Free Vibration of Pre-Tensioned Electromagnetic Nanobeams

Transcription

1 IOSR Journal of Mathmatics (IOSR-JM) -ISSN: , p-issn: X. Volum 3, Issu Vr. I (Jan. - Fb. 07), PP Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams M. Zaaria& Amira M. Al Harthy Dpartmnt of Mathmatics, Faculty of Scinc, Al-Baha Univrsity, Al-Aqiq, P.O. Box 988, Kingdom of Saudi Arabia. Abstract: Th transvrs fr vibration of lctromagntic nanobams subjctd to an initial axial tnsion basd on nonlocal strss thory is prsntd. It considrs th ffcts of nonlocal strss fild on th natural frquncis and vibration mods. Th ffcts of a small-scal paramtr at molcular lvl unavailabl in classical macro-bams ar invstigatd for thr diffrnt typs of boundary conditions: simpl supports, clampd supports and lastically constraind supports. Analytical solutions for transvrs dformation and vibration mods ar drivd. Through numrical xampls, ffcts of th dimnsionlss Hartmann numbr, nano-scal paramtr andpr-tnsion on natural frquncis ar prsntd and discussd. Kywords: Nanobam, Natural frquncy, Nonlocal strss, Pr-tnsiond, Vibration, mod, Hartmann numbr I. Introduction Rcntly, rsarch on dynamic bhavior ofnano-structurs has bcom a hot fild bcaus of th application prospcts of nano-lctromchanicalsystms (NEMSs) or nano-machin componnts. Although nano-structurs, such as nanobams and nano blts, hav bn proposd to hav practical applications, analysis in this fild has bn lacing in particular th dynamics of pr-tnsiond nano structurs. Th rlvant rsarchs on transvrs vibrations of axially moving macro-bams can b datd bac to Mot (965) constructd mathmatical modl of axially moving bams firstly basd on th Hamilton principl and also dtrmind th first thr natural frquncis and mods. His rsults wr confirmd by xprimnt (Mot and Nagulswaran, 966). Thr ar svral xcllnt and comprhnsiv survy paprs, notably Simpson (973) rsarchd th natural frquncy and mod functionof axially moving bams without pr-tnsion and clampd at both nds. Oz t al. (00) introducd axially moving bams with tim-dpndnt vlocity through multipl scal analysis. Liu and Zhang (007)prsntd th nonlinar vibration of viscolastic blts. Th bifurcation of transvrs vibration for axiallyacclrating moving strings was invstigatd by Chnand Wu (005).Yang and Chn (005) addrssd dynamic stability problm of axially moving viscolastic bams. Th nonlocal bam modls rcivd incrasing intrst in th past fw yars. Nonlocal continuum thoris rgardth strss stat at a point as a function of th strain stats of all points in th body, in contrast to th classical viwthat th strss stat at a givn point is uniquly dtrmind by th strain stat at that sam point. Eringn (97) has proposd a nonlocal continuum mchanics thory to dal with th small-scal structur problms whil in classical(local) lasticity, th matrial particls ar assumd to b continuously distributd and th strss tnsor at a rfrnc point is uniquly dtrmind by th strain tnsor through th algbraic rlation of Hoo s law, and, th matrial particls in nonlocal lasticity ar allowd to intract with low-rang forcs and th algbraic constitutiv quation is rplacd by a gnralizd constitutiv quation of intgral and diffrntial oprator typ. Prvious wor shows that th disprsion curvs obtaind by th nonlocal modl ar in xcllnt agrmnt with thos obtaind by th Born-Karman thory of lattic dynamics. Th dislocation cor and cohsiv strss prdictd by th nonlocal thory ar also clos to corrsponding stimats nown in th physics of solids (Eringn 97, 983; Eringn and Edln 97). Th nonlocal lasticity thory was applid in nano mchanics including lattic disprsion of lasticwavs, wav propagation in composits, dislocation mchanics, static dflction, fractur mchanics, and surfac tnsion fluids. (Rddy and Wang, 998;Pddison t al., 003; Zhang t al., 004; 005;Wang, 005; Lu t al., 006; Wang and Varadan, 006;Wang t al., 006; 008; Xu, 006; Lim and Wang,007; Bnzair t al., 008; Kumar t al., 008; Wangand Duan, 008). Th rcnt wor by Tounsi t al.(008) concludd that th scal cofficint was radius dpndnt. Vibration bhavior of bams has bn dvlopd for a long tim. Howvr, vry fw paprs considrnanobams with nonlocal ffcts. Th nano mchanical vibration of a lctromagntic nanobam is vry diffrnt from th classical continuum mchanics thory which dals with th macroscopic scal of a bam. In this papr, w attmpt to considr th nonlocal ffcts of a pr-tnsiond lctromagntic nanobam without axial motion and subsquntly study th transvrs vibration and th ffct of Hartmann paramtr of such ananobam. Th modl is dscribd by partial diffrntial quations in dimnsionlss quantitis such thatth analysis is mor gnral and distinctiv to dscrib th diffrnc btwn lctromagntic nano mchanics and classical mchanics. It is found that pr-tnsion and nonlocal strss play significant rols in th vibration DOI: / Pag

