Consider a rigid body rotating about a fixed axis in an
inertial reference frame. If its angular position as a function of time is , the angular velocity, acceleration, and jerk can be expressed as follows: •
Angular velocity, \omega(t)=\dot\theta(t)=\frac{\mathrm {d}\theta(t)} {\mathrm {d}t}, is the time derivative of . •
Angular acceleration, \alpha(t)=\dot\omega(t)=\frac{\mathrm {d}\omega(t)} {\mathrm {d} t}, is the time derivative of . • Angular jerk, \zeta(t) = \dot {\alpha}(t) =\ddot\omega(t) = \overset{...}{ \theta}(t), is the time derivative of . 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
vector, angular velocity , and one polar
vector, linear velocity . From this, the angular acceleration is defined as \boldsymbol{\alpha}(t) = \frac {\mathrm {d}} {\mathrm {d} t} \boldsymbol{\omega}(t)= \dot {\boldsymbol\omega}(t) and the angular jerk is given by \boldsymbol{\zeta}(t) = \frac {\mathrm {d}}{\mathrm {d}t}\boldsymbol{\alpha}(t)=\dot{\boldsymbol{\alpha}}(t) = \ddot{\boldsymbol{\omega}}(t) taking the angular acceleration from as \boldsymbol{\alpha} = \frac{d\boldsymbol{\omega}}{dt} = \frac{\mathbf r\times \mathbf a}{r^2} - \frac{2}{r}\frac{dr}{dt}\boldsymbol{\omega}, we obtain \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} replacing \frac{d\boldsymbol\omega}{dt} we can have the last item as \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}, and we finally get \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} or vice versa, replacing \left( \mathbf r \times \mathbf a \right) with \boldsymbol{\alpha} : \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} 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 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 , and an unbounded angular jerk in the driven wheel. Jerk does not preclude the Geneva drive from being used in applications such as movie projectors and
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 m/s), and the low friction. With
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). == In elastically deformable matter ==