Chaotic characteristics of electromagnetic emission signals during deformation and fracture of coal

°, Science DirectMININGSCIENCE ANDTECHNOLOGYELSEVIERMining Science and Technology 19(2009)0189-0193w.elsevier. com/locate/jcumtChaotic characteristics of electromagnetic emission signalsduring deformation and fracture of coalNIE Bai-sheng"2, HE Xue-qiu 2, LIU Fang-bin2, ZHU Cheng-wei 2, WANG Ping.2School of Resource and Safery Engineering, China University of Mining Technology, Beijing 100083, chinaState Key Lab of Coal Resources and Safe Mining, China University of Mining Technology, Beijing 100083, ChinaAbstract: Electromagnetic emission(EME) is a kind of physical phenomenon accompanying the process of deformation and fracure of loaded coal and rock and it is of importance in quantitatively analyzing its characteristics. This will reveal the process ofdeformation and fracture of coal and predicting dynamic disasters in coal mines. In this study, the G-P(Grassberger and Procacciaalgorithm, calculation steps of the(if only I dimension)correlation dimension of time series and the identification standards ofchaotic signals are introduced. Furthermore, the correlation dimensions of EME and the acoustic emission(AE)signals of timeseries during deformation and fracture of coal bodies are calculated and analyzed. The results show that the time series of pulsesnumber of EME and the time series of AE count rate are chaotic and that the saturation embedding dimensions of a k3 coal sampleare,respectively, 5 and 6. The results can be used to provide basic parameters for predicting of EME and AE time series.Keywords: coal and rock; electromagnetic emission; correlation dimension; chaotic characteristics1 Introductiontion of time variables that decide the long-term evo-lution of a system, all contain information onElectromagnetic emission(EME) is a kind of long-term evolution of all variables of a system.physical pheneon accompanying deformation and Therefore, the chaotic behavior of a system can befracture process of loaded coal and rock that might explored by any single variable of a time series thatbecome a new method for predicting dynamic coal decides the long-term evolution of a system.Chaosand rock disasters. At present, EME signals rules of and fractal theories were employed to probemmmm吗 EME characteristics during the process of defocoalcomplex pulse signals, questions arise how to distin-guish EME signals rules from these complex signals 2 Correlation dimension and calculationand use them in time series analysis, important tofurther enhancing prediction accuracy of dynamicSince the concept of fractals was proposed bydisasters. Nonlinear science, which is a discipline Mandelbrot in the 1970s, this theory has experienceddesigned to study common properties of nonlinear rapid development. The core of fractals isphenomena, has a broad field of application. The sci- self-similarity and its quantifiable characteristic isence comprises theories of dissipative construction, fractal dimension. There are a variety of methods tocoordination, mutation, chaos dynamics, fractals and figure out fractal dimensions, such as the Hausdoffso on. In particular, chaos dynamics has been or be- dimension, self-similarity dimension, box dimensiongins to be widely applied in many fields, such as in information dimension, multi-fractal dimension, tothe chaotic changes in acoustics, optics, turbulence name but a few. In our study, the correlation dimenand chemical reactions, as well as the chaotic charac- sion of EMe time series is introduced and calcu-teristics of earthquakes and"The Butterfly Effect"in lated! -s1.weather forecasting. The chaos studied by means ofThe fractal characteristic of time series signals cantime series analysis began with the theory of phase- beed hv the correlatinn dimension. The timespace reconstruction, suggested by Pachard Evolu- series中国煤化工tobe:Received 30 July 2008: accepted 20 October 2008CNMHGProjects 50427401 supported by the National Natural Science Foundation of China, and 2006BAK03 B06 by the National Eleventh Five-Year Key ScienceTechnology Project of ChinaCorresponding author. Tel: +86-10-82375620: E-mail address: bshnie@cumt. edu. crMining Science and TechnologyVol 19 No. 2x1,2:,……x(1 formation between phase pairs cannot be measuredAn m-dimensional sub-phase space is constructed. according to Eq (5), where C(r)=l and InC(r-oThen the first m data are taken according to thisWhen r is selected properly within some