非线性振动多尺度课件.pptx

1、这种方法是 Sturrock,Frieman,Nayfeh,Sandri 发展得一种奇异摄动法。它适合求解周期运动也可以用于耗散系统和其它场合。2 , 1 , 0 ,ntTnn22102210221100DDDTTTdtdTTdtdTTdtdTTdtd1.5 多尺度法多尺度法把解看成是T1, T2, T3 , 的函数。)()()()(2212121202022221212120102220211020202221022dtdTTdtdTTTdtdTTTdtdTTTdtdTTdtdTTTdtdTTTdtdTTTdtdTTTTTdtddtd22102210221100DDDTTTdtdTTdtdT

2、TdtdTTdtd)2(2 20212102021320221321210303202102022DDDDDDDDDDDDDDDDDDDDDDdtd)()()()(2212121202022221212120102220211020202221022dtdTTdtdTTTdtdTTTdtdTTTdtdTTdtdTTTdtdTTTdtdTTTdtdTTTTTdtddtd导数的简易计算22102210 DDDTTTdtd)2(2 )(1021210202221022DDDDDDDDDdtd),( ),(),()(2102221012100TTTxTTTxTTTxtx注意关系，精确度注意关系，精确度

3、 ),(12101nnTTTT),(20 xxfxpx 求解求解 设设 ),(),(),()(2102221012100TTTxTTTxTTTxtx)()( 02112020110001230222131120130320210002210 xDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDdtdxx)()( 02112020110001230222131120130320210002210 xDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDdtdxx )(2210 xxxtx )(2210 xxxtx)()(0211202011000 xDxDxD

4、xDxDxDx)2(2)2()2()2( 222 )2(2020211102202010120020120213020212210311020103203220212002020212102022xDDDxDDxDxDDxDxDxDDDxDDDxDDxDDxDDxDxDxDxDxDDDxDDxDdtxd)2(2 )(1021210202221022DDDDDDDDDdtd)(),(),(),( .)()(),() (),(),().(),()(),( ),(),(0110000100000002112020110000221000000000000000000 xDxDxxDxfxxxDxfx

5、DxfxDxDxDxDxDxxDxfxxxxDxfxDxfxDxxxDxfxxxxDxfxDxfxxf )(2210 xxxtx)()(0211202011000 xDxDxDxDxDxDx),(20 xxfxpx )(),(),(),() .( )2(2 )2(011000010002000221020020211102202010120020 xDxDxxDxfxxxDxfxDxfxxxpxDDDxDDxDxDDxDxD2200002201010010002220201 11020020000001011002(,)2(2)(,)(,)()D xp xD xD D xp xf x D xD

6、 xD D xDD D xp xf x D xf x D xxD xD xxx 对比系数对比系数 )(),(),( )2(2),(20011000010000202111022022000010120120020020 xDxDxxDxfxxxDxfxDDDxDDxpxDDxxfxDDxpxDxpxD)(),(),()2(2),(20011000010002200202111022000120010120020020 xDxDxxDxfxxxDxfxpxDDDxDDxDDxxfxpxDDxDxpxD02xxx 例1初始条件0)0(,)0(0 xax解：0020020 xpxD00),(),(2

7、1210iTiTeTTAeTTAxxxx2 120pxxxf2),(),(2000010120120 xDxfxDDxpxD0000001121211212101010),(),(),(),(iTiTiTiTiTiTeAiDeAiDeTTAieTTiADeTTAeTTADDxDD),(2000010120120 xDxfxDDxpxD02xxx 例1初始条件0)0(,)0(0 xaxxxx2 解：120p0020020 xpxD00),(),(21210iTiTeTTAeTTAxxxxf2),(0000iTiTeAiiAexD)(22),(0000000iTiTeAiiAexDxDxf0000

8、0000)( 2)( 2)( 2)( 2)( 2)( 2111111120120iTiTiTiTiTiTiTiTeiAAiDeiAAiDeAiAiDeiAAiDeAiiAeeAiDeAiDxpxD0011010iTiTeAiDeAiDxDD0000iTiTeAiiAexD)(22),(0000000iTiTeAiiAexDxDxf),(200010120120DxxfxDDxpxD01 AAD0),(),(21121TTATTTA0)()(dttAtdA考虑tCetA)(0),(),(21121TTATTTA1)(),(2021TeTaTTA00),(),(21210iTiTeTTAeTTAx

