Research ArticleNumerical Investigation of Pull-In Instability in a Micro-SwitchMEMS Device through the Pseudo-Spectral Method
P Di Maida and G Bianchi
Dipartimento di Scienze e Metodi dellrsquoIngegneria (DISMI) Universita degli Studi di Modena e Reggio Emilia Via G Amendola 242122 Reggio Emilia Italy
Correspondence should be addressed to P Di Maida pietrodimaidaunimoreit
Received 4 December 2015 Revised 10 May 2016 Accepted 5 October 2016
Academic Editor Julius Kaplunov
Copyright copy 2016 P Di Maida and G Bianchi 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
Apseudo-spectral approximation is presented to solve the problemof pull-in instability in a cantilevermicro-switch Aswell knownpull-in instability arises when the acting force reaches a critical threshold beyond which equilibrium is no longer possible Inparticular Coulomb electrostatic force is considered although the method can be easily generalized to account for fringe as well asCasimir effects A numerical comparison is presented between a pseudo-spectral and a Finite Element (FE) approximation of theproblem both methods employing the same number of degrees of freedom It is shown that the pseudo-spectral method appearsmore effective in accurately approximating the behavior of the cantilever near its tip This fact is crucial to capturing the thresholdvoltage on the verge of pull-in Conversely the FE approximation presents rapid successions of attractingrepulsing regions alongthe cantilever which are not restricted to the near pull-in regime
1 Introduction
Micro-Electro-Mechanical Systems (MEMS) form a ratherdiverse and inhomogeneous group of micro-devices aimedat sensing and actuating in a wide array of fields rangingfrom mechanical or electronic engineering to chemistry orbiology from micro-mechanics to micro-machining [1ndash5]The manufacturing technology is the common standgroundfor such devices which heavily relies on the different litho-graphic techniques borrowed from the technology of micro-electronics Indeed MEMS devices are mostly obtained froma silicon substrate It is observed that MEMS are reallyldquosystemsrdquo in the sense that they are often made up of severalfunctional parts joint together in the device (like piezo-and magneto-sensors [6]) Among MEMS micro-switchforms a distinct set with great application potential withspecial regard to phase shifters and Radio Frequency MEMS(RFMEMS)They are usually gathered in two groups namelycapacitor and metal-air-metal switches Besides they are fur-ther divided according to the actuationmethod electrostaticelectrothermal magnetostatic and piezoelectric among the
most common A study of magnetoelastic actuated micro-switch is given in [7 8] for the low-frequency asymptoticanalysis of energy scavengers In this paper we focus attentionon the pull-in instability of a capacitor micro-switch actuatedby electrostatic Coulomb force This particular applicationhas received extensive attention in the literature owing tothe importance of pull-in induced failures in applications Arecent review on the subject can be found in [9] A theoreticalanalysis of this problem within the static regime is providedin [10] and references therein Pull-in voltage in cantileverMEMS have been considered in [11ndash14] Failure mechanismsof MEMS include cracking [15ndash18] peeling of the cantilever[19ndash21] stiction to the substrate [22ndash24] and temperature[25 26] Besides micropolar theories are often preferredwhen dealing with micro- and nanodevices to incorporatethe scale effect [27 28] Spotlight is set on a pseudo-spectralapproximation of the problem which is compared with aFinite Element (FE) solution Spectral methods belong to thefamily of Galerkinrsquos (or Ritzrsquos) methods [29] Spectral meth-ods are often divided into two groups namely pseudo-spectralor interpolating The former group enforces the fulfillment of
Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2016 Article ID 8543616 6 pageshttpdxdoiorg10115520168543616
2 Modelling and Simulation in Engineering
L
t
b
D
x
y
y(x)
Figure 1 Micro-cantilever switch
the differential operator at a set of points termed nodes (this issometimes also named orthogonal collocation) For the lattergroup wherein the Galerkinrsquos method is properly placed theexpansion coefficients are obtained projecting the solutiononto the basis set [30]
The paper is structured as follows Section 2 sets forththe governing equations and the boundary condition for acantilever Section 3 introduces the pseudo-spectral methodA numerical comparison with the FE method is illustrated inSection 4 Finally conclusions are drawn in Section 5
2 Governing Equations
Let us consider a micro-cantilever switch device subjected toelectrostatic attractive force (Figure 1) The micro-cantileveracts as one armor of a capacitor under the electric potentialdifference 119881 Let 119905 denote the distance between the capac-itor armors We consider a plane problem and introducethe transverse displacement 119910(119909) for the cantilever Let usintroduce the dimensionless variables
120585 = 119909119871 119906 = 119910119905 (1)
Then the governing equation for the cantilever reads
where 119902119888 and 119902119890 are the Casimir and the electrostatic line-load being
119902119890 = 1205721205980 11988711988121199052 (1 minus 119906)2 (3)
Here 119881 stands for the electric potential difference actingbetween the capacitor armors (in the SI this is expressed involt ie 119881 = NmC where N stands for newton m formeter and C for electric charge expressed in Coulomb) 119887 isthe armors width 1205980 = 8854 sdot 10minus12 C2Nminus1mminus2 is theelectric permittivity (in vacuum) and 120572 is generally a func-tion of 119906 which takes into account the fringe effect For thesake of illustrating the method we neglect the Casimir force
contribution and assume 120572 independent of 119906 Then we canrewrite the governing equation (2) as
119906(119894V) = 119860(1 minus 119906)2 (4)
where prime denotes differentiation with respect to 120585 and thefollowing driving parameter is obtained
119860 = 1205721205980 119887119871411990531198641198681198812 ge 0 (5)
Under the attractive electrostatic force it is 0 le 119906(120585) le 119906max le1 and the boundary conditions (BCs) for the cantilever read
It is observed that integrating and making use of the last BCit may be deduced that
V101584010158401015840 (120585) = int1120585
119860V2 (120591)d120591 (9)
which shows that the shearing force is generally positive andit is zero only at 120585 = 1 The same argument may be applied toinfer that V10158401015840 is generally negative apart from the point 120585 = 1where it is zero and that V1015840 is generally negative although itvanishes at 120585 = 0 Consequently V is a monotonic decreasingfunction of 120585 and V(1) = Vmin The nonlinear fourth orderODE (7) may be integrated once [31 sect421] to give
2V1015840V101584010158401015840 minus (V10158401015840)2 = 2119860V
+ 43119861 (10)
It is observed that in the case 119861 = 0 (10) falls into theEmdem-Fowler class of nonlinear ODEs which in specialcases may admit closed form solutions [31 32] Evaluationat 120585 = 1 and making use of the BCs (8) give
119861 = minus32 119860Vmin
(11)
which shows that the situation 119861 = 0 is not relevant in thisproblem Besides it follows
2V1015840V101584010158401015840 minus (V10158401015840)2 = 2119860(1V
minus 1Vmin
) le 0 (12)
and evaluation at 120585 = 0 lends[V10158401015840 (0)]2 = 2119860( 1
Vminminus 1) (13)
Consideration of the sign for V1015840 and V101584010158401015840 yields the inequality
V10158401015840 le radic2119860( 1Vmin
minus 1V) (14)
Modelling and Simulation in Engineering 3
3 Pseudo-Spectral Method
The governing equations (7) may be numerically solvedthrough a pseudo-spectral approach [29 33] Accordingly a119899-degree polynomial function 119901119899(120585) is adopted to approxi-mate the function 119906(120585) on the interval [0 1] Since a collo-cation method is adopted the polynomial will be uniquelydetermined enforcing (7) at some 119899+1 predetermined points(nodes) This procedure results in a system of nonlinearalgebraic equations which may be solved through standardmethods such as the iterative Newton method The Jacobianof the system may be supplemented in closed form to thenumerical equation solver
Let 119901119899(120585) isin P119899 be the 119899-degree polynomial approxima-tion of V(120585) The collocation set is defined through the first119899 + 1 Gauss-Lobatto points
GL119899 = 120578119894 such that 1205780 = minus1 120578119899 = +1 1199011015840119899 (120578119894)= 0 119894 = 1 119899 minus 1 (15)
where 119901119899 stands for the 119899th degree Legendre polynomialEquation (7) evaluated at the interior nodes 120578119894 yields thesystem of (119899 minus 1) algebraic equations
The problem is now rewritten in matrix form To this aim letthe unknown column vector the vector of the square and thevector of the 119898th derivative
x = [119901119899 (120578119894)] y = [1199012119899 (120578119894)]
x(119898) = [119901(119898)119899 (120578119894)]
119894 = 0 119899 119898 isin N(18)
Making use of the (119899 + 1) times (119899 + 1) derivative matrix D (see[33 Chap7]) we have
x1015840 = Dx
x10158401015840 = DDx = D
2x (19)
and (16) and (17) may be rewritten through the derivativematrix as
yD4x + 119860Id = o (20)
where Id is the identity matrix Here D4 is D4 supplementedwith the BCs (8) that is the first row is set to zero apart fromthe first entry that is set to 1 the second the last-but-one and
00
minus02
minus04
minus06
minus08
minus10
minusu
00 02 04 06 08 10
120585
A = 12
A = 14
A = 16
A = 18
A = 20
Figure 2 Displacement 119906 at 119860 = 12 14 16 18 and 119860 = 20
the last rows are replaced by the first and the last rows of DD2 and D3 respectively Then we have
At the RHS of system (20) we have the zero vector o
supplemented with the BCs namely
o =[[[[[[[
100
