effect of flexural modes on squeeze film damping in mems cantilever resonators
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Effect of flexural modes on squeeze film damping in MEMS cantilever resonators
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2007 J. Micromech. Microeng. 17 2475
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IOP PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING
J. Micromech. Microeng. 17 (2007) 2475–2484 doi:10.1088/0960-1317/17/12/013
Effect of flexural modes on squeeze filmdamping in MEMS cantilever resonatorsAshok Kumar Pandey and Rudra Pratap
CranesSci MEMS Lab, Department of Mechanical Engineering, Indian Institute of Science,Bangalore-560012, Karnataka, India
E-mail: [email protected] and [email protected]
Received 14 May 2007, in final form 2 October 2007Published 1 November 2007Online at stacks.iop.org/JMM/17/2475
AbstractWe present an analytical model that gives the values of squeeze filmdamping and spring coefficients for MEMS cantilever resonators taking intoaccount the effect of flexural modes of the resonator. We use the exact modeshapes of a 2D cantilever plate to solve for pressure in the squeeze film andthen derive the equivalent damping and spring coefficient relations from theback force calculations. The relations thus obtained can be used for anyflexural mode of vibration of the resonators. We validate the analyticalformulae by comparing the results with numerical simulations carried outusing coupled finite element analysis in ANSYS, as well as experimentallymeasured values from MEMS cantilever resonators of various sizesvibrating in different modes. The analytically predicted values of dampingare, in the worst case, within less than 10% of the values obtainedexperimentally or numerically. We also compare the results with previouslyreported analytical formulae based on approximate flexural mode shapesand show that the current results give much better estimates of the squeezefilm damping. From the analytical model presented here, we find that thesqueeze film damping drops by 84% from the first mode to the second modein a cantilever resonator, thus improving the quality factor by a factor of 6 to7. This result has significant implications in using cantilever resonators formass detection where a significant increase in the quality factor is obtainedby using a vacuum.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
MEMS cantilevers are one of the most popular resonators usedin various applications of micro and nano technologies today.The simplicity of the structure, the ease of fabrication over awide range of dimensional variations, and the ease of excitationand resonance measurements are primary reasons for itspopularity. As a resonator, the most important characteristicsof a cantilever beam or plate are its resonant frequency andthe quality (Q) factor. In particular, the Q-factor—a measureof sharpness of the resonant peak—becomes the most crucialcharacteristic of the beam when one considers applicationsthat are based on the fine resolution of the resonant peakshift. In recent times there have been several studies wherecantilever resonators have been used for extremely small mass
detection such as those of biological molecules or viruses[1–3]. One of the key requirements in such applications isa high Q-factor of the resonator. For high Q-factor, onemust minimize damping and push the resonant frequencyas high as possible. In resonant devices operating underambient conditions, the squeeze film damping from trappedair is probably the most dominant mechanism of damping [4].Although one can package these devices in a vacuum to getrid of this damping, it is not always desirable or even practicalto vacuum seal all such devices. In such cases, it is imperativeto understand and model the squeeze film damping in thesestructures as accurately as possible. As we show in this study,such modeling can provide insights into reducing the effect ofsqueeze film damping significantly and thus help in increasingthe Q-factor.
0960-1317/07/122475+10$30.00 © 2007 IOP Publishing Ltd Printed in the UK 2475
A K Pandey and R Pratap
Figure 1. Schematic of a vibrating structure of length L, width W and thickness tb with (a) uniform and (b) non-uniform gap h variationalong the x-axis.
Squeeze film damping has been extensively studied fordifferent cases of parallel plate motion, rigid motion of non-perforated [5–9] and perforated structures [10–18], as well asflexible structures [17, 19–28] with thin to ultra-thin gaps andfrom smooth to rough surfaces [7, 29]. Here, we study theeffect of non-uniform gap variation between the oscillatingstructure and the fixed substrate as shown in figure 1(b) on thesqueeze film damping due to flexibility of the structure.
Since there are many elastic structures, such as cantileverbeams, membranes and thin plates, that oscillate with non-uniform displacement along their lengths even in the first modeof vibration, the analytical formulae, which are derived basedon the assumptions of uniform gap thickness, cannot giveaccurate results for these devices. Moreover, the degree ofnon-uniformity in the displacement increases even further ifa device oscillates in higher modes. To find the squeeze filmdamping for a system where the displacement is non-uniform,the elasticity equation that gives the dynamic displacement hasto be coupled with the conventional Reynolds equation [5, 6].