intervalC(r)=rwhere D is called the correlation dimension. And thenThe first point in this m-dimensional space is esD=Inc(r)tablished as 5. Removing x,, m data are taken inturn again asIn actual numerical calculations, r is usually as-2,巧,耳4(3) signed a specific value(small enough). If r is toosmall, i. e, lower than the vector difference caused byThe second point in the m-dimension is similarly environmental noise and measurement error, the calestablished as r2. Thus, a large number of points can culation using Eq (9)does not obtain the correlationdimension, but instead, the embedding dimension. Inpractice, m is usually valued from small to large,while D remains unchanged, i.e., the tangential path(4) in the double logarithm relation is In C(r)-InrTheslope of the best-fitted straight line is D, except whenThen a path curve is made by connecting, in turn,the slope is 0 or oo. Therefore, a scale limit is imposethese phase points f, 2, 5,",r, where m is in both the large and small directions when a scaletransformation is carried out for the actual system. Ifknown as the embedding dimension. Here, the em- the limit goes beyond this imposed scale that calledbedding dimension can be calculated from the corre- the non-feature scale area and eventually the problemlation between these phase points. It is known that, essentials cannot be reflected. Therefore, Eq- (9)isthe nearer the distance between phase points, the larger the extent of their correlation. Suppose Nmeaningful only on the non-feature scalepoints are created by a time series in3 Chaotic characteristics of acoustic emis-m-dimensional space, l.e., '2sion and EMEan arbitrary number r is established and for any dotpair (, r)the distance Ir -r, I is checked 3.1 Calculation steps of correlation dimensiongainst r, in particular whether this distance is lessand identification standards for chaotic timethan r. The proportion of dot pairs whose distance isless than r of the total number of dot pairs N is The question arises whether the time series of thetaken as C(r). The proportion can be expressedsignals of EME and acoustic emission(AE), releasedc()=1∑(1r-1Dduring the deformation and fracture of coal and rock,(5 are indeed chaotic time series We can use the corre-lation dimensions of the ae and eme time seriessignals as a basis for decision. In 1983, Grassbergerwhere a(x)is the Heaviside unit functionand Procaccia proposed the G-p( Grassberger and>0Procaccia's abbreviation) algorithm by which the6(x)(6) correlation dimension of a time series is calculatedx≤0he main steps of G-P algAny vector whose distance is less than r is called a t)Given a time series *, <2, x,,", x. and acorrelation vector, so C(r) represents the proportion small value mo, reconstruct a phase space accordingof all correlation vectors of all possible pairs Nto eqs. (2)to(4)Because calculating the distance Ir-r I by2)Calculate the correlation function according toEq2(5)Euclidean distance involves a large workload, it is3)When r is in the proper range, the dimension Dseldom used in practice. Actually, any distance that and the cumulative distribution function C(r) shouldsatisfies the axiom of distance is applicable. We de- satisfy a logarithmic linear relationship. Thus the es-fined the maximum component difference between timated yalue of the correlation dimension that cor-two vectors as the distance中国煤化工Ir-r -uIpeatCNMHnsion m>mo, re-nding estimatedIf the r selected is too large, the distances of all dot value of dimension D(m)no longer increases with anairs will not exceed it and thus the correlation in- increment of m over a specified error range. At thisNIE Bai-sheng et alChaotic characteristics of electromagnetic emission signalpoint, D is the correlation dimension of this time se- and the time series of AE count rate(received by 50ries that has chaotic characteristic If D increases with kHz ae probe)are reconstructed phase spaces, ac-the increment of m but does not converge to a stable cording to the steps in section 3. 1. The one-dimen-value, it indicates that this system is a random time sional time series are transformed into m-dimensionalsenesphase spaces. The relation In c(r)-ln r is presented in3.2 Calculation results and analysisFig. 