probabilistic analysis of micro-machined fixed-fixed beam for reliability

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Published in Micro & Nano LettersReceived on 24th March 2008Revised on 8th August 2008doi: 10.1049/mnl:20080010

ISSN 1750-0443

Probabilistic analysis of micro-machinedfixed–fixed beam for reliabilityMd. Fokhrul Islam1 M.A. Mohd. Ali1,2,3 B.Y. Majlis1,2

1Department of Electrical, Electronic and Systems Engineering, Faculty of Engineering and Built Environment,Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia2Institute of Microengineering and Nanoelectronics (IMEN), Universiti Kebangsaan Malaysia, 43600 Bangi,Selangor, Malaysia3Institute of Space Science (ANGKASA), Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, MalaysiaE-mail: [email protected]

Abstract: A realistic approach for modelling a fixed–fixed beam is presented. The use of probabilistic methods toassess the electromechanical behaviour of the beam under the presence of micro-machine manufacturing andprocess uncertainties is demonstrated. A finite-element model of the beam is constructed using thecommercial code ANSYS (10.0). In the standard approach of modelling, existing literature assumesdeterministic values for design parameters, however, fabrication of the device introduces some amount ofvariation in the design parameters. Here, the probabilistic approach is discussed to account for the variabilityin fabrication. Probabilistic analysis guides the design of the fixed–fixed beam to achieve a robust and reliabledesign in the most efficient way. The results of the probabilistic design approach on the performancecharacteristics such as maximum deflection and maximum stress will be discussed.

1 IntroductionMicro-electromechanical systems (MEMS) is a processtechnology used to create tiny integrated devices or systemsthat combine mechanical and electrical components. Theyare fabricated using integrated circuit batch processingtechniques and can range in size from a few micrometres tomillimetres. These devices or systems have the ability tosense, control and actuate on the microscale, and generateeffects on the macroscale [1]. Electrostatic MEMS is aspecial branch under micromechanics with a wide range ofapplication specific devices such as switches, micro-mirrorsand micro-resonators [2–4]. Modelling and simulation ofelectrostatic MEMS devices play an important role in thedesign phase in predicting device characteristics. The twomost important electromechanical features of a MEMSswitch are the pull-down voltage and the deflection. Boththese quantities are well known for imparting sharp instabilityin the behaviour of an elastically supported structure subjectedto parallel-plate electrostatic actuation [5].

The standard finite-element modelling approach isdeterministic. It considers input variables such as length,

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width and thickness of the beam as constant variables and theoutput parameters are obtained using both static and dynamicanalyses. Shanmugavalli et al. [6] and Voicu et al. [7] havereported extensive studies on the pull-in voltage analysis of amicro fixed-fixed beam. However, the effect of randomvariation in multiple input parameters is not considered forevaluation of sensitivity. The parameters of interest, such asdeflection and stress, can have a significant impact with smallvariations in input parameters such as length, width andthickness of the beam. Deterministic models do not capturethese variations as needed in many cases. In reality, thefabrication of these devices produces an amount of randomvariation in the geometric parameters and material properties.As these devices accumulate service, they undergo wear andtear which can partly be because of usage in harshenvironments. Furthermore, mishandling also contributes tochanges in model parameters [8]. Hence, deviations fromoptimum performance characteristics of the device are seen.

Uncertainty analysis is a technique by which one candetermine, with good approximation, whether a system willwork within raw specification limits when the parametersvary between their limits [9]. The manufacturing of structural

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components is generally associated with manufacturingimperfections. In general, the geometry of a component canbe reproduced only within certain finite tolerances. If theinfluencing variables are uncertain, a direct consequence isthat the response parameters are uncertain as well [10]. Inthe present work, probabilistic analysis is implemented fordeflection and stress analysis of a micro fixed–fixed beamby using ANSYSw for numerical simulation, byconsidering the uncertainty in geometric and materialproperties simultaneously.

2 Modelling of fixed–fixed beamA simple representation of an electrostatic pull-in device isshown in Fig. 1. It consists of two parallel conductiveplates forming a variable capacitor with an effective overlaparea A and separated by a gap spacing g. The bottom plateis fixed to the ground and the top plate is suspended overthe bottom plate and fixed at both ends.

The two most important elctromechanical features of thisdevice are the pull-down voltage and the deflection. Both ofthese quantities can be calculated by treating this device as amechanical spring. To calculate the pull-down voltage, onemust equate the force pulling down on the beam by theelectrostatic force between the two conductive plates [11]

Fdown ¼10AV 2

2 g2(1)

where 10 is the permittivity of air and V is the applied voltage.The force pushing up from the spring using Hooke’s Lawis [11]

Fup ¼ �k( g0 � g) (2)

Here, k is the spring constant and g0 is the original air gapbetween the two plates. For parallel-plate electrostaticactuation, when the gap reduces to two-thirds of theoriginal gap, the beam becomes unstable and experiences a‘pull-in’ effect. Equating the equations above where the gapis two-thirds of the original gap and solving for the pull-down voltage, we obtain [11]

Vpull�down ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi8 kg3

0

2710A

s(3)

The maximum deflection can also be calculated from the

Figure 1 Electrostatic beam configuration

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spring constant by [11]

d ¼ �F

k(4)

where d is the deflection and F is the force pushing down thespring. The effective spring constant k can be approximatedby [12]

k ¼32Et3w

l 3þ

8s 1� nð Þtw

l(5)

Here, E is the Young’s modulus of the moveable plate material,t the plate thickness, l the plate length, s the residual tensilestress and n is the Poisson’s ratio for the plate material.