There are various approaches which are proposed to getthe approximate solution of the coupled equation of elasticityand the Reynolds equation using numerical or semi-numericaltechniques. Neyfeh and Younis [19] have proposed a semi-analytical approach by first using perturbation techniquesto linearize the coupled equations and then solving thoseequations using finite-element techniques in order to calculatethe Q-factor of the structure. Hwang et al [20] haveproposed a numerical method to calculate the damping fordifferent modes of the oscillating structure by solving theReynolds equation. Since the computations involved areintense and time consuming, Hung and Senturia [21, 22] haveproposed a low-order model for fast dynamical simulation byextracting basis functions from a few FEM simulation resultsto calculate the damping. De and Aluru [24] have also usednumerical methods to solve the coupled equation involvingfluid–structure interaction. Although, all the above techniquesseem to yield good results, they are not easy to apply even fora simple structure like a microcantilever beam. To make theanalysis simple and accurate for various geometries, Mehneret al [26–28] have proposed a modal projection technique thatprovides an efficient method to compute squeeze film dampingparameters for flexible structures. Their approach involves theextraction of different modes using structural analysis andthen solving the Reynolds equation to calculate the dampingand spring forces corresponding to the different modes. Asimilar approach is followed in commercial FEM packagessuch as ANSYS [30] for reducing the time required for solvingcoupled equations. Among analytical approaches, Darlinget al [23] have proposed to use an approximate mode shape forthe first mode of vibration of a flexible structure for solving
the Reynolds equation. But this approximation gives about20–25% error. We have also used approximate mode shapesof a fixed–fixed beam in solving the Reynolds equation in oneof our earlier work [17]. Zhang et al [25] have derived anaccurate analytical model to account for the flexibility effectin a fixed–fixed beam oscillating in its first mode.
In this paper, we derive an analytical (compact) modelfor squeeze film effects in a cantilever resonator using exactmode shapes of the cantilever covering all bending modes ofvibration. Next, we evaluate the squeeze film damping inmicrocantilever beams of different lengths using the modalprojection technique available in ANSYS [30] for severalflexural modes of vibration. This method makes use of theelasticity equation to find the beam or the plate deflection.Then the Reynolds equation is solved to calculate the backpressure. To model the squeeze film damping using ANSYS,while considering the flexibility of the vibrating structure, weuse FLUID 136 elements to model the air-film and SOLID45 elements to model the structure. We then numericallyevaluate the damping ratio of the fluid-structure system.The rarefaction effect, which comes into play when the air-gap thickness between the vibrating structure and the fixedsubstrate becomes very small, is considered by using theeffective viscosity [31]. To validate and compare the analyticalas well as the numerical results, we perform experimentson some MEMS cantilever beams using Polytec’s scanninglaser vibrometer [32]. Using the half-width method [33], wecalculate the damping ratio corresponding to different modesfrom the frequency response curve. Finally, we comparethe experimental results with those from the numerical andthe analytical model for microcantilever resonators of lengthvarying from 150 µm to 350 µm.
2. Theory
Here, we briefly discuss the governing equations of fluid–structure interaction in squeeze film modeling due toelastic structural vibration. We also outline the numerical,experimental and analytical procedures to calculate thedamping ratio.
2.1. Numerical procedure
The theory of elastohydrodynamics used for determiningthe effect of flexibility on squeeze film damping involvesthe coupling of plate vibration equations and the Reynoldsequation under the assumptions of small strains anddisplacements. Under this condition, we obtain the linearequation of motion that governs the transverse deflection ofthe plate [19] subjected to a net pressure force p(x, y, t) perunit area due to the squeezing action of air in the gap between
2476
Effect of flexural modes on squeeze film damping in MEMS cantilever resonators
the oscillating plate and the fixed plate:
D
(∂4w
∂x4+ 2
∂4w
∂x2∂y2+
∂4w
∂x4
)+ ρmtb
∂2w
∂t2= −p(x, y, t),
(1)
where w(x, y, t) is the transverse deflection of point (x, y) on
the plate at time t, ρm is the density of the plate, D = Et3b
12(1−ν2)
is the flexural rigidity, tb is the plate thickness, E is Young’smodulus and ν is Poisson’s ratio. The linearized Reynoldsequation for the isothermal and the inertialess flow underthe assumptions of small displacement and the small pressurevariation p is given by [5]
h30
12µeff
(∂2p
∂x2+
∂2p
∂y2
)= h0
pa
∂p
∂t+
∂w
∂t, (2)
where pa is the ambient pressure, h0 the nominal air-gapthickness, h = h0 +w(x, y, t) is the variable air-gap thickness,µeff is the effective dynamic viscosity of air in the gapwhich accounts for the rarefaction effect through the Knudsennumber, Kn = λ
h0= 0.0064N/m
pah0[34]. Here, we use Veijola’s
model [31] for estimating the effective viscosity, which is givenby
µeff = µ
1 + 9.638Kn1.159 , (3)
where µ is the dynamic viscosity of air under the STPcondition. To obtain the pressure distribution corresponding todifferent modes of vibration, equations (1) and (2) are solvedsimultaneously with appropriate boundary conditions. Forsolving Reynolds equation, the pressure variation p is taken tobe zero on the open boundaries (free edges) while the pressuregradient is taken to be zero on the fixed side. In ANSYS, thestructural domain is modeled with SOLID45 elements whilethe fluid domain is modeled by FLUID136 elements.
We now outline the modal projection technique used inthis study to extract the damping and spring constants due tothe squeeze film flow. For further details, the reader shouldrefer to Mehner et al [27].
