Absement

Absement
When a particle moves, its motion can be described by the integrals of displacement, including absement, absity, abseleration, etc., as well as the derivatives of displacement, including velocity, acceleration, jerk, jounce, etc.
Common symbols	A
SI unit	m·s, m s

Classical mechanics
$\vec{F} = m\vec{a}$ Second law of motion
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Integrals and derivatives of displacement, including absement, as well as integrals and derivatives of energy, including actergy. (Janzen etal 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. 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^[1]^[2] (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 flown through it is linearly proportional to the absement of the gate, so it is also the same in both cases.^[3]

The word "absement" is a portmanteau of the words "absence" and "displacement". Similarly, "absition" is a portmanteau of the words "absence" and "position".^[4]^[5]

Occurrence in nature

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.

Flow-based musical instruments, such as the hydraulophone, exhibit this phenomenon inherently: whereas a piano produces note strength in proportion to the velocity at which the keys are struck, and the organ (a true tracker organ) produces note strength in proportion to how far down a key is pressed (displacement), the hydraulophone produces note strength in proportion to the time-integral of the distance down the water "key" is pressed. Thus pressing on the key (water jet) for a longer period of time will result in a buildup in sound level, as water accumulates in the sounding mechanism (reservoir), up to a certain maximum filling point beyond which the sound level reaches a maximum, or falls off (along with a slow decay when a water jet is unblocked).

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.^[2]

Applications

In addition to modeling fluid flow and for lagrangian modeling of electric circuits (Jeltsema 2012), absement is used in physical fitness and kinesiology to model muscle bandwidth, and as a new form of physical fitness training.^[6]^[7] 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).

Fluid flow in a throttle:

"A vehicle's distance travelled results from its throttle's absement. The further the throttle has been opened, and the longer it's been open, the more the vehicle's travelled." ^[8]

Relation to PID controllers

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^[9]

Quoting Bratland etal.:

"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 (Bratland 2014):

P = Position;
I = Absement;
D = Velocity.

Higher integrals

Just as displacement and its derivatives form kinematics, also displacement and its integrals form "Integral Kinematics" (Janzen etal 2014), giving rise to the ordered list of nth derivatives of displacement:

+4 Jounce
+3 Jerk
+2 Acceleration
+1 Velocity
=0 Displacement
-1 Absement
-2 Absity
-3 Abseleration
-4 Abserk

Absement and Absementom

Recent work in mechanics and memristors and memcapacitors further builds on the concept of absement, and assigns it the letter "a",^[10] and makes extensive use of plots such as the graph of absement as a function of displacement:

...amplitude of the sinusoidal displacement with period
T = 2π/ω,
and a0 = A/ω
is the value about which the analytic absement a(t) oscillates. [Pei etal 2015].

See Table 4, "Analytic Displacement and Absement" versus "Piecewise Continuous Displacement and Absement".

Strain absement

Strain absement is the time-integral of strain, and is used extensively in mechanical systems [Pei etal 2015] and memsprings:

quantity called absement which allows mem-spring models to display hysteretic response in great abundance.

References

↑ Jin-Song Pei, Joseph P. Wright, Michael D. Todd, Sami F. Masri, François Gay-Balmaz (2015). Understanding memristors and memcapacitors in engineering mechanics applications. Springer. Abstract: "for example, a new concept and state variable called “absement,” the time integral of deformation, emerge."
1 2 Dimitri Jeltsema (2012). Memory Elements: A Paradigm Shift in Lagrangian Modeling of Electrical Circuits. arXiv:1201.1032. Abstract: "Although time-integrated charge is a somewhat unusual quantity in circuit theory, it may be considered as the electrical analogue of a mechanical quantity called absement."
↑ Maya Burhanpurkar. Absement: Direct Evidence of the Time-Integral of Distance. Canada-Wide Science Fair 2014.
↑ "Absement, displacement, and velocity-sensitive music keyboard in which each key is a water jet", by Mann, Janzen, and Post, In Proceedings of the 14th annual ACM international conference on Multimedia, pp. 519-528. ACM, 2006.
↑ Amarashiki (2012-11-10). "LOG#053. Derivatives of position". The Spectrum Of Riemannium. Retrieved 2016-03-08.
↑ "Actergy as a Reflex Performance Metric: Integral-Kinematics Applications", Janzen etal., in Proceedings of the IEEE GEM 2014, pp. 311-2. DOI: 10.1109/GEM.2014.7048123
↑ "Integral Kinematics (Time-Integrals of Distance, Energy, etc.) and Integral Kinesiology", by Mann etal, in Proceedings of the IEEE GEM 2014, pp. 270-2.
↑ "What does the area under a distance-time graph correspond to?" Quara, 23 Jun 2014
↑ Bratland, Magne, Bjørn Haugen, and Terje Rølvåg. "Modal analysis of active flexible multibody systems containing PID controllers with non-collocated sensors and actuators." Finite Elements in Analysis and Design 91 (2014): 16-29.
↑ Pei, Jin-Song, Joseph P. Wright, Michael D. Todd, Sami F. Masri, and François Gay-Balmaz. "Understanding memristors and memcapacitors in engineering mechanics applications." Nonlinear Dynamics 80, no. 1-2 (2015): 457-489.

Classical mechanics SI units

Dimensions	—	L	L²	Dimensions	—	—	—
Linear/translational quantities				Angular/rotational quantities
T	time: $t$ s	absement: $A$ m s		T	time: $t$ s
—		distance: $d$ , position: $r$ , $s$ , $x$ , displacement m	area: $A$ m²	—		angle: $θ$ , angular displacement: $θ$ rad	solid angle: $Ω$ rad², sr
T⁻¹	frequency: $f$ s⁻¹, Hz	speed: $v$ , velocity: $v$ m s⁻¹	kinematic viscosity: $ν$ , specific angular momentum: $h$ m² s⁻¹	T⁻¹	frequency: $f$ s⁻¹, Hz	angular speed: $ω$ , angular velocity: $ω$ rad s⁻¹
T⁻²		acceleration: $a$ m s⁻²		T⁻²		angular acceleration: $α$ rad s⁻²
T⁻³		jerk: $j$ m s⁻³		T⁻³		angular jerk: $ζ$ rad s⁻³

M	mass: $m$ kg			ML²	moment of inertia: $I$ kg m²
MT⁻¹		momentum: $p$ , impulse: $J$ kg m s⁻¹, N s	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s	ML²T⁻¹		angular momentum: $L$ , angular impulse: $Δ L$ kg m² s⁻¹	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s
MT⁻²		force: $F$ , weight: $F g$ kg m s⁻², N	energy: $E$ , work: $W$ kg m² s⁻², J	ML²T⁻²		torque: $τ$ , moment: $M$ kg m² s⁻², N m	energy: $E$ , work: $W$ kg m² s⁻², J
MT⁻³		yank: $Y$ kg m s⁻³, N s⁻¹	power: $P$ kg m² s⁻³, W	ML²T⁻³		rotatum: $P$ kg m² s⁻³, N m s⁻¹	power: $P$ kg m²s⁻³, W

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