含壓電主動(dòng)層疊層夾持梁在電場(chǎng)作用下的解析解ppt課件

1、School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA趙壽根趙壽根航空科學(xué)與工程學(xué)院固膂力學(xué)研討所航空科學(xué)與工程學(xué)院固膂力學(xué)研討所School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 內(nèi)容：由彈性力學(xué)根本方程和力電耦合方程出發(fā)，內(nèi)容：由彈性力學(xué)根本方程和力電耦合方程出發(fā)，采用半逆解法得到

2、了含壓電自動(dòng)層復(fù)合資料夾持采用半逆解法得到了含壓電自動(dòng)層復(fù)合資料夾持梁在電場(chǎng)作用下的解析解，該解較為適宜提示復(fù)梁在電場(chǎng)作用下的解析解，該解較為適宜提示復(fù)合資料梁彎曲控制方面的作動(dòng)基理問(wèn)題，同時(shí)電合資料梁彎曲控制方面的作動(dòng)基理問(wèn)題，同時(shí)電勢(shì)沿壓電介質(zhì)的厚度方向由一高階多項(xiàng)式表示更勢(shì)沿壓電介質(zhì)的厚度方向由一高階多項(xiàng)式表示更為貼近真實(shí)情況。為貼近真實(shí)情況。Q 目的：為壓電復(fù)合梁的運(yùn)用者和智能構(gòu)造的外形目的：為壓電復(fù)合梁的運(yùn)用者和智能構(gòu)造的外形控制提供實(shí)際參考和自創(chuàng)?？刂铺峁?shí)際參考和自創(chuàng)。 School of Aeronautical Science and Engineering, BUAASch

3、ool of Aeronautical Science and Engineering, BUAAQ 概略：圖所示的含壓電層復(fù)合資料梁概略：圖所示的含壓電層復(fù)合資料梁(x方向也可稱方向也可稱之為之為1方向，方向，z方向也可稱之為方向也可稱之為3方向方向)。含壓電層復(fù)合資料梁School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA 簡(jiǎn)化：由于y方向(垂直于xz平面，也可稱之為2方向)可以自在伸展，加之電壓施加的方向?yàn)?向，因此該層合壓電梁在電場(chǎng)作

4、用下的問(wèn)題可以作為x-z平面內(nèi)的平面應(yīng)力問(wèn)題來(lái)求解。School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ (1) 平衡方程平衡方程Q 由前面的分析可得復(fù)合資料梁的平衡方程為：由前面的分析可得復(fù)合資料梁的平衡方程為： 0zxxzx0 xzxzz(1)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineerin

5、g, BUAAQ (2)協(xié)調(diào)方程：協(xié)調(diào)方程：Q 其協(xié)調(diào)方程為：其協(xié)調(diào)方程為：022222xzxzzxzx(3) 物理方程物理方程本體資料物理方程為：本體資料物理方程為： sxzszsxsssssszxszsxsssss44331313110000(2)(3)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA壓電層資料物理方程為：壓電層資料物理方程為： zaxzazaxaaaaaaxzazaxEddsssss0000033314433131

6、311 33333310EddDaxzazaxz(4)(5)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA(4) 幾何方程幾何方程 xwzuzwxuxzzx(5) 電學(xué)方程電學(xué)方程 0zDzzEz(6)(7)(8)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ (1) 應(yīng)力

7、函數(shù)的求取應(yīng)力函數(shù)的求取Q 由平衡方程式可知存在應(yīng)力函數(shù)由平衡方程式可知存在應(yīng)力函數(shù)使得：使得：zxxzxzzx22222(9)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA對(duì)于下層壓(+)電復(fù)合層，設(shè)其應(yīng)力函數(shù)用 a1表示 電勢(shì)用 a1表示 ，將(9)和(8)代入(5)式有： zxdzdDaaaz1332123321231將上式(10)代入(7)式中有： 0212332133331331zzxdzdaaa(10)(11)School