(i) First, the modal analysis of the elastic structure is carriedout using the finite-element method to extract the resonantfrequencies and the eigenvectors of different modes bysolving equation (1) without any forcing term. After finiteelement discretization, equation (1) can be written in thematrix form as
[M]{w} + [Ks]{w} = {F }, (4)
where [M] is the structural mass matrix, [Ks] is thestructural stiffness matrix and {F } is the nodal loadvector. If the displacement w(x, y, t) is written in termsof m eigenvectors φi(x, y) and a time-dependent modalcoordinate qi(t) as
w(x, y, t) =m∑
i=1
φi(x, y)qi(t) (5)
then the eigenvectors and the resonant frequencies can becalculated from the dynamic equilibrium equation:
[Ks]{φi} = ω2i [M]{φi}. (6)
Based on the orthogonal properties of eigenvectors, weget the following expression for the resonance frequenciescorresponding to different mode shapes:
{φi}T [Ks]{φi} = ω2i {φi}T [M]{φi}. (7)
Letting,
{φi}T [M]{φi} = M∗i and {φi}T [Ks]{φi} = K∗
i , (8)
we get
ω2i = K∗
i
M∗ , (9)
where M∗i and K∗ are the modal mass and stiffness of the
structure corresponding to the ith eigenvector φi and theresonance frequency ωi .
(ii) Now, the squeeze film model (which is meshed withFLUID136 elements) is excited by wall velocitiescorresponding to the values of the ith eigenvector. Theharmonic response analysis is carried out to compute thepressure response over the entire domain.
(iii) After integrating the element pressure, the nodal forcevector is calculated at each frequency as
{F } =∫
areaNT p(φi, qi) dA, (10)
where NT is the finite element shape function, qi is theith modal coordinate and φi is the ith eigenvector.
(iv) The total modal force corresponding to each mode iscalculated as
φT {F } = φT
∫area
NT p(φi, qi) dA. (11)
(v) The damping and spring coefficients of the squeeze filmare calculated from the real and imaginary parts of themodal forces:
Cij = φTi
∫NT Re{p(φj , qj )} dA
q,
Kij = φTi
∫NT Im{p(φj , qj )} dA
q.
(12)
The damping and spring coefficients of each mode due tothe squeeze film are the main diagonal entries Cii and Kii .Off-diagonal terms (i �= j) represent the fluidic cross-talkamong modes which occurs in the case of asymmetric airgap.
(vi) Finally, the damping ratio ξi and the spring ratio Kratio,which is defined as the ratio of the spring constant dueto the squeeze film to the structural stiffness K∗, arecalculated from the following expressions:
ξi = Cii
2ωiM∗i
, Kratio = Kii
K∗i
= Kii
ω2i M
∗i
, (13)
where M∗i is the modal mass corresponding to the
eigenfrequency ωi .(vii) Steps (ii) to (vi) are repeated to find the damping ratio and
the spring constant for the next eigenmode.
2.2. Experimental procedure
Figure 2 shows the experimental set-up used in theexperimentation to characterize the MEMS resonators. Itconsists of MSA 400 Micromotion Analyzer with thedisplacement and velocity decoder (a Polytec product tocharacterize the out-of-plane vibrations by scanning laserDoppler vibrometry (PSV) [32]), an optical microscope fittedwith a CCD camera which is clustered in the scanning headof the microscope, a laser beam (OFV 511) fed through the
2477
A K Pandey and R Pratap
Figure 2. Experimental set-up to characterize the out-of-planemotion of the MEMS devices.
Figure 3. Working principle of the test set-up.
camera port to create a laser spot on the structure, a junctionbox fitted with an internal function generator used to generatedc offset and ac signal for exciting the device under testing,and the vibrometer controller (OFV 3001) that measures theout-of-plane voltage/velocity/displacement.
The working principle of the set-up is schematicallyshown in figure 3. The experimentation involves electricalexcitation (say Vinput = Vdc + Vac, where Vac � Vdc) thatcauses the suspended structure to vibrate with respect to thesubstrate. The Vdc is selected such that it is �20% of thepull-in voltage of the structure to avoid the spring softeningeffect. Then, a small value of Vac is used to oscillate thestructure. The spot laser beam from the interferometer in thescanning head is positioned on a scan point on the object bymeans of mirrors and is scattered back. The back-scatteredlaser light interferes with the reference beam in the scanninghead. A photo detector records the interference. A decoderin the vibrometer outputs a voltage which is proportional tothe velocity of the scanned point parallel to the measurementbeam. The voltage is digitized and processed as the vibrometersignal. The output signal can be obtained as a velocity ordisplacement signal using the velocity or the displacementdecoder.
To measure the damping ratio and the damped naturalfrequency of the structure, we apply a pseudorandom signal ofvoltage Vinput during the experimentation. After averaging theFRF of the output signal over ten times to reduce the noise,we apply the half-width method [33] to calculate the qualityfactor in all modes and then the corresponding damping ratios.The expressions of the experimental quality factor Qexp andthe damping ratio ξexp are given by
Qexp ≈ 1
2ξ= fd
f2 − f1, ξexp ≈ 1
2Qexp, (14)
where f1 and f2 are frequencies at which the amplitude of theoutput signal is 1/
√2 times the maximum amplitude at the
resonance frequency fd . The procedure is repeated about tentimes, and finally the average value of Qexp is recorded. Here,we must mention that it is also possible to extract damping andcompute Q using modal analysis. In particular, experimentalmodal analysis becomes indispensable for damping extractionwhen there is considerable modal interaction (coupling) andthe modes overlap in the experimental data.
