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1、PPL/4 There are two differences between static and dynamic problems:1.Time-varying nature of the dynamic problems2.Coputing methodThe loads and the response vary with timeP(t)Internal forces and deflected shapes depend directly upon the given load and can be computed by force equilibrium.The result
2、are associated with inertia force resisting the accelerations。12.1 The features of dynamics and dynamic degrees of freedom1.The features of dynamics静力荷载-是指其大小、方向和作用位置不随时间而变化的荷载。动力计算的特点及动力自由度Static loads-magnitudes and position are not varying or varying very slowly with the time Dynamic loads-loads
3、of varying magnitudes and position with time 动力荷载-是指其大小、方向和作用位置随时间而变化的荷载。严格地说：实际荷载大多为动力荷载，但以荷载产生的影响来看，可分为两种情况：变化较慢，荷载对结构产生的影响与静荷载相差不变化较慢，荷载对结构产生的影响与静荷载相差不大，就归结为静荷载；大，就归结为静荷载；变化较快，荷载对结构产生的影响与静荷载相差变化较快，荷载对结构产生的影响与静荷载相差较大，就归结为动荷载；由它所引起的内力和变形都较大，就归结为动荷载；由它所引起的内力和变形都是时间的函数。是时间的函数。Dynamic loadsBlast loads
4、 caused by exploration Impact loadsCentrifugal loads of machineWind Earthquake Content of dynamical calculation 动力计算的内容动力计算的内容 To study the principle and method for calculate the responses of structure caused by dynamic loads.To calculate the response(over time),such as internal forces,displacement,
5、speed and acceleration of structure.1）确定结构本身的动力特性（固有自振频率、周期、振确定结构本身的动力特性（固有自振频率、周期、振型和阻尼等等）型和阻尼等等）natural vibration properties2）确定动力荷载和动力反应（即外力、内力、位移、速确定动力荷载和动力反应（即外力、内力、位移、速度与加速度）度与加速度）response caused by dynamic loads2.Categories of dynamic loadings 动荷载分类动荷载分类Prescribed loadingsP(t)t简谐荷载（按正余弦规律变化）简
6、谐荷载（按正余弦规律变化）PtOrdinary periodic excitationCentrifugal loads of machine一般周期荷载一般周期荷载Simple harmonic loading2.1 Periodic loadings 周期性荷载Random dynamic loadings 2.2 Short-duration impulsive loadings 冲击荷载冲击荷载trP2.3 Random dynamic loading 随机荷载：(非确定性荷载)PtP(t)ttrPEarthquakeWind地震荷载地震荷载风荷载风荷载blastcollision3.
7、Dynamic degrees of freedom(DOF)(xmy(x,t)x1.lumped mass method 集中质量法集中质量法 2.generalized coordinate method 广义坐标法广义坐标法3.finite element method 有限元法有限元法 一个体系的动力自由度是指任意时刻确定体系上全部质量所需要一个体系的动力自由度是指任意时刻确定体系上全部质量所需要的独立坐标的数目的独立坐标的数目.Numbers of independent movements(or coordinate)which are required to locate all
8、 the mass of a system fully at any time.Because the DOF of practical structure with distributed masses is infinite,it should be simplified.动力计算的自由度动力计算的自由度 把连续分布的质量集中为几个质点，将一个无限自把连续分布的质量集中为几个质点，将一个无限自由度的问题简化成有限自由度问题。由度的问题简化成有限自由度问题。1、集中质量法、集中质量法+m梁mmm梁mII2Im+m柱厂房排架水平振厂房排架水平振时的计算简图时的计算简图单自由度体系单自由度体系2
