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Orthogonal representation of random processes

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Orthogonal representation of random processes

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Content ORTHOGONAL REPRESENTATION OF RANDOM PROCESSES by Mohamed-Idris M. Traina A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillm ent of the Requirements for the Degree DOCTOR OF PHILOSOPHY;.',' (C ivil Engineering) May 1984 UMI Number: DP22172 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMT Dissertation Poblisterig UMI DP22172 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 9 0 0 0 7 This dissertation, written by Mohamed-Idris Traina under the direction of Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of requirements of the degree of <24 r u ? / 5 Jfi- — M y D O C T O R O F P H I L O S O P H Y DISSERTATION COMMITTEE Chairman ACKNOWLEDGEMENTS I wish to express my deepest thanks and appreciation to Professor Sami F. Masri, the chairman of my qualifying and dissertation commit tees, for the invaluable help, guidance, and constant encouragement during my graduate study at the:.University of Southern California I am also thankful to Professor R. K. M ille r for reviewing my work, making valuable comments and suggestions, and contributing to the subject matter of this dissertation. I am also indebted to Professor T. C. Cheng for his help as a committee member and fo r read ing and discussing this work. I would also lik e to acknowledge the help of Professor D. E. Hudson through the period of this research, and the help of Professor R. Kaplan for providing the data used in Chapter 5 and for reading this work and making valuable comments. The assistance of T. Dehghanyar and P. Liles in solving computer problems related to this work on various computer systems is much appreciated.. I would lik e to extend my thanks to Marje Cappellari whose patience, expertise, and typing s k ills were extremely helpful. My acknowledgements would not be complete without extending my thanks to my mother, Noriya, and my w ife, L a ila , for th e ir encourage ment and support, and to my children, Mahmoud and Iman, who were and s t i l l are great inspirations to me. This work was funded in part by the National Science Foundation. i i TABLE OF CONTENTS ACKNOWLEDGEMENTS............................................................................................................... i i LIST OF ILLUSTRATIONS................................................................................................... vi LIST OF TABLES.................................................................................................................. xiv ABSTRACT............................................................................... xv Chapter Page 1 INTRODUCTION .................................................................................... I 1.1 Background and Motivation,.. . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Assessment of the Problem........................................................ 3 1.3 Scope of the Work.......................................................................... 5 2 SPECTRAL DECOMPOSITION AND EXPANSION.......................................... 7 2.1 Direct Spectral Decomposition Method.............................. 7 2.1.1 Theoretical Background and Numerical Representation................................................................. 7 2.1.2 Analysis of Error in Surface F i t ........................ 12 2.1.3 Analytical Representation Using the Numerically Evaluated Eigenvectors................... 15 2.1.4 Use of Chebyshev Polynomials in Surface F i t ....................... 16 2.2 Spectral Decomposition as the Result of Least-Squares F i t ......................... 18 2.3 Galerkin Weighted Karhunen-Loeve Approach................. 23 2.4 Nonstationary Response of Linear Systems..................... 26 2.5 Extreme Values of the Nonstationary Response 30 2.6 Direct Analytical Representation........................................ 31 2.7 Improving the Estimate of Response..................................... 32 2.8 Remarks.............................................................................. 33 3 APPLICATIONS TO MODULATED WHITE NOISE PROCESSES................. 34 3.1 ^ ’Introduction...................................................................................... 34 3.2 Case 1....................................................... 36 3.2.1 Surface Approximation.................. 37 3.2.2 Transient Nonstationary Response.................... 44 i i i TABLE OF CONTENTS (Cont) Chapter Page 3 3.3 Case 2 .................................................. 44 3.3.1 Surface Approximation............................................... 49 3.3.2 Transient Nonstationary Response........................ 49 3.4 Case 3 .................................................. 55 3.5 A Note on tires Analytical F i t ................................................ 60 3.6 Improved Response Calculations.......................................... 65 4 APPLICATIONS TO REAL EARTHQUAKE RECORDS................................. 66 4.1 Introduction....................... 66 4.2 San Fernando Earthquake.......................................................... 68 4.2.1 Description...................................................................... 68 4.2.2 Surface F i t ...................................................................... 69 4.2.3 Transient Mean Square Response.......................... 75 4.3 Concluding Remarks..................................................................... 87 5 APPLICATIONS IN THE FIELD OF TURBULENT FLOW............................ 88 5.1 Introduction................................................................................... 88 5.2 Surface Approximation.............................................................. 89 5.3 Nonstationary Response of SDOF Harmonic O s c illa to rs ...................................................................................... 90 5.4 Use of Window in the Time Domain to Approximate the Covariance M a t r i x . . . . ...................................................... 105 6 SUMMARY AND CONCLUSIONS....................................................................... 106 Appendixes: A PROBABILISTIC STRUCTURAL DYNAMICS................................................ 110 A .1 Estimation of Covariance Matrix for a Nonstationary Random Process............................................................................... I l l A.2 Input-Output Relation of Linear MD0F Systems 113 B COMPUTER IMPLEMENTATION....................................................................... 117 B.l Program SEC21................................................................................. 118 B.2 Program SEC23................................................................................. 125 B.3 Program MSRSP................................................................................. ,131 i v TABLE OF CONTENTS (Cont) Appendixes Page C ADDITIONAL INFORMATION RELATED TO CHAPTER 3 ......................... 136 D ADDITIONAL INFORMATION RELATED TO CHAPTER 4 ......................... 151 E ADDITIONAL INFORMATION RELATED TO CHAPTER 5 ......................... 170 REFERENCES..........................................................'............................................. 176: v LIST OF ILLUSTRATIONS Figure Page 3.1 Case 1: Exact Covariance Matrix [C]........................................ .. 38 3.2 Case 1: Convergence of the Eigenvalues of [ C ]............ .. 39 3.3 Case 1: Rate of Convergence of [C^] to [C] and - Error Bounds on e, .............................................................................. .. 39 3.4 K Case 1: Exact (£^), A n a ly tic a lly Approximated (jj.j), and Error Ratio versus Number of C oefficients; i = 1 - 4 .................................................................. .. 41 3.5 Case 1: Exact ( p . ) , A n a ly tic a lly Approximated (£•)» and Error Ratio versus Number of Coefficients; i = 5 - 8 ................................................................ . . 42 3.6 Case 1: Diagonal Comparison Between [C] and [C^] ., . . 43 3.7 Case 1: Diagonal Comparison Between [C] and [C^], . . 44 3.8 Case 1: Approximate Covariance [Cg] Surface................... 45 3.9 Approximate Covariance (C-^q ] Surface...................................... 45 3.10 Approximate Covariance [ C ^ ] Surface...................................... 46 3.11 Case 1: Approximate Covariance [C-^] Surface Based on Using Analytical Expressions.................................... 46 3.12 Case 1; (k = 5) rri.s. Response of SDOF Harmonic O s cillato r (T = .5s - 4.s; .5s and 6 = .0 5 ) ................... 47 3.13 Case 1: (k = 10) m.s. Response of SDOF Harmonic O scillators (T = ,5s - 4 . s; .5s and £ = .0 5 ................... 48 3.14 Case 2: Exact Covariance M a tr ix ............................................. 50 3.15 Case 2: Convergence of the Eigenvalues of [ C ] .............. 51 3.16 Case 2: Rate of Convergence of [C. ] to [C] and Error Bounds on e. ............................................................................ 51 3.17 K Case 2: Eigenvectors (p .) of [C]; i = 1 - 8 ................. 52 ....... ...v.i.... LIST OF ILLUSTRATIONS (Cont) Figure Page 3.18 Case 2: Diagonal Comparison Between [C] and [C ^].............. 53 3.19 Case 2: Approximate Covariance [C .] Surface for k = 25, 50, 75, and 100......................... ............................................... 54 3.20 Case 2: Nonstationary Response of SDOF Harmonic O scillators w ith£ = .0 5........... 56 3.21 Case 3: Covariance Matrix [C] Surface....................................... 58 3.22 Case 3: A n a ly tic a lly Defined Eigenvectors <)>., i = 1 - 8 ....................................................................................................... 59 3.23 Case 3: Approximate Covariance Surface Based on Using 5 Terms of ( 2 . 4 3 ) . . ..................................................................... 61 '3.24 Case 3: Approximate Covariance Surface Based on Using 10 Terms of ( 2 .4 3 ) ....................................................................... 61 3.25 Case 3: Approximate Covariance Surface Based on Using 15 Terms of ( 2 .4 3 ) ....................................................................... 61 3.26 Case 3: Diagonal Comparison Between [C] and [C^] Based on Using k Terms of ( 2 .4 3 ) .................................................... 62 3.27 Case 3: Normalized Error Function fo r Estimating [C] by 5 Terms of ( 2 . 4 3 ) . . . . . ........................................................... 63 3.28 Case'3: Normalized Error Function fo r Estimating [C] by 10 Terms of ( 2 .4 3 ) ..................................................................... 63 3.29 Case 3: Normalized Error Functions fo r Estimating [C] by 15 Terms of ( 2 .4 3 ) ..................................................................... 63 3.30 Effect of Zero-ends on the Goodness of F i t ............................. 64 4.1 Covariance Matrix [C] for the San Fernando Earthquake.. 70 4.2 Convergence of the Eigenvalues of [ C ]......................................... 71 4.3 Rate of Convergence of [C^] to [ C ] ................................................. 71 4.4 Eigenvector No. 1....................................................................................... 72 4.5 Eigenvector No. 2 ....................................................................................... 72 LIST OF ILLUSTRATIONS (Cont) Figure Page 4.6 Diagonals of [C] and [C ^ ].......................................................................... 73 4.7 Approximate Covariance [C. ] Surface Based on Using k = 25 Eigenvectors................................................................................... 74" A 4.8 Approximate Covariance [C^] Surface Based on Using Approximate Analytical Expressions for k = 25 Eigenvectors........................................................................................................ 76 4.9 m.s. Response of a Harmonic O scillato r (T = 0.25 sec, £ = - 0 5 )............................................................................ 77 4.10 m.s. Response of a Harmonic O scillato r (T = .5 sec, £ ; = .0 5 ) ................................................................................. 77 4.11 m.s. Response of a Harmonic O scillato r (T = .75 sec, e = .0 5 ) .............................................................................. 78 4.12 m.s. Response of a Harmonic O scillato r (T = 1.0 sec, e = .0 5 ) .............................................................................. 78 4.13 m.s. Response o f a Harmonic O scillato r (T = 1.25 sec; % = . 0 5 ) ........................................................................... 79 4.14 m.s. Response of a Harmonic O scillato r j (T = 1.5 sec, £ = . 0 5 ) .............................................................................. 79 4.15 m.s. Response of a Harmonic O scillato r (T = 1 .75 sec, £ = . 0 5 ............................................................................. 80 4.16 m.s. Response of a Harmonic O s c illa to r (T = 2 .0 sec, £= .0 5 ) .......................................................................... 80 4.17 m.s. Response of a Harmonic O scillato r (T = 2.25 sec, e = .0 5 ) ............................................................................... 81 1.18 m.s. Response of a Harmonic O s c illa to r (T = 2.50 sec, 5 = .0 5 ) .......................................................................... 81 4.19 m.s. Response of a Harmonic O scillato r (T = 2.75 sec, 6 = .0 5 ) .......................................................................... 82 4.20 m.s. Response of a Harmonic O scillato r (T = 3.0 sec, C = .0 5 ) .......................................................................... 82 4.21 m.s. Response of a Harmonic O s c illa to r (T = 3.25 sec, K = .0 5 ) .......................................................................... 83 vi i i LIST OF ILLUSTRATIONS (Cont) F ig u re Page 4.22 m.s. Response of a Harmonic O s c illa to r (T = 3.50 sec, K = .0 5 } ............................................................................... 83 4.23 m.s. Response of a Harmonic O s c illa to r (T = 3.75 sec, £ = . 0 5 ) . ............................................................................. 84 4.24 m.s. Response of a Harmonic O s c illa to r (T = 4.0 sec, 5 = .0 5 ).............................................................. 84 4.25 Effect of Damping on the Nonstationary Response of Selected Harmonic O scillators Under the Effect of [C25] ................................................................................................................. 85 4.26 Exact and Approximate Extreme Values of the Nonstationary Root-Mean-Square Displacement Response of a SDOF O sc illa to r (^ = .0 5 ) .............................................................................. 86 4.27 Exact and Approximate Extreme Values of the Non stationary Root-Mean-Square Velocity of a SDOF O s c illa to r ( £ = .05) ................................................................................ 86 4.28 Extreme Values of the Nonstationary Root-Mean-Square Displacement Response of a SDOF O s c illa to r ( 5 = 0; .05; .10; and .1 5 ) ........................................ 86 4.29 Extreme Values of the Nonstationary Root-Mean-Square Velocity Response of a SDOF O s c illa to r ( g = 0; .05; .10; and . 1 5 ) .................................................................. 35 5.1 Sample Records from Group 1................................................................. gg 5.2 Sample Records from Group 2................................................................. 9q 5.3 Sample Records from Group 3................................................................. 90 5.4 Normalized Error Energy and Its Bounds for Groups 1,2, and 3 ......................................................................................... 92 5.5 Eigenvector No. 1............................................................................................. 93 5.6 Eigenvector No. 2 ........................................................................................... 93 5.7 Eigenvector No. 3 ............................................................................................ 94 5.8 Eigenvector No. 4 ............................................................................................ 94 5.9 Group 1: Exact and Approximate Covariance Surfaces 95 - , .. ■ ■ ■ —■ — i . i ... . .-I-/ LIST OF ILLUSTRATIONS (Cont) Figure Page 5.10 Group 2: Exact and Approximate Covariance S u rfa c e s .... 95 5.11 Group 3: Exact and Approximate Covariance S u rfa c e s .... 97 5.12 Group 1: Diagonal Comparison................................................................ 99 5.13 Group 2: Diagonal Comparison......................................... 