The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads
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- Ἰωάννης Λιακόπουλος
- 6 χρόνια πριν
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1 Journal of Mecancal Scence Tecnology (7 ( 8~8 wwwsprngerlnkcom/content/78-9 DOI 7/s--- Te unfed equatons to obtan te eact solutons of peoelectrc plane beam subjected to arbtrary loads Zang ang *, Gao Puyun, Dongu Wang Xue Department of Astronautcal Scence ngneerng, Scool of Aerospace Materals ngneerng, Natonal Unversty of Defense Tecnology, Hunan Cangsa, 7, Cna (Manuscrpt Receved February, ; Revsed Aprl 8, ; Accepted Aprl, Abstract Te unfed equatons to obtan te eact solutons for peoelectrc plane beam subjected to arbtrary mecancal electrcal loads wt varous ends supported condtons s founded by solvng functonal equatons Comparng ts general metod wt tradtonal tral-error metod, te most advantage s t can obtan te eact solutons drectly does not need to guess modfy te form of stress functon or electrc dsplacement functon repeatedly Frstly, te governng equaton for peoelectrc plane beam s derved Te general soluton for te governng equaton s epressed by s unknown functons Secondly, n terms of boundary condtons of te two longtudnal sdes of te beam, s functonal equatons are yelded Tese equatons are smplfed to derve te unfed equatons to solve te boundary value problems of peoelectrc plane beam Fnally, several eamples sow te correctness generalaton of ts metod Keywords: Functonal equaton; Peoelectrc plane beam; act soluton; Arbtrary load; lastcty teory Introducton Peoelectrc materals ave been wdely used as actuators sensors n deformaton vbraton control due to te couplng between mecancal electrcal felds Snce ts couplng caracterstcs, t s more complcated for analyss desgn of suc ntellgent structural system as compared wt tradtonal structural system Tat s te reason wy tere are so many numercal models for peoelectrc structure as can be seen n revew papers [-] Snce te elastcty solutons for smple form of peoelectrc structure can be regarded as te bencmark for verfyng te varous numercal models, t as also attracted many scentsts engneers to do wt ts problem Te necessty to guess modfy te form of stress functon electrc dsplacement functon to obtan te elastcty solutons for peoelectrc plane beam subjected to varous smple form of loads can be seen n almost eac paper tat deal wt tese tess or related topcs We only lst te paper tat publsed n recent years, suc as stress functon electrc dsplacement functon of qs ( ( n Ref [], q ( n Ref [], q (7 n Ref [], q ( n Ref [7], qs ( ( n Ref [8], q ( n Ref [9], q (9 n Ref [], Ts paper was recommended for publcaton n revsed form by Assocate dtor Maengyo Co * Correspondng autor Tel: , Fa: mal address: anglang8@gmalcom KSM & Sprnger q ( n Ref [], q ( n Ref [], q ( n Ref [] Usually, tey all need to guess te form of stress functon electrc dsplacement functon before carry out ter soluton procedure can only deal wt one specfc problem If te boundary condtons are canged, weter boundary condtons of te two longtudnal sdes or boundary condtons of te ends supported condtons, te prevous assumptons can not be used It s also te reason tat most of te model gven n tese papers are smply supported beam or cantlever beam For oter type of ends supported