AN INTRODUCTION TO DYNAMICS OF STRUCTURES Instructor’s & Student Guides This project, developed for the University Consortium on Instructional Shake Tables has been generously contributed by: Shirley J. Dyke Associate Professor Department of Civil Engineering Washington University in St. Louis ***** INSTRUCTOR’S GUIDE ***** INTRODUCTION TO DYNAMICS OF STRUCTURES A PROJECT DEVELOPED FOR THE UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES http://ucist.cive.wustl.edu/ Developed by: Mr. Juan Martin Caicedo (jc11@cive.wustl.edu) Ms. Sinique Betancourt Dr. Shirley J. Dyke (sdyke@seas.wustl.edu) Washington University in Saint Louis This project is supported in part by the National Science Foundation Grant Nos. DUE–9950340 and CMS–9733272. Additional support is provided by the Mid-America Earthquake Center (NSF EEC-9701785) and Washington University. Required Equipment: • Instructional Shake Table • Two Story Building • Three Accelerometers • MultiQ Board • Power Supply • Computer • Software: Wincon and Matlab INSTRUCTOR’S GUIDE 1 Washington University in St. Louis INSTRUCTOR’S GUIDE Introduction to Dynamics of Structures Structural Control & Earthquake Engineering Laboratory Washington University in Saint Louis Objective: The objective of this experiment is to introduce students to principles in structural dynamics through the use of an instructional shake table. Natural frequencies, mode shapes and damping ratios for a scaled structure are obtained experimentally. NOTE: If you do not have the Real-Time Workshop installed on your computer, you must add the following directory to the MATLAB path before proceeeding with this experiment (c:\matlabr11\toolbox\rtw). Contents of Instructor’s Guide 4.0 Experimental Procedure: Sample Results and Discussion 4.1 Random excitation and transfer function calculation Figure 1: Typical Recorded Time Histories. Figure 2: Typical Transfer Functions 4.2 Determination of mode shapes Figure 3: Diagram of Mode Shapes of Test Structure 4.3 Damping estimation 4.3.1 Exponential decay Figure 4: Free response of test structure in (a) Mode 1 and (b) Mode 2. 4.3.2 Half power bandwidth method Introduction to Dynamics of Structures 2 Washington University in St. Louis 4.0 Experimental Procedure: Sample Results and Discussion 4.1 Transfer function calculation ANSWER Figure 1 provides example acceleration records obtained from the ground, first and second floor for the white noise input. Please answer the following questions. • How many natural frequencies does the structure have? • What are the values of the natural frequencies? • Are these values the same in the two transfer functions? Why or why not? 0 50 100 150 -0.5 0 0.5 Ground Acceleration Time ( sec ) Acceleration (g) 0 50 100 150 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Time ( sec ) Displacement command Displacement (in) Figure 1. Typical time history records for a) shake table command signal, b) acceleration at ground level, c) first floor acceleration, and d) second floor acceleration. (b) Ground Acceleration Record (c) 1st Floor Acceleration Record (d) 2nd Floor Acceleration Record 0 50 100 150 -2 -1 0 1 2 First Floor Acceleration Time ( sec ) Acceleration (g) 0 50 100 150 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 First Floor Acceleration Time ( sec ) Acceleration (g) (a) Chirp Displacement Command to Shake Table Introduction to Dynamics of Structures 3 Washington University in St. Louis Figure 2 provides typical transfer functions for the test structure. The system has two natural frequencies. The natural frequencies of this structure are: 2 Hz and 5.8 Hz. The two values are the same in both plots. 4.2 Determination of Mode Shapes ANSWER The mode shapes of the test structure are shown in figure 3. The first mode has zero nodes and the second mode has one node. 0 2 4 6 8 10 -50 0 50 Amplitude (dB) Transfer Function Ground - First Floor Acceleration 0 2 4 6 8 10 -50 0 50 Fre q uenc y ( Hz ) Amplitude (dB) Transfer Function Ground - Second Floor Acceleration Figure 2. Sample transfer function plots for the test structure. Please do the following. • Sketch each of the mode shapes of the structure. • Obtain the number of nodes in each mode shape. • Does this result satisfy equation (35)? Explain. Test Structure First Mode Second Mode Figure 3. Diagram of mode shapes for the test structure. Introduction to Dynamics of Structures 4 Washington University in St. Louis 4.3 Damping estimation 4.3.1 Exponential decay ANSWER ANSWER To use the decrement method, the following calculations are performed. Please do the following. • What is the damping ratio obtained using this method? • Compare this damping ratio with that obtained in 4.3.2. 