66
G. Chakraborty and N. Jani
This equation can be written in another form, associated with the name of Van der
Pol.
m ¨
y − g
1 − y
2
˙
y + ky = 0
(19)
by substituting ˙
x = y.
If the oscillator is excited by harmonic excitation f 0 cos ωt, where ω ≈ ω 0 =
k
m
,
then the steady-state motion becomes periodic with a frequency equal to that of excitation. Thus, the response gets synchronized at ω. An interesting fact is that a small
change in excitation frequency can cause large change in response behaviour. This
happens when the excitation frequency falls outside the band in which ‘entrainment’
takes place.
4 Sources of Nonlinearity
Any vibrating system is generally nonlinear. Only under special operating conditions the effect of the nonlinear terms can be ignored. For example, the system is
adequately modelled as a linear system when the amplitude of vibration is small.
In resonant devices, this is not the case since the amplitude of vibration is kept to
a very high level in order to increase the sensitivity of the device. Furthermore, the
MEMS device is made by interconnecting different subsystems which have their
own dynamics [15]. Hence, modelling the nonlinear terms is a very difficult task
which can be achieved only with a detailed knowledge of the individual subsystems.
In what follows some common sources of nonlinearity are discussed.
The sources of nonlinearity can be broadly classified as following:
1. nonlinearity in mechanical structure,
2. nonlinearity in actuation system,
3. nonlinearity in sensing (measuring) devices and
4. nonlinearity in feedback and electrical circuit.
4.1 Nonlinearity in Mechanical Structure
The nonlinear terms appearing in the equation of motion of the structural component
of the resonant device (for example, beam, plate, wire, etc.) are of following types:
(i) stiffness nonlinearity,
(ii) damping nonlinearity and
(iii) nonlinearity due to external forces.
G. Chakraborty and N. Jani
This equation can be written in another form, associated with the name of Van der
Pol.
m ¨
y − g
1 − y
2
˙
y + ky = 0
(19)
by substituting ˙
x = y.
If the oscillator is excited by harmonic excitation f 0 cos ωt, where ω ≈ ω 0 =
k
m
,
then the steady-state motion becomes periodic with a frequency equal to that of excitation. Thus, the response gets synchronized at ω. An interesting fact is that a small
change in excitation frequency can cause large change in response behaviour. This
happens when the excitation frequency falls outside the band in which ‘entrainment’
takes place.
4 Sources of Nonlinearity
Any vibrating system is generally nonlinear. Only under special operating conditions the effect of the nonlinear terms can be ignored. For example, the system is
adequately modelled as a linear system when the amplitude of vibration is small.
In resonant devices, this is not the case since the amplitude of vibration is kept to
a very high level in order to increase the sensitivity of the device. Furthermore, the
MEMS device is made by interconnecting different subsystems which have their
own dynamics [15]. Hence, modelling the nonlinear terms is a very difficult task
which can be achieved only with a detailed knowledge of the individual subsystems.
In what follows some common sources of nonlinearity are discussed.
The sources of nonlinearity can be broadly classified as following:
1. nonlinearity in mechanical structure,
2. nonlinearity in actuation system,
3. nonlinearity in sensing (measuring) devices and
4. nonlinearity in feedback and electrical circuit.
4.1 Nonlinearity in Mechanical Structure
The nonlinear terms appearing in the equation of motion of the structural component
of the resonant device (for example, beam, plate, wire, etc.) are of following types:
(i) stiffness nonlinearity,
(ii) damping nonlinearity and
(iii) nonlinearity due to external forces.