1、,(1)ij jiifu,1 (2)2iji jj ik ik juuu u (3)ijijklklD (4)ijjiiinTuu0000 (5)iitiituuuu,(6)iij jiiiijjiRfuRnTdd0 (7)iiiiRv VRv S1=(8)NiIiIIua1=(9)NiIiIIvW bd0 (10)iIRW V()1,2,(10)IIWIN xx1 1,2,(11)0IIIxWINx 1,2,(12)iIiIRWINa2d2d0 (11)iiiiIiIRRVRVaa 1,2,(13)IIWIN441144d10 1,1d0 (14)021616xxwwxxwwxEIklxww
2、lplEI 221(5)(1)(15)24xx 22111 1(5)(1)(16)24a xxwa 2211(5)(1)11 (17)24xxRa 精确解配点法子域法伽辽金法最小二乘法1.00000.17880.17240.18380.17900.183210.0000.078360.067570.089290.078910.08304100.000.011340.009540.014530.011970.058181000.00.0010250.0009950.0015510.0012620.006068,dd0(18)iij jiiiijjiufuVunTSdddd(19)iiijijii
3、iiuu VVu f VuT S 1 nC22112211d dd d d dd d(20)ttiiijijttttiiiittuu V tV tu f V tuT S t 120iit tt tuu222111dd dd (21)tttiiiitttuuVuu t VT t ddd (23)ijijiiiiWVu f VuT S 21d0 (22)ttTWtdiiWu u V 21d0 (24)ttTWWtP1ddd (25)2ijklijkliiiiWDVf u VTu S 2211PPdd0 (26)ttttTtTtwuzx 22xwzx 2P0222001d(,)d21 d(,)d2l
4、xllEVq x t w xwEIxq x t w xx 22d TuwV201 d 2lAwx201d2lwIxx22P220042342300d(,)d(,)dllllwwEIxq x tw xxxwwwwEIq x txEIwxxxx 22112100dd d d dttltttltT tAw w x tAw w x t 44003232(,)0,00 xxx lx lwAwEIq x txlxwwxwwEIEIxxTLP()d (27)V C uTCP1()()d (28)2V Cu C uTTLP()d()d (29)VV C uC u (30)BuuTTTL1 (31)2 u Ku
5、u fBuuTTT0 (32)uKuf B uBuuT (33)ufKBuB0TCP()()d (34)V Cu C u TTTL11 (35)22 u Kuu fBuuBuuTTL (36)uKufBBuuTT (37)KB B ufB u21uk强制位移:11222 (38)ufkkufkk111211,(39)22ffufk2 f201 (40)uuB11222000 (41)010kkufkkuu11111,(42)22ffuk 11222 (43)ufkkuukk1000k取1111221001100020001000(1),200120012001fffuufkk21PLd (45
6、)ttTtL,1dd (46)2 uijiji jj iiiiuuVp uuS,0 0 10 2 0 ij jiijklklijiji jj ik ik jiiuijjifuxDxuuu uxuuxnp 0 uijjixnTx ijijklklijiijjijjDpnnT2121,11dd22ddd utijklijkliiijiji jj ittiiijjiitTDf uuuVtTu SnuuSt如图所示弹簧系统，在第一个和第三个自由度如图所示弹簧系统，在第一个和第三个自由度上施加外力，在第二个自由度上施加强制位移。用拉上施加外力，在第二个自由度上施加强制位移。用拉格朗日乘子法和罚函数法计算
7、第一个和第三个自由度格朗日乘子法和罚函数法计算第一个和第三个自由度的位移，并计算第二个自由度上的作用力的位移，并计算第二个自由度上的作用力20u强制位移:xuxxE 212122000d11dd(,)dd(1)22tttllltTtAuxEAuxf x t u xt2e1()(2)iiiuNuNuTeee1212 (3)NNuuNuc12c2 (4)2xxxxxl121111 (5)22NNTTTeeeT00TeTe01ddd(,)d2dd1d (6)2 lllEAxf x txxxTAxNNuuuNuN NuTTeeT00TeTe0ddd(,)dddd (7)lllEAxf x txxxTA
8、xNNuuNuN NuT1e11eT11eT12dddddd (8)2(,)d2EAlAllftNNKMN NfNeeeee (9)M uK ufee1121 (10)11126EAAllKMe21 (11)126AlMcossinsincos iiiiiiuuvvuvcossinsincosiiiiuuvveeiiuueeeeee00iijjuuuTuuueTeeTeeTeMT M TKT K TfT fcos,xxxyxzyxyyyzklzxzyzznnnnnnnk lnnn0T032210GPa7800kg/m200mm500mmEAL弹性模量 密 度 截 面 积 直杆长度 eee123
