第09章結(jié)構(gòu)動(dòng)力計(jì)算

1、內(nèi) 容垮塌后的彩虹橋垮塌后的彩虹橋 假設(shè)人體重量為假設(shè)人體重量為750N3000N3500N4500N6000N12500N新Tacoma橋均質(zhì)桿件mtPsinmEI單自由度單自由度系統(tǒng)振動(dòng)多自由度系統(tǒng)振動(dòng)連續(xù)（系統(tǒng)振動(dòng)0011ykymY mEIylyk11ym 3113lEIk ymykmaF 1111)(ymy ymyk 111111131113kEIlym yk11ym 011ykym mk112令：02yy tAtAycossin2100,0yyyyt ：初始條件0201,yAyA)sin(sincos00tatytyymEIyly002020,yytgyya其中：2Ta)(11tFy

2、kymP mk112令：mtFyyP)(2 mEIyl3113lEIk y)(tFPkyym )(tFPmtFyyP)(2 0, 00yyt：初始條件tytCtCystsincossin21通解：tFtFPsin)(簡(jiǎn)諧荷載：tAysin特設(shè)特解：tmFtmFysin1sin22222）（特FmFyst222max11)(t syty放大系數(shù)：tyystsin特則：）（解為：ttyystsinsintyystsin平穩(wěn)解為：tmFyysin2 則：tmSysin0, 000ymSvtFStP，00,0vyyyt：初始條件時(shí)當(dāng)：時(shí)刻作用ttmSySt)(sin時(shí)當(dāng)：時(shí)刻作用ttmdFdydStP

3、)(sin)(dtFmtyPt)(sin)(1)(0dtFmtvtytyPt)(sin)(1sincos)(000)(tFPtPF)(tFPdttdFdSP)()(tFPdtdFSP)(2)(maxt syty放大系數(shù)：0,0, 0)(0tFttFPP當(dāng)當(dāng))(tFP0t0PFmtFdtFmtyPPt）（cos1)(sin1)(000）（tyystcos1utFutttFPP0,0, 0)(0當(dāng)與當(dāng)mtutFdtFmtyPPucos)(cos)(sin1)(000)2(sin2sin2utuyyst)(tFP0t0PFurPrPrPttFttFtttF當(dāng)當(dāng),0,)(00)(tFP0t0PFrtm

4、EIyl3113lEIk y)(tFPrrrstrrstt,ttttytttttyytsinsin11),sin(1當(dāng)）（當(dāng)dtFmtyPt)(sin)(1)(020mFyPst011ykycym mcmk2112，令：022yyy tCey設(shè)解為：0222）（12mEIylkyym 3113lEIk ycy00,0yyyyt ：初始條件)sin(sincos000taetyytyeyrtrrrrtr）（12ri)cossin(21tCtCeyrrt211r時(shí)：令當(dāng)21r)sin(taeyrt稱為衰減系數(shù))sin(nrtntaeyn)2sin()(1nrTtntaeyn11ln21ln2nnn

5、nryyyy）（12時(shí)為臨界阻尼、當(dāng)12時(shí)為超阻尼、kykym 2m2y1m1y2m1m1y2ym 1ym 1R2R2y022212122ykykym 1111210PRRRR0222212PRRRR11111ykR 21212ykR 111ymRP 12121ykR 22222ykR222ymRP 動(dòng)力問題靜力問題)sin(),sin(2211tYytYy設(shè)：0)(0)(2222212121211211YmkYkYkYmk2111122212220kmkkkm頻：率方程0)()(2121122211222211122mmkkkkmkmk21兩個(gè)根：第二頻率稱為第一頻

6、率或基頻，其中：21幅值方程021211111ykykym 022212122ykykym 常數(shù)得：由：21212211)sin(),sin(YYyytYytYy1211112211121211mkkYYYYyy稱為第一振型，的：對(duì)應(yīng)122111222122mkkYY的為第二振型：對(duì)應(yīng)2m21Y1m11Y2m1m12Y22Y1m2m2k1k，計(jì)算自振頻率和振型、層間側(cè)移剛度分別為21kk2222212122111,kkkkkkkkk0)(2212111ykykkym 0221222ykykym 0)(222221221kmkmkk頻率方程：2m1m2k1k2y1y時(shí)討論當(dāng)：kkkmmm2121

