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1、CHAPTER 3.RESPONSE TO HARMONIC LOADING31 UNDAMPED SYSTEMBefore considering this viscously damped case,it is instructive to examine the behaviorof an undamped system as controlled bywhich has a complementary solution of the freevibrationParticular SolutionThe general solution must also include the pa
2、rticular solution which depends upon the form of dynamic loading.In this case of harmonic loading,it is reasonable to assume that the corresponding motion is harmonic and in phase with the loading;thus,the particular solution isThen we have,CHAPTER 3.RESPONSE TO HARMONIC LOADINGGeneral SolutionCHAPT
3、ER 3.RESPONSE TO HARMONIC LOADING32 SYSTEM WITH VISCOUS DAMPINGThe complementary solution of this equation is the damped freevibration responseThe particular solutionin which the cosine term is required as well as the sine term because,in general,theresponse of a damped system is not in phase with t
4、he loading.Then we have,In order to satisfy this equation for all values of t,it is necessary that each of the twosquare bracket quantities equal zero;thus,one obtainsCHAPTER 3.RESPONSE TO HARMONIC LOADINGThen we have the general solution,CHAPTER 3.RESPONSE TO HARMONIC LOADINGOf great interest,howev
5、er,is the steadystate harmonic response given by the second termThe ratio of the resultant harmonic response amplitude to the static displacement which would be produced by the force will be called the dynamic magnification factor D;thusIt is seen that both the dynamic magnication factor D and the p
6、hase angle vary with the frequency ratio and the damping ratio.Plots of D vs.and vs.Are shown in Figs.33 and 34,respectively,for discrete values of damping ratio,.CHAPTER 3.RESPONSE TO HARMONIC LOADINGFIGURE 32Variation of dynamic magnification factor with damping and frequency.FIGURE 34Variation of
7、 phase angle withdamping and frequency.CHAPTER 3.RESPONSE TO HARMONIC LOADINGExample E31.A portable harmonicloading machine provides an effective means for evaluating the dynamic properties of structures in the field.By operating the machine at two different frequencies and measuring the resulting s
8、tructuralresponse amplitude and phase relationship in each case,it is possible to determine the mass,damping,and stiffness of a SDOF structure.In a test of this type on a singlestory building,the shaking machine was operated at frequencies of =16 rad/sec and =25 rad/sec,with a force amplitude of500
9、lb 226.8 kg in each case.The response amplitudes and phase relationships measured in the two cases wereTo evaluate the dynamic properties from these data,With further algebraic simplification this becomesCHAPTER 3.RESPONSE TO HARMONIC LOADINGThen introducing the two sets of test data leads to the ma
10、trix equationwhich can be solved to giveThe natural frequency is given byTo determine the damping coefficient,CHAPTER 3.RESPONSE TO HARMONIC LOADINGThus with the data of the first testand the same result(within engineering accuracy)is given by the data of the second test.The damping ratio therefore
11、is33 RESONANT RESPONSEThe steadystate response amplitude of an undamped system tends toward infinity as the frequency ratio approaches unity.For low values of damping,it is seen in this same gure that the maximum steadystate response amplitude occurs at a frequency ratio slightly less than unity.Eve
12、n so,the condition resulting when the frequency ratio equals unity,i.e.,when the frequency of the applied loading equals the undamped natural vibration frequency,is called resonance.To nd the maximum or peak value of dynamic magnification factor,one must differentiate the above equation with respect
13、 to and solve the resulting expression for obtainingCHAPTER 3.RESPONSE TO HARMONIC LOADINGFor typical values of structural damping,say 0.10,the difference betweenAnd is small,the difference being onehalf of 1 percent for damping ratio=0.10 and 2 percent for damping ratio=0.20.For a more complete und
14、erstanding of the nature of the resonant response of a structure to harmonic loading,it is necessary to consider the general response,which includes the transient term as well as the steadystate term.At the resonant exciting frequency,the equation becomesAssuming that the system starts from rest,the
15、 constants areThen we have,CHAPTER 3.RESPONSE TO HARMONIC LOADINGFor the amounts of damping to be expected in structural systems,the term is nearly equal to unity;in this case,this equation can be written in the approximate form,For zero damping,this approximate equation is indeterminate;but when LH
16、ospitalsrule is applied,the response ratio for the undamped system is found to beCHAPTER 3.RESPONSE TO HARMONIC LOADINGFIGURE 33Response to resonant loading for atrest initial conditions.CHAPTER 3.RESPONSE TO HARMONIC LOADINGFIGURE 34Rate of buildup of resonant response from rest.CHAPTER 3.RESPONSE
17、TO HARMONIC LOADING34 ACCELEROMETERS AND DISPLACEMENT METERSAt this point it is convenient to discuss the fundamental principles on which the operation of an important class of dynamic measurement devices is based.These are seismic instruments,which consist essentially of a viscously damped oscillat
18、or as shown in Fig.35.The system is mounted in a housing which may be attached to the surface where the motion is to be measured.The response is measured in terms of the motion v(t)of the massThe equation of motion for this system is CHAPTER 3.RESPONSE TO HARMONIC LOADINGFIGURE 3-5Schematic diagram
19、of a typical seismometer.constantCHAPTER 3.RESPONSE TO HARMONIC LOADINGFIGURE 36 Response of seismometer to harmonic base displacement.CHAPTER 3.RESPONSE TO HARMONIC LOADING35 EVALUATION OF VISCOUSDAMPING RATIO In the foregoing discussion of the dynamic response of SDOF systems,it has been assumed t
20、hat the physical properties consisting of mass,stiffness,and viscous damping are known.While in most cases,the mass and stiffness can be evaluated rather easily using simple physical considerations or generalized expressions as discussed in Chapter 8,it is usually not feasible to determine the dampi
21、ng coefficient by similar means because the basic energyloss mechanisms in most practical systems are seldom fully understood.In fact,it is probable that the actual energyloss mechanisms are much more complicated than the simple viscous(velocity proportional)damping force that has been assumed in fo
22、rmulating the SDOF equation of motion.But it generally is possible to determine an appropriate equivalent viscousdamping property by experimental methods.A brief treatment of the methods commonly used for this purpose is presented in the following sections:FreeVibration Decay MethodThis is the simpl
23、est and most frequently used method of finding the viscousdamping ratio through experimental measurements.When the system has been set into free vibration by any means,the damping ratio can be determined from the ratio of two peak displacements measured over m consecutive cycles.CHAPTER 3.RESPONSE T
24、O HARMONIC LOADING Resonant Amplification MethodFIGURE 37 Frequencyresponse curve for moderately damped system.the actual maximum dynamic magnification factorCHAPTER 3.RESPONSE TO HARMONIC LOADING HalfPower(BandWidth)MethodAmplitudePeak ValueIt is evident that this method of obtaining the damping ra
25、tio avoids the need for obtaining the static displacement ;however,it does require that the frequencyresponse curve be obtained accurately at its peak and at the level .CHAPTER 3.RESPONSE TO HARMONIC LOADINGExample E34.Data from a frequencyresponse test of a SDOF system have been plotted in the following figure.The pertinent data for evaluating the damping ratio are shown.The sequence of steps in the analysis after the curve was plotted were as follows:Frequencyresponse experiment to determine damping ratioCHAPTER 3.RESPONSE TO HARMONIC LOADING