0 5 10 15 20 25 30 -1.5 -1 -0.5 0 0.5 1 1.5 Time ( sec ) Acceleration (g) Second Floor - First Mode 0 5 10 15 20 25 30 -1 -0.5 0 0.5 1 Time ( sec ) Acceleration (g) First Floor - First Mode 0 5 10 15 20 25 30 -4 -3 -2 -1 0 1 2 3 4 Time ( sec ) Acceleration (g) First Floor - Second Mode 0 5 10 15 20 25 30 -3 -2 -1 0 1 2 3 Time ( sec ) Acceleration (g) Second Floor - Second Mode Figure 4. Free response of test structure in (a) Mode 1 and (b) Mode 2. (a) First Mode Responses (b) Second Mode Responses Introduction to Dynamics of Structures 5 Washington University in St. Louis 4.3.2 Bandwidth method ANSWER y 1 0.363= y 2 0.348= ζ δ 2π y 1 y 2 ln 2π 0.363 0.348 ln 2π 6.716 10 3– ×== = = MODE 1: (using y-values from MATLAB plots) floor 1: y 1 0.498= y 2 0.469= ζ δ 2π y 1 y 2 ln 2π 0.498 0.469 ln 2π 9.549 10 3– ×== = = floor 2: y 1 1.985= y 2 1.920= ζ δ 2π y 1 y 2 ln 2π 1.985 1.920 ln 2π 5.299 10 3– ×== = = MODE 2: floor 1: y 1 1.778= y 2 1.721= ζ δ 2π y 1 y 2 ln 2π 1.778 1.721 ln 2π 5.186 10 3– ×== = = floor 2: Please do the following. • From the transfer functions obtained in 4.1 estimate the damping ratio using the half power bandwidth method described in 2.4.2. What is the damping ratio associated with each natural frequency? • Compare the damping values for each of the two modes. • Discuss the advantages and disadvantages of these two methods? Introduction to Dynamics of Structures 6 Washington University in St. Louis Using the bandwith method, the following calculations are performed. The computed damping values are approximately the same order of magnitude using both methods. The half-power bandwidth technique results in significant errors when the damping in the system is small because: 1) the actual peak in the transfer function is difficult to capture, and 2) interpolation is required to estimate the half-power points. On the other hand, the decrement technique is more effective for lightly damped systems. 5.0 References CHOPRA, A. K., Dynamics of Structures, Prentice Hall, N.J., 1995 HUMAR, J. L., Dynamics of Structures, Prentice Hall, N.J., 1990 PAZ, M., Structural Dynamics, Chapman & Hall, New York, 1997 f a 1.935= f b 2.035= MODE 1: ζ 1 f b f a – f b f a + 2.5%== (estimating values from plots) f a 5.73= f b 5.85= MODE 2: ζ 2 f b f a – f b f a + 1.04%== (estimating values from plots) INTRODUCTION TO DYNAMICS OF STRUCTURES A PROJECT DEVELOPED FOR THE UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES http://ucist.cive.wustl.edu/ Developed by: Mr. Juan Martin Caicedo (jc11@cive.wustl.edu) Ms. Sinique Betancourt Dr. Shirley J. Dyke (sdyke@seas.wustl.edu) Washington University in Saint Louis This project is supported in part by the National Science Foundation Grant No. DUE–9950340. Required Equipment: • Instructional Shake Table • Two Story Building • Three Accelerometers • MultiQ Board • Power Supply • Computer • Software: Wincon and Matlab Introduction to Dynamics of Structures 1 Washington University in St. Louis Introduction to Dynamics of Structures Structural Control & Earthquake Engineering Laboratory Washington University in Saint Louis Objective: The objective of this experiment is to introduce you to principles in structural dynam- ics through the use of an instructional shake table. Natural frequencies, mode shapes and damping ratios for a scaled structure will be obtained experimentally. 1.0 Introduction The dynamic behavior of structures is an important topic in many fields. Aerospace engineers must understand dynamics to simulate space vehicles and airplanes, while mechanical engineers must understand dynamics to isolate or control the vibration of machinery. In civil engineering, an understanding of structural dynamics is important in the design and retrofit of structures to with- stand severe dynamic loading from earthquakes, hurricanes, and strong winds, or to identify the occurrence and location of damage within an existing structure. In this experiment, you will test a small test building of two floors to observe typical dynamic behavior and obtain its dynamic properties. To perform the experiment you will use a bench-scale shake table to reproduce a random excitation similar to that of an earthquake. Time records of the measured absolute acceleration responses of the building will be acquired. 