8、 of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA由式(11)可設(shè)下壓電層的電勢(shì)分布函數(shù) a1為： 32211uzuzua式中z的取值范圍為： )5 . 0(5 . 0asshhzh設(shè)下壓電層的應(yīng)力函數(shù) a1表達(dá)式為： 362524232211zazxaxzazaxaxzaa(12)(13)(12)和(13)式代入(11)式有： 0226133533631uadad(13)式代入(9)式結(jié)合邊境條件有： 00)5 . 0(22)2(226226435

9、4152643aszzaxzazaxhhaxaaxazaazaazaxaaii式中： asihhz5 . 0(14)(15)結(jié)合(14)和(15)式有： 0052aa441)2(2ahhzaaas6333113adu從而得上層壓電復(fù)合層的應(yīng)力函數(shù) a1表達(dá)式為： 36242341)2(zaxzazaxzahhasa(16)(17)采用類似于分析下層壓電復(fù)合層的方法可得上層壓電復(fù)合層的電勢(shì)分布函數(shù) a2和應(yīng)力函數(shù) a2分別為： 32212vzvzva式中z的取值范圍為：sashzhh5 . 0)5 . 0(36242342)2(zbxzbzbxzbhhasa(18)(19)School of A

10、eronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA將(18)和(19)代入?yún)f(xié)調(diào)方程有： 6333113bdv從而由本體構(gòu)造的相容方程、 a1和 a2構(gòu)造的應(yīng)力函數(shù)為： 可設(shè)本體34232213zcxzczcxzcs(20)(21)(2) 應(yīng)力表達(dá)式的求取將式(17)、(19)和(21)代入(9)式有： (a) 下壓電層中的應(yīng)力和電場(chǎng) zaahhzaxaaasaxzazax446432)2(06222336316uzadEz(22)School of Aeron

11、autical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA(b) 上壓電層中的應(yīng)力和電場(chǎng) zbbhhzbxbbasaxzazax444432)2(06222336316uzbdEz(23)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA(b) 本體構(gòu)造中的應(yīng)力 zcczcxccsxzszsx3143220622(3)

12、位移表達(dá)式的求取將(22)(24)代入各自的資料本構(gòu)方程和變形幾何方程可得： (24)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA(a) 下壓電復(fù)合層的位移場(chǎng) zasahhsuzaddzaxaasuzaddzaxaasxwzuzwxuaasaaaaaaa4444442336313364313233631316431111112)2(6)622(6)622(25)前兩式可得： )(6)622()(6)622(1233631336431

13、31233631316431111xgzuzaddzzaxaaszfxuzaddxzaxaaswuaaaa第三式可得：9726332311187444244413113)2()()()(axaxadsazazahhszassxgzfaasaaa(26)(27)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA從而得位移場(chǎng)： 9726332311123363133643131874442444132336313164311136)622()

14、2()(6)622(axaxadszuzaddzzaxaaswazazahhszassxuzaddxzaxaasuaaaasaaaaa(28)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA(b) 上壓電復(fù)合資料的位移場(chǎng)同理可得上壓電復(fù)合層的位移場(chǎng)為： 9726332311123363133643132874442444132336313164311236)622()2()(6)622(bxbxbdszvzbddzzbxbbswbzbz

15、bhhszbssxvzbddxzbxbbsuaaaasaaaaa(29)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA(c) 本體構(gòu)造的位移 )2()622()622(314443213432113333zccszcxccszcxccsxwzuzwxusssssss前兩式可得： )()322()()62(34321334321133xgzzcxccszfxzcxccswussss(30)(31)School of Aeronautic

16、al Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA第三式可得： 7524116523132344144333)()(cxcxcsczczcszcszcsxgzfssss從而有： 752411432136521332314443211333)322()()62(cxcxcszzcxccsczczsczczcsxzcxccswusssssss(32)(33)School of Aeronautical Science and Engineering, BUAASchool of A

