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Determination of structural dynamic properties of three buildings in Vancouver, B.C., from ambient vibration… To, Ngok-Ming 1982

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THE DETERMINATION OF STRUCTURAL DYNAMIC PROPERTIES OF THREE BUILDINGS IN VANCOUVER, B.C., FROM AMBIENT VIBRATION SURVEYS By NGOK-MING £rO B.A.Sc, University of B r i t i s h Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © Ngok-Ming To, 1982 ' In present ing th is thes is i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e ;for .reference, and ".study. I fur ther agree that permission for extensive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my department or by h is or her repre-s e n t a t i v e s . It i s understood that copy or p u b l i c a t i o n of th is thes is for f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion . Ngok-Ming To Department of C i v i l Engineering The Un ivers i ty of B r i t i s h Columbia 2324 Main Mal l Vancouver, B . C . , Canada V6T 1W5 pate / g * / f t a w i J9$2 ABSTRACT The dynamic properties, namely the natural frequencies, mode shapes and damping values of three downtown Vancouver buildings have been obtained by means of ambient v i b r a t i o n surveys. The buildings surveyed include an asymmetrical h i - r i s e - the Harbour Centre, and two frame structures -Toronto Dominion Bank and IBM Towers. Three h i - r i s e buildings of i d e n t i c a l design - the Gage Residences - were also surveyed to examine t h e i r respec-t i v e dynamic properties. Two independent measurement systems were employed so that important data could be duplicated and s t a t i s t i c a l l y stable spectra obtained to deter-mine damping. Two damping estimation methods were used - the autocorrela-tion and p a r t i a l moment methods. A man excited test was also performed on the T/D Tower. The measured natural frequencies of the Harbour Center were much higher than published t h e o r e t i c a l values (64% higher i n one case). This suggests that the a n a l y t i c a l model used was too f l e x i b l e and may not have accounted for the i n t e r a c t i o n s of non-structural elements. In the case of the T/D and IBM Towers the frequencies were found to agree c l o s e l y with the r e s u l t s obtained from t h e o r e t i c a l analyses. The natural frequencies of the asymmetrical building - the Harbour Centre - were expected to be very close together (.01 Hz apart, i n some cases, as indicated by an e x i s t i n g a n a l y t i c a l study). A s p e c i a l smoothing process was employed to search for the close frequencies. Although t h i s search did not reveal frequencies spaced at 0.01 Hz, i t i s s t i l l possible that frequencies e x i s t at closer than .01 Hz or that they have very unequal f t -Fourier amplitudes at a .01 Hz spread. In either case, the smoothing process was not able to separate such frequencies. The r e s u l t s obtained for the Harbour Center E-W d i r e c t i o n suggest that the frequencies of the fundamental and second modes i n t h i s d i r e c t i o n may be equal, or nearly equal, and that the measured 'fundamental' mode shape'is not uniquely defined. The p a r t i a l moment method for damping estimates seems to provide stable r e s u l t s for a given length of record, and seems to be l i t t l e affected by the smoothing process. The low damping values obtained by t h i s method for the Harbour Centre, T/D and IBM Towers are t y p i c a l of damping determinations by ambient v i b r a t i o n t e s t s . Stable estimates of power spectral density or Fourier spectra were achieved e a s i l y i n th i s experimental programme. The autocorrelation method therefore could be confidently used to secure good damping estimates. The damping values obtained by t h i s method are comparable to the re s u l t s found from the p a r t i a l moment technique. The man-excited test r e s u l t s for the T/D Tower also provided damping estimates which were within the range of values determined by the p a r t i a l moment and autocorrelation methods. The natural frequencies and damping values (using only the p a r t i a l moment method) obtained for the three Gage Towers on the UBC Campus were almost the same; s t r u c t u r a l l y the Towers therefore may be regarded as being i d e n t i c a l . Ambient V i b r a t i o n Survey Page Table of Contents Abstract i . L i s t of Tables v L i s t of Figures v i Acknowledgements v i i i Chapter 1: Introduction 1 1.1 Object and Scope 1 1.2 L i t e r a t u r e Survey 2 Chapter 2: Background Theory for the Ambient V i b r a t i o n Survey 4 2.1 Introduction 4 2.2 The Measurement System and the Frequency Response Analysis 4 2.3 The Pulse Response Approach 5 2.4 The Fourier Transform, Relationships Between Pulse and Frequency Responses, Convolution and M u l t i p l i c a t i o n 6 2.5 Di s c r e t i z e d Functions and the Discrete Fourier Transform 6 2.6 D i r i c h l e t ' s Kernel, Windows, Smoothing 7 2.7 The Sampling Theorems 7 2.8 S t a t i s t i c a l Analysis of Spectra and the Idea of S t a b i l i t y 8 2.9 Damping Estimation Methods 9 Chapter 3: The Instrumentation System 11 3.1 Measurement Systems 11 3.2 C a l i b r a t i o n of the Earthquake Engineering Laboratory Measurement System 11 Chapter 4: Ambient V i b r a t i o n F i e l d Measurements 14 4.1 General 14 4.2 A) The Harbour Center 14 B) Toronto Dominion Bank and IBM Towers 20 C) Gage Residences 22 Chapter 5: Data and Dynamic S t r u c t u r a l Analyses Results 24 5.1 A) The Harbour Center 24 B) Toronto Dominion Bank and IBM Towers 34 C) Gage Residences 41 - i i i -Page Chapter 6: Discussion of Results and Conclusions 43 References 46 Figures 48 Appendices 84 Appendix A2 85 Appendix A3 91 Appendix A4 97 Appendix A5 102 Appendix B2 104 - i v -LIST OF TABLES Page 4.1 Published Natural Frequencies of the Harbour Center. 16 5.1 Wind Conditions for the Harbour Center (March 24, 1981). 25 5.2 Wind Conditions for the Harbour Center (May 12~, 1981) 26 and (December 19, 1981). 5.3 Measured Natural Frequencies of the Harbour Center. 30 5.4 Damping Values of the Harbour Center. 33 5.5 Wind Conditions for the T/D Tower ( A p r i l 11, 1981). 35 5.6(a) Measured Natural Frequencies of the T/D Tower. 36 5.6(b) A n a l y t i c a l Natural Frequencies of the T/D Tower 36 5.7 Wind Conditions for the IBM Tower. 37 5.8 Measured Natural Frequencies of the IBM Tower 38 5.9 A n a l y t i c a l Natural Frequencies of the IBM Tower. 38 5.10(a)Damping Value of the T/D Tower. 39 5.10(b)Damping Values of the T/D Tower (Man Excited Test). 39 5.11 Damping Values of the IBM Tower. 40 5.12 Gage Tower N-S Frequencies. 41 5.13 Gage Tower Damping Values. 42 - v -LIST OF FIGURES Page 2.1 Instrument C a l i b r a t i o n Curve. 48 2.6.1 Frequency Response of Two Window Functions. 49 4.1 Elevation of Harbour Center. 50 4.2 Harbour Center Mass D i s t r i b u t i o n and Mode Shapes. 51 4.3 Plan View of Harbour Center. 52 4.4 Seismometer Location on Typical O f f i c e Floor: Harbour Center. 53 4.5 Typical Floor Plan: T/D Tower. . 54 4.6 Typical Floor Plan: IBM Tower. 55 4.7 Typical Roof Plan of Gage Residence Towers. 56 5.1 Windows Resolution Tests. 57 5.2 Frequency I d e n t i f i c a t i o n : Harbour Center E-W. 58 5.3(A)(B) Harbour Centre Mode Shapes ( F l e x u r a l ) . 59 5.4 Harbour Center Mode Shapes ( T o r s i o n a l ) . 61 5.5(A)(B)(C)(D)(E)(F)(G) Autocorrelograms: Harbour Center. 62 5.6 Frequency I d e n t i f i c a t i o n : T/D Tower N-S. 69 5.7(A)(B)(C) Mode Shapes: T/D Tower. 70 5.8 Frequency I d e n t i f i c a t i o n : IBM Tower N-S. 73 5.9(A)(B)(C) Mode Shapes: IBM Tower. 74 5.10(A)(B) Autocorrelograms: T/D Tower. 77 5.11(A)(B)(C)(D)(E) Autocorrelograms: IBM Tower. 79 - v i -Page A.2.1 Noise i n Signal. 88 A.2.2 Fourier Transform of Noisy Data. 89 A.2.3 Smoothed Time Function. 90 A.3.1 Diagram of Maxwell Bridge C i r c u i t • 94 A.3.2 Seismometer C a l i b r a t i o n . 95 A.3.3 System C a l i b r a t i o n . 96 A.4.1(A) Typical Ambient Vibration Traces: Harbour Center. 98 A.4.1(B) Typical Ambient Vibration Traces: T/D Tower. 90 A.4.1.(C) Typical Ambient Vibration Traces: IBM Tower. 100 A.4.1(D) Typical Ambient Vibration Traces: Gage Towers. 101 A.5.1(A) Man-Excited Test: E-W T/D Tower Second Mode. 103 A.5.1(B) Man-Excited Test: E-W T/D Tower Third Mode. 103 A.5.1(C) Man-Excited Test: N-S T/D Tower Second Mode. 103 - v i i -ACKNOWLEDGEMENTS The author wishes to express sincere thanks to his advisors, Professors Cherry, Anderson and Nathan, and e s p e c i a l l y to Dr. Cherry, whose constant guidance provided the necessary a f f i r m a t i o n (and rejection) of concepts and procedures f o r the e n t i r e project. The generous assistance from the professors and research associates i n the UBC Department of Geophysics, and t h e i r permission to use both t h e i r equipment and computer programme l i b r a r y are much appreciated. In p a r t i c u -l a r , the f r i e n d l y discussions with Mr. Bob Meldrum helped clear away many of the obstacles encountered i n the instrument c a l i b r a t i o n and the general a p p l i c a t i o n of the measurement systems. The author i s g r a t e f u l to the owners of the Harbour Center, the Toronto Dominion and IBM Towers for permission to measure t h e i r buildings' v i b r a -t i o n s , and also to Unecon Engineering Consultants and McKenzie Snowball Skalbania & Assoc. Ltd. for providing the drawings of the b u i l d i n g s . The proper functioning of the dual measurement systems demands many competent operators. The graduate students and friends who spent long hours i n the f i e l d a s s i s t i n g i n the operation of the systems helped make t h i s project possible. Their friendship and s e l f l e s s contributions s h a l l remain the author's most valued personal gain i n completing t h i s research. The work was performed under contract (Contract No. 080-040/0-4410) with the National Research Council of Canada on behalf of the Council's Canadian National Committee for Earthquake Engineering. - v i i i -CHAPTER 1 1. INTRODUCTION 1.1 Object and Scope The purpose of t h i s research i s to determine the dynamic properties, namely the natural frequencies, mode shapes and damping values, of three Downtown Vancouver b u i l d i n g s : The Harbour Centre, IBM and the Toronto Dominion Bank Towers, by means of the ambient v i b r a t i o n survey method. Three highrise buildings of i d e n t i c a l design on the campus of the University of B r i t i s h Columbia have also been measured by the same method to compare the dynamic properties of " i d e n t i c a l " b u i l d i n g s . The ambient v i b r a t i o n survey i s a f u l l scale s t r u c t u r a l t e s t i n g method which measures the response of a structure to wind and c u l t u r a l noises and analyzes t h i s response to estimate the dynamic properties of the structure. The method may be applied to a structure f o r a comparatively short time duration just to ascertain i t s dynamic c h a r a c t e r i s t i c s , or i t may be part of a long term monitoring programme to compare the changes i n these character-i s t i c s between pre- and post-earthquake conditions. The structure need not be a bridge or a b u i l d i n g : long term ambient v i b r a t i o n monitoring of o f f -shore platforms has been performed to see i f any damage has been done to major s t r u c t u r a l members by comparing the frequencies between pre- and post-storm or earthquake recordings. In the past decade, enough tests have been c a r r i e d out such that both the v e r s a t i l i t y and the v a l i d i t y of the method have been established. To properly analyze the data gathered i n an ambient v i b r a t i o n survey, a background i n both c l a s s i c a l harmonic analysis and the s t a t i s t i c a l tech-niques describing a random process i s needed. Chapter 2 attempts to bring some of the relevant t h e o r e t i c a l considerations of these areas into focus. 2. The basic p r i n c i p l e s of communication engineering described i n Chapter 2 are applied to the c a l i b r a t i o n of the measurement system, some d e t a i l s of which are covered i n Chapter 3. Chapter 4 contains the actual measurement proced-ures adopted i n the f i e l d , and the problems encountered during the measure-ments. The concepts presented i n Chapter 2 are again put to use i n Chapter 5 i n which the actual analyses of the data are performed and the r e s u l t s are tabulated. Techniques which are s p e c i a l i z e d f o r a p p l i c a t i o n to a p a r t i c u l a r building w i l l be discussed i n t h i s chapter. Since one of the major a p p l i c a -tions of the ambient v i b r a t i o n survey i s to v e r i f y the mathematical models of structures, dynamic s t r u c t u r a l analyses of some of the tested structures were performed on the computer. The r e s u l t s of the computer analyses are compared to the survey r e s u l t s and discussed i n Chapter 5. The f i n a l chapter contains the various conclusions of the research. 1.2 L i t e r a t u r e Survey Some of the e a r l i e s t ambient v i b r a t i o n surveys of buildings were performed by Cherry and Brady (3) i n 1965 and Ward and Crawford (4) i n 1966. Since then many others (1,5,6,7,8,9,10) have contributed valuable experience and i n s i g h t to t h i s area of s t r u c t u r a l dynamics. The technique has also been extended to study the dynamic behaviour of bridges (11,12) and offshore platforms (2). The data a q u i s i t i o n and analyses of ambient v i b r a t i o n surveys involve random data c o l l e c t i o n procedures and Fourier or power spectrum analysis techniques. Cornelius Lanczos' two books (13, 14) o f f e r excellent informa-t i o n on harmonic analysis; relevant s t a t i s t i c a l analysis methods are con-tained i n a c l a s s i c l i t t l e book by Blackman and Tukey (15). 3. An empirical v e r i f i c a t i o n of the s t a t i s t i c a l s t a b i l i t y concept as applied to b u i l d i n g v i b r a t i o n studies i s given by Kircher and Shah (8), and a few meaningful comments on the s t a t i o n a r i t y and other assumptions necessary for power spectra analyses are given by Tukey (16). A very important dynamic property to be determined by ambient v i b r a t i o n surveys i s the damping of the structure. The man excited (17) and h a l f power bandwidth methods were among the f i r s t damping estimation methods to be used i n conjunction with the ambient v i b r a t i o n surveys. The pioneering work of Cherry and Brady (3) provided a rigorous mathematical basis for the autocorrelation method of damping estimation. D i g i t a l techniques that may be applied to power spectra to obtain the autocorrelation function may be found i n a paper by Toaka (18), and ideas pertinent to d i g i t a l f i l t e r i n g i n general may be found i n a book by Hamming (19). A method based on s p e c t r a l moments has also been used to f i n d the damping values (20). A comparison of three of these damping estimation methods (log decrement, h a l f power bandwidth and p a r t i a l moment) i s given by Durning and Engle (21). 4. CHAPTER 2 BACKGROUND THEORY FOR THE AMBIENT VIBRATION SURVEY 2.1 Introduction This chapter summarizes a p a r a l l e l in-depth mathematical background presented i n Appendix B2. It i s concerned with the following matters: i ) the c a l i b r a t i o n of the measurement system used i n the ambient v i b r a t i o n survey; i i ) the planning of the measurement procedures; i i i ) the subsequent analyses and i n t e r p r e t a t i o n of the acquired data. Quotations from C. Lanczos' "Applied Analysis" (13) and "Discourse on Fourier Series" (14), and also Blackman and Tukey's "The Measurement of Power Spectra" (15) w i l l not be separately referenced. 0 2.2 The Measurement System and the Frequency Reponse Analysis The measurement system can be represented as a l i n e a r time invariant t h black box which i s governed by the following n order, l i n e a r d i f f e r e n t i a l equations: ,n j n - l d y . d y . a — — + a , + a y = n , n n-1 , n-1 o dt dt r r - l b ± J i + b i x +  r d t r r ~ 1 a t 1 " 1 where x(t) i s input and y(t) output. There i s no simple experimental method to determine the a's and b's (which are constants) - so instead the input and output are observed and the system transfer function thereby obtained. 