《结构动力学》课程教学课件（讲稿）05 Response to impulsive loading

Wuhan University of Technology-Chapter5Responseto impulsiveloading5.1 General nature of impulsive loading5.2 Sine-wave impulse5.3 Rectangular impulse5.4 Triangular impulse5.5 Shock or response spectra5.6 Approximateanalysis of impulsive-load response6-1

6-1 Wuhan University of Technology 5.1 General nature of impulsive loading 5.2 Sine-wave impulse 5.3 Rectangular impulse 5.4 Triangular impulse 5.5 Shock or response spectra 5.6 Approximate analysis of impulsive-load response Chapter 5 Response to impulsive loading

Wuhan University of Technology5.1 General nature of impulsive loading The general solution must also include thetP(t)particularsolutionwhichdependsupontheformofdynamicloading..Suchaload consists of a singleprincipalimpulseofarbitraryform,asillustratedinFig51.Impulsiveorshockloadsfrequentlyareofgreatimportanceinthedesignofcertainclasses of structural systems, e.g., vehiclessuchastrucksorautomobilesortravelingcranes.Dampinghasmuchlessimportanceincontrollingthemaximumresponseofastructuretoimpulsiveloadsthanforperiodicorharmonicloadsbecausethemaximum response to a particular impulsive load will be reached in a veryshorttime,beforethedampingforcescanabsorbmuchenergyfromthestructure..For this reason only the undamped response to impulsive loads will beconsidered in this chapter.6-2

6-2 Wuhan University of Technology 5.1 General nature of impulsive loading • The general solution must also include the particular solution which depends upon the form of dynamic loading. • Such a load consists of a single principal impulse of arbitrary form, as illustrated in Fig. 51. Impulsive or shock loads frequently are of great importance in the design of certain classes of structural systems, e.g., vehicles such as trucks or automobiles or traveling cranes. • Damping has much less importance in controlling the maximum response of a structure to impulsive loads than for periodic or harmonic loads because the maximum response to a particular impulsive load will be reached in a very short time, before the damping forces can absorb much energy from the structure. • For this reason only the undamped response to impulsive loads will be considered in this chapter

Wuhan University of Technology> Suddenly Applied Constant Load[0,t0Fpot>0:POFpo sin o(t -t)dty(t) =(l-cosot)= y(1-cosot)二moJomoFPO=F,SStatic displacement under the load FpY stmo?6-3

6-3 Wuhan University of Technology  Suddenly Applied Constant Load 0 0, 0 ( ) , 0 p P t F t F t      t>0： 0 0 2 0 1 ( ) sin ( ) (1 cos ) (1 cos ) t P P st F y t F td t y t m m             0 2 P st P F y F m     Static displacement under the load FP FP(t) t FP0

Wuhan Universityof Technology2元T03元wtJ(t);质点围绕静力平衡位置作简谐振动[y(t)]maxβ==2Y st突加荷载所引起的最大位移比相应的静位移增大1倍6-4

6-4 Wuhan University of Technology max [ ( )] 2 st y t y    突加荷载所引起的最大位移比相应的静位移增大1倍 yst y(t) ωt 0 π 2π 3π 质点围绕静力平衡位置作简谐振动

Wuhan University of Technology> Suddenly Applied Load with a Limited DurationFp(t)当tuu阶段I(0<t≤u)：此阶段的荷载情况与突加荷载相同y(t) = ys, (1 -cos ot)6-5

6-5 Wuhan University of Technology  Suddenly Applied Load with a Limited Duration 0 0, 0 () , 0, p P t Ft F     当当0u 阶段I(0≤t≤u)：此阶段的荷载情况与突加荷载相同 FP(t) t P u   (1 cos ) st y ty t   

Wuhan阶段II (tzu)：位移自由振动：以阶段I终了时刻(tu)的位移和速度作起始Universityof Technolog.和起始速度，可按自由振动的解直接写出；按Duhamel积分计算：两个突加荷载的叠加。[" Fpo sino(t-t)dt+Oxsino(t -t)dty(t)=moJumoJFp [cos o(t- u) -cos ot]moou:2sinsin o(t -=y.S26-6

6-6 Wuhan University of Technology 阶段Ⅱ(t≥u)：  自由振动：以阶段I终了时刻(t=u)的位移和速度作起始 位移 和起始速度，可按自由振动的解直接写出；  按Duhamel积分计算；  两个突加荷载的叠加。 0 0 1 1 ( ) sin ( ) 0 sin ( ) u t P u yt F t d t d m m             0 [cos ( ) cos ] FP tu t m      2sin sin ( ) 2 2 st u u y t     

Wuhan University of Technology两个突加荷载的叠加Fp(t)y = y (1-cos ot)- y, (1-cos o(t -u)ou.2sinsin o(t -=yst26-7

6-7 Wuhan University of Technology  两个突加荷载的叠加 y y  t y   t u  st 1 cos  st 1 cos  FP(t) t FP FPu 2sin sin ( ) 2 2 st u u y t    

WuhanUniversityof TechnologyMax-ResponseofSystemys (1-cos ot)OT/2（加载持续时间大于半个自振周期）最大反应发生在阶段I，动力系数为β-2第二种情况是u<T/2最大反应发生在阶段II，动位移的最大值ouYmax = yst ×2 sin2ou元u动力系数β= 2sin2sin2T6-8

6-8 Wuhan University of Technology Max-Response of System 第一种情况是 u>T/2 （加载持续时间大于半个自振周期） 最大反应发生在阶段 I，动力系数为β=2 第二种情况是 u<T/2 最大反应发生在阶段II，动位移的最大值   1 cos 0  2sin sin ( ) 2 2 st st y t tu y t u u y t tu              max 2sin 2 st u y y    动力系数 2sin 2sin 2u u T     

WuhanUniversityofTechB1-2u元u2sinTTβ=1u2T221/21/6hnolog.Spectrumofmagnificationfactor动力系数β与结构的参数(T)和动荷参数(u)间的关系曲线>动力系数β与的数值取决于参数u/T，即短时荷载的动力效果取决于加载持续时间的长短。6-9

6-9 Wuhan University of Technology 1 2sin 21 2 2 u u T TuT          动力系数β与结构的参数(T)和动荷参数(u)间的关系曲线 T u β 1/6 1 1/2 2 Spectrum of magnification factor  动力系数β与的数值取决于参数u/T，即短时荷载的动力效果取 决于加载持续时间的长短

Wuhan University of Technology> Triangle LoadFp(t)Fp(t)0≤t≤ttzt,冲击荷载简化荷载Duhamel积分：sin o0≤t≤to1OSoF(t) =sin ot - sin o(t - t]- cos ottzt,6-10

6-10 Wuhan University of Technology  Triangle Load Duhamel积分： FP(t) t 冲击荷载 简化荷载          1 1 1 0 0 1 0 ( ) t t t t t t F F t                       1 1 1 1 1 sin sin cos 1 0 1 sin 1 cos ( ) t t t t t t t y t t t t t y t F t st st        P FP(t) t t1