99 5.14 Group 3: Diagonal Comparison............................................................ ' 100 5.15 Group 1: Analytic F it of £-|............................................ 101 5.16 Group 1: A nalytic F it of ............................. 101 5.17 Group 1: Analytic F it of £ 3 . . . ....................................................... 102 5.18 Group 1: A nalytic F it of £ ^ ............................................ 102 5.19 Group 1: A nalytical and Numerical Results of Nonstationary m .s. Response (T = .5s - 4 . 0 ; .5 and C = . 0 5 ) . . . . . .......................... 104 4.1 Procedure fo r Nonstationary Mean Value Measurement 114 4.2 Procedure fo r Nonstationary Autocorrelation Measurement........................................................................................................ 114 3.1 General Organization Chart for Program SEC21...,.................. 119 3.2 General Organization Chart fo r Program SEC23. ..................... 126 3.3 General Organization Chart for Program MSRSP,......................... 132 3.1 Case 1: Exact (p.^), Analytical ly Approximated (£_j), and Error Ratio versus Number of C o e ffic ie n ts ; i = 9 - 12.......................................................................................................... 137 1.2 Case 1: Exact (£.j) , A n a ly tic a lly Approximated, and Error Ratio versus Number of C o efficients (i = 13 - 1 5 ).................................... .............................................................. 138 1.3 Case 1: Normalized e rro r Function ([C] - [C. ]) Surface ■ ........................ 139 x LIST OF ILLUSTRATIONS (Cont) Figure Page C.4 Case 1: (k = 5) m.s. Response of SDOF Harmonic O s c illa to r (Tn = .25s - 3.75s; .5s and 5 = . 0 5 ) ................. 140 G. 5 Case 1: (k = 10) m.s. Response of SDOF Harmonic O s c illa to r (T^ = .25s - 3.75s; .5s and 5 = . 0 5 ) ................. 141 C .6 Case 1: (k = 15) m.s. Response of SDOF Harmonic O s c illa to r (T = .25s - 3.75s; ,5s and £ = . 0 5 ) ................. 142 C.7 Case 1: (k = 15) m.s. Response of SDOF Harmonic O s c illa to r (Tn = .5s - 4 .s ; .5s and £ = . 0 5 ) ........................ 143 C.8 Case 2: Additional Eigenvectors..................................................... 144 C .9 Case 2: Normalized Error Surface................... ............................... 145 c .io Case 2; Nonstationary Response of a SDOF Harmonic O s c ila to r ............................................................................. ............ * .......... 146 C .ll Case 3: A n a ly tic a lly Defined Eigevectors 4>.; i = 9 - 16............................................................................................... 147 : . i 2 Case 1: Nonstationary M.S. Response Under the Action of [C5] ( = 0 . 0 5 ) ..................................................................................... 148 : .i 3 Case 1: Nonstationary M.S. Response Under the Action of [C10]) = 0 .0 5 ) ........................................ ............................................ 149 : , H Case 1: Nonstationary M.S. Response Under the Action of [C15] ( - 0 .0 5 ) ................................................................................. 150 3.1 Sample Acceleration Records from San Fernando Earthquake..................................................... 156 3.2 Eigenvector No. 3 .................................. 157 13.3 Eigenvector No. 4 ........................... 157 0.4 Eigenvector No. 5 ................................ 158 0.5 Eigenvector No. 6 ...................................... 158 xi LIST OF ILLUSTRATIONS (Cont) Figure Page D.6 Eigenvector No. 7 ......................................................................................... 159 D.7 Eigenvector No. 8 ............................................................ 159 D.8 Eigenvector No. 9 ......................................................................................... 160 D.9 Eigenvector No. 10....................................................................................... 160 D.10 Eigenvector No. 11....................................................................................... 161 D .ll Eigenvector No. 12....................................................................................... 161 D.12 Eigenvector No. 13....................................................................................... 162 D.13 Eigenvector No. 14....................................................................................... 162 D.14 Eigenvector No. 15....................................................................................... 163 D.15 Eigenvector No. 16....................................................................................... 163 D. 16 Eigenvector No. 17....................................................................................... 164 D.17 Eigenvector No. 18....................................................................................... 164 D.18 Eigenvector No. 1 9 . . . . .............................................................................. 165 D.19 Eigenvector No. 20........................ 165 D.20 Eigenvector No. 21...................... 166 D.21 Eigenvector No. 22....................................................................................... 166 D.22 Eigenvector No. 23................. 167 D.23 Eigenvector No. 24....................................................................................... 167 D.24 Nonstationary m.s. Response of Harmonic O scillators (Tn = 0.25s -...............- 20s; .25 and C = 0 ) .................................... 168 D.25 Nonstationary m.s. Response of Harmonic O scillators (Tn = 2.25s -..............4.0s; 0.25 and £= 0).................................... 169 E.l Eigenvector No. 5 ......................................................................................... 171 E.2 Eigenvector No. 6 ......................................................................................... 171 XI 1 LIST OF ILLUSTRATIONS (Cont) Figure Page E.3 Eigenvector No. 7 .......................................................................................... 172 E.4 Eigenvector No. 8 .......................................................................................... 172 E.5 Eigenvector No. 9 .......................................................................................... 173 E.6 Eigenvector No. 10....................................................................................... 173 E.7 Group I: Analytical F it for £ g........................................................... 174 E.8 Group 1: Analytical F it fo r £ &...............................................................174 E.9 Group 1: Analytical and Numerical Results of Nonstationary m.s. Response (T = .25s - 3.75s; .5s and K = .0 5 )............................... ?........................................................ 175 XI 1 1 LIST OF TABLES Table Page B.l Brief Description of Subprograms Used by Program SEC21................................................................................................ 120 B.2 Brief Description of Subprograms Used by Program SEC23................................................................................................ 127 B.3 Brief Description of Subprograms Used by Program MSRSP................................................................................................ 133 „ D.l List of San Fernando Earthquake Records That Were Used to Generate [C ]................................................................................. 152 xiv ABSTRACT Results from nonstationary random data analyses are valuable only i f a s t a t is t i c a l ly s ig n ific a n t number of records is considered. Due to the large amount of data involved, a compact form to describe such processes is most desirable. Two techniques u t iliz in g orthogonal decomposition of the covariance kernel of the process are proposed to accomplish this task. The f i r s t method expands the covariance kernel into a f i n i t e and convergent sum of matrix-valued stages; the problem of analytical f i t is reduced to f i t t i n g a f i n i t e set of vectors re sulting from an eigenvalue formulation. The second method uses the R itz-G alerkin approach in conjunction with the Karhunen-Loeve expan sion to f i t a selected set of'vectors to the data-based covariance kernel. The coefficien ts of the expansion are obtained from a least-squares minimization process. The algorithms lead to a r e la tiv e ly simple, closed-form expres sion for the covariance kernel of lin e a r systems response, with a convenient grouping of terms re fle c tin g the input and system charac t e r is t ic s . An a n a ly tic a lly defined covariance function simulating modulated white noise, together with ensembles of measured records from strong ground shaking and a turbulent flow experiment, was used to establish the r e l i a b i l i t y , e ffic ie n c y , and lim its of the approaches. In addition to its data compression property, the work presented is a p o te n tia lly powerful tool for use in feature extraction (signature ana lysis) of nonstationary random processes. xv Chapter 1 INTRODUCTION 1.1 Background and Motivation The recorded data collected from actual earthquakes in the 1950s have made i t clear that the response of structures can be strongly influenced by the nonstationarity of the input loads (Bycroft, 1960; Wong and Trifunac, 1978). Real, individual records from a non stationary random process are lik e ly never to occur again due to the large set of random parameters involved. For an accurate analysis of dynamic systems response to nonstationary loading ( e .g ., earthquakes), i t is imperative that the evolutionary nature of the loads be properly represented as a nonstationary stochastic process. The growing recognition of th is fa c t and the significance of p ro b a b ilis tic approaches in engineering applications have stimulated a considerable amount of structural engineering research into the description, simulation, and response to nonstationary processes. Pertinent lite r a tu r e is lis te d in the references section. A study of the r e l i a b i l i t y of structural systems under non stationary loads can proceed along e ith e r of two lines: ( 1) in experimental studies of a family of physical models subjected to a large collection of d iffe r e n t excitations or ( 2 ) with construction of a proper mathematical model of a physical system which is then 1 subjected to a large co lle c tio n of loads. Both approaches re ly on the proper characterization of d iffe re n t excitatio n time h isto ries . Since the costs involved in such mathe matical or physical analyses are large, there are obvious advantages to studying the response of appropriate mathematical models under loads which embody the essential features of typical earthquake (or a sim ila r process) motions. In the p ro b a b ilis tic analysis of a sim p lified model, the designer is faced with the problem of choosing among several descriptions of the input (Masri et a l . , 1978): (1) design spectra, modified in various ways such as s ite conditions; ( 2 ) selected samples of recordec earthquakes normalized in some way; (3) a r t i f i c i a l earthquake records designed to match certain average ch a racteristics of real events; or (4) a p ro b a b ilis tic description of the input based on an analysis of recorded earthquake ground motions. The methods presented herein can be used to a n a ly tic a lly approximate the input and output of lin e a r dynamic systems whenever an ensemble of records (real or a r t i f i c i a l ) is available to represent the nonstationary input process. While i t is possible, in p rin c ip le , to use a condensed probabi l i s t i c representation of earthquake data such that actual earthquake records are replaced with "satisfactory" s ta tis tic a l models, i t is no': lik e ly that even the r e la tiv e ly large data base that currently exists can provide models with small variances. The inherent variations in source-to-receiver path ch a ra cteristics and source mechanisms con tained in the available data base in e v ita b ly lead to large variances 2 w ithin the reduced s t a t is t ic a l model. The usefulness of a s ta tis tic a l model fo r th is potential application must be judged subjectively on the basis of how small a variance in the model can be achieved, given the inherent v a r ia b ilit y of the earthquake process. Even i f the variance in the model is large, th is fa c t in i t s e l f may have important practical engineering consequences. Further reductions in the variance may be possible with detailed accounts of source-to-receiver path, s ite conditions, source mechanisms, e t c . , but previous attempts in th is direction have not produced overwhelming enthusiasm fo r th is approach. I t is important to re a liz e that this question re a lly cannot be resolved without actually applying the s t a t is t ic a l models and evaluating the results. An equally important use of a reduced s t a t is t ic a l model is in the attempt to quantify the uncertainty inherent in any design based on the lim ited data which currently exists. This is a problem of great importance in the design of structures to re s is t earthquake loads. The rational and objective evaluation of risks using a ll availab le s t a t is t ic a l data offers a major improvement over the subjective opinions of the few q u a lifie d experts upon whom many engineers currently depend. 1.2 Assessment of the Problem Some of the practical problems encountered in an a ly tic a ly struc tural dynamics studies are b r ie fly reviewed in appendix A. The response covariance of lin e a r dynamic systems to a nonstationary random process cahracterized by its covariance kernel can be evaluated 3 in a conceptually straightforward manner, but serious challenges to ;ome up with practical and convenient an a lytic solutions (see Eq. (A .11), appendix A) arise. Motivated by the discussion in appendix A, the work presented here is mainly an extension of the study made by Masri and M ille r (1982). A d ir e c t, two-dimensional f i t of Chebyshev polynomials with the input :ovariance kernel was determined and used to obtain an a n a ly tic expres sion for the response covariance of a lin e a r system. Instead of a d ir e c t, two-dimensional f i t , the two methods pre sented in chapter 2 involve the use of an orthogonal expansion. I t requires the solution o f an eigenvalue problem to determine the best information compression procedure. The f i r s t method (described in section 2 . 1 ) results in the evaluation of a set of numerically d eter mined vectors th a t, for a given number of terms, y ie ld the minimum representation error possible fo r the covariance m atrix. The analytical representation of the covariance kernel is reduced to Jetermining an analytical f i t fo r the numerically determined vectors. A set of co e ffic ien ts to best f i t a set of selected a n a ly tic func tions with the input covariance matrix is determined with the second method. The following are some features of the proposed methods: 1. The steps to determining the approximate response co- variance are completely independent of the steps involved in Drocessing the e xc itatio n . Once the input is characterized by the determined set of co e ffic ien ts and the set of selected an alytic func tions, no fu rth er modifications in these calculations are needed when 4 investigating the response of d iffe r e n t o s c illa to rs . 2. Nonstationary random vibration response calculations are reduced to the evaluation of simple algebraic expressions, whose evaluation does not require sophisticated techniques or fu rth e r approximations. 3. No assumptions are made regarding the nature of the co- variance kernel so that completely general excitation s may be con sidered. 4. The approximation introduced in the response calculations involves the approximation of the input covariance. No subsequent approximations have been made in deriving the m.s. (mean-square) response. 5. The methods are ideal fo r feature e xtractio n , since the information given by any number of terms in the expansion of ap- aroximating the covariance kernel of the input is optimum. 