CN104111172A - Null space differential operator and blind source separation based bearing combined fault diagnosis method - Google Patents

Abstract

The invention discloses a null space differential operator and blind source separation based bearing combined fault diagnosis method and belongs to the technical field of bearing fault diagnosis. By means of null space differential operators based on bearing fault features, bearing fault signals (random vibration, fault shock components and noises caused by rotation of a normal part of a bearing) are resolved into a series of local narrow-band signals (comprising components of fault features); subsequently, the obtained narrow-band signals and bearing signals are regarded as one group of observation signals; by means of a blind source separation algorithm, the observation signals are subjected to blind source separation to achieve separation of combined faults of the bearing; finally, the signals obtained from separation are subjected to Hilbert demodulation processing, and afterwards, fault features of the bearing are extracted to ultimately achieve combined fault diagnosis of the bearing. In new observation signals, the observation signals are more than source signals, such that a premise required by the blind source separation algorithm can be satisfied, and further, combined fault separation and feature extraction of the bearing are achieved.

Description

A kind of bearing combined failure diagnostic method separating with blind source based on kernel differentiating operator

Technical field

The present invention relates to a kind of bearing combined failure diagnostic method, relate in particular to a kind of bearing combined failure diagnostic method separating with blind source based on kernel differentiating operator, belong to bearing failure diagnosis technical field.

Background technology

Bearing is one of parts that are most widely used in rotating machinery, and the detection to its operating condition and fault diagnosis tool have very important significance.The fault vibration signal of bearing is the non-stationary signal of a quasi-representative and contains the undesired signals such as much noise, it is the nonlinear problem of a class complexity to its analyzing and processing, there is suitable difficulty, especially in the time of rolling bearing generation combined failure, influencing each other, disturbing between different faults, various faults feature be superimposed make fault signature complicated give accurately fault diagnosis bring great difficulty, the signal contacting in engineering reality is also often this type of signal, so the research of this type of signal is for engineering, application is extremely important.

In the analytical approach of signal, the most basic method comprises time-domain analysis and frequency-domain analysis.Time-domain analysis is the simplest and directly perceived, and the frequency-domain analysis information of reflected signal intension better.But for complicated non-stationary signal, the characteristic information that signal is rich in all can not be completely portrayed in simple time-domain analysis or frequency-domain analysis.Therefore Time-Frequency Analysis Method is arisen at the historic moment.Typical Time-Frequency Analysis Method has Fourier's variations in short-term, Wigner-Ville distribution, wavelet transformation, EMD to decompose etc., but general Time-Frequency Analysis Method is because its decomposition basis function single causes lacking certain adaptivity.

An adaptive signal decomposition algorithm based on narrow band signal and operator theory, this algorithm is finally named as kernel tracing algorithm (NSP) through continuous improvement.The core concept of NSP algorithm is that local narrow band signal is under the effect of unusual local linear operator " disappearance ", therefore, can extract by unusual local linear differentiating operator the local arrowband component of signal to be analyzed, and using the local narrow band signal obtaining as cell signal, superpose to approach original signal with it, and then realize the adaptive decomposition of signal.NSP algorithm has good robustness and adaptivity, has obtained good application in image processing field, but also comparatively rare in the application in mechanical fault diagnosis field.

In the last few years blind source separation algorithm be applied to mechanical fault diagnosis field and made some progress, but existing blind source separation algorithm exists certain deficiency, wherein topmost embodiment is that blind source separation algorithm must be greater than under the prerequisite of number of source signal effectively separation signal at observation signal number, and this is difficult to meet in engineering reality, in the time that being less than the number of source of trouble signal, the bearing fault signal number gathering cannot utilize the combined failure of blind source separation algorithm release bearing.

Summary of the invention

The object of the invention is in order to solve the feature extraction of bearing combined failure and the traditional blind source separation algorithm above-mentioned technical matters in the diagnosis of bearing combined failure, this method provides a kind of bearing combined failure diagnostic method separating with blind source based on kernel differentiating operator in conjunction with kernel tracing algorithm and blind source separation algorithm advantage separately.

For achieving the above object, the technical solution used in the present invention is a kind of bearing combined failure diagnostic method separating with blind source based on kernel differentiating operator, the method comprises according to fault vibration model utilizes the corresponding kernel differentiating operator of Matlab programming constructs, utilize the kernel differentiating operator based on bearing fault characteristics to decompose bearing combined failure signal to be analyzed, the narrow band signal that decomposition is obtained and bearing combined failure signal are regarded one group of observation signal as and are carried out blind source and separate, and carry out demodulation analysis and obtain fault signature separating rear signal.

