07
The mass-spring model of electrostatically actuated microelectromechanical sys-
tems (MEMS) or nanoelectromechanical systems (NEMS) is pervasive in the
MEMS and NEMS literature. Nonetheless a rigorous analysis of this model does
not exist. Here periodic solutions of the canonical mass-spring model in the vis-
cosity dominated time harmonic regime are studied. Ranges of the dimensionless
average applied voltage and dimensionless frequency of voltage variation are delin-
eated such that periodic solutions exist. Parameter ranges where such solutions fail
to exist are identied; this provides a dynamic analog to the static pull-in insta-
bility well known to MEMS/NEMS researchers.
KEY WORDS: MEMS; nanotechnology; electrostatics; periodic solutions;
saddle-node bifurcation; shooting method.
MATHEMATICS SUBJECT CLASSIFICATION (1991): 34C15; 34C60;
70K40.
1. INTRODUCTION
As the characteristic length of engineering systems approaches the micro
or nanometer scale the role of electrostatics grows correspondingly. Often
perceived as a nuisance in the macro-world, such as in the case of the
destruction of sensitive electronic circuits due to electrostatic discharge
(ESD),
4
electrostatic forces are increasingly being used to provide accurate,
1
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville,
AL 35899, USA. E-mail: ais@email.uah.edu
2
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA.
E-mail: pelesko@math.udel.edu
3
To whom correspondence should be addressed.
4
ESD is no laughing matter. Before the advent of nonammable anesthetics an errant
spark from a doctors scalpel would sometimes ignite the ether in a patients lungs,
rendering the operation a failure.
© 2007 Springer Science+Business Media, LLC Ai and Pelesko
controlled, stable locomotion for micro and nanoelectromechancical
systems (MEMS or NEMS). In this approach, voltage differences are
applied between mechanical components of the system. This induces a
Coulomb force between components which is varied in strength by vary-
ing the applied voltage. This technique is already employed in devices such
as accelerometers [4], optical switches [5], microgrippers [7], micro force
gauges [19], transducers [3], micro pumps [16] and nanotweezers [12].
In order to understand the operation of such devices researchers in the
MEMS and NEMS communities have relied upon idealized mathematical
models. The typical approach, rst introduced into the literature by Nathan-
son in 1967 [14], is to create a lumped mass-spring model. Here, the elastic
behavior of the system is represented by a linear spring while electrostatic
forces are computed using a simple parallel plate capacitor approximation.
This mass-spring model has persisted in the MEMS/NEMS literature and
has been rediscovered and discussed by numerous authors, [6, 8, 10, 12, 13,
15, 17, 18]. Nevertheless, the mathematical analysis of this canonical model
has remained primitive. Typically, authors have restricted their attention to
steady-state solutions [12], relied upon numerical simulation for dynamical
information [6], or have utilized perturbation methods to study approximate
dynamics in some region of parameter space [18].
In this paper, we begin to remedy this situation by providing a rig-
orous analysis of the viscosity dominated time harmonically forced mass-
spring MEMS/NEMS model. While this analysis does not capture the
dynamics of every possible MEMS or NEMS device, it is relevant for the
study of devices, such as micropumps [16], microgrippers [7], or nano-
tweezers [12], which operate in the viscosity dominated regime. In Sec-
tion 2, for the convenience of the reader we provide a brief derivation of
the model. In Section 3, we consider the situation where inertial forces
are completely negligible. In this case, the model is reduced to a nonlin-
ear rst order non-autonomous ordinary differential equation. We study
the existence of periodic solutions to this equation. We determine ranges
of the dimensionless applied voltage and dimensionless forcing frequency
for which such solutions exist. We show that outside of these ranges the
model has no solution. This is the dynamic analog of the static instability
well known to MEMS/NEMS researchers as the pull-in or snap-down
instability. In this instability, when a constant applied voltage is increased
beyond a certain critical voltage there is no longer a steady-state cong-
uration of the device where mechanical members remain separate. Here
in the dynamic situation, the device cannot be operated in an oscillatory
mode if the mean applied voltage is too large or the forcing frequency
is too small. In Section 4, we consider a situation where inertial forces
are small, but non-negligible. In this case, the model becomes a nonlinear Dynamics of Canonical MEMS/NEMS Model
Dashpot
Mass, m, at
Potential, V
u
L
Area, A
Grounded plate
Figure 1.
Sketch of the damped mass-spring system.
second order non-autonomous ordinary differential equation. Again we
investigate periodic solutions of this equation and determine criteria neces-
sary for such solutions to exist. Finally, in Section 5, we discuss the impli-
cations of our analysis for MEMS/NEMS device behavior.
