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Structural dynamic properties from ambient vibrations Topf, Ulf Andreas 1970

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STRUCTURAL DYNAMIC PROPERTIES FROM AMBIENT VIBRATIONS ULF ANDREAS TOPF Dipl.Ing. 9Technische U n i v e r s i t a t Hannover,1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH..COLUMBIA September 1970 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. ULP A. TOPF Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date September 1970 ABSTRACT Ambient v i b r a t i o n s o f a r e i n f o r c e d c o n c r e t e tower s t r u c t u r e were recordedand a n a l y z e d t o o b t a i n the n a t u r a l f r e q u e n c i e s , the a s s o c i a t e d mode shapes and an e s t i m a t e o f the e q u i v a l e n t v i s c o u s damping. The s t r u c t u r e i n v e s t i g a t e d c o n s i s t s o f f o u r c o n c r e t e w a l l p a n e l s , r i g i d l y connected a t v a r i o u s l e v e l s and c o n t a i n s a l i g h t p r e c a s t c o n c r e t e s t a i r w e l l . I t i s s i m i l a r t o t y p i c a l components o f l a r g e r s t r u c t u r e s , s u c h as s t a i r w e l l s and e l e v a t o r s h a f t s o r c o r e s . The g i v e n I n f o r m a t i o n s h o u l d be u s e f u l i n o f f e r i n g d e t a i l s o f the dynamic b e h a v i o u r o f t h i s type o f s t r u c t u r a l elements. The e x p e r i m e n t a l r e s u l t s are compared w i t h the t h e o r e t i c a l r e s u l t s o b t a i n e d from two- and t h r e e - d i m e n s i o n a l dynamic a n a l y s e s u s i n g m a t r i x methods a p p l i e d to l i n e a r e l a s t i c systems w i t h lumped masses. An e f f i c i e n t computer program t o f i n d the e i g e n v a l u e s and e i g e n v e c t o r s f o r t h i s type o f mathematical model i s d e s c r i b e d . - i i i -TABLE OF CONTENTS page Abstract i i L i s t of Tables i v L i s t of Figures v Acknowledgements v i CHAPTER 1. General 1.1. Introduction 1 1.2. Description of Structure and Si t e 4 CHAPTER 2. Experimental Program 2.1. Instrumentation 6 2.2. Test Procedure 7 2.3. Analysis of Data 11 CHAPTER 3. The o r e t i c a l Analysis 3.1. Computer Programs and Theory of Modal Analysis 20 3.2. Mathematical Models 22 CHAPTER 4. Comparison of Experimental and Theo r e t i c a l Results 4.1. Natural Frequencies 27 4.2. Mode Shapes 31 4.3 o Damping 31 CHAPTER 5. Conclusions 32 Bibliography 33 APPENDIX I i Seismometer C a l i b r a t i o n and Balancing 34 APPENDIX II : Output from Computer Analysis of Plane Frame Model and L i s t i n g of Masses f o r 3-dimensional Models 41 - i v -LIST OF TABLES s page Table 1: Hourly Average Wind Speed and D i r e c t i o n on Test Day 7 Table 2: Execution Times f o r Modal Analysis Program 26 Table 3 s Natural Frequencies i n Hz. 27 Table 4s Ratios of Natural Frequencies 28 Table 5° Damping i n Percent of C r i t i c a l 31 Table A: Seismometer Constants 39 - V -LIST OF FIGURES: F i g . Is View of Ladner Clocktower Looking N-E 3 F i g . 2: Locations of Transducers 5 F i g . 3: T y p i c a l Records of Wind and Man Excited Vibrations 10 F i g . 4 s Fourier Spectrum Pairs f o r T r a n s l a t i o n a l Modes 14-15 F i g . 5; Fourier Spectra f o r Torsional Frequency I d e n t i f i c a t i o n 17 F i g . 6: Response of a S-D-F System to Constant Power E x c i t a t i o n 19 F i g . 7s Mathematical Models f o r Dynamic Analysis 23 F i g . 8: Plan and Column S t i f f n e s s e s of Mathematical Models 24 F i g . 9; T r a n s l a t i o n a l Mode Shapes 30 F i g . A; Seismometer C a l i b r a t i o n C i r c u i t 34 F i g o B: Control Panel Wiring Scheme 36 F i g o C: Seismometer C a l i b r a t i o n Curves 38 F i g . Ds V e l o c i t y Response 40 - v i -ACKNOWLEDGEMENTS The c o n s t a n t h e l p and guidance o f my s u p e r v i s o r , P r o f e s s o r Dr. S o Cherry d u r i n g the r e s e a r c h program i s g r a t e f u l l y acknowledgedo Thanks are a l s o due to P r o f e s s o r D r . R . M . E l l i s and Mr..R. (Bob) Meldrum o f the Department o f Geophysics f o r the l o a n o f the i n s t r u m e n t a t i o n and t h e i r a s s i s t a n c e i n s e c u r i n g the e x p e r i m e n t a l d a t a . The a r c h i t e c t u r a l f i r m o f Thompson,Berwick,Pratt and P a r t n e r s k i n d l y s u p p l i e d the s t r u c t u r a l p l a n s and d e s i g n c a l c u l a t i o n s f o r the c l o c k t o w e r . The w r i t e r i s v e r y much inde b t e d t o Mr. R. Ian M i l l e r f o r h i s r e a d i n e s s to o f f e r a d v i c e d u r i n g the development o f the computer programs and t h e i r o f t e n f r u s t r a t i n g debugging procedure. The r e s e a r c h was made p o s s i b l e through a gr a n t from the N a t i o n a l Research C o u n c i l o f Canada. I t enabled the w r i t e r not o n l y t o deepen h i s knowledge o f s t r u c t u r a l e n g i n e e r i n g but a l s o to l e a r n to l o v e the b e a u t i f u l P r o v i n c e o f B r i t i s h Columbia. -1-1. GENERAL 1.1. INTRODUCTION T h i s t h e s i s p r e s e n t s the r e s u l t s o f a comparative e x p e r i m e n t a l and t h e o r e t i c a l a n a l y s i s o f a simple r e i n f o r c e d c o n c r e t e tower s t r u c t u r e . I t i s meant as a c o n t r i b u t i o n to the knowledge o f b a s i c dynamic c h a r a c t e r i s t i c s o f s t r u c t u r e s and t h e i r i d e a l i z a t i o n as a mathematical model. The t e c h n i q u e used f o r the e x p e r i m e n t a l program i s w e l l e s t a b l i s h e d [ l , 2 , 3 9 8 + ] and i s used to determine the n a t u r a l f r e q u e n c i e s o f v i b r a t i o n , m o d e shapes and the percentage o f e q u i v a l e n t v i s c o u s damping o f the s t r u c t u r e . I t i n v o l v e s f i e l d measurements o f the ambient v i b r a t i o n s o f the tower due to n a t u r a l ( wind,microtremors,etc.) and c u l t u r a l ( t r a f f i c , m a c h i n e v i b r a t i o n s , e t c o ) i n p u t s o u r c e s . The r e c o r d e d d a t a i s then a n a l y z e d by f i n i t e F o u r i e r t r a n s f o r m methods [9] t o y i e l d the d e s i r e d i n f o r m a t i o n . T h i s approach r e q u i r e s a r e l a t i v e l y c o n s t a n t power spectrum o f the i n p u t , w h i c h i s a t l e a s t the case f o r most o f the n a t u r a l s o u r c e s . I f t h i s can be assumed,the s t r u c t u r e i s e x c i t e d i n i t s n a t u r a l modes and a m p l i f i e s the resonant f r e q u e n c i e s p r o p o r t i o n a l to the r e l a t i v e modal d i s -placements. Chapter 2. o u t l i n e s the e x p e r i m e n t a l program and t h e v a r i o u s t e c h n i q u e s used to e x t r a c t the d e s i r e d d a t a . + Numbers i n b r a c k e t s r e f e r to b i b l i o g r a p h y numbers --2-T The next c h a p t e r d e s c r i b e s the m a t r i x a n a l y s i s o f t h r e e - , two- and one-dimensional mathematical models« The fundamentals o f the computer program used are g i v e n i n m a t r i x n o t a t i o n . A l l models are assumed to be l i n e a r l y e l a s t i c and are r e p r e s e n t e d by p r i s m a t i c members,the weight b e i n g lumped a t the nodes. The r e s u l t s o b t a i n e d from experiment and t h e o r y are found to be g e n e r a l l y i n good agreement,which c o n f i r m s the c o r r e c t n e s s o f the assumptions made f o r the mathematical models. -3-View o f Ladner c l o c k tower l o o k i n g N-E -4-1.2. DESCRIPTION OP STRUCTURE AND SITE The Ladner Tower at the U n i v e r s i t y of B r i t i s h Columbia was designed by Thompson,Berwick,Pratt and Partners,Architects and b u i l t of r e i n f o r c e d concrete by Smith Brothers and Wilson Ltd.,Contractors i n the summer of 1968 as a clocktower f o r the U. of B.C. campus. It i s located i n front of the Main Library on a paved plaza, surrounded by a park area.