结构动力学课件-1dyanmicsofstructures-ch1ch.ppt

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1、Dynamics of StructuresDynamics of Structures Junjie WangJunjie WangDept.of Bridge EngineeringDept.of Bridge Engineering2005.012005.01(a)simple harmonic;(b)complex;(c)impulsive;(d)longduration.FIGURE 11Characteristics and sources of typical dynamic loadings1.1 BACKGROUNDCHAPTER 1.OVERVIEW OF STRUCTUR

2、AL DYNAMICS(a)1999年台湾集集地震集鹿大桥破坏状态The Damages of Jilu Bridge(in Taiwan)in Jiji Earthquake of 1999 The Damages of Kobe Bridge(Japan)in Kobe Earthquake of 1995 CHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS(a)1999年台湾集集地震集鹿大桥破坏状态Sunshine Skyway BridgeTampa Bay,Florida(1980)Tasman BridgeDerwent River,Hobart,A

3、ustralia(1975)CHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS(a)1999年台湾集集地震集鹿大桥破坏状态1.2 ESSENTIAL CHARACTERISTICS OF A DYNAMIC PROBLEM timevarying nature of the dynamic problem inertial forces(more fundamental distinction)FIGURE 12Basic difference between static and dynamic loads:(a)static loading;(b)dynam

4、ic loading.CHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS(a)1999年台湾集集地震集鹿大桥破坏状态1.3 SOLUTIONS TO A DYNAMIC PROBLEMContinuous Models(partial differential equations;generalized displacement,sum of a series)FIGURE 13Sineseries representation of simple beam deflection.CHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS

5、(a)1999年台湾集集地震集鹿大桥破坏状态Discrete ModelsFIGURE 14Lumpedmass idealization of a simple beam.CHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS(a)1999年台湾集集地震集鹿大桥破坏状态FEMAthird method of expressing the displacements of any given structure in terms ofa nite number of discrete displacement coordinates,which combines c

6、ertain featuresof both the lumpedmass and the generalizedcoordinate proceduresFIGURE 15Typical finiteelement beam coordinates.CHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS1.4 FORMULATION OF THE EQUATIONS OF MOTION1.4.1 Direct Equilibration Using dAlemberts PrincipleThe equations of motion of any dynamic

7、 system represent expressions of Newtons second law of motion,which states that the rate of change of momentum of anymass particle m is equal to the force acting on it.This relationship can be expressedmathematically by the differential equationFor most problems in structural dynamics it may be assu

8、med that mass does notvary with time,in which case Eq.(13)may be writtenthe second term is called the inertial force resisting the acceleration of the mass.known as dAlemberts principleCHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS1.4.2 Principle of Virtual DisplacementsHowever,if the structural system i

9、s reasonably complex involving a number of interconnected mass points or bodies of finite size,the direct equilibration of all the forces acting in the system may be difficult.Frequently,the various forces involved may readily be expressed in terms of the displacement degrees of freedom,but their eq

10、uilibrium relationships may be obscure.In this case,the principle of virtual displacements can be used to formulate the equations of motion as a substitute for the direct equilibrium relationships.The principle of virtual displacements may be expressed as follows.If a system which is in equilibrium

11、under the action of a set of externally applied forces is subjected to a virtual displacement,i.e.,a displacement pattern compatible with the systems constraints,the total work done by the set of forces will be zero.1.4.3 Hamiltons principleCHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICS1.5 ORGANIZATION O

12、F THE TEXTCHAPTER 1.OVERVIEW OF STRUCTURAL DYNAMICSDYNAMICSPart ISDOFPart II Discrete MDOFPart IIIDistributed Parameter SystemsPart IV Random VibrationPart V Special FocusesBasic ConceptionsBasic MethodsStructural Properties(Mass,Damping,Stiffness)Selection of Dynamic DOFModal SuperpositionStep by S

