Surrogate model based prediction of transmission error characteristics based on generalized topography deviations

Neural networks have already been used for some time for the classification of acoustic questions [6]. In the context of gears, neural networks are used, among other things, for classification in the area of condition monitoring. Here they are used to monitor output variables of sensors and to detect deviations that indicate a change in the condition of the gearbox [7‐10].

Another use of neural networks for gear modeling is described by CABRERA CANO. In his work, a system of rigid differential equations describing a gearbox in one dimension is set up with the help of a neural network. In this way, the behavior of the original simulation model can be reproduced. The trained neural network is faster than the original simulation model by a factor of ten [11].

BAXMANN undertakes first attempts to calculate parameters describing the application behavior by optimizing the selection of the micro geometry of a bevel gear. He can prove that neural networks are in principle suitable to simulate the behavior of a gearpair and thus accelerate the calculation of the parameters by a factor of 2 [12].

The schematic structure of a neural network is shown in Fig. 1. The basic building block of a neural network is a neuron, see the left side of Fig. 1. A neuron receives n input parameters, each of which is multiplied by the weights w. These weighted input parameters are then summed together with the constant value b and given as input to an activation function, which calculates the output of this one neuron.

The structure of the network itself again consists of several neurons in one layer. These adjacent neurons together form a layer. Several layers one after the other result in the structure of the entire network, see center of Fig. 1. The last layer is called the output layer, the layer of the input parameters is called the input layer, and all other layers in between are called hidden layers. Neural networks with multiple hidden layers are therefore also referred to as deep neural networks, abbreviated DNN in the following [13].

DNN are characterized by various properties. These have a significant influence on the performance and quality of the calculation performed with the DNN. One important property is the number of layers, see Fig. 1. The lower the number of layers, the lower the computational effort of the DNN. However, the accuracy with which the input parameters can be classified decreases. In addition, further settings are possible. The setting of the learning rate describes in principle how fast the DNN converges. A low learning rate leads to low convergence. However, if the learning rate is set too high, the solution may end up fuzzy and the model may no longer converge. The batch size indicates with how many data points of all available variants the algorithm is trained simultaneously. A size of n = 32 has established itself as the standard value. If the value is chosen too large, the model may not be able to approximate new data. The number of epochs indicates how often the algorithm trains with the entire data set. The larger the number of epochs, the more accurate the DNN can become, but the longer the training takes [14].

The number of neurons per layer becomes clear from Fig. 1. The required number of neurons depends mainly on the number of input parameters and output parameters as well as on the number of layers. A too small number of neurons leads to the fact that the model cannot be approximated correctly, whereas a too large number of neurons leads to a high variance of the results and bad model performance [15].

Another possibility for optimization when using DNN is the normalization of the input data. The normalization of all input data to a value range from zero to one increases the performance, because the activation functions are designed for this value range [14].

Up to now, neural networks have been used in the field of gear technology mainly as a decision-making aid for classification problems. Here, the networks are used to infer damage in the gearbox on the basis of recorded data. In addition, there are already first successful attempts to model the behavior of gearbox defects using neural networks. However, these models are still one-dimensional models that do not calculate acoustic parameters.

If a gear is measured, a DS results for the flank. For each measuring point, the deviation of the real flank from the theoretical involute is specified [18]. The Σ DS is intended to represent the combined deviation of the wheel and pinion by combining the two individual DSs. The advantage of combining the deviations of both contact partners to a common sum deviation surface is that only one of the two gears has to be provided with a deviation in the simulation. The other wheel can be modeled with ideal involute. This reduces the number of necessary parameters for the substitute model by half, since only one deviation surface has to be described and not two. The accuracy of this method depends on the amount of the displacement of the contact line under load.

The Σ DS generated in this way is to be approximated by radial basis functions (RBF). BRIMMERS had found them to be suitable for this purpose [17]. In order to determine the optimal number of necessary supporting points of the RBF in tooth width (Nz) and height (Ns) direction, different resolutions are investigated. For this purpose, analytical combinations of the different standardized tooth flank modifications are created in a first step [19]. The variation range of the individual modification amounts is listed on the left side of Fig. 3.

The DS are then approximated by RBF with a different number of supporting points in height and width direction. The number of grid points is varied from two to ten. In the bottom center of Fig. 3, an analytical DS and the corresponding DS approximated by RBF are shown as examples. The difference between the two DSs is examined at 10,000 uniformly distributed points. For the accuracy of the approximation, the R² coefficient of determination known from statistics can be calculated and used for the evaluation. In the work of BRIMMERS this value is also used to determine the necessary degree of a surface polynomial for the approximation of the surface [17]. 1000 different variants are compared, which are generated with the Latin hypercube function of the Python package PYDOE in the given limits of the target modifications.

