Dynamic analysis of the influence of fiber orientation in composite laminated plates

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1 Vol. 7(1), pp. 1-8, March, 215 DOI: /JMER Artcle Numer: 54F88FB5192 ISSN Copyrght 215 Author() retan the copyrght of th artcle Journal of Mechancal Engneerng Reearch Full Length Reearch Paper Dynamc analy of the nfluence of fer orentaton n compote lamnated plate Adrana Dacenco*, Aroto Brtan Jorge and Patrca da Slva Mechancal Engneerng Inttute, Federal Unverty of Itajuá - UNIFEI, Campu Joé Rodrgue Seara, CP 5, CEP , Itajuá, MG, Brazl. Receved 22 Septemer, 213; Accepted 2 Octoer, 214. h paper evaluate numercally the dynamc ehavor of tructural compote lamnate materal n relaton to the angular change n fer layer of the lamnated compote. he ehavor of the materal modeled through fnte element method, where the Frt Order Shear Deformaton heory (FSD) ued whch mplemented on a rectangular element erendpty contanng eght node. he mathematcal modelng ha een mplemented ung the commercal avalale oftware MALAB. hrough numercal mulaton, t wll e pole to otan the natural frequence. And we wll preent a entvty analy wth repect to the fer orentaton parameter. Key word: Dynamc ehavor, compote lamnated plate, Frt Order Shear Deformaton heory (FSD), fer orentaton parameter. INRODUCION Compote materal have een ncreangly hghlghted n recent decade due to t advantage to the tradtonal engneerng materal (teel or alumnum), characterzed y t low denty aocated wth hgh trength/tffne relaton charactertc, and t ant-corroon properte. A compote materal can e defned when two or more dfferent materal are comned together to create a uperor and unque materal, to otan a et of properte that none of the component ndvdual feature (Mendonça, 25). here are dfferent clafcaton for compote materal avalale n the lterature. hey can e clafed accordng to morphology of the dpered phae n partcle renforced compote, fer renforced compote and compote tructural (Redd 1997). he compote tructural lamnate cont of a layer tac attached to each other wth the fer orented n dfferent drecton. And lamnate typcally cont of everal layer, often dentcal, varyng t gudelne to etter meet the degn requrement and manufacturng (Dacenco, 21). Compote materal are ncreangly preent n dfferent area of Engneerng where t neceary to tudy and analyze the ehavor of thee materal ecaue, t can e tated that the engneerng tructure are uject to dturance that affect tatc or dynamc repone charactertc and performance of the tructural ytem and control acton or vraton montorng natural frequence ecome neceary. In the cae of tructural compote ecome relevant perform the analy of tacng equence of layer of the *Correpondng author. E-mal:aadacenco@yahoo.com.r. Author() agree that th artcle reman permanently open acce under the term of the Creatve Common Attruton Lcene 4. Internatonal Lcene

2 2 J. Mech. Eng. Re. compote tructure n the context of attenuaton of vraton level, a hgh vraton are to e avoded wthn the tructural ntegrt and th can e analyzed to otan the functon of frequency repone of the compote. he ue of numercal method to ae the tructural talty have een wdely tuded and appled y Reddy (1997), Fara et al. (26), Lma et al. (29) and Dacenco et al. (213). Among thee method, the fnte element method ha een hown to e the mot utale, due to t charactertc of modelng flexlty and relatve eae of mplementaton numercal computatonal complex prolem n engneerng and n the lterature can e found a wde varety of theore ued n the formulaton of element appled to fnte compote materal. For the purpoe of th paper, the well-nown Frt- Order Shear Deformaton heory (FSD), propoed y Reddy (1997), uch theory preent gnfcant reult when worng wth moderately thc plate where mplemented n a rectangular Serendpty element contanng eght node and fve degree of freedom per node. Baed on what wa tated aove, the man ojectve of th artcle the mplementaton numercal computaton ung the fnte element method for compote plate and evaluate the nfluence of the tacng equence of the layer compote n ther dynamc ehavor n term of characterzaton of natural frequence and vraton ampltude. FINIE ELEMEN MODELING he mechancal ehavor of the compote tructure can e modeled y ung the FSD, n whch the dplacement at an artrary pont n uch a compote expreed a follow: x, t Azux, t U (1) In Equaton (1): x, t u x, t vx, twx, t U (2) 1 z A( z) 1 z 1 u x, yt, u x, t vx, t wx, t x, t x, t whereu x, t, v x, t, e w x t x y (3) (4), denote the dplacement n drecton x, y and z. repectvely., v w, are, repectvel the md- u, and x y plane dplacement and the cro-ecton rotaton n x and y drecton. he uual tran dplacement relaton are ued and the reultng tran are eparated n endng and tranvere hear tran, and, repectvel a follow:,,,,,,, D D u D u (5) x yzt z 1 xyt z xyt,,,,,,, D u D u (6) x yzt 2 xyt xyt Where x, t xx yy zz xy x, t. u x and yz zx xx, yy v y, w z, u y v x, zz yz v z w y u z w x.matrce D and zx,...,3 are formed y dfferental operator appearng n the tran dplacement relaton. Dcretzaton of the dplacement varale made y ung approprate nterpolaton functon. Hence, for the eght node rectangular plate element, the fve mechancal varale ncluded n vector u(x, t) are nterpolated from ther correpondng 4 nodal value through the followng relaton:,, t, t u N u (7) Where: u t t u t u t u and u t u v w 1to 8 x y xy. N, of dmenon 5 x 4, the matrx formed y the tandard erendpty eght-node hape nterpolaton functon formulated n local coordnate,, 1 ξ 1. And th wa llutrated n Fgure 1. By aocatng Equaton (1) to (4), the dplacement and tran feld are found to e expreed n term of the nodal value a follow: U x t AzN, ut, (8) x, t D zn, ut B,, zut (9) x, yzt,,, t, t D N u B u (1) Baed on the tre-tran relaton, the tran and netc energe of the compote plate element can e formulated

