Stresa, Italy, 26-28 April 2006
?TIMA Editions/DTIP 2006 -page- ISBN: 2-916187-03-0
INFLUENCE OF MICRO-CANTILEVER GEOMETRY AND GAP ON PULL-IN VOLTAGE
W.F. Faris*, H. M. Mohammed**, and M. M. Abdalla***, C. H. Ling**
*Department of Mechanical Engineering, Faculty of Engineering
International Islamic University Malaysia, P.O. Box 10, 50728 Kuala Lumpur, Malaysia
waleed@iiu.edu.my
**School of mechanical Engineering , University of Nottingham Jalan Broga43500 Semenyih,
Selangor Darul Ehsan Malaysia
Hazem.Dermdash@nottingham.edu.my
***Department of Aerospace Engineering, College of Engineering
Technical University of Delft, Delft, Netherland
M.M.Abdalla@lr.tudelft.nl
ABSTRACT
In this paper, we study the behaviour of a micro-
cantilever beam under electrostatic actuation using finite
difference method. This problem has a lot of applications
in MEMS based devices like accelerometers, switches
and others.
In this paper, we formulated the problem of a
cantilever beam with proof mass at its end and carried out
the finite difference solution. we studied the effects of
length, width, and the gap size on the pull-in voltage
using data that are available in the literature. Also, the
stability limit is compared with the single degree of
freedom commonly used in the earlier literature as an
approximation to calculate the pull-in voltage.
1. INTRODUCTION
Microsensors represent a large section of microsystems?
market. Microsensors offer the advantage of replacing
conventional sensors in a one to one fashion while saving
weight, energy and cost [3].
Several studies have investigated the behaviour of
electrostatically actuated microbeams in microsensors.
Yang et al.[9] modelled a clamped-clamped beam with
length 350 ?m. The model is test by passing four steps
electrostatic voltage of 21V, 22V, 25V and 30V. The
model is solved by finite differential method (FDM) then
compared with the results from the reduced model
generated by Karhunen-Loeve/ Galerkin approach. The
macromodel method saves more time (about 502.1 speed
up factor) and only contribute about 0.9% error.
Hu et al.[5] solved a model of microcantilever beam with
analytical Reileigh Riz method. The purpose of this paper
is to verify the validity of neglecting the higher terms in
the electrostatic force term. The result shows that it only
valid when the applied voltages are below the pull in
voltage. Hu also do a dynamic analysis of the beam
subjected to an AC bias using Runge Kutta method to
solve ordinary differential equation. Hu found that the
resonant frequencies decrease with the increasing
magnitude of applying voltage. Nayfeh and Younis[8]
presented a new approach to the modelling and
simulation of flexible microstructures under the effect of
squeeze-film damping. They applied perturbation
methods to the compressible Reynolds equation and
solved the equation using finite element method. The
results compared to which get from experiment are
acceptable. The further analysis was done on fully
clamped and clamped-free-clamped-free plate. Abdel-
Rahman et al.[1] modelled a plate clamped at both ends
and free along its width. They used shooting method to
solve and compared the results from previous results.
There are good agreement between them. At last, they
concluded that one may use ?
1
(6 x gap
distance/thickness) or axial force to stiffen the
microbeam. However, only ?
1
are used to tune the
relationship between the natural frequency and voltage.
Collenz et al.[2] developed an alternative approach based
on a sequential field-coupling (SFC) algorithm to deal
with strongly non-conservative electrostatic loads.
Collenz modelled a cantilever beam with length to gap
ratio of one (l/g =1). Collenz found that pull in cannot
occur because the gap is as big as the beam length,
therefore the beam tip cannot touch the underlying plane.
Hung and Senturia [6] focused on obtaining macromodels
based on global basis functions generation from an
approach that is mathematically equivalent to Karhunen-
Loeve/ Galerkin analysis of a small but representative
ensemble of dynamic FEM runs. They found that
macromodel speed up simulation by a factor of 37 over a
FDM with less than 2% error.
Faris and Abdalla [4] used Galerkin approximation
approach to solve MEMS based sensor under thermal
loading.
