Theoretical study of the effect of Casimir force, elastic boundary conditions andsize dependency on the pull-in instability of beam-type NEMSY. Tadi Benia, A. Koochib, M. Abadyanc,naFaculty of Engineering, University of Shahrekord, Shahrekord, IranbMechanical Engineering Group, Islamic Azad University, Naein Branch, Naein, IrancMechanical Engineering Group, Islamic Azad University, Ramsar Branch, Ramsar, Iranarticle infoArticle history:Received 30 September 2010Received in revised form16 November 2010Accepted 26 November 2010Available online 3 December 2010abstractInthispaper, thestaticpull-ininstabilityof beam-typenano-electromechanical systems(NEMS) istheoretically investigated considering the effect of Casimir attraction, elastic boundary conditions (BC)andsizedependency. Rotationalspringsareutilizedateachofthesupportedendsofthesimplyanddoubly supported beams to model an elastic BC. The modied couple stress theory is applied to examinethe size effects onthe instability of nanostructures. Inorder to solve the nonlinear constitutive equationofnano-beams, modied Adomian decomposition (MAD) as well as the numerical method is employed. Theresults reveal signicant inuences of Casimir attraction, elastic BC and size dependency on the pull-incharacteristics of NEMS. The obtained MAD solution agrees well with the numerical one.& 2010 Elsevier B.V. All rights reserved.1. IntroductionNano-electromechanical systems (NEMS) are increasinglybeingused in the development of advanced nano-devices such asswitches, actuators, probes, etc. [13]. A typical beam-type NEMSis constructed fromtwo conductive electrodes, where one ismovable and the other is xed (ground electrode). Application ofvoltage difference between the two causes the movable electrodetodeect towards the groundone as a result of electrostaticattraction. Thepull-ininstabilityoccurs whentheelectrostaticattraction exceeds the elastic restoring moment of the NEMS andleadstocontact betweenthetwoelectrodes. Thepull-inchar-acteristics,i.e.pull-involtageanddeectionofmicro-electrome-chanical systems (MEMS), have been studied for over two decadeswithout considerationof nanoscale effects [46]. Withthe decreaseindimensions, manyessential phenomenaappearatthemicro/nano-scales, whichare not important at macro-scales. Inthis paper,three effects of these phenomena are demonstrated andconsideredin simulation of pull-in instability of beam-type nano-actuator.Therst issuethat appearsat thenanoscaleistheeffect ofdispersion forces such as Casimir attraction. At small separations(typicallylessthanseveral micrometers), theCasimirforcecanhighly inuence the instability of NEMS. These forces canbeexplained by electromagnetic quantumvacuumuctuations exist-ing between two separated plates [7]. When the separationbetween the two surfaces is sufciently large, i.e. above the plasma(inmetals)/absorption(indielectrics)wavelengthofthesurfacematerial [810], the virtual photons emittedbyatoms of onesurfacedonotreachtheotherduringtheirlifetime[7,8]. Inthiscase, the interaction between the two surfaces is described by theCasimir force. Some researchers have studied the pull-in behaviorofelectromechanicalsystemshavingconsidered theeffectoftheCasimir force [1120]. Instability of micro-plates and micro-membranes under the Casimir force has been simulated by Batraet al. [1113] using the nite element method(FEM). Moghimi et al.