13、tep MethodsPartial Differential Equation of MotionAnalysis of Undamped Free VibrationAnalysis of Dynamic ResponseProbability TheoryRandom ProcessEigen ProblemStochastic Response of SDOF SystemsShipBridge CollisionStochastic Response of MDOF SystemsEarthquake EngineeringVehicleForces VibrationPART IS

14、INGLE DEGREE OF FREEDOM SYSTEMSCHAPTER 2.ANALYSIS OF FREE VIBRATION21 COMPONENTS OF THE BASIC DYNAMIC SYSTEMThe essential physical properties of any linearly elastic structural or mechanical system subjected to an external source of excitation or dynamic loading are its mass,elastic properties(exibi

15、lity or stiffness),and energyloss 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.A sketch of such a system is shown in Fig.21a.FIGURE 21Idealized SDOF system:(a)basic components;(b)forces in equilibrium.

16、CHAPTER 2.ANALYSIS OF FREE VIBRATION22 EQUATION OF MOTION OF THE BASIC DYNAMIC SYSTEMThe equation of motion for the simple system is most easily formulated by directly expressing the equilibrium of all forces acting on the mass using dAlemberts principle.The equation of motion is merely an expressio

17、n of the equilibrium of these forces as given byIn accordance with dAlemberts principle,the inertial force is the product of the mass and accelerationAssuming a viscous damping mechanism,the damping force is the product of the damping constant c and the velocityFinally,the elastic force is the produ

18、ct of the spring stiffness and the displacementCHAPTER 2.ANALYSIS OF FREE VIBRATION23 INFLUENCE OF GRAVITATIONAL FORCESFIGURE 22Influence of gravity on SDOF equilibrium.CHAPTER 2.ANALYSIS OF FREE VIBRATIONif the total displacement v(t)is expressed as the sum of the static displacement caused by the

19、weight W plus the additional dynamic displacement as shown in Fig.22c,i.e.,then the spring force is given bythen we have and noting that leads tonoting that does not vary with time,it is evident that and then we have It demonstrates that the equation of motion expressed with reference to the statice

20、quilibrium position of the dynamic system is not affected by gravity forces.For this reason,displacements in all future discussions will be referenced from the staticequilibrium positionCHAPTER 2.ANALYSIS OF FREE VIBRATION24 ANALYSIS OF UNDAMPED FREE VIBRATIONSIt has been shown in the preceding sect

21、ions that the equation of motion of a simple springmass system with damping can be expressed asThe solution of the above equation will be obtained by considering first the homogeneous form with the right side set equal to zero,i.e.,Follow the the theory of differential equations with constant coeffi

22、cients,the solution of the above equation can be obtained step by step,EginequationIf the damping is zero,i.e.,c=0,then one has,Frequency of free vibration of undamped system measured in radians/secondCHAPTER 2.ANALYSIS OF FREE VIBRATIONThe initial conditions are,Then one has,v(t)can be rewritten,in

23、 which,The solution of v(t)presents a simple harmonic motion.CHAPTER 2.ANALYSIS OF FREE VIBRATIONFIGURE 23 Undamped freevibration responseAngular velocity,or Circular frequencyCyclic frequency,usually referred to as the frequency of motion(cycles/sec)The period of motion(measured in seconds)Its reci

24、procal,CHAPTER 2.ANALYSIS OF FREE VIBRATION26 ANALYSIS OF DAMPED FREE VIBRATIONSThe motion equation is,Eginequation CriticallyDamped systemsIf the radical term in the above equation is set equal to zero,it is evident thatDefine Critical DampingCHAPTER 2.ANALYSIS OF FREE VIBRATIONUsing the initial co

25、nditions,Note that this free response of a criticallydamped system does not include oscillation about the zerodeection position;instead it simply returns to zero asymptotically in accordancewith the exponential term.However,a single zerodisplacement crossing would occur if the signs of the initial v

