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

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