Wuhan University of TechnologyChapter6Response to general dynamic loading:superpositionmethods6-1
6-1 Wuhan University of Technology Chapter 6 Response to general dynamic loading: superposition methods
Wuhan University of TechnologyContents6.1 Analysis through the time domain6.2 Analysis through the frequency domain6.3 Relationship between the time- and frequency-domaintransferfunctions6-2
6-2 Wuhan University of Technology 6.1 Analysis through the time domain 6.2 Analysis through the frequency domain 6.3 Relationship between the time- and frequency-domain transfer functions Contents
Wuhan University of Technology-6.1 Analysis through the time domain-FormulationofResponseIntegralUndamped System.The procedure described in Chapter 5for approximatingtheresponseofanundampedSDOFstructuretoshortdurationimpulsiveloadscanbe usedasthebasisfordevelopingaformulaforevaluating responsetoageneral dynamic loading. Consider an arbitrary general loadingp(t) as illustrated in Fig. 61 and, for the moment, concentrate on the intensityof loadingactin p() time. Tt--)ading acting during the interval oftimerepresedr a very shortduration impulseon tt p(+) dr cture, sothatEq.(521)canbeusedtoevaluatetheresultingresponse.6-3
6-3 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral Undamped System. The procedure described in Chapter 5 for approximating the response of an undamped SDOF structure to shortduration impulsive loads can be used as the basis for developing a formula for evaluating response to a general dynamic loading. Consider an arbitrary general loading as illustrated in Fig. 61 and, for the moment, concentrate on the intensity of loading acting at time . This loading acting during the interval of time represents a very shortduration impulse on the structure, so that Eq. (521) can be used to evaluate the resulting response
Wuhan Universityof Technology6.1 Analysis through the time domain-FormulationofResponseIntegraltp(0)dtdu(t)(t-t)≥0G-tResponsedu(t)Fig.6-1DerivationoftheDuhamel integral (undamped)6-4
6-4 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral Fig.6-1 Derivation of the Duhamel integral (undamped)
Wuhan Universityof Technology16.1 Analysis through the time domain-FormulationofResponseIntegralItshouldbenotedcarefullythatalthoughthisequationisapproximateforimpulses of finite duration, it becomes exactas the durationof loadingapproaches zero.Thus,for the differential time interval dt,the responseproduced by the impulse p(t) dr is exactly equal todv(t)= P(t)dtt≥tsino(t -t)mo6-5
6-5 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral It should be noted carefully that although this equation is approximate for impulses of finite duration, it becomes exact as the duration of loading approaches zero. Thus, for the differential time interval , the response produced by the impulse is exactly equal to ( ) ( ) sin ( ) p d dv t t t m