(Fig.1) The structure r i s e s 121.5 feet above the ground l e v e l (see Fig.2a) on a square plan of 13.5 x 13.5 f t . ( s e e Fig.2b). It i s founded 8 f t . below the ground l e v e l on a massive octagonal slab with a diameter of the circumscribed: c i r c l e of 32.4 f t . The slab r e s t s on a well graded sand deposit containing some s i l t and medium f i n e gravel? the standard penetration t e s t value i s approximately 100 blows/ft. The wall panels were cast i n s i t u and have the uniform cross-section shown i n Fig.2b up to elevation 101.2 f t . Above that l e v e l the walls are broken up into small columns of 9«7 f t . height to accomodate an observation platform. The top story consists of a closed box section with c i r c u l a r holes of 7 f t . diameter f o r the c l o c k d i a l s on a l l four faces. The s t a i r s and s t a i r l a n d i n g s were prefabricated of concrete with the same 4000 p s i . minimum 28-day strength as the s t r u c t u r a l parts of the tower. Heavily reinforced spandrels which incorporate the s t a i r l a n d i n g s connect and provide the necessary shear t r a n s f e r between the wall panels. - 5 -ELEVATION 4- 121.5' ~ (b) Typical Floor Plan ^Hatch <K ^ | r (c) Roof Plan clock level — M + 1 0 1 . 2 ' — A (observ. level) + 8 2 . 2 * Seismometer + 5 8 . 2 ' o Position Seismometer Orientation • J - 3 4 . 2 ' I STATION A (Roof) B D y Grade (a) Locations of Measuring Stations FIG. 2 LOCATIONS OF TRANSDUCERS 2. EXPERIMENTAL PROGRAM 2.1. INSTRUMENTATION The sensing instruments used f o r measuring the ambient v i b r a t i o n s were two Willmore Mk. II seismometers with 130 f t . long shielded cables. A control panel f o r c a l i b r a t i o n and balancing of the system incorporated a Maxwell impedance bridge and Geotech s o l i d s t a t e amplifiers,Model AS-330, with high- and low-cut f i l t e r s of 100 Hz. and 0.8 Hz r e s p e c t i v e l y . For, c a l i b r a t i o n and balancing procedures see Appendix I. A Tektronix two-channel o s c i l l o s c o p e with i n v e r t i n g input and a Sanborn heated st y l u s s i n g l e channel o s c i l l o g r a p h were employed to monitor the tape input while recording. The analogue s i g n a l was recorded by a PI 7-track LP Monitoring Recorder, Model PI 5107, at a speed of 15/64 i p s . A two way radio system proved very u s e f u l f o r communication between the recording crew on the ground and i n the tower. -7-2.2 TEST PROCEDURE The records used f o r the analysis were taken on May 4,1970, a day with l i g h t winds averaging 2-8 mph. Table 1 shows the hourly average windspeeds and d i r e c t i o n s as obtained from the Plant Science F i e l d Laboratory, University of B r i t i s h Columbia. Time | (DST) j D i r e c t i o n Average Speed (mph) Record No. [ 1 3 - 1 4 I SE 8 1,2,3 14-15 SE 8 4,5 15-16 S 5 6 I 16-17 1 SW 7 7,8,9 1 1 7 - 1 8 1 NW 3 10,11 1 1.8-19 J S 2 12 TABLE 1: Hourly average wind speed and d i r e c t i o n on test-day The Willmores were set at a resonant, frequency of about 1.2 cps. and damped to 55 percent of c r i t i c a l . To obtain a high s i g n a l to noise r a t i o , the tape input was kept close to the permiss-i b l e l e v e l of 2 Volts peak-to-peak by adjusting the a m p l i f i e r gain f o r each channel a f t e r observing the osci l l o s c o p e f o r some time before recording. Three s e r i e s of records were taken with the seismometers i n d i f f e r e n t l o cations and or i e n t a t i o n s . In the f i r s t s e r i e s of -8-records,one seismometer was kept stationary as a reference in the centre of station A (roof) in N-S direction (Position 1, see Fig<,2c ) „ The other seimometer was then moved from station B to stations C,L and E successively. A record of "both seismo-meter signals was taken for each setup. Assuming the base of the tower r i g i d l y fixed, this procedure determines 6 ratios of relative amplitudes,which are sufficient for defining the f i r s t three mode shapes of translation. The second series of records provided analogous information about the E-W direction. For the third series both seimometers were kept on the roof-level in different positions,yielding data to evaluate the torsional frequencies and the structural damping constants, as well as a relative calibration of the two seismometers. To record torsional motions,the two seismometers were placed in parallel locations along opposite wall faces (locations 4 and 5,Fig.2c). Man excitation was employed to obtain data for estimating the damping constants in the f i r s t translational mode from the 1 logarithmic decrement. Luring this part of the tests,the two seismometers were in perpendicular positions (locations 3 and 5 ) The synchronization of the swaying of the crew in the tower with the fundamental frequency was achieved by giving commands through a walkie-talkie from the ground level where an o s c i l l o -scope was operated to monitor the vibration signal from both -9-channels. An i n t e r e s t i n g beat-phenomenon was observed during these forced vibrations o A f t e r one d i r e c t i o n had been excited, the v i b r a t i o n gradually decayed on one channel,returning p e r i o d i c a l l y . By playing back both channels (see Fig.3b ) on a two channel o s c i l l o g r a p h , i t could be seen that a r o t a t i o n of the plane of v i b r a t i o n took place. A possible explanation f o r t h i s may be r e l a t e d to the f a c t that the structure has a r a d i a l symmetry of s t i f f n e s s . Thus the period of v i b r a t i o n in the excited fundamental mode i s the same f o r any d i r e c t i o n and a s l i g h t disturbance l i k e a wind gust,or the influence of the s t a i r w e l l which acts l i k e a s p i r a l i n s i d e the structure, may cause a r o t a t i o n of the plane of v i b r a t i o n . To reduce the e f f e c t of unwanted wind e x c i t a t i o n during these recordings,the a m p l i f i e r s were adjusted to low gain and an hour of the day was selected when the thermal wind was l i g h t , (see Table 1) As suggested i n [ 3] ,a r e l a t i v e c a l i b r a t i o n f o r the entir e system was c a r r i e d out. Both seismometers were set up i n se r i e s (positions 2 and 6,Fig.2c) on the roof l e v e l . By taking the r a t i o of the amplitudes of the Fourier spectra at the d i f f e r e n t f r e q u e n c i e s , r e l a t i v e c a l i b r a t i o n factors between the two seismometers could e a s i l y be obtained. It should be noted however,that these factors u s u a l l y vary with the frequency. Thus i t i s not possible to apply one fa c t o r as a constant m u l t i p l i e r to the data or the spectra. -11-2.3. ANALYSIS OF DATA The 7-track analogue tape which had been recorded at 15/64 ips. was played back at 15/16 ips. on a Hewlett-Packard Magnetic Data Recording System,3900 Series, and digitized at a rate of 130 samples/real sec./channel with one channel for each seismometer. 40,672 points/channel/record were digitized by an IBM 8092 d i g i t a l computer and an analogue-to-digital converter. This represents 312.9 sec. of real recording time per record in d i g i t a l form. To identify the digitized portion of the analogue record,a computer plot from each d i g i t a l record was made by plotting every 10th point at a scale of 100 points/inch. The rate of di g i t i z a t i o n had been chosen at 130 samples/sec. so that a folding frequency of 65 Hz. for the spectral analysis was available and any possible 60 Hz. e l e c t r i c a l noise could be identified in the Fourier spectra. The predominant frequencies of a vibration record can be visualized from the spikes of the Fourier spectrum. A Fourier spectrum is defined for a function f(x),not equal to zero for 0<x<T as: T F O ) = J f ( T ) . e~ l u > T dx o or,in terms of sines and cosines: T T F O ) = j" f(x)-coswx dx - i J f(x)-sin««x dx. o o The Fourier amplitude spectrum i s then given by the square root -12-o f the sum o f the squares o f the r e a l and imaginary p a r t s : V— ; ' if ; [J f ( T ) - c o s _ T d x ] 2 +[] f ( T ) . s i n w x d x ] 2 ' 0 0 A convenient method to c a r r y out t h i s i n t e g r a t i o n i n a d i g i t a l computer i s the Cooley-Tukey a l g o r i t h m [ 9 ] ,a f a s t F o u r i e r t r a n s f o r m . The program SPECTRA which was used i n the d a t a a n a l y s i s i s a s t a n d a r d program o f the U. o f B.C. C i v i l E n g i n e e r i n g program l i b r