Ideal Spring

Figure 7.4 depicts the ideal spring.

From Hooke's law, we have that the applied force is proportional to the
*displacement* of the spring:

where it is assumed that . The spring constant is sometimes called the

so that the impedance of a spring is

and the admittance is

This is the transfer function of a

The *frequency response* of the ideal spring, given the applied force
as input and resulting velocity as output, is

In this case, the amplitude response grows dB per octave, and the phase shift is radians for all . Clearly, there is no such thing as an ideal spring which can produce arbitrarily large gain as frequency goes to infinity; there is always some mass in a real spring.

We call
the *compression velocity* of the spring. In more
complicated configurations, the compression velocity is defined as the
difference between the velocity of the two spring endpoints, with positive
velocity corresponding to spring compression.

In circuit theory, the element analogous to the spring is the *capacitor*,
characterized by
, or
.
In an equivalent analog circuit, we can use the value
. The
inverse
of the spring stiffness is sometimes called the
*compliance*
of the spring.

Don't forget that the definition of impedance requires *zero
initial conditions* for elements with ``memory'' (masses and springs).
This means we can only use impedance descriptions for *steady
state* analysis. For a complete analysis of a particular system,
including the transient response, we must go back to full scale
Laplace transform analysis.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University