ambient vibration surveys of full-scale structures using personal ...

UNIVERSITY OF SOUTHERN CALIFORNIA DEPARTMENT OF CIVIL ENGINEERING

AMBIENT VIBRATION SURVEYS OF FULL-SCALE STRUCTURES USING PERSONAL COMPUTERS{ EXAMPLES FOR KAPRIELIAN HALL

by S.S. Ivanovi c and M.D. Trifunac

Report No. CE 95-05

Los Angeles, California July, 1995

ABSTRACT

An ambient vibration survey of a full-scale structure represents an ecient and accurate technique for detailed characterization of its three-dimensional dynamic response to wind and microtremor excitation. Fully developed in the early 1970's, this method now is becoming even more advantageous and convenient to use, due to the increased capabilities of personal computers (PC) and commercially available interface boards. In this report, we review the recent advances in ambient vibration testing techniques, illustrate a PC based system for ambient vibration testing, and show results of four tests of Kaprielian Hall, a four storey braced steel building on the main campus of the University of Southern California.

i

ACKNOWLEDGEMENT

Te authors thank A.A. Gladkov and E.I. Novikova for their contribution to the development of the ambient vibration testing system, and participation in the eld measurements, J. Qin for his participation in the eld measurements, and M.I Todorovska for editing the manuscript.

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TABLE OF CONTENTS

Page No. ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : i ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii TABLE OF CONTENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii I INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 I.1 Instrumentation for Ambient Vibration Testing : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 I.2 Description of Selected Transducers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2.1 Transducer as a Single Degree-of-freedom Oscillator : : : : : : : : : : : : : : : : : 4 I.2.2 Ranger Seismometer : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 I.2.3 CM-3KB Seismometer : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 I.2.4 \Old" Ranger Seismometer : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 I.2.5 Comparison of Recordings by Dierent Transducers : : : : : : : : : : : : : : : : : 20 I.3 Data Acquisition System : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 I.4 Cables and Adapters : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 II METHODOLOGY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 26 II.1 Time and Frequency Representation of Signals : : : : : : : : : : : : : : : : : : : : : : : : : : : 26 II.2 Choosing the Sampling Rate : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 II.3 Windowing in Time : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 II.4 Smoothing of the Spectra : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 II.5 Resolving Closely Spaced Frequencies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 II.6 System Transfer-function and Mode-shapes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 III FIELD PROCEDURE AND TYPICAL RESULTS : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 III.1 Selection of Measuring Points : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 III.2 Calibration Tests, Measurements and Other Details : : : : : : : : : : : : : : : : : : : : : 31 III.3 Typical Results and Analysis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 IV RESULTS OF FOUR TESTS OF KAPRIELIAN HALL : : : : : : : : : : : : : : : : : : : : : 45 IV.1 Description of the Building : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45 IV.2 The Experiments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 52 IV.2.1 First Experiment (June 5/6, 1993) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 52 IV.2.2 Second Experiment (January 19, 1994) : : : : : : : : : : : : : : : : : : : : : : : : : : : : 62 IV.2.3 Third Experiment (May 18, 1994) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 81 IV.2.4 Fourth Experiment (May 7, 1995) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 81 IV.2.4.1 Results of Ambient Vibration Excitation : : : : : : : : : : : : : : : : : : : : 89 IV.2.4.2 Results of Impulsive Excitation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 89 V CONCLUSIONS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :106 VI REFERENCES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :107 iii

