Analysis of axial vibration of non-uniform nanorods using boundary characteristic orthogonal polynomials

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1 mme.modares.ac.ir * sh.farughi@uut.ac.ir * () : : : - Analysisofaxialvibrationofnon-uniformnanorodsusingboundary characteristicorthogonalpolynomials SeyedMohammadHosseinGoushegir,ShirkoFaroughi * DepartmentofMechanicalEngineering,UrmiaUniversityofTechnology,Urmia,Iran. P.O.B ,Urmia,Iran,sh.farughi@uut.ac.ir ARTICLEINFORMATION ABSTRACT OriginalResearchPaper Received04November2015 Accepted11December2015 AvailableOnline30December2015 Keywords: nanorod nonlocalelasticitytheory Rayleigh-Ritzmethod boundarycharacteristicorthogonal polynomials Inthispaper,thelongitudinalvibrationofnanorodbasedonEringen snonlocalelasticitytheory was studied using Rayleigh-Ritz method. non-uniform nano-rod with variable cross-sectional area,densityandyoung smoduluswereconsidered.inthepresentwork,boundarypolynomials withorthogonalpolynomialswereusedasshapefunctionsintherayleigh-ritzmethodwhich causes the vibrational analysis to be computationally efficient and imposition of boundary conditions tobeeasier. Usingthe mentioned polynomialstheconvergencerateoftheobtained results was increased. All of the equations used in this study are nondimensionalized to reduce the number of effective parameters in the solution. The influence of the nonlocal and inhomogeneity parameters on the vibrational behavior of nanorod was investigated. The results were compared to available results in the literature and good agreement was achieved. The results showed that nanorod frequencies were dependent on the small scale effect, nonuniformity,andboundaryconditions.forinstance,anincreaseinfrequencyratiocausesthescale coefficient in all vibration modes to be increased, especially in higher modes. In addition, the frequencieswereincreasedbyincreasinginthelengthofthenanorod..[7-5].. 2 / 2MEMs/NEMs 1-1.[4-1] 1Carbonnano-tubes(CNTs) Pleasecitethisarticleusing: : S.M.H.Goushegir,Sh.Faroughi,Analysisofaxialvibrationofnon-uniformnanorodsusingboundarycharacteristicorthogonalpolynomials,ModaresMechanicalEngineering Vol.16No.1,pp ,2016(inPersian)

2 [12-685] 1. 2.[13].. [8] 3..[12-685] 4. [14].... ) (. [15].....[16].. : [1817] - 5 [2019]. - [21]. ( 6 ) ) (.. 1Smalllengthscaleeffect 2Nonlocalelasticitytheory 3clamped 4Continuummodel 5Hybridcontinuum-atomisticmechanics 6Local 7 [22].. 8 [2423]. 9. [27-25] [31-28] "1" L d(x) 12.[32] (1) (1 ) = + Fig. Non-uniform nanorod with variable cross-sectional area, density and Young smodulus 1 7Sizeeffect 8Nano-OptomechanicalSystems 9Nonlocalparameter 10Boundarycharacteristicorthogonalpolynomials 11Frequencyparameter 12Nonlocalconstitutiveequation

3 = [()()() + () ()()() (9) 2 ()() () ()()() + () ()()() (10) X = = = e 4 (11). =0 e 5. (10) (11) E (X)A (X) () X (X)A (X)U(X) + (X)A () (X)U(X) X = E (X) = (X) = A (X) (12) - 13). A (X) (X)E (X) X E (X) = 1 + X + X (X) = 1 + X + X A (X) = 1 + X - 13) U(X) = (X) 6 b N - (14) 2-Rayleigh-quotient 3-Admissibleshapefunction 4-Scalingeffectparameter 5-Internalcharacteristiclength 6-Thickeningcoefficient 7-Rayleigh-Ritzmethod G a = (e ). e (1) (1 ) = (2) E = = (3) m (,) A x (2) = (4) = (5) (5-3) ()() = 1 ()() (6) m ()()(6) = 0 (, ) = ()sin () ()() + 1 (()()() ) = 0 - (7) 1 (). () ()() () + 1 (()()() ) = 0 ()() () () = [()()()() + (7) - 8) () (()()()) - 8) ()() () 1-Weakform

4 Fig.2ConvergenceoffirstsixfrequencyparametersforC-C 1 nanorod Table 1 Comparison of first five vibration frequency parameter of C-C nanorodwithothersfordifferentvaluesofnonlocalparameter [35] [34] [33] µ Table 2 Comparison of first five vibration frequency parameter of C-F nanorodwithothersfordifferentvaluesofnonlocalparameter [35] [34] [33] µ "2 ". = 0.5.( = 0.4, = 0.1, = 0.3, = 0.2) "2 ". N Clamped-Clamped 1- = X (1 X) X =, = =,, b a, = (X)(X)X.[2831] (15) (16). [0,1] 2 (X) =, (17) = (18) 1 0-0,, = 1, = (19) (12) (14) [ ]{U} = [ ]{U} {U}= [ ] [ ] (20) [ ] = E A X X X = A + X A X X (21) (20) N 1 0 n m [34] [33] [35] Gram-Schmidtprocess 2Norm 3Maple

5 "3". =0.1 "3" ) =0. Fig.4Variationsoffundamentalfrequencyratiowithlengthofnanorod "3".. "4" = NLF LF 0.2 =0.1 FR 3 LF. (22) NLF "4 ".. 15 nm. "4". "5" 4.. 3Frequencyratio 4Scalingcoefficient (a) (b) (c) ) Fig.Variationsoffirstfivevibrationmodeswithfor,a)C-C,b)C-F 1 c)f- F 2 condition ( ) ) 3 ( ( 1-Clamped-Free 2-Free-Free

6 (a) ) Fig.6VariationsoffirsttwofrequencyparametersofC-Fnanorodwith parameters,sandp 6 Fig.7VariationsoffirsttwofrequencyparametersofC-Cnanorodwith parameters,qandr s p r q 7 (b) Fig.5Variationsoffirstthreefrequencyratioswithscalecoefficientfor,a)C- C,b)C-Fcondition ) ( 5 ( ( ( ) ( ( "6 " - s p r 0.2 b. [-0.5,0.5]. = 0, 0.4, 0.8 q "6".sp (12). s p

7 Fig.9VariationsoffirsttwofrequencyparametersofC-Fnanorodwith parameters,qands s q 9 s p ) "7 " - r q s 0.1 b. p q"7" q.r r r q "8 " () r p 0.2 b. q "8".rp. r p ) "9" - s q r 0.1 b. p s Fig.10Graphofvariationsinthefirstninefrequencyparameters withl/d forc-cnanorod L/d 10 "9 ". s q s q q. s (L/d) Fig.8VariationsoffirsttwofrequencyparametersofF-Fnanorodwith parameters,and 8 r p

8 Fig. 11 Variations of first two frequency parameters withfor different boundaryconditions 11 Fig.13Variationsoffourthmodeshapeofnanorodwith nm nm nm X=0..[36] "10 " s rqp "11 " 0.1. "11" Fig.12FirstthreemodeshapesofC-Cnanorod 12 L/d"10" s rqp

9 X - (nm) (Pa) (Pa) x X ( (Pa) (nm) ) = 0 = 1 - ( ( ) ) ) ) ) ) (nm ) ( ) ( ) x X x A () ". d(x) e E () E () FR G [ ] L LF m [ ] n N NLF (, ) U() (X) () () " s rqp "13 " - 4 ). ( -.. e0a q p s r L/d - 5 (nm) (nm ) x a ()

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