Biomedical Engineering Reference
In-Depth Information
system no longer remains in the undeformed state. It is interesting that
the
problem
is as symmetric as before (the forces are still on the same axis), but
now a slight perturbation in the initial configuration will evolve until the
system finds its equilibrium in the bent-plate state. Whether it bends to one
side or the other depends on the initial conditions. By the way, the problem is
an
extended
one: both space and time are relevant. In fact, we could think of
a situation in which the force was so strong that we broke the plate, creating
a most complex spatial pattern. But as long as we are interested in discussing
only whether the system is in the undeformed state or the bent state, we can
restrict ourselves to the study of the dynamics of the
spatial mode
=
A
sin
π
L
x
,
(5.1)
where
represents the deformation of the plate with respect to the plane
where the forces are acting. The situation where the forces are below the
critical value has
A
= 0 as a stable solution. Even without knowing the
details of the elastic properties in the problem, we can advance a little in our
modeling. The time evolution of this mode does not seem to be influenced
by any other higher modes of the problem, and therefore we can attempt to
model its dynamics in terms of only one variable. The dynamics can therefore
be written as
dA
dt
=
f
(
A
)
,
(5.2)
where
f
(
A
) depends on all the elastic properties of the plate, as well as the
forces. However, we know that since the problem is symmetric, the function
f
should satisfy the condition that
f
(
A
)=
A
). An expansion of this
function in a power series should therefore be, for small values of
A
,ofthe
form
−
f
(
−
f
(
A
)=
µA
+
cA
3
,
(5.3)
where
µ
and
c
depend on the parameters of the problem. We can advance
further in our qualitative modeling. We know that the transition as the ex-
ternal forces are increased can be captured by only the linear terms (since for
small values of
A
, the cubic term is negligible). Therefore,
µ
should depend on
the difference between the external forces and some critical value. Moreover,
when
µ>
0, we want the nonlinearities to be capable of stopping the growth
of
A
, and therefore
c
should satisfy the condition that
c<
0. In this way, we
find that for
µ<
0,
A
= 0 is stable, and when
µ>
0,
A
= 0 is an u
ns
table
solution, while the stability is transferred to
A
=
√
µ
and
A
=
−
√
µ
.The
moral of the tale (or at least one of them) is that even if the problem is an
extended one, the qualitative change from a nondeformed state to a deformed
one can be described in terms of a low-dimensional system of equations.
Other systems show changes that imply the appearance of motion: for
example, the establishment of oscillations in a pair of paper sheets as the
airflow between them exceeds a critical value, as shown in Fig. 5.2b. This
