Biomedical Engineering Reference
In-Depth Information
the plate (
h
o
) is noted. A load cell is attached above the oscillating plate to control the
squeezing force applied to the sample prior the application of the oscillation. The squeezing
force is simply applied to make sure that a good contact is established between the
oscillating plate and the sample. It will be shown below, however, that the results are
independent of that squeezing force.
The frequency response data is obtained using software that interfaces the results obtained
with the signal generator device (see also details in Figure 1). The transformed mechanical
ˆ
F
ˆ
impedance
Z
is then calculated from the force and velocity measured on the oscillating
ˆ
u
plate with the impedance head.
F
and
u
are the Fourier transformed variables and since
they have been transformed into the frequency domain, they, as well as the mechanical
impedance, are complex variables. The measured complex mechanical impedance at the
driving point can be defined as:
ˆ
ˆ
ˆ
ZZ
Z
(11)
meas
instrument
sample
ˆ
meas
ˆ
instrument
where
Z
is the measured impedance,
Z
is the instrument impedance that can be
ˆ
instrument
calculated simply as
Z im
because the instrument does not have any spring
mechanism or internal damping. From Equation (11) the sample impedance
plate
ˆ
sample
Z
can be
obtained by subtracting the instrument impedance from the measured impedance.
The rheological behavior of the sample can be described in terms of the viscous component
(
R
), which provides the damping of the oscillation and the elastic component (
S
), which
provides the sample elasticity (Figure 9). The relationship between the mechanical
impedance and the damping (viscous component) and stiffness (elastic component) of the
sample can be described by Equation (12) below:
S
ˆ
sample
Z
R
i
(
m
)
(12)
sample
sample
where is the angular frequency of the oscillation,
m
is the mass of the system, and
i
is
.
The mobility of the sample, can be plotted as a function of frequency to provide the
resonance spectrum of the sample. The resonance frequency,
f
res
of the sample, which is
obtained as the frequency at which the mobility is a maximum is directly related to the
stiffness and the mass of the system by Equation (13):
S
sample
f
(13)
res
m
A typical plot of Mobility versus frequency for different concentrations of xanthan gum, a
biopolymer that produces viscoelastic suspensions, is illustrated in Figure 10.
The higher is the concentration of xanthan gum the higher is its elasticity, which is clearly
illustrated in the Figure 10 by a shifting to the right of the resonance frequency. That shifting
of the frequency is a clear indication on increase in the stiffness of elasticity of the sample
with concentration (see Equation 13).
