estimation of pull-in instability voltage of euler ... · stretching as the source of the...

Int J Nano Dimens 6(5) 487-500 2015 (Special Issue for NCNC Dec 2014 IRAN)

487

ABSTRACT The static pull-in instability of beam-type micro-electromechanical systems is theoretically investigatedTwo engineering cases including cantilever and double cantilever micro-beam are considered Considering the mid-planestretching as the source of the nonlinearity in the beam behavior a nonlinear size-dependent Euler-Bernoulli beam modelis used based on a modified couple stress theory capable of capturing the size effect By selecting a range of geometricparameters such as beam lengths width thickness gaps and size effect we identify the static pull-in instability voltageBack propagation artificial neural network with three functions have been used for modeling the static pull-in instabilityvoltage of the micro cantilever beam The network has four inputs of length width gap and the ratio of height to scaleparameter of the beam as the independent process variables and the output is static pull-in voltage of microbeamNumerical data employed for training the network and capabilities of the model in predicting the pull-in instabilitybehavior has been verified The output obtained from the neural network model is compared with numerical results andthe amount of relative error has been calculated Based on this verification error it is shown that the back propagationneural network has the average error of 636 in predicting pull-in voltage of the cantilever micro-beam

Keywords Artificial neural networks Euler-Bernoulli Modified couple stress theory Nonlinear micro-beam Staticpull-in instability


DOI 107508ijnd201505006

Corresponding Author Mohammad HeidariEmail moh104337yahoocomTel (+98) 9131835779Fax (+98) 3833343096

5HFHLYHG)HEUXDU UHYLVHG-XO DFFHSWHG-XO DYDLODEOHRQOLQH$XJXVW

INTRODUCTIONMicro-electromechanical systems (MEMS) are

widely being used in todays technology Soinvestigating the problems referring to MEMS owns agreat importance One of the significant fields of studyis the stability analysis of the parametrically excitedsystems Parametrically excited micro-electromechanical devices are ever increasingly beingused in radio computer and laser engineering [1]Parametric excitation occurs in a wide range ofmechanics due to time dependent excitationsespecially periodic ones some examples are columnsmade of nonlinear elastic material beams with aharmonically variable length parametrically excitedpendulums and so forth Investigating stabilityanalysis of parametrically excited MEM systems is ofgreat importance In 1995 Gasparini et al [2] studiedon the transition between the stability and instability

Estimation of pull-in instability voltage of Euler-Bernoulli micro beam byback propagation artificial neural network

M Heidari

Mechanical Engineering Group Aligudarz Branch Islamic Azad University Aligudarz Iran

of a cantilevered beam exposed to a partially followerload Applying voltage difference between an electrodeand ground causes the electrode to deflect towardsthe ground At a critical voltage which is known as apull-in voltage the electrode becomes unstable andpulls-in onto the substrate The pull-in behavior ofMEMS actuators has been studied for over two decadeswithout considering the casimir force [35] Osterberget al [3 4] investigated the pull-in parameters of thebeam-type and circular MEMS actuators using thedistributed parameter models Sadeghian et al [5]applied the generalized differential quadrature methodto investigate the pull-in phenomena of micro-switchesA comprehensive literature review on investigatingMEMS actuators can be found in Ref [6] Furtherinformation about modeling pull-in instability of MEMShas been presented in Ref [7 8] The classicalcontinuum mechanics theories are not capable ofprediction and explanation of the size-dependentbehaviors which occur in micron- and sub-micron-scalestructures However some non-classical continuum

Research Paper


488

M Heidari

theories such as higher-order gradient theories andthe couple stress theory have been developed suchthat they are acceptably able to interpret the size-dependencies In the 1960s some researchers such asKoiter [9] Mindlin [10] and Toupin [11] introduced thecouple stress elasticity theory as a non-classic theorycapable to predict the size effects with the appearanceof two higher-order material constants in thecorresponding constitutive equations In this theorybeside the classical stress components acting onelements of materials the couple stress componentsas higher-order stresses are also available which tendto rotate the elements Utilizing the couple stress theorysome researchers investigated the size effects in someproblems [12]Employing the equilibrium equation ofmoments of couples beside the classical equilibriumequations of forces and moments of forces a modifiedcouple stress theory introduced by Yang Chong Lamand Tong [13] with one higher-order material constantin the constitutive equations Recently size-dependentnonlinear EulerBernoulli and Timoshenko beamsmodeled on the basis of the modified couple stresstheory have been developed by Xia et al [14] andAsghari et al [15] respectively Rong et al [16] presentan analytical method for pull-in analysis of clampedclamped multilayer beam Their method is Rayleigh-Ritz method and assumes one deflection shapefunction They derive the two governing equations byenforcing the pull-in conditions that the first andsecond order derivatives of the system energyfunctional are zero In their model the pull-in voltageand displacement are coupled in the two governingequations

This paper investigates the pull-in instability ofmicro-beams with a curved ground electrode under theaction of electric field force within the framework ofvon-Karman nonlinearity and the EulerBernoulli beamtheory The static pull-in voltage instability of clamped-clamped and cantilever micro-beam are obtained byusing MAPLE commercial software The effects ofgeometric parameters such as beam lengths widththickness gaps and size effect are discussed in detailthrough a numerical study The objective of this paperis to establish a neural network model for estimatingthe pull-in instability voltage of cantilever beams Morespecifically back propagation neural network is usedto construct the pull-in instability voltage Effectiveparameters influencing pull-in voltage and their levelsof training were selected through preliminarycalculations carried out on instability pull-in voltage

of micro-beam The network trained by the samenumerical data are then verified by some numericalcalculations different from those used in the trainingphase and the best model was selected based on thecriterion of having the least average values ofverification errors To the authors best knowledge noprevious studies which cover all these issues areavailable To the authors best knowledge no previousstudies which cover all these issues are available