9、eee123111110012EALALKKKMMMe4e411112102012EALALKMee130000010100000101EALKKe21010000010100000EALKeeee13241000010010010220001ALMMMMe41111111111112 21111EALK2211044221004410100001EALK2200002200200200002ALM012012uaa xa yvbb xb y012111iiijjjkkkxyauxyauxyauiijjkkiijjkkuN uN uN uvN vN vN v1()2 iiiiijkmjijki
10、jkNxyAx yx yyyxx注意下标轮换注意下标轮换,ij jk kie000000 iiijkjijkjkkuvNNNuuNNNvvuvuNu(,)ijjijN xy1ijkNNNeeeTeTddVVKB DBMN Neeee00eTeTeTeT00dddd SfSVVVVfN ffN TfB fB D111()d()niiifA f1001()d()2(0)fA ff11122111()d()()()()33fA fA fff111111cos()dcos()cos()1.675833cos()d2sin(1)1.6829121d?aaacaacacccb (12)KKufKKufaa
11、aaacc (13)K ufK u1aaaaacc (14)uKfK u111211121222221200 (15)00000nnjjjnnnnnnKKKufKKKufKuKKKuf 1 11121112122222121200 (16)0nnjjjjjnjjnnnnnnKKKufKKKufKKKKufKKKufjjKjjjK u(,)iiiixxiijjkkiijjkkxN xN xN xyN yN yN y三维问题？ee111eTT111111eTT111dd d ddd d d VVKB DBB DB JMN NN N J00111111eTeT111111111111eTeT0011
12、1111d d dd d dd d dd d d SffN f JfN T JfB JfB D J0 (1)MuKusin (2)tu0 (3)KM 2()0 (4)fKM1210 (5)nn0 (6)iiiKMKM0 (8)0iiijjjKMKMT1 (7)ii MT0 (9)ijji MTT (10)ijijijiij M K1 (12)niiiTT (11)MI K (13)KMMM (13)KM Mii1 (14)MK2自由度系统的例子TT()(15)K MTTTT()(16)K M 2T1T21()(17)niiinii (),(18)in (19)rKTT (20)IK1T (21
13、)I rAT22220 (22)maxnRxx AxAx1221 (23)Ir121max (24)iiI2min (25)iir1 (26)kkKxMx11111 (27)nkkiikinkkiikiaaxaxa1 (28)kkaa12111 (29)kkkka a a a11 122 (29)kkkknnaaa a1111 (30)knkkiiiiaxk 11 11 (31)kkax111 (32)kkxxTT1111TT1111()(33)kkkkkkkkkxKxxMxxxMxxMx21211112211()(34)kniiikkniiiaax1T1T (35)iiiikkjjkjiik
14、kiikkxx MxxxxMx11 (36)iikkxK MxT111T11 (37)iikkikiikkxKxxMx移 轴 法 (38)KM M (39)i1111 (40)knnikiiikiiiiaax怎么定？T KT MT1T1 (41)kkkkkkkkKP K PMP M Pk 112112,knkndiag kkkdiag mmmKM1111/2121 (42)kkkkMKPPP M (43)KMIT (44)kkP PIT1 (45)kkkkKP K P11cossin11sincos11kP(,)ki jK1(,)0ki jK11sin2cos2 (46)2kkkkijjjii
15、ijKKKK21arctan2 (47)4kijkkjjiiKKK112111 (49)kkiiiikiikijnkkiijjKKKKKK2111 (48)kijkkiijjKKKK112233()()()()(50)()()KOOOKOOOKK (51)kijkkiijjKKK221111122()10()10 (52)001kkkkOKKOKKP211222233()0()0()()(53)()()KOOKOOOOKK2221122242222233()()()()()()(54)()()()KOOOOKOOOOKOK11111111kP11(1)0 (55)(1)0 (56)kkkkij