condtons, suc as Fed end Fed end peoelectrc plane beam, tere are lttle report about t Huang [] only gves a metod of ow to solve te problem wen te peoelectrc beam acted by pressure load tat can be transferred nto snusodal seres need te value of functon wc represent te load must be equated to ero at te two ends of te beam Te assumpton about load n Ref [] does not accord wt practcal condtons Moreover, te ypotess of stress functon electrcal dsplacement functon n Ref [] are not sute for oter type of loads, suc as sear force or electrcal loads Ts paper consders te beavor of peoelectrc plane beam subjected to arbtrary mecancal electrcal loads Comparng ts general metod wt te metod gven n te paper tat lst n Refs [-], te most advantage s t can obtan te solutons drectly does not need to guess modfy te form of stress functon or electrcal dsplacement functon Furtermore, t can deal wt arbtrary mecancal
2 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 q ( q ( q ( q ( q ( ( w q ( ( u were Φ represent electrcal potental Te equlbrum equatons are σ σ σ σ, D D ( Fg Peoelectrc plane beam subjected to arbtrary mecancal electrcal loads From q ( - te followng compatblty equaton s obtaned electrcal loads, wc can not be realed n any open lterature to te best knowledge of te autors ε ε ε ( Teoretcal formulatons Consder a peoelectrc plane beam wt rectangular cross secton subjected to arbtrary loads as sown n Fg Suppose te wdt of beam s unt, te lengt egt are, respectvely, In plane, te consttutve equatons for peoelectrc materal can be epressed as Ref [] ε s s σ d ε s s σ d ε s σ d σ D d δ σ D d d δ σ were σ, σ, σ denote te stress components, ε, ε, ε are te stran components, u w dsplacement components, D D electrc dsplacement components, electrc feld components s, s, s s denote coeffcents of elastc complance d, d d are peoelectrcty coeffcents δ δ are delectrc mpermeablty coeffcents Consderng te followng boundary condtons of te two longtudnal sdes of te beam σ, q,, q σ, q,, q D, q, D, q σ σ were q ( (,,, (, ( ( q represent te arbtrary mecancal electrcal loads, respectvely Te knematc relaton are u w u w ε, ε, ε Φ Φ, ( By vrtue of q (, obtan Φ ε sσ sσ d Φ Φ ε sσ sσ d, ε sσ d Φ Φ D dσ δ, D dσ dσ δ Te stress components can be epressed by usng stress U as functon, U U U σ, σ, σ Substtutng q (8 nto q ( applyng q ( obtans U U U s s s s ( Φ Φ d ( d d Substtutng q (8 nto q (7 applyng q ( obtan U U d d d Φ Φ ( δ δ ( (7 (8 (9 ( Dfferentatng q ( wt respect to once com- Φ obtans bnng q (9 to elmnate Φ U U U a a a In ts secton, te concrete epressons of Append A ettng U ( U (,, ( a j are gven n (
3 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 Substtutng q ( nto qs (9 ( obtans U U U s ( s s s Φ Φ d ( d d U U d d d Φ Φ δ ( δ Integratng q ( wt respect to once obtans U U U s s s s Φ Φ d ( d d f ( ( ( ( ( were f ( are arbtrary dfferentable functon, ( f j ( denote te -t dervatve of f j ( Combnng qs ( ( obtans Φ U U U a a a a f ( Φ U U U a a a a f ( ( (7 Dfferentatng qs ( (7 wt respect to twce, respectvely, lettng tem equate to eac oter obtans U ( s s δ s δ d ( d d U sδ d ( s s δ sδ ( d d U sδ d sδ U sδ d were U (, U (, f ( s (8 f can be regarded as a assstant functon wen we derve q (8 take no effect ereafter q (8 s te governng equaton for peoelectrc beam Te caracterstc equaton of q (8 