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# 13.14: Angular Velocity

## Angular velocity $$\omega$$

It is useful to relate the rigid-body equations of motion in the space-fixed $$(\mathbf{\hat{x}}, \mathbf{\hat{y}},\mathbf{\hat{z}})$$ coordinate system to those in the body-fixed $$(\mathbf{\hat{e}}_1,\mathbf{\hat{e}}_2,\mathbf{\hat{e}}_3)$$ coordinate system where the principal axis inertia tensor is defined. It was shown in appendix $$19.4$$ that an infinitessimal rotation can be represented by a vector. Thus the time derivatives of these rotation angles can be associated with the components of the angular velocity $$\boldsymbol{\omega}$$, where the precession $$\omega_{\phi} = \dot{\phi}$$, the nutation $$\omega_{\theta} = \dot{\theta}$$, and the spin $$\omega_{\psi} = \dot{\psi}$$. Unfortunately the coordinates $$(\phi , \theta , \psi)$$ are with respect to mixed coordinate frames and thus are not orthogonal axes. That is, the Euler angular velocities are expressed in different coordinate frames, where the precession $$\dot{\phi}$$ is around the space-fixed $$\mathbf{\hat{z}}$$ axis measured relative to the $$\mathbf{\hat{x}}$$-axis, the spin $$\dot{\psi}$$ is around the body-fixed $$\mathbf{\hat{e}}_3$$ axis relative to the rotating line-of-nodes, and the nutation $$\dot{\theta}$$ is the angular velocity between the $$\mathbf{\hat{z}}$$ and $$\mathbf{\hat{e}}_3$$ axes and points along the instantaneous line-of-nodes in the $$\mathbf{\hat{e}}_3 \times \mathbf{\hat{z}}$$ direction. By reference to Figure $$13.13.1$$ it can be seen that the components along the body-fixed axes are as given in Table $$\PageIndex{1}$$.

Table $$\PageIndex{1}$$: Euler angular velocity components in the body-fixed frame
Precession $$\dot{\phi}$$ Nutation $$\dot{\theta}$$ Spin $$\dot{\psi}$$
$$\dot{\phi}_1 = \dot{\phi} \sin \theta \sin \psi$$ $$\dot{\theta}_1 = \dot{\theta} \cos \psi$$ $$\dot{\psi}_1 = 0$$
$$\dot{\phi}_2 = \dot{\phi} \sin \theta \cos \psi$$ $$\dot{\theta}_2 = -\dot{\theta} \sin \psi$$ $$\dot{\psi}_2 = 0$$
$$\dot{\phi}_3 = \dot{\phi} \cos \theta$$ $$\dot{\theta}_3 = 0$$ $$\dot{\psi}_3 = \psi$$

Note that the precession angular velocity $$\dot{\phi}$$ is the angular velocity that the body-fixed $$\mathbf{\hat{e}}_3$$ and $$\mathbf{\hat{z}} \times \mathbf{\hat{3}}$$ axes precess around the space-fixed $$\mathbf{\hat{z}}$$ axis. Table $$\PageIndex{1}$$ gives the Euler angular velocities required to calculate the components of the angular velocity $$\boldsymbol{\omega}$$ for the body-fixed $$(\mathbf{1}, \mathbf{2}, \mathbf{3})$$ axis system. Collecting the individual components of $$\boldsymbol{\omega}$$, gives the components of the angular velocity of the body, relative to the space-fixed axes, in the body-fixed axis system $$(1, 2, 3)$$

$\omega_1 = \dot{\phi}_1 + \dot{\theta}_1 + \dot{\psi}_1 = \dot{\phi} \sin \theta \sin \psi + \dot{\theta} \cos \psi \label{13.86}$

$\omega_2 = \dot{\phi}_2 + \dot{\theta}_2 + \dot{\psi}_2 = \dot{\phi} \sin \theta \cos \psi - \dot{\theta} \sin \psi \label{13.87}$

$\omega_3 = \dot{\phi}_3 + \dot{\theta}_3 + \dot{\psi}_3 = \dot{\phi} \cos \theta + \dot{\psi} \label{13.88}$

The angular velocity of the body about the body-fixed $$\mathbf{3}$$-axis, $$\omega_3$$, is the sum of the projection of the precession angular velocity of the line-of-nodes $$\dot{\phi}$$ with respect to the space-fixed $$\mathbf{x}$$-axis, plus the angular velocity $$\dot{\psi}$$ of the body-fixed 3-axis with respect to the rotating line-of-nodes.

Similarly, the components of the body angular velocity $$\boldsymbol{\omega}$$ for the space-fixed axis system $$(x,y,z)$$ can be derived to be

$\omega_x = \dot{\theta} \cos \phi + \dot{\psi} \sin \theta \sin \phi \label{13.89}$

$\omega_y = \dot{\theta} \sin \phi - \dot{\psi} \sin \theta \cos \phi \label{13.90}$

$\omega_z = \dot{\phi} + \dot{\psi} \cos \theta \label{13.91}$

Note that when $$\theta = 0$$ then the Euler angles are singular in that the space-fixed $$z$$ axis is parallel with the body-fixed 3 axis and there is no way of distinguishing between precession $$\dot{\phi}$$ and spin $$\dot{\psi}$$, leading to $$\omega_z = \omega_3 = \dot{\phi} + \dot{\psi}$$. When $$\theta = \pi$$ then the $$z$$ axis and 3 axis are antiparallel and $$\omega_z = \dot{\phi} - \dot{\psi} = -\omega_3$$. The other special case is when $$\cos \theta = 0$$ for which the Euler angle system is orthogonal and the space-fixed $$\omega_z = \dot{\phi}$$, that is, it equals the precession, while the body-fixed $$\omega_3 = \dot{\psi}$$, that is, it equals the spin. When the Euler angle basis is not orthogonal then equations \ref{13.86} - \ref{13.88} and \ref{13.89} - \ref{13.91} are needed for expressing the Euler equations of motion in either the body-fixed frame or the space-fixed frame respectively.

Equations \ref{13.86} - \ref{13.88} for the components of the angular velocity in the body-fixed frame can be expressed in terms of the Euler angle velocities in a matrix form as

$\begin{pmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \end{pmatrix} = \begin{pmatrix} \sin \theta \sin \psi & \cos \psi & 0 \\ \sin \theta \cos \psi & - \sin \psi & 0 \\ \cos \theta & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{pmatrix}$

Note that the transformation matrix is not orthogonal which is to be expected since the Euler angular velocities are about axes that do not form a rectangular system of coordinates. Similarly equations \ref{13.89} - \ref{13.91} for the angular velocity in the space-fixed frame can be expressed in terms of the Euler angle velocities in matrix form as

$\begin{pmatrix} \omega_x \\ \omega_y \\ \omega_z \end{pmatrix} = \begin{pmatrix} 0 & \cos \phi & \sin \theta \sin \phi \\ 0 & \sin \phi & \sin \theta \cos \phi \\ 1 & 0 & \cos \theta \end{pmatrix} \cdot \begin{pmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{pmatrix}$