INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING

J. Micromech. Microeng. 17 (2007) 139–146 doi:10.1088/0960-1317/17/1/018

Comments on the size effect on the

microcantilever quality factor in free air
space
Jiunn-Horng Lee1,2, Sheng-Ta Lee3, Chih-Min Yao2
and Weileun Fang1,3
1
MEMS Institute, National Tsing Hua University, Hsinchu, Taiwan
2
National Center for High-performance Computing, National Applied Research
Laboratories, Hsinchu, Taiwan
3
Power Mech. Eng. Dept., National Tsing Hua University, Hsinchu, Taiwan
E-mail: fang@pme.nthu.edu.tw

Received 19 August 2006, in final form 15 October 2006

Published 8 December 2006
Online at stacks.iop.org/JMM/17/139
Abstract
This study presents a numerical approach to investigate the size effect on the
quality factor associated with the first mode of microcantilever vibration in
1 atm air. The numerical simulation results are verified by experiments and
compared with the approximate analytical solutions. Bulk micromachined
cantilever arrays are employed as the test vehicles. Based on the
experimental and numerical results, this paper proposes a modification to the
existing approximate models for air damping analysis by taking into account
the geometry effects of the microcantilevers. The arrived semi-empirical
equation suggests that the quality factors of the microcantilevers are
approximately proportional to L−1.62 and b0.62 at a low kinetic Reynolds
number. Thus, the quality factor of the microcantilever resulting from the
free space air can be precisely predicted for design purposes.
(Some figures in this article are in colour only in the electronic version)

1. Introduction related to the free space air damping effect. Therefore, it is

important to predict the quality factor for the microcantilever
A structure will interact with the surrounding air when design.
operating in an ambient air environment. Due to the scale It remains a challenging task to derive an exact analytical
effect, the air effect on the microelectromechanical systems solution to predict the interaction between the air and
(MEMS) device cannot be ignored. The hydrodynamic force microcantilevers. The approximate analytical models have
exerted by air flow induces damping effect on the vibrating been presented in [5–7]. Most of them are based on
microstructure, and further influences the dynamic behavior the harmonic oscillating sphere theory [8], and model the
of the MEMS device. In general, the damping mechanisms microcantilevers as a sphere or a string of spheres. These
for MEMS are classified as squeeze film damping and free simplified approximate analytical models have been broadly
space damping. The squeeze film damping has attracted much applied to many areas including the development of micro
attention as it occurs in many MEMS devices such as mirrors, power generator [9, 10], study of air viscous damping effects
switches and resonators. Nevertheless, the free space damping by a Fabry–Perot micro-opto-mechanical device (FPMOD)
also plays an important role in various MEMS applications. [11], absolute pressure measurement [12], mass sensing
For instance, free vibrating microcantilevers have been resonators [13], measurement of liquid viscosity and density
widely explored in biosensors [1], chemical sensors [2], [14] and design of atomic force microscopy probes [15].
accelerometers [3] and scanning probe microscopes (SPMs) However, the geometry of the sphere-related models and the
[4]. The sensitivity and resolution of the microcantilever microcantilevers is different intrinsically. The validity of
devices strongly depend on their quality factors that are closely these simplified models needs to be further investigated. On

0960-1317/07/010139+08$30.00 © 2007 IOP Publishing Ltd Printed in the UK 139

J-H Lee et al

2. Modeling and analysis

(a) Clamped
Free end
There are various existing theoretical analyses that have been
reported to study the air damping of a vibrating microcantilever
[5, 7]. As a comparison, this study employs the numerical
approach to investigate the dynamics of a microcantilever
under the influence of ambient air.

