Fault diagnosis of machines operating in variable conditions using artificial neural network not requiring training data from a faulty machine

Highlights Abstract ▪ It is possible to train a neural network with only data from an undamaged machine. ▪ The order spectrum of the novel parameter rDPNS is proposed. ▪ The new method application for diagnosing unbalance and misalignment was analysed. ▪ Proposed architecture is resilient to overfitting without drop-outs and bagging. The fault diagnosis for maintenance of machines operating in variable conditions requires special dedicated methods. Variable load or temperature conditions affect the vibration signal values. The article presents a new approach to diagnosing rotating machines using an artificial neural network, the training of which does not require data from the damaged machine. This is a new approach not previously found in the literature. Until now, neural networks have been used for machine diagnosis in the form of classifiers, where data from individual faults were required. A new diagnostic parameter rDPNS (Relative Differences Product of Network Statistics) as a function of the machine's shaft order was proposed as a kind of new order spectrum independent of the machine's operating conditions. The presented work analyses the use of the proposed method to diagnose misalignment and unbalance. The results of an experiment carried out in the laboratory demonstrated the effectiveness of the proposed method.


Introduction
In the maintenance of rotating machinery, the vibration signal is the basis of condition monitoring systems. Dedicated signal analysis methods are applied depending on the technical object under investigation. An overview of methods for diagnosing planetary gears can be found in [1]. The vibration signal is often used in the diagnosis of rolling bearings, and several applications can be found in [2,3]. In a traditional condition monitoring approach following ISO 20816-1 [4], vibration measurement is required under fixed operating conditions of the machine. Unfortunately, this is often impossible to achieve in industrial conditions because a large group of machines operate exclusively in variable conditions. In addition to any damage, the values of the parameters obtained using classic methods of vibration signal analysis are also affected by the operating conditions, i.e. changes in rotational speed, load, or oil temperature [5,6]. Many studies have been conducted in which the problem of eliminating the impact of changes in rotational speed and load condition on the vibration signal was addressed [7][8][9]. One way to do this is to use synchronous methods [10][11][12], which allow for the elimination of spectrum blur resulting from variable rotational speed [5]. However, variable operating conditions also affect the values of spectral amplitudes [13,14].
Each operating condition factor affects the amplitude of diagnostic signals in a different way [15] and this requires the use of advanced signal analysis methods to separate the failure factors from the operating condition factors, given that the influence of these parameters can cause amplitude changes that can be interpreted by automatic monitoring systems as the presence of a fault. Vibration diagnostics can be carried out for fixed operating conditions, i.e. temperature, load, speed, to negate this effect. However, in industrial settings, in most cases ensuring fixed operating conditions is impossible to meet. The authors focused on the search for a method to diagnose machines operating at variable temperature and load.
In this work, a method based on the analysis of orders and artificial neural networks is proposed. A naturally occurring artificial intelligence model for diagnostic applications is that of classifier models. This is problematic, however, because when using neural networks as classifiers, learning data recorded for the machine to be diagnosed and for predicted faults are required [16][17][18][19][20][21][22][23][24]. There is work related to the reduction of damage data in learning vectors [25]. In industrial settings, acquiring measurement data for faults in a particular facility is difficult to achieve. Often, expensive expert inspections and preventive replacements are used to prevent faults from occurring, and this is especially true for machines that are unique in their design. As a result, the learning sets (necessary when using neural networks as classifiers) would contain a redundant representation of data from the undamaged machine relative to the data recorded during faults. There are review studies indicating that such unbalanced learning sets translate into poorer ability to classify underrepresented states [26][27][28] In excess of this, even for objects of the same type, vibration signals can vary considerably due to imperfect workmanship. In addition, any new unforeseen damage may be misclassified. It is also possible to search for a damage model when creating diagnostic tools [29][30][31][32]; however, damage models in most cases require calibration with measurement data.
In the approach proposed by the authors, the artificial neural network model is not directly responsible for classifying the state of the machine. The regressive artificial neural network is trained only with data recorded during the fault-free operation, creating a reference model of the correct operation of the machine. This is a new approach not previously found in the literature. Until now, neural networks have been used to diagnose machines in the form of classifiers, where data from individual faults were necessary. The authors emphasise that, as a result, the presented method allows diagnostics to be automated without the need for prior data collection from damaged machines.
The result of the applied method is the generation of values for a proposed diagnostic parameter rDPNS (Relative Differences Product of Network Statistics) as a function of the shaft order. This allows for damage identification according to the theory of classical vibroacoustic diagnostics, based on spectral analysis. This solves another problem of using artificial intelligence for diagnostics as there is no need to predict particular types of damage in advance.
The next section presents a method for analysis of measurement signals in order to obtain input data for the artificial neural network. Section 3 contains a description of the applied architecture of the artificial neural network, the training method, the implementation of the neural network, and the method for obtaining the spectrum of diagnostic parameters. Section 4 describes the verification of the proposed method in the laboratory. Section 5 presents the results of the conducted diagnostic experiment.

