壓電主動(dòng)層在梁振動(dòng)控制中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)院航空科學(xué)與工程學(xué)院School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 設(shè)如下圖復(fù)合資料梁，它由三部分組成分別為上設(shè)如下圖復(fù)合資料梁，它由三部分組成分別為上下下 ha厚度的壓電層，中間為厚度的壓電層，中間

2、為hs厚度的本體構(gòu)造層。厚度的本體構(gòu)造層。 School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 根據(jù)梁微元的分析，其橫向的振動(dòng)方程可以用下根據(jù)梁微元的分析，其橫向的振動(dòng)方程可以用下式表示：式表示： School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 其中：其中： Q為剪力，為剪力

3、， M為彎矩，為彎矩， f(x,t)為為單位長(zhǎng)為為單位長(zhǎng)度上的橫向載荷。度上的橫向載荷。0),(txfxQwA QxM(1)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 對(duì)于自在振動(dòng)令對(duì)于自在振動(dòng)令f(x,t)=0,從而上式從而上式(1)簡(jiǎn)化為：簡(jiǎn)化為： 022xMwA (2)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Sci

4、ence and Engineering, BUAAQ 對(duì)于彎矩對(duì)于彎矩M，可有下式得：，可有下式得： 111()22211111()222111222()222111111222()22222(3)sssasasssssasasshhh hasazh hhhhhh hasaijiijih hhhab iMB zdzB zdzB zdzwwwB E zeE zdzBE zzdzB E zeE zdzxxxwMMEx ( )()iaVH xH xlEhSchool of Aeronautical Science and Engineering, BUAASchool of Aeronautica

5、l Science and Engineering, BUAAQ 其中：其中： 333112121132322saassasBBMEhEhhh21()abijasaMBeEhh h 11( )()aaEEH xH xl為資料彈性模量為資料彈性模量 School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 其中其中 H(x)為海維賽德函數(shù)，其定義如下：為海維賽德函數(shù)，其定義如下： axaxaxH10)(Q 將將(3)式代入梁的振動(dòng)方程式代入梁的

6、振動(dòng)方程(2)式中有：式中有： 224422xEMxwMtwAiba(6) School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 將上將上(6)式化簡(jiǎn)有：式化簡(jiǎn)有： 224422xEdxwctwiQ 其中：其中： AMcaAMdb(7) School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BU

7、AAQ (1) 邊境條件邊境條件Q 當(dāng)當(dāng)x=0時(shí)：時(shí)： 0), 0(tw0), 0( tw(8) 當(dāng)當(dāng)x=L時(shí)：時(shí)：0),( tlw0),( tlw(9) 當(dāng)當(dāng)=0時(shí)時(shí)當(dāng)當(dāng)M=0時(shí)時(shí)當(dāng)當(dāng)Q=0時(shí)時(shí)School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ (2) 初始條件初始條件Q 當(dāng)當(dāng)t=0時(shí)：時(shí)： 0)0 ,(xw0)0 ,( xwSchool of Aeronautical Science and Engineering, BUAASch

8、ool of Aeronautical Science and Engineering, BUAAQ 4.1 分別變量法分別變量法Q 采用分別變量法求解上方程，為此設(shè)：采用分別變量法求解上方程，為此設(shè)：1)()(),(kkktqxWtxw(11) 式中式中 Wk(x)為模態(tài)方程。為模態(tài)方程。School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 對(duì)于懸臂梁其方式可用如下表達(dá)式表示：對(duì)于懸臂梁其方式可用如下表達(dá)式表示： )(sincos2)(

9、xshxAxchxlxWkkkkkk(12) 式中：式中： l597. 01l49. 12l5 . 23lkk)5 . 0( lchllshlAkkkkkcossinSchool of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 將將 W(x,t)表達(dá)式代入簡(jiǎn)化的振動(dòng)方程有：表達(dá)式代入簡(jiǎn)化的振動(dòng)方程有： 22211( )( )( )( )ikkkkkkkEW x q tW x q tdx (13) 方程式兩邊同時(shí)乘以方程式兩邊同時(shí)乘以 Wk(x),

