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