[14] have also applied FEMto simulate the inuence of the Casimirattraction on the dynamic pull-in behavior of nano-beams. Experi-mental investigation of the Casimir attraction has been performedbyBuksandRoukes[15,16]onthedynamicbehaviorsofpull-invariables in MEMS/NEMS. An one degree of freedomlumpedparametermodelhasbeenproposedbyLinandZhao[17,18]tosurveystictionofnano-actuatorsinthepresenceofelectrostaticandCasimir attractions. Ramezani et al. [19,20] used Greensfunction to investigate the pull-in parameters of cantileverbeam-typeactuators under Casimir forces. Further informationconcerningtheeffect of Casimirforceonthepull-ininstabilityofelectromechanicalsystemsanditsmodelingarepresentedinRefs. [2124].The second effect that appears at the submicron-scale is the sizedependencyofmaterial characteristics. Fromatheoreticalpointof view, the classical continuummechanics is not capable ofexplaining the size-dependent behaviors of ultra-small structures.Therefore, non-classical continuumtheories such as non-localelasticity[25] andcouplestress[26] areproposedtointerpretContents lists available at ScienceDirectjournalhomepage:www.elsevier.com/locate/physePhysica E1386-9477/$ - see front matter& 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physe.2010.11.033nCorresponding author.E-mail address: Abadyan@yahoo.com (M. Abadyan).Physica E 43 (2011) 979988the size-dependence behaviors. Some experimental works [2731]demonstratethatthesize dependencyisaninherentproperty ofconductive metals when the characteristic size of the structures isin the order of the internal material length scale. It has been shownthat torsional hardening of copper wire increases by a factor of 3 asthe wire diameter decreases from 170 to 12 mm [27]. An increasein plastic work hardening was caused by the thickness reductionduringa microbendingtest onnickel beams [28]. Inorder todeterminethelengthscaleparameterofmaterials, experimentalmethods such as micro/nano-indentation could be applied [29,30].The material lengthscale parameters of single crystal copperand those of polycrystalline copper have been evaluated to be 12and 5.84 mm, respectively [29,30]. Furthermore, atomistic simula-tion could be applied to evaluate the material length scaleparameter [31].Inbeam-typenanostructures, thecharacteristicsize(usuallythe beam thickness) is comparable with metal length scale para-meter [32]. Therefore, size dependency must be considered in thesimulation of the instability of these nanostructures [33]. Recently,anewmodiedcouplestresstheoryhasbeenproposedbyYanget al. [34]. According to their theory, two material constants in thecouplestresstheoryarereducedtoonlyonelengthscalepara-meter. In this view, this modied theory is applied to model micro-beams by many researchers [3539].Finally, the third important issue that must be considered in thesimulation of micro/nanoscale structures is characterization of realboundary conditions (BC). Supported BC characterization is impor-tant in the modeling of many micro/nanostructures such as opticalwaveguides [40], microscope probes [41] and switches [42]. It hasbeenobservedthattheBCofrealmicro/nanostructurescouldbemadeexiblebyrotation[42,43]. Themechanical responsesofbeam-type nano-devices have become varied under different BCs.Rinaldietal. [43]haveinvestigatedthecharacterizationofnon-classical support conditions of micro-cantilevers through electro-mechanical testing. Yunqiang et al. [44] have studied the BC effectin the static and dynamic responses of micro-plates. Owing to thelimitationsofmanufacturingtechniquesatthemicro/nanoscale,anyidealBCsuchasthesimplysupportedorclampedconditionwould be unacceptable and the boundary support conditions needto be theoretically quantied and experimentally validated [45].As far as the knowledge of the authors is concerned, none of thethree mentioned phenomena have contributed together in any ofthe pull-in models proposed by previous researchers. In this study,the modied couple stress theory is introduced to demonstrate thecombined effects of Casimir attraction, elastic BC and size depen-dency together on the pull-in behavior of a nano-beamfor the rsttime. The Euler beammodel is applied as a time-saving continuumapproachinobtainingconstitutivegoverningequations[4651].Rotational articial springs are usedat the supportedends to modelthe BCs of simply and doubly supported nano-beams. In order tosolvetheconstitutiveequationofnanostructures, modiedAdo-mian decomposition (MAD) is utilized. The MADresults arecomparedwiththenumericaldataaswellaswithotherresultsreported