a r y and i s based on the Cooley-Tukey a l g o r i t h m . To save computing time,SPECTRA a n a l y z e s the d a t a i n b l o c k s and averages the r e a l and imaginary p a r t s s e p a r a t e l y f o r each b l o c k b e f o r e n o r m a l i z i n g the a m p l i t u d e s . T h i s i s not q u i t e e x a c t , s i n c e the time s h i f t i s not accounted f o r i n the subsequent b l o c k s . To i n v e s t i g a t e the e r r o r i n t r o d u c e d by t h i s , the same p o r t i o n o f a r e c o r d was a n a l y z e d by computing the r e a l and imaginary p a r t s and n o r m a l i z i n g the amplitudes o f each b l o c k s e p a r a t e l y b e f o r e they were averaged. A maximum . d i f f e r e n c e o f o n l y 3 p e r c e n t was found,as compared to the method used i n SPECTRA. The same r e s u l t was a l s o found when e v a l u a t i n g the amplitude r a t i o s o f two seismometers. S e v e r a l p o r t i o n s o f a r e c o r d w i t h h i g h - f r e q u e n c y content were a n a l y z e d w i t h a bandwidth o f 0.032 Hz. No s i g n i f i c a n t peaks were e v i d e n t on the F o u r i e r s p e c t r a above 30 Hz. In o r d e r to save computer time and to reduce the amount o f output from the s p e c t r a l a n a l y s i s , i t was d e c i d e d to lower the f o l d i n g f requency to 32.5 Hz. T h i s was accomplished by c o n s i d e r i n g o n l y every second p o i n t o f the d i g i t a l d a t a and b r i n g i n g the - 1 3 -the sampling rate to 65 samples/sec. The average computer time f o r the analysis of 4096 points at a bandwidth of 0.064 Hz. was approximately 10.7 sec. on an IBM 360/67. To investigate the influence of the amplitude of the v i b r a t i o n s on the Fourier spectra,two portions of a record with high amplitudes induced by strong wind were analyzed and compared with the spectra of two low amplitude sections of the . same record. No dif f e r e n c e f o r the resonant frequencies and only l i t t l e e f f e c t on the mode shape r a t i o s was found. A s i g n i f i c a n t difference,however,could be noted i n the r e l a t i v e amplitudes of the s p e c t r a l peaks between the fundamental and the higher modes. It was found that the stronger winds excite v i b r a t i o n s mainly i n the fundamental mode whereas l i g h t e r winds seem to have an input spectrum which includes the frequency range of the higher modes. E a r l i e r i n v e s t i g a t o r s [ l ] have also mentioned t h i s r e s u l t . I t was further confirmed by the fac t that the spectra of the recordings f o r the E-W direction,taken at wind speeds between 2-5 mph.,quite c l e a r l y showed a peak fo r the t h i r d mode. In contrast,the spectra of the records f o r the N-S d i r e c t i o n with stronger winds from 8-15 mph. showed no t h i r d mode disti n g u i s h a b l e from the noiBe l e v e l . Thus two d i f f e r e n t sections with a low average amplitude from each record were chosen f o r the spectra used to determine the mode shapes. The mode shape r a t i o s were obtained by d i v i d i n g the Fourier c o e f f i c i e n t s of the resonant frequency of the RELATIVE AMPLITUDE RELATIVE AMPLITUDE 0.0 4.0 8.0 12.0 16.0 0.0 10.0 20.0 30.0 « . U " S I -l e v e l B,C„D and E by the F o u r i e r c o e f f i c i e n t s o f the r e f e r e n c e l e v e l A on the r o o f . These r a t i o s , t o g e t h e r w i t h a v a l u e o f 1 . 0 f o r the top l e v e l and 0 . 0 f o r the base by assuming the b u i l d i n g r i g i d l y f i x e d a t the ground l e v e l , w e r e n o r m a l i z e d w i t h r e s p e c t to the l a r g e s t r a t i o and y i e l d e d the mode shapes a t the resonant f r e q u e n c i e s . I t sho u l d be p o i n t e d o u t , t h a t by always u s i n g the r a t i o o f the same two seismometers,no c a l i b r a t i o n i s n e c e s s a r y , because the r a t i o o f the v e l o c i t y s e n s i t i v i t i e s o f the two seismometers i s a c o n s t a n t f o r a g i v e n frequency. T y p i c a l s p e c t r a which were used f o r freque n c y and mode shape i d e n t i f i -c a t i o n a re shown i n F i g . 4 . < To e s t a b l i s h the phase between the two seismometers,the r e a l p a r t s o f the r e s p e c t i v e F o u r i e r c o e f f i c i e n t s a re p r i n t e d out and compared f o r t h e i r s i g n . I f o f the same s i g n . t h e y are i n p h a s e , i f they are o f o p p o s i t e s i g n , t h e y are 180° out o f phase, p r o v i d e d the seismometers were s e t up i n the same d i r e c t i o n . Another,more cumbersome way o f e s t a b l i s h i n g the phase i s to form the sum and the d i f f e r e n c e o f the d i g i t i z e d r e c o r d s o f the two cha n n e l s . I f both a re i n phase,they must show a h i g h e r s p e c t r a l v a l u e f o r the sum than f o r the d i f f e r e n c e . C o n v e r s e l y , i f the d i f f e r e n c e y i e l d s a g r e a t e r v a l u e . t h e n they are 180° out o f phase. S i m i l a r l y , t o i d e n t i f y the t o r s i o n a l modes,the d i f f e r e n c e o f the r e c o r d s o f the two p a r a l l e l seismometers on the top l e v e l i n l o c a t i o n s 4 and 5 was taken t o e l i m i n a t e the t r a n s l a t i o n a l -17-(Q) data from single seismometer tn o UJ Q o O a. I I— 2.5 7.5 3.0 5.0 10.0 12.5 15.0 17.5 FREQUENCY-HZ T 20.0 5 FOURIER SPECTRA FOR TORSIONAL FREQUENCY IDENTIFICATION -18-modes. As a check,the sum was used to s u b t r a c t out the t o r s i o n a l modes. By comparing the two F o u r i e r s p e c t r a thus o b t a i n e d , t h e t o r s i o n a l f r e q u e n c i e s can be r e a d i l y i d e n t i f i e d as shown i n F i g . 5 . ( N o t e d i f f e r e n t s c a l e s on o r d i n a t e s ) . To o b t a i n an e s t i m a t e o f the amount o f e q u i v a l e n t v i s c o u s damping f o r the d i f f e r e n t modal resonances,two d i f f e r e n t methods were employed (see Ref. [ 5 ] ) . The f i r s t i s o n l y f e a s i b l e f o r modes o f up to 2 cps. I t con-s i s t s o f e x c i t i n g a mode by l e t t i n g a p e r s o n push a g a i n s t the s t r u c t u r e a t a s u i t a b l e e l e v a t i o n i n the d e s i r e d d i r e c t i o n a t the resonant f r e q u e n c y . The l o g a r i t h m i c decrement o f the amplitude decay from the o s c i l l o g r a p h r e c o r d o f the analogue s i g n a l then g i v e s the damping ^ by the formula l n ( A _ / A n + i ) = 2TC-( | / V l - f ) where An and A n+_ are two s u c c e s s i v e amplitudes o f v i b r a t i o n o f the s t r u c t u r e , a f t e r the e x c i t i n g f o r c e has been removed.' I f 0 . 2 , t h e n i t can be found w i t h s u f f i c i e n t a c c u r a c y from: l n ( A n / A n + _ ) = 2u-V? Another method consists o f measuring the bandwidth (see Fig.6) a t the h a l f power p o i n t s o f the F o u r i e r s p e c t r a l peaks a t <V0 , g i v i n g the v i s c o u s damping as <§ _ Aco/2 O J 0 . F o r a d e r i v a t i o n o f the c i t e d formulae above,see R e f . [ 5 ] < ; -19-P i g . 