]]]]]]]
(22)
4 Numerical Solution
In this section the pseudo-spectral approximation is com-pared with a Finite Element solution both methods employ-ing the same number of degrees of freedom which corre-sponds to 119899 = 20 for the order of the interpolating poly-nomial For the FE solution we introduce 10 nodes eachendowedwith 2 degrees of freedom that is one translationaland the other rotational In Figure 2 the system of algebraicequations (20) is solved for different values of the drivingparameter (voltage difference) 119860 with the FE method Itclearly appears that for 119860 = 20 the cantilever dimensionlesstip deflection is very close to 1 that is the cantilever is atthe verge of pull-in instability Figure 3 shows the productminusV(119894V)V2 which should be constant along the cantilever andequal to 119860 for the FE approximation It is evident thatthe quality of the numerical solution rapidly deterioratesnear the cantilever tip which is exactly where best accuracy
4 Modelling and Simulation in Engineering
30
25
20
15
10
5
0
A
00 02 04 06 08 10
120585
Figure 3 Plot of minusV2V(119894V) versus 120585 according to the FE approxima-tion this product should return the constant 119860
20
18
16
14
12
A
00 02 04 06 08 10
120585
Figure 4 Plot of minusV2V(119894V) versus 120585 according to the pseudo-spectralapproximation this product should return the constant 119860
is demanded to effectively capture the pull-in thresholdConversely Figure 4 plots the productminusV(119894V)V2 for the pseudo-spectral approximation The comparison between the twoplots is a striking example of the effectiveness of this methodin this kind of nonlinear problems Figure 5 plots the slopebending moment and shearing force along the cantileverbeam and it illustrates that BCs are well captured by thenumerical solution either FE or pseudo-spectral The plotis obtained by successive differentiation of the displacementfield and it is readily available for the spectral method wherepolynomial functions are employed Conversely obtainingthe corresponding curves for the FE approximation needssome extra care for curve fitting of the nodal displacementis first applied which is then successively differentiatedFigure 6 plots the applied line load density V(119894V) for the FEsolution near pull-in which corresponds to (the negativeof) the electrostatic Coulomb force minus119902119890 It is remarkablethat the electrostatic force appears highly oscillatory (notethat curve-fitting is employed to get a continuous plot)and it attains unphysical negative values As a comparisonFigure 7 describes the same behavior for 119860 = 12 that iswell below the threshold value for pull-in It appears thatthe electrostatic force is rather poorly approximated by the
5
0
minus5
minus10
(n)
00 02 04 06 08 10
120585
Figure 5 Slope (solid) bending moment (dashed) and shearingforce (dotted) near pull-inGraphs are indistinguishable between thepseudo-spectral and the FE solution
30
20
10
0
minus10
(i)
00 02 04 06 08 10
120585
Figure 6 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 20)
method even far from instability On the contrary Figures8 and 9 illustrate the same graphs as obtained from thepseudo-spectral method The smoothness of the solutionis remarkable The reason for the superior performance ofthe pseudo-spectral approximation seems to lie in the factthat it is more robust in dealing with little deviations of thecantilever tip displacement on the verge of contact Indeed itis well known that pseudo-spectral approximation guaranteeshigh precision and exponential convergence (under suitableassumptions see [29]) and this feature proves important insmoothly approximating the highly nonlinear behavior of theelectrostatic force
5 Conclusions
In this paper the pseudo-spectral method is adopted tonumerically solve the problem of pull-in instability in a can-tilever beam The beam constitutes one armor of a capacitorthe other armor being represented by a grounded flat surfaceAlthough only Coulomb electrostatic force is considered themethod is easily extended to deal with the fringe effect and
Modelling and Simulation in Engineering 5
20
15
10
5
0
(i)
00 02 04 06 08 10
120585
Figure 7 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 12)
20
15
10
5
(i)
00 02 04 06 08 10
120585
Figure 8 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 20)
the Casimir force Results may be especially relevant for can-tilever micro-switches inMEMS It is shown that the pseudo-spectral method compares very favorably with an equivalentFinite Element approximation equivalency being constitutedby an equal number of degrees of freedom in the methods Inparticular good approximation for the cantilever deflectionnear its tip is crucial to capturing the threshold voltage onthe verge of pull-in Indeed poor approximation leads to veryunphysical oscillatory attractionrepulsion forces along thecantilever It is further shown that the oscillatory behavioris not restricted to the near pull-in regime Finally it isemphasized that both methods exactly satisfy the boundaryconditions (BCs) It is remarked that the present analysis canbe extended to incorporate functionally graded cantilevers[34ndash36] and beam-plates [37 38] or to include viscoelasticeffects [39ndash42]
Competing Interests
The authors declare that they have no competing interests
12
10
8
6
4
(i)
00 02 04 06 08 10
120585
Figure 9 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 12)
References
[1] V A Salomoni C E Majorana G M Giannuzzi and AMiliozzi ldquoThermal-fluid flow within innovative heat storageconcrete systems for solar power plantsrdquo International Journalof Numerical Methods for Heat and Fluid Flow vol 18 no 7-8pp 969ndash999 2008
[2] A Nobili L Lanzoni and A M Tarantino ldquoExperimentalinvestigation and monitoring of a polypropylene-based fiberreinforced concrete road pavementrdquo Construction and BuildingMaterials vol 47 pp 888ndash895 2013
[3] V A Salomoni C E Majorana B Pomaro G Xotta andF Gramegna ldquoMacroscale and mesoscale analysis of concreteas a multiphase material for biological shields against nuclearradiationrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 38 no 5 pp 518ndash535 2014
[4] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[5] P Bisegna and R Luciano ldquoBounds on the overall properties ofcomposites with debonded frictionless interfacesrdquoMechanics ofMaterials vol 28 no 1ndash4 pp 23ndash32 1998
[6] A Nobili and A M Tarantino ldquoMagnetostriction of a hardferromagnetic and elastic thin-film structurerdquoMathematics andMechanics of Solids vol 13 no 2 pp 95ndash123 2008
[7] A Kudaibergenov A Nobili and L Prikazchikova ldquoOn low-frequency vibrations of a composite stringwith contrast proper-ties for energy scavenging fabric devicesrdquo Journal of Mechanicsof Materials and Structures vol 11 no 3 pp 231ndash243 2016
[8] J Kaplunov and A Nobili ldquoMulti-parametric analysis ofstrongly inhomogeneous periodic waveguideswith internal cut-off frequenciesrdquoMathematical Methods in the Applied Sciences2016
[9] W-M Zhang H Yan Z-K Peng and G Meng ldquoElectrostaticpull-in instability in MEMSNEMS a reviewrdquo Sensors andActuators A Physical vol 214 pp 187ndash218 2014
[10] Y Zhang and Y-P Zhao ldquoStatic study of cantilever beamstiction under electrostatic force influencerdquo Acta MechanicaSolida Sinica vol 17 no 2 pp 104ndash112 2004
[11] L C Wei A B Mohammad and N M Kassim ldquoAnalyticalmodeling for determination of pull-in voltage for an electro-static actuated MEMS cantilever beamrdquo in Proceedings of the
6 Modelling and Simulation in Engineering
5th IEEE International Conference on Semiconductor Electronics(ICSE rsquo02) pp 233ndash238 IEEE Penang Malaysia December2002
[12] 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
[13] 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
[14] A Ramezani A Alasty and J Akbari ldquoClosed-form solutionsof the pull-in instability in nano-cantilevers under electrostaticand intermolecular surface forcesrdquo International Journal ofSolids and Structures vol 44 no 14-15 pp 4925ndash4941 2007
[15] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[16] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics and AppliedMathematics vol 50 no 3 pp 435ndash456 1997
[17] R Luciano and J R Willis ldquoHashin-Shtrikman based FEanalysis of the elastic behaviour of finite random compositebodiesrdquo International Journal of Fracture vol 137 no 1ndash4 pp261ndash273 2006
[18] A Nobili E Radi and L Lanzoni ldquoA cracked infinite Kirchhoffplate supported by a two-parameter elastic foundationrdquo Journalof the European Ceramic Society vol 34 no 11 pp 2737ndash27442014
[19] L Lanzoni and E Radi ldquoThermally induced deformations in apartially coated elastic layerrdquo International Journal of Solids andStructures vol 46 no 6 pp 1402ndash1412 2009
[20] V Guidi L Lanzoni and A Mazzolari ldquoPatterning and mod-eling of mechanically bent silicon plates deformed throughcoactive stressesrdquoThin Solid Films vol 520 no 3 pp 1074ndash10792011
[21] N Tullini A Tralli and L Lanzoni ldquoIntefacial shear stressanalysis of bar and thin film bonded to 2D elastic substrateusing a coupled FE-BIE methodrdquo Finite Elements in Analysisand Design vol 55 pp 42ndash51 2012
[22] C H Mastrangelo Suppression of Stiction in MEMS vol 605 ofMRS Proceedings CambridgeUniversity Press Cambridge UK1999
[23] W Merlijn Van Spengen R Puers and I De Wolf ldquoA physicalmodel to predict stiction in MEMSrdquo Journal of Micromechanicsand Microengineering vol 12 no 5 pp 702ndash713 2002
[24] Z Yapu ldquoStiction and anti-stiction in MEMS and NEMSrdquo ActaMechanica Sinica vol 19 no 1 pp 1ndash10 2003
[25] G Xotta GMazzucco V A Salomoni C EMajorana andK JWillam ldquoComposite behavior of concrete materials under hightemperaturesrdquo International Journal of Solids and Structures vol64 pp 86ndash99 2015
[26] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashA Solids vol 46 pp 84ndash95 2014
