Application of Tribological Artificial Neural Networks in Machine Elements

The input parameters and the target values, in this case the film thickness and viscous friction, were normalised as per a standardised min–max normalisation function:where \(u\) and \(l\), where the upper and lower normalised unit values of \(1\) and \(-1\), respectively. \(x\) denotes the dimensional target input value and \(\widetilde{x}\) represents the final normalised input or output parameter of the ANN. The output layer of the ANN returns a normalised value, so therefore \({x}_{\text{max}},{x}_{\text{min}}\) must be stored as a variable to dimensionalise the output later.

The input training data set was split into three categories randomly: a training set, validation set, and test set. Each set contained \(70\%, 15\% \,and \,15\%\) of the training data, respectively. A limit of \(1000\) epochs was also implemented. During back propagation of the ANN the mean-squared error (MSE) was used to evaluate the network’s performance:where \(N\) is the number of data points being trained, validated, or tested. The coefficient of determination in line with [29], \({R}^{2}\) Eq. 22, was post-processed on the test data to evaluate the prediction quality of each ANN structure.where \({t}_{i}\) and \({y}_{i}\) are the target value and the predicted value, respectively. \(\overline{\text{y} }\) is the mean of the target sample. The numerator of the fraction in Eq. 22 is the residual sum of the squares, whilst the denominator is the total sum of the squares.

As suggested per [29], suitable activation functions of hyperbolic tangent (Eq. 23) and logistic sigmoid (LogSig) (Eq. 24) functions were used for the hidden layers. A rectilinear (ReLU) (Eq. 25) function was also evaluated.

A simple linear output activation function was used on the output. Several sizes of training data (\(600, 1000, 2000, \,{\text{and}}\, 5000\)) were tested to observe the influence over the prediction quality.

The ANN makes use of early stopping and regularisation in order to prevent statistical overfitting when conducting training [41]. The former improves generalisation of the trained network by observing the performance criteria of MSE of training data set relative to the validation training. Generalisation refers to the ANNs ability to adapt appropriately to previously unseen data drawn out of the training data set also known here as the validation set. If the validation set’s error increases, training is stopped early. The latter technique modifies the performance criteria by accounting for the change in the mean square of the network weights and biases (MSW) (Eq. 26). A performance adjustment factor, \(\gamma^{\prime}\), can be applied to reduce the weights and biases (Eq. 27) for any given propagation and reduce the likelihood of overfitting.

ANNs can only perform as well as the quality and quantity of the data that it is provided, therefore the generation of a many quality data points for the database is of high importance. To improve the timeliness of numerical data point generation explicit parallelisation can be implemented, i.e. use of multiple computational cores of the CPU. When using MBD solvers implicitly it is not possible to exploit the use of explicit parallelisation since a particular time step is dependent on the present time step as well as on data of the same time step. The ANN functionality in turn can be embedded with an implicit MBD computation environment to significantly improve the computation times without the need of explicit parallelisation despite benefitting from it indirectly.

One previously developed EHL film thickness database constructed via a Finite Element Method (FEM) solver covered a vast range of lubricants and material properties for relatively low speed conditions (\(<0.4\,\text{m}/\text{s}\) of 2D line contacts) [29]. However, if a similar database was to be used for a numerical EHL solver, the stability and ranges of their validity may be exceeded. Therefore, training databases produced via numeric solvers should be constrained in some manner to ensure its validity. Marian et al. used Latin Hypercube Sampling (LHS) for EHL film prediction as an effective method to generate a training database to cover a vast design space with many factors [29].

In this study, LHS was used but was constrained by the Greenwood Regime chart [42] as shown in Fig. 3. The regions of the chart are Isoviscous Rigid (IR), Isoviscous Elastic (IE), Piezo-Elastic (PE), and Piezo-Rigid (PR). These bounded regions are indicational and are in fact areas of transition from one regime to another but are used here as constraints for the numeric databases. The bounding regions classifications are dependent on the geometric, material, and rheological properties of the contact. To ascertain this, dimensionless elasticity and viscosity parameters, \({G}_{\text{e}}\) and \({G}_{\text{v}}\), respectively, are established.

The IR region can be described as a hydrodynamic region (lightly loaded contacts). Any data points generated via the LHS methodology that were found to be in this bounded region were redistributed with valid newly generated LHS EHL data points. The Piezoviscous Elastic (PE) regime prevails when the contact pressures are sufficiently high to cause elastic deformation of the bounding surfaces and a rise in lubricant viscosity. This signifies an EHL regime of lubrication and is the most prevalent regime for contact in gear pairs and sufficiently preloaded roller bearings. Piezo-rigid contacts also feature within the training dataset which refers to little to no elastic deformation of the bounding surfaces but there is a rise in lubricant viscosity.

