Page 68 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 55
It can be seen that the stiffness for this particular boundary conditions is
equal to the stiffness of an identical beam whose boundary conditions are
fixed-free, as illustrated in Fig. 1.30 (a).
This conclusion should not be very surprising within the context of the
stiffness that has been derived in Example 1.1, namely by means of
inversion of the compliance matrix. The first equation of Eqs. (1.7) can be
written with the aid of Eq. (1.16) as:

This equation, again, reflects the principle of superposition which indicates
that the total force being applied at the free end of a microcantilever is equal
to the algebraic sum of a force that needs to purely translate the free end by
with zero slope – the first term of Eq. (1.216) – and a force that
would simply rotate the free end by with zero deflection – the
second term in Eq. (1.216). This latter term has to be negative because the
real force that has to produce both and for a free end cantilever is
smaller than the force needed to only generate the same deflection, as in
Fig. 1.30 (b). This is the reason why the bending-related stiffness has to be
negative, as Example 1.1 has demonstrated.
However, individual springs, either linear or rotary, as the ones pictured
in Fig. 1.5 and utilized as equivalent lumped-parameter models of real,
distributed-parameter beams, have to be uniquely defined in terms of their
stiffnesses. Because stiffness depends on boundary conditions, it is expected
that two different sets of boundary conditions will generate two different
stiffnesses for the same physical spring. Conversely, one stiffness could not
possibly describe two different boundary conditions applied to the same
spring. It has been shown at the beginning of this chapter that axial and
torsional stiffnesses are simply calculated as algebraic inverses of their
corresponding compliances, and this relationship should also hold true for
bending-related stiffnesses as they define unique linear or rotary springs.
Indeed, the direct and cross stiffnesses of springs that model bending of
cantilevers are calculated as:

These expressions are clearly different from those of the stiffnesses
and that have been obtained in Example 1.1, Eq. (1.16),
through inversion of the compliance matrix, for the same fixed-free boundary
conditions. While the stiffness set of Eqs. (1.217) is utilized to individually