FREQUENCY ANALYSIS OF A DOUBLE-WALLED NANOTUBES SYSTEM

Transcription

1 Joural of Appled Matematcs ad Computatoal Mecacs 04, 3(4), 7-34 FREQUENCY ANALYSIS OF A DOUBLE-WALLED NANOTUBES SYSTEM Ata Cekot, Stasław Kukla Isttute of Matematcs, Czestocowa Uversty of Tecology Częstocowa, Polad ata.cekot@m.pcz.pl, staslaw.kukla@m.pcz.pl Abstract. I te preset paper, a problem of free vbrato of a double-walled aotubes system s cosdered. Te aotubes of te system are coupled by a arbtrary umber of elastc rgs. Te Gree s fuctos metod was used to obta a soluto to te problem. Numercal results are preseted grapcally. Keywords: aotubes system, free vbrato, Gree s fuctos Itroducto Te aostructures, especally carbo aotubes, ave may ovel electrcal, cemcal, mecacal ad termal propertes lke: almost perfect geometrcal structure ad extremely g stregt ad low mass desty. Tese propertes make te carbo aotubes a promsg tecology for future applcatos dfferet felds of aoscece ad aotecology. Tree dfferet teores of modellg te vbratoal beavor of aotubes: Euler-Beroull beam teory, te Tmoseko beam teory or te Reddy beam teory [] ave bee used te study of te propertes of sgle-walled, double- -walled ad mult-walled carbo aotubes. Te study of vbrato mult-walled carbo aotubes s a topc of papers [, 3]. Murmu, McCarty ad Adkar report te effect of logtudal magetc feld o te trasverse vbrato of double-walled carbo aotubes. Te paper [3] by Natsuk et al. s devoted to te vbrato caracterstcs of embedded double-walled carbo aotubes subected to axal pressure. Te autors cosder ts problem by usg a elastc cotuum mecacs model wt te va der Waals teracto betwee te er ad outer aotubes. Te soluto of te problems of free vbrato of double-walled aotubes ca be obtaed by applyg metods suc as tose used te classcal beam teores. Te explct expressos for te atural frequeces were preseted paper [4]. Te Bubov-Galerk ad Petrov-Galerk metod were appled to derve te atural frequeces.

2 8 A. Cekot, S. Kukla I te preset paper, a soluto to te free vbrato problem of a system of two aotubes coupled by a arbtrary umber of elastc rgs s preseted. Te soluto s obtaed by usg te Gree s fucto metod [5].. Equatos of moto of double-walled aotubes system Te goverg equatos of logtudal vbratos of te double-walled aotubes system are []: w w w w w η 4 x ρ 0 ρ η x 4 E I A H A ( e a) ( A A H ) = x x t x t x = (, ) (, ) δ c w x t w x t x x ( e a) c ( x, t) w ( x, t) x x w w 0 δ = x x () w w w w w η 4 x ρ 0 ρ η x 4 E I A H A ( e a) ( A A H ) = x x t x t x = (, ) (, ) δ c w x t w x t x x ( e a) c ( x, t) w ( x, t) x x w w 0 δ = x x were w s te dsplacemet, H x s te stregt of te logtudal magetc feld, ρ s te mass desty, E s te modulus of elastcty, A s te area of crosssecto of te -t aotubes, δ() deotes te Drac delta fucto, x, x are axal postos alog te aotubes, x, x, =,... are pots of te aotubes wc are oed by a -t elastc rg, c s a elastc rg costat, e 0 s a costat approprate to aotubes materal ad a s a teral caracterstc sze. Te fuctos w ( x, t ) satsfy te followg boudary codtos: () w x w (0, t) = (0, t) = 0, =, w x w ( L, t) = ( L, t) = 0, =, (3) (4)

3 Frequecy aalyss of a double-walled aotubes system 9 A sketc of te cosdered system of two aotubes coected by -elastc rgs s sow Fgure. Fg.. Scematc dagram of te double-walled aotubes system. Soluto of te problem Te partal dfferetal equatos () ad () wc gover te trasverse vbratos of te aotubes ca be solved assumg te soluto te form ωt w ( x, t) = W ( x) e =, (5) x W were ω s te crcular frequecy. Itroducg ew varables: =, W= L L to equatos ()-(), te followg o-dmesoal equatos ca be obtaed: 4 d W 4 F 4 PW d d W d = ( ) Ω = P γ W( ) W( ) δ( ) Pµ γ δ d d d W d W ( ) ( ) = (6)

