Tuesday, August 2, 2016

Moments of Inertia

Following are scalar moments of inertia. In general, the moment of inertia is a tensor, see below.

Description
Figure
Moment(s) of inertia

Point mass m at a distance r from the axis of rotation. A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved.
PointInertia.svg

 I = m r^2



Two point masses, M and m, with reduced mass μ and separated by a distance, x.
 I = \frac{ M m }{ M \! + \! m } x^2 = \mu x^2


Rod of length L and mass m, axis of rotation at the end of the rod. This expression assumes that the rod is an infinitely thin (but rigid) wire. This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate, with h = L and w = 0.
Moment of inertia rod end.svg
I_{\mathrm{end}} = \frac{m L^2}{3} \,\!  [1]



Rod of length L and mass m. This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the thin rectangular plate with axis of rotation at the center of the plate, with w = L and h = 0.
Moment of inertia rod center.svg
I_{\mathrm{center}} = \frac{m L^2}{12} \,\!  [1]



Thin circular hoop of radius r and mass m. This is a special case of a torus for b = 0 (see below), as well as of a thick-walled cylindrical tube with open ends, with r1 = r2 and h = 0.
Moment of inertia hoop.svg
I_z = m r^2\!
I_x = I_y = \frac{m r^2}{2}\,\!



Thin, solid disk of radius r and mass m. This is a special case of the solid cylinder, with h = 0. That I_x = I_y = \frac{I_z}{2}\, is a consequence of the Perpendicular axis theorem.
Moment of inertia disc.svg
I_z = \frac{m r^2}{2}\,\!
I_x = I_y = \frac{m r^2}{4}\,\!



Thin cylindrical shell with open ends, of radius r and mass m. This expression assumes that the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for r1 = r2.
Also, a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration.
Moment of inertia thin cylinder.png
I = m r^2 \,\!  [1]



Solid cylinder of radius r, height h and mass m.
This is a special case of the thick-walled cylindrical tube, with r1 = 0. (Note: X-Y axis should be swapped for a standard right handed frame).
Moment of inertia solid cylinder.svg
I_z = \frac{m r^2}{2}\,\!  [1]
I_x = I_y = \frac{1}{12} m\left(3r^2+h^2\right)



Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m. With a density of ρ and the same geometry I_z = \frac{1}{2} \pi\rho h\left({r_2}^4 - {r_1}^4\right), I_x = I_y = \frac{1}{12} \pi\rho h\left(3({r_2}^4 - {r_1}^4)+h^2({r_2}^2 - {r_1}^2)\right)
Moment of inertia thick cylinder h.svg
I_z = \frac{1}{2} m\left(r_1^2 + r_2^2\right) = m r_2^2 \left(1-t+\frac{1}{2}{t}^2\right)   [1] [2]
where t = (r2–r1)/r2 is a normalized thickness ratio;
I_x = I_y = \frac{1}{12} m\left[3\left({r_2}^2 + {r_1}^2\right)+h^2\right]



Tetrahedron of side s and mass m
Tetraaxial.gif
I_{solid} = \frac{m s^2}{20}\,\! I_{hollow} = \frac{m s^2}{12}\,\!



Octahedron (hollow) of side s and mass m
Octahedral axis.gif
I_z=I_x=I_y = \frac{5m s^2}{9}\,\!



Octahedron (solid) of side s and mass m
Octahedral axis.gif
I_z=I_x=I_y = \frac{m s^2}{5}\,\!



Sphere (hollow) of radius r and mass m. A hollow sphere can be taken to be made up of two stacks of infinitesimally thin, circular hoops, where the radius differs from 0 to r (or a single stack, where the radius differs from -r to r).
Moment of inertia hollow sphere.svg
I = \frac{2 m r^2}{3}\,\!  [1]



Ball (solid) of radius r and mass m. A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 to r (or a single stack, where the radius differs from -r to r).
Moment of inertia solid sphere.svg
I = \frac{2 m r^2}{5}\,\!  [1]



Sphere (shell) of radius r2, with centered spherical cavity of radius r1 and mass m. When the cavity radius r1 = 0, the object is a solid ball (above).
When r1 = r2, \left[\frac{{r_2}^5-{r_1}^5}{{r_2}^3-{r_1}^3}\right]=\frac{5}{3}{r_2}^2, and the object is a hollow sphere.
Spherical shell moment of inertia.png
I = \frac{2 m}{5}\left[\frac{{r_2}^5-{r_1}^5}{{r_2}^3-{r_1}^3}\right]\,\!  [1]



Right circular cone with radius r, height h and mass m
Moment of inertia cone.svg
I_z = \frac{3}{10}mr^2 \,\!  [3]
I_x = I_y = \frac{3}{5}m\left(\frac{r^2}{4}+h^2\right) \,\!  [3]



Torus of tube radius a, cross-sectional radius b and mass m.
Torus cycles.svg
About the vertical axis: \left(a^2 + \frac{3}{4}b^2\right)m  [4]
About a diameter: \frac{1}{8}\left(4a^2 + 5b^2\right)m  [4]



Ellipsoid (solid) of semiaxes a, b, and c with mass m

Ellipsoid 321.png
I_a = \frac{m (b^2+c^2)}{5}\,\!

I_b = \frac{m (a^2+c^2)}{5}\,\!

I_c = \frac{m (a^2+b^2)}{5}\,\!



Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the end of the plate)
Recplaneoff.svg
I_e = \frac {m h^2}{3}+\frac {m w^2}{12}\,\!