EXPERIMENTALPreliminaries

In the modified couple stress theory the strainenergy density u of a linear elastic isotropic materialin infinitesimal deformation is written as [17]

)321()(2

1 jimu ijijijij (1)

Where

ijijmmij 2 (2)

))()((21 T

ijijij uu (3)

ijij lm 22 (4)

))()((21 T

ijijij (5)

In which ij ij ijm and ij denote thecomponents of the symmetric part of stress tensor the strain tensor the deviatoric part of the couplestress tensor m and the symmetric part of the curvaturetensor respectively Also u and are thedisplacement vector and the rotation vector The twoLame constants and the material length scale parameterare represented by and l respectively The Lameconstants are written in terms of the Youngs modulusE and the Poissons ratio as

)21)(1( E and )1(2 EThe components of the infinitesimal rotation vector

i relate to the components of the displacement vectorfield iu as [18]

ii ucurl ))((2

1 (6)

For an EulerBernoulli beam the displacement fieldcan be expressed as

)(0)(

)( txwuux

txwztxuu zyx

(7)


489

Where u is the axial displacement of the centroid ofsections and w denotes the lateral deflection of the

beam The parameter xw stands in the angle of

rotation (about the y-axis) of the beam cross-sectionsAssuming the above displacement field afterdeformation the cross sections remain plane andalways perpendicular to the center line without anychange in their shapes It is noted that parameter zrepresents the distance of a point on the section withrespect the axis parallel to y-direction passing throughthe centroid

Governing Equation of MotionIn this section the governing equation and

corresponding classical and non-classical boundaryconditions of a nonlinear microbeam modeled on thebasis of the couple stress theory are derived Thecoordinate system and loading of an EulerBernoullibeam has been depicted in Fig 1 In this figure F(xt)and G(xt) refer to the intensity of the transversedistributed force and the axial body force respectivelyboth as force per unit length

It is noted that finite deflection w is permissible andonly it is needed that the slopes be very small Hereafterwe use Eq (8) for the axial strain instead of theinfinitesimal definition presented in Eq (3) Substitutionof Eqs (7) and (8) into (3)(5) yields the non-zerocomponents

Also combination of Eqs (6) and (7) gives [19]

0

zxy x

w (9)

Substitution of Eq (9) into (5) yields the followingexpression for the only non-zero components of thesymmetric curvature tensor

2

2

2

1

x

wyxxy

(10)

It is assumed that the components of strainsrotations and their gradients are sufficiently small Byneglecting the Poissons effect the substitution of Eq(8) into Eq (2) gives the following expressions for themain components of the symmetric part of the stresstensor in terms of the kinematic parameters

))(2

1( 2

2

2

x

w

x

wz

x

uEE xxxx

all other 0ij

(11)

Where E denotes the elastic modulus In order towrite the non-zero components of the deviatoric partof the couple stress tensor in terms of the kinematicparameters one can substitute Eq (10) into Eq (4) toget

2

22

x

wlmxy

(12)

Where and l are shear modulus and the materiallength scale parameter respectively To obtain thegoverning equations the kinetic energy of the beam Tthe beam strain energy due to bending and the changeof the stretch with respect to the initialconfiguration bsU and the increase in the stored energywith respect to the initial configuration due to theexistence of initially axial load isU and finally the totalpotential energy

isbs UUU are considered asfollows

dxdAt

w

xt

wz

t

uT

L

A

)()(2

1 222

0

(13a)

Fig 1 An EulerBernoulli loading and coordinate system

By assuming small slopes in the beam afterdeformation the axial strain ie the ratio of theelongation of a material line element initially in the axialdirection of its initial length can be approximatelyexpressed by the von-Karman strain as

22

22 )(

2

1)(

2

1

x

w

x

wz

x

u

x

w

x

u xxx

(8)


490

(13b)

Where 0N I and are the axial load area momentof inertia of the section about y axis and the massdensity respectively The work done by the externalloads acting on the beam is also expressed as

is the electrostatic force per unit length of the beam

The electrostatic force enhanced with first orderfringing correction can be presented in the followingequation [20]

])(

6501[)(2

)(2

20

B

wg

wg

BVtxFelec

(18)

Where 212120 108548

mNC is thepermittivity of vacuum V is the applied voltage g isthe initial gap between the movable and the groundelectrode and B is the width of the beam For clamped-clamped beam the boundary conditions at the endsare

0)(

0)(0)0(

0)0( dx

LdwLw

dx

dww (19)

For cantilever beam the boundary conditions at theends are

0)(

0)(

0)0(

0)0(3

3

2

2

dx

Lwd

dx

Lwd

dx

dww (20)

Table 1 shows the geometrical parameters andmaterial properties of micro-beam

(13c)

Where N and V represent the resultant axial andtransverse forces in a section caused by the classicalstress components acting on the section The resultantaxial and transverse forces are work conjugate to u andw respectively Also hP and hQ are the higher-orderresultants in a section caused by higher-order stressesacting on the section These two higher-orderresultants are work conjugate to

2)(21 xwxuxx a n d 22 xw respectively The parameter M is the resultant momentin a section caused by the classical and higher-orderstress components Now the Hamilton principle canbe applied to determine the governing equation

2

1

0)(t

t

dtWUT (14)

Where denotes the variation symbol By applying

Eqs (13) and (14) the governing equilibrium microbeam is derived as

)(2

2

2

2

4

4

txFt

wA

x

wN

x

wS

(15)

Where

L

dxx

w

L

EANN

0

20 )(

2 (16)

