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 introduced analogies of this sort in the 19th century. However, as electrical network analysis matured it was found that certain mechanical problems could more easily be solved through an electrical analogy. Theoretical developments in the electrical domain that were particularly useful were the representation of an electrical network as an abstract topological diagram (the circuit diagram) using the lumped element model and the ability of network analysis to synthesise a network to meet a prescribed frequency function.
This approach is especially useful in the design of mechanical filters—these use mechanical devices to implement an electrical function. However, the technique can be used to solve purely mechanical problems, and can also be extended into other, unrelated energy domains. Nowadays, analysis by analogy is a standard design tool wherever more than one energy domain is involved. It has the major advantage that the entire system can be represented in a unified, coherent way. Electrical analogies are particularly used by transducer designers, by their nature they cross energy domains, and in control systems, whose sensors and actuators will typically be domain-crossing transducers. A given system being represented by an electrical analogy may conceivably have no electrical parts at all. For this reason domain-neutral terminology is preferred when developing network diagrams for control systems.
Mechanical–electrical analogies are developed by finding relationships between variables in one domain that have a mathematical form identical to variables in the other domain. There is no one, unique way of doing this; numerous analogies are theoretically possible, but there are two analogies that are widely used: the impedance analogy and the mobility analogy. The impedance analogy makes force and voltage analogous while the mobility analogy makes force and current analogous. By itself, that is not enough to fully define the analogy, a second variable must be chosen. A common choice is to make pairs of power conjugate variables analogous. These are variables which when multiplied together have units of power. In the impedance analogy, for instance, this results in force and velocity being analogous to voltage and current respectively.
Variations of these analogies are used for rotating mechanical systems, such as in electric motors. In the impedance analogy, instead of force, torque is made analogous to voltage. It is perfectly possible that both versions of the analogy are needed in, say, a system that includes rotating and reciprocating parts, in which case a force-torque analogy is required within the mechanical domain and a force-torque-voltage analogy to the electrical domain. Another variation is required for acoustical systems; here pressure and voltage are made analogous (impedance analogy). In the impedance analogy, the ratio of the power conjugate variables is always a quantity analogous to electrical impedance. For instance force/velocity is mechanical impedance. The mobility analogy does not preserve this analogy between impedances across domains, but it does have another advantage over the impedance analogy. In the mobility analogy the topology of networks is preserved, a mechanical network diagram has the same topology as its analogous electrical network diagram.
Applications Mechanical–electrical analogies are used to represent the function of a mechanical system as an equivalent electrical system by drawing analogies between mechanical and electrical parameters. A mechanical system by itself can be so represented, but analogies are of greatest use in electromechanical systems where there is a connection between mechanical and electrical parts. Analogies are especially useful in analysing mechanical filters. These are filters constructed of mechanical parts but designed to work in an electrical circuit through transducers. Circuit theory is well developed in the electrical domain in general and in particular there is a wealth of filter theory available. Mechanical systems can make use of this electrical theory in mechanical designs through a mechanical–electrical analogy.Busch-Vishniac, p. 17
Mechanical–electrical analogies are useful in general where the system includes transducers between different energy domains.An energy domain pertains to a system or subsystem in which the energy and forces are all of a particular kind such as electrical, mechanical, acoustical, thermal, and so on. Another area of application is the mechanical parts of acoustic systems such as the pickup and tonearm of record players. This was of some importance in early phonographs where the audio is transmitted from the pickup needle to the horn through various mechanical components entirely without electrical amplification. Early phonographs suffered badly from unwanted resonances in the mechanical parts. It was found that these could be eliminated by treating the mechanical parts as components of a low-pass filter which has the effect of flattening out the passband.Darlington, p. 7
Electrical analogies of mechanical systems can be used just as a teaching aid, to help understand the behaviour of the mechanical system. In former times, up to about the early 20th century, it was more likely that the reverse analogy would be used; mechanical analogies were formed of the then little understood electrical phenomena.Care, pp. 74-77