2. FORMULATION OF THE CANONICAL MODEL
The system sketched in Fig. 1 represents a lumped approximation
of a typical electrostatically actuated MEMS/NEMS device. The governing
equation for this system is
m d
2
x
dt
2
= F
s
+ F
d
+ F
e
.
(2.1)
Here, x is the displacement of the top plate from the top wall and m is
the top plates mass. We assume that the bottom plate is held in place. The
forces acting on our system are the spring force, F
s
, a damping force rep-
resented by the dashpot in Fig. 1, F
d
, and the electrostatic force, F
e
, due
to the applied voltage difference between the plates. We assume that the
spring is a linear spring and follows Hookes law
F
s
= k(x l)
(2.2)
where l is the rest length of the spring and k is the spring constant. We
assume that damping is linearly proportional to the velocity, that is
F
d
= a dx
dt
(2.3) Ai and Pelesko
and compute the electrostatic force by treating the plates in Fig. 1 as in-
nite parallel plates. This yields
F
e
= 12
0
AV
2
(L x)
2
cos
2
(t ).
(2.4)
Here, 0
is the permittivity of free space, A is the area of the plates, V is
the average applied voltage, and
is the frequency at which the applied
voltage is varied. Inserting equations (2.2), (2.3) and (2.4) into equation
(2.1) yields
m d
2
x
dt
2
+ a dx
dt + k(x l) =
1
2 0
AV
2
(L x)
2
cos
2
(t ).
(2.5)
We recast this equation in dimensionless form by introducing a dimension-
less length scale
y
= x l
L
l
(2.6)
and dimensionless time scale
t
= kat .
(2.7)
Introducing equations (2.6) and (2.7) into equation (2.5) yields
1 2
d
2
y
dt
2
+ dy
dt + y = (1 y)
2
cos
2
(t),
(2.8)
where 2
= a
2
mk ,
= a
k ,
=
0
AV
2
2</b><i>k
(L l)
3
.
The dimensionless parameter
may be interpreted as a damping coef-
cient which measures the relative strength of the viscous damping force
as compared to the spring force. The dimensionless parameter
measures
the relative strength of electrostatic and elastic forces in our system. The
dimensionless parameter
is the ratio of damping and forcing time scales.
When damping effects dominate over inertial effects, we expect the
parameter
to be large. If inertial effects are completely negligible, we
send
and study the reduced model
d y
dt + y = (1 y)
2
cos
2
(t).
(2.9)
This situation is studied in the next section. In Section 4, we return to the
case where
is large, but inertial effects are not completely negligible in
(2.5). Dynamics of Canonical MEMS/NEMS Model
3. THE VISCOSITY DOMINATED CASE
In this section we study the 2
/-periodic solutions of equation (2.9).
We restrict our attention to solutions where y
<1 as y >1 implies that the
top plate in Fig. 1 has passed through the bottom plate. It is convenient
to introduce the following scalings:
u
(t) := [1 y(t/3)]
3
, := 2/3,
and hence the problem reduces to nding solutions u, 0
< u < 1, of the
periodic boundary value problem
u (t) = u
2
/3
(t) u(t) cos
2
(t/2),
(3.1)
u
(0) = u(2).
(3.2)
Since any solution u of (3.1)(3.2) satises u
(t) < 1 for t R (see Lemma
3.1 (i)), it sufces to study the positive solutions of (3.1)(3.2). We note
that, due to the non-differentiable term u
2
/3
at u
= 0, the initial value
problem of (3.1) at t
= with u() = 0 loses uniqueness on the side
t
for sufciently small > 0, and results in the existence of non-
negative solutions of (3.1)(3.2) which become 0 at t
= in [0, 2].
Although these solutions are not physically meaningful for (2.9), for com-
pleteness we include them in the following main result:
Theorem 3.1. For any >
0
:= 2/3, there exist a unique number 0
() (
16
6561 2
,
16
243 2
) and a unique number
b
() (4/27, 8/27) with
lim
b
() = 8/27 such that
(i)
If 0
< <
b
, or
=
b
, or
>
b
, then (3.1)(3.2) has exactly
two, one, no positive solutions, respectively, and does not have any
other non-negative solutions.
(ii)
If 0
<
0
, then (3.1)(3.2) has exactly two nonnegative solu-
tions: one is strictly positive, the other reaches zero only at t
=
in
[0, 2].
Remark 3.1. 0
= 2/3 is not optimal. Note that
16
243 2
< 4/27 if >
2
/3. For each >
0
, b
() is the bifurcation value resulting from the sad-
dle-node bifurcation of periodic solutions of (3.1). It can be shown that

b
() is a smooth function for (
0
, ). For each >
0
, the existence
of 0
() is due to the fact th