in the literature.2. Fundamentals of the modied couple stress theoryThe modied couple stress theory has been presented by Yanget al. [34] where the strain energy density is written asu r : em : v, 1wherethestresstensorr, straintensore, deviatoricpartofthecouple stress tensor mandsymmetric curvature tensor varedened by the following relations:r ltreI 2me, 2e 12 ru ruT , 3m2l2mv, 4v 12rhrhT, 5where l, mandl aretheLameconstant, shearmodulusandthematerial length scale parameter, respectively. The rotation vector his related to the displacement vector eld according to [35]h 12curlu: 6According to the basic hypotheses of the EulerBernoulli beams,the displacement eld has been assumed as [52]u zcX, v 0, wwX, 7where u, v and wrepresent the displacement along X, Y and Z axes,respectively, and the rotation angle cis related to the deection bycX @wX@X: 8With a small deformation in the elastic range, which isconsidered here using Eqs. (3), (7) and (8), it follows thatexxz@2wX@X2, eyyezzexyeyzezx0, 9and Eqs. (6)(8) giveyy@wX@X, yxyz0: 10Substituting Eq. (10) into (5), it follows thatwxy12@2wX@X2, wxxwyywzzwyzwzx0: 11By neglecting Poissons effect, so as to facilitate the formulationof the beamtheoryandbysubstitutingEq. (9) intoEq. (2), it is foundthatsxxEz@2w@X2 , syyszzsyzszxsxy0, 12where E is Youngs modulus. Same as above, bysubstituting Eq. (11)into Eq. (4) it is found thatmxyml2 @2w@X2 , mxxmyymzzmyzmxz0: 133. Governing equationFig. 1a andb shows simply supported(SS) anddoubly supported(DS) beams, respectively. The actuators are modeled by a beam oflengthLwith auniform rectangular crosssectionof widthB andthickness H, are suspended over a conductive substrate. Anarticial angularspringwithaspringstiffnessof K1yisusedtomodel the real BC in the SS nano-actuator. In the case of DS case,twospringswithrotational stiffnessof K1yandK2yareappliedatthesupportedends. Notethattheconsiderationofstretching(in xedxed beams) and that of axial tractions/forces along thebeamisbeyondthescopeofthisworkandtheseeffectswillbeconsidered in further publications.Inordertodevelopthegoverningequationofthebeams, theconstitutive material of the nano-actuator is assumed to be linearelastic, andonlythestaticdeectionof thenano-beamiscon-sidered.Weapplytheminimumenergyprinciple,whichimpliesequilibriumwhenthe free energy reaches a minimumvalue.Substituting Eqs. (9)(13) into Eq. (1), integrating and adding theelastic energy of the springs and by considering the work done byY. Tadi Beni et al. / Physica E 43 (2011) 979988 980external forces, the energy of the system can be written as [32]U 12_L0EI mAl2@2w@X2_ _2dX12K1y@w0@X_ _2SS12K1y@w0@X_ _212K2y@wL@X_ _2DS,___14aV _L0qXwXdX, 14bwhere I is the second moment of cross-sectional area around Z-axisand A is the cross-sectional area of the beam. Now, the HamiltonprinciplecanbeappliedtodeterminethegoverningequilibriumequationdVdU 0, 15where ddenotesthevariationsymbol.ByapplyingEqs.(14) and(15), the governing equilibrium micro/nano-beam is derived asEI mAl2d4wdX4 qX, 16with the following BC of the SS beam:w0 d3wLdX3d2wLdX20, 17EI mAl2d2w0dX2K1ydw0dX: 18Note that Eq. (17) implies zero displacement at the supportedend as well as zero force and moment at the free end. Furthermore,Eq. (18) reveals the moment balance at the supported end of thenano-beam.Similarly, the BC of the DS nano-beam is obtained asw0 wL 0, 19EI mAl2d2w0dX2K1ydw0dX, 20EI mAl2d2wLdX2K2ydwLdX: 21Notethat Eq. (16) revealsthesameconstitutivedifferentialequations with regard to SS and DS beams due to ignorance of axialtractions along the beam. Without omitting axial forces, anintegro-differential constitutiveequationisobtainedincaseof theDSbeam, which requires comprehensive efforts to be solvedanalytically. However, it should be mentioned that axial tractionscouldhavelargeinuenceonthepull-involtageatsomeranges[53] and will be discussed in further works.It should also be mentioned that the effect of nite kinematics