Nonlinear Oscillations Of Viscoelastic Microcantilever Beam . - Sharif
Scientia Iranica B (2021) 28(2), 785{794Sharif University of TechnologyScientia IranicaTransactions B: Mechanical ear oscillations of viscoelastic microcantileverbeam based on modi ed strain gradient theoryF. Taheran, M.T. Ahmadian , and K. FiroozbakhshSchool of Mechanical Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9567, Iran.Received 29 July 2019; received in revised form 18 December 2019; accepted 20 April 2020KEYWORDSMicrocantilever beam;Multiple time scalemethod;Nonlinear freevibration;Strain gradienttheory;Viscoelastic material.Abstract. A viscoelastic microcantilever beam is analytically analyzed based on themodi ed strain gradient theory. Kelvin-Voigt scheme is used to model beam viscoelasticity.By applying Euler-Bernoulli inextensibility of the centerline condition based on Hamilton'sprinciple, the nonlinear equation of motion and the related boundary conditions arederived from shortening e ect theory and discretized by Galerkin method. Inner damping,nonlinear curvature e ect, and nonlinear inertia terms are also taken into account. Inthe present study, the generalized derived formulation allows modeling any nonlinearcombination such as nonlinear terms that arise due to inertia, damping, and sti ness, as wellas modeling the size e ect using modi ed coupled stress or modi ed strain gradient theories.First-mode nonlinear frequency and time response of the viscoelastic microcantileverbeam are analytically evaluated using multiple time scale method and then, validatedthrough numerical ndings. The obtained results indicate that nonlinear terms have anappreciable e ect on natural frequency and time response of a viscoelastic microcantilever.Moreover, further investigations suggest that due to the size e ects, natural frequencywould drastically increase, especially when the thickness of the beam and the lengthscale parameter are comparable. The ndings elaborate the signi cance of size e ectsin analyzing the mechanical behavior of small-scale structures. 2021 Sharif University of Technology. All rights reserved.1. IntroductionMicrostructures have considerably drawn researchers'attention in recent years due to their wide variety ofapplications. Among di erent types of microstructures,microbeams are highlighted due to their numerousindustrial applications such as Microelectromechanical Systems (MEMS), capacitors, resonators, microswitches, atomic force microscopes, and biosensors.*. Corresponding author. Tel.: 98 21 66165503;Fax: 98 21 66000021E-mail address: Ahmadian@sharif.edu (M.T. Ahmadian)doi: 10.24200/sci.2020.54137.3612Nevertheless, it is not surprising that the investigatorfocuses on the dynamical analysis of microbeams [1].For instance, Ghommem and Abdelke [2] developeda nonlinear reduced-order model of an electricallyactuated microcantilever beam with a tip mass usingperturbation methods. They employed the resultantmodel as a resonant sensor for bio-mass detectionand sensing, analyzed the nonlinear dynamics ande ectiveness of the bio-mass sensor, and quanti ed themass of biological entities.On the contrary, numerous experimental resultshave proved the incapability of the classical theoryto accurately predict the mechanical behavior of microscale beams. Studies on microscale structuresshowed that the classical theory could only estimate
