結(jié)構(gòu)動力學(單自由度體系的振動分析)

1、自由振動自由振動-由初位移、初速度引起的由初位移、初速度引起的, ,在振動中無動荷載作用的振動。在振動中無動荷載作用的振動。一一. .運動方程及其解運動方程及其解EIl)(ty)(tym )()(11tymty )()(11tymtyk 0)()(2tyty 111121mmk一一. .運動方程及其解運動方程及其解EIl)(ty)(tym )()(11tymty )()(11tymtyk 0)()(2tyty 111121mmk其通解為其通解為tctctysincos)(21由初始條件由初始條件0)0(yy0)0(yy可得可得01yc /02yctytytysincos)(00sin0Ay c

2、os/0Ay)sin()(tAty22020yyA00tanyy二二. .振動分析振動分析其通解為其通解為tctctysincos)(21由初始條件由初始條件0)0(yy0)0(yy可得可得01yc /02yctytytysincos)(00sin0Ay cos/0Ay)sin()(tAty22020yyA00tanyy單自由度體系不計阻尼時的自由振動是簡諧振動單自由度體系不計阻尼時的自由振動是簡諧振動. .)2sin()sin()(tAtAty)2()2(sintytA2T自振周期自振周期2T自振周期自振周期21T自振園頻率自振園頻率( (自振頻率自振頻率) )與外界無關(guān)與外界無關(guān), ,體系

3、本身固有的特性體系本身固有的特性A 振幅振幅初相位角初相位角三三. .自振頻率和周期的計算自振頻率和周期的計算1.1.計算方法計算方法(1)(1)利用計算公式利用計算公式111121mmk11,WmgWststg2(2)(2)利用機械能守恒利用機械能守恒)(cos21)(21)(2222tmAtymtT)(sin21)(21)(2211211tAktyktUmaxmaxUT20202121kvymEI)(21)(21)(22tkytymtE2002cossin-21)(tvtymtE200sincosk21tvty20202121) t (kvymE=常數(shù)常數(shù)(3)(3)利用振動規(guī)律利用振動規(guī)

4、律)sin()(tAty)sin()(2tAty )sin()()(2tmAtymtI 位移與慣性力同頻同步位移與慣性力同頻同步. .211mAAk111kEIl)(ty2mAA幅值方程幅值方程mk1122.2.算例算例例一例一. .求圖示體系的自振頻率和周期求圖示體系的自振頻率和周期. .3117121mlEIm)221213221(111lllllllllEIEImlT127223EIlEIl=111=1ll/2l解解: :EIl3127例二例二. .求圖示體系的自振頻率和周期求圖示體系的自振頻率和周期. .3332231mlEIEIlmEIl31132EImlT32=1解解: :23lE

5、IEIllm/2EIEIll例三例三. .質(zhì)點重質(zhì)點重W,求體系的頻率和周期求體系的頻率和周期. .3113lEIkk解解: :EIkl11k111kk33lEIgWm/gWlEIk33例四例四. .求圖示體系的自振頻率和周期求圖示體系的自振頻率和周期. .222222max29)2(21)(21)2(21mllmlmlmT解解: :mk95lmEImlllkk)(t1.1.能量守恒能量守恒2222max25)2(21)(21kllklkUmaxmaxUT2.2.列幅值方程列幅值方程ml22ml22ml2lklk2A 0AM0222222222lklllmlmllkllml059222klml

6、mk95c-阻尼系數(shù)阻尼系數(shù) ( (damping coefficient ）)()(tyctR011ykycym )(ty)(tym )(11tyk)(tycmc2/022yyy tAety)(022221i)2(1mc 21D)cossin()(21tctcetyDDt小阻尼情況小阻尼情況00)0(,)0(vyyy02001,/)(ycyvcD)sin()(DDttAety20020)(DyvyA)/(tan000yvyDD011ykycym )(ty)(tym )(11tyk)(tycmc2/022yyy tAety)(0222臨界阻尼情況臨界阻尼情況)2(1mc )()(000ytyv

7、etyt超阻尼情況超阻尼情況)2(1mc )()(000tchytshyvetyccct12c)2(1mc 21D小阻尼情況小阻尼情況00)0(,)0(vyyy02001,/)(ycyvcD)sin()(DDttAety20020)(DyvyA)/(tan000yvyDD)cossin()(21tctcetyDDt1mcr2-臨界阻尼系數(shù)臨界阻尼系數(shù)mcccr2-阻尼比阻尼比 大量結(jié)構(gòu)實測結(jié)果表明，大量結(jié)構(gòu)實測結(jié)果表明，對于鋼筋混凝土和砌體結(jié)構(gòu)對于鋼筋混凝土和砌體結(jié)構(gòu) ，鋼結(jié)構(gòu)，鋼結(jié)構(gòu) 。各種壩體的。各種壩體的，拱壩，拱壩 ，重力壩（大頭壩），重力壩（大頭壩） ，土，土壩、堆石壩壩、堆石壩 。

8、05. 004. 003. 002. 02 . 003. 005. 003. 01 . 005. 02 . 01 . 0mc2/DDT221D小阻尼情況小阻尼情況)sin()(DDttAety20020)(DyvyA)/(tan000yvyDDttytycdyfEtDD)d()(10DDiiTTttiieAeAeAA)(1)sin()(DDttAetyDiiTAA1ln22D1ln21iiAAniiAAnln21niiBBnln21kN4 .160276.012ln421)/(102 .802.0104 .165311mNk) s (5 .04/2DT) s (4998.012DTT)s/1

9、(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/mcPROBLEMSPROBLEMS：3.A mass m is at rest,partially supported by a spring and 3.A mass m is at rest,partially supported by a spring and partially by stops.In the position shown,the spring force

10、 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 mass.Determine the motion of the mass. mass.Determine the motion of the mass. kmPROBLEMSPROBLEMS：5.A mass m5.

11、A 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 drops through a height h and sticks to m1 1 without rebound.Determine the subsequent motion y(t) measured wit

12、hout 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.For a system with damping ratio ,determine the number of 6.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.is zero.7.7.若系統(tǒng)作自由衰減振動，經(jīng)過若系統(tǒng)作自由衰減振動，