2.1. The existing theoretical analyses

As indicated in figure 1(a), the discussed vibrating
microcantilever has a length L, width b and thickness h.
L=length
In general, the hydrodynamic force induced by ambient air
b=width
z y will introduce added mass and added damping effects on the
x vibrating microcantilever [8, 19]. The added mass that is much
Microcantilever with thickness = h smaller than the mass of cantilever beam is ignored in this
study. Thus, the dynamic characteristic of the microcantilever
vibrating at its fundamental mode can be modeled as a damped
(b)
spring–mass system [4, 20], as shown in figure 1(b). The added
keq ceq
damping effect exerted on the cantilever beam is approximated
by an equivalent damping coefficient ceq. The equation
of motion of this damped single-degree-of-freedom (SDOF)
meq
u system shown in figure 1(b) is
meq ü + ceq u̇ + keq u = F (t) (1)
F(t)
in which keq = Ebh3/(4L3) is the equivalent stiffness, meq =
Sphere 0.24ρ bhbL is the equivalent mass and the undamped natural
(c) frequency ωn is



keq h E
ωn = = 3.52 2 , (2)
meq L 12ρb
where F(t), u, E and ρ b are the external force, displacement of
1 atm air
the tip, Young’s modulus and density of the cantilever beam,
Figure 1. Schematic illustrations of (a) the numerical model of the respectively. Equation (1) can be rewritten as [20]
cantilever beam vibrating in an ambient air environment, (b) an
equivalent damped SDOF system and (c) Bead model of the 1
microcantilever.
ü + 2ζ ωn u̇ + ωn2 u = F (t), (3)
meq
where ζ = ceq/ccr is the damping ratio and ccr = 2meqωn is the
the other hand, the numerical simulation of fluid-structure critical damping. The quality factor Q of the cantilever beam
interaction (FSI) has been extensively employed in the is then defined as [20]
macro world [16], and has been successfully applied to 1 meq ωn
some MEMS applications [17, 18] as well. The numerical Q= = . (4)
2ζ ceq
calculation that minimizes the approximations could not only
simulate the dynamic behavior of the microcantilevers more Thus, the quality factor Q can be determined by the bandwidth
accurately but also provide more insight into the problem of the frequency response or by the exponential decay of
of structure–fluid interaction through scientific visualization. vibration amplitude [20].
Thus, more innovative designs of micro structures can be Due to the complicated dynamics between the coupling
inspired. of structure and air, it is still a challenge to establish the
This study presents a numerical approach, as shown in analytical model to predict the quality factor of micromachined
figure 1(a), that exploits the FSI simulation to investigate the structures. Some simplified analytical approaches have been
size effect on quality factor associated with the first mode reported to estimate the quality factor instead. For example,
of microcantilever vibrating in 1 atm air. The numerical Newell [5] has applied Stokes’ law to yield the quality factor,
simulation results are verified by experiments and compared as pointed out in equation (5)
with the approximate analytical solutions. Moreover, based √
1 bh2 Eρb
on the experimental and numerical results, this paper proposes QNewell = , (5)
24 L2 µ
a semi-empirical expression to estimate the quality factors
of the microcantilevers for engineering design purpose. In where µ is the viscosity of air. Hosaka and Itao [7] have
application, the thermal oxide microcantilever arrays with employed a bead model shown in figure 1(c) to represent the
various dimensions were fabricated and characterized. cantilever beam as a string of spheres. The damping ratio is