2AlEIS (17)

If in Eq (15) N=0 then the model the of beam iscalled the linear equation (linear model) without theeffect of geometric nonlinearity The cross sectionalarea and length of beam are A and L respectively F(xt)

Table 1 Geometrical parameters and material propertiesof micro-beam

Materi al properties Geomet ri ca l dim ension s

E(GPa) )( mL )( mB )( mh )( mg

77 033 100-500 05-50 05-4 0-30

M Heidari

In the static case we have 0

and

dx

d

x

Hence Eq (15) is reduced to

])(

6501[)(2

])(2

[)(

2

20

2

2

0

204

42

B

wg

wg

BV

dx

wddx

dx

dw

L

EAN

dx

wdAlEI

L

(21)

A uniform micro-beam has a rectangular crosssection with height h and width B subjected to a givenelectrostatic force per unit length Let us consider thefollowing dimensionless parameters

EI

LN

L

xx

g

ww

EI

Al

B

g

EIg

LBV

I

AL

20

2

3

420

2

~~

6502

2

(22)


491

In the above equations the non-dimensionalparameter is defined the size effect parameter Also is non-dimensional voltage parameter The normalizednonlinear governing equation of motion of the beamcan be written as [21]

)~1()~1(~

~~)~

~(~

~)1(

22

22

1

0

4

wwxd

wdxd

xd

wd

xd

wd

(23)

Artificial neural networksArtificial NNs are non-linear mapping systems with

a structure loosely based on principles observed inbiological nervous systems In greatly simplified termsas can be seen from Fig 2 a typical real neuron has abranching dendritic tree that collects signals from manyother neurons in a limited area a cell body thatintegrates collected signals and generates a responsesignal (as well as manages metabolic functions) and along branching axon that distributes the responsethrough contacts with dendritic trees of many otherneurons The response of each neuron is a relativelysimple non-linear function of its inputs and is largelydetermined by the strengths of the connections fromits inputs In spite of the relative simplicity of theindividual units systems containing many neurons cangenerate complex and interesting behaviours [22]

A network is specialized to implement differentfunctions by varying the connection topology and thevalues of the connecting weights Complex functionscan be implemented by connecting units together withappropriate weights In fact it has been shown that asufficiently large network with an appropriate structureand property chosen weights can approximate witharbitrary accuracy any function satisfying certain broadconstraints Usually the processing units haveresponses like (see Fig 4)

Fig 2 A biological nervous systems

An ANN shown in Fig 3 is very loosely based onthese ideas In the most general terms a NN consistsof large number of simple processors linked byweighted connections By analogy the processingnodes may be called neurons Each node outputdepends only on information that is locally availableat the node either stored internally or arriving via theweighted connections Each unit receives inputs frommany other nodes and transmits its output to yet othernodes By itself a single processing element is notvery powerful it generates a scalar output with a singlenumerical value which is a simple non-linear functionof its inputs The power of the system emerges fromthe combination of many units in an appropriate way

)(i

iufy (24)

Where iu are the output signals of hidden layer tothe output layer )( iuf is a simple non-linear functionsuch as the sigmoid or logistic function This unitcomputes a weighted linear combination of its inputsand passes this through the non-linearity to produce ascalar output

In general it is a bounded non-decreasing non-linearfunction the logistic function is a common choiceThis model is of course a drastically simplifiedapproximation of real nervous systems The intent isto capture the major characteristics important in theinformation processing functions of real networkswithout varying too much about physical constraints

Fig 3 A layered feed-forward artificial NN

Fig 4 an artificial neuron model


492

imposed by biology The impressive advantages of NNsare the capability of solving highly non-linear andcomplex problems and the efficiency of processingimprecise and noisy data Mainly there are three typesof training condition for NNs namely supervisedtraining graded training and self-organization trainingSupervised training which is adopted in this study canbe applied as(1) First the dataset of the system including input andoutput values is established7KHGDWDVHWLVQRUPDOL]HGDFFRUGLQJWRWKHDOJRULWKP

7KHQWKHDOJRULWKPLVUXQ

(4) Finally the desired output values corresponding tothe input used in test phase [23]

Back propagation neural networkBack propagation neural network (BPN) developed

by Rumelhart [24] is the most prevalent of the supervisedlearning models of ANN BPN used the gradient steepestdescent method to correct the weight of theinterconnectivity neuron BPN easily solved theinteraction of processing elements by adding hiddenlayers In the learning process BPN the interconnectivethe weights are adjusted using an error convergencetechnique to obtain a desired output for a given inputIn general the error at the output layer in the BPN modelpropagates backward to the input layer through thehidden layer in the network to obtain the final desiredoutput The gradient descent method is utilized tocalculate the weight of the network and adjusts theweight of interconnectives to minimize the output errorThe formulas used in this algorithm are as follows1) Hidden layer calculation results

iii wxnet (25)

)( ii netfy (26)

Where ix and iw are input data and weights of theinput data respectively f is activation functionand iy is the result obtained from hidden layer2) Output layer calculation results

jkik wynet (27)

)( kk netfo (28)

Where jkw are the weights of output layers and kOis result obtained from output layer3) Activation functions used in layers are logsig tansigand linear

ineti enetf

1

1)( (logsig) (29)

i

i

net

net

ie

enetf

1

1)( (tansig) (30)

ii netnetf )( (linear) (31)

(4) Errors made at the end of one cycle

)1()( kkkkk ooote (32)

ijkiii weyye )1( (33)

Where kt is result expected from output layer ke isan error occurred at output layer and ie is the erroroccurred at hidden layer5) Weights can be changed using these calculatederror values according to Eqs (34) and (35)

jkikjkjk wyeww (34)

ijiiijij wxeww (35)

