Research Article Estimation of Static Pull-In Instability ...

Hindawi Publishing CorporationAdvances in Artificial Neural SystemsVolume 2013 Article ID 741896 10 pageshttpdxdoiorg1011552013741896

Research ArticleEstimation of Static Pull-In Instability Voltage of GeometricallyNonlinear Euler-Bernoulli Microbeam Based on ModifiedCouple Stress Theory by Artificial Neural Network Model

Mohammad Heidari1 Yaghoub Tadi Beni2 and Hadi Homaei2

1 Department of Mechanical Engineering Aligudarz Branch Islamic Azad University PO Box 159 Aligudarz Iran2 Faculty of Engineering University of Shahrekord PO Box 115 Shahrekord Iran

Correspondence should be addressed to Mohammad Heidari moh104337yahoocom

Received 11 September 2013 Accepted 22 November 2013

Academic Editor Ping Feng Pai

Copyright copy 2013 Mohammad Heidari et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this study the static pull-in instability of beam-type micro-electromechanical system (MEMS) is theoretically investigatedConsidering the mid-plane stretching as the source of the nonlinearity in the beam behavior a nonlinear size dependent Euler-Bernoulli beammodel is used based on a modified couple stress theory capable of capturing the size effect Two supervised neuralnetworks namely back propagation (BP) and radial basis function (RBF) have been used for modeling the static pull-in instabilityof microcantilever beamThese networks have four inputs of length width gap and the ratio of height to scale parameter of beamas the independent process variables and the output is static pull-in voltage of microbeam Numerical data employed for trainingthe networks and capabilities of the models in predicting the pull-in instability behavior has been verified Based on verificationerrors it is shown that the radial basis function of neural network is superior in this particular case and has the average errors of455 in predicting pull-in voltage of cantilever microbeam Further analysis of pull-in instability of beam under different inputconditions has been investigated and comparison results ofmodeling with numerical considerations show a good agreement whichalso proves the feasibility and effectiveness of the adopted approach

1 Introduction

Micro-electromechanical systems (MEMS) are widely beingused in todayrsquos technology So investigating the problemsreferring to MEMS owns a great importance One of thesignificant fields of study is the stability analysis of theparametrically excited systems Parametrically excitedmicro-electromechanical devices are ever being increasingly usedin radio computer and laser engineering [1] Parametricexcitation occurs in a wide range of mechanics due to time-dependent excitations especially periodic ones some exam-ples are columns made of nonlinear elastic material beamswith a harmonically variable length parametrically excitedpendulums and so forth Investigating stability analysis onparametrically excited MEM systems is of great importanceIn 1995 Gasparini et al [2] examined the transition betweenthe stability and instability of a cantilevered beam exposedto a partially follower load Applying voltage difference

between an electrode and ground causes the electrode todeflect towards the ground At a critical voltage which isknown as pull-in voltage the electrode becomes unstable andpulls in onto the substrate [3] The static pull-in behaviorof MEMS actuators has been studied for over two decadeswithout considering the Casimir force [4ndash6] Osterberg andSenturia [4] and Osterberg et al [5] investigated the pull-inparameters of the beam-type and circular MEMS actuatorsusing the distributed parameter models Beni et al [7]Koochi et al [8] and Ghalambaz et al [9] investigated theeffect of Casimir force on the pull-in behavior of beams-type NEMS Sadeghian et al [6] applied the generalizeddifferential quadrature method to investigate the pull-inphenomena of microswitches A comprehensive literaturereview on investigating MEMS actuators can be found in[10]Moghimi Zand andAhmadian [11] investigated the pull-in behavior of multilayer microplates using finite element

2 Advances in Artificial Neural Systems

method Different analytical numerical finite element meth-ods have been proposed to model the pull-in behavior ofmicrobeams [12ndash16] Further information about modelingpull-in instability of MEMS has been presented by Batraet al [17] and Lin and Zhao [18] The classical continuummechanics theory is not able to explain the size-dependentbehavior of materials and structures at submicron distancesTo overcome this problem nonclassical continuum theoriessuch as higher-order gradient theories and the couple stresstheory are developed considering the size effects [7] In 1960ssome researchers such as Koiter [19] Mindlin and Tiersten[20] and Toupin [21] introduced the couple stress elasticitytheory as a nonclassic theory capable to predict the size effectswith appearance of twohigher-ordermaterial constants in thecorresponding constitutive equations In this theory besidesthe classical stress components acting on elements of mate-rials the couple stress components as higher-order stressesare also available which tend to rotate the elements Utilizingthe couple stress theory some researchers investigated thesize effects in someproblems [22] Employing the equilibriumequation of moments of couples besides the classical equilib-rium equations of forces and moments of forces a modifiedcouple stress theory is introduced by Yang et al [23] with onehigher-order material constant in the constitutive equationsRecently size-dependent nonlinear Euler-Bernoulli andTim-oshenko beams modeled on the basis of the modified couplestress theory have been developed by Xia et al [24] andAsghari et al [25] respectively Rong et al [26] presentedan analytical method for pull-in analysis of clamped-clampedmultilayer beam Their method is the Rayleigh-Ritz methodand assumes one deflection shape functionThey derived thetwo governing equations by enforcing the pull-in conditionsthat the first- and second-order derivatives of the systemenergy functional are zero They investigated the static pull-in instability voltage and displacement of themultilayer beamwhichwere coupled in the two governing equations Artificialneural networks (ANNs) as one of the most attractivebranches in artificial intelligence have the potentiality tohandle problems such as modeling estimation predictiondiagnosis and adaptive control in complex nonlinear systems[27] As seen from pervious studies the researcher used theanalytical or numerical methods for static pull-in instabilityvoltage of microcantilever beams but in this study we usethe neural network to estimate the pull-in voltage Thisstudy investigates the pull-in instability of microbeams witha curved ground electrode under action of electric fieldforce within the framework of von Karman nonlinearityand the Euler-Bernoulli beam theory The static pull-ininstability voltage of cantilever microbeam is obtained byusing MAPLE commercial software The objective of thispaper is to establish the neural networkmodels for estimationof the pull-in instability voltage of cantilever beams Morespecifically two different types of neural networks backpropagation (BP) and radial basis function (RBF) are used toconstruct the pull-in instability voltage Effective parametersinfluencing pull-in voltage and their levels for training wereselected through preliminary calculations carried out oninstability pull-in voltage of microbeam Networks trained bythe same numerical data are then verified by some numerical

q(x)

x

L

g

h

z

z

b

yw(x)

V

Fixed substrate

Figure 1 An electrostatically actuated Euler-Bernoulli microbeam

calculations different from those used in training phase andthe best model was selected based on the criterion of havingthe least average values of verification errors To the authorsrsquobest knowledge no previous studies which cover all theseissues are available

2 Modeling the Microcantilever Based onthe Modified Couple Stress

Consider an electrostatically actuated microbeam of length119871 as shown in Figure 1 The microcantilever is under atransverse distributed electrical force 119902(119909) caused by the inputvoltage applied between the microbeam and the substrateThe cross section of the beam is here assumed to be rect-angular with width 119887 and height ℎ The governing equationfor the static behavior of a uniform homogeneous Euler-Bernoulli beam made of an isotropic linear elastic materialunder transverse distributed load 119902(119909)modeled based on themodified couple stress theory is written as [28]

1198781205974119908

1205971199094minus 119873

1205972119908

1205971199092+ 120588119860

1205972119908

1205971199052= 119902 (119909) (1)

where

119873 = 1198730+119864119860

2119871int

119871

0

(120597119908

120597119909)