6 Response o f a S-D-P system to c o n s t a n t power e x c i t a t i o n -20-3 . THEORETICAL ANALYSIS 3 d . COMPUTER PROGRAMS AND THEORY OF MODAL ANALYSIS A computer program was developed to f i n d the e i g e n v a l u e s and modeshapes o f l i n e a r l y e l a s t i c s t r u c t u r e s w i t h p r i s m a t i c members and lumped masses. There a r e two v e r s i o n s o f the program,one f o r 2-dimensional s t r u c t u r e s w i t h up t o 3 degrees o f freedom ( d - o - f ) p e r node,the o t h e r f o r 3-dimensional models w i t h up t o 6 d-o-f p e r node. F o r a s t r u c t u r e w i t h n d-o-f the s t r u c t u r e s t i f f n e s s m a t r ix[K ] i s o f the o r d e r n x n . D i r e c t l y from t h i s the m x m reduced f l e x i b i l i t y m a t r i x [F^ i s f o u n d , r e t a i n i n g o n l y those d-o-f which a re a s s o c i a t e d w i t h one o f the m masses. T h i s i s done by s o l v i n g m-times [K]{6}={P) ,where {P} i s a v e c t o r c o n t a i n i n g z e r o s except f o r a f o r c e o f magnitude 1 i n the row c o r r e s p o n d i n g to one o f the d-o-f t o be r e t a i n e d . Thus the reduced f l e x i b i l i t y m a t r i x i s generated column by column w i t h o u t i n v e r t i n g p a r t o f the matrix[K_. T h i s procedure i s o f p a r t i c u l a r advantage i f m i s s m a l l compared to n,which i s u s u a l l y the case,when no r o t a t i o n a l masses are b e i n g i n t r o d u c e d . I t i s even more ; pr o n o u n c e d , i f o n l y one t r a n s l a t i o n i s a s s o c i a t e d w i t h a mass. By c o n t r a s t , t h e c o n v e n t i o n a l way o f r e d u c i n g a m a t r i x by p a r t i t i o n i n g c o n s i s t s o f i n v e r t i n g a m a t r i x o f the o r d e r (n-m) x (n-m), which can be v e r y time consuming,if not i m p o s s i b l e f o r l a r g e v a l u e s o f (n-m). Knowing t h a t [ F * ] = [K*] ,where [K*] i s the reduced s t i f f n e s s matrix,we can proceed as f o l l o w s . I f a m a t r i x [K*]has the e i g e n v a l u e s X. and e i g e n v e c t o r s {&_J ,then these s a t i s f y the equa t i o n I f we p r e m u l t i p l y t h i s e q u a t i o n by --— [ K * ] " 1 and p o s t m u l t i p l y by [ M ] - 1 , t h e i n v e r s e o f the d i a g o n a l mass matrix,we o b t a i n [M -^T^HM - 0 • (2) I t f o l l o w s , t h a t the e i g e n v a l u e s o f e q u a t i o n (2) a r e the r e c i p r o c a l s o f those o f (1) w h i l e the e i g e n v e c t o r s remain the same. Thus the s m a l l e s t e i g e n v a l u e o f the o r i g i n a l problem (1) can be found by t a k i n g the i n v e r s e o f the l a r g e s t e i g e n v a l u e o f e q u a t i o n ( 2 ) . T h i s way i t i s p o s s i b l e t o s o l v e the dynamic e i g e n v a l u e problem wi t h o u t i n v e r t i n g p a r t o f the u s u a l l y l a r g e unreduced s t r u c t u r e s t i f f n e s s m a t r i x and y e t i n c o r p o r a t e the exact s t r u c t u r a l b e h a v i o r without r e s t r i c t i n g any d-o-f by assuming r i g i d g i r d e r s f o r the mathematical model. Another advantage i s t h a t , w i t h the s u b r o u t i n e used,the l a r g e s t e i g e n v a l u e s found from (2) are a l s o more a c c u r a t e than the s m a l l e s t e i g e n v a l u e s o f (1). F o r convenient s o l u t i o n , t h e unsymmetrical c o e f f i c i e n t determinant o f the frequency e q u a t i o n (2) i s co n v e r t e d i n t o the symmetrical form [ [ M ^ M * - ^ ! ] ] ^ ^ = 0 (3) where [ i ] i s the i d e n t i t y m a t r i x . The n a t u r a l f r e q u e n c i e s w± f o r each mode are then g i v e n as LO± =yi/Ai ' „ -22-3.2. MATHEMATICAL MODELS Three entirely different mathematical models (see Fig. 7) were derived directly from the structural drawings to investigate various methods of Idealizing a rather simple structure. One. common assumption was made for a l l three models: that the structure i s r i g i d l y fixed 1 f t . below ground l e v e l . This seemed jus t i f i e d by the very r i g i d foundation walls below that point. A uniform modulus of e l a s t i c i t y of 4000 k s i . was used for a l l models. It was calculated by using the formula E c = 57,000 ^TfcT where E c is the modulus of e l a s t i c i t y of normal aggregate concrete and fc' i s the compressive strength (both in ps i . ) . Shear deformations of the members were neglected. The f i r s t model (Fig.7a) i s a cantilever with 10 masses and varying stiffness over the height. Since the degree of shear transfer between the wallpanels through the spandrels could not be estimated readily,two extreme cases were considered to allow modelling as a simple cantilever. To establish an upper bound on the stiffness,complete shear transfer was assumed. The lower bound would be only the sum of the individual moments of inertia of the four wall panels acting parallel , without shear connection. This gives two different stiffness distributions for a simple cantilever in one plane. The derivation of the member stiffnesses i s shown in Fig.8 for FIG. 7 MATHEMATICAL MODELS FOR DYNAMIC ANALYSIS CANTILEVER LOWER BOUND: I = I *T UPPER BOUND: C*KK to 4 T »« <• T t i + A4<&* * Ag® 2 , PLANE FRAME SPACE FRAME — p f e — I - S j I " - I ' ' T _ I + I All moments of inertia as for wall panels in original structure ^ T a w T E S S E S O F the c a n t i l e v e r aa w e l l as f o r the o t h e r two models. Appendix I I l i s t s the member p r o p e r t i e s as used i n the computer a n a l y s i s . The next,more r e f i n e d model,shown i n Fig.7b i s the plane frame which i s modelled w i t h the same 10 lumped masses used f o r the c a n t i l e v e r . The w a l l p a n e l s and the shear t r a n s f e r r i n g s p a n d r e l s are r e p r e s e n t e d i n one plane as the columns and h o r i z o n t a l members o f an unsymmetric frame. S i n c e the columns do not have the w i d t h o f the o r i g i n a l w a l l s , t h e modelled s p a n d r e l s must be o f g r e a t e r l e n g t h than i n the r e a l s t r u c t u r e . To compensate f o r t h i s and s i n c e the g o v e r n i n g l o a d c a s e f o r these members i s shear t r a n s f e r , t h e y are g i v e n a g r e a t e r , bending s t i f f n e s s than the r e a l elements. The moment o f i n e r t i a i s c a l c u l a t e d so t h a t under the a c t i o n o f a u n i t shear they d e f l e c t the same amount as would the s h o r t e r , r e a l members. T h i s l e a d s to the formula Jnew = I o l d x ( L n e w / L o l d ) S S i m i l a r l y a c o r r e c t e d a r e a can be found t o y i e l d the same, a x i a l s t i f f n e s s o f the s p a n d r e l s i n the model and i n the r e a l s t r u c t u r e : Anew = A o l d x (^new/^old) Both m o d e l s , c a n t i l e v e r and plane frame,were s o l v e d by the p l a n e frame v e r s i o n o f the program d e s c r i b e d i n s e c t i o n 3.1. Each; j o i n t was g i v e n 3 degrees o f freedom ( d - o - f ) . The reduced m a t r i x was o f a s i z e 10 x 10 a c c o r d i n g to the h o r i z o n t a l d-o-f a t the j o i n t s w i t h a lumped mass. The t h i r d model i s a space frame,idealized as shown i n Fig.7c . Various numbers of masses were used,each of which was acting i n the two h o r i z o n t a l d i r e c t i o n s x and y. In the unreduced s t i f f n e s s matrix,each node has 6 degrees of freedom.3 trans= l a t i o n s and 3 r o t a t i o n s . Four columns represent the v e r t i c a l wall panels. The h o r i z o n t a l members simulate the action of the spandrels,with corrected axial,bending and t o r s i o n a l s t i f f n e s s e s as outlined on the foregoing page. To be able to take advantage of the concept of reducing the structure s t i f f n e s s matrix,the l e a s t possible number of masses should be used. The e f f e c t of decreasing the number of masses on the accuracy of the natural frequencies of the space frame was investigated. 72, 36 and 22 masses were used with the model which i s shown with 36 masses i n Fig.7c. The same member properties were used i n each case. The e f f e c t of the number of masses on the execution time of the computer programs f o r an IBM 360/67 i s demonstrated i n Table 2. 1 No. of masses No. of members No. of deg. of freedom Matrix-bandwidth CPU-Time (sec) Space frame 72 1 1 6 4 3 2 4 8 2 9 9 o l 36 1 1 6 4 3 2 4 8 1 2 5 » 8 2 2 1 1 6 4 3 2 4 8 8 4 . 7 Plane j frame 1 0 1 0 0 1 9 8 1 8 1 5 . 1 Canti=J -, QJ l e v e r J 1 0 3 0 6 7 » 0 TABLE 2 i Execution times f o r modal analysis program -27-4. COMPARISON OP EXPERIMENTAL AND THEORETICAL RESULTS As c o u l d be expected from the e s s e n t i a l l y symmetric s t r u c t u r e , no s i g n i f i c a n t d i f f e r e n c e was found between the E-W and N-S d i r e c t i o n s . Thus i n the f o l l o w i n g t a b l e s o n l y one d i r e c t i o n i s l i s t e d f o r the t r a n s l a t i o n a l modes. These r e s u l t s a re a p p l i c a b l e f o r any a x i s through the c e n t r e of the tower p l a n because o f the i n h e r e n t r a d i a l symmetry. 