W. Faris, H.M. Mohammed, M.M. Abdalla, and C. H. Ling
INFLUENCE OF MICRO-CANTILEVER GEOMETRY AND GAP ON PULL-IN VOLTAGE.
?TIMA Editions/DTIP 2006 -page- ISBN: 2-916187-03-0
Finite difference was not exploited in the solution of
MEMS based problems, though in our opinion it is quite
effective for a domain of problems which are easily
formulated mathematically.
2. MODELING OF A MICRO-BEAM
Figure 1 Cantilever Beam with a Lumped Mass at the End
Our model is a cantilever beam with a proof mass
suspended at the end as shown in Figure 1. We apply
Hamilton principle in developing models to analyze beam
behaviour. Hamilton principle is a consideration of the
motion of an entire system between two times, t
1
and t
2
.
The extended Hamilton?s principle:
2
1
()0
t
nc
t
TVWdt????+ =
?
, (,) 0,yxt? = 0 x L??, 1, 2ttt=
(1)
We start derive the beam bending equation by write the
kinetic energy expansion of the beam:
2
2
0
1 (,) 1 ( ,)
()
22
L
yxt yLt
Tmx dxM
tt
???? ??
=+
?? ??
?? ??
?
, (2)
and the strain energy (internal potential) is stated as:
2
2
2
0
1(,)
() ( )
2
L
yxt
Vt EIx dx
x
???
=
??
?
??
?
, (3)
The virtual work of the nonconservative distributed
forces is expressed as:
0
() ( ,) ( ,)nc
L
Wt fxt yxtdx??=
?
(4)
Substituting into Hamilton?s principle (1), we obtain:
22 2
(,) (,)
() (,) ()
yxt yxt
EI x f x t m x
xx
???? ?
?+=
??
??
(5a)
and the boundary conditions at 0x =
(,)
( , ) 0, 0,
yxt
yxt
x
?
==
?
(5b)
and at x L=
222
(,) (,) (,)
( ) 0, ( ) 0,
yxt yxt yxt
EI x EI x M
xx
??????
=?=
??
??
(5c)
In the rest of the paper, we assume the I(x) and m(x) are
constant over the length of the beam.
It is our interest to consider the effect of electrostatic
forces on the response of the microbeam. The
electrostatic load depends on the beam deflection as:
2
2
()
2( )
bV
fx
Gy
?
=
?
(6a)
Where: E = Young Modulus, m = mass per unit length of
beam, I = bh
3
/12 = cross section area moment of inertia, b
= width of beam, h = height of beam, G = gap distance, ?
= free space permittivity or dielectric constant of vacuum,
t = time.
As a result, equation (5) can be expressed as:
422
(,) (,)
2( )
yxt bV yxt
EI m
xGy t
???
?+=
??
(6b)
3. SYSTEM GOVERNING EQUATIONS
We consider a cantilever beam with a proof mass
suspended at the end, actuated by an electrostatic force
which consists of a DC component, V
p
and an AC
component v(t). We assume that the transverse deflection
of beam, y, is constant along the width of beam. The
beam equation and its boundary conditions at 0x = and
at x L= can be expressed as:
()
2
42
2
()
(,) (,)
2( )
p
bV vt
yxt yxt
EI m
xtGy
?+
??
+=
?
(7)
(,)
( , ) 0, 0, 0
yxt
yxt x
x
?
===
?
(7a)
232
(,) (,) (,)
0, 0,
yxt yxt yxt
EI EI M x L
x xt
???
=?==
(7b)
Where: E = Young Modulus, m = mass per unit length of
beam, I = bh
3
/12 = cross section area moment of inertia, b
= width of beam, h = height of beam, G = gap distance, ?
= free space permittivity or dielectric constant of vacuum,
t = time.
The microbeam deflection under an electric force is
composed of static component due to the DC voltage,
termed as ()
s
yx and dynamic component due to the AC
voltage, termed as (),uxt that is:
W. Faris, H.M. Mohammed, M.M. Abdalla, and C. H. Ling
INFLUENCE OF MICRO-CANTILEVER GEOMETRY AND GAP ON PULL-IN VOLTAGE.
