Earthquake Barcode from a Single-Degree-of-Freedom System

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1 Natural Sciece, 015, 7, Published Olie Jauary 015 i SciRes. Earthquake Barcode from a Sile-Deree-of-Freedom System Chei-Sha Liu 1, Chih-We Cha 1 Departmet of Civil Eieeri, Natioal Taiwa Uiversity, Taipei, Taiwa Cloud Computi ad System Iteratio Divisio, Natioal Ceter for Hih-Performace Computi, Taichu, Taiwa liucs@tu.edu.tw Received 4 November 014; revised 1 December 014; accepted 9 December 014 Copyriht 015 by authors ad Scietific Research Publishi Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). Abstract Earthquake is a violet ad irreular roud motio that ca severely damae structures. I this paper we subject a sile-deree-of-freedom system, cosisti of spri ad damper, to a earthquake excitatio, ad meawhile ivestiate the respose behavior from a ovel theory x = f x, t, by viewi the time-varyi sium fuctio of about the dyamical system ( ) f x ( f x). ( x( t), x ( t )) ad the secod compoet f = u ( ) t ζωx( t) ωx( t) of f, where u ( ) It ca reflect the characteristic property of each earthquake throuh t is a time-sampli record of the acceleratio of a roud motio. The barcode is formed by plotti Si : = si f x f x with respect to time. We aalyze the complex jumpi behavior i a ( ( ) ) barcode ad a essetial property of a hih percetae occupatio of the first set of dis-coectivity i the barcode from four stro earthquake records: 1940 El Cetro earthquake, 1989 Loma earthquake, ad two records of 1999 Chi-Chi earthquake. Throuh the comparisos of four earthquakes, we ca observe that stro earthquake leads to lare percetae of the first set of dis-coectivity. Keywords Earthquakes, Sile-Deree-of-Freedom System, Sium Fuctio, Barcode, Jumpi Behavior, The First Set of Dis-Coectivity, Scale Ivariace 1. Itroductio Earthquakes occur aroud the world frequetly. The most are so small that they caot be susceptible, ad oly How to cite this paper: Liu, C.-S. ad Cha, C.-W. (015) Earthquake Barcode from a Sile-Deree-of-Freedom System. Natural Sciece, 7,

2 C.-S. Liu, C.-W. Cha a few are lare eouh to cause a oticeable damae of buildi. Whe a structure is subjected to a earthquake motio, its base foudatio teds to move with the roud. Oe of the most importat applicatios of structural dyamics is by aalyzi the respose of structures to roud shaki caused by a earthquake, whose data are useful for establishi the structural respose spectra. The idetificatio ad evaluatio of roud motio parameters require access to the measuremets of stro roud motios i a actual earthquake [1]-[3]. The basic elemet of a acceleroraph is a trasducer elemet as described i [4] [5], which i its simplest form is a sile-deree-of-freedom mass-damper-spri system. Typically, the trasducer has atural frequecy f = 5 Hz ad dampi ratio ζ = 60% for moder aalo acceleroraphs, ad f = 50 Hz ad ζ = 70% for moder diit acceleroraphs [4]. A simple statioary represetatio of earthquake-iduced roud acceleratio was proposed by Kaai [6] ad Tajimi [7], based o the study of frequecy cotet of a umber of stro roud motio records. They suested that the roud acceleratio of the earth surface layer could be approximated by the absolute acceleratio of a simple oscillator with a cocetrated mass supported by a liear spri ad a dashpot. The spectrum property of a earthquake is reflected i its iteractio with the structures which have differet dampi ratios ad atural periods. I eeral, oe ca use the respose spectrum, power spectrum or the Fourier spectrum to characterize the spectrum property of a earthquake. I this paper we propose a ovel method to characterize the earthquake property whe it is exerted o a simple oe-deree-of-freedom system: where p( t) u ( t) ( ) + ω ( ) + ( ) = ( ) xt ζ xt ω xt pt, (1) = is the roud acceleratio due to a earthquake; ζ is the dampi ratio; ad ω is the atural frequecy. The roud motio produced by earthquake ca be very complicated. The exact form of earthquakes is subjected to a hih deree of ucertaity ad complexity. There are several ways to estimate the roud motio parameters, which are essetial for describi the importat characteristics of stro roud motio i a compact quatitative form. Because of the complexity of the earthquake roud motio, the idetificatio of a sile parameter that ca accurately describe all importat roud motio characteristics is impossible. I this paper we try to use the barcode ad the percetae of the first set of dis-coectivity i the barcode to reflect the characteristic property of a earthquake roud motio. It is kow that a barcode is a optical machie-readable represetatio of data relati to the object to which it is attached. A mai feature of the barcode is the iterveed black lies ad white lies with varyi spacis ad widths. Barcode is ubiquitously used i the idetificatio ad classificatio of huma made products. Here we will use the barcode to idetify the property of earthquake with the iput p( t) u ( t). A System Formulatio = i Equatio (1). To facilitate the formulatio we write a system of first-order ordiary differetial equatios (ODEs): Here =, = ( xy, ) = ( xx, ) ( t) x = f x,, t R, x R. () = y ad ( ) f = p t ω x ζω y, respec- x, ad two compoets of f are f 1 tively. As that doe by Liu [8], for Equatio () we ca defie a uit orietatio vector: : = x, x where x : = x x > 0 is the Euclidea orm of x, ad the dot betwee two vectors, say x y, deotes the ier product of x ad y. The we ca derive [9]-[11] f x f x x f x = x+, x (4) x x x x x d dt x x = f x. (3) (5) 19