[27] A Nobili ldquoOn the generalization of the Timoshenko beammodel based on the micropolar linear theory static caserdquoMathematical Problems in Engineering vol 2015 Article ID914357 8 pages 2015
[28] G Napoli and A Nobili ldquoMechanically induced Helfrich-Hurault effect in lamellar systemsrdquo Physical Review E vol 80no 3 Article ID 031710 2009
[29] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2nd edition 2000
[30] F Greco and R Luciano ldquoA theoretical and numerical stabilityanalysis for composite micro-structures by using homogeniza-tion theoryrdquo CompositesmdashPart B Engineering vol 42 no 3 pp382ndash401 2011
[31] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1st edition 1995
[32] P L Sachdev Nonlinear Ordinary Differential Equations andTheir Applications Marcel Dekker 1991
[33] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Springer 1992
[34] R Barretta L Feo R Luciano and F Marotti de SciarraldquoVariational formulations for functionally graded nonlocalBernoulli-Euler nanobeamsrdquo Composite Structures vol 129 pp80ndash89 2015
[35] R Barretta L Feo R Luciano and F Marotti de SciarraldquoA gradient Eringen model for functionally graded nanorodsrdquoComposite Structures vol 131 pp 1124ndash1131 2015
[37] R Barretta and R Luciano ldquoExact solutions of isotropicviscoelastic functionally graded Kirchhoff platesrdquo CompositeStructures vol 118 no 1 pp 448ndash454 2014
[38] A Apuzzo R Barretta and R Luciano ldquoSome analyticalsolutions of functionally graded Kirchhoff platesrdquo CompositesPart B Engineering vol 68 pp 266ndash269 2015
[39] L Dezi G Menditto and A M Tarantino ldquoHomogeneousstructures subjected to repeated structural system changesrdquoJournal of Engineering Mechanics vol 116 no 8 pp 1723ndash17321990
[40] L Dezi and A M Tarantino ldquoTime-dependent analysis ofconcrete structures with a variable structural systemrdquo ACIMaterials Journal vol 88 no 3 pp 320ndash324 1991
[41] L Dezi G Menditto and A M Tarantino ldquoViscoelastic het-erogeneous structures with variable structural systemrdquo Journalof Engineering Mechanics vol 119 no 2 pp 238ndash250 1993
[42] R Barretta L Feo and R Luciano ldquoTorsion of functionallygraded nonlocal viscoelastic circular nanobeamsrdquo CompositesPart B Engineering vol 72 pp 217ndash222 2015
the differential operator at a set of points termed nodes (this issometimes also named orthogonal collocation) For the lattergroup wherein the Galerkinrsquos method is properly placed theexpansion coefficients are obtained projecting the solutiononto the basis set [30]
The paper is structured as follows Section 2 sets forththe governing equations and the boundary condition for acantilever Section 3 introduces the pseudo-spectral methodA numerical comparison with the FE method is illustrated inSection 4 Finally conclusions are drawn in Section 5
2 Governing Equations
Let us consider a micro-cantilever switch device subjected toelectrostatic attractive force (Figure 1) The micro-cantileveracts as one armor of a capacitor under the electric potentialdifference 119881 Let 119905 denote the distance between the capac-itor armors We consider a plane problem and introducethe transverse displacement 119910(119909) for the cantilever Let usintroduce the dimensionless variables
120585 = 119909119871 119906 = 119910119905 (1)
Then the governing equation for the cantilever reads
where 119902119888 and 119902119890 are the Casimir and the electrostatic line-load being
119902119890 = 1205721205980 11988711988121199052 (1 minus 119906)2 (3)
Here 119881 stands for the electric potential difference actingbetween the capacitor armors (in the SI this is expressed involt ie 119881 = NmC where N stands for newton m formeter and C for electric charge expressed in Coulomb) 119887 isthe armors width 1205980 = 8854 sdot 10minus12 C2Nminus1mminus2 is theelectric permittivity (in vacuum) and 120572 is generally a func-tion of 119906 which takes into account the fringe effect For thesake of illustrating the method we neglect the Casimir force
contribution and assume 120572 independent of 119906 Then we canrewrite the governing equation (2) as
119906(119894V) = 119860(1 minus 119906)2 (4)
where prime denotes differentiation with respect to 120585 and thefollowing driving parameter is obtained
119860 = 1205721205980 119887119871411990531198641198681198812 ge 0 (5)
Under the attractive electrostatic force it is 0 le 119906(120585) le 119906max le1 and the boundary conditions (BCs) for the cantilever read
It is observed that integrating and making use of the last BCit may be deduced that
V101584010158401015840 (120585) = int1120585
119860V2 (120591)d120591 (9)
which shows that the shearing force is generally positive andit is zero only at 120585 = 1 The same argument may be applied toinfer that V10158401015840 is generally negative apart from the point 120585 = 1where it is zero and that V1015840 is generally negative although itvanishes at 120585 = 0 Consequently V is a monotonic decreasingfunction of 120585 and V(1) = Vmin The nonlinear fourth orderODE (7) may be integrated once [31 sect421] to give
2V1015840V101584010158401015840 minus (V10158401015840)2 = 2119860V
+ 43119861 (10)
It is observed that in the case 119861 = 0 (10) falls into theEmdem-Fowler class of nonlinear ODEs which in specialcases may admit closed form solutions [31 32] Evaluationat 120585 = 1 and making use of the BCs (8) give
119861 = minus32 119860Vmin
(11)
which shows that the situation 119861 = 0 is not relevant in thisproblem Besides it follows
2V1015840V101584010158401015840 minus (V10158401015840)2 = 2119860(1V
minus 1Vmin
) le 0 (12)
and evaluation at 120585 = 0 lends[V10158401015840 (0)]2 = 2119860( 1
Vminminus 1) (13)
Consideration of the sign for V1015840 and V101584010158401015840 yields the inequality
V10158401015840 le radic2119860( 1Vmin
minus 1V) (14)
Modelling and Simulation in Engineering 3
3 Pseudo-Spectral Method
The governing equations (7) may be numerically solvedthrough a pseudo-spectral approach [29 33] Accordingly a119899-degree polynomial function 119901119899(120585) is adopted to approxi-mate the function 119906(120585) on the interval [0 1] Since a collo-cation method is adopted the polynomial will be uniquelydetermined enforcing (7) at some 119899+1 predetermined points(nodes) This procedure results in a system of nonlinearalgebraic equations which may be solved through standardmethods such as the iterative Newton method The Jacobianof the system may be supplemented in closed form to thenumerical equation solver
Let 119901119899(120585) isin P119899 be the 119899-degree polynomial approxima-tion of V(120585) The collocation set is defined through the first119899 + 1 Gauss-Lobatto points
GL119899 = 120578119894 such that 1205780 = minus1 120578119899 = +1 1199011015840119899 (120578119894)= 0 119894 = 1 119899 minus 1 (15)
where 119901119899 stands for the 119899th degree Legendre polynomialEquation (7) evaluated at the interior nodes 120578119894 yields thesystem of (119899 minus 1) algebraic equations
The problem is now rewritten in matrix form To this aim letthe unknown column vector the vector of the square and thevector of the 119898th derivative
x = [119901119899 (120578119894)] y = [1199012119899 (120578119894)]
x(119898) = [119901(119898)119899 (120578119894)]
119894 = 0 119899 119898 isin N(18)
Making use of the (119899 + 1) times (119899 + 1) derivative matrix D (see[33 Chap7]) we have
x1015840 = Dx
x10158401015840 = DDx = D
2x (19)
and (16) and (17) may be rewritten through the derivativematrix as
yD4x + 119860Id = o (20)
where Id is the identity matrix Here D4 is D4 supplementedwith the BCs (8) that is the first row is set to zero apart fromthe first entry that is set to 1 the second the last-but-one and
00
minus02
minus04
minus06
minus08
minus10
minusu
00 02 04 06 08 10
120585
A = 12
A = 14
A = 16
A = 18
A = 20
Figure 2 Displacement 119906 at 119860 = 12 14 16 18 and 119860 = 20
the last rows are replaced by the first and the last rows of DD2 and D3 respectively Then we have
At the RHS of system (20) we have the zero vector o
supplemented with the BCs namely
o =[[[[[[[
100
]]]]]]]
(22)
4 Numerical Solution
In this section the pseudo-spectral approximation is com-pared with a Finite Element solution both methods employ-ing the same number of degrees of freedom which corre-sponds to 119899 = 20 for the order of the interpolating poly-nomial For the FE solution we introduce 10 nodes eachendowedwith 2 degrees of freedom that is one translationaland the other rotational In Figure 2 the system of algebraicequations (20) is solved for different values of the drivingparameter (voltage difference) 119860 with the FE method Itclearly appears that for 119860 = 20 the cantilever dimensionlesstip deflection is very close to 1 that is the cantilever is atthe verge of pull-in instability Figure 3 shows the productminusV(119894V)V2 which should be constant along the cantilever andequal to 119860 for the FE approximation It is evident thatthe quality of the numerical solution rapidly deterioratesnear the cantilever tip which is exactly where best accuracy
4 Modelling and Simulation in Engineering
30
25
20
15
10
5
0
A
00 02 04 06 08 10
120585
Figure 3 Plot of minusV2V(119894V) versus 120585 according to the FE approxima-tion this product should return the constant 119860
20
18
16
14
12
A
00 02 04 06 08 10
120585
Figure 4 Plot of minusV2V(119894V) versus 120585 according to the pseudo-spectralapproximation this product should return the constant 119860