140
 Comments on the size effect on the microcantilever quality factor in free air space

QHosaka 2.0

Displacement (µm)
140
Quality factor
QNewell
120 1.0
100
80
0
60
40
-1.0
20
0
160 -2.0
0 5e-005 0.00015 0.00025
200
Time (sec)
240
Length 280 Figure 3. Displacement-time history of the free end predicted from
35 40 45 numerical simulation.
(µm) 32010 15 20 25 30
Width (µm)
Thus, the damping ratio as well as the quality factor can be
Figure 2. The microcantilever quality factor predicted by Newell
determined from the exponential decay of vibration amplitude
(QNewell) and Hosaka (QHosaka) for different beam widths and lengths.
[20]. The CFD-ACE+ iteratively calculated between a fluid
solver that uses finite volume method (FVM) for Navier–
thus approximated by including the effect of flow interaction Stokes equations and a structure solver that uses finite element
among spheres, and can be expressed as method (FEM) for solid mechanics to solve the multiphysics
√
3πµb + 34 πb2 2ρa µω coupling problem.
ζHosaka = , (6) The numerical FSI simulation model is shown in
4ρb hb2 ωn
figure 1(a). Because of the high aspect ratio (length/thickness)
where ρ a and ω represent the density of air and the oscillating
of the thin film structure, the solid-shell element is used
frequency of the cantilever beam, respectively. Thus, the
to model the microcantilever. The Young’s modulus and
quality factor of the cantilever beam vibrating in its first mode
density of the SiO2 microcantilever, which were measured
is
by resonance test, are E = 73.38 GPa and ρ b = 2.49 ×
1 2ρb hb2 ωn 10−15 kg µm−3. The density and viscosity of the air are ρ a =
QHosaka = = √ . (7)
2ζHosaka 3πµb + 34 πb2 2ρa µω 1.20 kg m−3 and µ = 1.85 × 10−5 kg (m s)–1, respectively. The
ambient air pressure is 1 atm. To simulate the free vibration
As the kinetic Reynolds number Rk = ρ aωb2/4µ is small, the
of the cantilever beam, an end-load was specified to give a
second term in the denominator of equation (7) is smaller than
2 µm out-of-plane tip deflection as the initial condition. A
that of the first term, hence, equation (7) can be rewritten as
√ transient vibration was simulated by the software after the
2ρb hb2 ωn bh2 Eρb initial tip deflection of the cantilever was released. Figure 3
QHosaka = = 0.22 2 (8)
3πµb L µ shows a typical simulated displacement-time history of the
which has a similar form as QNewell. free end. The damping ratio ζ can be obtained after curve
Figure 2 shows the microcantilever quality factor fitting of the simulation results in figure 3 to the exponential
predicted by Newell (QNewell) and Hosaka (QHosaka) for decay curve. The quality factor of the cantilever beam can be
different beam widths and lengths. The scaling analysis on further determined by equation (4).
equations (1) and (2) indicates that the equivalent mass meq
is proportional to (Lhb) and the resonant frequency ωn is 3. Experiments
proportional to (h/L2). According to the scaling analysis on
equations (5) and (8), the quality factors of QNewell and Q Hosaka Bulk micromachined SiO2 cantilever beams were employed
are proportional to (bh2)/L2. Consequently, the equivalent as the test structures for experiment. The quality factors
damping coefficient ceq only varies with (L1) due to the scaling of the microcantilevers were determined from the measured
analysis on equation (4). In short, ceq is proportional to (L1) frequency responses of the test structures. The SiO2 cantilever
and (b0) based on the model of [5] and [7]. However, the beams were excited by a PZT transducer and characterized
equivalent viscous damping coefficient ceq should increase as using a laser Doppler vibrometer (LDV) system together with
the width of the cantilever beam increases because the drag a dynamic signal analyzer.
force exerted on the cantilever beam is increased as the frontal In this experiment, the SiO2 microcantilevers were
area of the cantilever beam increases [21]. Thus, this study fabricated by bulk micromachining. The fabrication process
employs the numerical simulation to discuss the validity of the flow is illustrated in figure 4. The (1 0 0) single crystal silicon
simplified model of [5] and [7]. substrate was first placed in a furnace to grow a 1.15 µm
thick thermal oxide film at 1050 ◦ C, as shown in figure 4(a).
After the photolithography, the thermal oxide was patterned
2.2. The present numerical simulation
by reactive ion etching (RIE), as shown in figure 4(b). Finally,
The study employed the CFD-ACE+ commercial software to the silicon substrate was wet etched anisotropically using
simulate the dynamic behavior of SiO2 microcantilevers. The tetramethyl ammonium hydroxide (TMAH) etchant. As
FSI function was applied to simulate the free vibration of the shown in figure 4(c), the SiO2 microcantilevers were freely
SiO2 microcantilever immersed in an ambient air environment. suspended on the substrate after bulk silicon etching.

141
J-H Lee et al

(a) Grow thermal oxide

(b) Pattern oxide with RIE

(c) Bulk wet etching

Figure 4. Fabrication process flow for the test microcantilever array.

