Analogical models are a method of representing a phenomenon of the world, often called the "target system" by another, more understandable or analysable system. They are also called dynamical analogies.
Two open systems have analog representations (see illustration) if they are black box isomorphic systems.
Explanation A simple type of analogy is one that is based on shared properties;Stanford Encyclopedia of Philosophy.Gentner, Dedre (1989), "Mechanisms of Analogical Learning", pp.199-241, in Stella Vosniadou and Andrew Ortony (eds.), Similarity and Analogical Reasoning, Cambridge: Cambridge University Press. and analogizing is the process of representing information about a particular subject (the analogue or source system) by another particular subject (the target system),"There is general agreement that analogical reasoning involves the transfer of relational information from a domain that already exists in memory (…source…) to the domain to be explained (…target…). Similarity is implicated in this process because a successful, useful analogy depends upon there being some sort of [perceived] similarity between the source domain and the target domain and because the perception of similarity is likely to play a major role in some of the key processes associated with analogical reasoning" (Vosniadou and Ortony, 1989, pp.6-7). in order "to illustrate some particular aspect (or clarify selected attributes) of the primary domain".Yeates (2004), p.71.
Analogical models, also called "analog" or "analogue" models, seek the analogous systems that share properties with the target system as a means of representing the world. It is often practicable to construct source systems that are smaller and/or faster than the target system so that one can deduce a priori knowledge of target system behaviour. Analog devices are therefore those in which may differ in substance or structure but share properties of dynamic behaviour (Truit and Rogers, p. 1-3).
For example, in analog electronic circuits, one can use voltage to represent an arithmetic quantity; operational amplifiers might then represent the arithmetic operations (addition, subtraction, multiplication, and division). Through the process of calibration these smaller/bigger, slower/faster systems are scaled up or down so that they match the functioning of the target system, and are therefore called analogs of the target system. Once the calibration has taken place, modellers speak of a one-to-one correspondence in behaviour between the primary system and its analog. Thus the behaviour of two systems can be determined by experimenting with one.
Creating an analogical model
Many different instruments and systems can be used to create an analogical model."An analogue model describes specific relationships between selected components of the "original" by creating analogies with the relationships that are displayed by components in some other "secondary domain" of a totally different medium." (Yeates, 2004, p.72). "Many important discoveries have been made when scientists commenced their work as if their theoretically postulated models of atoms, viruses, vitamins, hormones, and genes had actual, real world substantial existence. They proceeded as though each imaginary concept actually existed in precisely the form their theoretical speculation outlined; and, discarding any pretence of analogy, they proceeded with the view that the substantial, real world was exactly as they had theoretically described it. ... Consider the analogue model advanced to assist understanding of the behaviour of gases which suggests possible relationships between some theoretical activities of gas particles and some observable activities of billiard-balls. Achinstein (1964, p.332) reminds us that, despite thinking about gases in this useful way, "the physicist obviously supposes that molecules, not billiard balls, comprise gases" Yeates (2004, pp.71, 73)
A mechanical device can be used to represent mathematical calculations. For instance, the Phillips Hydraulic Computer MONIAC used the flow of water to model economic systems (the target system); electronic circuits can be used to represent both physiological and ecological systems. When a model is run on either an analog or digital computer this is known as the process of simulation.
Mechanical analogies
Any number of systems could be used for mapping electrical phenomena to mechanical phenomena, but two principle systems are commonly used: the impedance analogy and the mobility analogy. The impedance analogy maps force to voltage whereas the mobility analogy maps force to current.
The impedance analogy preserves the analogy between electrical impedance and mechanical impedance but does not preserve the network topology. The mobility analogy preserves the network topology but does not preserve the analogy between impedances. Both preserve the correct energy and power relationships by making power conjugate pairs of variables analogous.
Hydraulic analogy In a hydraulic analogy, a water integrator might perform the mathematical operation of integration.
Physiological analogies Francis Crick used the study of the visual system as a proxy for the study of awareness.
Formal analogies
"The same equations have the same solutions." -- Richard Feynman For example, the inverse-square laws of gravitation and electromagnetism can be described by analogous equations on a geometrical basis, almost without regard to the physical details about masses and charges. In population ecology, differential equations arise that are the same as those found in mechanics, albeit with different interpretations.Ginzburg and Colyvan 2004; Colyvan and Ginzburg 2010 Recursion requires a similarity within a situation; for example, Archimedes used the myriad to count the number of grains of sand on a beach by using the concept of myriad myriads.