2.3. Analytical model
We derive analytical formulae for the squeeze film dampingand spring constant for a cantilever beam using exact modeshapes of the beam oscillating in bending modes. We assumea 2D geometry, i.e., a plate, so that beams of varied planaraspect ratios (width/length) can be effectively dealt with. Theexact mode shapes of such a cantilever plate are given by [35],
�(x, y) = 1
βL
{cosh
(α
x
L
)− cos
(α
x
L
)
+ γ
[sinh
(α
x
L
)− sin
(α
x
L
)]}, (15)
where α = 1.875 104 for the first mode, 4.694 091 forthe second mode, 7.854 757 for the third mode, etc, andγ = −[cosh (α) + cos (α)]/[sinh (α) + sin (α)], and βL =cosh(α)−cos(α)+γ [sinh(α)−sin(α)]. Note that the bendingis assumed to be negligible along the width of the plate. Now, ifthe displacement amplitude at the tip is taken as the generalizedco-ordinate Z(t), then the displacement may be expressed as[33]
w(x, y, t) = �(x, y)Z(t) (16)
about the static equilibrium position. Assuming sinusoidalmotion Z(t) = δeiωt , the corresponding non-dimensional pres-sure distribution P (x, y, t) = p(x, y, t)/pa can be obtainedby solving the governing equation (given by equation (2))with the following boundary conditions:
P (x,W/2, t) = P (x,−W/2, t) = P (1, y, t) = 1,
∂P (0, y, t)
∂x= 0,
(17)
where pa is the ambient pressure. The normalized pressuredistribution under the vibrating flexible plate is obtained usingGreen’s function approach [23] and is given by
P (x, y, t) =∑
m,n=odd
16bm(−1)n+m
2 −1
π2mn
−iωδ eiωt
k2mn
/α2 + iω
× cosmπx
2Lcos
nπy
W, (18)
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Effect of flexural modes on squeeze film damping in MEMS cantilever resonators
where kmn = m2π2
4L2 + n2π2
W 2 , α2 is a constant = 12µeff
h20pa
, and bm is
given by
bm =[− mπ
2(−1)m−1
2
2α3γ(α4 − m4π4
16
)
+m2π2
4(α2 − m2π2
4
) [γ sin(α) + cos(α)]
+m2π2
4(α2 + m2π2
4
) [γ sinh(α) + cosh(α)]
]1
βL
. (19)
The total generalized reaction force F(t) on the movingplate is calculated by integrating the pressure distributionpaP (x, y, t) over the domain S = {(x, y)|0 � x �L,−W/2 � y � W/2}. So the net force after normalizing itwith LWpa is given by
F(t)
LWpa
= ftot =∑
m,n=odd
64b2m
π4m2n2
−iωδeiωt
k2mn/α
2 + iω. (20)
Now, the non-dimensional damping force fd and thespring force fs are calculated by separating ftot into imaginaryand real parts, respectively. Taking the absolute value of thenon-dimensional damping and spring force, we get
fd
δ= 64σ
π6
∑m,n=odd
(m2χ2/4 + n2)b2m
(mn)2{[m2χ2/4 + n2]2 + σ 2/π4} (21)
fs
δ= 64σ 2
π8
∑m,n=odd
b2m
(mn)2{[m2χ2/4 + n2]2 + σ 2/π4} , (22)
where σ = α2W 2ω = 12µeffW2ω
h20pa
is the well-known squeeze
number [5] that captures the compressibility effect, and χ = WL
is the aspect ratio of the cantilever structure. Generally, W isthe smallest dimension chosen out of length L and width W
of the rectangular structure (in this case, we have taken widthW as the smallest dimension). At the cut-off squeeze number,σcut-off , damping and spring forces become equal. If only oneterm is taken in the summation of equations (21) and (22), thenthe cut-off squeeze number is approximated by
σcut-off = π2
(χ2
4+ 1
). (23)
The corresponding analytical damping constant Ca andthe spring constant Ka are given by
Ca = paLW
ωh0
64σ
π6
∑m,n=odd
(m2χ2/4 + n2)b2m
(mn)2{[m2χ2/4 + n2]2 + σ 2/π4}(24)
Ka = paLW
h0
64σ 2
π8
∑m,n=odd
b2m
(mn)2{[m2χ2/4 + n2]2 + σ 2/π4} .
(25)
On substituting the value of squeeze number σ in the commonterm, we get the following expression for Ca ,
Ca = 768µeffLW 3
h30π
6
∑m,n=odd
(m2χ2/4 + n2
)b2
m
(mn)2{[m2χ2/4 + n2]2 + σ 2/π4} .
