# Engineering Acoustics/Mechanical Resistance

## Mechanical Resistance

For most systems, a simple oscillator is not a very accurate model.
While a simple oscillator involves a continuous transfer of energy
between kinetic and potential form, with the sum of the two remaining
constant, real systems involve a loss, or dissipation, of some of this
energy, which is never recovered into kinetic nor potential energy. The
mechanisms that cause this dissipation are varied and depend on many
factors. Some of these mechanisms include drag on bodies moving through
the air, thermal losses, and friction, but there are many others. Often,
these mechanisms are either difficult or impossible to model, and most
are non-linear. However, a simple, linear model that attempts to account
for all of these losses in a system has been developed.

## Dashpots

The most common way of representing mechanical resistance in a damped
system is through the use of a dashpot. A dashpot acts like a shock
absorber in a car. It produces resistance to the system\'s motion that
is proportional to the system\'s velocity. The faster the motion of the
system, the more mechanical resistance is produced.

As seen in the graph above, a linear relationship is assumed between the
force of the dashpot and the velocity at which it is moving. The
constant that relates these two quantities is $R_M$, the mechanical
resistance of the dashpot. This relationship, known as the viscous
damping law, can be written as:

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```
$F = R\cdot u$

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```
Also note that the force produced by the dashpot is always in phase with
the velocity.

The power dissipated by the dashpot can be derived by looking at the
work done as the dashpot resists the motion of the system:

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```
$P_D = \frac{1}{2}\Re\left[\hat{F}\cdot\hat{u^*}\right]= \frac{|\hat{F}|^{2}}{2R_{M}}$

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</center>
```
## Modeling the Damped Oscillator

In order to incorporate the mechanical resistance (or damping) into the
forced oscillator model, a dashpot is placed next to the spring. It is
connected to the mass ($M_M$) on one end and attached to the ground on
the other end. A new equation describing the forces must be developed:

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```
$F - S_Mx - R_Mu = M_Ma  \rightarrow  F = S_Mx + R_M\dot{x} + M_M\ddot{x}$

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```
It\'s phasor form is given by the following:

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```
$\hat{F}e^{j\omega t} = \hat{x}e^{j\omega t}\left[S_M + j\omega R_M + \left(-\omega ^2\right)M_M\right]$

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```
## Mechanical Impedance for Damped Oscillator

Previously, the impedance for a simple oscillator was defined as
$\mathbf{\frac{F}{u}}$. Using the above equations, the impedance of a
damped oscillator can be calculated:

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<center>
```
$\hat{Z_M} = \frac{\hat{F}}{\hat{u}} = R_M + j\left(\omega M_M - \frac{S_M}{\omega}\right) = |\hat{Z_M}|e^{j\Phi_Z}$

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```
For very low frequencies, the spring term dominates because of the
$\frac{1}{\omega}$ relationship. Thus, the phase of the impedance
approaches $\frac{-\pi}{2}$ for very low frequencies. This phase causes
the velocity to \"lag\" the force for low frequencies. As the frequency
increases, the phase difference increases toward zero. At resonance, the
imaginary part of the impedance vanishes, and the phase is zero. The
impedance is purely resistive at this point. For very high frequencies,
the mass term dominates. Thus, the phase of the impedance approaches
$\frac{\pi}{2}$ and the velocity \"leads\" the force for high
frequencies.

\
Based on the previous equations for dissipated power, we can see that
the real part of the impedance is indeed $R_M$. The real part of the
impedance can also be defined as the cosine of the phase times its
magnitude. Thus, the following equations for the power can be obtained.

```{=html}
<center>
```
$W_R = \frac{1}{2}\Re\left[\hat{F}\hat{u^{*}}\right] = \frac{1}{2}R_M|\hat{u}|^2 = \frac{1}{2}\frac{|\hat{F}|^2}{|\hat{Z_M}|^2}R_M = \frac{1}{2}\frac{|\hat{F}|^2}{|\hat{Z_M}|}cos(\Phi_Z)$

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</center>
```
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