Nonlinear Dynamics of Resonant Microelectromechanical System (MEMS): A Review
69
what is known as squeeze-film damping. As the amplitude of response increases, the
effect of nonlinearity also becomes significant. For example, the equation of motion
of a pined-pined beam vibrating near a fixed plate is
ρA
∂
2
w
∂t 2 + c
∂w
∂t
+ E I
∂
4
w
∂x 4 −
E A
2L
⎡
⎣
l
0
∂w
∂x
2
dx
⎤
⎦ ∂
2
w
∂x 2 =
b
2
−
b
2
p dy
(24)
where p(x, y, t) is the pressure distribution on the beam surface which has width b.
The pressure is calculated by solving the nonlinear version of Reynolds’s equation
·
w
3 p p
=
12μ
1 + 6K
∂ ( pw 0 )
∂t
(25)
where μ is the viscosity of the surrounding fluid and K is the Knudsen number
(=mean free path of the gas/w). When the amplitude of vibration is small the damping
is usually considered to be linear with damping coefficient, C squeeze =
b
3
d 3 lμ, where b,
d, l and μ are, respectively, the beam width, gap between plates, length of the plates
and air/fluid viscosity. When the amplitude of vibration increases, the nonlinearity
of damping is modelled by considering the damping coefficient as [16, 22]
C squeeze =
α
(d − w) 3
(26)
4.1.3 Nonlinear External Forces
As the dimensions of the structure are reduced, some kinds of interaction forces with
the nearby objects become pronounced which can introduce nonlinearity. For example, in scanning probe microscopy, the tip of the microcantilever often experiences
electrostatic interactions or attractive van der Waals forces from the surface under
observation. Most of the interaction forces are highly nonlinear function of the separating distance between tip and sample [23]. A very general model for tip–sample
interaction force is given by
F tip =
−
C
d α , for d > a 0
−
C
a
α
0
+ D(a 0 − d), for d ≤ a 0
(27)
Here C, D, d and a 0 are general attractive force parameters, tip–sample separation and
intermolecular distance at which contact is considered to be initiated. The parameters
C and D are functions of tip radius. The forces are dependent on material of tip and
sample. They are attractive (repulsive) for long (short) range.
69
what is known as squeeze-film damping. As the amplitude of response increases, the
effect of nonlinearity also becomes significant. For example, the equation of motion
of a pined-pined beam vibrating near a fixed plate is
ρA
∂
2
w
∂t 2 + c
∂w
∂t
+ E I
∂
4
w
∂x 4 −
E A
2L
⎡
⎣
l
0
∂w
∂x
2
dx
⎤
⎦ ∂
2
w
∂x 2 =
b
2
−
b
2
p dy
(24)
where p(x, y, t) is the pressure distribution on the beam surface which has width b.
The pressure is calculated by solving the nonlinear version of Reynolds’s equation
·
w
3 p p
=
12μ
1 + 6K
∂ ( pw 0 )
∂t
(25)
where μ is the viscosity of the surrounding fluid and K is the Knudsen number
(=mean free path of the gas/w). When the amplitude of vibration is small the damping
is usually considered to be linear with damping coefficient, C squeeze =
b
3
d 3 lμ, where b,
d, l and μ are, respectively, the beam width, gap between plates, length of the plates
and air/fluid viscosity. When the amplitude of vibration increases, the nonlinearity
of damping is modelled by considering the damping coefficient as [16, 22]
C squeeze =
α
(d − w) 3
(26)
4.1.3 Nonlinear External Forces
As the dimensions of the structure are reduced, some kinds of interaction forces with
the nearby objects become pronounced which can introduce nonlinearity. For example, in scanning probe microscopy, the tip of the microcantilever often experiences
electrostatic interactions or attractive van der Waals forces from the surface under
observation. Most of the interaction forces are highly nonlinear function of the separating distance between tip and sample [23]. A very general model for tip–sample
interaction force is given by
F tip =
−
C
d α , for d > a 0
−
C
a
α
0
+ D(a 0 − d), for d ≤ a 0
(27)
Here C, D, d and a 0 are general attractive force parameters, tip–sample separation and
intermolecular distance at which contact is considered to be initiated. The parameters
C and D are functions of tip radius. The forces are dependent on material of tip and
sample. They are attractive (repulsive) for long (short) range.