is demanded to effectively capture the pull-in thresholdConversely Figure 4 plots the productminusV(119894V)V2 for the pseudo-spectral approximation The comparison between the twoplots is a striking example of the effectiveness of this methodin this kind of nonlinear problems Figure 5 plots the slopebending moment and shearing force along the cantileverbeam and it illustrates that BCs are well captured by thenumerical solution either FE or pseudo-spectral The plotis obtained by successive differentiation of the displacementfield and it is readily available for the spectral method wherepolynomial functions are employed Conversely obtainingthe corresponding curves for the FE approximation needssome extra care for curve fitting of the nodal displacementis first applied which is then successively differentiatedFigure 6 plots the applied line load density V(119894V) for the FEsolution near pull-in which corresponds to (the negativeof) the electrostatic Coulomb force minus119902119890 It is remarkablethat the electrostatic force appears highly oscillatory (notethat curve-fitting is employed to get a continuous plot)and it attains unphysical negative values As a comparisonFigure 7 describes the same behavior for 119860 = 12 that iswell below the threshold value for pull-in It appears thatthe electrostatic force is rather poorly approximated by the
5
0
minus5
minus10
(n)
00 02 04 06 08 10
120585
Figure 5 Slope (solid) bending moment (dashed) and shearingforce (dotted) near pull-inGraphs are indistinguishable between thepseudo-spectral and the FE solution
30
20
10
0
minus10
(i)
00 02 04 06 08 10
120585
Figure 6 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 20)
method even far from instability On the contrary Figures8 and 9 illustrate the same graphs as obtained from thepseudo-spectral method The smoothness of the solutionis remarkable The reason for the superior performance ofthe pseudo-spectral approximation seems to lie in the factthat it is more robust in dealing with little deviations of thecantilever tip displacement on the verge of contact Indeed itis well known that pseudo-spectral approximation guaranteeshigh precision and exponential convergence (under suitableassumptions see [29]) and this feature proves important insmoothly approximating the highly nonlinear behavior of theelectrostatic force
5 Conclusions
In this paper the pseudo-spectral method is adopted tonumerically solve the problem of pull-in instability in a can-tilever beam The beam constitutes one armor of a capacitorthe other armor being represented by a grounded flat surfaceAlthough only Coulomb electrostatic force is considered themethod is easily extended to deal with the fringe effect and
Modelling and Simulation in Engineering 5
20
15
10
5
0
(i)
00 02 04 06 08 10
120585
Figure 7 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 12)
20
15
10
5
(i)
00 02 04 06 08 10
120585
Figure 8 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 20)
the Casimir force Results may be especially relevant for can-tilever micro-switches inMEMS It is shown that the pseudo-spectral method compares very favorably with an equivalentFinite Element approximation equivalency being constitutedby an equal number of degrees of freedom in the methods Inparticular good approximation for the cantilever deflectionnear its tip is crucial to capturing the threshold voltage onthe verge of pull-in Indeed poor approximation leads to veryunphysical oscillatory attractionrepulsion forces along thecantilever It is further shown that the oscillatory behavioris not restricted to the near pull-in regime Finally it isemphasized that both methods exactly satisfy the boundaryconditions (BCs) It is remarked that the present analysis canbe extended to incorporate functionally graded cantilevers[34ndash36] and beam-plates [37 38] or to include viscoelasticeffects [39ndash42]
Competing Interests
The authors declare that they have no competing interests
12
10
8
6
4
(i)
00 02 04 06 08 10
120585
Figure 9 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 12)
References
[1] V A Salomoni C E Majorana G M Giannuzzi and AMiliozzi ldquoThermal-fluid flow within innovative heat storageconcrete systems for solar power plantsrdquo International Journalof Numerical Methods for Heat and Fluid Flow vol 18 no 7-8pp 969ndash999 2008
[2] A Nobili L Lanzoni and A M Tarantino ldquoExperimentalinvestigation and monitoring of a polypropylene-based fiberreinforced concrete road pavementrdquo Construction and BuildingMaterials vol 47 pp 888ndash895 2013
[3] V A Salomoni C E Majorana B Pomaro G Xotta andF Gramegna ldquoMacroscale and mesoscale analysis of concreteas a multiphase material for biological shields against nuclearradiationrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 38 no 5 pp 518ndash535 2014
[4] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[5] P Bisegna and R Luciano ldquoBounds on the overall properties ofcomposites with debonded frictionless interfacesrdquoMechanics ofMaterials vol 28 no 1ndash4 pp 23ndash32 1998
[6] A Nobili and A M Tarantino ldquoMagnetostriction of a hardferromagnetic and elastic thin-film structurerdquoMathematics andMechanics of Solids vol 13 no 2 pp 95ndash123 2008
[7] A Kudaibergenov A Nobili and L Prikazchikova ldquoOn low-frequency vibrations of a composite stringwith contrast proper-ties for energy scavenging fabric devicesrdquo Journal of Mechanicsof Materials and Structures vol 11 no 3 pp 231ndash243 2016
[8] J Kaplunov and A Nobili ldquoMulti-parametric analysis ofstrongly inhomogeneous periodic waveguideswith internal cut-off frequenciesrdquoMathematical Methods in the Applied Sciences2016
[9] W-M Zhang H Yan Z-K Peng and G Meng ldquoElectrostaticpull-in instability in MEMSNEMS a reviewrdquo Sensors andActuators A Physical vol 214 pp 187ndash218 2014
[10] Y Zhang and Y-P Zhao ldquoStatic study of cantilever beamstiction under electrostatic force influencerdquo Acta MechanicaSolida Sinica vol 17 no 2 pp 104ndash112 2004
[11] L C Wei A B Mohammad and N M Kassim ldquoAnalyticalmodeling for determination of pull-in voltage for an electro-static actuated MEMS cantilever beamrdquo in Proceedings of the
6 Modelling and Simulation in Engineering
5th IEEE International Conference on Semiconductor Electronics(ICSE rsquo02) pp 233ndash238 IEEE Penang Malaysia December2002
[12] 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
[13] 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
[14] A Ramezani A Alasty and J Akbari ldquoClosed-form solutionsof the pull-in instability in nano-cantilevers under electrostaticand intermolecular surface forcesrdquo International Journal ofSolids and Structures vol 44 no 14-15 pp 4925ndash4941 2007
[15] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[16] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics and AppliedMathematics vol 50 no 3 pp 435ndash456 1997
[17] R Luciano and J R Willis ldquoHashin-Shtrikman based FEanalysis of the elastic behaviour of finite random compositebodiesrdquo International Journal of Fracture vol 137 no 1ndash4 pp261ndash273 2006
[18] A Nobili E Radi and L Lanzoni ldquoA cracked infinite Kirchhoffplate supported by a two-parameter elastic foundationrdquo Journalof the European Ceramic Society vol 34 no 11 pp 2737ndash27442014
[19] L Lanzoni and E Radi ldquoThermally induced deformations in apartially coated elastic layerrdquo International Journal of Solids andStructures vol 46 no 6 pp 1402ndash1412 2009
[20] V Guidi L Lanzoni and A Mazzolari ldquoPatterning and mod-eling of mechanically bent silicon plates deformed throughcoactive stressesrdquoThin Solid Films vol 520 no 3 pp 1074ndash10792011
[21] N Tullini A Tralli and L Lanzoni ldquoIntefacial shear stressanalysis of bar and thin film bonded to 2D elastic substrateusing a coupled FE-BIE methodrdquo Finite Elements in Analysisand Design vol 55 pp 42ndash51 2012
[22] C H Mastrangelo Suppression of Stiction in MEMS vol 605 ofMRS Proceedings CambridgeUniversity Press Cambridge UK1999
[23] W Merlijn Van Spengen R Puers and I De Wolf ldquoA physicalmodel to predict stiction in MEMSrdquo Journal of Micromechanicsand Microengineering vol 12 no 5 pp 702ndash713 2002
[24] Z Yapu ldquoStiction and anti-stiction in MEMS and NEMSrdquo ActaMechanica Sinica vol 19 no 1 pp 1ndash10 2003
[25] G Xotta GMazzucco V A Salomoni C EMajorana andK JWillam ldquoComposite behavior of concrete materials under hightemperaturesrdquo International Journal of Solids and Structures vol64 pp 86ndash99 2015
[26] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashA Solids vol 46 pp 84ndash95 2014
[27] A Nobili ldquoOn the generalization of the Timoshenko beammodel based on the micropolar linear theory static caserdquoMathematical Problems in Engineering vol 2015 Article ID914357 8 pages 2015
[28] G Napoli and A Nobili ldquoMechanically induced Helfrich-Hurault effect in lamellar systemsrdquo Physical Review E vol 80no 3 Article ID 031710 2009
[29] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2nd edition 2000
[30] F Greco and R Luciano ldquoA theoretical and numerical stabilityanalysis for composite micro-structures by using homogeniza-tion theoryrdquo CompositesmdashPart B Engineering vol 42 no 3 pp382ndash401 2011
[31] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1st edition 1995
[32] P L Sachdev Nonlinear Ordinary Differential Equations andTheir Applications Marcel Dekker 1991
[33] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Springer 1992
[34] R Barretta L Feo R Luciano and F Marotti de SciarraldquoVariational formulations for functionally graded nonlocalBernoulli-Euler nanobeamsrdquo Composite Structures vol 129 pp80ndash89 2015
[35] R Barretta L Feo R Luciano and F Marotti de SciarraldquoA gradient Eringen model for functionally graded nanorodsrdquoComposite Structures vol 131 pp 1124ndash1131 2015
[37] R Barretta and R Luciano ldquoExact solutions of isotropicviscoelastic functionally graded Kirchhoff platesrdquo CompositeStructures vol 118 no 1 pp 448ndash454 2014
[38] A Apuzzo R Barretta and R Luciano ldquoSome analyticalsolutions of functionally graded Kirchhoff platesrdquo CompositesPart B Engineering vol 68 pp 266ndash269 2015
[39] L Dezi G Menditto and A M Tarantino ldquoHomogeneousstructures subjected to repeated structural system changesrdquoJournal of Engineering Mechanics vol 116 no 8 pp 1723ndash17321990
