Page 307 - Mechanical design of microresonators _ modeling and applications

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Microcantilever and Microbridge Systems for Mass Detection

306 Chapter Six
z

∆m
3
1
x 2 a
l
Figure 6.12 Pointlike mass deposited on a microcantilever.

where m is the total mass of the microcantilever. Because the bending
stiffness at the free end of the microcantilever shown in Fig. 6.12 is
3EI y
k = (6.18)
b 3
l
and the effective mass which is located at the same point is
33
m = 140 m (6.19)
b
the bending resonant frequency in the presence of a deposited mass
ǻm can be expressed as

k b
Ȧ = (6.20)
b m + ǻm e
b
where ǻm e is the efficient deposited mass. It should be remembered that
all calculations pertaining to lumped-parameter modeling use the free
endpoint of a cantilever to locate both stiffness and mass. This is the
reason why an efficient (or equivalent) deposited mass, denoted by
ǻm e , which needs to be located at the free end, has to be calculated as
it corresponds to the real additional mass ǻm, which attaches at a
distance a, as illustrated in Fig. 6.12. By applying Rayleigh’s principle,
which has been utilized several times thus far in this book, it can be
shown that the effective deposited mass is

2
ǻm = f (a) ǻm (6.21)
e b
3
3 a 1 a
where f (a) =1 ෹ 2 l + 2 l 3 (6.22)
b
is the bending distribution function introduced in Eq. (2.63) which, in
the variant of Eq. (6.22), is the ratio of the beam deflection (or velocity)
at point 2 to the deflection (or velocity) at point 1 in Fig. 6.12.

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