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Absement
Absement
Absement
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Absement
When an object moves, its motion can be described by the integrals of displacement, including absement, absity, abseleration, abserk, etc., as well as the derivatives of displacement, including velocity, acceleration, jerk, jounce, etc.
Common symbols
A
SI unitmetre-second
In SI base unitsm·s
DimensionL T
Part of a series on
Classical mechanics
Second law of motion
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Fundamentals
Formulations
Core topics
Rotation
Scientists
Integrals and derivatives of displacement, including absement, as well as integrals and derivatives of energy, including actergy. (Janzen et al. 2014)

In kinematics, absement (or absition) is a measure of sustained displacement of an object from its initial position, i.e. a measure of how far away and for how long. The word absement is a portmanteau of the words absence and displacement. Similarly, its synonym absition is a portmanteau of the words absence and position.[1][2]

Absement changes as an object remains displaced and stays constant as the object resides at the initial position. It is the first time-integral of the displacement[3][4] (i.e. absement is the area under a displacement vs. time graph), so the displacement is the rate of change (first time-derivative) of the absement. The dimension of absement is length multiplied by time. Its SI unit is meter second (m·s), which corresponds to an object having been displaced by 1 meter for 1 second. This is not to be confused with a meter per second (m/s), a unit of velocity, the time-derivative of position.

For example, opening the gate of a gate valve (of rectangular cross section) by 1 mm for 10 seconds yields the same absement of 10 mm·s as opening it by 5 mm for 2 seconds. The amount of water having flowed through it is linearly proportional to the absement of the gate, so it is also the same in both cases.[5]

Occurrence in nature

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Whenever the rate of change f′ of a quantity f is proportional to the displacement of an object, the quantity f is a linear function of the object's absement. For example, when the fuel flow rate is proportional to the position of the throttle lever, then the total amount of fuel consumed is proportional to the lever's absement.

The first published paper on the topic of absement introduced and motivated it as a way to study flow-based musical instruments, such as the hydraulophone, to model empirical observations of some hydraulophones in which obstruction of a water jet for a longer period of time resulted in a buildup in sound level, as water accumulates in a sounding mechanism (reservoir), up to a certain maximum filling point beyond which the sound level reached a maximum, or fell off (along with a slow decay when a water jet was unblocked).[1] Absement has also been used to model artificial muscles,[6] as well as for real muscle interaction in a physical fitness context.[7] Absement has also been used to model human posture.[8]

As the displacement can be seen as a mechanical analogue of electric charge, the absement can be seen as a mechanical analogue of the time-integrated charge, a quantity useful for modelling some types of memory elements.[4]

Applications

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In addition to modeling fluid flow and for Lagrangian modeling of electric circuits,[4] absement is used in physical fitness and kinesiology to model muscle bandwidth, and as a new form of physical fitness training.[9][10] In this context, it gives rise to a new quantity called actergy, which is to energy as energy is to power. Actergy has the same units as action (joule-seconds) but is the time-integral of total energy (time-integral of the Hamiltonian rather than time-integral of the Lagrangian). Just as displacement and its derivatives form kinematics, so do displacement and its integrals form "integral kinematics".[9]

Relation to PID controllers

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PID controllers are controllers that work on a signal that is proportional to a physical quantity (e.g. displacement, proportional to position) and its integral(s) and derivative(s), thusly defining PID in the context of integrals and derivatives of a position of a control element in the Bratland sense:[11]

depending on the type of sensor inputs, PID controllers can contain gains proportional to position, velocity, acceleration or the time integral of position (absement)…

Example of PID controller:[11]

  • P, position;
  • I, absement;
  • D, velocity.

Strain absement

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Strain absement is the time-integral of strain,[3] and is used extensively in mechanical systems and memsprings. As Pei et al. describe:[3]

... [but] these newer models deserve deeper study, in part because of a little-studied quantity called absement which allows mem-spring models to display hysteretic response in great abundance.