2

119889119909

119878 = 119864119868 + 1205831198601198972

(2)

where 1198641198730 119868 and 119908(119909 119905) are Youngrsquos modulus in the beam

direction the axial load the second moment of inertia andthe transverse deflection In addition 119897 is a material lengthscale parameter and 120583 is the average shear modulus in theside-plane of the beam Letting 119897 = 0 the equation is reducedto one corresponding to the classical theory If in (1)119873 = 0then the model of beam is called the linear equation withoutthe effect of geometric nonlinearity The cross-sectional areaof beam is 119860 The electrostatic force enhanced with first-order fringing correction can be presented in the followingequation [29]

119902 (119909) =12057601198871198812

2(119892 minus 119908 (119909 119905))2[1 + 065

(119892 minus 119908)

119887] (3)

Advances in Artificial Neural Systems 3

where 1205760= 8854 times 10

minus12 C2Nminus1mminus2 is the permittivity ofvacuum 119881 is the applied voltage and 119892 is the initial gapbetween the movable and the ground electrode

In the static case it is clear that 120597120597120591 = 0 and 120597120597119909 =

119889119889119909 Hence (1) is reduced to

(119864119868 + 1205831198601198972)1198894119908

1198891199094minus [1198730+119864119860

2119871int

119871

0

(119889119908

119889119909)

2

119889119909]1198892119908

1198891199092

=12057601198871198812

2(119892 minus 119908)2[1 + 065

(119892 minus 119908)

119887]

(4)

For cantilever beam the boundary conditions at the ends are

119908 (0) = 0119889119908 (0)

119889119909= 0

1198892119908 (119871)

1198891199092

= 01198893119908 (119871)

1198891199093

= 0

(5)

Let us consider the following dimensionless parameters

120572 =1198601198712

2119868 120573 =

120576011988711988121198714

21198923119864119868 120574 = 065

119892

119887

120575 =1205831198601198972

119864119868 119908 =

119908

119892 119909 =

119909

119871 Γ =

11987301198712

119864119868

(6)

In the above equations the nondimensional parameter 120575 isdefined as the size effect parameter Also120573 is nondimensionalvoltage parameter The normalized nonlinear governingequation of motion of the beam can be written as [30]

(1 + 120575)1198894119908

119889119909minus Γ + 120572int

1

0

(119889119908

119889119909)

2

1198891199091198892119908

1198891199092

=120573

(1 minus 119908)2+

120574120573

(1 minus 119908)

(7)

3 Overview of Neural Networks

A neural network is a massive parallel system comprised ofhighly interconnected interacting processing elements ornodes The neural networks process through the interactionsof a large number of simple processing elements or nodesalso known as neurons Knowledge is not stored withinindividual processing elements but rather represented bythe strengths of the connections between elements Eachpiece of knowledge is a pattern of activity spread amongmany processing elements and each processing element canbe involved in the partial representation of many pieces ofinformation In recent years neural networks have becomea very useful tool in the modeling of complicated systemsbecause it has an excellent ability to learn and to generalize(interpolate) the complicated relationships between inputand output variables [27] Also the ANNs behave as a modelfree estimators that is they can capture and model complexinput-output relations without the help of a mathematicalmodel [31] In other words training neural networks forexample eliminates the need for explicit mathematical mod-eling or similar system analysis

31 Artificial Neural Network Models of Static Pull-In Insta-bility of Beam In this research backpropagation (BP) andradial basis function (RBF) neural networks have been usedformodeling the pull-in instability voltage ofmicrocantileverbeams The first ANN is very popular especially in thearea of online monitoring and manufacturing modeling asits design structure and operation are relatively simpleThe radial basis network has some additional advantagessuch as rapid learning and less error In particular mostRBFNs involve fixed basis functions with linearly unknownparameters in the output layer In contrast multilayer BPANNs comprise adjustable basis functions which results innonlinearly unknownparameters It is commonly known thatlinear parameters in RBFNmake the use of least squares errorbased updating schemes possible that have faster convergencethan the gradient-descent methods in multilayer BP ANNOn the other hand in practice the number of parametersin RBFN starts becoming unmanageably large only when thenumber of input features increases beyond 10 or 20 whichis not the case in our study Hence the use of RBFN waspractically possible in this research In this paper MATLABNeural Network Toolbox ldquoNNETrdquo was used as a platform tocreate the networks [32]

32 Backpropagation (BP) Neural Network The backprop-agation network (Figure 2) is composed of many intercon-nected neurons or processing elements (PE) operating inparallel and are often grouped in different layers

As shown in Figure 3 each artificial neuron evaluatesthe inputs and determines the strength of each through itsweighing factor In the artificial neuron the weighed inputsare summed to determine an activation level That is

net119896119895= sum

119894

119908119896

119895119894119900119896minus1

119894 (8)

and 119900119896minus1

119894is the output of the 119894th neuron in the (119896 minus 1)th

layer where net119896119895is the summation of all the inputs of the

119895th neuron in the 119896th layer and 119908119896

119895119894is the weight from the

119894th neuron to the 119895th neuron The output of the neuron isthen transmitted along the weighed outgoing connectionsto serve as an input to subsequent neurons In the presentstudy a hyperbolic tangent function 119891(net119896

119895) with a bias 119887

119895is

used as an activation function of hidden and output neuronsTherefore output of the 119895th neuron 119900119896

119895for the 119896th layer can be

expressed as

119900119896

119895= 119891 (net119896

119895) =

119890(net119896119895+119887119895) minus 119890

minus(net119896119895+119887119895)

119890(net119896119895+119887119895) + 119890

minus(net119896119895+119887119895)

(9)

Before practical application the network has to be trained Toproperly modify the connection weights an error-correctingtechnique often called backpropagation learning algorithmor generalized delta rule is employed Generally this tech-nique involves two phases through different layers of thenetwork The first is the forward phase which occurs whenan input vector is presented and propagated forward throughthe network to compute an output for each neuron Duringthe forward phase synaptic weights are all fixed The error


Inputlayer

layerlayer

First hidden

Output

Figure 2 Backpropagation neural network with two hidden layers

Activationfunction

Bias = 1

okminus11

okminus12

okminus1i

okminus1n

wkjn

okj

wkj2

wkj1

wj0 = bj

(f)wkji

Sumfunction(sum)

Figure 3 Architecture of an individual PE for BP network

obtained when a training pair consisting of both input andoutput is given to the input layer of the network is expressedby the following equation [33]

119864119901=1

2sum

119895

(119879119901119895minus 119874119901119895)2

(10)

where 119879119901119895is the 119895th component of the desired output vector

and 119874119901119895is the calculated output of 119895th neuron in the output

layer The overall error of all the patterns in the training set isdefined as mean square error (MSE) and is given by

119864 =1

119901

119899

sum

119901=1

119864119901 (11)

where 119899 is the number of input-output patterns in the trainingset The second is the backward phase which is an iterativeerror reduction performed in the backward direction fromthe output layer to the input layer In order to minimizethe error 119864 as rapidly as possible the gradient descentmethod adding a momentum term is used Hence the newincremental change of weight Δ119908119896

119895119894(119898 + 1) can be

Δ119908119896

119895119894(119898 + 1) = minus120578

120597119864

120597119908119896

119895119894

+ 120572Δ119908119896

119895119894(119898) (12)

where 120578 is a constant real number between 01 and 1 calledlearning rate 120572 is the momentum parameter usually set to

a number between 0 and 1 and 119898 is the index of iterationTherefore the recursive formula for updating the connectionweights becomes