(26)
Equation (26) gives the damping constant due to the squeezefilm in a cantilever structure in bending modes of vibration.Here, we point out that the effect of different modes comesthrough bm. If meff = ρLWtb
∫ 10 φ2(X) dX is the effective
mass of the beam oscillating in its nth resonant mode, and ωn
is the resonant angular frequency, then the damping ratio isdefined as [35]
ξ = Ca
2meffωn
. (27)
After substituting the expression of Ca in ξ , we get thefollowing expression of ξa
ξa = 384µeffW2
�1tbρωnh30π
6
∑m,n=odd
(m2χ2/4 + n2)b2m
(mn)2{[m2χ2/4 + n2]2 + σ 2/π4}(28)
which is obtained by substituting �1 = ∫ 10 φ2(X) dX (0.25 for
the first mode, 0.249 for the second mode and 0.243 for the
third mode, etc) and ωn = tbα2b
2L2
√E3ρ
, which is the undamped
resonant angular frequency corresponding to the nth mode ofthe beam with Young’s modulus E and density ρ. The valueof αb = 1.875, 4.694, 7.855 for the first, second and the thirdout-of-plane vibration modes of the cantilever beam.
2.3.1. Comparison with other analytical models. There areother simple analytical models that can be used, if appropriate,to predict the modal damping in cantilever resonators due tothe squeeze film. The closest among them is the 1D beammodel based on exact mode shapes of a 1D beam [6]. SinceMEMS cantilever resonators seem to have a fairly wide rangeof length to width aspect ratio, depending on their application,it is important to see how the proposed model compares withthe existing 1D models. In addition, it is also instructive tocompare the proposed model with the simplest known model ofrigid parallel plate motion. We now discuss these comparisonsbefore we get into a detailed discussion of comparisons amongthe numerical, experimental and the proposed analytical modelresults.
(i) 1D beam model. Under the conditions of incompressibleand non-inertial flow, the squeeze film dampingcoefficient with 1D flow assumption across the width ofthe beam, where the length L of the beam is very largecompared to its width W , is given by [6]
C1D = µeffLW 3
h30
meff
ρLWtb= µeffLW 3
h30
∫ 1
0φ2(X) dX,
(29)
where meff = ρLWtb∫ 1
0 φ2(X) dX and X = x/L. Thedamping ratio is given by
ξ1D = µeffW2
2ρtbωnh30
, (30)
where tb is the beam thickness and other parameters arethe same as defined earlier.Now, let us compare the 1D and 2D analytical models,which are given by ξ1D and ξd , with numerical resultsshown in figure 4. We have used cantilever beams ofdimensions W = 80 µm, tb = 4 µm, h0 = 1.4 µm,χ = [0.1, 1], E = 160 GPa, ρ = 2330 kg m−3, andcomputed the damping ratios in the first mode of vibration.
2479
A K Pandey and R Pratap
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-2
100
102
Aspect ratio, = W/L
Dam
ping
rat
io,
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
200
400
600
800
Aspect ratio, =W/L
% e
rror
in1D
mod
el
1D Anal2D Anal
Numerical
(a)
(b)
ξ
χ
χ
Figure 4. Comparison of 1D and 2D analytical damping ratios withnumerical results for W = 80 µm, tb = 4 µm, h0 = 1.4 µm,χ = [0.1, 1] in the first mode of vibration; (b) Percentage error in1D analytical model with respect to the 2D analytical model forχ = [0.1, 1].
We find that ξ1D holds good only if χ � 0.1 with anerror of <10%. The numerical and analytical results,which are based on 2D flow, match well with an errorof <5%. MEMS cantilever resonators with χ > 0.1 arequite common. We see here that for a cantilever resonatoreven with a small χ = 0.2, the 1D model gives as muchas 25% error.
(ii) Rigid parallel plate motion model. When thedisplacement is assumed to be uniform along the lengthand the width of the beam as shown in figure 1(a), then thedamping coefficient can be obtained from Blech’s formula[5] which is obtained by solving equation (2) analytically:
Crigid = 768µeffLW 3
h30π
6
∑m,n=odd
× (m2χ2 + n2)
(mn)2{[m2χ2 + n2]2 + σ 2/π4} . (31)
If we compare Ca from equation (24) with Crigid, we findthat there are two main differences: first, the term mχ inCrigid is replaced by mχ/2 in Ca which is because of the no-flow boundary condition on the fixed end of the cantileverstructure; second, the inclusion of an expression bm due to thenon-uniform gap variation in Ca (bm = 1 in Crigid becauseof uniform gap variation). On comparing Ca and Crigid, wefind that the effect of flexibility can also be modeled by takingan equivalent air-gap thickness heq(>h0) in the rigid plateformula (see Appendix).
3. Results and discussion
We now present the results of analytical, numerical andexperimental studies. The experimental results are used tovalidate the numerical and the analytical models. Since, theexperiments are carried out on resonators of different lengths,we first compare the results from all the three studies for
L L
W W
Figure 5. A picture of microcantilever beam taken from the Polytecscanning vibrometer showing the color bar along the beam length inthe first mode of vibration.
the first mode of vibration taking differing lengths of theresonators. Later, we also compare results for higher modesof vibration.
