結(jié)構(gòu)動(dòng)力學(xué)(單自由度體系的振動(dòng)分析)

1、自由振動(dòng)自由振動(dòng)-由初位移、初速度引起的由初位移、初速度引起的, ,在振動(dòng)中無動(dòng)荷載作用的振動(dòng)。在振動(dòng)中無動(dòng)荷載作用的振動(dòng)。一一. .運(yùn)動(dòng)方程及其解運(yùn)動(dòng)方程及其解EIl)(ty)(tym )()(11tymty )()(11tymtyk 0)()(2tyty 111121mmk一一. .運(yùn)動(dòng)方程及其解運(yùn)動(dòng)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(dòng)分析振動(dòng)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(jì)阻尼時(shí)的自由振動(dòng)是簡諧振動(dòng)單自由度體系不計(jì)阻尼時(shí)的自由振動(dòng)是簡諧振動(dòng). .)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(jì)算自振頻率和周期的計(jì)算1.1.計(jì)算方法計(jì)算方法(1)(1)利用計(jì)算公式利用計(jì)算公式111121mmk11,WmgWststg2(2)(2)利用機(jī)械能守恒利用機(jī)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(dòng)規(guī)律利用振動(dòng)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(diǎn)重質(zhì)點(diǎn)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(shí)測結(jié)果表明，大量結(jié)構(gòu)實(shí)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(dòng)，經(jīng)過若系統(tǒng)作自由衰減振動(dòng)，