10、對(duì)兩邊進(jìn)展對(duì)兩邊進(jìn)展0L積分，積分，再根據(jù)自然振型的正交性有：再根據(jù)自然振型的正交性有： School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 其中：其中： 222000000( )( )( )( )()( )( )()( )( )( )()( )( )()( )( )()( )(0)(likkkklikllikilliiiiEq tq tdW x dxxdE H xH x l W x dxdE H xH x l W xdEW x H x

11、H x l dxdEW x H xH x ldEW x H xH x l dxdE W lWdEW) lckk(14) School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 再將再將W(x,t)表達(dá)式代入復(fù)合梁的初始條件式表達(dá)式代入復(fù)合梁的初始條件式(8)有：有： (0)0kq(0)0kq(15) School of Aeronautical Science and Engineering, BUAASchool of Aeronauti

12、cal Science and Engineering, BUAAQ (1) 位移反響位移反響Q 為了使復(fù)合資料梁的振動(dòng)得到抑制，在設(shè)計(jì)控為了使復(fù)合資料梁的振動(dòng)得到抑制，在設(shè)計(jì)控制率時(shí)采用位移作為反響項(xiàng)，即改動(dòng)系統(tǒng)的剛度制率時(shí)采用位移作為反響項(xiàng)，即改動(dòng)系統(tǒng)的剛度特性，從而特性，從而Ei的表達(dá)式可設(shè)計(jì)為：的表達(dá)式可設(shè)計(jì)為： 20201( )( )( )22( )22( )( )lULliijijnijijnnnwwE tp qtqtpBedxBpexxw lBpeBpeW l T tx (16) 式中式中p為反響增益。為反響增益。 School of Aeronautical Science a

13、nd Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 將將(16)式代入式代入(14)式中有：式中有： 0)()(2)()(12nnnnijkkktTlWBpetTtT (17) School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ (2) 速度反響速度反響Q 為了使復(fù)合梁的振動(dòng)得到抑制，在設(shè)計(jì)控制率時(shí)為了使復(fù)合梁的振動(dòng)得到抑制，在設(shè)計(jì)控制率時(shí)應(yīng)

14、使方程出現(xiàn)阻尼項(xiàng)即采用速度作為反響項(xiàng)，從而應(yīng)使方程出現(xiàn)阻尼項(xiàng)即采用速度作為反響項(xiàng)，從而Ei的表達(dá)式可設(shè)計(jì)為：的表達(dá)式可設(shè)計(jì)為： nnnnijlxijxlxijlijlijLUitTlWBpetxwBpetxwtxwBpetxwBpetxwpBetqtqptE1202202023)()(22222)()()(18) School of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ 將將(18)式代入式代入(14)式中有：式中有： 0)()()(12tT

15、tTtTknknnkk (19) 其中： )()(2lWlWBdpeknijnkSchool of Aeronautical Science and Engineering, BUAASchool of Aeronautical Science and Engineering, BUAAQ (3) 加速度反響加速度反響Q 為了使復(fù)合資料梁的振動(dòng)得到抑制，在設(shè)計(jì)控為了使復(fù)合資料梁的振動(dòng)得到抑制，在設(shè)計(jì)控制率時(shí)采用加速度作為反響項(xiàng)，即改動(dòng)系統(tǒng)的質(zhì)制率時(shí)采用加速度作為反響項(xiàng)，即改動(dòng)系統(tǒng)的質(zhì)量特性，從而量特性，從而 Ei的表達(dá)式可設(shè)計(jì)為：的表達(dá)式可設(shè)計(jì)為： nnnnijlxijxlxijlijlijLUitTlWBpetxwBpetxwtxwBpetxwBpetxwpBetqtqptE122022220220224)()(22222)()()( (20) School of Aeronautical Science