Page 51 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_50_08/30/05

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

50 Chapter Two
The axial distribution function is determined by considering it as a
first-degree polynomial (with two unknown coefficients), which has to
satisfy the following boundary conditions that apply to the fixed-free
member of Fig. 2.1:
u (0) = u x u (l) =0 (2.7)
x
x
It follows that

x
f (x) =1— (2.8)
a l
The axial stiffness is found by combining Eqs. (2.3) through (2.6) as
l df (x) 2
a ฒ
k = E A(x) a dx (2.9)
0 dx
A similar procedure is applied to torsion, where the strain energy is

2
l M (x) l dș (x) 2
1
x
t
2ฒ
U = 2Gฒ I (x) = G I (x) dx dx (2.10)
t
t
0 t 0
In Eq. (2.10), M t (x) is the torsional moment, I t (x) is the torsional moment
of inertia, ș (x) is the angular deformation, and G is the shear modulus
x
of elasticity. The differential equation expressing the static equilibrium
in torsion is
2
d ș (x)I (x) =0 (2.11)
t
x
dx 2
The torsion angle at an abscissa x is expressed in terms of the torsion
angle at the free end by means of a distribution function f (x) as
t
ș (x) = f (x)ș (2.12)
x t x
By combining Eqs. (2.3) and (2.10) through (2.12), the torsional stiffness
is
l df (x) 2
t ฒ
k = G I (x) t dx (2.13)
t
0 dx
It can be shown that the torsion distribution function is identical to the
axial one [Eq. (2.8)]. Later in this chapter, inertia fractions are derived
that correspond to axial or torsional free vibrations. These inertia

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