1.3 Scope of the Work The theoretical basis and formulation of the two suggested proce dures are detailed in chapter 2. A study of the f e a s i b i li t y of the nethods to handle earthquake-like random processes is presented in chapter 3. The study was accomplished by processing d iffe re n t co- variance kernels associated with a r e la tiv e ly simple nonstationary random process, widely used in analytical dynamic studies involving nodulated white noise, d ig ita l approximation (Clough and Penzien, 1975; 'lasri and M ille r , 1982). Chebyshev polynomials and trigonometric functions are used with success, and the exact m.s. response of 5 various lin e a r o s c illa to rs is compared with that o f d iffe r e n t levels of approximation. In chapter 4, selected records from a p a r tic u la rly important earthquake (the San Fernando earthquake of 1971) were used to generate a covariance m atrix. The convergence of the method applied to the above covariance kernel was estimated in approximating both the input exc itatio n and the m.s. response. The powerfulness o f the algorithm in dealing with covariance matrices generated from actual experimental studies in the f ie ld of turbulent flow is investigated in chapter 5. A b r ie f summary and con clusions are presented in chapter 6 . In the appendixes, a method fo r estimating the covariance matrix from an ensemble of nonstationary records is discussed, along with some problems in analytical p ro b a b ilis tic structural dynamics (appendix A). A description of the computer implementation of the methods presented in sections 2.1 and 2.3 (appendix B), additional information re la tin g to chapter 3 (appendix C), and a l i s t of the acceleration records from the San Fernando earthquake used to construct the covariance matrix and some figures in chapter 4 are also included. F in a lly , additional data re la tin g to chapter 5 are presented in appendix E. CHAPTER 2 SPECTRAL .DECOMPOSITION AND EXPANSION 2.1, Direct Spectral Decomposition Method 2.1.1 Theoretical Background and Numerical Representation An a rb itra ry real and nonnull rectangular matrix R of the order (m x n) [R] = 11 21 R 1 2 ‘ , R In 2n ml R _ m n ( 2 . 1 ) is said to be separable i f i t can be expressed as the product of an Ti-length column: vector c_ and an n-length row vector i . e . , M i s sep arable i f i t can be w ritten in the form c1r ] Clr 2 . . . . Clr n [R] = c r C2r l c2r n V i c r m .n ( 2 . 2 ) Notice that a separable matrix has proportional rows and columns. A r e a lis tic matrix such as the covariance matrix of a nonstation ary random process is not expected to be separable, but can always be expressed as the sum of separable matrices, i . e . , a matrix of order • (rrrx h) can be expressed always as the sum of the products of m-length column vectors (c^ , i = l , . . ; , k ) and n-length row vectors (r^,i = 1 ,...,k) plus an error matrix [E^] of order (m x n) as in the following form k [R] = I c.r.. + [ E. ] i = l ~ 1~ 1 K = 1 [ r .] + [ E. ] (2.3) i= l 1 K where [r i ] = c ^ r . i = 1, 2, . . . , k (2.4) is the i ^*1 separable matrix in the sum. Assuming [R] is well approximated by k-terms of the expansion, we require (m + n)k elements to describe [R] compared to m x n elements required to store the matrix in its raw form. As.more terms are taken in the expansion, the elements of the error matrix are expected to get smaller, and an exact representation of [R] is guaranteed with a f in i t e number of terms in the expansion. The error matrix in this case w ill be exactly n u ll. Without loss of generality, assume that m, the number of rows of [R], is equal to or greater than n, the number of columns of [R], I f this is not the case, the original matrix is transposed before processing. 8 A solution to the representation of an a rb itra ry matrix that has the above features is based on the spectral decomposition,requiring solution of the eigenvalue problem as fo llow s(T reitel & Shanks, 1971): [R] = ^ p ^ - , + + • • • + = I (2.5) i = l 1-1-1 where A^, A • • •, An Rea^ and positive eigenvalues of the n-order symmetric, and positive d e fin ite matrix T [S] = [R] [R] (the superscript T denotes the matrix transpose) jd 1 , £ 2 >*-»£n Normalized column eigenvectors associated with the corresponding nonvanishing eigenvalues A|, X .. A^ of the m-order, symmetric, and positive semidefinite matrix [Q] = [R][R]^. i l l 5 S 2’ " ’^n Normalized row eigenvectors associated with the corresponding eigenvalues A^, A ^ , . . . ^ of [S] Note that in ( 2 .5 ) , the vectors must be normalized so that g £ .= 1.0 and £-j£i = 1 *° i = 1 » 2 , . . . , n ( 2 . 6 ) Each term of the spectral representation is the separable (m x n) matrix resulting from the m u ltip licatio n of the m-length column vector £.j into the n-length row vector £.j, weighted by the square root of the corresponding eigenvalue. I f a l l n terms associated with the n eigenvalues ^ . . . , are retained in the expansion, then the representation is exact. Since the spectral representation is going to be applied in th is study to covariance kernels of random processes which are neces s a r ily symmetric and conveniently square m atrices, the theory is s im p lifie d fo r th is class of matrices as follows: [c ] - ♦ [Ek] { 2 . 7 ) ^here 9 • • • Real and p o s itiv e eigenvalues of m atrix [C]. £-|, P2 , . . . ,£ k Normalized eigencolumns of [C] such that: pjp_i = 1.0 , i = 1 , 2 , . . . ,k [C] Square symmetric nonnull covariance m atrix of order (n x n). The above re s u lt is arrived a t since (1) Matrices [S] = [C ]T[C] and [Q] = [C ][C ]T are id en tic al in th is case. p (2) The eigenvalues of [C] are the square of the corres ponding eigenvalues of [C]. (3) The eigenvectors of [C] 2 and [C] are id e n tic a l. 10 — ■ 2 Since the exact representation in (2 .7 ) requires n terms (which is , fo r a large n, about twice the number of the d iffe r e n t elements in the m atrix), no d ire c t computing speed advantages are obtained, and the storage requirements are almost doubled. On the other hand, i f the expansion of (2 .7 ) is truncated by using only the most dominant k terms, then k x n terms are required fo r the representation, and as long as (k x n) < (n_+ ~ l ) x n? computer storage and speed advantages are expected. For example, i t was found (as w ill be discussed la t e r ) that 25 terms in the expansion (2 .7 ) gave a good representation fo r the covariance m atrix of order 501 from a set of records from the San Fernando earthquake (February 9, 1971). Hence, 25 x 501 = 12,525 terms were required to represent the m atrix, compared to ( 5 0 ^ + ) ) x 501 = 128,255 terms o r ig in a lly required. The savings in computer storage is over 90%. Further data compression is possible by a n a ly tic a lly representing the eigenvectors. For the mentioned example, about 160 Chebyshev c o e ffic ie n ts fo r each of the 25 highest eigenvectors gave m.s. response results that compare very well with the exact solution fo r the range of fundamental frequencies that are of in te re s t; hence, a to ta l of 25 x 160 c o e ffic ie n ts and 25 eigenvalues were required to describe the process. For some of the cases studied in Chapters 3 and 5, greater savings were possible and a to ta l o f about 100 Chebyshev c o e ffic ie n ts were good representa tion of a covariance m atrix of order 512 x 512. 11 / } This behavior is by no means acc id e n ta l, and, in general, c o v a ri ance matrices of nonstationary random processes have a tendency to behave as such since the eigenvalues are expected to be d iff e r e n t , and those of higher v ,ues w ill make the co n tribution of the cor responding terms dominant in the expansion. This is true because the truncation e rro r is uniquely defined by the eigenvalues, as w ill be shown. 2 .1 .2 Analysis of Error in Surface F it A convenient scalar-valued measure o f the truncation e rro r in Equation (2 .7 ) is given by the normalized e rro r energy which is defined to be: where is the sum o f the squares o f the elements o f the truncation e rro r m atrix lE^J. Jq is the sum o f the squares o f the elements of [C]. I t is proved that is uniquely defined by the eigenvalues o f [S] for the case o f a rectangular m atrix [R] (where [S] = [R)^[R]) in the following w a y (T re ite l & Shanks, 1971): A -. ‘ t’ A ft'f' a • a ^1 * ek = 1 - X]+ x^+ + k = 1 , 2 , . . . ,n ( 2 . 9 ) 12 The normalized erro r energy decreases as k increases, reaching the lim itin g value o f zero as k reaches n. Then; 0 <. ek < 1 (2.10) For the case o f symmetric square matrices (2. 9) becomes 2 2 2 A, + Ap A. e. = 1 - —--------- s k k = 1 ,2 ,. .. ,n (2.11) 2 2 2 At + A, + . . . + A l c n where the A's in th is case are the eigenvalues o f the o rig in a l covari ance matrix [C]. To investigate the convergence o f the normalized e rro r energy as i t approaches zero with k approaching n (the order o f the square matrix [C ]), we obtain an expression for the difference between suc cessive values of ej<. From (2 .1 1 ), x2 k+1 ek ' Gk+1 = x2 + x2 + ^ 2 k = 1 » 2 , . .. ,n -l ( 2 .12) Assuming the eigenvalues are d is tin c t and A, > A _ > > \ , I i n A |<+i decreases, and the difference (e^ - e|<+^) becomes progressively smaller. The dominant terms in the expansion (2 .7 ) are associated with the dominant eigenvalues of [C], and i f they are known, then i t is possible to find the number of terms required in the expansion to achieve a given degree of accuracy measured by the normalized error energy. 13 In the lim itin g case where a ll eigenvalues are equal, i t is seen that a ll terms of the expansion are equal, and each term is a matrix whose elements are those of [C] divided by n. The conver gence in this case is the slowest possible. Since the covariance matrices that we deal with in non- stationary random analysis are usually of high order, then i t is ad visable to cut down on computer storage and time by calculating only some of the highest eigenvalues. As a general practice, i t is advis able to s ta rt by calculating about (y^-) highest eigenvalues, eval uate the convergence, and then calculate the corresponding eigen vectors of those eigenvalues chosen to be used in the expansion to meet the required accuracy. I f i t is found that more eigenvalues need to be c a lcu lated , then it is possible to do that without re c a lc u la tin g the previously calculat ed ones by using one o f the many computer routines a v a ila b le . Suppose th at only X^ . . . .A^ are c a lc u la te d , then a lower bound normalized e rro r energy where (m £ k) becomes 2 2 2 X^ + X ? .+ A ■ 1 ■ x? + (2 - 13) An upper bound can also be calculated as: 2 2 2 A^ + A, + . . . + A„ (e ) - 1 - 4 -------- I------------- 1 (2.14) m max 12 , . 2 , ,2 1 2 •** lc 14 The above result is arrived at since an upper bound for the uncalculat ed eigenvalues is the smallest calculated eigenvalue and the lower bound is zero (0 s A ^+-j s A^). 2.1.3 Analytical Representation Using the liumerically Evaluated Eigenvectors An approximate analytical expression for the covariance matrix is determined by approximating a n a ly tic a lly the eigenvectors in equa tion (2 .7 ). The problem of f it t i n g the covariance matrix, a two- dimensional array, is thereby reduced to a problem of f it t i n g only k eigenvectors, which are one-dimensional arrays. The choice of the analytical function in the f i t is c r itic a l in this procedure. In addition to the requirement that the number of co e ffic ie n ts be as small as possible, and th e ir calculation simple, we require that they lead to convenient analytical expressions for the covariance kernels of the dynamic systems response. Among the various functions that can be employed to approximate the eigenvectors, the orthogonal functions are extremely a ttra c tiv e . Due to the orthogonality property, the computation of the coefficients is g reatly sim plified and the coefficien ts are independent, making i t possible (when a p rio ri knowledge of the order is lacking) to check several orders with the coefficien ts for the lower order remaining valid for the higher ones. The property of almost equal-error approximation related to Chebyshev polynomials, prevents high errors in , say, extremes of the data range to which approximation is required, against small ones 15 elsewhere, thus damping the approximation error which w ill be o s c il lating between two lim its that are almost equal. In the course of this study, several sets of orthogonal functions were trie d , including Bessel functions of d iffe re n t orders, spline functions, trigonometric functions, and Chebyshev polynomials. Among those trie d , trigonometric functions and Chebyshev polynomials proved convenient. I t should be noted, however, that convergence is only guaranteed by an in fin ite number of terms in the summations. 2.1.4 Use of Chebyshev Polynomials in Surface F it Equation (2.7) can be written as (2.15) £. can be expressed as (2.16) where the normalized values t^ are defined by: (2.17) The T's are Chebyshev polynomials defined as: Tn(£) = cos(n cos'1 O ' , ~ 1 <- K <. 1 (2.18) that satisfy the weighted orthogonality property 16 +1 / Tn(C)Tin(C)a1(C)d5 = " 0 , n f- m tt/ 2 , n = m ^ 0 (2.19) it , n - m = 0 where the weighting function a>(5) is defined as « ( 5) = V l ~ E 2 ( 2 . 20) H^j is the coefficien t of Chebyshev polynomial of order ( j) associated with eigenvector ( i ) , and m. is the number of c o e ffi- 1L cients used for the f i t of the i eigenvector. Using (2.1 9), H.. in (2.16) can be calculated from H. . = i j (2/tt)V1 . . j f 0 (l/irJ V .j j = 0 ( 2 . 21) where * / ' £ ,(€ )T ,(O u (C )d C (2.22) -1 ^ J Letting C = cos < p (2.23) V,-i = f M<J>)T.(<J>)d4> IJ I J (2.24) 17 and from (2 .1 5 ), we have (2.26) A where ( ^ ( t ^ ^ ) is the approximate covariance function evaluated in the arguments, and 0 t m ; i = 1, 2. k is the number of terms used | 1 iT iaX and the hat is used to indicate the use of approximate a n a ly tic expressions fo r the highest k eigenvectors, and [C] is the measured covariance kernel of the stochastic process. Note that the range of t^ , the argument of in (2 .1 6 ) s is normalized to be equal to the range of the square region covered by the covariance kernel. Note also th a t the number of co e ffic ien ts requirec k fo r the a n a ly tic a l representation is equal to I m.plus keigenvalues. i=l 1 2.2 Spectral Decomposition as the Result of Least Squares Expansion The maximization of the rate of convergence in the expansion of a covariance m atrix in the form ( 2.7 ) is of prime concern. The question of whether there are some base vectors that w ill give a fa s te r rate of convergence than th a t given by the eigenvectors of the covariance m atrix a ris e s . To in vestig ate this p o s s ib ilit y , le t us seek the vector £ that minimizes the e rro r energy J-j in the error m atrix [E^J[.__________________________________________________________________18_ [ E , ] = [Cx] - r i r [ ( 2 -27> . where [E-j] is the e rro r m atrix of order (n x n) re s u ltin g from the f i r s t stage estim ation of [C ] [C ] is the covariance m atrix of order (n x n ) . A _r^ is the n-length column vector to be found. and J1 * j , j , E f . i j (2.28) where E -| .jj is an element of [ E^ ]. Equation (2 .2 7 ) can be w ritte n in the form C x,ij = r7,i ■ rl.j + E l,ij < 2-M) where Cx ^ j, and r^ ^ are elements of [c ] and r^ , respectively. From (2 .2 8 ) and (2.29) 1 = i l l j l , (Cx.1d - r l , i r l , / <2 ' 30> To determine the elements of r^ th at minimizes the e rro r energy, n equations are obtained by d iff e r e n t ia tin g the above expression fo r J1 with respect to ^ (Jt = l , 2 , . . . , n ) and equating the result to zero : 19 '\ L _ = 3 j V f r 2 . 2C r r a r l,S. a r l,£ i = 1 j = l x . i j x . l j ' 1 ’ i liyJ + r? . r? .) 