3.2.2 Impulsive Vibration 3.2.2 Impulsive Vibration A serics impulsive excitation-Base Excitation mi0）+ct)+k)=城() ()+25)+x0)=-.() P()=-m,() 一系列冲古使产生的构 地面运动可表示为付四dr Integrate of a series impulsive excitation 3.2.2 Impulsive Vibration 3.2.2 Impulsive Vibration k0）=eoi-P(sin-rdr ()P(sin(-dr 7e0 mo o ()sinu-dr me) e0=e,mou-t 3.2.3 Arbitrary excitation 3.2.3 Arbitrary excitation Arbitrary excitation (随机漾励) Dynamic Equilibrium Equation w mass P( P(t) m++kx=p(t) c where a dot represents differentiation with respect to time D'Alambert's Equilibrium Equation: mc+kx=P(t) 7 7 冲击荷载下的自由振动 一系列冲击荷载产生的结构振动   .. 0 ( ) t d t g P x m   ＝ 地面运动 可表示为 3.2.2 Impulsive Vibration  A series impulsive excitation —— Base Excitation ( ) ( ) ( ) - ( ) mx t cx t kx t mx t    g No. 38 2 ( ) 2 ( ) ( ) - ( ) g x t x t x t x t      = 2 c m   k = m  P( ) - ( ) g t mx t  Integrate of a series impulsive excitation 3.2.2 Impulsive Vibration At moment, τ, the impulsive loading is P(τ), under this action, the responses of a SDOF is ；     ' ' ' ( )sin ( ) t P dx t e t d m            .. : ( )= - g if P m x      ' ' ' 0 ( )sin ( ) t t P x t e t d m                 ' .. ' ' 0 1 - sin ( ) t t x t e x t d g            Duhamel integral 3.2.2 Impulsive Vibration At moment, τ, the impulsive loading is P(τ), under this action, the responses of a SDOF is ；     ' ' ' ( )sin ( ) t P dx t e t d m            .. : ( )= - g if P m x      ' ' ' 0 ( )sin ( ) t t P x t e t d m                 ' .. ' ' 0 1 - sin ( ) t t x t e x t d g            Duhamel integral 3.2.2 Impulsive Vibration D’Alambert’s Equilibrium Equation: m  x   cx   kx  P(t) Pt() Pt() 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 41 Arbitrary excitation （随机激励） 3.2.3 Arbitrary excitation x k 0 c mass mx kxcx  Dynamic Equilibrium Equation  For a dynamic or time-dependant force p (t)，the equation of static equilibrium becomes one of dynamic equilibrium: where a dot represents differentiation with respect to time.  Equation must be satisfied at each instant of time during the time interval under consideration． mx cx kx p t      No. 42 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 42 3.2.3 Arbitrary excitation