Where ijw are the weights of the output layer

jkw and ijw are correction made in weights atthe previous calculation is learning ratio and is momentum term that is used to adjust the weightsIn this paper 90 and 90 are used2) Square error occurred in one cycle can be foundby Eq (36)

250 kk ote (36)

The completion of training the BPN relative error(RE) for each data and mean relative error (MRE) forall data is calculated according to Eqs (37) and (38)respectively

k

kk

t

otRE

)(100 (37)

n

i k

kk

t

ot

nMRE

1

)(1001 (38)

Where n is the number of data [25]

RESULTS AND DISCUSSIONStatic pull-in instability analysis

When the applied voltage between the twoelectrodes increases beyond a critical value theelectric field force cannot be balanced by the elasticrestoring force of the movable electrode and thesystem collapses onto the ground electrode Thevoltage and deflection at this state are known as thepull-in voltage and pull-in deflection which are ofutmost importance in the design of MEMS devices

M Heidari


493

The pull-in voltage of cantilever and fixed-fixed beamsis an important variable for analysis and design of micro-switches and other micro-devices Typically the pull-in voltage is a function of geometry variable such aslength width and thickness of the beam and the gapbetween the beam and the ground plane To study theinstability of the nano-actuator Eq (23) is solvednumerically and simulated To highlight the differencesbetween linear and nonlinear geometry model resultsof Euler-Bernoulli micro beam we first compare the pull-in voltage for a fixed-fixed and cantilever beams with alength of 100 m a width of 50 a thickness of 1 andtwo gap lengths For a small gap length of 05 (shownin Fig 5) we observe that linear and nonlinear geometrymodel gives identical results

However for a large gap length of 2 m (shown inFig 6) we observe that pull-in voltage for fixed-fixedbeam is significantly different

As shown in Fig 7 the difference in the pull-involtage is even larger when a gap length of 45 m isconsidered In Figs 8 9 and 10 pull-in voltage of fixed-free beams are shown It is evident that pull-in voltage

of fixed-fixed beam is larger than fixed-free beam Moreextensive studies for the cantilever beam with lengthsvarying from 100 to 500 and thicknesses varying from1 to 4are shown in Figs 11 and 12 The gap lengthsused vary from 5 to 30 For gaps smaller than 15 andlengths larger than 350 we observe that the pull-involtage obtained with linear and nonlinear geometrymodel are very close However for large gaps (such asthe 15 case) and for short beams (such as the 100 case)we observe that the difference in the pull-in voltageobtained with linear and nonlinear geometry model isnot negligible In Figs 13-14 we investigate the fixed-fixed beam example with lengths varying from 100 to500 and thickness varying from 05 to 2 We observethat for all cases the pull-in voltage obtained with linearmodel are in significant error (larger than 55) comparedto the pull-in voltages obtained with nonlinear geometrymodel When the gap increase the error in the pull-involtage with linear model increase significantlyFurthermore contrary to the case of cantilever beamsthe thickness has a significant effect on the error in thepull-in voltages

Fig 6 Comparison of linear and nonlinear geometry model results for a fixed-fixed beam with a gap 2 m



494


Fig 8 Comparison of linear and nonlinear geometry model results for a fixed-free beam with a gap 05 m

M Heidari



495

Fig 11 Gap vs pull-in voltage for cantilever beams with a thickness of 1 m For length=100 m the difference in pull-involtage between linear and nonlinear geometry model is significant when the gap is larger than 15 m For a length larger

than 350 m the pull-in voltages obtained with linear and nonlinear geometry model are identical





496

Fig 13 Gap vs pull-in voltage for fixed-fixed beams with a thickness of 05 m Observe the large difference in pull-involtage obtained from linear and nonlinear geometry model of beam

Fig 14 Gap vs pull-in voltage for fixed-fixed beams with a thickness of 2 m

M Heidari

The thinner the beam the larger the error Anotherobservation is that the length of the beam has little effecton the error in the pull-in voltage This observation isalso different from the case of cantilever beams Fromthe results it is clear the linear model is generally notvalid for the fixed-fixed beams case except when thegap is very small such as the 05 case as shown in Fig5 These figures represent that the size effect increasesthe pull-in voltage of the nano-actuators Fig 15 showsthe pull-in voltage vs size effect for the fixed-fixed beamwith gap 25

Modeling of Static Pull-in Instability Voltage of CantileverBeam Using Back Propagation Neural Network

Modeling of pull-in instability of micro-beam withBP neural network is composed of two stages trainingand testing of the networks with numerical data The

training data consisted of values for beam length (L)gap (g) width of beam (b) and (hl) and the

corresponding static pull-in instability voltage ( PIV )

total 120 such data sets were used of which 110 wereselected randomly and used for training purposes whilstthe remaining 10 data sets were presented to the trainednetworks as new application data for verification(testing) purposes Thus the networks were evaluatedusing data that had not been used for training TrainingTesting pattern vectors are formed each formed withan input condition vector and the corresponding targetvector Mapping each term to a value between -1 and 1using the following linear mapping formula

minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497

where N normalized value of the real variable1min N and 1max N minimum and maximum

values of normalization respectively R real value ofthe variable minR and maxR minimum and maximumvalues of the real variable respectively Thesenormalized data were used as the inputs and output totrain the ANN In other words the network has fourinputs of beam length (L) gap (g) width of beam (b)and (hl) ratio and one output of static pull-in voltage( PIV ) Fig 16 shows the general network topology formodeling the process Table 2 shows 10 numerical datasets have been used for verifying or testing networkcapabilities in modeling the process