2.0 Theory: Dynamics of Structures To understand the experiment it is necessary to understand concepts in dynamics of struc- tures. This section will provide these concepts, including the development of the differential equa- tion of motion and its solution for the damped and undamped case. First, the behavior of a single degree of freedom (SDOF) structure will be discussed, and then this will be extended to a multi degree of freedom (MDOF) structure. The number of degrees of freedom is defined as the minimum number of variables that are re- quired for a full description of the movement of a structure. For example, for the single story building shown in figure 1 we assume the floor is rigid compared to the two columns. Thus, the displacement of the structure is going to be completely described by the displacement, x, of the floor. Similarly, the building shown in figure 2 has two degrees of freedom because we need to describe the movement of each floor separately in order to describe the movement of the whole structure. [...]... range of operation To enable the shake table, one must depress the deadman button The shake table stops when the deadman button is released For a more detailed guide on how to operate the shake table see the “Bench-Top Shake Table User’s Guide” available in the University Consortium of Instructional Shake Tables web page (http://ucist.cive.wustl.edu/) Introduction to Dynamics of Structures 10 Washington... Discuss the advantages and disadvantages of these two methods? 5.0 References CHOPRA, A K., Dynamics of Structures, Prentice Hall, N.J., 1995 HUMAR, J L., Dynamics of Structures, Prentice Hall, N.J., 1990 PAZ, M., Structural Dynamics, Chapman & Hall, New York, 1997 Introduction to Dynamics of Structures 14 Washington University in St Louis ... δ- = -ζ = 2π 2π Introduction to Dynamics of Structures 8 (47) Washington University in St Louis Using equation (47) we can obtain the damping ratio ζ of the structure using the amplitude of the signal at two consecutive peaks in a free vibration record of displacement or acceleration 2.4.2 Half power bandwidth method The second method to obtain an estimation of the damping of a structure is the... response to be shaped with an exponential envelope as shown in figure 4 e –ζω n t term (exponential envelope) time 2π T ≅ ωd T Figure 4 Response of damped structures Introduction to Dynamics of Structures 5 Washington University in St Louis Summary: In this section you learned basic concepts for describing a single degree of freedom system (SDOF) In the followings section you will extend these concepts to. .. floor of the building 8 Click the third button “Plot transfer functions”, to calculate the transfer functions This will take a few seconds Two transfer functions will be computed, including • Ground excitation to first floor • Ground excitation to second floor Introduction to Dynamics of Structures 12 Washington University in St Louis When the plots are displayed another menu will appear This tool is... in each mode of the structure The sinusoidal excitation lasts for 30 seconds, and Introduction to Dynamics of Structures 13 Washington University in St Louis then the structure is in free vibration Then a record of the acceleration of the two floors of the structure will appear The next button on the menu, “Free Vibration Test” (See figure 14), will perform this test When you hit this button a control... 6 Diagram of mode shapes for a four degree of freedom structure Introduction to Dynamics of Structures 6 Washington University in St Louis 2.3 Frequency Domain Analysis The characteristics of the structural system can also be described in the frequency domain The Fourier transform of a signal x(t) is defined by X( f) = ∞ ∫–∞ x ( t )e – i2πft dt (36) and is related to the Fourier transform of the derivatives... Introduction to Dynamics of Structures 11 Washington University in St Louis 4.0 Experimental Procedure Important Notes: Safe Operation of the Shake Table • The “safety override” button on the power supply should ALWAYS remain in the off position • Turn the power supply off if you turn off or reboot the computer • The deadman switch must be depressed to excite the shake table Press this button and hold... damping for each mode Introduction to Dynamics of Structures 7 Washington University in St Louis 2.4.1 Exponential decay Using free vibration data of the acceleration of the structure one may obtain the damping ratio Figure 8 shows a free vibration record of a structure The logarithmic decrement, δ , between two peaks is defined as y1 δ = ln -y2 (42) where y 1 and y 2 are the amplitudes of the peaks Amplitude... case of multiple degree of freedom systems 2.2 Multiple degree of freedom systems A multiple degree of freedom structure and its equivalent dynamic model are shown in figure 5 The differential equations of motion of a multiple degree of freedom system is ·· · Mx + Cx + Kx = p ( t ) (34) where M, C and K are matrices that describe the mass, damping and stiffness of the structure, p(t) is a vector of external