An Exact Solution for the Differential Equation. Governing the Lateral Motion of Thin Plates. Subjected to Lateral and In-Plane Loadings

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1 Appled Mahemacal Sceces, Vol., 8, o. 34, A Eac Soluo for he Dffereal Equao Goverg he Laeral Moo of Th Plaes Subjeced o Laeral ad I-Plae Loadgs A. Karmpour ad D.D. Gaj Mazadara Uvers Deparme of Cvl ad Mechacal Egeerg Babol, P. O. Bo 484, Ira Absrac Oe of he powerful aalcal mehods o solve paral dffereal equaos s he Adoma decomposo mehod (ADM). Ths paper preses a ovel approach for he damc aalss of a fleble plae b usg he ADM, whch s a Boudar Value Problem (BVP). I hs regard, a geeral approach based o he geeralzed Fourer seres epaso s appled. The obaed aalcal soluo s smplfed erms of a gve orhogoal bass fucos ha hese fucos sasf he boudar codos of plae. For he frs me, we solved hs equao usg ADM ad compared he resuls wh hose of he modal classcal aalss wo cases o demosrae he vald of he prese sud. Kewords: The Adoma Decomposo Mehod (ADM); Th plaes; Boudar Value Problems (BVPs); Orhogoal bass fucos. Iroduco Mos scefc problems ad pheomea dffere felds of scece ad egeerg occur olearl. Ecep a lmed umber of hese problems, we E-mal address: al_k_sar7@ahoo.com (A. Karmpour) Correspodg auhor: Tel/fa: E-mal address: ddg_davood@ahoo.com (D.D. Gaj)

2 666 A. Karmpour ad D.D. Gaj ecouer dffcules fdg he eac aalcal soluos. Decomposo mehods provde he mos versale ools avalable olear aalss of egeerg problems. I cvl ad mechacal egeerg sceces, for desg of slabs ad ssems wh plae behavor, s mpora o kow he chage of defleco ad sress slabs uder he dffere loadgs. We use he classcal small defleco heor of h plaes suppled problems. Ths paper s devoed o he sud of recagular elasc plae ad her goverg dffereal equaos wh ADM [-3]. I fac, we used he geeral soluo of hs dffereal equao from he orhogoal fucos o sasf he comple boudar codos of plae, whch depeds o he aure of he suppored edges. There are resrcos for he eac aalcal soluo of plae ad here s o geeral soluo for a boudar codos, ad mosl umercal soluos are appled comple boudar codo ad shape of plae. Fall we successfull foud he geeral soluo of he dffereal equao goverg recagular plaes wh ADM, whch s he same as he classcal soluo.. Mahemacal modelg of he problem The basc dffereal equao of laeral moo for plaes wh forced, o damped moo ad subjeced o laeral ad -plae loadgs classcal Small Defleco heor of h plaes s obaed [,]: 4 [ ] (,, ) w w z w D w = p + + w + ρ h () whch s a varable coeffce fourh order parabolc paral dffereal equao, where w = w(, he defleco mddle surface of plae or he laeral plae dsplaceme, D s plae bedg sffess, p z (, s laeral loads per u area, ρ s he plae maeral des, ad order seres are -plae forces parallel o, drecos ad shear force, ad h s he plae hckess. I geeral, The Homogeeous Boudar Codos (HBCs) for recagular plae are a combao as he followgs (for eample for dreco ad a = a ) [,, 3]: 3 3 w w D[ + ( v) ] = 3 Two HBCs for free edge( a = a) w w () D[ + v ] = ad

3 Eac soluo for he dffereal equao 667 w = Two HBCs for fed edge( a = a) w (3) = or w = Two HBCs for smple edge( a = a) w w D[ + v ] = (4) whereν he Posso rao of plae maeral. Fg.. The recagular plae wh geeral loadgs I should be oed ha wo HBCs mus be sasfed a =, = a ad wo HBCs mus also be sasfed a =, = b. I a case for aalss of plae, we mus chose wo HBCs for each edge of recagular plae ha he choce as o whch of he wo equaos has o be sasfed depeds o he aure of he plae suppors. Also, s show ha a BVP cossg of a homogeeous dffereal equao wh homogeeous boudar codos ca be rasformed o a problem cossg of a homogeeous dffereal equao wh homogeeous boudar codos [4, 5]. The rasverse vbraos of plaes are sudes b a of he followg: fe eleme mehods, fe dfferece mehods, ad he modal aalss echque. Whle fe eleme ad fe dfferece mehods are he caegor of umercal echques, modal aalss s oe of he mos powerful aalcal meas, wh he capabl o jo wh he aforemeoed mehods order o aalze he damc characerscs of mechacal ssems.

