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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 . S c , U n i v e r s i t y 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 Civil  We accept  this  Engineering  t h e s i s as  to the r e q u i r e d  THE  conforming  standard  UNIVERSITY OF BRITISH COLUMBIA  ©  Ngok-Ming To, 1982 '  In p r e s e n t i n g t h i s  thesis in p a r t i a l fulfilment  of the requirements  an advanced degree a t  the U n i v e r s i t y o f B r i t i s h Columbia, I agree  L i b r a r y s h a l l make i t  freely  agree  a v a i l a b l e ;for .reference, and ".study. I  that permission for extensive  c o p y i n g of t h i s  purposes may be granted by the Head o f my department sentatives. financial  It  that  thesis for  this  scholarly  thesis  g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n .  Ngok-Ming To  Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2324 Main M a l l Vancouver, B . C . , Canada V6T 1W5  pate  /g*  / f t a w i  J9$2  the  further  or by h i s or her  i s understood t h a t copy or p u b l i c a t i o n of  for  reprefor  ABSTRACT  The  dynamic p r o p e r t i e s , namely the n a t u r a l f r e q u e n c i e s , mode shapes  damping v a l u e s of three downtown Vancouver b u i l d i n g s have been o b t a i n e d means of ambient v i b r a t i o n s u r v e y s .  The  b u i l d i n g s surveyed  asymmetrical h i - r i s e - the Harbour Centre, Toronto Dominion Bank and  IBM  Towers.  and  two  include  by  an  frame s t r u c t u r e s -  Three h i - r i s e b u i l d i n g s of  d e s i g n - the Gage Residences - were a l s o surveyed  and  identical  to examine t h e i r  respec-  t i v e dynamic p r o p e r t i e s . Two  independent measurement systems were employed so t h a t  data c o u l d be d u p l i c a t e d and mine damping. t i o n and the T/D The  Two  statistically  important  s t a b l e s p e c t r a obtained  to d e t e r -  damping e s t i m a t i o n methods were used - the a u t o c o r r e l a -  p a r t i a l moment methods.  A man  e x c i t e d t e s t was  a l s o performed  on  Tower. measured n a t u r a l f r e q u e n c i e s of the Harbour Center were much h i g h e r  than p u b l i s h e d  t h e o r e t i c a l values  (64% h i g h e r  t h a t the a n a l y t i c a l model used was  i n one  case).  too f l e x i b l e and may  T h i s suggests  not have accounted  f o r the i n t e r a c t i o n s of n o n - s t r u c t u r a l elements. In the case of the T/D  and  IBM  Towers the f r e q u e n c i e s were found to  agree c l o s e l y w i t h the r e s u l t s obtained The  from t h e o r e t i c a l  n a t u r a l f r e q u e n c i e s of the asymmetrical b u i l d i n g - the Harbour  Centre - were expected to be very c l o s e together  (.01  c a s e s , as i n d i c a t e d by an e x i s t i n g a n a l y t i c a l s t u d y ) . process  analyses.  was  employed to search  A special  f o r the c l o s e f r e q u e n c i e s .  s e a r c h d i d not r e v e a l f r e q u e n c i e s that f r e q u e n c i e s  Hz a p a r t , i n some  spaced at 0.01  Hz,  smoothing  Although  i t is still  possible  e x i s t a t c l o s e r than .01 Hz or that they have v e r y  ft-  this  unequal  F o u r i e r amplitudes  a t a .01 Hz s p r e a d .  In e i t h e r case, the smoothing  process was not able to separate such f r e q u e n c i e s . The  r e s u l t s o b t a i n e d f o r the Harbour Center E-W d i r e c t i o n suggest  that  the f r e q u e n c i e s o f the fundamental and second modes i n t h i s d i r e c t i o n may be e q u a l , o r n e a r l y e q u a l , and t h a t the measured 'fundamental'  mode shape'is  not u n i q u e l y d e f i n e d . The results  p a r t i a l moment method f o r damping e s t i m a t e s seems to p r o v i d e  stable  f o r a g i v e n l e n g t h o f r e c o r d , and seems to be l i t t l e a f f e c t e d by the  smoothing p r o c e s s .  The low damping v a l u e s o b t a i n e d by t h i s method f o r the  Harbour Centre, T/D and IBM Towers are t y p i c a l o f damping d e t e r m i n a t i o n s by ambient v i b r a t i o n  tests.  S t a b l e e s t i m a t e s o f power s p e c t r a l d e n s i t y o r F o u r i e r s p e c t r a were achieved e a s i l y i n t h i s e x p e r i m e n t a l programme. t h e r e f o r e c o u l d be c o n f i d e n t l y used  The a u t o c o r r e l a t i o n method  to secure good damping e s t i m a t e s .  damping v a l u e s o b t a i n e d by t h i s method a r e comparable to the r e s u l t s from  the p a r t i a l moment t e c h n i q u e .  The man-excited  The found  t e s t r e s u l t s f o r the T/D  Tower a l s o p r o v i d e d damping e s t i m a t e s which were w i t h i n the range o f v a l u e s determined  by the p a r t i a l moment and a u t o c o r r e l a t i o n methods.  The n a t u r a l f r e q u e n c i e s and damping v a l u e s ( u s i n g o n l y the p a r t i a l moment method) o b t a i n e d f o r 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 t h e r e f o r e may be regarded as b e i n g  identical.  Ambient V i b r a t i o n  Survey Page  T a b l e of Contents  Abstract  i.  L i s t of Tables  v  L i s t of F i g u r e s  vi  Acknowledgements  viii  Chapter 1:  1  Introduction  1.1  Object and  1.2  L i t e r a t u r e Survey  Chapter 2: 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Chapter  Background  1 2  Theory f o r the Ambient V i b r a t i o n Survey  Introduction  Measurement Systems C a l i b r a t i o n o f the Earthquake Measurement System  Chapter 4: 4.1 4.2  General A) The Harbour Center B) Toronto Dominion Bank and C) Gage Residences  6 6 7 7 8 9 11 11 11  IBM Towers  Data and Dynamic S t r u c t u r a l A n a l y s e s R e s u l t s A) B) C)  4 5  Engineering Laboratory  Ambient V i b r a t i o n F i e l d Measurements  Chapter 5:  4 4  The Measurement System and the Frequency Response Analysis The P u l s e Response Approach The F o u r i e r Transform, R e l a t i o n s h i p s Between P u l s e and Frequency Responses, C o n v o l u t i o n and M u l t i p l i c a t i o n D i s c r e t i z e d F u n c t i o n s and the D i s c r e t e F o u r i e r Transform D i r i c h l e t ' s K e r n e l , Windows, Smoothing The Sampling Theorems S t a t i s t i c a l A n a l y s i s o f S p e c t r a and the Idea o f S t a b i l i t y Damping E s t i m a t i o n Methods 3: The I n s t r u m e n t a t i o n System  3.1 3.2  5.1  Scope  The Harbour Center Toronto Dominion Bank and IBM Towers Gage Residences  - iii -  14 14 14 20 22 24 24 34 41  Page  Chapter 6:  D i s c u s s i o n o f R e s u l t s and C o n c l u s i o n s  43  References  46  Figures  48  Appendices  84  Appendix Appendix Appendix Appendix Appendix  A2 A3 A4 A5 B2  85 91 97 102 104  - iv -  LIST OF TABLES Page 4.1  P u b l i s h e d N a t u r a l F r e q u e n c i e s of the Harbour Center.  16  5.1  Wind C o n d i t i o n s f o r the Harbour Center  (March 24, 1981).  25  5.2  Wind C o n d i t i o n s f o r the Harbour Center and (December 19, 1981).  (May 12~, 1981)  26  5.3  Measured N a t u r a l F r e q u e n c i e s of the Harbour Center.  30  5.4  Damping Values of the Harbour Center.  33  5.5  Wind C o n d i t i o n s f o r the T/D Tower ( A p r i l 11, 1981).  35  5.6(a) Measured N a t u r a l F r e q u e n c i e s of the T/D Tower.  36  5.6(b) A n a l y t i c a l N a t u r a l F r e q u e n c i e s of the T/D Tower  36  5.7  Wind C o n d i t i o n s f o r the IBM Tower.  37  5.8  Measured N a t u r a l Frequencies of the IBM Tower  38  5.9  A n a l y t i c a l N a t u r a l F r e q u e n c i e s of the IBM Tower.  38  5.10(a)Damping Value of the T/D Tower.  39  5.10(b)Damping Values o f the T/D Tower (Man E x c i t e d T e s t ) .  39  5.11  Damping Values of the IBM Tower.  40  5.12  Gage Tower N-S F r e q u e n c i e s .  41  5.13  Gage Tower Damping V a l u e s .  42  - v -  LIST OF FIGURES Page 2.1  Instrument  Calibration  Curve.  2.6.1  Frequency  Response o f Two Window F u n c t i o n s .  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  P l a n View of Harbour Center.  52  4.4  Seismometer L o c a t i o n on T y p i c a l  4.5  Typical  Floor  Plan:  T/D Tower.  4.6  Typical  Floor  Plan:  IBM Tower.  4.7  Typical  Roof P l a n o f Gage Residence  5.1  Windows R e s o l u t i o n T e s t s .  5.2  Frequency  Identification:  48  Office Floor:  Harbour Center.  53  .  54 55  Towers.  56 57  Harbour Center E-W.  58  5.3(A)(B)  5.4  Harbour Centre Mode Shapes ( F l e x u r a l ) .  59  Harbour Center Mode Shapes ( T o r s i o n a l ) .  61  5.5(A)(B)(C)(D)(E)(F)(G) Autocorrelograms: 5.6  Frequency  Harbour Center.  Identification:  T/D Tower N-S.  62 69  5.7(A)(B)(C) Mode Shapes: 5.8  Frequency  T/D Tower.  Identification:  70 IBM Tower N-S.  73  5.9(A)(B)(C) Mode Shapes:  IBM Tower.  74  5.10(A)(B) A u t o c o r r e l o g r a m s : T/D Tower. 5.11(A)(B)(C)(D)(E) Autocorrelograms:  77  IBM Tower.  - vi-  79  Page  88  A.2.1  Noise i n S i g n a l .  A.2.2  Fourier  A.2.3  Smoothed Time F u n c t i o n .  A.3.1  Diagram of Maxwell Bridge  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)  T y p i c a l Ambient V i b r a t i o n  Traces:  Harbour  A.4.1(B)  T y p i c a l Ambient V i b r a t i o n  Traces:  T/D Tower.  90  A.4.1.(C)  T y p i c a l Ambient V i b r a t i o n  Traces:  IBM Tower.  100  A.4.1(D)  T y p i c a l Ambient V i b r a t i o n  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 T h i r d Mode.  103  A.5.1(C)  Man-Excited  Test:  N-S T/D Tower Second Mode.  103  Transform  of Noisy  89  Data.  90 Circuit  - vii-  94  •  Center.  98  ACKNOWLEDGEMENTS  The  author  wishes to express  C h e r r y , Anderson and guidance p r o v i d e d procedures The the UBC  the necessary  e s p e c i a l l y to Dr. a f f i r m a t i o n (and  Cherry, whose constant  r e j e c t i o n ) of concepts  and  f o r the e n t i r e p r o j e c t .  generous a s s i s t a n c e from the p r o f e s s o r s and Department of Geophysics,  equipment and lar,  Nathan, and  s i n c e r e thanks to h i s a d v i s o r s , P r o f e s s o r s  and  research associates i n  t h e i r permission  to use both  computer programme l i b r a r y are much a p p r e c i a t e d .  the f r i e n d l y d i s c u s s i o n s with Mr.  Bob  the o b s t a c l e s encountered i n the instrument  Meldrum helped  their  In  particu-  c l e a r away many of  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  Dominion and t i o n s , and  i s g r a t e f u l to the owners of the Harbour Center,  IBM  Towers f o r p e r m i s s i o n  a l s o to Unecon E n g i n e e r i n g  S k a l b a n i a & Assoc. The  proper  Ltd.  i n the f i e l d a s s i s t i n g project possible.  The  Toronto  to measure t h e i r b u i l d i n g s ' v i b r a C o n s u l t a n t s and McKenzie Snowball  f o r p r o v i d i n g the drawings of the b u i l d i n g s .  f u n c t i o n i n g of the d u a l measurement systems demands many  competent o p e r a t o r s .  the author's  the  The  and  f r i e n d s who  spent  long hours  i n the o p e r a t i o n of the systems helped make t h i s  T h e i r f r i e n d s h i p and  most v a l u e d  work was  graduate students  s e l f l e s s c o n t r i b u t i o n s s h a l l remain  p e r s o n a l g a i n i n completing  performed under c o n t r a c t  this  (Contract No.  research. 080-040/0-4410)  w i t h the N a t i o n a l Research C o u n c i l of Canada on b e h a l f of the Canadian N a t i o n a l Committee f o r Earthquake  - viii  -  Engineering.  Council's  1. CHAPTER 1 INTRODUCTION  1.1  Object and Scope The  purpose o f t h i s r e s e a r c h  namely the n a t u r a l f r e q u e n c i e s , Downtown Vancouver b u i l d i n g s :  i s to determine the dynamic p r o p e r t i e s ,  mode shapes and damping v a l u e s ,  o f three  The Harbour Centre, IBM and the Toronto  Dominion Bank Towers, by means o f the ambient v i b r a t i o n survey method. Three h i g h r i s e b u i l d i n g s o f i d e n t i c a l d e s i g n  on the campus o f the U n i v e r s i t y  of B r i t i s h Columbia have a l s o been measured by the same method to compare the dynamic p r o p e r t i e s o f " i d e n t i c a l " The  buildings.  ambient v i b r a t i o n survey i s a f u l l  s c a l e s t r u c t u r a l t e s t i n g method  which measures the response o f a s t r u c t u r e to wind and c u l t u r a l n o i s e s and a n a l y z e s t h i s response to e s t i m a t e the dynamic p r o p e r t i e s o f the s t r u c t u r e . The method may be a p p l i e d to a s t r u c t u r e f o r a c o m p a r a t i v e l y duration a long  short  time  j u s t to a s c e r t a i n i t s dynamic c h a r a c t e r i s t i c s , or i t may be p a r t o f  term m o n i t o r i n g programme to compare the changes i n these  i s t i c s between p r e - and post-earthquake c o n d i t i o n s . be a b r i d g e  or a b u i l d i n g : long  shore p l a t f o r m s  character-  The s t r u c t u r e need not  term ambient v i b r a t i o n m o n i t o r i n g o f o f f -  has been performed to see i f any damage has been done t o  major s t r u c t u r a l members by comparing the f r e q u e n c i e s storm or earthquake r e c o r d i n g s .  between p r e - and p o s t -  In the past decade, enough t e s t s have been  c a r r i e d out such t h a t both the v e r s a t i l i t y and the v a l i d i t y o f the method have been e s t a b l i s h e d . To  p r o p e r l y a n a l y z e 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 a n a l y s i s and the s t a t i s t i c a l niques d e s c r i b i n g a random process i s needed. some o f the r e l e v a n t  t h e o r e t i c a l considerations  tech-  Chapter 2 attempts to b r i n g o f these areas i n t o  focus.  2. The b a s i c p r i n c i p l e s of communication e n g i n e e r i n g d e s c r i b e d i n Chapter 2 are applied are  to the c a l i b r a t i o n of the measurement system,  covered i n Chapter 3.  ures adopted ments.  some d e t a i l s of which  Chapter 4 c o n t a i n s the a c t u a l measurement proced-  i n the f i e l d , and  the problems encountered  d u r i n g the measure-  The concepts presented i n Chapter 2 a r e a g a i n put to use i n Chapter  5 i n which the a c t u a l a n a l y s e s of the data are performed tabulated.  Techniques which are s p e c i a l i z e d  b u i l d i n g w i l l be d i s c u s s e d i n t h i s c h a p t e r .  and  the r e s u l t s a r e  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 Since one of the major a p p l i c a -  t i o n s of the ambient v i b r a t i o n survey i s to v e r i f y the mathematical of  s t r u c t u r e s , dynamic s t r u c t u r a l a n a l y s e s of some of the t e s t e d  were performed  on the computer.  models  structures  The r e s u l t s o f the computer a n a l y s e s are  compared to the survey r e s u l t s and d i s c u s s e d i n Chapter  5.  The f i n a l c h a p t e r c o n t a i n s the v a r i o u s c o n c l u s i o n s o f the r e s e a r c h .  1.2  Literature  Survey  Some of the e a r l i e s t performed  ambient v i b r a t i o n surveys of b u i l d i n g s were  by Cherry and Brady  (3) i n 1965  and Ward and Crawford  (4) i n 1966.  Since then many o t h e r s (1,5,6,7,8,9,10) have c o n t r i b u t e d v a l u a b l e e x p e r i e n c e and i n s i g h t to t h i s a r e a of s t r u c t u r a l The  dynamics.  technique has a l s o been extended  to study the dynamic behaviour o f  b r i d g e s (11,12) and o f f s h o r e p l a t f o r m s ( 2 ) . The data a q u i s i t i o n and a n a l y s e s of ambient v i b r a t i o n surveys i n v o l v e random d a t a c o l l e c t i o n procedures and F o u r i e r o r power spectrum techniques.  C o r n e l i u s Lanczos'  two books (13, 14) o f f e r e x c e l l e n t  t i o n on harmonic a n a l y s i s ; r e l e v a n t s t a t i s t i c a l tained i n a c l a s s i c l i t t l e  analysis informa-  a n a l y s i s methods are con-  book by Blackman and Tukey ( 1 5 ) .  3. An e m p i r i c a l 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  a p p l i e d to b u i l d i n g v i b r a t i o n s t u d i e s i s g i v e n by K i r c h e r and a few meaningful comments on the s t a t i o n a r i t y and necessary  f o r power s p e c t r a a n a l y s e s  A v e r y important surveys  other  Shah ( 8 ) , and  assumptions  are g i v e n by Tukey ( 1 6 ) .  dynamic p r o p e r t y to be determined by ambient  i s the damping of the s t r u c t u r e .  The man  power bandwidth methods were among the f i r s t  e x c i t e d (17)  and  vibration half  damping e s t i m a t i o n methods to  be used i n c o n j u n c t i o n w i t h the ambient v i b r a t i o n s u r v e y s . work of Cherry and  as  The  pioneering  Brady (3) p r o v i d e d a r i g o r o u s mathematical b a s i s f o r the  a u t o c o r r e l a t i o n method of damping e s t i m a t i o n .  Digital  techniques  that  be a p p l i e d to power s p e c t r a to o b t a i n the a u t o c o r r e l a t i o n f u n c t i o n may found  i n a paper by Toaka (18), and  g e n e r a l may  be found  ideas p e r t i n e n t to d i g i t a l  i n a book by Hamming ( 1 9 ) .  moments has a l s o been used to f i n d  (20).  p a r t i a l moment) i s g i v e n by Durning and  Engle  in  spectral  A comparison o f  three of these damping e s t i m a t i o n methods ( l o g decrement, h a l f power bandwidth and  be  filtering  A method based on  the damping v a l u e s  may  (21).  4. CHAPTER 2 BACKGROUND THEORY FOR THE AMBIENT VIBRATION SURVEY  2.1  Introduction T h i s c h a p t e r summarizes a p a r a l l e l i n - d e p t h mathematical  presented i n Appendix B2. i) vibration  I t i s concerned  background  w i t h the f o l l o w i n g m a t t e r s :  the c a l i b r a t i o n o f the measurement system used  i n the ambient  survey;  ii)  the p l a n n i n g o f the measurement  iii)  the subsequent a n a l y s e s and i n t e r p r e t a t i o n o f the a c q u i r e d d a t a .  Quotations  from C. Lanczos'  procedures;  " A p p l i e d A n a l y s i s " (13) and " D i s c o u r s e on  F o u r i e r S e r i e s " ( 1 4 ) , and a l s o Blackman and Tukey's "The Measurement o f Power S p e c t r a " (15) w i l l not be s e p a r a t e l y r e f e r e n c e d . 