The length L of the beam is varied from 150 µm to350 µm while the width W = 22 µm, the beam thicknesstb = 4 µm and the air-gap thickness between the suspendedstructure and the substrate h0 = 1.4 µm are kept the same forall the beams. The material of the beam is polysilicon. Therelevant mechanical properties of polysilicon are as follows[6]: ρsi = 2330 kg m−2 is the density, Esi = 160 GPa isYoung’s modulus, and ν = 0.22 is Poisson’s ratio. Thefluid medium is air. The relevant properties of air areρair = 1.2 kg m−2 is the density, µ = 1.8 × 10−5 N-s/m2 isthe dynamic viscosity under standard temperature and pressure(STP), and pa = 1.013 × 105 Pa is the surrounding pressure.Since the first resonant frequency of the beams of above-mentioned dimensions from the analytical beam models differfrom the numerically computed resonant frequencies from theplate equation by 0.5%, the assumption of neglecting the effectof Poisson’s ratio is found to be valid.
First, we compare some of the common characteristicnumbers to characterize the squeeze film flow. We use thecharacteristic flow length h = 1.4 µm of the device shown infigure 5:
(a) Knudsen number, Kn = λh
. It is defined as the ratio of themean free path λ of the air molecules to the characteristicflow length h. The mean free path of air molecules atambient pressure pa is about 65 nm. The Knudsen numbercorresponding to h = 1.4 µm is 0.046 which lies in theslip flow regime (0.01 < Kn < 0.1) [34]. So, the effectiveviscosity of air in the gap between the suspended structureand the fixed substrate is µeff = 1.42 × 10−5 N-s/m2
which is calculated from equation (3).(b) Squeeze number, σ = 12µeffW
2ω
pah20
. If f is the resonance
frequency in Hz, then for W = 22 µm, pa = 1.013 ×105 Pa, and the same values of h0 and µeff as mentionedabove, the squeeze number is σ = 4.16 × 10−7 × 2πf .The frequency limit below which σ < 1 is given byf = 3.8 × 105 Hz. Thus the compressibility effectcan be ignored if the frequency of vibration is less than0.38 MHz.
(c) Reynolds number, Re = ρah20ω
µeff. Reynolds number
corresponding to a resonance frequency f of the cantilever
2480
Effect of flexural modes on squeeze film damping in MEMS cantilever resonators
0 1 2 3 4 5 6 7 8 9
x 105
0
0.5
1
1.5
2
2.5x 10
-3
Frequency (Hz)
Mag
nitu
de (
m/s
/V)
First mode
Second mode
Third mode
Figure 6. Frequency response of a cantilever beam of lengthL = 350 µm, width W = 22 µm and thickness tb = 4 µm.
beam for the characteristic flow length h0, the air densityρa = 1.2 kg m−3, and the effective air viscosity µeff , is1.0 × 10−6 × f . The frequency limit corresponding toRe = 1 is thus f = 1 × 106 Hz. So, the inertial effect canbe ignored if the operating frequency is less than 1 MHz.The Reynolds equation of the form given by equation (2),which neglects the inertial effects, is valid for operatingfrequencies that are less than 1 MHz. Therefore, it canbe used to solve the squeeze film problem without anysignificant error from ignoring the inertial effects.
We perform experiments on five MEMS cantileverresonators of length varying between 150 µm and 350 µmwhile rest of the dimensions are kept the same. To performexperiments, we use a Polytec scanning laser vibrometer(PSV) and follow the procedure outlined in section 2.2. We usepseudorandom signals of different voltages Vinput = Vdc + Vac
to excite the resonators. Controlling the input voltage is verycritical as a large Vdc can cause the vibrating structure eitherto get stuck to the substrate if Vdc � Vpull-in or reduce theover-all stiffness of the structure due to the spring softeningeffect if Vdc < Vpull-in. Therefore, we keep the excitationvoltage in the range of 10% to 20% of the pull-in voltage.The maximum displacement corresponding to the input of 10–20% of the pull-in voltage is less than 6% of the nominal gapthickness. Since the oscillatory motion under small vibrationamplitude is governed by the linear equation of motion, theimportant design parameters such as the resonant frequencyand the damping ratio can be easily extracted from the formulaebased on the equation of linear oscillations. After applying theinput signal, we choose scan points on the top surface of thevibrating structure to measure the resonant frequencies andthe corresponding mode shapes from the frequency responsecurve of the structure as shown in figure 6. The experiment isrepeated ten times and then the average value of the resonantfrequency is recorded. Figure 6 is the frequency responsecurve (obtained from the velocity decoder) of the cantileverbeam of length 350 µm which shows three resonant peakscorresponding to the first three out-of-plane modes. Themode shapes shown as a subset of figure 6 are experimentallycaptured modes. Finally the damping ratio is calculated fromequation (14) by applying the half-width method.
The first resonance frequency, the correspondingReynolds’ number, the squeeze number and the Knudsen
0.5 1 1.5 2 2.5 3 3.5
x 104
0
1
2
3
4
5
6
Number of elements Ne
% E
rror
w.r.
t re
f
ref
ξ
ξ
Figure 7. Convergence of the computed damping ratio ξnum with thenumber of elements Ne. The results shown are for a cantilever beamof length 350 µm, vibrating in its first mode. Here the percentageerror is calculated with respect to the converged values ofξref = 0.454 corresponding to 30 456 elements.
