Modeling Acoustic Emission Signals Caused by Leakage.pdf

J Nondestruct Eval (2013) 32:6780DOI 10.1007/s10921-012-0160-x

Modeling Acoustic Emission Signals Caused by Leakagein Pressurized Gas Pipe

Saman Davoodi Amir Mostafapour

Received: 26 May 2012 / Accepted: 19 July 2012 / Published online: 9 November 2012 Springer Science+Business Media New York 2012

Abstract Leakage in high pressure pipes creates stresswaves which transmitted through the pipe wall. These wavescan be recorded by using acoustic sensor or accelerometerinstalled on the pipe wall. Knowing how these waves vi-brate pipe is very important in continuous leak source lo-cating process. In this paper the pipe radial displacementcaused by acoustic emission due to leakage is modeled an-alytically. The standard form of Donnells nonlinear cylin-drical shell theory is used to derive the motion equationof the pipe for simply supported boundary condition. Us-ing Galerkin method, the motion equation has been solvedand a system of nonlinear equations with 7 degrees of free-dom is obtained. A MATLAB code according to Runge-Kutta numerical method is generated to solve these equa-tions and derive the pipe radial displacement. To check thetheoretical results, acoustic emission testing with continu-ous leak source and linear array of two sensors positionedon two sides of the leakage source were carried out. Themajor noise of recorded signals was removed through thewavelet transform and filtering technique. For better analy-sis, fast Fourier transform (FFT) was taken from theoreticaland de-noised experimental results. Comparing the resultsshowed that the frequency which carried the most amountof energy is the same that expresses excellent agreement be-tween the theoretical and experimental results validating theanalytical model.

Keywords Acoustic emission Wavelet transform Donnells nonlinear theory Galerkin method

S. Davoodi () A. MostafapourMechanical Engineering Department, University of Tabriz,Tabriz, Irane-mail: [email protected]

1 Introduction

In the fields of micromechanics and seismology the defor-mation such as micro cracks has been formulated analyt-ically. Leak detection is one of the most important prob-lems in the oil and gas pipelines, where it can lead to fi-nancial losses, severe human and environmental impacts.The technique to monitor defects and abnormal vibrationsdue to machine failures is vitally important for the safetyof structures in the modern society. Acoustic emission (AE)has drawn a great attention because of its applicability toon-stream surveillance of structures. One important point isthe capability to acquire data very simply but with high sen-sitivity. The applicability is limited partly because the ac-curacy of solutions depends on the noise levels and partlybecause the phenomenon is usually irreproducible [1]. Pol-lock [2] has discussed the parameters which affect AE wavepropagation in structures. He noted that in gas filled vessels,attenuation can be very low and a number of Lamb wavemodes are experimentally observable. In liquid filled ves-sels the increased attenuation of signals due to its effect onthe waveform and the timing of the peak made source lo-cation more difficult. Elforjani and Mba [3] applied the AEtechnology to detect natural crack initiation and propagationon slow speed bearings which is one of the few publicationsthat address natural mechanical degradation on rotating ma-chine components. Shehadeh and et al. [4] concentrated onthe temporal aspects and on determining the arrival timesof propagating waves generated from a simulated source,hence identifying the AE wave speeds. Maji [5] has dis-cussed AE wave propagation in plates and beams. He notedthat it becomes more difficult to extract the arrival time of in-dividual frequency components from broad band transduc-ers and that it is preferable to use resonant transducers sothat specific frequencies dominate the recorded AE events.He found dominate peaks in the AE signals at around 100

