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Vibration monitoring plays a key role in the industrial machinery reliability since it allows enhancing the performance of the machinery under supervision through the detection of failure modes. Thus, vibration monitoring schemes that give information regarding future condition, that is, prognosis approaches, are of growing interest for the scientific and industrial communities. This work proposes a vibration signal prognosis methodology, applied to a rotating electromechanical system and its associated kinematic chain. The method combines the adaptability of neurofuzzy modeling with a signal decomposition strategy to model the patterns of the vibrations signal under different fault scenarios. The model tuning is performed by means of Genetic Algorithms along with a correlation based interval selection procedure. The performance and effectiveness of the proposed method are validated experimentally with an electromechanical test bench containing a kinematic chain. The results of the study indicate the suitability of the method for vibration forecasting in complex electromechanical systems and their associated kinematic chains.

Rotating machinery, like compressors, steam turbines, industrial fans, and so forth, is widely used in many industrial fields. Its reliability is an extensively investigated field, aimed at prolonging their life span and minimizing their maintenance cost. Maintenance programs try to avoid fatal breakdowns of machines and prevent production loss and human casualties. The early fault diagnosis is a challenging problem and has received more and more attention in recent years [

Vibration measurement is an effective, nonintrusive method to monitor machine condition during start-ups, shutdowns, and normal operation [

Nevertheless, the vibration signal is often a complex signal which contains stationary, nonstationary, and noisy components. Therefore, the appropriate signal processing techniques have to be applied in order to compute numerical fault indicators or features [

Clearly, the opportunity of conducting forecasting based on vibrations presents multiple benefits: (i) correct deviations before they affect the correct behavior of the system, (ii) anticipate the response towards failures, and (iii) assist the diagnosis algorithm in order to assess the future condition of the system [

Consequently, the forecasting of such vibration feature signals can be approached from the time series modeling point of view. Current modeling and forecasting techniques can be categorized into three classes, namely, model-based, data-driven, and hybrid prognostics approaches [

When facing vibration forecasting, the problem is how to decide for a given application the optimal forecasting horizon and the configuration of the models. In this regard, vibration signals usually present faster or at least equal dynamics in comparison with other magnitudes inherent to an electromechanical system, such as voltage, electric current, and temperature. Due to this fact, the modeling problem should be approached as a time series forecasting [

In this paper, the authors propose a study in order to optimize these three open issues to obtain a methodology for modeling the RMS of the vibrations from an electromechanical system and its associated kinematic chain. Therefore, the contributions of this work include a novel vibration forecasting method that utilizes the theory behind vibration signature under electromechanical failures to decompose the vibration signal in three frequency bands related to the vibration signature to enhance the forecasting accuracy and adaptability towards different dynamic contents. Then, the core of the methodology consists of the generation of three collaborative ANFIS models that are in charge of forecasting the evolution of specific spectral content of the vibration signal. The configuration of the model is optimized one step beyond the literature by analyzing cross-correlations between the signal and past instants, which are selected by using Genetic Algorithm (GA) optimization, to find the best input configuration. Complementarily, the paper includes a systematic study of the forecasting horizon affectation of the model configuration step, in order to establish a valid method for selecting the best forecasting horizon with regard to a proposed application. Finally, the proposed method is validated experimentally by means of vibration data extracted from a complex kinematic chain working under different failure conditions.

The paper is structured as follows: Section

The classical vibration condition monitoring schemes, focused on mechanical and electrical defects, are based on the consideration of specific vibrational frequency components [

Nevertheless, these characteristic frequencies of the vibration modes, apart from possible electrical and mechanical noise during the acquisition, are affected by the specific structure of the considered system (i.e., the kinematic chain) and the component deterioration stage. Indeed, the intensity of the impact produced between raceways and rolling elements, under a bearing defect, is attenuated throughout the propagation from the generation point to the transducer location. Therefore, the vibration measurement is done, generally, as close as possible to the mechanical element under test. Yet, though favoring the measurement of bearing faults vibration effects, the resulting spectrum contains additional vibration modes produced by the rest of the mechanical interactions. These additional vibration frequencies mask or complicate the comprehension of the signal.

The detection of one of the characteristic fault frequencies in the resulting spectrum should be interpreted as an existence of the corresponding fault in the component. However, the absence of clear characteristic fault frequencies should not be interpreted as a completely healthy condition. In this regard, a more general approach to the monitoring of the electromechanical components can be found in the literature as a four-stage process [

Main frequency zones in mechanical degradation.

