### 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}

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:

*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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Spring-Mass System