The contacts within two commonly used machine elements, a cylindrical element roller bearing and spur gear, were superimposed on the Greenwood chart. This ensured that the training data covered their respective points of operation. This further confirmed that the lubrication regimes of both case studies were well within the PE region. Maximum and minimum ranges for the input variables were chosen to reflect the ranges over which typical high-speed gears and bearing operations would cover. Table 2 shows the initial ranges of the training data set produced that the LHS methodology was implemented with. Further constraints on the numeric database were to ensure suitable Hertzian pressures over \(300\, {\text{MPa}}\) and below \(3.5\, {\text{GPa}}\) to ensure the numeric database stayed within the PR and PE regions. Training data sets for the central film thickness are openly available from Figshare [43].

Typically linear regression techniques can be significantly affected by multicollinearity of the input parameters. ANNs however are unaffected as it is an overparameterised methodology able to predict non-linear problems. When multicollinearity is apparent, the highly correlated input variables might require manipulation using methods to combine parameters, such as partial least squares, Principal Component Analysis (PCA), or ridge regression [44]. Figure 4 shows the absolute correlation factors between the input values for the EHL problem with the most significant bivariate correlation being between load and length with a magnitude of 0.23. This indicates a very weak correlation between the two parameters, but it must be noted that this may have been affected by the previous constraining of the input variables.

Interestingly in the study by Marian et al. [29], grouped non-dimensionalised parameters, denoted in Eq. 2 of the current study, were used as inputs and it was found to significantly impeded the ANNs ability to predict the central film thickness; the authors’ of the current manuscript drew similar conclusions. ANNs can use principal component analysis similar to linear regression in an effort to reduce the dimensionality of the input space of the problem. The output of PCA conducted on the selected inputs of the present study shown in Fig. 5 shows that each of the variables provide a similar contribution to variance and therefore any reduction in input parameters may impact the ANNs ability to predict the film thickness.

A simple shaft supported by two cylindrical roller bearings has been modelled in a flexible multi-body dynamic software. The shaft is radially loaded at the centre point between the two bearings using a static load application. The shaft is constrained to rotation about its central axis and lateral degrees of freedom. Bearing geometric parameters and operating conditions are shown in Table 3.

The dynamic simulation is run explicitly, and results required as inputs to the ANN are output. Results are generated for an individual rolling element in its orbit around the bearing. These include roller load per unit length, reduced radius of the roller-inner race contact, and contact entrainment velocity and are shown in Fig. 6. The bearing model accounts for the non-linear Hertzian force–deflection relationship. Peak roller loading occurs as the roller enters the lowest point in its orbit, due to the vertical direction of the radial load applied to the shaft. The loading pattern is cyclic in nature as the roller enters and exits the loaded region. Sufficient preload ensures that the roller and raceways remain in contact throughout operation, and an EHL contact is maintained throughout. The contact reduced radius and entraining velocity do not change throughout the rollers orbit as these are a function of the bearing geometry and operating speed, which remain constant.

The lubricant film prediction ANN has been embedded into an explicit workflow with a dynamic simulation of spur gear pair with microgeometry modifications. The relevant dimensions and operating conditions of the spur gear are displayed in Table 4. A common spur gear modification of a linear tip relief has been applied as it reduces mesh impact of the single to double contact during the meshing cycle [45].

Table 5 provides the lubricant parameters used in both the bearing and spur gear case study.

All the relevant input parameters required by the ANN such as reduced radius, entraining speed, and load across a meshing cycle were computed using a one degree of freedom dynamic model as shown in Fig. 7. A single meshing cycle consisted of \(198\) data points to be computed. The model considers tooth bending and tilt using a Weber–Benesheck method [47] as well as local deformations using a Hertzian model. Peak loading occurs during a period of the meshing cycle where the load is supported via a single flank pair. Cylindrical gears in automotive applications typically operate under a boundary to mixed EHL regime due to the contact pressures. Under mixed lubrication regimes, viscous friction is a predominate contributing factor to the overall friction.

The initial effort to generate a training database is of importance as the number of data points can influence the ANNs accuracy. During this study, the average time for a numerical data point to be generated was \(5.88\) s. This led to the database construction, using a single processor core, having a wall time between \(0.98\) h and \(8.16\) h for \(600\) to \(5000\) data points, respectively. Note the use of multiple processor cores could vastly improve the wall time over a single core.

To find the optimum structure a parameter study was completed using over \(500\) different ANN structures on the same input data. The number of hidden layers is varied between one and four, and the number of neurons between ten and twenty. The three documented activation functions were studied in the various hidden layer configurations including a combination of hyperbolic tangent with the final layer being a logistic tangent function. \({R}^{2}\) is reported only for the test data sets as this gives the best indication of the models performance and generalisation in comparison to the training and validation performances. The full training procedure completion time was also documented for each ANN structure tested.

Values for \({R}^{2}\) using \(600\) data points and different activation functions, number of layers, and neurons are presented in Tables 6, 7, 8, 9, 10, 11, and 12. The ReLU function performed significantly worse across all structures in comparison to the logistic sigmoid and hyperbolic tangent functions. However, it is worth noting that there is slight variation in the performance of a network with each new training session. The optimum number of layers was between two and three similar to tribological applications of ANNs [39].