26、elocity and displacement were different from each other.A very useful denition of the criticallydamped condition described above is that it represents the smallest amount of damping for which no oscillation occurs in the freevibration response.FIGURE 24Freevibration response with critical damping.CH

27、APTER 2.ANALYSIS OF FREE VIBRATION UndercriticallyDamped systemsIf damping is less than critical,that is,if c cc,it is apparent that the quantity under the radical sign in Eginequation is negative.To evaluate the freevibration response in this case,it is convenient to express damping in terms of a d

28、amping ratio which is the ratio of the given damping to the critical value,Then we have,Damping RatioDamped FrequencyCHAPTER 2.ANALYSIS OF FREE VIBRATIONAlternatively,this response can be written in the formIn whichNote that for low damping values which are typical of most practical structures,20%,t

29、he frequency ratio is nearly equal to unity.The relation between damping ratio and frequency ratio may be depicted graphically as a circle of unit radius as shown in the following Figure.FIGURE 25Relationship between frequency ratio and damping ratio.CHAPTER 2.ANALYSIS OF FREE VIBRATIONIt is of inte

30、rest to note that the underdamped system oscillates about the neutral position,with a constant circular frequency .FIGURE 26Freevibration response of undercriticallydamped system.CHAPTER 2.ANALYSIS OF FREE VIBRATIONThe true damping characteristics of typical structural systems are very complex and d

31、ifficult to define.However,it is common practice to express the damping of such real systems in terms of equivalent viscousdamping ratios which show similar decay rates under freevibration conditions.Consider any two successive positive peaksone obtains the socalled logarithmic decrement of dampingF

32、or low values of damping can be approximated bySufficient accuracy is obtained by retaining only the rst two terms in the Taylors series expansion on the right hand side,in which caseCHAPTER 2.ANALYSIS OF FREE VIBRATIONFIGURE 27Dampingratio correction factorThis graph permits one to correct the damp

33、ing ratio obtained by the approximate method.For lightly damped systems,greater accuracy in evaluating the damping ratio can be obtained by considering response peaks which are several cycles apart,say m cycles;thenwhich can be simplied for low damping to an approximate relationWhen damped free vibr

34、ations are observed experimentally,a convenient method for estimating the damping ratio is to count the number of cycles required to give a 50 percent reduction in amplitude.CHAPTER 2.ANALYSIS OF FREE VIBRATIONFIGURE 28Damping ratio vs.number of cycles required to reduce peak amplitude by 50 percent

35、.Example E21.A onestory building is idealized as a rigid girder supported by weightless columns,as shown in Fig.E21.In order to evaluate the dynamic properties of this structure,a freevibration test is made,in which the roof system(rigid girder)is displaced laterally by a hydraulic jack and then sud

36、denly released.During the jacking operation,it is observed that a force of 9,072 kg is required to displace the girder 0.508 cm.After the instantaneous release of this initial displacement,the maximum displacement on the first return swing is 0.406 cm and the period of this displacement cycle is T=1

37、.40 sec.From these data,the following dynamic behavioral properties are determined:CHAPTER 2.ANALYSIS OF FREE VIBRATIONFIGURE E21Vibration test of a simple building.(1)Effective weight of the girder:CHAPTER 2.ANALYSIS OF FREE VIBRATION(2)Undamped frequency of vibration:(3)Damping properties:(4)Ampli

38、tude after six cycles:CHAPTER 2.ANALYSIS OF FREE VIBRATION OvercriticallyDamped SystemsAlthough it is very unusual under normal conditions to have overcriticallydamped structural systems,they do sometimes occur as mechanical systems;therefore,it is useful to carry out the response analysis of an ove

39、rcriticallydamped system to make this presentation complete.In this case havingIt is easily shown that the response of an overcriticallydamped system is similar to the motion of a criticallydamped system;however,the asymptotic return to the zerodisplacement position is slower depending upon the amount of damping.