5. If we regard the input as composed of a superposition of s t r i c t l y periodic components (the Fourier technique), and i f we examine the responses due to these input components one by one, we have the frequency response approach for the black box. This concept can be extended to the treatment of non-periodic functions by the Fourier i n t e g r a l approach, namely: oo f ( t ) = ~ J F(w) e 1 W tdw 2.2.2 — 00 oo where F(w) = / f ( t ) e ~ i w t d t 2.2.3 —oo f ( t ) i s the non-period function i n the time domain and F(w) i s i t s Fourier spectrum i n the frequency domain. The laboratory c a l i b r a t i o n of the measurement system i s based on the frequency response approach. H i g h - f i d e l i t y reproduction of input i s evident over a wide frequency range for the measurement system. 2.3 The Pulse Response Approach For the l i n e a r , time-invariant system, the input can be modelled by a superposition of pulses, and the output as a superposition of pulse reponses. The response of such a system i s therefore: oo y(t) = / H(t-Q x ( Q d ? 2.3.6 —CO where H(t) i s the response to an impulse input at time t = 0. Equation 2.3.6 i s the convolution i n t e g r a l of H(t) and x ( t ) . A somewhat more det a i l e d physical argument i s provided i n Appendix B2.3. 6. 2.4 The Fourier Transform. Relationship Between Pulse and Frequency  Responses, Convolution and M u l t i p l i c a t i o n . A s l i g h t l y modified version of the Fourier Integral (equation 2.2.2) defines the Fourier transform: CO X(w) = y ^ — / x ( t ) e " 1 W t d t 2.4.1 — OO and the inverse transform: oo x(t) = y ^ — / X(w)e i w tdw 2.4.2 — 00 It can be proved that the complex frequency response of a l i n e a r system i s just the complex conjugate of the Fourier transform of i t s pulse response, m u l t i p l i e d by /STir. (See Appendix B2.4.) There i s another important aspect of the convolution i n t e g r a l (2.3.6): The Fourier Transform of a convolution of two functions i s the product of t h e i r Fourier Transforms. By the remarkable r e c i p r o c i t y of equations 2.4.1 and 2.4.2 the same rule applies for a convolution i n the frequency domain (see B2.4). 2.5 The Discrete Fourier Transform Although the Fourier Integral i s more general than the Fourier s e r i e s , we have to turn to the use of the Fourier series when we use the computer. The process of obtaining the Fourier series as a l i m i t i n g case of the Fourier i n t e g r a l ( d i s c r e t i z i n g ) w i l l give r i s e to a 'kernel' function which determines the 'focusing power' on the d i s c r e t i z e d frequencies, and also the amount of 'leakage' that r e s u l t s from i t s convolution with the true spec-trum. 7. 2.6 Windows, Smoothing Consider the f i n i t e time seri e s we sample as a product of two functions - the i n f i n i t e signal and a rectangular function that has a form of 0,0,0,1,1, 1,0,0,0 (there are (2N+1) non-zero terms f or the (2N+1) points that we have i n the d i s c r e t i z e d time s e r i e s ) . A function h(w) has these (2N+1) non-zero terms i n the frequency domain k=N ikw h(w) = E e which i s a geometric s e r i e s k=-N , , sin(N + 1/2)w for w within the fundamental or h(w) = * — . , w period s i n ^ h(w) i s a kernel function - or window, a f t e r Blackman and Tukey. This par-t i c u l a r window has a highly o s c i l l a t o r y nature, and when i t i s convolved with the Fourier Transform of the buil d i n g v i b r a t i o n s i g n a l , leakage w i l l occur. A smoothing process to reduce the secondary maxima may have to be applied to the transformed v i b r a t i o n records of the buil d i n g s . From the same standpoint, h i - f i d e l i t y reproduction of the input may not be the best p o l i c y i f we perform analyses i n the frequency domain. We may have to suppress the fundamental v i b r a t i o n modes of buildings to avoid leak-age. 2.7 The Sampling Theorems The r e s u l t s of the mathematics of sampling theorems may be stated as follows: ( i ) To avoid a l i a s i n g , a band-limited function must be sampled at higher than twice the fo l d i n g frequency. ( i i ) It i s necessary to sample at more than two samples per cycle for the highest frequency present so as to preserve the information of the o r i g i n a l function at the fold i n g frequency. 8. 2.8 S t a t i s t i c a l Analysis of Spectra and S t a b i l i t y There i s one major difference between harmonic analysis and the s t a t i s -t i c a l approach defining a random process, such as i s obtained from an ambient v i b r a t i o n survey (Jenkins (23)). Consider a f i n i t e s e r i e s of the sample autocovariance C^. ^ n-k C = — Z X X (zero mean process) n. £ — ^ 111c The raw spectral estimate I n(w^): n-1 I (w.) = — [c + 2 £ C.cos w.k] 2.8.3 n j ir L o k 2 k=l where w . = 2 TTi/n. The raw estimate I^(w^) w i l l fluctuate about the power sp e c t r a l density function at W j and no matter how long a sample we take i n the time domain, the variance of the f l u c t u a t i o n of In(w..) about the power sp e c t r a l density function does not decrease to zero. This i s a well observed phenomenon. Harmonic analysis of random num-bers produces a highly v a r i a b l e , spiked spectrum. (See Appendix A2 for physical i l l u s t r a t i o n ) . A modified procedure of harmonic analysis can lead to s t a b i l i z e d spec-t r a l estimates. In t h i s process the n terms of the sample X(t) are s p l i t i nto p sets of m terms so n=pm. By conducting a Fourier analysis f o r each set and by averaging these r e s u l t s at each frequency, we can make the variance as small as we wish. 9. For our ambient v i b r a t i o n survey we have to analyse the data using a reasonably fin e frequency r e s o l u t i o n , while ensuring a stable estimate of the spectra from which the damping values are obtained. These requirements demand that both 'm' and 'p' be large and a rather lengthy record has to be taken i n the f i e l d for t h i s purpose. It i s possible that long time records may v i o l a t e the s t a t i o n a r i t y assumption (Toaka (18)), but Tukey (16) recom-mends that the analyses of long records be judged on t h e i r own merit. They often produce very useful average spectra. Empirical proof of t h i s l a t t e r fact has been obtained from studies of building v i b r a t i o n s (Kircher and Shah (8)). The p r i n c i p l e of averaging spectra by using long records was adopted i n t h i s study. The use of two separate measurement systems i n the f i e l d (Chapter 4) i s based on t h i s p r i n c i p l e . 2.9 Damping Estimation Methods Two methods have been employed to evaluate the percentage of c r i t i c a l damping of the test structures: the autocorrelation and the p a r t i a l s p e c t r a l moment methods. (a) The Autocorrelation Method: The t h e o r e t i c a l background for t h i s method can be found i n Cherry and Brady (3). The autocovariance function for a function y(t) i s Cy(t): T/2 c y ( T ) - T t»¥ / y(t)y(t+t) dt. -T/2 When Cy(x) i s normalized by Cy(0) (the variance), i t i s termed the autocor-r e l a t i o n function. 10. For white noise input with constant s p e c t r a l density, the output auto-c o r r e l a t i o n function i s a cosinusoidal function with exponential decay. The c r i t i c a l damping r a t i o may be obtained from the autocorrelogram (the plot of autocorrelation function) by means of the log decrement method. In using the d i g i t a l process, attention has to be paid to the Fourier transform and reverse transform procedure. (See Toaka (18), and Appendix B2.9). (b) P a r t i a l Spectral Moment Method: Vanmarcke et a l (20) proposed the use of p a r t i a l s p e c t r a l moments to obtain natural frequency and damping estimates. The following are claimed to be the advantages of t h i s method: ( i ) For a given record length, estimates of s p e c t r a l moments may be expected to be much more r e l i a b l e than those of i n d i v i d u a l s p e c t r a l ordin-ates. ( i i ) Smoothing of the "raw" spectral estimates i s unnecessary. Applications of these methods to the data c o l l e c t e d are given i n Chapter 5. 11. Chapter 3  The Instrumentation System 3.1 Measurement Systems: Two separate measurement systems were employed i n the f i e l d . These are: ( i ) The Earthquake Engineering Laboratory ( C i v i l Engineering Dept.) System: This system co n s i s t i n g of four Ranger SD217 seismometers, a Teledyne SC201 signal conditioner with amplifiers and low pass displacement, v e l o c i t y and a cceleration f i l t e r cards ( a l l having a corner frequency of 100 Hz), a P h i l i p s ANA-L0G7 FM/AM tape recorder capable of recording i n 7 channels, and a Tektronix CRO which i s used for v i s u a l d i s p l a y . The s i g n a l conditioner has adjustable e l e c t r i c a l resistances which can a l t e r the t o t a l damping of each of the four seismometers. i i ) The UBC Geophysics System (on loan from the Department of Geophysics): This system i s made up of e i t h e r one or two Willmore Mark II seismome-t e r s , an a m p l i f i e r with band pass f i l t e r s (adjustable corner frequencies at 0.1 Hz or 0.8 Hz to 5 Hz or 12.5 Hz), an H.P. 3960 Instrumentation tape recorder, and a Brush 222 chart recorder for v i s u a l d i s p l a y . Since c a l i b r a t i o n procedures for the two systems are s i m i l a r , only the c a l i b r a t i o n of the Earthquake Engineering Laboratory System w i l l be described. 3.2 C a l i b r a t i o n of the Earthquake Engineering Laboratory Measurement System, a) The four Ranger seismometers were c a l i b r a t e d by using a WAVETEK model 111 voltage control generator (to provide the input signals) i n conjunction with a MAXWELL BRIDGE arrangement following the method described 1 2 . by K o l l a r and Russell (27). The same procedure was also employed and described by Cherry and Topf (7). The seismometer c h a r a c t e r i s t i c s (natural frequency, damping and seismometer constant) are tabulated i n Appendix A3, Table A3.1. b) Again using the WAVETEK generator for input to the s i g n a l conditioner, a separate c a l i b r a t i o n was car r i e d out on the signal conditioner and tape recorder as a u n i t . Output from the FM recording of the tape recorder was analysed. c) The r e s u l t s of these two c a l i b r a t i o n s were combined and analysed with a computer programme e s p e c i a l l y designed to handle t h i s c a l i b r a t i o n problem. The outputs of t h i s programme define the system frequency response (displacement, v e l o c i t y and acceleration) for each f i l t e r card. Since the v e l o c i t y f i l t e r cards were found to produce the best r e s u l t s i n the f i e l d , they were used throughout the measurement programme. A t y p i c a l v e l o c i t y response curves i s also shown i n Appendix A3, Figure A3.3. d) As an independent check on the above r e s u l t s a 'simulated shaking table' c a l i b r a t i o n was performed on the t o t a l measurement system. The WAVETEK 111 generator was connected to the MAXWELL Bridge through which the seismometer was operated. The seismometer s i g n a l was passed through the signal conditioner and then recorded on the FM tape recorder and system inputs and outputs compared. The system response obtained i n t h i s way was i n good agreement with the re s u l t s from the piece-meal c a l i b r a t i o n described i n a ) , b) and c ) . Thus, the use of the computer programme f or the piece-meal c a l i b r a t i o n was v e r i f i e d and adopted for system c a l i b r a t i o n . A flowchart of steps (a) to (d) i s shown below. 13. FLOW CHART OF CALIBRATION STEPS (b) Signal conditioner  and tape recorder  c a l i b r a t e d : Frequency response to sinusoidal input obtained. (a) Four Ranger SD217  seismometers  c a l i b r a t e d : Natural frequencies, damping values and seismometer constants obtained. (c) System frequency response obtained: by combining and analysing r e s u l t s of (a), (b), using computer program UNICAL. (d) Independent check: System response obtained by using 'Simulated Shaking Table' c a l i b r a t i o n method Procedures (a), (b), ( c ) , (the piece-meal c a l i b r a t i o n ) adopted for system c a l i b r a t i o n . 14. Chapter 4 Ambient V i b r a t i o n F i e l d Measurements 4.1 General The planning and execution of the measurement programme (i n c l u d i n g the layout of the instruments), and descriptions of the test structures are provided i n the following sections. Problems encountered i n the f i e l d are also noted. The presentation i s separated into three groups, according to the structures tested, as follows: A) The Sears Tower (or Harbour Centre) - Downtown Vancouver B) The Toronto Dominion Bank and the IBM Towers - Downtown Vancouver C) The Gage Residence, Towers A,B,C - University of B.C. Campus 4.2 A) The Harbour Centre: (performed on 24th March, 81, 12th May, 81 and 19th D e c , 81). The d e s c r i p t i o n of the Harbour Centre that follows has been taken from a paper by Tso and Bergmann (28): "The b u i l d i n g i s the main bui l d i n g of the Harbour Centre complex i n the downtown area of Vancouver (Author's Note: F i g . 4.1 - courtesy of Eng and Wright a r c h i t e c t s ) . It consists of a f i v e - s t o r e y substructure of which three are used for parking, shipping and receiving; a f i v e - s t o r e y section above grade which i s linked with an e x i s t i n g seven-storey bu i l d i n g to form together a large department store, a 21-storey o f f i c e b u i l d i n g above the store and an elevated observation tower with a revolving restaurant. The main b u i l d i n g having a 116' x 116' plan size i s separated from the surrounding construction above street l e v e l by expansion j o i n t s . The b u i l d -15. ing i s generally of reinforced concrete construction with the o f f i c e tower f l o o r s cast i n lightweight concrete. The perimeter basement walls and a system of i n t e r i o r shear walls create a very s t i f f base below str e e t l e v e l from which the tower ca n t i l e v e r s to a height of 455' The observation and restaurant f l o o r s are framed i n s t e e l supporting a lightweight concrete f l o o r deck." The same paper also reports the b u i l d i n g periods as obtained from a dynamic a n a l y s i s . These are reproduced i n Table 4.1, together with the corresponding frequency values. The t h e o r e t i c a l mode shapes for the f i r s t 3 l a t e r a l modes are also provided i n the Tso-Bergmann paper, and are duplicated i n F i g . 4.2. From the published r e s u l t s of the analysis of the Harbour Centre, the following became apparent: i ) The measurement programme must provide s u f f i c i e n t data to give a fine enough frequency r e s o l u t i o n to d i s t i n g u i s h close mode frequencies - as close as 0.01 Hz apart. i i ) There must also be enough blocks of data (with block lengths dependent on frequency r e s o l u t i o n desired) such that an averaging technique can be used to ensure a s a t i s f a c t o r y l e v e l of s t a b i l i t y ( s e c t i o n 2.8) for damping estimation. An estimate of the length of recording time needed f o r both ( i ) and ( i i ) showed that the battery powered Earthquake Engineering Laboratory measurement system could not meet the power requirement of long time record-ings. The decision was then made to introduce a separate (Geophysics) system having one Willmore Mark II seismometer ( s e c t i o n 3.1), for the 16. Table 4.1 PUBLISHED (28) NATURAL FREQUENCIES OF THE HARBOUR CENTER. Mode* Di r e c t i o n N-S Direc E-.tion •W T (SEC) f (Hz) T (SEC) f (Hz) 1 3.89s (TORSION) .26 4.16s (TORSION DOMINANT) .24 2 3.74s (FLEXURE) .27 3.53s (FLEXURE DOMINANT) .28 3 1.34s (TORSION) .75 1.39s (TORSION DOMINANT) .72 4 1.31s (FLEXURE) .76 1.27s (FLEXURE DOMINANT) .79 5 • 90s (TORSION) 1.11 • 91s (TORSION DOMINANT) 1.10 6 • 80s (FLEXURE) 1.25 .78s (FLEXURE DOMINANT) 1.28 *Modes are numbered according to decreasing order. 