Jerk

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

Look up jerk, jolt, surge, lurch in Wiktionary, the free dictionary.

In physics, jerk, also called jolt (esp. in British English), surge or lurch, is the rate of change of acceleration; more precisely, the derivative of acceleration with respect to time, the second derivative of velocity, or the third derivative of displacement. Jerk is described by the following equation:

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

where

$\vec a$ is acceleration,

$\vec v$ – velocity,

$\vec r$ – displacement

t

stands for time.

Yank is the analog of force with respect to jerk: mass times jerk, or equivalently, the derivative of force with respect to time (this is only true non-relativistically; since mass is velocity dependent in relativistic physics, force is usually written as the first derivative of the momentum, while yank would be the second derivative. For force, it can be shown that dp/dt reduces to the familiar ma when v << c). Jerk is a vector, and there is no generally used term to describe its scalar value.

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.

Jerk is used at times 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, for example going from zero to 100 percent instantaneously, result in theoretically infinite jerk. This can lead to premature tool wear and result in uneven lines of a cut. This is why modern motion controllers include such features as jerk limitation.

Higher derivatives of displacement are rarely necessary, and hence lack agreed names. The fourth derivative of position was considered in development of the Hubble Space Telescope's pointing control system, and called jounce. Many other suggestions have been made, such as jilt, jouse, jolt, and delta jerk. As more distinct terms that start with letters other than "j", snap, crackle, and pop have been proposed for the 4th, 5th, and 6th derivatives of displacement, respectively, somewhat facetiously.^[1]

1 Third order motion profile
2 Jerk systems
3 References
4 External links

[edit] 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 equations:

acceleration build-up, with constant jerk = maxJerk
move with constant acceleration, with jerk = 0
approaching the desired maximum velocity, with jerk = -maxJerk
move with constant velocity, with jerk = 0 and acceleration = 0
approaching the desired deceleration, with jerk = -maxJerk
move with constant deceleration, with jerk = 0
approaching the desired position, with jerk = maxJerk

For shorter motions, the maximum acceleration or maximum velocity may never be reached.

[edit] 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):

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

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:

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

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 $R A = R / A = 5 R / 3$ , and all capacitors are of equal size. The dominant frequency will be $1 / 2π R C$ . 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.