Page 35 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

34 Chapter One
x
k

m
f
c
Figure 1.30 Single-DOF mechanical system.

L R

i
e

C
Figure 1.31 Series LRC electric circuit.

force-current (or mass-capacitance) analogy. Both variants utilize the
same mechanical system, whereas the electrical counterpart is dif-
ferent.
Figure 1.30 illustrates a single-DOF system consisting of a mass
connected to a spring and to a dashpot (spring-damper combination)
and acted upon by an external force.
The differential equation describing the motion of the mass in
Fig. 1.30 is
2
d x dx
m + c + kx = F (1.118)
dt 2 dt
The series electric circuit of Fig. 1.31 is now considered, which is
composed of a voltage source e, an inductor having inductance L, a
resistor of resistance R, and a capacitor of capacitance C.
By applying Kirchhoff’s second (circuit) law, the source voltage can
be expressed as

di 1
e = L + Ri + Cฒ idt (1.119)
dt
The current i is related to the electrical charge q as

dq
i = (1.120)
dt

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