1,1 1 , j ' ^ X Cx , * j r l , j + Cx , U r l , i ] +2 r 1 ^ I r ] + [ r i , * j=1 i,J i=1 l , i = 0 H = 1, 2 , . . . , n (2.3 1) Since Cx ,£ i = Cx , U , (2-32) then r l ,A = jj-, Cx , £ j r l J / ^ r U j (2.33) or in m atrix form I ] = Ccx ^ l / l l l i (2.34) i j l - j is a scalar number, and the la s t equation can be put in the form t Cx3 ll = ( ^ t l ^ I l = ^ lll (2.35) which is recognized as an eigenvalue problem with \ 1 = rjr-| , being the eigenvalue corresponding to the e i g e n v e c t o r ^ . The above procedure can be applied to [E^] to get: = 12-2 + (2-36) 20 and minimizing J? = [ [ E9 . . with respect to the elements * i = l j = i of we get: [ E! ] r 2 - ( l 2£2 ) r 2 - (2.37) The above single-stage separation can be repeated k times to the resulting e rro r m atrix from the previous step, and i t is in d u ctively deduced that CEi_l3lli = ( l i l i ) l i = (2.38) - j where X .. = r ^ . is the highest eigenvalue of the e rro r matrix [E. -j ] . By comparison with ( 2 .7 ) , i t can be e a s ily shown that X .. is actually i . L equal to the i eigenvalue of [Cv] and that the least-squares expan- sion fo r [Cx] is term-by-term equivalent to the spectral representa- ;ion of [C ] . (Note th at in th is case, jr1 - is normalized such th at T r . r . _ \ \ —i - i ~ A i ■) W e note here th at the lowest i eigenvalues of [ E .] should vanish I t is also possible to determine the c o e ffic ie n t 0^ associated A/ith an a r b it r a r y nonnull vector Jj^for optimum representation in the least squares sense. Let [Cx] = + [E-, 3 (2.39) 21 then n n J i =-,Ji j= i ( Q 'x ’ i i ~ ( 2 ‘ 40 9 J n n 9aj i=l 1=1 " 2<Cx , i j ' a l 6l ,ie l , j )e l , i 6 l ,j ~ 0 ( 2 -41'9 a from which n n i | 1 j = - | C x,ij8 l ,iai ,j _ _ - - - I l 4 j i=] j=] * 1 (2.42) These operations can be repeated for E E3»*-* to 9et a 2 » a ^ • associated with which can be used to expand [C ]. Although convergence cannot be guaranteed, some idea of the general shape of eigenvectors might be of great value in choosing the §'s. In the following sections, a method based on the use of Karhunen-Loeve expansion u t i l iz i n g R itz-G alerkin approach requiring an a p rio ri choice of a suitable set of orthogonal functions is pre sented. The method determines the set of c o e ffic ie n ts corresponding to ttie chosen base functions, that describes the covariance matrix best. 22 \ ) 2.3 Galerkin Weighted Karhunen-Loeve Approach I t is shown from functional analysis (Courant and H ilb e r t , 1953 bhat according to Mercer's theorem (Mercer, 1909), a function ' x ^ l **"2^ which is symmetric, continuous, and non-negative d e fin ite in the square region 0 S t-j S t mQx and 0 S = t maxcan be expressed in an absolutely convergent expansion of the form Cx ( t l ’ t 2 ) = £ Xk ^ t l ) * (2.43) Where the <(>'s and the V s are the eigenfunctions and eignevalues of the integral equation: t . max W = X {j cx ( t 1 , t 2 )< f> k ( t 2 ) d t 2 (2.44) The main idea of the present method is to use the R itz-G a le rk in approach involving the choice of a s u ita b le complete set of orthogona functions (^ 's ) which is expected to lead to a compact set of c o e f f i cients that can be conveniently evaluated and w ill e x tra c t the essen t i a l features of the data. Let / x m <frk( t ) s 4>k ( t ) J a k ( t ) (2.45) \J " * J sJ or ♦k( t ) = £ * . ( t ) + (2.46) K j = l J J k 23 (m) th < { > k (t} is the approximate k eigenfunction of the in te g ra l equa tion (2.4 4) using m of the chosen set of functions (ip 1s ) and evaluated a t time t (0 s t s t ). v max7 a*j is the j th c o e ffic ie n t associated with the kth eigenfunction Equation (2 .4 4 ) can be w ritte n as: (m) , /"max , . , . ^k ^ 1 ^ = “Tm7 _ Cx^t l , t 2^ * ( t 2^d t 2 + e k ^ 1^ A U K (2.47) where the superscript (m) denotes the number of terms used in the eigenvector f i t . (m) th e k( t ) is the e rro r of estim ating the k eigenfunction using (m ) temis. of the set of base functions, at time t . Using the Galerkin method to determine the c o e ffic ie n ts ( a 's ) , jwe equate the weighted average e rro r to zero over th e domain 0 s t , s t 1 max t max , . / e k ( V W d t l = 0 r = m (2.48) S u bstituting equation (2 .4 7 ) into (2 .4 8 ), we get N a x 7mi i tmax i ^ ( ’M t 1) U k ( V - -T m J Cx ( t l ’ t 2 )<|)km ( V d t 2} d t l = 0 0 (2.49 ) 24 Replacing ^ ^ ( t ^ ) by its expansion, we get: m t|MX t t m . max max I k JHT i aj ! * r Ct,)E f Cx ( t , , t ),(, ( t 2 )dt Jdt = 0 J-1 n n (2.50) t max (2.51 ) and t + • max max G Then equation (2.47) can be w ritten as: xM l - /, Gr j ■ 0 < * ■ * > or in m atrix form A ^ [M ] a ; = [Gja (2.54) rfhich is an eigenvalue problem of order m that can be solved to get the eigenvalues , A>m>, A M and the correspondj ng eige„. 1 9 sectors a , a , . . . ,am. 25 An estimate of the ( t ' s ) is then a v a ila b le from: ’ £ , ( t ) ' r . p . . . • « i i r m % ] ( t ) " " & £ ( t ) ► s a<2 > • • 4 > ■ i n t * ) _ L , W • • ■ a (i»>- m (2.55) and the approximate covariance m atrix can be calculated from (2 .4 3 ). The powerfulness of the methods of orthogonal expansions as presented in th is chapter was tested in several fie ld s of random processes; the resu lts are given in Chapters 3, 4, and 5. 2.4 Nonstationary Response of lin e a r Systems Consider the nonstationary response y ( t ) of a lin e a r s in g le - jdegree-of-freedom system characterized by an impulse function h ( t ) , jnder the action of a nonstationary stochastic process [ x ( t ) ] whose covariance kernel is expressed in the form of (2 .2 6 ). Let [C ] represent the covariance kernel o f the nonstationary response then C y (t15t 2 ) = E [y(t-j ) y ( t 2 ) ] (2.5 6) where E denotes the expected value o f the q u a n tity between brackets and the process is assumed to have a zero mean t , t p E [ y ( t 1 ) y ( t 2 )] = / / h ( t 1- x 1 ) h ( t 2- ’:2 )Cx (T1 ,x 2 )dx2dx1 0 0 (2.5 7) 26 Replacing Cx ( ,t,,) by its representation as in (2 .2 6 ), the covar iance of the response w ill then be given by 2 k m -1 m -1 max ^ i_n H. .H i = l j=0 £=0 x < \ max h(tg - - 1 ^ (?2+l ))Tftfe 2)dc^| (2.58) Consider in Eq. (2.58) the typical integral 2t t— - i m ax t I . ( t ) = / h ( t- f - X k + l ) ) T i (c)dt ( 2 ‘ 59) This integral may be evaluated by expressing the T .(c ) as an i4^ J order polynomial in e T-(E) = I f .. ?k (2.60) ' k=0 where t h e f t ' s are given by the following recursion formulas: f 00 = 1 ’ f 10 = 0 ’ f l l = 1 27 f i + l ,0 f i -1 ,0 f i + l , k 2 f i ,k -l “ f i -1 ,k * f . , . = 2 f. . , i+ l ,k i ,k -l k - 0 k .= 1 , 2 , . . . , i -1 k = i , i +1 (2.61) Consider a l i g h t l y damped lin e a r SDOF o s c illa t o r with an impulse function 1 h ( t) e sin o > .t “ d d (2.62) where £ = r a tio o f c r i t i c a l damping 0)^ = natural frequency 03 . = 0 3 d? n V 1 - T Making use o f Eqs. (2.6 0) and ( 2 .6 2 ) , the value of the generic in te g ra l appearing in Eq. (2 .5 9 ) may be w ritte n out e x p lic it ly as where F . ( t ) and where , -(a-eiu t ) M * ) - -^e F i( t ) (2.63) ( f -1) max k=0 £=k p t n max b = - ^d^max \ = - l (2.64 (2.65 28 I „ _____ p = ° “ tan-1 (V l / e2 - 1) I t ! 8 = B (t) = co d( t - ) (2.66) Using (2.63): and ,(,2..54)' in (2.58) trie covariance of the response E [yft^ ) y ( t 2)] = S | - exp[2a-{<a ( t , + t 2)l 4“ d k m.-l m.-l I I I Mi i Hi £ Fj ( t l )F£( t 2) (2-67-) 1 = 1 j=0 £=0 1X- 3 1 * * which can be expressed in the form ! > k V V 1 ; E [y (t1 )y(-t2>] ^ (2.68) where V W = - ^ e x p [ 2 a - 5Mn( t 1+t 2) ] F ( t p F j f t ^ (2.69) t m 4cud The mean-square response is obtained by le ttin g t-j = t^ = t : k m.-l m.-l E [ / ( t ) ] = fx-l I H H Y£ ( t , t ) (27Q) i=l j=Q -9 , =0 13 1£, 3 ... ^ ‘70 } I t is seen from (2.70) that the mean square response is made up of two non-interacting groups of terms: (1) the H's representing the e x c ita tio n c h a ra c te ris tic s , and (2) t h e Y 's representing the dynamic system properties. The method given fo r the evaluation of the generic integral is the same as th at used by Masri and M ille r (1982). 2.5 Extreme Values of the Nonstationary Response 2 Letting Ymax(T ,£ ) represent the peak value of the nonstationary mean square response of the o s c illa to r of natural period T and small c r i t i c a l damping r a t i o n , then Y m :s ( T , 0 » the extreme value of the ma x displacement, and (T,£) the extreme value of the ve lo c ity l i l a X can be evaluated from (2.71) and max (2.72) 30 2. 6 Direct Analytical Representation In section 2. 3, a procedure for the a n a ly tic a l f i t of the covar iance matrix was introduced. The procedure reduces the problem to that of obtaining a suitable analytical expression fo r a set of vectors numerically evaluated to have the fastest rate of convergence possible. However, an an alytical expression could be evaluated d ir e c tly as follows: Let oo oo C x(tl*t2) = J, i, “i/iC W V where the 4>'s represent a complete set of orthonormal functions, and the a's are constants to be evaluated. Equation (2.73) can be put in the form oo oo [C ] = I I a $ £ (2.74; x i=l j=] 1 J where jp. is a column vector of length equal to the order of [C ] and « X evaluated at equal in te rv als over the domain of the orthogonality. Premultiplying by and postmultiplying by ^ , we get: oijj = [Cx]£ j (2.75] Despite the s im p lic ity of calculatin g the c o e ffic ie n ts a . , from ^ J (2 .7 5 ), the method of section 2.1 has the advantage of numerically providing the vectors of the fa stest rate of convergence possible and hence is much more suitable for signature analysis and simulation studies. 31 2.7 Error in the Nonstationary Response of Linear Systems The error in the m.s. response of a harmonic o s c illa to r resulting from the use of [C^] in place of the exact covariance matrix to char acterize the nonstationary excitation can be exactly evaluated by calculating the response of the o s c illa to r under the action of [ E^]. The determination of the exact error matrix [E^] requires the know ledge o fth e exact covariance matrix which, we assume, is not available The i*^ term of the expansion (2.15) can be regarded as the cross correlation matrix of the single record defined as (/" T 7 ). There fore, the contribution of each term in the expansion to the m.s. .response of any lin e ar system cannot be negative at any instant of time. I t then follows that E[yj"(t)] 5 E[y2 ( t ) ] (2.76) 2 where E[y ( t )] is the m.s. response of the system under the action of [C] at time t , and E[yk( t ) ] is the m.s. response of the system under the action of [Ck] at time t (k = 1 , 2 , . . . , n). An improved estimate fo r the nr.s. response of lin e ar systems underrthe action of [C] can be obtained by calculating th e ir re sponse under action of [Ck] , where ■lie II Ck»i j = T T ^ fT CM j ( 2 - 77) in which ||C|| and ||Ck || are the sums of the squares of the elements of [C] and [Ck] , respectively. 32 2 .$ Remarks 1. The methods of sections (2 .1 ) and (2 .3 ) w ill produce an e s s e n tia lly sim ila r re s u lt i f the size of the eigenvalue problem re sulting from the minimization process in the method of section (2.3 ) is equal to the number of terms used to approximate each of the eigne- vectors in the method of section ( 2 .1 ) , providing th at the same subset of base vectors are used in both methods. 2. Although m, the size of the eigenvalue problem of section (2 .3 ) can be greater than, equal to , or less than n, the size of the original covariance m atrix, fo r a successful choice of base functions i t is expect that m << n. 3. The off-diagonal elements of [M] in Section 2.3 w ill vanish i f the chosen base vectors are orthogonal. In ad d itio n , [M] w ill reduce to an id e n tity matrix i f the base vectors are orthonormalized. The choice of orthogonal base functions g re a tly s im p lifie s the eval uation of the elements of [G]. 33 CHAPTER 3 APPLICATIONS TO MODULATED WHITE NOISE PROCESSES 3.1 Introduction Consider a random process n ( t ) , a sample function of which n^(t) is established by assigning s t a t is t i c a l ly independent sampled values of a random variable n to successive ordinates spaced at equal in te r vals along the time abscissa and by assuming a lin e ar variatio n of the ordinates over each in te rv a l. The abscissa of the f i r s t point in the range t > 0 is a uniformly distributed random variable 0. Also assume that the random process n (t) has a zero mean and a Gaussian density function with variance given by - where Sg is a constant. The autocorrelation function Rn(x) fo r the random process n (t) is given by (Clough and Penzien, 1975) 2 a g = 2 7rSg/(At) (3.1) 2A t < | t | (3.2) 34 where x = t 2 - t^ , and the spectral density function of the process is: S0 ' S„(w) = ----------7 T [ 6 - 8 cos (wAt) + 2cos(2uAt)] n u t ) 4 i (3.3) Now, le t us consider the random process x ( t ) , a sample record xr ( t ) of which is obtained by xr ( t ) = g (t) n ( t ) n p( t) (3 .4 ) where g (t) is a determ inistic envelope function defined by g (t) = exp(-b-|t) - exp(-b2t) (3.5) in which b-| and b2 are positive constants; it( t ) is a boxcar function that has a value of unity in the range t . s t s t and vanishes els.e- J 3 min max - where in the t domain. A complete ensemble of such sample functions [x ( t ) ; r = 1, 2 , . . . ] can be obtained in a sim ilar manner. Function x ( t ) is defined in (3.4 ) is convenient for use in ana ly tic a l studies involving d ig ita l simulation of white noise. The covariance of the nonstationary random process x (t) is eas ily determined to be: cx( t i , t 2) = g (t-|)g (t2) n ( t i ) n ( t 2)Rn( t 2 - t-j) (3.6) I t is worth noting that as At -* 0, A t V * ^ 1 = 9 ^t l^ 9 ^t 2^n^t l^n^t 2^2nS0<s^t 2 = M (3.7) where <5(x) is the derac delta function, having a value of unity at i = 0 and a value of zero elsewhere. 35 The peak of this covariance kernel as defined by equation (3 .6 ) is d ir e c t ly proportional to the amplitude of the spectral density function Sg and is inversely proportional to the d ig tiz a tio n time step At. The width of the surface (c o rre la tio n time) is ±2At with respect to the diagonal time axis along which t-j = t 2 - Outside this band, the covariance surface reduces to a f l a t plane. As the d ig itiz a tio n rate of n (t) becomes progressively higher, the surface associated with C ( t ) becomes steeper, and the width of X . the non-zero-magnitude band becomes sm aller, reaching the lim itin g case when the d ig itiz a tio n rate becomes i n f i n i t e (At = 0) and the co- variance surface reduces to a f l a t plane with a modulated delta function along the diagonal (t-j = t 2 ). The orthogonal decomposition and expansion, as described in section 2 . 1 ’ is applied to the described nonstationary random pro cess. Two cases of d iffe r e n t bandwidths are considered. 3.2 Case 1 Consider a nonstationary random process o f the type discussed above and with: bj = 1.0; b2 = 0 .1 ; tmin = ° ' ’ ’ ‘ max * 20- ’ ; At = 1.0; and s0 = 1 . 0 . 36 3 .2 .1 Surface Approximation A covariance m atrix [C] o f the order (501 x 501 ) was constructed using equations (3 .6 ) and (3 .2 ) and is shown in Figure 3 .1 . The peak ;alue of the surface is approximately “2 .0 , occurring along the diag onal at t-j ~ The highest 20 eigenvalues were computed and p lo tte d in Figure 3.2. The th e o re tic a l bounds of the normalized e rro r energy were c a l culated from (2 .1 3 ) and (2 .1 4 ). Eigenvectors ck , i = 1 - 20 were also evaluated, and the 1 east-squares f i t of , i = 1-15 using 20 Chebyshev c o e ffic ie n ts was also determined. A measure of the goodness of f i t (E .R .) between .P.. and is defined as: . a k , rms value of (£^ - ) E- R* = rms value o f j>. J l ! > • ( * ) - P i U i r / M _ £= 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ ' / ? 2 y Z . P i U ) / M (3 .8 ) where k is the number of c o e ffic ie n ts used to estimate £• and M is the size of the estimated v e c to r, and rms value is the root mean square value. 37 C l t , , t 2 ) c( t, , t 0) 2.03 20.1 Figure 3 .1 . Case 1: Exact Covariance M atrix [C] 38 70: .... .....r",i,,"iM ■ ! 1 \... o r i 1 , 1 i i '" f » - « » I ■ » " 1 1 ........* > ' 1 k 30 Figure 3.2. Case l r Convergence o f the Highest 30 Eigenvalues o f Cc] Note: ^ and are only defined fo r k = 1 , 2 , . . . 1 - V i A \ > \ s " \ \ V \ £k \ % \ « V ' \ s \ ' \ V UPPER BOUND / 0 v lower\ BOUND - - ........................................ ............. /ACTUAL ERROR 1 ............................................ 1 ' 1 1 1 5 10 15 20 k Figure 3 .3 . Case 1: Rate o f Convergence o f [C^] to [C] and Error Bounds on 39 Plots of £ i and , i = 1 , 4 are given in Figure 3 .4 ; a graph of i.R. versus the number of Chebyshev c o e ffic ie n ts used in the f i t is ilso provided fo r each eigenvector. Figure 3.5 provides id e n tic a l in formation fo r £ . and , i = 5 - 8 . Additional s im ila r graphs fo r the test of the highest 15 eigenvectors are given in Appendix C. The actual normalized error, energy e|< was computed by adding the square of the elements of the e rro r m atrix ([C] - [Ck]) fo r k = 1, 2, 4. 5, . . . , 14 and d ivid in g by Jg. ^ a c t u a l = £ =1 £ = ] " Ck ( l ’ j ^ 2 /J o ( 3 -9 ) where N N 9 Jn = Z I C * ( i , j ) (3.10) u i=l j= l Figure 3.3 presents a graph of and its bounds based on the calcu latio n of the highest 20 vectors. The upper and lower bounds would p r a c t ic a lly coincide with the use of the highest 30 eigenvectors This is due to the fa c t that A^g is n e g lig ib le compared to A ^ . Figure 3.3 indicates that the f i r s t 10 (out of 501 ) eigenvectors embody prac t i c a l l y a ll of the s t a t is t i c a l information contained in [C]. Diagonal comparison between [C] and [C k] fo r k = 5, 10, and is ' given in Figure 3.6 a, b, and c, re s p e c tiv e ly . Diagonal comparison A between [C] and its a n a ly tic f i t [Ck] is also provided in Figure 3.7. 40 1. E.R (b) 0 1. <d) |' (f) (h) .2 2 20.1 .1 t 2 2 20.1 .1 t (e) -.2 20.1 0.1 t .2 20.1 0.1 t Figure 3.4. Case 1: Exact ( £ .) , A nalytically Approximated (£^)» and Error Ratio versus Number of Coefficents1 ; i = 1 - 4; p.(t); ---------p.(t) -------------------------- - ^ - T --------------- 1---------------------------------------------------------------------- (a) (b) 20.1 (d) 20.1 (e) 20.1 0 t 20.1 5 10 15 Figure 3.5. Case 1: Exact (p.), Analytically Approximated (p.), and Error 1 1 Ratio versus Number of Coefficients; i - 5 - 8 £. (t); ------£. ( t ) 20.1 F ig u re 3 .6 . Case 1: Diagonal Comparison Between [C] and [C^] C ( t , t ) ; C ( t , t ) - 1 . 5 * 20.1 20.1 F ig u re 3 . 