3 Design parameters of thedeterministic modelA micro fixed–fixed beam of dimensions given in Table 1 isdesigned using ANSYSw parametric design language.Maximum deflection is analysed using ANSYSw for theactual dimension of the beam and the analysis gives the

Table 1 Dimensions of the fixed–fixed beam

Parameters Nominal values

Geometric parameters

length of the beam (l ), mm 500

width of the beam (w), mm 75

thickness of the beam (t), mm 1

zero-voltage gap spacing ( g0), mm 1

Material properties of aluminium

Young’s modulus (E), MPa 70 000

Poisson’s ratio (n) 0.35

density (D), kg/mm3 2.7 � 10215

Figure 2 Equipotential contours of deflection

Micro & Nano Letters, 2008, Vol. 3, No. 3, pp. 95–100doi: 10.1049/mnl:20080010

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maximum deflection to be 0.287 mm. The resultant contoursof the deflection of the beam after the pull-down force arereached in the structure as shown in Fig. 2.

4 Probabilistic design systemThe probabilistic design system (PDS) analyses a componentor a system involving uncertain input parameters as shown inFig. 3. These input parameters could be anything ranging fromgeometry and material properties to different boundaryconditions. These parameters are defined as random inputvariables and are characterised by their distribution typeand variables (mean, standard deviation). The key outputsof the simulation are defined as random output parameters.During a probabilistic analysis, multiple analysis loops areexecuted to compute the random output parameters as afunction of the set of random input variables. The valuesfor the input variables are generated randomly using MonteCarlo simulation.

The analysis file containing the deterministic model(Fig. 2) gets executed or ‘loops through’ multiple timesduring the probabilistic analysis. In this deterministicmodel, six parameters were considered as random inputvariables. Table 2 illustrates the random input variables,

Figure 3 Schematic of the working of PDS


the distribution they are subjected to, and their distributionparameters. The discrete sampling characteristics andprobability density functions for the input variables areshown in Fig. 4. For carrying out uncertainty analysis of afixed–fixed beam, the deflection analysis is performed inANSYSw, according a 5% variation for each of these inputparameters assuming a Gaussian distribution for all thedimensional parameters and material properties.

5 Results and discussionThe intention of the probabilistic analysis is to demonstratethe use of probabilistic methods to guide the design processof the MEMS device to achieve a more reliable and robustdesign. For the probabilistic analysis, 100 latin hypercubesamples have been run and the corresponding statistics ofthe output parameters are given in Table 3. The histogramsof the output parameters shown in Fig. 5 illustrate thescatter induced in the output parameters because of thescatter of the input variables. The relative frequency shownin the histograms is equal to the number of samples withina certain interval, divided by the total number of samples(100 in this case).

Technical products are typically designed to fulfil certaindesign criteria based on the output parameters. Forexample, a design criterion is that the deflection will beabove or below a certain limit. The cumulative distributioncurve for the deflection is shown in Fig. 6. The line in themiddle is the probability P that the deflection remains lowerthan a certain limit value (g0 � g)-limit. The complement1.0 2 P is the probability that the deflection exceeds thislimit. For the stability of the beam, the deflection shouldremain within one-third of its original gap spacing [12].Hence, the reliability of the device is given by theprobability that the deflection falls within that range. Theupper and lower curves in Fig. 6 are the confidence boundsusing a 95% confidence level. The confidence boundsquantify the accuracy of the probability results.

Fig. 7 illustrates the sensitivity plots for the outputvariables with respect to the randomised inputs. Thesensitivities are given as absolute values (bar chart) andrelative to each other (pie chart). There are two important

Table 2 Input random variables for probabilistic design

Parameters Mean Standard deviation Minimum Maximum

length, mm 500 25 438 564

width, mm 75 3.8 66 87

thickness, mm 1.00 0.05 0.85 1.14

force, mm 0.200 0.010 0.172 0.225

Young’s modulus, MPa 70 000 3500 61 500 78 600

density (�10215 kg/mm3) 2.70 0.13 2.30 3.01

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Figure 4 Probability density function for input random variables

conclusions that can be derived from the sensitivity diagrams.Firstly, if the design is not sufficient (e.g. not reliableenough), these are the most important input variables thatmust be modified or controlled by quality assurancemeasures during manufacturing. It does not make sense tofocus on input variables of only minor importance or nosignificance at all. Here, the deflection is sensitive only tothree input variables. This is a reduction in the complexityof the problem from six input variables down to only three.This reduction in the complexity of the problem ensuresthat necessary design changes are identified in the most