Anglement

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Absement originally arose in situations involving valves and fluid flow, for which the opening of a valve was by a long, T-shaped handle, which actually varied in angle rather than position. The time-integral of angle is called "anglement" and it is approximately equal or proportional to absement for small angles, because the sine of an angle is approximately equal to the angle for small angles.[12]

Building on the concept of anglement, "Einstein's Lane Method"[citation needed] could be used to describe the spatiotemporal trajectory of systems involving rotational motion or angular displacement. Einstein’s Lane Method provides a framework for analyzing the curved paths of rotating objects or systems under the influence of gravitational fields, similar to how general relativity describes the bending of spacetime. This method would use angular displacements (rather than linear positions) as key variables, integrating them over time, much like the time-integral of angle in anglement.

Phase space: Absement and momentement

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In regard to a conjugate variable for absement, the time-integral of momentum, known as momentement, has been proposed.[13][14][15][16]

This is consistent with Jeltsema's 2012 treatment with charge and flux as the base units rather than current and voltage.[17]

References

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Absement

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Absement is a kinematic quantity representing the time of displacement, which measures the extent and duration of an object's deviation from its initial position. In the (SI), absement has dimensions of multiplied by time, typically expressed as meter-seconds (m·s). The term "absement," a portmanteau of "absence" and "displacement," was coined by Steve Mann, Ryan Janzen, and Mark Post in 2006 to describe a foundational measure in a of motion quantities that extends beyond standard derivatives like and . Mathematically, absement A(t)A(t) is defined as A(t)=∫0ts(τ) dτA(t) = \int_0^t s(\tau) \, d\tau, where s(t)s(t) is the displacement as a function of time, making it the zeroth-order integral in the calculus of motion. This positions absement as the counterpart to higher-order derivatives such as acceleration (second derivative of displacement) and jerk (third derivative), forming a complete spectrum from negative to positive orders of differentiation with respect to time. The first time derivative of absement recovers displacement, while further integration yields related quantities like absity (the double integral of displacement, with units m·s²). Absement was initially applied in the design of the , a fluidic where water jets serve as keys, allowing control of sound through absement, displacement, and for expressive polyphonic performance. In engineering mechanics, absement emerges as a in models of memory-dependent materials, such as mechanical analogs of memristors and memcapacitors, where it represents the accumulated deformation over time in systems exhibiting history-dependent behavior like shape-memory alloys or displacement-dependent dampers. These applications highlight absement's utility in capturing sustained effects in dynamic systems, though it remains a niche concept primarily explored in specialized kinematic and contexts, with recent extensions to and acoustics.

Definition and Fundamentals

Etymology and Overview

Absement is a portmanteau of "absence" and "displacement," coined to describe the time-integrated displacement of an object from its initial reference point, emphasizing the duration of separation from that origin. The term was first introduced in 2006 by Steve Mann, Ryan Janzen, and Mark Post in their research on the design of hydraulophones, fluid-based musical instruments that respond to various orders of motion. This naming convention parallels other kinematic terms like velocity and acceleration, extending the framework to negative orders of differentiation in calculus. Physically, absement provides an intuitive measure of not only how far an object moves but how long it stays displaced, capturing the cumulative effect of position over time. For example, opening a 1 mm for 10 seconds yields the same absement as opening it 5 mm for 2 seconds, both representing an equivalent total of sustained displacement. This highlights absement's role in quantifying prolonged deviations, distinguishing it from instantaneous position by incorporating temporal persistence. Initially proposed for niche applications in instrument engineering, absement has gained broader traction in dynamics research since 2006, appearing in contexts such as earthquake early warning systems and atmospheric boundary layer analysis. In everyday scenarios, like a door left slightly open, absement accumulates continuously as the door maintains its offset from the fully closed state, even without further movement.

Mathematical Formulation

Absement is defined as the time integral of the displacement function, quantifying the accumulated displacement of an object from its initial position over a specified interval. Mathematically, for a one-dimensional displacement x(t)x(t) measured from an initial reference position at time t0t_0, the absement A(t)A(t) at time tt (where t≥t0t \geq t_0) is given by A(t)=∫t0tx(τ) dτ.A(t) = \int_{t_0}^{t} x(\tau) \, d\tau. This formulation captures the cumulative "exposure" to displacement, with the lower limit t0t_0 often set to 0 for simplicity when starting from rest or a fixed reference. In multi-dimensional contexts, such as spatial motion in three dimensions, absement extends naturally to a vector form. For a position vector x⃗(t)\vec{x}(t)
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