10.4: Angular Momentum

For angular momentum, the summed particle equation is \[ \sum_{i=1}^N (\vec{M}_i + \vec{r}_i \times \vec{F}_i) \, = \, \sum_{i=1}^N \vec{r}_i \times \dfrac{d}{dt} (m_i \vec{v}_i) , \]

where \(\vec{M}_i\) is an external moment on the particle \(i\). Similar to the case for linear momentum, summed internal moments cancel. We have

\[ \sum_{i=1}^N (\vec{M}_i + \vec{r}_i \times \vec{F}_i) \, = \, \sum_{i=1}^N m_i \vec{r}_i \times \left[ \dfrac{\partial \vec{v}_o}{\partial t} + \vec{\omega} \times \vec{v}_o \right] + \sum_{i=1}^N m_i \vec{r}_i \times \left( \dfrac{\partial \vec{\omega}}{\partial t} \times \vec{r}_i \right) + \sum_{i=1}^N m_i \vec{r}_i \times (\vec{\omega} \times (\vec{\omega} \times \vec{r}_i)). \]The summation in the first term of the right-hand side is recognized simply as \(m \vec{r}_G\), and the first term becomes \[m \vec{r}_G \times \left[ \dfrac{\partial \vec{v}_o}{\partial t} + \vec{\omega} \times \vec{v}_o \right]. \]

The second term expands as (using the triple product)

\begin{align} \sum_{i=1}^N m_i \vec{r}_i \times \left( \dfrac{\partial \vec{\omega}}{\partial t} \times \vec{r}_i \right) \,\, &= \,\, \sum_{i=1}^N m_i \left( (\vec{r}_i \cdot \vec{r}_i) \dfrac{\partial \vec{\omega}}{\partial t} - \left( \dfrac{\partial \vec{\omega}}{\partial t} \cdot \vec{r}_i \right) \vec{r}_i \right) \\[4pt] &= \,\, \begin{Bmatrix} \sum_{i=1}^N m_i ((y_i^2 + z_i^2) \dot{p} - (y_i \dot{q} + z_i \dot{r}) x_i) \\[4pt] \sum_{i=1}^N m_i ((x_i^2 + z_i^2) \dot{q} - (x_i \dot{p} + z_i \dot{r}) y_i) \\[4pt] \sum_{i=1}^N m_i ((x_i^2 + y_i^2) \dot{r} - (x_i \dot{p} + y_i \dot{q}) z_i) \end{Bmatrix} \end{align}

Employing the definitions of moments of inertia,

\begin{align} I \,\, &= \,\, \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\[4pt] I_{yx} & I_{yy} & I_{yz} \\[4pt] I_{zx} & I_{zy} & I_{zz} \end{bmatrix} \quad \text{(inertia matrix)}\\[4pt] I_{xx} \,\, &= \,\, \sum_{i=1}^N m_i (y_i^2 + z_i^2) \nonumber \\[4pt] I_{yy} \,\, &= \,\, \sum_{i=1}^N m_i (x_i^2 + z_i^2) \nonumber \\[4pt] I_{zz} \,\, &= \,\, \sum_{i=1}^N m_i (x_i^2 + y_i^2) \nonumber \\[4pt] I_{xy} \,\, &= \,\, I_{yx} \,\, = \,\, -\sum_{i=1}^N m_i x_i y_i \quad \text{(cross-inertia)} \nonumber \\[4pt] I_{xz} \,\, &= \,\, I_{zx} \,\, = \,\, -\sum_{i=1}^N m_i x_i z_i \nonumber \\[4pt] I_{yz} \,\, &= \,\, I_{zy} \,\, = \,\, -\sum_{i=1}^N m_i y_i z_i, \nonumber \end{align}

the second term of the angular momentum right-hand side collapses neatly into \(I \partial \vec{\omega} / \partial t\). The third term can be worked out along the same lines, but offers no similar condensation:

\begin{align} \sum_{i=1}^N m_i \vec{r}_i \times ((\vec{\omega} \cdot \vec{r}_i) \vec{\omega} - (\vec{\omega} \cdot \vec{\omega}) \vec{r}_i) \,\, &= \,\, \sum_{i=1}^N m_i \vec{r}_i \times \vec{\omega}(\vec{\omega} \cdot \vec{r}_i) \\[4pt] &= \,\, \begin{Bmatrix} \sum_{i=1}^N m_i (y_i r - z_i q)(x_i p + y_i q + z_i r) \\[4pt] \sum_{i=1}^N m_i (z_i p - x_i r)(x_i p + y_i q + z_i r) \\[4pt] \sum_{i=1}^N m_i (x_i q - y_i p)(x_i p + y_i q + z_i r) \end{Bmatrix} \\[4pt] \quad \nonumber \\[4pt] &= \,\, \begin{Bmatrix} I_{yz} (q^2 - r^2) + I_{xz} pq - I_{xy} pr \\[4pt] I_{xz} (r^2 - p^2) + I_{xy} rq - I_{yz} pq \\[4pt] I_{xy} (p^2 - q^2) + I_{yz} pr - I_{xz} qr \end{Bmatrix} + \begin{Bmatrix} (I_{zz} - I_{yy}) rq \\[4pt] (I_{xx} - I_{zz}) rp \\[4pt] (I_{yy} - I_{xx}) qp \end{Bmatrix}. \end{align}

Letting \(\vec{M} = \{ K, \, M, \, N \} \) be the total moment acting on the body, i.e., the left side of Equation \(\PageIndex{1}\), the complete moment equations are

\begin{align} K \, = \, & I_{xx} \dot{p} + I_{xy} \dot{q} + I_{xz} \dot{r} \ + \\[4pt] &(I_{zz} - I_{yy})rq + I_{yz}(q^2 - r^2) + I_{xz}pq - I_{xy}pr \ + \nonumber \\[4pt] &m [y_G (\dot{w} + pv - qu) - z_G (\dot{v} + ru - pw)] \nonumber \end{align}

\begin{align} M \, = \, & I_{yx} \dot{p} + I_{yy} \dot{q} + I_{yz} \dot{r} \ + \\[4pt] &(I_{xx} - I_{zz})pr + I_{xz}(r^2 - p^2) + I_{xy} qr - I_{yz} qp \ + \nonumber \\[4pt] &m [z_G (\dot{u} + qw - rv) - x_G (\dot{w} + pv - qu)] \nonumber \end{align}

\begin{align} N \, = \, & I_{zx} \dot{p} + I_{zy} \dot{q} + I_{zz} \dot{r} \ + \\[4pt] &(I_{yy} - I_{xx})pq + I_{xy}(p^2 - q^2) + I_{yz}pr - I_{xz}qr \ + \nonumber \\[4pt] &m [x_G (\dot{v} + ru - pw) - y_G (\dot{u} + qw - rv)]. \nonumber \end{align}