Analyzer
LDV

Specimen Dynamic signal analyzer

PZT Power
amplifier
X-Y-Z stage
Vibration isolated table

Figure 5. Experiment setup to measure the dynamic response of the

microcantilever.
Figure 7. The predicted velocity field of air flow (a) along the beam
width and (b) along the beam length.
0.00E+00

-5.00E+00

4. Results and discussion

Mag. (db)

-1.00E+01

-1.50E+01 The typical simulation results are shown in figure 7.

-2.00E+01 Figure 7(a) shows the simulated velocity field of air flow
distributed in a particular y–z plane (i.e. along the beam
-2.50E+01 width). Figure 7(b) shows the velocity field of air flow
-3.00E+01 distributed in a particular x–z plane (i.e. along the beam
1.65E+04 1.75E+04 1.85E+04 length). Table 1 lists the quality factors of the microcantilevers
Frequency (Hz) determined from the experiments. The microcantilevers
Figure 6. Typical measured frequency response of a have length ranging from 160 µm to 320 µm and width
microcantilever. ranging from 10 µm to 45 µm. The comparison of the
measured quality factor Qexp with the predicted results from
The experimental setup is shown in figure 5. The Newell QNewell, Hosaka QHosaka and numerical simulation Qsim,
specimen was attached to a PZT transducer. The dynamic respectively are shown in both figures 8(a)–(c) and table 2.
signal analyzer (HP 35670A) generated harmonic signal Figure 8(c) shows that the numerical simulation results agree
through the power amplifier to drive the PZT transducer to well with the experimental results, whereas, figures 8(a),
excite the cantilever beams. The LDV system (Graphtec AT- (b) show that the analytical results of QNewell and QHosaka
3500) measured the dynamic response at each microcantilever have significant deviation from the experiment ones at some
tip, and the measured signal was analyzed by the dynamic particular dimensions of microcantilevers.
signal analyzer. Figure 6 shows one of the typical measured As shown in equation (5), the quality factor QNewell is
frequency responses of the microcantilevers from the dynamic linearly proportional to the width b of a microcantilever.
signal analyzer. The Lorentzian curve fitting for the frequency According to equation (7), the quality factor QHosaka becomes
responses with equation (9) [22] was performed to extract the approximately linearly proportional to the width b at low
quality factors of the microcantilevers, Rk. However, it is clearly shown in figure 8 that Qexp is not
D0 (ωn /ω) linearly proportional to the width of the cantilever beam. As a
Frequency response function =  , comparison, a surface fitting was applied to the experimental
1 + Q2 (ω/ωn − ωn /ω)2
results to determine the relationship between the quality factor
(9) and the in-plane dimensions of the microcantilever. This
where D0 is a constant. study proposes an improvement over the bead model shown

142
 Comments on the size effect on the microcantilever quality factor in free air space

Table 1. Measured quality factors Qexp for the test microcantilever array.
L\b 10 15 20 25 30 35 40 45
160 32.45 38.62 42.29 46.30 48.54 50.28 51.34 52.31
180 27.41 33.52 36.44 40.18 41.76 43.82 44.68 45.54
200 23.92 28.60 31.24 34.76 36.51 38.12 39.20 40.56
220 20.67 25.25 28.14 30.69 32.34 33.66 35.00 36.28
240 18.37 22.52 25.34 27.29 29.28 30.77 31.60 32.90
260 16.18 19.90 23.10 24.97 26.16 27.81 28.96 29.84
280 14.79 17.84 20.54 22.63 23.94 25.19 26.29 27.70
300 13.11 16.47 18.63 20.32 21.93 22.91 23.81 24.67
320 12.16 15.08 17.19 18.77 20.32 21.14 22.23 23.26

b = width, L = length, unit = µm.

