Development of predictive model for vibro-acoustic condition monitoring of lathe

2.1. Status of lathe

To perform the condition monitoring process, we record the vibration and acoustic data of any untested machine until its breakdown which may be due to common industrial failures like fatigue or wear and tear. The vibration and acoustic data obtained at the first breakdown point is used as the threshold value for performing the condition monitoring of similar machines. This might be a slight drawback in case of untested machines as the entire predicitive model can be applied only after the first breakdown. However, this threshold value can also be obtained from manufacturing catalogues or from an identical machine which is due for maintenance. For our experiment, six identical lathe machines were analyzed for vibration specific acceleration values (g-peak) and acoustic $L$ peak levels (dB). Considering the g-peak values along with their corresponding frequencies, a comparison was made with the standard Vibration Acceleration General Severity Chart [3]. The g-peak acceleration values obtained for all the six Lathe machines at idle running conditions and at a speed of 1800 rpm is shown in Table 1. The condition of the Lathe so obtained using the Vibration Velocity General Severity Chart [3] is used to classify the lathes into categories – good, fair, smooth and very smooth. This is also shown in Table 1. It is important to note that even though our experiments were done with mild steel as the material, the combination of the Acceleration and Velocity Severity charts is independent of the material being used. However, the classification will vary with the processing speed. But, for the purpose of development and comparison of the vibration and acoustic condition monitoring models, it is necessary to record the training data at a constant processing speed. Hence, Table 1 forms the backbone of the entire process and needs to be formulated for any machine prior to development of the models proposed below.

2.2. Selection of lathe

Machines B and C were observed to be the best machines out of the six lathes as they were the two lathes classified into the very smooth category. This can be seen from the Condition column in Table 1 and the lowest $L$-peak values in Table 2. As the $L$-peak values are similar for Machines B and C, the best machine was chosen to be Machine C due to its lower g-peak value. Here, the models are based on the training data from this machine so as to observe trends for a smooth machine until it fails and to ensure their applicability to any other identical machine irrespective of its condition. The vibro-acoustic condition monitoring of this lathe was carried out for a period of 5 months.

2.3. Selection of comparison standards

Parallel to the selection of the best lathe for condition monitoring, two other lathes (not from the above mentioned) were chosen – one with a faulty headstock bearing which required immediate replacement of the same (Machine 1) and the other for which overall maintenance was due (Machine 2). The results for both the machines were used as standards for prediction of failure states in the best lathe machine chosen (Machine C) and are given in Table 3. These values were considered as the thresholds for scheduling maintenance or replacement of bearings.

3.1. Overall acoustic results

The acoustic levels of machine C were measured through experiments carried out through duration of 157 days following the same experimental procedure as illustrated in Section 2. The noise level ($L$-peak) was measured for idle running, turning and facing operations for a spindle speed of 1800 rpm on Machine C and the results obtained are given in Table 4. The table shows the $L$-peak values obtained on not all but some of the representative days.

The depth of cut and feed given were 0.125 mm and 0.2 mm respectively. The facing and turning operations were performed on a mild steel material. Owing to these stated values and the fact that the recording of the $L$-peak values are instantaneous, the coolant system was switched off momentarily during measurement.

3.2. Overall vibration results

Similar to the acoustic experimentation, the g-peak acceleration values were measured for Machine C throughout the duration of 102 days, following the same experimental procedure as illustrated in Section 2. The last dates for both readings were the same. The coolant system was in continuous use while conducting the experiment. The main reason was to avoid excessive temperature rise during the cutting operations as well as to avoid tool damage and make the work on the lathe conducive to principles of production techniques. However, a completely isolated vibration signature of the coolant system was recorded while the lathe was not in use. The fundamental frequency corresponding to this coolant system was found to be equal to 175 Hz. The vibration signature for the same is shown in Fig. 4. Further analysis of results were carried out only after the fundamental frequency and all the harmonics of the coolant system were neglected from the signatures of the lathe machine for idle running, facing and turning.

Fig. 4Vibration signature of coolant system

The depth of cut and feed given were kept constant (at same values as mentioned above) for all the experiments. 300 frames were recorded by the accelerometer for each run and the results were captured and processed through the EDM software. The results obtained on some representative days are shown in Table 5.

3.3. Headstock bearing vibration results

The vibrations of the headstock bearings can be qualitatively identified from the vibratory signatures of the overall lathe machine. The theoretical fundamental frequencies corresponding to the bearing parts [18] can be obtained from the relations given below:

where FTF – fundamental train frequency, BPFI – frequency of inner ring, BPFO – ball pass frequency of outer ring, $S$ – revolutions per second, $Bd$ – ball/roller diameter, $Pd$ – pitch diameter, $Nb$ – number of balls/rollers.

