Biomedical Engineering Reference
In-Depth Information
position, such that (for small-amplitude oscillations) the motion about the
stable equilibrium position is given by
z = z
ω t ) (4-14)
where a and ω are the amplitude of vibration and resonant frequency,
respectively. The resonant frequency is related to the local stiffness by
+ a sin(
= k / m (4-15)
where m is the effective mass of the probe. Hence, the variation in the
stiffness field can be determined experimentally from the variation in the
resonant frequency:
2
ω
1/2
2
(
)
1/2
s 2
/ s 2
ω = ω S
1 −
s / s 0 +
− 1
(4-16)
Using the equipartition theorem gives the relationship between the
resonant frequency and amplitude of oscillation of the tip as a function of
temperature:
m ω
2
a 2
=
k T .
(4-17)
Hence, in thermal equilibrium the amplitude of tip oscillation should
vary reciprocally with the resonant frequency:
1/2
−2
(
)
1/2
s 2
/ s 2
a = a S
1 −
s / s 0 +
− 1
(4-18)
where ω S and a S are the frequency and amplitude of free vibration,
respectively.
The quantity k T is the natural energy scale for the environment, in
this case the indenter. The natural energy scale for the system is the
energy of deformation contained in the spring at the point of system
instability. The ratio of these two energy scales, ~ k T / A , provides a
measure of whether thermal, k T / A ≥ 1 , or mechanical, k T / A ≤ 1 , effects
are likely to dominate the behavior of the system. This becomes
important as the system approaches the instability point. The behavior of
the system in terms of the externally measured force, resonant frequency,
and thermal amplitude of vibration are shown in Fig. 4-6 ; the instability
point of the system at s = s 0 is clear in all three quantities.
 
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