The bearing fault model of vibration of wherein using is to set up according to the failure mechanism of bearing.Can be similar to and regard quality-spring-damper system as according to the fault vibration model of bearing fault mechanism bearing, be periodic decaying exponential function:

$x\left(t\right)=\underset{-\∞}{\overset{+\∞}{\mathrm{\Σ}}}{\mathrm{ke}}^{-\mathrm{nt}}\cos \mathrm{\ωt\δ}(t-k{T}_0)$

Wherein, can think decaying exponential function be bearing fault model of vibration fundametal component:

y(t)＝ke ^-ntcos(ωt)

Easily know that by dynamics above formula is second order differential equation y "+2ny'+ ω ²the approximate solution of y=0, i.e. y (t)=ke ^-nt(ω t) is in Second Order Differential Operator y "+2ny'+ ω to cos ²in the kernel of y=0.

For bearing fault signal model: y (t)=ke ^-nt(t) it is in differentiating operator d to ω to cos ²/ dt ²-2a (t) ' d/a (t) dt+ ω (t) ²+ 2 (a (t) '/a (t)) ²kernel in.Wherein a (t)=ke ^-nt.

Wherein kernel decomposition algorithm detailed step is as follows:

(1) input bearing combined failure signal S, stops threshold epsilon, λ ₂ ⁰, γ ⁰and λ ₁ ⁰initial value;

Make j=0, U _j=0, λ ^j ₁=λ ⁰ ₁, γ ^j=γ ⁰;

(2) be calculated as follows

$\widehat{\mathrm{\Φ}}=-{(A^{T}A+{\mathrm{\λ}}_2{M_2}^T{M}_2)}^{-1}{A}^T{D}_2(S-U);$

(3) be calculated as follows parameter

${\widehat{\mathrm{\λ}}}^{j+1}=\frac{1}{1+{\mathrm{\γ}}^j}\frac{S^{T}M{({{\mathrm{\λ}}_1}^j,{\mathrm{\γ}}^j,T)}^{T}S}{S^{T}M{({{\mathrm{\λ}}_1}^j,{\mathrm{\γ}}^j,T)}^{T}M({{\mathrm{\λ}}_1}^j,{\mathrm{\γ}}^j,T)S};$

(4) be calculated as follows

${\hat{U}}_{j+1}={(T^{T}T+(1+{\mathrm{\γ}}^j){{\mathrm{\λ}}_1}^{j+1}E)}^{-1}(T^T\mathrm{TS}+{\mathrm{\γ}}^j{{\mathrm{\λ}}_1}^{j+1}S);$

(5) be calculated as follows γ ^j+1:

${\mathrm{\γ}}^{j+1}=\frac{(S-{\hat{U}}_{j+1})S}{{\left|\right|S-{\hat{U}}_{j+1}\left|\right|}^2}-1;$

(6) judge whether to meet if meet, $\hat{U}={\hat{U}}_{j+1},$ ${\widehat{\mathrm{\λ}}}_1={{\widehat{\mathrm{\λ}}}_1}^{j+1},$ $\widehat{\mathrm{\γ}}={\mathrm{\γ}}^{j+1},$ $\widehat{\mathrm{\α}}={\widehat{\mathrm{\α}}}_j$ Output and U=S-R; Otherwise order; Otherwise make j=j+1 get back to step (3);

Wherein, $A=[A_{D_1(S-U)},A_{S-U}],$ $M_1={[{D_1}^T,E^T]}^T,$ $M_2=\left[\begin{array}{cc}{D}_2& \\ & E\end{array}\right],$ p and q can be from $\widehat{\mathrm{\Φ}}={[p^T,q^T]}^T$ Middle acquisition, $T=D_2+B_{\widehat{\mathrm{\Φ}}}{M}_1,$ $M({{\mathrm{\λ}}_1}^j,{\mathrm{\γ}}^j,T)={(T^{T}T+(1+\widehat{\mathrm{\γ}}){\mathrm{\λ}}_{1}E)}^{-1}.$

Matrix A _xthat diagonal element is the diagonal matrix of x vector.E is unit matrix, D ₁and D ₂represent single order and Second differential matrix, laGrange parameter, to be retained in T _ckernel in determine the parameter of quantity of information of S-R.