s λ λ λ λ ( s s δ s δ d ( d d sδ d ( s s δ sδ ( d d sδ d sδ sδ d Te general soluton of q (8 s (, ( λ ( λ ( λ ( λ ( λ ( λ U were (,,, are s arbtrary functons By usng qs (, ( (8 obtan σ λ λ λ ( ( λ λ ( λ ( ( λ λ ( λ ( ( σ λ λ ( λ ( ( λ λ ( λ ( ( λ λ ( λ ( ( σ λ λ ( λ ( ( λ λ ( λ ( ( λ λ ( λ ( ( denote te -t dervatve of ( ( ( ( ( were j j By applyng q (, q ( ntegratng q ( wt respect to trce obtans Φ a 7 λ λ ( ( a 7 λ ( λ ( a ( ( 7 λ λ f f f were f, f ( f ( are arbtrary functons Te dsplacement electrc dsplacement components can be yelded by usng q ( -, qs ( (7 as λ λ λ λ λ λ λ λ λ (9 were λ, λ, λ are caracterstc roots λ λ λ Te relatonsp between te caracterstc roots materal propertes can be obtaned by comparng q (9 wt q (8 of te coeffcents of λ as follows: u a λ λ a λ λ a λ λ d f d d f d g (
4 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 w a λ λ ( ( a λ ( λ ( a ( ( λ λ d f d f f ( ( D a λ ( λ ( ( a λ ( λ ( ( a λ ( λ ( ( δ f δ f ( δ f ( ( D a λ ( λ ( ( a λ ( λ ( ( a λ ( λ δ f δ f were f ( g ( are arbtrary functons By usng qs (, ( (8 (-(9 obtan ( ( d d f d d f ( f d f d f d g ( ( ( (7 (8 (9 ( δ f δ f δ f δ f ( q ( can be regarded as te quadratc algebra equaton wt respect to Usually, q ( can only ave two roots However, q ( must be satsfed for arbtrary value of n te regon [, ] Te only possble stuaton s te coeffcents of wt any degree equate to ero, wc mply tat δ δ f, f, f f ( Integratng q ( wt respect to twce obtan, f C C f C C δ 7 8 δ f C C C C Substtutng q ( nto q ( obtan ( g ( ( d d C C f ( δ d d C C dc dc7 δ ( ( Snce f ( g are te sngle varable functons of, respectvely, wc sow tat δ δ f d d C C d C d C C C 7 ( 9 g d d C C C C were (,,, ( ( C are unknown ntegral constants Solutons must satsfy te boundary condtons at te two longtudnal sdes of te beam eactly By means of q (, qs (, ( (9 yeld s functonal equatons ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( λ λ λ ( ( λ λ λ ( ( λ λ λ q( ( ( a λ λ ( ( a λ λ ( ( a λ λ qc ( ( ( a λ λ ( ( a λ λ ( ( a λ λ qc ( (7 (8 (9 ( ( (
5 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 87 were δ ( δ ( δ ( δ ( qc q C C C C, qc q C C C C Multplyng q (7 by λ ten mnus q (8 obtans λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ Multplyng q (9 by λ ten mnus q ( obtans λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ ( ( Replacng wt λ n q ( wt λ n q (, respectvely, obtans λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ q ( mnus q ( obtans ( ( ( ( λ λ λ λ λ λ ( ( λ λ λ λ λ λ λ λ ( ( ( ( ( ( ( ( ( ( ( ( q ( q ( λ( λ λ ( ( λ ( λ λ λ λ λ λ λ λ λ λ λ λq λ q λ Multplyng q (7 by λ ten plus q (8 obtan λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ Multplyng q ( by λ ten plus q ( obtan λ ( ( λ λ λ ( ( λ λ λ λ λ ( λq( q( λ λ λ λ λ (7 (8 (9 Replacng wt λ n q (8 wt λ n q (9, respectvely, obtan λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ (
6 88 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 λ λ λ ( ( ( λ λ ( λ λ λ λ ( λ λ λ λ ( λ λ λ ( ( λq λ q λ λ q ( mnus q ( obtans ( ( λ λ λ λ λ λ ( ( λ λ λ λ λ λ λ λ ( ( ( ( ( ( ( ( ( ( ( ( q ( q ( λ( λ λ ( ( λ ( λ λ λ λ λ λ λ λ λ λ λ λq λ q λ ( ( aλ aλ ( ( Multplyng q (7 by a λ ten mnus q ( obtans ( ( λ λ a λ a λ λ λ λqc aq a λ a λ ( ( aλ aλ ( Multplyng q (9 by a λ ten mnus q ( obtans ( ( λ λ a λ a λ λ λ λqc aq a λ a λ ( Replacng wt λ n q ( wt λ n q (, respectvely, obtan ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ q ( mnus q ( obtans ( ( λ λ λ λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ ( ( (7 q (7 mnus q (7 obtans ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ k ( (8 were
