CN106934125B - Residual life prediction method for trapezoidal noise distribution index model mechanical equipment - Google Patents

Residual life prediction method for trapezoidal noise distribution index model mechanical equipment Download PDF

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CN106934125B
CN106934125B CN201710109876.3A CN201710109876A CN106934125B CN 106934125 B CN106934125 B CN 106934125B CN 201710109876 A CN201710109876 A CN 201710109876A CN 106934125 B CN106934125 B CN 106934125B
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雷亚国
李宁波
李乃鹏
闫涛
林京
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Xian Jiaotong University
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Abstract

A method for predicting residual life of mechanical equipment by trapezoidal noise distribution index model includes such steps as creating trapezoidal noise distribution index model, then monitoring and collecting vibration signals of a bearing, a gear or a rotor in mechanical equipment in real time, extracting health state indexes from the vibration signals, determining fitting starting time, finally performing parameter estimation on a degradation model, and giving residual life estimation and probability distribution of a rolling bearing by adopting a random sampling method. That is, the initial value of the exponential model noise term is increased, and the health state index noise of the smooth operation stage is taken as the initial value of the model noise term, the experimental data of the accelerated life of the bearing are adopted to verify that the index model of trapezoidal noise distribution has higher precision and reliability for predicting the residual life compared with the traditional index model.

Description

Residual life prediction method for trapezoidal noise distribution index model mechanical equipment
Technical Field
The invention relates to the technical field of equipment residual life prediction, in particular to a residual life prediction method for trapezoidal noise distribution index model mechanical equipment.
Background
With the rapid development of manufacturing techniques and the continued expansion of the natural field of human exploration, many devices become more and more complex. Due to the complexity of the machinery and the effects of various operational factors (e.g., wear, external shock, load, operating environment), the performance and health of these devices will inevitably degrade and eventually fail. In the case of actual engineering equipment, once accidents caused by failures occur, the loss of personnel and property and even the damage to the environment are often immeasurable. Therefore, how to effectively evaluate the operation state of the mechanical equipment and prevent accidents caused by the failure of the mechanical equipment is a problem which needs to be solved urgently at present. From the beginning of degradation to complete failure of the mechanical device, a gradual degradation process is experienced. Therefore, if the abnormality can be found in time and the residual service life of the equipment can be predicted according to the monitoring information at the early stage of the degradation, the optimal time for maintaining the equipment is determined according to the abnormality, and the method has important significance for practically guaranteeing the operation safety, reliability and economy of the complex equipment.
The index model residual life prediction method is proposed by Gebraeel et al, university of Prime, USA, and attempts to describe the decline trend by using an index model and evaluate model parameters according to observation data so as to predict the health decline trend and residual life of equipment at a future time. In order to improve the accuracy of parameter evaluation, the index model is improved by Shixiansheng, Qinghua university and the like, and model parameters are evaluated by adopting a method combining expectation maximization and Bayesian updating, so that a better parameter evaluation effect is obtained. In an exponential prediction model, the accuracy of the model is a key factor influencing the prediction accuracy of the model, the above research work assumes that a noise term of the model obeys standard Brownian motion distribution of an initial value of 0, namely triangular noise distribution, and according to a large amount of data observation, the noise term of the model has larger fluctuation at the initial time of life prediction, and the fluctuation amplitude is larger and larger along with the increase of time. Therefore, the noise term of the conventional exponential model is assumed to be inconsistent with the actual situation, which results in the reduction of the prediction accuracy and reliability of the model.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to provide a method for predicting the residual life of mechanical equipment of an exponential model with trapezoidal noise distribution, and the prediction precision and reliability of the exponential model are improved.