I INTRODUCTION

The response of structures to dynamic loads (e.g. strong earthquake shaking, strong winds, explosions etc.) is often estimated using mathematical models. Mathematical models can vary in complexity, but they all represent some idealization of a real structure. Their success in representing the response of real structures is best evaluated by experiments on full-scale structures, such as ambient and forced vibration tests, and recording of earthquake response. Ambient vibration tests are often used to estimate the frequencies of vibration, modal damping, and mode shapes of full-scale structures, in the linear range of response. Conducted on a damaged structure, they can be used to estimate its small amplitude response characteristics. Their use for structural health monitoring, and detection and localization of damage warrants further development of both the eld experimentation procedures and of the methods of analysis for full-scale testing of structures. Modern forced and ambient vibration tests of full-scale structures have been performed for about sixty years. The forced vibration tests may require large forces to produce a useful signal to noise ratio. To excite the structure more eciently, the source of forced vibration (e.g. a shaker) is usually located on one of the top oors of the structure, and this results in larger amplitudes of response at the higher oors. Unfortunately, the paths of waves, caused by the shakers and propagating through the structure, do not coincide with those caused by earthquake or wind excitation. This dierence in wave paths has to be taken into account in the interpretation of the results. In contrast, the ambient vibration tests require lighter equipment and smaller number of operators. The sources for the \ambient loads" are e.g. wind, microtremors, microseisms and various local random and periodic sources. The United States Coast and Geodetic Survey started using ambient vibrations tests to measure fundamental periods of buildings in 1936. Crawford and Ward (1964) and Ward and Crawford (1966) were among the rst to show that this method can be used to determine some of the lowest frequencies and modes of vibration of full-scale structures. Trifunac (1970a,b) conducted ambient vibration tests of a 22-storey and a 39-storey steel frame building. Trifunac (1972) compared results of forced and ambient vibration experiments (conducted on a 22-storey steel frame and a 9-storey reinforced concrete building) and showed that both tests give consistent and comparable results. Udwadia and Trifunac (1973a) presented results of ambient vibration tests of three steel frame (22, 39 and 9-storey high) and one reinforced concrete (9-storey high) building, before and after an earthquake. They also analyzed the eects of interaction between soft soil and a sti structure immediately after and long after the earthquake. Trifunac et al. (1975) compared low level (linear) response to ambient noise with larger level response to forced vibration. Udwadia and Trifunac (1974) used ambient vibration tests to nd the preand post-earthquake apparent frequencies of full-scale structures. Fautch et al. (1975), Luco et al. (1976), Wong et al. (1976) and Moslem and Trifunac (1986) used ambient vibration tests to identify the three-dimensional nature of deformations associated with the apparent frequencies of response. Luco et al. (1986; 1987; 1988) and Wong et al. 1

(1988) used ambient vibration tests to resolve the con icting interpretations of the degree to which soil-structure interaction contributes to the system response, and the sources of observed nonlinear soil-structure response to strong earthquake excitation. Ambient and microtremor noise can also be used to identify the in situ properties of soil layers, by rst identifying the direction of wave approach and then inverting the computed phase and group velocities to estimate the velocity variation versus depth beneath the recording array, or by searching for characteristic site frequencies (Udwadia and Trifunac 1972, 1973b). The technological revolution we are witnessed in capability, portability and aordability of personal computers, digital data storage, interfaces, networking, etc. is oering possibilities for constant improvements in the quality, convenience and wider availability of eld instrumentation for ambient vibration testing. In 1993, a PC based system for data acquisition was commissioned by the second author from two graduate students, and since then has been used in earthquake engineering research and education at the University of Southern California (USC). This report gives and overview of this system and its sensors (Chapter I), reviews the standard methodology of data processing and analysis of ambient vibration test results (Chapter II), illustrates some typical results (Chapter III), and presents results of four tests conducted on the Civil Engineering Department oce building, Kaprielian Hall (4-storey steel building). The results of the four tests are compared and a possible explanation for the dierences in the natural frequencies is oered. I.1 Instrumentation for Ambient Vibration Testing

Ambient vibration testing of a structure consists of recording its response to ambient noise, e.g. wind, microtremors, microseisms..., at selected locations in the structure. The instrumentation is chosen so that it covers the amplitude and frequency ranges of interest, and may vary from one experiment to another. The choice of instrumentation depends also on the availability of certain types of equipment. In the early 1970's, when modern ambient vibration testing was launched, both analog and digital recording systems were available. In most of these experiments, velocity type transducers (Ranger seismometers) were used. The output from this transducer is voltage proportional to the relative velocity of the seismic mass. This voltage was ampli ed by a signal conditioner SC-201A and recorded by Lockheed Electronic Model 417 magnetic tape recorder. The analog data were later converted to digital form by an analog-to-digital (A/D) converter. For immediate visual inspection of the vibrations, the signals were simultaneously recorded on Mark 220 Brush Recorder (Trifunac 1970a,b, 1972). There were various sources of error that could arise at dierent phases of the test (Udwadia and Trifunac, 1973a,b, 1974). The eective generator constant of the transducer might vary by about 1-2% depending on the level of excitation and frequency 2