2 where m takes on the values 2, 3, 4, 5, 6,7,8, 9and 10, respectively. Then the tangential path wasx the EME and AE signals of a k3 coal sample un- fitted and the correlation dimension D calculated. Theer uniaxial compression were empirically tested by relation of D and the embedding dimension m isan electromagnetic emission experiment system'given in Fig 3.The results are shown in Fig. 1. The time series ofpulses number of EME (received by 50 kHz antenna)50 kHz日0412345.6(a) Pulses number of EME(b)Ring count rate of AEFig 2 InC(r)-lnr relation EME and AE signal series of K3 coal sample(The curves are from left to right: m=2, 3, 4, 5, 6, 7,8,9, 10)03004028(a) Pulses number of EME(b)Ring. down count rate of AeFig 3 m-D relation of the time series of EME and ae signal of 3 coal sampleAs can be seen from Fig 3, when m, the pulses known as the"saturated embedding dimensions".Atnumber of the Eme series is 5 and the count of the thisAE time series is 6, the correlation dimension no 0.27中国煤某化二series of pulseslonger increases with an increment in the embedding numbCN MH Gunt rate form thedimension. In that case, these embedding dimensionanted in finite timeare precisely the required embedding dimensions,Mining Science and TechnologyVol 19 No. 23.3 Fractal feature of EME and ae calculated bhe Hurst indexX(n,k)=E(x()-(X))(1snskwith the Hurst index, Wang et al. calculated the From Eq (10), with the aid of a logarithm trafractal dimensions during the deformation and fracformation we can obtainture of coal and rock-2. In order to verify our testresults, EME signals of the K3 coal sample were callog(R(k)/S())= Hd log k +loga(11)culated and verified by using the Hurst indexwhere a is a constant. i. e the hurst index H is theWhen investigating time series signals, Hurst dis- tangent of the relational curve of log(R(k)/S(k))andrelation between the range R(k)and standard devia- dimension D is 2- uction, it follows that the fractalcovered that for a time series (x(O, t=1, 2, .., N), the log k. From this detion s() is as followsBy fitting the EME and ae signals of the K3 coalR(k)/S(k)∝kb(10) sample, the Hurst index of ae signals is 0.9853 andthat of EME signals 0.9368, as shown in Fig. 4.here r(k)=max X(n, k)-min X (n, k)Therefore, the fractal dimension of AE signals is1.0147 and that of EME signals 1.0632. According tothe interpretation of the Hurst index, when H>1/2, thes()yk2(x()-(x2y2time series are not mutually independent but corre-lated instead, showing a growth trend. Therefore, thewhere (x)is the mean value of thesenes,EME and ae signals basically tend to increase duringdeformation and fracture of coal. Thus, by verifX(n,k) is the accumulated deviation of the time the hurst index, the time series of the Eme and AEsenessignals can be predictedX)4=∑x()(k=1,2,…,Mlog(RS=0.9853logk-0.0406log(RS=0.9368logk-00228051.5kFig 4 Fitting the Hurst index curve of EME and ae signals of the K3 coal sample4 Conclusionsthe accuracy of the calculations mentioned above1)The G-P algorithm, the calculation steps of cor- Acknowledgementsrelation dimension of time series and the identification standards of chaotic signal are introduced.Financial support for this work, provided by the2)By reconstructing phase spaces for the time se- National Natural Science Foundation of Chinaries of the pulses number of the EME and the AE (No 50427401 ), the National Eleventh Five-Year Keycount rate, the correlation dimension can be calcu- Science Technology Project(No 2006BAK03B06),lated, We have indicated that the two time series are the New Century Excellent Talent Program from thechaotic and that the saturation embedding dimensions Ministry of Education (No. NCET-07-0799), the Fokof the K3 coal sample are 5 and 6. These results can Ying-Tong Education Foundation for Young Teachersbe used to provide basic parameters for predicting in Higher Education Institutions of China(No. 111053EME and Ae time seriesand3)The Hurst indices of Eme and ae of the coal PlanV凵中国煤化工mywsample are calculated, which show that the EME andCNMHGAE signals tend to increase continuously, confirmingNIE Bai-sheng et al712-716References[7] Yi C X Nonlinear Sciend Their Applications inGeoscience. Beijing: China Meteorological Press, 1995.[] Wang E Y Srudy on Electromagnetic and Acoustic Emis- [8] LuJH, Lu JN, Chen S H Chaotic Time Series Analysission Efect during fracture of Coal or Rock Containingand Application. Wuhan: Wuhan University Press, 2002:Gas and its Application. Xuzhou: China University of11-71.(In Chinese)Mining and Technology Press, 1997.(In Chinese)[9] Grassberger P, Procaccia L. Measuring the strange at-[2] Nie B S Study on the Efecr of Stress and Electricity andactors. Physica D, 1983(9): 189-208dissertation]. 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