786F. Taheran et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 785{794lower sti ness values, unlike experimental ndings.The observations also indicated that the normalizeddeformation of microstructures depended on the size ofthe structure [3]. Due to the failure of classical theoryin justifying and explaining the mechanical response ofmicroscale structure, researchers proposed several nonclassical methods including Modi ed Strain Gradient(MSG) and Modi ed Coupled Stress (MCS) theories[4,5]. Mindlin [4] considered strain besides the rst andsecond strain derivatives to formulate strain gradienttheory. Fleck and Hutchinson [6] ignored the e ect ofthe second derivative of the strain and reformulatedand simpli ed the strain gradient theory. Lam et al.[7] modi ed the formulation derived from Fleck andHutchinson's study [6] by employing a non-classicalequilibrium equation (i.e., the equilibrium equationof the moment of couples) as well as the classicalequilibrium equations (i.e., equilibrium equations offorces and moment of forces) [8].A number of researchers utilized the strain gradient or the MCS theories to study the mechanicalbehavior of the microscale structures [9{13]. Chenand Li [14] studied buckling and post-buckling ofthe imperfect microbeam, analytically considering themid-plane stretching for di erent boundary conditions.They examined the size e ects through the modi edcoupled stress theory on the critical buckling, staticresponse, and dynamic properties of imperfect microbeams. Akg oz and Civalek [15] derived formulations to bend the linear elastic microbeam based onMSG theory for di erent boundary conditions andanalytically solved them. In addition, they studiedbuckling of an axially loaded microbeam and comparedthe calculated critical buckling load, which was basedon the modi ed strain gradient theory, via the MCStheory [16]. In another research, Miandoab et al.[17] calculated the length scale parameters along withYoung's modulus of polysilicone based on the MSGtheory by tting the experimental reported results ofstatic pull-in voltages to the predicted ones.Recently, the size e ects on nonlinear vibrationof viscoelastic microstructures have gained signi cance [18{22] since viscoelastic materials such as Polydimethylsiloxane (PDMS) [23] have been developedand used in manufactured microbeam biosensors dueto their compliance and biocompatibility [24]. Li etal. [25,26] formulated the governing equation of wavemotion of uid-conveying viscoelastic single-walled carbon nanotubes based on the nonlocal strain gradienttheory. They discussed the importance of nonlocal aswell as small-scale material parameters in the dispersion relation between the phase velocity and the wavenumber under magnetic eld. Attia and Mohamed [27]considered the size e ects based on the MCS theoryand derived a non-classical nonlinear continuum modelof an electrically actuated viscoelastic microbeams fordi erent boundary conditions. Their model accountsfor the nonlinearities due to axial residual stress,geometric nonlinearities, and electrical forcing withfringing e ect. They modeled the viscoelasticity ofthe microbeam using Maxwell scheme and solved theresulting equations by the di erential/integral quadrature model. In another research, they examined thee ects of viscoelastic relaxation time, material lengthscale parameter, and size on the bending response ofthe Euler-Bernoulli nanobeams using nonlocal couplestress model [28]. They also studied free vibrationof Functionally Graded (FG) viscoelastic nanobeamsand investigated the e ects of di erent parameters,including Poisson e ect, surface elasticity, materiallength-scale, and material damping gradient index.They assessed the size e ects based on MCS theoryand formulated the viscoelasticity e ect via the KelvinVoigt model [29].The oscillation of viscoelastic micro- and nanoscale Timoshenko beams based on fractional KelvinVoigt model was studied by Ansari et al. [30]. Theyemployed Galerkin and predictor-corrector methodsto analyze the impacts of length scale parameters,viscoelasticity coe cient, and fractional-order. Thenonlinear dynamic stability of a viscoelastic microbeamwas analyzed by Fu et al. [31]. They used numericalmethods to solve the formulations