content of the signal. Noise voltages might be introduced by the ampli cation procedure, and frequency modulation-demodulation errors might occur in the tape recording equipment. The noise in the A/D conversion could be reduced by improved resolution of the converter. Today, as a result of the major developments in the electronics and computer technologies, the equipment consisting of multiplexers, analog to digital (A/D) convertors and recorders (frequency modulated tape recorder and signal conditioner) can be all combined into one interface board (data acquisition board) in a personal computer. The frequency band and the gain for speci c recordings can all be set by the software for the data acquisition board, to values suitable to the characteristics of instruments and the levels of excitation. All the recorded digital data can be stored on the hard disk of a PC. Figures I.1.1, I.1.2 and I.1.3 show schematically an old analog, an old digital, and a new recording system with a data acquisition board. I.2 Description of Selected Transducers

For the tests illustrated in this report, three types of electrodynamic seismometers were used: (1) Ranger Seismometer SS-1, (2) an older model Ranger Seismometer, and (3) CM-3KB Seismometer. All these instruments can be adjusted to record either horizontal or vertical motions. In both cases, these transducers behave as a single degree-of-freedom oscillator. I.2.1 Transducer as a Single Degree-of-freedom Oscillator

The motion of the transducer, working as a single degree-of-freedom (SDOF) system (Novikova and Trifunac, 1991), can be described by the following dierential equation 1  + 2!  _ + !  = 0 x (I:2:1:1) 2 1

1

`0

where  is the angle of rotation from the equilibrium position (the relative motion of the transducer), ! and  are the natural frequency and the damping ratio of the transducer, ` is the generalized length of the pendulum and x is the absolute acceleration of the instrument housing. The complex transfer-function, C (!), between the instrument response  and the input motion x is:   1 ! 1   C (!) = (I:2:1:2)   ` ! ! ! 1 0 !1 + 2i !1  1

1

0

2

0

2

1

1

4

The phase shift of the output  with respect to the input displacement x is   Im[C (!)] 0

(!) = tan (I:2:1:3) Re[C (!)] where Im and Re represent the imaginary and real parts of a complex number. Introducing dimensionless frequency  = !=! , C (!) can be written as  1 C (!) = : (I:2:1:4) ` 1 0  + 2i  1

1

1

2 1

2 1

0

1 1

Suppose that the transducer is to measure the displacement of its support. That requires jj = jC (!)ei!tj = jC (!)j = const (I:2:1:5) jxj jei!tj This can be realized when  ! 1. If the transducer is to measure velocity, then the velocity response of the transducer needs to be proportional to the velocity of the support, i.e. jC (!)j=! = const. This can be achieved when  ! 1 and   1. To measure acceleration, it is required that jj=jxj = const., which requires  ! 0. Table I.2.1 and the plots in Fig. I.2.1.1 (Novikova and Trifunac, 1991) show amplitude and phase responses of displacement, velocity, and acceleration transducers. 1

1

1

I.2.2 Ranger Seismometer

The Ranger Seismometer can be used either for recording horizontal or vertical motion (Fig. I.2.2.1). In both positions, it works as a single degree-of-freedom oscillator. It is a spring-mass instrument with electromagnetic transduction. Its permanent magnet is the seismic mass while the coil is attached to the frame. The relationship between its major parts is shown schematically in Figure I.2.2.2. The mass is supported by two circular exures, which constrain it to move with one degree-of-freedom. When the seismometer is positioned vertically, the suspension spring is fully extended; when positioned horizontally, the spring is essentially not stressed. The basic natural period of the mass, exures and suspension springs can be extended by adding small rod magnets installed around the mass. Working as a single degree-of-freedom oscillator, the motion of a Ranger Seismometer is described by: mx + c(x 0 y_ ) + k(x 0 y) = 0 (I:2:2:1) where x stands for absolute motion of the mass and y is the motion of the instrument housing, m is the transducer mass, c is the damping constant and k is the stiness of the spring. 7

Fig. I.2.2.1 A Ranger Seismometer, model SS-1, with extended spring to measure vertical motion.

