Formula Sheet · Mass Moments of Inertia

Definitions

Moment of inertia about an axis

\[ I = \sum m_i r_i^2 = \int r^2\,dm \]

\(r\) = perpendicular distance to the axis. Units: \(\text{kg·m}^2\).

Radius of gyration

\[ k = \sqrt{I/m}, \qquad I = mk^2 \]

Plane rotation about a fixed axis or about \(G\): \(\sum M = I\alpha\), \(\ T = \tfrac12 I\omega^2\), \(\ H = I\omega\).
\(I_{xx} = \int (y^2 + z^2)\,dm\), \(I_{yy} = \int (z^2 + x^2)\,dm\), \(I_{zz} = \int (x^2 + y^2)\,dm\).
Thin plate in the \(xy\)-plane: \(I_{zz} = I_{xx} + I_{yy}\) (perpendicular-axis theorem, plates only).

Standard bodies (uniform, about axes through \(G\))

Body	Axis	\(\bar I\)
Slender rod, length \(l\)	perpendicular, through center	\(\tfrac1{12}ml^2\) (end: \(\tfrac13 ml^2\))
Thin ring, radius \(r\)	axis / diameter	\(mr^2\) / \(\tfrac12 mr^2\)
Thin disk, radius \(r\)	axis / diameter	\(\tfrac12 mr^2\) / \(\tfrac14 mr^2\)
Solid cylinder, \(r\), length \(h\)	axis / transverse	\(\tfrac12 mr^2\) / \(\tfrac1{12}m(3r^2 + h^2)\)
Thick tube, \(r_o\), \(r_i\)	axis	\(\tfrac12 m(r_o^2 + r_i^2)\)
Solid sphere, radius \(r\)	any diameter	\(\tfrac25 mr^2\)
Thin spherical shell	any diameter	\(\tfrac23 mr^2\)
Block \(a \times b \times c\)	parallel to \(c\)	\(\tfrac1{12}m(a^2 + b^2)\)
Thin plate \(a \times b\)	perpendicular / parallel to \(b\)	\(\tfrac1{12}m(a^2 + b^2)\) / \(\tfrac1{12}ma^2\)
Solid cone, base radius \(r\)	axis	\(\tfrac3{10}mr^2\)

Parallel axes and composite bodies

Parallel-axis theorem (one axis through \(G\))

\[ I = \bar I + m d^2, \qquad k^2 = \bar k^2 + d^2 \]

Between two axes that both miss \(G\): go through \(G\), \(I_B = I_A - m d_A^2 + m d_B^2\).
\(\bar I\) is the smallest moment of inertia among all parallel axes.
Composite: \(I_O = \sum (\bar I_i + m_i d_i^2)\); a hole is a part with negative mass (same density).
\(\bar x = \sum m_i x_i / \sum m_i\); then \(\bar I = I_O - m\bar d^2\).

Products of inertia

Definitions

\[ \begin{aligned} I_{xy} &= \textstyle\int xy\,dm \\ I_{yz} &= \textstyle\int yz\,dm \\ I_{zx} &= \textstyle\int zx\,dm \end{aligned} \]

Parallel-axis theorem

\[ I_{xy} = \bar I_{x'y'} + m\bar x\bar y \]

\(\bar x, \bar y\) signed coordinates of \(G\).

Products can be positive, negative or zero.
Plane of symmetry: the two products containing its normal coordinate vanish (\(xz\)-plane: \(I_{xy} = I_{yz} = 0\)).
Rod or plate parallel to the axes, and bodies of revolution about their axis: no products about their own centroidal axes.

