第09章结构动力计算

1、内 容垮塌后的彩虹桥垮塌后的彩虹桥 假设人体重量为假设人体重量为750N3000N3500N4500N6000N12500N新Tacoma桥均质杆件mtPsinmEI单自由度单自由度系统振动多自由度系统振动连续（系统振动0011ykymY mEIylyk11ym 3113lEIk ymykmaF 1111)(ymy ymyk 111111131113kEIlym yk11ym 011ykym mk112令：02yy tAtAycossin2100,0yyyyt ：初始条件0201,yAyA)sin(sincos00tatytyymEIyly002020,yytgyya其中：2Ta)(11tFy

2、kymP mk112令：mtFyyP)(2 mEIyl3113lEIk y)(tFPkyym )(tFPmtFyyP)(2 0, 00yyt：初始条件tytCtCystsincossin21通解：tFtFPsin)(简谐荷载：tAysin特设特解：tmFtmFysin1sin22222）（特FmFyst222max11)(t syty放大系数：tyystsin特则：）（解为：ttyystsinsintyystsin平稳解为：tmFyysin2 则：tmSysin0, 000ymSvtFStP，00,0vyyyt：初始条件时当：时刻作用ttmSySt)(sin时当：时刻作用ttmdFdydStP

3、)(sin)(dtFmtyPt)(sin)(1)(0dtFmtvtytyPt)(sin)(1sincos)(000)(tFPtPF)(tFPdttdFdSP)()(tFPdtdFSP)(2)(maxt syty放大系数：0,0, 0)(0tFttFPP当当)(tFP0t0PFmtFdtFmtyPPt）（cos1)(sin1)(000）（tyystcos1utFutttFPP0,0, 0)(0当与当mtutFdtFmtyPPucos)(cos)(sin1)(000)2(sin2sin2utuyyst)(tFP0t0PFurPrPrPttFttFtttF当当,0,)(00)(tFP0t0PFrtm

4、EIyl3113lEIk y)(tFPrrrstrrstt,ttttytttttyytsinsin11),sin(1当）（当dtFmtyPt)(sin)(1)(020mFyPst011ykycym mcmk2112，令：022yyy tCey设解为：0222）（12mEIylkyym 3113lEIk ycy00,0yyyyt ：初始条件)sin(sincos000taetyytyeyrtrrrrtr）（12ri)cossin(21tCtCeyrrt211r时：令当21r)sin(taeyrt称为衰减系数)sin(nrtntaeyn)2sin()(1nrTtntaeyn11ln21ln2nnn

5、nryyyy）（12时为临界阻尼、当12时为超阻尼、kykym 2m2y1m1y2m1m1y2ym 1ym 1R2R2y022212122ykykym 1111210PRRRR0222212PRRRR11111ykR 21212ykR 111ymRP 12121ykR 22222ykR222ymRP 动力问题静力问题)sin(),sin(2211tYytYy设：0)(0)(2222212121211211YmkYkYkYmk2111122212220kmkkkm频：率方程0)()(2121122211222211122mmkkkkmkmk21两个根：第二频率称为第一频

6、率或基频，其中：21幅值方程021211111ykykym 022212122ykykym 常数得：由：21212211)sin(),sin(YYyytYytYy1211112211121211mkkYYYYyy称为第一振型，的：对应122111222122mkkYY的为第二振型：对应2m21Y1m11Y2m1m12Y22Y1m2m2k1k，计算自振频率和振型、层间侧移刚度分别为21kk2222212122111,kkkkkkkkk0)(2212111ykykkym 0221222ykykym 0)(222221221kmkmkk频率方程：2m1m2k1k2y1y时讨论当：kkkmmm2121

7、,mkmk61803. 1,61803. 021618. 01;618. 1122122111YYYY1m2m2k1k2m1m2k1k2y1y时讨论当：2121,knkmnm22222221411211,411211mknnnmknnn4121;412112221121nYYnYY19;1109012221121YYYYn有设0002211222212122121211111nnnnnnnnnnnykykykymykykykymykykykym 021iPiiiRRRRnjiykRjijij, 2 , 1,iiiPymR )sin(tYy设：最小值称基频个根：有nn21( )12,iTiiin

8、iYY YY对应的振型向量nynm1m1y 0yKyM 02YMK 02MK频率方程为： 称为振型向量TnYYYY,21 为质量、刚度矩阵、 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,各杆均为1y2y2mEI1m1yllEI)(tP)(tq2y21111l1M121l 2222MP2)(tPltP )(2PMP1q22)(2ltqqMq1)(tqqPymymy11122211111 qPymymy22222221112 EIltPdxEIMMPP6)(5311EIl38322EIl6

10、532112EIltqdxEIMMqq8)(411EIl3311EIltPdxEIMMPP3)(8322EIltqdxEIMMqq24)(7422lEIi 1m2mlEI,基本体系1Z2Z11m2m1122221111m2m112k22kli6图2M1li6li6li611k21kli6图1M1li6li6li6li6li601212111PRZkZk02222121PRZkZk21148lik22224lik2211224likk1021PPRRilZZilZ122,2421221ilZilZ12,2422222112是否相等？与2112lEIi 1m2mlEI,基本体系1Z2Z112k22

11、kli6图2M1li6li6li611k21kli6图1M1li6li6li6li6li601212111PRZkZk02222121PRZkZk21148lik 22224lik2211224likk0121PPRRilZZ24221ilZ2421211121111m2m1122211111ymymy 222221112ymymy 0122211111ymymy 0222221112ymymy 02YMK已知： 0)(2iYMKi 0)(2jjYMK TTKKMM，又已知： )()(2)()(iTjiTjYMYYKYi )()(2)()(jTijTiYMYYKYj振型关于质量正交0)()(j

12、TiYMY 振型也关于刚度正交0)()(jTiYKY广义质量iiTiMYMY)()( 广义刚度iiTiKYKY)()(iiiMK 广义刚度矩阵广义质量，KYKYMYMYTT nnnnnnnYYYYYYYYYYYYY212222111211)()2()1(nMMMM00000021nKKKK00000021 niiinnYYYYy1)()()2(2) 1 (1令 0yKyM ( )TiiiYMyM可以得到：出现共振！时，或讨论：当0021DtPykykymsin121211111 tPykykymsin222212122 tYytYysin,sin2211平稳振动阶段设：222222121121

13、211211)()(PYmkYkPYkYmk2m2y1m1ytPsin2tPsin12121112122121222212112222212110)()()(PmkPkDPkPmkDkkmkmkD022011DDYDDY，2m2y1m1y)(2tP)(1tP 1)()()2(2) 1 (1YYYYYyniiinn令 )(tPyKyM 称为正则坐标称为几何坐标，y )(tPYKYM )(tPYYKYYMYTTT 广义荷载列阵，)()(tPtPKM nitPKMiiii, 2 , 1)( nitPMiiiii, 2 , 1)(1 或主振型叠加法称为主振型分解法 )0()0()(iiiTiiMyMY和可得初始条件已知： Yy 即可得到几何坐标：根据求出的正则坐