ⶌⶌⶌ Fast Romantics: Moments of inertia

Showing posts with label Moments of inertia. Show all posts

Wednesday, April 15, 2015

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.

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.

$I_{\mathrm{end}} = \frac{m L^2}{3} \,\!$ ^[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.

$I_{\mathrm{center}} = \frac{m L^2}{12} \,\!$ ^[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.

$I_z = m r^2\!$
$I_x = I_y = \frac{m r^2}{2}\,\!$

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.

$I_z = \frac{m r^2}{2}\,\!$
$I_x = I_y = \frac{m r^2}{4}\,\!$

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.

$I = m r^2 \,\!$ ^[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).

$I_z = \frac{m r^2}{2}\,\!$ ^[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 $I_z = \frac{1}{2} \pi\rho h\left({r_2}^4 - {r_1}^4\right)$ , $I_x = I_y = \frac{1}{12} \pi\rho h\left(3({r_2}^4 - {r_1}^4)+h^2({r_2}^2 - {r_1}^2)\right)$

$I_z = \frac{1}{2} m\left(r_1^2 + r_2^2\right) = m r_2^2 \left(1-t+\frac{1}{2}{t}^2\right)$ ^[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

$I_{solid} = \frac{m s^2}{20}\,\!$ $I_{hollow} = \frac{m s^2}{12}\,\!$

Octahedron (hollow) of side s and mass m

$I_z=I_x=I_y = \frac{5m s^2}{9}\,\!$

Octahedron (solid) of side s and mass m

$I_z=I_x=I_y = \frac{m s^2}{5}\,\!$

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).

$I = \frac{2 m r^2}{3}\,\!$ ^[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).

$I = \frac{2 m r^2}{5}\,\!$ ^[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.

$I = \frac{2 m}{5}\left[\frac{{r_2}^5-{r_1}^5}{{r_2}^3-{r_1}^3}\right]\,\!$ ^[1]

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

$I_z = \frac{3}{10}mr^2 \,\!$ ^[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.

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

$I_a = \frac{m (b^2+c^2)}{5}\,\!$

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

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

Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the end of the plate)

$I_e = \frac {m h^2}{3}+\frac {m w^2}{12}\,\!$

Thin rectangular plate of height h and of width w and mass m

$I_c = \frac {m(h^2 + w^2)}{12}\,\!$ ^[1]

Solid cuboid of height h, width w, and depth d, and mass m. For a similarly oriented cube with sides of length

, $I_{CM} = \frac{m s^2}{6}\,\!$

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

, $I = \frac{m s^2}{6}\,\!$ .

$I = \frac{m\left(W^2D^2+L^2D^2+L^2W^2\right)}{6\left(L^2+W^2+D^2\right)}$

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, ..., P_N and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.

$I=\frac{m}{6}\frac{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+1}\times\mathbf{P}_{n}\|((\mathbf{P}_{n+1}\cdot\mathbf{P}_{n+1})+(\mathbf{P}_{n+1}\cdot\mathbf{P}_{n})+(\mathbf{P}_{n}\cdot\mathbf{P}_{n}))}{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+1}\times\mathbf{P}_{n}\|}$

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:

$\rho(x,y) = \tfrac{m}{2\pi ab}\, e^{-((x/a)^2+(y/b)^2)/2}\,,$

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

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 e_x, e_y, e_z to obtain I_x, I_y, I_z respectively.

Solid sphere of radius r and mass m

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

Hollow sphere of radius r and mass m

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

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

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

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

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

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

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

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

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

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

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

Solid cylinder of radius r, height h and mass m

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

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

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

Description	Figure	Moment of inertia tensor

Pages

Wednesday, April 15, 2015

Moments of inertia

List of 3D inertia tensors