Biomedical Engineering Reference
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2π
f
0
m
Q
0
a
exc
A
mz
(
t
)+
z
(
t
)+
k
cant
(
z
(
t
)
−
d
)=
−
k
cant
z
(
t
−
t
0
)
(2.16)
As the cantilever is not excited with a specific externally set frequency, the cantilever
itself serves as the frequency determining element. Therefore, we make the ansatz
(Holscher et al., 2001; 2003)
z
(
t
0
)=
d
+
A
cos
(
2π
ft
)
(2.17)
and introduce it into Equation 2.17 into Equation 2.16. As a result we obtain a set of
two coupled trigonometric equations:
f
0
−
f
2
a
exc
cos
(
2π
ft
0
)=
(2.18a)
f
0
a
exc
A
1
Q
0
f
f
0
sin
(
2π
ft
0
)=
(2.18b)
The two equations can be decoupled with the assumption that the time shift
t
0
is set to a value corresponding to
t
0
90
◦
)
, which simultaneously cor-
responds to the by far most common choice for
t
0
. For this value, the solution of
Equation 2.18 is given by
=
1
/
(
4
f
0
)(=
f
=
f
0
(2.19a)
=
A
a
exc
Q
0
(2.19b)
This simple calculation demonstrates the very specific behavior of a self-driven
oscillator if the phase (or time) shift is set to 90
◦
. In this case, the cantilever oscillates
exactly with its eigenfrequency
f
0
. Due to this specific feature revealed by Equa-
tion 2.19a, we define that the cantilever is in
resonance
if this condition is fulfilled.
The linear relationship between the oscillation and excitation amplitude is described
by Equation 2.19b.
2.4.3 T
HEORY OF THE
AM-M
ODE
In the first step, we assumed that the cantilever vibrates far away from the sample
surface. Therefore, we neglected tip-sample forces in Equation 2.13 and finally got
the well-known theory of a driven-damped harmonic oscillator.
However, if the cantilever is brought closer toward the sample surface, the tip
senses the tip-sample interaction force
F
ts
, which changes the oscillation behavior of
the cantilever. However, since the mathematical form of realistic tip-sample forces
is highly nonlinear (see Figure 2.3b) this fact complicates the analytical solution of
the equation of motion Equation 2.12. For the analysis of DFM experiments, we
need to focus on steady-state solutions of the equation of motion with sinusoidal
cantilever oscillation. Therefore, it is advantageous to expand the tip-sample force
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