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\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}} \renewcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\t}[1]{\textsf{#1}} \newcommand{\m}[1]{\mathbb{#1}} \def\RR{\bf R} \def\bold#1{\bf #1} \def\mbf#1{\mathbf #1} \def\uv#1{\hat{\usf {#1}}} \def\dl#1{\underline{\underline{#1}}} \newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}} \def\bs#1{\usf #1}\] \[\renewcommand{\u}[1]{\boldsymbol{#1}} \newcommand{\usf}[1]{\mathsf{\u{\mathsf{#1}}}} \newcommand{\pr}[1]{\left( #1 \right)}\]

Angular Velocities: Relative angular velocity vector

Consider two rotation tensors $\u{R}_1 = \u{R}_1(t)$ and $\u{R}_2 = \u{R}_2(t)$, where $\u{R}_1$ transforms the basis vectors of the fixed reference frame ${\u{E}_i}_{i=1,2,3}$ to a new set of basis vectors ${\u{e}’_i}_{i=1,2,3}$ and $\u{R}_2$ transforms ${\u{e}’_i}_{i=1,2,3}$ to another set of basis vectors ${\u{e}_i}_{i=1,2,3}$. The composite rotation $\u{R} = \u{R}_2 R_1$ transfroms ${\u{E}_i}_{i=1,2,3}$ to ${\u{e}_i}_{i=1,2,3}$. The rotation tensors $\u{R}$, $\u{R}_1$, and $\u{R}_2$ can be defined by use of the representations \(\begin{align} \bs{R} &= \uv{e}_i \otimes \uv{E}_i \\ \bs{R}_1 &= \uv{e}'_i \otimes \uv{E}_i \\ \bs{R}_2 &= \uv{e}_i \otimes \uv{e}'_i \end{align}\)

Let us consider the following relative angular velocity tensor \(\begin{align} \hat{\bs{\Omega}}_{R_2} = \bs{\Omega}_R - \bs{\Omega}_{R_1}. \end{align}\)

Using the definition of the angular velocity tensors and the fact that \(\dot{\bs{R}} = \dot{\bs{R}}_2\bs{R}_1 + \bs{R}_2\dot{\bs{R}}_1\), we find that \(\begin{align} \hat{\bs{\Omega}}_{R_2} &= \dot{\bs{R}}\bs{R}^{\textsf{T}} - \dot{\bs{R}}_1\bs{R}_1^{\textsf{T}} \\ &= \dot{\bs{R}}_2\bs{R}_1\bs{R}_1^{\textsf{T}}\bs{R}_2^{\textsf{T}} + \bs{R}_2\dot{\bs{R}}_1\bs{R}_1^{\textsf{T}}\bs{R}_2^{\textsf{T}} - \dot{\bs{R}}_1\bs{R}_1^{\textsf{T}} \\ &= \dot{\bs{R}}_2\bs{R}_2^{\textsf{T}} + \bs{R}_2\bs{\Omega}_{R_1}\bs{R}_2^{\textsf{T}} - \bs{\Omega}_{R_1} \\ &= \dot{\bs{R}}_2\bs{R}_2^{\textsf{T}} + \bs{R}_2\bs{\Omega}_{R_1}\bs{R}_2^{\textsf{T}} + \bs{\Omega}_{R_1}^{\textsf{T}} \\ &= (\dot{\bs{R}}_2 + \bs{R}_2\bs{\Omega}_{R_1} + \bs{\Omega}_{R_1}^{\textsf{T}}\bs{R}_2)\bs{R}_2^{\textsf{T}}. \end{align}\)

However, \(\begin{align} \mathring{\bs{R}}_2 = \dot{\bs{R}}_2 + \bs{R}_2\bs{\Omega}_{R_1} + \bs{\Omega}_{R_1}^{\textsf{T}}\bs{R}_2, \end{align}\) where the corotational derivative \(\mathring{\bs{R}}_2\) is defined as \(\begin{align} \mathring{\bs{R}}_2 = \dot{R}_{2ik} \uv{e}'_i \otimes \uv{e}'_k, \end{align}\) where \(\begin{align} R_{2ik} = (\bs{R}_2 \uv{e}'_k) \cdot \uv{e}'_i \end{align}\) is the components of \(\bs{R}_2\) referred to the \(\{\uv{e}'_i\}_{i=1,2,3}\) basis. In other words, \(\mathring{\bs{R}}_2\) is the derivative of the tensor \(\bs{R}_2\) assuming that the \(\{\uv{e}'_i\}_{i=1,2,3}\) basis is fixed.