7 . Case 1: Diagonal Comparison Between [C] and [C^] Based on Using 20 Chebyshev C o e f f i c i e n t s to F i t k Eigenvectors C ( t , t ) ; . . . .C ( t . t ) ; C . ( t , t ) - C ( t , t ) CO Three-dimensional plots of [Cfc] fo r k = 5, 10, an 15, are given in Figures 3.8 - 3.10, respectively; corresponding e rro r surfaces ( [ ] - [C ])are given in Appendix C. The a n a ly tic a l approximation a . [C15] based on using 20 Chebyshev c o e ffic ie n ts fo r the f i t of each eigenvector is illu s tr a te d in Figure 3.11. 3.2.2 Transient Nonstationary Response The procedure indicated in equation (2.7 0) was used to a n a ly ti c a lly determine the tra n sie n t m.s. response of several lin e a r, v is cously damped SDOF harmonic o s c illa to rs under the action of the stochastic ex c ita tio n represented by [C]. A nalytical and numerical integration results based on using the exact [C ], approximate covar- * iance matrices [C^] and [C^], k = 5 , 10, and 15 are presented in Figures 3.12 and 3.13. Additional results are given in Appendix C. 3.3 Case 2 Consider a nonstationary random process of the same type as in case 1 , with the following parameters: b-j = 1.0; bp - 0 . 1 ; ‘ min 0-1 ’ ‘ max * At = 0.1 ; and S q = 1 . 0 . 44 . i l l M i s s s Figure 3.8. Case 1: Approximate Covariance [Cg] Surface -P * c n o.i Figure 3.9. Case 1: Approximate Covariance [C^g] Surface 0 . 1 o.l Figure 3.10. Case 1: Approximate Covariance [ C ^ Surface C T > 0.1 0.1 0.1 0.1 Figure 3.11. Case 1: Approximate Covariance [C-icl Surface Based on Using Analytical Expressions (20 Chebyshev coefficients) to approximate eigenvectors) (b) (c) (d) 2.5s (e) (f) 20 3.5s (h) > /. Figure 3.12. Case 1: (k = 5) m.s. Response of SDOF Harmonic Oscillators (T = ,5s - 4.0; .5s and£= .05') CCD; C C k3; — -CCk] 47 3E-2 2.E-3 (b) (d) (c) 2.75s (e) (f) 3.25s 3.75s Figure 3.13. Case 1: (k = 10) m.s. Response of SDOF Harmonic Oscillators (T = ,5s - 4.s; -5s and £ = .05) n . CC]; -------[C. ] ; -------[C. ] 3.3.1 Surface Approximation Note that this case is id en tic al to case 1 except the reduction of A t from 1.0 to 0 .1 . The e ffe c t of this change is the reduction of the band of the covariance m a trix , resu ltin g in extreme sharpness of the surface in the out-of-plane d ire c tio n . A covariance matrix [C] of order (501 x 501), describing the in te r val t • £ t ^ t > was constructed and is shown in Figure 3.14. mm max To achieve a degree of accuracy in the surface approximation, com parable to using 10 terms of the expansion (2 .7 ) fo r case 1, required i;he use of about 100 terms. Eigenvalue convergence is illu s tr a te d by Figure 3.15. The actual normalized error energy (computr fo r [C^], < = 1, 25, 50, 75, and 100), togetherrwith its th eo retic al bounds based on calculating the highest 100 eigenvalues is shown in Figure 3.16. Selected eigenvectors are given in Figure 3.17. Diagonal comparison between [C] and [C^] is illu s t r a te d by Figure 3.18 fo r some chosen values of k. Three-dimensional plots of the approximate covariance [C^] surface are given in Figure 3.19 fo r k = 25, 50, 7 5 ,and 100. 3 .3 .2 Transient Nonstationary Response The transient mean square response of lin e a r single-degree-of- ■ :reedom (SD0F) harmonic o s c illa to rs of various natural periods and a c r i t i c a l damping r a tio of ? = 0.05 was computed under the action of the nonstationary disturbance characterized by: 49 c (t,,tj t I c ( t , t 2) 2 . 20.1 (b) '1 20.1 Figure 3.14. Case 2: Exact Covariance M atrix 50 80 " X k 0 1 k 100 Figure 3.15. Case 2 : Convergence o f the Eigenvalues o f [ C] UPPER BOUND ACTUAL ERROR LOWER BOUND 100 Figure 3.16. Case 2: Rate o f Convergence o f [C^] to [C] and Error Bounds on 51 20.1 20.1 20.1 25 2 0 . Figure 3.17. Case 2; Eigenvector <£.) of [C] (a) (c) 25 20.1 20 J t 0.1 25. 20.1 0.1 t k = 20 k = 40 k = 60 k = 80 k = 100 0.1 t Figure 3.18. Case 2: Diagonal Comparison Between [C] and [Ck] C ( t , t ) ; C ( t , t ) 20.1 53 c.(t,,t~) \ 20.1 2 Figure 3.19. Case 2: Approximate Covariance [C^] Surface for k = 25 — 100; 25 ck( t 1 , t 2) 1. The exact covariance m atrix [C] 2. The approximate covariance m atrix [C^] , k = 25 3. The approximate covariance m atrix C l , |< = 50 K 4. The approximate covariance m atrix [C ] , k = 75 K 5. The approximate covariance m atrix [C ] , k = 100 K The results of these calcu latio n s are presented in Figure 3.20. Additional results are given in Appendix C. 3.4 Case 3 I t is noted from cases 1 and 2 th at the used d ig it iz a t io n rate is very high and a lower order of covariance m atrix [C] of about (51 x 51) is s u f f ic ie n t . This case describes the same type of process as in case 1, with the follow ing parameters: b-j = 1.0; 08j (b) V . V. .6 . .25 (c) (d) 20 1 . 2.5s (e) (f) 0 3.2 2.5 3.5 s (h) 0 20 0 t 20 t Figure 3.20. Case 2: Nonstationary Response of SDOF Harmonic Oscillators wi th £ = 0.05 C25; _ . . - C5o; --------- C75; ”' " C100; C 56 A covariance matrix [C] of order (51 x 51) at a sampling rate of 0.2 was constructed and is shown in Figure 3.21. The approach described in section (2 .3 ) was applied to this case with a subset of complete trigonometric functions being used as base functions (4>'s), namely: <lh(t) = 1 . 0 , 4>o(t) = sin -pr— ^ - p -------r ^ max min^ » p 3 ( t ) = cos-r-p— ^.(t) = sin-pp— ------ v max min' '• max min' ^ 24(1 ) S in ( t 13?r- t . ) i v max min' and , / + \ 1 3 IT t ’i»2 5 ( t ) ^ t - t . ) ( 3 n !) v max rm n' v ' Matrices [M] and [G] of order (25 x 25) were constructed using (2.51) and (2 .5 2 ), respectively. The eigenvalue problem, A25[M]a = [G ]a, (3.12; pcz was solved and all eigenvalues ( a *s) and eigenvectors U ’ s) were computed. The (j>'s) are then estimated using (2.5 5 ) and selected ones are shown in Figure 3.22. 57 o.l 0.1 (a) Figure 3.21. Case 3: Covariance M atrix [C] Surface 58 1 (a) 10.1 t (b) .1 10.1 t (d) .1 10.1 t 1 (c) 1 10.1 1 t (f) .1 10.1 t 10.1 1 t 1 (h) .1 10.1 t -1 Figure 3.22. Case 3: Analytically Defined Eigenvectors <j^., i = 1 - 8 59 Three-dimensional plots of [Ck ] fo r k., 5, 10, an 15, based on using k out of 25 eigenvalues (x 's ) and the corresponding a n a ly tic a lly expressed eigenvectors (_£'s) are given in Figures 3.23, 3.24, and 3.25, respectively. Diagonal comparison is given in Figure 3.26. The goodness of surface :fft is fu rth er illu s tr a te d by the three- dimensional views of the error surface ([C] t [C ^ ]), k = 5, 10, and 15, in Figures 3.27 to 3.29, respectively. Note that the ripples in the approximate surface decrease in ampli tude and increase in frequency as higher levels of approximation are used. 3.5 A Note on the Analytic F it In determining the analytic f i t fo r a vector that has a value of zero (or a constant) fo r many consecutive points, i t might be advan tageous to eliminate this portion from the f i t process. I t is demon strated in Figure 3.30, using o f case 2 (Figure 3.00a) as an ex ample, that a much better f i t (Figure 3.30b) can be obtained by con s id e rin g only the nonzero portion o f the eigenvector. Fifty.: Chebyshev co e ffic ien ts were used for both‘Figures 3.30b and 3.30c. Two more co e ffic ien ts are, of course, needed to store the index of the e lim in at ed portion and its value. The use of the above fact to determine an an alytic expression fo r the covariance matrix is only convenient i f a common portion of the eigenvectors hasa constant value. The inconvenience arises by the fact that several analytic expressions for the matrix are obtained- each of which is valid in a certain subregion. 60 Figure 3.23. Case 3: Approximate Covariance Surface Based on Using 5 Terms in (2.43) 2. . t -.O' 2 t 2 Figure 3.24. Case 3: Approximate Covariance Surface Based on Using 10 Terms in (2.43) 2.03 Figure 3.25. Case Approximate, Covariance Surface Based on Using 15 terms in (2.43) c(t.t) c.(t.t) - c(t,t) (a) -2.5 10.1 t .1 C (t,t) C, ( t , t ) c.(t,t) - c(t,t) (b) -2.5 2 C (t,t) C , ( t .t ) C . ( t .t ) - C (t .t ) 2 . A Figure 3.26. Case 3: Diagonal Comparison Between CC] arid CC^] Based on Using k Terms in (2.43) 62 - 1. .17 Figure 3.27, Case 3: Normalized Error Function for Estimating [C] by 5 terms in (2.43) .17 .17 t 2 Figure 3.28. Case 3: Normalized Error Functions for Estimating [C] by 10 terms in (2.43) .17 .17 t 2 Figure 3.29. Case 3: Normalized Error Function for Estimating [C] by 15 'Terms in (2.43) 63 .25 50 (a) -.2 5 20.1 .1 t .25 -.2 5 .1 20.1 t -.2 5 20.1 .1 t Figure 3.30. E ffect o f zero-ends on the Goodness o f F it . (a) Case 2 : 2.59 ; ^ E L 50 usin9 50 Chebyshev c o e ffic ie n ts to f-it part o f the vector; (c) P^q using 50 Chebyshev c o e ffic ie n ts to f i t the whole vector ( & ) 3.6 Improved Response Calculations Using section 2 .7 , [C^] was evaluated fo r case 1 fo r k = 5, 10, and 15. The results of calcu latin g the nonstationary response of har monic o s c illa to rs with natural periods 0 .5 , 1 .0 , 1 . 5 , . . . , 4.0 s, and c r i t i c a l damping ra tio of .05 under the action of [C^] are presented in Figures C.12, C.13, and C.14 fo r k = 5, 10, and 15, respectively. 65 Chapter 4 APPLICATIONS TO REAL EARTHQUAKE RECORDS 4.1 Introduction The network of strong motion recording devices has produced an invaluable c o lle c tio n o f data fo r engineering studies o f the charac t e r is t ic s of earthquakes and t h e ir e ffe c t on the environment and struc tures. This c o lle c tio n , however, does not encompass a ll conditions of recordings and s ite s , and thus does not represent a complete observa tion al basis fo r use in engineering design. M aintaining, improving, and expanding the existin g network of accelerograms worldwide become very important concerns, since accelerograms represent the best source jof information. Every r e lia b le recording of an earthquake helps to produce a clearer picture of the properties of futu re earthquakes and the potential damage to the environment and engineering structures. Due to the wide range of variatio n s in the c h a ra c te ris tic s of past earthquakes, i t is not possible to determine one acceleration record which would be adequate fo r a successful design. Nevertheless, an ensemble of acceleration records w ill b e tte r e x tra c t the important c h a ra cteristics and w ill lead to a more r e lia b le design. Many models of simulated earthquake ground shaking have been suggested in the lit e r a t u r e . Housner (1955) used a series of one- cycle sine-wave pulses to simulate an acceleration record. Randomly 66 d is trib u te d series of pulses were used by Goodman e t a l . (1955), Hudson (1956), Rosenblueth (1956), Bycroft (1960), and Rosenblueth and Bustamante (1962). Later models have recognized the nonstationary nature of earthquakes and used nonstationary random time series for the simulation (Bogdanoff e t a l . , 1961; Cornell, 1964; Shinozuka and Sato, 1967; Amin and Ang, 1968; Jennings et a l . , 1968). Gasparini and Vanmarcke (1976) developed a program to generate a s t a t i s t i c a l l y independent a r t i f i c i a l acceleration time h isto ry and jtry, by ite r a t io n , to match a specified response spectrum. Wong and Trifunac (1978) presented a method fo r constructing synthetic accelero grams which have a given amplitude spectrum duration. The s ite characteristics were used to develop the a r t i f i c i a l records. Further more, autoregressive/moving-average models have been used fo r the purpose of simulation (Chang et al ., 1979). The methods presented in th is work can be used a f te r a covariance matrix Has been generated from an ensemble of recorded or a r t i f i c i a l accelerograms. The covariance matrix is a convenient .way to describe the input and output of stru ctu ral systems. A new point o f view for the simulation of earthquakes can be achieved by try in g to simulate the dominant eigenvectors of the covariance m atrix ra th e r than the accelerograms. In the follow ing sections, the method presented in section 2.1 is applied to an ensemble of records from an actual earth quake. 67 4.2 San Fernando Earthquake 4 .2 .1 Descri ption The destructive San Fernando earthquake that struck the northern portion o f m etropolitan Los Angeles on February 9, 1971 with a magni tude of 6 .6 , was a major event from an engineering point of view, though i t was only a moderate shock in seismological terms (Hudson, 1971). The unprecedented amount of valuable data collected from the southern C a lifo rn ia strong-motion earthquake instrumentation network was s ig n ific a n t in the in te rp re ta tio n of the severe damage to many modern engineering structures. I t marked a major development in the f ie l d o f earthquake engineering (Jennings, 1971). A to ta l of 241 records were recovered from stations located be tween 8 and 369 km of the epicenter, with the m ajority from sites closer than 75 km. From the a v a ila b le 241 records, 66 records were chosen to form the covariance kernel o f the earthquake. The choice of representative records was based on such requirements as considering only the horizontal components co llected from accelerometers placed on the ground flo o rs of buildings (to elim inate m ag nification). Sample acceleration records are shown in Appendix D. D etails about the s ite s , records, and properties of the accelerographs can be found in the lit e r a t u r e . A covariance m atrix of the order (501 x 501) was formed to describe the f i r s t 20 seconds of the 66 records a t equal in te rv a ls of 0.04 second (the records are a v a ila b le at a d ig it iz a tio n rate of 0..02 second in te r v a ls ). The 66 records used to generate [C] are tabulated in Appendix D. 68 4 .2 .2 Surface f i t The general shape of the covariance kernel is illu s tr a te d by two three-dimensional views (see Figure 4 .1 ) . The highest 30 eigen values were also calculated (Figure 4.2). The actual normalized error energy and its lower and upper bounds are presented in Figure 4.3. The bounds were calculated at each stage of approximation ( i . e . , 1-30; 1) while the actual normalized e rro r energy was computed for only 1, 5, 10, 15, 20, 25, and 30 terms of the expansion to save CPU time. The 25 eigenvectors corresponding to the 25 highest eigenvalues were evaluated and a least-squares f i t using 160 Chebyshev coefficients was determined fo r each. In Figures 4 .4 and 4.5, the two highest exact eigenvectors (eigenvector numbers 1 and 2 ) are depicted in (a); th e ir least-squares f i t in ( b ) ; the e rro r of f i t in (c); and the e rro r r a tio (see Eq. (3.8)) of the f i t versus the number o f c o e ffic ie n ts in (d). The mismatch between the exact and approximate eigenvectors is due to the high frequency components used in the eigenvectors which were not detected by the order o f f i t used. Additional figu res describing eigenvectors are presented in Appendix D. The diagonal of the exact covariance m atrix [C] is shown in Figure 4 .6(a), while the diagonals of the approximate covariance matrices [C^], k=5, 10, 15, 20, and 25 are shown in the other graphs. Two three-dimensional views of the approximate covariance matrix [C 25] are shown in Figure 4.7; the corresponding views fo r the an a lytical approximation of the covariance m atrix [ C25-^ ^re 9 1ven Figure 4.8.. 69 11 C(t 1,t 2) I C ( t r t 2') (a) (b) Figure 4.1. Covariance M atrix [C] for the San Fernando Earthquake 70 3. 1 30 k Figure 4 .2 . Convergence o f the Eigenvalues of [C] Note: and are only defined fo r k = 1 , 2 , . . . UPPER BOUND £ EXACT ERROR LOWER BOUND 1 - 30 k Figure 4 .3 . Rate o f Convergence o f [C^] to [C] and Error Bounds oh 71 £.'•( t ) - . 2 5 V M v W ^ |/( (b ) - 7o .25 ^ ( t ) .25 ( c ) -.2 5 . 20s 0 1 E (d ) 160 E. ( c ) 20 r 1.0 (d) 0 1.60 1 0 N Figure 4.4. Eigenvector No. 1 Figure 4.5. Eigenvector No. 2 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ra tio versus no. of coefficien ts 72 .35E044 EXACT C ( t , t ) .35E4 - ck( t ,t ) 20s 0 k = 5 (b) * t 20s , 35E4 , 35E4 k = 10 t 20s ck( t ,t ) 0 35E4 ck( t , t ) k = 20 ck( t , t ) Figure 4 .6 . Diagonals o f [CJ and [C k] in (cm/sec/sec) 73 Ck(t] (t2 ) 2 Figure 4 .7 . (b) Approximate Covariance [C^] Surface Based on Using k = 25 eigenvectors 74 4 .2 .3 Transient mean square (m.s.) response Consider the transient m.s. response o f several SDOF harmonic o s c illa to rs with natural periods o f 0 .2 5 , 0 .5 , . . . , 4.0 seconds and a r a tio of c r i t i c a l damping o f 0.05 under the action of the nonstationary e x c ita tio n characterized by the covariance functions described above. The m.s. response o f those various o s c illa to rs was calculated during the period of e x c itatio n (20 seconds) under the action o f the exact covariance matrix [C] using Equation (2 .5 3 ). The approximate m.s. response was calculated by using [C2g] to approximate [C] in th is equation. The m.s. response under the action o f the a n a ly tic a lly described covariance m atrix [^25] using the 160 Chebyshev c o e ffic ie n ts to estimate each of the 25 eigenvectors was also calculated fo r the various o s c illa to rs described. The results of the m.s. response calculations are depicted in Figures 4.9 to 4.24. In each fig u re , there are three graphs of the m.s. response fo r a given period o s c illa tio n and a c r i t i c a l damping r a t io of 0 .0 5 . Graph (a) represents the m.s. response under the action of the exact covariance matrix [C ], graph (b) is the m.s. response under the action of [^05^ ’ an<^ 1S m*s * resPonse under the action of [Cgg]. The la s t response is calculated d ir e c tly from Equation (2 .7 0 ). The e ffe c t o f damping on the m.s. response of selected o s c il lato rs is shown in Figure 4.25. The displacement (Figure 4i26) and v elo city (Figure 4.27) extreme values fo r the forced vibration phase of 20 sec are also provided. The three curves in each figu re represent, 75 Ck^t 1 *^2 ^ t 2 (a) (b) Figure 4 .8 . Approximate Covariance [C .] Surface Based on Using Approximate A n a ly tic a l Expressions fo r k = 25 Eigenvectors 76 .12 .12 .12 E[y (t). 