结构动力学典型习题及答案

7-1（a)试求图示体系的自振频率与周期。解;485311EIl=δ;098.33mlEI=ω1=l/2mEIl/2l)(1tyl/2l/4;027.23EImlT=7-1（b)试求图示体系的自振频率与周期。解:;153611311EIl=δ;11153632mlEI=ω1=;531.03EImlT=l常数=EI2/l2/lm32/l16/l64/9l32/5l)(a1=2/l)(b求柔度系数：用位移法或力矩分配法求单位力作用引起的弯矩图（图a);将其与图b图乘，得;817.113mlEI=ω7-1（c)试求图示体系的自振频率与周期。解328mlEI=ω;22.23EImlT=lEI2/lm刚性杆m2lA2/A222ω⋅⋅Am2ωmA∑=0xFAlEImAmA322122=+ωωo222ω⋅⋅Am2/2ωmAAlEI3122ωmA由右面竖杆的平衡可求出铰处约束力。由水平杆的平衡：;828.23mlEI=ω7-1（d)试求图示体系的自振频率与周期。解;626.33EImlT=;732.13mlEI=ωmEIlEIlm∞=1EI3116lEIk=33112326mlEImlEImk===ω7-1（e)试求图示体系的自振频率与周期。解mkEIlm)2148(113112+==δωmEIl/2kkl/2kEIl21481311+=δmkEIlm)2148(11311+==δωmkEIlT)21481(23+=π7-3试求图示体系质点的位移幅值和最大弯矩值。已知解：EIlFyPst3=PF)(1tyωθ6.0=mEI=常数2ltFPθsin2llsty1=11δ2/llFP5625.1/1122=−=ωθµ位移幅值EIlFyAPst35625.1==µPFAm3375.02=θ2/lEIl31135=δlFMPd169.1max=PF3375.0PFlFP169.1解：7-4图示梁跨中有重量为20kN的电动机，荷载幅值P=2kN，机器转速400r/min，,梁长l=6m。试求梁中点处的最大动位移和最大动弯矩。24.1006.1mkNEI×=mEI=常数l/2tPθsinl/2923.1/1122−=−=ωθµmPyAst41110326.16−×−===µδµNmEIl/10245.41006.1486481773311−×=××==δ05.0=ξ)/1(115410245.410208.91273112sm=×××==−δω)/1(97.33s=ω)/1(888.41604002s=×=πθmNPlMd.10769.543max×==µ(b)阻尼比8726.14)1(12222=+−=βξβµmPyAst41110898.15−×===µδµmNPlMd.106178.543max×==µ解：7-5习题7-4结构的质量上受到突加荷载P=30kN作用，若开始时体系静止，试求梁中最大动位移。µδµ11PyAst==不计阻尼时，动力系数为2，利用上题数据，可得m02547.0210245.4103073=××××=−7-6某结构在自振10个周期后，振幅降为原来初始位移的10%（初位移为零），试求其阻尼比。解：0366.010ln1021=×=πξ8-1试求图示梁的自振频率和振型。m2mEIaaa解EIaEIa32231161;==δδEIa3211241−==δδ[][][]02=−mIδω0/1/122222111222111=−−ωδδδωδmmmm令21111ωδλm=03/14/12/11=−−−−λλ024/53/42=+−λλ181.0153.121==λλ3231352.2;931.0maEImaEI==ωω1=1=aa/2)(1ty)(2ty61.0/;277.3/22122111=−=xxxx{}{}=−=639.11;305.0121xX8-2.试求图示刚架的自振频率和振型解:EIl31134=δ令2111/1ωδλm=02/18/34/31=−−λλ032/9)2/1)(1(=−−−λλ1637.0336.121==λλ3231140.2;749.0mlEImlEI==ωωmlEImEIl1y2y12Xmω222Xmω1X2X11δ21δ1=12δ22δ1=1221212112XXmXm=+ωδωδ2222212212XXmXm=+ωδωδ02)1(211121=+−XXδδλ0)2(2112211121=−+XXλδδδδllEIl3211221==δδEIl32231=δ23.214/312111=−−=λXX897.014/322212−=−−=λXX{}{}−==1897.0;123.221XX{}=112X{}−=111X8-3.试求图示梁的自振频率和振型。MPaEcmINmgcml54102,82.68,1000,100×====731110662.0−×==EIlδs/175.384=ω按反对称振型振动m2/lEI2/l2/l2/lmm2/l2/l按对称振型振动=15l/323l/16=1l/2m2/l2/l=1l/45311100151.048−×==EIlδs/145.254=ωss/175.384/145.25421==ωω{}=112X{}−=111X8-4.试求图示刚架的自振频率和振型。EIl192311=δ3/856.13mlEI=ω按反对称振型振动按对称振型振动=1l/2=1l/8m2/lEI2/l2/l2/lmEI2/lEI2/lmEIml/8l/82/l2/lmm=13l/165l/32l/2=1EIl311=δ3/mlEI=ω31/mlEI=ω32/856.13mlEI=ω1y2y1y024)227)(60(222=−−−ωω8-5.试求图示刚架的自振频率和振型。mEI∞=1EI∞=1EIl2mEI2EI2EI2l2y311/60lEIk=32112/24lEIkk−==322/27lEIk=0222221122111=−−ωδωmkkkmk01044147224=+−ωω321/965.7mlEI=ω322/53.65mlEI=ω31/822.2mlEI=ω32/095.8mlEI=ω336.4;4612.022122111−==XXXX{}=168.211X{}−=231.012X8-6.试求图示刚架的自振频率和振型。设楼面质量分别为m1=120t和m2=100t,柱的质量已集中于楼面，柱的线刚度分别为i1=20MN.m和i2=14MN.m,横梁刚度为无限大。mNk/1051611×=mNkk/102162112×−==0222221122111=−−ωδωmkkkmk063062.7012.024=+−ωω221/1713.97s=ω559.1;5347.022122111−==XXXX{}=870.111X{}−=642.012X1ym1∞=1EI∞=1EIm22i2y2i1i1i4m4mmNk/1021622×=222/1287.537s=ωs/1885.91=ωs/1179.232=ω