?TIMA Editions/DTIP 2006 -page- ISBN: 2-916187-03-0
() () (),,,yxt y xt uxt
s
=+ (8)
As a result, the beam equation can be modified as
()
2
4 42
44 2 2
()
2( )
p
s
s
bV vt
y uu
EI m
x xtGyu
?+
??? ??
++ =
??
?? ? ??
??
(9)
To calculate static deflection,
s
y , we put time derivation
and the AC forcing term in equation (9) equal to zero and
obtain
2
4
42
2( )
p
s
s
bV
y
EI
x Gy
?
?
=
??
(10)
0, 0, at 0
s
s
y
yx
x
?
== =
?
(10a)
232
(,)
0, 0, at
ss
yyyxt
EIEIM xL
x xt
???
=?==
(10b)
To solve the eigenvalue problem, we set AC forcing term
in equation (9) equal to zero and use equation (10) to
eliminate the term representing equilibrium position.
22
42
242 2
2( ) 2( )
pp
ss
bV bV
uu
EI m
Gy x t Gy u
??
??
++=
?????
(11)
Expanding the nonlinear electrostatic force term by
Taylor series with respect to equilibrium position, 0u = ,
gives
22
2
34
12 3
...
2( )2()()()
pp
ssss
bV bV
uu
Gy u Gy Gy Gy
????
=+++
??
?? ? ? ?
??
(12)
Based on the small displacement assumption, the higher
order terms can be neglected, thus the nonlinear
electrostatic force can be linearized as
22
3
12
2( ) 2 ( ) ( )
pp
sss
bV bV
u
Gy u Gy Gy
????
=+
??
?? ? ?
??
(13)
Substituting equation (13) into equation (11) results in
2
42
42 3
0
()
p
s
bV
uu
EI m u
xtGy
?
??
+? =
?? ?
(14)
.
3. FINITE DIFFERENCE SOLUTION
Refer to the grid below, by assuming that at 0x = , 3i =
and at 0t = ,
4j =
. Hence, for 4,j ? ,
,
0
ij
y = since the
beam is not deflected at 0t ? .
Figure 2 Grid Point for the Dynamic Behavior Proble
()
2
,2, 1, 1, 2,42
,
,,4 ,3 ,2 ,12
464
2
11 56 114 104 35
12
t
ijij ij ij ij
ij
ijij ij ij ij
bvEI
yyyyy
h
Gy
m
yy y yy
k
?
?? ++
?? ? ?
??
??
? ?
? ?
??+? +
?
=?+ +
(15)
At x = 0,
,
0;
ij
y = (15a)
2, 1, 1, 2,
88 0
ij ij ij ij
yyyy
??++
??+ = (15b)
At x = L,
,2, 1, 1, 2,2
16 30 16 0
12
ijij ij ij ij
EI
yyyyy
h
?? ++
??
??
?+ ? + ? =
(15c)
2, 1, 1, 2,3
,,4 ,3 ,2 ,12
22
2
=1 5611410435
12
ij ij ij ij
ijij ij ij ij
EI
yyyy
h
M
yy y yy
k
??++
?? ? ?
? ?
? ?
?+ ? +
?+ +
(15d)
According to the assumption stated above, at 0x = ,
3i = , from the boundary condition (22a) and (22b), we
get :
3,
0
j
y = , (16a)
5,1, 2, 4,
88 0
jjjj
yy yy?+?=
(16b)
and from the beam equation at 0x = or 3i = ,
1, 2, 4, 5,
44 0
jjjj
yyyy??+= (16c)
Rearrange equation (16), we get the equation
for
1, 2, 3,
, and
jj j
yy y, which are
1, 2, 4, 5,
44-
j jjj
yyyy=+ (17a)
2,1 1, 4, 5,
1
8
8
jjj
yyyy? ?=+?
? ?
(17b)
3,
0
j
y =
, (17c)
W. Faris, H.M. Mohammed, M.M. Abdalla, and C. H. Ling
INFLUENCE OF MICRO-CANTILEVER GEOMETRY AND GAP ON PULL-IN VOLTAGE.
