Wuhan University of TechnologyChapter15Analysisof MDOFdynamicresponse:step-by-step methods15-1
15-1 Wuhan University of Technology Chapter 15 Analysis of MDOF dynamic response: step-by-step methods
Wuhan University of TechnologyContents15.1 Preliminary comments15.2 Incremental equations of motion15.3 Step-by-step integration:constant average acceleration method15.4 Step-by-step integration:linear acceleration method15-2
15-2 Wuhan University of Technology 15.1 Preliminary comments 15.2 Incremental equations of motion 15.3 Step-by-step integration: constant average acceleration method 15.4 Step-by-step integration: linear acceleration method Contents
WuhanUniversity of Technology15.1 Preliminary commentsTheonlygenerallyapplicableprocedureforanalysisofanarbitrarysetofnonlinearresponseequations,andalsoaneffectivemeansofdealingwithcoupledlinearmodalequations,isbynumericalstepbystepintegration.Theanalysis can becarried out as the exactMDOF equivalent of the SDOF stepbystepanalysesdescribed inChapter7.Theresponsehistoryisdivided into a sequenceof short,equaltimesteps,andduringeachsteptheresponseiscalculatedforalinearsystemhavingthephysical properties existing at the beginning of the interval.Attheend of the interval, thepropertiesaremodifiedto conformtothestateofdeformationandstressatthattimeforuseduringthesubsequenttimestepThus the nonlinearMDOF analysis is approximated as a sequence of MDOFanalysesofsuccessivelychanginglinearsystems.15-3
15-3 Wuhan University of Technology 15.1 Preliminary comments The only generally applicable procedure for analysis of an arbitrary set of nonlinear response equations, and also an effective means of dealing with coupled linear modal equations, is by numerical stepbystep integration. The analysis can be carried out as the exact MDOF equivalent of the SDOF stepbystep analyses described in Chapter 7. The response history is divided into a sequence of short, equal time steps, and during each step the response is calculated for a linear system having the physical properties existing at the beginning of the interval. At the end of the interval, the properties are modified to conform to the state of deformation and stress at that time for use during the subsequent time step. Thus the nonlinear MDOF analysis is approximated as a sequence of MDOF analyses of successively changing linear systems
WuhanUniversityof Technology福15.2Incremental equations of motionInthestepbystepanalysisofMDOFsystemsitisconvenienttouseanincrementalformulationequivalenttothatdescribedforSDOFsystemsinSection 76because theprocedurethen is equally applicable to either linear ornonlinearanalyses.Thustakingthedifferencebetweenvectorequilibriumrelationshipsdefinedfortimestoandt1=to+hgivestheincrementalequilibriumequation△fi+△fp+△fs=△pAfi=fi-fio=mAv△fp =fDi-fD。= co △iAfs=fst-fso=ko△v△p = P1- Po15-4
15-4 Wuhan University of Technology 15.2 Incremental equations of motion In the stepbystep analysis of MDOF systems it is convenient to use an incremental formulation equivalent to that described for SDOF systems in Section 76 because the procedure then is equally applicable to either linear or nonlinear analyses. Thus taking the difference between vector equilibrium relationships defined for times t0 and t1 = t0 + h gives the incremental equilibrium equation
Wuhan University of TechnologyV15.2 Incremental eguations of motionfsifDiTangentStiffnessTangentdampingkijoCijofsinnAfsiAfDiInJSiAverageAveragestiffnessdampingAijAuj2--1UjUjoU0fo(b)(a)FIGURE15-1Definitionof nonlinearinfluencecoefficients:(a) nonlinear viscous damping Ci; (b) nonlinear stiffness k15-5
15-5 Wuhan University of Technology FIGURE 15-1 Definition of nonlinear influence coefficients: (a) nonlinear viscous damping cij; (b) nonlinear stiffness kij 15.2 Incremental equations of motion