isnegligiblewhenL410g[19,20]; hence, nitekinematicsisnotconsidered in this study. This simplication is acceptable in mostcases [19,20]. Due to the large displacement/deformation consid-erations, thismodelshouldbeimprovedforshortbeamswithasmall H/g ratio. By rewriting the distribution force along the beam,q(X), asqX felecfCas, 22Eq. (16) is written as [46,47]EI mAl2d4wdX4 felecfCas, 23where felec and fCas are the electrostatic and Casimir forces per unitlength of the beam, respectively. In this equation, the constitutivematerial ofthenano-actuatorisassumedto belinearelastic andonly the static deection of nano-beam is considered. The electro-staticforceenhancedwithrstorderfringingcorrectioncanbepresented in the following equation [54]:felece0BV22gw210:65gwB_ _, 24where e08.8541012C2N1m2is the permittivity of vacuum,V is the applied voltage and g is the initial gap between the movableand theground electrode.Forultra-thin NEMS,thenite sizeandquantum effects must be considered when we calculate the surfacecharge distribution especially for narrow beams [55].In a realistic case, the Casimir interaction between two surfaceslargely depends on the dielectric properties of the surfaces and alsoonthegeometricparameters[8,10]. Consideringsomeidealiza-tions, the Casimir force per unit length of the actuator is [8]fCasp2_cB240gw4 , 25where _1.0551034J s is Plancks constant divided by 2p andc2.998108 m/sisthelightspeed. Whenactuatorsaresuf-cientlywiderthantheseparation, Eq. (25) providesacceptableresults [8]. In this study, only nano-actuators that are wider thanthe separation (g/Br1) are considered. One can use the substitu-tions, ww/g and xX/L, to transform Eq. (23) into the followingequation:1dd4^ wdx4a1 ^ wx4 b1 ^ wx2 gb1^ wx: 26In the above equations, the non-dimensional parameters, a, b, gand size effect parameter d are dened asa p2_cBL4240g5EI, 27b e0BV2L42g3EI, 28g 0:65gB, 29d mAl2EI: 30The BCs of the SS beam with the rotational spring are^ w000 K ^ wu0, ^ w0 ^ w001 ^ w0001 0, 31where K is dened byK K1yLEI mAl2 : 32Fig. 1Y. Tadi Beni et al. / Physica E 43 (2011) 979988 981Similarly the BCs of DS with the rotational spring at ends are^ w000 K1 ^ wu0, ^ w0001 K2 ^ wu1, ^ w0 ^ w1 0, 33where K1 and K2 are dened byK1K1yLEI mAl2 , K2K2yLEI mAl2 : 34It should be noted that special cases, i.e. beams with cantileverends, canbemodeledeasilybysettingthespringstiffnesstobeinnite (KN). Therefore, one can use the following BC in specialcases instead:^ wu0 ^ w0 ^ w001 ^ w0001 0, Cantilever C 35^ w0 ^ w1 ^ wu0 ^ wu1 0, ClampedClamped CC 36^ w0 ^ wu0 ^ w1 0, ^ w001 K ^ wu1,Clampedsimply supported CS 37Relations(26)(37)presentthegoverningequationofbeam-typenanostructures. Inorder tostudythepull-inbehavior ofnanostructures, Eq. (26) is solved numerically using MAPLE com-mercial software. Furthermore, MADisappliedtotheboundaryvalue problem, and the analytical results are compared with thoseof the numerical solution in the following section.4. Modied Adomian decompositionInordertoapplyMADforanalysisof thepull-ininstability,we use y1wtransformation to rewrite Eq. (26) into (38):d4ydx4 a1dy4xb1dy2xgb1dyx, 38with the following new BC of the simply supported beamy0 1, y000 Kyu0, y001 y0001 0, SS 39y0 y1 1, y000 K1yu0, y001 K2yu1, DS 40yu0 y001 y0001 0, y0 1, C 41y0 y1 1, yu0 yu1 0, CC 42y0 y1 1, yu0 0, y001 Kyu1: CS 43UsingtheADMmethod, thedeectionof thenano-beaminEq. (38) can be represented by (see Refs. [51,58])yx

1n 0ynx C1C212!C3x213!C4x311dL4a 1n 0Nn,3xb