16、iiijjjkkkkijiiijjjKKKKMMMMkkkijjjiikkkiiijjjKKKMMM0,kijkjjKK,(57)kkjjiiKK (58)sign22kkkkkiiiiijiiijkkkkkjjjjijjjijkkkkkiijjjjiikkkkkiijjKK MM KKK MM KKK MK MKKKK K210121012 K1cossin0sincos0001P41100.707030.7070.7070.7072K0.47772cos0sin010sin0cosP20.6340.32500.32530.62800.6282.366 K0.13413cossin0sinc
17、os0001P30.590100.08303.04380.6220.0830.6222.366K-0.536141000cossin0sincosP40.59010.04290.0720.04293.413600.07201.9963KT0.58580003.41420002.000 K (59)KQRT (60)Q KQRQ1 (61)kkkkkkKQ RKR Qk 1kK121kQ QQ1 (62)miiiqq1212mmqqqq里兹基向量里兹坐标TTTTTTTT()=(63)Kq Kqq Kq Mq Mqq MqT0 (64)KqMqqq Mq (65)KqMq=0q,iiqq (66)
18、iiiKM=0q (67)iiii (68)Kf2100.500141 010012000.5 KM100001f1K f711221217T7111712K KT291111129144M Mqq11221.34182.0008,2.4004,4.00321.34182.0008qq11220.8951.000.4470.000.8951.001232.04.06.0101 110f1K f51142651T721213K KT41 13113536M Mqq11220.7072.12,2.00,6.000.7076.36qq11220.7070.7070.7070.7070.7070.70
19、7KMKKTKLDL000012rXxxxkYMXT1kKXYkT1kLDL XY11kYMXT11kMXY1kKM1kk1YY111,2,kkiikiip11kkX,00 (1)(0),(0)MuCuKufuuuu (2)uq12 (3)sqqqq广义位移TTTT0000 (4),q Cqq fq Muq Mu()01sin()()d (6)(sincos)rrrrttrrrrtrrrrqetfeatbt22,1,2,(5)rrrrrrrqqqfrs (7)ttttXDXLf0L10 (8)nttXDX给定的初始条件递推矩阵1 (9)DPJP1,(10)00nnnnn JJJ1|1nJn 0
20、|1nJn 120 (11)0J1,2()max1 (12)iiDnJn 递推矩阵的谱半径nD023423411()26 (13)11()26tttttttttttttttOttttOt uuuuuuuuuu212 (14)12tttttttttttttuuuuuuu (15)ttttMuCuKuf (16)tttMuf222112 (17)2112ttttttttttMMCffKM uMC u20001 (18)2ttt uuuu0120322111,2,(19)2cccccttcKMC0u0u 0u 0030ttc uuuu01ccMMCTMLDL201()()tttttcccffKM u
21、MC u012tttttttttttccuuuuuuutttMuftttt()()01iitttiiiifuc McC220 (20)tttqqq22210 (21)11tttttttqqqtt (22)tttQDQTT22 (23)211110tttttttttqqqqttttQQD222221,21224 (24)2tt22222 (25)t cr2 (26)Ttt 2cr1 (27)Tt2cr1 (27)nTtt (28)ttttttttttuuuuuut12 (29)12ttttttttttttttuuuuuuuu21 (30)4 tttttttttuuuuu22(1)(31)12 t
22、tttttttttttttttttuuuuuuuuu未知量21111 (32)2tttttttttuuuuu11 (33)2tttttttttuuuuutt (34)ttttttttMuCuKuf2211111 (35)2112ttttttttttttttttMCK ufMuuuCuuu (36)ttttKuf (38)ttttKuf012324567111,12 (36)1,1,(1),2 cccctttcct ctct01023145(37)ttttttttttccccccccKKMCffMuuuCuuuKMC0u0u 0u TKLDLK02367tttttttttttttcccccuuu
23、uuuuuutttKuftt023145ttttttttttccccccffMuuuCuuutttt20 (39)ttttqq22211 (40)2 ttttttqqqtt q22,(41)ttttttqqqq222(1)(42)12 tttttttttttttqqqqtqqqtqqt22221112 tttttqttqt q22221112 ttttqttqt q222222112221 10(43)2 ttttttqtqt q (44)tttQDQ (45)10bcD2222222122 (46)1112 bhthtch21,24 (48)2bbc240 (49)1bcc2222144 (50)2101 (51)2th2111,(52)242cr21 (53)142 Ttt
