结构动力学-2(哈工大结构动力学)

1、运动方程运动方程运动方程的通解运动方程的通解)()(11tymty )()(11tymtyk 0)()(2tyty 令令111121mmk第二章第二章单自由度体系的振动分析单自由度体系的振动分析 EIl)(ty)(tym mkmtctctysincos)(21运动方程的特解运动方程的特解EIl)(ty)(tym mkm00)0(,)0(vyyytvtytysincos)(000)()(2tyty tctctysincos)(21运动方程的通解运动方程的通解令令cos/,sin00AvAy)sin()(tAty运动方程的特解运动方程的特解00)0(,)0(vyyytvtytysincos)(00

2、令令cos/,sin00AvAy)sin()(tAty22020vyA00tanvyAATty2T自振周期自振周期21T自振频率自振频率与外界无关与外界无关, ,体系本身固有的特性体系本身固有的特性自振频率和周期的计算自振频率和周期的计算1.1.利用计算公式利用计算公式111121mmk2TEIlEIl=111=1ll/2l例例1.1.求图示体系的自振周期求图示体系的自振周期. .3117121mlEIm)221213221(111lllllllllEIEImlT127223解解: :EIl3127(2)(2)利用机械能守恒利用机械能守恒常数)()(tUtT)(cos21)(21)(2222t

3、mAtymtT)(sin21)(21)(2211211tAktyktUmaxmaxUT例例2.2.求图示体系的自振频率和周期求图示体系的自振频率和周期. .222222max29)2(21)(21)2(21mllmlmlmT解解: :mk95lmEImlllkk)(t2222max25)2(21)(21kllklkUmaxmaxUT(3)(3)利用振动规律利用振动规律)sin()(tAty)sin()(2tAty )sin()()(2tmAtymtI 位移与惯性力同频同步位移与惯性力同频同步. .例例3.3.质点重质点重W,求体系的频率求体系的频率. .AlEIkAmA323解解: :gWm/

4、gWlEIk33EIkl2mAAkAAlEI332mAlmEImlllkkml22ml22ml2lklk2A例例4.4.求自振频率和周期求自振频率和周期. .解解: : 列幅值方程列幅值方程 0AM0222222222lklllmlmllkllml059222klml转动惯量转动惯量22292)2(mlmllmJ刚度系数刚度系数2522klllkllkk自振频率自振频率Jkmk112c-阻尼系数阻尼系数 ( (damping coefficient ）)()(tyctR011ykycym )(ty)(tym )(11tyk)(tycmc2/022yyy tAety)(0222011ykycym

5、 )(ty)(tym )(11tyk)(tycmc2/022yyy tAety)(022221i)2(1mc 21D)cossin()(21tctcetyDDt小阻尼情况小阻尼情况00)0(,)0(vyyy02001,/)(ycyvcD)sin()(DDttAety20020)(DyvyA)/(tan000yvyDD)2(1mc 21D)cossin()(21tctcetyDDt小阻尼情况小阻尼情况00)0(,)0(vyyy02001,/)(ycyvcD)sin()(DDttAety20020)(DyvyA)/(tan000yvyDD临界阻尼情况临界阻尼情况)2(1mc )()(000ytyv

6、etyt超阻尼情况超阻尼情况)2(1mc )()(000tchytshyvetyccct12cty1111mcr2-临界阻尼系数临界阻尼系数mcccr2-阻尼比阻尼比 大量结构实测结果表明，大量结构实测结果表明，对于钢筋混凝土和砌体结构对于钢筋混凝土和砌体结构 ，钢结构，钢结构 。各种坝体的。各种坝体的，拱坝，拱坝 ，重力坝（大头坝），重力坝（大头坝） ，土，土坝、堆石坝坝、堆石坝 。05. 004. 003. 002. 02 . 003. 005. 003. 01 . 005. 02 . 01 . 0it1itDTt)(tyiA1iADDT221D小阻尼情况小阻尼情况)sin()(DDttA

7、ety20020)(DyvyA)/(tan000yvyDDDDiiTTttiieAeAeAA)(1DiiTAA1ln22D阻尼测量阻尼测量DiiTAA1ln22D1ln21iiAAniiAAnln21kN4 .160276.012ln421)/(102 .802.0104 .165311mNk) s (5 .04/2DT) s (4998.012DTT)s/1 (57.122T)kg(5190/211km)kN(86.50 mgW)s/mN(36012mc)s/1 (89.1368005190102 .8252)s/1 (70.11)s (537.0/2T0257.02/mckN4 .1602

8、76.012ln421)/(102 .802.0104 .165311mNkPROBLEMSPROBLEMS：1.1.A heavy table is supported by flat steel legs.Its natural A heavy table is supported by flat steel legs.Its natural period in lateral vibration is 0.5sec.When a 50kg plate is period in lateral vibration is 0.5sec.When a 50kg plate is clamped

9、 to its surface.the natural period in lateral vibration clamped to its surface.the natural period in lateral vibration is lengthened to 0.75sec.What are the weight and the effective is lengthened to 0.75sec.What are the weight and the effective lateral stiffness of the table?lateral stiffness of the

10、 table?T=0.5secT=0.5secT=0.75secT=0.75secPROBLEMSPROBLEMS：2.2.An electromagnet weighing 400kg and suspended by a spring An electromagnet weighing 400kg and suspended by a spring having a stiffness of 100kg/cm.lifts 200kg of iron scrap. having a stiffness of 100kg/cm.lifts 200kg of iron scrap. Determ

11、ine the equation describing the motion when the electric Determine the equation describing the motion when the electric current is turned off and the scrap is dropped.current is turned off and the scrap is dropped.kkkPROBLEMSPROBLEMS：3.3.A mass m is at rest,partially supported by a spring and A mass

12、 m is at rest,partially supported by a spring and partially by stops.In the position shown,the spring force is partially by stops.In the position shown,the spring force is mg/2. At time t=0 the stops are rotated,suddenly releasing the mg/2. At time t=0 the stops are rotated,suddenly releasing the ma

13、ss.Determine the motion of the mass. mass.Determine the motion of the mass. kmPROBLEMSPROBLEMS：5.5.A mass mA mass m1 1 hangs from a spring k and is in static equilibrium. hangs from a spring k and is in static equilibrium. A second mass mA second mass m2 2 drops through a height h and sticks to m dr

14、ops through a height h and sticks to m1 1 without rebound.Determine the subsequent motion y(t) measured without rebound.Determine the subsequent motion y(t) measured from the static equilibrium position of mfrom the static equilibrium position of m1 1 and k. and k.1mk2mhPROBLEMSPROBLEMS：6.6.For a sy

15、stem with damping ratio ,determine the number of For a system with damping ratio ,determine the number of free vibration cycles required to reduce the displacement free vibration cycles required to reduce the displacement amplitude to 10% of the initial amplitude,the initial velocity amplitude to 10% of the initial amplitude,the initial velocity is zero.i