4 668 A. Karmpour ad D.D. Gaj Sce he 98s, sudes have llusraed ha he ADM ca be appled o deerme he soluo of a wde rage of lear ad olear, ordar or paral dffereal ad egral equaos. Ths mehod gves he soluo as a fe seres usuall covergg o a accurae soluo [6-6]. I rece ears, has bee appled o he problem of vbrao of srucural ad mechacal ssems wo ad hree space varables [7-] as well. I hs paper, he soluo of he goverg equao of a uform fleble plae s preseed whch akes he boudar codos of he problem o accou. For hs purpose, he al codos are epaded usg he eeded Fourer seres. The fal soluo s compared wh he resul from modal aalss. A comparso shows ha boh echques coverge o he same soluo as he seres approaches f. 3. Formulao wh ADM Usg he ADM for Eq. () ca be rewre operaor form as: ρ h[ Lw ] + D[ L 4w ] = Pz (,, + [ Lw ] + [ Lw ] + [ Lw ] (5) where he L, L 4, L, L, L operaors ad L verse are defed as follows: w w w w Lw =, L 4w = (6) w w w Lw =, Lw, = L w= ad σ L w = w(, τ ) dτ dσ (7) The geeral soluo of a homogeeous lear equao (), w(,, s a sum of a geeral soluo of he correspodg homogeeous equao u (, ad a parcular of he homogeeous equao v(, as follows: w (, = u(, + v(, (8) These wo erms wll be evaluaed separael he followg secos. 3.. Homogeeous problem I order o solve he homogeeous par, usg ADM, he oao of Ref. [4] s used hs seco. Neglecg he source erm o he rgh had sde of Eq. (5) ad roducg he, L operaor o boh sdes, he soluo of he homogeeous equao ca be wre as D L ( u) = [ L 4u] + [ Lu] + [ Lu] + [ Lu] (9)

5 Eac soluo for he dffereal equao 669 u(,, = f (, ) + g(, ) + D ( ) L { [ L 4u] [ Lu] [ Lu] [ Lu]} () B usg he Adoma decomposo mehod, u (, ca be epaded as a fe seres epaso erms of he (, compoes: u = u (, u (, () = I order o fd he compoes, u (,, subsuo of Eq. () o boh sdes of Eq. () elds: = u (, = f (, ) + g(, ) + L ( D L 4 L L L ) u (, () = Cosderg he decomposo mehod, u (, s assumed o be of followg form: u (,, = f (, ) + g(, ) (3) Alog wh he followg recurrece relao for u (, : u (, = L {( D L 4 L L L )[ u (, ]} (4) Thus he frs erms of he seres are u (, = f (, ) + g(, ) (5) u(, = L {( D L 4 L L L )[ u(, ]} 3 = ( D L 4 L L L )[ f (, ) + g(, )]! 3! u(, = L {( D L 4 L L L )[ u(, ]} 4 5 ( D) L 4 + ( ) L + = [ f (, ) + g(, )] ( ) ( ) 4! 5! L + L (6) (7)