3 C.-S. Liu, C.-W. Cha Equatios (4) ad (5) ca be put toether as ( ) f x x f f x x 3 d x x x x x x x =, T dt x ( f x) x x 0 3 x which is a Lie-type system edowi with the Lie-roup symmetry SO o (, 1) i the Mikowski space. (6) 3. Two Braches Solutios I order to develop a umerical scheme from Equatio (6), we suppose that the coefficiet matrix is costat with the pair f x, =, x x ( ab) bei costat i a small time stepsize. From Equatios (6) ad (7) we thus eed to solve a costat liear system: Let ad Equatio (8) becomes d x a b b a a bb x =. T dt 0 x a bb x z = a x, w= b x, y = x, (9) c = ab (10) 0, x = wa zb+ cyb (11) 0. 0, y = cw (1) At the same time, from the above equatios we ca derive the followi ODEs for z, w ad y: a = a Fortuately, the oriial ( ) z c0 a0 c 0 z d w 1 c0 c 0 w, dt = y 0 c0 0 y where dimesioal problem i Equatio (8) ca be reduced to a threedimesioal problem i Equatio (13). Depedi o the sium fuctio of ( ) ( a0 c0 ) ( ) (7) (8) (13) Si : = si = si x f f x, (14) there exist two differet types solutios of (z, w, y). Here, we do ot ive a detailed derivatio of the solutios for (z, w, y) ad the correspodi secod eeratio roup preservi scheme (GPS), but the reader ca refer [9]-[11]. 4. The Demostratio of Barcodes It is siificat that i Equatio (14) we have derived a sium fuctio to demad the alorithm ito two braches. Without havi the factor before f x i Equatio (14) oe has ( ) ( θ ) f x f x = f x 1 cos 0, where θ is the itersectio ale betwee x ad f; hece, it makes o sese to defie its sium fuctio. O the cotrast, by Equatio (14) we have 0

4 ( ( )) θ ( θ) C.-S. Liu, C.-W. Cha Si = si f x 1 cos = si cos, (15) which may be +1 or 1, depedi o x ad f by their itersectio ale θ. Whe θ is i the rae of π < θ < π or 3π 4< θ < 5π 4, the value of Si is Si = 1; otherwise, Si = +1. Thus we ca observe the time-varyi values of Si ad plot them as a barcode. I order to demostrate the use of barcode, let us ivestiate the followi examples. I all computatios ive below the iitial coditios are set to be x 0, x 0 = 0,0, uless specifyi otherwise. ( ( ) ( )) ( ) 4.1. Example 1 First we test a example with ω = ad ζ = 0.15, ad a harmoic force p(t) = cost is applied to Equat 0,100. The iput, the followi sium fuctio: tio (1) i a time iterval of [ ] { } 1 1 ( t) = f ( t) + f ( t) + x ( t) + y ( t) x( t) f ( t) + y( t) f ( t) Si si, (16) where f ( t) = y( t) ad f ( t) pt ( ) ω xt ( ) ζω yt ( ) 1 =, ad the resposes of x(t) ad y(t) are plotted i Fiure 1. It is iteresti that after 0.78 sec, Si = +1 for the harmoic iput. As compared with the closedform solutio, i Fiure 1(d) we plot the umerical error, which is quite accurate, ad is smaller tha the time stepsize h = used i this computatio. Fiure 1. Uder (a) a harmoic force, showi (b) the sium fuctio, (c) the resposes, ad (d) umerical error of displacemet. 1