It was found that \(600\) data points were sufficient to train a relatively accurate ANN for central film prediction; however, it must be noted in the current study more data points led to less variation in the \({R}^{2}\) values across all structures without necessarily overfitting. Although it must be noted that training times varied most significantly with increasing the number of data points rather than the network structure complexity. Figure 8a–d and Tables 9, 10, 11, and 12 demonstrate the influence of increasing the data points from \(600\) to \(1000, 2000,\, {\text{and}}\, 5000\) data points for a logistic sigmoid function. It was concluded during this study that \(1000 \,\text{to}\, 2000\) data points typically led to the ANNs ability to reliably predict the film thickness without leading to significant increases in training time. 5000 data points used with the most complex network structures exceeded five minutes with the aforementioned hardware and reached \(1000\) epochs.

In order to realise the potential of ANNs to predict complex multifaceted problems they require to be demonstrated into scenarios which can benefit from their implementation. In order to assess the ANNs ability and suitability for operation in dynamic machine element modelling a few case studies are presented below. Performances of the ANNs are to be compared relative to the numerical models and experimental data.

The analytical formula from Eq. 1, the numerical solution as described in Section III, and optimum-structured ANN were used to compute the EHL film thickness over a single meshing cycle of the spur gear and several cycles for the roller bearing. It is worth noting that the bearing and gear case studies operated within the range of the ANNs training dataset. The film thickness computed by each methodology for each of the machine element’s angular displacement is displayed in Fig. 9a and b. For both the bearing and spur gear applications Fig. 9a and b demonstrates clearly the ANNs (solid blue) ability to predict a similar central film thickness as the numerical method (long dash red) with minimal MSE error. The analytical (short dash green) method on the other hand over predicts the film by up to \(0.1\, \upmu\text{m}\) in the spur gear case study, in real terms this could allow for misinterpretation of little to no boundary interaction depending upon the surface roughness. Table 13 demonstrates the relative performance of the analytical and ANN against the numerical solution for the MSE. The computation time per data point was also documented for each of the case studies in Table 12 to demonstrate the timely solution of the spur gear and bearing film thickness using the analytical and ANN methods. The analytical solution is a factor of ~\(75\) faster than the ANN but is significantly less accurate as described by the MSE. The ANN represents approx. ~\(1500\) factor computation reduction over the numerical solution with marginal error. Despite significant computational gains, it must be noted for widespread adoption of EHL ANNs in MBD simulations and its ability to extrapolate for conditions in which the input range of the original training datasets is exceeded must be appropriately addressed. Whilst this is not the focus of the current study, it should be addressed in the future works to advance the use of this data-driven solution.

Throughout a gear meshing cycle, the contact between two flanks changes its slide-to-roll ratio (SRR). A typical maximum SRR of a spur gear is \(50\%\). One way to experimentally explore this contact-level phenomenon in a controlled manner is to use a ball-on-disc tribometer MTM2 (PCS Instruments, London, UK), where friction can be measured. Five tests were conducted, specifically varying the entrainment speed only, using a Group III base oil with a contact pair made of AISI 52,100 steel. Each of the discs were measured using Alicona Infinite Focus microscope (Alicona, Graz, Austria) at four different locations and the topographical parameters averaged.

A reduced order 2D EHL method was used to simulate the contact in discrete slices as per [48]. Table 14 shows the inputs into the model. The numerical methodology to compute the viscous and the deterministic Greenwood and Tripp methodology for boundary friction, as documented in Section IV and V, were used. A training data set of \(\text{10,000}\) was used due to the increased number of factors to consider in the design space. Moreover, to accommodate the slice-reduced order methodology, the load and length training input parameters of Table 2 were reduced to reflect the smaller contact lengths (\(0.01-10.0 \upmu {\text{m}}\)), whilst maintaining the similar load per unit length ratios. Separate viscous and boundary friction ANN’s were trained using the following ANN network structures, respectively, using hyperbolic tangent activation functions for the hidden layers:

Each ANNs training achieved an \({R}^{2}\) value of \(0.9985\) and \(0.9993\). The MTM friction results compared against the ANN, and numerical solutions are shown in Fig. 10. The Stribeck Oil Parameter is plotted to demonstrate the lubrication regimes experienced, mixed to EHL. One potential reason of both models overprediction of friction at the experiment’s higher entrainment speeds at the fixed SRR of 50% could be due to shear thinning.

The numerical and ANN viscous friction methodologies were also used in same spur gear example. As the gear flanks approach the pitch point the sliding velocity reduces towards zero. Therefore, the friction coefficient should be at its minimum. Both presented methodologies are capable of predicting this event as shown in Fig. 11. Literature has at points misused analytical lubricated friction formulas, such as those presented by Evans and Johnson [36] in gear applications where sliding–rolling motion is present. The ANN was ~ 1300 times faster at computing the friction compared the numerical solution.