Biomedical Engineering Reference
In-Depth Information
Equations 6.145 through 6.156 should be substituted in equation 6.144 to obtain
the governing equations for the ion transport and deformation dynamics. However,
such sophisticated detail will not be expanded in this chapter and will be left for
future work.
6.6.4
E
QUIVALENT
C
IRCUIT
M
ODELING
Newbury and Leo (2002) present a gray box model in the form of an equivalent
circuit. The model is based on the assumption of linear electromechanical coupling
among electric field, charge, strain, and applied stress. Under this assumption, an
impedance model of a cantilever transducer is derived and applied to the analysis
of transducer performance. This model enables simultaneous modeling of sensing
and actuation in the material. Furthermore, the model demonstrates the existence of
reciprocity between sensing and actuation.
The model is validated experimentally for changes in transducer length and
width. Experimental results demonstrate the improved tracking capability through
the use of model-based feedback. Recent publications by Newbury (2002) and
Newbury and Leo (2002) have produced a model of electromechanical coupling in
ionic polymer materials. The fundamental assumption in the model is that the
electromechanical coupling is a linear relationship among strain, stress, electric field,
and charge. Thus, they assume
STYdE
=
/
+
(6.157)
D T E
=+
ε
where
S
is the strain in the material
T
is the stress (Pa)
E
is the electric field (V/m)
D
is the charge density (C/m
2
)
The three material properties of the model are the elastic modulus
Y
(Pa), the electric
permitivity (F/m), and the linear coupling coefficient (m/V or C/N).
For a cantilevered sample of ionic polymer material, equation (6.157) can be
integrated over the volume of the transducer to produce a relationship among tip
displacement, tip force, voltage, and charge (Newbury, 2002). The only limiting
assumption in this derivation is that the stress acts at the surface of the bender
element. This assumption is consistent with other models of actuation in ionic
polymers (Nemat-Nasser and Li, 2000). Transforming the integrated equations into
the frequency domain produces an impedance relationship among voltage, force,
velocity, and current of the form:
v
f
=
Zj
()
ω
Zj
()
ω
i
u
11
12
(6.158)
Zj
()
ω
Zj
()
ω
12
22
