非線性振動(dòng)多尺PPT課件

1、)()()()(2212121202022221212120102220211020202221022dtdTTdtdTTTdtdTTTdtdTTTdtdTTdtdTTTdtdTTTdtdTTTdtdTTTTTdtddtd22102210221100DDDTTTdtdTTdtdTTdtdTTdtd第1頁(yè)/共39頁(yè))2(2 20212102021320221321210303202102022DDDDDDDDDDDDDDDDDDDDDDdtd)()()()(2212121202022221212120102220211020202221022dtdTTdtdTTTdtdTTTdtdTTTdtdT

2、TdtdTTTdtdTTTdtdTTTdtdTTTTTdtddtd第2頁(yè)/共39頁(yè)導(dǎo)數(shù)的簡(jiǎn)易計(jì)算22102210 DDDTTTdtd)2(2 )(1021210202221022DDDDDDDDDdtd第3頁(yè)/共39頁(yè)),( ),(),()(2102221012100TTTxTTTxTTTxtx注意關(guān)系，精確度 ),(12101nnTTTT),(20 xxfxpx 求解 設(shè) ),(),(),()(2102221012100TTTxTTTxTTTxtx)()( 02112020110001230222131120130320210002210 xDxDxDxDxDxDxDxDxDxDxDxDxD

3、xDxDxDxDxDdtdxx第4頁(yè)/共39頁(yè))()( 02112020110001230222131120130320210002210 xDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDxDdtdxx )(2210 xxxtx )(2210 xxxtx)()(0211202011000 xDxDxDxDxDxDx第5頁(yè)/共39頁(yè))2(2)2()2()2( 222 )2(2020211102202010120020120213020212210311020103203220212002020212102022xDDDxDDxDxDDxDxDxDDDxDDDxDDxDDxDD

4、xDxDxDxDxDDDxDDxDdtxd)2(2 )(1021210202221022DDDDDDDDDdtd第6頁(yè)/共39頁(yè))(),(),(),( .)()(),() (),(),().(),()(),( ),(),(0110000100000002112020110000221000000000000000000 xDxDxxDxfxxxDxfxDxfxDxDxDxDxDxxDxfxxxxDxfxDxfxDxxxDxfxxxxDxfxDxfxxf )(2210 xxxtx)()(0211202011000 xDxDxDxDxDxDx),(20 xxfxpx 第7頁(yè)/共39頁(yè))(),(),

5、(),() .( )2(2 )2(011000010002000221020020211102202010120020 xDxDxxDxfxxxDxfxDxfxxxpxDDDxDDxDxDDxDxD2200002201010010002220201 11020020000001011002(,)2(2)(,)(,)()D xp xD xD D xp xf x D xD xD D xDD D xp xf x D xf x D xxD xD xxx 對(duì)比系數(shù) 第8頁(yè)/共39頁(yè))(),(),( )2(2),(20011000010000202111022022000010120120020020 x

6、DxDxxDxfxxxDxfxDDDxDDxpxDDxxfxDDxpxDxpxD)(),(),()2(2),(20011000010002200202111022000120010120020020 xDxDxxDxfxxxDxfxpxDDDxDDxDDxxfxpxDDxDxpxD第9頁(yè)/共39頁(yè)02xxx 例1初始條件0)0(,)0(0 xax解：0020020 xpxD00),(),(21210iTiTeTTAeTTAxxxx2 120pxxxf2),(),(2000010120120 xDxfxDDxpxD第10頁(yè)/共39頁(yè)0000001121211212101010),(),(),()

7、,(iTiTiTiTiTiTeAiDeAiDeTTAieTTiADeTTAeTTADDxDD),(2000010120120 xDxfxDDxpxD02xxx 例1初始條件0)0(,)0(0 xaxxxx2 解：120p0020020 xpxD00),(),(21210iTiTeTTAeTTAxxxxf2),(0000iTiTeAiiAexD)(22),(0000000iTiTeAiiAexDxDxf第11頁(yè)/共39頁(yè)00000000)( 2)( 2)( 2)( 2)( 2)( 2111111120120iTiTiTiTiTiTiTiTeiAAiDeiAAiDeAiAiDeiAAiDeAiiA