6 67 A. Karmpour ad D.D. Gaj M u (, = L {( D L 4 L L L )[ u (, ] } ( ) D L 4 ( ) L + + = [ f (, ) + g(, )] + ( ) ( ) ( )! ( )! L + L + + u (, = {( ) L M }[ f (, ) + g(, )] ( )! ( + )! where D LM = L 4 L L L (9) D L M = ( ) L 4 + ( ) L + ( ) L + ( ) L () P LPL P L = () Operaor L P ca be ever operaor accordg up relaos. I fac, f (, ), g(, ) order seres are al codos or al dsplaceme ad veloc of plae ad he we ca be wre as: u(, ) = f (, ) () u (,,) = g (, ) (3) I should be oed ha he above fucos f (, ), g(, ) sasf he boudar codos of he problem. O he oher had, he geeral soluo of he homogeeous equao s also a sum of he u (, erms. I addo, f all u (, fucos sasf he boudar codos, he oe ma sae ha he sum of hem also sasfes he boudar codos. As show Eq. (8), (, fucos are deermed b applg he ( ) L M operaor o he fucos f (, ), g(, ). Ths ma lead o u (, fuco whch eher are zero or do o sasf he boudar codos a all. To preve hs dffcul he fucos f (, ), g(, ) are epaded erms of he kow orhogoal fuco φ (, ), φ (, ),... as a geeralzao of he Fourer seres epaso. φ, ), φ (, ),... ca be seleced o sasf he boudar codos ( before ad afer applg ( ) L M operaor (see Eq. (3)). As a resul, he fucos f (, ), g(, ) become f (, ) a φ (, ) k = j = u (8) = (4)

7 Eac soluo for he dffereal equao 67 g (, ) b φ (, ) = (5) k = j = Where he coeffces a b = = a b a, b are gve b he followg relaos: a f (, ) φ (, ) d d = (6) b g (, ) φ (, ) d d = = = (7) The bes se of fucos for he geeralzed Fourer seres epaso he case of our phscal problem s he se of egefucos of he followg self adjo ssem: LM φ (, ) = λφ (, ) (8) Prevous sudes dcae ha he egevalue problem defed Eq. (8) elds a fe se of real egevalues ad egefucos ( λ, φ (, )).These egefucos cosue he bass for he fe dmesoal Hlber space. Therefore, ever fuco h(, ) wh couous L M h(, ) whch sasfes he boudar codos of he ssem ha ca be epaded a absoluel ad uforml coverge seres he egefucos. Due o homogee of he egevalue problem, ol he shape of he egefucos s uque ad he amplude s arbrar. Accordg o Eq. (8), we ca ormalze he egefucos usg mass ad sffess operaors as follows: a b lh φ lh (, ) φ (, ) d d = (9) lh = = = a b lh k j φ lh (, ) L M. φ k j (, ) d d = (3) λ lh = k j = = Eqs.(37) lead also o L φ = λ φ (3) M Ad fall we oba L φ = ( λ ) φ (3) M Usg Eqs. () ad (3) ad subsug Eqs.(4) ad (5) o Eq. (8) ad embeddg he resul o Eq. () elds: u (,, ) = u(,, ) = f(, ) + g (, ) + = + + f (, ) + g(, ) ( )! ( + )! ( ) D L 4 ( ) L + = ( ) L + ( ) L

8 67 A. Karmpour ad D.D. Gaj = a φ (, ) + b φ (, ) + k = j= k = j= ( ) L a (, ) b φ (, ) + { } φ + M = ( )! k = j= ( + )! k = j= + ( ( λ ) ( ) ( ) λ = a ( ) + b ( ) φ (, ) k = j= = ( )! = ( + )! + ( λ ) b ( λ ) = a ( ) + ( ) φ (, ) k = j= = ( )! λ ( )! = + b u(, = a cos( λ ) + s( λ ) φ (, ) (33) k = j= λ 3.. Ihomogeeous problem Neglecg he frs wo erms o he rgh had sde of Eq. () ad roducg he L operaor o boh sdes of Eq. (6), he parcular soluo of he homogeeous equao ca be wre as D v (,, = ( ) L L 4 L L L v (,, + L Pz (, (34) A smlar procedure adoped he prevous seco ca be used o decompose he soluo b a fe sum of compoe epressed a seres form b v(, = v (, (35) = Ad v (, ca be deermed a smlar recurre procedure. Subsuo of Eq. (35) o boh sde of Eq. (34) gves v(, = L ( DL 4 L L L) v(, = = + L Pz (,, (36) ρ h The use of he decomposo mehod resuls v (, ) = L Pz(, ) (37) ρ h