0  2.2  The Measurement System and the Frequency Reponse A n a l y s i s The measurement system can be r e p r e s e n t e d as a l i n e a r time th  b l a c k box which i s governed by the f o l l o w i n g n  order, l i n e a r  invariant differential  equations: ,n j l d y . d y . a — — + a , + n ,n n-1 , n-1 dt dt n  -  a y = o r  ±Ji  b  r  where x ( t ) i s i n p u t and y ( t ) o u t p u t . t o determine  r - l  dt  r  +  i  b  r  ~  1  a t  x 1  "  +  1  There i s no simple e x p e r i m e n t a l method  the a's and b's (which a r e c o n s t a n t s ) - so i n s t e a d the i n p u t  and output a r e observed  and the system t r a n s f e r f u n c t i o n thereby  obtained.  5. If we  regard  the i n p u t as composed of a s u p e r p o s i t i o n of  p e r i o d i c components (the F o u r i e r t e c h n i q u e ) ,  and  due  have the frequency response  to these i n p u t components one  approach f o r the b l a c k  by one,  we  i f we  strictly  examine the  responses  box.  T h i s concept can be extended to the treatment o f n o n - p e r i o d i c  functions  by the F o u r i e r i n t e g r a l approach, namely: oo  f(t) = ~  J  F(w)  e  1 W t  dw  2.2.2  — 00  oo  where  F(w)  = /  f(t)e~  i w t  dt  2.2.3  —oo  f ( t ) i s the non-period f u n c t i o n i n the  time domain and  F(w)  i s i t s Fourier  spectrum i n the frequency domain. The  l a b o r a t o r y c a l i b r a t i o n of the measurement system i s based on  frequency response approach.  High-fidelity  reproduction  the  of i n p u t i s e v i d e n t  over a wide frequency range f o r the measurement system.  2.3  The  P u l s e Response Approach  For the l i n e a r , t i m e - i n v a r i a n t system, the i n p u t can be modelled by a s u p e r p o s i t i o n of p u l s e s , and reponses.  The  the output as a s u p e r p o s i t i o n of  pulse  response of such a system i s t h e r e f o r e : oo  y(t) = /  H(t-Q  x(Qd?  2.3.6  —CO  where H(t) 2.3.6  i s the response to an impulse input a t time t = 0.  i s the c o n v o l u t i o n  detailed  i n t e g r a l of H(t)  p h y s i c a l argument i s p r o v i d e d  and  x(t).  i n Appendix  Equation  A somewhat more B2.3.  6. 2.4  The F o u r i e r T r a n s f o r m . R e l a t i o n s h i p Between P u l s e and Responses, C o n v o l u t i o n 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  defines  the F o u r i e r  v e r s i o n of the F o u r i e r  Frequency  Integral (equation  2.2.2)  transform: CO  X(w)  = y ^ —  /  x(t)e"  1 W t  dt  2.4.1  — OO  and  the i n v e r s e  transform: oo  x(t) = y ^ —  /  X(w)e  iwt  dw  2.4.2  — 00  I t can be  proved t h a t the complex frequency response of a l i n e a r  i s j u s t the complex conjugate o f the F o u r i e r transform response, m u l t i p l i e d by  /STir.  (See  F o u r i e r Transform of a c o n v o l u t i o n  t h e i r F o u r i e r Transforms. and  2.4.2  the  (see  B2.4).  2.5  The  By  f u n c t i o n s i s the  i n the  amount of trum.  frequency domain  use  the  computer.  the F o u r i e r s e r i e s as a l i m i t i n g case of  F o u r i e r i n t e g r a l ( d i s c r e t i z i n g ) w i l l g i v e r i s e to a 'focusing  2.4.1  than the F o u r i e r s e r i e s ,  of the F o u r i e r s e r i e s when we  process of o b t a i n i n g  determines the  product of  the remarkable r e c i p r o c i t y o f e q u a t i o n s  I n t e g r a l i s more g e n e r a l  have to t u r n to the use The  of two  i n t e g r a l (2.3.6):  Transform  Although the F o u r i e r we  of the c o n v o l u t i o n  same r u l e a p p l i e s f o r a c o n v o l u t i o n  Discrete Fourier  of i t s pulse  Appendix B2.4.)  There i s another important aspect The  system  the  ' k e r n e l ' f u n c t i o n which  power' on the d i s c r e t i z e d f r e q u e n c i e s ,  'leakage' t h a t r e s u l t s from i t s c o n v o l u t i o n  w i t h the  and true  also spec-  the  7. 2.6  Windows, Smoothing C o n s i d e r the f i n i t e  time s e r i e s we sample as a product o f two f u n c t i o n s  - the i n f i n i t e  s i g n a l and a r e c t a n g u l a r f u n c t i o n t h a t has a form o f  0,0,0,1,1,  1,0,0,0 ( t h e r e a r e (2N+1) non-zero  p o i n t s t h a t we have i n the d i s c r e t i z e d  time  terms f o r the (2N+1)  series).  A f u n c t i o n h(w) has these (2N+1) non-zero  terms i n the f r e q u e n c y domain  k=N ikw h(w) = E  e  which i s a geometric  series  k=-N , , s i n ( N + 1/2)w h(w) = * — w sin ^  or  f o r w w i t h i n the fundamental . , period  h(w) i s a k e r n e l f u n c t i o n - or window, a f t e r Blackman and Tukey. ticular  This par-  window has a h i g h l y o s c i l l a t o r y n a t u r e , and when i t i s convolved  w i t h the F o u r i e r Transform o f the b u i l d i n g v i b r a t i o n s i g n a l , leakage occur.  A  smoothing  applied  to the transformed v i b r a t i o n r e c o r d s o f the b u i l d i n g s .  will  process t o reduce the secondary maxima may have to be  From the same s t a n d p o i n t , h i - f i d e l i t y  r e p r o d u c t i o n o f the i n p u t may not  be the best p o l i c y i f we perform a n a l y s e s i n the frequency domain. have t o suppress the fundamental  We may  v i b r a t i o n modes o f b u i l d i n g s to a v o i d l e a k -  age. 2.7  The Sampling The  Theorems  r e s u l t s o f the mathematics o f sampling  theorems may be s t a t e d as  follows: ( i ) To a v o i d a l i a s i n g , a b a n d - l i m i t e d f u n c t i o n must be sampled a t h i g h e r than twice the f o l d i n g (ii)  frequency.  I t i s n e c e s s a r y to sample a t more than two samples per c y c l e f o r  the h i g h e s t frequency present so as to p r e s e r v e the i n f o r m a t i o n o f the  8. o r i g i n a l f u n c t i o n a t the f o l d i n g  2.8  frequency.  S t a t i s t i c a l A n a l y s i s o f S p e c t r a and S t a b i l i t y There  i s one major d i f f e r e n c e between harmonic a n a l y s i s and the s t a t i s -  t i c a l approach  d e f i n i n g a random p r o c e s s , such as i s o b t a i n e d from an  ambient v i b r a t i o n survey ( J e n k i n s ( 2 3 ) ) .  Consider a f i n i t e  s e r i e s o f the  sample a u t o c o v a r i a n c e C^. C  The  raw s p e c t r a l e s t i m a t e  ^ n-k = — Z XX n. £ — ^ 111c  ( z e r o mean p r o c e s s )  I (w^): n  n-1 I  n  (w.) = — [c j ir o L  + 2 £  C.cos w.k] k 2  2.8.3  k=l where w . = 2 TTi/n. The  raw e s t i m a t e I^(w^) w i l l  f u n c t i o n a t W j and no matter  f l u c t u a t e about  how l o n g a sample we take i n the time domain,  the v a r i a n c e o f the f l u c t u a t i o n o f I (w..) about n  f u n c t i o n does not decrease  physical  phenomenon.  a highly variable,  Harmonic a n a l y s i s o f random num-  s p i k e d spectrum.  (See Appendix A2 f o r  illustration).  A m o d i f i e d procedure tral  the power s p e c t r a l d e n s i t y  to z e r o .  T h i s i s a w e l l observed bers produces  the power s p e c t r a l d e n s i t y  estimates.  o f harmonic a n a l y s i s can l e a d  to s t a b i l i z e d  spec-  In t h i s p r o c e s s the n terms o f the sample X ( t ) a r e s p l i t  i n t o p s e t s o f m terms so n=pm.  By c o n d u c t i n g a F o u r i e r a n a l y s i s f o r each  set and by a v e r a g i n g these r e s u l t s a t each v a r i a n c e as s m a l l as we w i s h .  frequency, we can make the  9. For our ambient v i b r a t i o n survey we  have to a n a l y s e the d a t a u s i n g a  r e a s o n a b l y f i n e frequency r e s o l u t i o n , w h i l e e n s u r i n g a s t a b l e e s t i m a t e o f the s p e c t r a from which the damping v a l u e s are o b t a i n e d . demand t h a t both  'm'  taken i n the f i e l d may  and  These  requirements  'p' be l a r g e and a r a t h e r l e n g t h y r e c o r d has  f o r t h i s purpose.  I t i s p o s s i b l e t h a t l o n g time r e c o r d s  v i o l a t e the s t a t i o n a r i t y assumption  (Toaka  ( 1 8 ) ) , but Tukey (16) recom-  mends t h a t the a n a l y s e s of long r e c o r d s be judged o f t e n produce v e r y u s e f u l average  to be  on t h e i r own  merit.  They  spectra.  E m p i r i c a l proof of t h i s l a t t e r f a c t has been o b t a i n e d from s t u d i e s of b u i l d i n g v i b r a t i o n s ( K i r c h e r and s p e c t r a by u s i n g long r e c o r d s was  Shah ( 8 ) ) . adopted  separate measurement systems i n the f i e l d  The  p r i n c i p l e of a v e r a g i n g  i n t h i s study.  The  use of  two  (Chapter 4) i s based  on  this  methods have been employed to e v a l u a t e the percentage  of  critical  principle.  2.9  Damping E s t i m a t i o n Methods Two  damping of the t e s t  s t r u c t u r e s : the a u t o c o r r e l a t i o n and  the p a r t i a l  spectral  moment methods.  (a)  The A u t o c o r r e l a t i o n Method: The  t h e o r e t i c a l background f o r t h i s method can be found  i n Cherry  and  Brady ( 3 ) . The  autocovariance f u n c t i o n for a f u n c t i o n y ( t ) i s Cy(t):  c  y( ) T  T  t»¥  T/2 /  y(t)y(t+t) dt. -T/2  When Cy(x) i s n o r m a l i z e d by Cy(0) relation  function.  ( t h e v a r i a n c e ) , i t i s termed the  autocor-  10. For white n o i s e  input with constant s p e c t r a l density,  correlation function i s a cosinusoidal  the output a u t o -  function with exponential  decay.  The  c r i t i c a l damping r a t i o may be o b t a i n e d from the a u t o c o r r e l o g r a m ( t h e p l o t o f autocorrelation function) In u s i n g  by means o f the l o g decrement method.  the d i g i t a l p r o c e s s , a t t e n t i o n has to be p a i d  t r a n s f o r m and r e v e r s e  to the F o u r i e r  t r a n s f o r m procedure. (See Toaka (18), and Appendix  B2.9).  (b)  Partial  S p e c t r a l Moment Method:  Vanmarcke e t a l (20) proposed the use o f p a r t i a l s p e c t r a l moments t o obtain natural  frequency and damping e s t i m a t e s .  The f o l l o w i n g a r e claimed  to be the advantages o f t h i s method: (i)  For a given record  l e n g t h , e s t i m a t e s o f 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 o f i n d i v i d u a l s p e c t r a l o r d i n ates. (ii)  Smoothing o f the "raw" s p e c t r a l e s t i m a t e s i s unnecessary.  Applications Chapter 5.  o f these methods to the d a t a c o l l e c t e d a r e g i v e n i n  11. Chapter 3 The  3.1  System  Measurement Systems: Two  (i)  Instrumentation  separate measurement systems were employed i n the f i e l d .  The Earthquake E n g i n e e r i n g L a b o r a t o r y  These a r e :  ( C i v i l E n g i n e e r i n g Dept.) System:  T h i s system c o n s i s t i n g of f o u r Ranger SD217 seismometers, a Teledyne SC201 s i g n a l c o n d i t i o n e r w i t h a m p l i f i e r s and and  a c c e l e r a t i o n f i l t e r c a r d s ( a l l having  a c o r n e r frequency  P h i l i p s ANA-L0G7 FM/AM tape r e c o r d e r capable a T e k t r o n i x CRO has  low pass d i s p l a c e m e n t , o f 100  velocity Hz),  a  of r e c o r d i n g i n 7 c h a n n e l s ,  which i s used f o r v i s u a l d i s p l a y .  The  and  signal conditioner  a d j u s t a b l e e l e c t r i c a l r e s i s t a n c e s which can a l t e r the t o t a l damping o f  each of the f o u r seismometers. ii)  The UBC  Geophysics System (on l o a n from the Department of  T h i s system i s made up o f e i t h e r one t e r s , an a m p l i f i e r w i t h band pass f i l t e r s 0.1  Hz o r 0.8  r e c o r d e r , and  Hz  to 5 Hz or 12.5  a Brush 222  o r two  Geophysics):  Willmore Mark I I seismome-  (adjustable corner frequencies at  Hz), an H.P.  3960 I n s t r u m e n t a t i o n  tape  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  f o r the two  systems are s i m i l a r , o n l y  c a l i b r a t i o n of the Earthquake E n g i n e e r i n g L a b o r a t o r y  System w i l l  the  be  described.  3.2  C a l i b r a t i o n of the Earthquake E n g i n e e r i n g L a b o r a t o r y Measurement System, a)  model 111  The  f o u r Ranger seismometers were c a l i b r a t e d by u s i n g a WAVETEK  voltage c o n t r o l generator  ( t o p r o v i d e the i n p u t s i g n a l s ) i n  c o n j u n c t i o n w i t h a MAXWELL BRIDGE arrangement f o l l o w i n g the method d e s c r i b e d  12.  by K o l l a r and R u s s e l l ( 2 7 ) .  The same procedure was a l s o employed and  d e s c r i b e d by Cherry and Topf ( 7 ) .  The seismometer  characteristics  (natural  frequency, damping and seismometer  c o n s t a n t ) a r e t a b u l a t e d i n Appendix A3,  Table A3.1. b)  Again u s i n g the WAVETEK generator f o r i n p u t t o the s i g n a l  c o n d i t i o n e r , a s e p a r a t e c a l i b r a t i o n was c a r r i e d out on the s i g n a l c o n d i t i o n e r and tape r e c o r d e r as a u n i t . the  Output  from the FM r e c o r d i n g o f  tape r e c o r d e r was a n a l y s e d . c)  The r e s u l t s o f these two c a l i b r a t i o n s were combined and a n a l y s e d  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 o u t p u t s o f t h i s programme d e f i n e the system f r e q u e n c y response  ( d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n ) f o r each f i l t e r velocity f i l t e r  c a r d s were found to produce  they were used throughout  d)  Since the  the best r e s u l t s i n the f i e l d ,  the measurement programme.  response c u r v e s i s a l s o shown i n Appendix  card.  A typical velocity  A3, F i g u r e A3.3.  As an independent check on the above r e s u l t s a ' s i m u l a t e d s h a k i n g  t a b l e ' c a l i b r a t i o n was performed  on the t o t a l measurement system.  WAVETEK 111 g e n e r a t o r was connected seismometer  was o p e r a t e d .  to the MAXWELL  The seismometer  The  Bridge through which the  s i g n a l was passed through the  s i g n a l c o n d i t i o n e r and then r e c o r d e d on the FM tape r e c o r d e r and system i n p u t s and outputs compared.  The system response o b t a i n e d i n t h i s way was  i n good agreement w i t h the r e s u l t s from the piece-meal c a l i b r a t i o n d e s c r i b e d in  a ) , b) and c ) .  Thus, the use o f the computer programme f o r the p i e c e -  meal c a l i b r a t i o n was v e r i f i e d and adopted  f o r system c a l i b r a t i o n .  A f l o w c h a r t o f s t e p s (a) to (d) i s shown below.  13. FLOW CHART OF CALIBRATION STEPS  (b) S i g n a l c o n d i t i o n e r and tape r e c o r d e r calibrated: Frequency response to s i n u s o i d a l input o b t a i n e d .  (a)  Four Ranger SD217 seismometers calibrated: Natural frequencies, damping v a l u e s and seismometer c o n s t a n t s obtained.  (c) System frequency response o b t a i n e d : by combining and a n a l y s i n g r e s u l t s of ( a ) , ( b ) , u s i n g computer program UNICAL.  (d) Independent check: System response o b t a i n e d by u s i n g '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 f o r system calibration.  14. Chapter 4 Ambient V i b r a t i o n F i e l d Measurements  4.1  General The  layout  planning  of the  provided  the  e x e c u t i o n of the measurement programme ( i n c l u d i n g  instruments),  i n the  a l s o noted.  and  d e s c r i p t i o n s of the  following sections.  The  structures  and  presentation  t e s t e d , as  test structures  Problems encountered i n the  Sears Tower ( o r Harbour Centre) - Downtown Vancouver  B)  The  Toronto Dominion Bank and  C)  The  Gage Residence, Towers A,B,C  4.2  A)  19th  Dec,  the IBM  - U n i v e r s i t y of B.C.  Campus  Harbour C e n t r e : (performed on 24th March, 81,  and  Bergmann  12th  May,  81  and  building  i s the main b u i l d i n g of the Harbour Centre complex  Wright a r c h i t e c t s ) .  which three  been taken from  (28):  i n the downtown area of Vancouver (Author's Note: and  to  81).  a paper by Tso  Eng  are  Towers - Downtown Vancouver  d e s c r i p t i o n o f the Harbour Centre t h a t f o l l o w s has  "The  field  follows:  The  The  are  i s separated i n t o three groups, a c c o r d i n g  A)  The  the  - courtesy  of  I t c o n s i s t s of a f i v e - s t o r e y s u b s t r u c t u r e  of  are used f o r p a r k i n g ,  shipping  and  F i g . 4.1  receiving; a five-storey  s e c t i o n above grade which i s l i n k e d w i t h an e x i s t i n g s e v e n - s t o r e y b u i l d i n g to form together a l a r g e department s t o r e , a 21-storey o f f i c e b u i l d i n g above the  s t o r e and  an e l e v a t e d  The  main b u i l d i n g having a 116'  surrounding c o n s t r u c t i o n  observation x 116'  tower w i t h a r e v o l v i n g plan  restaurant.  s i z e i s separated from  above s t r e e t l e v e l by expansion j o i n t s .  the The  build-  15. ing  i s g e n e r a l l y of r e i n f o r c e d c o n c r e t e c o n s t r u c t i o n w i t h the o f f i c e  f l o o r s cast i n lightweight concrete.  The  perimeter  tower  basement w a l l s and  system of i n t e r i o r shear w a l l s c r e a t e a v e r y s t i f f base below s t r e e t from which the tower c a n t i l e v e r s to a h e i g h t of 455' and  The  a  level  observation  r e s t a u r a n t f l o o r s are framed i n s t e e l s u p p o r t i n g a l i g h t w e i g h t  concrete  f l o o r deck." The  same paper a l s o r e p o r t s the b u i l d i n g p e r i o d s as o b t a i n e d  dynamic a n a l y s i s . corresponding The provided  These are reproduced  frequency  i n Table 4.1,  together  from a  with  the  values.  t h e o r e t i c a l mode shapes f o r the f i r s t i n the Tso-Bergmann paper, and  3 l a t e r a l modes a r e a l s o  are d u p l i c a t e d i n F i g .  4.2.  From the p u b l i s h e d r e s u l t s of the a n a l y s i s o f the Harbour Centre,  the  f o l l o w i n g became apparent: i)  The measurement programme must p r o v i d e s u f f i c i e n t data  f i n e enough frequency c l o s e as 0.01 ii)  Hz  to g i v e a  r e s o l u t i o n to d i s t i n g u i s h c l o s e mode f r e q u e n c i e s - as  apart.  There must a l s o be enough b l o c k s o f data  dependent on frequency  (with block  lengths  r e s o l u t i o n d e s i r e d ) such t h a t 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 e s t i m a t i o n . An e s t i m a t e o f the l e n g t h o f r e c o r d i n g time needed f o r both ( i ) and (ii)  showed that the b a t t e r y powered Earthquake E n g i n e e r i n g  measurement system c o u l d not meet the power requirement ings.  