结构动力学-2z.ppt

1,一. 无阻尼体系,运动方程,运动方程的通解,令,二阶线性齐次常微分方程,第三章 单自由度体系,3.1 无阻尼自由振动,2,运动方程的特解,设初始条件为,令,3,其中,自振周期,自振频率,,与外界无关,体系本身固有的特性,4,自振频率和周期的计算,1.利用计算公式,,例1.求图示体系的自振周期.,解:,5,(2)利用机械能守恒,例2.求图示体系的自振频率和周期.,解:,6,(3)利用振动规律,位移与惯性力同频同步.,7,8,阻尼:使振动衰减的作用.,阻尼产生原因：材料的内摩擦,连接点、支承面等处的外 摩擦及介质阻力等.,阻尼力：,在振动分析当中用于代替阻尼作用的阻碍振动的力粘滞阻尼理论假定阻尼力大小与速度成正比,方向与速度相反c-----阻尼系数 (damping coefficient ）,运动方程及其解,令,运动方程,设,特征方程,3.2 有阻尼自由振动,9,根为,令,方程的通解为,小阻尼情况,由初始条件,10,临界阻尼情况,超阻尼情况,11,-----临界阻尼系数,-----阻尼比,大量结构实测结果表明，对于钢筋混凝土和砌体结构 ，钢结构 。

各种坝体的，拱坝 ，重力坝（大头坝） ，土坝、堆石坝 12,周期延长,计算频率和周期可不计阻尼,小阻尼情况,振动是衰减的,对数衰减率,13,阻尼测量,2.刚度系数,14,3.有阻尼周期,4.重量,5.阻尼系数,6.若质量增加800kg,周期和阻尼比,,15,,PROBLEMS：,1.A heavy table is supported by flat steel legs.Its natural period in lateral vibration is 0.5sec.When a 50kg plate is clamped to its surface.the natural period in lateral vibration is lengthened to 0.75sec.What are the weight and the effective lateral stiffness of the table?,,,,T=0.5sec,T=0.75sec,16,PROBLEMS：,2.An electromagnet weighing 400kg and suspended by a spring having a stiffness of 100kg/cm.lifts 200kg of iron scrap. Determine the equation describing the motion when the electric current is turned off and the scrap is dropped.,17,PROBLEMS：,3.A mass m is at rest,partially supported by a spring and partially by stops.In the position shown,the spring force is mg/2. At time t=0 the stops are rotated,suddenly releasing the mass.Determine the motion of the mass.,18,PROBLEMS：,5.A mass m1 hangs from a spring k and is in static equilibrium. A second mass m2 drops through a height h and sticks to m1 without rebound.Determine the subsequent motion y(t) measured from the static equilibrium position of m1 and k.,19,PROBLEMS：,6.For a system with damping ratio ,determine the number of free vibration cycles required to reduce the displacement amplitude to 10% of the initial amplitude,the initial velocity is zero.,7.若系统作自由衰减振动‘，经过m个周期振幅正好减至原来的一半，试求系统的阻尼比。

8.已知：为测阻尼系数c，用千斤顶使横梁产生侧移25mm，突然放开，经5周期后测得振幅为7.12mm，计算c。