9、cceeTaxiTT01)(200 复数共轭关系复数共轭关系 )Re(2ZZZBAABBABA)()(AfAfBAABBAABBBAA)(0120120 xpxD00),(),(2112111iTiTeTTAeTTAx001110iTiTeAieiAxD001111110iTiTeAiDeAiDxDD0000iTiTeAiiAexD001101iTiTeADeADxD002121021iTiTeADeADxD0),(000 xxDxf)(2)(2)(),( 0000111111110000iTiTiTiTeADeADeAieiAxDDxxDxf0022020iTiTeAiDeAiDxDD001

10、111110iTiTeAiDeAiDxDD0022020iTiTeAiDeAiDxDD002121021iTiTeADeADxD0),(00 xDxxf)(2)(2)(),( 000011110110000iTiTiTiTeADeADeAieiAxDxDxxDxf)(),(),( )2(20110000100002021110220220 xDxDxxDxfxxxDxfxDDDxDDxpxDcceADiAAiDADAiDeADeADeAieiAeAiDeAiDeADeADeAiDeAiDxDxDxxDxfxxxDxfxDDDxDDxpxDiTiTiTiTiTiTiTiTiTiTiT000000

11、00000)2222()(2)(2)(2)()(2)(),(),()2(21122111111122212111110110000100002021110220220cceADiAAiDADAiDeADeADeAieiAeAiDeAiDeADeADeAiDeAiDxDxDxDxxfxxDxxfxDDDxDDxpxDiTiTiTiTiTiTiTiTiTiTiT00000000000)2222 ()( 2)( 2)( 2)()( 2)(),(),()2(2112211111112221211111011000100020211102202201)(),(2021TeTaTTA11111)(2)(2

12、2)(2)(2)(2222222022011120202201111221111TTTTTeTaiDeTaiAAiDeTaeTaiDeTaiAAiDADAiDADiAAiD0)(2)(221120220111TTeTaiDeTaiAAiD消除永年项0)(2)(221120220111TTeTaiDeTaiAAiD消除永年项)(2)(211120220111TTeTaDeTiaAAD)()(xqyxpdxdy)()()(cdxexqeydxxpdxxp)()(2)(21)()(2)(2121120220211120220111111TaTTaDTiaeTadTeeTaDTiaeATdTTdTcc

13、eTaTTaDTiaexiTT01)()(2)(2121120220100),(),(2112111iTiTeTTAeTTAxcceeTaxiTT01)(200cceTaTTaDTiaexiTT01)()(2)(21211202201120220)(2)(211TTaDTiaeT此项会使x1发散，所以0)(2)(20220TaDTia20002iTeaacceeTaxiTT01)(200cceTaexiTT01)(211)2cos(2)()cos()(2)2cos(2)()()()()()()(),(),(),()(200202120002212002212002212021022210121

14、001101201010120101tteaOTeTaTTeaOcceTaeeeaOcceTaeeeeaOcceTaeeeTaTTTxTTTxTTTxtxtTTiTTiTiTTiTTiTTiTiTTiTTdtdxxxdtxd)1 (222)(),(),( )2(2),(200110001000202111022022000010120120020020 xDxDxDxxfxxDxxfxDDDxDDxpxDDxxfxDDxpxDxpxD例2初始条件0)0(,)0(0 xax0020020 xpxDdtdxxxdtxd)1 (22200),(),(21210iTiTeTTAeTTAx120p),(

15、200010120120DxxfxDDxpxD0011010iTiTeAiDeAiDxDD0000iTiTeAiiAexD0020000)1 (),(xDxxDxfcceiAeAAAADieAieAAAADieiAeAAAADieAeAAeAAeAieAiiAeeAiDeAiDeAiiAeeAAeeAiDeAiDeAiiAexeAiDeAiDxDxeAiDeAiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiT0000000000000000000000000033213321332133223311211201100201112

16、0120)2()2()2()()(22)() (1 22)(1 ()(2)1 ()(2),(200010120120DxxfxDDxpxD0011010iTiTeAiDeAiDxDD0000iTiTeAiiAexD0020000)1(),(xDxxDxf00000000000000033213131313321)2(2)(2 )(2)2(iTiTiTiTiTiTiTiTiTiTiTiTiTiTiTeiAAAAADiAeAiAAeAAiiAeeADiAeAiAAeAAiiAeiAeDeAAAieAAAieAieAiDeAieAAAADicceiAeAAAADixpxDiTiT0033211201