[40] L Dezi and A M Tarantino ldquoTime-dependent analysis ofconcrete structures with a variable structural systemrdquo ACIMaterials Journal vol 88 no 3 pp 320ndash324 1991
[41] L Dezi G Menditto and A M Tarantino ldquoViscoelastic het-erogeneous structures with variable structural systemrdquo Journalof Engineering Mechanics vol 119 no 2 pp 238ndash250 1993
[42] R Barretta L Feo and R Luciano ldquoTorsion of functionallygraded nonlocal viscoelastic circular nanobeamsrdquo CompositesPart B Engineering vol 72 pp 217ndash222 2015
The governing equations (7) may be numerically solvedthrough a pseudo-spectral approach [29 33] Accordingly a119899-degree polynomial function 119901119899(120585) is adopted to approxi-mate the function 119906(120585) on the interval [0 1] Since a collo-cation method is adopted the polynomial will be uniquelydetermined enforcing (7) at some 119899+1 predetermined points(nodes) This procedure results in a system of nonlinearalgebraic equations which may be solved through standardmethods such as the iterative Newton method The Jacobianof the system may be supplemented in closed form to thenumerical equation solver
Let 119901119899(120585) isin P119899 be the 119899-degree polynomial approxima-tion of V(120585) The collocation set is defined through the first119899 + 1 Gauss-Lobatto points
GL119899 = 120578119894 such that 1205780 = minus1 120578119899 = +1 1199011015840119899 (120578119894)= 0 119894 = 1 119899 minus 1 (15)
where 119901119899 stands for the 119899th degree Legendre polynomialEquation (7) evaluated at the interior nodes 120578119894 yields thesystem of (119899 minus 1) algebraic equations
The problem is now rewritten in matrix form To this aim letthe unknown column vector the vector of the square and thevector of the 119898th derivative
x = [119901119899 (120578119894)] y = [1199012119899 (120578119894)]
x(119898) = [119901(119898)119899 (120578119894)]
119894 = 0 119899 119898 isin N(18)
Making use of the (119899 + 1) times (119899 + 1) derivative matrix D (see[33 Chap7]) we have
x1015840 = Dx
x10158401015840 = DDx = D
2x (19)
and (16) and (17) may be rewritten through the derivativematrix as
yD4x + 119860Id = o (20)
where Id is the identity matrix Here D4 is D4 supplementedwith the BCs (8) that is the first row is set to zero apart fromthe first entry that is set to 1 the second the last-but-one and
00
minus02
minus04
minus06
minus08
minus10
minusu
00 02 04 06 08 10
120585
A = 12
A = 14
A = 16
A = 18
A = 20
Figure 2 Displacement 119906 at 119860 = 12 14 16 18 and 119860 = 20
the last rows are replaced by the first and the last rows of DD2 and D3 respectively Then we have
At the RHS of system (20) we have the zero vector o
supplemented with the BCs namely
o =[[[[[[[
100
]]]]]]]
(22)
4 Numerical Solution
In this section the pseudo-spectral approximation is com-pared with a Finite Element solution both methods employ-ing the same number of degrees of freedom which corre-sponds to 119899 = 20 for the order of the interpolating poly-nomial For the FE solution we introduce 10 nodes eachendowedwith 2 degrees of freedom that is one translationaland the other rotational In Figure 2 the system of algebraicequations (20) is solved for different values of the drivingparameter (voltage difference) 119860 with the FE method Itclearly appears that for 119860 = 20 the cantilever dimensionlesstip deflection is very close to 1 that is the cantilever is atthe verge of pull-in instability Figure 3 shows the productminusV(119894V)V2 which should be constant along the cantilever andequal to 119860 for the FE approximation It is evident thatthe quality of the numerical solution rapidly deterioratesnear the cantilever tip which is exactly where best accuracy
4 Modelling and Simulation in Engineering
30
25
20
15
10
5
0
A
00 02 04 06 08 10
120585
Figure 3 Plot of minusV2V(119894V) versus 120585 according to the FE approxima-tion this product should return the constant 119860
20
18
16
14
12
A
00 02 04 06 08 10
120585
Figure 4 Plot of minusV2V(119894V) versus 120585 according to the pseudo-spectralapproximation this product should return the constant 119860
is demanded to effectively capture the pull-in thresholdConversely Figure 4 plots the productminusV(119894V)V2 for the pseudo-spectral approximation The comparison between the twoplots is a striking example of the effectiveness of this methodin this kind of nonlinear problems Figure 5 plots the slopebending moment and shearing force along the cantileverbeam and it illustrates that BCs are well captured by thenumerical solution either FE or pseudo-spectral The plotis obtained by successive differentiation of the displacementfield and it is readily available for the spectral method wherepolynomial functions are employed Conversely obtainingthe corresponding curves for the FE approximation needssome extra care for curve fitting of the nodal displacementis first applied which is then successively differentiatedFigure 6 plots the applied line load density V(119894V) for the FEsolution near pull-in which corresponds to (the negativeof) the electrostatic Coulomb force minus119902119890 It is remarkablethat the electrostatic force appears highly oscillatory (notethat curve-fitting is employed to get a continuous plot)and it attains unphysical negative values As a comparisonFigure 7 describes the same behavior for 119860 = 12 that iswell below the threshold value for pull-in It appears thatthe electrostatic force is rather poorly approximated by the
5
0
minus5
minus10
(n)
00 02 04 06 08 10
120585
Figure 5 Slope (solid) bending moment (dashed) and shearingforce (dotted) near pull-inGraphs are indistinguishable between thepseudo-spectral and the FE solution
30
20
10
0
minus10
(i)
00 02 04 06 08 10
120585
Figure 6 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 20)
method even far from instability On the contrary Figures8 and 9 illustrate the same graphs as obtained from thepseudo-spectral method The smoothness of the solutionis remarkable The reason for the superior performance ofthe pseudo-spectral approximation seems to lie in the factthat it is more robust in dealing with little deviations of thecantilever tip displacement on the verge of contact Indeed itis well known that pseudo-spectral approximation guaranteeshigh precision and exponential convergence (under suitableassumptions see [29]) and this feature proves important insmoothly approximating the highly nonlinear behavior of theelectrostatic force
5 Conclusions
In this paper the pseudo-spectral method is adopted tonumerically solve the problem of pull-in instability in a can-tilever beam The beam constitutes one armor of a capacitorthe other armor being represented by a grounded flat surfaceAlthough only Coulomb electrostatic force is considered themethod is easily extended to deal with the fringe effect and
Modelling and Simulation in Engineering 5
20
15
10
5
0
(i)
00 02 04 06 08 10
120585
Figure 7 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 12)
20
15
10
5
(i)
00 02 04 06 08 10
120585
Figure 8 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 20)
the Casimir force Results may be especially relevant for can-tilever micro-switches inMEMS It is shown that the pseudo-spectral method compares very favorably with an equivalentFinite Element approximation equivalency being constitutedby an equal number of degrees of freedom in the methods Inparticular good approximation for the cantilever deflectionnear its tip is crucial to capturing the threshold voltage onthe verge of pull-in Indeed poor approximation leads to veryunphysical oscillatory attractionrepulsion forces along thecantilever It is further shown that the oscillatory behavioris not restricted to the near pull-in regime Finally it isemphasized that both methods exactly satisfy the boundaryconditions (BCs) It is remarked that the present analysis canbe extended to incorporate functionally graded cantilevers[34ndash36] and beam-plates [37 38] or to include viscoelasticeffects [39ndash42]
Competing Interests
The authors declare that they have no competing interests
12
10
8
6
4
(i)
00 02 04 06 08 10
120585
Figure 9 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 12)
References
[1] V A Salomoni C E Majorana G M Giannuzzi and AMiliozzi ldquoThermal-fluid flow within innovative heat storageconcrete systems for solar power plantsrdquo International Journalof Numerical Methods for Heat and Fluid Flow vol 18 no 7-8pp 969ndash999 2008
[2] A Nobili L Lanzoni and A M Tarantino ldquoExperimentalinvestigation and monitoring of a polypropylene-based fiberreinforced concrete road pavementrdquo Construction and BuildingMaterials vol 47 pp 888ndash895 2013
[3] V A Salomoni C E Majorana B Pomaro G Xotta andF Gramegna ldquoMacroscale and mesoscale analysis of concreteas a multiphase material for biological shields against nuclearradiationrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 38 no 5 pp 518ndash535 2014
[4] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[5] P Bisegna and R Luciano ldquoBounds on the overall properties ofcomposites with debonded frictionless interfacesrdquoMechanics ofMaterials vol 28 no 1ndash4 pp 23ndash32 1998
[6] A Nobili and A M Tarantino ldquoMagnetostriction of a hardferromagnetic and elastic thin-film structurerdquoMathematics andMechanics of Solids vol 13 no 2 pp 95ndash123 2008
[7] A Kudaibergenov A Nobili and L Prikazchikova ldquoOn low-frequency vibrations of a composite stringwith contrast proper-ties for energy scavenging fabric devicesrdquo Journal of Mechanicsof Materials and Structures vol 11 no 3 pp 231ndash243 2016
[8] J Kaplunov and A Nobili ldquoMulti-parametric analysis ofstrongly inhomogeneous periodic waveguideswith internal cut-off frequenciesrdquoMathematical Methods in the Applied Sciences2016
