21 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Jerk (physics) | 3/4 | https://en.wikipedia.org/wiki/Jerk_(physics) | reference | science, encyclopedia | 2026-05-05T11:15:07.559806+00:00 | kb-cron |
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{\displaystyle {\begin{aligned}{\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^{2}r}{dt^{2}}}{\boldsymbol {\omega }}-{\frac {2}{r}}{\frac {dr}{dt}}{\frac {d{\boldsymbol {\omega }}}{dt}}\end{aligned}}}
replacing
d
ω
d
t
{\displaystyle {\frac {d{\boldsymbol {\omega }}}{dt}}}
we can have the last item as
−
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r
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ω
d
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{\displaystyle {\begin{aligned}-{\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{aligned}}}
, and we finally get
ζ
=
r
×
j
r
2
+
v
×
a
r
2
−
4
r
3
d
r
d
t
(
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+
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2
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{\displaystyle {\begin{aligned}{\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^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}
or vice versa, replacing
(
r
×
a
)
{\displaystyle \left(\mathbf {r} \times \mathbf {a} \right)}
with
α
{\displaystyle {\boldsymbol {\alpha }}}
:
ζ
=
r
×
j
r
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+
v
×
a
r
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−
4
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ω
{\displaystyle {\begin{aligned}{\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^{2}r}{dt^{2}}}{\boldsymbol {\omega }}\end{aligned}}}
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 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 α, and an unbounded angular jerk ζ 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 ==
An elastically deformable mass deforms under an applied force (or acceleration); the deformation is a function of its stiffness and the magnitude of the force. If the change in force is slow, the jerk is small, and the propagation of deformation is considered instantaneous as compared to the change in acceleration. The distorted body acts as if it were in a quasistatic regime, and only a changing force (nonzero jerk) can cause propagation of mechanical waves (or electromagnetic waves for a charged particle); therefore, for nonzero to high jerk, a shock wave and its propagation through the body should be considered. The propagation of deformation is shown in the graphic "Compression wave patterns" as a compressional plane wave through an elastically deformable material. Also shown, for angular jerk, are the deformation waves propagating in a circular pattern, which causes shear stress and possibly other modes of vibration. The reflection of waves along the boundaries cause constructive interference patterns (not pictured), producing stresses that may exceed the material's limits. The deformation waves may cause vibrations, which can lead to noise, wear, and failure, especially in cases of resonance.
The graphic captioned "Pole with massive top" shows a block connected to an elastic pole and a massive top. The pole bends when the block accelerates, and when the acceleration stops, the top will oscillate (damped) under the regime of pole stiffness. One could argue that a greater (periodic) jerk might excite a larger amplitude of oscillation because small oscillations are damped before reinforcement by a shock wave. One can also argue that a larger jerk might increase the probability of exciting a resonant mode because the larger wave components of the shock wave have higher frequencies and Fourier coefficients.