9 Eac soluo for he dffereal equao 673 ad for (,, he recurre relao becomes v v(, = L DL 4 L L L v (, (38) For, epadg v (, a smlar wa as used for Eqs. (4) ad (5), we oba ρ h P (, = N φ (, ) (39) z k = j = Where he coeffces N ( ) a b = = are provded b he followg relao: N ( Pz (, φ (, ) dd ρ h = (4) Usg Eqs. (3), (37) ad (39), we ca rewre Eq. (38) as follows: D ( ) L 4 ( ) L + + v (,, = ( ) φ (, )( L ) N ( k j = = + ( ) L + ( ) L I + ( ) + M φ k = j = I ( ) = ( ) L (, )( L ) N ( + + λ φ k = j= I + () = ( ) ( ) (, )( L ) N ( + I order o fd v (, ), we eed o evaluae ( L ) N ( ) I ( ) s wre as: τ σ (4). For hs purpose, (4) I ( = L N ( = N ( σ) dσ dτ = N ( σ) dτ dσ = N ( σ)( σ) dσ Ad usg he same approach for I ( ) leads o I () ( L = ) N () = L I () = I ( σ )( σ) dσ σ = ( σ ) N ( τ)( σ τ) dτdσ σ = N ( τ )( σ τ)( σ) dτdσ

10 674 A. Karmpour ad D.D. Gaj = N ( τ )( σ τ)( σ) dσdτ τ τ = N ( τ ) ( σ τ )( σ ) dσdτ I + (, Therefore oe ca wre, for ) + + = = 3 3 σ ( τ) ( ) = N ( τ ) dτ = N ( σ) dσ 3! 3! (43) ( σ ) I () ( L ) N () N ( σ ) dσ (44) ( + )! Cosequel he fal form of he parcular soluo of he homogeeous equao usg Eqs. (35), (4) ad (44), s as follows: + + = λ φ k = j = I + ( ) + ( σ ) (,, ) ( ) ( ) (, ) ( ) = k = j = ( + )! v (,, ( ) ( ) (, )( L ) N ( v = λ φ N σ dσ + [( σ) λ ] = φ N ( σ) ( ) ( ) dσ k = j= λ ( )! = + v(, = ( ) N ( σ )s(( σ) λ ) dσ φ k = j= λ (45) 3.3. Geeral soluo of dffereal equao of laeral moo of h plae Geeral soluo of he plae equao ha s homogeeous dffereal equao s he sum of he soluo of he homogeeous problem gve b Eq. (33) ad soluo of he homogeeous problem gve b Eq. (45). Therefore he fal soluo ca be wre as b a cos( ω + s( ω + ω w (,, = (, ) φ (46) k = j = ( ) N ( σ)s (( σ) ω ) dσ ω where ω = ( λ ) (47)

11 Eac soluo for he dffereal equao 675 ad k, j =,,3,... he fudameal mode of fleural vbrao s a sgle se wave he X ad Y drecos, respecvel ad frequec of he plae. ω s he h mode aural 4. Eample The accurac of he decomposo mehod s eamed b wo cases of plae. The resuls are he compared wh kow eac soluos. 4.. Smpl suppored recagular plae whou laeral ad -plae loadgs As a frs case, a recagular plae s cosdered wh cosa D s plae bedg sffess, ρ s he plae maeral des ad h s he plae hckess, whch s smpl suppored a each four edges. The correspodg dffereal equao of hs case s 4 w (,, ) D. w (,, ) + ρ h = (48) Wh he boudar codos hs case are [,, 3] w( =, = w(, = (49) w( = a, = w( a, = (5) w(, =, = w(,, = (5) w(, = b, = w(, b, = (5) Accordg he Eqs. (9), (3) ad (3), he soluo hs case, frs he geeralzed Fourer seres epaso fucos are deermed ha hese fucos sasf he boudar codos before ad afer applg he L M operaor. For hs case we oba: D LM = L 4 (53) k π. j π. φ (,) = s( )s( ) (54) ab a b Fall hs case o deermed he aural frequec usg Eqs. (3), (47), (53) ad (54) we obaed:

