# Engineering Acoustics/Electro-Mechanical Analogies

## Why Circuit Analogs?

Acoustic devices are often combinations of mechanical and electrical
elements. A common example of this would be a loudspeaker connected to a
power source. It is useful in engineering applications to model the
entire system with one method. This is the reason for using a circuit
analogy in a vibrating mechanical system. The same analytic method can
be applied to Electro-Acoustic
Analogies.

## How Electro-Mechanical Analogies Work

An electrical circuit is described in terms of its potential (voltage)
and flux (current). To construct a circuit analog of a mechanical system
we define flux and potential for the system. This leads to two separate
analog systems. The Impedance Analog denotes the force acting on an
element as the potential and the velocity of the element as the flux.
The Mobility Analog equates flux with the force and velocity with
potential.

                         Mechanical   Electrical Equivalent
  ---------------------- ------------ -----------------------
  **Impedance Analog**                
  Potential:             Force        Voltage
  Flux:                  Velocity     Current
  **Mobility Analog**                 
  Potential:             Velocity     Voltage
  Flux:                  Force        Current
                                      

For many, the mobility analog is considered easier for a mechanical
system. It is more intuitive for force to flow as a current and for
objects oscillating the same frequency to be wired in parallel. However,
either method will yield equivalent results and can also be translated
using the dual (dot) method.

## The Basic Elements of an Oscillating Mechanical System

**The Mechanical Spring:**

:   ![](resistor11.jpg "fig:resistor11.jpg")

The ideal spring is considered to be operating within its elastic limit,
so the behavior can be modeled with Hooke\'s
Law.
It is also assumed to be massless and have no damping effects.

$$F=-cx, \$$

**The Mechanical Mass**

In a vibrating system, a mass element opposes acceleration. From
Newton\'s Second Law:

$$F=mx^{\prime\prime}=ma=m\frac{du}{dt}$$

$$F=K\int\,u dt$$

**The Mechanical Resistance**

:   ![](Dashpot.png "fig:Dashpot.png")

The dashpot is an ideal viscous damper which opposes velocity.

$$F=R u\displaystyle$$

**Ideal Generators**

The two ideal generators which can drive any system are an ideal
velocity and ideal force generator. The ideal velocity generator can be
denoted by a drawing of a crank or simply by declaring $u(t)=f(t)$, and
the ideal force generator can be drawn with an arrow or by declaring
$F(t)=f(t)$

**Simple Damped Mechanical Oscillators**

:   ![](Forced_Oscillator.PNG‎ "fig:Forced_Oscillator.PNG‎")

In the following sections we will consider this simple mechanical system
as a mobility and impedance analog. It can be driven either by an ideal
force or an ideal velocity generator, and we will consider simple
harmonic motion. The m in the subscript denotes a mechanical system,
which is currently redundant, but can be useful when combining
mechanical and acoustic systems.

## The Impedance Analog

**The Mechanical Spring**

In a spring, force is related to the displacement from equilibrium. By
Hooke\'s Law,

$$F(t)=c_m \Delta x = c_m \int_{0}^{t} u( \tau )d \tau$$

The equivalent behaviour in a circuit is a capacitor:

$$V(t)=\frac{1}{C}\int_{0}^{t} \,i(\tau) d\tau$$

**The Mechanical Mass**

The force on a mass is related to the acceleration (change in velocity).
The behaviour, by Newton\'s Second Law, is:

$$F(t)=m_ma=m_m\frac{d}{dt}u(t)$$

The equivalent behaviour in a circuit is an inductor:

$$V(t)=L\frac{d}{dt}i(t)$$

**The Mechanical Resistance**

For a viscous damper, the force is directly related to the velocity

$$F=R_m u\displaystyle$$

The equivalent is a simple resistor of value $R_m\displaystyle$

$$V=R i\displaystyle$$

**Example:**

Thus the simple mechanical oscillator in the previous section becomes a
series RCL Circuit:

![](RLC_series_circuit_v1.svg "RLC_series_circuit_v1.svg")

The current through all three elements is equal (they are at the same
velocity) and that the sum of the potential drops across each element
will equal the potential at the generator (the driving force). The ideal
voltage generator depicted here would be equivalent to an ideal force
generator.

**IMPORTANT NOTE**: The velocity measured for the spring and dashpot is
the relative velocity ( velocity of one end minus the velocity of the
other end). The velocity of the mass, however, is the absolute velocity.

**Impedances:**

  Element               Impedance
  --------- ----------- -------------------------------------------------
  Spring    Capacitor   $Z_c = \frac{V_c}{I_c} = \frac{c_m}{j \omega }$
  Mass      Inductor    $Z_m = \frac{V_m}{I_m} = j \omega m_m$
  Dashpot   Resistor    $Z_d = \frac{V_m}{I_m} = R_m$

## The Mobility Analog

Like the Impedance Analog above, the equivalent elements can be found by
comparing their fundamental equations with the equations of circuit
elements. However, since circuit equations usually define voltage in
terms of current, in this case the analogy would be an expression of
velocity in terms of force, which is the opposite of convention.
However, this can be solved with simple algebraic manipulation.

**The Mechanical Spring**

$$F(t)= c_m \int u(t)d t$$

The equivalent behavior for this circuit is the behavior of an inductor.

$$\int V dt=\int L \frac{d}{dt} i(t) dt$$

$$i=\frac{1}{L}\int\,V dt$$

**The Mechanical Mass**

$$F=m_ma=m_m\frac{d}{dt}u(t)$$

Similar to the spring element, if we take the general equation for a
capacitor and differentiate,

$$\frac{d}{dt}V(t)=\frac{d}{dt}\frac{1}{C}\int \,i(t) dt$$

$$i(t)=C\frac{d}{dt}V(t)$$

**The Mechanical Resistance**

Since the relation between force and velocity is proportionate, the only
difference is that the mechanical resistance becomes inverted:

$$F=\frac{1}{r_m} u=R_m u$$

$$i=\frac{1}{R}V$$

**Example:**

The simple mechanical oscillator drawn above would become a parallel RLC
Circuit. The potential across each element is the same because they are
each operating at the same velocity. This is often the more intuitive of
the two analogy methods to use, because you can visualize force
\"flowing\" like a flux through your system. The ideal voltage generator
in this drawing would correspond to an ideal velocity generator.

![](RLC_parallel_circuit.png‎ "RLC_parallel_circuit.png‎")

**IMPORTANT NOTE:** Since the measure of the velocity of a mass is
absolute, a capacitor in this analogy must always have one terminal
grounded. A capacitor with both terminals at a potential other than
ground may be realized physically as an inverter, which completes all
elements of this analogy.

**Impedances:**

  Element               Impedance
  --------- ----------- --------------------------------------------------
  Spring    Inductor    $Z_c = \frac{V_m}{I_m} = \frac{j \omega}{c_m}$
  Mass      Capacitor   $Z_m = \frac{V_c}{I_c} = \frac{1}{j \omega m_m}$
  Dashpot   Resistor    $Z_d = \frac{V_m}{I_m} = r_m = \frac{1}{R_m}$

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