17、20)2 (0221AAAAD 消去长期项消去长期项 0221AAAAD),(exp),(212121TTiTTaA 设设 ),(),(2121TTTTa为实数为实数 0)8121(0exp81exp21expexp0231131121aaTaTaiiaiaiTiaiTaAAAAD081210311aaTaTa1)(14)(222TeTcaT 初始条件为初始条件为 0) 0(,) 0(0 xaxteaaaTcTca) 14(1414)(,)(1420220222000000021122cos()cos( )iTiTiTiTiixAeAeae eae eaTTaOcos( )xaO 一次近似解一

18、次近似解 cTc)(2 一次近似一次近似 1)(14)(222TeTcaTcceiAxpxDiT033120120cceiAeTTBxiTiT003321181),()(),(),( )2(201100010002021110220220 xDxDxDxxfxxDxxfxDDDxDDxpxD200000001),( 2),(xxDxxfDxxxDxxf1)(14)(222TeTcaT二次近似时二次近似时 ),(exp),(212121TTiTTaAcceADeTTBiDxDDiTiT0033121111083),( ),(2),()2(00212212102021cceTTAiDeTTADxD

19、DDiTiTcceAeTTiBeTTADxDxDiTiTiT0003321211011083),(),()8383()(1 )8181)()( 2221432)(),(),()2(20000000000000000000033133123333221331101100010002021110220220iTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTeAiBeAeDeAiBeAeDAeAeeiABeeiABeiAeiAeAeAecceAiDeADcceADeBiDxDxDxDxxfxxDxxfxDDDxDDxpxDcceAiBeeADeAAAeAAAeiABeA

20、AeAAAeAieAiDeADeADeBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiT)83)(2121)81)(22432000000000000033122223322222213311220220cceAiBeeADeAeAAAeiABeeAAAeAieAiDeADeADeBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiT)83)(21 )81)(2(22432000000000000033122223322222213311220220cceAiBeeADeAAAeAAAeiABeAAeAAAeAieAiDeADeADeBiDxpxDiTiTiTiTi

21、TiTiTiTiTiTiTiTiT)83)(2121)81)(22432000000000000033122223322222213311220220cceAAieBAeADAeABeiAeADAeAiBeAeDAAeAABeAieAABeAiAeABeiAeADeAiDADBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiT0000000000000000032212553231233132234553233122112202208383)83)(21 (41221441243)22(cceAiBeeADeAeAAAeiABeeAAAeAieAiDeADeADe

22、BiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiT)83)(21 )81)(2(22432000000000000033122223322222213311220220cceAAieBAeADAeABeiAeADAeAiBeAeDAAeAABeAieAABeAiAeABeiAeADeAiDADBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiT0000000000000000032212553231233132234553233122112202208383)83)(21 (41221441243)22(NSTcceiBADAAeAABiAADAA

23、ABiABAiAAiDADBiDNSTcceAAeBiAeADAeiBeADAAeAABiABeAiAeAiDADBiDxpxDiTiTiTiTiTiTiTiT00000000)(21 ()83412422(83)(21 ()4124()22(13221232222113221213222211220220NSTcceBAAiADAABiAADAAABiABAiAAiDADBiDxpxDiT0)21 ()21 (412422121232222112202200000003333320323200088cossinsin3321cos11671111lnsin1sin 3 1648163216i

24、TiTiTiTiTiTiixAeAeBeBeA eA eaabataaaabtat2通过消除久期项，即可得到解表达式解的最终表达式为(取实部)：例例：用多尺度求Duffing方程自由振动的二次近似解 301xxx 23001210122012,0 xxT T Tx T T TxT T T解解：设： 方程右端为： 3201232200132230013()3()xxxxx xxx x 200023011010022202201 1100200102223D xxD xxD D xxD xxD D xD xD D xx x 003223011123iTiTAD xxiA A eA eccT 213

25、0AiA AT0000000000000033332012233223355335532324433883283228iTiTiTiTiTiTiTiTiTiTiTiTiTiTAAx xAeAeeeA eA eAAA eA eA eA eA A eA A eA AeAA e000000000223202221112535324223532452211342 3283283933228848iTiTiTiTiTiTiTiTiTAAAAAD xxiieeieTTTTA eA A eA AeccAAAAiA AeiA A eA eccTTT 00035232450222152132888iTiTiTAD xxiA AeA AeA eccT 消除长期项，则二阶可解性条件：322322152081516AiA ATAiA AT 00354522116464iTiTxA AeA ecc 下一步，求A12iAae0224120003158256aaaatt代入一阶、二阶可解性条件并分离实部虚部可解的： 2325000022400020121cos1cos3cos53232102431518256inixaaaaaatTt