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
Wuhan Universityof Technology117.2 Beam flexure: elementary caseSumming all forces acting vertically leads to the first dynamic equilibriumrelationshipoV(r,t)drl - fi(a,t) da = 0V(r,t) +p(r,t) dr-[v(r,t)+or2u(r,t)fi(r,t) da=m(r) daOt2av(r,t)a2v(r,t)=p(c,t) -m(a)Ot2ar17-6
17-6 Wuhan University of Technology Summing all forces acting vertically leads to the first dynamic equilibrium relationship 17.2 Beam flexure: elementary case
Wuhan Universityof Technology117.2 Beam flexure: elementary caseThe second equilibrium relationship is obtained by summing moments aboutpointAontheelasticaxis.Afterdroppingthetwosecondordermomenttermsinvolving the inertia and applied loadings, one getsaM(r,t)M(a,t) +V(r,t)da-M(r,t) +OrBecauserotational inertiaisneglected,this equationsimplifiesdirectlytothestandardstaticrelationshipbetweenshearandmomentM(r,t) = V(a,t)ara2M(r,t)a2v(r,t)p(a,t)mr0r2Ot217-7
17-7 Wuhan University of Technology The second equilibrium relationship is obtained by summing moments about point A on the elastic axis. After dropping the two secondorder moment terms involving the inertia and applied loadings, one gets 17.2 Beam flexure: elementary case Because rotational inertia is neglected, this equation simplifies directly to the standard static relationship between shear and moment
Wuhan Universityof Technology17.2 Beam flexure: elementary casewhich,uponintroducingthebasicmomentcurvaturerelationshipM=EI%8x2a2a2v(r,t)a2v(r,t)EI(C)p(r,t)ma0r2Ot20r2Thisisthepartialdifferential equationof motionfortheelementarycaseofbeamflexure.Thesolutionofthisequationmust,ofcourse,satisfytheprescribed boundary conditions at x = O and x = L.17-8
17-8 Wuhan University of Technology which, upon introducing the basic momentcurvature relationship 17.2 Beam flexure: elementary case This is the partial differential equation of motion for the elementary case of beam flexure. The solution of this equation must, of course, satisfy the prescribed boundary conditions at x = 0 and x = L
Wuhan Universityof Technology17.3 Beam flexure: including axial-force effectsp(x,t)u(x.t)N(L)N(O)q(x)(a)Mar,.) + CMcadxtp(x,t)dxoxV(x.t)M(x,1)N(x)+dNour.DdxN(x)axaVir.D dxV(x, 1) +oxJi(x,)dxv(x.t)(b)FIGURE17-2Beamwithstaticaxial loadinganddynamiclateralloading:(a)Beamdeflectedduetoloadings;(b)resultantforcesactingondifferentialelement.17-9
17-9 Wuhan University of Technology 17.3 Beam flexure: including axial-force effects FIGURE 17-2 Beam with static axial loading and dynamic lateral loading: (a) Beam deflected due to loadings; (b) resultant forces acting on differential element
Wuhan University of Technology17.3 Beam flexure:includingaxial-force effectsTheline ofactionof theaxialforcechanges withthebeamdeflection sothatthemomenteguilibriumeguationnowbecomesaM(r,t)Ou(r,t)M(r,t) + V(r,t) dr+N(r)0dadrOrFrom which the vertical section force V (x, t) is found to bew(r,t)aM(r,t)V(a,t)=-N(carOrThepartialdifferentialequationofmotion,includingaxialforceeffects,isa2a2v(c,t)aa2v(a,t)Ov(r,t)EIN(rm(a)p(r,t)at20r20r2OrOr17-10
17-10 Wuhan University of Technology The line of action of the axial force changes with the beam deflection so that the momentequilibrium equation now becomes 17.3 Beam flexure: including axial-force effects From which the vertical section force V (x, t) is found to be The partial differential equation of motion, including axialforce effects, is