68 J Nondestruct Eval (2013) 32:6780

and 160 kHz. The AE energy is partly reflected and trans-mitted when it encounters a boundary between the steel pipewall and the fluid contained in the pipe. The partitioning be-tween transmitted and reflected waves depends on the angleof incidence and relative material acoustic impedances. So ifthe two materials are well matched in acoustic impedance,a large portion of energy will transmitted [6]. Fuller et al.[7] have also studied the free waves in fluid-filled cylindri-cal shells and noted that behavior of these waves depends onthe thickness of wall shell and the ratio of shell material den-sity to contained fluid density. Acoustic emission waves inlarge structures propagate in more than one mode with dif-ferent travelling speed. So the determination of arrival timesfor different modes and their rates of attenuation are im-portant. Watanabe et al. [8] have located pin-hole sourcesusing auto-correlation and cross-correlation methods, andthey have noted that cross-correlation is more effective forlarge pipelines with a high acoustic damping, whereas auto-correlation is more effective for short pipes with low acous-tic damping. This method is more useful for leak locating inpipes. Cross correlation is a technique for measuring timedifferences between two AE signals recorded at transduc-ers. If the speed of wave propagation was defined we can lo-cate the leakage source. There are many algorithms for leaklocating in one, two or three dimensions based on time dif-ference estimation. Knowledge of how the pipe wall vibratesby acoustic emission resulting from leakage is a key parame-ter for leak detection and location. Goncalves and Batista [9]considered simply supported circular cylindrical shells filledwith incompressible fluid. To model the shell Sanders non-linear shell theory and a novel mode expansion were used.It was found that the presence of a dense fluid increases thesoftening characteristics of the frequency amplitude relationwhen compared with the results for the same shell in vac-uum. Mayers and Wrenn [10] used both Donnells nonlinearshallow shell theory and the Sanders-Koiter nonlinear shelltheory to study shell vibrations.

In this present work modeling of AE signals and theirvibration characteristics due to leakage in pressurized pipeis studied. Donnells non-linear theory for cylindrical shellis used to modeling the pipe vibration and deriving of mo-tion equation of the pipe in radial direction and by Galerkinmethod a system of nonlinear ODEs will be derived. Acous-tic emission testing with a point source and a linear array oftransducers are carried out to survey the accuracy of analyt-ical results. Wavelet transform is taken from experimentalresults to omit the environmental noises and reconstruct theAE signals without noise.

2 Analytical Model

A circular cylindrical shell with radius R, length L andthickness h is considered as a pipe. A cylindrical coordi-

nate system (O;x, r, ) is used in which O is origin placedat the center of one end of shell. The displacements of shellpoints in the axial, circumferential and radial directions areu, v, w respectively.

Donnells non-linear theory for cylindrical shell is used toderive the motion equation (the displacement in the radialdirection) [11].

D4w + chw + hw= f p + (1/R)(2F/x2)

+ 1R2

(2F

22w

x2 2

2F

x

2w

x+

2F

x22w

2

)(1)

where D represents the flexural rigidity defined as D =Eh3/12(1 2), c: damping factor, E: Youngs modulus, :the mass density and f : the force that is applied to the pipesurface caused by the jet of the leaking gas, w = w/t ,w = 2w/t2 and F is in-plane stress function. More infor-mation in this matter can be obtained by [12].

w(x, , t) is the displacement in radial direction of pointson shell which is expanded by using the linear shell eigen-modes [12]:

w(x, , t)

=2

k=1

[Am,n(t) cos(n) + Bm,n(t) sin(n)

]sin(mx)

+3

m=1A(2m1),0(t) sin((2m1)x) (2)

where m denotes the axial wave number (equal to the num-ber of half-waves along the shell.), n is the circumferentialwave number, and m is the wave number which dependson boundary conditions. In this study the boundary condi-tion for two ends of a simply supported pipe is as bellow:

w = 0, Mx = D[2w

x2+

2w

R22

]= 0

at x = 0,L

Nx = 0, = 0 at x = 0,L

(3)

where Mx and Nx are the bending moment and the ax-ial force per unit length. For simply supported shell: m =mx/L.