20s 20s 20s Figure 4.9. M.s. response ofa harmonic oscillator (T = 0.25 sec,£ = 0.05) in cm 120 . 120 E[y (t) 20s 20s 0 t 2 Figure 4.10. M.s. response of a harmonic oscillator (T = 0.50 sec, £ = 0.05) in cm (a) under the action of [C.]; (b) under the action of (c) under the action of [ C ] I j j £[y ( t ) ] EC y (t)3 20s 20s Figure 4.11. M .s. response of a harmonic o scillator (T =0.75 sec,£ = 0.05) in cm ' 15.0 15.0 E[y (t)] 20s 20s 20s Figure 4.12. M.s. response of a harmonic oscillator (T = 1.0 sec, £= 0.05) in cm ' (a) under the action of [ C]; (b) under the action of [ C__]; 12. E[y ( t ) ] 20s 20s 20s Figure 4.13. M.s. response of harmonic oscillator (T = 1.25 sec,£ = 0.05) in cm ' 15. 15. ' 205 0 t 20s 0 Figure 4.14. M.s. response of a harmonic oscillator (T = 1.5 sec, £= 0.05) in cm ' a) under the action of [ C ]: (b) under the action o f [ C J; (c) under the action.cf [C„ ] 20s 20. 20. .(b) E[y (t)] E[y (t)] 20 s 20s 20s Figure 4.15. M.s. response of a harmonic oscillator (T = 1.75 sec,£ = 0.05) in cm ' 10. 10. 10. E[y (t)3 20s 20s Figure 4.16. M.s. response of a harmonic oscillator (T = 2.0 sec, £ - 0 .0 5 )in cm ' (a) under the action of [C]; (b) under the action of [C ^ ; (c) under the action of [ C25] c o o E[y t ] E [y ( t ) ] 20s 20s 20s Figure 4.17., M.s. response of a harmonic oscillator (T = 2.25 sec, £ = 0.05) i in cm E[y ( t ) ] E[y ( t ) ] 00 E[y (t)] in cm Figure 4.18. M u s. response of a harmonic oscillator (T = 2.5 sec, £ = 0.05) (a) under the action of [C]; (b) under the action of [C ]; (c) under the action of [ Coc] 25 70. E[y ( t ) ] 2 0 s 20s- 2 0 s Figure 4 . 1 9 . M . s . response o f a harmonic o s c i l l a t o r (T * 2 . 7 5 -s e c ,? = 0 . 0 5 ) in cm' 100 100 E[y ( t ) j ' t 20s 0 t ' '2<3s 0 F ig u re 4 . 2 0 . M.s. response o f a harmonic o s c i l l a t o r (T = 3 . 0 sec, £ » 0 . 0 5 ) in cm' ( a ) under the a c tio n o f [C] ; (b ) under the 20$ ac tio n o f [ C ^ J ; ( c ) under the a c tio n o f CC?^] C O ro 150 e r(t) ] 20s 20s 20s Figure 4.21. M.s. response of a. harmonic oscillator (T = 3.25 sec 200. 200. 200. E [y ( t ) ] E[y (t)3 i t 20s 0 t 20s 0 Figure 4.22. M.S. response of a harmonic oscillator (T = 3.50 sec, £ = 0.05) in cm ' a) under the action of [ C ]; (b) under the action of [C^; (c) under the action o f[ £ ] 20s 200. E[y (t)] 200. 200. (a) (b) ' E[y2( t ) ] \/\l ■ I i .................... |- 0 , — r —r o t 20s 0 t 20s 0 Figure 4.23. M.s. response of a harmonic oscillator (T = 3.75 sec, £ = 0.05) in cm 2 20s 200 E [y ( t) ] 200 . 200 E[y (tfi Figure 4.24. M.s. response of a harmonic oscillator (T = 4.0 sec, £ = q. 05) , r —2 (a) under the action of [C]; (b) under the action of [C ]; (c) under tfv action of [C ] 4b 25 20s • i 4 .05 Figure 4.25. Effect of Oamping on the Nonstationary Re sponse of Selected Harmonic Oscillators Under the Effect of [ C l 2 (m.s. response in cm ) .05 0 0 t 20s 13 170 4 20s T = 0.25s £ = .10 (b) .05 .15 20s 0 20s 5 « 0.15 ^Os 70 0 20s / ' / % ® 0 . 0 5 max P e r i o d T cm/sec S - 0 . 0 5 •— ■ — ■ — ■ — ■ — * P e r i o d T C C. Ak ■ — Cl F i g u r e 4 . 2 6 . E x a c t and A p p r o x i mate Extrem e Values o f th e Non stationary Root-Mean-Square Dis p la c e m e n t Response o f a SDOF O s c i l l a t o r ( K ■ 0 . 0 5 ) C h ck F i g u r e 4 . 2 7 . E x a c t and A p p r o x i mate Extreme V a lu e s o f th e Non stationary Root-Mean-Square V e l o c i t y o f a SDOF O s c i l l a t o r ( . 0 5 ) F i g u r e 4 . 2 8 . Extrem e V a lu e s o f t h e N o n s t a t i o n a r y Root-Mean-Square D is p la c e m e n t Response o f a SDOF O s c i l l a t o r (5 a 0 , . 0 5 , . 1 0 , and . 1 5 ) 0 . 0 5 P e r i o d T cm/sec P e r io d T F i g u r e 4 . 2 9 . Extrem e V alues o f t h e N o n s t a t i o n a r y Root-Mean-Square V e l o c i t y Response o f a SDOF O s c i l l a t o r (£ = o , . 0 5 , . 1 0 , and . 1 5 ) 86' resp ectively, the extreme displacement and velo city under the action of 1. the exact covariance m atrix [ C ] ; 2 . the approximate covariance matrix [^25]» /\ 3. the a n a ly tic a lly expressed covariance matrix [C25] * The displacement and v e lo c ity spectra calculated under the action of tCgg] with various c r i t i c a l damping ra tio s (0, 0 .0 5 , 0 .1 0 , and 0.15 are given in Figures 4.28 and 4 .2 9 , re s p e c tiv e ly . Additional m.s. response results are given in appendix D. 4 .3 Concluding Remarks 1. The 25-term approximation o f the covariance m atrix, using the exact or approximate eigenvectors has led to m.s. responses that compared very well w ith the exact re s u lts . The erro r in m.s. response calculations is higher fo r o s c illa to rs with higher frequency (lower natural p e rio d ), as was expected. 2. Since 66 records were used to construct [C ], i t was expected that Ag? = Agg = = ^Q-]^ = 0 (from section 2.3 and the d e fin itio n o | [C ]). This fa c t was used to v e rify the computer program, "SEC21. 1 3. A study of the eigenvectors of covariance matrices of many other earthquakes might reveal important trends and information that could lead to e ffe c tiv e earthquake simulation techniques. 87 Chapter 5 APPLICATIONS IN THE FIELD OF TURBULENT FLOW 5.1 Introduction The flow of flu id s and gases a t high speeds is turbulent in nature, and objects tra velin g in those media at high speeds ( e .g ., a i r c r a f t , m is siles, submarines, e t c .) are subject to random forces. These forces can best be described as nonstationary random processes due to the many variables th a t are not under control and can only be described in a s ta t is t ic a l sense. There is a trend in modern applications technology in which the travel of objects through gaseous or liq u id media at high speeds is required. This growing trend has made in vestigating the response of those objects to the high-speed environment a very active area of research. Only thorough understanding of these forces and th e ir effe cts on the tra v e lin g bodies can lead to safe, economic, and r e lia b le design. In th is chapter, the e ffic ie n c y of the orthogonal expansion tech nique is tested , using a large amount of valuable data obtained from an experimental turbulent flow over a p la te . The data is divided into three groups. Each group has 500, 1024-data-point records repre senting the flow at a section; 100 records each were obtained from fiv e anemometers which were used to measure the ve lo c ity at various 88 locations in the section. Four sample records of time versus v e lo city from each of the three groups are presented in Figures 5.1 to 5.3. The tran sient nature o f the records from groups 1 and 2 and the sta- t io n a r it y o f the records from the th ird group are indicated. The value of the v elo city at the s t a r t o f the experiment is constant fo r the ''ecords of groups 1 and 2, but i t varies in the records of group 3. To ensure the r e l i a b i l i t y of data without p lo ttin g a ll records, the records from the same anemometer in each group were used to con struct a covariance m atrix. This was found to have the same general oattern and amplitudes fo r each two sets of records obtained from d iffe re n t anemometers but in the same group, The use o f a ll 1024 data points in each record results in a square covariance m atrix o f the same order; t h is , however, is deemed p ro h ib itiv e ly large. Thus, the size o f the covariance matrices was lim ited to 512. This choice retains most o f the information because of the high d ig it iz a tio n rate of the data (2000 points per second) and the nature o f the experiment. A covariance matrix [CJ ] of order (512 x512) was constructed to represent the nonstationary process defined by the 500 records in each group (the superscript J denotes the group number). 5.2 Surface Approximation 1 2 Some of the highest eigenvalues were computed fo r [C ] , [C ] , and [C ] , and the th eo retic al bounds fo r the normalized e rro r energy were calculated from Equations (2.13) and (2 .1 4 ). The actual normalized erro r energy was computed fo r the selected levels from lEJr V 1E5j v - o - 4 1E 5' V o - 1E5' v - n .1E5 - „ : (a ) V .5E4 j — ---------------;------------ v (a) fa ) t . t t min max . 1E5 ' _____i , : (b) t . t t mm max • 5E4 i J A - ___________________ : (b) v; t . t t • min max [ ‘ (b)_ t . t t 0 m in m ax . .1E5 " (C) t. t t" min max .5E4 J ------------- — v ; t . t t min max t . t t 0 min max . 1E5 - — — v : t . t ; min max . 5E4- — _ ■ : t . t t mi n max (d ) "t. t t ” t, t t U t t t min max min max m in . max Fig u re 5.1. Sample Records from Group 1 F ig u r e 5.2. Sample Records from Group 2 Figure 5.3 Sample Records from Group : ' o J c, J nr actual m C J (5.1 ) where nr actual t h t H is related to the J group and the m level of approximation; is the approximate covariance m atrix fo r group J, based on using m exact eigenvectors; and 512 512 i i [c°(k,z)r . (5 .2 ) k=l £=1 The results of these calculatio ns are presented in Figure 5.4. Note the extremely fa s t rate of convergence fo r group 2 in which about 90% of the s t a t is t ic a l information (as measured by the normalized e rro r energy) is embodied in the highest eigenvector. Group 3 has a much slower rate of convergence, so more eigenvalues and eigenvectors were computed fo r th is case to obtain comparable re s u lts . The highest 1 2 3 four eigenvectors of [C ] , [C ] , and [C ] are presented in Figures 5.5 to 5 .8 ; additional figures are given in appendix E. The highest eigen vectors were found to flu c tu a te in the time domain w ith amplitudes proportional to th at o f corresponding records. Approximate covariance matrices [C^] using k exact eigenvectors to estimate the actual covariance matrices [CJ ] were compute fo r J = 1, 2, and 3. Plots o f the exact and approximate surfaces are given in Figures 5.9 to 5.11, where the same scale is used fo r exact and approximate matrices from the same group. Diagonal comparison between 91 1 . e k 0 30 k 1 1 . (a) n SW . U ,TrV mi y im»m*y* (b) 30 Lower bound on Upper bound on ek Actual e. Group 1 Group 2 1 e 50 k 1 (c) Group 3 Figure 5 .4 . Normalized Error Energy and It s Bounds fo r Groups 1, 2, and 3 92 .25 -.2 5 -.2 5 max max .25 (b) -.2 5 -.2 5 max max .25 .25 (c) — • 25 ....... I"’ 0 t t max Figure 5 .5 . Eigenvector No. 1 o f -.2 5 max Finure 5 .6 . Eigenvector No. 2 o1 .25 .25 (a) (a) -.2 5 max max ,25 A .25 0 . ,25 .25 0 max i 3 o f t n (c) max £4 o f [(£] £3 o f [CT] max £ 4 of [CJ] max Figure 5 .7 . Eigenvector No. 3 o f Figure 5 .8 . Eigenvector No, [C ] , J = 1, 2, and 3 [Cu] , J = 1, 2, 4 of and 3. 94 .153E7 - . 3E6 C3 o (ta t) (a) j t ( t v t z ) .153E7 -.3E6 (c) .1 53E7 . 3E6 .153E7 - . 3E6 1 (d) Figure 5.9. Group 1: Exact and Approximate Covariance Surfaces cn f cz ( t l f t 2 ) .25E7 ■JE6 C30^1 ’V t ,25E7 - . 1 E6 . C .25E7 t - .1 E6 '2 (c) Figure 5.10. Group 2: Exact and Approximate Covariance Surfaces .25E7 - . 1 E 5 C T ) I ^ ^ ‘1*^2^ C50 1 ’^2 — . t. \ t (c) -•.5E5 C 5 0 ( t | .tg ). -.5E5 X t 1 ^ Figure 5.11. Group 3: Exact and ADDroximate Covariance Surfaces -* < J [C°] and [C^] fo r selected values of k are given in Figures 5.12 to 5.14 (J = 1, 2, 3, re s p e c tiv e ly ). Analytical expressions using 50 Chebyshev c o e ffic ie n ts were evaluated fo r the highest 20 eigenvectors of group 1. The exact eigen vectors ( £ .) and the vector of error in f i t are plo tted in Figures 5.15 to 5.18 fo r i = 1 , . . . , 4 . The e rro r ra tio (E .R .) versus the number of Chebyshev c o e ffic ie n ts (N) used in the le a s t squares f i t fo r eigen vectors is also graphed. The e rro r ra tio (E .R .) is defined as rms value of (£.j - ) rms value of"jET [ p - U ) - — --------------------------------------------- (5 .3 ) / I £ i 2U)/m V H=1 1 where k is the number o f c o e ffic ie n ts used to estimate and M is the size of the vector. In the follow ing section, the tra n s ie n t mean square response of various o s c illa to rs is calculated for group 1 , which was chosen as an example because i t is intermediate in the convergence rate between group 2 and group 3. 5.3 Nonstationary Response of SDOF Harmonic •O scillato rs The tra n sien t mean square response o f various SDOF harmonic o s c i l lators with natural periods o f 0 .2 5 s -4 .0 s ; 0.25s ( i . e . , from 0.25s to 4.0s in increments of 0.25s) and a c r i t i c a l viscous damping r a tio of 0.05 was computed during the action of the stochastic e x c ita tio n ___________________________________________ 98 E.R.(k) = . 25E7 (a) - . 1E7 max . 25E7 - . 1E7 t t 0 m a x . (b) - . T E 7 max . 25E7. 20 (b) - JE7 max . 25E7 - . 1E 7 max [c) Figure 5.12. Group 1: Diagonal Comparison Between [CJ ] and [CJ] J = 1 k [CJ]; .........[ C ^ ; tC^’j - [CJ] -. 1E7 max Figure 5.13. Group 2: Diagonal Comparison Between [C ] and [C^] J = 2 99 - 5E6 J k = 10 . 5E6- k = 20 n/A>vVirrA'^ r vVVyV y l< A ^ V vs' ^ ' ^ v .• ■ -/-,v "vjv. w-, _ •*. f. .. - ,> - • w v . ** y> V v\ / V * ' ’ •- \ 1 Y <^V*' ■ > **% '— • * * '*•- V * t -. 5E6 *a) - . 5E6 (hi C t t • max 0 t t - max .5E6 j k = 30 .5E6 . k = 40 ■A '*-' . :' ’-vV.v' r\A’^ v ^ r ''X ^ v * V ~ '-V •''v-A*t * ■ T L '*--. v**-' ^ -.'•■-.y,-v ' » - .'5E6 (C) - 5Ffi (d) 0 E C x ° 1 ^ . 5E6 ^ k = 45 - 5E6 k = 50 -,vVr 1 (e) -. 5E6 ( f) . . . . . . y 0 max t t 1 max Figure 5.14. Group 3: Diagonal Comparison Between [C^] and [C^] k 100 (a) -.25 -.25 max max .25 —1 -.25 (b) (b) -.25- max max .25 V^V1 (c) -.25 max max (d) (d) Figure 5.16. Group 1: Analytic Fit of £; Figure 5.15. Group 1: Analytic F it . of £j (a) exact eigenvector _ p ; (b) approximate eigenvector £^; (c) error in f i t ; (d) Error ratio convergence 101 .25 - (a) m ax max 25 (b) -.25 -.25 max 25“ (c) max max (d) 10 20 30 40 50 10 20 30 40 50 Figure 5.17. Group 1: Analytic F it Figure 5.18. Group 1: Analytic Fi o f £ 3 o f £ 4 (a) exact; (b) approximate eigenvector ; (c) error in f i t ; (d) error ra tio c.o rute.cg.en.ee. 102 1 represented by [C ] in the follow ing manner: a. Numerical integratio n results were obtained using the exact 1 covariance m atrix [C ] from Equation (2.57) b. Numerical in te g ratio n results were also evaluated fo r [C^q] c. A nalytical results fo r the m.s. response were obtained using Equation (2 .7 0 ). The results of these calculatio ns are illu s tr a te d in Figure 5.19 for o s c illa to rs with natural periods of 0 .5 - 4 .0 ; 0 .5 s ; the rest o f the results are presented in appendix E. Results of comparable accuracy fo r group 2 can be obtained by using only four eigenvectors, as can be seen from the normalized error energy (see Figure 5 .4 ). While the use of th is approximation level y ield s poor results with high frequency o s c illa to r s , a very good approximation is achieved with Tow frequency ones. This is a ttr ib u ta b le to the concentration of high frequency content in the eigenvectors th a t were neglected in the recovering of the m atrix. Moreover, although the a n a ly tic a l results from fig u re part (c) poorly attempt to approximate results from part (b' s.. ^ . fo r low frequency o s c illa to r s , the two curves p r a c tic a lly coincide fo r high frequency o s c illa to r s . This is due to the fa c t th a t the n e g li gence of higher Chebyshev polynomial c o e ffic ie n ts , representing higher frequency content of the eigenvectors, have a noticeable e ffe c t on the high frequency o s c illa to rs only. 103 '5E4 5E3 (b) (a) max max 2E6 (d) (c) max max ■ 25E6 3.0s (f) (e) max max 4E6 4.0s 3.5s (h) ■ 3 t t max Figure 5.19. Group 1: Analytical and Numerical Results of Nonstationary m.s. Response (T^ =^.5s - 4.s;.5s and£ = .05) max 104 5.4 Use of Window in the Time Domain to Approximate the Covariance M atrix Records from nonstationary stochastic processes are obtained from sensitive measuring devices th at are triggered by low amplitudes o f the neasured quantity. Experimenters would also s ta r t t h e ir measurements some time before and a f te r the in te rv al of t h e ir concern. As a re s u lt, iiost o f the information is concentrated in a central part from the records and consideration of other parts in constructing the covariance matrix of the process is not advisable. For example, i t is seen from the sample records of group 2 and the corresponding covariance m atrix th at a large portion of the covariance matrix representing this group has n e g lig ib le amplitudes compared to those in the active portion. The use of a square window to zoom in on the core of the covariance m atrix w ill lead to great savings in storage and computer tim e, since a great reduction in the order o f the eigenvalue problem is at hand in th is case. Another advantage is expected in th is case by the use of a window which is the absence of the h a r d - t o - f it extremes from the eigenvectors (Figures 5-5 through 5 .8 ). 