In order to achieve the purpose, the invention adopts the technical scheme that:
a residual life prediction method for mechanical equipment of an exponential model of trapezoidal noise distribution comprises the following steps:
step 1, establishing an exponential model of trapezoidal noise distribution:
y＝aebt+ε (1)
wherein y is index of index model health state, a, b are index model trend term parameters, epsilon is index model noise term, obedience mean is 0, variance is
Figure BDA0001234128080000021
Is the normal distribution of (a), t is the time t from the start of the fitting0The time of the start of the operation,
Figure BDA0001234128080000022
is an initial value of the noise term of the exponential model, sigma2Is an exponential model noise term variance diffusion coefficient;
step 2, monitoring and acquiring vibration signals of a bearing, a gear or a rotor in mechanical equipment in real time, calculating a health state index sequence, and determining a fitting starting moment t0；
Step 3, the index sequence of the health state is processed according to the fitting starting time t0Two subsequences are divided: health state index sequence Y ═ Y in steady operation stage1′,y′2,y3′,…,y′k]And the index model health state index observed value sequence Y ═ Y1,y2,y3,…,yk](ii) a Health state index sequence Y ═ Y in steady operation stage1′,y′2,y3′,…,y′k]For fitting the starting time t0Pre-health status indicator subsequence, yiIs' tiThe' health status indicator at the time point,ti' is the working time, where ti′＜t0(ii) a Index model health state index observed value sequence Y ═ Y1,y2,y3,…,yk]For fitting the starting time t0Post-health status indicator subsequence, yiIs ti+t0Health status indicator at time, tiTo start from fitting time t0Operating time of start, where ti＜tk，tkTo start from fitting time t0The last working moment of the start, k is the number of observation moments;
step 4, estimating parameters in the exponential model of the trapezoidal noise distribution so as to obtain the exponential model of the trapezoidal noise distribution;
step 5, obtaining the final working time t by an exponential model of trapezoidal noise distributionkThereafter, the health status index y is predicted at each timepObey mean value of aebtVariance is σ2(t-tk) Is normally distributed, i.e.
yp～N(aebt,σ2(t-tk)) (12)
Where t > tk；
Step 6, let t equal tk+Δt,tk+2 Δ t, …, according to ypIs performed by the probability density function ofsSub-randomly sampling to obtain NsHealth state index prediction sequence Y of individual index modelp＝[yp1,yp2,yp3,…]I.e. NsThe failure time EOL of each degraded track is the first prediction index y in the index model health state index prediction sequencepExceeding the failure threshold yhAt the moment of time, i.e.
EOL＝Inf(yp＞yh) (13)
The residual life RUL is the failure time EOL and the fitting starting time t0Last working moment t of startkA difference of (i) that
RUL＝EOL-tk(14)
To NsCounting the residual service life RUL of the strip degradation track, and drawing the residualAnd (3) performing frequency distribution histogram of the service life RUL and fitting a corresponding probability density function, determining a 95% remaining service life confidence interval, and giving an estimated value of the remaining service life.
The step 4 comprises the following specific steps:
4.1) estimating initial value of noise term of exponential model
Figure BDA0001234128080000041
Firstly, performing median filtering on a health state index sequence in a stable operation stage to obtain a health state index sequence trend in the stable operation stage; subtracting the trend of the health state index sequence in the steady operation stage obtained by the median filtering from the health state index sequence in the steady operation stage to obtain the health state index sequence noise in the steady operation stage, and calculating the variance of the health state index sequence noise as the initial value of a noise item
Figure BDA0001234128080000042
(ii) an estimate of (d);
4.2) estimating the index model trend term parameters a, b by using a maximum likelihood method:
because of the exponential model noise term
Figure BDA0001234128080000043
Index model health status index
Figure BDA0001234128080000044
Probability density function of index model health status index at a certain moment:
wherein θ ═ (a, b);
likelihood function L (θ):
log-likelihood function l (θ):
Figure BDA0001234128080000047
the maximum likelihood function is therefore the minimum residual sum:
Figure BDA0001234128080000051
obtaining the estimated values of the index model trend item parameters a and b through a numerical optimization algorithm;
4.3) estimating the exponential model noise term variance diffusion coefficient sigma2：
Transforming the exponential model noise term of equation (2) into
ε～N(0,σ2(t+t0)) (8)
Wherein
Thus, it is possible to provide
Figure BDA0001234128080000053
Figure BDA0001234128080000054
Solving to obtain an exponential model noise term variance diffusion coefficient sigma according to the two formulas of (9) and (11)2And further obtaining an exponential model of trapezoidal noise distribution.
The invention has the beneficial effects that: the invention solves the problem that the noise item assumption in the traditional index prediction model does not conform to the actual situation, changes the original triangular noise distribution into trapezoidal noise distribution, namely, increases the initial value of the index model noise item, and takes the health state index sequence noise in the stable operation stage as the initial value of the index model noise item. Compared with the traditional index model, the model can better reflect the real decline trend of mechanical equipment, and improves the prediction precision and reliability of the residual service life.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a graph of vibration signals over the life of a test bearing.
FIG. 3 is a diagram of the selection result of the effective value index and the regression starting time of the bearing to be tested.
FIG. 4 shows the state at time tkWhen the time is 5000s, the exponential model of the trapezoidal noise distribution is compared with the fitting of the traditional exponential model noise term, the graph (a) is the fitting of the exponential model noise term of the trapezoidal noise distribution, and the graph (b) is the fitting of the traditional exponential model noise term.