1 7 . express purpose of obtaining long time-history records of the b u i l d i n g v i b r a t i o n s . These long recordings serve to provide: a) stable Fourier spectra for damping estimates. b) an independent check of the natural frequencies obtained by the Earthquake Engineering Laboratory system. Later, (12th May), two Willmore Mark II seismometers were employed i n the Geophysics system to re-examine the phase r e l a t i o n s h i p of the bui l d i n g vibrations o r i g i n a l l y measured, and the Geophysics system became, as far as i d e n t i f y i n g b u i l d i n g f l e x u r a l and t o r s i o n a l frequencies was concerned, a "back-up" for the Earthquake Engineering Laboratory system. A second and f i n a l check on the phase r e l a t i o n was c a r r i e d out (Dec. 19) using a portable Geophysics system employing two Willmore Mark II seismometers, a signal conditioner and a Teledyne Geotech MCR-600/Microcorder with a Brush 222 chart recorder for v i s u a l d i s p l a y . These checks were c a r r i e d out because of apparent p e c u l i a r i t i e s i n some of the mode shapes derived from the f i r s t test s e r i e s . This i s discussed i n a l a t e r section. Although the Willmore Mark II seismometers are not as ' s e n s i t i v e ' as the Ranger SD217 seismometers (since they have a lower 'g' - generator constant - value than the Ranger SD217, see Appendix A3), there were several advantages i n using the Geophysics system as "back-up": • The added recording capacity of the Geophysics system enabled important measurements to be duplicated and allowed for the p o s s i b i l i t y of securing v i b r a t i o n records over a long time period. • Since the low and high corner frequencies of the amplifier are adjust-able (see section 3.1), i f , f o r example, the measurements were made on a day with sudden wind gusts, the low cut-off frequency could be adjusted from .1 to .8 Hz, so as to suppress the fundamental mode, thereby enhancing the 18. detection of the higher modal frequencies. (Strong wind excites p r i m a r i l y the fundamental frequency of high r i s e buildings - Cherry and Topf (7); see also section 2.5 on suppressing the fundamental mode). The high corner frequency could also be adjusted to avoid high frequency noise i f i t was found to be present. 9 The system could be operated by one person and was more portable than the Earthquake Engineering Laboratory system. Re-checking f i e l d measure-ments could be made more e a s i l y with t h i s system than the Earthquake Engineering Laboratory system. Although the double system measurement programme required more manpower i n the f i e l d , i t had a f l e x i b i l i t y not possible with the single system programme. The actual placement of the instruments for the Harbour centre was as follows: (method of placement was s i m i l a r for a l l b u i l d i n g s ) . a) The Geophysics system (hereafter referred to as System 2): The two Willmore seismometers f i r s t were placed side by side and i n the same d i r e c t i o n on the roof (machine room l e v e l ) of the Harbour Centre for the purpose of performing a " c o l l o c a t i o n c a l i b r a t i o n " , which was used to v e r i f y the r e l a t i v e magnitude and phase of the seismometer outputs (when they were measuring the same input) both before and a f t e r the ambient measurements were recorded. The seismometers then were placed on the roof (or an upper f l o o r i n the other buildings, since the upper f l o o r s have reasonable v i b r a t i o n amplitudes for a l l modes of i n t e r e s t ) i n pairs as shown i n F i g . 4.3. To d i s t i n g u i s h between the t o r s i o n a l and f l e x u r a l frequencies, the two paired seismometers were placed at positions 1 and 2 res p e c t i v e l y f o r about 30 minutes and then at positions 1' and 2' for a further 30 minutes. Two 19. a d d i t i o n a l p a i r s of half-hour records were taken with seismometers simultan-eously located on the roof i n the positions 3 or 4 and at the observation deck l e v e l d i r e c t l y above the elevator shaft (shown i n F i g . 4.3), pointing i n the same d i r e c t i o n as 3 or 4 r e s p e c t i v e l y . To i s o l a t e the t o r s i o n a l and f l e x u r a l frequencies, as well as to ascer-t a i n the phase r e l a t i o n s h i p of v i b r a t i o n a l motion on d i f f e r e n t f l o o r s , simultaneous recordings, a f t e r d i g i t i z a t i o n , were added and subtracted i n turn before being transformed into the frequency domain. Torsional or out of phase signals between f l o o r s w i l l be enhanced a f t e r subtraction i s performed; f l e x u r a l or i n phase signals between f l o o r s w i l l be enhanced a f t e r addition i s performed. Enhancement (or lack of i t ) i s v i s i b l e from the r e s u l t i n g change i n amplitude of the Fourier Spectra spikes correspond-ing to the frequencies i s o l a t e d . On March 24, 1981 only one Willmore Mark II seismometer was employed for the sole pupose of obtaining stable spectral estimates (see section 2.8). The instrument was placed at the west edge of the roof pointing North (same as p o s i t i o n 1) for 30 minutes, then at p o s i t i o n l 1 for an a d d i t i o n a l half-hour. b) The Earthquake Engineering Laboratory System: (hereafter r e f e r r e d to as  System 1). Af t e r c o l l o c a t i o n c a l i b r a t i o n (see (a) above) on the 27 f l o o r (on an upper f l o o r for other b u i l d i n g s ) , the four Ranger seismometers were placed on various f l o o r s to define the mode shapes. Ambient v i b r a t i o n measurements were taken r e l a t i v e to a reference seismometer located on an upper f l o o r (23 f l o o r f or the Harbour Centre). The remaining three instruments were placed at various l e v e l s to record storey movements at the same time and i n the same d i r e c t i o n as t h i s reference point v i b r a t i o n . The phase r e l a t i o n s h i p s 20. of the storey movements r e l a t i v e to the reference point were found by a process of addition and subtraction of simultaneous d i g i t i z e d s i g n a l s . The r e l a t i v e displacement amplitudes of the building were found by comparing the spectral amplitudes at the natural frequencies i d e n t i f i e d on the Fourier spectra, which were determined from a Fourier Transform of simultaneous recordings at various f l o o r s , with the corresponding amplitudes of the reference f l o o r . The phase and r e l a t i v e displacements define the mode shapes. In F i g . 4.4 positions 1, 2 represent the locations of seismometers i n two perpendicular d i r e c t i o n s on a p a r t i c u l a r f l o o r . For c e r t a i n selected data a check of the r e l a t i v e phase r e l a t i o n s h i p between f l o o r s was performed by examining the signs of the r e a l parts of the amplitude c o e f f i c i e n t s (7) of the Fourier Transform. To e s t a b l i s h the phase r e l a t i o n s h i p between the four seismometer s i g -nals, i n order to i s o l a t e the f l e x u r a l and t o r s i o n a l frequencies, the four instruments were assembled as shown i n F i g . 4.4, on the 27 f l o o r (on an upper f l o o r for the other b u i l d i n g s ) , a f t e r the mode shape measurements had been completed. The same addi t i o n and subtraction methods described i n (a) were then applied to the simultaneous s i g n a l s . A f i n a l c o l l o c a t i o n c a l i b r a t i o n , s i m i l a r to the i n i t i a l one described e a r l i e r , was then performed. The r e s u l t s of the measurements and methods of analysis are discussed i n the next chapter. (B) Toronto Dominion Bank and the IBM Towers: (Measurements of the T/D Tower were performed on A p r i l 11, 1981; measurements of the IBM Tower were performed on A p r i l 12, 1981). These buildings are the two "black towers" situated i n the Downtown core of the Ci t y of Vancouver. The Toronto Dominion Bank Tower i s located 21. on the south side of Georgia Street at G r a n v i l l e . It i s a 33-storey s t e e l structure with a 3 bay x 5 bay moment r e s i s t i n g d u c t i l e space frame capable of withstanding a l l of the design horizontal loads. The building's plan dimensions are 150' x 110' and i t r i s e s to a height of about 445*. There i s a machine room on the roof, and a mechanical room on the 15 and 16 f l o o r l e v e l s above ground. The IBM Tower i s situated on north side of Georgia Street at G r a n v i l l e . It i s also a s t e e l structure with a moment r e s i s t i n g d u c t i l e space frame. The bu i l d i n g layout consists of 3 bays by 4 bays having plan dimensions of 110' and 120' re s p e c t i v e l y . It has 21 storeys and i s about 320 feet t a l l . 1) The Toronto Dominion Bank Tower: Measurement System 2 (Geophysics) was located on the 28 f l o o r . The seismometers were located i n positions 1 and 2, F i g . 4.5 for the determination of the t o r s i o n a l and f l e x u r a l frequencies. System 1 (Earth-quake Engineering Lab), a f t e r c o l l o c a t i o n c a l i b r a t i o n on the 26 f l o o r , was set up on that same f l o o r with seismometers placed i n positions s i m i l a r to 1 and 2. Measurements for mode shape determination were than taken on various f l o o r s i n the two perpendicular d i r e c t i o n s indicated by 4 and 5, adopting the instrument placement pattern used i n the Harbour Centre test described e a r l i e r i n t h i s chapter. The reference l o c a t i o n was established on the 28 f l o o r for the f i r s t 6 set of readings; i t was then s h i f t e d to the 18 f l o o r to overcome a r e s t r i c t i o n on cable lengths. Simultaneous recordings of the 18 and 28 fl o o r s were obtained to provide continuity i n the measurement programme. After 12 sets of readings were obtained with System 1, the connecting cable for seismometer No. 3 broke. The two f i n a l sets of read-ings and the f i n a l c o l l o c a t i o n tests were completed with only 3 seismometers 22. (seismometers 1, 2, 4). 2) The IBM Tower: System 2 was placed on the 18 f l o o r with seismometers i n the positions shown i n F i g . 4.6. Instrument positions 1 and 2 provided information for determining the frequencies i n two perpendicular d i r e c t i o n s . Collocation c a l i b r a t i o n for System 1 was performed on the 18 f l o o r and then recordings were taken with instruments located s i m i l a r to 1 and 2 on the 18 f l o o r . Although the cable for seismometer No. 3 was temporarily repaired for these t e s t s , i t s s h i e l d was subsequently found to be d i s c o n t i n -uous. As a r e s u l t , the f i r s t two sets of readings from t h i s seismometer turned out to be saturated with noise and of l i t t l e p r a c t i c a l value. How-ever, since a f i n a l c o l l o c a t i o n c a l i b r a t i o n was performed at the end of the t e s t , and since the frequency i d e n t i f i c a t i o n readings were duplicated by System 2, a useful set of data for the v i b r a t i o n analysis was a v a i l a b l e . The broken s h i e l d was remedied i n the f i e l d following the second set of readings. Seismometer No. 2 suffered a malfunction on the 9th data set, so for the 9th to the 14th set of recordings, and for the f i n a l c o l l o c a t i o n , there were only 3 usable seismometers a v a i l a b l e i n System 1. The mode shape recording positions on a t y p i c a l f l o o r are indicated by 3, 4 i n F i g . 4.6. The r e s u l t s of these measurements are also presented i n the next chapter. (C) Gage Residences: Towers A,B,C (measurements performed on May 21, 1981) The Gage Towers are 3 high-rise residences situated on the UBC campus. 23. They are of i d e n t i c a l construction: square reinforced concrete buildings having plan dimensions of about 83' x 83', each structure i s 18-storeys and r i s e s to a height of about 153'. These buildings have a well developed system of r i g i d shear walls and are therefore very s t i f f . The purpose of t h i s measurement programme was to compare the dynamic properties - natural frequencies and damping values - of three " i d e n t i c a l " b u i l d i n g s . Only System 2 was used for t h i s purpose. The seismometers were located on the roof of each of the three buildings i n two perpendicular d i r e c t i o n s , as represented by positions 1 and 2 i n F i g . 4.7. Measurements were preceded and followed by c o l l o c a t i o n c a l i b r a t i o n s . The r e s u l t s of these tests are also reported i n the next chapter. V 24. Chapter 5 Data and Dynamic St r u c t u r a l Analyses Results 5.1 General The procedures for analyzing the data c o l l e c t e d , and the r e s u l t s of the experiments and of the t h e o r e t i c a l analyses of the mathematical models of the structures under consideration, are presented i n three groups i n t h i s chapter as follows: A) The Harbour Centre B) The Toronto Dominion and IBM Towers C) The three i d e n t i c a l Gage Towers on the UBC Campus. 5.2 (A) The Harbour Centre: The testing of t h i s structure was undertaken on three separate occa-sions. The f i r s t tests was performed on March 24, 1981, using both System 1 (Earthquake Engineering) and System 2 (Geophysics). Only one Willmore Mark II seismometer was used with System 2 i n t h i s p a r t i c u l a r t e s t . (For more d e t a i l s see Chapters 3 & 4). The wind conditions during t h i s t e s t i n g period were as reported i n Table 5.1 (unless otherwise stated, wind information has been obtained from the Vancouver Harbour Weather Observing Station: Coal Harbour at Stanley Park) 25. Table 5.1 WIND CONDITIONS FOR THE HARBOUR CENTER (MARCH 24, 1981) TIME AVG SPEED* mph DIRECTION **MEASUREMENT SYSTEM NUMBER 7 to 8 pm 12.7 E 1 (Earthquake Engineering) and, 2 (Geophysics) 8 to 9 pm 12.7 ESE 1 and, 2 9 to 10 pm 12.7 SE- 1 and, 2 10 to 11 pm 13.8 ENE 1 11 to 12-midnight 13.8 ENE 1 12 to 1 am 12.7 ENE 1 1 to 2 am 4.6 ESE 1 2 to 3 am 6.9 ESE 1 3 to 4 am 4.6 ESE 1 * AVG SPEED OF WIND ONE MINUTE TO THE HOUR EACH HOUR ** SHOWS WHICH SYSTEM(S) WAS (WERE) IN USE AT THAT TIME. SYSTEM 1 = Earthquake Engineering SYSTEM 2 = Geophysics (one seismometer o n l y ) . 2 6 . The second test was performed on May 12, 1981, with System 2 (2 Willmore Mark II seismometers were employed). The purpose of t h i s test was to check the phase r e l a t i o n s h i p of the motions between the observation deck of the restaurant tower and some lower f l o o r s of the o f f i c e b u i l d i n g , and also to v e r i f y the to r s i o n and f l e x u r a l frequencies obtained from the f i r s t t e s t . The check for phase r e l a t i o n s h i p was repeated a second time on 19th December 1981. The equipment used f o r t h i s test has been described i n Chapter 4. The wind conditions during these two check tests are given i n Table 5.2. Table 5.2 Wind Conditions For The Harbour Center  (May 12, 1981) and (Dec. 19, 1981) TIME AVG SPEED mph DIRECTION MEASUREMENT SYSTEM NO. DATE TESTED 1 to 2 pm 6.9 WSW 2* May 12, 1981 2 to 3 pm 6.9 W 2 3 to 4 pm 6.9 WSW 2 4 to 5 pm 3.5 w 2 5 to 6 pm 4.6 W 2 6 to 7 pm 4.6 WSW 2 10 to 11 am 3.5** W 2* Dec. 19, 1981 11 to noon 4.6 S 2 noon to 1 pm 4.6** ssw 2 * 2 Seismometers i n System 2 ** With strong gusts up to 12.7 mph 2 7 . Samples of t y p i c a l analog traces obtained using System 2 are shown i n Appendix A4. The recorded analog data were played back through an analog f i l t e r (low pass corner frequency at 5 Hz - there were no s i g n i f i a n t f r e -quencies above 3Hz) and d i g i t i z e d using an AN5800 A-D converter system modi-f i e d for simultaneous sample and hold, a Kennedy 8108 9-track d i g i t a l tape transport with Kennedy B203 buffered formatter. The sampling rate was 39.075 Hz with a Nyquist frequency i n excess of 19.5 Hz. The d i g i t i z e d data had equal time i n t e r v a l s of .0256 seconds and were recorded on tapes compa-t i b l e with the UBC Computer f a c i l i t e s . The d i g i t i z e d data were analyzed to determine (Step 1) the natural frequencies and mode shapes of the structure and (Step 2) the corresponding damping values according to the following procedures: Step (1): Data were f i r s t analysed by a modified version of UBC C i v i l Engineering programme l i b r a r y 'spectra' (Cherry and Topf ( 7 ) ) . From t h i s new version of 'spectra', a suitable bandwidth was chosen for the damping estimation procedure, which i s outlined i n Step (2). To decide on the proper bandwidths to use, we r e c a l l that i n Chapter 4 i t was noted that frequencies as close as .01 Hz must be separated to i s o l a t e the possible close v i b r a t i o n modes of the Harbour Centre. We note also from F i g . 