7 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 89 k ( ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ ( ( λ( λ λ ( ( λ ( λ λ λq λ q λ λq λ q λ (9 Replacng wt λ n q ( wt λ n q (, respectvely, obtan ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ q ( mnus q ( obtans ( ( λ λ λ λ ( ( λ λ λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ aλ aλ ( λ λ a λ a λ a λ a λ λ λ ( aλ aλ ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ q ( mnus q ( obtans ( ( ( ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ k ( ( were k ( ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ ( ( λ( λ λ ( ( λ ( λ λ λq λ q λ λq λ q λ ( Replacng wt λ n q ( wt λ n q (, respectvely, obtan ( λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ λ a λ a λ λ λ ( aλ aλ a λ a λ λ λ ( aλ aλ ( ( λqc λ aq λ a λ a λ q ( mnus q ( obtans ( ( ( λ ( λ ( ( a8 λ λ λ λ ( ( a8 λ λ λ λ k ( ( (7 were
8 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 k ( ( λ λ ( λ aq qc a λ a λ ( λ λ ( λ aq qc a λ a λ (8 qs (8, ( (7 are te unfed equatons to solve te boundary value problem of peoelectrc plane beam acted by arbtrary mecancal electrcal loads We can construct (, ( ( n terms of k ( (,, Once (, ( ( are constructed, ( can be constructed by replacng wt λ n q ( ntegratng q ( wt respect to trce ( ( λ a8 λ λ a8 λ λ λ qc λ d d d aλ aλ a q λ d d d a λ a λ A A A (9 can be constructed by ntegratng q ( wt respect to trce λ λ λ ( λ λ λ λ( λ λ λ λ λ λ( λ λ λ λ λ λ( λ λ λ λ λ q λ d d d λ q λ d d d λ A A A (7 can be constructed by ntegratng q ( wt respect to trce ( λ λ λ λ ( λ λ λ λ λ (7 λ λ λ λ( λ λ λ λ λ q λ d d d λ q λ d d d λ A A A were A j are unknown constants Te epressons of σ, σ, σ, u, w, D, D Φ can be epressed by usng ( (,,, at present Snce q ( are satsfed, te reamng unknown constants can be determned by usng te boundary condtons of two ends of te beam obtan te epressons of stress, dsplacement, electrcal dsplacement electrcal potental fnally From te soluton procedure mentoned above, ow to construct (, ( ( s te key pont to fgure out te problem Te conclusons gven below can be confrmed by substtutng te epressons of loads epressons of (, ( ( nto qs (8, ( (7 Wen te plane beam acted by loads m qj( qj qj ( j, (7 (, ( ( can be constructed as m m m ( A, ( A, ( A (7 were A,,, m are unknown constants Wen te plane beam acted by loads A m qj( qj qj ( j, (7 (, ( ( can be constructed as m m m ( A, ( A, ( A (7 were A,,, m are unknown constants Wen te plane beam acted by loads A
9 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 m qj ( qj qj ( j, (7 (, ( ( can be constructed as m m m ( A, ( A, ( A (77 were A,,, m are unknown constants Wen te plane beam acted by loads A sn( δ or cos( δ (, (78 q q q q j j j j j were can be constructed as ( δ s a known coeffcent,, δ δ δ δ δ δ ( A cos( F sn( ( A cos( F sn ( ( A cos( F sn ( (79 were A,,,,, F,, F, F are unknown constants Smlarly, f we need to deal wt A sn( δ or cos( δ (, (8 q q q q j j j j j we can smply let te polynomal part of q (79 wt degree of aganst for followng type