The forces per unit length in the axial (Nx ) and circum-ferential (N ) directions, as well as the shear force (Nx ),are given by:

Nx = 1R2

2F

2, N =

2F

x2, Nx = 1

R

2F

x

(4)

J Nondestruct Eval (2013) 32:6780 69

The strain-displacement relations are:

(1 2)Nx

Eh= w

R+ 1

2

(w

x

)2+

2

(w

R

)2

+ ux

+ R

(v

)

(1 2) N

Eh= w

R+

2

(w

x

)2+ 1

2

(w

R

)2

+ ux

+ 1R

v

(1 2)Nx

Eh= 2(1 )

[1R

w

x

w

+ 1

R

u

+ v

x

]

(5)

In these equations is Poisson ratio. According to Hamil-tons principle we can derive the motion equations as:

Nx

+ Nx

= R2h uN

+ Nx

+ Q

= R2h v Qx

+ Q

N = hR2 w + fBx

+ Bx

RQx = 0Bx

+ B

RQ = 0

(6)

where Bx , B , Bx are equivalent static couples, Qx , Q areequivalent static shearing and = x/R, u = u/R, v = v/R,w = w/R. Donnells theory represents a simplification ofthe theory. There are two assumptions in Donnells theoryto solve Eqs. (6). It is argued that in the equations of motion(6) the transverse shearing stress resultant Q make a negli-gible contribution to the equilibrium forces in the circumfer-ential direction and hence Q may be neglected. Secondlyit is argued that v has no effect on the relationship betweencurvature and the displacements. So we can rewrite Eqs. (6)as:

2u

2+ K0

2u

2+ K 0

2v

+ w

= R

2

c2u

K 02u

+ K0

2v

2+

2v

2+ w

= R

2

c2v

u

v

4w = R2

c2w + Rf

K

(7)

where = h2/12R2, K 0 = (1+)/2,K0 = (1)/2, K =Eh/(1 2). By some calculations we can obtain Eq. (1)from Eqs. (7).

The perturbation pressure is calculated by usingPaidousis and Dennis model [13] and the method of vari-ables separation:

pr = f Lm

In(mr/L)

I n(mR/L)

(

t

)2w (8)

where f is the gas density. Pipelines and piping system areimportant in the infrastructure of modern society. Pipelinesnetworks frequently cross highly populated regions water,oil and gas supplies or natural reserves. So knowing the vi-bration behavior of the gas and fluid-filled pipes caused byleakage is very important to detect the leakage and preventthe explosion, environmental pollution and saving energiesand water supplies. This model can be used to model thefluid and gas filled pipe vibrations and be useful in severalindustrial fields.

Applied force to the pipe caused by leaking fluid is con-sidered as follows:

f = F(x x0)( 0)eit (9)where x0, 0 denote leak source coordinates and is the fre-quency of applied force caused by leaking gas. By replacingw in the right hand side of Eq. (2), a partial difference equa-tion (PDE) for in-plane stress function is defined as:

F = Fh + Fp (10)where Fp and Fh denote particular and homogeneous solu-tions of in-plane stress function. To determine Fp and Fh,Mathematica software is used [12].

By replacing w(x, , t), F , f and pr in right hand sideof Eq. (1) and by using the Galerkin technique with weight-ing functions, the partial differential equation of motionis solved and a system of non-linear ordinary differentialequations with 7 degree of freedom is obtained in Math-ematica. By using the Galerkin method, seven second or-der ordinary, coupled non-linear differential equations areobtained. The Galerkin projection of Eq. (1) has been per-formed by using the Mathematica computer software. TheRunge-Kutta method was used to solve the system of equa-tions (Appendix).

In this model the non-linear interaction among linearmodes of the chosen basis involves only the asymmetricmodes having a given n value, and all the axisymmetricmodes is considered. Then the obtained theoretical resultswere compared with de-noised experimental results. Liuet al. [14] and Kandasamy et al. [15] used the computer sim-ulation source or a point excitation source (a small piezo-electric disc) in their experimental set-up for theoretical re-sults validation and in some other researches [16] a linearinteraction among the modes were used to study the vibra-tion behavior of pipe caused by simulated leak source. Inour previous work [11] we studied the validation of the an-alytical model by noisy experimental results. The boundaryconditions were as:

u = 0 at x = 0,L and v = 0 at x = 0,L (11)In this study we used a non-linear model with a differ-ent boundary conditions (as mentioned in Eqs. (3) and the


model wsa solved per these boundary conditions. Then theanalytical results were compared with the de-noised ex-perimental results per a specific range of frequencies. Thenoise of the experimental results were suppressed by wavelettransform.