FIG. 5 is an exponential model of the trapezoidal noise distribution at time tkWhen the prediction result is 5000s, the graph (a) is an effective value prediction trend graph, and the graph (b) is a remaining life prediction distribution graph.
FIG. 6 is a diagram of a conventional exponential model at time tkWhen the prediction result is 5000s, the graph (a) is an effective value prediction trend graph, and the graph (b) is a remaining life prediction distribution graph.
Fig. 7 is a graph of the residual life estimation and predicted 95% confidence interval results from the prediction start time of the exponential model of trapezoidal noise distribution.
Fig. 8 is a graph of the residual life estimation and prediction 95% confidence interval results of the conventional exponential model from the prediction start time.
Detailed Description
The present invention is described in further detail below with reference to the attached drawings.
As shown in fig. 1, a method for predicting the remaining life of mechanical equipment by using an exponential model of trapezoidal noise distribution includes the following steps:
step 1, establishing an exponential model of trapezoidal noise distribution:
y＝aebt+ε (1)
wherein y is index of index model health state, a, b are index model trend term parameters, epsilon is index model noise term, obedience mean is 0, variance is
Figure BDA0001234128080000061
Is the normal distribution of (a), t is the time t from the start of the fitting0The time of the start of the operation,is an initial value of the noise term of the exponential model, sigma2Is an exponential model of the variance diffusion coefficient of the noise term because of the variance of the noise term
Figure BDA0001234128080000063
The exponential model is an initial value, linearly increases along with time t and is in trapezoidal distribution, so the exponential model is called as an exponential model of trapezoidal noise distribution;
step 2, monitoring and acquiring vibration signals of a bearing, a gear or a rotor in mechanical equipment in real time, calculating a health state index sequence, and determining a fitting starting moment t0；
Step 3, the index sequence of the health state is processed according to the fitting starting time t0Two subsequences are divided: health state index sequence Y ═ Y in steady operation stage1′,y′2,y3′,…,y′k]And the index model health state index observed value sequence Y ═ Y1,y2,y3,…,yk](ii) a Health state index sequence Y ═ Y in steady operation stage1′,y′2,y3′,…,y′k]For fitting the starting time t0Pre-health status indicator subsequence, yiIs' ti' health State index at time, ti' is the working time, where ti′＜t0(ii) a Index model health state index observed value sequence Y ═ Y1,y2,y3,…,yk]For fitting the starting time t0Post-health status indicator subsequence, yiIs ti+t0Health status indicator at time, tiTo start from fitting time t0Operating time of start, where ti＜tk，tkTo start from fitting time t0The last working moment of the start, k is the number of observation moments;
and 4, estimating parameters in the exponential model of the trapezoidal noise distribution to further obtain the exponential model of the trapezoidal noise distribution, wherein the specific steps are as follows:
4.1) estimating initial value of noise term of exponential model
Figure BDA0001234128080000071
In order to eliminate the initial value of noise term caused by the change of factors such as working conditions, loads and the like in the stable operation stage of mechanical equipment
Figure BDA0001234128080000072
Estimating the generated interference, and firstly carrying out median filtering on the health state index sequence in the stable operation stage to obtain the health state index sequence trend in the stable operation stage; subtracting the trend of the health state index sequence in the steady operation stage obtained by the median filtering from the health state index sequence in the steady operation stage to obtain the health state index sequence noise in the steady operation stage, and calculating the variance of the health state index sequence noise as the initial value of a noise item
Figure BDA0001234128080000073
(ii) an estimate of (d);
4.2) estimating the index model trend term parameters a, b by using a maximum likelihood method:
because of the exponential model noise term
Figure BDA0001234128080000074
Index model health status index
Probability density function of index model health status index at a certain moment:
wherein θ ═ (a, b);
likelihood function L (θ):
Figure BDA0001234128080000082
log-likelihood function l (θ):
Figure BDA0001234128080000083
the maximum likelihood function is therefore the minimum residual sum:
Figure BDA0001234128080000084
obtaining the estimated values of the index model trend item parameters a and b through a numerical optimization algorithm;
4.3 estimating the exponential model noise term variance diffusion coefficient σ2：
Transforming the exponential model noise term of equation (2) into
ε～N(0,σ2(t+t0)) (8)
Wherein
Figure BDA0001234128080000085
Thus, it is possible to provide
Figure BDA0001234128080000086
Solving to obtain an exponential model noise term variance diffusion coefficient sigma according to the two formulas of (9) and (11)2Obtaining an exponential model of trapezoidal noise distribution;
step 5, obtaining the final working time t by an exponential model of trapezoidal noise distributionkThereafter, the health status index y is predicted at each timepObey mean value of aebtVariance is σ2(t-tk) Is normally distributed, i.e.