5 C.-S. Liu, C.-W. Cha 4.. Example The with ω = 3 ad ζ = 0.1 3, ad uder a radom force ( ) 5 R, p t i = (17) where R i are radom umbers betwee [ 1, 1], the iput, the Si ad the resposes are plotted i Fiure. It is iteresti that the values of Si are radomly jumpi betwee +1 ad 1 i some itervals of time Aalyzi the Barcode uder Earthquakes With ω = π 1.5 ad 0.0 u t, which is adjusted to p( t) = 0.98u ( t) m sec as the iput to Equatio (1), the barcode ad the resposes are plotted i Fiure 3. As show i Fiure 3(b), the time history of Si looks like a barcode, which is iteresti that the values of Si are frequetly jumpi betwee +1 ad 1 with a certai structure. Now we aalyze the Si ad the leth as show i Fiure 4 withi a short time iterval of t [ 0, ], which is starti from the iitial coditios ( x( 0 ), x ( 0) ) = ( 0.1, 0.01), where we take ω = ad ζ = 0.01 ad p( t) = 50u ( t) bei the iput. First we ote that the leth as overed by Equatio (5) has the followi property: ζ =, ad uder the El Cetro earthquake ( ) i Fiure. Uder (a) a radom force, showi (b) the sium fuctio, ad (c) the resposes.

6 C.-S. Liu, C.-W. Cha where si ( c 0 ) = si ( ) ( ) < 0 Fiure 3. Uder (a) the El Cetro earthquake, showi (b) the barcode, ad (c) the resposes. < d < si ( c0 ) 0 x 0, (18) > dt > f x. The, as a schematic plot show i Fiure 5, there are two dis-coected sets of f x f x i the hyper-plae (x, f): 1 f x > f x, (19) 1 f x< f x. (0) We may call them the first set of dis-coectivity ad the secod set of dis-coectivity, respectively. Clearly, the first set of dis-coectivity is a subset of si(c 0 ) = +1, ad the secod set of dis-coectivity is a subset of si(c 0 ) = 1. If Equatio () satisfies 3

7 C.-S. Liu, C.-W. Cha Fiure 4. Uder the El Cetro earthquake, (a) showi two sium fuctios, ad (b) the leth with a small time iterval. Fiure 5. A schematic plot of the area of Si = +1, Si = 1, si(c 0 ) = +1, ad si(c 0 ) = 1 i the plae. ( ( ) ) ( ) ( ) ( ) ( ) f x t, t x t > 0, ad si f x f x = 1, t t, the x t, t. (1) Uder the first assumptio the case i Equatio (0) is impossible because it cotradicts to ( ) f( x( t0), t0) x ( t0) > 0. The uder the coditio of ( ) si f x f x = 1, it is always 1 f x > f x, t t 0, () 4