In conclusion, the first three spectral zones are of major interest from the diagnosis and the vibration monitoring point of view. The forecasting method is intended to take advantage from this spectral decomposition in order to isolate each frequency band and gain forecasting performance by modeling each band, from

There are different techniques to design forecasting models. Yet, the primarily used schemes are frequently based on the concept of hybrid forecasting, which means that the method integrates different techniques in order to take advantage of each one involved. In this topic, one of the most important hybrid systems is ANFIS [

The neural-fuzzy architecture can be divided into five different layers, as shown in Figure

Adaptive network-based Fuzzy Inference System scheme with two inputs, four membership functions, two rules, and one single output.

The input layer,

The ANFIS structure allows the consideration of a data-driven modeling approach with adaptive rule changing capability and fast convergence rate and does not require extensive experience about the process to include such patterns in the fuzzy rules.

The objective of the proposed method is to model and forecast the evolution curve of the vibration’s RMS signal during the start-up and the thermal stabilization of an electromechanical actuator. The main point is to obtain a forecasting model of the RMS capable of giving the future value with enough resolution taking into consideration the dynamics of different failures occurring to the system. After the output of the method, the presence of the failure can be anticipated by knowing the future value of the RMS and a simply statistical characterization as a diagnosis method.

Therefore, the challenge is to develop a vibration model with enough performance, in terms of forecasted signal error and forecasting horizon that considers the dynamics modes generated by different faulty scenarios of different elements of the electromechanical actuator. The general block diagram of the method is shown in Figure

Block diagram of the proposed method.

The training procedure corresponds to the off-line learning process in order to do the following: (i) tuning of the structure of the forecasting models, (ii) analysis of the optimum inputs delays, and (iii) identification of the longest forecasting horizon.

The first step,

First filter, band

Second filter, band

Third filter, band

The next step,

Prior to the model generation, the identification of the best past intervals to enhance the forecasting performance is proposed as

The resulting correlation coefficient is obtained by (

It is common to find modeling situations, especially with vibration signals, in which it is necessary to deal with different training sets or operating conditions. In these situations, the search for the optimal intervals turns into an iterative process that tries to localize those intervals with a consistent correlation, and once identified, select the maximum common range that accumulates the highest correlation.

The following step,

Although each model presents the same quantity of inputs, the delayed samples

In order to evaluate the performance of the models, classical statistical metrics have been used, which are RMSE defined in (

One of the primary characteristics of the proposed method is that it is prepared to work continuously online, that is, receiving new data from the machine as shown in Figure

The test bench used to obtain experimental vibrations is shown in Figure

Electromechanical test bench used for experimental validation of the method.

Vibration signal from the perpendicular plane of the motor axis is acquired using a triaxial accelerometer, LIS3L02AS4, mounted on a board with the signal conditioning and antialiasing filtering. Sampling frequency is set to 3 kHz for vibration acquisition. The data retrieved by the DAS is stored in a regular computer (PC).

Four scenarios have been considered, that is, the healthy condition, HC, a bearing failure (BF) condition, a half-broken rotor bar (HBRB) condition, and full-broken rotor bar (FBRB). The detail of the failures is shown in Figure

Detail of the failures produced in the test bench. (a) corresponds to the bearing failure, BF. (b) corresponds to the

Two different datasets with the same length have been acquired; the first one is used to train the proposed method, and the second dataset is used to validate the performance of the method. In this regard, periodical acquisitions of 30 seconds containing 90 ksamples are obtained from the accelerometer; the acquisitions are temporally spaced at 30 seconds. The average duration of the experiments for all operating conditions is configured to be 3600 seconds.

The selected duration allows obtaining the complete response of the vibrations with regard to the thermal stability of the electromechanical system. This is done in order to consider the thermal evolution of the vibrations as part of the system and facilitate the representation of the failure. The proposed approach brings the experimental validation closer to the real behavior of complex industrial machinery, in which the behavior of the vibration presents a nonconstant dynamic from the starting point till its steady state. As a result, the modeling of the vibration considering the thermal evolution represents an additional challenge to the validation of the proposed method. Furthermore, Figure

Evolution of the acceleration signal versus the motor temperature.

As can be appreciated in Figure

The vibration signal filtering and the corresponding windowed RMS calculation are carried out by means of FIR filters and considering the rotating speed, the sampling frequency, and, also, the theoretical background with regard to mechanical failures. As a result, the cut-off frequencies of the filters in order to isolate the three primary frequency bands,

Design characteristics of the three digital filters used to decompose the vibration signal. Calculations made for a sampling frequency of 3 kHz.

