Jerk (physics)

In physics, jerk, also known as jolt (especially in British English), surge and 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 position. Jerk is defined by any of the following equivalent expressions:

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

where

$\vec a$ is acceleration,

$\vec v$ is velocity,

$\vec r$ is position

t is time.

Jerk is a vector, and there is no generally used term to describe its scalar magnitude (e.g., "speed" as the scalar magnitude for velocity).

The SI units of jerk are metres per second cubed (metres per second per second per second, m/s³ or m·s⁻³). There is no universal agreement on the symbol for jerk, but j is commonly used. ȧ, Newton's notation for the derivative of acceleration, can also be used, especially when "surge" or "lurch" is used instead of "jerk" or "jolt".

If acceleration can be felt by a body as the force (hence pressure) exerted by the object bringing about the acceleration on the body, jerk can be felt as the change in this pressure. For example a passenger in an accelerating vehicle with zero jerk will feel a constant force from the seat on his or her body; whereas positive jerk will be felt as increasing force on the body, and negative jerk as decreasing force on the body.

Note also the existence of yank—the derivative of force with respect to time.

Applications

Normally concerning forces, speed and acceleration are used for analysis. For example, the "jerk" produced by falling from outer space to the Earth is not particularly useful given the gravitational acceleration changes very slowly. Sometimes the analysis has to extend to jerk for a particular reason.

Jerk is often used in engineering, especially when building roller coasters.^{[citation needed]} Some precision or fragile objects — such as passengers, who need time to sense stress changes and adjust their muscle tension or suffer conditions such as whiplash — can be safely subjected not only to a maximum acceleration, but also to a maximum jerk.^{[citation needed]} Even where occupant safety isn't an issue, excessive jerk may result in an uncomfortable ride on elevators, trams and the like, and engineers expend considerable design effort to minimize it. Jerk may be considered when the excitation of vibrations is a concern. A device that 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 cuts. This is why modern motion controllers include jerk limitation features.

In mechanical engineering, jerk is considered, in addition to velocity and acceleration, in the development of cam profiles because of tribological implications and the ability of the actuated body to follow the cam profile without chatter.

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 seven segments defined by the following:

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

Explanation

Jerk without calculus

Jerk can be a difficult to conceptualize when it is defined in terms of calculus. When a force (push or pull) is applied to an object, that object starts to move. As long as the force is applied, the object will continue to speed up. When described in these terms, we are oversimplifying slightly. We think along the lines that there is no force on the object, then all of the sudden there is a force on the object. We do not think about how long it takes to apply the force.

However, in truth, the application of force does not instantly happen. A change always happens over time. Jerk is the change in acceleration over time. Typically, the time of contact where a force is applied is a split second.

If you push on a wall, it takes a fraction of a second before you apply the full push. Your fingertips will squoosh slightly as you begin to push. How long the squooshing takes determines the jerk. If you push on a wall very slowly, you can actually feel your push increasing. In such a case, the jerk is very low, because the change in force is happening over a relatively long time of several seconds. Jerk happens when a force is applied and removed. But the whole time a force is acting consistently on an object, there is no jerk. (This is because the acceleration is constant when there is a constant force.)

How quickly the force starts its push or pull determines the yank and subsequently the jerk. In most applications, it is not important how quickly the force is applied, and thus we typically think of forces being applied instantaneously. In this way, jerk can seem counter-intuitive, because it is not something people experience qualitatively on a daily basis.

If you take a piece of paper in your hands, holding it on both ends, you can pull very hard and not tear the paper. But if you put your hands close together and then jerk on the paper, you can tear it. You do not even have to pull hard. That's because you are using a high jerk (high acceleration in short time) to tear the paper. The paper's fibers require a small amount of time to respond to your pull and pull back. (In Rheology, this is called the relaxation time.) If you pull too fast, the paper simply snaps in half. You can similarly jerk on silly putty and snap it in half. The speed at which you pull, and not how hard, determines whether the silly putty snaps or stretches. To be clear, the key is how long it takes to go from zero acceleration to the full application of acceleration. This is not the same thing as how long you pull on the paper, but rather how long the snapping of the paper side to side lasts.

Jerk Equations (without Calculus)

y - yank (force per unit time)

m - mass of object

j - jerk (acceleration per unit time)

Δt - change in time

E - energy of collision

y = ΔF / Δt
j = Δa / Δt
j = y / m
j = ΔF / (m * Δt)

The higher the force or acceleration, the higher the jerk. The shorter the time of change in acceleration, such as a rollercoaster 'whipping' around a corner, the higher the jerk.

Jerk and impulse

An important concept to consider, but not to confuse with jerk, is impulse. An impulse is a force applied over time, which causes a change in momentum. This concept is relevant to discussions of jerk, because the impulse of crash (the force multiplied by time) equals the energy of a crash. Therefore, punching quickly and immediately pulling a hand back has a much shorter contact time than a punch with follow through. Thus, this style of punching results in a much higher amount of force, making a deadlier punch. Impulse and jerk are inter-related, because both concepts show us how large forces acting in short time frames can wreak havoc.

Characteristic impulse equation is E = F * Δt. Where E is the energy spent moving an object.

An application of impulse is determining the kick of a gun. When a gun fires, the size of the bullet and the length of the barrel determine how long the bullet is in the gun after firing. This determines the time of the impulse. Because the energy release due to the bullet is constant, the Force of firing the bullet (the kick) is F = E / Δt. So, the smaller the bullet (lower E), or longer barrel (increase Δt) will decrease the kick. This is why 22 rifles hardly kick; they have a full length barrel and a small powder charge in the bullets.

Impulse considers how long a force is applied to an object.

Jerk considers how quickly a force can begin to act on an object.

See also

• Shock (mechanics)
• Jounce (or snap)
• Terminal velocity

References

• Sprott JC (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850839-5. 
• Sprott JC (1997). "Some simple chaotic jerk functions" (PDF). Am J Phys 65 (6): 537–43. Bibcode 1997AmJPh..65..537S. doi:10.1119/1.18585. http://sprott.physics.wisc.edu/pubs/paper229.pdf. Retrieved 2009-09-28. 
• Blair G (2005). "Making the Cam" (PDF). Race Engine Technology (010). http://www.profblairandassociates.com/pdfs/Camshaft%20RET%20010.pdf. Retrieved 2009-09-29. 

External links

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