[9] W-M Zhang H Yan Z-K Peng and G Meng ldquoElectrostaticpull-in instability in MEMSNEMS a reviewrdquo Sensors andActuators A Physical vol 214 pp 187ndash218 2014
[10] Y Zhang and Y-P Zhao ldquoStatic study of cantilever beamstiction under electrostatic force influencerdquo Acta MechanicaSolida Sinica vol 17 no 2 pp 104ndash112 2004
[11] L C Wei A B Mohammad and N M Kassim ldquoAnalyticalmodeling for determination of pull-in voltage for an electro-static actuated MEMS cantilever beamrdquo in Proceedings of the
6 Modelling and Simulation in Engineering
5th IEEE International Conference on Semiconductor Electronics(ICSE rsquo02) pp 233ndash238 IEEE Penang Malaysia December2002
[12] 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
[13] 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
[14] A Ramezani A Alasty and J Akbari ldquoClosed-form solutionsof the pull-in instability in nano-cantilevers under electrostaticand intermolecular surface forcesrdquo International Journal ofSolids and Structures vol 44 no 14-15 pp 4925ndash4941 2007
[15] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[16] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics and AppliedMathematics vol 50 no 3 pp 435ndash456 1997
[17] R Luciano and J R Willis ldquoHashin-Shtrikman based FEanalysis of the elastic behaviour of finite random compositebodiesrdquo International Journal of Fracture vol 137 no 1ndash4 pp261ndash273 2006
[18] A Nobili E Radi and L Lanzoni ldquoA cracked infinite Kirchhoffplate supported by a two-parameter elastic foundationrdquo Journalof the European Ceramic Society vol 34 no 11 pp 2737ndash27442014
[19] L Lanzoni and E Radi ldquoThermally induced deformations in apartially coated elastic layerrdquo International Journal of Solids andStructures vol 46 no 6 pp 1402ndash1412 2009
[20] V Guidi L Lanzoni and A Mazzolari ldquoPatterning and mod-eling of mechanically bent silicon plates deformed throughcoactive stressesrdquoThin Solid Films vol 520 no 3 pp 1074ndash10792011
[21] N Tullini A Tralli and L Lanzoni ldquoIntefacial shear stressanalysis of bar and thin film bonded to 2D elastic substrateusing a coupled FE-BIE methodrdquo Finite Elements in Analysisand Design vol 55 pp 42ndash51 2012
[22] C H Mastrangelo Suppression of Stiction in MEMS vol 605 ofMRS Proceedings CambridgeUniversity Press Cambridge UK1999
[23] W Merlijn Van Spengen R Puers and I De Wolf ldquoA physicalmodel to predict stiction in MEMSrdquo Journal of Micromechanicsand Microengineering vol 12 no 5 pp 702ndash713 2002
[24] Z Yapu ldquoStiction and anti-stiction in MEMS and NEMSrdquo ActaMechanica Sinica vol 19 no 1 pp 1ndash10 2003
[25] G Xotta GMazzucco V A Salomoni C EMajorana andK JWillam ldquoComposite behavior of concrete materials under hightemperaturesrdquo International Journal of Solids and Structures vol64 pp 86ndash99 2015
[26] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashA Solids vol 46 pp 84ndash95 2014
[27] A Nobili ldquoOn the generalization of the Timoshenko beammodel based on the micropolar linear theory static caserdquoMathematical Problems in Engineering vol 2015 Article ID914357 8 pages 2015
[28] G Napoli and A Nobili ldquoMechanically induced Helfrich-Hurault effect in lamellar systemsrdquo Physical Review E vol 80no 3 Article ID 031710 2009
[29] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2nd edition 2000
[30] F Greco and R Luciano ldquoA theoretical and numerical stabilityanalysis for composite micro-structures by using homogeniza-tion theoryrdquo CompositesmdashPart B Engineering vol 42 no 3 pp382ndash401 2011
[31] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1st edition 1995
[32] P L Sachdev Nonlinear Ordinary Differential Equations andTheir Applications Marcel Dekker 1991
[33] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Springer 1992
[34] R Barretta L Feo R Luciano and F Marotti de SciarraldquoVariational formulations for functionally graded nonlocalBernoulli-Euler nanobeamsrdquo Composite Structures vol 129 pp80ndash89 2015
[35] R Barretta L Feo R Luciano and F Marotti de SciarraldquoA gradient Eringen model for functionally graded nanorodsrdquoComposite Structures vol 131 pp 1124ndash1131 2015
[37] R Barretta and R Luciano ldquoExact solutions of isotropicviscoelastic functionally graded Kirchhoff platesrdquo CompositeStructures vol 118 no 1 pp 448ndash454 2014
[38] A Apuzzo R Barretta and R Luciano ldquoSome analyticalsolutions of functionally graded Kirchhoff platesrdquo CompositesPart B Engineering vol 68 pp 266ndash269 2015
[39] L Dezi G Menditto and A M Tarantino ldquoHomogeneousstructures subjected to repeated structural system changesrdquoJournal of Engineering Mechanics vol 116 no 8 pp 1723ndash17321990
[40] L Dezi and A M Tarantino ldquoTime-dependent analysis ofconcrete structures with a variable structural systemrdquo ACIMaterials Journal vol 88 no 3 pp 320ndash324 1991
[41] L Dezi G Menditto and A M Tarantino ldquoViscoelastic het-erogeneous structures with variable structural systemrdquo Journalof Engineering Mechanics vol 119 no 2 pp 238ndash250 1993
[42] R Barretta L Feo and R Luciano ldquoTorsion of functionallygraded nonlocal viscoelastic circular nanobeamsrdquo CompositesPart B Engineering vol 72 pp 217ndash222 2015
Figure 3 Plot of minusV2V(119894V) versus 120585 according to the FE approxima-tion this product should return the constant 119860
20
18
16
14
12
A
00 02 04 06 08 10
120585
Figure 4 Plot of minusV2V(119894V) versus 120585 according to the pseudo-spectralapproximation this product should return the constant 119860
is demanded to effectively capture the pull-in thresholdConversely Figure 4 plots the productminusV(119894V)V2 for the pseudo-spectral approximation The comparison between the twoplots is a striking example of the effectiveness of this methodin this kind of nonlinear problems Figure 5 plots the slopebending moment and shearing force along the cantileverbeam and it illustrates that BCs are well captured by thenumerical solution either FE or pseudo-spectral The plotis obtained by successive differentiation of the displacementfield and it is readily available for the spectral method wherepolynomial functions are employed Conversely obtainingthe corresponding curves for the FE approximation needssome extra care for curve fitting of the nodal displacementis first applied which is then successively differentiatedFigure 6 plots the applied line load density V(119894V) for the FEsolution near pull-in which corresponds to (the negativeof) the electrostatic Coulomb force minus119902119890 It is remarkablethat the electrostatic force appears highly oscillatory (notethat curve-fitting is employed to get a continuous plot)and it attains unphysical negative values As a comparisonFigure 7 describes the same behavior for 119860 = 12 that iswell below the threshold value for pull-in It appears thatthe electrostatic force is rather poorly approximated by the
5
0
minus5
minus10
(n)
00 02 04 06 08 10
120585
Figure 5 Slope (solid) bending moment (dashed) and shearingforce (dotted) near pull-inGraphs are indistinguishable between thepseudo-spectral and the FE solution
30
20
10
0
minus10
(i)
00 02 04 06 08 10
120585
Figure 6 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 20)
method even far from instability On the contrary Figures8 and 9 illustrate the same graphs as obtained from thepseudo-spectral method The smoothness of the solutionis remarkable The reason for the superior performance ofthe pseudo-spectral approximation seems to lie in the factthat it is more robust in dealing with little deviations of thecantilever tip displacement on the verge of contact Indeed itis well known that pseudo-spectral approximation guaranteeshigh precision and exponential convergence (under suitableassumptions see [29]) and this feature proves important insmoothly approximating the highly nonlinear behavior of theelectrostatic force
5 Conclusions
In this paper the pseudo-spectral method is adopted tonumerically solve the problem of pull-in instability in a can-tilever beam The beam constitutes one armor of a capacitorthe other armor being represented by a grounded flat surfaceAlthough only Coulomb electrostatic force is considered themethod is easily extended to deal with the fringe effect and
Modelling and Simulation in Engineering 5
20
15
10
5
0
(i)
00 02 04 06 08 10
120585
Figure 7 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 12)
20
15
10
5
(i)
00 02 04 06 08 10
120585
Figure 8 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 20)
the Casimir force Results may be especially relevant for can-tilever micro-switches inMEMS It is shown that the pseudo-spectral method compares very favorably with an equivalentFinite Element approximation equivalency being constitutedby an equal number of degrees of freedom in the methods Inparticular good approximation for the cantilever deflectionnear its tip is crucial to capturing the threshold voltage onthe verge of pull-in Indeed poor approximation leads to veryunphysical oscillatory attractionrepulsion forces along thecantilever It is further shown that the oscillatory behavioris not restricted to the near pull-in regime Finally it isemphasized that both methods exactly satisfy the boundaryconditions (BCs) It is remarked that the present analysis canbe extended to incorporate functionally graded cantilevers[34ndash36] and beam-plates [37 38] or to include viscoelasticeffects [39ndash42]
Competing Interests
The authors declare that they have no competing interests
12
10
8
6
4
(i)
00 02 04 06 08 10
120585
Figure 9 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 12)
References
[1] V A Salomoni C E Majorana G M Giannuzzi and AMiliozzi ldquoThermal-fluid flow within innovative heat storageconcrete systems for solar power plantsrdquo International Journalof Numerical Methods for Heat and Fluid Flow vol 18 no 7-8pp 969ndash999 2008
