Nonlinear Dynamics of Resonant Microelectromechanical System (MEMS): A Review
67
4.1.1 Nonlinearity in Stiffness Terms
The stiffness or restoring force term in the equation of motion of the structure can
be nonlinear because of (i) nonlinearity in strain–displacement relationship at large
amplitude (geometric nonlinearity) [16], (ii) nonlinear constitutive relations of the
material (material nonlinearity) and (iii) impact causing sudden change in the system
behaviour [17].
In deriving the equation of motion of a vibrating structure, one needs the relationship between stress and strain components (constitutive equations) which can be
mathematically expressed as
σ i j = σ i j ( kl ) ; i, j, k, l = 1, 2, 3
Together with the relationship between strain and displacement gradient. In a linear
system, both these are expressed as linear relations, namely,
σ i j =
k
l
C i jkl kl , where C i jkl = C jikl = C i jlk = C kli j
and i j =
1
2
∂u i
∂x j
+
∂u j
∂x i
(20)
where u i (i = 2, 3) are the components of displacement field.
However, during large amplitude oscillation, significant deviations from the aforesaid relations occur. Even if the material nonlinearity can be ignored, the nonlinear
relation between i j and
∂u i
∂x j
terms shows non-trivial effects. For example, the nondimensional equation of motion of a MEMS beam fixed at both ends can be written
as [16]
∂
2
w
∂t 2 +
∂
4
w
∂x 4 −
Al
2
2I
∂
2
w
∂x 2
l
0
∂w
∂x
2
dx = 0, 0 ≤ x ≤ l
(21)
Here, even if the constitutive relations are assumed to be linear, the nonlinear relationship between strain and displacement, i.e. xx =
∂u
∂x
+
1
2
(
∂w
∂x
)
2 , where u(x, t) and
w(x, t) are axial and transverse displacements, respectively, couples the longitudinal
and transverse motion of the beam. The net effect is stretching of the neutral axis
of the beam during large deformation, a fact which is often ignored during small
amplitude oscillation.
In a cantilever beam, the above-mentioned mid-plane stretching does not take
place but the geometrical nonlinear terms exist. The equation of motion can be
written as [18]
∂ 2 w
∂t 2 +
∂ 4 w
∂s 4 +
∂
∂s
∂w
∂s
∂
∂s
∂w
∂s
∂ 2 w
∂s 2
+
∂
∂s
⎡
⎣ ∂w
∂s
s
1
∂
∂s
⎛
⎝
s
0
∂w
∂s
∂ 2 w
∂s∂t
ds
⎞
⎠ ds
⎤
⎦ = 0
(22)
where s is the distance measured from the fixed end of the beam.
67
4.1.1 Nonlinearity in Stiffness Terms
The stiffness or restoring force term in the equation of motion of the structure can
be nonlinear because of (i) nonlinearity in strain–displacement relationship at large
amplitude (geometric nonlinearity) [16], (ii) nonlinear constitutive relations of the
material (material nonlinearity) and (iii) impact causing sudden change in the system
behaviour [17].
In deriving the equation of motion of a vibrating structure, one needs the relationship between stress and strain components (constitutive equations) which can be
mathematically expressed as
σ i j = σ i j ( kl ) ; i, j, k, l = 1, 2, 3
Together with the relationship between strain and displacement gradient. In a linear
system, both these are expressed as linear relations, namely,
σ i j =
k
l
C i jkl kl , where C i jkl = C jikl = C i jlk = C kli j
and i j =
1
2
∂u i
∂x j
+
∂u j
∂x i
(20)
where u i (i = 2, 3) are the components of displacement field.
However, during large amplitude oscillation, significant deviations from the aforesaid relations occur. Even if the material nonlinearity can be ignored, the nonlinear
relation between i j and
∂u i
∂x j
terms shows non-trivial effects. For example, the nondimensional equation of motion of a MEMS beam fixed at both ends can be written
as [16]
∂
2
w
∂t 2 +
∂
4
w
∂x 4 −
Al
2
2I
∂
2
w
∂x 2
l
0
∂w
∂x
2
dx = 0, 0 ≤ x ≤ l
(21)
Here, even if the constitutive relations are assumed to be linear, the nonlinear relationship between strain and displacement, i.e. xx =
∂u
∂x
+
1
2
(
∂w
∂x
)
2 , where u(x, t) and
w(x, t) are axial and transverse displacements, respectively, couples the longitudinal
and transverse motion of the beam. The net effect is stretching of the neutral axis
of the beam during large deformation, a fact which is often ignored during small
amplitude oscillation.
In a cantilever beam, the above-mentioned mid-plane stretching does not take
place but the geometrical nonlinear terms exist. The equation of motion can be
written as [18]
∂ 2 w
∂t 2 +
∂ 4 w
∂s 4 +
∂
∂s
∂w
∂s
∂
∂s
∂w
∂s
∂ 2 w
∂s 2
+
∂
∂s
⎡
⎣ ∂w
∂s
s
1
∂
∂s
⎛
⎝
s
0
∂w
∂s
∂ 2 w
∂s∂t
ds
⎞
⎠ ds
⎤
⎦ = 0
(22)
where s is the distance measured from the fixed end of the beam.