Filter | Cut-off frequency 1 | Cut-off frequency 2 | Order |
---|---|---|---|

| 0 Hz | 120 Hz | 248 |

| 120 Hz | 350 Hz | 136 |

| 350 Hz | 1000 Hz | 120 |

Finally, the RMS feature is calculated for each filter output; the results are 3 RMS signals that summarize the information regarding the vibration of the electromechanical actuator under a concrete failure condition and are the target signals to be modeled. For this experimental application, the temporal window is configured to be

RMS of each filter output for all operating conditions: (a) HS, (b) BF, (c) FBRB, and (d) HBRB.

Finally, to understand the fundamentals of the method, Figure

RMS of the vibration signal during the thermal stabilization period. Note that the different failures can be identified during this period, and as it was expected, the BF case shows the highest vibration.

The selection of the forecasting horizon is a crucial step that needs to be faced before the modeling. Most of the related literature selects the forecasting horizon simply by application requirements. Yet, the procedure to analyze the behavior of the target signal to find the optimal forecasting horizon is not established in the literature. It is proposed, then, in this work, to select the optimal horizon by analyzing the performances of the method at different horizon values.

The proposed analysis interval comprises from

Study of the affectation of the forecasting horizon to the forecasting performance.

The general results show that as it was expected in time series forecasting applications in which the forecasted signal does not present a periodic pattern, the lowest error can be found in the initial area that comprises

In specific conclusions, it should be noticed that low frequency dynamics are easier to be modeled independently of the selected forecasting horizon, since they represent the tendency of the vibration, and thus they suffer slower changes among time. It should be observed how error increases in the higher frequency bands, since fast dynamics are more punctual and closely related to adjacent samples of the current vibration value.

The proposed method begins once the forecasting horizon is selected and the RMS of the three filter outputs is calculated. The structure of the three models,

At this point, the problem derives to the selection of the best past values,

Due to the fast dynamics of the acquired vibration signal, the delayed signals of more than 20 acquisitions are considered to be nonsignificant for the application; therefore, the total number of delayed signals is set to 600. The amount of necessary correlation to consider a strong relation between the target signal and its self-delayed signals is not defined. Thus, the experimental results show that useful intervals are considered with a correlation threshold above 40% (

Correlation coefficient for each target RMS. (a) corresponds to the correlation in the first band,

The correlation of the target signal in

The correlation of

Finally,

In the proposed ANFIS modeling structure, each input is normalized with the min–max method in order to obtain a range from 0 to 1. The inputs are fuzzified by means of three generalized bell-shaped membership functions. The model is trained for 15 epochs by means of the classical hybrid learning algorithm, which is the combination of the least-squares method and the backpropagation gradient descent method. The resulting past value indexes after 20 generations of the GA are shown in Table

Selected past index as a result from the constrained GA based optimization.

| | |
---|---|---|

| 217 | 389 |

| 285 | 397 |

| 247 | 348 |

As a result, the forecasting models have been trained and validated using the validation set. The performance of the model for all operating conditions is quantitatively analyzed by means of RMSE, MAE, and MAPE coefficients, shown in Table

Error achieved by the model with the four different sets used in the experimental validation.

| HC | | 0.014 | | HC | | 0.020 | | HC | | 0.007 |

| 3.654% | | 4.120% | | 1.212% | ||||||

| 0.1 | | 0.127 | | 0.075 | ||||||

| |||||||||||

| BF | | 0.010 | | BF | | 0.014 | | BF | | 0.005 |

| 1.699% | | 1.213% | | 0.784% | ||||||

| 0.09 | | 0.097 | | 0.065 | ||||||

| |||||||||||

| HBRB | | 0.099 | | HBRB | | 0.017 | | HBRB | | 0.005 |

| 1.625% | | 2.245% | | 0.745% | ||||||

| 0.084 | | 0.116 | | 0.063 | ||||||

| |||||||||||

| FBRB | | 0.018 | | FBRB | | 0.014 | | FBRB | | 0.007 |

| 3.397% | | 2.815% | | 1.002% | ||||||

| 0.118 | | 0.105 | | 0.073 | ||||||

| |||||||||||

| Mean | | 0.013 | | Mean | | 0.016 | | Mean | | 0.006 |

| 2.5937% | | 2.5981% | | 0.936% | ||||||

| 0.0982 | | 0.111 | | 0.069 |

The modeling of each frequency band vibration presents remarkable results for all operating conditions, as it achieves a MAPE lower than 5% in all operating conditions. Nevertheless, the healthy condition presents a dynamic that is more difficult to model in comparison with the others. The behavior of the vibration in healthy condition presents a steady pattern since there is an absence of a strong signature related to the failure such as the BF case. This healthy pattern should seem a priori easier to be modeled; however, as the model is being forced to learn different dynamics, it dedicates more efforts to those datasets that are similar to each other.