7、,mkmk61803. 1,61803. 021618. 01;618. 1122122111YYYY1m2m2k1k2m1m2k1k2y1y時(shí)討論當(dāng)：2121,knkmnm22222221411211,411211mknnnmknnn4121;412112221121nYYnYY19;1109012221121YYYYn有設(shè)0002211222212122121211111nnnnnnnnnnnykykykymykykykymykykykym 021iPiiiRRRRnjiykRjijij, 2 , 1,iiiPymR )sin(tYy設(shè)：最小值稱基頻個(gè)根：有nn21( )12,iTiiin

8、iYY YY對(duì)應(yīng)的振型向量nynm1m1y 0yKyM 02YMK 02MK頻率方程為： 稱為振型向量TnYYYY,21 為質(zhì)量、剛度矩陣、 KM2m2ylEI,1m1ylEI,21111l1M121l 2222M122211111ymymy 222221112ymymy 0122211111ymymy 0222221112ymymy EIl3311EIl38322EIl6532112000021212221121121yymmyy 2m2ylEI,1m1ylEI,22ym 11ym )sin()sin(2211tYytYy)sin()sin(2222212111tYmymtYmym 2221

9、12YmYm，0)1(0)1(222221121221212111YmYmYmYm22222211122)()(YmYmY12222111121)()(YmYmY2Y1Y222Ym112Ym01122221212122111mmmm212mEI1m1yllEI)(tP)(tq2y1m2mlEI,各桿均為1y2y2mEI1m1yllEI)(tP)(tq2y21111l1M121l 2222MP2)(tPltP )(2PMP1q22)(2ltqqMq1)(tqqPymymy11122211111 qPymymy22222221112 EIltPdxEIMMPP6)(5311EIl38322EIl6

10、532112EIltqdxEIMMqq8)(411EIl3311EIltPdxEIMMPP3)(8322EIltqdxEIMMqq24)(7422lEIi 1m2mlEI,基本體系1Z2Z11m2m1122221111m2m112k22kli6圖2M1li6li6li611k21kli6圖1M1li6li6li6li6li601212111PRZkZk02222121PRZkZk21148lik22224lik2211224likk1021PPRRilZZilZ122,2421221ilZilZ12,2422222112是否相等？與2112lEIi 1m2mlEI,基本體系1Z2Z112k22

11、kli6圖2M1li6li6li611k21kli6圖1M1li6li6li6li6li601212111PRZkZk02222121PRZkZk21148lik 22224lik2211224likk0121PPRRilZZ24221ilZ2421211121111m2m1122211111ymymy 222221112ymymy 0122211111ymymy 0222221112ymymy 02YMK已知： 0)(2iYMKi 0)(2jjYMK TTKKMM，又已知： )()(2)()(iTjiTjYMYYKYi )()(2)()(jTijTiYMYYKYj振型關(guān)于質(zhì)量正交0)()(j

12、TiYMY 振型也關(guān)于剛度正交0)()(jTiYKY廣義質(zhì)量iiTiMYMY)()( 廣義剛度iiTiKYKY)()(iiiMK 廣義剛度矩陣廣義質(zhì)量，KYKYMYMYTT nnnnnnnYYYYYYYYYYYYY212222111211)()2()1(nMMMM00000021nKKKK00000021 niiinnYYYYy1)()()2(2) 1 (1令 0yKyM ( )TiiiYMyM可以得到：出現(xiàn)共振！時(shí)，或討論：當(dāng)0021DtPykykymsin121211111 tPykykymsin222212122 tYytYysin,sin2211平穩(wěn)振動(dòng)階段設(shè)：222222121121

13、211211)()(PYmkYkPYkYmk2m2y1m1ytPsin2tPsin12121112122121222212112222212110)()()(PmkPkDPkPmkDkkmkmkD022011DDYDDY，2m2y1m1y)(2tP)(1tP 1)()()2(2) 1 (1YYYYYyniiinn令 )(tPyKyM 稱為正則坐標(biāo)稱為幾何坐標(biāo)，y )(tPYKYM )(tPYYKYYMYTTT 廣義荷載列陣，)()(tPtPKM nitPKMiiii, 2 , 1)( nitPMiiiii, 2 , 1)(1 或主振型疊加法稱為主振型分解法 )0()0()(iiiTiiMyMY和可得初始條件已知： Yy 即可得到幾何坐標(biāo)：根據(jù)求出的正則坐