The evaluation of the R² value for these 1000 variants is shown on the right side of Fig. 3. The first diagram shows the mean R² value for all variants as a function of the supporting points of the RBF in the height direction (Ns) and width direction (Nz). It can be seen that in the width direction, no further improvement of the R² value can be achieved from a number of Nz = 3 and in height direction from a number of Ns = 6 grid points.

In order to build a learning model or a model converging against real behavior (regression), there must be examples available for the range in which the model should function. The boundaries of this space are shown on the left side of Fig. 3. In the optimal case, all variants are evenly distributed in the space to be developed. A Latin hypercupe experimental design is used for this. The variant space, which provides the basis for the testing of the different models in the further course, is computed from 3000 variants. The library PYDOE is used to create the test plan [20].

For the variants defined in Chap. 4.2, ZAKO3D (see Chap. 1) calculations are performed and single-number characteristic values are calculated from the values of TE. In this specific ZAKO3D simulation, the surrounding structure is not considered. The used characteristic values are:

The calculation and application of these characteristic values for TE is described by WILLECKE [21]. The TE, which is used as a basis for the calculation of the characteristic values, usually has a constant component. For the calculation of some characteristic values, such as the RMS value or the fluctuation width, this constant component is eliminated before the calculation of the characteristic value. Thus, only the fluctuation component that also leads to an acoustic excitation of the structure is considered. The calculated characteristic values are stored together with the corresponding parameters p_i for all variants in a common HDF5 file. This format allows a fast and efficient data access for programs in different programming languages [22].

The data set generated in Chap. 4.3 is used as a basis to test different modeling approaches and to select a suitable approach for the application. To avoid having to implement, train and test each approach individually, the commercial software OPTIMUS from the manufacturer NOESIS is used to generate an initial overview. Here, the data set can be loaded and then a large number of models can be tested automatically. The R² value is used to evaluate how well the respective model is suited to represent the data set. For this purpose OPTIMUS performs a cross-validation with the data set. Since the training and evaluation of the large number of models requires some time, only the fluctuation width and the first tooth mesh order were used as result parameters in the first step. The results of the calculation are shown in Table 2.

The R² in Table 2 indicates the lowest error for the deep neural networks (DNN). Therefore, these will be further investigated in the following. The settings used by OPTIMUS are: Normalization of the input data, a group size of 32, an epoch count of 100, a layer count of 3 with 64 neurons per layer, and a learning rate of 0.001 (see Sect. 2.1 for an explanation of the settings).

The model, which is used for the further investigations, is built up in PYTHON based on the library KERAS as part of the total library TENSORFLOW [16, 23]. The library KERAS is chosen for the implementation and the construction of the model, because powerful DNN can be built with it and the complexity and the extent of the modeling can be increased granularly [16].

The construction and training of the model is divided into several sections. In the first step, the data must be prepared for training the model. DNN work most efficiently if the data to be processed are normalized, see Chap. 2.1. The normalization of the data set is done with the library SCIKIT-LEARN [24].

In the second step, the structure of the model is defined. First, the input layer is defined. The model should work with a fixed number of input parameters (the n_S = 18 grid points, see Chap. 4.1). Afterwards additional hidden layers are added to the model. These hidden layers are mainly characterized by their neuron number and the activation function of the contained neurons. Different topologies and hyper-parameters were tested here. The results are described in Chap. 5.3. The last layer added is the output layer. This receives as many neurons as output variables to be computed. After the topology of the network is defined, the model is compiled. Here the learning rate and the loss function are defined. A variation of the learning rate shows that a higher value than 0.001 leads to a slightly accelerated learning process, but the achieved loss value is higher at the end. Thus, 0.001 is used as the learning rate. A square error function is used as the loss function, since this is a regression problem and not a classification problem.

The third step is the training of the model. The training data set used for this is described in Chap. 4.2. A group size of 32 is chosen, because the state of the art recommends it, see Chap. 2.1. The training is always performed in blocks of 1000 epochs each, called “training unit” in the following. Afterwards, the result of the loss function is considered and, if necessary, the next 1000 epochs are trained. One “training unit” takes about 40 s. The number of necessary “training units” varies depending on the topology of the DNN.

