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Condition Monitoring Diagnosis Method of Aircraft Engine Rotating Details N.I.Bouraou, P.I.Marchuk and A.N.Tyapchenko National Technical University of Ukraine P.O.Box 117, Kiev, 04070, Ukraine Fax: (38044) 241-77-02 Contact E-mail: nadya@burau.inec.kiev.ua

Abstract

Presented work is dedicated to development of the new passive-active low-frequency automatic vibroacoustical early fault diagnosis method of aircraft gas-turbine engine units (compressor stages, turbine stages), which contain the rotating details (blades, disks, shafts, etc.). The proposed method is condition monitoring and can be used while engine is operating. The method is based on testing object non-stationary narrow-band vibratory excitation with the variable central frequency, and signal processing of the engine vibro and acoustical noise in the low-frequency range (0-50 kHz). As the testing object the blade is considered at the presence and the absence of the fatigue crack. In this paper the theoretical correlation and spectral researches are given for the testing object non-stationary oscillations without and with the above mentioned fault. For nondestructive evaluation of the relative fault size the feature set, components of which are higher and basic harmonics spectral amplitudes ratios is used. The evaluation is produced by using the generalized likelihood ratio (GLR) method. The estimations statistics are computed.

Keywords: vibroacoustical diagnosis, fault, non-stationary oscillations, correlation and spectral analysis, nondestructive evaluation, likelihood ratio equation, estimations statistics

Introduction

For diagnosis and evaluation of the testing object fatigue cracks the low-frequency vibroacoustical methods of free and forced oscillations can be used. As known, the forced oscillations vibroacoustical diagnosis method consists of excitation of object resonant oscillations and processing of the object radiated acoustical noise. For the rotating bladed systems of the gas-turbine engines the above mentioned excitation can be provided by using rotor rotation with the constant rotation frequency. However, such excitation of the objects resonant oscillations is possible, if the objects natural frequencies are known. For the case when objects natural frequencies are unknown, but natural frequencies range is known, we propose to use the non-stationary excitation with the variable frequency [1]. Such excitation is proposed to provide by the respective (in the natural frequencies range) change of the rotor rotation frequency. Thus, the developed diagnosis method of aircraft engine blades fatigue cracks includes the following procedures: a) the excitation of the object oscillations by using the non-stationary narrow-band vibration excitation with the variable frequency, and b) signal processing of engine vibro and acoustical noise in the low-frequency range. Presented investigation of the object oscillations at the above mentioned excitation will allow to explain many phenomena appeared during the method usage, to choose signal processing methods and optimal features, to evaluate parameters of the fault, etc.

Testing object and vibratory excitation models

As the testing object model we consider the one-mass oscillatory system with the single degree of freedom without damping. At the presence of the fatigue crack the object has different rigidity at the stretching and the compression, therefore, the testing object can be described by oscillatory system model with the piecewise-linear restitution characteristic [1]. In this case the oscillatory system impulsive response g(t) can be presented in series Fourier :

(1)

where J is the relative rigidity changing at the fault presence, it is the relative crack size function; k is the harmonic number of the main frequency w ₀ of the system with fault; frequency w _* is the same as of the system without fault.
Assuming that the instantaneous frequency rotation of the engine shaft at the engine running start is: where w _r0 is the initial frequency rotation; b is the velocity of the frequency variation, the non-stationary narrow-band vibratory excitation model is considered in the form:

(2)

where A₀ is the constant quantity; x (t) is the stationary random process with the zero mean and the correlation function Kx (t ); Q ₀ is the uniform distributed on [0;2p] initial phase, it is statistically independent of the process x (t) .
By using the correlation function of the stationary random process x (t) in the form: where the correlation function of the vibratory excitation (2) at the statistically independent x (t) and Q ₀ can be obtained in the form:

(3)

where .
The spectral density of the above mentioned excitation is obtained by expression [2]: where K_p(t₁,t₂) is the correlation function of the process P(t); t_k is the observation duration.
By using expression (3) and assuming that w _r0 =0 and t_{2 ³} t₁ we will obtain the excitation spectral density in the following form:

(4)

where                     B_j are functions of the Fresnel integrals C(u_ri) and S(u_ri);

Correlation functions of the object non-stationary oscillations

The autocorrelation function (ACF) of object oscillations is obtained by expression [2]: where g(· ) is the object impulsive response; K_p(· ) is the correlation function of the process P(t).
By using expressions (1) and (3) we can obtain the ACF of the cracked object model non-stationary oscillations in the form:

(5)

where F_o, F_kand y_k are functions of the Fresnel integrals C(u_lj) and S(u_lj);

At the absence of the fault we consider the testing object model as a linear oscillatory system with the following impulsive response: At this case we obtain the ACF in the form:

(6)

where F_* and y_*are functions of the Fresnel integrals C(u_lj) and S(u_lj);
The cross-correlation function (CCF) of the object input and output process is defined by expression [2]: After mathematical transformation the CCF can be presented at the fault presence in the form:

(7)

where D_o, D_k,j_o and j_k are functions of the C (u_o), S(u_o), C(u_i2) and S(u_i2).
For the object without fault the CCF is obtained in the following form:

(8)

where D_* and j_* are functions of the C (u_*o), S(u_*o), C(u_*2) and S(u_*2)
The received expressions (7) and (8) can be presented in the form of the oscillatory process with the frequency-linear modulation:

(9)

where A(t₁,t₂) is the CCF envelope; g (t₂) is the CCF phase.
At the fault presence we can obtain the CCF envelope from expression (7):

(10)

where: and at the fault absence we can obtain the CCF envelope from expression (8):

(11)

The received correlation functions analytical expressions are new. They allow to investigate the ACF and CCF dependencies on the excitation parameters and the testing object parameters at the both linear and piecewise-linear restitution characteristics cases, and these dependencies can be used for fault diagnosis and evaluation.
By using the received analytical expressions (5)- (8) the ACF and CCF are computed for the following conditions: b = 190 s^-2, it corresponds to instantaneous frequency changing of engine shaft rotation D f_p=300 Hz at the t=10 s; s ²_p = 1; a = 0.01. The natural frequency of the testing object linear model is f_* = 600 Hz; the 8 harmonics of the series (1) are taken into consideration for the piecewise-linear model at the relative rigidity changing J = 0.05. Fresnel integrals calculation errors are less than 10^-8, the N=10⁴ points are computed for the given time intervals. . The dependencies of the ACF on the variable t₁ and t₂ =40- t₁ ( i.e. in the section, which is orthogonal to t₁ =t₂) are presented on the Fig.1. The dependencies CCF on the variable t₁ and t₁=5-t₂ are presented on the Fig.2; the graphs of envelope of the above mentioned functions at the variable t₂ (t₁=0) are presented on the Fig.3. The dependencies of the CCF on the t₁ and t₂ =5- t₁ for the object linear model (Fig.2а) and the high-frequency component of the expression (7) for the object piecewise-linear model at the J =0.05 (Fig.2c) are simiral. The given on the Fig.2b common dependence K_px(t₁, t₂) for model with fault is the sum of the frequency-linear modulated oscillation, which caused by a₀ , and the high-frequency non-stationary component S₁ , the intensity of which is 5 times less .

Fig 1: Dependence of the ACF on the t1 at the t2 = 40 - t1; a) Linear model b)Piecewise linear model.

Fig 2: Dependence of the CCF on the t1 and t2 = 5 - t1 for (a)object model with out fault; (b)object model with fault; (c)High frequency component ĺ1 of the CCF for object model with fault.

Fig 3:Dependence of the CCf envelope on the t2 at the t1 = 0 for (a) object mdel without fault;(b) object model with fault; .


The presented results show clearly the difference of the considered functions at the fault presence from the same ones at the fault absence, and they can be used for faults diagnosis and evaluation.