The  d e c i s i o n was  system having  Laboratory  o f l o n g time r e c o r d -  then made to i n t r o d u c e a separate  (Geophysics)  one Willmore Mark I I seismometer ( s e c t i o n 3.1),  f o r the  16.  Table 4.1 PUBLISHED (28) NATURAL FREQUENCIES OF THE HARBOUR CENTER.  Direc . t i o n E-•W  Direction N-S  Mode*  T (SEC)  f (Hz)  f (Hz)  T (SEC)  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 a r e numbered a c c o r d i n g  to d e c r e a s i n g  order.  17.  express purpose of o b t a i n i n g l o n g t i m e - h i s t o r y r e c o r d s o f the b u i l d i n g vibrations.  These long r e c o r d i n g s serve to p r o v i d e :  a)  s t a b l e F o u r i e r s p e c t r a f o r damping e s t i m a t e s .  b)  an independent check of the n a t u r a l f r e q u e n c i e s o b t a i n e d by the Earthquake E n g i n e e r i n g L a b o r a t o r y system.  L a t e r , (12th May),  two Willmore Mark I I 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 o f the b u i l d i n g v i b r a t i o n s o r i g i n a l l y measured, and the Geophysics system became, as f a r 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 f r e q u e n c i e s was "back-up"  f o r the Earthquake E n g i n e e r i n g L a b o r a t o r y system.  f i n a l check on the phase r e l a t i o n was  concerned, a A second and  c a r r i e d out (Dec. 19) u s i n g a p o r t a b l e  Geophysics system employing two Willmore Mark I I seismometers, a c o n d i t i o n e r and a Teledyne Geotech MCR-600/Microcorder chart recorder f o r v i s u a l d i s p l a y .  signal  w i t h a Brush 222  These checks were c a r r i e d out because o f  apparent p e c u l i a r i t i e s i n some o f the mode shapes d e r i v e d from the f i r s t test series.  This i s discussed i n a l a t e r  section.  A l t h o u g h the W i l l m o r e Mark I I seismometers a r e not as ' s e n s i t i v e ' as the Ranger  SD217 seismometers ( s i n c e they have a lower 'g' - g e n e r a t o r  c o n s t a n t - v a l u e than the Ranger  SD217, see Appendix A3), t h e r e were s e v e r a l  advantages i n u s i n g the Geophysics system as •  "back-up":  The added r e c o r d i n g c a p a c i t y o f the Geophysics system enabled important  measurements to be d u p l i c a t e d and a l l o w e d f o r the p o s s i b i l i t y o f s e c u r i n g v i b r a t i o n r e c o r d s over a l o n g time p e r i o d . •  Since the low and h i g h c o r n e r f r e q u e n c i e s of the a m p l i f i e r are a d j u s t -  a b l e (see s e c t i o n 3.1), i f ,  f o r example,  the measurements were made on a day  with sudden wind g u s t s , the low c u t - o f f f r e q u e n c y c o u l d be a d j u s t e d from to  .8 Hz, so as to suppress the fundamental mode, t h e r e b y enhancing the  .1  18. d e t e c t i o n of the h i g h e r modal f r e q u e n c i e s . the  ( S t r o n g wind e x c i t e s  primarily  fundamental frequency of h i g h r i s e b u i l d i n g s - Cherry and Topf ( 7 ) ; see  a l s o s e c t i o n 2.5 on s u p p r e s s i n g the fundamental mode).  The h i g h c o r n e r  frequency c o u l d a l s o be a d j u s t e d to a v o i d h i g h frequency n o i s e i f i t was found to be p r e s e n t . 9 the  The system c o u l d be operated by one person and was more p o r t a b l e t h a n Earthquake E n g i n e e r i n g L a b o r a t o r y system.  Re-checking f i e l d measure-  ments c o u l d be made more e a s i l y w i t h t h i s system than the Earthquake E n g i n e e r i n g L a b o r a t o r y system. A l t h o u g h the double system measurement programme r e q u i r e d 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 p o s s i b l e w i t h the s i n g l e  system  programme. The a c t u a l placement o f the i n s t r u m e n t s f o r the Harbour c e n t r e was follows:  a)  (method  of placement was  similar  as  for a l l buildings).  The Geophysics system ( h e r e a f t e r r e f e r r e d to as System 2 ) : The two Willmore seismometers f i r s t were placed s i d e by s i d e and i n the  same d i r e c t i o n on the r o o f (machine room l e v e l ) of the Harbour Centre f o r the  purpose of p e r f o r m i n g 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 o f the seismometer o u t p u t s (when they were measuring the same i n p u t ) both b e f o r e and a f t e r the ambient measurements were r e c o r d e d . The seismometers then were p l a c e d on the r o o f ( o r an upper f l o o r i n the other b u i l d i n g s , s i n c e the upper f l o o r s have r e a s o n a b l e v i b r a t i o n amplitudes for  a l l modes o f i n t e r e s t ) i n p a i r s 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 f r e q u e n c i e s , the two  p a i r e d seismometers were p l a c e d a t p o s i t i o n s 1 and 2 r e s p e c t i v e l y f o r about 30 minutes and then at p o s i t i o n s 1' and 2' f o r a f u r t h e r 30 minutes.  Two  19. a d d i t i o n a l p a i r s of h a l f - h o u r r e c o r d s were taken w i t h seismometers e o u s l y l o c a t e d on the r o o f i n the p o s i t i o n s 3 or 4 and  at the  deck l e v e l d i r e c t l y above the e l e v a t o r s h a f t (shown i n F i g .  simultan-  observation 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 f r e q u e n c i e s , as w e l l as to a s c e r -  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 simultaneous r e c o r d i n g s , a f t e r d i g i t i z a t i o n , were added and t u r n b e f o r e being  transformed  i n t o the frequency  domain.  floors,  subtracted i n  T o r s i o n a l or  out  o f phase s i g n a l s between f l o o r s w i l l be enhanced a f t e r s u b t r a c t i o n i s performed; f l e x u r a l or i n phase s i g n a l s between f l o o r s w i l l be a f t e r a d d i t i o n i s performed.  enhanced  Enhancement (or l a c k of i t ) i s v i s i b l e  the r e s u l t i n g change i n amplitude o f the F o u r i e r S p e c t r a s p i k e s i n g to the f r e q u e n c i e s On  March 24,  1981  The  instrument  correspond-  isolated. o n l y one  Willmore Mark I I seismometer was  f o r the s o l e pupose of o b t a i n i n g s t a b l e s p e c t r a l e s t i m a t e s 2.8).  from  was  employed  (see s e c t i o n  p l a c e d a t the west edge of the r o o f p o i n t i n g North  (same as p o s i t i o n 1) f o r 30 minutes, then at p o s i t i o n l  1  f o r an a d d i t i o n a l  half-hour.  b)  The  Earthquake E n g i n e e r i n g  Laboratory  System: ( h e r e a f t e r r e f e r r e d to as  System 1 ) . 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 (see (a) above) on the 27 f l o o r (on  an  upper f l o o r f o r o t h e r b u i l d i n g s ) , the f o u r Ranger seismometers were p l a c e d on v a r i o u s f l o o r s to d e f i n e the mode shapes. were taken r e l a t i v e  to a r e f e r e n c e  f l o o r f o r the Harbour C e n t r e ) . at v a r i o u s l e v e l s  The  Ambient v i b r a t i o n measurements  seismometer l o c a t e d on an upper f l o o r remaining  three instruments  were p l a c e d  to r e c o r d s t o r e y movements at the same time and  same d i r e c t i o n as t h i s r e f e r e n c e p o i n t v i b r a t i o n .  The  (23  i n the  phase r e l a t i o n s h i p s  20. of the s t o r e y movements r e l a t i v e to the r e f e r e n c e p o i n t were found by a process of a d d i t i o n and s u b t r a c t i o n of simultaneous d i g i t i z e d  signals.  The  r e l a t i v e displacement amplitudes o f the b u i l d i n g were found by comparing s p e c t r a l amplitudes at the n a t u r a l f r e q u e n c i e s i d e n t i f i e d on the s p e c t r a , which were determined  from a F o u r i e r Transform o f  the  Fourier  simultaneous  r e c o r d i n g s a t v a r i o u s f l o o r s , w i t h the c o r r e s p o n d i n g amplitudes o f the reference f l o o r . shapes. i n two  The  In F i g . 4.4  phase and positions  r e l a t i v e d i s p l a c e m e n t s d e f i n e the mode 1, 2 r e p r e s e n t the l o c a t i o n s of  p e r p e n d i c u l a r 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 .  seismometers  For c e r t a i n  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 by examining  selected performed  the s i g n s o f the r e a l p a r t s of the amplitude c o e f f i c i e n t s  of the F o u r i e r  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 f o u r seismometer n a l s , i n o r d e r to i s o l a t e the f l e x u r a l and i n s t r u m e n t s were assembled upper  floor  sig-  t o r s i o n a l f r e q u e n c i e s , the f o u r  as shown i n F i g . 4.4,  on the 27 f l o o r  (on an  f o r the o t h e r b u i l d i n g s ) , a f t e r the mode shape measurements had  been completed.  The  were then a p p l i e d  same a d d i t i o n and  to the simultaneous  s u b t r a c t i o n methods d e s c r i b e d i n (a) signals.  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 e a r l i e r , was  (7)  then  one d e s c r i b e d  performed.  The r e s u l t s o f the measurements and methods o f a n a l y s i s a r e d i s c u s s e d i n the next c h a p t e r .  (B)  Toronto Dominion Bank and the IBM Towers:  Tower were performed performed  on A p r i l  on A p r i l 11, 1981;  12,  (Measurements o f the  T/D  measurements o f the IBM Tower were  1981).  These b u i l d i n g s a r e the two core of the C i t y of Vancouver.  " b l a c k towers" s i t u a t e d i n the Downtown The  Toronto Dominion Bank Tower i s l o c a t e d  21. on the south s i d e o f G e o r g i a  Street at Granville.  I t i s a 33-storey  s t r u c t u r e w i t h 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 of  w i t h s t a n d i n g a l l o f the d e s i g n h o r i z o n t a l l o a d s .  capable  The b u i l d i n g ' s p l a n  dimensions a r e 150' x 110' and i t r i s e s to a h e i g h t o f about 445*. a machine room on the r o o f , and a mechanical  steel  There i s  room on the 15 and 16 f l o o r  l e v e l s above ground. The  IBM Tower i s s i t u a t e d on n o r t h s i d e o f G e o r g i a S t r e e t a t G r a n v i l l e .  I t i s a l s o a s t e e l s t r u c t u r e w i t h a moment r e s i s t i n g d u c t i l e space frame. The b u i l d i n g l a y o u t c o n s i s t s o f 3 bays by 4 bays having p l a n dimensions o f 110'  and 120' r e s p e c t i v e l y .  1)  The Toronto  I t has 21 s t o r e y s and i s about 320 f e e t  Dominion Bank Tower:  Measurement System 2 (Geophysics) The  tall.  was l o c a t e d on the 28 f l o o r .  seismometers were l o c a t e d i n p o s i t i o n s 1 and 2, F i g . 4.5 f o r the  d e t e r m i n a t i o n o f the t o r s i o n a l and f l e x u r a l f r e q u e n c i e s .  System 1 ( E a r t h -  quake E n g i n e e r i n g L a b ) , 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 t h a t same f l o o r w i t h seismometers p l a c e d i n p o s i t i o n s s i m i l a r to 1  and  2.  Measurements f o r mode shape d e t e r m i n a t i o n were than  taken on v a r i o u s  f l o o r s i n the two p e r p e n d i c u l a r d i r e c t i o n s i n d i c a t e d by 4 and 5, a d o p t i n g the instrument  placement p a t t e r n used i n the Harbour Centre  e a r l i e r i n t h i s chapter. f l o o r f o r the f i r s t to  The r e f e r e n c e l o c a t i o n was e s t a b l i s h e d on the 28  6 s e t o f r e a d i n g s ; i t was then s h i f t e d  overcome a r e s t r i c t i o n on c a b l e l e n g t h s .  18 and 28 f l o o r s were o b t a i n e d programme.  test described  to the 18 f l o o r  Simultaneous r e c o r d i n g s o f the  to p r o v i d e c o n t i n u i t y i n the measurement  A f t e r 12 s e t s o f r e a d i n g s were o b t a i n e d w i t h System 1, the  c o n n e c t i n g c a b l e f o r seismometer No. 3 broke.  The two f i n a l  sets of read-  i n g s and the f i n a l c o l l o c a t i o n t e s t s were completed w i t h o n l y 3 seismometers  22. (seismometers 1, 2,  2)  The  IBM  Tower:  System 2 was  p l a c e d on the 18 f l o o r w i t h seismometers i n the p o s i t i o n s  shown i n F i g . 4.6. determining  4).  Instrument p o s i t i o n s 1 and  the f r e q u e n c i e s i n two  perpendicular  C o l l o c a t i o n c a l i b r a t i o n f o r System 1 was then r e c o r d i n g s were taken w i t h instruments the 18 f l o o r .  Although  r e p a i r e d f o r these uous.  As a r e s u l t ,  2 provided  the f i r s t two  performed on the 18  subsequently  of l i t t l e  ever, s i n c e 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 s i n c e the frequency  readings. for  broken s h i e l d was Seismometer No.  the 9th to the 14th  2 on  temporarily  found  to be d i s c o n t i n -  from t h i s  seismometer  p r a c t i c a l value.  How-  performed at the end  of  the  i d e n t i f i c a t i o n r e a d i n g s were d u p l i c a t e d by available.  remedied i n the f i e l d f o l l o w i n g the second s e t of 2 s u f f e r e d a m a l f u n c t i o n on the 9th data  s e t of r e c o r d i n g s , and  there were o n l y 3 usable  and  3 was  System 2, a u s e f u l s e t of data f o r the v i b r a t i o n a n a l y s i s was The  floor  l o c a t e d s i m i l a r to 1 and  s e t s of readings  turned out to be s a t u r a t e d w i t h n o i s e and  t e s t , and  directions.  the c a b l e f o r seismometer No.  t e s t s , i t s s h i e l d was  information for  f o r the f i n a l  seismometers a v a i l a b l e i n System  s e t , so  collocation,  1.  The mode shape r e c o r d i n g p o s i t i o n s on a t y p i c a l f l o o r a r e i n d i c a t e d by 3, 4 i n F i g . The  4.6.  r e s u l t s of these measurements a r e a l s o presented  i n the  next  chapter.  (C)  Gage Residences:  Towers A,B,C  (measurements performed on May  21,  1981) The  Gage Towers a r e 3 h i g h - r i s e r e s i d e n c e s s i t u a t e d on the UBC  campus.  23. They are of i d e n t i c a l c o n s t r u c t i o n : having p l a n dimensions of about 83' r i s e s to a h e i g h t system of r i g i d The  of about 153'.  shear w a l l s and  square r e i n f o r c e d c o n c r e t e x 83',  buildings  each s t r u c t u r e i s 1 8 - s t o r e y s  and  These b u i l d i n g s have a w e l l developed are  therefore very  purpose of t h i s measurement programme was  properties - natural frequencies  and  stiff. to compare the dynamic  damping v a l u e s - of three  "identical"  buildings. Only System 2 was  used f o r t h i s purpose.  on the roof of each of the as r e p r e s e n t e d  three b u i l d i n g s i n two  by p o s i t i o n s 1 and  2 in Fig.  Measurements were preceded and The  The  followed  r e s u l t s of these t e s t s a r e a l s o r e p o r t e d  seismometers were l o c a t e d perpendicular  directions,  4.7. by c o l l o c a t i o n c a l i b r a t i o n s . i n the next  V  chapter.  24. Chapter  5  Data and Dynamic S t r u c t u r a l A n a l y s e s R e s u l t s  5.1  General  The  procedures  experiments  f o r a n a l y z i n g the d a t a c o l l e c t e d , and  the r e s u l t s o f the  and of the t h e o r e t i c a l a n a l y s e s o f the mathematical  models o f  the s t r u c t u r e s under c o n s i d e r a t i o n , a r e presented i n t h r e e groups i n t h i s chapter as  follows:  A)  The Harbour  B)  The Toronto Dominion and  C)  The  5.2  (A) The Harbour C e n t r e : The  sions.  Centre  three i d e n t i c a l  IBM  Towers  Gage Towers on the UBC  t e s t i n g o f t h i s s t r u c t u r e was The  (Earthquake  first  t e s t s was  performed  E n g i n e e r i n g ) and  I I seismometer was  Campus.  undertaken  on t h r e e s e p a r a t e occa-  on March 24,  System 2 ( G e o p h y s i c s ) .  1981,  u s i n g both System 1  Only one Willmore Mark  used with System 2 i n t h i s p a r t i c u l a r  test.  ( F o r more  d e t a i l s see Chapters 3 & 4 ) .  The wind c o n d i t i o n s d u r i n g t h i s t e s t i n g p e r i o d  were as r e p o r t e d i n Table 5.1  ( u n l e s s otherwise s t a t e d , wind i n f o r m a t i o n has  been o b t a i n e d from the Vancouver Harbour Weather Observing Harbour a t S t a n l e y Park)  S t a t i o n : Coal  25.  T a b l e 5.1 WIND CONDITIONS FOR THE HARBOUR CENTER (MARCH 24, 1981)  AVG SPEED* mph  TIME  DIRECTION  **MEASUREMENT SYSTEM NUMBER  7 to 8 pm  12.7  E  8 to 9 pm  12.7  ESE  9 to 10 pm  12.7  10 to 11 pm  13.8  ENE  1  11 to 12midnight  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 t o 4 am  4.6  ESE  1  SE-  1  (Earthquake Engineering) and, 2 (Geophysics) 1 and, 2 1 and, 2  * 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 E n g i n e e r i n g SYSTEM 2 = Geophysics (one seismometer o n l y ) .  26.  The second t e s t was performed on May 12, 1981, w i t h System 2 (2 Willmore Mark I I seismometers were employed). the  phase r e l a t i o n s h i p  The purpose o f t h i s t e s t was to check  o f the motions between the o b s e r v a t i o n deck o f the  r e s t a u r a n t tower and some lower f l o o r s o f the o f f i c e b u i l d i n g ,  and a l s o to  v e r i f y the t o r s i o n and f l e x u r a l f r e q u e n c i e s o b t a i n e d from the f i r s t The check f o r phase r e l a t i o n s h i p 1981.  The equipment  test.  was repeated a second time on 19th December  used f o r t h i s t e s t has been d e s c r i b e d i n Chapter 4.  The wind c o n d i t i o n s d u r i n g these two check t e s t s a r e g i v e n i n T a b l e 5.2. T a b l e 5.2 Wind C o n d i t i o n s F o r The Harbour Center (May 12, 1981) and (Dec. 19, 1981)  AVG SPEED mph  TIME  DIRECTION  MEASUREMENT SYSTEM NO.  1 to 2 pm  6.9  WSW  2*  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  DATE TESTED  May  10 to 11 am 11 to noon noon to 1 pm  12, 1981  3.5**  W  2*  4.6  S  2  ssw  2  Dec.  4.6**  * 2 Seismometers i n System 2 ** With s t r o n g g u s t s up to 12.7 mph  19, 1981  27. Samples o f t y p i c a l a n a l o g t r a c e s o b t a i n e d u s i n g System 2 a r e shown i n Appendix A4. filter  The  r e c o r d e d analog data were played back through an  (low pass c o r n e r frequency a t 5 Hz - there were no s i g n i f i a n t  quencies above 3Hz) fied  analog  and d i g i t i z e d u s i n g an AN5800 A-D  f o r simultaneous  sample and h o l d , a Kennedy 8108  t r a n s p o r t with Kennedy B203 b u f f e r e d f o r m a t t e r . 