4 30 A. Cekot, S. Kukla 4 4 Ω F 4 QW 4 d W d W rs ( ) = d d 3 = sq γ W W δ µ s Q d d d W d W γ ( ) ( ) δ = (7) were: L L 4 4 ρa L c L e0a EI ρ A Ω = ω, γ =, µ=, s=, r=, E I E I L E I ρ A =, p Q= A A ( µ pqs) 3 4 =, = ( Ω µ ). F q P P= ( µ q), x η A H L q=, E I 4 ( µ Ω r) s pq F =, Takg to accout equato (5) equatos (3)-(4) te followg boudary codtos for te fucto W( ) ca be obtaed W (0) = W (0) = 0, =, (8) W ( L ) = W ( L ) = 0, =, (9) Te soluto of te problem (6)-(9) ca be preseted te form [5]: P W ( ) = γ W ( ) W ( ) G (, ) = Pµ γ G d W d W ( ) ( ), = d d W ( ) = sq 3 γ W ( ) W ( ) G (, ) = µ s Q d d d W d W γ ( ) ( ) G, = (0) () were (, ) G η are te Gree s fuctos. Tese fuctos satsfy te dfferetal equatos

5 Frequecy aalyss of a double-walled aotubes system 3 4 G G 4 F 4 PΩ G = δ η 4 4 G G rs Q F 4 G 4 = Ω ( ) δ( η ) () (3) ad te boudary codtos aalogous to te codtos (8)-(9): G G ( η ) G = 0 0, = = 0 ( L η ) G = L, = = 0 (4) (5) Te dervato of te Gree s fuctos as bee preseted paper [6]. Te equatos (0)-() are satsfed for all values of depedet varables,, partcularly for =, =, ( =,,... ) : P W ( ) = γ W ( ) W ( ) G (, ) = µ γ d W d W P ( ) ( ) G, = d d sq W ( ) = 3 γ W ( ) W ( ) G (, ) = µ sq d W d W γ ( ) ( ) G, = d d (6) (7) Smlarly, te secod order dervatve of te fuctos W( ) ad W( ) determed by usg equatos (0)-() satsfy te equatos: W ( ) P G γ W = W (, ) = µ γ d W d W G P ( ) ( ), = d d (8)

6 3 A. Cekot, S. Kukla W ( ) G sq 3 γ W W = µ s Q d d (, ) = d W d W G γ ( ) ( ), = (9) Itroducg a ew costat W W = ( ) ( ) µ V W W to equatos (6)-(9), after some trasformatos we obta te system of equatos wc ca be wrtte te form: were ( γ δ ) = A V = 0, =,... sq sq (, ) (, 3 ) G G A = G PG µ (, ) P (, ) Ts system of equatos as a soluto f ad oly f t satsfes te codto det γ A δ = 0 (0) Ts equato s solved umercally wt respect to atural frequeces of te Ω. 3. Numercal results Te system of double-walled carbo aotubes wt detcal pyscal propertes ad te same legt was aalyzed. Te aotubes of te system are coected by two elastc rgs at = 0.3 ad = 0.7 of te frst aotube ad = 0.3 ad = 0.7 of te secod aotube. Te stregt of te logtudal magetc feld H x= 0, was assumed. Te computatos for four dfferet values of te odmesoal rgs costats: c= c= ; 0; 0; 50, ave bee performed. Te four dmesoless atural vbrato frequeces as fuctos of parameter µ were calculated for ts system ad tese are plotted Fgure. Te Maple package [7] was used for calculatos. Te curves te Fgure sowed tat for te frst two egefrequeces Ω, Ω, te values of te rgs costats are partcularly relevat. Moreover, t s

7 Frequecy aalyss of a double-walled aotubes system 33 observed tat a crease of te parameter caracterzed te aotubes (materal ad teral caracterstc sze) causes a decrease of te frequeces for all cosdered values of te rgs costats. Fg.. Te frst four dmesoless atural vbrato frequeces as a fucto of µ Coclusos I ts paper te Gree s fucto metod was appled to solve te problem of free vbrato of double-walled aotubes coupled by -elastc rgs. Smplysupported boudary codtos were employed ts study. It s observed tat as te parameter µ creases, te frequeces of te aotubes decrease. I te preseted example te umber of aotubes ad couplg rgs was lmted to two,

8 34 A. Cekot, S. Kukla but te approac ca be used to solve te problems of systems cosstg of may aotubes coupled by elastc rgs. Refereces [] Asar R., Sama S., Small scale effect o vbratoal respose of sgle-walled carbo aotubes wt dfferet boudary codtos based o olocal beam models, Commu. Nolear Sc. Numer. Smulat. 0, 7, [] Murmu T., McCarty M.A., Adkar S., Vbrato respose of double-walled carbo aotubes subected to a exterally appled logtudal magetc feld: a olocal elastcty approac, Joural of Soud ad Vbrato 0, 33, [3] Natsuk T., Le X. W., S J.X., N Q.Q., Vbrato caracterstcs of embedded double-walled carbo aotubes subected to a axal pressure, 8 Iteratoal Coferece o Composte Materals. [4] Elsakoff I., Petaras D., Fudametal atural frequeces of double-walled carbo aotubes, Joural of Soud ad Vbrato 009, 3, [5] Duffy D.G., Gree s Fuctos wt Applcatos, Capma & Hall/CRC, 00. [6] Cekot A., Kukla S., Frequecy aalyss of a double-aobeam-system, Joural of Appled Matematcs ad Computatoal Mecacs 04, (3), 3-3. [7] Rcards D., Advaced Matematcal metods wt Maple, Cambrdge Uversty Press, 009.