In the systems sciences system equivalence is the behavior of a parameter or component of a

system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...

in a way similar to a parameter or component of a different system. Similarity means that mathematically the parameters and components will be indistinguishable from each other. Equivalence can be very useful in understanding how

complex system A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication ...

s work.

Overview

Examples of equivalent systems are first- and second-

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

(in the

independent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...

)

translational Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

, electrical, torsional, fluidic, and

caloric Caloric is a brand of kitchen appliances, which dates back to 1903. History Caloric Corporation began as the Klein Stove Company in Philadelphia in 1890. The Caloric brand was introduced in 1903. It was reorganized in 1946 as the Caloric Stove C ...

systems. Equivalent systems can be used to change large and expensive mechanical, thermal, and fluid systems into a simple, cheaper electrical system. Then the electrical system can be analyzed to validate that the

system dynamics System dynamics (SD) is an approach to understanding the nonlinear behaviour of complex systems over time using stocks, flows, internal feedback loops, table functions and time delays. Overview System dynamics is a methodology and mathematical ...

will work as designed. This is a preliminary inexpensive way for engineers to test that their complex system performs the way they are expecting. This testing is necessary when designing new complex systems that have many components. Businesses do not want to spend millions of dollars on a system that does not perform the way that they were expecting. Using the equivalent system technique, engineers can verify and prove to the business that the system will work. This lowers the risk factor that the business is taking on the project. The following is a chart of equivalent variables for the different types of systems : : Flow variable: moves through the system : Effort variable: puts the system into action : Compliance: stores energy as potential : Inductance: stores energy as kinetic : Resistance: dissipates or uses energy The equivalents shown in the chart are not the only way to form mathematical analogies. In fact there are any number of ways to do this. A common requirement for analysis is that the analogy correctly models energy storage and flow across energy domains. To do this, the equivalences must be compatible. A pair of variables whose product is power (or energy) in one domain must be equivalent to a pair of variables in the other domain whose product is also power (or energy). These are called power conjugate variables. The thermal variables shown in the chart are not power conjugates and thus do not meet this criterion. See

mechanical–electrical analogies Mechanical–electrical analogies are the representation of mechanical systems as electrical networks. At first, such analogies were used in reverse to help explain electrical phenomena in familiar mechanical terms. James Clerk Maxwell introduc ...

for more detailed information on this. Even specifying power conjugate variables does not result in a unique analogy and there are at least three analogies of this sort in use. At least one more criterion is needed to uniquely specify the analogy, such as the requirement that impedance is equivalent in all domains as is done in the

impedance analogy The impedance analogy is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially ...

Examples

; Mechanical systems :Force

F = -kx = c\frac = m\frac

; Electrical systems :Voltage

V = \frac = R\frac = L\frac

All the fundamental variables of these systems have the same functional form.

Discussion

The system equivalence method may be used to describe systems of two types: "vibrational" systems (which are thus described - approximately - by harmonic oscillation) and "translational" systems (which deal with "flows"). These are not mutually exclusive; a system may have features of both. Similarities also exist; the two systems can often be analysed by the methods of Euler, Lagrange and Hamilton, so that in both cases the energy is quadratic in the relevant degree(s) of freedom, provided they are linear. Vibrational systems are often described by some sort of wave (partial differential) equation, or oscillator (ordinary differential) equation. Furthermore, these sorts of systems follow the capacitor or spring analogy, in the sense that the dominant degree of freedom in the energy is the generalized position. In more physical language, these systems are predominantly characterised by their potential energy. This often works for solids, or (linearized) undulatory systems near equilibrium. On the other hand, flow systems may be easier described by the hydraulic analogy or the diffusion equation. For example, Ohm's law was said to be inspired by Fourier's law (as well as the work of C.-L. Navier).T Archibald, "Tension and potential from Ohm to Kirchhoff," ''Centaurus'' 31 (2) (1988), 141-163 Other laws include Fick's laws of diffusion and generalized transport problems. The most important idea is the flux, or rate of transfer of some important physical quantity considered (like electric or magnetic fluxes). In these sorts of systems, the energy is dominated by the derivative of the generalized position (generalized velocity). In physics parlance, these systems tend to be kinetic energy-dominated. Field theories, in particular electromagnetism, draw heavily from the hydraulic analogy.

References

{{Reflist

External links

A simulation using a hydraulic analog as a mental model for the dynamics of a first order system

System Analogies
Engs 22 — Systems Course, Dartmouth College. Applied mathematics Dynamical systems Systems engineering Systems theory Equivalence (mathematics)

Overview

Examples

Discussion

See also

References

Further reading

External links