39.075 Hz w i t h a N y q u i s t  The  frequency i n excess of 19.5  fre-  c o n v e r t e r system modi9-track d i g i t a l  tape  sampling  rate  Hz.  digitized  The  was data  had equal time i n t e r v a l s of  .0256 seconds and were r e c o r d e d on tapes compa-  t i b l e w i t h the UBC  facilites.  Computer  The d i g i t i z e d d a t a were analyzed  to determine  f r e q u e n c i e s and mode shapes of the s t r u c t u r e and  (Step 1) the n a t u r a l  (Step 2) the  corresponding  damping v a l u e s a c c o r d i n g to the f o l l o w i n g procedures: Step ( 1 ) :  Data were f i r s t  a n a l y s e d by a m o d i f i e d v e r s i o n of UBC  E n g i n e e r i n g programme l i b r a r y new  ' s p e c t r a ' ( C h e r r y and  v e r s i o n of ' s p e c t r a ' , a s u i t a b l e bandwidth was  Topf  (7)).  Civil From t h i s  chosen f o r the damping  e s t i m a t i o n procedure, which i s o u t l i n e d i n Step ( 2 ) . To d e c i d e on the proper bandwidths to use, we i t was  noted  t h a t f r e q u e n c i e s as c l o s e as  recall  t h a t i n Chapter  .01 Hz must be s e p a r a t e d to  i s o l a t e the p o s s i b l e c l o s e v i b r a t i o n modes of the Harbour C e n t r e . a l s o from F i g . 2.6.1  4  t h a t by u s i n g the Hanning window to reduce  We  note  leakage  the  width of the main lobe of the k e r n e l f u n c t i o n i s a c t u a l l y widened, hence the power to r e s o l v e c l o s e f r e q u e n c i e s i s much reduced. both the window and  i t s t r a n s f o r m to be narrow.  In r e a l i t y ,  i m p o s s i b l e s i t u a t i o n and we must seek a compromise. continuous v a r i a b l e , the use of r e s u l t i n both a window and possible.  J.F.  I d e a l l y we  would  like  t h i s i s an  In the domain o f a  ' p r o l a t e s p h e r o i d a l f u n c t i o n s ' are known to  i t s t r a n s f o r m width which a r e l i m i t e d as much as  K a i s e r worked out the window weights  a b l e domain, as approximations  i n the d i s c r e t e  vari-  f o r the continuous v a r i a b l e s , to be used  on  28.  the F o u r i e r c o e f f i c i e n t s .  These windows a r e known as the K a i s e r - B e s s e l  windows and the formulae a r e c o n t a i n e d  i n Hamming's book ( 1 9 ) .  Two such windows have been programmed i n t o ' s p e c t r a ' : K a i s e r - B e s s e l 2sample (K-B2) and 3-sample (K-B3) c o n v o l u t i o n windows i n the domain.  The weights have been c a l c u l a t e d from the formula:  frequency (from Hamming  (19))  2N s i n h [ ot/l - ( — ) ] 2  W(w) = a Io( a)  A - (—)  2  wa  where: N  i s the h a l f w i d t h o f the even window f u n c t i o n  wa  is — N  a  i s a constant  cx  to c o n t r o l the "shape" o f the window.  For the 2-  sample (K-B2) window a = 1.333; f o r the 3-sample (K-B3) a = 2.5. a = 1.333 w i l l g i v e a k e r n e l ( i n the frequency main l o b e and h i g h e r  i  ( va )  o  W(w)  domain) o f narrower  s i d e l o b e l e v e l s than a = 2.5.  = i + I [isainl]2  '  , n=l  L  L  n!  J  a r e the weights used f o r the c o n v o l u t i o n i n the frequency domain.  To t e s t the f u n c t i o n s o f the r e c t a n g u l a r (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  content  time i n p u t w i t h  o f 0.26 and 0.27 Hz was mixed with random n o i s e .  the known i n p u t f r e q u e n c i e s t i o n was F o u r i e r transformed,  (.26,  .27 Hz) were e q u a l .  frequency  The magnitudes o f  The generated  func-  and then passed through the f o u r windows t o  examine t h e i r r e s p e c t i v e r e s o l v i n g powers. The  e f f e c t s o f the d i f f e r e n t r e s o l v i n g powers are shown i n F i g .  5.1.  A l t h o u g h 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. p o s s i b l e t h a t i f the two  f r e q u e n c i e s i n the i n p u t s i g n a l were d i f f e r e n t i n  magnitude ( f o r example i f one o t h e r ) , the box-car may  fail  frequency  was  100  times  s t r o n g e r than  t o t a l l y to r e s o l v e them because the  from the h i g h s i d e l o b e s of the box-car may t o t a l l y swamped ( H a r r i s ( 2 6 ) ) .  the  leakage  cause the weaker s i g n a l to be  In t h i s i n s t a n c e , the Hanning window f a i l e d  to r e s o l v e the c l o s e f r e q u e n c i e s (bandwidth used throughout the t e s t s i s .004  Hz);  but the K-B2  and  were o b v i o u s l y h i g h e r  K-B3  windows were s u c c e s s f u l .  than the random n o i s e l e v e l .  windows have been used to separate tudes i n a r a t i o of up The  K-B2  .0048 Hz).  or K-B3,  c l o s e f r e q u e n c i e s having  When a ' g o o d - q u a l i t y '  the smoothing process  dure can be found  quencies  i n Harris (26).  "close-frequency separated  window i s used, such as Hanning,  f r e q u e n c i e s a r e separated  100:1.  Hz.  Hence, when more than one  made to use  of the Harbour Center  d e t a i l s of t h i s  "overlap"  proce-  percentage o v e r l a p b u i l t i n t o the  new  There may  still  .01 Hz,  f r e q u e n c i e s may  two  fre-  be a p o s s i b i l i t y that o r t h a t the amplitude  be' i n a r a t i o of more  two coeffi-  than  they a r e not worth s e p a r a t i n g (Tukey ( 1 6 ) ) , s i n c e a  f i n e r e s o l u t i o n spectrum may d e c i s i o n was  The  data  incorpora-  j u s t d e s c r i b e d d i d not show any  by l e s s than  to these  In e i t h e r case  The  to  40%.  search"  by 0.01  c i e n t s corresponding  K-B3  data ( u s i n g a band-  t h a t r e s u l t s w i l l cause the data  use of the d a t a .  'Spectra' programme i s about The  and  unequal magni-  i n c l u d e d f o r a n a l y s i s , an " o v e r l a p " read procedure was  ted to make e f f i c i e n t  levels  to a 100:1. ( H a r r i s ( 2 6 ) ) .  taper o f f at the ends of each data b l o c k . b l o c k was  signal  In f a c t , K-B2  f o u r windows were a p p l i e d to the Harbour Centre  width of  The  .009  data.  be  s t a t i s t i c a l l y unstable.  Therefore  Hz as the bandwidth i n the subsequent  the analyses  30. The d a t a r e c o r d e d w i t h System 1 on the 27* f l o o r o f the Harbour 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 t o r s i o n a l  frequencies.  Centre  Simulta-  neous r e c o r d i n g s were added and s u b t r a c t e d , and the r e l a t i v e growth and decay o f the out-of-phase  and in-phase  signals  i n the East-West  direction  can be e a s i l y d e t e c t e d , as shown i n F i g . 5.2. The  f r e q u e n c i e s i n the North-South  direction  F o u r i e r s p e c t r a o b t a i n e d v i a the same p r o c e s s . s e l e c t e d from v i b r a t i o n North-South is basically  direction  were determined  The t o r s i o n a l  from t h e  signals  r e c o r d s i n which they were more s i g n i f i c a n t  i n this case).  The method f o r e s t a b l i s h i n g  were (the  the phase  as d e s c r i b e d f o r the f l e x u r a l mode d e t e r m i n a t i o n s . Table 5.3 MEASURED NATURAL FREQUENCIES OF THE HARBOUR CENTER  DIRECTION  MODE 1  MODE 2  MODE 3  NORTH SOUTH  0.47  1.47  2.19  EAST WEST  0.46  1.50  1.65  TORSION DOMINANT  0.72  1.33  2.17  The measured f r e q u e n c i e s f o r the Harbour Centre a r e l i s t e d 5.3.  The mode shapes c o r r e s p o n d i n g  5.3(a) and (b) ( f l e x u r a l )  to these f r e q u e n c i e s a r e shown i n F i g s .  and F i g . 5.4 ( t o r s i o n a l ) .  these shapes were determined  i n Table  The manner i n which  has been d e s c r i b e d i n Chapter 4.  * I t should be noted t h a t the f l o o r numbering system f o r the Harbour Center does not i n c o r p o r a t e the number 13. In a l l our f i g u r e s , t a b l e s and d i s c u s s i o n s we have chosen to i g n o r e s u p e r s t i t i o n and number each f l o o r s u c c e s s i v e l y , i n c l u d i n g 13. Hence the f l o o r a c t u a l l y numbered 28 i n the b u i l 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 F i g u r e 5.3(a) and elsewhere i n t h i s thesis. S i m i l a r l y , i n o t h e r f l o o r s beyond the 13th f l o o r .  31. The mode shapes a r e based  on d a t a gathered on March 24, 1981 and Dec.  19, 1981. The d a t a from the May 12, 1981 measurements were omitted due to a d i g i t i z i n g hardware m a l f u n c t i o n which l e d to u n c e r t a i n t i e s 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 d e r i v e d from data secured on March 24, 1981. A 'normal'  f i r s t mode d i d not seem to e x i s t -  t h a t i s , t h e r e was a zero c r o s s i n g i n the mode c o r r e s p o n d i n g to the fundamental  frequency and a d d i t i o n a l zero c r o s s i n g s i n the modes f o r each  c e s s i v e frequency ( F i g . normal t h i r d mode.  5 . 3 ( b ) ) ; f o r example, the second mode l o o k e d l i k e a  To ensure  that t h i s was not the r e s u l t o f an instrument  o r i e n t a t i o n o r d a t a r e d u c t i o n e r r o r a f u r t h e r t e s t was performed 1981.  suc-  The data from t h i s t e s t was used  on Dec. 19,  to check the phase r e l a t i o n s h i p  between the o b s e r v a t i o n deck and the 27 f l o o r , and a l s o the o b s e r v a t i o n deck and  the 25 f l o o r o n l y .  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 w i t h the March 24, 1981 data i n the N-S and t o r s i o n a l tions.  In the E-W d i r e c t i o n ,  f o r the fundamental  direc-  mode shape, a r e v e r s e i n  phase o c c u r r e d i n the Dec. 19 data as compared to the March 24 d a t a , and a minor d i f f e r e n c e i n the t h i r d mode shape ( F i g .  5.3(b)) was a l s o  noted.  I t i s p o s s i b l e t h a t t h i s unusual o c c u r r e n c e can be a t t r i b u t e d to the e x i s t e n c e o f repeated f r e q u e n c i e s (fundamental uniqueness  o f t h e i r a s s o c i a t e d mode shapes.  d i s c u s s e d i n the f i n a l Step ( 2 ) :  section of this  and second) and the non-  T h i s e x p l a n a t i o n w i l l be  thesis.  Using the bandwidth d e f i n e d i n Step ( 1 ) , two t e c h n i q u e s were  a p p l i e d to s e l e c t e d  s e t s o f data used  values: i)  The p a r t i a l moment method  ii)  The a u t o c o r r e l o g r a m method.  i n Step (1) to o b t a i n t h e damping  32. Since these methods have been d i s c u s s e d a t some l e n g t h i n Chapter  2 and  Appendix B2 o n l y a few a d d i t i o n a l remarks need be made here. The a u t o c o r r e l o g r a m method can be a p p l i e d to i n d i v i d u a l w e l l - d e f i n e d s p e c t r a l peaks f i l t e r e d  out from the F o u r i e r s p e c t r a .  Since t h i s method  o p e r a t e s i n the time domain, the process o f t r a n s f o r m / i n v e r s e - t r a n s f o r m may cause  some problems i f no a t t e n t i o n i s p a i d to the l i m i t a t i o n s o f the f i l -  t e r i n g process ( s e c t i o n B2.9). does not d i f f e r  The b a s i c concept o f t h i s f i l t e r i n g  process  from the ' g l o b a l smoothing' procedure d i s c u s s e d i n Appendix  A2. The  p a r t i a l moment method does not depend as much on a " w e l l - d e f i n e d "  s p e c t r a l peak, as i t i n v o l v e s the area and a l s o the f i r s t  and second  t r a l moments o f the power s p e c t r a d e n s i t y p l o t w i t h i n c e r t a i n frequency l i m i t s .  spec-  specified  I t i s known to be more s t a b l e (20) than methods which  depend on the o r d i n a t e s o f the s p e c t r a l d e n s i t y ; smoothing p r o c e s s e s a r e not needed to r e f i n e the damping e s t i m a t e s o b t a i n e d from "raw" s p e c t r a l  esti-  mates . The % c r i t i c a l damping  £ determined  Centre a r e r e p o r t e d i n Table 5.4  by these methods f o r the Harbour  The damping v a l u e s o b t a i n e d by both  methods i n g e n e r a l a r e comparable i n terms o f t h e i r o r d e r o f magnitude. percentage  l a g i s an important c o n s i d e r a t i o n i n the a u t o c o r r e l o g r a m method  ( 1 8 ) , and s u f f i c i e n t l a g time to produce a u t o c o r r e l o g r a m i s recommended.  a minimum o f 5 peaks i n the  To achieve t h i s g o a l the l a g time f o r some  o f the lower f r e q u e n c i e s was i n c r e a s e d ( s e e note below t a b l e ) . The ratios  The  p a r t i a l moment method may be s e n s i t i v e to the c u t o f f frequency  ( s e c t i o n B2.9) but e f f o r t s have been made to keep the frequency  r a t i o s v e r y c l o s e to u n i t y .  33. Table  5.4  DAMPING VALUES FOR HARBOUR CENTER  5 (% CRITICAL DAMPING) FREQ (Hz)  DIRECTION AUTOCORRELOGRAM  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  * NOTE:  PARTIAL MOMENT  F i g . 5.5(a) to (g) f o r Autocorrelograms  Damping v a l u e based on 30% Lag (Time Lag Number = 600) Instead of the 15% Lag Shown i n F i g . 5.5(b).  The v a l u e s o f K f o r the f l e x u r a l and t o r s i o n a l f r e q u e n c i e s have been obtained  from the r e c o r d s which gave the h i g h e s t s i g n a l to n o i s e r a t i o f o r  the p a r t i c u l a r frequency The earlier  percentage  concerned.  l a g f o r the autocorrelograms  As mentioned  the v a l u e s o f $1 , fi^ (see s e c t i o n B2.9 b) f o r the p a r t i a l moment  method a r e chosen to be c l o s e to u n i t y , and & = a  are w e l l separated limits.  i s about 15%.  there i s no d i f f i c u l t y  Q^.  As the d i f f e r e n t modes  i n s e l e c t i n g the frequency  34. The p a r t i a l moment method a l s o g i v e s ( a s output) a n a t u r a l frequency upon the s p e c t r a l moment c a l c u l a t i o n s . listed  i n Table 5 . 4 ,  based  T h i s output frequency, which i s  can be compared to the i n p u t frequency, which comes  from the v i s u a l i n s p e c t i o n o f the s p e c t r a l p l o t .  In the case o f the Harbour  Centre t h e r e were no s i g n i f i c a n t d i f f e r e n c e s between the i n p u t and output natural frequencies.  T h i s i n d i c a t e s t h a t the (damping e s t i m a t e s )  results  o b t a i n e d by t h i s method a r e s a t i s f a c t o r y .  (B)  IBM Tower and T/D Tower:  Step 1:  A bandwidth o f 0.02 Hz was chosen a f t e r performing the process as  d e s c r i b e d i n Step 1 f o r the Harbour Centre, i)  The T/D Tower: The wind c o n d i t i o n s on the day o f measurement were as r e p o r t e d i n Table  35. T a b l e 5.5 Wind C o n d i t i o n s F o r t h e 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 a n a l y s i s o f the r e c o r d e d d a t a r e s u l t e d i n the F o u r i e r s p e c t r a in Fig.  5.6 from which the n a t u r a l f r e q u e n c i e s f o r the T/D Tower were  established.  The r e s u l t s a r e g i v e n i n T a b l e  5.6(a).  36. Table 5.6(a) Measured N a t u r a l F r e q u e n c i e s  o f T/D Tower  MODE 3 FREQ (Hz)  DIRECTION  MODE 1 FREQ (Hz)  MODE 2 FREQ (Hz)  NORTH-SOUTH (flexural)  .26  .75  1.89  EAST-WEST (flexural)  .24  .68  1.41  TORSIONAL  .83  .99  1.21  The mode shapes a s s o c i a t e d w i t h some o f these 5.7(a) t o 5 . 7 ( c ) .  f r e q u e n c i e s a r e shown i n F i g s .  I t can be seen t h a t the c r o s s - o v e r  p o i n t s f o r 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), a lower than t y p i c a l b u i l d i n g e l e v a t i o n .  5.7(b) a r e l o c a t e d a t  T h i s i s l i k e l y due t o the e x i s t -  ance o f m e c h a n i c a l f l o o r s a t the 15 and 16 s t o r e y e l e v a t i o n s , r e s u l t i n g i n a s o f t e n i n g o f the s t r u c t u r e a t these The  levels.  t h e o r e t i c a l n a t u r a l f r e q u e n c i e s and mode shapes o f t h i s b u i l d i n g  were a v a i l a b l e and p r o v i d e d University. corresponding  by P r o f e s s o r Jean-Guy B e l i v e a u o f Sherbrooke  The t h e o r e t i c a l f r e q u e n c i e s a r e g i v e n i n T a b l e 5.6(b) and the mode shapes i n F i g . 5.7.  I t can be seen that there i s good  agreement between the measured and computed fundamental f l e x u r a l and  that the low e x p e r i m e n t a l  c r o s s over p o i n t s noted above a r e confirmed by  analysis. Table 5.6(b) A n a l y t i c a l Natural Frequencies  DIRECTION  frequencies  MODE 1 Hz  o f T/D Tower  MODE 2 Hz  MODE 3 Hz  NORTH-SOUTH (flexural)  .27  .58  .97  EAST-WEST (flexural)  .22  .48  .77  37. ( i i ) The IBM Tower: Wind C o n d i t i o n s d u r i n g the ambient t e s t s were as shown i n Table 5.7  T a b l e 5.7 Wind C o n d i t i o n s f o r the IBM Tower ( A p r i l 12, 1981)  TIME  Noon -  AVG SPEED mph  DIRECTION  5.8  ESE  1 pm 1 - 2 pm  1 2 1 2  below  2.3  2 - 3 pm  MEASUREMENT SYSTEM USED  WSW  1 2  10.4  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  1  2.3  7 - 8 pm  10.4  E  1  8 - 9 pm  9.2  E  1  38. F i g . 5.8 p r e s e n t s the F o u r i e r s p e c t r a f o r the IBM Tower. quencies  identified  from  ponding  mode shapes a r e drawn i n F i g . 5.9.  The n a t u r a l f r e -  these s p e c t r a a r e summarized i n Table 5.8.  Corres-  T a b l e 5.8 Measured N a t u r a l F r e q u e n c i e s o f IBM Tower  The  DIRECTION  MODE 1 FREQ (Hz)  MODE 2 FREQ (Hz)  MODE 3 FREQ (Hz)  NORTH-SOUTH (flexural)  0.38  1.20  2.14  EAST-WEST (flexural)  0.42  1.30  2.00  TORSIONAL  1.43  1.76  2.48  UBC C i v i l E n g i n e e r i n g Computer programme MAC.FRAME was used to  a n a l y z e a c o n c e p t u a l model o f the IBM Tower.  The Tower was s e t up as a 3-  d i m e n s i o n a l 21 s t o r e y frame s t r u c t u r e , w i t h 3 degrees The  s t i f f n e s s o f the frames were c a l c u l a t e d  the members and f l o o r s .  from  o f freedom per f l o o r .  the s t r u c t u r a l drawings o f  The r e s u l t i n g f r e q u e n c i e s and mode shapes o b t a i n e d  from these a n a l y s e s a r e d e t a i l e d  i n Table 5.9 and F i g . 5.9 r e s p e c t i v e l y .  