixz], [ixy, iyy, iyz], [ixz, iyz, izz]]

if __name__ == '__main__':
    import sys
    if len(sys.argv) < 2:
        print "Inertia matrix as:"
        print "[[ixx, ixy, ixz], [ixy, iyy, iyz], [ixz, iyz, izz]]"
        print "Cube inertia of cube mass 1.0 and all dimensions 1.0:"
        print get_cube_inertia_matrix(1.0, 1.0, 1.0, 1.0)
        print "Sphere inertia of mass 1.0 and radius 1.0:"
        print get_sphere_inertia_matrix(1.0, 1.0)
        print "Cylinder inertia of mass 1.0 and radius 1.0 and height 1.0:"
        print get_cylinder_inertia_matrix(1.0, 1.0, 1.0)
    elif len(sys.argv) == 3:
        print get_sphere_inertia_matrix(float(sys.argv[1]),
                                  float(sys.argv[2]))
    elif len(sys.argv) == 4:
        print get_cylinder_inertia_matrix(float(sys.argv[1]),
                                    float(sys.argv[2]),
                                    float(sys.argv[3]))
    elif len(sys.argv) == 5:
        print "Cube inertia matrix:"
        print get_cube_inertia_matrix(float(sys.argv[1]),
                                float(sys.argv[2]),
                                float(sys.argv[3]),
                                float(sys.argv[4]))
    else:
        print "Get the inertia matrix of simple shapes."
        print "Usage:"
        print "For a sphere:"
        print sys.argv[0] + " mass radius"
        print "For a cylinder:"
        print sys.argv[0] + " mass radius height"
        print "For a cube:"
        print sys.argv[0] + " mass x y z"
        print "\nFor example:\n" + sys.argv[0] + "1.0 1.0 1.0 1.0"
        print "[[0.16666666666666666, 0.0, 0.0], [0.0, 0.16666666666666666, 0.0], [0.0, 0.0, 0.16666666666666666]]"

Hi,

Thanks for the script, very useful.

I believe there is a mistake in the cylinder calculation at izz = (1.0 / 12.0) * radius**2 * mass. It should be 1.0/2.0 and not divided by 12.0. Double checked with SolidWorks.

I believe there is a mistake in the cylinder calculation at izz = (1.0 / 12.0) * radius**2 * mass. It should be 1.0/2.0 and not divided by 12.0. Double checked with SolidWorks.

Good catch! Editing it right now.

Hi,

Thanks for the script and the links!

According to the link you provide, for the computation of the inertia of a cylinder, don't you have a 4* in front of the height for ixx and iyy?

ixx = (1.0 / 12.0) * (3.0 * radius**2 + height**2) * mass -> ixx = (1.0 / 12.0) * (3.0 * radius**2 + 4*height**2) * mass
iyy = (1.0 / 12.0) * (3.0 * radius**2 + height**2) * mass ->  iyy = (1.0 / 12.0) * (3.0 * radius**2 + 4*height**2) * mass

?