Wuhan University of TechnologyChapter17Partial differential eguations ofmotion17-1
17-1 Wuhan University of Technology Chapter 17 Partial differential equations of motion
Wuhan University of TechnologyContents17.1 Introduction17.2 Beam flexure: elementary case17.3 Beam flexure: including axial-force effects17.4 Beam flexure: including viscous damping17.5 Beam flexure: generalized support excitations17.6 Axial deformations: undamped17-2
17-2 Wuhan University of Technology 17.1 Introduction 17.2 Beam flexure: elementary case 17.3 Beam flexure: including axial-force effects 17.4 Beam flexure: including viscous damping 17.5 Beam flexure: generalized support excitations 17.6 Axial deformations: undamped Contents
Wuhan Universityof Technology17.1IntroductionThediscretecoordinatesystemsdescribedinPartTwoprovideaconvenientandpracticalapproachtothedynamicresponseanalysisofarbitrarystructuresHowever,thesolutionsobtainedcanonlyapproximatetheiractualdynamicbehavior because themotions are represented bya limited numberofdisplacementcoordinates.Theprecisionoftheresults canbe madeasrefined as desiredbyincreasingthenumberofdegreesoffreedomconsideredintheanalyses.Inprinciple,however,aninfinitenumberofcoordinateswouldberequiredtoconvergetothe exactresultsforanyreal structurehaving distributed properties;hencethisapproachtoobtaininganexactsolutionismanifestlyimpossible17-3
17-3 Wuhan University of Technology 17.1 Introduction The discretecoordinate systems described in Part Two provide a convenient and practical approach to the dynamicresponse analysis of arbitrary structures. However, the solutions obtained can only approximate their actual dynamic behavior because the motions are represented by a limited number of displacement coordinates. The precision of the results can be made as refined as desired by increasing the number of degrees of freedom considered in the analyses. In principle, however, an infinite number of coordinates would be required to converge to the exact results for any real structure having distributed properties; hence this approach to obtaining an exact solution is manifestly impossible
Wuhan University of Technology17.1IntroductionTheformalmathematicalprocedureforconsideringthebehaviorofaninfinitenumberofconnectedpointsisbymeansofdifferentialequationsinwhichtheposition coordinates are takenas independent variablesInasmuchastimeisalsoanindependentvariableinadynamicresponseproblem, the formulation of the equations of motion in this way leads topartialdifferentialequations.Differentclassesof continuoussystemscanbeidentified inaccordancewiththenumberof independentvariablesrequiredtodescribethedistributionoftheirphysical properties.17-4
17-4 Wuhan University of Technology 17.1 Introduction The formal mathematical procedure for considering the behavior of an infinite number of connected points is by means of differential equations in which the position coordinates are taken as independent variables. Inasmuch as time is also an independent variable in a dynamicresponse problem, the formulation of the equations of motion in this way leads to partial differential equations. Different classes of continuous systems can be identified in accordance with the number of independent variables required to describe the distribution of their physical properties
Wuhan UniversityofTechnology17.2 Beam flexure: elementary caseU(x,t)p(x,t)EI(x),m(x)dxL(a)tp(x,t)dxV(x,t)+aM(x.dxM(x.0) +M6axv(x,dxV(x,t) +oxi(x,t)dxdx(b)FIGURE17-1Basicbeamsubjectedtodynamicloading(a)beampropertiesandcoordinates;(b)resultantforcesactingondifferential element.17-5
17-5 Wuhan University of Technology 17.2 Beam flexure: elementary case FIGURE 17-1 Basic beam subjected to dynamic loading: (a) beam properties and coordinates; (b) resultant forces acting on differential element