Vibration signals analysis method
The proposed method for diagnosing rotary machines operating under variable loads is based on vibration acceleration, rotational speed, current intensity, and temperature measurements, with these signals being recorded synchronously. Then, the vibration signals are retested against the rotational speed signal of one of the shafts of the system. For this purpose, the order analysis method was used. In the first phase, the signal from the tachometer is subjected to the interpolation procedure using a cascaded integrator-comb CIC filter. Then, based on the filtered signal from the tachometer, Analysis of orders also allows for observation of individual orders over time. By monitoring the amplitudes of characteristic orders, it is possible to obtain information about the technical condition of the tested object. However, a change in the amplitude value may also be caused by a change in the system load [34]. Thus, the order analysis makes it possible to compensate for changes in the frequency domain caused by variable loads. However, the question of the impact of the load on the signal amplitude remains. Therefore, this work attempted to develop a method that gives results independent of working conditions. The measurement of the load moment requires specialised apparatus, and in most cases is impossible to carry out in industrial conditions. However, assuming that the motor driving the system is powered by a constant frequency and amplitude voltage, which often occurs in industrial conditions, any change in speed will be caused by a changing load. Therefore, the rotational speed signal in this work will be used not only for resampling by the order analysis method but also for describing the impact of the load on vibration signals.
Another indirect measurement method of the load moment is the measurement of the current supplying the drive motor [35].
In the proposed method, the waveforms of both the current intensity and the rotational speed were taken as data describing the load change. Fig. 1 shows a signal processing algorithm for building vectors that teach an artificial neural network. At the input of the processing algorithm, signals of the vibration acceleration and the tachometer with a length of 30s are given.
Then, the order analysis procedure is carried out using the conditions. The next step is to determine the moving mean for N subsequent elements. Averaging was used to reduce data dispersion. The data prepared were input vectors to the artificial neural network. The authors similarly prepared the data in their previous work [15].

3.1.Network architecture
In order to take into account the relationship between the values of the order spectrum and the operating conditions of the machine, it was decided to use a deep neural network as a universal method, effectively approximating the complex relationships between the data [36][37][38]. In regression problem where the number of output variables is greater than the number of input variables, a common approach is to use an architecture built from multiple neural networks, where the output variables are divided between several networks so that each of the subnetworks has more inputs than outputs. Such an approach has proved its effectiveness in solving inverse problems in tomography [39][40][41]. In the paper [15], results were published in which data on working conditions were treated as input data and attempts were made to recreate the values of the amplitudes Such an approach allows us to build a single network with a large number of input variables in relation to the number of outputs, which allows us to obtain satisfactorily small estimation errors. In addition, thanks to the reversal of the estimation method, the need to use synthetic output data was removed, making the system resistant to the phenomenon of overfitting and eliminating the need to use bagging, which multiplied the number of neural networks needed to train the network. A diagram of the architecture of used multilayer fullyconnected perceptron is presented in Fig. 2. The fundamental consideration in designing a neural network is its relatively small size to enable implementation in continuous monitoring systems. The number of input variables depends on the number of order amplitudes that we want to observe in the diagnosis process. In the case studied, the mesh order is No. 72, so 100 orders in two axes were analysed in order to observe the mesh modulation.
Model of the network and the training process was implemented in R programming language via Keras API.

