《结构动力学》课程教学课件（讲稿）01 Overview of structural dynamics（2/2）

Wuhan University of TechnologyChapter1Overview of Structural Dynamics2-1

2-1 Wuhan University of Technology Chapter 1 Overview of Structural Dynamics

Wuhan University of TechnologyContents1.1Methods of discretizationLumped-mass procedureGeneralizeddisplacementsThe finite element concept1.2 Formulation of the equations of motionDirect equilibration using d'Alembert's principlePrinciple of virtual displacementsVariational approach2-2

2-2 Wuhan University of Technology 1.1 Methods of discretization Lumped-mass procedure Generalized displacements The finite element concept 1.2 Formulation of the equations of motion Direct equilibration using d’Alembert’s principle Principle of virtual displacements Variational approach Contents

Wuhan University of Technology1.1 Methods of discretization-Lumped-massprocedurep(t)Inertialforces2-3

2-3 Wuhan University of Technology 1.1 Methods of discretization – Lumped-mass procedure

Wuhan University of Technology1.1 Methods of discretization-Lumped-massprocedureBecause the mass of the beam is distributed continuously alongits length, the displacements and accelerations must be defined foreach point along the axis if the inertial forces are to be completelydefined;In this case, theanalysis must be formulated in terms of partialdifferential equations because position alongthe spanas well astimemustbetakenasindependentvariables2-4

2-4 Wuhan University of Technology  Because the mass of the beam is distributed continuously along its length, the displacements and accelerations must be defined for each point along the axis if the inertial forces are to be completely defined;  In this case, the analysis must be formulated in terms of partial differential equations because position along the span as well as time must be taken as independent variables. 1.1 Methods of discretization – Lumped-mass procedure

Wuhan University of Technology1.1 Methods of discretization-Lumped-massprocedurep(t)m,m2m3Lumped-massidealizationof asimplebeam2-5

2-5 Wuhan University of Technology 1.1 Methods of discretization – Lumped-mass procedure Lumped-mass idealization of a simple beam

Wuhan University of Technology1.1 Methods of discretization-Lumped-massprocedureHowever, if one assumes the mass of the beam to beconcentratedatdiscretepointsasshowninFig,theanalyticalproblem becomesgreatlysimplebecause inertialforcesdeveloponly at these mass points;In this case, it is necessary to define the displacements andaccelerationsonlyatthesediscretelocations;Thenumberofdisplacementcomponentswhichmustbeconsidered in order to represent the effects of all significant inertialforcesofastructuremaybetermedthenumberofdynamicdegreesoffreedomofthestructure.2-6

2-6 Wuhan University of Technology 1.1 Methods of discretization – Lumped-mass procedure  However, if one assumes the mass of the beam to be concentrated at discrete points as shown in Fig., the analytical problem becomes greatly simple because inertial forces develop only at these mass points;  In this case, it is necessary to define the displacements and accelerations only at these discrete locations;  The number of displacement components which must be considered in order to represent the effects of all significant inertial forces of a structure may be termed the number of dynamic degrees of freedom of the structure

Wuhan University of Technology1.1 Methods of discretization-Lumped-massprocedureIf the three masses in the system of the Figure are fullyconcentratedandareconstrainedsothatthecorrespondingmasspoints translate only in a vertical direction, this would be called athree-degree-of-freedom (3 DOF) system;If these masses are not fully concentrated so that they possessfinite rotational inertia, the rotational displacements of the threepointswill alsohavetobe considered,inwhichcasethesystemhas 6 DOF;If axial distortions of thebeamare significant,translationdisplacementsparallelwiththebeamaxiswillalsoresultgivingthesystem 9 DOF;2-7

2-7 Wuhan University of Technology 1.1 Methods of discretization – Lumped-mass procedure  If the three masses in the system of the Figure are fully concentrated and are constrained so that the corresponding mass points translate only in a vertical direction, this would be called a three-degree-of-freedom (3 DOF) system;  If these masses are not fully concentrated so that they possess finite rotational inertia, the rotational displacements of the three points will also have to be considered, in which case the system has 6 DOF;  If axial distortions of the beam are significant, translation displacements parallel with the beam axis will also result giving the system 9 DOF;

Wuhan Universityof Technology1.1 Methods of discretization-Lumped-massprocedure If the structure can deform in three-dimensional space, eachmasswill have6DOF;thenthesystemwillhave18DOFHowever,ifthemassesarefullyconcentratedsothatnorotationalinertia is present,thethree-dimensionalsystemwillthenhave 9 DOF.2-8

2-8 Wuhan University of Technology 1.1 Methods of discretization – Lumped-mass procedure  If the structure can deform in three-dimensional space, each mass will have 6 DOF; then the system will have 18 DOF;  However, if the masses are fully concentrated so that no rotational inertia is present, the three-dimensional system will then have 9 DOF

Wuhan University of Technology1.1 Methods of discretization-GeneralizeddisplacementsThe lumped-mass idealization described above provides a simplemeansoflimitingthenumberof degreesof freedomthatmustbeconsidered in conducting adynamic analysis of an arbitrarystructuralsystem;The lumping procedure is most effective in treating systems inwhich a large proportion of the total mass actually is concentratedatafew discretepoints;Then the mass of the structure which supports theseconcentrationscanbeincludedinthelumps,allowingthestructureitselftobeconsideredweightless2-9

2-9 Wuhan University of Technology 1.1 Methods of discretization – Generalized displacements  The lumped-mass idealization described above provides a simple means of limiting the number of degrees of freedom that must be considered in conducting a dynamic analysis of an arbitrary structural system;  The lumping procedure is most effective in treating systems in which a large proportion of the total mass actually is concentrated at a few discrete points;  Then the mass of the structure which supports these concentrations can be included in the lumps, allowing the structure itself to be considered weightless

Wuhan University of Technology1.1 Methods of discretization-GeneralizeddisplacementsHowever, in cases where the mass of the system is quiteuniformlydistributedthroughout,analternativeapproachtolimitingthenumberofdegreesoffreedommaybepreferable;This procedure is based on the assumption that the deflectedshape of thestructure can be expressed asthe sum of aseriesofspecified displacement patterns;Thesepatternsthenbecomethedisplacementcoordinatesofthe structure.2-10

2-10 Wuhan University of Technology 1.1 Methods of discretization – Generalized displacements  However, in cases where the mass of the system is quite uniformly distributed throughout, an alternative approach to limiting the number of degrees of freedom may be preferable;  This procedure is based on the assumption that the deflected shape of the structure can be expressed as the sum of a series of specified displacement patterns;  These patterns then become the displacement coordinates of the structure