  
3 2
(a) 140 QNewell (ceq )Lee = A1 (3πµb) + A2 πb 2ρa µω
Quality factor

120 Qexp 4
   A3
100 L b
80 × , (10)
b L
60
40 where A1, A2 and A3 are constants, A1 is the geometry factor
20 for Stokes drag force from a sphere to a square plate, A2 is
0
160 the geometry factor for Basset history force from a sphere to
200 a square plate, Lb is the no. of sphere in the cantilever beam,
 b A3
Length 240 L
= Geff is the geometry effective factor of the cantilever
(µm) 280 45 beam.
30 35 40
32010 15 20 25
Width (µm) Substituting equation (10) into equation (4), the
(b) QHosaka expression of the quality factor becomes
140 Qexp
Quality factor

120
(0.24ρb hbL)ωn
100 QLee =   √   L   b A3 .
80 A1 (3πµb) + A2 34 πb2 2ρa µω b L
60 (11)
40
20
0 The geometry factors were determined to be A1 = 0.69, A2 =
160 0.33 and A3 = 0.38 after surface fitting of equation (11) to the
200 measurement results in figure 8. Therefore,
240
Length
280 (0.24ρb hbL)ωn
(µm) 35 40 45 QLee = 
32010 15 20 25 30  √   L   b 0.38 .
Width (µm) 0.69(3πµb) + 0.33 34 πb2 2ρa µω b L
(c)
140 (12)
Qsim
Quality factor

120 Qexp
100 Note that equation (12) is a semi-empirical expression yielded
80 from experiment over the measurement range of 0.10 < Rk <
60 10.60. The variations of the quality factors QLee and Qexp are
40
less than 3.33%, as shown in figure 9. If the A2 term in
20
0 equation (10) is ignored at low Rk, the present equivalent
160 damping coefficient (ceq)Lee of the microcantilever is
200 proportional to L0.62 and b0.38, and the quality factor QLee
240 of the cantilever beam is proportional to (b0.62h2)/L1.62. As
Length
280
(µm) 40 45 a comparison, the equivalent damping coefficient ceq in [5]
320 25 30 35
10 15 20 Width (µm) and [7] is proportional to (L1) and (b0), and the quality
factors of QNewell and Q Hosaka are proportional to (bh2)/L2.
Figure 8. Comparison of predicted and measured quality factors,
(a) QNewell versus Qexp, (b) QHosaka versus Qexp and (c) Qsim versus In conclusion, the approximate analytical models of [5] and
Qexp. [7] overestimate the beam length effect on air damping but
underestimate the beam width effect on air damping. The
geometry effective factor Geff that is proportional to (b/L)0.38
in figure 1(c) [7] by incorporating some geometry effects of plays an important role in this regard. As the beam width b
the microcantilever. Thus, the equivalent damping coefficient increases or beam length L decreases, the influence of the A2
established in this study has the form term in denominator needs to be taken into consideration. In

143
144

J-H Lee et al
Table 2. Comparison of measured quality factors Qexp with analytical results (QNewell, QHosaka) and numerical simulation results (Qsim).
10 15 20 25 30 35 40 45
Error (%) Error (%) Error (%) Error (%) Error (%) Error (%) Error (%) Error (%)
L\b EQN EQH EQS EQN EQH EQS EQN EQH EQS EQN EQH EQS EQN EQH EQS EQN EQH EQS EQN EQH EQS EQN EQH EQS
160 −51.86 71.93 14.7 −39.31 87.70 11.1 −26.11 101.56 9.7 −15.64 105.83 5.1 −3.44 113.08 4.4 8.77 119.07 3.6 21.74 125.54 3.5 34.41 130.53 3.1
180 −54.94 66.55 14.7 −44.75 78.85 9.0 −32.24 95.06 9.2 −23.20 99.08 4.3 −11.30 109.00 5.2 −1.39 113.05 3.5 10.52 120.48 4.0 22.00 126.02 3.9
200 −58.19 59.11 12.8 −47.55 76.36 10.5 −35.98 92.83 10.7 −28.08 96.17 5.3 −17.83 104.79 5.5 −8.18 110.65 4.7 2.04 116.86 4.8 10.95 119.63 3.5
220 −60.04 55.88 13.4 −50.89 70.53 9.5 −41.26 83.90 8.0 −32.68 91.85 5.1 −23.35 100.49 5.6 −14.05 107.66 5.5 −5.54 112.09 4.7 2.51 114.99 3.5
240 −62.22 50.52 12.1 −53.73 65.19 8.4 −45.19 77.43 6.4 −36.39 88.35 5.2 −28.86 94.09 4.2 −20.99 99.74 3.4 −12.09 107.15 4.3 −5.02 109.60 3.0
260 −63.41 48.21 12.7 −55.38 63.17 9.2 −48.79 70.74 4.3 −40.77 81.34 3.0 −32.15 92.05 5.0 −25.53 96.01 3.3 −18.27 101.00 3.0 −10.76 105.97 3.0
280 −65.52 41.99 10.1 −57.12 60.26 9.2 −50.34 69.91 5.4 −43.61 77.73 2.4 −36.05 87.01 3.9 −29.10 93.37 3.7 −22.37 98.36 3.4 −17.11 99.21 1.4
300 −66.13 41.50 11.5 −59.50 54.04 6.6 −52.28 66.94 5.0 −45.32 77.02 3.2 −39.22 83.13 3.2 −32.08 91.23 4.1 −25.33 97.44 4.5 −18.93 102.03 4.4
320 −67.85 35.69 8.6 −61.14 50.40 5.4 −54.57 62.30 3.3 −47.95 72.46 1.5 −42.32 78.35 1.9 −35.34 87.46 3.5 −29.73 91.72 3.0 −24.42 94.63 2.1