It is as follows that blind source detachment process wherein adopts eigenmatrix approximately joint diagonalization algorithm to realize concrete steps:

(1) narrow band signal and bearing combined failure signal are formed to one group of observation signal;

(2), to the processing of above-mentioned observation signal prewhitening, obtain observing matrix and albefaction matrix after albefaction;

(3) above-mentioned gained signal carries out associating diagonalization processing;

(4) obtain estimate source signal;

Technique effect of the present invention is: utilizing kernel tracing algorithm is the stack of a series of local narrow band signal that comprises fault signature by bearing combined failure signal decomposition adaptively, and regards the narrow band signal of gained and bearing combined failure signal as one group of observation signal.In new observation signal, the number of observation signal is greater than source signal number, so just meets the required hypotheses of blind source separation algorithm, and then realizes separation and the feature extraction of bearing combined failure.

Brief description of the drawings

Fig. 1 is the bearing combined failure diagnostic method process flow diagram separating with blind source based on kernel differentiating operator of the present invention.

Fig. 2 is the time domain waveform figure of centre bearer Internal and external cycle combined failure experimental signal of the present invention.

Fig. 3 is the demodulation spectrogram of centre bearer Internal and external cycle combined failure experimental signal of the present invention.

Fig. 4 utilizes kernel tracing algorithm Internal and external cycle combined failure to be decomposed to the time-domain diagram of 3 narrow band signals that obtain for 3 times in the present invention.

Fig. 5 is the time domain waveform figure after the present invention separates Internal and external cycle combined failure.

Fig. 6 is the demodulation spectrogram after the present invention separates Internal and external cycle combined failure.

Embodiment

Below in conjunction with the drawings and specific embodiments, the invention will be further described.

Fig. 1 is the bearing combined failure diagnostic method process flow diagram separating with blind source based on kernel differentiating operator of the present invention.

Below in conjunction with process flow diagram, the bearing combined failure diagnostic method principle separating with blind source based on kernel differentiating operator is elaborated.

(1) set up bearing fault model of vibration according to the failure mechanism of bearing.Can be similar to and be regarded as quality-spring-damper system by the known rolling bearing fault vibration mode of dynamics, the kinetic model of quality-spring-damper system is:

my″+cy′+ky＝0

Wherein m is spring-mass, and c is ratio of damping, and k is spring constant, makes ω _n ²=k/m, n=c/2m, definition damping ratio ξ=n/ ω _n, in underdamping situation, ξ < < 1, so above formula can be rewritten as following form:

y″+2ξωy'+ω ²y＝0

The solution of above formula is:

y(t)＝ke ^-ξωtcos(ωt)

Be that bearing fault model of vibration is the decaying exponential function occurring in the cycle, can be expressed as:

$x\left(t\right)=\underset{-\&infin;}{\overset{+\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{ke}}^{-\mathrm{nt}}\cos \mathrm{\&omega;t\&delta;}(t-k{T}_0)$

The effective constituent of model be exponential damping signal:

y(t)＝ke ^-ntcos(ωt)

(2) according to the corresponding kernel differentiating operator of bearing fault Construction of A Model.The effective constituent (decaying exponential function) of easily being known bearing fault model by (1) is second order differential equation y "+2 ξ ω y'+ ω ²the solution of y=0 is that the fault vibration model of bearing is in Second Order Differential Operator y "+2 ξ ω y'+ ω ²in the kernel of y.For bearing fault signal model: y (t)=ke ^{-ξ ω t}(ω t) easily proves that it is in differentiating operator d to cos ²/ dt ²-2a (t) ' d/a (t) dt+ ω (t) ²+ 2 (a (t) '/a (t)) ²kernel in.Wherein a (t)=ke ^{-ξ ω t}.

(3) input original signal S, stops threshold epsilon, λ ₂ ⁰, γ ⁰and λ ₁ ⁰initial value;

Make j=0, U _j=0, λ ^j ₁=λ ⁰ ₁, γ ^j=γ ⁰;

(4) be calculated as follows

$\widehat{\mathrm{\&Phi;}}=-{(A^{T}A+{\mathrm{\&lambda;}}_2{M_2}^T{M}_2)}^{-1}{A}^T{D}_2(S-U);$

(5) be calculated as follows parameter

${\widehat{\mathrm{\&lambda;}}}^{j+1}=\frac{1}{1+{\mathrm{\&gamma;}}^j}\frac{S^{T}M{({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)}^{T}S}{S^{T}M{({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)}^{T}M({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)S};$