(a)
(b)
(c)
Figure 8. Pressure distribution (in Pa) on the bottom surface of thecantilever beam due to the squeeze film in (a) the first mode, (b) thesecond mode and (c) the rigid motion of a cantilever beam of length350 µm. Note that the frequency of excitation corresponding torigid motion is taken as 43 kHz. The light line in the middle islocation of the axes as output by ANSYS.
number for the beams of different lengths are listed in table 1.From table 1, we find that the Reynolds number and thesqueeze number are small compared to 1, so the inertia and thecompressibility effect can be safely neglected in the numericalor the analytical modeling of the squeeze-film effect.
To model the effect of squeeze film numerically oranalytically due to inertialess flow, one needs to solvethe Reynolds equation, which is given by equation (2).Since equation (2) is the linearized Reynolds equation, thedisplacement of the air gap should be very small, preferablyless than 10% of the nominal gap thickness h0.
To numerically estimate the damping ratio due to squeezefilm for the cantilever beam oscillating in the first or the highermodes, the coupled equation of elasticity and the Reynoldsequation are solved using the modal projection techniqueexplained in section 2.1.
To ensure convergence, we first do several simulationsfor a beam of length 350 µm. We find that the numericaldamping ratio for the first mode of vibration is within 1% ofthe converged value of ξref (corresponding to 30 456 elements)if the fluid volume is meshed with more than 5000 elements(see figure 7).
The effect of vibrational mode shape on squeeze filmdamping is a manifestation of the effect of the mode shape
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A K Pandey and R Pratap
Table 1. Experimental values of the first natural frequency of the cantilever beams of different lengths L and their corresponding values ofRe, Kn and σ .
Length L 1st natural freq. Knudsen no. Reynolds no. Squeeze no.
S.No (in µm) f1(in kHz) Kn = λ
h0Re = ρah2
0ω
µeffσ = 12µeffW
2ω
pah20
1 150 240 0.0451 0.25 0.632 200 133 0.0451 0.14 0.353 250 80 0.0451 0.08 0.214 300 50 0.0451 0.05 0.135 350 43 0.0451 0.04 0.11
Table 2. Comparison between the experimental and the numerical results for the different beam lengths in the first mode of vibration. Theexperimental value ξexp is the mean value of ten independent measurements and σexp is the standard deviation.
Length Experimental Numerical % Error Analytical % Error
(in µm) result ξexp ± σexp result ξnum
∣∣∣ ξexp−ξnum
ξexp
∣∣∣ × 100% result ξd
∣∣∣ ξexp−ξd
ξexp
∣∣∣ × 100%
150 0.071 ± 0.002 0.074 3.93% 0.07 1.68%200 0.125 ± 0.001 0.135 8.00% 0.13 4.00%250 0.201 ± 0.005 0.22 9.45% 0.21 4.47%300 0.320 ± 0.002 0.33 3.12% 0.31 3.12%350 0.415 ± 0.003 0.45 8.43% 0.42 1.69 %
Table 3. Comparison of analytical models for the approximate modeshape ξ[23] and rigid mode ξrigid with the experimental results ξexp of thecantilever beam of different lengths.
Length Experimental Analytical Analytical % Error % Error
(in µm) result ξexp result ξ[23] result ξrigid
∣∣∣ ξexp−ξ[23]ξexp
∣∣∣ × 100%∣∣∣ ξexp−ξrigid
ξexp
∣∣∣ × 100%
150 0.071 0.055 0.18 22.75% 150.0%200 0.125 0.10 0.44 20.00% 238.5%250 0.201 0.16 0.87 20.40% 278.3%300 0.320 0.24 1.28 25.00% 276.5%350 0.415 0.34 1.7 18.10 % 304.8%
Table 4. Comparison between the damping ratios in three different modes of a cantilever beam of length 350 µm.
Transverse Frequency Reynolds Squeeze Experimental Numerical Analyticalmode (in kHz) number (Re) number (σ ) result (ξexp) result (ξnum) result (ξ2d)
1st 43 0.04 0.11 0.415 ± 0.002 0.45 0.4222nd 245 0.25 0.64 0.066 ± 0.002 0.072 0.0673rd 690 0.72 1.80 0.027 ± 0.002 0.025 0.024
on the pressure distribution in the fluid film. Figures 8(a) and(b) clearly show the difference in the pressure variation in thefluid film in the first two modes of flexural vibration. Forcomparison, we also plot the pressure distribution in the mostcommonly used parallel plate motion in 8(c). If we comparethe pressure distribution in all the three cases, we observethat the maximum back pressure on the bottom surface of themoving structure in case (c) is more uniform along the beamlength than that in cases (a) and (b) where it is localized onlyin a small portion. This difference in the pressure distributionultimately shows up in the computed damping ratios that aretabulated and compared in tables 2, 3 and 4.
3.1. Damping in the first mode
In table 2, we compare the experimental, numerical and theanalytical results for the first mode of the cantilever resonatorsof length varying from 150 µm to 350 µm. In comparing thethree results, we compute the percentage error with respect tothe experimental results. Although experimental results are
not free from errors, we make this choice because both thenumerical computation and the analytical formulae are meantto be predictive models with their own sets of idealizations.As is evident from table 2, the numerical results match verywell with the experimental results in all the cases and give anerror of less than 10%. This validates the numerical procedurebased the modal projection technique. Similarly, the analyticalresults match very well with the experimental results for allbeams, the error being below 10%. It is illustrative to comparethe analytical results with those obtained from Blech’s formulaassuming rigid motion. This comparison is shown in table 3.This table clearly shows that the elastic flexure plays asignificant role in the damping ratio, and hence in the Q-factor calculation. Assuming rigid motion gives increasinglyerroneous values of ξ as the length of the resonator increases.