119908119896

119895119894(119898 + 1) = 119908

119896

119895119894(119898) + Δ119908

119896

119895119894(119898 + 1) (13)

These corrections can be made incrementally (after eachpattern presentation) or in batch mode In the latter casethe weights are updated only after the entire training patternset has been applied to the network With this method theorder in which the patterns are presented to the network doesnot influence the training This is because of the fact thatadaptation is done only at the end of each epoch And thuswe have chosen this way of updating the connection weights

33 Radial Basis Function (RBF) Neural Network The con-struction of a radial basis function (RBF) neural network inits most basic form involves three entirely different layersA typical RBFN with 119873 input and 119872 output is shown inFigure 4The input layer is made up of source nodes (sensoryunits) The second layer is a single hidden layer of highenough dimension which serves a different purpose in afeed-forward networkThe output layer supplies the responseof the network to the activation patterns applied to theinput layer The input units are fully connected though unit-weighed links to the hidden neurons and the hidden neuronsare fully connected by weighed links to the output neuronsEach hidden neuron receives input vector X and compares itwith the position of the center ofGaussian activation functionwith regard to distance Finally the output of the 119895th-hiddenneuron can be written as

119860119895= exp(minus

10038171003817100381710038171003817X119894minus C119895

10038171003817100381710038171003817

2

1198782) (14)

where X119894is an 119873-dimensional input vector C

119895is the vector

representing the position of the center of the 119895th hiddenneuron in the input space and 119878 is the standard deviation orspread factor of Gaussian activation function The structureof a radial basis neuron in the hidden layer can be seen inFigure 5Output neurons have linear activation functions andform a weighted linear combination of the outputs from thehidden layer

119884119896=

119867

sum

119895=1

119908119896119895119860119895 (15)

where119884119896is the output of neuron 119896119867 is the number of hidden

neurons and 119908119896119895

is the weight value from the 119895th hiddenneuron to the 119896th output neuron Basically the RBFN has theproperties of rapid learning easy convergence and less errorand generally possess following characteristics

(1) It may require more neurons than the standard feed-forward BP networks

(2) It can be designed in a fraction of the time that it takesto train the BP network

(3) It has excellent ability of representing nonlinear func-tions RBFN is being used for an increasing number ofapplications proportioning a very helpful modelingtool


X1

X2

XN

X WC S Y

Y1

YM

W11

WM1

WM2

Network inputs

Input layer Hidden layer Output layer

Network outputs

Figure 4 Radial basis function neural network

1

b

a

||dist||

a = f(||dist|| times b)

||dist|| = X minus C

C1 C2 C3

X1

X2

X3

n10

05

00

minus0833 +0833

a = radbas(n)

a

n

Radial basis function

Figure 5 Radial basis neuron

4 Results and Discussion

41 Static Pull-In Instability Analysis In order to obtaindifferent pull-in instability parameters and output features fortraining and testing of neural networks a series of numericalanalysis was performed on a MAPLE package At first somepreliminary calculations were carried out to determine thestable domain of the pull-in instability parameters and alsothe different ranges of pull-in variables Based on preliminarycalculations results beam length (119871) width of beam (119887) gap(119892) and (ℎ119897) ratio were chosen as independent input param-etersThe total data obtained fromMAPLE calculations is 120which forms the neural networksrsquo training and testing setsIn the considered case study it is assumed that the cantileverbeam is made of silicon in 110 direction that is the length ofthe beam is along the 110 direction of silicon crystal with 119864 =

1692GPa Also the side planes of the beam are considerednormal to the 110 direction with average in-plane Poissonrsquosratio and shear modulus equal to ] = 0239 and 120583 = 658GPa[29]

42 Modeling of Static Pull-In Instability of Cantilever BeamUsing Neural Networks Themodeling of pull-in instability ofmicrobeam with BP and RBF neural networks is composedof two stages training and testing of the networks withnumerical data The training data consisted of values forbeam length (119871) gap (119892) width of beam (119887) and (ℎ119897) andthe corresponding static pull-in instability voltage (119881PI) Atotal of 120 such datasets were used of which 110 wereselected randomly and used for training purposes whilstthe remaining 10 datasets were presented to the trainednetworks as new application data for verification (testing)purposes Thus the networks were evaluated using data

Output layer

Hidden layer

Input layer

L

gVPI

b

hl

Figure 6 General ANN topology

that had not been used for training Trainingtesting patternvectors are formed each with an input condition vector andthe corresponding target vector Map each term to a valuebetweenminus1 and 1 using the following linearmapping formula

119873 =(119877 minus 119877min)

lowast

(119873max minus 119873min)

(119877max minus 119877min)+ 119873min (16)

where 119873 normalized value of the real variable 119873min = minus1

and119873max = 1 minimum andmaximum values of normaliza-tion respectively 119877 real value of the variable 119877min and119877max minimum and maximum values of the real variablerespectively These normalized data were used as the inputand output to train the ANN In other words the networkhas four inputs of beam length (119871) gap (119892) width of beam (119887)and (ℎ119897) ratio and one output of static pull-in voltage (119881PI)Figure 6 shows the general network topology for modelingthe process

43 BP Neural Network Model The size of hidden layer(s) isone of the most important considerations when solving theactual problems with using multilayer feed-forward networkHowever it has been shown that the BP neural network withone hidden layer can uniformly approximate any continuousfunction to any desired degree of accuracy given an adequatenumber of neurons in the hidden layer and the correctinterconnection weights [34] Therefore one hidden layerwas adopted for the BP model To determine the number ofneurons in the hidden layer a procedure of trial and errorapproach needs to be done As such some attempts havebeenmade to study the network performance with a differentnumber of hidden neurons Hence a number of candidatenetworks are constructed each trained separately and theldquobestrdquo network was selected based on the accuracy of thepredictions in the testing phase It should be noted that if thenumber of hidden neurons is too large the ANN might beovertrained giving spurious values in the testing phase If toofew neurons are selected the function mapping might not beaccomplished due to undertrainingThree functions namelynewelm newff and newcf have been used for creating the BPnetworks Table 1 shows that 10 numerical datasets have beenused for verifying or testing network capabilities in modelingthe process

Therefore the general network structure is supposed tobe 4-119899-1 which implies 4 neurons in the input layer 119899


Table 1 Beam geometry and pull-in voltage for verification analysis

Test number 119871 (120583m) 119887 (120583m) ℎ119897 119892 (120583m) 119881PI (volt)1 75 05 4 05 01792 100 5 6 1 2443 125 10 8 15 7314 150 20 10 2 16825 175 25 12 25 26786 200 30 14 3 40277 225 35 16 35 53848 250 40 18 4 68019 275 45 20 45 845310 300 50 22 5 10362

neurons in the hidden layer and 1 neuron in the output layerThen by varying the number of hidden neurons differentnetwork configurations are trained and their performancesare checked The results are shown in Table 2

For training problem equal learning rate andmomentumconstant of 120578 = 120572 = 09 were used Also error stoppingcriterion was set at 119864 = 001 which means that trainingepochs continued until the mean square error fell beneaththis value Both the required iteration numbers and mappingperformances were examined for these networks As the errorcriterion for all networks was the same their performancesare comparable As a result from Table 2 the best networkstructure of BP model is picked to have 8 neurons in thehidden layer with the average verification errors of 636in predicting 119881PI by newelm function Table 3 shows thecomparison of calculated and predicted values for static pull-in voltage in verification cases After 1884 epochs the MSEbetween the desired and actual outputs becomes less than001 At the beginning of the training the output from thenetwork is far from the target value However the outputslowly and gradually converges to the target value with moreepochs and the network learns the inputoutput relation ofthe training samples