of loads sn ( δ or cos( δ (, (8 q q q q j j j j j In te case wen te loads are complcated, suc as tey are not contnuous n one of te longtudnal sde, we can translate functons, wc represent te loads, nto Fourer seres as q H H j were η η cos ψ sn (, j j jn n jn n n η j q j d nπ η jn q cos, j H n d H n ψ jn q sn j H n d (, ( ( can be constructed as (8 (8 ( A ncos( H n Fncos( H n n n ( A ncos( H n Fncos( H n n n ( A ncos( H n Fncos( H n n n (8 were A,,, m, n, F n, n, F n, n F n are unknown constants Analogcally, we can deal wt dscontnuous dstrbuton of qj ( ( j, qj ( ( j, by only cange polynomal part of q (8 wt degree of aganst, respectvely A Applcatons In ts secton, varous boundary condtons, ncludng two longtudnal sdes of te beam two ends supported condtons of te beam, are consdered to valdatng te correctness generalaton of ts metod In Secton, te soluton procedure based on te dea gven n Secton s presented n detal Ten te solutons for tree addtonal eamples are gven As te loads become complcated, te fnal epressons become very lengt, In Sectons, 7 te numercal results are gven n te form of surface Te materal constants are all derved from Ref [] For all te numercal eamples, are taken as m m, respectvely Hnged end roller end beam subjected to unform sear force Ts eample s used to compare te solutons wt te results gven n Ref [] Te boundary condtons of te two longtudnal sdes are,,, q (, q ( q q q q q Te boundary condtons of two ends of beam are u, w, w ( σ d, Φ ( σ d ( σ, d ( ( D d, D d Step : Constructng te form of (, ( ( used, are constructed as (8 (8 n terms of te type of load In ts case, q (7 s ( A, ( A, ( A (87
10 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 Step : Substtutng q (87 nto te unfed equatons of qs (8, ( (7 obtan tree lnear algebra equatons wt respect to For te same reason mentoned below q (, te coeffcents of wt any degree must be equated to ero to satsfy te unfed equatons It means tat te unfed equatons wll provde lnear algebra equatons wt respect to unknown constants A, A A, wc wll sow us te relatonsp between ( A A ( A n ts case Step : Substtutng q (87 nto qs (9-(7 to obtan te eplct epressons of (, ( ( It s noted tat at present te epressons of ( (,,, are all eplct polynomal epressons Step : Substtutng ( (,,, nto qs (-(9 combnng qs (, (-( obtan te eplct epressons of σ, σ, σ, u, w, D, D Φ wt some unknown constants We can verfy tat boundary condtons of te two longtudnal sdes of te beam ave been satsfed Step : Te remanng unknown constants can be determned by usng te two ends supported condtons of te beam obtan te fnal solutons By usng te soluton procedure gven above, te epressons for stress, dsplacement, electrcal dsplacement electrcal potental are ( ( q q σ, σ, σ (88 q ( dδ dδ D, D δ (89 q ( dδ dδ dδ Φ δδ (9 q ( sδ dd ( w δ (9 q ( sδ d( qs ( u δ q sδδ δd δd ( d d δ δ (9 We can confrm tat qs (88-(9 satsfy qs (8 (8 ettng te coeffcents of peoelectrcty delectrc mpermeablty equate to ero, te soluton degenerate to te Hnged end Roller end ortotropc plane beam subjected to unform sear force as can be seen n Ref [], wc sow te correctness of ts metod Hnged end roller end beam subjected to unform electrc dsplacement Te boundary condtons of te two longtudnal sdes are q (, q ( q q, q, q, q (9 Te ends supported condtons are te same to q (8 By usng te