and investigatethe e ects of creep quantity, geometric nonlinearity,environmental, and inner damping with symmetricelectrostatic load on the principal region of instability. However, they did not take the length scaleparameters into account. Fu and Zhang [32] examinedthe electromechanical dynamic buckling behavior ofa viscoelastic microbeam. They also discussed thee ects of length scale parameters via MCS theoryto analyze the pull-in phenomenon of viscoelasticmicrobeams with electronic actuation [33]. Ghayeshand Farokhi studied the dynamics of Euler-Bernoulliviscoelastic microcantilevers such as stability and bifurcations [34,35], internal resonances and modal interactions [36], and large-amplitude oscillation [37,38]and investigated the resonant response of an imperfectTimoshenko microbeam considering extensibility [39].They employed the MCS theory to evaluate the e ectsof length scale size in these studies and numericallysolved equations. In another study, Ghayesh [40]numerically analyzed the nonlinear mechanics of anelastically supported viscoelastic FG microcantileverconsidering energy transfer via internal resonance andmotion complexity. He also employed the MCS theoryto investigate the size e ects.A review of the studies on the vibration ofnonlinear microcantilevers clearly shows that straingradient size e ects are not considered in modelingthe nonlinear viscoelastic microcantilevers and theresultant formulations of full nonlinear models based on
F. Taheran et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 785{794the MCS theory are numerically solved. In the presentstudy, the general nonlinear equation of motion for aviscoelastic microcantilever beam was expressed basedon the modi ed strain gradient theory representedby Lam et al. [7] and solved analytically for therst time. The nonlinear curvature e ect and inertiaterms were also taken into account while modelingthe shortening e ect. The equation was analyticallysolved through multiple time scale method, and timevariant nonlinear rst-mode frequency of the microcantilever was consequently derived. According tothe obtained results, the signi cance of size e ectson the vibration amplitude, damping time, and natural frequency of microcantilever became more highlighted.2. Model development and analytical solutionA typical rectangular beam of width b, thickness h, andlength L was considered in this study. For a nonlinearcantilever, the elongation of an element with length ds(as depicted in Figure 1) can be de ned as follows [41]:qds ds(1)e (1 u0 )2 v0 2 1;dswhere u and v are the longitudinal and lateral deections of the cantilever, respectively, prime denotesderivative to x, and z represents the distance of thecross-section from the neutral axis.An inextensible beam was taken into account tomodel the shortening e ect [41]:p1e 0 ) u0 1 v0 2 1 v022) u s0x 12 v02 dx;(2)Referring to Figure 1 [41]:v0tan :(3)1 u0787Di erentiating Eq. (3) and using Eq. (2) wouldyield [41]:v00 (1 u0 ) v0 u00) 0 v00 12 v00 v02 :(4) 0 (1 u0 )2 v0 2Hamilton's principle was employed to derive theequation of motion, according to which the kinetic andpotential energy of vibrating homogenous microcantilever with a cross-section A (i.e., A bh) and density could be calculated.The kinetic energy of the microcantilever can beobtained as shown in the following [41]:1 L 2 2(5)s A u v dx:2 0By substituting u into v0 from Eq. (2) in Eq. (5), thefollowing equation can be obtained [41]:T ( )1@ x 1 02 2T s0L As v dx Av 2 dx:2@t 0 2(6)The density of strain energy of a linear elasticsolid based on the MSG theory derived by Lam et al.