105 Chapter 6 SUMMARY AND CONCLUSIONS The use of orthogonal functions in the ch a ra cterizatio n and description o f random processes was investigated. Several approaches were presented in chapter 2 fo r the compact a n a ly tic a l representation af the covariance kernels of such processes. The method described in section 2.1 expands the covariance kernel into a f i n i t e and convergent sum of matrix-valued stages, and an a n a ly tic a l expression fo r the covariance kernel is obtained using a f i n i t e set of vectors determined from an eigenvalue form ulation. I t was shown in section 2.2 th a t th is set of vectors is the most e f f i c i e n t discrim inatory set o f axes and .the rate of convergence is the fa s te s t possible. The information given by any number of the highest eigenvectors and the corresponding eigen values is optimum compared to any other set of axes. The suggested compact representation leads to big savings in computer storage. The method discussed in section 2.3 uses the R itz-G a le rk in approach in conjunction with the Karhunen-Loeve expansion to f i t a complete set of vectors to the data-based covariance kernel. .The co e ffic ie n ts of the expansion are obtained from a minimization process requiring the solution of an eigenvalue problem o f an order lower than that of the covariance m atrix. The response o f lin e a r systems under Iche e ffe c t of such random processes characterized by th e ir covariance 106 xernel is investigated in section 2 .4 , and a r e la tiv e ly simple, closed- form expression with a convenient grouping of terms re fle c tin g the input and system ch a ra cteristics was obtained for the nonstationary response. The app lication of these methods to an a n a ly tic a lly defined covariance matrix that has been extensively used in a n a ly tic a l random v ib ratio n studies was considered in Chapter 3. This covariance matrix has a controlled degree o f sharpness in the d ire c tio n normal to the diagonal. Two processes of a r e la tiv e ly smooth and an extremely sharp (knife-edge) covariance kernel were considered. I t was demonstrated that the rate of convergence is much fa s te r fo r the smoother covariance m atrix. I f a large magnitude peak exists in both the diagonal and the normal d ire c tio n (pyramid shape), then a square window th at includes the peak may be considered as representative o f the process. This s itu a tio n was met in chapter 5 in which real data from a turbulent flow experiment were considered. Real earthquake data from the p a r tic u la r ly important earthquake at San Fernando were processed in Chapter 4. The covariance m atrix in th is case has a large number of peaks. A 25-term approximation was a good representation of the process in the sense th at approximate m.s. response calculations compared very well with the exact results for a wide range o f natural frequencies of lin e a r systems. The techniques described are quite general and powerful. The follow ing are some of th e ir important features: 1. The steps involved in determining the approximate response covariance are completely independent of the steps involved in 107 processing the e x c ita tio n . Once the input is characterized by th^ determined set o f c o e ffic ie n ts and the set o f selected functions, no fu rth e r modifications in these calculations are needed when investigating the response of d iffe r e n t o s c illa to r s . 2. Nonstationary random vib ratio n response calculatio ns are reduced to simple algebraic expressions, whose evaluation does not require the use of sophisticated techniques or fu rth e r approxima tions . 3. No assumptions are made regarding the nature of the covariance kernel so th a t completely general exc itatio n s may be considered. 4. The approximation introduced in the response calculatio ns involved the approximation of the input covariance. No subsequent approxi mations have been made in deriving the m.s. response. 5. The methods are ideal fo r feature e x tra c tio n , as the information given by any number of terms in the expansion fo r approximating the covariance kernel of the input is optimum. 6. I f a cro ss-co rrelatio n m atrix [S] is constructed fo r a given record _r, then, from the d e fin itio n of the cro ss -c o rre latio n matrix and from section 2 .3 , i t is easy to show th at the only non-vanishing eigenvalue o f [S] w ill be = r^ r and th a t the corresponding eigenvector w ill be p, = — r. 7. I f a covariance m atrix of order N was constructed using M < N records, then a g en eralizatio n of the above statement leads to AM+1 = AM+2 = = AN = ° ‘ 108 This fa c t was used to check the computer program in chapter 4 in which 66 records were used to construct [C] and Agy was found to be equal to zero. 8. The association of the higher eigenvalues with the lesser zero-crossings eigenvectors is an important feature as i t reduces the number of c o e ffic ie n ts th a t are required fo r the a n a ly tic a l representation of eigenvectors. 9. I t is easy to show, by simple physical reasoning, th a t the con trib u tio n o f each term in the representation of [C] (as in Equa tio n (2 .1 5 )) to the mean square response of lin e a r systems cannot be negative a t any moment in time. I t follows that i f the expansion is truncated at a level k, then the response is always underestimated i f / 0. 109 APPENDIX A PROBABILISTIC STRUCTURAL DYNAMICS 110 APPENDIX A PROBABILISTIC STRUCTURAL DYNAMICS A .1 Estimation of Covariance M atrix fo r aTNonstationary Random Process Consider a c o lle c tio n of N sample records x ^ t ) ; O s t s T ^ ^ i = 1, 2 , . . . ,N, from a nonstationary random process [ X ( t ) ] . An e s t i mate of the nonstationary mean at any time t can be calculated from Dx(t) j , *,<*> <A-1> Note th at the true mean is defined as: N p ( t ) = E [x . ( t ) ] = lim I x . ( t ) (A .2) 1 N-*» i=l 1 and that y „ ( t) is an unbiased estimated of u ( t ) for a ll t , indepen- X X dent of N . The variance of the estimate is 1 ° X ( t ) var[Ox(t)] - E[<Sx(t) - Mx( t ) P ] = (A.3) 7 where a ( t ) is the variance associated with the underlying nonstation- ary process [ x ( t ) ] (Bendat and P ie rs o l, 1971). Assuming N to be a s t a t i s t i c a l l y s ig n ific a n t number, we proceed to define the covariance function at a r b itr a r y fixed values of t-| and t 2 (in the range OS ^ ^ Tmax and OS t ^ S Tmax) t0 be: 111 cx(tr t 2) = E[{x^ti ) - - ux(t2)] = E [x(t])x(t2 )] - u x (ti)yx (t2 ) = Rx (t i> t 2^ " ^ x ^ l ^ x ^ ) vhere Rx ( t - j , t 2 ) is the nonstationary au to c o rre la tio n function o f the process which can be estimated from , N Rx^t l ’ t 2^ = N " i ?1 xi ^ ^ xi Ct2 ) (A .5) md an estimate o f the covariance function is obtained from cx ( t i ’V = ' ^ x ^ i ^ x ^ 1 ^ (A*6 ) A recommended procedure to evaluate the nonstationary covariance :ernel is to hold t-j fixed and vary t^ over the range o f in te re s t, l.et t-j = t and t 2 = t - x where x is a fixe d time delay value. Then 1 N Cx ( t , t - x ) J x . ( t ) x . ( t - x ) - ux ( t ) y x ( t - x ) (A .7) 112 For each fixed delay value t and each record x . ( t ) , calculate and store the product x.j ( t ) x . (t--x ). Repeat for a ll N records and perform the ensemble averaging to estimate Cx( t , t - x ) . The whole operation is repeated for every d iffe re n t t and x o f concern. Figures A.l and A.2 illu s tr a te a recommended procedure (Bendat and Pierson, 1971) for the measurement of nonstationary mean and autocorrelation functions, respectively. Covariance function com putation is then calculated using (A.6). A.2 Input-Output Relation of Linear MDOF Systems Consider a lin e a r multi-degree-of-freedom (MDOF) dynamic system governed by the matrix d iffe r e n tia l equation: x ( t) = [ A ( t ) ] x ( t ) + [ B ( t ) ] y ( t ) , (A .8) where x ( t) is the state vector whose i n i t i a l condition is xU q) = Xq. The solution of Eq. (A .8) is: t x ( t ) = C<i>(t, t p ) H^g + [ [<p ( t , T ) ] [ B ( t ) ] u ( t ) d t , (A .9) t 0. where [<j>] is the system tra n s itio n matrix. For tim e-in variant sys tems, the general solution of Eq. (A .9) sim plifies to: x ( t) = eAtxQ + / eA^ ' T^[B]y(x)dT (A .10) t 0 Thus, i f xQ = 0, the covariance matrix for the response: W V = EC x(t1 )xT( t 2 )] 113 Multiple store memory Ensemble averaging circuit / Add and \ \divide by N) Figure A . I . Procedure fo r Nonstationary Mean Value Measurement *; (t - t) Multiplier Time delay generator, r Multiple store memory Ensemble averaging circuit Figure A .2. Procedure for Nonstationary Autocorrelation Measurement 114 can be related to the e x c ita tio n covariance matrix: where t-j and t^ are w ithin the range in which the input process ( u ( t ) ) is defined I t is clear from Eq. ( A . I I ) that once the covariance matrix for the input process (U) is defined, the covariance of the output process (X) of a given system can be found in a straightforw ard manner by performing the indicated operations. However, while there are no conceptually challenging problems in implementing Eq. (A. 11) operations, there are some p ractical problems which lim it the u t i l i t y of th is formal procedure in typcial app lica tions encountered in an a lytic a l structural dynamic studies dealing with earthqakes (or s im ila r processes). W e note the following: 1. Although the a v a i l a b i l i t y in the recent past of a large number of recorded earthquake ground motion (Hudson, 1976, 1979)made it possible to determine a data-based covariance m atrix corresponding to a s t a t i s t i c a l l y s ig n ific a n t ensemble of c ertain classes of earth quakes in certain s ite s , so fa r no larg e-scale uniform processing of the basic data to provide much s t a t i s t i c a l l y useful representation has been carried through. 2 . Some earthquake simulation studies have resulted in models of various complexity which can approximate seismic exc itatio n s and can be used d ir e c tly in Eq. (A .11) to compute the response covariance. 115 However, many approximate analytical representations of the excitation covariance kernel, when used in Eq. (A.1 1 ), may require the use of numerical integration and may also involve a s ig n ific a n t amount of e ffo r t to .develop closed-form expressions for the response covariance. An indi,cation o f the e ffo r t involved, despite customary sim plifying assumptions regarding the form of the excitation covariance kernel, may be found in the nonstationary random vibration studies of Fung (1955), Bogdanoff and Goldberg (1959), Caughey and Stumpf (1961), Lin (1967), Barnoski and Maurer (1973), Vanmarcke (1976), Masri (1978), and Gasparini and DebChaudhury (1980). 116 APPENDIX B COMPUTER IMPLEMENTATION APPENDIX B COMPUTER IMPLEMENTATION This appendix describes the following programs: 1. SEC21: An implementation o f the method presented in Section 2.1 . 2. SEC23: An implementation o f the method presented in Section 2.3 . 3. MSRSP: An implementation o f the calcu latio n o f m.s. response and spectra of lin e a r systems. All programs are w ritten in FORTRAN language. A very b r ie f description o f each program is given. 3.1 Program SEC21 Program SEC21 is a FORTRAN program th a t implements the method af Section 2 .1 . I t performs the following steps: 1. - Reads input data 2. Reads an existin g [C] or generates i t as defined in Chapter 3 , 3. Computes desired eigenvalues and corresponding eigenvectors. 4. Calculates the theo re tic a l bounds fo r the normalized erro r energy 5. Evaluates convergence o f [C^] to [C ] 6. Computes Chebyshev c o e ffic ie n ts fo r the f i t of eigenvectors. * 7. Evaluates convergence o f [C^] to [C ] An organization chart and a b r ie f description of the program -'ollow 118 SEC21 FORMC HEEB IN DAT CHECK2 EHVLOP TRED1 T I N V I T X N R M L SERIES C H E B Y C H B F I T ■ j E Q L I N C P O L C H B F I N D C 1 FLAGR Figure B.1 . General Organization Chart fo r Program SEC21 119 SORT R U R I T 1 FIT FIT! 80 ' COVX80 WINDOW P L T M O D EIG180 RWRIT6 C O V N FITMOD RWRIT3 S O M E I G C H E C K ! TRBAKl T R I D I B Table B .l. B rief Description of Subprograms Used by Program SEC21 120 No-.- Subprogram Name Needed Subprogram Purpose 1 SUB.CHBFIT FINDC!,EQLINC,XNRML Performs Chebyshev f i t of a data vector with series up to a given order 2 SUB. CHEBY — Performs Lagrangian interpolation, computes coefficients arid converts Chebyshev series to its equivalent power series 3 SUB.CHECK! - Constructs [C^] for desired levels of k and computes the actual normalized error, energy 4 SUB.CHECK2 SERIES,POLCHB Constructs a matrix [Cfc] from the analytical expressions for k eigenvectors (stored in a disk f il e ) to compare with [C] 5 FUNC.COVN Computes the auto-correlation function of a random Gaussian sampled data of fin ite dura tion; given the power spectral density of the process 6 FUNC.C0VX80 COVN,ENVLOP,WINDOW Computes a covariance matrix [C] of a stochastic process of the type defined: in Chapter 3 7 SUB.EIG180 RWRIT3 ,SOMEIG Reads or computes and writes required eigen values and corresponding eigenvectors of[C ] 8 FUNC.ENVLOP - Defines an exponential envelope h -» Ho h ~ * 9 SUB.EQLINC SORT,FLAGR Uses Lagrangian interpolation of a given order to redefine the given vector at equal increments of a given length No. Subprogram Name Needed Subprogram Purpose 10 SUB.FINDC1 Integrates a tabulated function to find Chebyshev series by using Simpson's rule n SUB.FIT CHBFITyXNRMLjSORT, EQLINC,CHEBY,SERIES Performs one-dimensional f i t of a tabulated function (not necessarily at equal increments) 12. SUB.FIT.180 RWRIT6,FITM0D Reads or computes and writes Chebyshev coeffi cients for the f i t of selected eigenvectors 13 SUB.FITMOD FIT Selects eigenvectors to be fitte d and calls f i t 14 SUB.FLAGR Evaluates an interpolating polynomial of a given degree and argument using the tabulated vector 15 SUB.FORMC RWRIT1,C0VX80 Generates^ covariance matrix [C] of the type defined in Chapter 3 or reads an arbitrary covariance matrix from an existing f ile 16 SUB.INDAT - Reads input data and flags from an existing disk f ile 17 SUB.NEEB - Computes theoretical error bounds for all stages of expansion based on knowing some eigenvalues 18 SUB.PLTMOD - Plots desired eigenvectors h-* ro ro 19 FUNC.POLCHB - Evaluates Chebyshev polynomial of given order and argument Name Subprogram Name Needed Subprogram Purpose 20 SUB.RWRIT1 - Reads or writes dbvariance matrix [C ] 21 SUB.RWRIT3 - Reads or writes eigenvectors 22 SUB.RWRIT6 - Reads or writes Chebyshev coefficients for the f i t of k eigenvectors 23 FUNC. SERIES Evaluates a one-dimensional Chebyshev series, given the coefficients and an unnormalized argument 24 SUB.SOMEIG TRED1,TRIDIB,TINVIT, TRBAK1. Calculates a given number of eigenvalues and corresponding eigenvectors of a real symmetric matrix starting at a given index, 25 SUB.SORT - Sorts two vectors in ascending values 26 SUB.TINVIT An EISPAK* routine: Finds those eigenvectors of a tridiagonal symmetric matrix correspond ing to specified eigenvalues, using inverse iterations 27 SUB.TRBAK1 An EISKPAK routine:* Forms the eigenvectors of a real symmetric matrix by back-transforming those of the corresponding symmetric tridiagonal matrix determined by TRED1. 1 — js y CO * B. T. Smith et a l . , Matrix Eigensystem Routines, New York: Springer-Verlag,. 1974). EISPACK Guide, 2nd ed. (Berlin, Heidelberg, Name Subprogram Name Needed Subprogram Purpose 28 29 SUB.TRED1 SUB.TRIDIB 30 FUNC.WINDOW 31 SUB.XNRML An EISPAK routine: Reduces a real symmetric matrix to a symmetric tridiagonal matrix using orthogonal sim ilarity transformations An EISPAK:routine: Finds those eigenvalues of a tridiagonal symmetric matrix between specified boundary indices, using bisection Computes a rectangular window Normalizes a data vector to lie in the interval between -1 and +1 ro B .2 Program SEC23 Program SEC23 is a FORTRAN program that implements the method of Section 2 .3, I t performs the following steps:: 1. Reads input data 2. Reads an existin g [C] or generates i t as in Chapter 3. 