For our experiments, the bearing specifications and calculated frequencies are shown in Table 6.

The vibration signatures of Machine C (for idle running, turning and facing operations) were scanned for g-peaks corresponding to calculated FTF, BPFI and BPFO. The overall vibration signature of the Machine C zoomed in the range 0 to 800 Hz is shown in Fig. 5. The peak at 175 Hz is to be neglected, as it corresponds to the coolant system as shown before. The next significant peak occurs at around 318 Hz as pointed out in Fig. 5. This falls around the theoretical value of BPFI. As there is no significant peak around the values of BPFO and FTF, we can establish that the outer race and the ball bearings unlike the inner race are in good condition. Further, the results of the g-peak values corresponding to a specific Day No. is tabulated in Table 7.

Fig. 5Overall vibration signature

4.1. Normal equations method

In case of Normal Equations model, the historical data or the training set is fed to the learning algorithm which in turn assists a hypothesis function [22] to predict the number of days till the threshold value is reached at which the lathe needs maintenance.

The hypothesis function, $h\left(\theta \right)$ developed is shown below:

where $X_1$ – idle run data, $X_2$ – turning data, $X_3$ – facing data. Each $X_i$ ($i=$1, 2, 3) may be the acoustic, vibratory or threshold value for which the $h\left(\theta \right)$ is the corresponding value of Day No. The squared error function $J\left(\theta \right)$ [22] is defined as shown below:

where $y^i$ is the value of the Day No. in the training set.

To estimate the threshold value of $h\left(\theta \right)$ corresponding to the threshold values of $X_{i^{\text{'}}s}^{}$ the function $J\left(\theta \right)$ has to be optimized to get the values of ${\theta }_{i^{\text{'}}s}$ ($i=$ 1, 2, 3). Solving for optimal values of ${\theta }_{i^{\text{'}}s}$ we get the Normal equation [22] which gives a closed form solution to the multi variate regression model:

where $X$ is the design matrix consisting of all $X_{i^{\text{'}}s}^{}$ and $X^T$ is the transpose of $X$.

4.2. Artificial neural networks

This section deals with the development of predictive models using Artificial Neural Networks (ANN) [19-20]. As opposed to the traditional mathematic and statistical models, ANN is a robust tool that can capture non-linear relationships with better accuracy and can automatically adjust their weights to optimize their behaviour. The basic architecture of every ANN consists of three layers-input layer, hidden layer and output layer.

In our case, a Feedforward Backpropagation Neural Network model [19-20, 23-24] has been implemented for training and simulating the historical data. It has two layers of hidden network (Fig. 6) and is trained using the Levenberg-Marquardt algorithm. The input to the network is the training data obtained for Idle running, Turning and Facing operations and the output is the corresponding value of day number. The Mean Sqaured Error function is used as the performance function for performing input to output mapping with higher accuracy. Once the network is trained and the validation performance curve has reached its minima, the network is simulated using the threshhold values to obtain the number of days to failure.

For training the input data, all the neurons are sigmoidal in the three layered architecture. Any other activation functions have not been used because there was no need to worry about over fitting, given the amount of data we had. The number of layers in the neural networks was tuned as a hyper-parameter to the algorithm and the best results were obtained using the three layered structure. A ten fold cross validation set database was used. The application of both the algorithms has been explained in Section 5 for Acoustic and Vibration analysis.

Fig. 6Neural network architecture

5.1. Normal equation model

In order to develop the overall acoustic prediction model, the training set ($m=$157) shown in Table 4 is fed to the machine learning algorithm discussed in Section 4.1. The hypothesis function obtained by optimizing the squared error function is shown below:

The threshold values for $X_{i^{\text{'}}s}^{}$ are substituted from Table 3 to get the threshold value of $h\left(\theta \right)$ which is the Day No. at which maintenance is to be carried out. A value of 156.69 is obtained by subtracting the duration of the experiment from the threshold value of $h\left(\theta \right)$. The significance of this value in the hypothesis function is discussed in Section 6. The overall vibration prediction model is developed using the training set ($m=$ 102) shown in Table 5 as input to the machine learning algorithm discussed in Section 4. The hypothesis function obtained by optimizing the squared error function is shown below:

Similar to the procedure mentioned before, the threshold values for $X_{i^{\text{'}}s}^{}$ are substituted from Table 3 to get the threshold value of $h\left(\theta \right)$ which is the Day No. at which maintenance is to be carried out. A value of 164.66 is obtained by subtracting the duration of the experiment from the threshold value of $h\left(\theta \right)$. The significance of this value as well as the negative coefficient of $X_i$ in the hypothesis function is discussed in Section 6. In order to develop the vibration prediction model for the headstock bearings, the training set ($m=$ 102) shown in Table 7 is fed to the machine learning algorithm discussed in Section 4. The hypothesis function obtained by optimizing the squared error function is shown below:

The threshold values for $X_{i^{\text{'}}s}^{}$ are substituted from Table 3 to get the threshold value of $h\left(\theta \right)$ which is the Day No. at which maintenance is to be carried out. A value of 178.59 is obtained by subtracting the duration of the experiment from the threshold value of $h\left(\theta \right)$. The significance of this value as well as the negative coefficient of $X_i$ in the hypothesis function is discussed in Section 6.