Table 5.9 A n a l y t i c a l N a t u r a l F r e q u e n c i e s o f the IBM Tower  DIRECTION  MODE 1 Hz  MODE 2 Hz  MODE 3 Hz  N-S flexural  0.37  1.09  1.98  E-W flexural  0.48  1.39  2.51  Torsional  1.37  3.79  There i s good agreement between the measured and computed f r e q u e n c i e s .  39. Step 2: ( i ) T/D Tower The damping v a l u e s determined 5.10(a),  f o r the T/D Tower a r e g i v e n i n Table  5.10(b). T a b l e 5.10(a) Damping V a l u e s o f T/D Tower  FREQ Hz  DIRECTION  £ (% CRITICAL DAMPING) AUTOCORRELOGRAM* PARTIAL MOMENT 2.64  0.26  N-S (flexural)  0.75  N-S (flexural)  1.89  N-S (flexural)  0.24  E-W (flexural)  4.12  2.15  0.68  E-W (flexural)  1.54  2.42  1.82  1.76  2.92  Torsion  0.99  3.58  *Percentage Lag i s About 15% T a b l e 5.10(b) Damping V a l u e s o f T/D Tower ( M a n - E x c i t e d ) * *  FREQ Hz  DIRECTION  E, (% CRITICAL DAMPING) (LOG DECREMENT METHOD)  0.67  E-W (flexural)  2.4  1.40  E-W (flexural)  1.9  0.74  N-S (flexural)  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 f r e q u e n c i e s (as d e f i n e d e a r l i e r i n t h i s c h a p t e r ) from the p a r t i a l moment method a r e p r a c t i c a l l y the same as the i n p u t n a t u r a l frequencies. The damping v a l u e s from the a u t o c o r r e l o g r a m s , p a r t i a l moment and manexcited  t e s t s a r e t y p i c a l o f the r e s u l t s  from such t e s t s ;  the d i f f e r e n t  methods y i e l d comparable v a l u e s . Step 2; ( i i ) IBM Tower The damping v a l u e s determined 5.11.  f o r the IBM Tower a r e r e p o r t e d i n T a b l e  F i g s . 5.11(a) - (e) show some o f the a u t o c o r r e l o g r a m v s . Lag time  plots. Table 5.11 Damping V a l u e s o f IBM Tower  FREQ (Hz)  C (% CRITK:AL DAMPING)  DIRECTION  AUTOCORRELOGRAM **  PARTIAL MOMENT  0.38  N-S (flexural)  1.20  N-S (flexural)  2.14  N-S (flexural)  0.42  E - W (flexural)  3.62  3.12  1.30  E - W (flexural)  3.01  2.27  2.00  E - W (flexural)  1.43  TORSIONAL  2.37  2.52  1.76  TORSIONAL  3.61  3.54  **Percentage  3.35  2.19  0.78  1.16  1.12  Lag i s About 15%  41. (C)  Gage Residences The measured n a t u r a l f r e q u e n c i e s f o r the three towers a r e summarized i n  Table 5.12. Table 5.12 Gage Towers N-S F r e q u e n c i e s  DIRECTION  0.34 (flexural)  0.34 (flexural)  0.35 (flexural)  0.62 (flexural)  0.62 (flexural)  0.61 (flexural)  0.03 (flexural)  0.85 (flexural)  0.93 (flexural)  1.17 (torsion)  1.17 (torsion)  1.18 (torsion)  1.26 (torsion)  1.26 (torsion)  1.23 (torsion)  0.57 (flexural)  0.57 (flexural)  0.56 (flexural)  0.78 (flexural  0.79 (flexural)  0.79 (flexural)  1.23 (flexural)  1.21 (flexural)  1.18 (flexural)  NORTH-SOUTH  •TOWER C' Hz  EAST-WEST  TORSION  TOWER B Hz  TOWER A Hz  In the N-S d i r e c t i o n the data d e f i n i n g the f l e x u r a l f r e q u e n c i e s had a r e l a t i v e l y low s i g n a l to n o i s e r a t i o such t h a t a m e a n i n g f u l d e t e r m i n a t i o n o f the damping by the p a r t i a l moment method was not p o s s i b l e .  However, f o r the  t o r s i o n a l modes the s i g n a l to n o i s e r a t i o was s i g n i f i c a n t and the damping v a l u e s from the p a r t i a l moment method f o r these Table 5.13.  frequencies are l i s t e d i n  A l s o r e p o r t e d a r e the E-W damping v a l u e s , which were o b t a i n e d  from data w i t h a good s i g n a l to n o i s e  ratio.  42. Table 5.13 Gage Towers Damping V a l u e s  NORTH-SOUTH EAST-WEST  TOWEII C  TOWI:R B  TOWER A  DIRECTION  K  FREQ  1.17  0.85  1.18  0.61  0.99  1.26  1.09  1.23  1.09  0.57  4.39  0.57  2.53  0.56  2.24  0.78  2.70  0.79 .  2.46  0.79  2.43  1.23  0.58  1.21  1.00  1.18  2.47  FREQ  FREQ  Hz  %  Hz  1.17  1.33  1.26  %  K  Hz  %  There are some s m a l l d i f f e r e n c e s i n the n a t u r a l f r e q u e n c i e s and v a l u e s f o r the 3 Gage Towers, but i n g e n e r a l t h e i r dynamic reasonable  agreement w i t h one another.  damping  p r o p e r t i e s are i n  Hence they can be c o n s i d e r e d as  b u i l d i n g s of i d e n t i c a l d e s i g n and c o n s t r u c t i o n .  43. Chapter 6  D i s c u s s i o n o f R e s u l t s and C o n c l u s i o n s The p r i n c i p l e r e s u l t s o b t a i n e d from t h i s i n v e s t i g a t i o n i z e d as  may be summar-  follows:  • The n a t u r a l  f r e q u e n c i e s o f the Harbour C e n t r e , T/D and IBM Towers were  determined from ambient .38Hz r e s p e c t i v e l y Hz r e s p e c t i v e l y  v i b r a t i o n measurements to be 0.47 Hz, .26 Hz and  i n the North-South d i r e c t i o n and .46 Hz, .24 Hz and .42  i n the East-West d i r e c t i o n .  In the case o f the T/D and IBM  towers the f r e q u e n c i e s were found to agree c l o s e l y w i t h the r e s u l t s o b t a i n e d 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% h i g h e r i n the N-S and 64% h i g h e r 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  Hz E-W) o b t a i n e d from the Tso-Bergmann model. torsional Hz).)  results  (.27 Hz N-S;  (The measured  .28  fundamental  f r e q u e n c y (.72 Hz) i s 177% h i g h e r than the t h e o r e t i c a l  r e s u l t (.26  The Tso-Bergmann model appears to be too f l e x i b l e and may n o t account  for interactions  o f n o n - s t r u c t u r a l 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 s e c u r i n g , from the measured d a t a i n the  E-W d i r e c t i o n o f the Harbour Center s t r u c t u r e ,  first  and second mode  shapes having the u s u a l o r normal appearances. It  was found from measurements taken on two s e p a r a t e o c c a s i o n s that t h e  first  2 mode shapes c o u l d not be u n i q u e l y determined.  first  test  (March 24, 1981) y i e l d e d  The d a t a from the  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. first  The data from the second t e s t  (December 19, 1981) p r o v i d e d a  mode having a normal shape and a second mode r e s e m b l i n g a t h i r d mode  configuration.  44. Although a c l o s e frequency s e a r c h d i d not f i n d any f r e q u e n c i e s separated by .01 Hz, i t i s s t i l l  p o s s i b l e t h a t two v e r y c l o s e f r e q u e n c i e s e x i s t a t  f i n e r than 0.01 Hz o r t h a t the amplitude  c o e f f i c i e n t s o f f r e q u e n c i e s separa-  ted by 0.01 Hz may have a r a t i o h i g h e r than 100:1 (Chapter 5 ) , such t h a t one s i g n a l would be swamped by the o t h e r . not worth s e p a r a t i n g even i f they e x i s t  In e i t h e r case, c l o s e f r e q u e n c i e s a r e (16), since a f i n e 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 u n s t a b l e . T h e o r e t i c a l l y , i f two f r e q u e n c i e s a r e e q u a l , then t h e i r mode shapes cannot  be u n i q u e l y d e f i n e d ( 2 9 ) .  In the case o f the Harbour Centre  West d i r e c t i o n we may have two v e r y c l o s e l y spaced  East-  f r e q u e n c i e s ( t h e funda-  mental and second modes) such t h a t they i n t e r f e r e w i t h each o t h e r , g i v i n g a s i n g l e mode shape which i s not unique. that the observed  second  Then the l o g i c a l e x t e n s i o n would be  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 f o r 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  f o r a g i v e n l e n g t h o f r e c o r d , and seems to be l i t t l e  by the smoothing p r o c e s s .  affected  The damping v a l u e o b t a i n e d by t h i s method f o r the  Harbour Center fundamental mode (N-S) i s 1.11% whereas those f o r the T/D and IBM  Towers a r e 2.64% and 3.35% r e s p e c t i v e l y .  are t y p i c a l o f r e s u l t s determined  These low v a l u e s o f damping  by ambient v i b r a t i o n  tests.  F o r an e x p e r i m e n t a l programme such as t h i s one, s t a b l e e s t i m a t e s o f power s p e c t r a l d e n s i t y or F o u r i e r s p e c t r a can be e a s i l y a c h i e v e d .  Thus the  a u t o c o r r e l a t i o n method can be a p p l i e d to the d a t a and good damping  estimates  can be o b t a i n e d . and  The damping v a l u e s o b t a i n e d f o r the  Harbour Centre, T/D  IBM Towers by t h i s method a r e comparable to the r e s u l t s o b t a i n e d  the p a r t i a l moment method.  As w e l l , the man-excited  t e s t damping  from  results  f o r the T/D Tower a r e a l s o w i t h i n the same range o f v a l u e s as the p a r t i a l moment and a u t o c o r r e l a t i o n methods.  45. • The " i d e n t i c a l " Gage Towers b u i l d i n g s have v e r y s i m i l a r ties. and,  The r e s p e c t i v e f r e q u e n c i e s are p r a c t i c a l l y  dynamic  proper-  the same i n a l l c a s e s ,  f o r the most p a r t , the damping v a l u e s a r e s u f f i c i e n t l y c l o s e to warrant  c o n s i d e r i n g the 3 b u i l d i n g as d y n a m i c a l l y e q u i v a l e n t .  46.  REFERENCES  (1)  F.G.  Udwadia and  of F u l l  Scale  (2)  Sheldon Rubin, "Ambient V i b r a t i o n Survey o f O f f s h o r e P l a t f o r m " , Eng. Mech., June 1980, V o l . 106.  ASCE,  (3)  S. 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Toaka, " D i g i t a l F i l t e r i n g o f Ambient Response Data", ASCE-EMD S p e c 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 " ,  Prentice-Hall,  1977.  (20) E.H. Vanmarcke and R.N. Iascone, " E s t i m a t i o n of Dynamic C h a r a c t e r i s t i c s of Deep Ocean Tower S t r u c t u r e s " , MIT Sea Grant P r o j e c t , June 1972. (21) P. Durning and D. E n g l e , " V i b r a t i o n I n s t r u m e n t a t i o n System Measures an O f f s h o r e P l a t f o r m ' s Response to Dynamic Loads", ASCE-EMD, S p e c i a l t y Conference, March 1976. (22) S.O. R i c e (1945), "Mathematical A n a l y s i s o f Random Noise", B e l l System T e c h n i c a l J o u r n a l , P a r t I = V o l . 23, 1944, pg. 282-332, P a r t I I = V o l . 24, pg. 46-156. (23) G.M. J e n k i n s , "General C o n s i d e r a t i o n s i n the A n a l y s i s of S p e c t r a " , Technometries, May 1961, V o l . 3, No. 2. (24) T . J . U l r y c h and T.N. Bishop, "Maximum Entropy S p e c t r a l A n a l y s i s and A u t o r e g r e s s i v e Decomposition", Reviews o f Geophysics and Space P h y s i c s , Feb. 1975, V o l . 13, No. 1. (25) R.A.  Wiggins,  " I n t e r p o l a t i o n o f D i g i t i z e d Curves", BSSA, V o l . 66,  1976.  (26) F . J . H a r r i s , "On the Use of Windows f o r Harmonic A n a l y s i s With the D i s c r e t e F o u r i e r Transform", P r o c e e d i n g s , IEEE, V o l . 66, No. 1, Jan, 1978. (27) F. K o l l a r and R.D. R u s s e l l , "Seismometer A n a l y s i s Using an c u r r e n t Analog", BSSA, V o l . 56, Dec. 1966, pg. 1193-1205.  Electric  (28) W.K. Tso and R. Bergmann, "Dynamic A n a l y s i s o f an Unsymmetrical High R i s e B u i l d i n g " , Canadian J o u r n a l o f C i v i l E n g i n e e r i n g , 3, 107 (1976). (29) L. M e i r o v i t c h , "Elements o f V i b r a t i o n A n a l y s i s " , McGraw-Hill,  1975.  48.  Velocity S e n s i t i v i t y (Horizontal)' 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  FREQUENCY , Hz  FIG.  2.1  INSTRUMENT C A L I B R A T I O N CURVE  50  100  49.  -0.4'  0  -i; 0.1  i  0.2  •  i  i  0.3  0.4  •  0.5  NON-DIMENSIONALIZED VARIABLE NOTE:- R e c t a n g u l a r window i s e q u i v a l e n t to. D i r i c h l e t ' s kerne f u n c t i o n w i t h N=5. Only h a l f the frequency response f u n c t i o n s a r e p l o t t e d , ( n o n - d i m e n s i o n a l i z e d v a r i a b l e ) from 0 t o 0.5 because o f t h e i r symmetry.  FUNCTIONS: AFTER HAHiUNG ( 1 9 ) .  50.  OBSERVATION DECK  MECHANICAL ROOM,ROOF  FIG,  A-1  ELEVATION HARBOUR  OF  CENTER  i  i  i  i  i 7  3  -  iH : n r n ' ^.ym™.  T i n •• i ;  i-»ii-aaaj . '=—p» i-  ;::nna 'tfD :bnn;-jjtrT !;-nrici. innn"nnh. im^ :  51.  /V- S  BLBVAJ/ON  DYMAMIC  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  o f N-S  MOOB.L.  direction.  1 ' In  *  ^i  \  *M  \ MOOS. / T/'374J  Lateral  FIG.  4.2  HARBOUR SHAPES  MOOS. Pi./S/r  MOO£. 3 7>-OHOJ-  mode shapes.(N-S)  CENTER (FROM  7  MASS  REF.  DISTRIBUTION  28)  AND  MODE  52. N  ®  ELEVATOR SHAFT  STAI R WELL 4 ©  © . ®  J®  ©^  © ©  1  SHOWS SENSITIVE AXIS OF INSTRUMENT  FIG.  4-3  PLAN  SHOWING  VIEW  OF  HARBOUR  SEISMOMETER  CENTER  LOCATIONS.  53.  V  ^3 E LE VATO R SHAFT  STAIR WELL  SEISMOMETER  I G , HA  (FLEXURAL 8 TORSIONAL FREQ.)  SEISMOMETER ~L0£AT.M ON TYPI GAll"' OFF ICE FLOOR IR CENTRE. ' '  0  }  APPROX. POSITION OF ELEVATOR SHAFT  1  1  0  ©  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  PLAN  OF  ROOF  SHOWING  OF  GAGE  RESIDENCE  SEISMOMETER  TOWERS  LOCATIONS.  (TYPICAL)  FOURIER SPECTRUM RELATIVE AMPLITUDE  RELATIVE AMPLITUDE  FREQUENCY,Hz  2 3 IE  TORSION DOM IN A N T { SIGN A L  SUBTRACTED)  tU Ul  a.  I'  cc  3  4  5  O  u.  I  i  I  6 7 8 9 FREQUENCY. Hz  1 12  r  10  1  13  ui o 3  1.0  F L E X U R A L DOMINANT ( SIGN A L  ADDED)  a.  < ui 0.5 > < _J Ul  I 1(3  n'512.  2  3  4  5  6 7 8 9 10 FREQUENCY, Hz  FREQUENCY IDENTIFICATION:  II  HARBOUR CENTRE (E-1!)  12  13  59.  FIG.  MODE I  MODE 2  0.47Hz  1.47Hz  5.3(A) f  HARBOUR  CENTER  ™  >  '  MODE  MODE 3 2. 19Hz  SHAPES  DOMINANT.  (NORTH-SOUTH)  FLEXURAL  60.  MARCH 24,1981 DATA DECEMBER 19,1981 DATA  OBSERVATION  ROOF FLOOR 27  1/  DECK  \  y  \  ^  co  \  —  c. O ? 11 C  / /  1 Q J  i  1I7 i  /  J  1 i *-*5 i 1  / /  /  o  /  11  1 1  Q  1  /  / \  \ \  (  \  1  \  \  MODE I 0.46Hz 5.3(B)  HARBOUR DOMINANT  \ \  i  FIG.  i I i  \  \  •  MODE 2 1.50Hz CENTER  NODE  SHAPE  MODE 3 1.65Hz (EAST-WEST)  FLEXURAL  61.  62.  63.  64.  65.  FN = 1.65 Hz = nat'l freq.  FIG,  5,5  '  69.  *J|ooo  *  uibXCiXQ: UJ  _ I _ J O O O _| u.  U. L U — r - H  i 6  1  1  7  8  r 9  10  II  12  13  FREQUENCY,Hz  F L E X U R A L DOMINANT (SIGNALS ADDED)  1  r  1  r  6  7  8  9  10  II  12  13  F R E Q U E N C Y , Hz  UJ  l.0  , TORSIONAL DOMINANT (SIGNAL SUBTRAHTFn)  n  o 3  -J  2 <  0 . 5 H  ui  "i  1  1  r  "I  1  1  1  1  1  1-  6  7  8  9  10  II  12  13  F R E Q U E N C Y , Hz .5.6  FREQUENCY I D E N T I F I C A T I O N : TORONTO DOMINION TOWER ( N - S )  70.  Theoretical E x p e r i mental FLOOR  FIG.  MODE I  MODE 2  MODE 3  0.27Hz(Th.) 0.26Hz(Ex.)  0.58Hz(Th.) 0.75Hz(Ex.)  0.97Hz(Th.)  5.7(A)  TORONTO C  DOMINION  (NOPJH - SOUTH)  BANK -  TOWER  F L E X U RA L  MODE " " ~ \  SHAPES^_I3  71.  Theoretical Experimental  FLOOR 29 28 27  / /  //  /1  1 1  h25  !  ll I 1  1  1 /  1  16  /  / //  1 1 1 1 1 1 1 1j  18  /I  l \ 1  1  22  hi  A /1 f l I rfV l\ 7^  l]  i  j  4-  i  ' 10  //  /  \  1/  \ \ \ \  1-8  \\ \' \  1/  6  1  ll  h-4-  \  /  MODE I 0.22Hz(Th.) 0.24Hz (Ex.) FIG.  5.7(g)  TORONTO C  \  MODE 2  MOOE 3  0.48Hz(Th.) 0.68Hz(Ex.)  0.77Hz(Th.) 1.41 Hz(Ex.)  DOMINION -  BANK  TOWER  FLEXURAL/.  MODE  SHAPES  72.'  FIG.  5.70%) "  TORONTO  DOMINION  (EXPERIMENTAL)  -  BANK  TOWER  TORSIONAL.  MODE  SHAPE  FREQUENCY.Hz I.'O-i  TRANSLAT10NAL  DOMINANT (SIGNAL ADDED)  0.5H  i  1  1  r  6 7 8 9 FREQUENCY, Hz  10  II  12  13  TORSIONAL DOMINANT (SIGNAL SUBTRACTED) I.O-i  0.5H  i  1  1  1  r — r  6 7 8 9 10 FREQUENCY, Hz FIG.  5,a  FREQUENCY  IDENTIFICATION:  IBM TOWER  II  (N-S)  12  13  74.  Theoretica I Experi mental  FLOOR  MODE 1 2 0.37Hz(Th.) 0.38Hz(Ex.) FIG.  5.9(A)  IBM  TOWER  MODE l.09Hz(Th.) l.20Hz(Ex.) MODE  SHAPES- -  MODE 3 l.98Hz(Th.) ' 2. 14 Hz (Ex.)' f NORTH-SOUTH)-  FLEXURAL.  75.  Theoretical Experimental  FLOOR  MODE I 0.48Hz(Th.) 0.42Hz(Ex.) FIG.  5.3(f)  IBM  TOWER  MODE 2 l.39Hz(Th.) l.30Hz(Ex.) MODE ' S H A P E S  MODE 3 2.5IHz(Th.)  - . (EAST-WEST).  -  FLEXURAL.  76.  —  Theoretical Experimental  FLOOR  MODE I l.37Hz(Th.) 1.43Hz (Ex.)  MODE 2 3.79Hz(Th.)  77.  TIME C " „ F I G T .  5..10(A)  L A G NUMBER  TORONTO D O M I N I O N  TOWEROE-W)  AUTOCO RRELATI Ol)l-D AMP LN G - E S T I M A T I O N .  78.  FN = 1.20Hz = nat'l freq.  1.04-  <  on o o _J UJ  on. oc o o o i-  <  — 1.04—  50  150  100 TIME  i-  FIG.5,  200  250  LAG 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.  '""  300  0  50  100  150  200  250  TIME LAG N U M B E R FIG,  5 . 1 1 CD)  IBM  TOWER  (E-W)  (®QMf  BAJION  DAMPING  ESTIMATION.  FIG,  5.11(E)  IBM  TOWER  (TORSION)  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 NOISE AND  A trial  A2  SMOOTHING TECHNIQUES  run of the ambient v i b r a t i o n survey w i t h the Earthquake  E n g i n e e r i n g Research L a b o r a t o r y measurement system (see Chapter conducted  to a s s e s s the manpower and  time requirements  In the process n o i s e problem was  encountered.  developed  to e l i m i n a t e the n o i s e c o n t a m i n a t i o n  data w i l l  serve to i l l u s t r a t e  a few important  i n the  The  lytic  n o i s e i n our time  phenomenon which may  pound person generating  on the smooth course of the p h y s i c a l f u n c t i o n we  proper  I f we  process  t r y to measure.  Figure  t r y to expand these  The way  run.  and p u l s e s cannot be conceived a n a l y t i c a l p r o p e r t i e s and  as will  to be d e f i n e d *  process of the s e r i e s w i l l  The  superimposed  (those p o i n t i n g upwards) resemble -  f u n c t i o n s s i n c e they do not possess  require a l i m i t  spe-  In our case n o i s e i s  p r o v i d e s an example of a n o i s y s i g n a l recorded d u r i n g the t r i a l  the p u l s e s mentioned i n s e c t i o n B2.3,  run-  s i g n a l s as a t y p i c a l l y nonana-  d e s t r o y the t r u e s i g n a l .  The m a j o r i t y of ' p o s i t i v e ' needles  the  points.  present i n the form of peaks of v a r y i n g h e i g h t s and d i r e c t i o n ,  A2.1  technique  d u r i n g the a n a l y s i s of  n i n g past the seismometers, or by the r e c o r d i n g instrument observe  was  field.  smoothing  I t does not matter whether n o i s e i s c r e a t e d by a 200  c i o u s peaks - we  3)  "needles" i n an i n f i n i t e  fail:  s e r i e s , the  limit  the s e r i e s w i l l not converge anywhere.  t h a t the n o i s e a f f e c t s our a n a l y s i s i s obvious  in Fig.  