Page 70 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_69_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 69
M (x) = F x (2.73)
z
y
for a beam loaded with a tip force F z , we can find the angle ș y (x) by
integrating the first of Eqs. (2.72), considering that
ș (l) =0 (2.74)
y
and its equation is

2
2
F (l — x )
z
ș (x) = (2.75)
y
2EI
y
By substituting ș (x) of Eq. (2.75) into the second of Eqs. (2.72) and by
y
carrying out the necessary integration with the boundary condition
u (l) =0 (2.76)
z
the deflection u z (x) can be written into the generic form of Eq. (2.62)
where the distribution function is
GA(2l + x)(l — x) +6țEI y
f sh (x) = (l — x)
b 2 (2.77)
2l(GAl +3țEI )
y
and the tip deflection is

l 2 ț
u = u (0) = F l + GA) (2.78)
z z z ( 3EI
y
Equation (2.78) enables us to find the shearing-dependent stiffness as

F 3EGI A
sh z y
k b,e = = 2 (2.79)
u
z l(GAl +3țEI )
y
The effective mass which is placed at the free extremity is determined
by following Rayleigh’s procedure, which has been outlined previously,
and by using the distribution function of Eq. (2.77). Its equation is
2
2 2
2 2 4
2
3(140ț E I +77țEGI Al +11G A l )
sh y y
m b,e = m (2.80)
2
140(GAl +3țEI ) 2
y
where m is the mass of the microcantilever.
The corresponding resonant frequency is found by means of the
stiffness given in Eq. (2.79) and the effective mass of Eq. (2.80) 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.