As a consequence, the healthy state presents a different dynamic with regard to the faulty scenarios and therefore the modeling error slightly increases, around 2% of error increase in terms of MAPE. The learning ability of the model is finite; thus, there should be a trade-off between the number of different dynamics that the model needs to consider and the performance achieved by the model versus a single set. Furthermore, regarding the models, the best performance is achieved by

Performance of the statistical error metrics applied to the combination of all model outputs. Also, the performance of the forecasting without the frequency decomposition is also shown in the table.

Method | Single modeling | |||||
---|---|---|---|---|---|---|

RMSE | MAPE | MAE | RMSE | MAPE | MAE | |

HC | 0.013 | 1.69% | 0.135 | 0.203 | 3.45% | 0.129 |

BF | 0.020 | 2.27% | 0.125 | 0.275 | 6.77% | 0.265 |

HBRB | 0.022 | 2.82% | 0.133 | 0.478 | 6.92% | 0.184 |

FBRB | 0.025 | 3.44% | 0.139 | 0.394 | 8.93% | 0.192 |

Results of the forecasting method. (a) Combined output applied to HC. (b) Combined output applied to HBRB. (c) Combined output applied to FBRB. (d) Combined output applied to BF versus the output of the single model method to the same scenario.

The results show that when the outputs of the three models are combined, the obtained signals outperform in terms of error the individual results of all models. In this regard, low error is achieved in all the considered scenarios. The method is capable of obtaining a MAPE lower than 2% in all the cases studied. Also the MAE shows the stability of the output by maintaining the absolute error near 0.125 in all cases. Additionally, as can be seen in the figure, and by terms of RMSE, the proposed method presents a smooth response that presents a marginal number of outliers.

To conclude the results, the performance of the proposed method is tested versus a forecasting approach in which the vibration signal is not decomposed. The performance achieved by the second method is named single modeling that can be also seen in Table

The main limitation of the single model approach is that, despite the fact that the model is able to learn the first scenario, HC, with virtually fewer errors than the proposed method, it is not capable of modeling additional dynamics contained in the other scenarios. For example, as it can be seen in Figure

This paper presents a novel vibration signal modeling methodology applied to a kinematic chain under different failure conditions. First, it should be pointed that the RMS values of the vibration have been proved to be a representative feature regarding the health condition of the electromechanical actuator.

Furthermore, there are three important aspects in this new method having a strong implication in the vibration forecasting field. The first one is the application of a signal decomposition strategy. It has been proved that signal decomposition allows a better characterization of the signal dynamics and improves the forecasting accuracy by reducing the amount of dynamic that each model should face. The second is the optimal model configuration by means of the proposed analysis of the prediction horizon and time delays. This analysis leads to an optimal configuration of the optimization and thus the simplification of the process assuring the convergence to a reliable and robust forecasting. The third is the calculation of the forecasting value by means of the linear combination of the models. How the combination of the three individual models concludes in a better performance when facing the original forecasting method has been seen; this fact is evidenced when the proposed method is compared with the classical approach with only one model.

For the experimental validation of the method, four different operating conditions have been considered which represent an important range of system condition possibilities. Under these experimental conditions, the proposed prognosis methodology achieves reliable results with an error lower than 2% in all the exposed scenarios. Moreover, considering the analyzed model error parameters, the prognosis methodology shows still enough dynamic range to include additional patterns.

The aim of the study has been to propose a new reliable prognosis methodology for diagnosis analysis under multiple failure conditions. The study is based on different motor faults scenario. Therefore, it should be pointed that the results obtained in this work suggest that this methodology may be also useful for any other component in the kinematic chain.

A limitation of this study is that the affectation of other sources of information, such as the temperature of the motor or the stator current, has not been considered. This additional information should be correlated with the vibration of the system and thus may help to increase the response and adaptation capabilities of the models towards new scenarios. In this regard, further research can be focused on investigating information management methods such as information compression or topology preservation to exploit and add other information available in the kinematic chain in the design of the forecasting models.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported in part by the Spanish Ministry of Education, Culture, and Sport under Grant FPU13/00589. The authors also wish to acknowledge financial support from the Generalitat de Catalunya (GRC MCIA, Grant no. SGR 2014-101).