8

q

By introducing ! = mk { natural frequency and  = CCCR { damping ratio, where CCR = 2m! is the critical damping, equation (I.2.2.1) becomes z + 2! z_ + ! z = 0y (I:2:2:2) where z(t) = x(t) 0 y(t) is the relative displacement of the mass. The solution forz(t) i!t in the case of harmonic excitation y(t) = y e is z (t) = y(t)C (!) (I:2:2:3) where   ! !0 C (!) = (I:2:2:4)   ! ! 1 0 !0 + 2i !0 The electrical output signal V is proportional to the relative velocity z_ of the mass V = Gz_ (I:2:2:5) where G is the generator constant G = B`: (I:2:2:6) In eqn (I.2.2.6), B is the magnetic induction and ` is the length of the wire in the working coil. The sensitivity of the transducer related to the input displacement,Sd(!), can be written as Gz_ Gi!  V Sd (!) = = = Gi!C(!) = (I:2:2:7) y y 1 0  + 2i where  = !!0 . The amplitude of Sd(!) and the phase angle are 0

0

2 0

0

0

2

2

0

3

2

jSd j =

G!0 3

(I:2:2:8)

f(1 0 2 )2 + (2)2 g1=2 

10  (I:2:2:9) 2 Similarly, the sensitivity related to the input velocity, Sv (!), and its amplitude and phase are z_ G V Sv (!) = = G = GC (!) = (I:2:2:10) y_ y_ (1 0  ) + 2i "d = tan01

2

2

2

jSv j =

G2

f(1 0 2 )2 + (2)2 g1=2

"v = tan01



02  2

10

(I:2:2:11) (I:2:2:12)

and for the sensitivity related to the input acceleration G!  z_ GC (!) V Sa = = G = = y y i! i(1 0  ) 0 2i 1

2

jSa j =

G

!0 f(1 0 2 )2 + (2)2 g1=2



2

(I:2:2:13) (I:2:2:14)

10  (I:2:2:15) 2 As it can be seen from the graphs of amplitudes and phases in Fig. I.2.1.1, this type of transducer can be de ned as a velocity-meter for the range of frequencies! > ! , and for damping ratio   1. "a = tan01

2

0

Typical Speci cations for SS-1 Ranger Seismometers: 1s Natural Period, Tn Coil Resistance, Rc 5500 Ohms Critical Damping Resistance 6500 Ohms Calibration Coil Resistance 100 Ohms Generator Constant, G 340 Volts/m/s Total mass travel 61 mm Mass weight 1.45 kg 0

The fraction of critical damping is adjusted to  0:7 by choosing the external resistance approximately equal to the coil resistance. Based on these characteristics, the Ranger Seismometer is a velocity type transducer. A model SS-1 seismometer is 5 inches in diameter, 11 inches long and weighs 9 pounds (Fig. I.2.2.1 and I.2.2.2). I.2.3 CM-3KB Seismometer

The CM-3KB seismometer can be used for recording either horizontal (Fig. I.2.3.1a,b) or vertical motions (Fig. I.2.3.2). It works as a single degree-of-freedom oscillator. It is shown in Figure I.2.3.2 in con guration for measuring vertical motion. The transducer consists of a permanent magnet, a pendulum (3), a coil (7) attached to the pendulum (it is not visible in the gures), and a systems for regulation of the period (5) and adjustment of the equilibrium position (6). When the instrument is used to measure horizontal motion, the whole system (except the base plate) is rotated by 90 about the longitudinal axis and is balanced by using regulator screws (5) and (6). The same pendulum with the same case is then used to induce voltage proportional to the velocity of horizontal motion. 11

12 Fig. I.2.3.1a A CM-3KB seismometer (side view from right) in position to measure horizontal motion.

13 Fig. I.2.3.1b A CM-3KB seismometer (side view from left) in position to measure horizontal motion.

For detailed and accurate results, calibration test can be carried out for each new position of the instrument. An electrical impulse (e.g. a step-function) from an outside source, inducted by the calibration coil, forces the pendulum to move. The response of the transducer resulting from that motion can be used for identi cation of the transducer parameters. Changing the orientation of this instrument from horizontal to vertical position is more time consuming than for the Ranger SS-1 Seismometer Typical Speci cations for CM-3KB Seismometers Natural Period, Tn 2s Coil Resistance, Rc 1600 { 2400 Ohms Critical Damping Resistance 35 { 53 Ohms Calibration Coil 56 { 84 Ohms Generator Constant G 135 Volts/m/s Mass Weight 7.1 kg 0