12 676 A. Karmpour ad D.D. Gaj k j D ω = λ = π [( ) + ( ) ] (55) a b ρ h Tha hs soluo for aural frequec s eacl he same as he soluo of Ref. [, ]. 4.. Smpl suppored plae uder -plae loadg As a secod case, he cosas ad boudar codos are smlar o prevous case for a recagular plae, bu hs case we have -plae loadg parallel o dreco. The correspodg dffereal equao of hs case s 4 w (,, ) w D. w (,, ) + ρ h + [ ] = (56) ρ h I hs case, we have a orhogoal fuco smlar o prevous case: k π. j π. φ (,) = s( )s( ) (57) ab a b Bu operaor L M chage o hs form: D LM = L 4 L (58) Fall hs case o deermed he aural frequec usg Eqs. (3), (47), (58) ad (57) we obaed: D k π jπ k π ω = λ = [( ) + ( ) ] + ( ) (59) ρ h a b D a Tha hs soluo for aural frequec s eacl he same as he soluo of Ref. [, ]. 5. Coclusos I hs leer Adoma decomposo mehod has bee successfull used o oba he plaes equao. The resuls obaed b decomposo mehod are ecelle agreeme wh classcal mehod. Bu usg he Adoma decomposo mehod s based upo he orhogoal fucos, so developg he mehod for dffere applcaos s o eas ad fdg hese orhogoal fucos are also dffcul. For eample free ed of plae or fed ed of plae fdg he orhogoal fucos ha sasfed he boudar codos are dffcul bu hs possble.

13 Eac soluo for he dffereal equao 677 Refereces [] R. Szlard, Theores ad Applcaos of Plae Aalss, Joh Wle & Sos Ic., New Jerse 4. [] J.S. Rao, Damcs of Plaes, Narosa Publshg House, New Delh, 999. [3] A. C. Ugural, Sresses Plaes ad Shells, d ed., McGraw-Hll Ic., 999. [4] L. Merovch, Prcples ad Techques of Vbrao, Prece Hall Ic., New Jerse 997. [5] L. Merovch, Aalcal Mehods Vbrao, Macmlla, New York, 967. [6] G. Adoma, Solvg Froer problems of Phscs: The Decomposo Mehod, Kluwer Academc, Boso, MA, 994. [7] G. Adoma, R. Rach, Aalcal soluo of olear boudar-value problems several dmesos b decomposo, Joural of Mahemacal Aalss ad Applcaos, 74 (993), [8] G. Adoma, R. Rach, Equal of paral soluos he decomposo mehod for lear or olear paral dffereal equaos, Compuers & Mahemacs wh Applcaos, 9 (99), 9-. [9] G. Adoma, R. Rach, A furher cosderao of paral soluos he decomposo mehod, Compuers & Mahemacs wh Applcaos, 3 (99), [] G. Adoma, R. Rach, N. Shawagfeh, O he aalcal soluo of he Lae- Emde equao, Foudaos of Phscs Leers, 8 (995), 6-8. [] A. M. Wazwaz, Cosruco of solar wave soluos ad raoal soluos for he KdV equao b Adoma decomposo mehod, Chaos, Solos, & Fracals, (), [] A.M. Wazwaz, A ew algorhm for solvg dffereal equaos of Lae- Emde pe, Appled Mahemacs ad Compuao, 8 (), [3] G. Adoma, Soluo of phscal problems b decomposo, Compuers & Mahemacs wh Applcaos, 7 (994),

14 678 A. Karmpour ad D.D. Gaj [4] D. Kaa, A ew approach o he elegraph equao: A applcao of he decomposo mehod, Bulle of he Isue of Mahemacs Academa Sca 8 (), [5] D. Kaa, A applcao of he decomposo mehod o secod order wave equaos, Ieraoal Joural of Compuer Mahemacs, 75 (), [6] A.M. Wazwaz, A ew algorhm for calculag Adoma polomals for olear operaors, Appled Mahemacs ad Compuao, (), [7] A.M. Wazwaz, Aalc reame for varable coeffce fourh order parabolc paral dffereal equaos, Appled Mahemacs ad Compuao, 3 (), 9 7. [8] A. Sadgh, D.D. Gaj, Aalc Treame of Lear ad Nolear Schrödger Equaos: Sud wh homoop-perurbao ad Adoma Decomposo Mehods, Phscs Leers A, 37 (8), [9] A. Sadgh, D.D. Gaj, Eac soluos of Laplace equao b homoopperurbao ad Adoma decomposo mehods, Phscs Leers A, 367 (7), [] A. Sadgh, D.D. Gaj, A Sud o Oe Dmesoal Nolear Thermoelasc b Adoma Decomposo Mehod, World Joural of Modelg ad Smulao, 4() (8), 9-5. Receved: December 8, 7