Introduction to Mechanics for Engineersby Andy Ruina & Rudra Pratap21 Elementary introduction to 3D rigid-object dynamics

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Table II. Methods for calculating velocity and acceleration

Some facts about path coordinates
The path of a particle is ${\accentset{\rightharpoonup}{\boldsymbol{r}}}(t)$ .

	$\displaystyle\quad\hat{\boldsymbol{e}}_{t}\equiv{d{\accentset{\rightharpoonup}% {\boldsymbol{r}}}(s)\over ds},\quad\hat{\boldsymbol{e}}_{t}={d{\accentset{% \rightharpoonup}{\boldsymbol{r}}}(t)\over dt}{dt\over ds}={{\accentset{% \rightharpoonup}{\boldsymbol{v}}}\over v},\quad{\accentset{\rightharpoonup}{% \boldsymbol{\kappa}}}\equiv{d\hat{\boldsymbol{e}}_{t}\over ds}={d\hat{% \boldsymbol{e}}_{t}\over dt}{1\over v},\quad$
	$\displaystyle\hat{\boldsymbol{e}}_{n}={{\accentset{\rightharpoonup}{% \boldsymbol{\kappa}}}\over\|{\accentset{\rightharpoonup}{\boldsymbol{\kappa}}}\|% },\quad\boldsymbol{e}_{b}\equiv\hat{\boldsymbol{e}}_{t}\times\hat{\boldsymbol{% e}}_{n},\quad\rho={1\over\|{\accentset{\rightharpoonup}{\boldsymbol{\kappa}}}\|}.$

Summary of the direct differentiation method

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Figure 21.64:

In the direct differentiation method, using moving frame $\cal B$ , we calculate ${\accentset{\rightharpoonup}{\boldsymbol{v}}}_{P}$ by using a combination of the product rule of differentiation and the facts that

$\dot{\hat{\boldsymbol{\imath}}^{\prime}}={\accentset{\rightharpoonup}{% \boldsymbol{\omega}}}_{\cal B}\times\hat{\boldsymbol{\imath}}^{\prime}$ , $\dot{\hat{\boldsymbol{\jmath}}^{\prime}}={\accentset{\rightharpoonup}{% \boldsymbol{\omega}}}_{\cal B}\times\hat{\boldsymbol{\jmath}}^{\prime}$ , and $\dot{\hat{\boldsymbol{k}}^{\prime}}={\accentset{\rightharpoonup}{\boldsymbol{% \omega}}}_{\cal B}\times\hat{\boldsymbol{k}}^{\prime}$ ,

as follows:

	$\displaystyle{\accentset{\rightharpoonup}{\boldsymbol{v}}}_{P}$	$\displaystyle=\frac{d}{dt}{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{P}=% \frac{d}{dt}\left[{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{O^{\prime}/O}% +{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{P/O^{\prime}}\right]$
		$\displaystyle=\frac{d}{dt}\left[(x\hat{\boldsymbol{\imath}}+y\hat{\boldsymbol{% \jmath}}+z\hat{\boldsymbol{k}})+(x^{\prime}\hat{\boldsymbol{\imath}}^{\prime}+% y^{\prime}\hat{\boldsymbol{\jmath}}^{\prime}+z^{\prime}\hat{\boldsymbol{k}}^{% \prime})\right]$
		$\displaystyle=(\dot{x}\hat{\boldsymbol{\imath}}+\dot{y}\hat{\boldsymbol{\jmath% }}+\dot{z}\hat{\boldsymbol{k}})+(\dot{x}^{\prime}\hat{\boldsymbol{\imath}}^{% \prime}+\dot{y}^{\prime}\hat{\boldsymbol{\jmath}}^{\prime}+\dot{z}^{\prime}% \hat{\boldsymbol{k}}^{\prime})+$
		$\displaystyle\left[x^{\prime}({\accentset{\rightharpoonup}{\boldsymbol{\omega}% }}_{\cal B}\times\hat{\boldsymbol{\imath}}^{\prime})+y^{\prime}({\accentset{% \rightharpoonup}{\boldsymbol{\omega}}}_{\cal B}\times\hat{\boldsymbol{\jmath}}% ^{\prime})+z^{\prime}({\accentset{\rightharpoonup}{\boldsymbol{\omega}}}_{\cal B% }\times\hat{\boldsymbol{k}}^{\prime})\right]$

but stop short of identifying these three groups of three terms as

{\accentset{\rightharpoonup}{\boldsymbol{v}}}_{P}={\accentset{\rightharpoonup}% {\boldsymbol{v}}}_{O^{\prime}/O}+\dot{\accentset{\rightharpoonup}{\boldsymbol{% r}}}_{rel}+{\accentset{\rightharpoonup}{\boldsymbol{\omega}}}_{\cal B}\times{% \accentset{\rightharpoonup}{\boldsymbol{r}}}_{P/O}.

