Mechanical Impedance Analysis

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Mechanical Impedance Analysis

Impedance analysis is commonly used to analyze electrical circuits [109]. By means of equivalent circuits, we can use the same analysis methods for mechanical systems.

For example, referring to Fig.L.9, the Laplace transform of the force on the spring is given by the so-called voltage divider relation:^L.2

$\displaystyle F_k(s) = F_{\mbox{ext}}(s) \frac{R_k(s)}{R_m(s)+R_k(s)} = F_{\mbox{ext}}(s) \frac{\frac{k}{s}}{ms+\frac{k}{s}}$

Similarly, the Laplace transform of the force on the mass

is given by

$\displaystyle F_m(s) = F_{\mbox{ext}}(s) \frac{R_m(s)}{R_m(s)+R_k(s)} = F_{\mbox{ext}}(s) \frac{ms}{ms+\frac{k}{s}}. \protect$

(L.1)

As a simple application, let's find the motion of the mass , after time zero, given that the input force is an impulse at time 0:

$\displaystyle f_{\mbox{ext}}(t)=\delta(t) \;\leftrightarrow\; F_{\mbox{ext}}(s)=1$

Then, by the ``voltage divider'' relation Eq. $\,$ (L.1), the Laplace transform of the mass force

after time 0 is given by

$\displaystyle F_m(s) = \frac{ms}{ms+\frac{k}{s}} = \frac{s^2}{s^2+\frac{k}{m}} \isdef \frac{s^2}{s^2+\omega_0^2},$

where we have defined $\omega_0^2\isdef k/m$ . The mass velocity Laplace transform is then

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Physical Audio Signal Processing
    Introduction to Lumped Models
       One-Port Network Theory
          Mechanical Impedance Analysis