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Design at Resonance of Mechanical Microsystems
Design at Resonance of Mechanical Microsystems 37
in (t) out (t)
INPUT L SYSTEM L OUTPUT
In (s) Out (s)
Figure 1.35 Simple system with input (driving) and output (response) functions.
original function f(t). It is beyond the scope of this book to go into detail
with the Laplace transform and its related aspects, but it should be
mentioned briefly that this transform enables approaching and solving
integral differential equations (such as the one introduced previously
in defining the behavior of mechanical and electrical systems) that are
transformed into algebraic equations in the Laplace s domain. The most
usual Laplace transforms that are connected to differentiation and in-
tegration are
2
d f (t) 2 df (t) F(s)
വ = s F(s) വ = sF(s) വ f (t) dt = s (1.127)
dt 2 dt
where the first two of Eqs. (1.127) are valid when f(0) = 0 and df(t)/
dx = 0 (for t = 0).
The Laplace transform is directly utilized in defining the transfer
function of a system. Consider a system that is defined by specific
properties and is acted upon by an input function (also called a driving
function). The system input interaction results in an output function
(or response function) as sketched in Fig. 1.35.
The system properties can be conveniently characterized by means
of the transfer function TF, which is defined as the ratio of the Laplace
transform of the output function to the Laplace transform of the input
function:
വ out (t) Out (s)
TF = = (1.128)
വ in (t) In (s)
Example: Determine the transfer function of the single-DOF system of
Fig. 1.30 by considering Fig. 1.36, which indicates that the input function is
the force f(t), and the output is the displacement x(t).
It has been shown that the time-domain behavior of the simple mechanical
system is governed by the differential equation
2
d x(t) dx(t)
m 2 + c + kx(t) = f (t) (1.129)
dt dt
By applying the Laplace transform to Eq. (1.129) and by taking into account
the rules outlined in Eqs. (1.127), the following equation is obtained that
describes the system behavior into the Laplace s domain:
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