Therefore the general network structure is supposedto be 4-n-1 which implies 4 neurons in the input layern neurons in the hidden layer and 1 neuron in theoutput layer Then by varying the number of hiddenneurons different network configurations are trainedand their performances are checked For trainingproblem equal learning rate and momentum constantof 90 were used [25] Also error stoppingcriterion was set at E=001 which means trainingepochs continued until the mean square error fellbeneath this value

BP Neural Network ModelThe size of hidden layer(s) is one of the most

important considerations when solving actual problemsusing multi-layer feed-forward network However it hasbeen shown that BP neural network with one hidden

layer can uniformly approximate any continuousfunction to any desired degree of accuracy given anadequate number of neurons in the hidden layer andthe correct interconnection weights [26] Therefore onehidden layer was adopted for the BP model To determinthe number of neurons in the hidden layer a procedureof trail and error approach needs to be done As suchattempts have been made to study the networkperformance with a different number of hidden neuronsHence a number of candidate networks are constructedeach of trained separately and the best network wereselected based on the accuracy of the predictions inthe testing phase It should be noted that if the numberof hidden neurons is too large the ANN might be over-trained giving spurious values in the testing phase Iftoo few neurons are selected the function mappingmight not be accomplished due to under-training Threefunctions namely newelm newff and newcf [27] havebeen used for creating of BP networks Then byvarying the number of hidden neurons differentnetwork configurations are trained and theirperformances are checked The results are shown inTable 3 Both the required iteration numbers andmapping performances were examined for thesenetworks As the error criterion for all networks wasthe same their performances are comparable As aresult from Table 3 the best network structure of BPmodel is picked to have 8 neurons in the hidden layerwith the average verification errors of 636 inpredicting PIV by newelm function

Fig 15 Pull-in voltage vs size effect for fixed-fixed beam with gap 25 m a thickness of 1 m length 300 m andwidth 05 m for nonlinear geometry model


498

Table 3 The effects of different number of hidden neurons on the BP network performance

N o of h idde n

ne urons E poch A ve ra ge er ror in PIV ( )

w it h new el m f uncti on

A vera ge e rr or in PIV ( )

w ith n ew cf functi on


w it h new f f f unc tion

4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari

Fig 16 General ANN topology

Table 2 Beam geometry and pull-in voltage forverification analysis

T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2

Table 4 shows the comparison of calculated andpredicted values for static pull-in voltage in verificationcases After 1884 epochs the MSE between the desiredand actual outputs becomes less than 001 At thebeginning of the training the output from the networkis far from the target value However the output slowlyand gradually converges to the target value with more

epochs and the network learns the inputoutput relationof the training samples Fig 17 shows the pull-in voltageevaluated by the modified couple stress theory respectto length of the beam and with hl = 4 g=105 m b=50 m and h=294 The pull-in voltage of themicro-cantilever versus parameter hl for bg=50 isdepicted in Fig 18 with three BP functions

Table 4 Comparison of PIV calculated and predicted by the BP neural network model with three functions

Test No

PIV (volt) PIV (vol t) PIV (volt )

Calculated BP model (newelm)

Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499

Fig 17 Comparing the theoretical and BP neural networks pull-in voltages for silicon 110

Fig 18 Comparing of pull-in voltage of the micro-cantilever versus parameter hl and bg=50 with BP models

increasing of gap length the pull-in voltage issignificantly increased For both fixed-fixed andcantilever beams by increasing of thickness of beamsthe pull-in voltage is significantly increased For bothfixed-fixed and cantilever beams by increasing oflength of beams the pull-in voltage is significantlydecreased By using modified couple stress theory itis found that the dimensionless pull-in voltage ofMEMS increases linearly due to the size effect Thisemphasizes the importance of size effect considerationin design and analysis of MEMS When the ratio ofhl increases the pull-in voltage predicted by modifiedcouple stress theory and ANN is constantapproximately The conclusion above indicates thatthe geometry of the beam has significant influenceson the electro-static characteristics of micro-beamsthat can be designed to tailor for the desiredperformance in different MEMS applications

CONCLUSIONThe primary contributions of the paper are

summarized as follows The BP neural network iscapable of constructing model using only numericaldata describing the static pull-in instability behaviorThe results show that newelm function is more accuratethan newff and newcf functions Also the Levenberg-Marquardt training is faster than other trainingmethods For cantilever beams length has a significanteffect on the error in pull-in voltages while for fixed-fixed beams the length has little effect on the error Onthe other hand for fixed-fixed beams thickness hassignificant effect on the error in pull-in voltage whilefor cantilever beams it has little effect The static pull-in instability voltage of clampedclamped andcantilever beam are compared For both clampedclamped and cantilever beams the pull-in voltage innonlinear geometry beam model is bigger than linearmodel For both fixed-fixed and cantilever beams by


500

microscale beams Static bending post buckling and freevibration Int J Eng Sci 48 2044-2053

[15] Asghari M Rahaeifard M Kahrobaiyan M H AhmadianM T (2011) On the size-dependent behavior offunctionally graded micro-beams Mater Design 32 1435-1443

[16] Rong H Huang Q A Nie M Li W(2004) An analyticalmodel for pull-in voltage of clampedclamped multilayerbeams Sens Actuators A 116 15-21

[17] Yang F Chong A C M Lam D C C Tong P (2002)Couple stress based strain gradient theory for elasticity IntJ Solids and Struc 39 2731-2743

[18] Shengli K Shenjie Z Zhifeng N Kai W (2011) Thesize-dependent natural frequency of BernoulliEuler micro-beams J Eng Sci 46 427-437

[19] Ma H M Gao X L Reddy J N (2008) A microstructure-dependent Timoshenko beam model based on a modifiedcouple stress theory J Mech and Physics of Solids 5633793391

