Wuhan University of Technology7.2 Piecewise exact methodThesimpleststepbystepmethodforanalysisofSDOFsystemsisthesocalled"piecewiseexact"method,whichisbasedontheexactsolutionoftheequationofmotionforresponseofa linearstructuretoaloadingthatvarieslinearlyduringadiscretetimeinterval.In using this method,the loading historyis divided into time intervals, usuallydefinedbysignificant changes of slope in the actual loading history;betweenthesepoints,it isassumedthattheslopeoftheloadcurveremainsconstantAlthoughtheresponseexpressionderivedfortheselinearlyvaryingload stepsis exact, it must be recognized that the actual loading history is onlyapproximated by the constant slope steps.Thusthecalculatedresponsegenerallyisnotanexactrepresentationofthetrue response to the real loading; however, the error can be reduced to anyacceptablevaluemerelybyreducingthelengthofthetimestepsandthusbetterapproximatingtheloading.Ifdesired,thelengthofthetimestepscanbevariedfromoneintervaltothenextinordertoachievethebestpossibletoftheloadinghistorybythesequenceofstraightlinesegments;however,forreasonsofcomputationalefficiencyitiscustomarytouseaconstanttimestep7-6
7-6 Wuhan University of Technology 7.2 Piecewise exact method The simplest stepbystep method for analysis of SDOF systems is the socalled “piecewise exact” method, which is based on the exact solution of the equation of motion for response of a linear structure to a loading that varies linearly during a discrete time interval. In using this method, the loading history is divided into time intervals, usually defined by significant changes of slope in the actual loading history; between these points, it is assumed that the slope of the load curve remains constant. Although the response expression derived for these linearly varying load steps is exact, it must be recognized that the actual loading history is only approximated by the constant slope steps. Thus the calculated response generally is not an exact representation of the true response to the real loading; however, the error can be reduced to any acceptable value merely by reducing the length of the time steps and thus better approximating the loading. If desired, the length of the time steps can be varied from one interval to the next in order to achieve the best possible t of the loading history by the sequence of straight line segments; however, for reasons of computational efficiency it is customary to use a constant time step
Wuhan University of Technology7.2 Piecewise exact methodUi+14 U(r)tpt)Actualα=Po-PiAssumed00p(n)= Po+αThUUj+1totit1to→t=1-1oT=1-10(b)Responsehistory(a)LoadinghistoryFIGURE 7-1 Notation for piecewise exact analysis.7-7
7-7 Wuhan University of Technology 7.2 Piecewise exact method FIGURE 7-1 Notation for piecewise exact analysis
Wuhan University of Technology17.2 Piecewise exact methodThe assumed linearly varying loading during the time step is given byp(T)=Po+Q Tmi+ci+ku=po+αT(T) = Uh(T) +Up(T)Then the damped freevibration response, as shown byEq.(248),is givenbyVh(T)=exp(-&WT)[AcOsWpT+BsinWpTac(PO+αT)Up(T) =7-8
7-8 Wuhan University of Technology 7.2 Piecewise exact method The assumed linearly varying loading during the time step is given by Then the damped freevibration response, as shown by Eq. (248), is given by
Wuhan University of Technology7.2 Piecewise exact methodThe displacement during the time step is(T)=Ao+AT+A2exp(wT) cosWpT+Aexp(-SwT)sinWpTIn which250VoAow203aA1 =w2A2 = uo - AoA3=+swA2Uoww27-9
7-9 Wuhan University of Technology 7.2 Piecewise exact method The displacement during the time step is In which
Wuhan University of Technology7.2 Piecewise exact methodSimilarly, the velocity during the time step is found to bei(T)=A+(WpA3-&wA2)exp(-wT)cOSWpT-(WDA2+wA3)exp(-SwT)sinWpTFor situations where the applied loading may be approximated wellby a series of straight line segments, this piecewise exact methodundoubtedly is the most efficient means of calculating the response ofaSDOFsystem.However,itmustalwaysberememberedthattheloadingbeingconsidered is only an approximation of the true loading history, whichusuallyisasmoothlyvaryingcurve,andthatthesteplengthsmustbechosensoastoachieveanacceptableapproximationofthetrueresponse history.7-10
7-10 Wuhan University of Technology 7.2 Piecewise exact method Similarly, the velocity during the time step is found to be For situations where the applied loading may be approximated well by a series of straight line segments, this piecewise exact method undoubtedly is the most efficient means of calculating the response of a SDOF system. However, it must always be remembered that the loading being considered is only an approximation of the true loading history, which usually is a smoothly varying curve, and that the step lengths must be chosen so as to achieve an acceptable approximation of the true response history