Page 39 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

38 Chapter One

f (t) x (t)
MECHANICAL
L L
SYSTEM
F (s) X (s)
Figure 1.36 Transfer function diagram of a mass-spring-damper system.

2
(ms + cs + k)X(s) = F(s) (1.130)
Equation (1.130) enables us to express the transfer function of the mechan-
ical system as
X(s) 1
TF = = (1.131)
F(s) 2
ms + cs + k
Recalling the mechanical-electrical analogies discussed previously in
this section, we emphasize that a more mathematically sound definition
of the analogy states that two different systems are analogous if they
4
have the same transfer function (Ogata ), and this is an extremely
important feature that allows unitary mathematical treatment of
physically different systems. For instance, an analog of the mechanical
system of Fig. 1.30 is the series electrical system sketched in Fig. 1.31.
By following an approach similar to the one taken for the mechanical
systems, it can be shown that the transfer function of the electrical
system is
Q(s) 1
TF = =
E(s) Ls + Rs +1 C (1.132)
2
/
In other words, and strictly speaking, the mechanical and electrical
systems are analogous when the following conditions are satisfied:

1
m싀L c싀R k싀 (1.133)
C
There are microresonator systems, as will be shown shortly, that are
formed as serial connections between simple subsystems (units), and
this system is schematically illustrated in Fig. 1.37.
It is assumed that the transfer functions of all component subsystems
TF , TF , . . . , TF are known, and the aim is to find a global (system)
1
n
2
transfer function to be equivalent to the component subsystems in a
manner that will preserve the input Laplace-transformed I(s) and
output O(s) functions. Figure 1.37 also indicates that the output from
an intermediate subsystem is the input to the immediately following
one, and therefore

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