Dynamical analogies Dynamical analogies establish analogies between systems in different energy domains by means of comparison of the system dynamic equations. There are many ways such analogies can be built, but one of the most useful methods is to form analogies between pairs of power conjugate variables. That is, a pair of variables whose product is power. Doing so preserves the correct energy flow between domains, a useful feature when modelling a system as an integrated whole. Examples of systems that require unified modelling are mechatronics and audio electronics.Busch-Vishniac, p. 18
The earliest such analogy is due to James Clerk Maxwell who, in 1873, associated mechanical force with electrical voltage. This analogy became so widespread that sources of voltage are still today referred to as electromotive force. The power conjugate of voltage is electric current which, in the Maxwell analogy, maps to mechanical velocity. Electrical impedance is the ratio of voltage and current, so by analogy, mechanical impedance is the ratio of force and velocity. The concept of impedance can be extended to other domains, for instance in acoustics and fluid flow it is the ratio of pressure to rate of flow. In general, impedance is the ratio of an effort variable and the flow variable that results. For this reason, the Maxwell analogy is often referred to as the impedance analogy, although the concept of impedance was not conceived until 1886 by Oliver Heaviside, some time after Maxwell's death.
Specifying power conjugate variables still does not result in a unique analogy, there are multiple ways the conjugates and analogies can be specified. A new analogy was proposed by Floyd A. Firestone in 1933 now known as the mobility analogy. In this analogy electrical impedance is made analogous to mechanical mobility (the inverse of mechanical impedance). Firestone's idea was to make analogous variables that are measured across an element, and make analogous variables that flow through an element. For instance, the across variable voltage is the analogy of velocity, and the through variable current is the analogy of force. Firestone's analogy has the advantage of preserving the topology of element connections when converting between domains. A modified form of the through and across analogy was proposed in 1955 by Horace M. Trent and is the modern understanding of through and across.
+Comparison of various power conjugate analogies for electrical, mechanical, rotational, and fluid flow domainsBusch-Vishniac, pp. 18-20Impedance analogy (Maxwell)Mobility analogy (Firestone)Through and across analogy (Trent)Effort or across power conjugatesV, F, T, pV, u, ω, QV, u, ω, pFlow or through power conjugatesI, u, ω, QI, F, T, pI, F, T, Q where V is voltage F is force T is torque p is pressure I is electric current u is velocity ω is angular velocity Q is volumetric flow rate
Table of equivalents +Table of equivalents under the through and across systemOlson, pp. 27-29Through variableAcross variableEnergy storage 1Energy storage 2Energy dissipationElectricalCurrent (I)Voltage (V)Capacitor (C)Inductor (L)Resistor (R)Mechanical linearForce (F)Velocity (u)Spring (K)Mass (M)Damper (B)Mechanical rotationalTorque (T)Angular velocity (ω)Torsion spring (κ)Moment of inertia (I)Rotary damperHydraulicVolume flowPressure (p)TankMassValve
Hamiltonian variables The Hamiltonian variables, also called the energy variables, are those variables which when time-differentiated are equal to the power conjugate variables. The Hamiltonian variables are so called because they are the variables which usually appear in Hamiltonian mechanics. The Hamiltonian variables in the electrical domain are charge () and flux linkage () because (Faraday's law of induction) and
In the translational mechanical domain, the Hamiltonian variables are distance displacement () and momentum () because (Newton's second law of motion) and
There is a corresponding relationship for other analogies and sets of variables.Busch-Vishniac, p. 21 The Hamiltonian variables are also called the energy variables. The integrand of a power conjugate variable with respect to a Hamiltonian variable is a measure of energy. For instance, and are both expressions of energy.Borutzky, pp. 27-28
Practical uses Maxwell's analogy was initially used merely to help explain electrical phenomena in more familiar mechanical terms. The work of Firestone, Trent and others moved the field well beyond this, looking to represent systems of multiple energy domains as a single system. In particular, designers started converting the mechanical parts of an electromechanical system to the electrical domain so that the whole system could be analyzed as an electrical circuit. Vannevar Bush was a pioneer of this kind of modelling in his development of analogue computers, and a coherent presentation of this method was presented in a 1925 paper by Clifford A. Nickle.Care, p. 76
From the 1950s onward, manufacturers of mechanical filters, notably Collins Radio, widely used these analogies in order to take the well -developed theory of filter design in electrical engineering and apply it to mechanical systems. The quality of filters required for radio applications could not be achieved with electrical components. Much better quality resonators (higher Q factor) could be made with mechanical parts but there was no equivalent filter theory in mechanical engineering. It was also necessary to have the mechanical parts, the transducers, and the electrical components of the circuit analyzed as a complete system in order to predict the overall response of the filter.
Harry F. Olson helped popularise the use of dynamical analogies in the audio electronics field with his book dynamical analogies first published in 1943.Libbey, p. 13
Non-power-conjugate analogies