2.6.1 that by using the Hanning window to reduce leakage the width of the main lobe of the kernel function i s a c t u a l l y widened, hence the power to resolve close frequencies i s much reduced. I d e a l l y we would l i k e both the window and i t s transform to be narrow. In r e a l i t y , t h i s i s an impossible s i t u a t i o n and we must seek a compromise. In the domain of a continuous v a r i a b l e , the use of 'prolate spheroidal functions' are known to r e s u l t i n both a window and i t s transform width which are l i m i t e d as much as possible. J.F. Kaiser worked out the window weights i n the d i s c r e t e v a r i -able domain, as approximations for the continuous v a r i a b l e s , to be used on 28. the Fourier c o e f f i c i e n t s . These windows are known as the Kaiser-Bessel windows and the formulae are contained i n Hamming's book (19). Two such windows have been programmed into 'spectra': Kaiser-Bessel 2-sample (K-B2) and 3-sample (K-B3) convolution windows i n the frequency domain. The weights have been calculated from the formula: (from Hamming (19)) 2N sinh [ ot/l - (—) 2] W(w) = a Io( a) A - (—) 2 wa where: N i s the h a l f width of the even window function cx wa i s — N a i s a constant to control the "shape" of the window. For the 2-sample (K-B2) window a = 1.333; for the 3-sample (K-B3) a = 2.5. a = 1.333 w i l l give a kernel ( i n the frequency domain) of narrower main lobe and higher side lobe l e v e l s than a = 2.5. i ( a ) = i + I [isainl]2 o v ' L , L n! J n=l W(w) are the weights used for the convolution i n the frequency domain. To test the functions of the rectangular (box-car) window, the Hanning and K-B2 and K-B3 windows, a known d i s c r e t i z e d time input with frequency content of 0.26 and 0.27 Hz was mixed with random noise. The magnitudes of the known input frequencies (.26, .27 Hz) were equal. The generated func-ti o n was Fourier transformed, and then passed through the four windows to examine the i r respective resolving powers. The e f f e c t s of the d i f f e r e n t resolving powers are shown i n F i g . 5.1. Although the box-car window does seem to have the best r e s o l u t i o n , i t i s 29. possible that i f the two frequencies i n the input signa l were d i f f e r e n t i n magnitude (for example i f one frequency was 100 times stronger than the other), the box-car may f a i l t o t a l l y to resolve them because the leakage from the high side lobes of the box-car may cause the weaker signal to be t o t a l l y swamped (Harris (26)). In t h i s instance, the Hanning window f a i l e d to resolve the close frequencies (bandwidth used throughout the tests i s .004 Hz); but the K-B2 and K-B3 windows were successful. The signa l l e v e l s were obviously higher than the random noise l e v e l . In f a c t , K-B2 and K-B3 windows have been used to separate close frequencies having unequal magni-tudes i n a r a t i o of up to a 100:1. (Harris (26)). The four windows were applied to the Harbour Centre data (using a band-width of .0048 Hz). When a 'good-quality' window i s used, such as Hanning, K-B2 or K-B3, the smoothing process that r e s u l t s w i l l cause the data to taper o f f at the ends of each data block. Hence, when more than one data block was included for analysis, an "overlap" read procedure was incorpora-ted to make e f f i c i e n t use of the data. The d e t a i l s of t h i s "overlap" proce-dure can be found i n Harris (26). The percentage overlap b u i l t into the new 'Spectra' programme i s about 40%. The "close-frequency search" just described did not show any two f r e -quencies separated by 0.01 Hz. There may s t i l l be a p o s s i b i l i t y that two frequencies are separated by less than .01 Hz, or that the amplitude c o e f f i -cients corresponding to these frequencies may be' i n a r a t i o of more than 100:1. In either case they are not worth separating (Tukey (16)), since a f i n e r e s o l u t i o n spectrum may be s t a t i s t i c a l l y unstable. Therefore the d e c i s i o n was made to use .009 Hz as the bandwidth i n the subsequent analyses of the Harbour Center data. 30. The data recorded with System 1 on the 27* f l o o r of the Harbour Centre were u t i l i z e d to i d e n t i f y the f l e x u r a l and to r s i o n a l frequencies. Simulta-neous recordings were added and subtracted, and the r e l a t i v e growth and decay of the out-of-phase and in-phase signals i n the East-West d i r e c t i o n can be e a s i l y detected, as shown i n F i g . 5.2. The frequencies i n the North-South d i r e c t i o n were determined from the Fourier spectra obtained v i a the same process. The t o r s i o n a l signals were selected from v i b r a t i o n records i n which they were more s i g n i f i c a n t (the North-South d i r e c t i o n i n t h i s case). The method for e s t a b l i s h i n g the phase i s b a s i c a l l y as described for the f l e x u r a l mode determinations. Table 5.3 MEASURED NATURAL FREQUENCIES OF THE HARBOUR CENTER DIRECTION MODE MODE MODE 1 2 3 NORTH - 0.47 1.47 2.19 SOUTH EAST - 0.46 1.50 1.65 WEST TORSION 0.72 1.33 2.17 DOMINANT The measured frequencies f o r the Harbour Centre are l i s t e d i n Table 5.3. The mode shapes corresponding to these frequencies are shown i n Fi g s . 5.3(a) and (b) ( f l e x u r a l ) and F i g . 5.4 ( t o r s i o n a l ) . The manner i n which these shapes were determined has been described i n Chapter 4. * It should be noted that the f l o o r numbering system for the Harbour Center does not incorporate the number 13. In a l l our fi g u r e s , tables and discus-sions we have chosen to ignore s u p e r s t i t i o n and number each f l o o r succes-s i v e l y , including 13. Hence the f l o o r a c t u a l l y numbered 28 i n the buil d i n g i s , i n f a c t , c a l l e d the 27th f l o o r i n Figure 5.3(a) and elsewhere i n t h i s t h e s i s . S i m i l a r l y , i n other f l o o r s beyond the 13th f l o o r . 31. The mode shapes are based on data gathered on March 24, 1981 and Dec. 19, 1981. The data from the May 12, 1981 measurements were omitted due to a d i g i t i z i n g hardware malfunction which led to uncertainties i n the d i g i t i z e d data. Some p e c u l i a r i t i e s were noted i n the E-W mode shapes derived from data secured on March 24, 1981. A 'normal' f i r s t mode did not seem to e x i s t -that i s , there was a zero crossing i n the mode corresponding to the funda-mental frequency and ad d i t i o n a l zero crossings i n the modes for each suc-cessive frequency ( F i g . 5.3(b)); for example, the second mode looked l i k e a normal t h i r d mode. To ensure that t h i s was not the r e s u l t of an instrument o r i e n t a t i o n or data reduction error a further test was performed on Dec. 19, 1981. The data from t h i s test was used to check the phase r e l a t i o n s h i p between the observation deck and the 27 f l o o r , and also the observation deck and the 25 f l o o r only. The phase r e l a t i o n s h i p s between these f l o o r s were found to agree with the March 24, 1981 data i n the N-S and t o r s i o n a l d i r e c -t i o n s . In the E-W d i r e c t i o n , for the fundamental mode shape, a reverse i n phase occurred i n the Dec. 19 data as compared to the March 24 data, and a minor difference i n the t h i r d mode shape ( F i g . 5.3(b)) was also noted. It i s possible that t h i s unusual occurrence can be a t t r i b u t e d to the existence of repeated frequencies (fundamental and second) and the non-uniqueness of t h e i r associated mode shapes. This explanation w i l l be discussed i n the f i n a l section of t h i s t h e s i s . Step (2): Using the bandwidth defined i n Step (1), two techniques were applied to selected sets of data used i n Step (1) to obtain the damping values: i ) The p a r t i a l moment method i i ) The autocorrelogram method. 32. Since these methods have been discussed at some length i n Chapter 2 and Appendix B2 only a few ad d i t i o n a l remarks need be made here. The autocorrelogram method can be applied to i n d i v i d u a l well-defined spectral peaks f i l t e r e d out from the Fourier spectra. Since t h i s method operates i n the time domain, the process of transform/inverse-transform may cause some problems i f no attention i s paid to the l i m i t a t i o n s of the f i l -t e ring process (section B2.9). The basic concept of t h i s f i l t e r i n g process does not d i f f e r from the 'global smoothing' procedure discussed i n Appendix A2. The p a r t i a l moment method does not depend as much on a "well-defined" spectral peak, as i t involves the area and also the f i r s t and second spec-t r a l moments of the power spectra density plot within c e r t a i n s p e c i f i e d frequency l i m i t s . It i s known to be more stable (20) than methods which depend on the ordinates of the spectral density; smoothing processes are not needed to re f i n e the damping estimates obtained from "raw" spectral e s t i -mates . The % c r i t i c a l damping £ determined by these methods f or the Harbour Centre are reported i n Table 5.4 The damping values obtained by both methods i n general are comparable i n terms of t h e i r order of magnitude. The percentage lag i s an important consideration i n the autocorrelogram method (18), and s u f f i c i e n t lag time to produce a minimum of 5 peaks i n the autocorrelogram i s recommended. To achieve t h i s goal the lag time for some of the lower frequencies was increased (see note below t a b l e ) . The p a r t i a l moment method may be sens i t i v e to the cut o f f frequency r a t i o s (section B2.9) but e f f o r t s have been made to keep the frequency r a t i o s very close to unity. 33. Table 5.4 DAMPING VALUES FOR HARBOUR CENTER FREQ (Hz) DIRECTION 5 (% CRITICAL DAMPING) AUTOCORRELOGRAM PARTIAL MOMENT 0.46 E-W 2.20 0.69 1.48 E-W 0.85 1.25 1.65 E-W 0.62 0.74 0.47 N-S 1.18 1.11 1.47 N-S 1.59 1.18 2.19 N-S 1.36 0.72 TORSIONAL DOMINANT 2.47* 1.10 1.33 TORSIONAL DOMINANT 0.86 1.60 See F i g . 5.5(a) to (g) for Autocorrelograms * NOTE: Damping value based on 30% Lag (Time Lag Number = 600) Instead of the 15% Lag Shown i n F i g . 5.5(b). The values of K f o r the f l e x u r a l and t o r s i o n a l frequencies have been obtained from the records which gave the highest signal to noise r a t i o for the p a r t i c u l a r frequency concerned. The percentage lag for the autocorrelograms i s about 15%. As mentioned e a r l i e r the values of $1 , fi^ (see section B2.9 b) for the p a r t i a l moment method are chosen to be close to unity, and &a= Q^ . As the d i f f e r e n t modes are well separated there i s no d i f f i c u l t y i n sel e c t i n g the frequency l i m i t s . 34. The p a r t i a l moment method also gives (as output) a natural frequency based upon the spectral moment c a l c u l a t i o n s . This output frequency, which i s l i s t e d i n Table 5.4, can be compared to the input frequency, which comes from the v i s u a l inspection of the spectral p l o t . In the case of the Harbour Centre there were no s i g n i f i c a n t differences between the input and output natural frequencies. This indicates that the (damping estimates) r e s u l t s obtained by t h i s method are s a t i s f a c t o r y . (B) IBM Tower and T/D Tower: Step 1: A bandwidth of 0.02 Hz was chosen a f t e r performing the process as described i n Step 1 for the Harbour Centre, i ) The T/D Tower: The wind conditions on the day of measurement were as reported i n Table 35. Table 5.5 Wind Conditions For the T/D Tower ( A p r i l 11, 1981) TIME AVG SPEED mph DIRECTION MEASUREMENT SYSTEM USED 1 - 2 pm 1.2 W 1 2 2 - 3 pm 4.6 S 1 2 3 - 4 pm 5.8 S 1 2 4 - 5 pm 4.6 SSW 1 5 - 6 pm 5.8 SSE 1 6 - 7 pm 6.9 SSE 1 7 - 8 pm 6.9 SE 1 8 - 9 pm 5.8 SW 1 9 - 10 pm 6.9 ESE 1 10 - 11 pm 9.2 E 1 11 - 12 Midnight 10.4 ESE 1 An analysis of the recorded data resulted i n the Fourier spectra i n F i g . 5.6 from which the natural frequencies for the T/D Tower were established. The r e s u l t s are given i n Table 5.6(a). 36. Table 5.6(a)  Measured Natural Frequencies of T/D Tower DIRECTION MODE 1 FREQ (Hz) MODE 2 FREQ (Hz) MODE 3 FREQ (Hz) NORTH-SOUTH ( f l e x u r a l ) .26 .75 1.89 EAST-WEST ( f l e x u r a l ) .24 .68 1.41 TORSIONAL .83 .99 1.21 The mode shapes associated with some of these frequencies are shown i n Fig s . 5.7(a) to 5.7(c). It can be seen that the cross-over points for the 2nd mode i n both the N-S and E-W d i r e c t i o n s ( F i g s . 5.7(a), 5.7(b) are located at a lower than t y p i c a l b u i l d i n g elevation. This i s l i k e l y due to the e x i s t -ance of mechanical f l o o r s at the 15 and 16 storey elevations, r e s u l t i n g i n a softening of the structure at these l e v e l s . The t h e o r e t i c a l natural frequencies and mode shapes of t h i s building were av a i l a b l e and provided by Professor Jean-Guy Beliveau of Sherbrooke University. The t h e o r e t i c a l frequencies are given i n Table 5.6(b) and the corresponding mode shapes i n F i g . 5.7. It can be seen that there i s good agreement between the measured and computed fundamental f l e x u r a l frequencies and that the low experimental cross over points noted above are confirmed by an a l y s i s . Table 5.6(b)  A n a l y t i c a l Natural Frequencies of T/D Tower DIRECTION MODE 1 Hz MODE 2 Hz MODE 3 Hz NORTH-SOUTH ( f l e x u r a l ) .27 .58 .97 EAST-WEST ( f l e x u r a l ) .22 .48 .77 37. ( i i ) The IBM Tower: Wind Conditions during the ambient tests were as shown i n Table 5.7 Table 5.7 Wind Conditions f o r the IBM Tower ( A p r i l 12, 1981) TIME AVG SPEED mph DIRECTION MEASUREMENT SYSTEM USED Noon -1 pm 5.8 ESE 1 2 1 - 2 pm below 2.3 1 2 2 - 3 pm 10.4 WSW 1 2 3 - 4 pm 10.4 w 1 4 - 5, pm 11.5 w 1 5 - 6 pm 6.9 w 1 6 - 7 pm below 2.3 1 7 - 8 pm 10.4 E 1 8 - 9 pm 9.2 E 1 38. F i g . 5.8 presents the Fourier spectra f o r the IBM Tower. The natural f r e -quencies i d e n t i f i e d from these spectra are summarized i n Table 5.8. Corres-ponding mode shapes are drawn i n F i g . 5.9. Table 5.8 Measured Natural Frequencies of IBM Tower DIRECTION MODE 1 FREQ (Hz) MODE 2 FREQ (Hz) MODE 3 FREQ (Hz) NORTH-SOUTH ( f l e x u r a l ) 0.38 1.20 2.14 EAST-WEST ( f l e x u r a l ) 0.42 1.30 2.00 TORSIONAL 1.43 1.76 2.48 The UBC C i v i l Engineering Computer programme MAC.FRAME was used to analyze a conceptual model of the IBM Tower. The Tower was set up as a 3-dimensional 21 storey frame structure, with 3 degrees of freedom per f l o o r . The s t i f f n e s s of the frames were calculated from the s t r u c t u r a l drawings of the members and f l o o r s . The r e s u l t i n g frequencies and mode shapes obtained from these analyses are de t a i l e d i n Table 5.9 and F i g . 5.9 re s p e c t i v e l y . Table 5.9 A n a l y t i c a l Natural Frequencies of the IBM Tower DIRECTION MODE 1 Hz MODE 2 Hz MODE 3 Hz N - S f l e x u r a l 0.37 1.09 1.98 E - W f l e x u r a l 0.48 1.39 2.51 Torsional 1.37 3.79 There i s good agreement between the measured and computed frequencies. 39. Step 2: ( i ) T/D Tower The damping values determined for the T/D Tower are given i n Table 5.10(a), 5.10(b). Table 5.10(a) Damping Values of T/D Tower FREQ Hz DIRECTION £ (% CRITICAL DAMPING) AUTOCORRELOGRAM* PARTIAL MOMENT 0.26 N - S ( f l e x u r a l ) 2.64 0.75 N - S ( f l e x u r a l ) 1.76 1.82 1.89 N - S ( f l e x u r a l ) 2.92 0.24 E - W ( f l e x u r a l ) 4.12 2.15 0.68 E - W ( f l e x u r a l ) 1.54 2.42 0.99 Torsion 3.58 *Percentage Lag i s About 15% Table 5.10(b)  Damping Values of T/D Tower (Man-Excited)** FREQ Hz DIRECTION E, (% CRITICAL DAMPING) (LOG DECREMENT METHOD) 0.67 E - W ( f l e x u r a l ) 2.4 1.40 E - W ( f l e x u r a l ) 1.9 0.74 N - S ( f l e x u r a l ) 1.5 ** See Appendix A5 40. F i g . 