2 Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams bhavior of a lctromagntic nanobam. Thir ffcts ar analyzd and discussd in dtail in a fw numrical xampls. II. Problm dfinition and modling Considr a pr-tnsiond nanobam with th lngth L, initial axial tnsion P at th nds. Th nd boundary conditions ar arbitrary and will b spcifid in various cass ofstudy. Th forc quilibrium for an lmnt of th nano bam. For vibration of a nanobam, th bnding rotation angl with rspct to x-axis is dnotd as θ. Bcaus only small dformation is considrd for linar vibration, w hav whr w is th transvrs dformation. W considr also a constant magntic fild of strngth acts in th dirction of th.y Using Ohm s: whr is th lctrical conductivity, is th intnsity vctor of th lctric fild, is th magntic prmability, and is th displacmnt vctor, nglcting th ffct of th intnsity vctor Th pondr motiv forc has non-vanishing componnt dirctions, givn by Th quilibrium quation of an lmnt with rspct to th z-axis can b obtaind basd on th D Almbrt Principl (Fung,965) as whrq is th shar forc, x is th axial, N is th intrnal axial forc, is th lin dnsity and t is tim. whr M isth bnding momnt. ughout, w hav (Yang and Lim, 008) which can also b xprssd as For such a nanobam with a constant, xtrnal axial tnsion P at th nds, w hav According to Eringn (983), th nonlocal strss in a D domain can b approximatly govrnd by ascondordr diffrntial quation whr (i, j=, ) ar th nonlocal strsss, (i, j=,) th classical local strsss, a constant dpndnton matrial, and a an intrnal charactristic lngth,.g., for lattic paramtr, C-C bond lngth. For ananobam, th govrning quation abov with rspct to th nutral axis can b rducd to an ordinary diffrntial quation as whr indicats th nonlocal normal strsss whil th classical local normal strsss along th x-axis.from Eq. (8), th nonlocal normal strsss can b solvd and xprssd in an infinit sris as (Lim andwang, 007) whr E is th Young s modulus and z is th transvrs coordinat. Intgrating Eq. (0) abov with rspct to thdistanc from th nutral axis and ovr th cross-sctional ara, th nonlocal bnding momnt isgovrnd by whr EI is th flxural stiffnss. From Eqs. (8) and(), th following govrning diffrntial quation ofmotion for a nanobam subjctd to an initial axial tnsion P can b drivd as For gnrality, dimnsionlss formulation is adoptd using th following paramtrs DOI: / Pag

3 Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams whr is th Hartmann numbr. In dimnsionlss quantitis, Eq. (3) thn bcoms, dropping th astriss for convninc whr, and. From Eqs. (8) and (), th nonlocal bnding momntfor th non-dimnsional form can b xprssd as For linar fr vibration of a nanobam, th mods ar harmonic in tim. Hnc th timdpndnttransvrs dformation of th nanobam can b rprsntd by whr is th dimnsionlss amplitud of vibrationand n=,, dnots th vibration modnumbr. Substituting Eq. (6) into Eqs. (4), and (5) th govrning quation is transformd into th frquncydomain as Eq. (7) can b factorizd as whr, and ar th roots of th charactristic quation whr and Sinc Eq. (0) is a fourth-ordr polynomial in trms of th four roots ar dnotd by, rspctivly. Bcaus only linar fr vibration isconcrnd, th suprposition of th four solutionswith rspct to ach root is also a solution of Eq. (9). Hnc whr ar four arbitrary constants ofintgration DOI: / Pag

4 Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams associatd with Eq. (0) which is afourth-ordr ordinary diffrntial quation. III. Applications and discussion To illustrat th ffcts of nonlocal strss and initial axial tnsion on th fr vibration frquncy of a nanobam, th following xampls for various boundary conditions ar prsntd and discussd. 3. Simply supportd nanobams For a nanobam simply supportd at both nds, th boundary conditions for th bnding momnts and displacmnts ar Substitut Eq. (6) into Eq. (3) and simplify th rsults. Furthr substituting Eqs. () and () into th rsults obtaind abov yilds whr For an arbitrary th cofficints ineq. (4) can b obtaind as Thrfor, th n-mod amplitud of vibration from Eqs. () and (5) is and th corrsponding tim-dpndnt displacmnt from Eq. (6) is whr is an arbitrary constant. For nontrivialsolution of Eq. (), th dtrminant of th cofficintmatrix must b zro n in n in i n i n i i which yilds a charactristic quation as i 4 n i 4 n 0, DOI: / Pag