The DNN is investigated by examining the influence of the number of hidden layers, the number of neurons per layer and the activation functions used. As a first step, the settings from NOESIS OPTIMUS are used. Since it could not be determined which activation function is used by OPTIMUS, different activation functions are tested here. In Fig. 4 the ratio between the predicted and the actually calculated value is shown on the left side for the two characteristic values fluctuation width and first tooth meshing order. The RMS value as well as the parabolic correlations and the skewness behave similarly, but are not shown here for reasons of space.

The training data set itself is shown on the left. In a perfect prediction, all points, each representing a variant, would lie on the line of origin in the middle. This is not the case. To better visualize the deviation of the prediction from the ideal result, a regression line is placed through the points, highlighting the deviation from the original line. In the middle of Fig. 4, the same characteristic values and the same model are shown. Here, however, not the training data set but the test data set was predicted.

It can be seen that the deviations between the actual values and the predicted values are increasing. On the right side, a prediction of the training data set is shown again. Here, however, the activation function in the model has been changed from Softmax to Softplus. The prediction accuracy of the characteristic values of the test data set is already better with this activation function than the prediction of the characteristic values of the training data set with the Softmax activation function. In addition to the softmax and softplus activation functions, elu, relu and tanh functions were also tested. However, these showed a lower quality of prediction. After the softplus activation function was determined to be promising for the application, the number of neurons per layer is varied.

The result of varying the number of neurons is shown in Fig. 5. It can be seen that a higher number of neurons produces a significant improvement in the prediction quality. Likewise, no deterioration due to an overfit can be detected. In the area of the low amplitudes of the first tooth meshing order, outliers can still be seen. The reason for this could be that there are not enough examples of geometry in the training data set that produce a very low first tooth meshing order. This is also supported by the fact that only very few points in the test data set have such a low first tooth meshing order. Both training data set and test data set should in principle have a similar frequency distribution in the boundaries of the variant space.

In the following, it will therefore be test whether a network with more neurons and layers can in principle predict the training data set better than a network with fewer neurons and layers if the training is sufficiently long. In Fig. 6, the upper diagrams in the left and middle columns show the self-prediction of the training data set.

It was trained with a factor ten low learning rate (0.0001 instead of 0.001) until the loss value did not change. No differences can be seen between the two networks in the self-prediction. However, the network with two hidden layers and 64 neurons per layer trained significantly longer than the network with three hidden layers and 256 neurons per layer.

The prediction of the test data set in the lower two diagrams shows an improvement in the mapping when examined closely. Likewise, for the network with two hidden layers and 64 neurons, an overfit already occurs. On the right side of Fig. 6, the model is shown in a state in which it has been trained only up to a loss value L = 0.0085. The “trained out” version on the left shows a loss value of L = 0.0005. It can be seen here that despite the higher loss value, the test data set is better predicted, i.e. there is less overfit.

For the network with three hidden layers and 256 neurons, an overfit could in principle already be present. However, the test data set still shows a better prediction accuracy than the other two models.

The advantage of calculating the excitation with the DNN can be illustrated by an example. For a deviation surface from the micro geometry C_α = 7 µm, C_Hα = 5 µm, C_β = 7 µm, C_Hβ = 5 µm, C_αa = 5 µm, sC_αa = 0.9 the characteristic values of the total angular misalignment are determined with ZAKO3D and with the DNN. ZAKO3D calculates f_z1 = 0.098 µm for the first tooth meshing order. The DNN calculates f_z1 = 0.099 µm. The value range in the variant space for the first tooth meshing order extends from f_z1 = 0.003 µm to f_z1 = 0.521 µm. A deviation of ∆f_z1 = 1 nm in the prediction is therefore to be regarded as small. However, relevant differences occur in the calculation duration. While ZAKO3D needs t_s = 551 s to perform the simulation the DNN has the result after t_s = 0.026 s and is thus 21,192 times faster. Since the calculation of the DNN can be parallelized better than parallel running ZAKO3D calculations, even higher speed advantages can be achieved on a computer with a 12 core CPU with a large number of calculations. For example, the DNN computes the 480 variants of the test data set with t_s = 0.038 s ~ 1,000,000 times faster than ZAKO3D with t_s = 10 h. This speed advantage allows the user new application areas of tooth contact analysis. It would thus be possible to simulative pair all wheels with all pinions for a production lot and thus calculate the pairing setup that ensures the best excitation behavior. In this way, the optimum can be extracted from existing resources. Since the original ZAKO3D simulation model had been validated with real measurements and the DNN predicts the same results as the original model, it can be assumed, that the results of the DNN also match the measurements.