We would calculate ${\accentset{\rightharpoonup}{\boldsymbol{a}}}_{P}$ similarly and would get a formula with 15 non-zero terms (3 for each term in the ‘five-term’ acceleration formula).

\begin{array}[]{l|||||||||||c|c|c|}\mbox{\bf Method}&\mbox{\bf Position}&\mbox% {\bf Velocity}&\mbox{\bf Acceleration}\\ \hline\cr\hline\cr\hline\cr\hline\cr&&&\\ \vtop{\raggedright In general, as measured relative to the fixed frame $\cal F$.\@add@raggedright}&{\accentset{% \rightharpoonup}{\boldsymbol{r}}}\quad\mbox{or}\quad{\accentset{% \rightharpoonup}{\boldsymbol{r}}}_{P}\quad\mbox{or}\quad{\accentset{% \rightharpoonup}{\boldsymbol{r}}}_{P/O}&{\accentset{\rightharpoonup}{% \boldsymbol{v}}}\quad\mbox{or}\quad{\accentset{\rightharpoonup}{\boldsymbol{v}% }}_{P}\quad\mbox{or}\quad{\accentset{\rightharpoonup}{\boldsymbol{v}}}_{P/\cal F% }&{\accentset{\rightharpoonup}{\boldsymbol{a}}}\quad\mbox{or}\quad{\accentset{% \rightharpoonup}{\boldsymbol{a}}}_{P}\quad\mbox{or}\quad{\accentset{% \rightharpoonup}{\boldsymbol{a}}}_{P/\cal F}\\ \hline\cr&&&\\ \mbox{Cartesian Coordinates}&r_{x}\hat{\boldsymbol{\imath}}+r_{y}\hat{% \boldsymbol{\jmath}}+r_{z}\hat{\boldsymbol{k}}&\begin{array}[]{c}\quad v_{x}% \hat{\boldsymbol{\imath}}+v_{y}\hat{\boldsymbol{\jmath}}+v_{z}\hat{\boldsymbol% {k}}\\ =\dot{r}_{x}\hat{\boldsymbol{\imath}}+\dot{r}_{y}\hat{\boldsymbol{\jmath}}+% \dot{r}_{z}\hat{\boldsymbol{k}}\end{array}&\begin{array}[]{c}\quad a_{x}\hat{% \boldsymbol{\imath}}+a_{y}\hat{\boldsymbol{\jmath}}+a_{z}\hat{\boldsymbol{k}}% \\ =\dot{v}_{x}\hat{\boldsymbol{\imath}}+\dot{v}_{y}\hat{\boldsymbol{\jmath}}+% \dot{z}_{z}\hat{\boldsymbol{k}}\\ =\ddot{r}_{x}\hat{\boldsymbol{\imath}}+\ddot{r}_{y}\hat{\boldsymbol{\jmath}}+% \ddot{r}_{z}\hat{\boldsymbol{k}}\end{array}\\ &&&\\ \hline\cr&&&\\ \vtop{\raggedright Polar Coordinates/ \hfil\break Cylindrical Coordinates % \@add@raggedright}&R\hat{\boldsymbol{e}}_{\scriptscriptstyle\!R}+z\hat{% \boldsymbol{k}}&\begin{array}[]{c}\quad v_{R}\hat{\boldsymbol{e}}_{% \scriptscriptstyle\!R}+v_{\theta}\hat{\boldsymbol{e}}_{\scriptscriptstyle\!% \theta}+v_{z}\hat{\boldsymbol{k}}\\ =\dot{R}\hat{\boldsymbol{e}}_{\scriptscriptstyle\!R}+R\dot{\theta}\hat{% \boldsymbol{e}}_{\scriptscriptstyle\!\theta}+\dot{z}\hat{\boldsymbol{k}}\end{% array}&\begin{array}[]{c}a_{R}\hat{\boldsymbol{e}}_{\scriptscriptstyle\!R}+a_{% \theta}\hat{\boldsymbol{e}}_{\scriptscriptstyle\!\theta}+a_{z}\hat{\boldsymbol% {k}}\\ =(\ddot{R}-R\dot{\theta}^{2})\hat{\boldsymbol{e}}_{\scriptscriptstyle\!R}+(R% \ddot{\theta}+2\dot{R}\dot{\theta})\hat{\boldsymbol{e}}_{\scriptscriptstyle\!