Nonlinear Dynamics of Resonant Microelectromechanical System (MEMS): A Review
63
Case 3: 3ω ≈ ω n . It is also possible to get large amplitude response because of
‘superharmonic resonance’.
When the oscillator is excited by force with two or more different frequencies then
in addition to above, large amplitude oscillation may take place due to ‘combination
resonance’.
3.2 System with Parametric Excitation
The effect of nonlinear terms in a parametrically excited system is to limit the oscillation amplitude to a finite value when the excitation causes instability to a linear
system. For example, if the nonlinearity is of duffing type as in the following equation
of motion:
m ¨
x + c ˙
x + k(1 + 2ε cos 2ωt)x + αx
3
= 0
( 8 )
one or two limit cycles may exist depending on the value of
ω
ω n
, ζ =
c
2
√
mk
and the sign
of α. For a duffing-type nonlinearity, the limit cycle is stable if this is unique. For the
system where two limit cycles exist, the one with higher amplitude of oscillation is
stable. The instability region for a nonlinear system is shown in Fig. 4. The parametric
space is divided into three regions. In the region I, both linear and nonlinear equations
predict the same steady-state response which decays to insignificantly small value.
In region II, the nonlinear equation predicts a stable limit cycle. However, the steadystate amplitude depends on initial condition in region III. For some initial conditions,
the predictions of the nonlinear equation and the linear equations are identical, while
for a different set of initial conditions a large amplitude of oscillation is predicted
when the nonlinear term is present.
Fig. 4 Regimes for stable
and unstable behaviours of
parametrically excited
system
63
Case 3: 3ω ≈ ω n . It is also possible to get large amplitude response because of
‘superharmonic resonance’.
When the oscillator is excited by force with two or more different frequencies then
in addition to above, large amplitude oscillation may take place due to ‘combination
resonance’.
3.2 System with Parametric Excitation
The effect of nonlinear terms in a parametrically excited system is to limit the oscillation amplitude to a finite value when the excitation causes instability to a linear
system. For example, if the nonlinearity is of duffing type as in the following equation
of motion:
m ¨
x + c ˙
x + k(1 + 2ε cos 2ωt)x + αx
3
= 0
( 8 )
one or two limit cycles may exist depending on the value of
ω
ω n
, ζ =
c
2
√
mk
and the sign
of α. For a duffing-type nonlinearity, the limit cycle is stable if this is unique. For the
system where two limit cycles exist, the one with higher amplitude of oscillation is
stable. The instability region for a nonlinear system is shown in Fig. 4. The parametric
space is divided into three regions. In the region I, both linear and nonlinear equations
predict the same steady-state response which decays to insignificantly small value.
In region II, the nonlinear equation predicts a stable limit cycle. However, the steadystate amplitude depends on initial condition in region III. For some initial conditions,
the predictions of the nonlinear equation and the linear equations are identical, while
for a different set of initial conditions a large amplitude of oscillation is predicted
when the nonlinear term is present.
Fig. 4 Regimes for stable
and unstable behaviours of
parametrically excited
system