3 Wavelet Transform

Wavelet transform is one of the more useful mathematicaltransform in signal processing domain. Regarding the na-ture of multi-resolution analysis, this transform plays veryimportant role in most processing usages. Important pointsof wavelet transform in non-stationary can be obtained by[17, 18]. In practical conditions, AE signals resulted fromleakage are very complicated. Recognition of main sig-nals from noises like flow, surrounding and used equipmentnoises is very difficult by using mentioned acoustic tech-nique, so there needs other new methods for analysis of non-stationary signals [19]. Wavelet transform (WT) is an ana-lytical time-frequency method which is considered in anal-ysis of AE signals. Qi [20] has also used the WT to analyzeAE signals, this time by decomposing the signal into differ-ent wavelet levels, each level representing one componentof the decomposed AE signal within a certain frequencyrange. Also, the energy of each level of the decomposedAE signal is carried with it, thus providing a comprehen-sive means of inspecting the AE source. Kalogiannakis [21]used WT for tracing the AE waves resulted from defects incomposite structures. Kishimoto et al. [22] defined that WTtogether with Gabor wavelet is a useful method for analy-sis of propagated signals. On the other hand a source lo-cating algorithm according to Gabor wavelet provided simi-lar results with cross correlation technique. Inoue et al. [23]showed that a three-dimensional plot of the magnitude ofthe WT in time-frequency space has peaks whose locationsindicate the arrival times of each component of the wave.Most authors have observed that the velocity of propagationis frequency-dependent. Rosa et al. [24] studied the possibil-ity of using wavelets and wavelet packets to detect and char-acterize alarm signals produced by termites. A set of syn-thetics have been modeled by mixing the real acquired tran-sients with computer generated noise processes. Identifica-tion has been performed by means of analyzing the impulseresponses of three sensors undergoing natural excitations.The conventional methods and wavelet transform methodsare based on the premise that the individual wave modes canbe recognized. For example, Bayray and Rauscher [25] havecompared the windowed Fourier transform and the wavelettransform for identifying AE wave mode and found that theygave similar results in helping to identify fast extensionalwave and slow flexural wave propagation. For cases wherethe geometry and material make modes difficult to distin-guish, simple frequency filtering may not be as effective,

Fig. 1 Continuous leak source

and so this work uses a wavelet packet transform [26] andthen a Butterworth filter [27] together with a threshold cross-ing technique to fix the arrival time(s). This technique willbe compared with results obtained using threshold crossing,cross correlation and Gabor wavelet techniques. With theavailability of advanced computing resources and data stor-age and transmission capability, recording and anad analysisof the complete signal waveforms is becoming the preferredanalysis approach. Though the signals captured by sensorsare affected by the medium of propagation and the sensorcharacteristics, the signals still contain information aboutthe nature of the source. The use of wavelet transform fornoise suppression in acoustic emission detection is widelyused [28]. We used Dauechies wavelet (db4) in this research.Such wavelet needs a few calculations that is one of its im-portant advantages.

4 The Experimental Set-Up

Experiments were carried out with a linear array of sensorson steel pipe (ASTM A 106/99) of nominal length of 5.6 m,


Fig. 2 Sensors lay out

Fig. 3 Equipment lay out

7.35 mm wall thickness and external diameter of 169 mm.Two ends of the pipe were capped by 10 mm thicknessplates. An opening was formed in distance of 258.5 cm fromone end of the pipe. Simulated continuous leak source with0.8 and 0.6 mm diameter hole according to Fig. 1 was fit-ted the opening and pipe was then filled by air with 5 barpressure [29]. To prevent pressure loss, the regulator wasinstalled in the inlet of the pipe. Two R15a sensors were

mounted on both sides of leak source by adhesive tape andvacuum grease couplant as shown in Fig. 2. As it mentionedin Eq. (10), we study the vibration behavior of the pipe an-alytically per the resonance frequency. Therefore we usedresonance sensor of R15a. AE signals were pre-amplifiedusing a PAC1120A operating at 60 dB amplification andthe amplifier output was filtered between 0.01 and 0.4 MHz.The AE data being captured at a sampling rate of 10 MHz


Fig. 4 AE raw signals for testno. 1 at (a) S1, (b) S2

Table 1 Sensors position

Test No. Nozzle Hole Dia.(mm)

X1(cm)

X2(cm)

1 0.6 100 202 0.8 100 203 0.6 214 1024 0.8 214 102

for a record length of 6 ms. The sensor distances from leaksource location are shown in Table 1. The equipment lay outis shown in Fig. 3.