F o u r i e r t r a n s f o r m of about 4000 p o i n t s of the n o i s y data (both  A2.2. needles  86. and  tape n o i s e ) i n one b l o c k w i l l g i v e us F o u r i e r c o e f f i c i e n t s t h a t 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 . B e s i d e s the p o s s i b l e a l i a s i n g e f f e c t  ( s e c t i o n B2.7) o f those f r e q u e n c y  components h i g h e r than the Nyquist frequency, leakage ( s e c t i o n B2.6) may cause much d i s t o r t i o n i n the spectrum. There a r e two approaches  to d e a l w i t h t h i s n o i s e problem:  " i n the l a r g e "  ( o r g l o b a l smoothing) and smoothing  (neighbourhood  smoothing) (See Lanczos  A)  smoothing  " i n the s m a l l "  (13)):  Smoothing i n the Large: The F o u r i e r technique i s a powerful t o o l i n t h i s r e s p e c t . The non-  convergent  p a r t of the F o u r i e r s e r i e s r e p r e s e n t s the n o n a n a l y t i c a l nature o f  the n o i s e .  An i d e a l l y random n o i s e would have a F o u r i e r spectrum which has  no p r e f e r e n c e f o r any frequency and would have an average random f l u c t u a t i o n s f o r a l l f r e q u e n c i e s .  amplitude  with  F o r our case we know a p p r o x i m a t e l y  the maximum frequency t h a t we want f o r b u i l d i n g v i b r a t i o n s , hence we can simply omit any h i g h e r frequency i n the s e r i e s . range o f i n t e r e s t n o i s e i s s t i l l  But w i t h i n the frequency  p r e s e n t ; and we do not know what e x a c t l y  a l i a s i n g does to the s i g n a l . B)  Smoothing i n the S m a l l : T h i s neighbourhood i)  technique i s v e r y a t t r a c t i v e to us because:  these " g l a r i n g " mistakes a r e v a s t l y d i f f e r e n t  f rom  the a c t u a l  signal. ii)  they occupy o n l y one d i g i t i z e d i n the d i g i t i z e d  p o i n t and hence a r e easy t o d e t e c t  form o f the d a t a .  Smoothing i n the s m a l l i s thus done u s i n g a programme w i t h a " d e t e c t o r " 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  87. (Wiggins (25)) to generate a new  p o i n t to r e p l a c e the n e e d l e .  The  Wiggins'  technique makes d u a l use of a p o i n t as i n the L a g r a n g i a n i n t e r p o l a t i o n t e c h n i q u e , but uses a c u b i c i n t e r p o l a t i o n f u n c t i o n and some e x t r a c r i t e r i a ensure the new  p o i n t f o l l o w s the course of the p h y s i c a l f u n c t i o n i n g e n e r a l .  The method has been found to g i v e good e m p i r i c a l The A2.3.  smoothed t r i a l  the same concept i s used i n d i g i t a l of s e v e r a l modes.  reported i n section  results.  run s i g n a l o b t a i n e d by t h i s method i s shown i n F i g .  Though we d i d not a p p l y smoothing  from a spectrum  " i n the l a r g e " i n t h i s i n s t a n c e ,  filtering  of a s i n g l e frequency peak  Some d e t a i l s o f t h i s f i l t e r i n g  trum seems extremely v a r i a b l e and peaky when d i f f e r e n t d a t a a r e compared.  process i s  B2.9.  There i s a l s o another i n t e r e s t i n g a s p e c t i n F i g . A2.2.  B2.8).  The n o i s e spec-  p l o t s o f the n o i s y  T h i s t u r n s out to be a w e l l - o b s e r v e d phenomenon ( S e c t i o n  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  p o i n t s used i s much more than 4000.  of  our main o b j e c t i v e s and the f i e l d measurement programmes subsequent  the t r i a l The  to  To decrease t h i s v a r i a b i l i t y i s one  run have been m o d i f i e d to a c h i e v e s t a b l e source of the n o i s e i n t h i s t r i a l  a recording channel. measurements.  T h i s channel was  run was  to  results. found to be l o c a l i z e d i n  not employed i n a l l subsequent  1.0  u _| !^ ^ M  ^  i  0.5H  >  = < o  U.  J  UJ  cr  0.00  D  1 0  0  2  0  0  30.0  10.0  SO.O  60.0  70.0  BD.O  90.0  FREQUENCY,  100.0  .1)0.0  110.0  130.0  110.0  ISO.O  160.0  Hz 00  FIG.  A2.2"  FOURIER' (4000  TRANSFORM-OF  POINTS  IN  ONE  NOTSV BLOCK)  DATA  0.9 0.8_ 0.7_ 0.6_  TIME  (POINTS  FIG.  A 2 . 3 ; SMOOTHED  AT  .025  SEC.... INTERVALS)' "...  TIMEiFUNCTION.  •  91. Appendix A3 CALIBRATION PROCEDURES AND RANGER AND  K o l l a r and  CHARACTERISTICS OF  THE  WILLMORE MARK I I SEISMOMETERS.  R u s s e l l (27) have observed t h a t the e l e c t r o m e c h a n i c a l  of the seismometer can be i d e a l i z e d as an e q u i v a l e n t c i r c u i t a n a l o g ) , and Bridge,  t h a t the measurement made a t the output  F i g u r e A3.1,  (electrical  t e r m i n a l s of a Maxwell  can be used to determine the c h a r a c t e r i s t i c s of  seismometer under s i n u s o i d a l ground motions i n the same frequency the c o n t r o l l e d i n p u t  "MAIN" and  the  range as  voltage.  By u s i n g the WAVETEK 111 s u i t a b l e incremental  system  as the i n p u t o s c i l l a t o r , known i n p u t s ( o f  frequencies)  can be  "SUBSTITUTE" c i r c u i t s .  fed i n t o the c i r c u i t  K o l l a r and  through  R u s s e l l d e f i n e the  the  impedance  of the seismometer as  S  Rg  V o l t a g e main Voltage s u b s t i t u t e  B where — £ — a r e known q u a n t i t i e s , and E  V o l t a g e main and V o l t a g e  s u b s t i t u t e are  R  measured. Plotting  the v a l u e s of Zg vs i n p u t frequency  l i n e a r , i n p u t frequency phase s h i f t , The low  see  = output  frequency  s e c t i o n B2.2), we  h i g h frequency  asymptote (from  about  at the low .15  to  shown i n F i g .  frequency  .4 Hz  end  to 6 Hz).  The  resonance i s d e f i n e d as  Zg^.  i s called  . OA  The  the the  frequencies  impedance Zg at the p o i n t where the  s i o n s of the l i n e a r p o r t i o n s meet i s d e f i n e d as Z  a  A3.2:  i n the f i g u r e ) , while  asymptote i s s i m i l a r l y d e f i n e d f o r the h i g h e r  (from about 3 Hz  are  w i t h an amplitude change and  o b t a i n the r e s u l t s  l i n e a r p o r t i o n of the curve  frequency  (as the d e v i c e s  exten-  peak impedance a t  At the low frequency 2 1  o  2  asymptote:  ° 1  w  re: 8  1  w  c o n s t a n t at  i s the seismometer g e n e r a t o r  the low frequency  asymptote  = n a t u r a l frequency o  m  = mass o f magnet i n seismometer  V  is  the v a l u e of Zg a t frequency w  i n the low frequency  1  asymptote  range of f r e q u e n c i e s . At the h i g h frequency g  2 2  =  Z j  1  asymptote:  . ! .  where: Zg  1  i s the v a l u e of Zg a t frequency w  asymptote range of f r e q u e n c i e s , g c o n s t a n t at the h i g h frequency The average o f g^,  2  1  i n the h i g h  frequency  i s the seismometer  generator  asymptote.  g i v e s the v a l u e o f 'g', the seismometer  c o n s t a n t ; the n a t u r a l frequency W  i s o b t a i n e d by a geometric  q  generator mean  method ( F i g . A3.2). The damping v a l u e s  5 f o r the seismometer i s :  1  Z  OA.  , Z  z  have been d e f i n e d e a r l i e r  so and are shown i n F i g . A3.2.  uU  The c a l c u l a t e d v a l u e s of 'g', £' and ° ' 1  are t a b u l a t e d i n t a b l e A3.1 tions) •  'w  ' f o r a l l f o u r seismometers o  ( f o r seismometers i n h o r i z o n t a l p o s i -  93. TABLE A3.1  ^ \ SEISMOMETER ^v. CHARACTER^^ISTICS SEISMOMETER^. NO.  N a t°u r a l frequency Hz W  Generator constant v/m/ s  n  Internal damping % critial  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 I I seismometers  i n the Geophysics system a r e v e r y  s i m i l a r and have the f o l l o w i n g common c h a r a c t e r i s t i c s : W  = 1 Hz ( n a t u r a l  g  = 185.5 v/m/s.(generator c o n s t a n t )  q  £  i n t  freq.)  = ( i n t e r n a l damping o f  seismometer)  = .01 ( 1 % o f c r i t i c a l ) U s i n g these v a l u e s and the e l e c t r i c a l r e s i s t a n c e v a l u e s i n F i g . A3.1, together w i t h the frequency response f o r the SC201 and the tape r e c o r d e r as i n p u t t o the computer programme UNICAL*, the system response curve can be obtained.  A t y p i c a l curve r e p r e s e n t a t i v e o f a l l f o u r Ranger seismometers i s  shown i n F i g . A3.3.  (SYSTEM v e l o c i t y response i s d e f i n e d i n s e c t i o n B2.2)  * UBC Department of Geophysics Programme  Library  OUTPUT VOLTAGE  SUBSTITUTION INPUjT V O L T A G E ( W A V E T E K  R R  = 30.05 fi  R  D  = 9 . 5 0 7 kft  C  n  = .178 mF  R  B  = 611 kfi to 611 kJJ  R  £  = 9.502 kfi  III)  MAIN  INPUT  INPUT  R R R = C L  c  Z  S  -JLJi (about 2000 fi) R_  -  W  R R R B = ——— R E  B  V o l t a g e main .. _ — . (impedance of seismometer) substitute 2  Voltage  F i g . A3.1  Diagram of Maxwell Bridge Used i n the  Calibration  95.  =  1  I I I I I I I  1  1  I I I I I I I  1 1 1 1 1 11  Zso  J/  \  GEOM MEAN  \  (0.85 X |.I5)2 =0.9  | AVG = I.OOHz \ ( 0 . 7 7 X | . 3 0 ) I = 1.0 0 J 9  —  Zs  Ranger Se ismometer # 4 E/Q LAB -  1 .1  IIIIIII 0.5  1  I  I I I I I I I  5  I I I I I I I  1  10  50  FREQUENCY , Hz  FIG.  A3.2  SEISMOMETER  CALIBRATION  .  96.  10 =  tn \ g \ tn  1  I I I I I I I  1  I I I I I I I  1  1 1 111 M  +J  i-H  O >  j,^»o—o-o-o-oo< 1  0  0  0—0  CO  z o  a  io  3  or  —  J  o _i  UJ  >  Seismometer 1 — Horizontal, Velocity F i l t e r , Computer Generated  UJ to  |  0  2  1  10 0.1  I I I I I I I 0.5  1  I I I I I I I  I  5 FREQUENCY  FIG.  A3.3  ,  10 Hz  SYSTEM €ALIBRATIQM. V  1  I I I I I I I 50  100  APPENDIX A4 TYPICAL AMBIENT VIBRATION TRACES  100-mv 10  Chart  Date:  December  Rate:  10  Sensitivity: FIG.  A4;i  (A)  ;  TYPICAL  1981  mm/s  100 mv/ d i v i s i o n  AMBIENT  HARBOUR  19,  mm  VIBRATION  CENTRE  TRACES  oo  10  Chart  Date:  April  Rate:  25  Sensitivity: FIG.A4.1  (B)  TYPICAL TORONTO  11,  mm  1981  mm/s  100 m v / d i v i s i o n AMBIENT  DOMINION  VIBRATION TOWER  TRACES  18 Floor  "5  JM  I B  —I—I  1—I—I—I—t—H-+-H—^—^—r—h  I  M  iii  H—I—I—I—I—I—H—I—I—I--H—I—I—I—I—I—\—H  -i—I  BRUSH  1  1—I—I  1  ACCUCMART  M—I—I  H 1—I  1 1  1—I—1—I—I—  0) Pi  18 Floor  100 mvT  T  I  iiLuiliitiMi 10  Chart  Date:  April  Rate:  25  Sensitivity:  F I G . A M  ( •  m  mm  12, 1981  mm/s o o  100 m v / d i v i s i o n  TYPICAL IBM  iillli!  AMBIENT VIBRATION TOWER  TRACES  I I I- I I I+  I 4- 4 -I--I--I—-4 • 4-  Roof  4-—I 4 — 1 1 - I 4-4 - — 4 — I I — 4- + — — •(— -( Gouid Inc.. Inatrunwnl Syttvma DMiJon Cleveland, Ohio  Primed in U S A  Roof  •  i  <  i  i—t  <  . —> • < ..(....(-...4...,  *  t  i-i  -i  | 4 .-• I—4-4 I-H 4—1—I 1 1 4 4 10 mm Date Chart Rate Sensitivity  May 21, 1981 5 mm/s 100 mv/division (D)  FIG.  A4.1  (D)  Gage Towers  TYPICAL GAGE  AMBIENT  TOWERS  VIBRATION  TRACES  102.  APPENDIX A5 MAN-EXCITED TEST RESULTS  : - T l : : : : ! : : : : ) - : : : i . : • : : : ! : : : • : ! : : : . : -:::":!•--..;:::•!'-.;.::••::::•:;:-•:  -.1:.:  100 mv  -  :— :  •  ...:  itiitifc,:  * ; :  - J  \\ ••• •  -  \ ;-/ \ \  A  — — mm (C)  FIG.  1 /  -7 \  HY^i  —y -. \r -V : ••:\ I  Itiivii H - . r : ^  /\ /\  V /'  ::  1  !. .. :  ••I  ii;  d:ti  "i ^  s\  i:::ijit:  H J:..:•:;;;  ; :  -  :  ....i—  '  ..:.:+T:-.  =  •i - tr: i i'i i iii• ii i tii.tp i !:ti! it:ii" l  :-„-V-  ;  ::  -_.:1-:T-::  j: r  :  : i  I  J  :  ;  rr: •  •  ;  :;;  : :i ' : . . :-,•-•:.:.:••:.•::.  7~:::;Ti^.;i::;:|7-:i';:; ir-~q:::;i-;::.;.;.:. :  N-S Second F l e x u r a l Mode. 2nd mode f r e q . •= 0.74 Hz Damping = 1.5% c r i t i c a l .  A5.1  MAN-EXCITED  TEST  RESULTS  -  T / D TOWER  (APRIL  11, 81)  104. APPENDIX B2 DETAILED BACKGROUND THEORY FOR  B2.1  THE  AMBIENT VIBRATION SURVEY  Introduction  T h i s appendix p r o v i d e s i)  some mathematical background f o r :  the c a l i b r a t i o n of the measurement system used i n the ambient vibration  survey,  ii)  the p l a n n i n g of the measurement procedures  iii)  the subsequent a n a l y s e s  and  used i n the f i e l d ,  i n t e r p r e t a t i o n of the a c q u i r e d  Much of the d i s c u s s i o n has been h e a v i l y drawn and Lanczos'  'Applied A n a l y s i s ' (13) and  a l s o Blackman and t i o n s from these The  Tukey's 'The  reproduced  data.  from  C.  'Discourse on F o u r i e r S e r i e s ' (14),  Measurement of Power S p e c t r a '  (15);  f o r a b a s i c understanding  and  quota-  sources w i l l not be s e p a r a t e l y r e f e r e n c e d i n t h i s  d i s c u s s i o n s are intended  and,  chapter.  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 A n a l y s i s  The measurement system used to a c q u i r e data c o n s i s t s of sensors mometers), a s i g n a l c o n d i t i o n e r and  a tape r e c o r d e r .  (seis-  Some d e t a i l s of  f u n c t i o n a l p a r t s of the system were d i s c u s s e d i n Chapter 3.  We  are  the  only  concerned here w i t h the g e n e r a l c h a r a c t e r i s t i c s which are common to mechanisms i n communication We box  has  regard two  engineering.  the system as a 'black box'  ends:  t i o n s f o r the b l a c k  the i n p u t and box:  of unknown s t r u c t u r e .  the output.  We  This black  make the f o l l o w i n g assump-  105. i)  I t Is l i n e a r : a)  With an a r b i t r a r y i n p u t x ( t ) , the output i s y ( t ) . When the i n p u t is  b)  otx(t), where a i s a c o n s t a n t , the output i s oty(t).  The  s u p e r p o s i t i o n p r i n c i p l e h o l d s ; the simultaneous  two  i n p u t s generates the sum  any mutual ii)  a p p l i c a t i o n of  o f the c o r r e s p o n d i n g outputs  without  interference.  The b l a c k box has unknown components t h a t do not change w i t h time; is,  t h e i r p h y s i c a l c h a r a c t e r i s t i c s are time  iii)  The output f o l l o w s the i n p u t .  iv)  The b l a c k box d i s s i p a t e s  invariant.  energy.  A known l i n e a r d i f f e r e n t i a l e q u a t i o n r e l a t i n g the i n p u t x ( t ) and output y ( t ) has the a  ,n d y . — — + a , n , n n-1 dt  ,n-l d y , r + , n-1 dt  b  dt  r  We  r  dt^  r  d  a y = o^  bx  +  1  note t h a t the f i r s t  such t h a t 4^  w i l l correspond  t  two  assumptions  to -—, — ^ dt d  t  are inherent i n t h i s  w i l l correspond to  equation  etc.. dt*  2  so that the s u p e r p o s i t i o n p r i n c i p l e a p p l i e s ; a l s o the a^,  b^,  are a l l c o n s t a n t s i n s t e a d of being f u n c t i o n s o f time, showing the time pendence o f the p h y s i c a l c h a r a c t e r i s t i c s of the b l a c k  c o n s t a n t s a's and b ' s . are observed, and the £L  . is  So i n s t e a d , the i n p u t and ratio  . taken,  inde-  box.  There i s no simple e x p e r i m e n t a l method to determine  output -— —input  the  form:  fLx + _! i L _ 2 L  b  that  the v a r i o u s  the output o f the  system  106.  This r a t i o defines We  can  regard  the  t r a n s f e r f u n c t i o n of the  a generalized  s t r i c t l y p e r i o d i c components. nique:  i n t o a sum  of pure s i n e and  assumptions t h a t the  I f we  s i g n a l as a s u p e r p o s i t i o n  in a finite cosine  system i s l i n e a r and  then be a m o d i f i e d  box).  T h i s i s p o s s i b l e because of the F o u r i e r  an a r b i t r a r y f u n c t i o n d e f i n e d  resolved  will  p h y s i c a l input  system ( b l a c k  v e r s i o n of the  i n t e r v a l can always  functions.  With the  techbe  fundamental  unchanging w i t h time, the  sum  of  output  of these components.  have an i n p u t harmonic f u n c t i o n 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 y ( t ) = F(w) where F(w) F(w)  i s a complex number having the = A(w)  e  -  i  9  (  w  system at frequency  w.  form:  )  i s an amplitude and  f u n c t i o n s of  0(w)  ,is the  phase s h i f t , both of which  are  w.  whence, / - N  »y  y ( t ) = A(w) and  2.2.1  i w t  i s the complex frequency response f o r the  F(w) where A(w)  e  be:  v  e  i (wt -  6(w))  y ( t ) i s the response of the b l a c k box  at frequency w.  We  note t h a t  the  iwt response to the p e r i o d i c i n p u t e quency w but  with a modified  i s a p e r i o d i c output of the  amplitude A(w)  and  a phase s h i f t  As a simple i l l u s t r a t i o n of t h i s procedure c o n s i d e r  same f r e 9(w).  the e q u a t i o n of  motion f o r a l i n e a r system composed of a l i n e a r s p r i n g with s t i f f n e s s k a l i n e a r viscous  damper of c o e f f i c i e n t C.  and  I f the i n p u t 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) + k y ( t ) = X ( t )  2.2.1(a)  106. This r a t i o defines We  can  regard  the t r a n s f e r f u n c t i o n of the system ( b l a c k a generalized  s t r i c t l y p e r i o d i c components. nique:  p h y s i c a l input  i n t o a sum  of pure s i n e and  assumptions that the  If we  in a finite cosine  system i n l i n e a r and  w i l l then be a m o d i f i e d  s i g n a l as a s u p e r p o s i t i o n  T h i s i s p o s s i b l e because of the F o u r i e r  an a r b i t r a r y f u n c t i o n d e f i n e d  resolved  box).  v e r s i o n of the  i n t e r v a l can always  functions.  With the  techbe  fundamental  unchanging with time, the  sum  of  output  of these components.  have an input harmonic f u n c t i o n 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 y ( t ) = F(w) where F(w) F(w)  i s a complex number having the = A(w)  e  "  i  8  (  w  w.  form:  0(w)  i s the phase s h i f t , both of which  are  w.  whence, ^  system at frequency  )  i s an amplitude and  f u n c t i o n s of  v  y ( t ) = A(w) and  2.2.1  i w t  i s the complex frequency response f o r the  F(w) where A(w)  e  be:  .  e  i (wt v  v  6(w)) "  y ( t ) i s the response of the b l a c k box  at frequency w.  We  note t h a t  the  iwt response to the p e r i o d i c i n p u t e quency w but  with a m o d i f i e d  i s a p e r i o d i c output of the same f r e -  amplitude A(w)  and  a phase s h i f t  As a simple i l l u s t r a t i o n of t h i s procedure c o n s i d e r  6(w).  the e q u a t i o n o f  motion f o r a l i n e a r system composed of a l i n e a r s p r i n g with s t i f f n e s s k a l i n e a r viscous  damper of c o e f f i c i e n t C.  and  I f the i n p u t 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 From e q u a t i o n  2.2.1  system.  above y(t) = F ( w ) X e  2.2.1(b)  i W t  o  S u b s t i t u t i n g 2.2.1(b) i n t o 2.2.1(a)  , . . ,v  x iwt  (ciw + k) F(w)e F(w) The  =  = e  iwt  1  k+icw  amplitude of F(w)  is  A(w):  A(w)  =  |F(w)| =  2.2.1(c) (k*f 2w ) 2  1 / 2  C  and  the phase s h i f t  9(w) i s : 0(w)  and  F(w)  = A(w)e~  i 0 ( w )  w i t h A(w)  = tan" ^)  2.2.1(d)  1  and  0(w)  g i v e n by 2.2.1(c) and  2.2.1(d)  respectively.  