Thin rectangular plate of height h and of width w and mass m
Recplane.svg
I_c = \frac {m(h^2 + w^2)}{12}\,\!  [1]



Solid cuboid of height h, width w, and depth d, and mass m. For a similarly oriented cube with sides of length s, I_{CM} = \frac{m s^2}{6}\,\!
Moment of inertia solid rectangular prism.png
I_h = \frac{1}{12} m\left(w^2+d^2\right)
I_w = \frac{1}{12} m\left(h^2+d^2\right)
I_d = \frac{1}{12} m\left(h^2+w^2\right)



Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis. For a cube with sides s, I = \frac{m s^2}{6}\,\!.
Moment of Inertia Cuboid.svg
I = \frac{m\left(W^2D^2+L^2D^2+L^2W^2\right)}{6\left(L^2+W^2+D^2\right)}



Triangle with vertices at the origin and at P and Q, with mass m, rotating about an axis perpendicular to the plane and passing through the origin.



I=\frac{m}{6}(\mathbf{P}\cdot\mathbf{P}+\mathbf{P}\cdot\mathbf{Q}+\mathbf{Q}\cdot\mathbf{Q})



Plane polygon with vertices P1, P2, P3, ..., PN and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.
Polygon Moment of Inertia.svg
I=\frac{m}{6}\frac{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+1}\times\mathbf{P}_{n}\|((\mathbf{P}_{n+1}\cdot\mathbf{P}_{n+1})+(\mathbf{P}_{n+1}\cdot\mathbf{P}_{n})+(\mathbf{P}_{n}\cdot\mathbf{P}_{n}))}{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+1}\times\mathbf{P}_{n}\|}



Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. a stands for side length.
I=\frac{ma^2}{24}[1 + 3\cot^2(\tfrac{\pi}{n})]  [5]



Infinite disk with mass normally distributed on two axes around the axis of rotation with mass-density as a function of x and y:
\rho(x,y) = \tfrac{m}{2\pi ab}\, e^{-((x/a)^2+(y/b)^2)/2}\,,

Gaussian 2D.png

I = m (a^2+b^2) \,\!



Uniform disk about an axis perpendicular to its edge.
 I = \frac {3mR^2} {2}[6]


List of 3D inertia tensors

This list of moment of inertia tensors is given for principal axes of each object.
To obtain the scalar moments of inertia I above, the tensor moment of inertia I is projected along some axis defined by a unit vector n according to the formula:
\mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n}\equiv n_i I_{ij} n_j\,,
where the dots indicate tensor contraction and we have used the Einstein summation convention. In the above table, n would be the unit Cartesian basis ex, ey, ez to obtain Ix, Iy, Iz respectively.




Solid sphere of radius r and mass m
Moment of inertia solid sphere.svg

I =
\begin{bmatrix}
  \frac{2}{5} m r^2 & 0 & 0 \\
  0 & \frac{2}{5} m r^2 & 0 \\
  0 & 0 & \frac{2}{5} m r^2
\end{bmatrix}



Hollow sphere of radius r and mass m

Moment of inertia hollow sphere.svg


I =
\begin{bmatrix}
  \frac{2}{3} m r^2 & 0 & 0 \\
  0 & \frac{2}{3} m r^2 & 0 \\
  0 & 0 & \frac{2}{3} m r^2
\end{bmatrix}



Solid ellipsoid of semi-axes a, b, c and mass m

Ellipsoide.png


I =
\begin{bmatrix}
  \frac{1}{5} m (b^2+c^2) & 0 & 0 \\
  0 & \frac{1}{5} m (a^2+c^2) & 0 \\
  0 & 0 & \frac{1}{5} m (a^2+b^2)
\end{bmatrix}



Right circular cone with radius r, height h and mass m, about the apex

Moment of inertia cone.svg


I =
\begin{bmatrix}
  \frac{3}{5} m h^2 + \frac{3}{20} m r^2 & 0 & 0 \\
  0 & \frac{3}{5} m h^2 + \frac{3}{20} m r^2 & 0 \\
  0 & 0 & \frac{3}{10} m r^2
\end{bmatrix}



Solid cuboid of width w, height h, depth d, and mass m

180x



I =
\begin{bmatrix}
  \frac{1}{12} m (h^2 + d^2) & 0 & 0 \\
  0 & \frac{1}{12} m (w^2 + d^2) & 0 \\
  0 & 0 & \frac{1}{12} m (w^2 + h^2)
\end{bmatrix}



Slender rod along y-axis of length l and mass m about end

Moment of inertia rod end.svg



I =
\begin{bmatrix}
  \frac{1}{3} m l^2 & 0 & 0 \\
  0 & 0 & 0 \\
  0 & 0 & \frac{1}{3} m l^2
\end{bmatrix}



Slender rod along y-axis of length l and mass m about center

Moment of inertia rod center.svg



I =
\begin{bmatrix}
  \frac{1}{12} m l^2 & 0 & 0 \\
  0 & 0 & 0 \\
  0 & 0 & \frac{1}{12} m l^2
\end{bmatrix}



Solid cylinder of radius r, height h and mass m

Moment of inertia solid cylinder.svg


I =
\begin{bmatrix}
  \frac{1}{12} m (3r^2+h^2) & 0 & 0 \\
  0 & \frac{1}{12} m (3r^2+h^2) & 0 \\
  0 & 0 & \frac{1}{2} m r^2\end{bmatrix}



Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m

Moment of inertia thick cylinder h.svg


I =
\begin{bmatrix}
  \frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2) & 0 & 0 \\
  0 & \frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2) & 0 \\
  0 & 0 & \frac{1}{2} m ({r_1}^2 + {r_2}^2)\end{bmatrix}

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