8 C.-S. Liu, C.-W. Cha because the two sets of dis-coectivity are dis-coected, ad from the first set to the secod set it must o to ( ( ) ) si f x f x = + 1 i some time iterval. The usi Equatios (5) ad () we have d 1 x > f > 0, (3) dt which meas that the leth rows with time. Thus, Equatio (1) is prove. For a system bei excited by earthquake the value of Si is ot always +1 as that show i Fiure 1(b) for the harmoic iput. As show i Fiure 3(b) the values of Si are varyi betwee 1 to +1 ad +1 to 1 very fast, ad the the Si will retur to +1 aai as show i Fiure 4(a); otherwise, by Equatio (1) the system will respod ustably ad thus the displacemet will ted to ifiity. I order to compare the values of si(c 0 ) with the values of Si we plot them i Fiure 4(a) with solid lie ad dashed lie, respectively, where we make a sliht shift of the dashed lie dowward for clear. O the other had, for a distictio the first set of dis-coectivity is filled by solid black poits. As show i Fiure 5, i the plae the areas of si(c 0 ) = +1 ad si(c 0 ) = 1 are coected while that the sets of Si = +1 ad Si = 1 are dis-coected as metioed i the above. Whe si(c 0 ) chaes from +1 to 1, for example from poit f 1 to poit f i the fiure, the Si must jump from 1 to +1. This jumpi behavior is termed a cross jumpi. O the other had, we also fid the co-jumpi behavior which is happeed whe both the Si ad si(c 0 ) are jumpi from 1 to +1 simultaeously. The above two jumpi behaviors are collected as a coupled-jumpi, which reders a quite complex structure of the barcode as show i Fiure 3(b). From Fiure 4(a) we ca observe the followi iteresti pheomea: 1) While the first set of dis-coectivity is a subset of si(c 0 ) = +1, the secod set of dis-coectivity is a subset of si(c 0 ) = 1. ) I a time iterval of the state with Si = 1, the state is either i the first set of dis-coectivity or i the secod set of discoectivity. 3) From oe first set of dis-coectivity to aother first set of dis-coectivity there must accompay a jump from Si = 1 to Si = +1. This also holds for the secod set of dis-coectivity. 4) From oe first set of dis-coectivity to oe secod set of dis-coectivity, or vice-versa, there must accompay a jump from Si = 1 to Si = +1. For some cases this jumpi oly happes at oe time poit. 5) Whe si(c 0 ) jumps from +1 to 1 as remarked i Fiure 4(a) by the symbols + ad (i.e., the leth is decreased), ad if the Si is i the state of 1, the the Si will jump from 1 to +1 as remarked i Fiure 4(a) by the symbols ad +. Ideed i Fiure 4(a) we ca observe that there are four times to happe the cross jumpi behavior i the iterval of t [ 0, ]. Besides the first oe, the other three times are marked by the Arabic umbers, 3 ad 4 i Fiure 4(a). Besides, there is also a co-jumpi as remarked i Fiure 4(a). The proof of 5) is obvious by viewi Fiure 5 that the first set of Si = 1 caot directly jump to the secod set of Si = 1. Whe f 1 oes to f, the Si chaes from 1 to +1. I Fiure 6 we compare the barcodes with 1) ω = π 1.5, 0.0 ω = π 1.5, 0.0 ζ = ad ( ) ( ) ad 4) ω = π, 0.0 p t = 4.9u t m sec, 3) ω = π 1.5, 0.05 ζ = ad ( ) ( ) p t = 0.98u t m sec The First Set of Dis-Coectivity uder Earthquakes ζ = ad ( ) ( ) p t = 0.98u t m sec, ) ζ = ad ( ) ( ) p t = 0.98u t m sec, I this example we let the Chi-Chi earthquake be the iput to Equatio (1). With ω = π 1.5 ad ζ = 0.0, ad uder the Chi-Chi earthquake u ( t) recorded at the statio of CHY08, we adjust it to be p( t) = 0.98u ( t) m sec as show i Fiure 7(a). The barcode ad the resposes are plotted i Fiure 7(b) ad Fiure 7(c). I Fiure 8 we compare the barcodes with 1) ω = π 1.5, ζ = 0.0 ad p( t) = 0.98u ( t) m sec, ) ω = π 1.5, 0.0 ζ = ad ( ) ( ) ad 4) ω = π 0.5, 0.0 p t = 4.9u t m sec, 3) ω = π 1.5, 0.05 ζ = ad ( ) ( ) p t = 0.98u t m sec. ζ = ad ( ) ( ) I Fiure 9(a) for the Chi-Chi earthquake with differet scali factor f r i p( t) fu ( t) p t = 0.98u t m sec, = 9.8 r as bei the iput to Equatio (1), ad with ω = π 1.5 ad ζ = 0.0, we plot the percetae i the first set of dis-coectivity i Fiure 9(b), which is ear to forty eiht percetae, ad at the same time we also display the umbers 5