[2] A Nobili L Lanzoni and A M Tarantino ldquoExperimentalinvestigation and monitoring of a polypropylene-based fiberreinforced concrete road pavementrdquo Construction and BuildingMaterials vol 47 pp 888ndash895 2013
[3] V A Salomoni C E Majorana B Pomaro G Xotta andF Gramegna ldquoMacroscale and mesoscale analysis of concreteas a multiphase material for biological shields against nuclearradiationrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 38 no 5 pp 518ndash535 2014
[4] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[5] P Bisegna and R Luciano ldquoBounds on the overall properties ofcomposites with debonded frictionless interfacesrdquoMechanics ofMaterials vol 28 no 1ndash4 pp 23ndash32 1998
[6] A Nobili and A M Tarantino ldquoMagnetostriction of a hardferromagnetic and elastic thin-film structurerdquoMathematics andMechanics of Solids vol 13 no 2 pp 95ndash123 2008
[7] A Kudaibergenov A Nobili and L Prikazchikova ldquoOn low-frequency vibrations of a composite stringwith contrast proper-ties for energy scavenging fabric devicesrdquo Journal of Mechanicsof Materials and Structures vol 11 no 3 pp 231ndash243 2016
[8] J Kaplunov and A Nobili ldquoMulti-parametric analysis ofstrongly inhomogeneous periodic waveguideswith internal cut-off frequenciesrdquoMathematical Methods in the Applied Sciences2016
[9] W-M Zhang H Yan Z-K Peng and G Meng ldquoElectrostaticpull-in instability in MEMSNEMS a reviewrdquo Sensors andActuators A Physical vol 214 pp 187ndash218 2014
[10] Y Zhang and Y-P Zhao ldquoStatic study of cantilever beamstiction under electrostatic force influencerdquo Acta MechanicaSolida Sinica vol 17 no 2 pp 104ndash112 2004
[11] L C Wei A B Mohammad and N M Kassim ldquoAnalyticalmodeling for determination of pull-in voltage for an electro-static actuated MEMS cantilever beamrdquo in Proceedings of the
6 Modelling and Simulation in Engineering
5th IEEE International Conference on Semiconductor Electronics(ICSE rsquo02) pp 233ndash238 IEEE Penang Malaysia December2002
[12] 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
[13] 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
[14] A Ramezani A Alasty and J Akbari ldquoClosed-form solutionsof the pull-in instability in nano-cantilevers under electrostaticand intermolecular surface forcesrdquo International Journal ofSolids and Structures vol 44 no 14-15 pp 4925ndash4941 2007
[15] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[16] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics and AppliedMathematics vol 50 no 3 pp 435ndash456 1997
[17] R Luciano and J R Willis ldquoHashin-Shtrikman based FEanalysis of the elastic behaviour of finite random compositebodiesrdquo International Journal of Fracture vol 137 no 1ndash4 pp261ndash273 2006
[18] A Nobili E Radi and L Lanzoni ldquoA cracked infinite Kirchhoffplate supported by a two-parameter elastic foundationrdquo Journalof the European Ceramic Society vol 34 no 11 pp 2737ndash27442014
[19] L Lanzoni and E Radi ldquoThermally induced deformations in apartially coated elastic layerrdquo International Journal of Solids andStructures vol 46 no 6 pp 1402ndash1412 2009
[20] V Guidi L Lanzoni and A Mazzolari ldquoPatterning and mod-eling of mechanically bent silicon plates deformed throughcoactive stressesrdquoThin Solid Films vol 520 no 3 pp 1074ndash10792011
[21] N Tullini A Tralli and L Lanzoni ldquoIntefacial shear stressanalysis of bar and thin film bonded to 2D elastic substrateusing a coupled FE-BIE methodrdquo Finite Elements in Analysisand Design vol 55 pp 42ndash51 2012
[22] C H Mastrangelo Suppression of Stiction in MEMS vol 605 ofMRS Proceedings CambridgeUniversity Press Cambridge UK1999
[23] W Merlijn Van Spengen R Puers and I De Wolf ldquoA physicalmodel to predict stiction in MEMSrdquo Journal of Micromechanicsand Microengineering vol 12 no 5 pp 702ndash713 2002
[24] Z Yapu ldquoStiction and anti-stiction in MEMS and NEMSrdquo ActaMechanica Sinica vol 19 no 1 pp 1ndash10 2003
[25] G Xotta GMazzucco V A Salomoni C EMajorana andK JWillam ldquoComposite behavior of concrete materials under hightemperaturesrdquo International Journal of Solids and Structures vol64 pp 86ndash99 2015
[26] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashA Solids vol 46 pp 84ndash95 2014
[27] A Nobili ldquoOn the generalization of the Timoshenko beammodel based on the micropolar linear theory static caserdquoMathematical Problems in Engineering vol 2015 Article ID914357 8 pages 2015
[28] G Napoli and A Nobili ldquoMechanically induced Helfrich-Hurault effect in lamellar systemsrdquo Physical Review E vol 80no 3 Article ID 031710 2009
[29] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2nd edition 2000
[30] F Greco and R Luciano ldquoA theoretical and numerical stabilityanalysis for composite micro-structures by using homogeniza-tion theoryrdquo CompositesmdashPart B Engineering vol 42 no 3 pp382ndash401 2011
[31] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1st edition 1995
[32] P L Sachdev Nonlinear Ordinary Differential Equations andTheir Applications Marcel Dekker 1991
[33] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Springer 1992
[34] R Barretta L Feo R Luciano and F Marotti de SciarraldquoVariational formulations for functionally graded nonlocalBernoulli-Euler nanobeamsrdquo Composite Structures vol 129 pp80ndash89 2015
[35] R Barretta L Feo R Luciano and F Marotti de SciarraldquoA gradient Eringen model for functionally graded nanorodsrdquoComposite Structures vol 131 pp 1124ndash1131 2015
[37] R Barretta and R Luciano ldquoExact solutions of isotropicviscoelastic functionally graded Kirchhoff platesrdquo CompositeStructures vol 118 no 1 pp 448ndash454 2014
[38] A Apuzzo R Barretta and R Luciano ldquoSome analyticalsolutions of functionally graded Kirchhoff platesrdquo CompositesPart B Engineering vol 68 pp 266ndash269 2015
[39] L Dezi G Menditto and A M Tarantino ldquoHomogeneousstructures subjected to repeated structural system changesrdquoJournal of Engineering Mechanics vol 116 no 8 pp 1723ndash17321990
[40] L Dezi and A M Tarantino ldquoTime-dependent analysis ofconcrete structures with a variable structural systemrdquo ACIMaterials Journal vol 88 no 3 pp 320ndash324 1991
[41] L Dezi G Menditto and A M Tarantino ldquoViscoelastic het-erogeneous structures with variable structural systemrdquo Journalof Engineering Mechanics vol 119 no 2 pp 238ndash250 1993
[42] R Barretta L Feo and R Luciano ldquoTorsion of functionallygraded nonlocal viscoelastic circular nanobeamsrdquo CompositesPart B Engineering vol 72 pp 217ndash222 2015
Figure 7 Line load density 119902119890 on the cantilever beam near pull-inaccording to the FE approximation (119860 = 12)
20
15
10
5
(i)
00 02 04 06 08 10
120585
Figure 8 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 20)
the Casimir force Results may be especially relevant for can-tilever micro-switches inMEMS It is shown that the pseudo-spectral method compares very favorably with an equivalentFinite Element approximation equivalency being constitutedby an equal number of degrees of freedom in the methods Inparticular good approximation for the cantilever deflectionnear its tip is crucial to capturing the threshold voltage onthe verge of pull-in Indeed poor approximation leads to veryunphysical oscillatory attractionrepulsion forces along thecantilever It is further shown that the oscillatory behavioris not restricted to the near pull-in regime Finally it isemphasized that both methods exactly satisfy the boundaryconditions (BCs) It is remarked that the present analysis canbe extended to incorporate functionally graded cantilevers[34ndash36] and beam-plates [37 38] or to include viscoelasticeffects [39ndash42]
Competing Interests
The authors declare that they have no competing interests
12
10
8
6
4
(i)
00 02 04 06 08 10
120585
Figure 9 Line load density 119902119890 on the cantilever beam near pull-inaccording to the pseudo-spectral approximation (119860 = 12)
References
[1] V A Salomoni C E Majorana G M Giannuzzi and AMiliozzi ldquoThermal-fluid flow within innovative heat storageconcrete systems for solar power plantsrdquo International Journalof Numerical Methods for Heat and Fluid Flow vol 18 no 7-8pp 969ndash999 2008
[2] A Nobili L Lanzoni and A M Tarantino ldquoExperimentalinvestigation and monitoring of a polypropylene-based fiberreinforced concrete road pavementrdquo Construction and BuildingMaterials vol 47 pp 888ndash895 2013
[3] V A Salomoni C E Majorana B Pomaro G Xotta andF Gramegna ldquoMacroscale and mesoscale analysis of concreteas a multiphase material for biological shields against nuclearradiationrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 38 no 5 pp 518ndash535 2014
[4] G Dinelli G Belz C E Majorana and B A SchreflerldquoExperimental investigation on the use of fly ash for lightweightprecast structural elementsrdquo Materials and Structures vol 29no 194 pp 632ndash638 1996
[5] P Bisegna and R Luciano ldquoBounds on the overall properties ofcomposites with debonded frictionless interfacesrdquoMechanics ofMaterials vol 28 no 1ndash4 pp 23ndash32 1998
[6] A Nobili and A M Tarantino ldquoMagnetostriction of a hardferromagnetic and elastic thin-film structurerdquoMathematics andMechanics of Solids vol 13 no 2 pp 95ndash123 2008
[7] A Kudaibergenov A Nobili and L Prikazchikova ldquoOn low-frequency vibrations of a composite stringwith contrast proper-ties for energy scavenging fabric devicesrdquo Journal of Mechanicsof Materials and Structures vol 11 no 3 pp 231ndash243 2016
[8] J Kaplunov and A Nobili ldquoMulti-parametric analysis ofstrongly inhomogeneous periodic waveguideswith internal cut-off frequenciesrdquoMathematical Methods in the Applied Sciences2016
[9] W-M Zhang H Yan Z-K Peng and G Meng ldquoElectrostaticpull-in instability in MEMSNEMS a reviewrdquo Sensors andActuators A Physical vol 214 pp 187ndash218 2014
[10] Y Zhang and Y-P Zhao ldquoStatic study of cantilever beamstiction under electrostatic force influencerdquo Acta MechanicaSolida Sinica vol 17 no 2 pp 104ndash112 2004