3 Dacenco et al. 3 Fgure 1. Serendpty famly element ued n the fnte element formulaton of lamnated compote plate: (a) local coordnate, () coordnate. n term of the natural varale of tran feld and the mechancal materal properte. After, Lagrange equaton are ued, conderng the nodal dplacement and rotaton a generalzed coordnate, to otan the followng elementary ma and tffnee matrce, repectvely: n z e N, A z A z N, det J d d dz (11) 1 z z 1 1 n e B,, z C B,, z det J d d dz 1 z 1 z z 1 1 (12) 1 1 n z e 1 z z 1 1 B, C B, det J d d (13) det ndcate the determnant of the Jacoan of the tranformaton from the n-plane phycal varale x, yto the natural varale,, and matrce C and C In Equaton (11) to (13) J repreent, repectvel the orthotropc endng and hear elatc matrce of the th layer, whch are contructed accordng to the Clacal Lamnate heory (CL) a follow: C C (14) C C (15) Where C and C are, repectvel the endng and hear elatc property matrce of the th layer, referred to t prncpal orthotropc ax, and and are the aocated rotaton matrce. From the elementary matrce computed for each element of the fnte element meh, the gloal equaton of moton are contructed, accountng for the node connectvt ung tandard fnte element aemlng procedure, Huener et al. (1982). After aemlng, the gloal equaton of moton n the tme doman can e wrtten a follow: Mq t Kqt f t (16) nelem nelem Where e and e are the gloal FE e1 e1 ma and tffne matrce. Symol ndcate matrx aemlng and f t q t the vector of gloal d.o.f. the vector of generalzed external load. he equaton of moton n the tme doman (16) can e ued to perform varou dynamc analyze uch a the calculaton of repone tme, egenvalue and egen vector, and frequency repone. In the frequency

4 4 J. Mech. Eng. Re. doman the aove equaton n (16) tae the form: K 2 M Q F, (18) F U, cq Y (19) NxN whch M, K (, ) R repectvely repreent the ma matrx (ymmetrc, potve-defnte) and the tffne matrx (ymmetrc and nonnegatve defnte.) N N Q() R and F () R repreent, repectvel the dplacement vector and the vector of external force. c Y () R the vector of repone, and the vector f U () R reduced from external force. And the R Nxf cxn c R are the matrce that matrce and allow to chooe among degree-of-freedom of the fnte element model the degree of freedom where the force are appled exctaton, and the degree of freedom whch are calculated ytem repone, repectvely. Expreon (17) yeld the followng expreon for the complex dynamc tffne matrx: Z 2, K G, K M (2) e Defned the complex tffne, the next tep to olve the ytem n the frequental doman, whch can e done y uldng dynamc flexlty matrx or array of Frequency Repone Functon (FRF): -1, cz, H (21) SENSIIVIY ANALYSIS FOR DYNAMIC RESPONSES OF FINIE DIFFERENCES he gloal fnte element matrce appearng n fnte element modelng etalh the dependence of the repone of the ytem wth repect to a et of degn parameter. Such functonal dependence can e expreed a follow: (Lma et al., 29): p r r M, K p v (22) Where r and p degnate vector of tructural repone and degn parameter, repectvely. he entvty of the repone wth repect to a gven parameter p, evaluated for a gven et of value of the degn parameter p can e etmated y fnte dfference and then defned a the followng partal dervatve: r p p r lm Δp M p Δp, K p Δp rm p, K p Δp Δp (23) Where p an artrary varaton tendng to zero, appled to the current parameter value p whle all other parameter are ept unchanged. he entvty of the repone wth repect to p can e etmated numercally y fnte dfference y calculatng ucceve repone correpondng to p p e p p p, a follow: r p p r M p Δp, K p Δp r M p, K p Δp Δp (25) h procedure effcent when t come to mall tructure and ther ue qualtatvely decre the degree of nfluence of dfferent degn parameter on the dynamc repone: the larger the ampltude of the entvty functon wth repect to a gven degn parameter, the greater the nfluence of th parameter on the dynamc repone. NUMERICAL SIMULAIONS Here, we preent two dfferent numercal applcaton were mplemented n the programmng envronment Matla. he frt attempt to evaluate the FSD accuracy of the theory n otanng the natural frequence of compote tructure. he econd and thrd numercal applcaton how the effect of orentaton and tacng equence of layer n the lo factor and natural frequence vraton of lamnated compote tructure. In the two numercal applcaton t condered a flat plate compote lamnate, ung the FE model of a mple upported quare L x = L y =.16 m, compote plate a hown n Fgure 2 llutrate the model compoed y a total numer of 64 fnte element and 225 node. he followng mply upported oundary condton are appled on the quare compote plate (Correa et al., 2): u = w = ψ z = ς x = ς z = n y = and y = L e u = w = ψ z = ς y = ς z = n x = and x = Lx. he compote plate cont of 5 layer of the ame thcne h/5 (h=a/128m).he real value of the materal properte charactertc of each layer are E1 172, 4GPa, E2 E3 6, 89GPa, G12 G13 3, 45GPa, G23 1, 38GPa, 12 13,25,, , 1566 g m. Frt applcaton In th frt applcaton wll e condered fve layer lamnated orented ( / 9 / / 9 / ) and (9 /