4.1. NATURAL FREQUENCIES Table 3 g i v e s the n a t u r a l f r e q u e n c i e s o f the v a r i o u s mathemat-i c a l models as d e s c r i b e d under 3.2. and the r e s u l t s o f the ambient v i b r a t i o n survey. MODE CANTILEVER PLANE SPACE FRAME AMBIENT Lower bound Upper bound FRAME 22 masses 36 masses 72 masses VIBRATION TEST TRANSLATION i 0.91 1.93 2.07 1.98 1.95 1.95 1.78 TRANSLATION 2 3.32 3.92 9.17 8.55 8.47 8.47 7.52 TRANSLATION 3 7.45 15.62 17.55 16.95 16.95 16.95 15.38 fc 1 - - - 3.91 3.89 3.89 3.76 JRSIC 2 - - - 9.53 9.52 9.52 5.65 r 3 - - - 15.87 15.87 15.87 10.57 TABLE 3 : N a t u r a l f r e q u e n c i e s i n Hz. The r e s u l t s o f Ta b l e 3 are n o r m a l i z e d as frequen c y r a t i o s i n Table 4 on the f o l l o w i n g page. - 2 8 -IMODE CANTILEVER PLANE SPACE FRAME AMBIENT Lower bound Upper bound FRAME 2 2 masses 3 6 masses 7 2 masses VIBRATION TEST ION 1 1 , 0 0 1 . 0 0 1 . 0 0 1 . 0 0 1 . 0 0 1 . 0 0 1 . 0 0 SLAT 2 3 o 6 7 2 . 0 4 4 . 3 8 4 . 3 2 4 . 3 3 4 . 3 3 4 . 2 2 . . TRAN 3 8 . 1 0 8 . 1 1 8 . 3 7 8 . 5 6 8 . 6 6 8 . 6 6 8 . 6 6 1 - - - 1 . 0 0 LOO 1 . 0 0 1 . 0 0 RSIO: 2 - - -- 2 . 4 4 2 . 4 4 2 . 4 4 1 . 5 0 o EH 3 - - - 4 . 0 6 4 . 0 8 4 . 0 8 2 . 8 6 TABLE 4 t R a t i o s o f n a t u r a l f r e q u e n c i e s TRANSLATIONS I t can be seen t h a t a l l models.except the lower bound on the c a n t i l e v e r g i v e r e s u l t s which are i n good a g r e e -ment w i t h the fundamental frequ e n c y observed i n the ambient v i b r a t i o n t e s t . T a b l e 4 shows t h a t the r a t i o s o f the f r e q u e n c i e s o f the more a c c u r a t e plane frame and space frame models are a l s o i n v e r y good agreement w i t h the e x p e r i m e n t a l r e s u l t s . T h i s means t h a t almost any d i f f e r e n c e f o r a l l the f r e q u e n c i e s c o u l d have been e l i m i n a t e d by u s i n g a lower modulus of e l a s t i c i t y E i n the computer models. In t h e a n a l y s i s p r e s e n t e d , a v a l u e o f E = 4 0 0 0 k s i . was used. S i n c e o n l y d a t a on the 28day s t r e n g t h o f the c o n c r e t e were a v a i l a b l e , t h e a c t u a l s t r e n g t h a t the date o f the t e s t had to be e x t r a p o l a t e d , t h u s i n v o l v i n g some u n c e r t a i n t y . I t would be d e s i r a b l e f o r f u t u r e i n v e s t i g a t i o n s t o c a r r y out n o n - d e s t r u c -t i v e t e s t s t o g i v e exact v a l u e s f o r a dynamic modulus t o be used f o r the a n a l y t i c a l models. I t s hould be noted t h a t f o r a l l models the transf o r m e d member s e c t i o n s ( i . e i n c l u d i n g r e i n f o r c i n g s t e e l ) were used. A p r e l i -minary a n a l y s i s u s i n g p l a i n c o n c r e t e s e c t i o n s produced f r e q u e n c i e s which were up t o 5 p e r c e n t lower than those l i s t e d i n Table 3 . TORSION: The fundamental t o r s i o n a l frequency o f the space frame models i s o n l y 3 . 8 p e r c e n t h i g h e r than the v a l u e from the ambient v i b r a t i o n t e s t s , b u t f o r the h i g h e r modes d i f f e r e n c e s o f up t o 41 p e r c e n t are e v i d e n t . Hence the r a t i o s o f the t h e o r e t i c a l t o r s i o n a l f r e q u e n c i e s do not compare f a v o u r a b l y w i t h the t e s t r e s u l t s . No apparent r e a s o n c o u l d be found i n the a n a l y t i c a l model f o r t h i s d i s c r e p a n c y . The space frame model i s b e l i e v e d t o be q u i t e a c c u r a t e , a s can be seen from the t r a n s l a t i o n a l modes and f r e q u e n c i e s and the f i r s t t o r s i o n a l f requency. -30-STATION Mode I Mode 2 Mode 3 MODAL C O E F F I C I E N T S I s t MODE 2 n d MODE 3 r d MODE Station Space Frame Experiment Space Frame Experiment Space Frame Experiment A 1. 0 0 1.00 1.00 i .;oo__ l.OC 0 . 6 4 B 0. 78 0 .77 6. 13 - 0 . 0 7 " - 0 . 8 1 - 1 . 0 0 C 0 .62 0 . 5 8 - 0.22 - 0 . 3 7 - 0 . 6 1 - 0 . 6 1 D 0 . 3 6 0 .38 - 0 . 5 2 - 0 . 5 8 0.28_ 0.51 E 0 . 15 0.20 - 0 .38 - 0 . 4 3 0 .65 0 . 7 3 Ground 0 . 0 0 — 0 . 0 0 — 0 . 0 0 — FIG. 9 TRANSLATIONAL MODE S H A P E S -51-4.2. MODE SHAPES The comparison of the normalized mode shapes i n Pig.8 shows excellent agreement f o r the f i r s t t r a n s l a t i o n a l mode. The second and t h i r d mode shapes show some deviations at c e r t a i n levels,hut generally the a n a l y t i c a l r e s u l t s are f a i r l y consistent with the experiment. The table below the pl o t t e d mode shapes l i s t s the modal amplitudes of the p l o t . 4.3. DAMPING Table 6 l i s t s the percentage of equivalent viscous damping obtained from man induced v i b r a t i o n s f o r the f i r s t mode and for the f i r s t three modes as evaluated from the Fourier s p e c t r a l peaks. The t o r s i o n a l damping constants were only derived from the bandwidth of the modal resonances. ! LATION J 1 st Mode 2 nd Mode 3 rd Mode ! LATION FOURIER 1 SPECTRA 2.7 0.3 0.2 UJ EH MAN EXCITATION 3.2 -ITORSION FROM 1 FOURIER SPECTRA 0.5 0.7 0.4 TABLE 5: Damping i n percent of c r i t i c a l The r e s u l t f o r the f i r s t t r a n s l a t i o n a l mode i s i n good agreement f o r the two methods,although i t seems somewhat high fo r a structure of the type under consideration. The maximum displacements associated with the damping constants of Table 5 fo r wind and man e x c i t a t i o n were 0.1 and 0.3 mm r e s p e c t i v e l y . -32-5. CONCLUSIONS The ambient v i b r a t i o n s u r v e y r e p r e s e n t s a simple and i n e x p e n s i v e method t o o b t a i n the resonant frequencies,mode shapes and damping c o e f f i c i e n t s o f s t r u c t u r e s a t a low s t r e s s l e v e l . To determine the h i g h e r modes i t seems n e c e s s a r y to work w i t h l i g h t winds as e x c i t i n g f o r c e . F o r f u t u r e p r o j e c t s , i f no a b s o l u t e v a l u e s are d e s i r e d , t h e use o f a c c e l e r o m e t e r s as t r a n s d u c e r s s h o u l d be c o n s i d e r e d . They o f f e r the advantage o f a more compact d e s i g n , a l t h o u g h they are more s u s c e p t i b l e t o a c c i d e n t a l damage. Due to an i n c o m p a t i b i l i t y between the r e c o r d e r i n p u t and the output o f the a v a i l a b l e Brush c a r r i e r a m p l i f i e r s , e x i s t i n g a c c e l e r o m e t e r s c o u l d not be used w i t h the PI magnetic tape r e c o r d e r . A h i g h speed computer i s almost e s s e n t i a l to e v a l u a t e the amount o f d a t a n e c e s s a r y f o r a meaningful sample s i z e . Because o f the s i m p l i c i t y o f the s t r u c t u r e i t was p o s s i b l e t o c o n s t r u c t r e l i a b l e mathematical models f o r the a n a l y s i s w i t h d i g i t a l computers. The concept o f the reduced f l e x i b i l i t y m a t r i x a l l o w e d m o d e l l i n g w i t h a h i g h degree o f a c c u r a c y . I t i s shown t h a t v e r y good r e s u l t s can be o b t a i n e d even w i t h a s m a l l number o f lumped masses. -33-BIBLIOGRAPHY: Crawford and Ward: Determination of the Natural Periods of Buildingso B u l l e t i n Seismological Society of America, Vol.54,No.6,Dec.1964. Ward and Crawford: Wind Induced Vibrations and B u i l d i n g Modes. B u l l e t i n Seismological Society of America,Vol.56, No.4,Aug.1966. R.R.Blandford,V.R.Lamore and J.Aunon: S t r u c t u r a l Analysis of M i l l i k a n L i b r a r y from Ambient Vib r a t i o n s . Feb.1968. Published by Earth Sciences,a Teledyne Co. P.Kollar and R.D.Russell: Seismometer Analysis Using an E l e c t r i c Current Analog. B u l l e t i n Seismological Society of America,Vol.56,No.6,1966. W.Hurty and M.Rubinstein: Dynamics of Structures. Prentice H a l l Inc. 1964. J.Blume,N.Newmark and L.Corning: Design of Mu l t i s t o r y Reinforced Concrete Buildings f o r Earthquake Motions. Portland Cement Assoc.,Skokie,Illinois,1961. S.Cherry : Basic Dynamic P r i n c i p l e s of Response of Linear Structures to Earthquake Ground Motions. Proceedings Symposium on Earthquake Engineering,University of B r i t i s h Columbia,1965. S.Cherry and A.G.Brady : Determination of S t r u c t u r a l Dynamic Proporties by S t a t i s t i c a l Analysis of Random Vibr a t i o n s . Proceedings 3rd World Conference on Earthquake Engineering, New Zealand,1965. V.W.Cooley and J.W.Tukey : An Algorithm f o r the Machine