System equivalence

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

Overview

Examples of equivalent systems are first- and second-order (in the independent variable) translational, electrical, torsional, fluidic, and caloric systems.

Equivalent systems are mostly 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 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.

Chart of equivalent variables for the different types of systems

System type Flow variable Effort variable Compliance Inductance Resistance
Mechanical x, dx/dt, d2x/dt2 F = force spring (k) mass (m) damper (c)
Electrical i = current V = voltage capacitance (C) inductance (L) resistance (R)
Thermal qh = heat flow rate T = change in temperature object (C) inductance (L)[1] conduction and convection (R)
Fluid qm = mass flow rate,

qv = volume flow rate

p' = pressure, h = height tank (C) mass (m) valve or orifice (R)
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

Examples

Mechanical systems
Force
Electrical systems
Voltage

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).[2][3][4] 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

1. Bosworth, R.C.L. (31 January 1948). "Thermal Mutual Inductance". Nature. 161: 166–167. doi:10.1038/161166a0.
2. G. S. Ohm (1827). Die galvanische Kette, mathematisch bearbeitet [The galvanic circuit investigated mathematically] (PDF) (in German). Berlin: T. H. Riemann.
3. B. Pourprix, "G.-S. Ohm théoricien de l'action contiguë," Archives internationales d'histoire des sciences 45(134) (1995), 30-56
4. T Archibald, "Tension and potential from Ohm to Kirchhoff," Centaurus 31 (2) (1988), 141-163