(6) be calculated as follows

${\hat{U}}_{j+1}={(T^{T}T+(1+{\mathrm{\&gamma;}}^j){{\mathrm{\&lambda;}}_1}^{j+1}E)}^{-1}(T^T\mathrm{TS}+{\mathrm{\&gamma;}}^j{{\mathrm{\&lambda;}}_1}^{j+1}S);$

(7) be calculated as follows γ ^j+1:

${\mathrm{\&gamma;}}^{j+1}=\frac{(S-{\hat{U}}_{j+1})S}{{\left|\right|S-{\hat{U}}_{j+1}\left|\right|}^2}-1;$

(8) judge whether to meet if meet, $\hat{U}={\hat{U}}_{j+1},$ ${\widehat{\mathrm{\&lambda;}}}_1={{\widehat{\mathrm{\&lambda;}}}_1}^{j+1},$ $\widehat{\mathrm{\&gamma;}}={\mathrm{\&gamma;}}^{j+1},$ $\widehat{\mathrm{\&alpha;}}={\widehat{\mathrm{\&alpha;}}}_j$ Output and U=S-R; Otherwise order; Otherwise make j=j+1 get back to step (5);

Wherein, $A=[A_{D_1(S-U)},A_{S-U}],$ $M_1={[{D_1}^T,E^T]}^T,$ $M_2=\left[\begin{array}{cc}{D}_2& \\ & E\end{array}\right],$ p and q can be from $\widehat{\mathrm{\&Phi;}}={[p^T,q^T]}^T$ Middle acquisition, $T=D_2+B_{\widehat{\mathrm{\&Phi;}}}{M}_1,$ $M({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)={(T^{T}T+(1+\widehat{\mathrm{\&gamma;}}){\mathrm{\&lambda;}}_{1}E)}^{-1}.$

(9) gained narrow band signal and former bearing fault-signal are formed to one group of new observation signal;

(10), to the processing of above-mentioned observation signal prewhitening, obtain observing matrix and albefaction matrix after albefaction;

(11) above-mentioned signal is carried out to associating diagonalization processing and obtain source signal;

(12) above-mentioned gained source signal carried out to Hilbert demodulation process and then realize the diagnosis of bearing combined failure;

Fig. 2 is the time domain waveform figure of centre bearer Internal and external cycle combined failure experimental signal of the present invention.Motor speed R=1496r/min, the large footpath D=80mm of bearing, path d=35mm, rolling body number is Z=8, contact angle α=0.Sample frequency is 15369Hz, and sampling number is 4096 points, and inner ring characteristic frequency is 123.72Hz, and outer ring characteristic frequency is 78.72Hz.In figure, can find out under noise effect, impact not obvious.

Fig. 3 is the demodulation spectrogram of centre bearer Internal and external cycle combined failure experimental signal of the present invention.Can see and not only comprise inner ring characteristic frequency but also comprise outer ring characteristic frequency.

Fig. 4 utilizes kernel tracing algorithm Internal and external cycle combined failure to be decomposed to the time-domain diagram of 3 narrow band signals that obtain for 3 times in the present invention.

The time domain waveform figure of Fig. 5 after to be the present invention separate Internal and external cycle combined failure, can see and impact that rule is obvious more.

Fig. 6 is the demodulation spectrogram after the present invention separates Internal and external cycle combined failure.From figure, can clearly find that Internal and external cycle fault characteristic frequency and frequency multiplication have been separated clearly.

Claims (2)

1. the bearing combined failure diagnostic method separating with blind source based on kernel differentiating operator, it is characterized in that: the method comprises according to fault vibration model utilizes the corresponding kernel differentiating operator of Matlab programming constructs, utilize the kernel differentiating operator based on bearing fault characteristics to decompose bearing combined failure signal to be analyzed, the narrow band signal that decomposition is obtained and bearing combined failure signal are regarded one group of observation signal as and are carried out blind source and separate, and carry out demodulation analysis and obtain fault signature separating rear signal;

The bearing fault model of vibration of wherein using is to set up according to the failure mechanism of bearing; Can be similar to and regard quality-spring-damper system as according to the fault vibration model of bearing fault mechanism bearing, be periodic decaying exponential function:

$x\left(t\right)=\underset{-\&infin;}{\overset{+\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{ke}}^{-\mathrm{nt}}\cos \mathrm{\&omega;t\&delta;}(t-k{T}_0)$