44 RBF Neural Network Model Spread factor (119878) valueof Gaussian activation functions in the hidden layer is theparameter that should be determined by trial and error whenusing MATLAB neural network toolbox for designing RBFnetworks It has to be larger than the distance betweenadjacent input vectors so as to get good generalization butsmaller than the distance across thewhole input spaceThere-fore in order to have a network model with good generaliza-tion capabilities the spread factor should be selected between05 and 534 For training the RBF network at first a guess ismade for the value of spread factor in the obtained intervalAlso the number of radial basis neurons is originally set asone At each iteration the input vector that results in loweringthemost network training error is used to create a radial basisneuronThen the error of the new network is checked and ifit is low enough the training stops Otherwise the next neu-ron is addedThis procedure is repeated until the error goal isachieved or the maximum number of neurons is reached Inthe present case it was found by trial and error that 20 hidden

neurons with the spread factor of 3 can give a model whichhas the best performance in the verification stage Table 4shows the effect of the number of hidden neurons on theRBF network performance It is clear that adding of hiddenneuronsmore than 20makes the training error (MSE) smallerbut deteriorates networkrsquos generalization capabilities with theincrease of average verification errors instead of decreasingTherefore the optimum number of radial basis neurons is20 The selected network has the average errors of 455 inresponse to the 10 input verification calculations (Table 1)for 119881PI with newrbe function Table 5 lists output valuespredicted by the RBF neural model and the calculated onesin verification (testing) phase Two functions namely newrbeand newrb have been used for creating of RBF networks

5 Selection of the Best Model

The simplest approach to the comparison of some differentnetworks is to evaluate the error function by using the datawhich is independent of that used for training [35] Hencethe selection of the corresponding ldquobest networkrdquo is carriedout based on the accuracy of predicting the process outputsin verification stage From Tables 2 and 4 it is concluded thatRBFNmodel with the total average error of 455 in compar-ison with 636 for BP model has superior performance andtherefore is picked as the best model Figure 7 illustrates thenumerical and predicted 119881PI by BP and RBF neural networksin verification stage Figures 8 and 9 compare the pull-involtages evaluated by the modified couple stress theory withℎ119897 = 4 119892 = 105 120583m 119887 = 50 120583m and ℎ = 294 120583m withthe results of the BP and RBF neural networks respectivelyThe pull-in voltages of the microcantilever versus parameterℎ119897 for 119887119892 = 50 are depicted in Figure 10 with threeBP functions The same conditions have been used for twofunctions of RBFN in Figure 11 As a further step to study thecapabilities of each network in fitting all points in the inputspace a linear regression between the network output and thecorresponding target (numerical) values was performed Inthis case the entire data set (training and verification) wasput through the trained networks and regression analysis wasconducted The networks have mapped the output very wellThe correlation coefficients (119877) are also given as a criterion ofcomparison The amounts of 119877 are 0989 for BP model and0997 for RBF model in simulating 119881PI

6 Conclusions and Summary

In this paper by using the modified couple stress theorythe size-dependent behavior of an electrostatically actu-ated microcantilever has been investigated In this studythe pull-in instability of geometrically nonlinear cantilevermicrobeams under an applied voltage is investigated Twosupervised neural networks have been used for the staticpull-in instability voltage of microcantilever beams Basedon the results of each network with some data set differentfrom those used in the training phase it was shown thatthe RBF neural model has superior performance than BPnetwork model and can predict the output in a wide range ofmicrobeam conditions with reasonable accuracy In sum the


Table 2 The effects of different numbers of hidden neurons on the BP network performance

Number of hidden neurons Epoch Average error in 119881PI ()with newelm function

Average error in 119881PI ()with newcf function

Average error in 119881PI ()with newff function

4 18914 1231 1027 12305 4970 1438 1838 20196 1783 819 1165 12757 3984 972 939 11178 1884 636 828 10149 2770 1339 1186 199810 2683 1167 1640 1548

Table 3 Comparison of 119881PI calculated and predicted by the BP neural network model with three functions

Testnumber

119881PI (volt) 119881PI (volt) 119881PI (volt)Calculated BP model (newelm) Error () Calculated BP model (newff) Error () Calculated BP model (newcf) Error ()

1 0179 0190 656 0179 0191 712 0179 0193 8292 244 261 728 244 252 339 244 259 6283 731 732 016 731 754 316 731 770 5394 1682 1921 1424 1682 1829 875 1682 1921 14245 2678 2822 539 2678 2816 519 2678 2876 7416 4027 4217 474 4027 4603 1431 4027 4399 9257 5384 5713 612 5384 5749 679 5384 5796 7678 6801 7160 529 6801 7839 1527 6801 8214 20789 8453 8939 576 8453 8671 259 8453 9251 94510 10362 11203 812 10362 12053 1632 10362 11674 1267

Table 4 The effects of different numbers of hidden neurons on theRBF network performance (119878 = 3)

Number ofhiddenneurons

Average error () inpredicting 119881PI withnewrbe function

Average error () inpredicting 119881PI withnewrb function

16 887 7318 966 107720 455 60822 1232 147124 1289 151626 1378 1624

following items can also bementioned as the general findingsof the present research

(1) The BP and RBF neural networks are capable of con-structing models using only numerical data describ-ing the static pull-in instability behavior

(2) RBF neural network which possesses the privilegesof rapid learning easy convergence and less errorwith respect to BP network has better generalizationpower and is more accurate for this particular caseThis selection was done according to the resultsobtained in the verification phase

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10Test number

Pull-

in v

olta

ge (v

olt)

NumericalRBFNBPN

Figure 7The comparison of119881PI between verification and numericalresults

(3) The results show that the newelm function is moreaccurate than newff and newcf functions Also theLevenberg-Marquardt training is faster than othertraining methods

(4) The results have demonstrated the applicability andadaptability of the RBNN for analysis of instabilitystatic pull-in voltage of cantilever beams and alsothe newrbe function is more accurate and faster thannewrb function


Table 5 Comparison of 119881PI calculated and predicted by the RBF neural network model

Test number 119881PI (volt) 119881PI (volt)Calculated RBF model (newrbe) Error () Calculated RBF model (newrb) Error ()

1 0179 019 849 0179 018 3782 244 247 134 244 270 10973 731 768 512 731 781 6934 1682 1688 038 1682 1727 2715 2678 2747 259 2678 2776 3696 4027 4196 420 4027 4320 7297 5384 5480 179 5384 5496 2098 6801 7064 388 6801 7506 10389 8453 8543 107 8453 8840 45910 10362 12089 1667 10362 11375 978

0102030405060708090

100

50 100 150 200 250 300Cantilever length

Pull-

in v

olta

ge (v

olt)

Numericalnewelm BP function

newff BP functionnewcf BP function

Figure 8 Comparing the theoretical and BP neural networks pull-in voltages for silicon 110

0102030405060708090


Pull-

in v

olta

ge (v

olt)

Numericalnewrbe RB functionnewrb RB function

Figure 9 Comparing the theoretical and RBF neural networks pull-in voltages for silicon 110

(5) For cantilever beams by increasing of gap length thepull-in voltage is significantly increased

(6) For cantilever beams by increasing of length of beamsthe pull-in voltage is significantly decreased