?TIMA Editions/DTIP 2006 -page- ISBN: 2-916187-03-0
Solving equation (15) for 4 2,iN?? ? we get
()
2
44
-2, -1, 1, 2,24
1
2
,
,
635
42
11 56 114 10412
,-4 ,-3 ,-2 ,-12
12
bv EI
t
yyyy
ij ij ij i j
h
Gy
y ij
ij
EI M
m
yy y yhk
ij ij ij ij
k
?
???+
++
?
=
+
??+?
??
??
??
??
??
??
??
??
??
??
(18)
In addition, for 2,iN=?we have to consider another
two boundary condition to find
1,Nj
y
?
and
,Nj
y . From
equation (15c) and equation (15d), we get
1, ,4, 3, 2,
1
16
16 30
Nj NjNj Nj Nj
y yyyy
? ???
??
??
=?++
(19a)
22
, 4, 3, 1,
11 563
2, 4 2, 32
2
114 104 35
12
2, 2 2, 1 2,
yy y y
Nj Nj Nj Nj
yy
Nj NjMh
y
kEI
Nj Nj Nj
=?+
???
?
?? ??
+
+?+
?? ?? ?
??
??
(19b)
To facilitate in programming, we assume that
A
4
EI
h
=
,
2
B
2
t
bv?
=
,
2
C
12
m
k
=
,
2
D
12
M
k
=
and
3
F
2
EI
h
=
. So
the equations describe
,ij
y for 4j ? is expressed
below.
5,1, 2, 4,
88
jjjj
yyyy=?+ (20a)
2, 1, 4, 5,
1
4
4
jjjj
yyyy??=?+
??
(20b)
3,
0
j
y = (20c)
for 4 2,iN?? ?
()
()
2
44
-2, -1, 1, 2,
1
,
,
635
11 56 114 104
,-4 ,-3 ,-2 ,-1
Ay y y y
ij ij ij i j
Gy
ij
y
ij
AC
Cy y y y
ij ij ij ij
B??
??
???+
????++
??
?
=
??
+
??+?
??
(20d)
1
16 30
,1, 4, 3, 2,
16
yyyyy
NjNj Nj Nj Nj
??
=?++
??????
??
(20e)
, 4, 3, 1,
2, 4 2, 3 2, 2 2, 1 2,
22
11 56 114 104 35
Nj Nj Nj Nj
N jNj Nj NjNj
y yyy
D
yy y yy
F
???
?? ?? ?? ?? ?
=
+
?+
?+ ? +
(20f)
.
4. RESULTS AND ANALYSIS
In this paper, we study the behavior of a microbeam
under a nonlinear electrostatic force. First, we present a
numerical procedure to solve the static behaviour or
boundary value problem of a microbeam under DC
electrostatic force. We are interested in finding the pull-in
voltages corresponding to static deflection at different
beam lengths.
Second, we determine the natural frequencies and mode
shapes of microbeam under DC electrostatic force.
Equations (10) describe the microbeam static deflection
under static electrostatic force. We use finite difference
method to solve the problem numerically for
s
y .
2
4
42
2( )
p
s
s
bV
y
EI
x Gy
?
?
=
??
(21)
0, 0, at 0
s
s
y
yx
x
?
== =
?
(21a)
232
(,)
0, 0, at
ss
yyyxt
EI EI M x L
x xt
???
=?==
(21b)
We initially set
0
s
y as zero and iterate until it converge
to a value within 0.1%. Considering a cantilever beam
with a mass suspended at the end made of silicon. The
geometries and material parameter are given in table 1.
Parameters Values
Length (L) 300 ?m
Width (W) 50 ?m
Thickness (t) 3 ?m
Initial Gap Distance (G) 3 ?m
Young Modulus (E) 160 x 10
9
kg/m
2
Density (?) 2330 kg/m
3
Free Space Permittivity (?) 8.8541878 x 10
-12
(F/m)
Table 1 The Geometric and Material Parameters of the
Microbeam
We solve equations (21) for a range of electrostatic forces
by changing the applied voltage. Then, we will change
the length (L) to see its effect on static with respect to
applied voltages (V
p
).
Figure 3 shows the deflection along the microbeam with
dimensions L = 300 ?m, W = 50 ?m, t = 3 ?m, G = 3
?m at voltage = 10 V. The beam is fixed at one end. The
more distance the point from the fixed end is, the larger
will be the deflection.