soluton metod gven above, te solutons are σ, σ, σ (9, D ( D q q (9 q δ δ ( Φ δδ (9 qd q ( δd δd w δ δδ q ( δd δd δd qd ( δ δ δ (97 We can confrm tat qs (9-(97 satsfy qs (9 (8 Te epresson for u are very lengt wll not lst ere for concse It s noted tat wen peoelectrc beam acted by unform electrcal dsplacement te components of w arse but te components of stress equate to ero, wc can be utled for deformaton control Fed end free end beam subjected to unform sear force Te boundary condtons of te two longtudnal sdes are,,, q (, q ( q q q q q Te boundary condtons of two ends of beam are u d w, w,, Φ d ( σ d ( σ d ( σ d ( D d Te solutons are ( (,, (98 (99 q σ, σ q ( ( σ ( q ( δd δd( ( D, D δ ( q ( d Φ δ ( ( δ ( q s d w δ ( δ ( ( q s dd δ (
11 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 We can confrm tat qs (-( satsfy qs (98 (99 Fed end fed end beam subjected to unform sear force Te boundary condtons of te two longtudnal sdes are,,, q (, q ( q q q q q Te boundary condtons of two ends of beam are u, w, u, w dw d dw,, Φ d ( ( D d, D d ( ( (a (b Fg Dstrbuton of σ σ for wole peoelectrc beam Te solutons are ( ( q q σ, σ, σ ( q ( δd δd D, D δ (7 q ( δd δddδ Φ δδ (8 q ( sδ dd ( w δ (9 We can confrm tat qs (-(9 satsfy qs ( ( Fed end roller end beam subjected to lnear sear force Te numercal results for dstrbuton of σ σ n wole peoelectrc beam are gven n Fg (a (b, respectvely, wt loads parameters taken as q q Pa It s noted from Fg (a te ma value of σ take place near te regon of coordnate (, But for σ, te ma value arses near te regon of Fed end nged end beam subjected to quadratc electrc dsplacement Te boundary condtons of te two longtudnal sdes are q, q, q, q q (, q ( q q q Te boundary condtons of two ends of beam are ( Te boundary condtons of te two longtudnal sdes are q, q, q, q q( q q, q ( Te boundary condtons of two ends of beam are ( u, w, u, w d w d, Φ, ( σ d ( ( D d, D d ( u, w, u d w d, Φ ( σ d ( σ, d ( ( D d, D d ( Te numercal results for dstrbuton of w u n wole peoelectrc beam are gven n Fg (a (b, respectvely, wt loads parameters taken as q q q C m We can fnd from Fg (a tat te ma value of w do not arses at te center of te beam also not te bottom upper surface of te beam As to te dstrbuton of u, te cange nterval of ts value are te same to w, but te ma value of u take place at te borderlne of te beam
12 8 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 Te ends supported condtons are te same to q ( Te numercal results for dstrbuton of D D n wole peoelectrc beam are gven n Fg (a (b, respectvely, wt loads parameters taken as q Pa Comparng Fg (a wt (b, te dstrbuton rule of D D s totally dfferent Te dstrbuton of D can be regarded as antsymmetry wt respect to te center poston (, of beam Te dstrbuton of D can be treated as antsymmetry aganst te as lne of te beam (a Conclusons Te formulae presented n ts paper can consder peoelectrc plane beam subjected to arbtrary mecancal electrcal loads wt varous ends supported condtons Comparng ts general metod wt tradtonal tral--error metod, te most advantage s t can obtan te eact solutons drectly does not need to guess modfy te form of stress functon or electrcal dsplacement functon amples sow te correctness generalaton of ts metod Fg Dstrbuton of w u for wole peoelectrc beam Fg Dstrbuton of D 7 Fed end fed end