5.10(a) and (b) show two autocorrelograms f o r the T/D Tower. Again, the output frequencies (as defined e a r l i e r i n t h i s chapter) from the p a r t i a l moment method are p r a c t i c a l l y the same as the input natural frequencies. The damping values from the autocorrelograms, p a r t i a l moment and man-excited tests are t y p i c a l of the res u l t s from such tests; the d i f f e r e n t methods y i e l d comparable values. Step 2; ( i i ) IBM Tower The damping values determined for the IBM Tower are reported i n Table 5.11. Figs. 5.11(a) - (e) show some of the autocorrelogram vs. Lag time p l o t s . Table 5.11  Damping Values of IBM Tower FREQ (Hz) DIRECTION C (% CRITK :AL DAMPING) AUTOCORRELOGRAM ** PARTIAL MOMENT 0.38 N - S ( f l e x u r a l ) 3.35 1.20 N - S ( f l e x u r a l ) 2.19 0.78 2.14 N - S ( f l e x u r a l ) 1.16 0.42 E - W ( f l e x u r a l ) 3.62 3.12 1.30 E - W ( f l e x u r a l ) 3.01 2.27 2.00 E - W ( f l e x u r a l ) 1.12 1.43 TORSIONAL 2.37 2.52 1.76 TORSIONAL 3.61 3.54 **Percentage Lag i s About 15% 41. (C) Gage Residences The measured natural frequencies f o r the three towers are summarized i n Table 5.12. Table 5.12  Gage Towers N-S Frequencies TOWER A Hz TOWER B Hz •TOWER C' Hz DIRECTION 0.34 ( f l e x u r a l ) 0.34 ( f l e x u r a l ) 0.35 ( f l e x u r a l ) NORTH-SOUTH 0.62 ( f l e x u r a l ) 0.62 ( f l e x u r a l ) 0.61 ( f l e x u r a l ) NORTH-SOUTH 0.03 ( f l e x u r a l ) 0.85 ( f l e x u r a l ) 0.93 ( f l e x u r a l ) NORTH-SOUTH 1.17 (torsion) 1.17 (torsion) 1.18 (torsion) TORSION 1.26 (torsion) 1.26 (torsion) 1.23 (torsion) TORSION 0.57 ( f l e x u r a l ) 0.57 ( f l e x u r a l ) 0.56 ( f l e x u r a l ) EAST-WEST 0.78 ( f l e x u r a l 0.79 ( f l e x u r a l ) 0.79 ( f l e x u r a l ) EAST-WEST 1.23 ( f l e x u r a l ) 1.21 ( f l e x u r a l ) 1.18 ( f l e x u r a l ) EAST-WEST In the N-S d i r e c t i o n the data defining the f l e x u r a l frequencies had a r e l a t i v e l y low signal to noise r a t i o such that a meaningful determination of the damping by the p a r t i a l moment method was not possible. However, for the to r s i o n a l modes the signal to noise r a t i o was s i g n i f i c a n t and the damping values from the p a r t i a l moment method for these frequencies are l i s t e d i n Table 5.13. Also reported are the E-W damping values, which were obtained from data with a good s i g n a l to noise r a t i o . 42. Table 5.13  Gage Towers Damping Values DIRECTION TOWER A TOWI :R B TOWEI I C NORTH-SOUTH FREQ Hz % FREQ Hz K % FREQ Hz K % NORTH-SOUTH 1.17 1.33 1.17 0.85 1.18 0.61 NORTH-SOUTH 1.26 0.99 1.26 1.09 1.23 1.09 EAST-WEST 0.57 4.39 0.57 2.53 0.56 2.24 EAST-WEST 0.78 2.70 0.79 . 2.46 0.79 2.43 EAST-WEST 1.23 0.58 1.21 1.00 1.18 2.47 There are some small differences i n the natural frequencies and damping values for the 3 Gage Towers, but i n general t h e i r dynamic properties are i n reasonable agreement with one another. Hence they can be considered as buildings of i d e n t i c a l design and construction. 43. Chapter 6 Discussion of Results and Conclusions The p r i n c i p l e r e s u l t s obtained from t h i s i n v e s t i g a t i o n may be summar-ized as follows: • The natural frequencies of the Harbour Centre, T/D and IBM Towers were determined from ambient v i b r a t i o n measurements to be 0.47 Hz, .26 Hz and .38Hz res p e c t i v e l y i n the North-South d i r e c t i o n and .46 Hz, .24 Hz and .42 Hz resp e c t i v e l y i n the East-West d i r e c t i o n . In the case of the T/D and IBM towers the frequencies were found to agree c l o s e l y with the r e s u l t s obtained from t h e o r e t i c a l analyses. However, the Harbour Centre ambient r e s u l t s (.47 Hz i n N-S; .46 Hz i n E-W) were found to be 74% higher i n the N-S and 64% higher i n the E-W d i r e c t i o n than the t h e o r e t i c a l r e s u l t s (.27 Hz N-S; .28 Hz E-W) obtained from the Tso-Bergmann model. (The measured fundamental t o r s i o n a l frequency (.72 Hz) i s 177% higher than the t h e o r e t i c a l r e s u l t (.26 Hz).) The Tso-Bergmann model appears to be too f l e x i b l e and may not account for interactions of non-structural s t i f f n e s s elements. • D i f f i c u l t i e s were encountered i n securing, from the measured data i n the E-W d i r e c t i o n of the Harbour Center structure, f i r s t and second mode shapes having the usual or normal appearances. It was found from measurements taken on two separate occasions that the f i r s t 2 mode shapes could not be uniquely determined. The data from the f i r s t test (March 24, 1981) yielded a fundamental mode shape which had a second mode appearance and a second mode shape which had a t h i r d mode appearance. The data from the second test (December 19, 1981) provided a f i r s t mode having a normal shape and a second mode resembling a t h i r d mode configuration. 44. Although a close frequency search did not fi n d any frequencies separa-ted by .01 Hz, i t i s s t i l l possible that two very close frequencies e x i s t at f i n e r than 0.01 Hz or that the amplitude c o e f f i c i e n t s of frequencies separa-ted by 0.01 Hz may have a r a t i o higher than 100:1 (Chapter 5), such that one sig n a l would be swamped by the other. In either case, close frequencies are not worth separating even i f they exist (16), since a fine r e s o l u t i o n spec-trum may be s t a t i s t i c a l l y unstable. T h e o r e t i c a l l y , i f two frequencies are equal, then t h e i r mode shapes cannot be uniquely defined (29). In the case of the Harbour Centre East-West d i r e c t i o n we may have two very c l o s e l y spaced frequencies (the funda-mental and second modes) such that they i n t e r f e r e with each other, giving a sin g l e mode shape which i s not unique. Then the l o g i c a l extension would be that the observed second mode i s a c t u a l l y the t h i r d mode, and so on. • The p a r t i a l moment method for damping c a l c u l a t i o n s does seem to have good s t a b i l i t y for a given length of record, and seems to be l i t t l e affected by the smoothing process. The damping value obtained by t h i s method for the Harbour Center fundamental mode (N-S) i s 1.11% whereas those for the T/D and IBM Towers are 2.64% and 3.35% re s p e c t i v e l y . These low values of damping are t y p i c a l of r e s u l t s determined by ambient v i b r a t i o n t e s t s . For an experimental programme such as t h i s one, stable estimates of power sp e c t r a l density or Fourier spectra can be e a s i l y achieved. Thus the autocorrelation method can be applied to the data and good damping estimates can be obtained. The damping values obtained for the Harbour Centre, T/D and IBM Towers by t h i s method are comparable to the r e s u l t s obtained from the p a r t i a l moment method. As well, the man-excited test damping r e s u l t s for the T/D Tower are also within the same range of values as the p a r t i a l moment and autocorrelation methods. 45. • The " i d e n t i c a l " Gage Towers buildings have very s i m i l a r dynamic proper-t i e s . The respective frequencies are p r a c t i c a l l y the same i n a l l cases, and, f o r the most part, the damping values are s u f f i c i e n t l y close to warrant considering the 3 building as dynamically equivalent. 46. REFERENCES (1) F.G. Udwadia and M.D. Trifunac "Ambient Vibration Tests of F u l l Scale Structures", 1974, 5th WCEE. (2) Sheldon Rubin, "Ambient Vibration Survey of Offshore Platform", ASCE, Eng. Mech., June 1980, Vol. 106. (3) S. Cherry and Brady, "Determination of Structural Damping Properties by S t a t i s t i c a l Analysis of Random Vibrations", Proceedings, 3rd World Conference on Earthqukae Engineering, Vol. I I , 1965. (4) H.S. Ward and R. Crawford, "Wind-induced Vibrations and Building Modes", BSSA, Vol. 56, August 1966. (5) R.R. Blandford, V.R. McLamore and J. Aunon, "Str u c t u r a l Analysis of M i l i k a n Library from Ambient Vibrations" Teledyne Co., February 1968. (6) H.S. Ward, "Dynamic C h a r a c t e r i s t i c s of a Multistorey Concrete B u i l d -ing", NRC, Proceedings, V o l . 43 August 1969, Vol. 45 A p r i l 1970. (7) W.A. Topf and S. Cherry, "Structural Dynamic Properties from Ambient Vibr a t i o n s " , Proceedings, 3rd European Symposium on Earthquake Engineering, Sophia 1970, Pg. 447-457. (8) C.A. Kircher, H.C. Shah, "Ambient Vibration Study of 6 Similar H i r i s e Apartment Buildings", National Science Foundation, Report. No. 14, Stanford University, 1975. (9) J . Petrovski, R.M. Stephen, "Ambient and Forced Vibration Studies of a Multistorey Triangular-shaped Building", ASCE-EMD, Specialty Confer-ence, March 1976. (10) W.A. Dalgliesh and J.H. Rainer, "Measurements of Wind Induced Displacements and Accelerations of a 57-Storey Building i n Toronto, Canada", Proceedings, 3rd Colloquium on I n d u s t r i a l Aerodynamics i n Aachen, Germany, June 1978. (11) V.R. McLamore, "Ambient Vibration Survey of Chesapeake Bridge", Teledyne Geotronics Report, March 1970. (12) J.H. Rainer and A. Vanselst, "Dynamic Properties of Lion's Gate Suspension Bridge", ASCE-EMD Specialty Conference,-< U.C.L.A., March 1976. (13) C. Lanczos, "Applied Analysis", Prentice-Hall Inc., N.J., 1964. (14) C. Lanczos, "Discourse on Fourier Series", Hafner Publishing Co., N.Y., 1966. (15) R.B. Blackman and J.W. Tukey, "The Measurement of Power Spectra", Dover, 1958. 47. (16) J.W. Tukey, "Comments on Jenkins and Parzen Papers", Technometrlcs, 1961. (17) D.E. Hudson, W.O. Keightley and N.N. Nielsen, "A New Method for the Measurement of the Natural Periods of Buildings", BSSA, Vol. 1, Feb. 1964. (18) G.T. Toaka, " D i g i t a l F i l t e r i n g of Ambient Response Data", ASCE-EMD Spec i a l t y Conference, March 1976. (19) R.W. Hamming, " D i g i t a l F i l t e r s " , P r e n t i c e - H a l l , 1977. (20) E.H. Vanmarcke and R.N. Iascone, "Estimation of Dynamic C h a r a c t e r i s t i c s of Deep Ocean Tower Structures", MIT Sea Grant Project, June 1972. (21) P. Durning and D. Engle, "Vibration Instrumentation System Measures an Offshore Platform's Response to Dynamic Loads", ASCE-EMD, Specialty Conference, March 1976. (22) S.O. Rice (1945), "Mathematical Analysis of Random Noise", B e l l System Technical Journal, Part I = V o l . 23, 1944, pg. 282-332, Part II = V o l . 24, pg. 46-156. (23) G.M. Jenkins, "General Considerations i n the Analysis of Spectra", Technometries, May 1961, V o l . 3, No. 2. (24) T.J. Ulrych and T.N. Bishop, "Maximum Entropy Spectral Analysis and Autoregressive Decomposition", Reviews of Geophysics and Space Physics, Feb. 1975, Vol. 13, No. 1. (25) R.A. Wiggins, "Interpolation of D i g i t i z e d Curves", BSSA, Vol. 66, 1976. (26) F.J. Harris, "On the Use of Windows for Harmonic Analysis With the Discrete Fourier Transform", Proceedings, IEEE, Vol. 66, No. 1, Jan, 1978. (27) F. K o l l a r and R.D. Ru s s e l l , "Seismometer Analysis Using an E l e c t r i c current Analog", BSSA, Vol. 56, Dec. 1966, pg. 1193-1205. (28) W.K. Tso and R. Bergmann, "Dynamic Analysis of an Unsymmetrical High Rise Bu i l d i n g " , Canadian Journal of C i v i l Engineering, 3, 107 (1976). (29) L. Meirovitch, "Elements of Vibration Analysis", McGraw-Hill, 1975. 48. V e l o c i t y S e n s i t i v i t y (Hor izonta l ) ' Seismometer & Channel No. 2(SC201-Vel F i l t e r : 0.8-100 Hz) and P h i l i p s Ana-Log 7 Tape Recorder O.I 0.5 I 5 10 50 100 FREQUENCY , Hz F I G . 2 . 1 INSTRUMENT C A L I B R A T I O N CURVE 49. - 0 .4 ' -i; i • i i • 0 0.1 0.2 0 .3 0.4 0 .5 NON-DIMENSIONALIZED VARIABLE NOTE:- Rectangular window i s equivalent to. D i r i c h l e t ' s kerne function with N=5. Only h a l f the frequency response functions are plo t t e d , (non-dimensionalized variable) from 0 to 0.5 because of t h e i r symmetry. FUNCTIONS: AFTER HAHiUNG ( 1 9 ) . OBSERVATION DECK MECHANICAL ROOM,ROOF F I G , A-1 E L E V A T I O N O F H A R B O U R C E N T E R 50. i i i i i 7 3 -iH : n r n ' T i n •• ; i ^.ym™. i-»ii-aaaj . '=—p» i-;::nna 'tfD :bnn;-jjtrT !;-nrici. innn":nnh. im^ 51. /V- S BLBVAJ/ON DYMAMIC MOOB.L. Mass d i s t r i b u t i o n f o r e x c i t a t i o n of N-S d i r e c t i o n . 1 ' In * ^ i \ *M \ MOOS. / MOOS. 7 MOO£. 3 T/'374J Pi./S/r 7>-OHOJ-L a t e r a l mode shapes.(N-S) F I G . 4 . 2 H A R B O U R C E N T E R M A S S D I S T R I B U T I O N A N D MODE S H A P E S ( F R O M R E F . 2 8 ) N 52. ® 4 © J ® STAI R WELL © . ® ELEVATOR SHAFT ©^ © © 1 SHOWS SENSITIVE AXIS OF INSTRUMENT F I G . 4 - 3 P L A N V I E W O F H A R B O U R C E N T E R S H O W I N G S E I S M O M E T E R L O C A T I O N S . 53. V E LE VATO R SHAFT STAIR WELL ^ 3 SEISMOMETER (FLEXURAL 8 TORSIONAL FREQ.) I G , HA SEISMOMETER ~ L 0 £ A T . M ON TYPI GAll"' OFF ICE FLOOR IR CENTRE. ' ' 0 } APPROX. POSITION OF ELEVATOR SHAFT 1 0 1 © 4-5 TYPICAL FLOOR PLAN - T/D TOWER •SHOWING SEISMOMETER LOCATIONS. 55. APPROX. POSITION OF ELEVATOR SHAFT N . 4-6 TYPICAL FLOOR PLAN OF IBM TOWER . SHOWING SEISMOMETER LOCATIONS. FIG. 4 - 7 P L A N O F R O O F O F G A G E R E S I D E N C E TOWERS ( T Y P I C A L ) S H O W I N G S E I S M O M E T E R L O C A T I O N S . FOURIER SPECTRUM RELATIVE AMPLITUDE RELATIVE AMPLITUDE FREQUENCY,Hz 2 3 IE t-U U l a. cc O u. TORSION DOM IN ANT { SIGN A L S U B T R A C T E D ) I' I i I r 3 4 5 6 7 8 9 10 FREQUENCY. Hz 1 1 12 13 ui o 3 a. < ui > < _ J U l 1.0 0.5 F L E X U R A L DOMINANT ( SIGN A L ADDED) I 1(3 2 3 4 5 6 7 8 9 10 II 12 13 FREQUENCY, Hz n'512. FREQUENCY IDENTIFICATION: HARBOUR CENTRE (E-1!) 59. M O D E I 0 . 4 7 H z M O D E 2 1 . 4 7 H z M O D E 3 2 . 19Hz F I G . 5 . 3 ( A ) H A R B O U R C E N T E R MODE S H A P E S ( N O R T H - S O U T H ) F L E X U R A L f ' ™ > D O M I N A N T . 60. MARCH 24,1981 DATA DECEMBER 19,1981 DATA OBSERVATION DECK ROOF \ 1/ FLOOR \ y 27 ^ — \ c o c. O ? 1 C 1 1 Q / / i J 1 7 / / i I 1 5 J / i *-* i / / 1 o 11 / /  1 Q \ 1 \ \ \ \ 1 \ i \ \ \ \ ( i I i • MODE MODE MODE I 2 3 0.46Hz 1.50Hz 1.65Hz F I G . 5 . 3 ( B ) H A R B O U R C E N T E R N O D E S H A P E ( E A S T - W E S T ) F L E X U R A L D O M I N A N T 61. 6 2 . 63. 64. 65. F I G , 5 , 5 ' FN = 1.65 Hz = n a t ' l freq. * J |ooo * uibXCiXQ: UJ _I_JOOO _| U. L U — r - H u. 69. i 1 1 r 6 7 8 9 10 II F R E Q U E N C Y , H z 12 13 F L E X U R A L DOMINANT (SIGNALS ADDED) 1 r 1 r 6 7 8 9 10 II FREQUENCY, Hz 12 13 l . 0 n , TORSIONAL DOMINANT (SIGNAL SUBTRAHTFn) UJ o 3 - J 0 . 5 H 2 < ui "i 1 1 r "I 1 1 1 1 1 1-6 7 8 9 10 II 12 F R E Q U E N C Y , Hz 13 .5.6 FREQUENCY IDENTIFICATION: TORONTO DOMINION TOWER (N-S) 70. Theoret ica l Experi mental FLOOR MODE MODE MODE I 2 3 0.27Hz(Th.) 0.58Hz(Th.) 0.97Hz(Th.) 0.26Hz(Ex.) 0.75Hz(Ex.) F I G . 5 . 7 ( A ) T O R O N T O D O M I N I O N B A N K TOWER MODE S H A P E S ^ _ I 3 C ( N O P J H - S O U T H ) - F L E X U RA L " " ~ \ 7 1 . Theoretical Experimental FLOOR 29 28 27 h25 22 18 16 h i 4-10 1-8 6 h-4-l] A /1 / / // f l I / 1 1 1 ! rf-V l\ 1 1 ll I 1 l \ 1 / 7^  /I 1 1 1 1 1 1 1 / //  j 1 / 1 j i ' // i \ / 1/ \ \ \ \ 1/ \ \ \ ' \ ll 1 \ / \ MODE I 0.22Hz(Th.) 0.24Hz (Ex.) MODE 2 0.48Hz(Th.) 0.68Hz(Ex.) F I G . 5 .7(g) T O R O N T O D O M I N I O N B A N K TOWER MODE S H A P E S C - F L E X U R A L / . MOOE 3 0.77Hz(Th.) 1.41 Hz(Ex.) 72.' F I G . 5 . 7 0 % ) T O R O N T O D O M I N I O N B A N K T O W E R MODE S H A P E " ( E X P E R I M E N T A L ) - T O R S I O N A L . FREQUENCY.Hz I . ' O - i 0 . 5 H TRANSLAT10NAL DOMINANT (SIGNAL ADDED) i 1 1 r 6 7 8 9 10 II FREQUENCY, Hz 12 13 I . O - i 0 . 5 H TORSIONAL DOMINANT (SIGNAL SUBTRACTED) i 1 1 1 r — r 6 7 8 9 10 II FREQUENCY, Hz 12 13 F I G . 5,a FREQUENCY I D E N T I F I C A T I O N : IBM TOWER ( N - S ) 74. Theoretica I Experi mental FLOOR MODE MODE MODE 1 2 3 0.37Hz(Th.) l .09Hz(Th.) l .98Hz(Th.) 0.38Hz(Ex.) l .20Hz(Ex.) ' 2. 14 Hz (Ex.)' F I G . 5 . 9 ( A ) I B M TOWER MODE S H A P E S - - f N O R T H - S O U T H ) - F L E X U R A L . 75. Theoret ical Experimental FLOOR MODE MODE MODE I 2 3 0 .48Hz(Th.) l . 39Hz (Th . ) 2 . 5 IHz (Th . ) 0.42Hz(Ex.) l . 30Hz (Ex . ) F I G . 5 . 3 ( f ) I B M TOWER MODE ' S H A P E S - . ( E A S T - W E S T ) . - F L E X U R A L . 76. — Theoretical Experimental FLOOR MODE MODE I 2 l .37Hz(Th.) 3 .79Hz(Th.) 1.43Hz (Ex.) 77. TIME LAG NUMBER C " „ F I G T . 5..10(A) TORONTO DOMINION T O W E R O E - W ) AUTOCO RRELATI Ol)l-D AMP LN G - E S T I M A T I O N . 7 8 . 1 . 0 4 -FN = 1.20Hz = n a t ' l freq. < on o o _J UJ on. oc o o o i-< — 1.04— 50 i- F I G . 5 , 100 150 2 0 0 250 T IME L A G N U M B E R IBM T Q W H J f E S ^ A U T O C O R R E L A T I O N DAMPING " ESTIMATION. ' " " 3 0 0 0 50 100 150 2 0 0 250 TIME LAG NUMBER F I G , 5 . 1 1 CD) IBM TOWER ( E - W ) (®QMf B A J I O N D A M P I N G E S T I M A T I O N . F I G , 5 . 1 1 ( E ) I B M T O W E R ( T O R S I O N ) A U T O C O R R E L A T I O N D A M P I N G E S T I M A T I O N . 