Mobility analogy

The mobility analogy, also called admittance analogy or Firestone 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 in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole. The mathematical behaviour of the simulated electrical system is identical to the mathematical behaviour of the represented mechanical system. Each element in the electrical domain has a corresponding element in the mechanical domain with an analogous constitutive equation. All laws of circuit analysis, such as Kirchhoff's laws, that apply in the electrical domain also apply to the mechanical mobility analogy. The mobility analogy is one of the two main mechanical–electrical analogies used for representing mechanical systems in the electrical domain, the other being the impedance analogy. The roles of voltage and current are reversed in these two methods, and the electrical representations produced are the dual circuits of each other. The mobility analogy preserves the topology of the mechanical system when transferred to the electrical domain whereas the impedance analogy does not. On the other hand, the impedance analogy preserves the analogy between electrical impedance and mechanical impedance whereas the mobility analogy does not.

Applications The mobility analogy is widely used to model the behaviour of mechanical filters. These are filters that are intended for use in an electronic circuit, but work entirely by mechanical vibrational waves. Transducers are provided at the input and output of the filter to convert between the electrical and mechanical domains. Another very common use is in the field of audio equipment, such as loudspeakers. Loudspeakers consist of a transducer and mechanical moving parts. Acoustic waves themselves are waves of mechanical motion: of air molecules or some other fluid medium.

Elements Before an electrical analogy can be developed for a mechanical system, it must first be described as an abstract mechanical network. The mechanical system is broken down into a number of ideal elements each of which can then be paired with an electrical analogue. The symbols used for these mechanical elements on network diagrams are shown in the following sections on each individual element. The mechanical analogies of lumped electrical elements are also lumped elements, that is, it is assumed that the mechanical component possessing the element is small enough that the time taken by mechanical waves to propagate from one end of the component to the other can be neglected. Analogies can also be developed for distributed elements such as transmission lines but the greatest benefits are with lumped-element circuits. Mechanical analogies are required for the three passive electrical elements, namely, resistance, inductance and capacitance. What these analogies are is determined by what mechanical property is chosen to represent voltage, and what property is chosen to represent current. In the mobility analogy the analogue of voltage is velocity and the analogue of current is force. Mechanical impedance is defined as the ratio of force to velocity, thus it is not analogous to electrical impedance. Rather, it is the analogue of electrical admittance, the inverse of impedance. Mechanical admittance is more commonly called mobility, hence the name of the analogy.

Resistance

The mechanical analogy of electrical resistance is the loss of energy of a moving system through such processes as friction. A mechanical component analogous to a resistor is a shock absorber and the property analogous to inverse resistance (conductance) is damping (inverse, because electrical impedance is the analogy of the inverse of mechanical impedance). A resistor is governed by the constitutive equation of Ohm's law,

i = v G {\displaystyle i=vG}

The analogous equation in the mechanical domain is,

F = u R m {\displaystyle F=uR_{\mathrm {m} }}

where, G = 1/R is conductance R is resistance v is voltage i is current Rm is mechanical resistance, or damping F is force u is velocity induced by the force. Electrical conductance represents the real part of electrical admittance. Likewise, mechanical resistance is the real part of mechanical impedance.

Inductance

The mechanical analogy of inductance in the mobility analogy is compliance. It is more common in mechanics to discuss stiffness, the inverse of compliance. A mechanical component analogous to an inductor is a spring. An inductor is governed by the constitutive equation,

v = L d i d t {\displaystyle v=L{\frac {di}{dt}}}

The analogous equation in the mechanical domain is a form of Hooke's law,

u = C m d F d t {\displaystyle u=C_{\mathrm {m} }{\frac {dF}{dt}}}

where, L is inductance t is time Cm = 1/S is mechanical compliance S is stiffness The impedance of an inductor is purely imaginary and is given by,

Z = j ω L {\displaystyle Z=j\omega L}

The analogous mechanical admittance is given by,

Y m = j ω C m {\displaystyle Y_{\mathrm {m} }=j\omega C_{\mathrm {m} }}

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