EQN = error of QNewell, EQH = error of QHosaka, EQS = error of Qsim.

 Comments on the size effect on the microcantilever quality factor in free air space

Table 3. Comparison of predicted and measured quality factors QLee and Qexp for a new microcantilever family.
L (µm) 142 162 182 202 222 242 262 282 302
QLee 39.37 33.28 28.60 24.90 21.93 19.49 17.47 15.77 14.32
Qexp 39.26 32.49 28.16 25.19 22.35 20.04 17.67 15.86 14.62
Error (%) 0.28 2.43 1.55 −1.14 −1.88 −2.72 −1.12 −0.57 −2.04

frequency responses of the microcantilevers. The numerical

QLee
simulations determine the quality factors using the free
Qexp
vibration model of the microcantilevers. The numerical
55 simulation results are in good agreement with the experimental
Quality factor

50 results. In addition, the velocity field of air flow is available

45
40 from the simulation. It provides valuable information while
35 designing the micromachined structures. In summary, the
30
25 numerical FSI computation is a promising method to predict
20 the dynamic characteristics of microstructures in an ambient
15 45
10 40 air environment. Based on the experimental and numerical
35
160 180
200 220 25
30 results, this study has also modified the existing approximate
240 260 20 models for air damping analysis by taking into account the
280 300 15 Width (µm)
Length (µm) 32010 geometry effects of the microcantilevers. The quality factors
Figure 9. Variation of quality factors QLee and Qexp with the width of the microcantilevers are approximately proportional to
and length of the microcantilever. L−1.62 and b0.62 at low Rk. Thus, the air damping and
quality factor of microcantilever can be precisely predicted by
40
the proposed semi-empirical expression for design purpose.
QLee However, the physical meaning of these factors and the full
Quality factor

