Jerk (physics)

This article is about the physics concept of jerk. For other terms of jerk, see Jerk (disambiguation).

In physics, jerk, jolt (especially in British English), surge or lurch, is the rate of change of acceleration; that is, the derivative of acceleration with respect to time, the second derivative of velocity, or the third derivative of displacement. Jerk is defined by the following equation:

${\vec {j}}={\frac {\mathrm {d} {\vec {a}}}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}{\vec {v}}}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} ^{3}{\vec {r}}}{\mathrm {d} t^{3}}}$

where

${\vec {a}}$ is acceleration,

${\vec {v}}$ is velocity,

${\vec {r}}$ is displacement

${\mathit {t}}$ is time.

Jerk is a vector, and there is no generally used term to describe its scalar magnitude.

The units of jerk are metres per second cubed (m/s³). There is no universal agreement on the symbol for jerk, but j is commonly used.

Related concepts

Denny Burgos is a JERK.

Applications

Jerk is often used in engineering, especially when building roller coasters. Some precision or fragile objects — such as passengers, who need time to sense stress changes and adjust their muscle tension, or suffer, e.g., whiplash — can be safely subjected not only to a maximum acceleration, but also to a maximum jerk. Jerk may be considered when the excitation of vibrations is a concern. A device which measures jerk is called a "jerkmeter."

Jerk is also important to consider in manufacturing processes. Rapid changes in acceleration of a cutting tool can lead to premature tool wear and result in uneven lines of a cut. This is why modern motion controllers include jerk limitation features.

Third order motion profile

In motion control, a common need is to move a system from one steady position to another (point-to-point motion). Following the fastest possible motion within an allowed maximum value for speed, acceleration, and jerk, will result in a third-order motion profile as illustrated below:

The motion profile consists of up to 7 segments defined by the following:

1. acceleration build-up, with maximum positive jerk
2. constant acceleration (zero jerk)
3. acceleration ramp-down, approaching the desired maximum velocity, with maximum negative jerk
4. constant speed (zero jerk, zero acceleration)
5. deceleration build-up, approaching the desired deceleration, with maximum negative jerk
6. constant deceleration (zero jerk)
7. deceleration ramp-down, approaching the desired position at zero velocity, with maximum positive jerk

If the initial and final positions are sufficiently close together, the maximum acceleration or maximum velocity may never be reached.

Jerk systems

A jerk system is a system whose behavior is described by a jerk equation, which is an equation of the form (Sprott, 2003):

${\frac {\mathrm {d} ^{3}x}{\mathrm {d} t^{3}}}=f\left({\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}},{\frac {\mathrm {d} x}{\mathrm {d} t}},x\right)$

For example, certain simple electronic circuits may be designed which are described by a jerk equation. These are known as jerk circuits.

One of the most interesting properties of jerk systems is the possibility of chaotic behavior. In fact, certain well-known chaotic systems such as the Lorenz attractor and the Rössler map are conventionally described as a system of three first-order differential equations, but which may be combined into a single (although rather complicated) jerk equation.

An example of a jerk equation is:

${\frac {\mathrm {d} ^{3}x}{\mathrm {d} t^{3}}}+A{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\frac {\mathrm {d} x}{\mathrm {d} t}}-|x|+1=0$.

Where A is an adjustable parameter. This equation has a chaotic solution for A=3/5 and can be implemented with the following jerk circuit:

In the above circuit, all resistors are of equal value, except $R_{A}=R/A=5R/3$, and all capacitors are of equal size. The dominant frequency will be $1/2\pi RC$. The output of op amp 0 will correspond to the x variable, the output of 1 will correspond to the first derivative of x and the output of 2 will correspond to the second derivative.

References

• Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850839-5 and ISBN 978-0-19-850839-7.
• Am. J. Phys., Vol. 65, No. 6, Pg. 538, June 1997

External links

• What is the term used for the third derivative of position?, description of jerk in the Usenet Physics FAQ.