1n 0Nn,2xbg 1n 0Nn,1x_ _:44In Eq. (44), the function Nn,k, which approximates the nonlineartermykn, is determinedthroughthe MADs polynomials (seeRefs. [51,58]) asN0,kyk0N1,kky1yk10N2,kky2yk10kk12!y21yk20N3,kky3yk10kk1y1y2yk20kk1k23!y31yk30:::::::45Substituting relation (45) into Eq. (44), we obtainy01y1C1C2x12!C3x213!C4x314!11dabbgx4y211d4a2bbg14!C1x415!C2x516!C3x617!C4x7_18!11dabbgx8_y3C214!11d10a3bbgx42C1C25!11d10a3bbgx52C1C32C226!11d10a3bbgx62C1C46C2C37!11d10a3bbgx718!C111d24a2bbg2_2C111d2a2bbg10a3bbg8C2C46C2311d10a3bbg_x819!C211d24a2bbg2_10C211d2a2bbg10a3bbg20C3C411d10a3bbg_x9110!C311d24a2bbg2_30C311d2a2bbg10a3bbg20C2411d10a3bbg_x10111!C411d24a2bbg2_Table 1Variation in typical NEMS deection obtained by MAD. Analytical solution converges to the numerical solution as the number of the selected terms increases.Case Numerical MAD2 terms 3 terms 4 terms 5 terms 6 termsSS NEMS (abd0.5, g/B1, K30) 0.1732 0.1251 0.1573 0.1685 0.1720 0.1729DS NEMS (ab20, g/B1, d0.5, K1K230) 0.1669 0.1150 0.1413 0.1572 0.1615 0.1653Table 2Geometrical parameters and material properties of nano-beam of Table 3.Case Material properties Geometrical dimensionsE (GPa) n L (mm) B (mm) H (mm) g (mm)Narrow beam 77 0.33 300 0.5 1 2.5Wide beam 77 0.33 300 50 1 2.5Y. Tadi Beni et al. / Physica E 43 (2011) 979988 98235C411d2abbg20a3bbg_x11112!11d3a2bbg4a2bbg2_7011d3abbg210a3bbg_x12 : 46Therefore, the solution of Eq. (38) can be summarized to^ wx C1C2x12!C3x213!C4x314!11dabbg_4a2bbgC110a3bbgC21x415!11d2C1C210a3bbg4a2bbg_ x516!11d2C1C32C2210a3bbg_4a2bbgC3x617!11d 2C1C46C2C310a3bbg4a2bbgC4_ x718!C111d24a2bbg2_2C111d2a2bbg10a3bbgTable 3Pull-in voltage comparison of cantilever beamof Table 2. Casimir force is neglected.Case Pull-in voltage (V)Ref.[5]Ref.[56]Ref.[57]Ref.[20]Ref.[49]Numerical MADNarrowbeam1.23 1.20 1.21 1.29 1.21 1.24 1.27Wide beam 2.27 2.25 2.27 2.37 2.16 2.27 2.31Fig. 2Fig. 3Y. Tadi Beni et al. / Physica E 43 (2011) 979988 9838C2C46C2311d10a3bbg11d24a2bbgabbg_x819!C211d24a2bbg2_10C211d2a2bbg10a3bbg20C3C411d10a3bbg_x9110!C311d24a2bbg2_30C311d2a2bbg10a3bbg20C2411d10a3bbg_x10111!C411d24a2bbg2_35C411d2abbg20a3bbg_x11Fig. 4Fig. 5 Fig. 6Y. Tadi Beni et al. / Physica E 43 (2011) 979988 984112!11d3a2bbg4a2bbg2_70abbg210a3bbg_x12 , 47whereconstantsC1,C2,C3andC4canbedetermined bysolvingtheresulting algebraic equations fromBC at xLb, i.e. using Eqs. (39)(43).With any given a, b, d and g/B, Eq. (47) can be used to obtain thepull-inparametersof thebeam-typeNEMS. Thepull-involtage(bPI) of the nano-beams can generally be determined via plottingbvs. w . Similarly, for freestandingbeam, critical valueof theCasimir force (aC) can be obtained by plotting a vs. w .Inorder to verify the obtained series, typical beam-type NEMS isnumerically simulated and the results are compared with those ofthe MAD. Table 1 shows the comparison between numericalsolutionandthoseoftheMADones. Asseen, ahigheraccuracycanbeobtainedbyevaluatingmoretermsoftheseriessolutionw (x). Byusing6terms, theglobalerrorislessthan1%, whichiswithin the excellent range for most engineering applications.Therefore, 6termsareselectedinthefollowingsectionfortheconvenience of calculations and acceptable error.For comparison with the literature, the pull-in voltage of typicalcantilever micro-actuators (ad0, KN) is calculated. Geome-try and constitutive material of the beams are identied in Table 2.Acomparisonbetweenpull-involtages obtained by MADandthoseof the literature [5,20,49,56,57] is presented in Table 3; It revealsthat the difference between MAD and numerical values is withinthe range of those of other methods presented in the literature.5. Results and discussion5.1. Simply supported beamFig. 2depictsthevariationin bPIofSSnano-beamvs. springstiffness (K) and Casimir force (a) without considering size effect.As seen, enhancing the spring stiffness leads to an increase in pull-in voltage of the SS nano-actuator. On the other hand, the Casimirforce decreases the pull-involtage of a nano-structure. At anarbitrary K value, an increase in a leads to a decrease in bPI. Finally,Fig. 7 Fig. 8Y. Tadi Beni et al. / Physica E 43 (2011) 979988 985when areachesitscriticalvalue, i.e. aC, no bPIisobtained. Thismeans that for a large value of Casimir attraction, the beambecomes unstable and attaches to the ground plane even withoutapplying voltage difference.Fig. 3 depicts variation in bPIvs. K and a at d0.5. By comparingFigs. 