Page 57 - Mechanical design of microresonators _ modeling and applications

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Basic Members: Lumped- and Distributed-Parameter Modeling and Design

56 Chapter Two
means of equations of the type (2.1), lumped-parameter inertia prop-
erties need to be established, in addition to the lumped-parameter
stiffnesses already discussed. As mentioned previously, only the direct
linear bending stiffness (as the algebraic inverse of the corresponding
compliance) is important in terms of bending, in addition to the axial-
and torsion-related stiffness. As a consequence, inertia properties will
be sought that are associated with axial, torsional, and direct linear
bending free vibrations.
The procedure enabling calculation of these inertia fractions that are
going to be located at the free end of a fixed-free micromember follows
5
the Rayleigh principle. This principle, as mentioned by Timoshenko or
6
Thomson, states that the velocity distribution over a freely vibrating
member is identical to the static deformation distribution of the same
member. This statement permits calculation of the so-called equivalent
(or effective) inertia fraction [either translatory (therefore a mass
fraction) or rotary (therefore a mechanical moment of inertia)] by
equating the kinetic energy of the real distributed-parameter system to
that of the lumped-parameter inertia particle which is located at the
free end of the micromember.

Axial vibrations In axial loading, this principle translates to the follow-
ing relationships between the generic point deformation/velocity and
the free endpoint deformation/velocity, by means of the axial distribu-
tion function f a (x):
du (x) du x
x
u (x) = f (x)u x dt = f (x) dt (2.36)
a
x
a
The kinetic energy of the distributed-parameter fixed-free microrod of
Fig. 2.1 undergoing free vibrations is
l
1 du (x) 2
x
ȡ
T = 2 ฒ A(x) dx (2.37)
a dt
0
The kinetic energy of the effective mass is
du
1 a,e( dt ) 2
x
T a,e = 2 m (2.38)
The distribution function corresponding to free axial vibrations of a mi-
crocantilever is explicitly formulated later in this chapter when we
show that constant- and variable-cross-section configurations have dif-
ferent distribution functions.

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