0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)



Figure 10 Comparing pull-in voltage of the microcantilever versusparameter ℎ119897 and 119887119892 = 50 with BP models

0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)


Figure 11 Comparing pull-in voltage of the microcantilever versusparameter ℎ119897 and 119887119892 = 50 with RBF models

(7) When the ratio of ℎ119897 increases the pull-in voltagepredicted by modified couple stress theory and ANNis constant approximately

The conclusion indicates that the geometry of beam hassignificant influences on the electrostatic characteristics of


microbeams that can be designed to tailor the desired per-formance in different MEMS applications

References

[1] I Khatami M H Pashai and N Tolou ldquoComparative vibrationanalysis of a parametrically nonlinear excited oscillator usingHPM and numerical methodrdquoMathematical Problems in Engi-neering vol 2008 Article ID 956170 11 pages 2008

[2] A M Gasparini A V Saetta and R V Vitaliani ldquoOn thestability and instability regions of non-conservative continuoussystem under partially follower forcesrdquo Computer Methods inApplied Mechanics and Engineering vol 124 no 1-2 pp 63ndash781995

[3] Y T Beni M R Abadyanb and A Noghrehabadi ldquoInvestiga-tion of size effect on the pull-in instability of beam type NEMSunder Van Der Waals attractionrdquo Procedia Engineering vol 10pp 1718ndash1723 2011

[4] P M Osterberg and S D Senturia ldquoM-test a test chip forMEMS material property measurement using electrostaticallyactuated test structuresrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 107ndash118 1997

[5] P M Osterberg R K Gupta J R Gilbert and S D SenturialdquoQuantitative models for the measurement of residual stressPoisson ratio and Youngrsquos modulus using electrostatic pull-in ofbeams and diaphragmsrdquo in Proceedings of the Solid-State Sensorand Actuator Workshop pp 184ndash188 Hilton Head Island SCUSA June 1994

[6] H Sadeghian G Rezazadeh and P M Osterberg ldquoApplicationof the generalized differential quadrature method to the studyof pull-in phenomena of MEMS switchesrdquo Journal of Microelec-tromechanical Systems vol 16 no 6 pp 1334ndash1340 2007

[7] Y T Beni A Koochi and M Abadyan ldquoTheoretical study ofthe effect of Casimir force elastic boundary conditions andsize dependency on the pull-in instability of beam-type NEMSrdquoPhysica E vol 43 no 4 pp 979ndash988 2011

[8] A Koochi A S Kazemi Y T Beni A Yekrangi andM Abady-an ldquoTheoretical study of the effect of Casimir attraction on thepull-in behavior of beam-type NEMS using modified Adomianmethodrdquo Physica E vol 43 no 2 pp 625ndash632 2010

[9] M Ghalambaz A Noghrehabadi M Abadyan Y T Beni AR N Abadi and M N Abadi ldquoA new power series solution onthe electrostatic pull-in instability of nano-cantilever actuatorsrdquoProcedia Engineering vol 10 pp 3708ndash3716 2011

[10] A Y Salekdeh A Koochi Y T Beni andM Abadyan ldquoModel-ing effect of three nano-scale physical phenomena on instabilityvoltage of multi-layerMEMSNEMSmaterial size dependencyvan der waals force and non-classic support conditionsrdquo Trendsin Applied Sciences Research vol 7 no 1 pp 1ndash17 2012

[11] MMZand andMTAhmadian ldquoCharacterization of coupled-domain multi-layer microplates in pull-in phenomenon vibra-tions and dynamicsrdquo International Journal of Mechanical Sci-ences vol 49 no 11 pp 1226ndash1237 2007

[12] K Das and R C Batra ldquoSymmetry breaking snap-through andpull-in instabilities under dynamic loading of microelectrome-chanical shallow archesrdquo SmartMaterials and Structures vol 18no 11 Article ID 115008 2009

[13] M Mojahedi M M Zand and M T Ahmadian ldquoStatic pull-inanalysis of electrostatically actuated microbeams using homo-topy perturbation methodrdquo Applied Mathematical Modellingvol 34 no 4 pp 1032ndash1041 2010

[14] S A Tajalli M M Zand and M T Ahmadian ldquoEffect of geo-metric nonlinearity on dynamic pull-in behavior of coupled-domain microstructures based on classical and shear deforma-tion plate theoriesrdquo European Journal of Mechanics vol 28 no5 pp 916ndash925 2009

[15] S Pamidighantam R Puers K Baert and H A C TilmansldquoPull-in voltage analysis of electrostatically actuated beamstructures with fixed-fixed and fixed-free end conditionsrdquo Jour-nal of Micromechanics and Microengineering vol 12 no 4 pp458ndash464 2002

[16] S Chowdhury M Ahmadi and W C Miller ldquoA closed-formmodel for the pull-in voltage of electrostatically actuated can-tilever beamsrdquo Journal ofMicromechanics andMicroengineeringvol 15 no 4 pp 756ndash763 2005

[17] R C Batra M Porfiri and D Spinello ldquoReview of model-ing electrostatically actuated microelectromechanical systemsrdquoSmartMaterials and Structures vol 16 no 6 pp R23ndashR31 2007

[18] W-H Lin and Y-P Zhao ldquoPull-in instability of micro-switchactuators model reviewrdquo International Journal of NonlinearSciences and Numerical Simulation vol 9 no 2 pp 175ndash1842008

[19] W T Koiter ldquoCouple-stresses in the theory of elasticity I and IIrdquoKoninklijke Nederlandse Akademie Van Weteschappen Series Bvol 67 pp 17ndash44 1964

[20] R D Mindlin and H F Tiersten ldquoEffects of couple-stresses inlinear elasticityrdquo Archive for Rational Mechanics and Analysisvol 11 no 1 pp 415ndash448 1962

[21] R A Toupin ldquoElastic materials with couple-stressesrdquo Archivefor Rational Mechanics and Analysis vol 11 no 1 pp 385ndash4141962

[22] A Anthoine ldquoEffect of couple-stresses on the elastic bending ofbeamsrdquo International Journal of Solids and Structures vol 37 no7 pp 1003ndash1018 2000

[23] F Yang A C M Chong D C C Lam and P Tong ldquoCouplestress based strain gradient theory for elasticityrdquo InternationalJournal of Solids and Structures vol 39 no 10 pp 2731ndash27432002

[24] W Xia L Wang and L Yin ldquoNonlinear non-classical micro-scale beams Static bending postbuckling and free vibrationrdquoInternational Journal of Engineering Science vol 48 no 12 pp2044ndash2053 2010

[25] M Asghari M Rahaeifard M H Kahrobaiyan and M TAhmadian ldquoThe modified couple stress functionally gradedTimoshenko beam formulationrdquo Materials and Design vol 32no 3 pp 1435ndash1443 2011

[26] H Rong Q-A Huang M Nie andW Li ldquoAn analytical modelfor pull-in voltage of clamped-clamped multilayer beamsrdquoSensors and Actuators A vol 116 no 1 pp 15ndash21 2004

[27] J A Freeman and D M Skapura Neural Networks AlgorithmsApplications and Programming Techniques Computation andNeural Systems Addision-Wesley Boston Mass USA 1991

[28] Y T Beni andM Heidari ldquoNumerical study on pull-in instabil-ity analysis of geometrically nonlinear Euler-Bernoulli microb-eam based on modified couple stress theoryrdquo InternationalJournal of Engineering and Applied Sciences vol 4 no 4 pp 41ndash53 2012