W. Faris, H.M. Mohammed, M.M. Abdalla, and C. H. Ling
INFLUENCE OF MICRO-CANTILEVER GEOMETRY AND GAP ON PULL-IN VOLTAGE.
?TIMA Editions/DTIP 2006 -page- ISBN: 2-916187-03-0
Figure 3 Static Deflection of Microbeam
( L = 300 ?m, W = 50 ?m, t = 3 ?m, G = 3 ?m and
Voltage = 10 V)
Figure 4 shows the variation of the maximum
static deflection of microbeam,
max
()
ss
WYyxL== =,
with applied voltages for L=200 ?m, 225 ?m, 250 ?m,
275 ?m and 300 ?m. Other geometry dimensions are
remaining same. The dashed line represents the stability
limit, W
max
= /3G predicted by the single-degree-of-
freedom spring-mass model. As applied voltage
increases, the maximum deflection increases. The slope
of maximum deflection,
max p
yV??
, increases as the
voltage increases and finally approach infinity when it
reaches pull in voltage. Beside this, maximum deflection
is also larger for longer beam. This relationship is linear
at low voltage and becomes increasingly nonlinear when
the voltage applied is increased. The slope of maximum
deflection,
max p
yV??, is larger for longer beam at
specific voltage. As a result, microbeam with shorter
length can sustain larger voltage before it collapses.
Variations of the maximum deflection with
voltage for various thicknesses are shown in figure 5. As
voltage increases, static deflection increases. At specific
voltage, maximum deflection increasing as thickness is
reduced. The slope of maximum deflection,
max p
yV??
,
increases more rapidly for thin beam. This proves that the
pull in voltage is larger for thicker beam.
Figure 4 Variation of maximum Deflection with applied
Voltage until Pull-In for L=200 ?m, 225 ?m, 250 ?m,
275 ?m and 300 ?m.
Figure 5 Variation of the Maximum Deflection with
Voltage until Pull-In for t=2 ?m, 2.5 ?m, 3 ?m, 3.5 ?m
and 4 ?m.
Gap distance has influence on the static deflection too.
As represent in Figure 6, static deflection will be reducing
when the gap between microbeam and support is
increased.
.
W. Faris, H.M. Mohammed, M.M. Abdalla, and C. H. Ling
INFLUENCE OF MICRO-CANTILEVER GEOMETRY AND GAP ON PULL-IN VOLTAGE.
?TIMA Editions/DTIP 2006 -page- ISBN: 2-916187-03-0
Figure 6 Variation of the Maximum Deflection with
Voltage for until Pull-In for G=2 ?m, 2.5 ?m, 3 ?m, 3.5
?m and 4 ?m.
.
5. CONCLUSIONS
In this paper, we formulate the problem of a cantilever
beam with proof mass at the end which is used in many
applications, most notably is MEMS based accelerometer.
We used Hamiltonian principle as a tool of this
formulation. Also, we explained the solution using finite
difference technique. We solved the static part of the
problem and decided the pull-in voltage for different
geometric parameters of the beam. The parameters
studied were beam length, thickness, width, and gap. The
finite difference method proved to be effective and easy
to code for solving such problems. The length and
thickness were found to have a great influence on the
pull-in voltage while the width was found of no influence
on the pull-in voltage. The gap also was found to have a
significant influence on the pull-in results as well.
Also, it was found that the stability limits from the single
degree of freedom approximation is quite erroneous
compared to full geometry calculations.
.
6. ACKNOWLEDGEMENT
The first author would like to thank the Research Center
at IIUM for the financial support to carry out this work..
7. REFERENCES
[1] Abdel-Rahman, E. M., Younis, M.I., and Nayfeh, A. H., ?
Characterization of the Mechanical Behavior of an Electrically
Actuated Microbeam,? J. Micromech. Microeng. 12 (2002) 759-
766.
[2] Collenz, A., De Bonal, F., Gugliotta, A., and Soma, A.,
?large Deflections of Microbeams under Electrostatic Loads,? J.
Micromech. Microeng. 14 (2004) 365-373.