9 C.-S. Liu, C.-W. Cha Fiure 6. Uder the El Cetro earthquake, compari the barcodes uder differet scali factor, dampi ratio ad atural period. of coupled-jumpi as discussed above with respect to the scali factor f r i Fiure 9(c). It ca be see that the umber is either 4 or 5. I the plae as show i Fiure 5, the first set of dis-coectivity with Si = 1 is oly i oe quarter. But uder a stro earthquake, the hih occupatio of the first set of dis-coectivity is ear to fifty percetae as show i Fiure 9(b). I Fiure 10, uder the Chi-Chi earthquake (CHY080) as show i (a), we plot the first set of dis-coectivity which is ear to forty ie percetae as show i Fiure 10(b), ad the umber of coupled-jumpi is plotted i Fiure 10(c), which is either 6 or 7. I Fiure 11(a) for the El Cetro earthquake (ELC180) with differet scali factor f r i p( t) = 9.8 fu r ( t), ad with ω = π ad ζ = 0.0, we plot the first set of dis-coectivity which is ear to forty six percetae as show i Fiure 11(b), ad the umber of coupled-jumpi is plotted i Fiure 11(c), which is 0 or 1 for most scali factors. I Fiure 1(a) for the Loma Prieta earthquake (47379 Gilroy Array # 1) with differet scali = 9.8 r, ad with ω = π ad ζ = 0.0, we plot the first set of dis-coectivity which is ear to forty six percetae as show i Fiure 1(b), ad the umber of coupled-jumpi is plotted i Fiure 1(c), which is betwee 1 to 7. It is surprisi that the percetae of the first set of discoectivity ad the umber of coupled-jumpi are almost costat for each earthquake, which seems to be scale-ivariat. I the plae as show i Fiure 5, the factor f r i p( t) fu ( t) 6

10 C.-S. Liu, C.-W. Cha Fiure 7. Uder the Chi-Chi earthquake (CHY08) i (a), (b) showi the barcode, ad (c) the resposes. first set of dis-coectivity with Si = 1 is oly i oe quarter. But uder stro earthquakes, the hih occupatio of the first set of dis-coectivity is ear to fifty percetae. The hih percetae of occupatio demostrates the importace of the first set of dis-coectivity i the stro earthquake motio. As compari the above four earthquakes we ca also observe that stroer earthquake leads to larer percetae of the first set of dis-coectivity. Fially, uder the same scali factor f r = 0.5 i p( t) = 9.8 fu r ( t) ad the same ω = π ad ζ = 0.0, i Fiure 13 we compare the relative percetaes of first set of dis-coectivity for the El Cetro earthquake (ELC180), the Loma Prieta earthquake (47379 Gilroy Array #1), ad two records (CHY08, CHY080) of Chi-Chi earthquake. The relative percetae is a time history of the time-duratio of the first set of dis-coectivity, whose time is dividi by the total time of earthquake. This fiure reflects that the Chi-Chi earthquake has larer values tha that of the El Cetro ad the Loma Prieta earthquakes, ad CHY080 is larer tha CHY08. This ca also be explaied by calculati the power of earthquake with the -orm of earthquake, of which the power value of CHY080 is , of CHY08 is ad of the El Cetro earthquake is The Loma Prieta earthquake (47379 Gilroy Array #1) leads to a close curve with that produced by the El Cetro earthquake, whose power is Coclusios For a sile-deree-of-freedom system subjected to a earthquake excitatio, we have ivestiated the respose 7

11 C.-S. Liu, C.-W. Cha Fiure 8. Uder the Chi-Chi earthquake (CHY08), compari the barcodes uder differet scali factor, dampi ratio ad atural period. by viewi the sium f x f x The mai idea i this paper is that the earthquake will make the fuctio f hihly complex i time, eve the liear system is simple i x. Uder this situatio, we expect that the sium fuctio bei a time varyi fuctio will disclose the complexity of earthquake, such that we ca study its structure. The barcode was formed by plotti the sium fuctio Si with respect to the time history of a earthquake. We have aalyzed the jumpi behavior from two famous earthquake iputs of 1940 El Cetro earthquake ad 1999 Chi-Chi earthquake. Several barcodes were used to reveal a quite complex structure of earthquake, ad we have demostrated the importace of the first set of dis-coectivity, whose hih percetae occupatio is very iteresti whe oe cosiders the liear system uder a stro earthquake. Throuh the comparisos of four earthquake iputs, we have observed that stroer earthquake leads to larer percetae of the first set of dis-coectivity. The coupled-jumpi behavior may happe due to the radom property of behavior from a ewly developed theory about the dyamical system of x = f( x,t) fuctio of ( ). 8