3.2.Network training
In industrial environments, data from damaged machines are difficult to access or scarce compared to data from undamaged machines because an industry strives to repair such machines as quickly as possible. Preventive replacement of parts is also practised (e.g. in the energy industry) to avoid the effects of a fault in the form of downtime, high costs, or disasters [42,43].
Another problem is the multitude of different types of damage that can occur in real machines, which will characterise a different nature of machine malfunctions. For these reasons, using classical machine learning methods is significantly more difficult, as they require collecting data sets containing predetermined types of damage. In addition, a small number of data cases with damage in training sets (in relation to fault-free Eksploatacja i Niezawodność -Maintenance and Reliability Vol. 25, No. 3, 2023 data) may negatively affect the model training process. In order to address these two fundamental problems, a revised approach to network training was applied, whereby the teaching set for a given neural network contained only data from an undamaged machine. In this way, the learned network will be a mathematical model of the undamaged machine. In general, the mechanism allowing for detecting faults consists in observing the differences between the measured machine The data set from undamaged machine was divided into the training and test parts in an 80/20 percent ratio, and MSE (mean square error) was assumed as a function of the loss in the training process.

3.3.Using a trained network for diagnostics
The applied model allows for direct detection of a potential fault by observing the error scatter of the learned network. It is worth emphasising that while the technique based on comparing the error scatter allows for relatively simple detection of potential defects, it has limited identification capabilities. To detect and identify the defect, the authors propose to carry out an additional procedure allowing for the assessment of the discrepancy of network results for individual spectrum orders. The algorithm of this procedure is presented in Table 1. Table 1. Procedure for determining the diagnostic parameter for individual orders.

3.4.Diagnostic parameter developed
As part of this work, the diagnostic parameter Relative Differences Product of Network Statistics (rDPNS) was developed (1), which is determined for each order separately: where 2 ( ), 2 ( ), ( ), ( ) are the variances and means calculated according to the procedure in Table 1 and ReLU is

Stage
Step Action The use of a measure whose values fall within the range [0,1] allows for the use of the effect size scale. It was decided to use the Cohen [44] scale mainly used for qualitative variables, which also works perfectly for other coefficients [15].
The Cohen scale is as follows: with the classic spectral analysis of the vibration signal. For example, changes in the value for order 1 will be caused by unbalance of the machine shaft. Changes in the value for order 2 will be caused by shifts in placement between rigidly connected machines [45]. However, for couplings, it will be an order corresponding to the number of coupling gears.

Experimental validation
In order to validate the proposed method of diagnosis, an experiment was carried out on a laboratory stand for diagnosing planetary gears.

4.1.Rig design
The laboratory stand (Fig. 3)    A three-axis vibration acceleration sensor PCB 356B08 and a temperature sensor LM35 are mounted on the transmission housing. The rotational speed was measured using a laser tachometer, while the current intensity was measured using an ACS714 sensor. The recording of the measurement signals and the signal processing algorithm were carried out in a dedicated application built in the LabVIEW environment. Fig. 4 shows a photo of the laboratory stand.

4.2.Experimental methodology
In order to verify the effectiveness of the trained network in diagnosis, faults were introduced in the laboratory bench.
Measurements were carried out for four states of the drive system, without damage, for misalignment between the transmission and the braking motor, unbalance, and simultaneous unbalance and misalignment. where the washers were mounted is marked on Fig. 3 (d1).
Unbalance (S2) was introduced by placing an additional mass (3g) on the output shaft coupling (Fig. 3)

Results
The results were analysed using order analysis alone, and the dependence of the spectrum of orders on load and temperature is presented. Next, the results of the neural network for the data from the experiment are presented, and the values of the proposed rDPNS parameter in the order function are presented.