C a l c u l a t i o n of Complex Fourier Series. Math.Comp.,Vol.19, A p r i l 1965. -34-APPENDIX I : SEISMOMETER CALIBRATION AND BALANCING I t can be shown t h a t a seismometer may be r e p r e s e n t e d by an e q u i v a l e n t c i r c u i t (see F i g . A below and Ref. [4]). CURRENT DRIVEN SEISMOMETER EQUIVALENT CIRCUIT MAXWELL BRIDGE OSCILLATOR F I G . A SEISMOMETER CALIBRATION CIRCUIT The v a l u e s o f the v a r i o u s components o f the e q u i v a l e n t c i r c u i t a r e r e l a t e d t o the seismometer c o n s t a n t s . s p r i n g c o n s t a n t U,damping c o n s t a n t D.mass M,transducer c o n s t a n t g and t o the ground a c c e l e r a t i o n y. R Q and L_ are the r e s i s t a n c e and i n d u c t a n c e o f the c o i l and the s w i t c h i s analogous t o a clamp used t o prevent the mass from swinging. As K o l l a r and R u s s e l l [ 4 ] o b s e r v e , t h e e l e c t r o m a g n e t i c seismometer and the e q u i v a l e n t c i r c u i t are i n d i s t i n g u i s h a b l e by measurements made a t the output t e r m i n a l s . The c a l i b r a t i o n o f the seismometers i n v o l v e s the d e t e r m i n a t i o n o f i t s response t o s i n u s o i d a l ground motions i n the d e s i r e d frequency range. For the c a l i b r a t i o n the clamped seismometer i s placed i n the 'unknown* p o s i t i o n of the Maxwell impedance bridge and the bridge i s balanced i n the usual manner f o r XMAIN" input. The balance condition i s independent of the frequency and gives the values of R and L . c c K o l l a r and Russell have shown that with the seismometer un= clamped,the r a t i o of detector outputs f o r 'MAIN' and *' SUBSTITUTION ' inputs i s (R E»Z g)/(R R-R B) from which Z Q may be determined as a function of w . The pos i t i o n s of the reson-ant peak and the asymptotes of a logarithmic p l o t of Z f F i g . C ) against cu ,together with the known suspended mass M,determine the values of U,D and g. They also show that a p o t e n t i a l v applied to the ' MAIN ' input of the bridge produces the same r e s u l t as a current generator v/R-r, i n parallel with Z_,which, comparing with the equivalent c i r c u i t , i s equivalent to a ground ac c e l e r a t i o n (g« v)/(M«R-g). Fig.B shows a sketch of the seismic control panel used. S 1 , S 2 and S 3 represent switches? A and B denote the two seismometers. BRIDGE BALANCE PROCEDURE : Clamp Seismometer S 1 = K S 2 = K S 3 = A or B Connect 1 cps. 30 V p.p. sine wave to "MAIN' terminals of bridge and adjust R-g to get a minimum d e f l e c t i o n on the oscilloscope which i s connected to the v SANBORN ' WHITE and - 3 6 -6REEN c o n t a c t s through i n p u t 1 and i n v e r t e d i n p u t 2, W H I T E ^ 0 = ® ^ ^ ] SANBORN | S 3 | _f ivBLACK K TAPE RECORDER WPUT SUBSTITUTION NPUT FIG. B CONTROL PANEL WIRING SCHEME DETERMINATION OP SEISMOMETER CONSTANTS : S 1 = K S 2 = K S 3 = A o r B Connect o s c i l l a t o r a l t e r n a t i n g l y t o XMAIN' and xSUBSTITUTION' i n p u t . * SANBORN ' ; WHITE = I n v e r t e d i n p u t 2 GREEN = Input 1 Then unclamp seismometer,set p e r i o d a d j u s t e r t o 3 and l e v e l ] O s c i l l o s c o p e -37-the instrument which preferably should be set up i n a l o c a t i o n with a low noise l e v e l . The attenuation of the o s c i l l a t o r should be between 2 and 5 V p.p. according to the noise l e v e l . I f a high noise l e v e l i s present 9no c l e a r readings can be taken at a high gain s e t t i n g of the o s c i l l o s c o p e which w i l l be. required with a small input s i g n a l . S t a r t i n g with an o s c i l l a t o r frequency of 0.1 Hz. and increas-ing the frequency i n steps s u i t a b l e f o r a log-scale,the d i f f e r e n t i a l output on the o s c i l l o s c o p e should be read a l t e r -n a t i v e l y f o r * SUBSTITUTION' and *MAIN' input. The c a l i b r a t i o n curves f o r seismometer A and B are shown i n Fig.C. Prom Pig.A the t r a n s f e r function of the equivalent c i r c u i t i s found ass (l/R)+(l/5u,i)+(3wc) with R = g 2/D L = g2/U C = M/g2 K o l l a r and Russel [5] have shown that V /V = z -V M A I N / V S U B ^s RR « RB For high frequencies ( c o » t o ) ; ( 1 ) M (2) At resonance ( c o = co ) s Z„—o- R and I Z„l = R = ( 3 ) and also c o = YU/M' (4) For low frequencies ( t o ^ - t o ) ; Z a — o j w L and |Z g| = (5) -39-With equations (1) through (5) on page 36 and M =4.75 kg the following constants f o r the seismometers were determined from Pig.C : Seismometer A B (R R. R B)/R E 401 390 co 0 (rad/sec) 7.29 7.92 U (Newtons/m) 252 298 g g (Volt/m/sec) 188 206 g^ (Volt/m/sec) 196 222 D (Newton/m/sec) 1.15 1.30 TABLE A s Seismometer constants DETERMINATION OE VELOCITY RESPONSE OP COMPLETE SYSTEM. S 1 = 1,2,3,4 S 2 = S S 3 = A or B O s c i l l a t o r connected to XMAIN ' input. Oscillograph connected to o s c i l l a t o r to measure V i n p U ^ ? * s l a t e r connected to PLAY-BACK-OUTPUT of taperecorder a f t e r s i g n a l i s on tape and can be played back to measure output. K o l l a r and Russell[5] have also shown that the v e l o c i t y resp-onse of a system i s given by: (V o u t/V i n)-(^-M.R B)/g = (VOLTS/m/sec) The v e l o c i t y response of the two seismometers,including the amp l i f i e r s i s p l o t t e d i n Pig. D. -41-APPENDIX II: Output from Computer Analysis of Plane Frame Model and Listing of Masses for 3-dimensional Models. R F S N O o 77C818 .****$********$$*:&*: $ S I G T O P F P R I O = V * * L A S T S I G N O N WA S • U N I V E R S I T Y O F B C C O M P U T I N G C E N T R E M T S ( A N 120) 233 26: 2 5 07-22-70 : * T H I S J O B S U B M I T T E D T H R O U G H F R O N T D E S K R E A D E R * * * * * * * * * * * * * * * * * * * * 23:26:21 07-22-70 U S E R " T O P F 1 1 $ L I S D A T A 1 2 3 4 S I G N E D O N A T 23:26:26 O N 07-22-70 J O I N T 2 36 N O S - A N D M A S S E S F O R 36 M A S S - S P A C E F R A M E 5 6 __7_ 8 9 10 1 2 6 20. 22 17.f 7 23 18, 17, 10 17 . 26 17.5 11 17 . 27 17.5 14 17. 30 17.5 15 31 17.25 17.5 18 17.5 34 17.5 19 17.5 35 17.5 38 17 .5 39 17 .5 42 17.5 54 19.5 55 18.55 58 17.6 70 19. 72 19. 74 16. 43 17.5 59 17.6 76 16. 46 17.5 62 24. 47 64 18. 5 24. 50 19.5 66 31 , 51 19.5 68 31. 11 12 13 14 15 16 J O I N T N O S . A N D M A S S E S F O R 72 M A S S - S P A C E F R A M E 72 17 1 2 18 5 10 . 6 10 . 7 10 . 8 8.5 9 8.5 10 8, 5 11 8. 5 12 8. 5 * It- 19 1 3 8. 5 14 8. 5 15 8.5 16 8.75 17 8 .75 18 8.75 19 8.75 20 8.75 t •-' 20 21 8.75 22 8.7 5 23 8 .75 24 8.75 25 '8. 75 26 8. 75 27 8.75 28 8 .75 21 29 8.7 5 30 8.75 31 8.75 32 8.75 33 8.75 34 8.75 35 8.7 5 36 8. 75 22 37 8. 75 38 8.7 5 39 8.75 40 8.7 5 41 3.75 42 8 .75 4 3 8.75 44 8.75 V '2 3 45 8.75 46 8,75 47 8.75 48 9. 75 49 •9. 75 50 9.75 51 9.75 52 9.75 24 53 9.7.5 54 9.75 55 9.75 56 • 8.8 57 ; s.s 58 8.8 59 8.8 60 8. 8 • i t. 25 61 12. 62 12, 63 12. 64 12, 65 15.5 66. 15.5 67 15.5 68 15.5 26 69 9.5 70 9.5 71 9.5 72 9.5 73 8. 74 8. 75 8. 76 8. E N D O F F I L E m 1 $ SIG m H ; i RFS NO. 770967 UNIVERSITY OF 8 C COMPUTING CENTRE MTS(AN120) 17:27:53 07-27-70 ••REMINDER—ALL TAPE MOUNTS MUST HAVE RACK NUMBERS** J M : * * * * * * * * * * * * * * * * * * THIS JOB SUBMITTED THROUGH FRONT DESK READER ******************** $SIG TOPE PRI0=V P=500 T=2QQ C0PIES=5 **LAST S I G N O N WAS: 10:33:41 07-27-70 USER "TOPE" SIGNED ON AT 17:27:54 ON 07-27-70 $LIS 2DRES 1 $RUN OSEISMO 7=*DUMMY* 2 EXECUTION BEGINS 3 * * * * * * * * * * # * * * $ * * * * * * ifdfc******** * * * * * * * # * * # 3 ? t * * * * &***********:£* 4 5 6 8 9 10 CLOCK TOWER - PLANE FRAME ANALYSIS AS A FRAME-MODEL E=4000 KSI ********#* *************************************** "MODULUS " O F ~ELASf I C I Y Y" ~ 4000. (KSI) IGRID = 1 NO. OF MEMBERS100 NO. OF D-O-F OF REDUCED MATRIX 10 11 NO. OF ELASTIC SUPPORTS 0 12 IDIM= 0 NFIX= 1 13 N 8 AY = 2 NSTOR= 32 14 NO. OF JOINTS CONST R AINE D= 33 16 < 17 20 INPUT JOINT DATA 21 JOINT HOR VERT ROT X (FT) Y (FT) I1 22 • 23 1 0 0 "6 ' 0 . 0 24 2 0 0 0 6.167 0 . 0 25 3 0 0 0 12.334 0.0 26 4 0 0 0 0.0 2.167 27 • 5 1 1 1 6. 167 2.167 28 6 1 1 1 12. 334 2.167 29 7 1 r "I 0.0 5.167 30 8 1 •1 1 6.167 5 .167 31 9 0 0 0 12.334 5. 167 32 10 0 0 0 0.0 8.167 33 1 1 1 1 1 6.167 8.167 34 12 1 1 1 12.334 8.167 35 13 1 1 1 0.0 i1.167 36 14 1 1 1 6. 