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1、CHAPTER 5.RESPONSE TO IMPULSIVE LOADING5.1 GEGERAL NATURE OF IMPUSIVE LOADINGFIGURE 51 Arbitrary impulsive loadingRelatively short durationDamping has much less importance in controlling the maximum responseOnly the undamped response to impulsive loads will be consideredFIGURE 52 Half-sine-wave impu
2、lseFIGURE 53 Rectangular impulseFIGURE 54Triangular impulse5.2 SINEWAVE IMPULSECHAPTER 5.RESPONSE TO IMPULSIVE LOADINGFor impulsive loads which can be expressed by simple analytical functions,closed form solutions of the equations of motion can be obtained.the single half-sine-wave impulsetwo phases
3、:forced-vibration phasefree-vibration phasePhase I:Assuming the system starts from restSince it is indeterminate for ,LHospitals rule must be applied to obtain a useable expression for this special case.Taking this action,one obtainsPhase II:The free-vibration motion which occurs during this phase,d
4、epends on the displacement and velocity existing at the end of Phase I;Eq.(2-33)in its response-ratio form this free-vibration response is shown to beFor ,requiring once again the use of LHospitals rule leading toCHAPTER 5.RESPONSE TO IMPULSIVE LOADINGCHAPTER 5.RESPONSE TO IMPULSIVE LOADINGthe maxim
5、um value of response depends on the ratio of the load duration to the period of vibration of the structureFIGURE 55 Response ratios due to half-sine pulseCHAPTER 5.RESPONSE TO IMPULSIVE LOADING5.3 RECTANGULAR IMPULSEPhase I.The suddenly applied load which remains constant during this phase is called
6、 a step loading.The particular solution to the equation of motion for this case is simply the static deflectionUsing this result,the general response-ratio solution,in which the complementary free-vibration solution constants have been evaluated to satisfy the at-rest initial conditions,is easily fo
7、und to beCHAPTER 5.RESPONSE TO IMPULSIVE LOADINGPhase II.Taking the vector sum of the two orthogonal components in this expression givesshowing that the maximum response to the rectangular impulse varies as a sine function for CHAPTER 5.RESPONSE TO IMPULSIVE LOADING5.4 SHOCK OR RESPONSE SPECTRAthe m
8、aximum response depends only on the ratio of the impulse duration to the natural period of the structure.It is useful to plot the maximum value of response ratio as a function of for various forms of impulsive loading,commonly known as displacement-response spectra.FIGURE 56 Displacement-response sp
9、ectra(shock spectra)for three types of impulseCHAPTER 5.RESPONSE TO IMPULSIVE LOADING5.5 APPROXIMATE ANALYSIS OF IMPULSIVELOAD RESPONSEA convenient approximate procedure for evaluating the maximum response to a short-duration impulsive load,which represents a mathematical expression of this second c
10、onclusion,may be derived as follows.The impulse-momentum relationship for the mass m may be writtenThe response after termination of loading is the free vibrationCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODS6.1 ANALYSIS THROUGH THE TIME DOMAIN Formulation of Response IntegralUn
11、damped SystemFIGURE 61Derivation of the Duhamelintegral(undamped)CHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODSDuhamel integral equation,unit-impulse response functionCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODSUnder-Critically-Damped System Numerical Eval
12、uation of Response IntegralUndamped SystemTo develop these procedures,use is made of the trigonometric identityCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODSCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODSCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSI
13、TION METHODSUnder-Critically-Damped SystemCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODSFIGURE E62 Response of water tower to blast loadCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODS6.2 ANALYSIS THROUGH THE FREQUENCY DOMAIN The Fourier transform of Convoluti
14、on integral form 12f tfftd 120tf tfftd 12f tftft 1221ftftftft 12i tFfftdedt 12i tFfftedt dCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODStztzdtdz ()2222i tizi ziiftedtfz edzfz edz eFe 1212iFfeFdFF 11ftF 22ftF 1212ftftFF 12i tf tFFed 12f tfftdCHAPTER 6.RESPONSE TO GENERAL DYNAMIC
15、LOADING:SUPERPOSITION METHODS 0tv tph td 0()()ti tv tHPed 1sintDDh tetm 220011()2Hmi ()i tHh t edt 1()2i th tHedCHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODS Discrete Fourier Transform(DFT)To be practical,the time history of an action or a response must be presented in discrete
16、 time series,so the Fourier Transform must be carried out for discrete value series,that is the DFT.tTNt(),0,1,2,1mxx mtmNm is the order number of sampling value.0()cossinkkkx tAktBkt2ttT2ttN t022()cossinkkkktktx tABN tN tTriangular SeriesDenote the sampling number is N,and the equal time increment is ,then the duration of the discrete time series is CHAPTER 6.RESPONSE TO GENERAL DYNAMIC LOADING:SUPERPOSITION METHODS/2022()cossinNkkkktktx tABN tN t2pN/2022cossinNmkkkkmkmxABNNtmtTotally 2(N/2+1)c
