《结构动力学》课程教学课件（讲稿）02 Analysis of free vibration

Wuhan University of TechnologyPart ISingle-degree-of-freedom systems(SDOF)Chapter2Analysis of free vibrations3-1

3-1 Wuhan University of Technology Chapter 2 Analysis of free vibrations Part I Single-degree-of-freedom systems (SDOF)

Wuhan University of TechnologyContents2.1 Introduction2.2 Components of the basic dynamic system2.3 Equation of motion of the basic dynamic system2.4 Influence of gravitational forces2.5 Influence of support excitation2.6 Analysisof undampedfree vibrations2.7Damped free vibrations2.8 Review and problems3-2

3-2 Wuhan University of Technology 2.1 Introduction 2.2 Components of the basic dynamic system 2.3 Equation of motion of the basic dynamic system 2.4 Influence of gravitational forces 2.5 Influence of support excitation 2.6 Analysis of undamped free vibrations 2.7 Damped free vibrations 2.8 Review and problems Contents

Wuhan University of Technology2.1 IntroductionThis chapter will discuss thefree vibration of SDOF system;Theknowledgeof partial differential equationstudied inadvancedmathematicsarebasicallyreviewed.Inthediscussionoffreevibrationsofstructures,threescenariosare involved:(a) Critically-damped systems;(b) Undercritically-damped systems;(c)Overcritically-dampedsystems.3-3

3-3 Wuhan University of Technology 2.1 Introduction  This chapter will discuss the free vibration of SDOF system;  The knowledge of partial differential equation studied in advanced mathematics are basically reviewed.  In the discussion of free vibrations of structures, three scenarios are involved: (a) Critically-damped systems; (b) Undercritically-damped systems; (c) Overcritically-damped systems

Wuhan University of Technology2.2 Components of the basic dynamic systemThe essential physical properties of any linearly elasticstructural or mechanical system subjected to an externalexcitation ordynamic loading are its mass, elastic properties(flexibility or stiffness), and energy-loss mechanism or damping;InthesimplestmodelofaSDOFsystem,eachoftheseproperties is assumed to be concentrated in a single physicalelement.3-4

3-4 Wuhan University of Technology 2.2 Components of the basic dynamic system  The essential physical properties of any linearly elastic structural or mechanical system subjected to an external excitation or dynamic loading are its mass, elastic properties (flexibility or stiffness), and energy-loss mechanism or damping; In the simplest model of a SDOF system, each of these properties is assumed to be concentrated in a single physical element

Wuhan Universityof Technology2.2Componentsof the basicdynamicsystemA sketch of such a system is shown in following Figy(t)Cp(t)m500000kBasiccomponentsofaidealizedSDoFsystem3-5

3-5 Wuhan University of Technology 2.2 Components of the basic dynamic system A sketch of such a system is shown in following Fig. Basic components of a idealized SDOF system y ( t ) c m p ( t ) k

Wuhan University of Technology2.2 Components of the basic dynamic system The entire mass m of this system is included in the rigidblock whichis constrained by rollers so that it can moveonly in simple translation; thus, the single displacementcoordinate x(t) completely defines its position;The elasticresistanceto displacement is provided bytheweightless spring of stiffness k, while the energy-lossmechanism is represented by the damper c;The external dynamic loading producing the response of thissystem is the time-varying force p(t);3-6

3-6 Wuhan University of Technology  The entire mass m of this system is included in the rigid block which is constrained by rollers so that it can move only in simple translation; thus, the single displacement coordinate x(t) completely defines its position;  The elastic resistance to displacement is provided by the weightless spring of stiffness k, while the energy-loss mechanism is represented by the damper c;  The external dynamic loading producing the response of this system is the time-varying force p(t); 2.2 Components of the basic dynamic system

Wuhan University of TechnologyL2.3 Equation of motion of the basic dynamicsystemTheeguationofmotionforthesimplesysteminFig.ismosteasilyformulated bydirectlyexpressingtheequilibrium ofally(t)forces acting on the mass usingd'Alembert's principle;fb(t)fi(t)p(t)The forces acting in thefs(t) 4directionofthedisplacementdegreeoffreedomaretheappliedForcesinequilibriumofaload p(t) and the three resistingidealizedSDOFsystemforces resulting from the motioni.e., the inertial force fi(t), thedamping force f,(t), and thespring force fs(t).3-7

3-7 Wuhan University of Technology 2.3 Equation of motion of the basic dynamic system Forces in equilibrium of a idealized SDOF system  The equation of motion for the simple system in Fig. is most easily formulated by directly expressing the equilibrium of all forces acting on the mass using d’Alembert’s principle;  The forces acting in the direction of the displacement degree of freedom are the applied load p(t) and the three resisting forces resulting from the motion, i.e., the inertial force fI(t), the damping force f D(t), and the spring force fS(t). y ( t ) fD ( t ) fS ( t ) fI ( t ) p ( t )

Wuhan Universityof Technology62.3 Equation of motion of the basic dynamiesystemTheequationofmotionismerelyanexpressionoftheequilibriumof these forces as given byfi(t)+ fb(t)+ fs(t)= p(t)Inaccordancewithd'Alembert'sprinciple,theinertialforceistheproduct of the mass and acceleration.f,(t) = mi(t)Assumingaviscousdampingmechanism,thedampingforceistheproduct of the damping constant c and the velocityf,(t) =ci(t)3-8

3-8 Wuhan University of Technology 2.3 Equation of motion of the basic dynamic system  The equation of motion is merely an expression of the equilibrium of these forces as given by  In accordance with d'Alembert's principle, the inertial force is the product of the mass and acceleration.  Assuming a viscous damping mechanism, the damping force is the product of the damping constant c and the velocity () () () () IDS f t f t f t pt    () () I f t my t   () () Df t cy t  

WuhanUniversityofTechnologyC2.3 Equation of motion of the basic dynamicsystemThe elastic force is the product of the spring stiffness and thedisplacementfs(t) = ky(t)TheequationofmotionforthisSDOFsystemisrewrittenasmi(t) +ci(t) + ky(t) = p(t)3-9

3-9 Wuhan University of Technology  The elastic force is the product of the spring stiffness and the displacement 2.3 Equation of motion of the basic dynamic system  The equation of motion for this SDOF system is rewritten as () () Sf t ky t  my t cy t ky t p t   () () () ()   

Wuhan University of TechnologyC1.3 Equation of motion of the basic dynamicsystemTo introduce an alternative formulation procedure, it is instructiveto develop this same equation of motion by a virtualwork approach If the mass is given a virtual displacement y compatible with thesystem's constraints, the total work done by the equilibrium systemof forces must equal zero as shown by-f(t)Sy-f,(t)Sy-fs(t)Sy+p(t)Sy=03-10

3-10 Wuhan University of Technology 1.3 Equation of motion of the basic dynamic system  To introduce an alternative formulation procedure, it is instructive to develop this same equation of motion by a virtualwork approach.  If the mass is given a virtual displacement δy compatible with the system's constraints, the total work done by the equilibrium system of forces must equal zero as shown by () () () () 0 IDS    f t y f t y f t y pt y   