beam subjected to unform pressure force Te boundary condtons of te two longtudnal sdes are q (, q ( q, q, q q, q (b (a (b D for wole peoelectrc beam ( References [] A Benjeddoudvances n peoelectrc fnte element modelng of adaptve structural elements a survey, Computers Structures, 7 ( ( 7- [] D A Saravanos P R Heylger, Mecancs computatonal models for lamnated peoelectrc beams plates sellsppled Mecancs Revews, ( (999 - [] H A Irsck, Revew on statc dynamc sape control of structures by peoelectrc actuaton, ngneerng Structures, (7 ( - [] D J Huang, H J Dng W Q Cen unfed soluton for an ansotropc functonally graded peoelectrc beam subject to snusodal transverse loads, Journal of Intellgent Materal Systems Structures, (8 (9 - [] D J Huang, H J Dng W Q Cen, Statc analyss of ansotropc functonally graded magneto-electro-elastc beams subjected to arbtrary loadng, uropean Journal of Mecancs A/Solds, 9 ( ( -9 [] D J Huang, H J Dng W Q Cennalyss of functonally graded lamnated peoelectrc cantlever actuators subjected to constant voltage, Smart Materals Structures, 7 (9 (8 - [7] D J Huang, H J Dng W Q Cennalytcal soluton for functonally graded magneto-electro-elastc plane beams, Internatonal Journal of ngneerng Scence, ( (7 7-8 [8] T T Zang Z F S, Bendng beavor of - multlayered peoelectrc curved actuators, Smart Materals Structures, ( (7 - [9] Z F S, Bendng beavor of peoelectrc curved actuator, Smart Materals Structures, ( ( 8-8 [] H J Xang Z F S, lectrostatc analyss of functonally graded peoelectrc cantlevers, Journal of Intellgent Materal Systems Structures, 8 ( (7 7-9 [] Z F S, H J Xang B F Spencer, act analyss of
13 Zang et al / Journal of Mecancal Scence Tecnology (7 ( 8~8 8 mult-layer peoelectrc/composte cantlevers, Smart Materals Structures, (9 ( 7-8 [] Y Cen Z F S, act solutons of functonally gradent peotermoelastc cantlevers parameter dentfcaton, Journal of Intellgent Materal Systems Structures, ( ( -9 [] H J Xang Z F S, Statc analyss for mult-layered peoelectrc cantlevers, Internatonal Journal of Solds Structures, (9 (8-8 [] T Yu, Z Zong, Bendng analyss of a cantlever peoelectrc functonally graded beam, Scence n Cna Seres G: Pyscs, Mecancs & Astronomy, ( ( [] D J Huang, H J Dng H M Wangnalytcal soluton for fed-end ortotropc beams subjected to unform load, Journal of Zejang Unversty (ngneerng Scence, ( ( - Append a a a a a a a a d sδ δ ( d d dδ ( s s δ d ( d d δ ( d d dδ sδ δ ( d d dδ δ δ ( d d dδ sδ d ( d d δ ( d d dδ ( s s δ ( d d δ ( d d dδ sδ δ ( d d dδ δ δ ( d d d δ a ( s d a λ ( s d a λ d a s d a s d a d a s d a s d a d λ a ( λ ( λ λ a ( λ ( λ λ a (A (A (A a a ( s da λ s da d λ a a ( s da λ s da d λ a a ( s da λ s da d λ a a ( d δa λ δa δ λ a a ( d δa λ δa δ λ a a ( d δa λ δa δ λ a a ( d δa λ ( d δa λ δ λ a a ( d δa λ ( d δa λ δ λ a a ( d δa λ ( d δa λ δ λ a a7 aλ a λ a a7 aλ a λ a a7 aλ a λ λ( λ λ λ( a8 a8, a8 a8 λ( λ λ λ( λ λ λ( λ λ λ( a8 a8, a8 a8 λ( λ( aλ aλ λ λ a8, a8, a8 aλ aλ λ (A (A (A (A7 (A8 Zang ang receved te B MS degrees n cvl engneerng from ogstcal ngneerng Unversty, Congqng, Cna n 7, respectvely He s currently a doctor cdate at te Department of Astronautcal Scence ngneerng, Natonal Unversty of Defense Tecnology Hs researc focuses on elastc analyss control of peoelectrc structures
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