Documentation of the rotational motion of reference frames #3619
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eggrobin
reviewed
Apr 27, 2023
pleroy
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eggrobin
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Apr 28, 2023
| \end{equation} | ||
| Injecting these expressions in the derivative formula (\ref{eqndv}) makes it possible to compute the trihedron of the derivatives $\pa{\dot{\vf}, \dot{\vn}, \dot{\vb}}$ of (\ref{eqnfbn}). The angular velocity is then written: | ||
| \[ | ||
| \VectorSymbol{\gw} = \pascal{\scal{\dot{\vn}}{\vb}} \vf + \pascal{\scal{\dot{\vb}}{\vf}} \vn + \pascal{\scal{\dot{\vf}}{\vn}} \vb |
| \end{equation} | ||
| Injecting these expressions in the second derivative formula (\ref{eqnddv}) makes it possible to compute the trihedron of the second derivatives $\pa{\ddot{\vf}, \ddot{\vn}, \ddot{\vb}}$ of (\ref{eqnfbn}). The angular acceleration is then written: | ||
| \begin{align*} | ||
| \dot{\VectorSymbol{\gw}} &= \pascal{\scal{\ddot{\vn}}{\vb}} \vf + \pascal{\scal{\dot{\vn}}{\dot{\vb}}} \vf + \pascal{\scal{\dot{\vn}}{\vb}} \dot{\vf} \\ |
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Document the derivation of 𝝎̇.
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The derivation of 𝝎̇ is rather obvious once the formula for 𝝎 is known. It's just the formula for the derivative of products.
| &+ \pascal{\scal{\ddot{\vf}}{\vn}} \vb + \pascal{\scal{\dot{\vf}}{\dot{\vn}}} \vb + \pascal{\scal{\dot{\vf}}{\vn}} \dot{\vb} | ||
| \end{align*} | ||
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| \section*{Jerk} |
eggrobin
approved these changes
May 1, 2023
documentation/Rotational Motion.tex
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| \pderiv{\vq}{\VectorSymbol{\gG}} \deriv{t}{\vq} + | ||
| \sum{k = 1}[n] \pderiv{\vq_k}{\VectorSymbol{\gG}} \deriv{t}{\vq_k} = | ||
| \pderiv{\vq}{\VectorSymbol{\gG}} \dot{\vq} + | ||
| \sum{k = 1}[n] \pderiv{\vq_k}{\VectorSymbol{\gG}} \dot{\vq_k} |
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Put the dot over the q, rather than sticking out over the index: \dot{\vq}_k
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#3358, #3615.