Spectral densities of the testing object non-stationary oscillations

We consider the getting of the non-stationary narrow-band process realization P(t) through the oscillatory system with the impulsive response (1). The transfer functions of the above mentioned system elements are:
By using expression (4) we will obtain the spectral density of the cracked object non-stationary oscillations in the form [2]:

(12)

where H^*_r(w₁); is the conjugated to .
At the absence of the fault the spectral density of the object non-stationary oscillations can be presented in the form:

(13)


It can be seen from expressions (12) and (13) that differences of the considered models spectral densities are caused by the rigidity changing at the fault presence, therefore, these differences will serve as the fault features and can be used for fault diagnosis and evaluation.
By using (12) and (13) the power spectral densities (PSD) are computed for the above mentioned conditions at the w ₁=qw _r(t), where q is the harmonic number of the rotating rotor instantaneous frequency, and w ₂=lw₀ or w ₂=lw_*, where l is the harmonic number of the considered object model natural frequency; t_k=30s. The PSD dependencies of the object non-stationary oscillations at the fault absence and presence on harmonic number l are presence on the Fig.4, they show that PSD components of higher harmonics are unequal zero at the fault presence. As the fault feature we propose to use the multidimensional vector, components of which are ratios of the PSD of higher and basic harmonics. The proposed feature set do not depend on the excitation parameters. It can be seen from Fig.5 that the increment of the relative rigidity changing J at the fault presence brings about to the increasing of the above mentioned features that can be used for fault size evaluation by using the acoustical signal measurements and processing.

Fig 4: Power spectral densities of the testing object model oscillations; without fault (a) and with fault (b) at the q = 1.

Fig 5: Dependencies of the features on the relative rigidity changing

Nondestructive fault size evaluation

It can be seen from the above mentioned investigation that the PSD higher and basic harmonics ratios are equal practically to the ratios of the series (1) higher and basic harmonics spectral amplitudes.
The using feature set, the components of which are higher and basic (k=1) harmonics spectral amplitudes ratios, is:

(14)

According to the practical measurements we consider the spectral amplitude of the basic and any r-harmonics at the i-measuring to be the Gaussian random value x _ik :

(15)

The means m_k can be approximated by the first order polinomial for small J =0.01.....0.1 :

m1 = m 01 + m 11J and mr = m 0r + m 1rJ ,

(16)


the amplitude estimation variance is used in the form: where N₀ is the white noise intensity; t is the Gaussian impulse duration.
The evaluation of the value J will be produced by using the GLR method [3]. The probability density w(h ) of the independent random values ratio h_ir = x_ir /x_i1 is obtained by the integral: By using the expression (15) the probability density w(h _ir,J ) at the i-measuring can be presented in the form:

(17)

where
The maximum likelihood equation for the independent elements selection is:
and after the mathematical transformations (17) by using (16) we can write for each component of the set (14):

(18)

where

The received equation (18) is represented as the n-sum of the J second and first order polinomials ratios, where n is the measurements number.
For the single measurement (n = 1) the equation (18) solution relatively is the quadratic equation solution. At the n > 1 the estimations have been obtained by the numerical solution of the (18). The table gives the calculated results of the dependence of the relative rigidity changing estimation variance on the measuring noise variance for the each component (K=10) of the set (14) at the n =1, J =0.05. At the n =2 the results of the calculation are coincided practically to the table mentioned results (the variance decrement of the second, forth and sixth harmonics from 1% to 1.5% in the s _i² mentioned intervals), and variance changing regularities are coincided for both n =1 and n =2 cases. As we can see from the given results that the estimations variances by the higher odd harmonics exceed considerably the ones by the higher even harmonics in the whole interval of the measuring noise intensity values mentioned. Moreover, the estimation variances increase with the increment of the using higher harmonic number k that is caused by the decrement of the signal-to-noise ratio at the measurements with the equal variance quantity s _i² for the all k harmonics. The received results allow to eliminate the higher odd harmonics features.
The following fault estimations analysis is given for the n =1 and for the feature set, which consists of the higher even harmonics only:

(19)

By computing the fault estimation for each of the components of the set (19) we can form the following estimations set of the relative rigidity changing J :

which allows to obtain the mean m and variance D of the estimation for the present measurement
The dependencies of the lgD on the measuring noise intensity at the J =0.05 are presented on the Fig.6 for the following feature sets formed from (19):

Fig 6: Dependencies of the estimation variance on the measuring noise intensity for the considered feature sets


It can be seen from Fig.6 that estimation by the set ₁ has least variance, and the precision of the evaluation reduces by the all mentioned sets at the measuring noise intensity increment.

	10-10	10-9	10-8	10-7	10-6
2	3.35x10-8	3.34x10-8	3.32x10-8	3.07x10-8	1.09x10-8
3	1.92x10-5	1.27x10-5	2.12x10-5	7.13x10-3	5.72x10-1
4	3.18x10-8	1.82x10-8	9.62x10-8	2.25x10-5	2.36x10-3
5	3.27x10-6	4.82x10-4	6.02x10-2	3.27	7.06x101
6	6.53x10-10	1.7x10-6	2.11x10-4	2.02x10-2	1.30>
7	2.93x10-4	4.08x10-2	2.40	5.55x101	7.39x102
8	1.93x10-6	2.38x10-4	2.27x10-2	1.43	3.72x101
9	8.92x10-3	6.91x10-1	2.14x101	3.25x102	3.71x103
10	9.11x10-5	8.96x10-3	6.47x10-1	2.03x101	3.10x102
Table 1: Dependence of the estimation variance on the noise variance for each componenet of set (14)


Since the increment of the relative rigidity changing J is the same as of the signal-to-noise ratio increment for the each noise intensity quantity, that is expedient to use the estimation variance and mean square ratio [4] to analyse the estimations statistical quality: The dependencies of the Q₁=20lgQ on the evaluating quantity and on the noise intensity are presented on the Fig.7 for the feature sets ₁ (Fig.7a), ₂ (Fig.7b) and ₃ (Fig.7c). Presented results allow to choose the feature set using which the value of the ratio Q will not exceed the threshold quantity for the given experimental condition.

Fig 7:Dependencies Q1 on the evaluating parameter quantity and on the measuring noise intensity for feature sets Y1 (a), Y2 (b) and Y3 (c).

Conclusions

In this paper the piecewise-linear system oscillations are investigated at the non-stationary narrow-band vibratory excitation. The above mentioned system is used as the blade model at the fault presence for the developed vibroacoustical diagnosis method of the rotating details faults of the aircraft gas-turbine engine. The new analytical dependencies of the object oscillations auto- and cross-correlation functions are received and evaluated at the various sections. The spectral analysis of the testing object oscillations is given, on the basis of the which the fault features is chosen and investigated. For fault size evaluation the maximum likelihood equation analytical expression is received and its solutions are explored. The estimations statistics analysis given here allowed to choose the new feature sets, components of which are ratios of the higher even and basic harmonics spectral amplitudes. The precision (statistical quality) of the fault size evaluation is estimated by using the variance and mean square ratio.
The received results are common and can be used for the vibroacoustical condition monitoring and fatigue fault evaluation of the aircraft engine blades.

References

1. N.I. Bouraou, L.M. Gelman. Low-frequency vibroacoustical method of forced oscillations. In Proceedings of the 2^nd International Conference on Computer Method and Inverse Problems in Nondestructive Testing and Diagnostics, Minsk, 20 - 23 October 1998. - DGZfP, Berlin, 1998, pp. 33-40.
2. Julius S. Bendat, Allan G. Piersol. Random data: Analysis and Measurement Procedures, 1986 by John Willy & Sons, Inc., New York.
3. Levin B.Z. Teoreticheskie osnovy radiotehniki, Moskva, Sovetskoe radio, 1975.
4. S.Lawrence Marple, Jr. Digital Spectral Analysis with Applications. Prentice-Hall.Inc.,1987.

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