[29] R K Gupta Electrostatic pull-in test structure design for in-situmechanical property measurements of microelectromechanicalsystems [PhD thesis] Massachusetts Institute of Technology(MIT) Cambridge Mass USA 1997


[30] J Zhao S Zhou BWang and XWang ldquoNonlinearmicrobeammodel based on strain gradient theoryrdquo Applied MathematicalModelling vol 36 no 6 pp 2674ndash2686 2012

[31] D Gao Y Kinouchi K Ito and X Zhao ldquoNeural networks forevent extraction from time series a back propagation algorithmapproachrdquo Future Generation Computer Systems vol 21 no 7pp 1096ndash1105 2005

[32] H Demuth and M Beale ldquoMatlab Neural Networks ToolboxUserrsquos Guide The Math Worksrdquo 2001 httpwwwmathworkscom

[33] H Shao and G Zheng ldquoConvergence analysis of a back-prop-agation algorithm with adaptive momentumrdquoNeurocomputingvol 74 no 5 pp 749ndash752 2011

[34] H Zhang W Wu and M Yao ldquoBoundedness and convergenceof batch back-propagation algorithm with penalty for feedfor-ward neural networksrdquo Neurocomputing vol 89 pp 141ndash1462012

[35] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



method Different analytical numerical finite element meth-ods have been proposed to model the pull-in behavior ofmicrobeams [12ndash16] Further information about modelingpull-in instability of MEMS has been presented by Batraet al [17] and Lin and Zhao [18] The classical continuummechanics theory is not able to explain the size-dependentbehavior of materials and structures at submicron distancesTo overcome this problem nonclassical continuum theoriessuch as higher-order gradient theories and the couple stresstheory are developed considering the size effects [7] In 1960ssome researchers such as Koiter [19] Mindlin and Tiersten[20] and Toupin [21] introduced the couple stress elasticitytheory as a nonclassic theory capable to predict the size effectswith appearance of twohigher-ordermaterial constants in thecorresponding constitutive equations In this theory besidesthe classical stress components acting on elements of mate-rials the couple stress components as higher-order stressesare also available which tend to rotate the elements Utilizingthe couple stress theory some researchers investigated thesize effects in someproblems [22] Employing the equilibriumequation of moments of couples besides the classical equilib-rium equations of forces and moments of forces a modifiedcouple stress theory is introduced by Yang et al [23] with onehigher-order material constant in the constitutive equationsRecently size-dependent nonlinear Euler-Bernoulli andTim-oshenko beams modeled on the basis of the modified couplestress theory have been developed by Xia et al [24] andAsghari et al [25] respectively Rong et al [26] presentedan analytical method for pull-in analysis of clamped-clampedmultilayer beam Their method is the Rayleigh-Ritz methodand assumes one deflection shape functionThey derived thetwo governing equations by enforcing the pull-in conditionsthat the first- and second-order derivatives of the systemenergy functional are zero They investigated the static pull-in instability voltage and displacement of themultilayer beamwhichwere coupled in the two governing equations Artificialneural networks (ANNs) as one of the most attractivebranches in artificial intelligence have the potentiality tohandle problems such as modeling estimation predictiondiagnosis and adaptive control in complex nonlinear systems[27] As seen from pervious studies the researcher used theanalytical or numerical methods for static pull-in instabilityvoltage of microcantilever beams but in this study we usethe neural network to estimate the pull-in voltage Thisstudy investigates the pull-in instability of microbeams witha curved ground electrode under action of electric fieldforce within the framework of von Karman nonlinearityand the Euler-Bernoulli beam theory The static pull-ininstability voltage of cantilever microbeam is obtained byusing MAPLE commercial software The objective of thispaper is to establish the neural networkmodels for estimationof the pull-in instability voltage of cantilever beams Morespecifically two different types of neural networks backpropagation (BP) and radial basis function (RBF) are used toconstruct the pull-in instability voltage Effective parametersinfluencing pull-in voltage and their levels for training wereselected through preliminary calculations carried out oninstability pull-in voltage of microbeam Networks trained bythe same numerical data are then verified by some numerical

q(x)

x

L

g

h

z

z

b

yw(x)

V

Fixed substrate

Figure 1 An electrostatically actuated Euler-Bernoulli microbeam

calculations different from those used in training phase andthe best model was selected based on the criterion of havingthe least average values of verification errors To the authorsrsquobest knowledge no previous studies which cover all theseissues are available

2 Modeling the Microcantilever Based onthe Modified Couple Stress

Consider an electrostatically actuated microbeam of length119871 as shown in Figure 1 The microcantilever is under atransverse distributed electrical force 119902(119909) caused by the inputvoltage applied between the microbeam and the substrateThe cross section of the beam is here assumed to be rect-angular with width 119887 and height ℎ The governing equationfor the static behavior of a uniform homogeneous Euler-Bernoulli beam made of an isotropic linear elastic materialunder transverse distributed load 119902(119909)modeled based on themodified couple stress theory is written as [28]

1198781205974119908

1205971199094minus 119873

1205972119908

1205971199092+ 120588119860

1205972119908

1205971199052= 119902 (119909) (1)

where

119873 = 1198730+119864119860

2119871int

119871

0

(120597119908

120597119909)

2

119889119909

119878 = 119864119868 + 1205831198601198972

(2)

where 1198641198730 119868 and 119908(119909 119905) are Youngrsquos modulus in the beam

direction the axial load the second moment of inertia andthe transverse deflection In addition 119897 is a material lengthscale parameter and 120583 is the average shear modulus in theside-plane of the beam Letting 119897 = 0 the equation is reducedto one corresponding to the classical theory If in (1)119873 = 0then the model of beam is called the linear equation withoutthe effect of geometric nonlinearity The cross-sectional areaof beam is 119860 The electrostatic force enhanced with first-order fringing correction can be presented in the followingequation [29]

119902 (119909) =12057601198871198812

2(119892 minus 119908 (119909 119905))2[1 + 065

(119892 minus 119908)

119887] (3)


where 1205760= 8854 times 10




(119864119868 + 1205831198601198972)1198894119908

1198891199094minus [1198730+119864119860

2119871int

119871

0

(119889119908

119889119909)

2

119889119909]1198892119908

1198891199092

=12057601198871198812

2(119892 minus 119908)2[1 + 065

(119892 minus 119908)

119887]

(4)


119908 (0) = 0119889119908 (0)

119889119909= 0

1198892119908 (119871)

1198891199092

= 01198893119908 (119871)

1198891199093

= 0

(5)


120572 =1198601198712

2119868 120573 =

120576011988711988121198714

21198923119864119868 120574 = 065

119892

119887

120575 =1205831198601198972

119864119868 119908 =

119908

119892 119909 =

119909

119871 Γ =

11987301198712

119864119868

(6)


(1 + 120575)1198894119908

119889119909minus Γ + 120572int

1

0

(119889119908

119889119909)

2

1198891199091198892119908

1198891199092

=120573

(1 minus 119908)2+

120574120573

(1 minus 119908)

(7)






net119896119895= sum

119894

119908119896

119895119894119900119896minus1

119894 (8)

and 119900119896minus1






119895) with a bias 119887

119895is



expressed as

119900119896

119895= 119891 (net119896

119895) =

119890(net119896119895+119887119895) minus 119890

minus(net119896119895+119887119895)

119890(net119896119895+119887119895) + 119890

minus(net119896119895+119887119895)

(9)