[3] Gad El-Hak, ?The MEMS Handbook,? CRC Press, Boca
Raton, Florida.
[4] Faris, W. and Abdalla, M. ?Effect of Temperature and
geometric Nonlinearity in MEMS-Based Diaphragm Devices
Modelling? Fifth EUROMECH Nonlinear Dynamics
Conference (ENOC 2005), August 7-12, 2005, Eindhoven,
Netherland.
[5] Hu, Y. C., Chang, C.M., and Huang, S.C.,? Some Design
Considerations on the Electrostatically Actuated
Microstructures,? Sensors and Actuators A 112 (2004) 155-161.
[6] Hung, E. S. and Senturia, S. D,? Generating Efficient
Dynamical Models for Microelectromechanical Systems from
Few Finite Element Simulation Runs,? IEEE J. Microelect.
Syst., 8(3) (1999).
[7] Lim, M. K., Du, H., Su, C., Jin, W.L.,?A Micromachined
Piezoresistive Accelerometer with High Sensitivity: Design and
Modelling,? Microelectronic Engineering 49 (1999) 263-272.
[8] Nayfeh, A. H. and Younis, M. I.,? A New Approach to the
Modeling and Simulation of Flexible Microstructures Under the
Effect of Squeeze Film Damping,? J. Micromech. Microeng
14(2004) 170-181.
[9] Yang, Y-J., Cheng, S-Y, and Shen, K-Y,? macromodeling of
Coupled-Domain MEMS Devices with Electrostatic and
Electrothermal Effects,?J. Micromech. Microeng. 14 (2004)
1190-1196.
Appendix
One-Dimensional Pull-In Voltage Analysis
When a voltage is applied between two
electrodes, the electrostatic force between them pulls
them together. The restoring force of the beam, arising
from the beam?s stiffness, resists the electrostatic force.
When the voltage is increased, a point is reached where
the electrostatic force equals the spring restoring force of
the beam. If this point, known as pull-in voltage, is
passed, the beam will snap to the substrate.
In one-dimensional geometries, the pull-in
phenomenon is greatly simplified and the basic
dependencies are easily visible. Now, the displacement
field is just a scalar y, and the mechanical force,
m
F and
electric force,
e
F are algebraic functions of u. The pull-in
position is determined by the equality of the forces and
their derivatives,
me
FF=
(1)
W. Faris, H.M. Mohammed, M.M. Abdalla, and C. H. Ling
INFLUENCE OF MICRO-CANTILEVER GEOMETRY AND GAP ON PULL-IN VOLTAGE.
?TIMA Editions/DTIP 2006 -page- ISBN: 2-916187-03-0
me
F F
yy
??
=
??
(2)
Dividing the two equations we obtain
me
me
F F
KK
=
(3)
where we have defined the electric and mechanical spring
constants, e
e
F
K
y
?
=
?
and m
m
F
K
y
?
=
?
respectively and it is
valid for all voltage driven electro-mechanical systems. It
may equally well be written as
e
m
F
y
F
?=
(4)
where
mm
yK F? =
describes the nonlinearity of the
elasticity. For linear cases
1? = by definition. Equation
(4) provides a starting point for the pull-in iteration
scheme.
In the one-dimensional case the electric field has
an analytical solution, ()EVGy=?, where G is the
initial aperture between the mass and the ground level.
The electric energy now yields
2
1
2
V
A
Gy
?? =
?
(5)
where A is the area of the capacitor. The electric force
and spring constant with fixed potential may be obtained
by differentiations,
()
2
2
1
,
2
e
V
FA
y
Gy
?
??
==
?
?
and
()
22
32e
V
KA
y
Gy
?
??
==
?
?
(6)
Inserting the above formulas in Equation (4) and
solving for the pull-in position gives 3yG= . The
relative displacement of 1/3 is a characteristic value for
this one-dimensional case. Also more complicated cases
have values independent of the scale and of the material
parameters.
The pull-in voltage can be obtained from
equation (1) by equating electric and mechanical spring
constants,
em
K K=
.
()
2
3
23
PI
m
V
A K
G
? =
(7)
3
8
27
m
PI
GK
V
A?
=
(8)