MotionIntegralsDerivativesAbsementActergy

In kinematics, absement (or absition) is a measure of sustained

displacement Displacement may refer to: Physical sciences Mathematics and Physics * Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...

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 A portmanteau word, or portmanteau (, ) is a blend of words Absement changes as an object remains displaced and stays constant as the object resides at the initial position. It is the first time-

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...

of the displacement (i.e. absement is the area under a displacement vs. time graph), so the displacement is the rate of change (first time-

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

) of the absement. The

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

of absement is length multiplied by

time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...

. Its

SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...

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 The metre per second is the unit of both speed (a scalar quantity) and velocity (a vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the speed of a body covering a distance of one metre in ...

(m/s), a unit of

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

, 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.

Occurrence in nature

Whenever the rate of change ′ of a quantity is proportional to the displacement of an object, the quantity 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 A hydraulophone is a tonal acoustic musical instrument played by direct physical contact with water (sometimes other fluids) where sound is generated or affected hydraulically."Fluid Melodies: The hydraulophones of Professor Steve Mann" In Wate ...

, 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). Absement has also been used to model artificial muscles, as well as for real muscle interaction in a physical fitness context. Absement has also been used to model human posture. As the displacement can be seen as a mechanical analogue of

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...

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

Applications

In addition to modeling fluid flow and for Lagrangian modeling of electric circuits, absement is used in physical fitness and kinesiology to model muscle bandwidth, and as a new form of physical fitness training. 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:

Relation to PID controllers

PID controller A proportional–integral–derivative controller (PID controller or three-term controller) is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuou ...

s 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 Example of PID controller (Bratland 2014): * P, Position; * I, Absement; * D, Velocity.

Higher integrals

Just as displacement and its derivatives form kinematics, so do displacement and its integrals form "integral kinematics" (Janzen ''et al.'' 2014), giving rise to the ordered list of ''n''-th derivatives of displacement:

Strain absement

Strain absement is the time-integral of

strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...

, and is used extensively in mechanical systems and memsprings: : a quantity called absement which allows mem-spring models to display hysteretic response in great abundance.

Anglement

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.

Phase space: Absement and momentement

In regards to a conjugate variable for absement, the time-integral of momentum, known as momentement, has been proposed. This is consistent with Jeltsema's 2012 treatment with charge and flux as the base units rather than current and voltage.Jeltsema, D. (2012). Memory elements: A paradigm shift in Lagrangian modeling of electrical circuits. IFAC Proceedings Volumes, 45(2), 445-450.

References

External links

* {{Classical mechanics derived SI units Motion (physics) Vector physical quantities