Inputlayer

layerlayer

First hidden

Output


Activationfunction

Bias = 1

okminus11

okminus12

okminus1i

okminus1n

wkjn

okj

wkj2

wkj1

wj0 = bj

(f)wkji

Sumfunction(sum)



119864119901=1

2sum

119895

(119879119901119895minus 119874119901119895)2

(10)




119864 =1

119901

119899

sum

119901=1

119864119901 (11)


119895119894(119898 + 1) can be

Δ119908119896

119895119894(119898 + 1) = minus120578

120597119864

120597119908119896

119895119894

+ 120572Δ119908119896

119895119894(119898) (12)



119908119896

119895119894(119898 + 1) = 119908

119896

119895119894(119898) + Δ119908

119896

119895119894(119898 + 1) (13)



119860119895= exp(minus

10038171003817100381710038171003817X119894minus C119895

10038171003817100381710038171003817

2

1198782) (14)


119895is the vector


119884119896=

119867

sum

119895=1

119908119896119895119860119895 (15)


neurons and 119908119896119895






X1

X2

XN

X WC S Y

Y1

YM

W11

WM1

WM2

Network inputs


Network outputs


1

b

a

||dist||



C1 C2 C3

X1

X2

X3

n10

05

00

minus0833 +0833

a = radbas(n)

a

n







Output layer

Hidden layer

Input layer

L

gVPI

b

hl



119873 =(119877 minus 119877min)

lowast


(119877max minus 119877min)+ 119873min (16)





















4 18914 1231 1027 12305 4970 1438 1838 20196 1783 819 1165 12757 3984 972 939 11178 1884 636 828 10149 2770 1339 1186 199810 2683 1167 1640 1548


Testnumber


1 0179 0190 656 0179 0191 712 0179 0193 8292 244 261 728 244 252 339 244 259 6283 731 732 016 731 754 316 731 770 5394 1682 1921 1424 1682 1829 875 1682 1921 14245 2678 2822 539 2678 2816 519 2678 2876 7416 4027 4217 474 4027 4603 1431 4027 4399 9257 5384 5713 612 5384 5749 679 5384 5796 7678 6801 7160 529 6801 7839 1527 6801 8214 20789 8453 8939 576 8453 8671 259 8453 9251 94510 10362 11203 812 10362 12053 1632 10362 11674 1267





16 887 7318 966 107720 455 60822 1232 147124 1289 151626 1378 1624




0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10Test number

Pull-

in v

olta

ge (v

olt)

NumericalRBFNBPN







1 0179 019 849 0179 018 3782 244 247 134 244 270 10973 731 768 512 731 781 6934 1682 1688 038 1682 1727 2715 2678 2747 259 2678 2776 3696 4027 4196 420 4027 4320 7297 5384 5480 179 5384 5496 2098 6801 7064 388 6801 7506 10389 8453 8543 107 8453 8840 45910 10362 12089 1667 10362 11375 978

0102030405060708090

100


Pull-

in v

olta

ge (v

olt)




0102030405060708090


Pull-

in v

olta

ge (v

olt)





0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)




0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)







References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




where 1205760= 8854 times 10




(119864119868 + 1205831198601198972)1198894119908

1198891199094minus [1198730+119864119860

2119871int

119871

0

(119889119908

119889119909)

2

119889119909]1198892119908

1198891199092

=12057601198871198812

2(119892 minus 119908)2[1 + 065

(119892 minus 119908)

119887]

(4)


119908 (0) = 0119889119908 (0)

119889119909= 0

1198892119908 (119871)

1198891199092

= 01198893119908 (119871)

1198891199093

= 0

(5)


120572 =1198601198712

2119868 120573 =

120576011988711988121198714

21198923119864119868 120574 = 065

119892

119887

120575 =1205831198601198972

119864119868 119908 =

119908

119892 119909 =

119909

119871 Γ =

11987301198712

119864119868

(6)


(1 + 120575)1198894119908

119889119909minus Γ + 120572int

1

0

(119889119908

119889119909)

2

1198891199091198892119908

1198891199092

=120573

(1 minus 119908)2+

120574120573

(1 minus 119908)

(7)






net119896119895= sum

119894

119908119896

119895119894119900119896minus1

119894 (8)

and 119900119896minus1






119895) with a bias 119887

119895is



expressed as

119900119896

119895= 119891 (net119896

119895) =

119890(net119896119895+119887119895) minus 119890

minus(net119896119895+119887119895)

119890(net119896119895+119887119895) + 119890

minus(net119896119895+119887119895)

(9)



Inputlayer

layerlayer

First hidden

Output


Activationfunction

Bias = 1

okminus11

okminus12

okminus1i

okminus1n

wkjn

okj

wkj2

wkj1

wj0 = bj

(f)wkji

Sumfunction(sum)



119864119901=1

2sum

119895

(119879119901119895minus 119874119901119895)2

(10)




119864 =1

119901

119899

sum

119901=1

119864119901 (11)


119895119894(119898 + 1) can be

Δ119908119896

119895119894(119898 + 1) = minus120578

120597119864

120597119908119896

119895119894

+ 120572Δ119908119896

119895119894(119898) (12)



119908119896

119895119894(119898 + 1) = 119908

119896

119895119894(119898) + Δ119908

119896

119895119894(119898 + 1) (13)



119860119895= exp(minus

10038171003817100381710038171003817X119894minus C119895

10038171003817100381710038171003817

2

1198782) (14)


119895is the vector


119884119896=

119867

sum

119895=1

119908119896119895119860119895 (15)


neurons and 119908119896119895






X1

X2

XN

X WC S Y

Y1

YM

W11

WM1

WM2

Network inputs


Network outputs


1

b

a

||dist||



C1 C2 C3

X1

X2

X3

n10

05

00

minus0833 +0833

a = radbas(n)

a

n







Output layer

Hidden layer

Input layer

L

gVPI

b

hl



119873 =(119877 minus 119877min)

lowast


(119877max minus 119877min)+ 119873min (16)





















4 18914 1231 1027 12305 4970 1438 1838 20196 1783 819 1165 12757 3984 972 939 11178 1884 636 828 10149 2770 1339 1186 199810 2683 1167 1640 1548


Testnumber


1 0179 0190 656 0179 0191 712 0179 0193 8292 244 261 728 244 252 339 244 259 6283 731 732 016 731 754 316 731 770 5394 1682 1921 1424 1682 1829 875 1682 1921 14245 2678 2822 539 2678 2816 519 2678 2876 7416 4027 4217 474 4027 4603 1431 4027 4399 9257 5384 5713 612 5384 5749 679 5384 5796 7678 6801 7160 529 6801 7839 1527 6801 8214 20789 8453 8939 576 8453 8671 259 8453 9251 94510 10362 11203 812 10362 12053 1632 10362 11674 1267





16 887 7318 966 107720 455 60822 1232 147124 1289 151626 1378 1624




0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10Test number

Pull-

in v

olta

ge (v

olt)

NumericalRBFNBPN







1 0179 019 849 0179 018 3782 244 247 134 244 270 10973 731 768 512 731 781 6934 1682 1688 038 1682 1727 2715 2678 2747 259 2678 2776 3696 4027 4196 420 4027 4320 7297 5384 5480 179 5384 5496 2098 6801 7064 388 6801 7506 10389 8453 8543 107 8453 8840 45910 10362 12089 1667 10362 11375 978

0102030405060708090

100


Pull-

in v

olta

ge (v

olt)




0102030405060708090


Pull-

in v

olta

ge (v

olt)





0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)




0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)