5.2. Artificial neural networks

In order to develop the overall acoustic prediction model, the acoustic training set shown in Table 4 is used as input to the network and the corresponding day number as output. The network is trained and the validation performance of 0.0011343 is achieved at 1000 epoch as shown in Fig. 7. The network is then simulated using the threshold values from Table 3 and a value of 153.0457 is obtained as output which is the day number at which maintenance is to be carried out.

In order to develop the overall vibration prediction model, the historical data shown in Table 5 is used to train the network. Best validation performance of 0.001389 is achieved at 1000 epochs as shown in Fig. 8. The network is then simulated using the threshold values from Table 3 and an output value of 164.5762 is obtained. Similarly, the vibration prediction model for the headstock bearings is developed using the training set shown in Table 7. Best validation performance of 0.00014865 is obtained at 1000 epochs as shown in Fig. 9. Using the threshold values as shown in Table 3, the network is simulated and an output value of 180.853 days is obtained. The significance of all these values is discussed in Section 6.

Fig. 7Performance curve for acoustic predictive model

Fig. 8Performance curve for vibration predictive model

Fig. 9Performance curve for headstock bearing vibration predictive model

6.1. Trends and results

The following figures (Figs. 10-13) show the variation in the vibration signatures and acoustic level from the start to the end of experiment. The orange plot in Figs. 10, 11 and 12 is the vibration signature obtained at the start (Day 1) of the experiment wheras the blue plot shows the vibration signature obtained at the last day (Day 102) of the experiment. Fig. 13 shows us the increase in Decibel values over the entire duration of experiment for Idle Running, Turning and Facing operations. From the graphs it is clear that there is a general increase in the acceleration g-peak values and the $L$-peak values over a period of time.

Fig. 10Vibration signature – Idle running

Fig. 11Vibration signature – Turning

Fig. 12Vibration signature – Facing

6.2. Analysis of trends and inferences

The results obtained in Section 5 give us the duration after which the maintenance of Lathe and replacement of faulty parts needs to be carried out. For the overall acoustic prediction, the Normal Equations model and the Artificial Neural Networks model yield values 156.69 and 153.0457 days which signifies that if the prediction algorithms give these results today, then the next maintenance will be due in about 153 to 156 days from now. Hence, scheduling a maintenance on or before the 153rd day will probably prevent a possible breakdown of the machine. Firstly, the lesser value is taken here assuming that an earlier date is only reassuring the safety of the machine. Secondly, this inference is valid only for the lathe machine chosen in our experiment and the given training set. However, this procedure or prediction model can be applied to other lathe machines to predict failures or to schedule maintenance. Similarly, for the overall vibration prediction, the normal equations model and the ANN model give us values – 164.66 and 164.7 respectively which show the durations after which maintenance is to be scheduled. Again, these values signify that maintenance scheduled on or before the 164th day will probably prevent deterioration of the lathe. The values of 178.59 or 180.53 obtained from the two models discussed which is obtained from the bearing vibration analysis can be used as a good prediction of the no. of days till the next headstock bearing replacement for the lathe machine.

Fig. 13Acoustic signatures for all processes

In case of the Normal Equations method, a multi-variable model was used to build the prediction algorithm. However it must be noted that to visualize the effect of change of only one of the parameters in the multi-variable model, a new hypothesis function which includes only that parameter, i.e. a uni-variable model must be formulated. Hence, the negative coefficients of $X_1$ in the hypothesis functions in equations do not imply a negative dependence of $h\left(\theta \right)$ with $X_1$. For instance, in order to account for the dependence of only idle running operation on the prediction value, instead of substituting the values of variables $X_2$ and $X_3$ as null in the multi-variable model, we formulate a uni-variable model using the historical data so obtained for idle running. The uni-variable model formulated using the g-peak values to account for only idle running operation gives us the following form of the hypothesis function:

This shows that only if the dependence of more than one parameter is to be incorporated, a multi – variable model is adopted. Moreover, if it is needed to study the effect of only one parameter, for example the turning operation on the lathe machine’s performance, then it is incorrect to use a multi-variable model equation by inputting a null value for the other $X_{i^{\text{'}}s}$.