Biomedical Engineering Reference
In-Depth Information
command input into its own second-order open-loop transfer function. The outputs
of the independent transfer functions are the individual deflection time responses
(one for each portion of the curved beam). Those outputs are collected into a
deflection time response matrix; each column of that matrix represents the beam
deflection at a certain simulation time.
Figure 7.28 depicts the beam deflection as a function of time for the hypothetical
case initially described in support of figure 7.21. The darker line in the figure
represents the final deflection vector command (solution to the nonlinear large-angle
deflection equation).
7.4.4
V
ARIABLE
M
OMENTS
The previous sections have all constrained the input moment to a constant value.
The model developed, however, will work equally well under variable moment
conditions. To model the beam deflection properly under variable moment condi-
tions, the moment at each time step is used to create a final deflection command
vector at each time step. That variable command vector replaces the constant step
command used as an input to the open-loop transfer functions.
In order to develop the variable moment deflection plot shown in figure 7.29, a
square wave—switching from +0.5 to -0.5 and back every second for 10 sec—was
used as the moment input. The resultant motion is the expected “flapping” motion
seen during hardware tests.
The beam model presented in the last sections is only an initial model. In order
to upgrade the fidelity of this model, additional work in three areas must be performed
as outlined next.
7.4.4.1
Moment Modification
The simplifying assumption inherent to the implementation of the large-angle deflec-
tion equation (7.5) is that the moment, which produces the deflection, is simply the
moment at the wall. Though this assumption is reasonable for a first-order model,
it is easy to appreciate the errors induced by making such an assumption. Shahinpoor
(2000, 2002e, 2002f) has shown that the electric field produces a unique moment
upon each segment of the beam.
The moment produced by an electric field can be approximated using a parabola
with a maximum value near the wall and a minimum value (zero) at the beam tip.
Figure 7.30 presents a hypothetical moment.
The current model implementation reduces the complex moment model to a
single composite moment at a given distance from the wall. The equivalence of those
representations is presented in figure 7.31. The moments at the wall for both cases
in figure 7.31 are equal, so the deflection of both beams would be identical. Yet,
even a cursory analysis of the steady-state deflection of each of those cases would
show that the beams would indeed deflect very differently.
The future work that must be undertaken in this area involves dividing the beam
into numerous small segments, calculating the deflection of each segment due to the