167 11.167 37 15 o 0 0 12.334 11.167 38 16 0 0 0 0.0 14.167 39 17 1 1 1 6. 167 14.167 40 18 1 1 1 12.334 14.167 41 19 1 1 1 0.0 17.167 42 20 1 1 1 6. 167 17.167 • 43 21 0 0 0 12.334 17.167 44 22 0 0 0 0.0 20.167 45 23 1 1 1 6.167 20.167 A 46 24 1 1 1 12.334 20.167 47 25 1 " 1 1 6.0 23.167 48 26 1 1 1 6.167 23.167 •- 49 27 0 0 0 12.334 23.167 50 51 52 53 54 55 28 29 30 31 32 3 3 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 .1 0 0.0 6. 167 12.334 0.0 6.167 12.334 26.167 26.167 26.167 29.167 29. 167 29.167 u i .J> u>,ro o sO vO vO|vO CO O J ro o o ro cn ojro en o o e o ! o o W . H o] OJ f-* U J O jOJ 0* .p. - J i -p- —J O O O CD 00 00 ro NJ ro :cn cn cn o o o O o o vJ3 00 -si ,0 U l -P cc CD ro ,00 co -vi u i ' j > UJ o r-1 It-1 C O i-* O t-> ro 0 o o o o U> t— o 0J o o o ro cn o o o OJ t—' |OJ cn •f- -si 00 00 00 OO 00 00 OJ 0J OJ o o o O O O O O \0 U J ro i—• o >xi oo 00 03 -vi! -vj - s i (-* O vO'CD -si o* o o O .—' f-1 I— O O i - 1 t-'.t-' o ro cn Ojro & o o e o I o d o O J O|0J i-« o 0J O ( O J o -p- -J l-p- -si -si -s) -sl'-sl -vj -vl -j -vl -vj J> .p. .p. -vi o- u i -p- oJ ro - V J "vj -vj . ~vj - V J U l -P1 U ) M H o H Hn- o o I—1 li-" i — o o o ro o o ro cn o 0 0 0 ; 0 « A U J ^ - O ! O J •— o 0J O , W cr j> -vj |4> - v j - J -sliCn O Cn •—'CO 00 00 vO vO OD;OO co co O co -vi cn cn cn O'Cr* cn cn v0 00 -JO Ul -P-o o o ro cn o ro o o O 0 0 s 0 OJ H O W H O OJ O ,0J 0s 0s- O Ch cn cn cr u i u i u i ro ro ro co oo oo. oo oo co u i J > OJ ro f-- o cn cn cn • o u i u i O t— t— o o o O »-• i i—• I-' O ro cn O ro cn o o o o ' • o ly H O U) H u) cn OJ cn ^ . - j -vi U l U l U l i U l u i u i vO vC vO cn o cn ~vl —j —J -vl -vj vo co -si cn ui -P-•ui u i u i U l U l u i -~j cn u i J> u i ro Q I—' r— r-> t— O O O O >— I—' h-" ' O ro cn C ro cn o 0J '—1 o U J H OJ cn OJ cn U l U l Ul i u i U l u i OJ UJ O J ! O O O - V J - - j -v i , - j cn cn OJ ro r - ' o vo oo u i u i -p-l-p- -P* -P-O vO CC -~J o o o o O t— I-1 I— O f\> cn O --o cn O o e © « © O J >— O,0J oJ cn O J cn -P- — i ;J> - j -vj -j -v); .p. .p. 4> cn cn cn o cn cn cn u i -P- u> ro 4>- ^  -P- -P- -P-O O h-* i—' H-• O ro cn o ro cn o e o e o o e W H O ' W i— o O J cn |OJ cn 4S -vj -vl ^ ^ O J 0J OJ I—' t—• I—'00 oo co cn cn u i u i u i u i i — O vO co -j cn O J O J O J OJ OJ OJ •X) oo cn u i -P-o o o ro cn o ro o o O 0 O 0 O 0 O J cn u) Cn 4> -vj J> -J U> 0J 0J 0J 0J u> u i u i u i ro ro ro O cn cn cn cn cn -Vl -V| -vl ~J -vj cn cn cniCn cn cn -v| -v| -vi|-vl -J -vj O cn cn cn cn O -vj -vj -J : -vj -vj -v| Cn Cn CMcn cn O -j -J -Ji-J -si -sj Cn cn cMo cn cn -vj "-J -J -vj -J -v Cn cn cn cn cn cn -vj -.1 -vj;-J "-j -vl cn cn cn ! cn cn cn -J —i "si -J -vj Cn cn cn cn cn Cn - J - j -s) -^ i - J Cn cn cn cn cn cn -s| -J -v|, —J -VJ —J cn cn cn cn Cn cn —J "Si -vj -vj "Si -J 116 117 118 119 120 121 94 95 96 97 98 99 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 0.0 6.167 12.334 0.0 6. 167 12.334 111.834 111.834 111.834 122.500 122.500 122.500 / 122 123 O U T P U T J O I N T D A T A NU=198 124 { C 001 N G . N U M B E R S ) 126 J O I N T N O ( I , 1) N D U ,2) N O ( 1,3) ' 1 127 J 128 1 0 0 0 J 129 2 0 0 0 1 1 3 0 3 0 0 0 t 1 3 1 4 0 0 0 J ' 132 5 1 2 3 1 133 6 4 5 6 1 1 3 4 7 7 8 9 I 135 8 10 11 12 ' f 136 9 0 0 0 1 ' 137 10 0 0 0 138 11 13 14 15 " : 139 12 16 17 18 I ! 140 13 19 20 21 141 14 22 23 24 142 15 0 0 0 143 16 0 0 0 144 17 2 5 26 27 < 145 18 28 29 30 146 19 31 32 33 147 2 0 34 35 36 - 148 21 0 0 0 • >• 149 22 0 0 0 150 23 37 38 39 151 24 40 41 42 152 25 43 44 45 153 26 46 47 48 154 27 0 0 0 155 28 0 0 0 156 29 49 50 51 157 30 52 53 54 r 158 31 55 56 57 * 159 3 2 58 59 60 160 3 3 0 0 0 161 34 0 0 0 162 3 5 61 62 63 163 36 64 65 66 164 3 7 67 68 69 165 3 8 70 71 72 166 39 0 0 0 167 40 0 0 0 168 41 73 74 75 169 42 76 77 78 ! 170 43 79 80 81 ! 171 44 82 83 84 1 172 45 0 0 o ! 173 46 0 0 0 174 47 85 86 87 > > 175 48 88 89 90 176 49 91 92 93 J I r 177 50 94 95 96 178 51 0 0 0 179 52 0 0 0 •> 180 53 97 98 9 9 181 54 100 10 1 102 L 182 55 103 104 105 1.83 56 106 107 108 \ 184 57 0 0 0 > 185 58 0 0 0 186 59 109 110 111 187 60 112 113 114 188 61 115 116 117 189 6 2 118 119 120 1 90 63 0 0 0 - 191 64 0 '0 0 192 65 121 122 123 193 66 124 12 5 126 194 67 127 128 129 195 68 130 131 132 196 69 0 0 0 197 70 0 0 0 ' 198 71 -133 134 135 199 72 136 137 138 200 73 139 140 141 201 74 142 143 144 202 75 0 0 0 203 76 0 0 0 204 77" 145 "146 147 205 78 148 149 150 - 206 79 151 152 153 • 2 07 80 154 155 156 208 81 0 0 0 > 209 82 0 0 0 ""210" 83 .157 158 159 211 84 160 161 162 212 85 163 164 165 2 13 86 166 167 168 214 87 0 0 0 215 88 169 170 171 2 16 89 172 173 ~~ 174 217 90 175 176 177 218 91 178 179 180 219 92 181 182 183 220 93 184 185 186 221 94 187 188 189 222 95 190 19 1 '192 223 96 193 194 195 224 97 0 0 0 225 98 196 197 198 226 99 0 0 0 227 2 28 I N P U T MEMB~ER D A T A ' 229 M E M B E R JNL JNG KL KG A R E A 1IN2) I (IN4 ) 230 231 1 1 7 1 1 1477.00 33370.00 232 2 7 13 1 1 1477.00 33370.00 233 3 13 19 1 1 1477.00 33370.00 ... 2 34" 4 "19 2 5 ~ 1 1 1477.00 33370.00 235 5 25 31 1 1 1435.00 32560.00 236 6 31 37 1 1 1435.00 32560.00 j 237 2*3 8 239 240 241 10 11 12 37 43 49 55 61 67 43 49 55 61 67 73 1435.00 1435.00 1421.00 1421.00 1400.00 1400.00 32560.00 32560.00 31830.00 31830.00 31220.00 31220.00 7T 243 13 73 79 1 1 1400.00 31220.00 \ 2 44 14 79 85 1 1 1400.00 31220.00 .. _245 ...... 15 ._ . _ 8 5 88 1 1 1400.00 31220.00 246 16 8 8 91 1 1 1400.00 31220.00 247 17 91 94 1 1 1030.00 7780.00 248 18 3 6 1 1 1477.00 33370.00 x 249 19 6 12 1 1 1477.00 33370.00 250 20 12 18 1 1 1477.00 33370.00 a. 251 21 18 24 1 1 1477.00 33370.00 2 52 22 24 30 1 1 1435 .00 " 3 2 560.00 253 23 30 36 1 1 1435,00 32560.00 2 54 24 36 42 1 1 1435.00 32560.00 255 25 42 48 1 1 1435 .00 32560.00 256 26 48 54 .1 1 1421.00 31830.00 r 257 27 54 60 1 1 1421.00 31830.00 x 2 58 28 60 66 " 1 1 1400". 00 31220 .00 " 259 29 66 72 1 1 1400.00 31220.00 260 30 72 78 1 1 1400.00 31220.00 261 31 78 84 1 1 1400.00 31220.00 2 62 3 2 84 90 1 1 1400.00 31220.00 -' 263 33 90 93 1 1 1400 .00 31220.00 264 34 93 96 " l " 1 1030.00 7780.00 265 35 2 5 1 1 2954.00 3808000.00 266 36 5 8 1 1 2954.00 3808000.00 267 37 8 11 1 1 2954.00 3808000.00 268 38 11 14 1 1 2954.00 3808000.00 >• 269 39 14 17 1 1 2954.00 3808000.00 270 40 17 20 1 1 2954.00 "3808000.00 271 41 20 23 1 1 2954.00 3808000.00 272 42 23 26 1 1 2954.00 3808000.00 2 73 43 26 29 1 1 2870.00 3722000.00 2 74 44 29 32 1 1 2870 .00 3722000.00 275 3 2 35 1 2870.00 3722000.00 2 76 46 3 5 38 ± 1 2870 .00 3722000.00 277 47 3 8 41 1 1 2870 .00 3722000.00 278 48 41 44 1 1 2870.00 3722000.00 279 49 44 47 1 1 2870.00 3722000.00 280 50 47 50 1 1 28 70 .00 3722000.00 281 51 50 53 1 1 2842.00 3644000.00 2 82 52 53 56 1 1 2 842 .00 3644000.00 283 53 56 59 1 1 2842.00 3644000.00 284 54 59 62 1 1 2842.00 3644000.00 285 .5 5 62 65 1 1 2.800 .00 3578000.00 286 56 65 68 1 1 2800.00 3578000.00 2 87 57 68 71 1 1 2800.00 3578000.00 288 5 8 71 74 1 1 28 00.00 3.578000.00 2 89 59 74 77 1 1 2800.00 3578000.00 2 90 60 77 80 1 1 2800 .00 3578000.00 291 61 80 83 1 1 2800.00 3578000.00 2 92 62 83 86 1 1 2800.00 35780 00.00 2 93 63 86 89 1 1 2800 .00 3578000.00 294 64 89 92 1 1 2800.00 3578000.00 295 65 92 95 1 1 20 60.00 15540.00 L 296 66 95 98 1 1 2860 .00 9340000.00 j li ( 2 97 67 5 6 1 1 1000 . 0 0 150000.00 298 68 7 8 1 1 1000 . 0 0 150000.00 299 69 11 12 1 1 1000 . 0 0 150000 . 0 0 I 300 70 13 14 1 1 1000.00 150000.00 301 71 17 18 1 1 663.00 91000 . 0 0 302 72 19 20 1 1 663.00 91000.00 J 303 73 23 24 1 1 663.00 91000.00 304 74 25 26 1 1 663.00 91000.00 305 75 29 30 1 1 663.00 91000.00 306 76 31 32 " 1 1 663 .00 91000.00 307 77 35 36 1 1 6 63.00 91000.00 3 08 78 37 38 1 1 663 .00 91000.00 309 79 41 42 1 1 663.00 91000.00 310 80 43 44 1. 1 663.00 91000 . 0 0 311 81 47 48 1 1 6 63.00 91000 . 0 0 312 8 2 49 """50" 1 1 663.00 91000 . 0 0 " " • 313 83 53 54 1 1 663.00 91000.00 314 84 5 5 56 1 1 663 .00 91000 . 0 0 1 315 85 59 60 1 1 663 . 0 0 91000.00 3 16 86 61 62 1 1 663 . 0 0 91000.00 r 317 87 65 66 1 1 663 . 0 0 91000 . 0 0 318 88 67 68 1 1 6 63.00 91000.00 319 89 71 72 1 1 663 .00 91000 .00 320 90 73 74 1 1 663.00 91000.00 321 91 77 78 1 1 663.00 91000.00 322 92 79 80 1 1 663.00 91000.00 323 93 83 84 1 1 6 63. 