Table 3 Output variable characteristics in the probabilisticdesign simulations

Parameters Mean Standarddeviation

Minimum Maximum

deflection,mm

0.295 0.070 0.151 0.568

stress, mm 1.029 0.137 0.736 1.400

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efficient way. Secondly, if the design is satisfactory, there isusually the need to reduce the manufacturing costs withoutsacrificing reliability. In this case, the manufacturingtolerances of the insignificant or the less important inputparameters can be relaxed since they have no impact. Forthe geometry parts described by those parameters,expensive quality assurance measures can be abandoned.

If the input parameter, LENGTH or THICKNESS shouldbe modified in the design here, then the question that stillremains is how this should be done. To answer this question,the so-called scatter plots are useful. Fig. 8 illustrates thescatter plots of the deflection and stress as a function ofthe change of the LENGTH and the THICKNESS of thebeam, respectively. The trendlines describe the amount ofscatter in the deflection and the stress because of the scatter ofLENGTH and THICKNESS, respectively. The deviationsfrom the trendline are because of the scatter in the other inputvariables. The trendlines can be used to estimate to whatextent the scatter of the deflection or stress could be reduced ifthe scatter of LENGTH or THICKNESS is reduced.


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Figure 5 Histogram distribution of output variables deflection and stress

Figure 6 Probability curve of the deflection

Figure 8 Deflection and stress as a function of LENGTH andTHICKNESS

Figure 7 Sensitivities of the deflection and stress


6 ConclusionsProbabilistic methods provide a computationally efficientmethod to treat uncertainties in engineering calculations. Inthis work, we have demonstrated the use of the Monte Carloapproach to analyse the effect of parameter uncertainty ondeflection and stress analysis for a fixed–fixed beam. It wasobserved that the changes in length and thickness tend to bethe most influencing parameters, which need to be tightlycontrolled. In addition, probabilistic methods are capable ofidentifying where reductions of the manufacturing costs arepossible. The objective of the study is to show how FEM andprobabilistic design can be used to simulate the effects ofmanufacturing tolerances on the behaviour of the device. Theprobabilistic methods can be used to quantify a more reliableand robust design and an analysis procedure for the MEMSdevice. This provides guidance for necessary design changesin a most efficient way.

7 AcknowledgmentThe authors would like to thank the Ministry of Science,Technology and Innovation (MOSTI) of Malaysia forsupporting this work under the eScienceFund 03-01-02-SF0254.

8 References

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[2] REZVANIAN O., ZIKRY M.A., BROWN C., KRIM J.: ‘Surfaceroughness, asperity contact and gold RFMEMSswitch behaviour’, J. Micromech. Microeng., 2007, 17,pp. 2006–2015

[3] WEE PING C., SATOSHI I . , SHINSUKE S.: ‘Strength andreliability analysis of MEMS micromirror’, http://www.fml.t.u-tokyo.ac.jp/�izumi/papers/Master_H15MEMS.pdf,accessed February 2008

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[4] MAWARDI A., PITCHUMANI R.: ‘Design of microresonatorsunder uncertainty’, J. Microelectromech. Syst., 2005, 14,(1), pp. 63–69

[5] SOBOYEJO A.B.O., BHALERAO K.D., SOBOYEJO W.O.: ‘Reliabilityassessment of polysilicon MEMS Structures underMechanical Fatigue Loading’, J. Mater. Sci., 2003, 38,pp. 4163–4167

[6] SHANMUGAVALLI M., UMA G., VASUKI B., UMAPATHY M.: ‘Designand simulation of MEMS devices using interval analysis’,J. Phys.: Conf. Ser., 2006, 34, pp. 601–605

[7] VOICU R., TIBEICA C., BAZU M.: ‘Design parametersoptimization using process variations of the pull-in voltagefor MEMs’. Proc. 25th Int. Conf. Microelectronics (MIEL2006), Belgrade, Serbia and Montenegro, 14–17 May 2006

[8] VUDATHU S.P., DUGANAPALLI K.K., LAUR R., KUBALINSKA D.,GERSTNER A.B.: ‘Parametric yield analysis of MEMS via

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statistical methods’. DTIP of MEMS & MOEMS, Stresa,Italy, 26–28 April 2006

[9] VAMSHI K., MANOJ K., YING F., JAMES B.: ‘Probabilistic designsystem (PDS): arealistic approach of finite element modellingfor capacitive micro-machined ultrasonic transducers(cMUTS)’. ECNDT 2006 – Tu.4.7.4

[10] STEFAN R., PAUL L., DALE O.: ‘Quality based designand design for reliability of micro electro mechanical systems(MEMS) using probabilistic methods’ (ANSYS, Inc.)

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probabilistic analysis of micro-machined fixed-fixed beam for reliability

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