9、个自由度个自由度水平振动时的计算体系水平振动时的计算体系3个自由度个自由度4个自由度个自由度m1m2m32个自由度个自由度构架式基础顶板简化成刚性块构架式基础顶板简化成刚性块u(t)v(t)(t)y2y1自由度与质量数不一定相等自由度与质量数不一定相等2个自由度个自由度12.2 Free Vibration of Single-Degree of Freedom System单自由度体系的自由振动单自由度体系的自由振动1、Simple structuremkThe pergolaThe elevated water tankThe one-story workshop They can be
10、idealized as a concentrated or lumped mass m supported by a massless structure with stiffness k in the lateral direction.we will assume the lateral motion of these structures is small in the sense that the supporting structure deform within their linear elastic limit.支撑杆件的变形在材料的线弹性范围内支撑杆件的变形在材料的线弹性范
11、围内无阻尼自由振动方程2.Undamped free vibration structure It is disturbed from its static equilibrium position and then allowed to vibrate without any external dynamic excitation.y设某一时刻t，质点离开平衡位置y2.1 Equation of motion)(tky弹簧力与位移方向相反 惯性力-与运动加速度方向相反)(tym Inertia forceElastic resisting force0kyym is equal to the
12、 product of mass times its acceleration and acting in a direction opposite to the acceleration.达朗贝尔原理指出：将惯性力包括在内时，体系在每一瞬时都处于平衡。Stiffness method 刚度法刚度法 常用于刚架结构常用于刚架结构0kyym It is based DAlemberts principle which state that with inertia force included,a system is in equilibrium at each time instant.y质点
13、在 t 时刻的位移y 就等于质点在惯性力作用下的静位移flexibility method 柔度法常用于梁式结构柔度法常用于梁式结构1k1惯性力ymFI)(ymFyI 刚度法和柔度法的方程是相通的3.Solution to the differential equation0 kyym 0ymky mk02yy 自振圆频率tCtCtycossin)(21微分方程的解Natural circular frequency of vibrationthe homogeneous differential equation00(0)(0)vyyy已知初始条件：0201yCvC，tytvtycossin
14、)(00y0ty-ytycosyt0Tvvtvsin)(sin)(tatyyt0)sin(taA-AT00122020tanvyvya；The result may be written tytvtycossin)(004.Connatural characteristic of structure2TThe period21TfCyclic frequencyfT22Circular frequencymk自振圆频率的表达式 expression of circular frequencym1WgstgW-质点的重量st-重力引起震动方向的静位移kmT2自振周期的表达式 expression
15、 of natural vibration period1、自振周期只与结构的质量和刚度有关，外力只能影响振幅a,而不能影响自振周期的大小。2、要改变结构的振动频率，需要改变质量和刚度。3、自振周期T、频率和振幅是结构动力特性的重要数量标志。ggWmst222例1：求图示体系的水平自振周期（忽略杆件轴线变形）周期mT211LEILLLLEI3322113EILmT323mLEIEA例1a：求图示体系的竖向自振周期：(考虑轴向变形）mLEIEA周期mT211EALEAL11EAmLT2例2：求图示体系的自由振动频率329221LLmL/32L/3P=12L/939221111LLEI)9232(
16、L解：质点作竖直方向的振动，用柔度法求解是方便的)9232(L34243LEI1m自振频率34432mLEI例3：求图示结构的自振频率mEIEILL1m自振频率：3748mLEI解：质点只在水平方向产生振动，用柔度法计算自振频率。在质点水平方向加单位力，其水平位移即为柔度系数。1p8383L85L图M)8532(858521LLL1 132 3()2838LLLEI)8332(838321LLL3748LEI例3：求图示结构的自振频率mEIEIp8383L85L图M1p图1ML取基本体系计算计算位移会简化31 12 51 37()2383848LLLL LEIEI 31487EImmL自振频率：例4：求图示结构的自振频率取基本体系计算：31121315121323()()()2423643322423641536LLLLLLLLEIEIEIEIL/2L/2LP=1L6413L323MP=14L1M31153623EImmL自振频率：例5：求图示结构的自振圆频率LLLEImII2I解：简化为单自由度体系，在水平方向振动。mk自振频率339mLEI刚度系数=1k33LEI324LEI312LEI333912243LEILEIEIEIk