Page 100 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_99_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 99

2 ʌ RN 2 ʌ EAu 2
r
U = = (2.195)
2EA R
The elementary kinetic energy is

2
ȡARu ˙ d́
r
dT = (2.196)
2
The total kinetic energy of the radially vibrating ring is calculated by
integrating Eq. (2.196) over the whole circumference, which gives
2
T = ʌȡARu ˙ (2.197)
r
For conservative vibrations the sum of strain and kinetic energies is a
constant, which means that its time derivative is zero:

d(T + U )
=0 (2.198)
dt
By combining Eqs. (2.195), (2.197), and (2.198) the following equation
is obtained:

2
ü + Ȧ u =0 (2.199)
r a r
where the resonant frequency is

1 E
Ȧ = (2.200)
a R ȡ
5
According to Timoshenko, the first resonant frequency of a circular
ring which vibrates torsionally (see Fig. 2.37) is
1 E I x
Ȧ = (2.201)
t R ȡ I t
where I is the moment of inertia of the ring’s cross section with respect
x
to the x axis (which is the axis located in the plane of the ring, as shown
in Fig. 2.37), whereas I t is the torsional moment of inertia of the ring’s

z
x

deformed x-axis
Figure 2.37 Circular ring undergoing free torsional vibrations.

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