5 Dacenco et al. 5 Fgure 2. Mah of fnte element. Fgure 3. Vraton ampltude for dfferent tacng equence. ale 1. Natural frequence for the frt three mode of vraton to ( / 9 / /9 / ). Mode Natural frequency (Hz) ale 2. Natural frequence for the frt three vraton mode of the (9 / /9 / /9 ). Mode Natural frequency (Hz) / 9 / / 9 ) wth the ame thcne. Fgure 3 how vraton ampltude of the lamnated tructure. he reult hown n ale 1 and 2 clearly how the nfluence of orentaton of the fer on the dynamc ehavor, nce the ymmetry of the varaton layer alter gnfcantly the frequence of vraton. Second applcaton he dcretzed y fnte element, the geometrcal

6 6 J. Mech. Eng. Re. Fgure 4. Compote lamnate llutratng the degn parameter. Fgure 5. Sentvty of the FRF otaned wth repect to for %. charactertc and condton of lamnated compote contour of the plate are the ame a thoe n the example ecton earler. It preent the calculaton of entvte of the FRF of the compote plate lamnated accordng to Equaton (25) ung ucceve varaton of fer orentaton, correpondng to 1% of the nomnal confguraton (9 / /9 / /9 ). Fgure 4 llutrate the compote lamnate. Fgure 5 to 7 how the repone functon frequency calculated y fnte dfference. he larger the ampltude of the entvty functon wth repect to a gven degn parameter, the greater the nfluence of th parameter on dynamc repone. In th cae, t oerved that the orentaton of the three fve layer how a gnfcant nfluence on the dynamc ehavor of compote plate. Concluon he numerou numercal mulaton conducted to evaluate the performance of modelng procedure

7 Dacenco et al. 7 Fgure 6. Sentvty of the FRF otaned wth repect to for %. Fgure 7. Sentvty of the FRF otaned wth repect to 5 for 5 5 1%. developed a a tool for analy and degn lamnated compote tructure to how mportant apect of dynamc ehavor of the ame n term of the parameter entvte orentaton of the fer n the layer. Conflct of Interet he author have not declared any conflct of nteret.

8 8 J. Mech. Eng. Re. ACKNOWLEDGEMEN he author are deeply grateful to the followng organzaton: Mna Gera State Agency FAPEMIG for the fnancal upport to ther reearch actvte. REFERENCES Correa VMF, Gome MAA, Suleman A, Soare CMM, Soare CAM (2). Modellng and Degn of Adaptve Compote Structure. Comput. Method Appl. Mech. Eng. 185: de Lma AMG, Dacenco AA, Côrrea EO (29). Fnte Element Modelng of Compote Sandwch Plate wth Vcoelatc Layer. In: 2 th Internatonal Congre of Mechancal Engneerng (COBEM), Gramado, RS. Dacenco AA, Lma AMGD, Corrêa EO (213). Fnte element reducton trategy for compote andwch plate wth vcoelatc layer. Mater. Re. 16(2): Fara AW (26). Modelagem por Elemento Fnto de Placa Compota dotada de Senore e Atuadore Pezoelétrco: mplementação computaconal e avalação numérca f. Dertação (Engenhara Mecânca) Faculdade de Engenhara Mecânca, Unverdade Federal de Uerlânda. Huener KH, hornton EA (1982). he fnte element method for engneer. New Yor: John Wley & Son. Mendonça PR (25). Matera Compoto and Etrutura - Sanduíche: Projeto e Anále 1.ed. Manole. Reddy JN (1997). Mechanc of lamnated compote plate: theory and analy. 2nd ed. CRC Pre.