Finally, we remark that the exact mode-shape-basedanalytical model presented here does better than theapproximate mode-shape-based models as we should expect.For example, the model proposed by Darling et al [23] based ona quadratic approximation of the mode shape predicts damping
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Effect of flexural modes on squeeze film damping in MEMS cantilever resonators
0 0.2 0.4 0.6 0.8 1-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Normalized length
Nor
mal
ized
dis
plac
emen
t
First mode
Second mode
Third mode
0 0.2 0.4 0.6 0.8 11.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2
= W/L
heq
/h0
First mode
Second mode
Third mode
(a) (b )
χ
Figure 9. (a) Mode shapes of a cantilever beam for the first, second and the third modes of vibration; (b) variation of the ratio of anequivalent air-gap thickness heq to the original air-gap thickness h0 with the aspect ratio of the cantilever resonator for the first, second andthe third modes.
values that are in error by 18% to 25% when compared with theexperimental values listed here for the five resonators.
3.2. Damping in higher modes
One of the main motivating factors for deriving the analyticalmodel presented in this paper is to predict the damping inhigher modes of flexural vibration. In table 4, we tabulateexperimental, numerical and analytical results for the first threeflexural modes of vibration of the 350 µm long resonator.We have also included the flow characteristic numbers, Reand σ , in this table for reference. Although the analyticalmodel as well as the numerical model are obviously capableof predicting damping in even higher modes, we were limitedin our ability to excite the higher modes experimentally. Theresults from the comparison of the three modes show theefficacy of the analytical as well as the numerical model.
If we compare the damping ratios for the first three modes,we find that the damping effect reduces by 84% and 94%,respectively, in the second and the third modes of vibration.This result is significant because it shows that the Q-factor ofa resonator can go up by a factor of 6 to 7 even in the ambientconditions if one uses the second mode rather than the firstmode of vibration. This observation has implications in thetiny mass detection applications of the cantilever resonator[3]. It is clear that the analytical model captures the flexibilityeffect in the higher modes equally well for the cantileverresonator. Finally, as a caveat, we state that the analyticalformula presented here does not include inertia effect and,therefore, should be used only as long as Re < 1.
4. Conclusions
We have presented analytical, numerical and experimentalstudy of the effect of elastic flexibility of MEMS cantileverresonators on the squeeze film damping. The analytical modelpresented here is derived using exact flexural mode shapesof a 2D cantilever elastic plate. The compact formula thusobtained for computing the squeeze film damping in MEMScantilever resonators is valid for all flexural modes and beamgeometries with a large range of aspect ratios. The model,however, assumes inertialess flow conditions. The numericalstudies are based on a coupled fluid–structure finite elementmodel in ANSYS that incorporates modal projection technique
for including the effect of elastic flexure. The analytical andnumerical results are compared with each other as well as withexperimental values of the squeeze film damping obtainedfrom direct measurements of frequency response spectra ofMEMS cantilever beams of varying lengths and in differentmodes of vibration. The results show that both analyticalmodel and the numerical model predict the squeeze filmdamping values within 10% of the experimentally obtainedvalues. The analytical model is also compared with otheranalytical models based on approximate mode shapes or 1Dbeam mode shape and is found to do much better in predictingsqueeze film damping.
These results show that both the analytical model and thenumerical model can be used very effectively to account forthe effect of elastic mode shapes of MEMS resonators on thesqueeze film damping predictions. In particular, the analyticalmodel can be used as a very convenient tool by designers tocompute the squeeze film damping in MEMS resonators.
Acknowledgments
This work is partially supported by a grant from theDepartment of Science and Technology, Government ofIndia. We greatly acknowledge the help of Professor EnakshiBhattacharya and Mr Soma Bhat of IIT Madras, India forproviding the fabricated structure. We also thank Professor VR Sonti for the discussion on experimental modal analysis.
Appendix
For incompressible and inertialess flow, if we compare Ca andCrigid, we find the expression of heq as
heq = h0
∑ 1(mn)2(m2χ2+n2)∑ b2
m
(mn)2(m2χ2+n2)
1/3
, (A.1)
where m and n are odd numbers. For a very thin cantileverbeam, heq is about 1.6 times h0 for the first, second and the thirdmodes as shown in figure 9. Note that σ and µeff are calculatedbased on the original h = h0 while h0 in the denominator ofequation (31) is calculated from heq (equation (A.1)). Thevariation of the ratio heq/h0 with respect to the aspect ratioof the cantilever beam for different mode shapes are shown infigure 9. Clearly, the ratio depends on the mode of vibration
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A K Pandey and R Pratap
as well as on the aspect ratio χ in a non-intuitive manner.However, it is easy to compute heq from equation (A.1) for agiven resonator and then use Blech’s formula for computingthe squeeze film damping. Unfortunately, there is no savingin effort by following this route.
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