tem  T h i s frequency response approach to the a n a l y s i s of a 'black box'  sys-  i s e s p e c i a l l y s u i t a b l e f o r the l a b o r a t o r y environment, i n which we  can  c o n t r o l the p e r i o d i c  i n p u t to the system and  observe the o u t p u t .  I f by comparing the output to the input we A(w)e"  i G ( w )  obtain a transfer function  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 l a r g e range of frequency w, l i t y reproduction  of the i n p u t .  we  then have a system t h a t has  This i s also s u f f i c i e n t  proof  high-fide-  that  system i s l i n e a r , s i n c e an harmonic i n p u t remains an harmonic output unchanged frequency, although with m o d i f i e d The  amplitude and  with  phase.  above d i s c u s s i o n forms the b a s i s of the l a b o r a t o r y c a l i b r a t i o n of  the measurement systems ( f o r d e t a i l s r e f e r to Chapter 3 ) . v e l o c i t y response curve ( F i g . 2.1), various  shifted  our  frequencies  A typical  system  which d e f i n e s the system's response a t  to v e l o c i t y input t r a n s m i t t e d  by the sensors  at the same  109. the system w i l l  then be a s u p e r p o s i t i o n o f p u l s e r e s p o n s e s .  This process i s  v e r y s i m i l a r to the e l e c t r i c a l e n g i n e e r s ' technique o f u s i n g the H e a v i s i d e U n i t Step F u n c t i o n as a b u i l d i n g b l o c k to c o n s t r u c t i n p u t s i g n a l s . p u l s e response method i s more v e r s a t i l e ,  s i n c e i t does not presuppose  d i f f e r e n t i a b i l i t y o f the i n p u t s i g n a l as does H e a v i s i d e ' s method. advantage  Yet the the  The  dis-  i s that the p u l s e cannot be c o n c e i v e d as a l e g i t i m a t e f u n c t i o n i n  the proper sense, s i n c e we have to impose a l i m i t  process f o r i t s d e f i n i t i o n  (see f u r t h e r d i s c u s s i o n of t h i s p o i n t i n Appendix  A2).  The p u l s e response method f o r a system proceeds as Let the i n p u t s i g n a l be x ( t ) , and  follows:  the response of the system (which i s  a c h a r a c t e r i s t i c o f t h a t system) be k ( t ) .  Then the response y ( t ) a t t = 0  is: 0 y(0) =  /  x(T)k(T)dT  — oo I n t e g r a t i o n l i m i t s i n d i c a t e t h a t the output f o l l o w s the i n p u t . Since the p h y s i c a l c h a r a c t e r i s t i c s o f the system a r e time  invariant,  f o r the same phenomenon a t any l a t e r time moment t , the i n p u t w i l l  be  x ( t + x ) , and the response y ( t ) : 0 y(t) =  /  x(t +  x)k(x)dT  — 00  i n t r o d u c e the new  variable t +  x = K  t y(t) -  /  x(0  k(£rt) d£  2.3.1  — 00  I f we we  l e t the i n p u t f u n c t i o n be the u n i t impulse a t time moment  5=0,  have y(t) = k(-t)  k ( t ) becomes the p u l s e response of the system.  2.3.2 Let H ( t ) be the response o f  the system a t time t to the u n i t p u l s e i n p u t a p p l i e d a t t = 0, then from  110. k ( t ) = H(-t)  2.3.3  But H ( t ) i s zero f o r a l l n e g a t i v e  time moments s i n c e the output  follows  the i n p u t , we have f o r a l l p o s i t i v e t : k(t) =0,  ( t > 0)  With 2.3.4 we can extend  2.3.4  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  — C O In the g e n e r a l case requirement  2.3.4 i s not n e c e s s a r y ,  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 c o n v o l u t i o n i n t e g r a l o f f u n c t i o n s H ( Q  and x( £) .  B2.4 The F o u r i e r Transform, R e l a t i o n s h i p Between P u l s e and Frequency Responses, C o n v o l u t i o n and M u l t i p l i c a t i o n  So f a r two d i f f e r e n t approaches have been used ( a c c o r d i n g to two d i f f e r e n t p h i l o s o p h i e s ) to o b t a i n the responses same s e t o f assumptions. responses  are related.  I n t u i t i v e l y we know t h a t the p u l s e and frequency To be a b l e to compare these responses,  convert e i t h e r the p u l s e response response two  o f the same system w i t h the  to the time domain.  to the frequency  we have to  domain, or the frequency  The F o u r i e r t r a n s f o r m i s the l i n k between the  domains. A s l i g h t l y m o d i f i e d v e r s i o n o f the F o u r i e r i n t e g r a l ( e q u a t i o n 2.2.2)  d e f i n e s the F o u r i e r t r a n s f o r m :  oo X(w) = — and  the i n v e r s e  /  x(t)e"  i w t  dt  2.4.1  dw  2.4.2  transform:  00 x(t) = — —  /  X(w)e  1Wt  These two o p e r a t i o n s have remarkable r e c i p r o c i t y .  111. Assume t h a t the x ( t ) i n s e c t i o n 2.3  can be s y n t h e s i z e d from i t s  p e r i o d i c components, from 2.4.2:  oo x(t) = We  )  X(w)e  dw  can c o n s i d e r t h a t x ( t ) can be modelled as the s u p e r p o s i t i o n o f a c o n t i n u -  ous  s e t of s i n u s o i d s having f r e q u e n c i e s of d i f f e r e n t  1 /2TT  w's  and amplitudes o f  iwt X(w)dw.  (Changing e  First  iwt  we  to e  l o o k at the i n p u t of e  iwx  f o r the  i n the i n t e g r a l  2.3.5:  integration)  oo y(t) =  /  k(x - t ) e  i W X  dx  — CO let  5 = x - t y(t)  /"  =  I c C Q e ^ ^ d S  — 00 = e  The  iwt  t  J  oo w  „ iwC,, k( 5)e dK  — CO  integral i s just  the complex conjugate o f the F o u r i e r Transform of  k( £ ) , the p u l s e response, m u l t i p l i e d by K*(w)  =  /  k(Qe  i W ?  V^2"TT.  I t i s a complex number K*(w):  d£  — 00 and can be expressed i n terms of an amplitude A(w) y ( t ) = /2ri K * ( w ) e v -i6(w) iwt = A(w)e 'e.  and phase e  ^ ^  w  ) '  ± W t  v  =  . i(wt A(w)e  6(w))  y ( t ) i s the response i n s e c t i o n B2.2,  and A(w)e  ^^^ ^ w  i  s  j u s t F(w)  i n equa-  t i o n 2.2.1. T h e r e f o r e the complex frequency response F(w)  i n s e c t i o n B2.2  i s just  the complex c o n j u g a t e o f the F o u r i e r Transform o f i t s p u l s e response ( s e c t i o n B2.3), m u l t i p l i e d  by /2ir.  There i s y e t another important a s p e c t o f the c o n v o l u t i o n i n t e g r a l  112.  00 y(t) =  /  H(t -  Qx(?)d5  — 00 Taking the F o u r i e r Transform of y ( t )  CO  Y(w) where Y(w) using  = — —  /  y(t)e"  i w t  gives:  dt  2.4.3  i s the F o u r i e r Transform of y ( t ) .  2.4.1,  S u b s t i t u t i n g 2.4.3  i n t o 2.3.6,  2.4.2  00 00 Y(w)  =  J y/2~TT  =  f  = X(w) =  [-i—  5)xU)e"  i W t  dt  d5  X  ( 0 e  i  w  5  [JL_ VTTV  f  H(t -  _ l w ( t  "  5 )  dt]d?  -OO 2.4.4  00 /  H(t -  £)e~  ~  i w ( t  5 )  dt ]  -oo  i s the F o u r i e r Transform of H(t -  £).  We  that the F o u r i e r Transform of a c o n v o l u t i o n  thus have the important r e s u l t  of two  2.3.6) i s the product of t h e i r F o u r i e r Transforms. r e c i p r o c i t y of equations 2.4.1 convolution  Qe  H'(w)  •PLTI  and  H(t -  -0°  V2TT / -  where H'(w)  /  -oo -oo  and  2.4.2, the  functions By the  (as i n e q u a t i o n  remarkable  same r u l e a p p l i e s f o r a  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 c o n v o l u t i o n  v i s u a l i z e the two discretized  processes.  A n t i c i p a t i n g the next s e c t i o n s , we  focus  d i s c r e t i z e d f u n c t i o n s i s j u s t the p o i n t by  point  m u l t i p l i c a t i o n of the two  (15).  domain, i t w i l l be h e l p f u l i f we  "overlapping"  functions.  A convolution,  on  hand, can be v i s u a l i z e d i n 3 ways as suggested by Blackman and The  can on  functions.  A product o f two  other  i n the other  the Tukey  author p r e f e r s the moving weight f u n c t i o n concept of weighted  i n t e g r a t i o n : one  of the f u n c t i o n s i n the c o n v o l u t i o n may  moving weight, and  be regarded as a  i n t e g r a t i o n i s performed as the w e i g h t i n g f u n c t i o n moves  p o i n t by p o i n t over the  second f u n c t i o n .  The  r o l e s of the  f u n c t i o n s may  be  113. interchanged.  A pictorial  representation  o f the moving weight i n t e g r a t i o n  process can be found i n Hamming's book ( 1 9 ) .  B2.5 D i s c r e t i z e d F u n c t i o n s  The  discussions  and the D i s c r e t e F o u r i e r  i n the p r e v i o u s  Transform  s e c t i o n s have focused  c o n t i n u o u s f u n c t i o n s , and the F o u r i e r  on analog o r  i n t e g r a l i s the l i m i t  o f the F o u r i e r  s e r i e s f o r the case when the base time o f a n a l y s i s i n c r e a s e s Though the F o u r i e r i n t e g r a l i s more g e n e r a l  to i n f i n i t y .  than the F o u r i e r s e r i e s (which  demands s t r i c t p e r i o d i c i t y o f the f u n c t i o n ) , we have to t u r n to the use o f the F o u r i e r s e r i e s , s i n c e the computer o n l y works w i t h d i s c r e t i z e d f u n c t i o n s and  the c o n t i n u o u s " d e n s i t y "  incompatible  with d i g i t a l  s e r i e s as a l i m i t i n g  f u n c t i o n that the F o u r i e r  equipment.  The p r o c e s s o f o b t a i n i n g  case o f the F o u r i e r  the purpose o f d i g i t a l  i n t e g r a l produces i s  integral will  help  the F o u r i e r  us understand  smoothing.  Let us s t a r t w i t h a p e r i o d i c time f u n c t i o n w i t h p e r i o d 2ir f ( t + 2TT) = f ( t ) This 2TT(N+1). The  f u n c t i o n e x i s t s between the l a r g e l i m i t s o f t = -2irN and t = The f u n c t i o n i s repeated 2N+1 times as N tends to i n f i n i t y . Fourier analysis gives  the F o u r i e r  spectrum F(w) ( e q u a t i o n  2.2.3):  2TT(N+1)  F(w)  = / -2ITN  f(t)e  1  W  C  dt  However, because o f the p e r i o d i c i t y , we need i n t e g r a t e over o n l y one p e r i o d , and  shift  the p e r i o d i c f u n c t i o n to account f o r the c o n t r i b u t i o n s  i n the  2TT f(t)e o s h i f t i n g g(w) to the next p e r i o d w i l l g i v e e ^ ^ g ( w ) and so on, 7W  * .'. F(w) =  r -2irNwi , -2n(N+l)wi , [e + e ' + v  e  2irNwi  , .  Jg(w) n  d t and  114. The  terms i n the square b r a c k e t  r e p l a c e d by  the  sin  F(w)  x  TT(2N+1) W  sin I  r  =  (2.5.1)  IT w  -ir(2N+l)w>! . . . — — Jg(w) sm TIW m  I f N i s l a r g e , the f i r s t w = 0,  ±1, ±2  At  f a c t o r puts a v e r y s t r o n g weight on i n t e g e r s of  these  frequencies  (w = k) the continuous  changes more and more to a l i n e spectrum. complete s i n c e the l i n e spectrum has i s small. evaluate  The F(w) K+£  But  TI f v . /, N F(w)dw - g(k)  f J  and  the change i s never q u i t e  a f i n i t e w i d t h of k+e  k-e,  w:  spectrum  and  k-e where e  f a c t o r ( 2 . 5 . 1 ) i s a h i g h l y o s c i l l a t o r y f u n c t i o n , and at w between k+e  |J k-e  be  identity:  sin  . and  i s a geometric p r o g r e s s i o n which can  f o r small  i f we  e,  s i n ir(2N+l)w , ^ dw  £  -e 2 „ . = — g(k) J £  r  J  TT  s i n n(2N+l)w , — dw W  O  = g(k) |  Si(-)  where Si(°°) i s the s i n e i n t e g r a l of  TT(2N+1)W.  AS N becomes l a r g e , Si(«°) =  2'  k+e .-. / k-e  F(w)dw = g(k)  (2.5.2)  The  c o n t i n u o u s spectrum i s thus reduced to the i n t e g e r f r e q u e n c i e s  w=k,  and  the g(k)  C .  Since the complex F o u r i e r s e r i e s i s :  v a l u e s g i v e the c o e f f i c i e n t s of the complex F o u r i e r s e r i e s  f(t) = I  C e  ikivt  k  k=-°° and  C  k = YW^ ic  z  II  f(t)e"  i k l T t  dt =  ^g(w)  •  — IT  f o r our  case from (k-e) c  k =  at  T^  to (k+e)  we  have, f o r e q u a t i o n  ( k )  I t i s the Ck t h a t we  t r y to  evaluate.  2.5.2,  115. There a r e two important a s p e c t s c o n c e r n i n g  the ' k e r n e l ' f u n c t i o n 2.5.1,  namely: i)  The continuous spectrum w i l l  be changed to a l i n e spectrum w i t h  width of 2e i n the process o f the D i s c r e t e F o u r i e r Transform.  e will  decrease i f N i n c r e a s e s , i n c r e a s i n g the focus on the frequency w=k; hence the f u n c t i o n 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 .  T h i s i s a source o f problem,  s i n c e the time f u n c t i o n f ( t ) may not be p e r i o d i c and r e p e a t i n g  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 a t the j u n c t i o n s o f the r e p e t i t i o n . N £ C, e k=-N  i  Gibbs o s c i l l a t i o n s w i l l  cause the s e r i e s  k  t  to f l u c t u a t e so that the  K  s e r i e s may not converge to f ( t ) a t any p o i n t . (or  Thus the c o r r e s p o n d i n g raw  unsmoothed) spectrum w i l l be o f l i t t l e v a l u e  w i l l be  and the subsequent a n a l y s i s  misleading.  ii)  The k e r n e l f u n c t i o n w i l l determine the f o c u s i n g power and the  amount o f 'leakage' t h a t r e s u l t s from i t s c o n v o l u t i o n trum.  I t i s therefore very  w i t h the t r u e  spec-  important t h a t we take a c l o s e r l o o k a t these  functions.  B2.6  D i r i c h l e t ' s K e r n e l , Windows, Smoothing  D i r i c h l e t i n v e s t i g a t e d the v a l i d i t y o f the F o u r i e r expansion f o r a wide c l a s s o f f u n c t i o n s , and examined the f i n i t e fn(x)  =  y Q a  +  a  i  c  o  s  x  +  + b^ s i n x + at  a  n  c  o  s  s e r i e s o f n terms: n  x  bn s i n nx  a f i x e d p o i n t x i n the s e r i e s by s u b s t i t u t i n g the i n t e g r a l s f o r  c o e f f i c i e n t s a, and b. :  116.  a^=—^j  f ( t ) cos k t d t  v a r i a b l e t i s to d i s t i n g u i s h i t from the f i x e d p o i n t x i n the s e r i e s .  -ir b, = - / k IT  f ( t ) s i n kt dt  -IT  By u s i n g  trigonometric  i d e n t i t i e s and g e o m e t r i c a l  series (for details refer  to Lanczos (14) pg. 129-131) we can form f n (x) = i  /  f ( t ) C n ( t - x) d t  - ir  where Cn(t - x) i s a f i n i t e s e r i e s o f n terms. Cn(5)  i s the ' D i r i c h l e t K e r n e l '  and has the form  r ( r> - s i n (n + 1/2) g " ZTT s i n (1/2 £) n  C  N  (  Q  As n i n c r e a s e s  2  to i n f i n i t y , f n ( x ) should  which can be made a r b i t r a r i l y  small.  , , ' 6  a  approach f ( x ) w i t h an e r r o r  This requires a strong  focusing  power  o f Cn(£) i n the immediate neighbourhood o f £ = 0, such t h a t e v e r y t h i n g  else  i s blotted out. The requirements f o r the performance o f the k e r n e l f u n c t i o n (which i s even) a r e : IT  i)  lim /  /Cn(0/dC = 0  which guarantees t h a t the k e r n e l  f u n c t i o n b l o t s out e v e r y t h i n g  except i n the  immediate neighbourhood o f t = x (£ = 0) ii)  lim  +e / Cn( g)d£ = 1  which g i v e s the proper weight to the f u n c t i o n f ( t ) a t t = x f o r the i n t e g r a tion. These two requirements o f the k e r n e l f u n c t i o n resemble those f o r a u n i t pulse or d e l t a f u n c t i o n except t h a t  f o r a u n i t pulse  l i m i t i n g p r o c e s s which r e q u i r e s the width 2e t o approach  there zero.  i s an e x t r a  117. Note the s t r o n g kernel  t o the D i r i c h l e t ' s  ( 2 . 5 . 1 )  the o n l y d i f f e r e n c e l i e s i n the f a c t o r o f 2IT i n the s i n e  ( 2 . 6 . 1 ) ;  integral.  resemblance o f the k e r n e l o f  The D i r i c h l e t k e r n e l does not have the f a c t o r i n the s i n e  t i o n of Cn(£), sin  whereas 2IT i s present  in  func-  ( 2 . 5 . 1 ) :  2TT(N + 1 / 2 )w  w sin  ("2') 2  IT  T h i s apparent d i s c r e p a n c y D i r i c h l e t k e r n e l was obtained  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 : t h e from the F o u r i e r s e r i e s a n a l y s i s , which  o p e r a t e s i n the fundamental i n t e r v a l o f -IT t o TT (or -SL to I) but the k e r n e l ( 2 . 5 . 1 )  was a r e s u l t o f the continuous F o u r i e r spectrum becoming a l i n e  spectrum, assuming t h a t the o r i g i n a l time f u n c t i o n i n the fundamental p e r i o d (0  to 2tr) was repeated N times. Since  pictorial  the k e r n e l  f u n c t i o n r e c u r r s c o n s t a n t l y i n harmonic a n a l y s i s , a  representation  e f f e c t of i t s operation. rectangular  o f the D i r i c h l e t ' s k e r n e l (Fig.  2 . 6 . 1 ) .  window" i s e q u i v a l e n t  ( 2 . 6 . 1 )  show t h e  In the f i g u r e , the "unmodified for N = 5 .  to D i r i c h l e t ' s k e r n e l  f u n c t i o n i s symmetric, so o n l y h a l f t h e k e r n e l i s p l o t t e d . we found that the f o c u s i n g  will  The  In s e c t i o n B 2 . 5  power i s r e l a t e d to the width o f the main lobe o f  t h i s k e r n e l ; a l s o the o s c i l l a t i o n s beyond the main lobe a r e n o t s m a l l enough to ensure the predominance o f the frequency a t £ = 0 .  The i n s u f f i c i e n t  f o c u s i n g power o f the k e r n e l w i l l mean t h a t we can a p p l y c l a s s of f u n c t i o n s which a r e s u f f i c i e n t l y will  fail  to a p p l y .  I f the f i n i t e  p e r i o d i c and the d i s c r e t e F o u r i e r described  i n section  B 2 . 5 ) ,  smooth, otherwise e q u a t i o n 2 . 5 . 2  time s e r i e s t h a t we a n a l y s e i s nontransform  regards i t as p e r i o d i c ( a s  t h e 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  f a i l u r e o f the u n i f o r m convergence o f the F o u r i e r inaccurate r e s u l t s .  i t o n l y t o the  cause  s e r i e s , and we have  118. The  following  the k e r n e l  i s a c o n v e n i e n t p h y s i c a l i n t e r p r e t a t i o n o f the r o l e o f  function.  Suppose the f i n i t e functions:  time s e r i e s t h a t we sampled i s a product o f two  the i n f i n i t e  s i g n a l t h a t e x i s t s i n the f i e l d , and a  f u n c t i o n t h a t has a form o f 0,0,0,1,1 zero terms f o r the (2N + 1) p o i n t s  rectangular  1,0,0,0 ( t h e r e a r e (2N + 1) non  that we have i n the d i s c r e t i z e d time  series). A f u n c t i o n 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: h(w)  »  k=N V ikw -iNw , - i [ N - l ] w , iNw =2. e =e + e . . . .+e k=-N 1  We sum t h i s geometric  progression:  i(N+l/2)w _ h  (  w  )  Iw72  =  e  h(w)  and  we o b t a i n  =  -i(N+l/2)w  =fw72  - e  ("'" ( - * - / ^ ) ) sin | s  J  n  N+  w  f  o  rw  within  the fundamental  period  the f a m i l i a r k e r n e l .  For our purpose, i n the study o f b u i l d i n g v i b r a t i o n s , we choose to do the a n a l y s i s i n the frequency domain. the  time domain o f the r e c t a n g u l a r  a convolution Fourier  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  f u n c t i o n and the time s i g n a l now becomes  i n the frequency domain o f the k e r n e l  transform of b u i l d i n g v i b r a t i o n s i g n a l .  