[20] Gupta R K (1997) Electrostatic pull-in test structure designfor in-situ mechanical property measurements ofmicroelectromechanical systems PhD DissertationMassachusetts Institute of Technology (MIT) Cambridge MA

[21] Zhao J Zhou S Wanga B Wang X (2012) Nonlinearmicrobeam model based on strain gradient theory ApplMathemat Modell 36 2674-2686

[22] Freeman J A Skapura D M (1992) Neural networksalgorithms applications and programming techniquesAddision-Wesley

[23] Gao D Kinouchi Y Ito K Zhao Z (2005) Neuralnetworks for event extraction from time series a backpropagation algorithm approach Future Gener Comp Sys21 1096-1105

[24] Rumelhart D E Hinton G E Williams R J (1986)Learning representations by back propagating error Nature323 533-536

[25] Zhang H Wei W Mingchen Y (2012) Boundednessand convergence of batch back-propagation algorithm withpenalty for feedforward neural networks Neurocomputing89 141-146

[26] Hongmei S Gaofeng Z (2011) Convergence analysisof a back-propagation algorithm with adaptive momentumNeurocomputing 74 749-752

[27] Demuth H Beale M (2001) Matlab Neural NetworksToolbox Users Guide The Math Works Inc http

wwwmathworkscom

REFERENCES[1] Khatami I Pashai M H Tolou N (2008) Comparative

vibration analysis of a parametrically nonlinear excitedoscillator using HPM and numerical method MathematProblems in Eng 2008 1-11

[2] Gasparini A M Saetta A V Vitaliani R V (1995) on thestability and instability regions of non-conservativecontinuous system under partially follower forces ComputMeth Appl Mech Eng 124 63-78

[3] Osterberg P M Senturia S D (1997) M-TEST A testchip for MEMS material property measurements usingelectrostatically actuated test structures JMicroelectromech Syst 6 107-118

[4] Osterberg P M Gupta R K Gilbert J R Senturia S D(1994) Quantitative models for the measurement of residualstress poisson ratio and youngs modulus using electrostaticpull-in of beams and diaphragms Proceedings of the Solid-State Sensor and Actuator Workshop Hilton Head SC

[5] Sadeghian H Rezazadeh G Osterberg P (2007) Applicationof the generalized differential quadrature method to the studyof pull-in phenomena of MEMS switches IEEEASME JMicro Electro Mech Sys 16 1334-1340

[6] Salekdeh Y A Koochi A Beni Y T Abadyan M (2012)Modeling effect of three nano-scale physical phenomenaon instability voltage of multi-layer MEMSNEMS Materialsize dependency van der waals force and non-classic supportconditions Trends in Appl Sci Res 7 1-17

[7] Batra R C Porfiri M Spinello D (2007) Review ofmodeling electrostatically actuated microelectromechanicalsystems Smart Mater Struct 16 R23-R31

[8] Lin W H Zhao Y P (2008) Pull-in instability of micro-switch actuators Model review Int J Nonlinear Sci NumerSimulation 9 175-184

[9] Koiter W T (1964) Couple-stresses in the theory ofelasticity I and II Proceed Koninklijke NederlandseAkademie van Wetenschappen Series B 6717-6744

[10] Mindlin R D Tiersten H F (1962) Effects of couple-stresses in linear elasticity Archive for Rational MechAnalysis 11 415-448

[11] Toupin R A (1962) Elastic materials with couple-stressesArchive for Rational Mech Analysis 11 385414

[12] Anthoine A (2000) Effect of couple-stresses on theelastic bending of beams Int J Solids and Struc 37 1003-1018


[14] Xia W Wang L Yin L (2010) Nonlinear non-classical

M Heidari

How to cite this article (Vancouver style)

Heidari M (2015) Estimation of pull-in instability voltage of Euler-Bernoulli micro beam by back propagationartificial neural network Int J Nano Dimens 6(5) 487-500DOI 107508ijnd201505006URL httpijndirarticle_15293_1117html


488

M Heidari

theories such as higher-order gradient theories andthe couple stress theory have been developed suchthat they are acceptably able to interpret the size-dependencies In the 1960s some researchers such asKoiter [9] Mindlin [10] and Toupin [11] introduced thecouple stress elasticity theory as a non-classic theorycapable to predict the size effects with the appearanceof two higher-order material constants in thecorresponding constitutive equations In this theorybeside the classical stress components acting onelements of materials the couple stress componentsas higher-order stresses are also available which tendto rotate the elements Utilizing the couple stress theorysome researchers investigated the size effects in someproblems [12]Employing the equilibrium equation ofmoments of couples beside the classical equilibriumequations of forces and moments of forces a modifiedcouple stress theory introduced by Yang Chong Lamand Tong [13] with one higher-order material constantin the constitutive equations Recently size-dependentnonlinear EulerBernoulli and Timoshenko beamsmodeled on the basis of the modified couple stresstheory have been developed by Xia et al [14] andAsghari et al [15] respectively Rong et al [16] presentan analytical method for pull-in analysis of clampedclamped multilayer beam Their method is Rayleigh-Ritz method and assumes one deflection shapefunction They derive the two governing equations byenforcing the pull-in conditions that the first andsecond order derivatives of the system energyfunctional are zero In their model the pull-in voltageand displacement are coupled in the two governingequations