By its characteristics, this instrument is a velocity type transducer. It weighs about 7.5 kg (16.5 pounds) and has dimensions 9:2 2 6:8 2 4 inches. The sensitivities of the CM-3KB seismometer, related to the input displacement and velocity, are

where x0 : x_ 10 : ks : `0 : Ssk : T: I0 :

S `2 T 2 x0 = sk 0 2 I0 ks 4

(I:2:3:1)

S `2 T x_ 10 = sk 0 I0 ks 2

(I:2:3:2)

sensitivity related to displacement (m) sensitivity related to velocity (m/s) mass moment of inertia of the pendulum (8:5 2 100 kgm ) length of the pendulum (meters) ` = 0:2486T where T is natural period of the pendulum sensitivity of the coil period of the current in the coilV amplitude of the current (I = R , where V is the electrical output signal and R is the resistance of the coil). 3

0

2 0

0

0

15

2

16 Fig. I.2.4.1 An Earth Sciences (Teledyne) Ranger Seismometer, oriented to measure horizontal motion.

Equations (I.2.3.1) and (I.2.3.2) correspond to the expressions forC (!) and C !! from Table I.2.1 derived for a single degree-of-freedom oscillator, and to equations (I.2.2.8) and (I.2.2.11) for jSdj and jSv j for a Ranger SS-1 Seismometer. ( )

I.2.4 \Old" Ranger Seismometer

Figure I.2.4.1 shows an Earth Sciences (Teledyne) Ranger Seismometer used extensively in ambient vibration tests in the late 1960's and through the 1970's. Its construction and principles of operation are essentially the same as those of a SS-1 seismometer, except that it has slightly smaller damping ( 0:6 of critical). Its natural period and gain (generator constant) are very similar to those of the SS-1 model. I.2.5 Comparison of Recordings by Dierent Transducers

Figures I.2.5.1, I.2.5.2 and I.2.5.3 show ratios of Fourier amplitude spectra of simultaneous calibration test of three dierent instruments (the instruments are placed next to each other and it is assumed that they are excited by exactly the same motion). Clearly, all three instruments can be used in the same experiment. To eliminate the eect of dierent instrument responses, all results are corrected (normalized) by using \spectral correction ratios" from such calibration tests. I.3 Data Acquisition System

The multi-channel Digital Data Acquisition System shown in Fig. I.3.1 was designed to collect data from dierent types of transducers. The signals (voltages) from the transducers are digitized by the board, and are stored on the hard disk drive of a Personal Computer. The sampling frequency, ampli cation factor, duration of recording, and order of channels can be chosen by the user. The National Instruments AT-M10-16X is a high performance and multifunction analog, digital and timing input/output board for IBM PC AT (and compatibles) and EISA personal computers. The board has a 10 s and 16-bit sampling rate A/D converter, that can monitor a single input channel, or can scan through 16 single ended or through 8 dierential channels. It is expandable with National Instruments multiplexing products (in our case with a AMUX-64T board) up to 64 channels, at a programmable gain of 1, 2, 5, 10, 20 or 100 for unipolar or bipolar input, and range up to plus-minus 10 Volts. The maximum data acquisition rate without data loss is 100,000 samples per second. Internal and external triggering and sampling are supported. The analog input section of the AT-MIO-16X can be controlled by software. There are three dierent input modes. To measure signals from the transducers output, that are

oating (isolated) outputs, \Referenced Single Ended" (RSE) is the recommend input 20

con guration of the board. RSE means that all input signals are referenced to a common ground point, that is also tied to the analog input ground of the Board. The accuracy of measurements with AT-M10-16X can be seriously aected by environmental noise. For best results, the board should be shielded with a grounded metal screen placed around the whole board, and it should be placed as far as possible from the hard disk drive. By doing this carefully, the noise level can be kept less than the63 units, out of 32,768 units of the full range of input voltage. Switch Box

A simple switch-box can be designed to provide connection between the transducers and the data acquisition board. Additional features of the switch-box may be to send the calibration signal to the particular transducer and to provide power supply 6( 12 Volts DC) for other types of transducers. An optional calibration signal can be created by the box. The excitation voltage (either from an external signal generator, or from an internal battery) then goes to the particular transducer (Fig. I.3.2a and b). At the same time, the transducer response can be recorded. Data Acquisition Programs

