Page 254 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_253_08/30/05

Resonant Micromechanical Systems

Resonant Micromechanical Systems 253
The expressions of Eqs. (5.63) and (5.64) can be found in Beer and Johnston, 3
for instance.
The potential energy of the spring system corresponding to the resonator
of Fig. 5.26 can be expressed as

U = U + U + U + U (5.65)
b,x b,y b,z t
where U b,x , U b,y , and U b,z are elastic energy terms produced through bending
of various legs about the x, y, and z axis, respectively, whereas U t is the elastic
energy produced through torsion of the four legs. The four different potential
energy terms of Eq. (5.65) are expressed next. The U b,x potential energy is

1 l 2 2 1 l 2 2
U = k u + ș x) + k u í ș x) (5.66)
b,x 2 l,o( z 2 2 l,o( z 2
where k l,o is the linear stiffness of a leg corresponding to out-of-the-plane
bending and is expressed for fixed-free boundary conditions as
3
Ew t
1 1
k = (5.67)
l,o 3
4l 1
The U b,y potential energy is similarly expressed as
l 2 l 2
1 2 1 2
u +
u í
U b,y = 2 k l,o( z 2 ș y) + 2 k l,o( z 2 ș y) (5.68)
The U b,z potential energy term is calculated as
l 2
1 2
U =4 k ș z) (5.69)
b,z 2 l,i( 2
where k l,i is the linear stiffness of a root leg corresponding to the in-plane
bending and is given by
3
Ew t
1 1
k l,i = (5.70)
4l 3
1
Eventually, the torsional potential energy term of Eq. (5.65) is
t ( 1 2 1 t y)
2
U =2 2 k ș + 2 k ș (5.71)
t x
By applying Lagrange’s equations, the differential equations which govern
the motion of the plate of Fig. 5.26 are

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