84. APPENDICES A2 A3 A4 A5 B2 85. Appendix A2  NOISE AND SMOOTHING TECHNIQUES A t r i a l run of the ambient v i b r a t i o n survey with the Earthquake Engineering Research Laboratory measurement system (see Chapter 3) was conducted to assess the manpower and time requirements i n the f i e l d . In the process noise problem was encountered. The smoothing technique developed to eliminate the noise contamination during the analysis of the data w i l l serve to i l l u s t r a t e a few important points. It does not matter whether noise i s created by a 200 pound person run-ning past the seismometers, or by the recording instrument generating spe-cious peaks - we observe noise i n our time signals as a t y p i c a l l y nonana-l y t i c phenomenon which may destroy the true s i g n a l . In our case noise i s present i n the form of peaks of varying heights and d i r e c t i o n , superimposed on the smooth course of the physical function we t r y to measure. Figure A2.1 provides an example of a noisy signal recorded during the t r i a l run. The majority of 'positive' needles (those pointing upwards) resemble -the pulses mentioned i n section B2.3, and pulses cannot be conceived as proper functions since they do not possess a n a l y t i c a l properties and w i l l require a l i m i t process to be defined* If we t r y to expand these "needles" i n an i n f i n i t e s e r i e s , the l i m i t process of the series w i l l f a i l : the series w i l l not converge anywhere. The way that the noise a f f e c t s our analysis i s obvious i n F i g . A2.2. The Fourier transform of about 4000 points of the noisy data (both needles 86. and tape noise) i n one block w i l l give us Fourier c o e f f i c i e n t s that w i l l not diminish: the "frequency components" of the peaks are spread out from zero Hz to i n f i n i t y . Besides the possible a l i a s i n g e f f e c t (section B2.7) of those frequency components higher than the Nyquist frequency, leakage (section B2.6) may cause much d i s t o r t i o n i n the spectrum. There are two approaches to deal with t h i s noise problem: smoothing " i n the large" (or global smoothing) and smoothing " i n the small" (neighbourhood smoothing) (See Lanczos (13)): A) Smoothing i n the Large: The Fourier technique i s a powerful t o o l i n t h i s respect. The non-convergent part of the Fourier series represents the nonanalytical nature of the noise. An i d e a l l y random noise would have a Fourier spectrum which has no preference for any frequency and would have an average amplitude with random flu c t u a t i o n s f o r a l l frequencies. For our case we know approximately the maximum frequency that we want for building v i b r a t i o n s , hence we can simply omit any higher frequency i n the s e r i e s . But within the frequency range of i n t e r e s t noise i s s t i l l present; and we do not know what exactly a l i a s i n g does to the s i g n a l . B) Smoothing i n the Small: This neighbourhood technique i s very a t t r a c t i v e to us because: i ) these "g l a r i n g " mistakes are v a s t l y d i f f e r e n t f rom the actual s i g n a l . i i ) they occupy only one d i g i t i z e d point and hence are easy to detect i n the d i g i t i z e d form of the data. Smoothing i n the small i s thus done using a programme with a "detector" to select the needles from the s i g n a l , and an i n t e r p o l a t i o n technique 8 7 . (Wiggins (25)) to generate a new point to replace the needle. The Wiggins' technique makes dual use of a point as i n the Lagrangian i n t e r p o l a t i o n tech-nique, but uses a cubic i n t e r p o l a t i o n function and some extra c r i t e r i a to ensure the new point follows the course of the physical function i n general. The method has been found to give good empirical r e s u l t s . The smoothed t r i a l run signal obtained by t h i s method i s shown i n F i g . A2.3. Though we did not apply smoothing " i n the large" i n t h i s instance, the same concept i s used i n d i g i t a l f i l t e r i n g of a single frequency peak from a spectrum of several modes. Some d e t a i l s of t h i s f i l t e r i n g process i s reported i n section B2.9. There i s also another i n t e r e s t i n g aspect i n F i g . A2.2. The noise spec-trum seems extremely variable and peaky when d i f f e r e n t plots of the noisy data are compared. This turns out to be a well-observed phenomenon (Section B2.8). The v a r i a b i l i t y of the spectrum w i l l not decrease even i f the number of points used i s much more than 4000. To decrease t h i s v a r i a b i l i t y i s one of our main objectives and the f i e l d measurement programmes subsequent to the t r i a l run have been modified to achieve stable r e s u l t s . The source of the noise i n t h i s t r i a l run was found to be l o c a l i z e d i n a recording channel. This channel was not employed i n a l l subsequent measurements. 1.0 u _| !^  ^ M i 0 . 5 H ^ > = < o J U. UJ cr 0.0-0 D 1 0 0 2 0 0 30.0 10.0 SO.O 60.0 70.0 BD.O 90.0 100.0 .1)0.0 110.0 130.0 110.0 ISO.O 160.0 F R E Q U E N C Y , Hz F I G . A 2 . 2 " F O U R I E R ' T R A N S F O R M - O F N O T S V D A T A ( 4 0 0 0 P O I N T S I N O N E B L O C K ) 00 0.9 0.8_ 0.7_ 0.6_ T I M E ( P O I N T S A T . 0 2 5 S E C . . . . I N T E R V A L S ) ' " . . . • F I G . A 2 . 3 ; S M O O T H E D T I M E i F U N C T I O N . 91. Appendix A3 CALIBRATION PROCEDURES AND CHARACTERISTICS OF THE RANGER AND WILLMORE MARK II SEISMOMETERS. K o l l a r and Russell (27) have observed that the electromechanical system of the seismometer can be i d e a l i z e d as an equivalent c i r c u i t ( e l e c t r i c a l analog), and that the measurement made at the output terminals of a Maxwell Bridge, Figure A3.1, can be used to determine the c h a r a c t e r i s t i c s of the seismometer under sinusoidal ground motions i n the same frequency range as the c o n t r o l l e d input voltage. By using the WAVETEK 111 as the input o s c i l l a t o r , known inputs (of suitable incremental frequencies) can be fed into the c i r c u i t through the "MAIN" and "SUBSTITUTE" c i r c u i t s . K o l l a r and Russell define the impedance of the seismometer as Voltage main S Rg Voltage substitute B where — £ — are known qua n t i t i e s , and Voltage main and Voltage substitute are R E measured. P l o t t i n g the values of Zg vs input frequency (as the devices are l i n e a r , input frequency = output frequency with an amplitude change and a phase s h i f t , see section B2.2), we obtain the r e s u l t s shown i n F i g . A3.2: The l i n e a r portion of the curve at the low frequency end i s c a l l e d the low frequency asymptote (from about .15 to .4 Hz i n the f i g u r e ) , while the high frequency asymptote i s s i m i l a r l y defined for the higher frequencies (from about 3 Hz to 6 Hz). The impedance Zg at the point where the exten-sions of the l i n e a r portions meet i s defined as Z . The peak impedance at OA resonance i s defined as Zg^. At the low frequency asymptote: 2 2 ° 1 o 1 w re: 81 i s the seismometer generator constant at the low frequency asymptote w o = natural frequency m = mass of magnet i n seismometer V i s the value of Zg at frequency w1 i n the low frequency asymptote range of frequencies. At the high frequency asymptote: g 2 2 = Z j 1 . ! . where: Z g 1 i s the value of Zg at frequency w1 i n the high frequency asymptote range of frequencies, g 2 i s the seismometer generator constant at the high frequency asymptote. The average of g^, gives the value of 'g', the seismometer generator constant; the natural frequency Wq i s obtained by a geometric mean method ( F i g . A3.2). The damping values 5 for the seismometer i s : 1 zso Z , Z have been defined e a r l i e r and are shown i n F i g . A3.2. OA. u U The calculated values of 'g', 1 £' and 'w ' for a l l four seismometers ° ' o are tabulated i n table A3.1 (for seismometers i n horizontal p o s i -tions) • 93. TABLE A3.1 ^ \ SEISMOMETER ^v. CHARACTER-^ ^ I S T I C S SEISMOMETER^. NO. W ° n Natural frequency Hz Generator constant v/m/ s Internal damping % c r i t i a l 1 1.57 219.8 5.4 2 .90 239.0 7.9 3 .82 231.5 8.1 4 .87 237.0 6.5 The 2 Willmore Mark II seismometers i n the Geophysics system are very sim i l a r and have the following common c h a r a c t e r i s t i c s : Wq = 1 Hz (natural freq.) g = 185.5 v/m/s.(generator constant) £ i n t = ( i n t e r n a l damping of seismometer) = .01 (1% of c r i t i c a l ) Using these values and the e l e c t r i c a l resistance values i n F i g . A3.1, together with the frequency response for the SC201 and the tape recorder as input to the computer programme UNICAL*, the system response curve can be obtained. A t y p i c a l curve representative of a l l four Ranger seismometers i s shown i n F i g . A3.3. (SYSTEM v e l o c i t y response i s defined i n section B2.2) * UBC Department of Geophysics Programme Library O U T P U T V O L T A G E SUBSTITUTION INPUT INPUjT VOLTAGE ( W A V E T E K III) MAIN INPUT R = 30.05 fi R R D =9 .507 kft C n = .178 mF R B = 611 kfi to 611 kJJ R £ = 9.502 kfi R R R = -JLJi (about 2000 fi) C R_ L c - W B R R R B Voltage main . . _ Z = ——— 2 — . (impedance of seismometer) S R E Voltage subst i tu te F i g . A3.1 Diagram of Maxwell Bridge Used i n the C a l i b r a t i o n 95. = 1 I I I I I I I 1 I I I I I I I Zso \ GEOM MEAN \ (0.85 X |.I5)2 =0.9 \ ( 0 . 7 7 X | . 3 0 ) I = 1.0 1 1 1 1 1 1 11 9 | AVG = I.OOHz 0 J J / Zs — 1 I I I I I I I Ranger Se 1 I I I I I I I ismometer # 4 E/Q LAB -1 I I I I I I I .1 0.5 I 5 10 50 FREQUENCY , Hz F I G . A 3 . 2 S E I S M O M E T E R C A L I B R A T I O N . 96. 10 tn \ g \ tn +J i-H O > CO z o a i o 3 or o _i UJ > UJ to | 0 2 10 = 1 I I I I I I I 1 I I I I I I I j,^»o—o-o-o-oo< 1 1 1 1 1 1 M 1 0 0 0—0 — J Seismometer 1 — Hor izonta l , V e l o c i t y F i l t e r , Computer Generated 1 I I I I I I I 1 I I I I I I I 1 I I I I I I I 0.1 0.5 I 5 10 FREQUENCY , Hz 50 100 F I G . A3.3 S Y S T E M V € A L I B R A T I Q M . APPENDIX A4 TYPICAL AMBIENT VIBRATION TRACES 100-mv 10 mm Date: December 19, 1981 C h a r t R a t e : 10 mm/s S e n s i t i v i t y : 100 mv/ d i v i s i o n oo F I G . A 4 ; i ( A ) ; T Y P I C A L A M B I E N T V I B R A T I O N T R A C E S H A R B O U R C E N T R E 10 mm D a t e : A p r i l 11, 1981 C h a r t R a t e : 25 mm/s S e n s i t i v i t y : 100 m v / d i v i s i o n FIG.A 4 . 1 ( B ) T Y P I C A L A M B I E N T V I B R A T I O N T R A C E S T O R O N T O D O M I N I O N TOWER 18 F l o o r "5 B —I—I 1—I—I—I—t—H-+-H—^—^—r—h 0) Pi 18 F l o o r 100 mv T T iiLuiliitiMi I J M i i i H—I—I—I—I—I—H—I—I—I--H—I—I—I—I—I—\—H I M B R U S H A C C U C M A R T -i—I 1 1—I—I 1 M— I— I 1—I 1—I—1—I—I— m I H 1 1 iillli! 10 mm Date : A p r i l 12, 1981 C h a r t R a t e : 25 mm/s S e n s i t i v i t y : 100 m v / d i v i s i o n o o F I G . A M ( • T Y P I C A L A M B I E N T V I B R A T I O N T R A C E S IBM T O W E R Roof Roof • i < i i—t < . —> • < ..(....(-...4..., * t i - i - i I I I- I I I+ I 4 - 4 -I-I-I—-4 • 4-Gouid Inc.. Inatrunwnl Syttvma DMiJon -4—I 4—1 1 - I 4-4 - 4— I—I — 4- +—(•—(-  Cleveland, Ohio Primed in U S A | 4 .-• I—4-4 I-H 4—1—I 1 1 4 4 10 mm Date Chart Rate Se n s i t i v i t y May 21, 1981 5 mm/s 100 mv/division (D) Gage Towers F I G . A 4 . 1 ( D ) T Y P I C A L A M B I E N T V I B R A T I O N T R A C E S G A G E TOWERS 102. APPENDIX A5 MAN-EXCITED TEST RESULTS 100 mv -.1:.: : - T l : : : : ! : : : : ) - : : : i . : • : : : ! : : : • : ! : : : . : -:::":!•--..;:::•!'-.;.::••::::•:;:-•: - : — : HY^i . . . : • itiitifc,: - Itiivii H - . r : ^ "i ^  * : ; — -. r -V : ••: - J \ ;-/ \ -7 \ y \ \ I /\ /\ s\ \\ \ 1 / V / ' A i:::ijit: —— ; H: :J:..:•:;;; : ' ....i— •i -l tr: i i'i i iii• ii ;i :tii.tp ;i ;!:ti! : ; ;it:ii" ••• • 1 ! . . . d:ti = . . : . : + T : - . :-„-V- : : j : r : : J - - : : i ' : . . -_.:1-:T-:: i I r r : • • : - , • - • : . : . : • • : . • : : . - mm : ••I :ii; 7 ~ : : : ; T i ^ . ; i : : ; : | 7 - : i ' ; : ; : i r - ~ q : : : ; i - ; : : . ; . ; . : . (C) N-S Second F l e x u r a l Mode. 2nd mode freq. •= 0.74 Hz Damping = 1.5% c r i t i c a l . F I G . A 5 . 1 M A N - E X C I T E D T E S T R E S U L T S - T / D TOWER ( A P R I L 1 1 , 8 1 ) 104. APPENDIX B2 DETAILED BACKGROUND THEORY FOR THE AMBIENT VIBRATION SURVEY B2.1 Introduction This appendix provides some mathematical background f o r : i ) the c a l i b r a t i o n of the measurement system used i n the ambient v i b r a t i o n survey, i i ) the planning of the measurement procedures used i n the f i e l d , and, i i i ) the subsequent analyses and i n t e r p r e t a t i o n of the acquired data. Much of the discussion has been heavily drawn and reproduced from C. Lanczos' 'Applied Analysis' (13) and 'Discourse on Fourier Series' (14), and also Blackman and Tukey's 'The Measurement of Power Spectra' (15); quota- tions from these sources w i l l not be separately referenced i n t h i s chapter. The discussions are intended for a basic understanding of the problems encountered; hence the approach w i l l not n e c e s s a r i l y be mathematically rigorous. B2.2 The Measurement System and the Frequency Response Analysis The measurement system used to acquire data consists of sensors ( s e i s -mometers), a signal conditioner and a tape recorder. Some d e t a i l s of the functional parts of the system were discussed i n Chapter 3. We are only concerned here with the general c h a r a c t e r i s t i c s which are common to mecha-nisms i n communication engineering. We regard the system as a 'black box' of unknown structure. This black box has two ends: the input and the output. We make the following assump-tions f o r the black box: 105. i ) It Is l i n e a r : a) With an a r b i t r a r y input x ( t ) , the output i s y ( t ) . When the input i s otx(t), where a i s a constant, the output i s oty(t). b) The superposition p r i n c i p l e holds; the simultaneous a p p l i c a t i o n of two inputs generates the sum of the corresponding outputs without any mutual interference. i i ) The black box has unknown components that do not change with time; that i s , t h e i r physical c h a r a c t e r i s t i c s are time i n v a r i a n t . i i i ) The output follows the input. iv) The black box di s s i p a t e s energy. A known l i n e a r d i f f e r e n t i a l equation r e l a t i n g the input x(t) and the output y(t) has the form: ,n ,n-l d y . d y , a — — + a , r + a y = n , n n-1 , n-1 o^ dt dt b fLx + b _! i L _ 2 L + b x r d t r r d t ^ 1 We note that the f i r s t two assumptions are inherent i n t h i s equation such that 4^ w i l l correspond to -—, — ^ w i l l correspond to e t c . . d t d t d t 2 d t * so that the superposition p r i n c i p l e applies; also the a^, b^, are a l l constants instead of being functions of time, showing the time inde-pendence of the physical c h a r a c t e r i s t i c s of the black box. There i s no simple experimental method to determine the various constants a's and b's. So instead, the input and the output of the system are observed, and the r a t i o output . . -—£L—- is taken, input 1 0 6 . This r a t i o defines the transfer function of the system (black box). We can regard a generalized physical input s i g n a l as a superposition of s t r i c t l y periodic components. This i s possible because of the Fourier tech-nique: an a r b i t r a r y function defined i n a f i n i t e i n t e r v a l can always be resolved into a sum of pure sine and cosine functions. With the fundamental assumptions that the system i s l i n e a r and unchanging with time, the output w i l l then be a modified version of the sum of these components. If we have an input harmonic function x(t) i n the complex form: iwt x(t) = cos wt + i s i n wt = e where w i s an a r b i t r a r y e x c i t a t i o n frequency, the output w i l l be: y(t) = F(w) e i w t 2.2.1 where F(w) i s the complex frequency response for the system at frequency w. F(w) i s a complex number having the form: F(w) = A(w) e - i 9 ( w ) where A(w) i s an amplitude and 0(w) ,is the phase s h i f t , both of which are functions of w. whence, / - N » y v i (wt - 6(w)) y(t) = A(w) e and y(t) i s the response of the black box at frequency w. We note that the iwt response to the periodic input e i s a periodic output of the same f r e -quency w but with a modified amplitude A(w) and a phase s h i f t 9(w). As a simple i l l u s t r a t i o n of t h i s procedure consider the equation of motion for a l i n e a r system composed of a l i n e a r spring with s t i f f n e s s k and a l i n e a r viscous damper of c o e f f i c i e n t C. If the input e x c i t a t i o n i s X(t) = iwt X^e , where X q i s the amplitude of X ( t ) , t h i s becomes Cy(t) + ky(t) = X(t) 2.2.1(a) 106. This r a t i o defines the transfer function of the system (black box). We can regard a generalized physical input signal as a superposition of s t r i c t l y periodic components. This i s possible because of the Fourier tech-nique: an a r b i t r a r y function defined i n a f i n i t e i n t e r v a l can always be resolved into a sum of pure sine and cosine functions. With the fundamental assumptions that the system i n l i n e a r and unchanging with time, the output w i l l then be a modified version of the sum of these components. If we have an input harmonic function x(t) i n the complex form: iwt x(t) = cos wt + i s i n wt = e where w i s an a r b i t r a r y e x c i t a t i o n frequency, the output w i l l be: y(t) = F(w) e i w t 2.2.1 where F(w) i s the complex frequency response for the system at frequency w. F(w) i s a complex number having the form: F(w) = A(w) e " i 8 ( w ) where A(w) i s an amplitude and 0(w) i s the phase s h i f t , both of which are functions of w. whence, ^ v . i (wt - 6(w)) y(t) = A(w) e v v " and y(t) i s the response of the black box at frequency w. We note that the iwt response to the periodic input e i s a periodic output of the same f r e -quency w but with a modified amplitude A(w) and a phase s h i f t 6(w). As a simple i l l u s t r a t i o n of t h i s procedure consider the equation of motion for a l i n e a r system composed of a l i n e a r spring with s t i f f n e s s k and a l i n e a r viscous damper of c o e f f i c i e n t C. If the input e x c i t a t i o n i s X(t) = iwt X^e , where X q i s the amplitude of X ( t ) , t h i s becomes Cy(t) + ky(t) = X(t) 2.2.1(a) 107. where y(t) i s the response of the l i n e a r system. From equation 2.2.1 above y(t) = F ( w ) X o e i W t 2.2.1(b) Substituting 2.2.1(b) into 2.2.1(a) , . . , v x iwt iwt (ciw + k) F(w)e = e F(w) = 1 k+icw The amplitude of F(w) i s A(w): A(w) = |F(w)| = 2.2.1(c) ( k * f C 2 w 2 ) 1 / 2 0(w) = t a n " 1 ^ ) 2.2.1(d) and the phase s h i f t 9(w) i s : and F(w) = A ( w ) e ~ i 0 ( w ) with A(w) and 0(w) given by 2.2.1(c) and 2.2.1(d) res p e c t i v e l y . This frequency response approach to the analysis of a 'black box' sys-tem i s e s p e c i a l l y suitable for the laboratory environment, i n which we can c o n t r o l the periodic input to the system and observe the output. If by comparing the output to the input we obtain a transfer function A ( w ) e " i G ( w ) with: A(w) = constant, independent of w 9(w) = constant, m u l t i p l i e d by w = a w f o r a large range of frequency w, we then have a system that has high-fide-l i t y reproduction of the input. This i s also s u f f i c i e n t proof that our system i s l i n e a r , since an harmonic input remains an harmonic output with unchanged frequency, although with modified amplitude and s h i f t e d phase. The above discussion forms the basis of the laboratory c a l i b r a t i o n of the measurement systems (for d e t a i l s r e fer to Chapter 3). A t y p i c a l system v e l o c i t y response curve ( F i g . 