5 Results and Discussion

The raw AE signals captured at sensors S1 and S2 for testNo. 1 are shown in Fig. 4 respectively. Varying the amountof leaking gas will change the signal amplitude. On the otherhand the acoustic properties of the signal transmitter struc-ture can affect the amplitude of the generated AE signals.The leakage source propagates acoustic signals in a widerange of frequencies. In addition to main AE signals gen-erated by leakage, noises like surrounding, flow and equip-ment noise are transmitted through the pipe wall and cap-tured by acoustic emission sensors mounted on the pipewall. In practical situations recognition of main AE sig-nal from noises are very difficult. For better analysis Fast


Fourier Transform (FFT) was taken from raw AE signals.As can be seen in Fig. 5 AE signals and surrounding noisespropagate in a wide range of frequencies. To study the ver-ification of the used analytical model, it is necessary todetection the main AE signals and suppressing the noises.

Fig. 5 Frequency domain of recorded signal for test no. 1 at (a) S1,(b) S2

Wavelet transform is a kind of analyzing ways for measure-ment of time and frequency of the signal. It can be used toanalyze pipeline leak signals and filter the noise appearedin the signal. Captured AE signals by sensors split into anapproximation and a detail in wavelet transform. The ap-proximation is then itself split into a second level approxi-mation and detail and the process is repeated. Noises usu-ally occur in details. Choosing the appropriate level of de-composition is very important. A 6 layers decomposition isapplied in this research for experimental results. By filter-ing the approximation of layer 6, more noises were removedand the filtered approximation was split into an approxima-tion and a detail in order to remove more noises. At last wereconstructed the main AE signals without noise. Figure 6indicates the reconstructed AE main signals. To obtain theradial displacement of the pipe theoretically, the resonantfrequency of the AE sensor was used and the applied forceto the pipe due to gas leaking was measured by dynam-ics of gases. So varying the hole size, caused different ra-dial displacement. The Runge-Kutta numerical method wasused to solve the system of equations. Replacing the me-chanical properties of the pipe and gas in the equations andusing ODE tool of MATLAB software the pipe vibrationof the different points was calculated. In our experimentalset-up there were two major types of noises. One of themis surrounding noises especially compressor noises and an-other one is the reflected AE waves which were capturedby AE sensors. The first type (caused by compressor) wassuppressed by isolating the compressor from experimentalset-up environment in a separate room. Moreover before do-ing the main test, a pre-test with a cap on the leak source

Fig. 6 De-noised signals afterWT for test no. 1 at (a) S2,(b) S1


Fig. 7 Theoretical results fortest no. 1 at (a) S1, (b) S2

was carried out and a pre-threshold proportional to the envi-ronmental noises was set to suppress these unwanted noises.So only the AE signals produced by leakage were recordedby mounted acoustic emission sensors. The pipe boundariesreflected some AE waves in a wide range of frequenciesand amplitudes. These reflected waves lost a majority of

their energies. So AE signal with lower energy interferedwith initially waves. In this study, for each sensor the cap-tured signals were narrowband filtered at center frequenciesof 150 KHz with a determined bandwidth. By this filteringsome components of reflected waves were removed. In ad-dition to eliminate more of reflected waves, a threshold ac-


Fig. 8 FFT results of test no. 1for sensor S1 (a) experimental,(b) theoretical

cording to maximum amplitude of raw AE signal was setso the waves which crossed this threshold were recorded. InFig. 7 the predicted signals of test no. 1 are shown. To surveythe vibration characteristics of AE signals the time domainof signals is changed to the frequency domain. The greatestamount of AE energy is carried out per specific frequencyranges. By fast Fourier transform, the frequency domain ofthe modeled signals was calculated. The theoretical modelcalculated the amplitude of the AE signals in meter (m) but

the sensors used in the experimental set-up measured theamplitude of the AE waves in volt (V). Therefore the scalesof the theoretical and experimental results are different. Sothe frequencies which carried the most amounts of energiesare comparable.

