depicts the ideal spring.
The ideal spring characterized by
From Hooke's law
, we have that the applied force
is proportional to the
of the spring:
where it is assumed that
The spring constant
is sometimes called the stiffness
spring. Taking the Laplace transform
so that the impedance
of a spring is
and the admittance
This is the transfer function
of a differentiator
. We can say that
the ideal spring differentiates the applied force (divided by
produce the output velocity
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
per octave, and the phase
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.
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
In an equivalent analog circuit, we can use the value
of the spring stiffness is sometimes called the
of the spring.
Don't forget that the definition of impedance
for elements with ``memory'' (masses and springs).
This means we can only use impedance descriptions for steady
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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