2.1), which defines the system's response at various frequencies to v e l o c i t y input transmitted by the sensors at the same 109. the system w i l l then be a superposition of pulse responses. This process i s very s i m i l a r to the e l e c t r i c a l engineers' technique of using the Heaviside Unit Step Function as a b u i l d i n g block to construct input s i g n a l s . Yet the pulse response method i s more v e r s a t i l e , since i t does not presuppose the d i f f e r e n t i a b i l i t y of the input s i g n a l as does Heaviside's method. The d i s -advantage i s that the pulse cannot be conceived as a legitimate function i n the proper sense, since we have to impose a l i m i t process for i t s d e f i n i t i o n (see further discussion of t h i s point i n Appendix A2). The pulse response method for a system proceeds as follows: Let the input signal be x ( t ) , and the response of the system (which i s a c h a r a c t e r i s t i c of that system) be k ( t ) . Then the response y(t) at t = 0 i s : 0 y(0) = / x ( T ) k ( T ) d T — oo Integration l i m i t s i n d i c a t e that the output follows the input. Since the physical c h a r a c t e r i s t i c s of the system are time i n v a r i a n t , f o r the same phenomenon at any l a t e r time moment t, the input w i l l be x(t+x), and the response y ( t ) : 0 y(t) = / x(t + x)k(x)dT — 0 0 introduce the new v a r i a b l e t + x = K t y(t) - / x ( 0 k(£rt) d£ 2.3.1 — 0 0 If we l e t the input function be the unit impulse at time moment 5=0, we have y(t) = k(-t) 2.3.2 k(t) becomes the pulse response of the system. Let H(t) be the response of the system at time t to the unit pulse input applied at t = 0, then from 110. k(t) = H(-t) 2.3.3 But H(t) i s zero for a l l negative time moments since the output follows the input, we have f or a l l p o s i t i v e t: k( t ) =0, ( t > 0) 2.3.4 With 2.3.4 we can extend the upper l i m i t of 2.3.1 to i n f i n i t y : oo y(t) = / x ( Q k(£ " t) dC 2.3.5 —CO In the general case requirement 2.3.4 i s not necessary, and making use of 2.3.3 we have: CO y(t) = / H(t - Z)x(Z)dZ 2.3.6. —oo 2.3.6 i s known as the convolution i n t e g r a l of functions H(Q and x( £) . B2.4 The Fourier Transform, Relationship Between Pulse and Frequency Responses, Convolution and M u l t i p l i c a t i o n So far two d i f f e r e n t approaches have been used (according to two d i f f e r e n t philosophies) to obtain the responses of the same system with the same set of assumptions. I n t u i t i v e l y we know that the pulse and frequency responses are r e l a t e d . To be able to compare these responses, we have to convert either the pulse response to the frequency domain, or the frequency response to the time domain. The Fourier transform i s the l i n k between the two domains. A s l i g h t l y modified version of the Fourier i n t e g r a l (equation 2.2.2) defines the Fourier transform: oo X(w) = — / x ( t ) e " i w t d t 2.4.1 and the inverse transform: 00 x(t) = — — / X(w)e 1 W tdw 2.4.2 These two operations have remarkable r e c i p r o c i t y . 111. Assume that the x(t) i n section 2.3 can be synthesized from i t s periodic components, from 2.4.2: oo x(t) = ) X(w)e dw We can consider that x(t) can be modelled as the superposition of a continu-ous set of sinusoids having frequencies of d i f f e r e n t w's and amplitudes of 1 iwt X(w)dw. F i r s t we look at the input of e i n the i n t e g r a l 2.3.5: /2TT iwt iwx (Changing e to e for the integration) oo y(t) = / k(x - t ) e i W X d x — CO l e t 5 = x - t y ( t ) = / " I c C Q e ^ ^ d S — 00 oo iwt t w „ iwC,, = e J k( 5)e dK — CO The i n t e g r a l i s just the complex conjugate of the Fourier Transform of k( £), the pulse response, m u l t i p l i e d by V^2"TT. It i s a complex number K*(w): K*(w) = / k ( Q e i W ? d £ — 00 and can be expressed i n terms of an amplitude A(w) and phase e ^ ^ w ) ' y(t) = /2ri K*(w)e ± W t v -i6(w) iwt = A(w)e v 'e. . i(wt - 6(w)) = A(w)e y(t) i s the response i n section B2.2, and A(w)e ^^^ w^ i s just F(w) i n equa-tion 2.2.1. Therefore the complex frequency response F(w) i n section B2.2 i s just the complex conjugate of the Fourier Transform of i t s pulse response (section B2.3), m u l t i p l i e d by /2ir. There i s yet another important aspect of the convolution i n t e g r a l 112. 00 y(t) = / H(t - Qx(?)d5 — 0  Taking the Fourier Transform of y(t) gives: CO Y(w) = — — / y ( t ) e " i w t d t 2.4.3 where Y(w) i s the Fourier Transform of y ( t ) . Substituting 2.4.3 into 2.3.6, using 2.4.1, 2.4.2 0  0  Y(w) = J / H(t - 5 ) x U ) e " i W t d t d5 y/2~TT - o o -oo = f X ( 0 e i w 5 [ J L _ f H(t - Q e _ l w ( t " 5 ) d t ] d ? V/2-TT -0° VTTV -OO = X(w) H'(w) 2.4.4 0  where H'(w) = [-i— / H(t - £ ) e ~ i w ( t ~ 5 ) d t ] •PLTI -oo and i s the Fourier Transform of H(t - £). We thus have the important r e s u l t that the Fourier Transform of a convolution of two functions (as i n equation 2.3.6) i s the product of t h e i r Fourier Transforms. By the remarkable r e c i p r o c i t y of equations 2.4.1 and 2.4.2, the same rule applies for a convolution i n the frequency domain. Because of the s i g n i f i c a n c e of the correspondence of m u l t i p l i c a t i o n i n one domain to convolution i n the other domain, i t w i l l be h e l p f u l i f we can v i s u a l i z e the two processes. A n t i c i p a t i n g the next sections, we focus on d i s c r e t i z e d functions. A product of two d i s c r e t i z e d functions i s just the point by point m u l t i p l i c a t i o n of the two "overlapping" functions. A convolution, on the other hand, can be v i s u a l i z e d i n 3 ways as suggested by Blackman and Tukey (15). The author prefers the moving weight function concept of weighted in t e g r a t i o n : one of the functions i n the convolution may be regarded as a moving weight, and integration i s performed as the weighting function moves point by point over the second function. The r o l e s of the functions may be 113. interchanged. A p i c t o r i a l representation of the moving weight i n t e g r a t i o n process can be found i n Hamming's book (19). B2.5 D i s c r e t i z e d Functions and the Discrete Fourier Transform The discussions i n the previous sections have focused on analog or continuous functions, and the Fourier i n t e g r a l i s the l i m i t of the Fourier series for the case when the base time of analysis increases to i n f i n i t y . Though the Fourier i n t e g r a l i s more general than the Fourier series (which demands s t r i c t p e r i o d i c i t y of the function), we have to turn to the use of the Fourier s e r i e s , since the computer only works with d i s c r e t i z e d functions and the continuous "density" function that the Fourier i n t e g r a l produces i s incompatible with d i g i t a l equipment. The process of obtaining the Fourier series as a l i m i t i n g case of the Fourier i n t e g r a l w i l l help us understand the purpose of d i g i t a l smoothing. Let us st a r t with a periodic time function with period 2ir f ( t + 2TT) = f ( t ) This function e x i s t s between the large l i m i t s of t = -2irN and t = 2TT(N+1). The function i s repeated 2N+1 times as N tends to i n f i n i t y . The Fourier analysis gives the Fourier spectrum F(w) (equation 2.2.3): 2TT(N+1) F(w) = / f ( t ) e 1 W C dt -2ITN However, because of the p e r i o d i c i t y , we need integrate over only one period, and s h i f t the periodic function to account for the contributions i n the 2TT f ( t ) e dt and o s h i f t i n g g(w) to the next period w i l l give e ^ 7 W^g(w) and so on, * r -2irNwi , -2n(N+l)wi , 2irNwi n , . .'. F(w) = [e + e v ' + e Jg(w) 114. The terms i n the square bracket i s a geometric progression which can be replaced by the i d e n t i t y : s i n TT(2N+1) W s i n IT w (2.5.1) . x r s i n -ir(2N+l)w>! . . and F(w) = I . m — — Jg(w) s m TIW If N i s large, the f i r s t factor puts a very strong weight on integers of w: w = 0, ±1, ±2 At these frequencies (w = k) the continuous spectrum changes more and more to a l i n e spectrum. But the change i s never quite complete since the l i n e spectrum has a f i n i t e width of k+e and k-e where e i s small. The factor (2.5.1) i s a highly o s c i l l a t o r y function, and i f we evaluate F(w) at w between k+e and k-e, for small e, |- K + £ TI f v . /, N f £ s i n ir(2N+l)w , J F(w)dw - g(k) J ^ dw k-e -e 2 „ . r £ s i n n(2N+l)w , = — g(k) J — dw TT J W O = g(k) | Si(-) where Si(°°) i s the sine i n t e g r a l of TT(2N+1)W. AS N becomes large, Si(«°) = 2' k+e .-. / F(w)dw = g(k) (2.5.2) k-e The continuous spectrum i s thus reduced to the integer frequencies at w=k, and the g(k) values give the c o e f f i c i e n t s of the complex Fourier s e r i e s C . Since the complex Fourier series i s : f(t) = I C ke k=-°° ikivt Ck = YW^ f ( t ) e " i k l T t d t = ^g(w) and ic z II • — IT for our case from (k-e) to (k+e) we have, for equation 2.5.2, ck = T ^ ( k ) It i s the Ck that we t r y to evaluate. 115. There are two important aspects concerning the 'kernel' function 2.5.1, namely: i ) The continuous spectrum w i l l be changed to a l i n e spectrum with width of 2e i n the process of the Discrete Fourier Transform. e w i l l decrease i f N increases, increasing the focus on the frequency w=k; hence the function f ( t ) i n the i n t e r v a l [0,2ir] w i l l have to be repeated many times to the l e f t and r i g h t as N becomes l a r g e r . This i s a source of problem, since the time function f ( t ) may not be periodic and repeating i t many times w i l l cause d i s c o n t i n u i t i e s to occur at the junctions of the r e p e t i t i o n . N i k t Gibbs o s c i l l a t i o n s w i l l cause the series £ C, e to fluctuate so that the k=-N K series may not converge to f ( t ) at any point. Thus the corresponding raw (or unsmoothed) spectrum w i l l be of l i t t l e value and the subsequent analysis w i l l be misleading. i i ) The kernel function w i l l determine the focusing power and the amount of 'leakage' that r e s u l t s from i t s convolution with the true spec-trum. It i s therefore very important that we take a closer look at these functions. B2.6 D i r i c h l e t ' s Kernel, Windows, Smoothing D i r i c h l e t investigated the v a l i d i t y of the Fourier expansion for a wide cl a s s of functions, and examined the f i n i t e series of n terms: fn(x) = y aQ + a i c o s x + a n c o s n x + b^ s i n x + bn s i n nx at a fixed point x i n the series by s u b s t i t u t i n g the i n t e g r a l s for c o e f f i c i e n t s a, and b. : 116. a^=—^j f ( t ) cos kt dt variable t i s to d i s t i n g u i s h - i r i t from the fixed point x i n the s e r i e s . b, = - / f ( t ) s i n kt dt k IT -IT By using trigonometric i d e n t i t i e s and geometrical series ( f o r d e t a i l s r e f e r to Lanczos (14) pg. 129-131) we can form fn (x) = i / f ( t ) Cn(t - x) dt - ir where Cn(t - x) i s a f i n i t e s e r i e s of n terms. Cn(5 ) i s the ' D i r i c h l e t Kernel' and has the form rn( r> - s i n (n + 1/2) g , , C N ( Q " ZTT s i n (1/2 £) 2 ' 6 a As n increases to i n f i n i t y , fn(x) should approach f(x) with an error which can be made a r b i t r a r i l y small. This requires a strong focusing power of Cn(£) i n the immediate neighbourhood of £ = 0, such that everything else i s blotted out. The requirements for the performance of the kernel function (which i s even) are: IT i ) l im / /Cn(0/dC = 0 which guarantees that the kernel function blots out everything except i n the immediate neighbourhood of t = x (£ = 0) + e i i ) l im / Cn( g)d£ = 1 which gives the proper weight to the function f ( t ) at t = x for the integra-t i o n . These two requirements of the kernel function resemble those f or a unit pulse or delta function except that for a unit pulse there i s an extra l i m i t i n g process which requires the width 2e to approach zero. 117. Note the strong resemblance of the kernel of ( 2 . 5 . 1 ) to the D i r i c h l e t ' s kernel ( 2 . 6 . 1 ) ; the only difference l i e s i n the factor of 2IT i n the sine i n t e g r a l . The D i r i c h l e t kernel does not have the factor i n the sine func-t i o n of Cn(£), whereas 2IT i s present i n ( 2 . 5 . 1 ) : s i n 2TT(N + 1 / 2 )w w si n ("2') 2 IT This apparent discrepancy l i e s i n the d i f f e r e n t d e r i v a t i o n s : the D i r i c h l e t kernel was obtained from the Fourier series a n a l y s i s , which operates i n the fundamental i n t e r v a l of -IT to TT (or -SL to I) but the kernel ( 2 . 5 . 1 ) was a r e s u l t of the continuous Fourier spectrum becoming a l i n e spectrum, assuming that the o r i g i n a l time function i n the fundamental period ( 0 to 2tr) was repeated N times. Since the kernel function recurrs constantly i n harmonic a n a l y s i s , a p i c t o r i a l representation of the D i r i c h l e t ' s kernel ( 2 . 6 . 1 ) w i l l show the e f f e c t of i t s operation. ( F i g . 2 . 6 . 1 ) . In the fi g u r e , the "unmodified rectangular window" i s equivalent to D i r i c h l e t ' s kernel for N = 5 . The function i s symmetric, so only h a l f the kernel i s plotted. In section B 2 . 5 we found that the focusing power i s related to the width of the main lobe of t h i s kernel; also the o s c i l l a t i o n s beyond the main lobe are not small enough to ensure the predominance of the frequency at £ = 0 . The i n s u f f i c i e n t focusing power of the kernel w i l l mean that we can apply i t only to the class of functions which are s u f f i c i e n t l y smooth, otherwise equation 2 . 5 . 2 w i l l f a i l to apply. If the f i n i t e time series that we analyse i s non-periodic and the di s c r e t e Fourier transform regards i t as periodic (as described i n section B 2 . 5 ) , the d i s c o n t i n u i t i e s thus a r i s i n g w i l l cause f a i l u r e of the uniform convergence of the Fourier s e r i e s , and we have inaccurate r e s u l t s . 