In Figs. 8, 9, 10, 11 the FFT results of theoretical and ex-perimental data for test nos. 1 and 2 are shown. Comparisonthe results showed that the maximum amount of AE energycarried per the same frequencies. In Table 2 the results of an-


Fig. 9 FFT results of test no. 1for sensor S2 (a) experimental,(b) theoretical

alytical modeling and experimental is shown for some tests.As can be seen the mean error between analytical modelingand experimental results is less than 7 %. So the experimen-tal results confirm performance of the developed analyticalmodel.

6 Conclusion

In this paper, modeling of AE signals and their vibrationcharacteristics due to leakage in pressurized pipe is studied.Donnells non linear theory has been used to obtain the ra-dial displacement of the pipe. The simulated continuous leak

source, used in acoustic emission testing, propagated wavesin a wide range of frequencies. Unwanted noises in additionto main AE signals propagate through the pipe wall in a widerange so suppressing these noises is very important. Wavelettransform is used to suppress these surrounding noises. Thetheoretical model has been solved numerically per the reso-nant frequency of the AE sensor used in acoustic emissiontesting and the radial displacement of the pipe has been cal-culated. To verification of the model, we study the frequencyranges which carry the maximum energy of the AE signals.Fast Fourier Transform (FFT) was taken from theoreticaland de-noised experimental results. Comparing the results


Fig. 10 FFT results of testno. 2 for sensor S1(a) experimental, (b) theoretical

showed that the frequency which carried the most amountof energy is the same that expresses excellent agreement be-tween the theoretical and experimental results validating theanalytical model.

Appendix: Non-Linear Equations Obtainedin Mathematica Software

The system of non-linear ordinary differential equationswhich is obtained in Mathematica are given by:

A1,n(t) + 21,n1,nA1,n(t) + 21,nA1,n(t) + h1A31,n(t) + h1A1,n(t)B21,n(t)+ h2A1,n(t)A22,n(t) + h3A1,n(t)B22,n(t) + h4A2,n(t)B1,n(t)B2,n(t)+ h5A1,n(t)A1,0(t) + h6A1,n(t)A3,0(t) + h7A1,n(t)A5,0(t) + h8A1,n(t)A21,0(t)+ h9A23,0(t)A1,n(t) + h10A1,n(t)A25,0(t)h11A1,n(t)A1,0(t)A3,0(t)+ h12A1,n(t)A3,0(t)A5,0(t) = Feit

m1 = hL/2, 21,n =L

2

[D

(2

L2+ n

2

R2

)2+ Eh

4

R2L4

/(2

L2+ n

2

R2

)2]/m1

1,n = chL2 /(21,nm1)

(12)


Fig. 11 FFT results of testno. 2 for sensor S2(a) experimental, (b) theoretical

B1,n(t) + 21,n1,nB1,n(t) + 21,nB1,n(t) + h1B31,n(t) + h1B1,n(t)A21,n(t)+ h2B1,n(t)B22,n(t) + h3B1,n(t)A22,n(t) + h4B2,n(t)A1,n(t)A2,n(t)+ h5B1,n(t)A1,0(t) + h6B1,n(t)A3,0(t) + h7B1,n(t)A5,0(t) + h8B1,n(t)A21,0(t)+ h9B1,n(t)A23,0(t) + h10B1,n(t)A25,0(t) + h11B1,n(t)A1,0(t)A3,0(t)+ h12B1,n(t)A3,0(t)A5,0(t) = 0 (13)

A2,n(t) + 22,n2,nA2,n(t) + 22,nA2,n(t) + k1A32,n(t) + k1A2,n(t)B22,n(t) + k2A2,n(t)A21,n(t)+ k3A2,n(t)B21,n(t) + k4A1,n(t)B1,n(t)B2,n(t) + k5A2,n(t)A1,0(t) + k6A2,n(t)A3,0(t)+ k7A2,n(t)A5,0(t) + k8A2,n(t)A21,0(t) + k9A2,n(t)A23,0(t) + k10A2,n(t)A25,0(t)+ k11A2,n(t)A1,0(t)A3,0(t) + k12A2,n(t)A1,0(t)A5,0(t) = 0

m2 = hL/2, 22,n =L

2

[D

(42

L2+ n

2

R2

)+ 16Eh

4

R2L4

/(42

L2+ n

2

R2

)2]/m2

2,n = chL2 (22,nm2)

(14)


Table 2 Experimental andanalytical results Test Theoretical main

power spectraldensity freq.