12 C.-S. Liu, C.-W. Cha Fiure 9. Uder (a) the Chi-Chi earthquake (CHY08) with differet scali factor, showi (b) the percetae i the first set of dis-coectivity, ad (c) the umber of coupled-jumpi. Fiure 10. Uder (a) the Chi-Chi earthquake (CHY080) with differet scali factor, showi (b) the percetae i the first set of dis-coectivity, ad (c) the umber of coupled-jumpi. 9

13 C.-S. Liu, C.-W. Cha Fiure 11. Uder (a) the El Cetro earthquake (ELC180) with differet scali factor, showi (b) the percetae i the first set of dis-coectivity, ad (c) the umber of coupled-jumpi. Fiure 1. Uder (a) the 1989 Loma Prieta earthquake (47379 Gilroy Array #1) with differet scali factor, showi (b) the percetae i the first set of dis-coectivity, ad (c) the umber of coupled-jumpi. 30

14 C.-S. Liu, C.-W. Cha Fiure 13. Compari the relative percetaes of first set of dis-coectivity for the El Cetro earthquake (ELC180), two records (CHY08, CHY080) of Chi-Chi earthquake, ad the Loma Prieta earthquake (47379 Gilroy Array #1). earthquakes, ad its umber seems scale-idepedece. This is just a iitial study by correlati the percetae of the first set of dis-coectivity to the earthquake, ad there are still may works to study the barcode ad the use of it to forecast the itesity of earthquake before the rapid comi of mai shock impulse. I the future, we will aalyze the sial of barcode for may earthquake records, ad thus we ca correlate the itesity ad frequecy cotet of a earthquake to the first set of dis-coectivity. Refereces [1] Loh, C.H., Lee, Z.K., Wu, T.C. ad Pe, S.Y. (000) Groud Motio Characteristics of the Chi-Chi Earthquake of 1 September Earthquake Eieeri ad Structural Dyamics, 9, [] Sokolov, V.Y., Loh, C.H. ad We, K.L. (00) Compariso of the Taiwa Chi-Chi Earthquake Stro-Motio Data ad Groud-Motio Assessmet Based o Spectral Model from Smaller Earthquakes i Taiwa. Bulleti of the Seismoloical Society of America, 9, [3] Sokolov, V.Y., Loh, C.H. ad We, K.L. (003) Evaluatio of Hard Rock Spectral Models for the Taiwa Reio o the Basis of the 1999 Chi-Chi Earthquake Data. Soil Dyamics ad Earthquake Eieeri, 3, [4] Chopra, A.K. (1995) Dyamics of Structures: Theory ad Applicatios to Earthquake Eieeri. Pretice-Hall, New Jersey. [5] Kramer, S.L. (1996) Geotechical Earthquake Eieeri. Pretice-Hall, New Jersey. [6] Kaai, K. (1957) Semi-Empirical Formula for the Seismic Characteristics of the Groud. Bulleti of the Earthquake Research Istitute, Uiversity of Tokyo, 35, [7] Tajimi, H. (1960) A Statistical Method of Determii the Maximum Resposes of a Buildi Structure duri a Earthquake. Proceedi of the d World Coferece o Earthquake Eieeri, Japa, [8] Liu, C.-S. (001) Coe of No-Liear Dyamical System ad Group Preservi Schemes. Iteratioal Joural of No-Liear Mechaics, 36, [9] Liu, C.-S. ad Jhao, W.S. (014) The Secod Lie-Group SO o (,1) Used to Solve Ordiary Differetial Equatios. Joural of Mathematic Research, 6, [10] Liu, C.-S. (014) Disclosi the Complexity of Noliear Ship Rolli ad Duffi Oscillators by a Sium Fuctio. CMES: Computer Modeli i Eieeri & Scieces, 98, [11] Liu, C.-S. (015) A Novel Lie-Group Theory ad Complexity of Noliear Dyamical Systems. Commuicatios i Noliear Sciece ad Numerical Simulatio, 0,

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