[7] can be expressed as follows:(1) (1)% ij "ij pi i ijk ijk msij sij ;(7)where the components of the classical strain tensor("), dilatation gradient vector ( ), deviatoric stretchgradient tensor ( (1) ), and symmetric part of therotation gradient tensor ( s ) are de ned as [7]:1"ij (ui;j uj;i ) ;2@i ("11 "22 "33 );@i1(1) ijk ("ij;k "jk;i "ki;j )3(8)(9) 1 (" 2"mk;m )15 ij mm;k jk ("mm;i 2"mi;m ) ki ("mm;j 2"mj;m ) ;Figure 1. Coordinate system of a de ectedmicrocantilever.(10)1(11)sij (eipq "qj;p ejpq "qi;p ) ;2where ui presents the components of the displacementvector u; and ij and eijk denote Kronecker delta andalternating tensor, respectively. For the viscoelasticmaterial, the classical Kelvin-Voigt model is used. Inthis regard, the stress tensor ( ) and the correspondinghigher-order stresses (p; (1) ; ms ) relations are:
788F. Taheran et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 785{794 tot E" C " vis ;(12)ptot p pvis 2 k l02 2 C l02 ;(13)(1)(1)tot (1) vis 2 k l12 (1) 2 C l02 (1) ;(14) K xvx ; v ; LLsEI; AL4 KC CIL2 K; SC CI S;S 1 K 26l2 01 h1l2 02(1 )L 120 l1225 h 4 l15 L 2 2 ! l2h 2 ! @3@ x 2 @ "@ @ v x ss s @ x @ x 1 0 S @ v @ 2 v @ 3 v @ x @ x 2 @ x 3@2@ x @ @2@ x @ @ @ v @ 2 v @ 4 v @ x @ x @ x 2 @ x 4 0"@ v @ 2 v @ 4 v @ x @ x 2 @ x 4 @ 3 @ v @ @ v @ 2 v K @ x 3 @ x @ x @ x @ x 2@ 2 @ v @ 2 v @ 3 v @ x 2 @ x @ x 2 @ x 3 @ v @ @ v @ 2 v @ x @ x @ x @ x 2@@ 2 v @ v @ 3 v @ x @ x @ @ x @ x 3 (19)( v000 v 000 v 0 2 v 0 v 00 2 ) v j10 0(v 00 v 00 v 0 2 ) v 0 j10 0#m 1;'m ( x) cosh (zm x ) cos (zm x ) ;(18)where l is the area moment of inertia. The Hamiltonprinciple can be used to achieve the normalized formof the nonlinear di erential equation of motion andthe related boundary conditions. It can be observedthat due to the coupling e ects of large deformation,strain gradient theory, and viscoelasticity, di erentnonlinearities are incorporated.@ 5 v @ 7 v @ 6 v @ 2 v @ 4 v S 4 S 4 K 6K 62@@ x @ x @@ x @@ x @ @ v @ @ v @ 2 v S @ x @ x @ x @ x @ x 2 @ v @ @ v @ 2 v @ x @ x @ x @ x 20 0 x 0!v v ) x 1 ! v 00 v 000 0 :(20)The Galerkin method was employed so that thefollowing mode shapes could be considered for thecantilever beam:1Xv 'm ( x) qm ( ) ;(21) !C ; ;E2 K EIL2 K;S EI S; 2 t; @4@ x 3 @@@ 2 v @ v @ 5 v @ x @ x @ @ x @ x 5mstot ms msvis 2 k l22 s 2 C l22 s ;(15)where E is the elastic modulus, C is the dampingcoe cient, and l0 ; l1 ; l2 are new material constantsknown as length scale parameters. Moreover, k and C are elastic and viscoelastic shear moduli related toPoisson's ratio ( ), as shown in the following:CE k ; C :(16)2(1 )2(1 )The variation of works in terms of nonconservative forces can be formulated as: Wvis s ( vis " pvis vis mvis )dV:(17)The dimensionless parameters presented below areutilized to derive the normalized form of the di erentialequations: !# @ 2 v 2 @ v @ 3 v dsds@ x @@ x @ x @ 2cos (zm ) cosh (zm )(sin (zm x ) sinh (zm x )) ;sin (zm ) sinh (zm )(22)cos(zm ) cosh(zm ) 1 0:(23)The shape functions are normalized with the orthogonality condition of Eq. (24): 'm ( x); 'n ( x) Z 10'm ( x)'n ( x)dx mn : (24)By substituting Eq. (21) into Eq. (19) and takingthe inner product of the resulting equation with '1(simply mentioned as '), the rst-mode shape timedependent equation can be expressed as the followingdi erential equation:2 g 2q 2 !q !2 q g1 q3 g2 qq 2 g3 q2 q 4 qq! 0;(25)where the coe cients ! and gi can be obtained as:
F. Taheran et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 785{794 1!2 Sa 2;Ka 4g1 Sa 5;Ka"1 !g2 q0g2 g3 a3 ; 6g4 Sa 7;Ka(26)a1 a2 a3 a4 a5 0Z 100Z 10Z 10Z sZ s10 020' dsds dx ;hidx ;a7 3a50Z 10hi' 2'0 '00 '000 '0 2 'IV dx ;hi' 2'0 '00 'V '0 2 'V I dx :q 2" !q !2 q "N [q( )] 0;i!T0 :(33)!2 A3 e3i!T0(28)q02@q0 i!T0 CC ; (34) (i!) A3 e3i!T0 A2 Ae@T0!2 i!T0 CC ;A3 e3i!T0 3A2 Aewhere CC denotes Complex Conjugate. By substituting Eq. (34) into Eq. (32), we have: !2 (g2 g3 ) 2i g4 A3 e3i!T0g13g1 ! (g2 3g3 ) 2i g4 2A A ei!T0 CC:(35)In case the secular terms of Eq. (35) are equal to zero,q1 can be calculated as:!2 (g2 g3 ) 2i g4 3 3i!T0Ae CC: (36)8!2If the polar form of A is used to omit secular terms ofEq. (35), we have:gq1 11(T1 ) ei (T1 ) !2(0 ! g4 3 0 )4!