### Mechanical Impedance Analysis

Impedance analysis is commonly used to analyze electrical circuits [110]. By means of equivalent circuits, we can use the same analysis methods for mechanical systems.

For example, referring to Fig.7.9, the Laplace transform of the force on the spring is given by the so-called voltage divider relation:8.2

Similarly, the Laplace transform of the force on the mass is given by

 (8.1)

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:

Then, by the voltage divider'' relation Eq.(7.1), the Laplace transform of the mass force after time 0 is given by

where we have defined . The mass velocity Laplace transform is then

Thus, the impulse response of the mass oscillates sinusoidally with radian frequency , and amplitude . The velocity starts out maximum at time , which makes physical sense. Also, the momentum transferred to the mass at time 0 is ; this is also expected physically because the time-integral of the applied force is 1 (the area under any impulse is 1).
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