peaky because o f the v a r i o u s  The r e s u l t i s l i k e l y  to be  modes p r e s e n t ; hence the requirement o f a  smooth f u n c t i o n i s n o t met (see s e c t i o n B2.5). i n t e g r a t i o n o f the c o n v o l u t i o n trum, because the k e r n e l  f u n c t i o n and the  We have the moving weight  g i v i n g us a f a l s e r e p r e s e n t a t i o n  f u n c t i o n does n o t focus p r o p e r l y  o f the spec-  and the secondary  maxima o f the k e r n e l , when m u l t i p l i e d by the o t h e r peaks present and i n t e -  119. grated, w i l l  c o n t r i b u t e to the e s t i m a t e o f the F o u r i e r 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.  T h i s i s the u n f o r t u n a t e 'leakage'  The k e r n e l of the r e c t a n g u l a r f u n c t i o n — Tukey — tions.  problem.  or window, a f t e r Blackman and  i s c e r t a i n l y not s u i t a b l e f o r frequency a n a l y s i s of b u i l d i n g V a r i o u s smoothing techniques have been proposed  requirements  s e t out i n s e c t i o n B2.5.  vibra-  to a c h i e v e the  two  Among these techniques are the  Hanning and Hamming windows, which can be a p p l i e d e i t h e r as a m u l t i p l i c a t i o n i n the time domain or a c o n v o l u t i o n i n the frequency domain. may  i n c r e a s e the f o c u s i n g power o f the main l o b e ; o t h e r s may  h e i g h t o f the wide o s c i l l a t i o n s thus r e d u c i n g leakage. window w i l l depend on the  Some windows suppress  the  The c h o i c e of the  situation.  A d i s c u s s i o n on the a c t u a l c h o i c e o f the windows used i n the d a t a anal y s i s of the b u i l d i n g s measured, f o r the purpose t i o n , was We  g i v e n i n Chapter  are now  input s i g n a l (see s e c t i o n B2.2)  may  h i g h f i d e l i t y r e p r o d u c t i o n o f the  not be good f o r us.  render the r e s u l t i n g  Thus the i d e a of f i l t e r i n g suggested the system  For b u i l d i n g  spectrum  c o n v o l u t i o n w i t h the  erroneous  due  to l e a k a g e .  the data b e f o r e a n a l y s e s are performed  by Blackman and Tukey ( p r e w h i t e n i n g ) . frequency response  We  buildings.  of our equipment suppresses  This i s e s s e n t i a l l y a f i l t e r i n g  is  see from Fig.2.1  that  the amplitude  the frequency i n the range which u s u a l l y i n c l u d e s the fundamental of  vibra-  f r e q u e n c y u s u a l l y i s the most dominant, and a h i g h -  f i d e l i t y r e p r o d u c t i o n of the s i g n a l and a subsequent k e r n e l f u n c t i o n may  separa-  5.  i n a p o s i t i o n to see why  t i o n s the fundamental  o f c l o s e frequency  process and may  of  frequency  h e l p to  enhance the d e t e c t i o n o f h i g h e r modes and weaker s i g n a l s than the fundamental.  120.  B2.7  The Sampling Theorems  The mathematics of the sampling  theorem have been w e l l documented; good  p r e s e n t a t i o n s of the theory can be found Hamming ( 1 9 ) .  i n Blackman and  Tukey (15) and  R.W.  Some r e s u l t s r e l e v a n t to a n a l y s e s t h a t f o l l o w are quoted  here: i)  A b a n d - l i m i t e d f u n c t i o n extending  e q u a l l y spaced  p o i n t s with a s p a c i n g  c y c l e occur i n the h i g h e s t frequency i n g to  i s called  from t = - « to  i s sampled a t  At such t h a t a t l e a s t two present.  The  frequency  the Nyquist or f o l d i n g frequency.  f u n c t i o n i s sampled a t lower  0 0  samples per  f  correspond-  I f the b a n d - l i m i t e d  than twice the f o l d i n g frequency,  i t will  the h i g h e r f r e q u e n c i e s present i n the f u n c t i o n to f o l d back, c a u s i n g  cause  the  problem of a l i a s i n g . ii)  At the f o l d i n g frequency  i t s e l f , i n f o r m a t i o n cannot be  transmitted  because the s i n e f u n c t i o n w i l l become i d e n t i c a l l y z e r o , (see Hamming, We  have to sample a t more than two  frequency  present  a t the f o l d i n g  B2.8  (19))  samples per c y c l e f o r the h i g h e s t  so as to p r e s e r v e the i n f o r m a t i o n of the o r i g i n a l f u n c t i o n  frequency.  S t a t i s t i c a l A n a l y s i s of S p e c t r a and  There i s a v e r y important t i o n Survey t h a t we  the Idea of  Stability  a s p e c t of the a n a l y s i s i n the Ambient V i b r a -  have y e t to d e a l w i t h .  T h i s i s the randomness of  the  data. Since the measurements a r e of random s i g n a l s , we sample we measure as one  should r e g a r d each  r e a l i z a t i o n of a p o p u l a t i o n ( p r o c e s s ) governed  by  121. an unknown s t a t i s t i c a l  distribution.  These observed  v a l u e s , X^, have a  mean v a l u e E(X ) = u, assumed t o be a c o n s t a n t ( u s u a l l y z e r o ) , and a fc  variance a : 2  00 a  = E(X -y) = /  2  (X -y)p(X )dX  — 00 where p(X ) i s the p r o b a b i l i t y d e n s i t y f u n c t i o n o f the e r r o r s .  I f the pro-  fc  b a b i l i t y d i s t r i b u t i o n of X distribution  fc  i s Gaussian,  then  p and a  2  c h a r a c t e r i z e the  completely.  However, the X  fc  we measure a r e a l s o f u n t i o n s o f time:  Then c o n s e c u t i v e v a l u e s o f X ( t ) a r e c o r r e l a t e d .  To account  X  t  = X(t).  f o r this corre-  l a t i o n , we have to assume f u r t h e r t h a t the random s e r i e s i s s t a t i o n a r y , so t h a t the number o f c a l c u l a t i o n s t o be done to o b t a i n the a u t o c o v a r i a n c e s may be rendered  realistic.  S t a t i o n a r i t y means t h a t a l l s t a t i s t i c a l p r o p e r t i e s  depend on the time d i f f e r e n c e s ( o r l a g s ) r a t h e r than on the d i f f e r e n t instants.  Thus a p a r t i c u l a r a u t o c o v a r i a n c e \  E i s the expected  " ^  X i  i  X  +  k "  like:  ^  value operator.  Then the v a l u e s o f y^, w i t h tion  = E{(  y may l o o k  time  \i and a  2  w i l l determine the Gaussian  distribu-  completely. A s t a t i o n a r y random process w i t h zero mean can be r e p r e s e n t e d by ( R i c e  (22)): n X(t) =  I  A i S i n ( w i t + <|>i)  i=l where  Ai  =  random  amplitude  <j)i  =  random phase angle th  wi  =  frequency o f the i  contributing  sinusoid.  s u c c e s s i v e v a l u e s o f A i a r e m u t u a l l y independent and have zero mean. mean square of A i i s the v a r i a n c e A i . 2  The mean square  The  (or variance) of  122. X(t) i s o  =  2  I  Ai  2  i=l A f u n c t i o n G(w)  whose v a l u e a t wi i s equal to Ai /Aw, f o r wi a t e q u a l 2  i n t e r v a l s of Aw, i s : G(w)Aw = A i  2  As the number of s i n u s o i d s i n the random process becomes l a r g e , the v a r i a n c e w i l l become equal to the area under the continuous known as the power s p e c t r a l d e n s i t y f u n c t i o n . importance  f u n c t i o n G(w),  G(w)  expresses  which i s  the  of s i n u s o i d s with f r e q u e n c i e s w i t h i n some s p e c i f i e d  relative  frequency  band. The area under the continuous together w i t h G(w),  curve G(w)  i s the v a r i a n c e , which,  c h a r a c t e r i z e s a s t a t i o n a r y Gaussian  there must be a r e l a t i o n between G(w) F o u r i e r Transforms  of one  and  process.  Hence  ( i n e q u a t i o n 2.8.1).  They a r e  another:  CO  G(w)  =\ I  Y  k  cos kw dk  — co  2.8.2  I f we n o r m a l i z e both s i d e s o f 2.8.2 G(w) 1 r \ . ,. —-—- = — I cos kw dk a — °° a  —  rG(w) •> [ • J w i l l determine o  by  a , 2  the d i s t r i b u t i o n c o m p l e t e l y and w i l l  reflect  the b a s i c  2  v a r i a b l e s 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 s t a t i o n a r y zero mean Gaussian  process. There i s one major d i f f e r e n c e between harmonic a n a l y s i s and statistical  approach ( J e n k i n s ( 2 3 ) ) .  Consider a f i n i t e  r e p r e s e n t a t i o n of the sample a u t o c o v a r i a n c e C^ ( c f . ^ C^ = —  i n 2.8.2) In  series i n 2.8.1):  n-k £  X  A F o u r i e r Transform G(w)  the  t t+k X  of n C^'s  (Wj):  ( f o r a zero mean p r o c e s s ) w i l l g i v e the  'raw'  s p e c t r a l estimate ( c f .  123. 2.8.3 where It  would seem t h a t In(w^) w i l l approximate G(w)  i n c r e a s e s , but i t i s not so. will  f l u c t u a t e about G(w)  matter  how  mean periodogram  take i n the time domain, though the  numbers (as an a r t i f i c i a l s p i k e d spectrum.  phenomenon.  k  to z e r o .  Harmonic a n a l y s i s of random  (An i n t e r e s t i n g p h y s i c a l i l l u s t r a t i o n o f t h i s phenomenon on Noise  analysis).  i s w e l l known t h a t f o r a Gaussian d i s t r i b u t i o n w i t h zero mean and  u n i t v a r i a n c e , the independent y  does not decrease  source of white n o i s e ) produces a h i g h l y v a r i a b l e ,  i s g i v e n i n the Appendix A2  3  resulting  does tend to the s p e c t r a l d e n s i t y , the v a r i a n c e of the  T h i s i s a w e l l observed  y  In(w.) J  (see J e n k i n s ( 2 3 ) ) , which means t h a t no  f l u c t u a t i o n s of In(w^) about G(w^)  It  as n  I t has been shown t h a t the raw e s t i a m t e  a t w^  long a sample we  i n 2.8.2  °  b  X  e  2  k  v a r i a b l e s i n t h i s d i s t r i b u t i o n , y^,  y^  y  y  = v y  2  + v + l 2 2  y  which r e p r e s e n t s a c h i - s q u a r e d i s t r i b u t i o n w i t h k degrees i n c r e a s e s , X^ becomes r e l a t i v e l y l e s s v a r i a b l e .  of freedom.  As  k  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 o r n e a r l y p o s i t i v e s p e c t r a l e s t i m a t e i s i t s e q u i v a l e n t number of c h i - s q u a r e degrees to  of freedom.  Tables are  g i v e the r a t i o o f i n d i v i d u a l s p e c t r a l v a l u e to i t s average  available  v a l u e exceed-  ed, with g i v e n p r o b a b i l i t y , f o r d i f f e r e n t v a l u e s of k - the c h i - s q u a r e degree of freedom.  For the In(w_.) mentioned above, f o r l a r g e n the  distri-  b u t i o n of In(Wj) i s a m u l t i p l e o f c h i - s q u a r e d i s t r i b u t i o n w i t h k = 2, independent  of the a c t u a l s i z e of n.  T h i s means t h a t In(w^) i s so v a r i a b l e 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 t r a l estimates.  procedure o f harmonic a n a l y s i s can l e a d to s t a b i l i z e d T h i s i s the simple  device of s p l i t t i n g  sample X ( t ) i n t o p s e t s o f m terms so t h a t n = pm. a n a l y s i s f o r each s e t and by averaging  spec-  the n terms o f t h e  By c o n d u c t i n g  a Fourier  these r e s u l t s a t each frequency, we  have lm,r(w.) r=l We can make the v a r i a n c e will  i n c r e a s e the e q u i v a l e n t  i n c r e a s e the s t a b i l i t y , the  chi-square  but i n c r e a s e d  frequency r e s o l u t i o n :  r e s o l u t i o n i n each  of Im(w^) as small as we wish.  degrees o f freedom and thus  stability  determination  will  sacrificing  subseries. have to a n a l y s e  f i n e frequency r e s o l u t i o n , (about  while using a s t a b l e estimate are o b t a i n e d .  i s a t t a i n e d by  process  a s m a l l number 'm' w i l l decrease the frequency  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 using a reasonably  This  the data  .004 Hz i n one case)  o f the s p e c t r a from which the damping  values  Tukey (16) suggests t h a t f o r a p p l i c a t i o n s s i m i l a r to the of s t r u c t u r a l p r o p e r t i e s a very s t a b l e estimate  i s necessary.  These requirements w i l l mean t h a t both 'm' and 'p' a r e l a r g e and a r a t h e r lengthy  r e c o r d has to be taken i n the f i e l d .  I t i s p o s s i b l e that long  time  r e c o r d s may v i o l a t e the s t a t i o n a r i t y assumption (Toaka ( 1 8 ) ) , but Tukey (16) recommends t h a t the a n a l y s e s  o f l o n g r e c o r d s be judged on t h e i r own m e r i t .  They o f t e n produce v e r y u s e f u l average An for  spectra.  e m p i r i c a l v e r i f i c a t i o n o f the u s e f u l n e s s  of analyzing  b u i l d i n g v i b r a t i o n has been done by K i r c h e r and Shah ( 8 ) .  long  records  The r e s u l t  was t h a t a f t e r 10 a v e r a g i n g s the average s p e c t r a were s t a b l e and d i d not change upon more a v e r a g i n g .  A c o r r e l a t i o n method was employed to study t h e  125. change i n the v a l u e s o f averaged found  s p e c t r a a f t e r 10 a v e r a g i n g s , and i t was  that e x c e l l e n t s t a b i l i t y was a c h i e v e d . The p r i n c i p l e o f a v e r a g i n g s p e c t r a by u s i n g long r e c o r d s (from 30  minutes to 40 minutes to p r o v i d e data f o r a t l e a s t 10 averages) in  t h i s study.  The use o f 2 s e p a r a t e measurement systems i n the f i e l d  (Chapter 4) i s based  B2.9  was adopted  on t h i s  principle.  Damping E s t i m a t i o n Methods Two methods have been employed to e v a l u a t e the percentage  of c r i t i c a l  damping o f the t e s t s t r u c t u r e s : the a u t o c o r r e l a t i o n and the p a r t i a l  spectral  moment methods. a)  The A u t o c o r r e l a t i o n method: The  t h e o r e t i c a l background f o r t h i s method ( C h e r r y and Brady ( 3 ) ) can  be summarized as f o l l o w s : For a l i g h t l y damped s i n g l e degree o f freedom o s c i l l a t o r having a n a t u r a l frequency w and a s m a l l damping r a t i o  £ the system response f u n c t i o n  is h(x): - Ejwx h(x) =  s i n [A-B,  6  w/l-  E  h( T) = 0 f or  2  WT],  T > 0  2  x< 0  (The c o n d i t i o n o f output  f o l l o w s the i n p u t )  The a u t o c o v a r i a n c e f u n c t i o n * f o r a f u n c t i o n y ( t ) i s Cy( x): , T/2 Cy(x) - Ida, £ J  T+oo  1  -x/2  y ( t ) y ( t + x)dt  For white n o i s e i n p u t w i t h c o n s t a n t s p e c t r a l d e n s i t y Go, the output variance i s  *Note: The a u t o c o r r e l a t i o n f u n c t i o n i s the a u t o c o v a r i a n c e f u n c t i o n normalized by C y ( 0 ) , the v a r i a n c e .  autoco-  126.  Cy(T) = *2_ [e-^ (cos 2gw  / r ^ w T  W T  3  +  §  A -  s i n / 1 - 5* WT) ]  e  This i s a c o s i n u s o i d a l f u n c t i o n w i t h e x p o n e n t i a l decay.  The decay o f the  envelope o f the a u t o c o v a r i a n c e e s t i m a t e o f the output response to log  e s t i m a t e the c r i t i c a l damping r a t i o  can be  £ o f each mode o f the system  used  by the  decrement method. To s e p a r a t e each mode i n the frequency domain a t t e n t i o n must be g i v e n  to  the shortcomings  of the d i g i t a l  process.  The  filtering  out of c e r t a i n  f r e q u e n c i e s i n the frequency domain i s s i m i l a r to a p p l y i n g a window to the i n f i n i t e F o u r i e r s e r i e s , ( s e c t i o n B2.6).  I f the t r u n c a t e d s e r i e s i s t r a n s -  formed back to the time domain u n i f o r m convergence  o f 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 a t the b e g i n n i n g and end of the time s e r i e s , where the n o n - p e r i o d i c time s e r i e s i s t r e a t e d as p e r i o d i c by the d i s c r e t e t r a n s f o r m (see s e c t i o n B2.5).  T h i s phenomenon i s caused  c o n v o l v i n g the k e r n e l f u n c t i o n o f the f i l t e r data window.  The r e s u l t  Fourier  by  w i t h the o r i g i n a l r e c t a n g u l a r  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 a t the d i s c o n t i -  nuities . Some k i n d o f 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  filter,  i n some f a s h i o n (use a  f o r example) i n the frequency domain.  trapezoidal  A f t e r t r a n s f o r m a t i o n back to  the time domain, a c e r t a i n number of p o i n t s at the b e g i n n i n g and end of the time s e r i e s are d i s c a r d e d (see Toaka ( 1 8 ) ) . adopted b)  i n the computer programme used  Partial  T h i s smoothing process has been  f o r t h i s damping e s t i m a t i o n method,  S p e c t r a l Moment Method:  Vanmarcke e t a l (20) proposed  the use of p a r t i a l s p e c t r a l moments to  o b t a i n n a t u r a l frequency and damping e s t i m a t e s .  127.  The  s p e c t r a l moments a r e d e f i n e d as CO  CO  Xo = / w°G(w)dw = / G(w)dw '0 0 where X  0  i s the z e r o t h s p e c t r a l moment, G(w) i s d e f i n e d as the one-sided  ( f o r p o s i t i v e f r e q u e n c i e s o n l y ) power s p e c t r a l d e n s i t y f u n t i o n o f the zero mean s t a t i o n a r y Gaussian process  i n question.  S i m i l a r l y , f i r s t and second s p e c t r a l moments a r e d e f i n e d as CO  X  = / w G(w)dw 0 x  1  oo X  = / w G(w)dw 0 2  1  F o r v e r y l i g h t damping for a)  finding the n a t u r a l frequncy n  b)  L  X0  wn:  J  the v a l u e o f 5  r W  [£ * .15] the f o l l o w i n g s i m p l i f i c a t i o n s can be used  r  1+  =  wa wn wb u b = —rwb  8  2  3  0  These a r e v a l i d  l U-  -All  and wa; wb a r e the c u t o f f f r e q u e n c i e s i s o l a t e d modal peak,  f o r the s p e c i a l case o f fia -  !Tft>.  f o r the  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 c l a i m s the f o l l o w i n g : 1)  For a g i v e n r e c o r d l e n g t h T, e s t i m a t e s  expected to be much more r e l i a b l e nates.  This f o l l o w s from the f a c t  o f s p e c t r a l moments may be  than those o f i n d i v i d u a l s p e c t r a l o r d i t h a t the area under the s p e c t r a l curve,  which i s p r o p o r t i o n a l to the t o t a l power, has a much s m a l l e r v a r i a n c e the power spectrum e s t i m a t e  (see a l s o U l r y c h and Bishop ( 2 4 ) ) .  than  128. 2)  Smoothing of the "raw"  p a r t i a l s p e c t r a l moments and  s p e c t r a l e s t i m a t e s i s unnecessary; parameters based  upon them may  estimated  be expected  to  change v e r y l i t t l e as a r e s u l t of smoothing. D i s c u s s i o n s of the a p p l i c a t i o n s a r e g i v e n i n Chapter  5.  of these methods to the d a t a c o l l e c t e d  

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