2and3, itisfoundthatwitharbitraryspringstiffnessandCasimirforce, presenceofthesizeeffect(da0)produceshighervalues of bPI. This effect cannot be modeled by the classicalcontinuum theory.5.2. Doubly supported beamFig. 4 depicts the effect of elastic BC on the pull-in voltage of DSbeam at ad0. Neglecting both the Casimir force as well as thesizeeffect is acommonpracticeinMEMSliterature. As seen,stiffening the BC increases the pull-in voltage of the beam.Fig. 5 shows the variation in aC for freestanding beam ignoringthe size effect (b0 and d0). When the gap between the groundand movable electrode is sufciently small, instability occurswithoutapplicationofvoltageduetotheCasimirforce(a4aC).This phenomenon must be considered in designing the minimumgapof beam-type NEMS to ensure that the NEMS does not adhere tothe substrate as a result of intermolecular force. Fig. 5 also revealsthat stiffening the BC increases aC of the beam.Figs. 6 and 7 depict the effect of size dependency (d0.5) on bPIand aCof DS actuators, respectively. As seen fromthese gures, thesize dependency increases the values of bPI and aC of DS actuators.This is attributedto the increase inthe rigidity of the beamconstitutive material, due to the size effect.5.3. Case studiesIn order to thoroughly investigate the pull-in behavior of NEMS,three special cases including clampedclamped (K1K2N),clampedhinged(K1N, K20)andhingedhinged(K1K20)are investigated. Fig. 8ac shows the centerline deection ofclampedclamped, clampedhinged and hingedhinged nano-beams, respectively. Increase in voltage from zero to bPI, increaseswfrom its initial value to pull-in deection. While the maximumdeectionof clampedclampedandclampedhingedbeams occursat x0.5, the maximumdeection of the clampedhinged beam isobtainedatx0.578. Itisobservedthatclampedclampedandhingedhingedstructures havethehighest andthelowest bPIvalues among the three BC, respectively.This isattributed to thehighestelasticrigidityoftheclampedclampedandexibilityofthe hingedhinged boundary conditions.EffectoftheCasimirforceonthepull-inbehaviorofclampedclamped, clampedhinged and hingedhinged actuators is illustratedin Fig. 9ac respectively. These gures represent the obtainedresults for various a values and depict that the intermolecular forcedecreasesthepull-indeectionandvoltageofthenano-actuators.These gures also reveal that the beamhas an initial deection due toFig. 9Y. Tadi Beni et al. / Physica E 43 (2011) 979988 986thepresenceof intermolecular forceevenwithout applicationofvoltage (b0).Fig. 10acshows thestrongsizedependencyof thepull-involtage in clampedclamped, clampedhinged and hingedhingednano-beams, respectively. As seenfromthese gures, the size effectgreatly inuences bPI of nanostructures. With increase in d from 0to0.5, bPIincreases morethantwiceinthecaseof clampedclampedandhingedhingedstructures. Ontheotherhand, thepull-in deection is less sensitive to the size effect in comparison tothe pull-in voltage.6. ConclusionsIn this article, the modied couple stress theory, in conjunctionwiththe MADsolvingmethod, is introducedtoinvestigate the effectof the Casimir attraction, elastic boundaryconditions andsizedependency on nonlinear pull-in behavior of the supported beam.Results reveal that the Casimir force decreases the pull-in voltageand deection of beamat submicron scales. On the other hand, sizeeffect cangreatly increasethepull-inparametersof nano-beams.The pull-indeectionof NEMS is less sensitive to the size effect thanthe pull-involtage. Furthermore, theinstability of NEMSstronglydepends on the type of applied BC. This emphasizes the importanceof characterizingreal BCindesignandanalysis of NEMS. Therelative error of MAD solution with respect to the numerical one iswithin the acceptable range for most engineering applications andcanbe reducedby selecting more series terms. The proposedanalytical methodavoids time-consuming numerical iterationsandmakesparametricstudiespossible. 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