00 91000.00 3 24 94 85 86" "T " " 1 663 .00 91000.00 325 95 8 8 89 1 1 > 850.00 120000.00 326 96 89 90 1 1 850.00 120000.00 327 328 329 3 30 3 31 332 97 98 99 1 0 0 91 92 94 95 92 93 95 "96" 850.00 8 50.00 850 .00 120000.00 120000.00 120000.00 120000 ".00 OUTPUT MEMBER DATA 3 33 335 336 337 338 339 340 341 342 343" 344 345 346 347 348 349 350 351 352 353 354 3 5 5 " 356 3 57 ( C O D I N G N U M B E R S ) MEMBER HL 1 2 _3_ 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 0 7 19 31 43 5 5 67 79 91 10 3 115 127 139 151 163 169 178 0 4 1 6 28 VL 8 20 32 44 56 " 6 8 80 92 104 1 16 128 140 152 164 170 179 _ 9 -5-17 29 RL 6 9 21 33 45 _ 5 L 81 93 105 117 129 141 153 165 171 180 0 6 18 30 HG "l 19 31 43 55 67 ' 7"9 91 103 115 127 13 9 151 163 169 T78" 187 4 16 28 40 VG 20 32 44 56 _ 6 8 80 92 104 116 128 140 152 164 170 T T T 188 5 17 29 41 RG 9 21 33 45 5 7 69_ 81 93 105 117 129 141 153 165 171 180 189 _6 "~1 8" 30 42 -p. -p- -p- -P- -P• -P-I—• t—' *—' H - ' t—» t—' -vi cn ui ,-ps u> ro co co -si -vl -v| -v| H-* o i cc -j cr co -I -J;o O yi co oo -j^o cn ui cn O .J> 'Co fu cn co oo -j cn cn ui -v| >—* Ul vC U> - v l oo oo -J co ro cn CO GO -J \0 W vj -J cn ui O -P- 00 -j o ui t-* Ul vO vD co -vi -j cn cn O -P- co,ro cn o J N J > J > , .JS -P" t-> i— O O O O H - O sO'oo - J Cn —J -J - v | ] - v j -VJ —J ui -p- w;ro i - 4 o J> -p- OJU> ru v0 US -s)!*-1 Ul vO ui -P- u>|U) ro ro O -P- coiro cn o ui -P- OJ;OJ ro ro >-* ui vO!OJ ->i ui 4> 4 s u i ro ro ro cn o;-P- cc ro ui -P- -P-IOJ ro ro UJ - v l I — 1 Ul vO 0J ui 4 s -p-1 u) UJ ro -P- oo ro;cn o 4> 4>. 4> -P> :4> -P- 4> O O OjO o o Ul -P- U) i|>0 h-* O cn cn cn: cn cn cn <X) 00 -j;cn ui _p-K-J i\0 CO - v j U J -4 H o 1—• ro *~* UO 00 -vl 4 s 00 ro'i-* ro UJ t-" j v0 Go —J ui vO uiiro UJ -P* cn vO v£) CO -P-icn o >— (—* r—• ivO *£> CD -vi H-" ui-si i—1 ro f—* t—* . v0 J 3 00 00 ro 0 , 0 0 ro O J OJ OJ OJ OJ vO vO vO O *x> 00 -vj cn OJ OJ v0 \0 ui J> cn cn cn ;cn ui ui Ui ro f 10 vo co cn 'ui ui -P- 4- u) Cn -vj 4* iui ro OJ cn ui ui ;-P- -p- OJ ->i 00 ui ;o OJ 4> cn ui 'ui j -p- -p- OJ co vO o- -j -p- ui -si cn ui ui -P- -P* ro cn -si! jn ui ro -j cn ui; ui -P- -P-OJ --i 00 ui 0 s OJ - j cn ui ui J> 4" 4 s 00 1 cn -vi -P* OJ UJ OJ OJ OJ OJ \Q ! S 0 CO 00 OJ ro t-" O vO co ui ui ui'ui ui ui cn uii-P- OJ NJ OJ ro 1— 'o o o i—* co vO cn OJ ro t-^'t- o ro v0 !o --j UJ ro ro:(-< o ro u> O'I— cc j—> j— ' ^ —*; > t—* OJ L J ro 1—» O OJ O i-\ 0 0 ^0 OJ OJ roih--P- ro'a5 o UJ OJ ro ro ui ro w o 0 J OJ OJ 0 J UJ OJ co co 00 i 00 00 00 -j cn ui j.p- UJ ro ui ui -P-.p- -P- -P-o sO co -j cn sO 00 oa -j - J cn -P- ui ro ,UJ O >JD 00 00 ;-vi -vi cn ui Cn OJl-p1 i—* ro vO 00 00 -vj -si cn cn - j 4s ui ro OJ v0 SO 00 ' C O - s i - - j - j -P- ui ro OJ o •.o sO 00100 -J -j 00 ui cn, OJ -P- !-• s0 --O co 00 -s! —j vo cn - j Jp- ui ro OJ OJ u> OJ OJ OJ CO 00 "si - V J -vj -si o sO co -vi cn j> ^  -P- -P- -P- -P-oi -Mil :M H o ui -tn 4s ;uj OJ ro co v0 cn -vi -P- ui ui ui J S . u 1 OJ ro vO O - j 00 Ui tn cn ui J> OJ OJ ro o t-> 00,sO cn - j O Ol J>'^ W w •-• cc so, cn - j -p-cn ui ui -P" OJ u> ro VJO O'-J co ui o cn ui, -P- OJ OJ OJ o t— 00 cn 0 J OJ 0 J 0 J OJ -s) -vl -si -J -si ui -P* OJ ro i*~* OJ OJ OJ OJ OJ OJ CC - v ) ; cn Ul ro I - H ro OJ o co N H H CO UJ > H ' W O Ul W H H 00 -P> ui ro.ui o cn no ro !—>'•-• u i ro OJ o J5 OJ ro ro t— cn OJ -P-' I\J -p-N N H H sO •-j 4n ui ro UJ ui 0 J U) 0 J 0 J OJ UJ cn cn cn 'cn cn cn vO 00 - j cn Ul J> OJ OJ OJ 'OJ ro ru - J O s^ (jj ro ui o 00 !cn 4» ru -si cn -jn !0J ro cn t-1 v 0 1 -si ui OJ -si o ui OJ ro •>! w O 00 » -P co -j cn, jn U J ro 4- ui O' co cn 4^  00 - v j Oi-p- ui ro Ul Cn t— \ 0 - v j ui 00 -si cn,ui O J ro cn -si ro , 0 00 cn UJ 0 J OJ ul OJ 0 J cn Cn cn cn ui Ui OJ ro 1—• o v0 00 no r\j ro ro ro ro cn ui -P- UJ ro O 00 -vl cn ui -P-O co cn j> ro O o 00 -si cn ui 4> >-• -si 'Ul O J i - 4 O J) -J O Ul -P ro o co cn -P- ro H O CO vj O Ui ro o 00 cn 4> ro H O CD -j OUl OJ I - " -vl Ul OJ (-1 o s O - - j cn ui J> M o B cr J> f 418 82 91 92 93 94 95 96 419 83 97 98 99 100 101 102 420 84 10 3 104 105 106 107 108 421 35 109 1 10 111 1.12 1 13 114 422 86 115 116 117 118 119 120 t 423 87 121 122 123 124 12 5 126 J 424 88 127 128 129 130 131 132 425 89 133 134 135 136 137 13 3 4 26 90 139 140 141 142 143 144 427 91 145 146 147 148 149 150 428 92 15 1 152 153 154 155 156 429 93 157 158 159 160 161 162 43C 94 163 164 165 166 167 163 431 95 169 170 171 17 2 173 174 432 96 17 2 173 174 175 176 17 7 43 3 " 97 17 8" 179 180 181 182 183 434 98 18 1 182 183 184 185 186 435 99 187 188 189 190 191 192 r 436 100 190 191 192 193 194 195 437 438 HALF BAND WIDTH NB = 18 439 440 RETAINED DEGREES OF FREEDOM 441 I ND( I ) JNO( I ) N 442 1 1 14 22 443 2 1 26 46 444 3 1 3 8 70 445 4 1 " "50" " 94 446 5 1 62 118 447 6 1 74 142 - 448 7 1 89 172 449 8 1 92 181 450 9 1 95 190 """451 . . . . . . i o 1 98 196 452 GRAVITY= 32. 20 NO . OF SPECTRUM CASES(NDISP)= 1 DAMPING PERCENTAGE= 0.0 453 VERTICAL V I BRAT ION = 0 NMASS= 10 I 454 LUMPED WEIGHTS ! ^55 1 68.000 2 70. 000 3 70.000 4 70.000 5 70.000 6 78.000 456 7 92.000 a 6.2.000 9 38.000 10 28.000 7 """45 7 "~~ EIGEN VALUES 458 41403.135 28368.513 15263.330 7345. 513 3 250.492 1716.552 1185.738 372.853 103 .410 5.396 459 PERIODS(SEC.) 460 0.005 0. 00 7 0.009 0.013 0.019 0.027 0.032 0.0 57 0. 109 0.477 461 EIGEN VECTORS 462 1 -0.951 1 .000 -0.995 -0.913 0. 548 0.024 0.287 -0.143 -0.057 0.016 463" 2 1.000 -6". 55 7* -0.199 -0.962 1.000 0.056~ 0.721 -0.428 -0.189 0.063 464 3 -0.983 -0 . 138 1.000 0. 703 0.314 0. 050 0.748 -0.63 5 -0.332 0.133 465 4 0. 838 0. 767 -0.531 0.970 -0 .743 0.007 0.271 -0. 638 -0.441 0.222 466 467 468 "469 470 471 5 6 7 " 8 9 10 -0.598 0.280 _-Q.G77 0 .0 2 1 -0.001 0.001 -0.965 0.60 1 -0.196 " 6.059 -0.00 3 0. 002 -0.625 0.941 _-_0.45.9_ 0.169 -0 .0 10 0.007 •0.674 •0.946 1.000 -0. 509 0 .04 7 -0. 02 2 -0. 822 0. 177 0.921 -0.794 0. 146 -0.044 -0.031 -0. 024 0_. 045 0.082 •0.998 1.000 -0.381 -0.725 -Oj.311 1 .000 0.012 -0 .409 •0.413 -0.021 0.520 1.000 -0. 607 •0.957 •0.485 -0.447 -0.282 0.017 0.701 1 .000 472 PARTICIPATION FACTORS 473 -21.279 17.526 -14.253 -7.659 14.209 474 SPECTRAL DISPLACEMENTS! I NCH_ OR CM) _ _ 47 5 * " " " o 7 o 00 " """' 0 .000 6 . OOO" 0.001 6 . 0 0 1 477 MODAL FORCES 478 1 453.403 392.881 317.822 156.744 174.638 47.351 '""6. ooY 25.414 8.285 0.003 53 .322 -4. 30 7 " 0.04 2 55.160 -19.521 0.153 98.996 0.32 5 0.439 0. 590 6.759 0.887 1.000 2.255 2.932 MAX. PROBA 3.243 729 479 2 -491.02 8 -225.349 65.505 17C.069 327.889 61.047 137.818 170.176 339.769 13.06 6 774 A80 3 482.885 -55.611 -328.905 -124.238 102.892 54.329 143.000 252.316 598.008 27.738 903 481 4 -411.296 310.058 174.563 -171.413 -243.514 7.845 51.894 253.635 794.422 46.272 1041 482 5 293.537 -390.388 205.696 119.195 -269.503 -34.055 -72.922 164.069 873.610 67.744 1081 483 6 -153.123 270.848 -344.989 186.207 64.837 -28.958 -154.534 9.381 897.010 101.776 1046 484 7 49.663 -104.089 198 .258 -232.287 396.730 64.860 -79.586 -271. 793 666.692 161.349 904 485 8 -9.288 21. 212 -49.272 79.643 -230.544 79.444 169.343 -352. 155 -27.712 139.936 4 9 T 486 9 0 .234 -0.609 1.84,2 -4.553 26.058 -591. 881 1.297 131 .105 -686.072 100.236 921 487 . 10 _-0.156 0.366 -0.910 1.557 -5.814 437.073 -31.305 152.211 -720.738 83.257 861 489 MODAL SHEARS ' . ^ ^ p-, 0 B A 490 1 453.403 392.881 317.822 156.744 174.638 25.414 53.322 55.160 98.996 3.243 729 491 2 -37. 624 167.532 383.327 326.813 502.526 86.461 191.139 225.337 438.764 16.309 907 492 3 445.261 111.922 54.422 202.575 605.418 140.790 334.139 477.652 1036.772 44.047 1434 493 4 33.965 421.980 228.985 31.161 361.905 148.635 386.033 731.288 1831.195 90.319 2104 494 5 327.502 31. 592. 434.681 150.357 __ 92.402 114.581 _ 313.110 895.357 2704.804 158.063 2929 495 6 174.379 302.440 89.692 336.564 157.239 " 85.622 158.577 904.739 3601.815 259.839 3762 496 7 224.042 198.351 287.950 104.277 553.969 150.483 78.991 632.946 4268.507 421.188 4395 497 8 214.754 219.563 238.678 183. 920 323.425 229.926 248.334 280.791 4240.795 561. 125 4333 498 9 214.989 218.954 240.520 179.367 349.483 -361.954 249.631 411.896 3554.723 661.361 3707 499 10 214.832 219,320 239.610 180.924 343.669 75.119 218.326 564.107 2833.986 744.618 3043 500_ _ EXECUTION TERMINATED 501 SSINK PREVIOUS END OF FILE A S IG 

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