Beyond velocity and acceleration: jerk, snap and higher derivatives (2016)

Jerk is also very important for road or rail track design. If you imagine needing to make a 90 degree bend, the "obvious" way to do it is by rounding off the corner with a circular radius.

But if you do that, it means the vehicle goes from having 0 sideways acceleration to experiencing 100% of the centripetal acceleration to move an object on a circular path (a = v^2 / r) instantaneously.

As an occupant of the car, that means you go from sitting comfortably to suddenly being thrown sideways.

It's much more comfortable if you ease into the turn, with the track design considering the rate of change of acceleration. If the designer didn't consider jerk you would definitely notice.

One thing that is strange is that we can easily imagine the first two derivatives: position we can just imagine a static point, velocity we can imagine a constant speed i.e. a straight line on a position-time graph, acceleration we just imagine a parabola, but jerk is somehow conceptually indistinguishable. The difference between a point, a line, and a parabola are stark, the third order jerk is not so easy to distinguish, instead still just looking like the parabola.

I've always wondered why this is, why curves in general are perceptually similar if scaled correctly, while a straight line is so clearly different. Perhaps it is because our perceptions developed to distinguish between inertial and non inertial reference frames?

> American people want to have the gut feeling when riding a lift, especially an express lift in those high buildings. In difference in Asian countries the lift ride must be smooth as possible. They don't like to have the feeling of riding a lift at all. In Europe it is something in between

How representative are these stated preferences actually of the population. I'd imagine that the individual preferences vary greatly from person to person and also change with age.

The terms jerk and snap while perhaps known in the rare space of elevator purchasing aren’t generally used terms in most fields. I’m surprised that’s in any way controversial ?

The terms are also understood in economics as prudence, and template. Albeit, not widely used.

Statics, dynamics, and two semesters of physics are required courses for every engineer. None of them incorporate the teaching of jerk or snap.

This is breaking my brain a little, any eli5?

Why bother with giving words for something which is longer than their mathemathical definition? The word can be unknown, but if the function is unknown the word is useless?

When it comes to a measured time-series, the function is always discrete and arbitrary; there is no 'curve' like that generated by a function well-defined on the reals, and so there is no real closed-form derivative. In this context derivatives are not equations so much as derived data, reducing two points to one. And this process is recursive, such that you can take 4 points, reduce the two adjacent ones to 1 point, and then reduce those two into 1 point. In fact for 2^n points you can get the nth derivative in this way. The utility of this data is highly questionable in almost every context, but its available for every time-series. (One application that comes to mind is a kind of checksum, where you recursively derive a time series and stop when local neighbors go to 0, and you're left with a sparse list of high-order numbers that in some sense characterize the series.)

Snap could be the acceleration of "influencers" sharing an article? Basically, how fast it spreads from one to many

DDoS attacks, high inflection from a lot of disparate sources.

>I wish designers of vehicles - particularly cars, trains and busses, would work to minimize jerk, snap and crackle.

They do.

>Turns out if you minimize those, you get a far more comfortable ride. It matters far more than acceleration.

They know that this is the case. And put a lot of effort into making sure your car has the desired feel.

Besides your comfort these considerations are extremely important for the durability analysis for the vehicle.

>Finite element models of the whole system (tyres and suspension components and flexing elements of the vehicle body and road/track) can quickly allow analysis of the jerk, snap and crackle, and allow tuning of damping and drive system control loops to make a far more comfortable ride.

Finite element simulations are undesirable, they are extremely calculation expensive for those kind of large models and somewhat unsuitable. They are used in crash tests.

For the application you described multi body systems are used, where the car is decomposed into its functional components, which can be modeled either as stiff or flexible. With that you have a reasonably accurate model of a car which you can use to test on a virtual test track.

Basically every competent car manufacturer is doing this.