yp～N(aebt,σ2(t-tk)) (12)
Where t > tk；
Step 6, let t equal tk+Δt,tk+2 Δ t, …, according to ypIs performed by the probability density function ofsSub-randomly sampling to obtain NsHealth state index prediction sequence Y of individual index modelp＝[yp1,yp2,yp3,…]I.e. NsThe failure time EOL of each degraded track is the first prediction index y in the index model health state index prediction sequencepExceeding the failure threshold yhAt the moment of time, i.e.
EOL＝Inf(yp＞yh) (13)
The residual life RUL is the failure time EOL and the fitting starting time t0Last working moment t of startkA difference of (i) that
RUL＝EOL-tk(14)
To NsAnd counting the residual service life RUL of the strip degradation track, drawing a frequency distribution histogram of the residual service life RUL, fitting a corresponding probability density function, determining a 95% residual service life confidence interval, and giving a residual service life estimation value.
The present invention will be described in detail with reference to examples.
Example (b): the invention is verified by taking the experimental data of the accelerated life of the rolling bearing as an example.
The accelerated life test of the rolling bearing is completed on a PRONOSTIA test table, and the bearing works under a high-load condition by loading the bearing in air pressure, so that the bearing can be degraded from a normal state to complete failure within several hours. In the experiment, the bearing rotating speed is 1800rpm, and the load is 4 kN. An acceleration sensor is adopted to sample the bearing vibration signal, the sampling frequency is 25.6kHz, the data length is 2560, the sampling duration is 0.1s each time, and the sampling interval is 10 s. When the vibration amplitude exceeds 20g, the bearing completely fails. The life-cycle vibration signal for the experimental bearing is shown in figure 2.
Extracts the effective value from the vibration signal as the health status indicator and determines the fitting start time, as shown in figure 3,
selecting a time tkThe model is evaluated when the time is 5000s, the noise distribution is shown as a solid line in fig. 4, parameters are estimated by using a trapezoidal noise distribution index model and a traditional index model respectively, and the obtained 95% confidence intervals of the fitted noise term are shown as fig. 4(a) and fig. 4(b) respectively.
The prediction results of the trapezoidal noise distribution index model are shown in fig. 5. FIG. 5(a) is a prediction expectation and interval for the effective value, FIG. 5(b) is a predicted remaining life profile, the remaining life is estimated at 7840s, the 95% confidence interval is [7160,8310] s, and the true remaining life is 8000 s. FIG. 6 shows the predicted results of a conventional exponential model with a residual life estimate of 6570s, a 95% confidence interval of [5530,7320] s, and a true residual life of 8000 s. The comparison result shows that the trapezoidal noise distribution index model is more accurate in prediction and higher in precision.
FIG. 7 shows a trapezoidal noise distribution exponent model from tkThe residual life estimate and 95% confidence interval predictions started at 1000s and compared to the conventional exponential model predictions (fig. 8). As can be seen from the figure, the trapezoidal noise distribution index model and the traditional index model have undesirable predicted fluctuation in the early prediction period, and both methods gradually converge to the true value along with the time. But the convergence rate of the trapezoidal noise distribution index model is higher, and the prediction precision is higher; and the 95% confidence interval obtained by the trapezoidal noise distribution index model is smaller, and the estimation of the residual life is more reliable
The advantage of the method in the service life prediction of mechanical equipment is verified by adopting bearing accelerated life experimental data. The method improves the traditional index model, increases the initial value of the distribution variance of the noise items of the index model, and enables the distribution variance to be more consistent with the rule of degradation data of mechanical equipment, thereby improving the accuracy and reliability of the index model for predicting the service life of the mechanical equipment.
The method for predicting the residual service life of the mechanical equipment by the trapezoidal noise distribution index model is not limited to the residual service life prediction of the mechanical equipment, and can also be applied to the residual service life prediction of other electronic elements. A large number of research works prove that the method is suitable for predicting the residual life of various electromechanical products with exponential decay forms. The practitioner only needs to make appropriate adjustments to the corresponding steps of the method to suit the application requirements of different products, and it should be noted that modifications and variations can be made without departing from the spirit of the invention.