@ 3g1 !2 (g2 3g3 ) 2 :@T18!A (30)(31) i!T0 CC ;A2 Ae@ 2 q0 @T02(27)where T0 and T1 " are fast and slow time scales,respectively, that characterize the motions happeningat the natural frequencies and related shifts due tononlinearities. Then, in terms of the power of ",Eq. (28) can be rewritten as follows:@ 2 q0 ! 2 q0 0;@T02 @q0 2 @T0 02 2i!A 22i ! A where N is the summation of nonlinear terms:2 g 2N g1 q3 g2 qq 2 g3 q2 q 4 qq:(29)!To calculate the periodic solution of q, the expansion in terms of " can be written as:q q0 (T0 ; T1 ) "q1 (T0 ; T1 ) ; @ 2 q1 2 ! q1 @T02The multiscale perturbation method was employed to solve Eq. (25). For weak nonlinearities inthe microcantilever beam, Eq. (25) can be rewrittenas:"0 !2 g4 2 @q0q: (32)! 0 @T0q02 ' '00 3 4'0 '00 '000 '0 2 'IV dx ;Z 1@ 2 q0@T02g1 q0 3While solving Eq. (31), q0 can be determined as follows:q0' '0 2 'V 4'0 '00 'IV 6'00 2 '000 3'0 '000 2a6 3a4g3 q02@q0@T0 i!T0 CC;q03 A3 e3i!T0 3A2 Ae''V I dx ;' '0 @q0 2@T02 !Therefore, we have:''IV dx ;Z 1 q0 A (T1 ) ei!T0 A (T1 ) ewhere ai is derived as Eq. (27):Z 12@ 2 q12 q 2 @ q0 !1@T02@T0 @T1789Calculating82 :andfrom Eq. (37) would result in:2 20 (4!2 g4 42!)e2 !"0 2 (g2 3g3 ) 3g1 !4 g42 !"(37)lng4 204!2 g4g 2 4 0 ln 4!2 :2 2 !"0 e!(38)
790F. Taheran et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 785{794By substituting Eq. (38) into Eq. (33) andEq. (36), with consideration of Eq. (30) and Eq. (21),the nonlinear frequency and response of a viscoelasticmicrocantilever based on the MSG theory may beexpressed as shown in the following:8!NL ! v ' :cos (!NL )3 32" !2 g1" g4 316!2sin (3!NL ) !2 (g2 g3 ) cos (3!NL )(39)Figure 3. Linear and nonlinear frequencies of aviscoelastic microcantilever.In the absence of damping, 0, the outcome ofEq. (37) and Eq. (38) would be:( 02 3g1 ! 8(!g2(40)3g3 ) 2 "0if l0 l1 l2 0, the obtained equation wouldbe the same as that reported by Nayfeh and Nayfeh [42]for elastic microcantilever via classical beam theory.8!NL ! v ' :cos (!NL ) " 332!2g1!2 (g2 g3 )cos (3!NL ) Figure 4. Linear and nonlinear maximum amplitudes offree vibration of a viscoelastic microcantilever.(41)3. Results and discussionIn this section, the rst-mode nonlinear frequencyand time response of the viscoelastic microcantileverbeam are investigated considering the dimensionlessparameters, namely 0:01, 0:17, !1 3:5160,and h l 5. Constants ai were evaluated for therst-mode shape of the microcantilever beam, and forMSG formulation, equal length scale parameters areconsidered (i.e., l0 l1 l2 l) [7,43].The numerical results obtained from Eq. (28) werecompared with the analytical ndings from Eq. (38)plotted in Figure 2. The outcomes indicate thatthe two-term expansion of Eq. (38) models the timeresponse fairly with the maximum error of 5%.Figure 2. Numerical and two-term expansion modelresponse of a viscoelastic microcantilever at the free end.Figure 5. Nonlinear frequency of viscoelasticmicrocantilever for di erent values of h l and ClassicalTheory (CT).The rst mode linear and nonlinear natural frequencies of a viscoelastic microcantilever with respectto the MSG theory are depicted in Figure 3 whichshows that nonlinear frequency approaches the linearone asymptotically over time. The dimensionless maximum amplitudes of linear and nonlinear vibrations arecompared the results of which are shown in Figure 4.As observed, the nonlinear phenomena have a loweramplitude at a xed frequency, implying that nonlineare ect causes higher strength in the structure.The e ects of length scale parameters on the rstmode frequency and the amplitude of viscoelastic microcantilever beam vibration are presented in Figures 5and 6. As the thickness of the microcantilever crosssection approaches the length scale parameters, thenonlinear frequency evaluated by MSG theory wouldincrease (Figure 5). The same phenomenon for this