Experimental mainpower spectraldensity freq.

Error %

0.6 mm Dia. Hole and 20 cmsensor-source Distance

111 115 3.4


133 127 4.7


119 117 1.8


127 119 6.7


135 129 4.6


115 120 4.4


130 122 6.5


118 114 3.5

B2,n(t) + 22,n2,nB2,n(t) + 22,nB2,n(t) + k1B32,n(t) + k1B2,n(t)A22,n(t)+ k2B2,n(t)B21,n(t) + k3B2,n(t)A21,n(t) + k4B1,n(t)A1,n(t)A2,n(t)+ k5B2,n(t)A1,0(t) + k6B2,n(t)A3,0(t) + k7B2,n(t)A5,0(t) + k8B2,n(t)A21,0(t)+ k9B2,n(t)A23,0(t) + k10B2,n(t)A25,0(t) + k11B2,n(t)A1,0(t)A3,0(t)+ k12B2,n(t)A1,0(t)A5,0(t) = 0 (15)

A1,0(t) + 21,01,0A1,0(t) + 21,0A1,0(t) + l1A1,0(t)A21,n(t) + l1A1,0(t)B21,n(t)+ l2A1,0(t)A22,n(t) + l2A1,0(t)B22,n(t) + l3A21,n(t) + l3B21,n(t) + l4A22,n(t) + l4B22,n(t)+ l5A3,0(t)A21,n(t) + l5A3,0(t)B21,n(t) + l6A3,0(t)A22,n(t) + l6A3,0(t)B22,n(t)+ l7A5,0(t)A22,n(t) + l7A5,0(t)B22,n(t) = Feit

m1,0 = hL, 21,0 =L

m1,0

(D4

L4+ Eh

R2

)

1,0 = chL/(21,0m1,0)

(16)

A3,0(t) + 23,03,0A3,0(t) + 23,0A3,0(t) + n1A3,0(t)A21,n(t) + n1A3,0(t)B21,n(t)+ n2A3,0(t)A22,n(t) + n2A3,0(t)B22,n(t) + n3A21,n(t) + n3B21,n(t) + n4A22,n(t)+ n4B22,n(t) + n5A1,0(t)A21,n(t) + n5A1,0(t)B21,n(t) + n6A1,0(t)A22,n(t)+ n6A1,0(t)B22,n(t) + n7A5,0(t)A21,n(t) + n7A5,0(t)B21,n(t) = Feit

m3,0 = hL, 23,0 =L

m3,0

(81D4

L4+ Eh

R2

)

3,0 = chL/(23,0m3,0)

(17)

A5,0(t) + 25,05,0A5,0(t) + 25,0A5,0(t) + p1A5,0(t)A21,n(t) + p1A5,0(t)B21,n(t)+ p2A5,0(t)A22,n(t) + p2A5,0(t)B22,n(t) + p3A21,n(t) + p3B21,n(t) + p4A22,n(t)+ p4B22,n(t) + p5A1,0(t)A22,n(t) + p5A1,0(t)B22,n(t) + p6A3,0(t)A21,n(t)+ p6A3,0(t)B21,n(t) = Feit

m5,0 = hL, 25,0 =L

m5,0

(625D4

L4+ Eh

R2

)

5,0 = chL/(25,0m5,0)

(18)


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Modeling Acoustic Emission Signals Caused by Leakage in Pressurized Gas PipeAbstractIntroductionAnalytical ModelWavelet TransformThe Experimental Set-UpResults and DiscussionConclusionAppendix: Non-Linear Equations Obtained in Mathematica SoftwareReferences

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