《结构动力学》课程教学课件（讲稿）15 Analysis of MDOF dynamic response（step-by-step methods）

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

Wuhan Universityof Technology15.2Incremental equations of motiondfpCidvdfm△v+co△v+ko△v=ApTheincrementalforceexpressionsontheleftsideofEg.(154)areonlyapproximationsbecauseof theuseof initial tangentvaluesforcoandko.However,accumulationoferrorsduetothisfactorwill beavoided iftheaccelerationatthebeginningof eachtime step is calculated fromthetotalequilibriumof forcesatthat time,as was mentioned indiscussingtheSDOFcase.15-6

15-6 Wuhan University of Technology The incremental force expressions on the left side of Eq. (154) are only approximations because of the use of initial tangent values for c0 and k0. However, accumulation of errors due to this factor will be avoided if the acceleration at the beginning of each time step is calculated from the total equilibrium of forces at that time, as was mentioned in discussing the SDOF case. 15.2 Incremental equations of motion

Wuhan University of Technology15.3 Stepbystep integration:-constantaverageaccelerationmethodFortheanalysisofgeneralMDOFsystems,theconstantaverageaccelerationassumptionhastheveryimportantadvantagethatitprovidesanunconditionallystable integration procedure. Any method that is only conditionally stable mayrequireuseofextremelyshorttimestepstoavoidinstabilityinthehighermoderesponses,and suchinstabilitywill causetheanalysisto"blowup"eveniftheunstablemodesmakeno significantcontributiontotheactual dynamicresponsebehavior.Theincrementalpseudostaticequilibriumequationwillbestatedaskev=ApTheeffectivestiffnessmatrixinthiscaseisgivenby4.2k.=ko+原mCo15-7

15-7 Wuhan University of Technology 15.3 Stepbystep integration: - constant average acceleration method For the analysis of general MDOF systems, the constant average acceleration assumption has the very important advantage that it provides an unconditionally stable integration procedure. Any method that is only conditionally stable may require use of extremely short time steps to avoid instability in the higher mode responses, and such instability will cause the analysis to “blow up” even if the unstable modes make no significant contribution to the actual dynamic response behavior. The incremental pseudostatic equilibrium equation will be stated as The effective stiffness matrix in this case is given by

Wuhan University of Technology15.3 Stepbystep integration:- constant average acceleration methodTheincrementaleffectiveloadvectorisasfollows:v0+2voAp。=△p+2covo+mThe stepbystep analysis is carried out using Eq. (155) by first evaluating个from themass,damping,and stiffness properties determined from theconditions at the beginning of the time step and also evaluating p.from thedamping property as well as the velocity and acceleration vectors at thebeginningofthetimestepcombinedwiththeload incrementspecifiedforthestep.Thenthesimultaneousequations[Eq.(155)]aresolvedforthedisplacement increment △y, usually using Gauss or Choleski decomposition; itshouldbenoted that thechanging values ofk,and C,ina nonlinear analysisrequirethatthedecompositionbeperformedforeachtimestep,andthisisamajorcomputationaleffortforasystemwithverymanydegreesoffreedom15-8

15-8 Wuhan University of Technology The incremental effective load vector is as follows: The stepbystep analysis is carried out using Eq. (155) by first evaluating from the mass, damping, and stiffness properties determined from the conditions at the beginning of the time step and also evaluating from the damping property as well as the velocity and acceleration vectors at the beginning of the time step combined with the load increment specified for the step. Then the simultaneous equations [Eq.(155)] are solved for the displacement increment , usually using Gauss or Choleski decomposition; it should be noted that the changing values of k0 and c 0 in a nonlinear analysis require that the decomposition be performed for each time step, and this is a major computational effort for a system with very many degrees of freedom. 15.3 Stepbystep integration: - constant average acceleration method

Wuhan University of Technology-15.3 Stepbystep integration:-constantaverageaccelerationmethodWhenthedisplacement increment has been calculated,thevelocity incrementisgivenbythefollowingexpression,whichisanalogoustoEq.(724c)butisbased ontheconstantaverageaccelerationassumption2△V-2voAv:hToavoid accumulation oferrors,asnoted before,theinitial accelerationvectoriscalculateddirectlyfromtheconditionof equilibriumatthebeginningof thestep; thus,Vo = m-1 [po -fD。-fso]15-9

15-9 Wuhan University of Technology When the displacement increment has been calculated, the velocity increment is given by the following expression, which is analogous to Eq. (724c) but is based on the constant average acceleration assumption To avoid accumulation of errors, as noted before, the initial acceleration vector is calculated directly from the condition of equilibrium at the beginning of the step; thus, 15.3 Stepbystep integration: - constant average acceleration method

Wuhan University of Technology15.4 Stepbystep integration:-linearaccelerationmethodComparativenumericaltestshavedemonstratedthatthelinearaccelerationmethod gives better results using any specified step length that does notapproach the integration stability limit.Incertaintypesof structures,notablymultistorybuildingsthataremodelledwithonedegreeoffreedomperstoryforplanarresponseorwiththreedegrees offreedomforgeneralthreedimensional response,thereislittledifficultyinadoptingatimestepthat ensuresstabilityintheresponseof eventhehighestmodes.Insuchsituations,thelinearaccelerationversionoftheabovedescribedprocedure is recommended, replacing Eqs. (155), (156a, b), and (157) by theirlinearaccelerationeguivalentsasfollows:15-10

15-10 Wuhan University of Technology 15.4 Stepbystep integration: - linear acceleration method  Comparative numerical tests have demonstrated that the linear acceleration method gives better results using any specified step length that does not approach the integration stability limit.  In certain types of structures, notably multistory buildings that are modelled with one degree of freedom per story for planar response or with three degrees of freedom for general threedimensional response, there is little difficulty in adopting a time step that ensures stability in the response of even the highest modes.  In such situations, the linear acceleration version of the abovedescribed procedure is recommended, replacing Eqs. (155), (156a, b), and (157) by their linear acceleration equivalents as follows: