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

Angular Velocities: Time derivative of angular velocity

Let the components of the angular velocity vector $\usf{w}(\tau)$ w.r.t the co-rotational basis $\usf{e}_i(\tau)$ be $w_i$. That is,

\[\begin{align} \usf{w}(\tau)&=w_i\usf{e}_i(\tau) \end{align}\]

The time derivative of $\usf{w}(\tau)$ is \(\begin{align} \dot{\usf{w}}(\tau)=w'_i(\tau)\usf{e}_i(\tau)+w_i\usf{e}'_i(\tau) \label{eq:omegatimeder} \end{align}\)

We showed in the time derivative co-rotational section that that \(\begin{align} \usf{e}'_i(\tau) &=\usf{w}(\tau)\times \usf{e}_i(\tau)\\ &=\usf{W}(\tau) \usf{e}_i(\tau).\\ \end{align}\)

Using the above result in $\eqref{eq:omegatimeder}$, we get \(\begin{align} \dot{\usf{w}}(\tau)&=w'_i(\tau)\usf{e}_i(\tau)+w_i(\usf{w}(\tau)\times \usf{e}_i(\tau))\\ &=w'_i(\tau)\usf{e}_i(\tau)+\usf{w}(\tau)\times( w_i\usf{e}_i(\tau))\\ &=w'_i(\tau)\usf{e}_i(\tau)+\usf{w}(\tau)\times \usf{w}(\tau)\\ &=w'_i(\tau)\usf{e}_i(\tau) \end{align}\)

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