5.1.Order analysis
In the case of machines operating under variable load or rotational speed, synchronous methods are used. One popular tool is the order spectrum, however order spectrum analysis may not be a sufficient tool to assess technical conditions in high load variability. Fig. 6 shows the spectrum of vibration acceleration orders for the system without damage (black) and for the system with misalignment.  The oil temperature also influences the amplitude of vibrations generated by the gearbox. Fig. 8 shows the spectra of the orders as a function of the temperature measured on the gear body. A significant impact on the vibration acceleration amplitude values is visible, especially in the meshing band (64-80 orders).

5.2.Scatter plots of errors
The error scatter of individual network output values was analysed. Fig. 9 shows a graph of the error scatter of network output values for the test data compared to the error scatter for the training data, while Fig. 10 compares analogous graphs for data with the considered faults introduced. Fig. 9 shows that introducing new faultless data into the network results in the fact that the observable scatter of estimation errors remain at a level similar to that observed for the training data. This means that the network has correctly learnt to recognise faultless operation states, and the model is not overfitted. Fig. 9. Scatter of network output estimation error for faultless data. The output was subject to standardisation. Data from the test set (black), data from the training set (grey).
On the other hand, as shown in Fig. 10 two states from one another. It is also interesting that while the unbalance alone increases the temperature estimation error, in the case of co-existence with the misalignment, the unbalance seems to cause a decrease in the temperature estimation error in relation to the error when the misalignment occurs on its own. Fig. 10. Scatter of network output estimation error for fault data. The output values were subject to standardisation. Fault-free -test data (black), unbalance (blue), misalignment (red), misalignment and unbalance (orange).

5.3.Neural network
This section presents the graphs of the rDPNS parameter as a function of the order, determined following the algorithm presented in Table 1. Fig.11 shows the spectrum for the misalignment state (S1). According to the theory of diagnostics, the misalignment causes an increase in order amplitude corresponding to the number of claws (4). The rDPNS parameter value for order 4 is in the range of 0.3-0.5, which indicates a medium effect according to the Cohen scale.
Significant values (> 0.5) are adopted by rDPNS in the range of meshing (68-76 orders) because misalignment also affects the way the gears mesh [46]. Fig. 11. Spectrum of orders of parameter rDPNS for state S1 (misalignment).  For the state S3 corresponding to misalignment and unbalance, the values indicating a large effect occur both for order 1 and for orders from the mesh band (Fig. 13). The order spectrum of the parameter rDPNS looks very similar as in the case of S1; however, there are symptoms related to misalignment (order 4 and orders 68-76) and unbalance (order 1). Fig. 13. Spectrum of orders of parameter rDPNS for state S3 (misalignment and unbalance).

Conclusions
This study addressed the diagnosis of machines operating in In order to identify emerging faults, a procedure for analysing the neural network response was proposed, allowing us to generate the rDPNS diagnostic parameter in the form of a normalised spectrum. The normalised spectrum allows for use in automatic monitoring systems. The rDPNS parameter proposed in the paper allows for the determination of the size of the potential damage effect on the characteristics of each order.
As a result, it allows us to obtain a spectrum of orders of the rDPNS parameter, which can be interpreted in accordance with the theory of vibroacoustic diagnostics. This spectrum is resistant to interference introduced by variable operating conditions (e.g. load, oil temperature).
In order to verify the correctness of the proposed method, an experiment was carried out on a laboratory stand, and the possibility of detecting misalignment, unbalance and misalignment and unbalance at the same time was analysed. In the case of misalignment, a large effect of the rDPNS parameter failure was observed for orders corresponding to this damage.
Similarly, in the case of unbalance, a large effect was observed for the rDPNS corresponding to order 1. However, in the case of simultaneous misalignment and unbalance, a spectrum of orders of the rDPNS parameter was obtained similar to the state of misalignment alone. However, there is also a large effect on the parameter corresponding to order 1.
The conducted experiment proves that the presented method allows for the potential identification of a wide range of various types of faults without the need to take into account -at the system design stage -which faults are to be captured. In addition, by presenting the results in the form of a standardised spectrum, the result of the analysis is intuitive for diagnostics specialists or interpretable by automatic systems, which is another aspect facilitating the implementation of such a technique in industrial conditions. Further experiments will address the potential of using other network models to reduce the computational complexity of the entire procedure.