5 Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams (8) Th n-mod vibration mod and transvrs dformation can b solvd to th xtnt of an arbitrary constant Th analysis abov can b dscribd clarly through numrical xampls. For instanc, taing and th roots for satisfyingeqs. (8) and (8) can b obtaind as th intrcptsof th horizontal axis in Fig. whr th dtrminant Eq. (7) vanishs. It is obvious that thr ar infinit mods of frquncy, which ma th dtrminant zro. Th first intrcpt with th horizontal axis is th fundamntal frquncy; th scond intrcpt is th scond mod frquncy, and so on. Following th numrical procdur abov, th rlationship btwn and th dimnsionlss nano scal paramtr τ can b obtaind as shown in Fig.. W can find that th fundamntal and th scond mod frquncis rduc with th incrasing τ. Hnc, th natural frquncis rduc whn th strongr nonlocal strss ffct is prsnt. It is also obvious that th frquncis incras with th dimnsionlss pr-tnsion P. Obviously, τ and P affct vry much th natural vibration frquncis. 3.. Clampd nanobams Th problm of a clampd, pr-tnsiond nano bam is prsntd in th following xampl. Thclampd boundary conditions ar From th abov quation, th rsult can b dducd by Eqs. (6) and () i n n n in which yilds i n i n i i i 4 n i 4 n C C C C n 0, (30) DOI: / Pag

6 Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams whr Hnc, th n-mod amplitud of vibration is and th corrsponding tim-dpndnt displacmnt is For nontrivial solution of matrix Eq. (8), th dtrminant of th cofficint matrix must b zro, or (34) Analogously, from Eqs. (0) and (34), w cansolv th unnown quantitis in Eqs. (3) and (33). Tosolv th problm numrically, th rlationship btwn and is prsntd in Fig. 3 for two valus of P. Again, it is obvious that frquncy dcrass with an incras in whil it incrass with anincras in P Nanobams with lastically constraind nds In this xampl, w considr a spcial supporting condition for nanobams with lastically constraind nds (Xi, 007). Th support conditions may b formulatd with th following boundary conditions DOI: / Pag

7 Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams whr is th dimnsionlss stiffnss of thlastically constraind nds in which is th physicalstiffnss of th lastic constraint. If approachs 0, ths nds dgnrat to simply supports as discussdin subsction 3., whil if approachs infinity, thydgnrat to clampd ons in subsction 3.. Substituting Eqs. (6) and () into Eq. (35) yilds whr A = n i n i i i n n in i n i i 4 n i i i i n C C C C n C n For th following solutions of cofficints ar obtaind by solving Eq. (37): 4 n whr Thn th n-mod amplitud of vibration can b obtaind as (39) and th corrsponding tim-dpndnt displacmnt is shown in Eq. (40): DOI: / Pag

8 Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams (39) For nontrivial solution of Eq. (36), th dtrminant of th cofficint matrix must b zro, or Eq. (40) (40) Th rlationship btwn th natural frquncis and nanoscal paramtr is Fig. 4 for = 0.. Again, w obsrv similar ffcts of and whr incrass in and caus thfrquncis to dcras and incras, rspctivly. IV. Conclusion In this papr, w concludd that th transvrs fr vibration of lctromagntic nanobam is significantly influncd by th xistnc of a pr-tnsion and th dimnsionlss nano scal paramtr. Thr numrical xampls ar prsntd which includ simply supportd nano bams, clampd nanobams and nano bams with lastically constraind nds. In th numrical xampls, w find that th first two modfrquncis drop quicly with incrasing dimnsionlss nano scal paramtr. On th contrary, thfirst two mod frquncis incras with incrasing pr-tnsion. Th ffcts ar similar for th thr xampls invstigatd. Rfrncs []. Mot, C.D.Jr., 965. A study of band saw vibrations. Journal of th Franlin Institut, 79(6): [doi:0.06/ (65)9073-5] []. Mot, C.D.Jr., Nagulswaran, S., 966. Thortical and xprimntal band saw vibrations. ASME Journal of Enginring Industry, 88():5-56. [3]. Simpson, A., 973. Transvrs mods and frquncis of bams translating btwn fixd nd supports. Journal of Mchanical Enginring Scinc, 5(3):59-64.[doi:0.43/JMES_JOUR_973_05_03_0] [4]. Oz, H.R., Padmirli, M., Boyaci, H., 00. Non-linar vibrations and stability of an axially moving bam with tim-dpndnt vlocity. Intrnational Journal of Non-Linar Mchanics, 36():07-5. [doi:0.06/s (99) ] [5]. Liu, Y.Q., Zhang, W., 007. Transvrs nonlinar dynamical charactristic of viscolastic blt. Journal of Bijing Univrsity of Tchnology, 33():0-3. [6]. Chn, L.Q., Wu, J., 005. Bifurcation in transvrs vibration ofaxially acclrating viscolastic strings. Acta Mchanic a Solida Sinica, 6(): [7]. Yang, X.D., Chn, L.Q., 005. Dynamic stability of axially moving viscolastic bams with pulsating spd. Applid Mathmatics and Mchanics, 6(8): DOI: / Pag

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