17、eronautical Science and Engineering, BUAAQ (1) 位移邊境條件位移邊境條件Q (a) 當(dāng)當(dāng)x=0時(shí)：時(shí)：0000333xsssxwwu(34)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA(b) 當(dāng)z=+0.5hs和z=-0.5hs時(shí)：sshxsshxsswuwu5 . 0335 . 011sshxsshxsswuwu5 . 0335 . 022School of Aeronautical

18、Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAA (2) 應(yīng)力邊境條件 當(dāng)z=+0.5hs+ha和z=-0.5hs-ha時(shí)：00)5 . 0(ashhzaxzaz在設(shè)應(yīng)力函數(shù)時(shí)已思索該邊境條件。 (3) 力邊境條件 當(dāng)壓電復(fù)合梁只遭到電場(chǎng)作用時(shí)，那么有：aaaaaaaaaaaaaaaaaaaaaaaahhhaxhhhhhaxsxhhhaxzhhhhhaxzsxzhhhaxhhhhhaxsxdzzdzzdzzdzdzdzdzdzdz5 . 0)5 . 0(5 . 05 . 05 .

19、05 . 05 . 0)5 . 0(5 . 05 . 05 . 05 . 05 . 0)5 . 0(5 . 05 . 05 . 05 . 0000 (3) 電學(xué)邊境條件 (a) 當(dāng)z=-0.5hs和+0.5hs時(shí)：05 .01ashhza0)5.0(2ashhza(b) 當(dāng)z=+0.5hs+ha和z=-0.5hs-ha時(shí)：Uashhza5.01Uashhza)5 . 0(2將式(22)(24)代入式(37)，將式(28)、(29)和(33)代入式(35)，將式(33)代入式(34)，將式(12)和(18)代入式(38)和(39)有： 01a02a04a05a07a08aAs2h)U)h+)(h

20、h+h(hs6d+(UAda11aas2aas11231313aAda31as331123163)Uh+(hs6d-2A)Uh+)(hdd-(ssd3hA2h)Uh+)(hhh6(h+(AsddhAs2h)Uh+)(hhh6(h+(ssdh-2hp)U+)(hsd-(ssdh3h-as333133a1311312saas2aas1123133sa11aas2aasa1311331s23133a111331as9a01b02b04b05b07b08b99ab As2h)U)h+)(hh+h(hs6d+(UAda11aas2aas11231313bAdab31as3311231663)Uh+(hs

21、6d-01c02c03c05c06c07c)Ud-)(sh+(h2d-23133a11as314cAu)Uh+(hs6d-as112311a3311s2a33113a3311a3s2313s33a113sa112)U)/hsh12hs8hsh6hdh-s(hs (uA2h)Uh+(hs)dh+h6(h+(Uh-ss2h2)Uh+(hsd3haas112312aass33a11113aas112312s3uAv)Uh+(hs6d-as112311a3311s2a33113a3311a3s2313s33a113sa112)U)/hsh12hs8hsh6hdh-s(h-s(vA2h)Uh+(hs)d

22、h+h6(h+(Uh-ss2h2)Uh+(hsd3haas112312aass33a11113aas112312s3v其中： 331111311311211211323133112)6126)(sshshshhshhshdsAappspspasaSchool of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 如圖如圖1所示的復(fù)合資料梁，其幾何參數(shù)為壓電層所示的復(fù)合資料梁，其幾何參數(shù)為壓電層厚度為厚度為ha=0.5mm，本體構(gòu)造層厚度為，本體構(gòu)造層厚度為 hs=1.0mm，其長(zhǎng)度其長(zhǎng)度l=100mm，施加的電壓值為，施加的電壓值為U=100V。所。所用的資料的力學(xué)和電學(xué)性能參數(shù)如下表所示。計(jì)用的資料的力學(xué)和電學(xué)性能參數(shù)如下表所示。計(jì)算所得和有限元計(jì)算所得復(fù)合資料梁中的位移分算所得和有限元計(jì)算所得復(fù)合資料梁中的位移分布和應(yīng)力分布如以下圖所示。布和應(yīng)力分布如以下圖所示。 School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engin