Wherein, can think decaying exponential function be bearing fault model of vibration fundametal component:

y(t)＝ke ^-ntcos(ωt)

Easily know that by dynamics above formula is second order differential equation y "+2ny'+ ω ²the approximate solution of y=0, i.e. y (t)=ke ^-nt(ω t) is in Second Order Differential Operator y "+2ny'+ ω to cos ²in the kernel of y=0;

For bearing fault signal model: y (t)=ke ^-nt(t) it is in differentiating operator d to ω to cos ²/ dt ²-2a (t) ' d/a (t) dt+ ω (t) ²+ 2 (a (t) '/a (t)) ²kernel in; Wherein a (t)=ke ^-nt;

Wherein kernel decomposition algorithm detailed step is as follows:

(1) input bearing combined failure signal S, stops threshold epsilon, λ ₂ ⁰, γ ⁰and λ ₁ ⁰initial value;

Make j=0, U _j=0, λ ^j ₁=λ ⁰ ₁, γ ^j=γ ⁰;

(2) be calculated as follows

$\widehat{\mathrm{\&Phi;}}=-{(A^{T}A+{\mathrm{\&lambda;}}_2{M_2}^T{M}_2)}^{-1}{A}^T{D}_2(S-U);$

(3) be calculated as follows parameter

${\widehat{\mathrm{\&lambda;}}}^{j+1}=\frac{1}{1+{\mathrm{\&gamma;}}^j}\frac{S^{T}M{({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)}^{T}S}{S^{T}M{({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)}^{T}M({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)S};$

(4) be calculated as follows

${\hat{U}}_{j+1}={(T^{T}T+(1+{\mathrm{\&gamma;}}^j){{\mathrm{\&lambda;}}_1}^{j+1}E)}^{-1}(T^T\mathrm{TS}+{\mathrm{\&gamma;}}^j{{\mathrm{\&lambda;}}_1}^{j+1}S);$

(5) be calculated as follows γ ^j+1:

${\mathrm{\&gamma;}}^{j+1}=\frac{(S-{\hat{U}}_{j+1})S}{{\left|\right|S-{\hat{U}}_{j+1}\left|\right|}^2}-1;$

(6) judge whether to meet if meet, $\hat{U}={\hat{U}}_{j+1},$ ${\widehat{\mathrm{\&lambda;}}}_1={{\widehat{\mathrm{\&lambda;}}}_1}^{j+1},$ $\widehat{\mathrm{\&gamma;}}={\mathrm{\&gamma;}}^{j+1},$ $\widehat{\mathrm{\&alpha;}}={\widehat{\mathrm{\&alpha;}}}_j$ Output and U=S-R; Otherwise order; Otherwise make j=j+1 get back to step (3);

Wherein, $A=[A_{D_1(S-U)},A_{S-U}],$ $M_1={[{D_1}^T,E^T]}^T,$ $M_2=\left[\begin{array}{cc}{D}_2& \\ & E\end{array}\right],$ p and q can be from $\widehat{\mathrm{\&Phi;}}={[p^T,q^T]}^T$ Middle acquisition, $T=D_2+B_{\widehat{\mathrm{\&Phi;}}}{M}_1,$ $M({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)={(T^{T}T+(1+\widehat{\mathrm{\&gamma;}}){\mathrm{\&lambda;}}_{1}E)}^{-1};$

Matrix A _xthat diagonal element is the diagonal matrix of x vector; E is unit matrix, D ₁and D ₂represent single order and Second differential matrix, laGrange parameter, to be retained in T _ckernel in determine the parameter of quantity of information of S-R;

It is as follows that blind source detachment process wherein adopts eigenmatrix approximately joint diagonalization algorithm to realize concrete steps:

(1) narrow band signal and bearing combined failure signal are formed to one group of observation signal;

(2), to the processing of above-mentioned observation signal prewhitening, obtain observing matrix and albefaction matrix after albefaction;

(3) above-mentioned gained signal carries out associating diagonalization processing;

(4) obtain estimate source signal.