F. Taheran et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 785{794Figure 6. Nonlinear maximum amplitude for di erentvalues of h l and Classical Theory (CT).791Figure 8. Maximum amplitude of nonlinear freevibration of a viscoelastic microcantilever by Modi edCoupled Stress (MCS) theory, Modi ed Strain Gradient(MSG) theory, and classical theory.more precisely than MCS theory, and MCS theorycould calculate it more accurately than MSG theory.4. ConclusionFigure 7. Nonlinear frequency of a viscoelasticmicrocantilever by Modi ed Coupled Stress (MCS) theory,Modi ed Strain Gradient (MSG) theory, and classicaltheory.amplitude is depicted in Figure 6. The results indicatedthat as thickness approached the length scale, thevibration amplitude drastically decreased. Moreover,the hardening e ect of considering the length scaleparameters on the microcantilever behavior would leadto higher nonlinear frequency of the microcantileverand lower amplitude of the vibration. Figures 5 and6 indicate that the e ect of the length scale increasedas the thickness-to-length scale parameter ratio of thebeam reduced; therefore, the di erence between theresults of the derived model and predictions of theclassical one cannot be ignored.In Figures 7 and 8, the frequencies and maximumamplitudes of viscoelastic microcantilever beam versustime are compared based on the modi ed strain gradient, MCS, and classical theories. It can be observedthat both MCS and MSG theories estimate higher sti ness for the beam than classical estimations; however,the modi ed strain gradient theory could improve thehardness much more than the MCS theory. In otherwords, the values of natural frequency calculated bythe MSG theory are higher than those modeled byMCS theory, and the values estimated by MCS theoryare higher than those estimated by the classical theory.However, the maximum amplitude of the free vibrationwas reversed, implying that the classical method couldcalculate the maximum amplitude of the vibrationA number of investigations have been conducted tomodel the nonlinear dynamics of microcantilevers.However, the size-dependent behavior of viscoelasticmicrocantilevers in the framework of strain gradienttheory has not been analytically studied so far. Inthis paper, the nonlinear dynamics of the viscoelastic Euler-Bernoulli microcantilever beam via modi edstrain gradient theory was investigated using centerlineinextensibility. To this end, the shortening e ect theorywas utilized to develop the equation of motion for themicrocantilever beam. The Kelvin-Voigt scheme wasalso employed to model the viscoelastic behavior of thebeam materials. The equation of motion was derivedusing Hamilton's principle, and the regular mode shapefunctions of a microcantilever were used to derive timeresponse equation through the Galerkin method.Through the perturbation method, i.e., the multiple time scales method, both nonlinear natural frequency and time response of the microcantilever wereanalytically obtained based on the modi ed strain gradient theory. The resultant formulations showed timedependency of the vibration amplitudes and nonlinearnatural frequencies. This formulation allows modelingany nonlinearity arising from inertia, damping, andsti ness and directly derives analytical solutions for therst time.The numerical results were measured to validatethe ndings. The rst-mode nonlinear frequency andmaximum amplitude of free vibration were comparedusing di erent methods such as the modi ed straingradient, modi ed couple stress, and classical theoriesfor various length scale parameters. The obtainedresults indicated that when the thickness of the beamwas in order of the material length scale, the di erencebetween the results of the classical and non-classicaltheories was considerable. However, as the thickness-