% \theta}+\ddot{z}\hat{\boldsymbol{k}}\end{array}\\ &&&\\ \hline\cr&&&\\ \mbox{Path Coordinates}&\mbox{not used}&v\hat{\boldsymbol{e}}_{t}&\begin{array% }[]{c}a_{t}\hat{\boldsymbol{e}}_{t}+a_{n}\hat{\boldsymbol{e}}_{n}\\ \\ =\dot{v}\hat{\boldsymbol{e}}_{t}+(v^{2}/\rho)\hat{\boldsymbol{e}}_{n}\end{% array}\\ &&&\\ \hline\cr&&&\\ \vtop{\raggedright\vskip-21.68121ptUsing data from a moving frame $\cal B$ % with origin at $O^{\prime}$ and angular velocity relative to the fixed frame of ${\accentset{\rightharpoonup}{% \boldsymbol{\omega}}}_{\cal B}$. The point $P^{\prime}$ is glued to $\cal B$ and instantaneously coincides with $P$. \@add@raggedright}&{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{O^{% \prime}/O}+{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{P/O^{\prime}}&\begin% {array}[]{c}{\accentset{\rightharpoonup}{\boldsymbol{v}}}_{P^{\prime}/\cal F}+% {\accentset{\rightharpoonup}{\boldsymbol{v}}}_{P/\cal B}=\\ \\ \underbrace{\dot{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{O^{\prime}/O}+% \ {\accentset{\rightharpoonup}{\boldsymbol{\omega}}}_{\cal B}\times{\accentset% {\rightharpoonup}{\boldsymbol{r}}}_{P^{\prime}/O^{\prime}}}_{{\accentset{% \rightharpoonup}{\boldsymbol{v}}}_{P^{\prime}/{\cal F}}}+\underbrace{{}^{\cal B% }{\dot{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{P/O^{\prime}}}}_{{% \accentset{\rightharpoonup}{\boldsymbol{v}}}_{P/\cal B}}\end{array}&\begin{% array}[]{c}\\ {\accentset{\rightharpoonup}{\boldsymbol{a}}}_{P^{\prime}/\cal F}+{\accentset{% \rightharpoonup}{\boldsymbol{a}}}_{P/\cal B}+2{\accentset{\rightharpoonup}{% \boldsymbol{\omega}}}_{\cal B}\times{\accentset{\rightharpoonup}{\boldsymbol{v% }}}_{P/\cal B}\qquad=\\ \\ \overbrace{\ddot{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{O^{\prime}/O}+{% \accentset{\rightharpoonup}{\boldsymbol{\omega}}}_{\cal B}\times{\accentset{% \rightharpoonup}{\boldsymbol{\omega}}}_{\cal B}\times{\accentset{% \rightharpoonup}{\boldsymbol{r}}}_{P^{\prime}/O^{\prime}}+\dot{\accentset{% \rightharpoonup}{\boldsymbol{\omega}}}_{\cal B}\times{\accentset{% \rightharpoonup}{\boldsymbol{r}}}_{P^{\prime}/O^{\prime}}}^{{\accentset{% \rightharpoonup}{\boldsymbol{a}}}_{P^{\prime}/\cal F}}\\ \\ +\underbrace{{}^{\cal B}{\ddot{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{P% /O^{\prime}}}}_{{\accentset{\rightharpoonup}{\boldsymbol{a}}}_{P/\cal B}}+2{% \accentset{\rightharpoonup}{\boldsymbol{\omega}}}_{\cal B}\times{\accentset{% \rightharpoonup}{\boldsymbol{v}}}_{P/\cal B}\\ \\ \mbox{`the 5-term acceleration formula'}\end{array}\\ &&&\\ \hline\cr\end{array}

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Basic flow chart for solving dynamics problems.

The methods (boxes 1-8) depend on the goal (ellipses a-d). Statics (b) only uses the solution route $1\rightarrow 2\rightarrow 4\rightarrow 5\rightarrow b$ . Dynamics uses other boxes as needed. At first reading this chart shows you the logic of the subject. Later the flow chart in this box should become self-evident.