It's designed for in the road/track, not the vehicle. For train tracks in the UK the recommended max jerk is 0.35 mm/s/s/s. The jerk is limited by using 'Euler spiral' sections to join up the straights and the curves. Travelling along an Euler spiral at constant speed means you feel constant jerk laterally, so can be scaled to keep the jerk below any arbitrary value.

Do you have proof for that, or is this like audiophiles asking for gold connectors because "they make the sound better"?

The color printing industry has Snap, Gracol and Swop. Yes, the Gracol logo is a blackbird perched on CMYK dots.

https://duckduckgo.com/?q=snap+gracol+swop

> Lore has it that Snap, Crackle, and Pop are named after the three elves on Kellogg’s Rice Krispies cereal boxes.

Surely this is true. Is there any likely alternative explanation?

this is the same as dimensions

Yes. That is the point of the article.

> Jerk and snap can be observed in various areas of physics and engineering. In physics and engineering jerk and snap should always be considered when vibration occurs and particularly when this excitation induces multi-resonant modes of vibration. They should also be considered at all times when a transition occurs such as: start up and shutdown; take-off and landing; and accelerating and decelerating.

> Acceleration without jerk is just a static load, and therefore constant acceleration alone could never cause vibration. In a machine shop, a toolmaker can damage the mill or the job if the setup starts vibrating. This vibration happens because of jerk and snap.

> In mechanical engineering it is important in automotive design to ensure that the cam-follower does not jump off the camshaft. It is also important in manufacturing processes as rapid changes in acceleration of a cutting tool can lead to premature tool wear and result in an uneven and rough surface finish.

> In civil engineering railway train tracks and roads should be designed for a smooth exit from a straight section into a curve, and it is common to use a transition called a clothoid, which is part of a Cornu spiral (also referred to as an Euler spiral). When a clothoid is implemented the change in acceleration is not abrupt and the levels of jerk and possibly snap are significantly reduced. If the transition between different radii of curvature is sudden, the transition is uncomfortable for passengers and potentially dangerous as it could cause the car or train to be thrown off the road or track. With good physics design engineers are attempting to produce a gradual jerk and constant snap, which gives a smooth increase in radial acceleration, or preferably a zero snap, constant jerk, and linear increase in radial acceleration. Just as road and railway engineers design out jerk and snap using the clothoid transition so, too, do roller coaster designers when they design loops and helices for the roller coasters [11, 12].

Jerk (how fast acceleration changes) is useful. I've found being a passenger in newer electric buses to pose more challenges than ICE buses because EVs can change their acceleration so rapidly. While their maximum acceleration isn't very high, they can go from standstill to accelerating in a split second, toppling anyone standing unless they hold on to something. ICEs need more time to reach maximum acceleration. In other words, EVs jerk more.

Here is a video of a guy that tried to automate a grinding machine by installing an electric motor. Initially the movement was very unsatisfactory, it was not smooth, or very jerky. He then received an upgraded motor that included a “jerk control” feature and the movement of his machine became smooth.

It came as a surprise to me but it seems like jerk is something that can be felt in real life.

https://youtu.be/FPhNc6GwX1o?si=8cf7wU14puB8lsaa

If you're driving along and want to stop for the traffic lights, you start decelerating. The car in front of you slams the brakes and leaves less space then you anticipated. You now need to decelerate faster. That's negative jerk. If you apply the change in deceleration instantaneously, you will also experience jerkiness in your braking (= way to remember what this derivative is called).

Yes. The higher derivates are useful in many cases, both as sought properties and observations. The invariances implied by relativity,(the trivial notion that the universe behaves the same regardless of where you select the center), mean that most laws are defined on the second derivative. Taylor approximations are useful to approximate something locally, but properties of the system over wider regions generally need to account for the higher derivatives. You can see this in e.g. simulating a system over time requires that the derivatives at the borders of the valid taylor approximation region to be included as diracs.

Or in other words, you can approximate exp(x) as a set of first order taylor approximations that each covers a small window to arbitrary precision, but the combination of them is still has well defined higher derivatives that are not 0.