This paper investigates the pull-in instability ofmicro-beams with a curved ground electrode under theaction of electric field force within the framework ofvon-Karman nonlinearity and the EulerBernoulli beamtheory The static pull-in voltage instability of clamped-clamped and cantilever micro-beam are obtained byusing MAPLE commercial software The effects ofgeometric parameters such as beam lengths widththickness gaps and size effect are discussed in detailthrough a numerical study The objective of this paperis to establish a neural network model for estimatingthe pull-in instability voltage of cantilever beams Morespecifically back propagation neural network is usedto construct the pull-in instability voltage Effectiveparameters influencing pull-in voltage and their levelsof training were selected through preliminarycalculations carried out on instability pull-in voltage

of micro-beam The network trained by the samenumerical data are then verified by some numericalcalculations different from those used in the trainingphase and the best model was selected based on thecriterion of having the least average values ofverification errors To the authors best knowledge noprevious studies which cover all these issues areavailable To the authors best knowledge no previousstudies which cover all these issues are available

EXPERIMENTALPreliminaries

In the modified couple stress theory the strainenergy density u of a linear elastic isotropic materialin infinitesimal deformation is written as [17]

)321()(2

1 jimu ijijijij (1)

Where

ijijmmij 2 (2)

))()((21 T

ijijij uu (3)

ijij lm 22 (4)

))()((21 T

ijijij (5)

In which ij ij ijm and ij denote thecomponents of the symmetric part of stress tensor the strain tensor the deviatoric part of the couplestress tensor m and the symmetric part of the curvaturetensor respectively Also u and are thedisplacement vector and the rotation vector The twoLame constants and the material length scale parameterare represented by and l respectively The Lameconstants are written in terms of the Youngs modulusE and the Poissons ratio as

)21)(1( E and )1(2 EThe components of the infinitesimal rotation vector

i relate to the components of the displacement vectorfield iu as [18]

ii ucurl ))((2

1 (6)

For an EulerBernoulli beam the displacement fieldcan be expressed as

)(0)(

)( txwuux

txwztxuu zyx

(7)


489








0

zxy x

w (9)


2

2

2

1

x

wyxxy

(10)


))(2

1( 2

2

2

x

w

x

wz

x

uEE xxxx

all other 0ij

(11)


2

22

x

wlmxy

(12)



dxdAt

w

xt

wz

t

uT

L

A

)()(2

1 222

0

(13a)



22

22 )(

2

1)(

2

1

x

w

x

wz

x

u

x

w

x

u xxx

(8)


490

(13b)




])(

6501[)(2

)(2

20

B

wg

wg

BVtxFelec

(18)

Where 212120 108548


0)(

0)(0)0(

0)0( dx

LdwLw

dx

dww (19)


0)(

0)(

0)0(

0)0(3

3

2

2

dx

Lwd

dx

Lwd

dx

dww (20)


(13c)



2

1

0)(t

t

dtWUT (14)



)(2

2

2

2

4

4

txFt

wA

x

wN

x

wS

(15)

Where

L

dxx

w

L

EANN

0

20 )(

2 (16)

2AlEIS (17)





77 033 100-500 05-50 05-4 0-30

M Heidari


and

dx

d

x


])(

6501[)(2

])(2

[)(

2

20

2

2

0

204

42

B

wg

wg

BV

dx

wddx

dx

dw

L

EAN

dx

wdAlEI

L

(21)


EI

LN

L

xx

g

ww

EI

Al

B

g

EIg

LBV

I

AL

20

2

3

420

2

~~

6502

2

(22)


491


)~1()~1(~

~~)~

~(~

~)1(

22

22

1

0

4

wwxd

wdxd

xd

wd

xd

wd

(23)






)(i

iufy (24)






492






iii wxnet (25)

)( ii netfy (26)


jkik wynet (27)

)( kk netfo (28)


ineti enetf

1

1)( (logsig) (29)

i

i

net

net

ie

enetf

1

1)( (tansig) (30)






jkikjkjk wyeww (34)

ijiiijij wxeww (35)



250 kk ote (36)


k

kk

t

otRE

)(100 (37)

n

i k

kk

t

ot

nMRE

1

)(1001 (38)




M Heidari


493








494



M Heidari



495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




489








0

zxy x

w (9)


2

2

2

1

x

wyxxy

(10)


))(2

1( 2

2

2

x

w

x

wz

x

uEE xxxx

all other 0ij

(11)


2

22

x

wlmxy

(12)



dxdAt

w

xt

wz

t

uT

L

A

)()(2

1 222

0

(13a)



22

22 )(

2

1)(

2

1

x

w

x

wz

x

u

x

w

x

u xxx

(8)


490

(13b)




])(

6501[)(2

)(2

20

B

wg

wg

BVtxFelec

(18)

Where 212120 108548


0)(

0)(0)0(

0)0( dx

LdwLw

dx

dww (19)


0)(

0)(

0)0(

0)0(3

3

2

2

dx

Lwd

dx

Lwd

dx

dww (20)


(13c)



2

1

0)(t

t

dtWUT (14)



)(2

2

2

2

4

4

txFt

wA

x

wN

x

wS

(15)

Where

L

dxx

w

L

EANN

0

20 )(

2 (16)

2AlEIS (17)





77 033 100-500 05-50 05-4 0-30

M Heidari


and

dx

d

x


])(

6501[)(2

])(2

[)(

2

20

2

2

0

204

42

B

wg

wg

BV

dx

wddx

dx

dw

L

EAN

dx

wdAlEI

L

(21)


EI

LN

L

xx

g

ww

EI

Al

B

g

EIg

LBV

I

AL

20

2

3

420

2

~~

6502

2

(22)


491


)~1()~1(~

~~)~

~(~

~)1(

22

22

1

0

4

wwxd

wdxd

xd

wd

xd

wd

(23)






)(i

iufy (24)






492






iii wxnet (25)

)( ii netfy (26)


jkik wynet (27)