3. Constructs [G] and [M] using given base functions (V.s) in an an alytic form 4. Solves the eigenvalue problem [M]a = A[G] j* 5. Determines the shape functions ( V s ) from (2.55) 6. Evaluates the convergence of [C^] to [C] An organization chart and-a b rie f description of the program follow; 125 ( SEC23 ) ENVLOP F I N D G M FWRJ INOAT FORMC READ1 COVX8D W I N D O W F I N D G G j F G R J F I N D A V COVN [ I G E N N O R M S H J A C O B I 4 j G M P R D | PLTPHI S H A P E COMPFN . BESJ POL-CHB PLTMOD G J P L T 3 F O R M C H H COVXHT G M S U B CONVRG GMNORM F O R M E P / " Y Figure B.2. General Organization Chart fo r Program SEC23 126 Table B.2. B rief Description of Subprograms Used by Progam SEC23 127 No. Name Needed Subprogram Purpose 1 SUB.BESJ - Evaluates Bessel function of a given order at a given argument value 2 FUNC.COMPFN BESJ,P0LCHB Analytically defines the base functions ( xp' s) 3 SUB.CONVRG GMNORM.FORMEP Evaluates the convergence of the analytic expres sion -for the. covariance matrix to the exact one 4 FUNC.COVN Computes the auto-correlation function of random Gaussian sampled data of fin ite duration; given the power spectral density of the process 5 FUNC.C0VX80 ENVLOP, WINDOW, COVN Computes covariance matrix [ C ] o f a nonstationary process of the type1 defined in Chapter 3 6 FUNC.COVXHT SHAPE Constructs the approximate analytic matrix at given arguments 7 SUB.EIGEN JACOBI,GMPRD Solves an eigenvalue problem 8. FUNC.ENVLOP - Defines an exponential envelope 9 FUNC.FGRJ C0X80,C0MPFN Evaluates an element of [G] 10 SUB.FINDAV FINDGM,FINDGG, EIGEN Constructs and solves Ma = A[G]a ro ro 129 Needed No • Name Subprogram Purpose 11. SUB. FINDGG FGRJ, Constructs [G] Constructs [M] Evaluates an element of [M] 12. SUB.FINDGM FMRJ 13. FUNC.FMRJ 14. SUB. FO RM C C0VX80 15. SUB.FORMCH COVXHT 16 SUB.FORMEP FORMCH.GMPRD 17 SUB.GJPLT3 18 SUB.GMNORM 19 SUB.GMPRD 20 SUB.GMSUB 21 SUB.INDAT 22 SUB.JACOBI 23 SUB.NORMSH 24 SUB.PLTPHI SHAPE,GJPLT3 Generates a covariance matrix as defined in Chapter 3 or reads an a rb itra ry existing one. Forms [C^] Forms the error function matrix ([ C] - 2-dimensional p lo ttin g Computes the norm of a general matrix General matrix m u ltip licatio n General matrix subtraction Reads input data and flags Solves an eigenvalue problem by the Jacobi method Orthonormalizes a given vector No. Name Needed Subprogram Purpose 25 SUB. PLTMOD GJPLT3 Plots the (a 's ) i f desired 26 FUNC.POLCHB - Evaluates a Chebyshev polynomial of given order and argument 27 SUB.READ1 Reads eigenvectors (as base vectors) resulting from program SEC21 NOTE: This procedure was used for test purposes 28 FUNC.SERIES POLCHB Evaluates a one-dimensional Chebyshev series given the co efficien ts and an argument that is not normalized 29 FUNC.SHAPE COMPFN Numerically evaluates the a n a ly tic a lly defined vectors (< |> !s) 30 FUNC.WINDOW - Computes a rectangular window CO o B.3 Program MSRSP Program MSRSP evaluates numerical in te g ra tio n re s u lts o f the m.s. response o f lin e a r damped systems to a stochastic process ch a ra c te r ized by a given covariance m atrix [C ], or by Chebyshev c o e ffic ie n ts re s u ltin g from the f i t o f eigenvectors. The program is also capable o f c a lc u la tin g the spectra o f the process. The program does the fo llo w in g : 1. Reads input data co n sisting o f: a . Linear system c h a ra c te r is tic s , desired values o f dam ping,ratios, and natural frequencies b. Covariance m atrix i f numerical • in te g ra tio n resu lts are desired c . Chebyshev c o e ffic ie n ts corresponding to eigenvectors to be used together w ith the eigenvalues d. Flags 2. I f normal in te g ra tio n re s u lts are d esired , EXACT is c a lle d to loop over the given values o f damping ra tio s and natural periods, and to c a lc u la te the spectra o f the process 3. I f a n a ly tic a l re s u lts corresponding to (2 .7 0 ) are d e s ire d , ANLTC is c a lle d upon to loop over the given values o f damping ra tio s and natural p erio d s, and to c a lc u la te the spectra o f the process. An o rgan ization chart and a b r ie f d escrip tio n of the program f o l 1o w .. 131 covy Figure EXACT T D N T R P Y I J EAD GJPLT3 ANLTC SPECT B .3 . General O rg anization Chart fo r Program MSRSP 132 Table B.3. B r ie f D escription of Subprogram Used by Program MSRSP 133 No. Name Subprograms Called Purpose 1 SUB.ANLTC YIJ A n aly tically evaluates the m.s. response of damped systems from (2.70) a fte r reading eigen values and Chebyshev coefficien ts for the f i t o f corresponding eigenvectors 2 SUB.COVY - Computes the response covariance matrix of a damped SDOF lin e a r system 3 SUB.EXACT TDNTRP,C0VY,H1, H2,H3 Finds the exact m.s. response o f lin e a r sys tems under a stochastic process of given covariance matrix using d irect numerical integration 4 FUNC.FI - Evaluates (2.64) given an argument and a subscript 5 FUNC.H1 6 7 FUNC.H2 FUNC.H3 ) > To avoid re p e titiv e calculation of integration regions those functions are used only to compute regions not covered in the previous step 8 SUB.SPECT Finds the extreme values of response for each call to exact for given damping ra tio and natural period, and plots the spectra 1 — GJ -P > No. Name Subprograms Called Purpose 9 FUNC.TDNTRP _ Performs lin e a r two-dimensional interpolation given corner values to redefine [C3 NOTE: In a ll cases considered in this study, exact points o f the covariance matrix were used; no interpolation of [C] was performed. 10 SUB.YLJ FI Evaluates (2.69) at given arguments and subscripts O O On APPENDIX C ADDITIONAL INFORMATION RELATED TO CHAPTER 3 136 (a) (b) 20.1 (c) 20.1 20 (e) (f) 20’. 1 (g) (h) 20.1 Figure C .l. Case l'.Exact ( £ .) , Analytically Approximated (_£.), and Error Ratio versus Number of Coefficients; i = 9 - 1 2 £ . ( t ) ; ------£ . ( t ) 137 .2 .2 20.1 1.0 E.R. (b) 0.0 N 2 (c) .2 20.1 .1 t a (d) 0.0 10 20 15 5 .2 .2 20.1 .1 t Figure C.2. Case 1: Exact ( £ .) , Analytically Approximated (p^), and Error Ratio versus Number of Coeffiicents; i = 13 -15; £ . ( t ) ; . . . £ , ( t ) 138 ■ 1 s t, S 20. 1; i = 1,2 k = 1 5 Figure C.3. Case 1: Normalized Error Function ([C] - [Ck] ) Surface Scaled by the Peak o f [ C] . (See Figure 3.1 for magnitude appreciation) CO I5E"4" IE-2 75s 25s (b) 1.75s 1.25s (c) 2.25s 2.75s i f ) 3.25s 3.75s (h) Figure C.4. Case 1: (k = 5) m.s. Response of SDOF Harmonic Oscillators (T = .25s - 3.75s; ,5s and £ = .05) [C]; ...........[C, ] ; .............[C, ] 140 15E4 75s 25s (b) (a) 1.75s 1.25s (c) 2.75s 2.25s (f) (e) 3.75s 3.25s M Figure C.5. Case 1: (k = 10) m.s. Response of SDOF Harmonic Oscillators T = .25s - 3.75s; .5s and E , = .05) n [C] 141 . 15E-4 25as (a) 20 t 0 J E - 2 75s (b) .06 1.75s (d) 0 0 20 t 01 1.25s (c) 0 20 0 t 25 2.25s 0 0 20 t 6 2.75s i f ) 0 20 0 t 1 3.25s 3.75s (h) 0 20 0 t Figure C.6. Case 1: (k = 15) m.s. Response of SDOF Harmonic Oscillators (T = .25s - 3.75s; .5s and £ = .05) n EC]; ....... (ck3 ; -----------[Ck] . 3E-2 . 2 E-3 (b) 20 12 1.5s (d) 0 0 0 20 0 20 t t 8 2.5s 3.0s i f ) (e) 0 0 20 t 1. 2.2 3.5s 4.0s (h) 0 0 20 20 t 0 t Figure C.7. Case 1: (k = 15) m.s. Response of SDOF Harmonic Oscillators (T = .5s - .4s; ,5s and B , =.05) n [C] ; CCk] ; [Ck] 143: Figure C.8. Case 2: Additional Eigenvectors (£.) 7 Figure C.9. Case 2: Normalized Error [C] - [C, ] Surface Scaled by the Peak of [C] (See Figure 3 J 4 fo r magnitude appreciation) H — » -p* c n .6E-3 - 04 75s (b) 0 0.15 1.25s (d) (c) 0 rr» T»|. 20 0 20 t 0 t 0.8 .> 2.75s 2.25s .(O (e) 20 0 20 0 t t 2.0 3.25s 20 0 t .0 3.75s 0 20 0 t Figure C.10. Case 2: Nonstationary Response of SDOF Harmonic Oscillators wi th £ = .05 r r c ■ ■ c -. 25’ — 50’ ----------75’ -------------100' 1146 * 1 1 (b) 1 TO M 1 1 1 (d) 1 10.1 1 ( f ) -1 .1 10.1 t 1 (e ) 10.1 1 .1 10.1 t 1 (h ) 1 Figure C -.ll. Case 3: A n a ly tic a lly Defined Eigenvectors <p. , i = 9 - 1 6 ' 1 147. 0 . 2 E T 3 J 1 .Os (b) (a) 20 0 20 t 0,12 0 .0 3 - 1.5s (d) (c) 20 20 0 t 0.4J 0. 3.0s 2.5s (f) (e) 20 0 20 t 0 t 2 . 2 0 0 20 20 0 t 0 t Figure C.12. Case 1: Nonstationary M.S. Response Under the Action of [C ] (£ = 0.05);____[C ];--------[C ] 1.48 0.2E-3 0.5s (a) 0 0.3E-2 1 .Os (b) 0 0 t 0.03 (c) 0 t 0.12 2.0s (d) 0 t 0.4 2.5s (e) 0 .8 3.0s (f) 0 2 0 0 t 20 2.2 (h) Figure C.13. Case 1: Nonstationary M.S. Response Under the Action of CCD;. CC1Q]( C = 0.05); [c io] 149' 0.2E-3 3E-2 i 1 .Os (a) (b) 0.03 1.5s (c) (d) 0 0 t 20 t 20 0.4 ( f) (e) 2 . 2 ' (h) 20 Figure C.14. Case_1: Nonstationary M.S. Response Under the Action APPENDIX D ADDITIONAL INFORMATION RELATED TO CHAPTER 4 151 Table D . l . L is t o f San Fernando Earthquake Records that Used to Generate [C] were Log No. Address ........... . Location Other Identifying Numbers ..................1 ..................................... ' 2 U D O O • p ''-' Los Angeles 1901 Avenue of the Stars Sub-basement N 46°W S* = 7.6 cm/g S 44°W S* = 7.6 cm/g 71.007 Castaic Old Ridge Route N 21° E S =8.1 cm/g N 69°W S* = 7.6 cm/g 71.009 Lake Hughes Array Sta. 12 N 21° E S* =7.6 cm/g N 69°W S* = 7.6 cm/g 71.012 Lake Hughes Array Sta. 9 N 21°E S* = 7.6 cm/g N 69° W S* = 7.6 cm/g 71.018 Pasadena Caltech Seismological Laboratory South S* = 1.9 cm/g East S* = 1.9 cm/g 71.019 Pasadena Caltech Athenaeum East S = 1.82 cm/g North S a 1.79 cm/g 71.022 Pasadena Caltech Mi 11ikan Library East Basement S* = 1.9 cm/g North S* = 1.9 cm/g 71.024 Los Angeles 15250 Ventura Blvd. Basement N 11°E S =1.95 cm/g N 79°W S - 1.78 cm/g 71.032 h -+ in s Pasadena Jet Propulsion Laboratory Basement S 8 L° W S* =1.9 cm/g S 82° E S* = 1.9 cm/g Table D.l (Cont) Log No. Address Location 0ther Identifying Num bers 1 -2 71.036 Los Angeles 3838 Lahkershim Blvd. Sub-basement North S* = 1.9 cm/g West S* = 1.9 cm/g 71.046 Los Angeles 3710 W ilshire Blvd. Basement West S* = 7.6 cm/g South S* = 7.6 cm/g 71.048 Los Angeles 4680 W ilshire Blvd. Basement N 15°E . S* =7.6 cm/g N 75°W S* =7.6 cm/g 71.053 Los Angeles 4867 Sunset Blvd. Basement S 89°W S* =7.6 cm/g S Q1°E S* = 7.6 cm/g 71.056 Los Angeles 3470 W ilshire Blvd. Sub-basement North S* =7.6 cm/g West S* =7.6 cm/g 71.059 Los Angeles Water & Power Bldg. Basement N 5Q°W S* =7.6 cm/g S 40°W S* =7.6 cm/g 71.060 Los Angeles 445 Figueroa St. Sub-basement N 52°W S* = 7.6 cm/g S 38°W S* = 7.6 cm/g 71.062 Santa F elicia Dam Outlet Works S 82°W S* = 7.6 cm/g S 08° E S* = 7.6 cm/g 71. 065 Lake Hughes Array Station 4 S 21° W S* = 1.9 cm/g S 69° E S* = 1.9 cm/g Table D.l (Cont) Log No. Address Location Other Id en tifyin g Numbers 1 2 71.150 Arcadsia Santa Anita Reservoir N 03°E S* = 7.6 cm/g N 87 °W S*= 7.6 cm/g'.' 71.066 Carbon Canyon Dam Carbon Canyon Dam S 40°W S* =1.9 cm/g S 50°E S* = 1.9 cm/g 71.069 Los Angeles G r iffith Park Observatory Moon Room South S* = 1.9 cm/g West S* = 1.9 cm/g 71.096 Los Angeles 3407 Sixth St, Basement South S* =7.6 cm/g East S* = 7.6 cm/g 71.102 Glendale Municipal Service S 70°E S* = 15.2 cm/g S 20°W S* = 15.2 cm/g 71.105 Los Angeles 2011 Zonal Ave. Basement S 28°W S* = 7.6 cm/g S 62°E S* =7.6 cm/g 71.108 Los Angeles 3345 W ilshire Blvd. Basement South S* = 7.6 cm/g East S* = 7.6 cm/g 71.119 Palos Verdes Estates 2516 Via Tejon S 25°E S* =3.8 cm/g N 65°E S* =3.8 cm/g 71.126 Los Angeles 1800 Century Park East Basement (P-3) S 36°E S = 1.98 cm/g N 54°E S = 1.80 cm/g .01 4 5 » Table D.l (Cont) Log No. Addresss Location Other Id en tifyin g Numbers . . . . 1 ’ ...... 2 71.129 Los Angeles 2500 W ilshire Blvd. Basement N 29°E S = 1.96 cm/g N 61 0 W S = 2.00 cm/g 71.135 Los Angeles 15910 Ventura Blyd. Basement S 09°W S =2.00 cm/g S 810 E S = 2.09 cm/g 71.152 Lake Hughes Array Sta. 1 N 21° E S* =7.6 cm/g S 69° E S* = 7.6 cm/g 71.156 Los Angeles Hollywood Storage Basement East S = 13.3 cm/g South S = 12.4 cm/g 71.175 Fairmont Reservoir N 34°W S = 17.4 cm/g N 56° E S = 16.7 cm/g 71.153 San Dimas Puddingstone Reservoir N 55°E S* = 7.6 cm/g N 35°W S*= 7.6 cm/g *Nominal • s e n s itiv ity . I— 4 CJ1 U1 -90 i— ex o o -250 r o 250 TIME (Seconds) Fiqure D .l. Sample Acceleration (in cm/sec/sec) Record from San Fernando Earthquake * » . J . _ ^ . /U\ 71 1 no rnmnnnonf cnilth .25 -.25 .25 .25 P3(t ) V * ) (a) -.25 20s 0 .25 ,v ^ ( t ) (b) 1.0 ~~r~ 20s -.25 .25 ^ (t) (c) 20s -.25. E.R. (d) E.R. 0 (a) 20s (b) M — — |*- 2Cs (d) 10 N 160 10 N Figure D.2. Eigenvector No. 3 Figure D.3. Eigenvector No. 4 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio versus number of coefficients '1 5 7 ~ r led .25- Vtr - . 2 5 .25 %<t} (b) - . 2 5 - . 2 5 .25 ^ I p ' ^ (t) (c) E.R. - . 2 5 1.0 I E.R. (d) 0 10 N 160 . 10 (b) ' r 20s (c) 20s (d) N Figure 0.4. Eigenvector No. 5 Figure D.5. Eigenvector No. 6 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio vs no. of coefficients 160 158 M y W 20s .0 E.R 0 Figure D.6 . Eigenvector No. 7 Figure D.7.. Eigenvector No. 8 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio versus no. of coefficients 159 • .25 -.25 -.25 ' I >’» ■ > ■ » w w-v 'l*** 20s 20s .25 .25 (b) (b) t.25 -.25 20s 20s .25 .25 -9 -.25 -.25 20s 20s (d) 160 Figure D.8. Eigenvector No. 9 160 no. of coeffi icents .25 .25 -.2 5 20s .25 -.25 -.25 20s .25 -n -.25 20s 20s 160 10 N 160 Figure D . n . Eigenvector No. 12 Figure D.1Q Eigenvector No. 11 161 no. of coefficients .25 £l3(tK -.25 U .25 A . . £ u (t) -.25' (b) e,o(t) % 4«*> -.25 20s 0 .25 - |- r fT f I r r * , rr| r . o r prrr p rw | « m r. | T T r. |» t 20s .25 (c) * ( . ' ' - " I > . p n . |» 1.0 E.R. 20s (d) 10 N Figure D.12. Eigenvector No. 13 160 , 10 N 160 Figure D.13. Eigenvector No. 14 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio versus no. of coefficients 162 0 t 20s 0 t 20s .25 .25 -16 -.25 -.25 20s 1.0 1.0 R (d) (d) 0 160 10 N Figure D.14. Eigenvector No. 15 Figure D.15. Eigenvector No. 16 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio versus no. of coefficients 163 V t} ' ■Vt} — 18 e , , ( t ) E.R. E.R. 0 (d) 0 (d) 10 N 1 G O Figure D.17. Eigenvector No. 18 10 N 160 Figure 0.16 Eigenvector No. 17 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio versus no. of coefifcients 164 .25 £ 19( ‘ > -.25 .25 .25 f,V ^W £20{t) -.25 "20s 0 .25 — r 20s £1 9 < t > -.25 e, „ (t )' -.25 ** r". '20s 0 t .25 —f.-.*i •—■[- 20s £2o^^ E.R. 0 1,0 E.R. 0 10 N 160^ 10 N Figure D.18. Eigenvector No. 19 Figure D.19. Eigenvector No. 20 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio versus no. of coefficients i- 160 165 f / £-22 1 .25 .25 (a) '20s -.25 .25 £21(t) -.25 .25 f t 2.2 2 ( t ) <b) »tr)f-r.. rrt» * V ir . . , « •* ,, . • . . t . 0 t 20s 21 ( t ) -.25 . 1.0 -.25 .25 % (t) (c) *r*T ,> I-’ * - ’*- - 20s E.R. -.25 1.0 E.R. (d) 0 (a) 20s (b) 20s 7,‘irh!' (c) 20 (d) 10 N Figure D. 20. Eigenvector No. 21 160 10 N 150 Figure 0.2! Eigenvector No. 22 (a) exact; (b) least-squares f i t ; (c) error in f i t ; (d) error ratio versus no. of coefficients 166 e„„(t) fpV‘|*T ^ T ^ 10 N 160 Figure D.23.- Eigenvector No. 24 1 0 N 16 0 Figure D. 22.. Eigenvector No. 23 (a) exact; (b) least-squarees f i t ; (c) error in f i t ; (d) error ratio versus __no. of. coefficients 167 -J Figure 0.24. Nonstationary m.s. Response of Harmonic^Oscillators (T - .25s - 2 .s ; .25s and £ • 0) in cm n__________________ _— — . — — — --------------- ----------------- 160 E[y ( t ) (b) 20s 20s 155-' 260 " = 2.75s E[y (t)3 (d) 20s 340 " 380 = 3.25s ety‘ (t); 20s 20s 450 - 467 j 20s I 3 t 20s 0 t Figure D.25. Nonstationary m.s. Response of Harmonic .psciliators APPENDIX E ADDITIONAL INFORMATION RELATED TO CHAPTER 5 170 .25 * t max (b) -.2 5 25 0 t t max 25 .25 (c) (c) 25 -.2 5 0 t 0 t, t t, max ina x Figure E .l.^ Eigenvector No. 5 or Figure E.2. Eigenvector No. 6 of! [CJ ] , J = 1, 2, and 3; tCu] , J = 1, 2, and 3 171 : .25 (a) ( a ) -.25 -.25 max max .25 .25 (b) (b ) -.25 -.25 max max 25J .25 (c) (c) -.25 ■ .25 max m ax figure £.3. Eigenvector No. 7 of fC ] Figure E.4. Eigenvector No. 8 of [C ] 172 .25 (a) (a) -.25 0 t t max max .25 - (b> (b) -.25 -.25 t t o max max (c) -.25 -.25 0 t 0 t t, max max Figure E.5. Eigenvector No. 9 for [C^] Figure E.6. Eigenvector No. 10 for [CJ] J = 1, 2, and 3 J = 1, 2, and 3 173 .25 5 (a) (a) -.25 -.25 t t 0 t 0 t max 25 5 (b) (b) -.25 -.25 t t 0 t 0 t max max .25 ..25 (c) -.25 -.25 t 0 t max max .R. (d) (d) 0 Figure E.8. Group 1: Analytic Fit Figure E.7. Group 1: Analytic Fit for £ 5 2 , (a) exact eigenvector £•; (b) approximate eigenvector^ .(c) error in f i t ; (d) error ratio convergence , 174 25 75s 25s (b) 0 0 t t max max 1.75s 25s (d) (c) t 0 t max max 2E5 1E6 2.75s 2.25s •'« • ** # I* < t S 2 u V* d **» * ' $y? (f) 0 max max . 5E6 .3E6 3.75s (h) (g ) 0 t t max Figure E.9. 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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Traina, Mohamed-Idris M (author)

Core Title Orthogonal representation of random processes

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, civil,OAI-PMH Harvest

Language English

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-40378

Unique identifier UC11345867

Identifier DP22172.pdf (filename),usctheses-c17-40378 (legacy record id)

Legacy Identifier DP22172.pdf

Dmrecord 40378

Document Type Dissertation

Rights Traina, Mohamed-Idris M.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA