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Resonant Micromechanical Systems
Resonant Micromechanical Systems 265
2
Ȧ z F 0
r 0
2
x ˙˙ = í íȦ x + sin(Ȧt )
i ʌȝ x ˙ r i m i
i i
(5.95)
2
Ȧ z F
2
x ˙˙ = í r 0 íȦ x + 0 sin(Ȧt )
i +1 ʌȝ x ˙ r i +1 m i +1
i +1 i +1
Because the sensing is performed electrostatically, the microdevice displace-
ment can be assessed at any time by Eq. (5.90) in terms of the measured
capacitance, namely,
gC i
x = í l (5.96)
i İl 0x
z
By combining Eqs. (5.94) and (5.95), the following two equations are pro-
duced:
2
Ȧ z F
2
x ˙ = x ˙ + ǻt (1 íȕ ) í r 0 íȦ x + 0 sin(Ȧt )
i +1 i 1 ʌȝ x ˙ r i m i
i i
2
Ȧ z F 0
r 0
2
+ ǻt ȕ í íȦ x + sin(Ȧt )
1 ʌȝ x ˙ r i +1 m i +1
i +1 i +1
(5.97)
2
Ȧ z
(ǻt) 2 r 0 2 F 0
x = x + ǻtx ˙ + (1 íȕ ) í íȦ x + sin(Ȧt )
i +1 i i 2 2 ʌȝ x ˙ r i m i
i i
2
(ǻt) 2 Ȧ z 2 F 0
r 0
+ ȕ í íȦ x + sin(Ȧt )
2 2 ʌȝ x ˙ r i +1 m i +1
i +1 i +1
By starting with zero-displacement and zero-velocity initial conditions, it is
possible to determine the series Ș i by working with Eqs. (5.97) in this se-
quence:
1. Determine the coefficient of dynamic viscosity Ș i+1 as a function of known
displacements x i and x i+1 , the velocity dx/dt at moment i, and the dynamic
viscosity coefficient at the previous time moment Ș i from the second of
Eqs. (5.97).
2. Determine the velocity dx/dt at moment i + 1 from the first of Eqs. (5.97).
The average coefficient of dynamic viscosity can eventually be determined
as the arithmetic average of the n values of Ș i .
The motion about the x direction can be enabled, as illustrated in
Fig. 5.40b, in “parallel-plate” transduction where the mobile plate moves
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