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: $$ F = R\cdot u $$

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

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:

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

It's phasor form is given by the following:

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

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:

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

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.

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

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