[11] L C Wei A B Mohammad and N M Kassim ldquoAnalyticalmodeling for determination of pull-in voltage for an electro-static actuated MEMS cantilever beamrdquo in Proceedings of the
6 Modelling and Simulation in Engineering
5th IEEE International Conference on Semiconductor Electronics(ICSE rsquo02) pp 233ndash238 IEEE Penang Malaysia December2002
[12] 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
[13] 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
[14] A Ramezani A Alasty and J Akbari ldquoClosed-form solutionsof the pull-in instability in nano-cantilevers under electrostaticand intermolecular surface forcesrdquo International Journal ofSolids and Structures vol 44 no 14-15 pp 4925ndash4941 2007
[15] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[16] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics and AppliedMathematics vol 50 no 3 pp 435ndash456 1997
[17] R Luciano and J R Willis ldquoHashin-Shtrikman based FEanalysis of the elastic behaviour of finite random compositebodiesrdquo International Journal of Fracture vol 137 no 1ndash4 pp261ndash273 2006
[18] A Nobili E Radi and L Lanzoni ldquoA cracked infinite Kirchhoffplate supported by a two-parameter elastic foundationrdquo Journalof the European Ceramic Society vol 34 no 11 pp 2737ndash27442014
[19] L Lanzoni and E Radi ldquoThermally induced deformations in apartially coated elastic layerrdquo International Journal of Solids andStructures vol 46 no 6 pp 1402ndash1412 2009
[20] V Guidi L Lanzoni and A Mazzolari ldquoPatterning and mod-eling of mechanically bent silicon plates deformed throughcoactive stressesrdquoThin Solid Films vol 520 no 3 pp 1074ndash10792011
[21] N Tullini A Tralli and L Lanzoni ldquoIntefacial shear stressanalysis of bar and thin film bonded to 2D elastic substrateusing a coupled FE-BIE methodrdquo Finite Elements in Analysisand Design vol 55 pp 42ndash51 2012
[22] C H Mastrangelo Suppression of Stiction in MEMS vol 605 ofMRS Proceedings CambridgeUniversity Press Cambridge UK1999
[23] W Merlijn Van Spengen R Puers and I De Wolf ldquoA physicalmodel to predict stiction in MEMSrdquo Journal of Micromechanicsand Microengineering vol 12 no 5 pp 702ndash713 2002
[24] Z Yapu ldquoStiction and anti-stiction in MEMS and NEMSrdquo ActaMechanica Sinica vol 19 no 1 pp 1ndash10 2003
[25] G Xotta GMazzucco V A Salomoni C EMajorana andK JWillam ldquoComposite behavior of concrete materials under hightemperaturesrdquo International Journal of Solids and Structures vol64 pp 86ndash99 2015
[26] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashA Solids vol 46 pp 84ndash95 2014
[27] A Nobili ldquoOn the generalization of the Timoshenko beammodel based on the micropolar linear theory static caserdquoMathematical Problems in Engineering vol 2015 Article ID914357 8 pages 2015
[28] G Napoli and A Nobili ldquoMechanically induced Helfrich-Hurault effect in lamellar systemsrdquo Physical Review E vol 80no 3 Article ID 031710 2009
[29] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2nd edition 2000
[30] F Greco and R Luciano ldquoA theoretical and numerical stabilityanalysis for composite micro-structures by using homogeniza-tion theoryrdquo CompositesmdashPart B Engineering vol 42 no 3 pp382ndash401 2011
[31] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1st edition 1995
[32] P L Sachdev Nonlinear Ordinary Differential Equations andTheir Applications Marcel Dekker 1991
[33] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Springer 1992
[34] R Barretta L Feo R Luciano and F Marotti de SciarraldquoVariational formulations for functionally graded nonlocalBernoulli-Euler nanobeamsrdquo Composite Structures vol 129 pp80ndash89 2015
[35] R Barretta L Feo R Luciano and F Marotti de SciarraldquoA gradient Eringen model for functionally graded nanorodsrdquoComposite Structures vol 131 pp 1124ndash1131 2015
[37] R Barretta and R Luciano ldquoExact solutions of isotropicviscoelastic functionally graded Kirchhoff platesrdquo CompositeStructures vol 118 no 1 pp 448ndash454 2014
[38] A Apuzzo R Barretta and R Luciano ldquoSome analyticalsolutions of functionally graded Kirchhoff platesrdquo CompositesPart B Engineering vol 68 pp 266ndash269 2015
[39] L Dezi G Menditto and A M Tarantino ldquoHomogeneousstructures subjected to repeated structural system changesrdquoJournal of Engineering Mechanics vol 116 no 8 pp 1723ndash17321990
[40] L Dezi and A M Tarantino ldquoTime-dependent analysis ofconcrete structures with a variable structural systemrdquo ACIMaterials Journal vol 88 no 3 pp 320ndash324 1991
[41] L Dezi G Menditto and A M Tarantino ldquoViscoelastic het-erogeneous structures with variable structural systemrdquo Journalof Engineering Mechanics vol 119 no 2 pp 238ndash250 1993
[42] R Barretta L Feo and R Luciano ldquoTorsion of functionallygraded nonlocal viscoelastic circular nanobeamsrdquo CompositesPart B Engineering vol 72 pp 217ndash222 2015
5th IEEE International Conference on Semiconductor Electronics(ICSE rsquo02) pp 233ndash238 IEEE Penang Malaysia December2002
[12] 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
[13] 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
[14] A Ramezani A Alasty and J Akbari ldquoClosed-form solutionsof the pull-in instability in nano-cantilevers under electrostaticand intermolecular surface forcesrdquo International Journal ofSolids and Structures vol 44 no 14-15 pp 4925ndash4941 2007
[15] A M Tarantino ldquoOn the finite motions generated by a modeI propagating crackrdquo Journal of Elasticity vol 57 no 2 pp 85ndash103 1999
[16] A M Tarantino ldquoNonlinear fracture mechanics for an elasticBell materialrdquo The Quarterly Journal of Mechanics and AppliedMathematics vol 50 no 3 pp 435ndash456 1997
[17] R Luciano and J R Willis ldquoHashin-Shtrikman based FEanalysis of the elastic behaviour of finite random compositebodiesrdquo International Journal of Fracture vol 137 no 1ndash4 pp261ndash273 2006
[18] A Nobili E Radi and L Lanzoni ldquoA cracked infinite Kirchhoffplate supported by a two-parameter elastic foundationrdquo Journalof the European Ceramic Society vol 34 no 11 pp 2737ndash27442014
[19] L Lanzoni and E Radi ldquoThermally induced deformations in apartially coated elastic layerrdquo International Journal of Solids andStructures vol 46 no 6 pp 1402ndash1412 2009
[20] V Guidi L Lanzoni and A Mazzolari ldquoPatterning and mod-eling of mechanically bent silicon plates deformed throughcoactive stressesrdquoThin Solid Films vol 520 no 3 pp 1074ndash10792011
[21] N Tullini A Tralli and L Lanzoni ldquoIntefacial shear stressanalysis of bar and thin film bonded to 2D elastic substrateusing a coupled FE-BIE methodrdquo Finite Elements in Analysisand Design vol 55 pp 42ndash51 2012
[22] C H Mastrangelo Suppression of Stiction in MEMS vol 605 ofMRS Proceedings CambridgeUniversity Press Cambridge UK1999
[23] W Merlijn Van Spengen R Puers and I De Wolf ldquoA physicalmodel to predict stiction in MEMSrdquo Journal of Micromechanicsand Microengineering vol 12 no 5 pp 702ndash713 2002
[24] Z Yapu ldquoStiction and anti-stiction in MEMS and NEMSrdquo ActaMechanica Sinica vol 19 no 1 pp 1ndash10 2003
[25] G Xotta GMazzucco V A Salomoni C EMajorana andK JWillam ldquoComposite behavior of concrete materials under hightemperaturesrdquo International Journal of Solids and Structures vol64 pp 86ndash99 2015
[26] FMarotti de Sciarra andM Salerno ldquoOn thermodynamic func-tions in thermoelasticity without energy dissipationrdquo EuropeanJournal of MechanicsmdashA Solids vol 46 pp 84ndash95 2014
[27] A Nobili ldquoOn the generalization of the Timoshenko beammodel based on the micropolar linear theory static caserdquoMathematical Problems in Engineering vol 2015 Article ID914357 8 pages 2015
[28] G Napoli and A Nobili ldquoMechanically induced Helfrich-Hurault effect in lamellar systemsrdquo Physical Review E vol 80no 3 Article ID 031710 2009
[29] J P Boyd Chebyshev and Fourier Spectral Methods DoverPublications 2nd edition 2000
[30] F Greco and R Luciano ldquoA theoretical and numerical stabilityanalysis for composite micro-structures by using homogeniza-tion theoryrdquo CompositesmdashPart B Engineering vol 42 no 3 pp382ndash401 2011
[31] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1st edition 1995
[32] P L Sachdev Nonlinear Ordinary Differential Equations andTheir Applications Marcel Dekker 1991
[33] D Funaro Polynomial Approximation of Differential Equationsvol 8 of Lecture Notes in Physics Springer 1992
[34] R Barretta L Feo R Luciano and F Marotti de SciarraldquoVariational formulations for functionally graded nonlocalBernoulli-Euler nanobeamsrdquo Composite Structures vol 129 pp80ndash89 2015
[35] R Barretta L Feo R Luciano and F Marotti de SciarraldquoA gradient Eringen model for functionally graded nanorodsrdquoComposite Structures vol 131 pp 1124ndash1131 2015
[37] R Barretta and R Luciano ldquoExact solutions of isotropicviscoelastic functionally graded Kirchhoff platesrdquo CompositeStructures vol 118 no 1 pp 448ndash454 2014
[38] A Apuzzo R Barretta and R Luciano ldquoSome analyticalsolutions of functionally graded Kirchhoff platesrdquo CompositesPart B Engineering vol 68 pp 266ndash269 2015
[39] L Dezi G Menditto and A M Tarantino ldquoHomogeneousstructures subjected to repeated structural system changesrdquoJournal of Engineering Mechanics vol 116 no 8 pp 1723ndash17321990
[40] L Dezi and A M Tarantino ldquoTime-dependent analysis ofconcrete structures with a variable structural systemrdquo ACIMaterials Journal vol 88 no 3 pp 320ndash324 1991
[41] L Dezi G Menditto and A M Tarantino ldquoViscoelastic het-erogeneous structures with variable structural systemrdquo Journalof Engineering Mechanics vol 119 no 2 pp 238ndash250 1993
[42] R Barretta L Feo and R Luciano ldquoTorsion of functionallygraded nonlocal viscoelastic circular nanobeamsrdquo CompositesPart B Engineering vol 72 pp 217ndash222 2015