35 Qexp applicable range of the semi-empirical equation need to be

further investigated.
30

25
Acknowledgments

20 This research is based on work supported by the National

Science Council of Taiwan under grant of NSC-94-2212-E-
15 007-026. The authors would like to thank the Central Regional
MEMS Research Center of the National Science Council, the
10
160 180 200 220 240 260 280 300 Semiconductor Research Center of the National Chiao Tung
Length (µm) University and the National Nano Device Laboratory of NSC
for providing the fabrication facilities.
Figure 10. Comparison of predicted and measured quality factors
QLee and Qexp for a new microcantilever family.
References
other words, the quality factor QLee is no longer simplified to
be proportional to (b0.62h2)/L1.62 as Rk increases. [1] Ziegler C 2004 Cantilever-based biosensors Anal. Bioanal.
Chem. 379 946–59
To demonstrate the validity of equation (12), the SiO2 [2] Lavrik N V, Sepaniak M J and Datskos P G 2004 Cantilever
microcantilever array, with thickness h = 0.97 µm, width transducers as a platform for chemical and biological
b = 25 µm and length L ranging from 142 µm to 302 µm, sensors Rev. Sci. Instrum. 75 2229–53
were fabricated and characterized. Figure 10 and table 3 show [3] Brandl M and Kempe V 2001 High performance
the quality factors of the microcantilevers determined from accelerometer based on CMOS technologies with low cost
add-ons IEEE MEMS 2001 (Interlaken, Switzerland) pp 6–9
the experiments, and predicted by equation (12) as well. It [4] Chen G Y, Warmack R J, Thundat T, Allison D P and
indicates that the results predicted by equation (12) have only Huang A 1994 Resonance response of scanning force
less than 3% deviation with the measurements. In summary, microscopy cantilevers Rev. Sci. Instrum. 65 2532–7
equation (12) can be employed to accurately predict the quality [5] Newell W E 1968 Miniaturization of tuning forks
factor of microcantilevers. Science 161 1320–6
[6] Blom F R, Bouwstra S, Elwenspoek M and Fluitman J H J
1992 Dependence of the quality factor of micromachined
5. Conclusions silicon beam resonators on pressure and geometry
J. Vacuum Sci. Technol. B 10 19–26
The size effect on the first mode quality factors of [7] Hosaka H and Itao K 1999 Theoretical and experimental study
on airflow damping of vibrating microcantilevers Trans.
microcantilevers in free air space has been investigated by ASME. J. Vib. Acoust. 121 64–9
experiments and numerical simulations. The experiments [8] Landau L D and Lifshitz E M 1987 Fluid Mechanics 2nd edn
extracted the quality factors by measuring the dynamic (New York, NY: Pergamon)

145
J-H Lee et al

[9] Mizuno M and G Chetwynd D 2003 Investigation of a cantilevers for atomic force microscopy Rev. Sci.
resonance microgenerator J. Micromech. Instrum. 67 3583–90
Microeng. 13 209–16 [16] Bathe K J and Zhang H 2004 Finite element developments for
[10] Kulah H and Najafi K 2004 An electromagnetic micro power general fluid flows with structural interactions Int. J. Numer.
generator for low-frequency environmental vibrations IEEE Methods Eng. 60 213–32
MEMS 2004 (Maastricht, Netherlands) pp 237–40 [17] Athavale M M, Yang H Q and Przekwas A 1999 Coupled
[11] Nieva P M, McGruer N E and Adams G G 2006 Air viscous fluid-thermal-structure simulations in microvalves and
damping effects in vibrating microbeams Proc. SPIE 6169 microchannels Proc. MSM’99 (San Juan, Puerto Rico)
148–59 pp 570–3
[12] Bianco S et al 2006 Silicon resonant microcantilevers for [18] Giridharan M G, Stout P, Yang H Q, Athavale M, Dionne P
absolute pressure measurement J. Vac. Sci. Technol. and Przekwas A 2001 Multi-disciplinary CAD system for
B 24 1803–9 MEMS J. Model. Simul. Microsyst. 2 43–50
[13] Vignola J F, Judge J A, Jarzynski J, Zalalutdinov M, [19] Hosaka H, Itao K and Kuroda S 1995 Damping characteristics
Houston B H and Baldwin J W 2006 Effect of viscous loss of beam-shaped micro-oscillators Sensors Actuators A
on mechanical resonators designed for mass detection Appl. 49 87–95
Phys. Lett. 88 041921 [20] Thomson W T 1993 Theory of Vibration with Applications
[14] Shih W Y, Li X, Gu H, Shih W-H and Aksay I A 2001 4th edn (Englewood Cliffs, NJ: Prentice Hall)
Simultaneous liquid viscosity and density determination [21] White F M 2002 Fluid Mechanics 5th edn (New York, NY:
with piezoelectric unimorph cantilevers J. Appl. Phys. McGraw-Hill)
89 1497–505 [22] Patil S and Dharmadhikari C V 2002 Investigation of the
[15] Walters D A, Cleveland J P, Thomson N H, Hansma P K, electrostatic forces in scanning probe microscopy at low
Wendman M A, Gurley G and Elings V 1996 Short bias voltages Surf. Interface Anal. 33 155–8

146