Page 52 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_51_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 51
fractions are determined in terms of distribution functions that are
identical to the one expressed in Eq. (2.8).
The bending presents the characteristic of effect coupling, as deflec-
tion or rotation can be produced by both a moment and a force. The
strain energy in bending about the y axis is expressed as

2
l M 2 (x) E l d u (x) 2
1
U = 2Eฒ b,y dx = 2ฒ I (x) z dx (2.14)
b,y I (x) y 2
0 y 0 dx
where M b,y (x) is the bending moment about the y axis, I y (x) is the
moment of inertia calculated about the same axis, and u (x) is the
z
deflection about the z axis. The differential equation for the bending
static equilibrium is
4
d I (x)u (x) =0 (2.15)
y
z
dx 4
The deflection at a generic point on the micromember length is ex-
pressed in terms of the free end deflection and rotation, as well as two
distribution functions: the linear one f (x) and the rotary one f (x) in the
l
r
form:
u (x) = f (x)u + f (x)ș y (2.16)
z
l
r
z
The two distribution functions of Eq. (2.16) are determined as third-
degree polynomials (with four unknown coefficients each) by imposing
the following boundary conditions:
u (0) = u z ș (0) = ș y u (l) =0 ș (l) =0 (2.17)
y
z
z
y
The bending-related distribution functions are found to be
3x 2 2x 3 2x 2 x 3
f (x) =1— + f (x) = x — + (2.18)
l 2 3 r l 2
l l l
We show later in this chapter, when deriving inertia fractions, that
other distribution functions need to be utilized when quantifying the
mass fraction (called the effective mass) that corresponds to bending.
By combining Eqs. (2.3) and (2.14) through (2.16), the three bending-
related stiffnesses (direct linear, direct rotary, and cross) are expressed as

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.