Programming the AT-M10-16X involves writing onto and reading from the various registers on the board. A program named \64 GEN" was written in MicrosoftC to accomplish general 64 channels data acquisition. First, it sets the number of channels being used. After that the program can run optional routines which deal with the gain of the particular channels. The acquisition rate in is same for all the channels. Then, the program requests input for the duration of the data acquisition session in minutes, and after inputting the value and pressing the button, the system starts to collect data. The data are stored in binary format in le SEIS in the working directory. The size of each recorded data sample is 16 bits. The data are stored channel by channel, cycle after cycle. When converted to the decimal system, the amplitude of the signal is in special units. To get the data in volts (for example for gain of 100) the value of one unit is 3.05 Volts. For the experiments described in this report, the gain was set to 100 for all the channels, and the sampling frequency was set to 400 or 800/s. The length of the recording sessions varied, and was typically 1, 3 or 5 min.

22

23 Fig. I.3.2a The “black box” housing the AMUX-64T multiplexer for 64 channels (front view).

24 Fig. I.3.2b The box housing AMUX -64T multiplexer (rear view, can hande up to 64 channels).

I.4 Cables and Adapters

Sets of 7-lead and 4-lead cables with adapters were designed (Fig. I.3.1) to take signals from the transducers (Fig. I.2.2.1, I.2.2., I.2.3.1a,b, I.2,3.2 and I.2.4.1) to the switch-box (Fig. I.3.2. a and b). The lengths of the cables vary from 40 to 600 feet to enable one to create adjustable con gurations which are suitable for various test needs.

25

II METHODOLOGY

A transducer converts the mechanical motion of a point on the structure (function of time) into an electrical (analog) signal. The A/D convertor in the data acquisition board samples the analog signal at equally spaced time intervals, 1t, and converts it into digital form for further processing by software. This chapter reviews the basic theory on which the analysis of ambient vibration measurements is based on, leading to the frequencies and mode shapes of the structure. II.1 Time and Frequency Representation of Signals

Let x(t) be the signal recorded at a measuring point, andX (!) be its Fourier Transform. These two representations of the (continuous) signal are related by X (!) =

Z1

x(t)e0i!t dt

01 (II:1:1) Z1 1 i!t x(t) = 2 01 X (!)e d! The most ecient way to compute X (!) is by the Fast Fourier Transform (FFT) algorithm (the number of oating point operations for the FFT is of the order ofN log N , while for the discrete Fourier Transform is of the order ofN , where N is the number of samples of the discretized x(t)). If sampled at equally spaced time intervals 1t (i.e. with sampling frequency fs = 1=1t), the number of samples and the length of the signal L are related by L = N 1t (II:1:2) For example, if fs = 400 samples/s, N = 24; 000 samples corresponds to L = 60 s. The FFT algorithms require that N is a power of 2 (Oppenheim and Shafer, 1975; Udwadia and Trifunac, 1977). Some convenient powers of 2 and correspondingN and L (evaluated for sampling rate fs = 400 samples/s, i.e. for 1t = 0:0025 s) are: 14 : N = 2 = 16; 384 L = 40:96 s 15 : N = 2 = 32; 768 L = 81:92 s = 1:36 min 16 : N = 2 = 65; 536 L = 163:84 s = 2:73 min 17 : N = 2 = 131; 072 L = 327:68 s = 5:46 min Setting N to a power of 2 is achieved by truncation or by zero padding the recorded signal. 2

14 15 16 17

26

2

II.2 Choosing the Sampling Rate

To avoid aliasing, the signal that is sampled has to be bandlimited, and the sampling frequency, fs, has to be large enough. If f is the maximum frequency contained in the signal, then the signal should be sampled so that the Nyquist frequency 1 fN = fs (II:2:1) 2 is grater than f , which is equivalent to (II:2:2) 1t < 2f1 max

max

max

II.3 Windowing in Time

In reality, the Fourier Transform is computed from a nite length segment of the signal. The truncation is equivalent to multiplication of the original signal (possibly of in nite length) by a square window functiond(t) (see Fig. II.1) (0 t