8、eeAiDeAiDxpxD0011010iTiTeAiDeAiDxDD0000iTiTeAiiAexD)(22),(0000000iTiTeAiiAexDxDxf),(200010120120DxxfxDDxpxD第12頁(yè)/共39頁(yè)01 AAD0),(),(21121TTATTTA0)()(dttAtdA考慮tCetA)(0),(),(21121TTATTTA1)(),(2021TeTaTTA00),(),(21210iTiTeTTAeTTAxcceeTaxiTT01)(200第13頁(yè)/共39頁(yè) 復(fù)數(shù)共軛關(guān)系 )Re(2ZZZBAABBABA)()(AfAfBAABBAABBBAA)(第14頁(yè)

9、/共39頁(yè)0120120 xpxD00),(),(2112111iTiTeTTAeTTAx001110iTiTeAieiAxD001111110iTiTeAiDeAiDxDD0000iTiTeAiiAexD001101iTiTeADeADxD002121021iTiTeADeADxD0),(000 xxDxf)(2)(2)(),( 0000111111110000iTiTiTiTeADeADeAieiAxDDxxDxf0022020iTiTeAiDeAiDxDD第15頁(yè)/共39頁(yè)001111110iTiTeAiDeAiDxDD0022020iTiTeAiDeAiDxDD002121021iTi

10、TeADeADxD0),(00 xDxxf)(2)(2)(),( 000011110110000iTiTiTiTeADeADeAieiAxDxDxxDxf)(),(),( )2(20110000100002021110220220 xDxDxxDxfxxxDxfxDDDxDDxpxD第16頁(yè)/共39頁(yè)cceADiAAiDADAiDeADeADeAieiAeAiDeAiDeADeADeAiDeAiDxDxDxxDxfxxxDxfxDDDxDDxpxDiTiTiTiTiTiTiTiTiTiTiT00000000000)2222()(2)(2)(2)()(2)(),(),()2(2112211111

11、1122212111110110000100002021110220220第17頁(yè)/共39頁(yè)cceADiAAiDADAiDeADeADeAieiAeAiDeAiDeADeADeAiDeAiDxDxDxDxxfxxDxxfxDDDxDDxpxDiTiTiTiTiTiTiTiTiTiTiT00000000000)2222 ()( 2)( 2)( 2)()( 2)(),(),()2(2112211111112221211111011000100020211102202201)(),(2021TeTaTTA11111)(2)(22)(2)(2)(2222222022011120202201111221

12、111TTTTTeTaiDeTaiAAiDeTaeTaiDeTaiAAiDADAiDADiAAiD0)(2)(221120220111TTeTaiDeTaiAAiD消除永年項(xiàng)第18頁(yè)/共39頁(yè)0)(2)(221120220111TTeTaiDeTaiAAiD消除永年項(xiàng))(2)(211120220111TTeTaDeTiaAAD)()(xqyxpdxdy)()()(cdxexqeydxxpdxxp)()(2)(21)()(2)(2121120220211120220111111TaTTaDTiaeTadTeeTaDTiaeATdTTdTcceTaTTaDTiaexiTT01)()(2)(2121

13、120220100),(),(2112111iTiTeTTAeTTAx第19頁(yè)/共39頁(yè)cceeTaxiTT01)(200cceTaTTaDTiaexiTT01)()(2)(21211202201120220)(2)(211TTaDTiaeT此項(xiàng)會(huì)使x1發(fā)散，所以0)(2)(20220TaDTia20002iTeaa第20頁(yè)/共39頁(yè)cceeTaxiTT01)(200cceTaexiTT01)(211)2cos(2)()cos()(2)2cos(2)()()()()()()(),(),(),()(2002021200022120022120022120210222101210011012010

14、10120101tteaOTeTaTTeaOcceTaeeeaOcceTaeeeeaOcceTaeeeTaTTTxTTTxTTTxtxtTTiTTiTiTTiTTiTTiTiTTiTT第21頁(yè)/共39頁(yè)dtdxxxdtxd)1 (222)(),(),( )2(2),(200110001000202111022022000010120120020020 xDxDxDxxfxxDxxfxDDDxDDxpxDDxxfxDDxpxDxpxD例2初始條件0)0(,)0(0 xax第22頁(yè)/共39頁(yè)0020020 xpxDdtdxxxdtxd)1 (22200),(),(21210iTiTeTTAeTTA

15、x120p),(200010120120DxxfxDDxpxD0011010iTiTeAiDeAiDxDD0000iTiTeAiiAexD0020000)1 (),(xDxxDxf第23頁(yè)/共39頁(yè)cceiAeAAAADieAieAAAADieiAeAAAADieAeAAeAAeAieAiiAeeAiDeAiDeAiiAeeAAeeAiDeAiDeAiiAexeAiDeAiDxDxeAiDeAiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiT00000000000000000000000000332133213321332233