Therefore, the relative angular velocity tensor is \(\begin{align} \hat{\bs{\Omega}}_{R_2} = \bs{\Omega}_R - \bs{\Omega}_{R_1} = \mathring{\bs{R}}_2 \bs{R}_2^{\textsf{T}}. \end{align}\) The relative angular velocity vector is \(\begin{align} \hat{\bs{\omega}}_{R_2} = \bs{\omega}_R - \bs{\omega}_{R_1} \end{align}\). Its components can be computed as \(\begin{align} (\hat{\bs{\omega}}_{R_2})_i = -\frac{1}{2}\epsilon_{ijk}(\hat{\bs{\Omega}}_{R_2})_{jk}. \end{align}\)

Example: Relative angular velocity vector

To illustrate the convenience of relative angular velocity result, let us consider an example. Suppose we have two rotations $\bs{R}_1$ and $\bs{R}$, where \(\begin{align} \bs{R}_1 &= \cos\psi(\bs{I}-\uv{E}_3\otimes\uv{E}_3) + \sin\psi (*\uv{E}_3) + \uv{E}_3\otimes\uv{E}_3 \\ \bs{R} & = \bs{R}_2\bs{R}_1. \end{align}\)

Here, the relative rotation tensor $\bs{R}_2$ is chosen to correspond to a rotation through an angle $\theta$ about \(\uv{e}'_2 = \cos\psi \uv{E}_2 - \sin\psi \uv{E}_1\): \(\begin{align} \bs{R}_2 = \cos\psi(\bs{I}-\uv{e}'_2\otimes\uv{e}'_2) + \sin\psi (*\uv{e}'_2) + \uv{e}'_2\otimes\uv{e}'_2. \end{align}\)

The tensor $\bs{R}_1$ defines a transformation consisting of a rotation through an angle $\psi$ about $\uv{E}_3$. This rotation transforms \(\uv{E}_i\) to \(\uv{e}'_i\), where \(\begin{align} \uv{e}'_1 &= \cos\psi \uv{E}_1 + \sin\psi \uv{E}_2, \\ \uv{e}'_2 &= \cos\psi \uv{E}_2 - \sin\psi \uv{E}_1, \\ \uv{e}'_3 &= \uv{E}_3. \end{align}\)

In addition, $\bs{R}$ consists of the rotation $\bs{R}_1$ followed by a rotation through an angle $\theta$ about \(\uv{e}'_2\).

We can calculate \(\bs{\omega}_{R_1}\) from \(\bs{R}_1\) as \(\begin{align} \bs{\omega}_{R_1} = \dot{\psi} \uv{E}_3. \end{align}\) We use the relative angular velocity vector to calculate \(\bs{\omega}_{R}\). To do this, we first need to write \(\bs{R}_2\) with respect to an appropriate basis. The appropriate basis is \(\uv{e}'_i\otimes\uv{e}'_k\): \(\begin{align} {R_2} = \cos\psi(\uv{e}'_1\otimes\uv{e}'_1+\uv{e}'_3\otimes\uv{e}'_3) - \sin\psi (\uv{e}'_3\otimes\uv{e}'_1-\uv{e}'_1\otimes\uv{e}'_3) + \uv{e}'_2\otimes\uv{e}'_2. \end{align}\) Calculating the corotational rate of this tensor, we take its derivative, keeping \(\uv{e}'_i\) fixed: \(\begin{align} \mathring{\bs{R}}_2 = -\dot{\theta}\sin\psi(\uv{e}'_1\otimes\uv{e}'_1+\uv{e}'_3\otimes\uv{e}'_3) - \dot{\theta}\cos\psi (\uv{e}'_3\otimes\uv{e}'_1-\uv{e}'_1\otimes\uv{e}'_3). \end{align}\) Hence, \(\begin{align} \hat{\bs{\Omega}}_{R_2} = \dot{\theta} (\uv{e}'_1\otimes\uv{e}'_3-\uv{e}'_3\otimes\uv{e}'_1) \end{align}\) and \(\begin{align} \hat{\bs{\omega}}_{R_2} = \dot{\theta} \uv{e}'_2. \end{align}\)

Combining the expressions for \(\bs{\omega}_{R_1}\) and \(\hat{\bs{\omega}}_{R_2}\), we arrive at an expression for the angular velocity vector for \(\bs{R}\): \(\begin{align} \bs{\omega}_{R} = \dot{\theta} \uv{e}'_2 + \dot{\psi} \uv{E}_3. \end{align}\)

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