结构动力学课件—2dyanmicsofstructures-ch3

1、,CHAPTER 3. RESPONSE TO HARMONIC LOADING,31 UNDAMPED SYSTEM,Before considering this viscously damped case, it is instructive to examine the behavior of an undamped system as controlled by,which has a complementary solution of the freevibration,Particular Solution,The general solution must also inclu

2、de the particular 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 is,Then we have,CHAPTER 3. RESPONSE TO HARMONIC LOADING,Gener

3、al Solution,CHAPTER 3. RESPONSE TO HARMONIC LOADING,32 SYSTEM WITH VISCOUS DAMPING,The complementary solution of this equation is the damped freevibration response,The particular solution,in which the cosine term is required as well as the sine term because, in general, the response of a damped syst

4、em is not in phase with the loading.,Then we have,In order to satisfy this equation for all values of t, it is necessary that each of the two square bracket quantities equal zero; thus, one obtains,CHAPTER 3. RESPONSE TO HARMONIC LOADING,Then we have the general solution,CHAPTER 3. RESPONSE TO HARMO

5、NIC LOADING,Of great interest, however, is the steadystate harmonic response given by the second term,The 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; thus,It is seen that both th

6、e dynamic magnication factor D and the phase 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 LOADING,FIGURE 3-2 Variation of dynamic magnification fa

7、ctor with damping and frequency.,FIGURE 3-4 Variation of phase angle with damping and frequency.,CHAPTER 3. RESPONSE TO HARMONIC LOADING,Example E31. A portable harmonicloading machine provides an effective means for evaluating the dynamic properties of structures in the field. By operating the mach

8、ine at two different frequencies and measuring the resulting structuralresponse 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 frequ

9、encies of =16 rad/sec and =25 rad/sec, with a force amplitude of 500 lb 226.8 kg in each case. The response amplitudes and phase relationships measured in the two cases were,To evaluate the dynamic properties from these data,With further algebraic simplification this becomes,CHAPTER 3. RESPONSE TO H

10、ARMONIC LOADING,Then introducing the two sets of test data leads to the matrix equation,which can be solved to give,The natural frequency is given by,To determine the damping coefficient,CHAPTER 3. RESPONSE TO HARMONIC LOADING,Thus with the data of the first test,and the same result (within engineer

11、ing accuracy) is given by the data of the second test. The damping ratio therefore is,33 RESONANT RESPONSE,The 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

12、steadystate response amplitude occurs at a frequency ratio slightly less than unity. Even 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

13、 value of dynamic magnification factor, one must differentiate the above equation with respect to and solve the resulting expression for obtaining,CHAPTER 3. RESPONSE TO HARMONIC LOADING,For typical values of structural damping, say 0.10, the difference between,And,is small, the difference being one

14、half of 1 percent for damping ratio= 0.10 and 2 percent for damping ratio= 0.20.,For a more complete understanding 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

15、 term. At the resonant exciting frequency, the equation becomes,Assuming that the system starts from rest, the constants are,Then we have,CHAPTER 3. RESPONSE TO HARMONIC LOADING,For the amounts of damping to be expected in structural systems, the term is nearly equal to unity; in this case, this equ

16、ation can be written in the approximate form,For zero damping, this approximate equation is indeterminate; but when LHospitals rule is applied, the response ratio for the undamped system is found to be,CHAPTER 3. RESPONSE TO HARMONIC LOADING,FIGURE 3-3 Response to resonant loading for at-rest initia

17、l conditions.,CHAPTER 3. RESPONSE TO HARMONIC LOADING,FIGURE 3-4 Rate of buildup of resonant response from rest.,CHAPTER 3. RESPONSE TO HARMONIC LOADING,34 ACCELEROMETERS AND DISPLACEMENT METERS,At this point it is convenient to discuss the fundamental principles on which the operation of an importa

18、nt class of dynamic measurement devices is based. These are seismic instruments, which consist essentially of a viscously damped oscillator 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 ter

19、ms of the motion v(t) of the mass,The equation of motion for this system is,CHAPTER 3. RESPONSE TO HARMONIC LOADING,FIGURE 3-5 Schematic diagram of a typical seismometer.,constant,CHAPTER 3. RESPONSE TO HARMONIC LOADING,FIGURE 3-6 Response of seismometer to harmonic base displacement.,CHAPTER 3. RES

20、PONSE TO HARMONIC LOADING,35 EVALUATION OF VISCOUSDAMPING RATIO,In the foregoing discussion of the dynamic response of SDOF systems, it has been assumed that the physical properties consisting of mass, stiffness, and viscous damping are known. While in most cases, the mass and stiffness can be evalu

21、ated rather easily using simple physical considerations or generalized expressions as discussed in Chapter 8, it is usually not feasible to determine the damping coefficient by similar means because the basic energyloss mechanisms in most practical systems are seldom fully understood. In fact, it is

22、 probable that the actual energyloss mechanisms are much more complicated than the simple viscous (velocity proportional) damping force that has been assumed in formulating the SDOF equation of motion. But it generally is possible to determine an appropriate equivalent viscousdamping property by exp

23、erimental methods. A brief treatment of the methods commonly used for this purpose is presented in the following sections:,FreeVibration Decay Method,This is the simplest 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 TO HARMONIC LOADING,Resonant Amplification Method,FIGURE 3-7 Frequency-response c