792F. Taheran et al./Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 785{794to-material length scale ratio of the beam increased,the results of the non-classical assumption converged tothose of the classical theory. These ndings highlightedthe signi cance of the size e ect in analyzing themechanical behavior of the small-scale structures.Acknowledgement(1) ijk c k ijThis paper is supported by INSF.(1)ijkNomenclature! AbCdsdVEeeijkhlLl0 ; l1 ; l2msijpiq( )TT0 ; T1uvxyzsij ij Wvis"ij'( x)iMicrobeam cross-section areaMicrobeam widthDamping coe cientMicrobeam element lengthMicrobeam element volumeElastic modulusMicrobeam element elongationComponents of alternating tensorMicrobeam thicknessArea moment of inertiaMicrobeam lengthLength scale parametersComponents of higher-order stress dueto the rotation gradient tensorComponents of higher-order stress dueto the dilatation gradient vectorThe time part of the microbeamresponseThe kinetic energy of the microbeamFast and Slow dimensionless timescalesLongitudinal de ection of themicrobeamLateral de ection of the microbeamDistance from the clamped end of themicrobeamThe lateral axis of coordinate systemThe distance of the cross-section fromthe neutral axisComponents of the symmetric part ofthe rotation gradient tensorComponents of Kronecker delta tensorVariation of works due to nonconservative forcesComponents of classical strain tensorShape function of the microbeamComponents of dilatation gradientvectorDotPrimeComponents of deviatoric stretchgradient tensorViscoelastic shear modulusElastic shear modulusMicrobeam material densityComponents of the classical stresstensorComponents of higher-order stress dueto deviatoric stretch gradient tensorPoisson's ratioDimensionless frequencyThe density of strain energy of themicrobeamDi erentiation with respect to timeDi erentiation with respect to xReferences1. Zand, M.M. and Ahmadian, M.T. \Application ofhomotopy analysis method in studying dynamic pullin instability of microsystems", Mechanics ResearchCommunications, 36(7), pp. 851{858 (2009).2. Ghommem, M. and Abdelke , A. \Nonlinear reducedorder modeling and e ectiveness of electricallyactuated microbeams for bio-mass sensing applications", International Journal of Mechanics and Materials in Design, 15(1), pp. 125{143 (2019).3. McFarland, A.W. and Colton, J.S. \Role of materialmicrostructure in plate sti ness with relevance tomicrocantilever sensors", Journal of Micromechanicsand Microengineering, 15(5), pp. 1060{1067 (2005).4. Mindlin, R.D. \Second gradient of strain and surfacetension in linear elasticity", International Journal ofSolids and Structures, 1(4), pp. 417{438 (1965).5. Koiter, W.T. \Couple stresses in the theory of elasticity, I and II", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series B, pp.6717{6744 (1964).6. Fleck, N.A. and Hutchinson, J.W. \A reformulation ofstrain gradient plasticity", Journal of the Mechanicsand Physics of Solids, 49(10), pp. 2245{2271 (2001).7. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., andTong, P. \Experiments and theory in strain gradientelasticity", Journal of the Mechanics and Physics ofSolids, 51(8), pp. 1477{1508 (2003).8. Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T.,and Firoozbakhsh, K. \Strain gradient formulation offunctionally graded nonlinear beams", InternationalJournal of Engineering Science, 65, pp. 49{63 (2013).9. Rahaeifard, M., Ahmadian, M., and Firoozbakhsh, K.\Vibration analysis of electrostatically actuated nonlinear microbridges based on the modi ed couple stresstheory", Applied Mathematical Modelling, 39(21), pp.6694{6704 (2015).