118. The following i s a convenient physical i n t e r p r e t a t i o n of the r o l e of the kernel function. Suppose the f i n i t e time seri e s that we sampled i s a product of two functions: the i n f i n i t e s i g n a l that e x i s t s i n the f i e l d , and a rectangular function that has a form of 0,0,0,1,1 1,0,0,0 (there are (2N + 1) non zero terms for the (2N + 1) points that we have i n the d i s c r e t i z e d time s e r i e s ) . A function h(w) has these (2N + 1) non zero c o e f f i c i e n t s i n the frequency domain: k=N » V ikw -iNw , -i[N-l]w , iNw h(w) =2. e = e + e 1 J . . . .+e k=-N We sum t h i s geometric progression: i(N+l /2)w _ -i(N+l /2)w h ( w ) = I w 7 2 = fw72 e - e h(w) = (s"'"n ( N +-*-/^) w ) f o r w within the fundamental period s i n | and we obtain the f a m i l i a r kernel. For our purpose, i n the study of bui l d i n g v i b r a t i o n s , we choose to do the analysis i n the frequency domain. Thus the o r i g i n a l m u l t i p l i c a t i o n i n the time domain of the rectangular function and the time s i g n a l now becomes a convolution i n the frequency domain of the kernel function and the Fourier transform of bu i l d i n g v i b r a t i o n s i g n a l . The r e s u l t i s l i k e l y to be peaky because of the various modes present; hence the requirement of a  smooth function i s not met (see section B2.5). We have the moving weight inte g r a t i o n of the convolution giving us a f a l s e representation of the spec-trum, because the kernel function does not focus properly and the secondary maxima of the kernel, when m u l t i p l i e d by the other peaks present and i n t e -119. grated, w i l l contribute to the estimate of the Fourier c o e f f i c i e n t s i n the r e s u l t i n g spectrum. This i s the unfortunate 'leakage' problem. The kernel of the rectangular function — or window, a f t e r Blackman and Tukey — i s c e r t a i n l y not suitable for frequency analysis of bu i l d i n g v i b r a -t i o n s . Various smoothing techniques have been proposed to achieve the two requirements set out i n section B2.5. Among these techniques are the Hanning and Hamming windows, which can be applied either as a m u l t i p l i c a t i o n i n the time domain or a convolution i n the frequency domain. Some windows may increase the focusing power of the main lobe; others may suppress the height of the wide o s c i l l a t i o n s thus reducing leakage. The choice of the window w i l l depend on the s i t u a t i o n . A discussion on the actual choice of the windows used i n the data ana-l y s i s of the buildings measured, for the purpose of close frequency separa-t i o n , was given i n Chapter 5. We are now i n a p o s i t i o n to see why high f i d e l i t y reproduction of the input signa l (see section B2.2) may not be good for us. For bu i l d i n g v i b r a -tions the fundamental frequency usually i s the most dominant, and a high-f i d e l i t y reproduction of the signal and a subsequent convolution with the kernel function may render the r e s u l t i n g spectrum erroneous due to leakage. Thus the idea of f i l t e r i n g the data before analyses are performed i s suggested by Blackman and Tukey (prewhitening). We see from Fig.2.1 that the system frequency response of our equipment suppresses the amplitude of the frequency i n the range which usually includes the fundamental frequency of buildings. This i s e s s e n t i a l l y a f i l t e r i n g process and may help to enhance the detection of higher modes and weaker signals than the fundamen-t a l . 1 2 0 . B2.7 The Sampling Theorems The mathematics of the sampling theorem have been well documented; good presentations of the theory can be found i n Blackman and Tukey (15) and R.W. Hamming (19). Some r e s u l t s relevant to analyses that follow are quoted here: i ) A band-limited function extending from t = -« to 0 0 i s sampled at equally spaced points with a spacing At such that at l e a s t two samples per cycle occur i n the highest frequency present. The frequency f correspond-ing to i s c a l l e d the Nyquist or f o l d i n g frequency. If the band-limited function i s sampled at lower than twice the f o l d i n g frequency, i t w i l l cause the higher frequencies present i n the function to f o l d back, causing the problem of a l i a s i n g . i i ) At the fo l d i n g frequency i t s e l f , information cannot be transmitted because the sine function w i l l become i d e n t i c a l l y zero, (see Hamming, (19)) We have to sample at more than two samples per cycle for the highest frequency present so as to preserve the information of the o r i g i n a l function at the fo l d i n g frequency. B2.8 S t a t i s t i c a l Analysis of Spectra and the Idea of S t a b i l i t y There i s a very important aspect of the analysis i n the Ambient Vibra-t i o n Survey that we have yet to deal with. This i s the randomness of the data. Since the measurements are of random sig n a l s , we should regard each sample we measure as one r e a l i z a t i o n of a population (process) governed by 121. an unknown s t a t i s t i c a l d i s t r i b u t i o n . These observed values, X^, have a mean value E(X f c) = u, assumed to be a constant (usually zero), and a variance a 2: 0  a 2 = E(X -y) = / (X -y)p(X )dX — 00 where p(X f c) i s the p r o b a b i l i t y density function of the e r r o r s . If the pro-b a b i l i t y d i s t r i b u t i o n of Xfc i s Gaussian, then p and a 2 characterize the d i s t r i b u t i o n completely. However, the Xfc we measure are also funtions of time: X t = X ( t ) . Then consecutive values of X(t) are correlated. To account f o r t h i s corre-l a t i o n , we have to assume further that the random series i s stationary, so that the number of c a l c u l a t i o n s to be done to obtain the autocovariances may be rendered r e a l i s t i c . S t a t i o n a r i t y means that a l l s t a t i s t i c a l properties depend on the time differences (or lags) rather than on the d i f f e r e n t time instants. Thus a p a r t i c u l a r autocovariance y may look l i k e : \ = E { ( X i " ^ X i + k " ^ E i s the expected value operator. Then the values of y^, with \i and a 2 w i l l determine the Gaussian d i s t r i b u -t i o n completely. A stationary random process with zero mean can be represented by (Rice ( 2 2 ) ) : n X(t) = I Ai Sin (wit + <|>i) i = l where Ai = random amplitude <j)i = random phase angle th wi = frequency of the i contributing sinusoid. successive values of A i are mutually independent and have zero mean. The mean square of Ai i s the variance A i 2 . The mean square (or variance) of 122. X(t) i s o 2 = I A i 2 i = l A function G(w) whose value at wi i s equal to Ai 2/Aw, for wi at equal i n t e r v a l s of Aw, i s : G(w)Aw = A i 2 As the number of sinusoids i n the random process becomes large, the variance w i l l become equal to the area under the continuous function G(w), which i s known as the power sp e c t r a l density function. G(w) expresses the r e l a t i v e importance of sinusoids with frequencies within some s p e c i f i e d frequency band. The area under the continuous curve G(w) i s the variance, which, together with G(w), characterizes a stationary Gaussian process. Hence there must be a r e l a t i o n between G(w) and ( i n equation 2.8.1). They are Fourier Transforms of one another: CO G(w) = \ I Y k cos kw dk 2.8.2 — co If we normalize both sides of 2.8.2 by a 2, G(w) 1 r \ . ,. —-—- = — I — cos kw dk a — °° a rG(w) •> [ • J w i l l determine the d i s t r i b u t i o n completely and w i l l r e f l e c t the basic o 2 variables i n the s t a t i s t i c a l d i s t r i b u t i o n of a stationary zero mean Gaussian process. There i s one major di f f e r e n c e between harmonic analysis and the s t a t i s t i c a l approach (Jenkins (23)). Consider a f i n i t e series representation of the sample autocovariance C^ ( c f . i n 2.8.1): ^ n-k C^ = — £ X t X t + k (for a zero mean process) A Fourier Transform of n C^'s w i l l give the 'raw' spectral estimate ( c f . G(w) i n 2.8.2) In (Wj): 123. where 2.8.3 It would seem that In(w^) w i l l approximate G(w) i n 2.8.2 as n increases, but i t i s not so. It has been shown that the raw estiamte In(w.) J w i l l fluctuate about G(w) at w^  (see Jenkins (23)), which means that no matter how long a sample we take i n the time domain, though the r e s u l t i n g f l uctuations of In(w^) about G(w^) does not decrease to zero. This i s a well observed phenomenon. Harmonic analysis of random numbers (as an a r t i f i c i a l source of white noise) produces a highly v a r i a b l e , spiked spectrum. (An i n t e r e s t i n g physical i l l u s t r a t i o n of t h i s phenomenon i s given i n the Appendix A2 on Noise a n a l y s i s ) . It i s well known that for a Gaussian d i s t r i b u t i o n with zero mean and unit variance, the independent variables i n t h i s d i s t r i b u t i o n , y^, y^y y 3 y k ° b e y which represents a chi-square d i s t r i b u t i o n with k degrees of freedom. As k increases, X^ becomes r e l a t i v e l y l e s s v a r i a b l e . A convenient d e s c r i p t i o n of the s t a b i l i t y of any p o s i t i v e or nearly p o s i t i v e s p e c t r a l estimate i s i t s equivalent number of chi-square degrees of freedom. Tables are a v a i l a b l e to give the r a t i o of i n d i v i d u a l s p e c t r a l value to i t s average value exceed-ed, with given p r o b a b i l i t y , for d i f f e r e n t values of k - the chi-square degree of freedom. For the In(w_.) mentioned above, for large n the d i s t r i -bution of In(Wj) i s a multiple of chi-square d i s t r i b u t i o n with k = 2, independent of the actual s i z e of n. This means that In(w^) i s so v a r i a b l e mean periodogram does tend to the spectral density, the variance of the X 2 = v 2 + v 2 + k y l y2 3 124. that i n no s t a t i s t i c a l sense does In(w^) converge to G(w^), no matter how large n i s . A modified procedure of harmonic analysis can lead to s t a b i l i z e d spec-t r a l estimates. This i s the simple device of s p l i t t i n g the n terms of the sample X(t) into p sets of m terms so that n = pm. By conducting a Fourier analysis for each set and by averaging these r e s u l t s at each frequency, we have We can make the variance of Im(w^) as small as we wish. This process w i l l increase the equivalent chi-square degrees of freedom and thus w i l l increase the s t a b i l i t y , but increased s t a b i l i t y i s attained by s a c r i f i c i n g the frequency r e s o l u t i o n : a small number 'm' w i l l decrease the frequency r e s o l u t i o n i n each subseries. For our ambient v i b r a t i o n a n a l y s i s , we w i l l have to analyse the data using a reasonably fi n e frequency r e s o l u t i o n , (about .004 Hz i n one case) while using a stable estimate of the spectra from which the damping values are obtained. Tukey (16) suggests that for applications s i m i l a r to the determination of s t r u c t u r a l properties a very stable estimate i s necessary. These requirements w i l l mean that both 'm' and 'p' are large and a rather lengthy record has to be taken i n the f i e l d . It i s possible that long time records may v i o l a t e the s t a t i o n a r i t y assumption (Toaka (18)), but Tukey (16) recommends that the analyses of long records be judged on t h e i r own merit. They often produce very useful average spectra. An empirical v e r i f i c a t i o n of the usefulness of analyzing long records for b uilding v i b r a t i o n has been done by Kircher and Shah (8). The r e s u l t was that a f t e r 10 averagings the average spectra were stable and did not change upon more averaging. A c o r r e l a t i o n method was employed to study the r=l lm,r(w.) 125. change i n the values of averaged spectra a f t e r 10 averagings, and i t was found that excellent s t a b i l i t y was achieved. The p r i n c i p l e of averaging spectra by using long records (from 30 minutes to 40 minutes to provide data for at least 10 averages) was adopted i n t h i s study. The use of 2 separate measurement systems i n the f i e l d (Chapter 4) i s based on t h i s p r i n c i p l e . B2.9 Damping Estimation Methods Two methods have been employed to evaluate the percentage of c r i t i c a l damping of the test structures: the autocorrelation and the p a r t i a l s p e c t r a l moment methods. a) The Autocorrelation method: The t h e o r e t i c a l background for t h i s method (Cherry and Brady (3)) can be summarized as follows: For a l i g h t l y damped sing l e degree of freedom o s c i l l a t o r having a natural frequency w and a small damping r a t i o £ the system response function i s h(x): - Ejwx h(x) = 6 s i n [A-B,2 WT], T > 0 w/l- E2 h( T) = 0 f or x < 0 (The condition of output follows the input) The autocovariance function* for a function y(t) i s Cy( x): , T/2 Cy(x) - Ida, £ J y ( t ) y ( t + x)dt T+oo 1 -x/2 For white noise input with constant spectral density Go, the output autoco-variance i s *Note: The autocorrelation function i s the autocovariance function normalized by Cy(0), the variance. 126. Cy(T) = *2_ [e-^ W T(cos / r ^ w T 2gw3 + § sin/1 - 5* WT) ] A - e This i s a cosinusoidal function with exponential decay. The decay of the envelope of the autocovariance estimate of the output response can be used to estimate the c r i t i c a l damping r a t i o £ of each mode of the system by the log decrement method. To separate each mode i n the frequency domain attention must be given to the shortcomings of the d i g i t a l process. The f i l t e r i n g out of c e r t a i n frequencies i n the frequency domain i s s i m i l a r to applying a window to the i n f i n i t e Fourier s e r i e s , (section B2.6). If the truncated series i s trans-formed back to the time domain uniform convergence of the i n f i n i t e s e r i e s i s l o s t , p a r t i c u l a r l y at the beginning and end of the time s e r i e s , where the non-periodic time series i s treated as periodic by the d i s c r e t e Fourier transform (see section B2.5). This phenomenon i s caused by convolving the kernel function of the f i l t e r with the o r i g i n a l rectangular data window. The r e s u l t i s Gibb's o s c i l l a t i o n s , e s p e c i a l l y at the d i s c o n t i -n u i t i e s . Some kind of smoothing i s needed to compensate f o r the o s c i l l a t i o n s . One technique i s to taper the f i l t e r i n some fashion (use a trapezoidal f i l t e r , f o r example) i n the frequency domain. After transformation back to the time domain, a c e r t a i n number of points at the beginning and end of the time series are discarded (see Toaka (18)). This smoothing process has been adopted i n the computer programme used for t h i s damping estimation method, b) P a r t i a l Spectral Moment Method: Vanmarcke et a l (20) proposed the use of p a r t i a l s p ectral moments to obtain natural frequency and damping estimates. 1 2 7 . The spectral moments are defined as CO CO Xo = / w°G(w)dw = / G(w)dw ' 0 0 where X 0 i s the zeroth s p e c t r a l moment, G(w) i s defined as the one-sided ( f o r p o s i t i v e frequencies only) power spectral density funtion of the zero mean stationary Gaussian process i n question. S i m i l a r l y , f i r s t and second spectral moments are defined as CO X = / wxG(w)dw 1 0 oo X = / w2G(w)dw 1 0 For very l i g h t damping [£ * .15] the following s i m p l i f i c a t i o n s can be used for finding a) the natural frequncy wn: n LX0 J b) the value of 5 2 r = r1 + l U- -All wa W 8 3 wn and wa; wb are the cut o f f frequencies f o r the 0 wb i s o l a t e d modal peak, u b = — r -wb These are v a l i d for the s p e c i a l case of fia - !Tft>. These s i m p l i f i e d formulas have been used i n t h i s i n v e s t i g a t i o n . Vanmarcke claims the following: 1) For a given record length T, estimates of sp e c t r a l moments may be expected to be much more r e l i a b l e than those of i n d i v i d u a l s p ectral o r d i -nates. This follows from the fact that the area under the spectral curve, which i s proportional to the t o t a l power, has a much smaller variance than the power spectrum estimate (see also Ulrych and Bishop (24)). 128. 2) Smoothing of the "raw" s p e c t r a l estimates i s unnecessary; estimated p a r t i a l s p e c t r a l moments and parameters based upon them may be expected to change very l i t t l e as a r e s u l t of smoothing. Discussions of the applications of these methods to the data c o l l e c t e d are given i n Chapter 5. 

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