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
Wuhan University of Technology6.1 Analysis through the time domain-FormulationofResponseIntegralTheentireloadinghistorycanbeconsideredtoconsistofasuccessionofsuchshortimpulses,eachproducing itsowndifferential responseoftheformofEg(61).Forthislinearlyelasticsystem,thetotal responsecanthenbeobtainedby summing all the differential responses developed during the loading history,thatis,byintegratingEq.(61)asfollows:v(t)=t≥0p(t)sino(t-t)dt6-6
6-6 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral The entire loading history can be considered to consist of a succession of such short impulses, each producing its own differential response of the form of Eq. (61). For this linearly elastic system, the total response can then be obtained by summing all the differential responses developed during the loading history, that is, by integrating Eq. (61) as follows: 0 1 ( ) ( )sin ( ) 0 t vt p t d t m
Wuhan Universityof Technology6.1 Analysis through the time domain-FormulationofResponseIntegralThisrelation,generallyknownastheDuhamelintegral equation,canbeusedtoevaluatetheresponseofanundampedSDOFsystemtoanyformofdynamicloadingp(t);however,forarbitraryloadingstheevaluationmustbeperformednumericallyusingproceduresdescribedsubsequentlyEquation(62)canalsobeexpressedinthegeneral convolutionintegralform:v(t) = I" p(t)h(t -t)dtt≥01h(t-sinw(t-t)mo6-7
6-7 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral This relation, generally known as the Duhamel integral equation, can be used to evaluate the response of an undamped SDOF system to any form of dynamic loading p(t); however, for arbitrary loadings the evaluation must be performed numerically using procedures described subsequently. Equation (62) can also be expressed in the general convolution integral form: 0 ( ) ( ) ( ) 0 t vt p ht d t 1 ht t ( ) sin ( ) m
Wuhan Universityof Technology6.1 Analysis through the time domain-FormulationofResponseIntegralThefunctionisknown astheunitimpulseresponsefunction becauseit expressestheresponseof theSDOF systemtoapureimpulseofunitmagnitudeapplied attimet= T.GeneratingresponseusingtheDuhamel orconvolution integralisonemeansofobtainingresponsethroughthetimedomain.Itisimportanttonotethatthisapproachmaybeappliedonlytolinearsystemsbecausetheresponseisobtainedbysuperpositionofindividual impulseresponses.6-8
6-8 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral The functionis known as the unitimpulse response function because it expresses the response of the SDOF system to a pure impulse of unit magnitude applied at time . Generating response using the Duhamel or convolution integral is one means of obtaining response through the time domain. It is important to note that this approach may be applied only to linear systems because the response is obtained by superposition of individual impulse responses
Wuhan Universityof Technology6.1 Analysis through the time domain-FormulationofResponseIntegralIn Eqs. (61) and (62) it has been tacitly assumed that the loading was initiatedat timet=O and thatthestructurewasat restat that time.Foranyotherspecified initial conditions (0) + 0 and (0) + 0. , the additional free vibrationresponse must be added to this solution; thus, in generalv(O)sinot +v(O)cosot+p(t)sino(t-t)dtv(t)0maTheShouldthenonzeroinitialconditionsbeproducedbyknownloadingp(t)fort<o,thetotal responsegivenbythis equationcouldalsobefoundthroughEq.(62)by changing thelower limit of the integral from zero to minus infinity.6-9
6-9 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral In Eqs. (61) and (62) it has been tacitly assumed that the loading was initiated at time t = 0 and that the structure was at rest at that time. For any other specified initial conditions and , the additional free vibration response must be added to this solution; thus, in general 0 (0) 1 ( ) sin (0) cos ( )sin ( ) v t vt t v t p t d m The Should the nonzero initial conditions be produced by known loading p(t) for t<0, the total response given by this equation could also be found through Eq. (62) by changing the lower limit of the integral from zero to minus infinity
Wuhan University of Technology6.1 Analysis through the time domain-FormulationofResponseIntegralUnderCriticallyDampedSystem.ThederivationoftheDuhamelintegral equationwhichexpressestheresponseofaviscouslydampedsystemtoageneraldynamicloadingisentirely equivalentthatfortheundampedcase,exceptthatthe freevibration response initiatedbythe differential load impulsep(r) dr experiences exponential decay. By expressing Eq. (249) in terms oft = T rather than t, and substituting zero for u(0) and p(t) dt /m for i(0). oneobtainsthedampeddifferentialresponsep(t)dtdv(t)=sino,(t-t) [exp[-Eo(t-t)mop6-10
6-10 Wuhan University of Technology 6.1 Analysis through the time domain - Formulation of Response Integral UnderCriticallyDamped System. The derivation of the Duhamel integral equation which expresses the response of a viscously damped system to a general dynamic loading is entirely equivalent that for the undamped case, except that the free vibration response initiated by the differential load impulse experiences exponential decay. By expressing Eq. (249) in terms of rather than t, and substituting zero for and for one obtains the damped differential response ( ) ( ) sin ( ) exp ( ) D D p d dv t t t m