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Nonlinear oscillations of viscoelastic microcantilever beam based on modi ed strain gradient theory . nonlinear curvature e ect, and nonlinear inertia terms are also taken into account. In the present study, the generalized derived formulation allows modeling any nonlinear . Introduction Microstructures have considerably drawn researchers' .
Introduction 1.1 Background in Nonlinear Oscillations Many phenomena associated with nonlinear oscillations, such as synchronizations, bifurcation phenomena, almost periodic oscillations, and chaotic oscillations, occur in nonlinear systems. In order to analyze the phenomena, we model the systems that exhibit the oscillations by nonlin-ear .
Chapter 7 Natural oscillations of non-linear oscillators 71 7.1 Pendulum oscillations 71 7.2 Oscillations described by the Duffing equation 72 7.3 Oscillations of a material point in a force field with the Toda potential 75 7.4 Oscillations of a bubble in fluid 77 7.5 Oscillations of species strength described by the Lotka-Volterra equations 81
The nonlinear oscillations manifest themselves in various ways, depending on the initial conditions, and have a rich phenomenology. The study of neutrinos from these astrophysical sources therefore demands careful consideration of these nonlinear e ects. In this thesis, we put forward a framework to study nonlinear avor oscillations of neutrinos.
onlinear Rabi oscillations. Time-domain numerical simulations are run to observe both regimes. An analytical model is proposed to describe each regime and the transition from nonlinear to linear oscillations. This work confirms that the strong coupling between a single emitter and a
oscillations of nonlinear states have yet to be introduced. We report the Rabi oscillations of azimuthons— . 1 Introduction Rabi oscillations were introduced in quantum mechanics,1 but by now are widely investigated in a variety of optical and pho-tonic systems that include fibers,2,3 multimode waveguides,4-6
Nonlinear oscillations of gas bubbles in liquids: steady-state solutions Andrea Prosperetti California Institute of Technology, Pasadena, California 91109 The nonlinear oscillations of a spherical gas bubble in an incompressible, viscous liquid subject to the action of a sound field are investigated by means of an asymptotic method.
Microcantilever based disposable viscosity sensor for serum and blood plasma measurements Onur Cakmaka, Caglar Elbukenb,1, Erhan Ermekb, Aref Mostafazadehb, Ibrahim Barisc, B. Erdem Alacaa, Ibrahim Halil Kavaklic,d, Hakan Ureyb, a Koc University, Mechanical Engineering, Rumeli Feneri Yolu, 34450 Sariyer, Istanbul, Turkey bKoc University, Electrical and Electronic Engineering, Rumeli Feneri .
Studi Pendidikan Akuntansi secara keseluruhan adalah sebesar Rp4.381.147.409,46. Biaya satuan pendidikan (unit cost) pada Program Studi Akuntansi adalah sebesar Rp8.675.539,42 per mahasiswa per tahun. 2.4 Kerangka Berfikir . Banyaknya aktivitas-aktivitas yang dilakukan Fakultas dalam penyelenggaraan pendidikan, memicu biaya-biaya dalam penyelenggaraan pendidikan. Biaya dalam pendidikan .