Claims (1)

1. A residual life prediction method for mechanical equipment of an exponential model of trapezoidal noise distribution is characterized by comprising the following steps:
step 1, establishing an exponential model of trapezoidal noise distribution:
y＝aebt+ε (1)
wherein y is index of index model health state, a, b are index model trend term parameters, epsilon is index model noise term, obedience mean is 0, variance is
Figure FDA0002196874980000011
Is the normal distribution of (a), t is the time t from the start of the fitting0The time of the start of the operation,
Figure FDA0002196874980000012
is an initial value of the noise term of the exponential model, sigma2Is an exponential model noise term variance diffusion coefficient;
step 2, monitoring and acquiring vibration signals of a bearing, a gear or a rotor in mechanical equipment in real time, calculating a health state index sequence, and determining a fitting starting moment t0；
Step 3, the index sequence of the health state is processed according to the fitting starting time t0Two subsequences are divided: smooth operation stage health status indicator sequence Y ═ Y'1,y′2,y′3,…,y′k]And index modelHealth status indicator observation sequence Y ═ Y1,y2,y3,…,yk](ii) a Smooth operation stage health status indicator sequence Y ═ Y'1,y′2,y′3,…,y′k]For fitting the starting time t0Pre-health status indicator subsequence, yiIs' ti'health State index at time, t'iIs working time, wherein t'i＜t0(ii) a Index model health state index observed value sequence Y ═ Y1,y2,y3,…,yk]For fitting the starting time t0Post-health status indicator subsequence, yiIs ti+t0Health status indicator at time, tiTo start from fitting time t0Operating time of start, where ti＜tk，tkTo start from fitting time t0The last working moment of the start, k is the number of observation moments;
step 4, estimating parameters in the exponential model of the trapezoidal noise distribution so as to obtain the exponential model of the trapezoidal noise distribution;
step 5, obtaining the final working time t by an exponential model of trapezoidal noise distributionkThereafter, the health status index y is predicted at each timepObey mean value of aebtVariance is σ2(t-tk) Is normally distributed, i.e.
yp～N(aebt,σ2(t-tk)) (12)
Where t > tk；
Step 6, let t equal tk+Δt,tk+2 Δ t, …, according to ypIs performed by the probability density function ofsSub-randomly sampling to obtain NsHealth state index prediction sequence Y of individual index modelp＝[yp1,yp2,yp3,…]I.e. NsThe failure time EOL of each degraded track is the first prediction index y in the index model health state index prediction sequencepExceeding the failure threshold yhAt the moment of time, i.e.
EOL＝Inf(yp＞yh) (13)
The residual life RUL is the failure time EOL and the fitting starting time t0Last working moment t of startkA difference of (i) that
RUL＝EOL-tk(14)
To NsCounting the residual service life RUL of the strip degradation track, drawing a frequency distribution histogram of the residual service life RUL, fitting a corresponding probability density function, determining a 95% residual service life confidence interval, and giving a residual service life estimated value;
the step 4 comprises the following specific steps:
4.1) estimating initial value of noise term of exponential model
Figure FDA0002196874980000021
Firstly, performing median filtering on a health state index sequence in a stable operation stage to obtain a health state index sequence trend in the stable operation stage; subtracting the trend of the health state index sequence in the steady operation stage obtained by the median filtering from the health state index sequence in the steady operation stage to obtain the health state index sequence noise in the steady operation stage, and calculating the variance of the health state index sequence noise as the initial value of a noise item
Figure FDA0002196874980000022
(ii) an estimate of (d);
4.2) estimating the index model trend term parameters a, b by using a maximum likelihood method:
because of the exponential model noise term
Figure FDA0002196874980000031
Index model health status index
Probability density function of index model health status index at a certain moment:
Figure FDA0002196874980000033
wherein θ ═ (a, b);
likelihood function L (θ):
Figure FDA0002196874980000034
log-likelihood function l (θ):
Figure FDA0002196874980000035
the maximum likelihood function is therefore the minimum residual sum:
Figure FDA0002196874980000036
obtaining the estimated values of the index model trend item parameters a and b through a numerical optimization algorithm;
4.3) estimating the exponential model noise term variance diffusion coefficient sigma2：
Transforming the exponential model noise term of equation (2) into
ε～N(0,σ2(t+t0)) (8)
Wherein
Thus, it is possible to provide
Figure FDA0002196874980000042
Figure FDA0002196874980000043
Solving to obtain an exponential model noise term variance diffusion coefficient sigma according to the two formulas of (9) and (11)2To obtain an estimated value ofAn exponential model of the trapezoidal noise distribution is generated.
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