**1. Introduction**

Jerk—the time derivative of acceleration—is an important consideration for many applications in science and engineering. For example, jerk has long been used as a design factor to ensure ride comfort, e.g., in amusement rides [1–7], ships [8–10], lifts/elevators [11] and buses [12], and there are many reasons to believe that the relevance of jerk—and higher derivatives of displacement—will increase. A number of ISO standards also refer to jerk [13–19].

Displacement, velocity and acceleration are well known concepts for everyone who has studied physics at secondary level, whereas jerk—the time derivative of acceleration—and higher derivatives are rarely mentioned, let alone discussed, even in university physics or engineering textbooks. This omission was pointed out three decades ago by Sandin [20], who only found one reference to jerk [21] in a dozen reviewed text books. In addition, in an earlier article for university physics teachers [22], Schot presented the concept and also discussed the radial and tangential components of jerk. However, despite this early effort for jerk to be included in text books, jerk is not yet discussed in physics and engineering courses, except for a few textbooks [23–25]. The lack of detail in textbooks also contributes to some confusion concerning terminology.

Higher derivatives have been discussed by Thompson in a conference presentation [26], where he argued that since immediate acceleration onsets have a detrimental effect on equipment, acceleration should be ramped up by placing a limit on jerk. He then proposed an alternative strategy, claimed to be even better, which is ramping up jerk by placing a limit on its higher derivatives of snap, crackle and pop. However, the details of his studies are beyond the scope of this work.

Jerk and higher derivatives of acceleration are relevant for understanding the impact of motion and vibrations in a wide range of applications, as reviewed in this paper [1–12,27–195]

The considerable interest is also reflected in the large number of downloads (>100,000) of a 2016 paper on jerk and higher derivatives [1].

To support secondary school teachers and university lecturers who would like to introduce the concept of jerk, as well as higher derivatives, in their science or engineering courses, we have conducted a thorough systematic review, from terminology and historical background to standards, measurements and current applications. The articles are divided, based on the applications of jerk, into twenty-one categories. Each article is ranked against where it is published using Scientific Journal Rankings (SJR)—Scimago index, from the quartiles Q1 to Q4, where Q1 is occupied by the top 25% of journals, Q2 by the top 25% to 50% and so on. We used N/A for thesis, reports, and conferences and journals where we were unable find any information with regards to their quality. Other sources are also included in the reference list. The rationale for including these sources is the scarcity of research on jerk. It is then the responsibility of the reader to evaluate the articles against their own objective. A number of papers have been included which use the concept of jerk, even if they don't explicitly mention the term or only refer to it briefly.

#### *Background*

There is a limited number of good textbooks about the history of mechanics. The best are probably Szabo [196] (in German) and Dugas [197] (in French and translated into English [198]). Although these books contain a lot of interesting information, neither covers jerk or higher derivatives.

Newton's second law of motion is commonly written as **F** = *m***a**, i.e., "force equals mass times acceleration", although Newton did not use the term acceleration in his equations but stated that the rate of change in momentum is equal to the applied force. The concept of acceleration was formalized by Pierre Varignon (1654–1722) [199]. Before the second world war German engineers, including Melchior [200] and Schlobach [201], pointed at applications of jerk for handheld machines.

One aspect that is frequently overlooked is terminology. By standardizing terminology, misunderstandings could be avoided, communication between scientists from different disciplines could be enabled, and searches on the global network would be facilitated. The term "jerk" for the first derivative of acceleration seems to be most widely used. However, alternative terms like "acceleration onset rate" are also used, e.g., Whinnery et al. [202], ISO 2041 [13] and a report by NASA [94].

The higher derivatives are less familiar, and different terminologies have been used by different groups. The term "jounce" is sometimes used to describe the fourth derivative of position, and in some Japanese articles it is referred to as "jerk-dot" [148,173,175].

In this work, we will use the more common terms "snap", "crackle" and "pop" for the 4th–6th derivatives (named after pictorial characters on Kellogs' Corn Flakes packages from the thirties). For the 7th–10th derivatives, the terms "lock", "drop", "shock" and "put" have been proposed informally, although we have been unable to find them in the literature.

#### **2. Method**

Articles in the period from 2015 to 2020 with the term "jerk" mentioned in the title were looked up in an electronic database (Scholar.google.com), excluding citations and patents. There were 550 results. The initial 550 articles was refined by excluding 129 based on title, non-English language and duplication. Of the remaining 421 articles, 147 articles were excluded in a more detailed subsequent assessment that included a review of the abstracts and conclusions, and their source, i.e., low-quality journals/conferences. Of the remaining 274 remaining articles, 139 articles that referred to jerk in the context of chaotic systems and nonlinear dynamics were excluded. Terms signaling this context include jerk system, hyper-jerk system, chaotic jerk-system, jerk attractors, jerk circuit, jerk dynamics, jerk map, jerk function, jerk oscillators and jerk equations, and traffic jerk model (n = 17) articles that

used jerk in the context of weightlifting were removed, since this refers to a weightlifting technique rather than the jerk itself.

In addition to the articles emerging from this database search, the authors were aware of 84 articles before 2015, as well as articles where jerk was not mentioned in the title. These articles have been included in the review. The inclusion-exclusion criteria flow chart is given in Figure 1.

**Figure 1.** Systematic review 'inclusion-exclusion criteria' flow chart.

After reviewing the collected literature (n = 202) and those articles that the author were aware of, the applications of jerk were categorised into 20 categories.