References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Inputlayer

layerlayer

First hidden

Output


Activationfunction

Bias = 1

okminus11

okminus12

okminus1i

okminus1n

wkjn

okj

wkj2

wkj1

wj0 = bj

(f)wkji

Sumfunction(sum)



119864119901=1

2sum

119895

(119879119901119895minus 119874119901119895)2

(10)




119864 =1

119901

119899

sum

119901=1

119864119901 (11)


119895119894(119898 + 1) can be

Δ119908119896

119895119894(119898 + 1) = minus120578

120597119864

120597119908119896

119895119894

+ 120572Δ119908119896

119895119894(119898) (12)



119908119896

119895119894(119898 + 1) = 119908

119896

119895119894(119898) + Δ119908

119896

119895119894(119898 + 1) (13)



119860119895= exp(minus

10038171003817100381710038171003817X119894minus C119895

10038171003817100381710038171003817

2

1198782) (14)


119895is the vector


119884119896=

119867

sum

119895=1

119908119896119895119860119895 (15)


neurons and 119908119896119895






X1

X2

XN

X WC S Y

Y1

YM

W11

WM1

WM2

Network inputs


Network outputs


1

b

a

||dist||



C1 C2 C3

X1

X2

X3

n10

05

00

minus0833 +0833

a = radbas(n)

a

n







Output layer

Hidden layer

Input layer

L

gVPI

b

hl



119873 =(119877 minus 119877min)

lowast


(119877max minus 119877min)+ 119873min (16)





















4 18914 1231 1027 12305 4970 1438 1838 20196 1783 819 1165 12757 3984 972 939 11178 1884 636 828 10149 2770 1339 1186 199810 2683 1167 1640 1548


Testnumber


1 0179 0190 656 0179 0191 712 0179 0193 8292 244 261 728 244 252 339 244 259 6283 731 732 016 731 754 316 731 770 5394 1682 1921 1424 1682 1829 875 1682 1921 14245 2678 2822 539 2678 2816 519 2678 2876 7416 4027 4217 474 4027 4603 1431 4027 4399 9257 5384 5713 612 5384 5749 679 5384 5796 7678 6801 7160 529 6801 7839 1527 6801 8214 20789 8453 8939 576 8453 8671 259 8453 9251 94510 10362 11203 812 10362 12053 1632 10362 11674 1267





16 887 7318 966 107720 455 60822 1232 147124 1289 151626 1378 1624




0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10Test number

Pull-

in v

olta

ge (v

olt)

NumericalRBFNBPN







1 0179 019 849 0179 018 3782 244 247 134 244 270 10973 731 768 512 731 781 6934 1682 1688 038 1682 1727 2715 2678 2747 259 2678 2776 3696 4027 4196 420 4027 4320 7297 5384 5480 179 5384 5496 2098 6801 7064 388 6801 7506 10389 8453 8543 107 8453 8840 45910 10362 12089 1667 10362 11375 978

0102030405060708090

100


Pull-

in v

olta

ge (v

olt)




0102030405060708090


Pull-

in v

olta

ge (v

olt)





0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)




0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)







References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




X1

X2

XN

X WC S Y

Y1

YM

W11

WM1

WM2

Network inputs


Network outputs


1

b

a

||dist||



C1 C2 C3

X1

X2

X3

n10

05

00

minus0833 +0833

a = radbas(n)

a

n







Output layer

Hidden layer

Input layer

L

gVPI

b

hl



119873 =(119877 minus 119877min)

lowast


(119877max minus 119877min)+ 119873min (16)





















4 18914 1231 1027 12305 4970 1438 1838 20196 1783 819 1165 12757 3984 972 939 11178 1884 636 828 10149 2770 1339 1186 199810 2683 1167 1640 1548


Testnumber


1 0179 0190 656 0179 0191 712 0179 0193 8292 244 261 728 244 252 339 244 259 6283 731 732 016 731 754 316 731 770 5394 1682 1921 1424 1682 1829 875 1682 1921 14245 2678 2822 539 2678 2816 519 2678 2876 7416 4027 4217 474 4027 4603 1431 4027 4399 9257 5384 5713 612 5384 5749 679 5384 5796 7678 6801 7160 529 6801 7839 1527 6801 8214 20789 8453 8939 576 8453 8671 259 8453 9251 94510 10362 11203 812 10362 12053 1632 10362 11674 1267





16 887 7318 966 107720 455 60822 1232 147124 1289 151626 1378 1624




0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10Test number

Pull-

in v

olta

ge (v

olt)

NumericalRBFNBPN







1 0179 019 849 0179 018 3782 244 247 134 244 270 10973 731 768 512 731 781 6934 1682 1688 038 1682 1727 2715 2678 2747 259 2678 2776 3696 4027 4196 420 4027 4320 7297 5384 5480 179 5384 5496 2098 6801 7064 388 6801 7506 10389 8453 8543 107 8453 8840 45910 10362 12089 1667 10362 11375 978

0102030405060708090

100


Pull-

in v

olta

ge (v

olt)




0102030405060708090


Pull-

in v

olta

ge (v

olt)





0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)




0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)







References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



















4 18914 1231 1027 12305 4970 1438 1838 20196 1783 819 1165 12757 3984 972 939 11178 1884 636 828 10149 2770 1339 1186 199810 2683 1167 1640 1548


Testnumber


1 0179 0190 656 0179 0191 712 0179 0193 8292 244 261 728 244 252 339 244 259 6283 731 732 016 731 754 316 731 770 5394 1682 1921 1424 1682 1829 875 1682 1921 14245 2678 2822 539 2678 2816 519 2678 2876 7416 4027 4217 474 4027 4603 1431 4027 4399 9257 5384 5713 612 5384 5749 679 5384 5796 7678 6801 7160 529 6801 7839 1527 6801 8214 20789 8453 8939 576 8453 8671 259 8453 9251 94510 10362 11203 812 10362 12053 1632 10362 11674 1267





16 887 7318 966 107720 455 60822 1232 147124 1289 151626 1378 1624




0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10Test number

Pull-

in v

olta

ge (v

olt)

NumericalRBFNBPN







1 0179 019 849 0179 018 3782 244 247 134 244 270 10973 731 768 512 731 781 6934 1682 1688 038 1682 1727 2715 2678 2747 259 2678 2776 3696 4027 4196 420 4027 4320 7297 5384 5480 179 5384 5496 2098 6801 7064 388 6801 7506 10389 8453 8543 107 8453 8840 45910 10362 12089 1667 10362 11375 978

0102030405060708090

100


Pull-

in v

olta

ge (v

olt)




0102030405060708090


Pull-

in v

olta

ge (v

olt)





0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)




0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)







References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1 0179 019 849 0179 018 3782 244 247 134 244 270 10973 731 768 512 731 781 6934 1682 1688 038 1682 1727 2715 2678 2747 259 2678 2776 3696 4027 4196 420 4027 4320 7297 5384 5480 179 5384 5496 2098 6801 7064 388 6801 7506 10389 8453 8543 107 8453 8840 45910 10362 12089 1667 10362 11375 978

0102030405060708090

100


Pull-

in v

olta

ge (v

olt)




0102030405060708090


Pull-

in v

olta

ge (v

olt)





0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)




0

20

40

60

80

100

120

5 10 15 20 25 30 35 40hl

Pull-

in v

olta

ge (v

olt)







References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





References












































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in

















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article Estimation of Static Pull-In Instability ...

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Transcript of Research Article Estimation of Static Pull-In Instability ...