16、112112011002011120120)2()2()2()()(22)() (1 22)(1 ()(2)1 ()(2),(200010120120DxxfxDDxpxD0011010iTiTeAiDeAiDxDD0000iTiTeAiiAexD0020000)1(),(xDxxDxf第24頁(yè)/共39頁(yè)00000000000000033213131313321)2(2)(2 )(2)2(iTiTiTiTiTiTiTiTiTiTiTiTiTiTiTeiAAAAADiAeAiAAeAAiiAeeADiAeAiAAeAAiiAeiAeDeAAAieAAAieAieAiDeAieAAAADiccei

17、AeAAAADixpxDiTiT003321120120)2 (0221AAAAD 消去長(zhǎng)期項(xiàng) 第25頁(yè)/共39頁(yè)0221AAAAD),(exp),(212121TTiTTaA 設(shè) ),(),(2121TTTTa為實(shí)數(shù) 0)8121(0exp81exp21expexp0231131121aaTaTaiiaiaiTiaiTaAAAAD081210311aaTaTa1)(14)(222TeTcaT第26頁(yè)/共39頁(yè)初始條件為 0) 0(,) 0(0 xaxteaaaTcTca) 14(1414)(,)(1420220222000000021122cos()cos( )iTiTiTiTiixAeAe

18、ae eae eaTTaOcos( )xaO一次近似解 cTc)(2一次近似 1)(14)(222TeTcaT第27頁(yè)/共39頁(yè)cceiAxpxDiT033120120cceiAeTTBxiTiT003321181),()(),(),( )2(201100010002021110220220 xDxDxDxxfxxDxxfxDDDxDDxpxD200000001),( 2),(xxDxxfDxxxDxxf1)(14)(222TeTcaT二次近似時(shí) ),(exp),(212121TTiTTaAcceADeTTBiDxDDiTiT0033121111083),( ),(2),()2(0021221

19、2102021cceTTAiDeTTADxDDDiTiTcceAeTTiBeTTADxDxDiTiTiT0003321211011083),(),(第28頁(yè)/共39頁(yè))8383()(1 )8181)()( 2221432)(),(),()2(20000000000000000000033133123333221331101100010002021110220220iTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTeAiBeAeDeAiBeAeDAeAeeiABeeiABeiAeiAeAeAecceAiDeADcceADeBiDxDxDxDxxfxxDxxfxDDD

20、xDDxpxDcceAiBeeADeAAAeAAAeiABeAAeAAAeAieAiDeADeADeBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiT)83)(2121)81)(22432000000000000033122223322222213311220220第29頁(yè)/共39頁(yè)cceAiBeeADeAeAAAeiABeeAAAeAieAiDeADeADeBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiT)83)(21 )81)(2(22432000000000000033122223322222213311220220cceAiBeeADeAAAeAAAe

21、iABeAAeAAAeAieAiDeADeADeBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiT)83)(2121)81)(22432000000000000033122223322222213311220220第30頁(yè)/共39頁(yè)cceAAieBAeADAeABeiAeADAeAiBeAeDAAeAABeAieAABeAiAeABeiAeADeAiDADBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiT0000000000000000032212553231233132234553233122112202208383)83)(21 (41221

22、441243)22(cceAiBeeADeAeAAAeiABeeAAAeAieAiDeADeADeBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiT)83)(21 )81)(2(22432000000000000033122223322222213311220220第31頁(yè)/共39頁(yè)cceAAieBAeADAeABeiAeADAeAiBeAeDAAeAABeAieAABeAiAeABeiAeADeAiDADBiDxpxDiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiTiT0000000000000000032212553231233132234553233122

23、112202208383)83)(21 (41221441243)22(NSTcceiBADAAeAABiAADAAABiABAiAAiDADBiDNSTcceAAeBiAeADAeiBeADAAeAABiABeAiAeAiDADBiDxpxDiTiTiTiTiTiTiTiT00000000)(21 ()83412422(83)(21 ()4124()22(13221232222113221213222211220220第32頁(yè)/共39頁(yè)NSTcceBAAiADAABiAADAAABiABAiAAiDADBiDxpxDiT0)21 ()21 (412422121232222112202200000003333320323200088cossinsin3321cos11671111lnsin1sin 3 1648163216iTiTiTiTiTiTiixAeAeBeBeA eA eaabataaaabtat2通過(guò)消除久期項(xiàng)，即可得到解表達(dá)式解的最終表達(dá)式為(取實(shí)部)：第33頁(yè)/共39頁(yè)例：用多尺度求Duffin