)( kk netfo (28)


ineti enetf

1

1)( (logsig) (29)

i

i

net

net

ie

enetf

1

1)( (tansig) (30)






jkikjkjk wyeww (34)

ijiiijij wxeww (35)



250 kk ote (36)


k

kk

t

otRE

)(100 (37)

n

i k

kk

t

ot

nMRE

1

)(1001 (38)




M Heidari


493








494



M Heidari



495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




490

(13b)




])(

6501[)(2

)(2

20

B

wg

wg

BVtxFelec

(18)

Where 212120 108548


0)(

0)(0)0(

0)0( dx

LdwLw

dx

dww (19)


0)(

0)(

0)0(

0)0(3

3

2

2

dx

Lwd

dx

Lwd

dx

dww (20)


(13c)



2

1

0)(t

t

dtWUT (14)



)(2

2

2

2

4

4

txFt

wA

x

wN

x

wS

(15)

Where

L

dxx

w

L

EANN

0

20 )(

2 (16)

2AlEIS (17)





77 033 100-500 05-50 05-4 0-30

M Heidari


and

dx

d

x


])(

6501[)(2

])(2

[)(

2

20

2

2

0

204

42

B

wg

wg

BV

dx

wddx

dx

dw

L

EAN

dx

wdAlEI

L

(21)


EI

LN

L

xx

g

ww

EI

Al

B

g

EIg

LBV

I

AL

20

2

3

420

2

~~

6502

2

(22)


491


)~1()~1(~

~~)~

~(~

~)1(

22

22

1

0

4

wwxd

wdxd

xd

wd

xd

wd

(23)






)(i

iufy (24)






492






iii wxnet (25)

)( ii netfy (26)


jkik wynet (27)

)( kk netfo (28)


ineti enetf

1

1)( (logsig) (29)

i

i

net

net

ie

enetf

1

1)( (tansig) (30)






jkikjkjk wyeww (34)

ijiiijij wxeww (35)



250 kk ote (36)


k

kk

t

otRE

)(100 (37)

n

i k

kk

t

ot

nMRE

1

)(1001 (38)




M Heidari


493








494



M Heidari



495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




491


)~1()~1(~

~~)~

~(~

~)1(

22

22

1

0

4

wwxd

wdxd

xd

wd

xd

wd

(23)






)(i

iufy (24)






492






iii wxnet (25)

)( ii netfy (26)


jkik wynet (27)

)( kk netfo (28)


ineti enetf

1

1)( (logsig) (29)

i

i

net

net

ie

enetf

1

1)( (tansig) (30)






jkikjkjk wyeww (34)

ijiiijij wxeww (35)



250 kk ote (36)


k

kk

t

otRE

)(100 (37)

n

i k

kk

t

ot

nMRE

1

)(1001 (38)




M Heidari


493








494



M Heidari



495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




492






iii wxnet (25)

)( ii netfy (26)


jkik wynet (27)

)( kk netfo (28)


ineti enetf

1

1)( (logsig) (29)

i

i

net

net

ie

enetf

1

1)( (tansig) (30)






jkikjkjk wyeww (34)

ijiiijij wxeww (35)



250 kk ote (36)


k

kk

t

otRE

)(100 (37)

n

i k

kk

t

ot

nMRE

1

)(1001 (38)




M Heidari


493








494



M Heidari



495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




493








494



M Heidari



495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




494



M Heidari



495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




495







496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




496



M Heidari







minminmax

minmaxmin

)(

)()(N

RR

NNRRN

(39)


497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




497









498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




498


N o of h idde n







4 18 91 4 1 2 31 1 0 27 1 23 0

5 4 97 0 1 4 38 1 8 38 2 01 9

6 1 78 3 8 19 1 1 65 1 27 5

7 3 98 4 9 72 93 9 1 11 7

8 1 88 4 6 36 82 8 1 01 4

9 2 77 0 1 3 39 1 1 86 1 99 8

1 0 2 68 3 1 1 67 1 6 40 1 54 8

M Heidari



T e st N o

L (micro m )

b (micro m ) lh

g (microm )

P IV( vo lt )

1 7 5 0 5 4 0 5 0 1 7 9

2 1 0 0 5 6 1 2 4 4

3 1 2 5 1 0 8 1 5 7 3 1

4 1 5 0 2 0 1 0 2 1 6 8 2

5 1 7 5 2 5 1 2 2 5 2 6 7 8

6 2 0 0 3 0 1 4 3 4 0 2 7

7 2 2 5 3 5 1 6 3 5 5 3 8 4

8 2 5 0 4 0 1 8 4 6 8 0 1

9 2 7 5 4 5 2 0 4 5 8 4 5 3

1 0 3 0 0 5 0 2 2 5 1 0 3 6 2




Test No



Error () Calculated

BP m odel (newff)

Error () Calculated

BP mod el (n ewcf)

Error ()

1 0179 0190 656 0179 0191 712 0 1 79 0193 829 2 244 261 728 244 2 5 2 339 2 44 259 628 3 731 732 016 731 7 5 4 316 7 31 770 539 4 1682 1921 1424 1682 1829 875 1 682 1921 1424 5 2678 2822 539 2678 2816 519 2 678 2876 741 6 4027 4217 474 4027 4603 1431 4 027 4399 925 7 5384 5713 612 5384 5749 679 5 384 5796 767 8 6801 7160 529 6801 7839 1527 6 801 8214 2078 9 8453 8939 576 8453 8671 259 8 453 9251 945

10 10362 11203 812 10362 1 2053 1632 10362 11674 1267


499







500















wwwmathworkscom
















M Heidari




499







500















wwwmathworkscom
















M Heidari




500















wwwmathworkscom
















M Heidari



estimation of pull-in instability voltage of euler ... · stretching as the source of the...

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