The inertia tensor

Tensor, angular momentum and kinetic energy

\[ \Imat = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{xy} & I_{yy} & -I_{yz} \\ -I_{xz} & -I_{yz} & I_{zz} \end{bmatrix}, \quad \Hvec = \Imat\wvec, \quad T = \tfrac12\wvec^\mathsf{T}\Imat\wvec = \tfrac12\wvec\cdot\Hvec \]

About a fixed point \(O\) or about \(G\). Spin \(\omega\khat\): \(\Hvec = (-I_{xz}\ihat - I_{yz}\jhat + I_{zz}\khat)\,\omega\).
\(\Hvec\) fixed in a body turning at constant \(\wvec\): \(\sum\Mvec = \wvec \times \Hvec\) (bearing moment).
Transfer: \(\Imat_O = \Imat_G + m\left(|\bar\rvec|^2\,\mathbf{1} - \bar\rvec\,\bar\rvec^\mathsf{T}\right)\), with \(\bar\rvec\) the position of \(G\).

Any axis and rotated axes

Axis through \(O\) along the unit vector \(\uvec\)

\[ \begin{aligned} I_{Oa} = \uvec^\mathsf{T}\Imat_O\uvec &= I_{xx}u_x^2 + I_{yy}u_y^2 + I_{zz}u_z^2 \\ &\quad - 2I_{xy}u_xu_y - 2I_{yz}u_yu_z - 2I_{zx}u_zu_x \end{aligned} \]

Axes turned \(\theta\) about \(z\)

\[ \begin{aligned} I_{x'x'} &= \tfrac{I_{xx} + I_{yy}}{2} + \tfrac{I_{xx} - I_{yy}}{2}\cos 2\theta - I_{xy}\sin 2\theta \\ I_{x'y'} &= \tfrac{I_{xx} - I_{yy}}{2}\sin 2\theta + I_{xy}\cos 2\theta \end{aligned} \]

In general \(\Imat' = R\,\Imat R^\mathsf{T}\) (rows of \(R\) = new unit vectors). The trace does not change.
Block about its body diagonal through \(G\): \(\dfrac{m}{6}\dfrac{a^2b^2 + b^2c^2 + c^2a^2}{a^2 + b^2 + c^2}\); a cube: \(\tfrac16 ma^2\) about every axis through \(G\).

Principal axes

In the plane (\(z\) principal)

\[ \tan 2\theta_p = \frac{2I_{xy}}{I_{yy} - I_{xx}} \] \[ I_{\max,\min} = \tfrac{I_{xx} + I_{yy}}{2} \pm \sqrt{\left(\tfrac{I_{xx} - I_{yy}}{2}\right)^2 + I_{xy}^2} \]

In 3D

\[ \left(\Imat - I\,\mathbf{1}\right)\uvec = \mathbf{0}, \quad I^3 - J_1 I^2 + J_2 I - J_3 = 0 \]

\(J_1\) = trace, \(J_2\) = sum of the \(2\times2\) principal minors, \(J_3 = \det\Imat\). Software: numpy.linalg.eigh, MATLAB eig.

Mohr's circle: \(X = (I_{xx}, I_{xy})\) turns through \(2\theta\) about \(C\) as the axes turn through \(\theta\).
An axis of symmetry, or the normal to a plane of symmetry, is principal.
Dynamic balance: the shaft is a principal axis through \(G\). Free spin is stable about the max and min axes, unstable about the intermediate one.

Common mistakes

Distance to a point, not to the axis. \(r\) in \(\int r^2\,dm\) is the perpendicular distance to the axis.
Parallel axes that both miss \(G\). \(I_B \ne I_A + m\,\overline{AB}^2\); go through \(G\).
Adding a hole. Subtract its \(\bar I + md^2\), with the plate's density.
Dropping signs of products. \(I_{xy} = m\bar x\bar y\) with signed coordinates.
Product signs in the tensor. The off-diagonal entries are \(-I_{xy}\), \(-I_{yz}\), \(-I_{zx}\).
A non-unit axis vector. Normalise \(\uvec\) before \(\uvec^\mathsf{T}\Imat\uvec\).
Which principal axis is which. \(\tan 2\theta_p\) gives two axes \(90^\circ\) apart; test one in \(I_{x'x'}\).
Static balance is not enough. \(G\) on the shaft does not make \(\Hvec\) parallel to \(\wvec\).