Editing Jerk (physics) (section)

== In rotation ==
[[File:Chronogrammes croix malte 4 branches complet EN.svg|thumb|upright=1.3|Timing diagram over one revolution for angle, angular velocity, angular acceleration, and angular jerk]]

Consider a rigid body rotating about a fixed axis in an [[inertial frame of reference#Newton's inertial frame of reference|inertial reference frame]]. If its angular position as a function of time is {{math|''θ''(''t'')}}, the angular velocity, acceleration, and jerk can be expressed as follows:

* [[Angular velocity]], <math qid=Q161635>\omega(t)=\dot\theta(t)=\frac{\mathrm {d}\theta(t)} {\mathrm {d}t}</math>, is the time derivative of {{math|''θ''(''t'')}}.
* [[Angular acceleration]], <math qid=Q186300>\alpha(t)=\dot\omega(t)=\frac{\mathrm {d}\omega(t)} {\mathrm {d} t}</math>, is the time derivative of {{math|''ω''(''t'')}}.
* Angular jerk, <math>\zeta(t) = \dot {\alpha}(t) =\ddot\omega(t) = \overset{...}{ \theta}(t)</math>, is the time derivative of {{math|''α''(''t'')}}.

Angular acceleration equals the [[torque]] acting on the body, divided by the body's [[moment of inertia]] with respect to the momentary axis of rotation. A change in torque results in angular jerk.

The general case of a rotating rigid body can be modeled using kinematic [[screw theory]], which includes one axial [[Pseudovector|vector]], angular velocity {{math|'''Ω'''(''t'')}}, and one polar [[Pseudovector|vector]], linear velocity {{math|'''v'''(''t'')}}. From this, the angular acceleration is defined as

<math display="block">\boldsymbol{\alpha}(t) = \frac {\mathrm {d}} {\mathrm {d} t} \boldsymbol{\omega}(t)= \dot {\boldsymbol\omega}(t)</math>

and the angular jerk is given by
<math display="block">\boldsymbol{\zeta}(t) = \frac {\mathrm {d}}{\mathrm {d}t}\boldsymbol{\alpha}(t)=\dot{\boldsymbol{\alpha}}(t) = \ddot{\boldsymbol{\omega}}(t)</math>

taking the angular acceleration from [[Angular acceleration#Particle in three dimensions]] as
<math display="block">
\boldsymbol{\alpha} = \frac{d\boldsymbol{\omega}}{dt}
= \frac{\mathbf r\times \mathbf a}{r^2} - \frac{2}{r}\frac{dr}{dt}\boldsymbol{\omega}</math>, we obtain

<math display="block">\begin{align}
\boldsymbol{\zeta} = \frac{d \boldsymbol{\alpha} }{dt} =
\frac{1}{r^2}\left( \mathbf r \times \frac{d \mathbf a}{dt} + \frac{d \mathbf r}{dt} \times \mathbf a \right)
- \frac{2}{r^3}\frac{dr}{dt}\left( \mathbf r \times \mathbf a \right)\\
\\
+ \frac{2}{r^2}\left(\frac{dr}{dt} \right)^2 \boldsymbol{\omega}
- \frac{2}{r}\frac{d^2r}{dt^2} \boldsymbol{\omega}
- \frac{2}{r}\frac{dr}{dt} \frac{d \boldsymbol{\omega} }{dt}
\end{align}</math>

replacing <math>\frac{d\boldsymbol\omega}{dt}</math> we can have the last item as
<math display="block">\begin{align}
- \frac{2}{r}\frac{dr}{dt} \frac{d\boldsymbol\omega}{dt}
&= - \frac{2}{r}\frac{dr}{dt} \left( \frac{\mathbf r\times \mathbf a}{r^2} - \frac{2}{r}\frac{dr}{dt}\boldsymbol\omega \right)\\
\\
&= - \frac{2}{r^3}\frac{dr}{dt} \left( \mathbf r\times \mathbf a \right)
+ \frac{4}{r^2} \left(\frac{dr}{dt} \right)^2 \boldsymbol\omega
\end{align}</math>, and we finally get

<math display="block">\begin{align}
\boldsymbol{\zeta} = \frac{\mathbf r\times \mathbf j}{r^2}
+ \frac{\mathbf v\times \mathbf a}{r^2}
- \frac{4}{r^3}\frac{dr}{dt}\left( \mathbf r \times \mathbf a \right)
+ \frac{6}{r^2}\left(\frac{dr}{dt} \right)^2 \boldsymbol{\omega}
- \frac{2}{r}\frac{d^2r}{dt^2} \boldsymbol{\omega} 
\end{align}</math>

or vice versa, replacing <math>\left( \mathbf r \times \mathbf a \right)</math> with <math> \boldsymbol{\alpha} </math>:
<math display="block">\begin{align}
\boldsymbol{\zeta} = \frac{\mathbf r\times \mathbf j}{r^2}
+ \frac{\mathbf v\times \mathbf a}{r^2}
- \frac{4}{r}\frac{dr}{dt} \boldsymbol{\alpha} 
- \frac{2}{r^2}\left(\frac{dr}{dt} \right)^2 \boldsymbol{\omega}
- \frac{2}{r}\frac{d^2r}{dt^2} \boldsymbol{\omega} 
\end{align}</math>

[[File:Animiertes Prinzip Malteserkreuzgetriebe 3D.gif|thumb|upright|Animation showing a four-position external [[Geneva drive]] in operation]]

For example, consider a [[Geneva drive]], a device used for creating intermittent rotation of a driven wheel (the blue wheel in the animation) by continuous rotation of a driving wheel (the red wheel in the animation). During one cycle of the driving wheel, the driven wheel's angular position {{mvar|θ}} changes by 90 degrees and then remains constant. Because of the finite thickness of the driving wheel's fork (the slot for the driving pin), this device generates a discontinuity in the angular acceleration {{mvar|α}}, and an unbounded angular jerk {{mvar|ζ}} in the driven wheel.

Jerk does not preclude the Geneva drive from being used in applications such as movie projectors and [[Cam (mechanism)|cams]]. In movie projectors, the film advances frame-by-frame, but the projector operation has low noise and is highly reliable because of the low film load (only a small section of film weighing a few grams is driven), the moderate speed (2.4&nbsp;m/s), and the low friction.<!-- [[File:Cames conjuguees rotation intermittente un sixieme de tour.svg|thumbnail|left|Double cam, one sixth per rotation]]
[[File:Cames conjuguees rotation intermittente un tiers de tour.svg|thumb|left|Double cam, one third per rotation]] -->
{{multiple image
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<!-- Image 1 -->
| image1 = Cames conjuguees rotation intermittente un sixieme de tour.svg
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| caption2 ={{sfrac|1|3}} per revolution

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With [[Cam (mechanism)|cam drive]] systems, use of a dual cam can avoid the jerk of a single cam; however, the dual cam is bulkier and more expensive. The dual-cam system has two cams on one axle that shifts a second axle by a fraction of a revolution. The graphic shows step drives of one-sixth and one-third rotation per one revolution of the driving axle. There is no radial clearance because two arms of the stepped wheel are always in contact with the double cam. Generally, combined contacts may be used to avoid the jerk (and wear and noise) associated with a single follower (such as a single follower gliding along a slot and changing its contact point from one side of the slot to the other can be avoided by using two followers sliding along the same slot, one side each).{{clear}}