2. a kind of bearing combined failure diagnostic method separating with blind source based on kernel differentiating operator according to claim 1, is characterized in that: (1) sets up bearing fault model of vibration according to the failure mechanism of bearing; Can be similar to and be regarded as quality-spring-damper system by the known rolling bearing fault vibration mode of dynamics, the kinetic model of quality-spring-damper system is:

my″+cy′+ky＝0

Wherein m is spring-mass, and c is ratio of damping, and k is spring constant, makes ω _n ²=k/m, n=c/2m, definition damping ratio ξ=n/ ω _n, in underdamping situation, ξ < < 1, so above formula can be rewritten as following form:

y″+2ξωy'+ω ²y＝0

The solution of above formula is:

y(t)＝ke ^-ξωtcos(ωt)

Be that bearing fault model of vibration is the decaying exponential function occurring in the cycle, can be expressed as:

$x\left(t\right)=\underset{-\&infin;}{\overset{+\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{ke}}^{-\mathrm{nt}}\cos \mathrm{\&omega;t\&delta;}(t-k{T}_0)$

The effective constituent of model be exponential damping signal:

y(t)＝ke ^-ntcos(ωt)

(2) according to the corresponding kernel differentiating operator of bearing fault Construction of A Model; "+2 ξ ω y'+ ω that easily know that by (1) effective constituent of bearing fault model is second order differential equation y ²the solution of y=0 is that the fault vibration model of bearing is in Second Order Differential Operator y "+2 ξ ω y'+ ω ²in the kernel of y; For bearing fault signal model: y (t)=ke ^{-ξ ω t}(ω t) easily proves that it is in differentiating operator d to cos ²/ dt ²-2a (t) ' d/a (t) dt+ ω (t) ²+ 2 (a (t) '/a (t)) ²kernel in; Wherein a (t)=ke ^{-ξ ω t};

(3) input original signal S, stops threshold epsilon, λ ₂ ⁰, γ ⁰and λ ₁ ⁰initial value;

Make j=0, U _j=0, λ ^j ₁=λ ⁰ ₁, γ ^j=γ ⁰;

(4) be calculated as follows

$\widehat{\mathrm{\&Phi;}}=-{(A^{T}A+{\mathrm{\&lambda;}}_2{M_2}^T{M}_2)}^{-1}{A}^T{D}_2(S-U);$

(5) be calculated as follows parameter

${\widehat{\mathrm{\&lambda;}}}^{j+1}=\frac{1}{1+{\mathrm{\&gamma;}}^j}\frac{S^{T}M{({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)}^{T}S}{S^{T}M{({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)}^{T}M({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)S};$

(6) be calculated as follows

${\hat{U}}_{j+1}={(T^{T}T+(1+{\mathrm{\&gamma;}}^j){{\mathrm{\&lambda;}}_1}^{j+1}E)}^{-1}(T^T\mathrm{TS}+{\mathrm{\&gamma;}}^j{{\mathrm{\&lambda;}}_1}^{j+1}S);$

(7) be calculated as follows γ ^j+1:

${\mathrm{\&gamma;}}^{j+1}=\frac{(S-{\hat{U}}_{j+1})S}{{\left|\right|S-{\hat{U}}_{j+1}\left|\right|}^2}-1;$

(8) judge whether to meet if meet, $\hat{U}={\hat{U}}_{j+1},$ ${\widehat{\mathrm{\&lambda;}}}_1={{\widehat{\mathrm{\&lambda;}}}_1}^{j+1},$ $\widehat{\mathrm{\&gamma;}}={\mathrm{\&gamma;}}^{j+1},$ $\widehat{\mathrm{\&alpha;}}={\widehat{\mathrm{\&alpha;}}}_j$ Output and U=S-R; Otherwise order; Otherwise make j=j+1 get back to step (5);

Wherein, $A=[A_{D_1(S-U)},A_{S-U}],$ $M_1={[{D_1}^T,E^T]}^T,$ $M_2=\left[\begin{array}{cc}{D}_2& \\ & E\end{array}\right],$ p and q can be from $\widehat{\mathrm{\&Phi;}}={[p^T,q^T]}^T$ Middle acquisition, $T=D_2+B_{\widehat{\mathrm{\&Phi;}}}{M}_1,$ $M({{\mathrm{\&lambda;}}_1}^j,{\mathrm{\&gamma;}}^j,T)={(T^{T}T+(1+\widehat{\mathrm{\&gamma;}}){\mathrm{\&lambda;}}_{1}E)}^{-1};$

(9) gained narrow band signal and former bearing fault-signal are formed to one group of new observation signal;

(10), to the processing of above-mentioned observation signal prewhitening, obtain observing matrix and albefaction matrix after albefaction;

(11) above-mentioned signal is carried out to associating diagonalization processing and obtain source signal;

(12) above-mentioned gained source signal carried out to Hilbert demodulation process and then realize the diagnosis of bearing combined failure.