60
G. Chakraborty and N. Jani
It is therefore required to have large excitation amplitude F 0 in order to increase the
sensitivity for small value of m.
This limitation can be avoided if the oscillator is excited by parametric excitation
rather than direct excitation. For the simplest situation, the following equation of
motion is obtained:
m ¨
x + c ˙
x + k (1 + 2ε cos 2ωt) x = 0
( 6 )
The response in this case shows two distinct behaviour depending on the values
of the parameters ε, ω/ω n and ζ. The amplitude either increases exponentially if
certain inequalities ε > f (ω/ω n , ζ) are satisfied or gets reduced to zero when ε <
f (ω/ω n , ζ). Thus, the system can be conceived to be either in 0-state corresponding
to low amplitude of oscillator or in the 1-state which corresponds to high amplitude of
oscillation. When the natural frequency is changed, the system may be easily brought
from one state to other. Next by proper adjustment of the excitation frequency ω, the
system can be restored to its original state. The measured value of ω required for
that purpose is used to obtain the change in the mass or stiffness of the oscillator.
For example, when the excitation frequency ω is nearly equal to the natural frequency ω n , primary parametric resonance in an undamped system (c = 0) occurs,
if 1 −
ω
ω n
2 < ε < 1 +
ω
ω n
2
, i.e. within a narrow frequency zone of width,
1 − ε <
ω n
ω
2 < 1 + ε or approximately 1 −
ε
2
<
ω n
ω
< 1 +
ε
2
. The state of the
oscillator changes when ω n is changed by amount
εω
2
≈
εω n
2
.
3 Nonlinearities in MEMS
The response amplitudes of resonant MEMS devices are generally large and hence
the linear approximations are often insufficient to predict their correct behaviour.
Also nonlinearities are sometimes introduced deliberately to gain some advantages.
Presence of nonlinearities makes the analysis complicated because of two reasons,
(a) the model becomes quite complex and (b) unlike linear systems there is no generic
behaviour of system response. Every nonlinear system behaves differently depending
on the type of nonlinearity, excitation or initial conditions. Thus, a complete discussion of nonlinear dynamics of MEMS devices will be too lengthy to be considered
in a review of modest length.
Resonant MEMS devices can be broadly classified as shown in Fig. 1.
Nonlinear analysis of the system is different for different subsystem. Further each
of the subsystem can be classified according to the mode of excitation in the following
manner:
1. system with direct excitation,
2. system with parametric excitation,
3. system with combined direct and parametric excitations and
4. system with self-excitation.
G. Chakraborty and N. Jani
It is therefore required to have large excitation amplitude F 0 in order to increase the
sensitivity for small value of m.
This limitation can be avoided if the oscillator is excited by parametric excitation
rather than direct excitation. For the simplest situation, the following equation of
motion is obtained:
m ¨
x + c ˙
x + k (1 + 2ε cos 2ωt) x = 0
( 6 )
The response in this case shows two distinct behaviour depending on the values
of the parameters ε, ω/ω n and ζ. The amplitude either increases exponentially if
certain inequalities ε > f (ω/ω n , ζ) are satisfied or gets reduced to zero when ε <
f (ω/ω n , ζ). Thus, the system can be conceived to be either in 0-state corresponding
to low amplitude of oscillator or in the 1-state which corresponds to high amplitude of
oscillation. When the natural frequency is changed, the system may be easily brought
from one state to other. Next by proper adjustment of the excitation frequency ω, the
system can be restored to its original state. The measured value of ω required for
that purpose is used to obtain the change in the mass or stiffness of the oscillator.
For example, when the excitation frequency ω is nearly equal to the natural frequency ω n , primary parametric resonance in an undamped system (c = 0) occurs,
if 1 −
ω
ω n
2 < ε < 1 +
ω
ω n
2
, i.e. within a narrow frequency zone of width,
1 − ε <
ω n
ω
2 < 1 + ε or approximately 1 −
ε
2
<
ω n
ω
< 1 +
ε
2
. The state of the
oscillator changes when ω n is changed by amount
εω
2
≈
εω n
2
.
3 Nonlinearities in MEMS
The response amplitudes of resonant MEMS devices are generally large and hence
the linear approximations are often insufficient to predict their correct behaviour.
Also nonlinearities are sometimes introduced deliberately to gain some advantages.
Presence of nonlinearities makes the analysis complicated because of two reasons,
(a) the model becomes quite complex and (b) unlike linear systems there is no generic
behaviour of system response. Every nonlinear system behaves differently depending
on the type of nonlinearity, excitation or initial conditions. Thus, a complete discussion of nonlinear dynamics of MEMS devices will be too lengthy to be considered
in a review of modest length.
Resonant MEMS devices can be broadly classified as shown in Fig. 1.
Nonlinear analysis of the system is different for different subsystem. Further each
of the subsystem can be classified according to the mode of excitation in the following
manner:
1. system with direct excitation,
2. system with parametric excitation,
3. system with combined direct and parametric excitations and
4. system with self-excitation.