Forming an analogy Electrical systems are commonly described by means of a circuit diagram. These are network diagrams that describe the topology of the electrical system using a specialised graph notation. The circuit diagram does not try to represent the true physical dimensions of the electrical components or their actual spatial relationship to each other. This is possible because the electrical components are represented as ideal lumped elements, that is, the element is treated as if it is occupying a single point (lumped at that point). Non-ideal components can be accommodated in this model by using more than one element to represent the component. For instance, a coil intended for use as an inductor has resistance as well as inductance. This can be represented on the circuit diagram as a resistor in series with an inductor.Chan, pp. 2-3 Thus, the first step in forming an analogy of a mechanical system is to describe it as a mechanical network in a similar way, that is, as a topological graph of ideal elements.Busch-Vishniac, p. 17 Alternatively, more abstract representations to the circuit diagram are possible, for instance the bond graph. In an electrical network diagram, limited to linear systems, there are three passive elements: resistance, inductance, and capacitance; and two active elements: the voltage generator, and the current generator.The five-element scheme can be extended to active devices such as transistors by the use of two-port networks containing dependent generators provided the transistor is operating in a substantially linear region. The mechanical analogs of these elements can be used to construct a mechanical network diagram. What the mechanical analogs of these elements are depends on what variables are chosen to be the fundamental variables. There is a wide choice of variables that can be used, but most commonly used are a power conjugate pair of variables (described below) and the pair of Hamiltonian variables derived from these.Busch-Vishniac, pp. 18, 21
There is a limit to the applicability of this lumped element model. The model works well if the components are small enough that the time taken for a wave to cross them is insignificant, or equivalently, if there is no significant phase difference in the wave either side of the component. What amounts to significant depends on how accurate the model is required to be, but a common rule of thumb is to require components to be smaller than one sixteenth of a wavelength.Kleiner, p. 69 Since wavelength decreases with frequency, this puts an upper limit on the frequency that can be covered in this kind of design. This limit is much lower in the mechanical domain than the equivalent limit in the electrical domain. This is because the much higher propagation speeds in the electrical domain lead to longer wavelengths (mechanical vibrations in steel propagate at about 6,000 m/s,Myers, p. 136 electromagnetic waves in common cable types propagate at about White, p. 93). For instance, traditional mechanical filters are only made up to around 600 kHzCarr, pp. 170-172 (although MEMS devices can operate at much higher frequencies due to their very small size). In the electrical domain, on the other hand, the transition from the lumped element model to the distributed element model occurs in the hundreds of megahertz region.Froehlich & Kent, vol. 6, p. 434
In some cases it is possible to continue using a topological network diagram even when components needing a distributed element analysis are present. In the electrical domain, a transmission line, a basic distributed element component, can be included in the model with the introduction of the additional element of electrical length.Joines et al., pp. 69-71 The transmission line is a special case because it is invariant along its length and hence the full geometry need not be modelled.Radmanesh, p. 214 Another way of dealing with distributed elements is to use a finite element analysis whereby the distributed element is approximated by a large number of small lumped elements. Just such an approach was used in one paper to model the cochlea of the human ear.Fukazawa & Tanaka, pp. 191-192 Another condition required of electrical systems for the application of the lumped element model is that no significant fields exist outside the component since these can couple to other unrelated components.Agarwal & Lang, pp. 9-11 However, these effects can often be modelled by introducing some virtual lumped elements called strays or parasitics.Semmlow, p. 405 An analog of this in mechanical systems is vibration in one component being coupled to an unrelated component.Sen, pp. 29, 41
Power conjugate variables The power conjugate variables are a pair of variables whose product is power. In the electrical domain the power conjugate variables chosen are invariably voltage (v) and current (i). Thus, the power conjugate variables in the mechanical domain are analogs. However, this is not enough to make the choice of mechanical fundamental variables unique. The usual choice for a translational mechanical system is force (F) and velocity (u) but it is not the only choice. A different pair may be more appropriate for a system with a different geometry, such as a rotational system.Busch-Vishniac, pp. 18-19
Even after the mechanical fundamental variables have been chosen, there is still not a unique set of analogs. There are two ways that the two pairs of power conjugate variables can be associated with each other in the analogy. For instance the associations F with v and u with i can be made. However, the alternative associations u with v and F with i are also possible. This leads to two classes of analogies, the impedance analogies and the mobility analogies.Busch-Vishniac, p. 19 These analogies are the dual of each other. The same mechanical network has analogs in two different electrical networks. These two electrical networks are the dual circuits of each other.Eargle, p. 5
Hamiltonian variables The Hamiltonian variables, also called the energy variables, are those variables which are conjugate according to Hamilton's equations:
Further, the time derivatives of the Hamiltonian variables are the power conjugate variables.