UDK 531.225:621.824:519.233.4:004.032.26 Original scientific article/Izvirni znanstveni članek ISSN 1580-2949 MTAEC9, 48(1)81(2014) DETERMINATION OF THE NOTCH FACTOR FOR SHAFTS UNDER TORSIONAL STRESS WITH ARTIFICIAL NEURAL NETWORKS UPORABA UMETNIH NEVRONSKIH MREŽ ZA DOLOČANJE FAKTORJA ZAREZNEGA UČINKA NA GREDEH, OBREMENJENIH S TORZIJSKIMI NAPETOSTMI Murat Tolga Ozkan1, Cengiz Eldem1, Ismail Sahin1 !Gazi University, Faculty of Technology, Department of Industrial Design Engineering, 06500 Ankara, Turkey tozkan@gazi.edu.tr, mtozkan06@yahoo.com Prejem rokopisa — received: 2013-04-01; sprejem za objavo - accepted for publication: 2013-04-22 When designing machine equipment, geometrical figures or discontinuities such as notches, holes, steps and curves can occur. Sudden cross-section changes, discontinuities and force flows cause concentrations, particularly in the stress area. Stress concentrations may be formed due to dimensional features of a material or directions of applied forces. Such stress concentrations are considered as they have a notch effect on the material. The notch effect may lead to a breaking and distortion of a material. In this study, a mathematical model estimating the notch-factor values for a grooved round bar in torsion, a round shaft with a transverse hole in torsion and a round shaft with a shoulder fillet in torsion, using artificial neural networks (ANN) is introduced. The model estimates the notch factor using shaft dimensions, torque and corner rounding values. The ANN model developed in the study quickly and accurately estimates the notch-factor values, otherwise obtained from the catalogues with complicated analytical calculations. In this model, the uncertainties occurring in analytical calculations and the calculation errors were eliminated, thus long calculation times were saved as well. The results reviewing the performance of the ANN model developed for a grooved round bar in torsion, a round shaft with a transverse hole in torsion and a round shaft with a shoulder fillet in torsion were quite good. In the study, a multiple regression analysis of the data was also performed, but no conclusion evaluating the data was obtained. Keywords: shafts, notch-sensitivity factor, torsion, artificial neural network, statistical analysis Pri konstruiranju strojnih delov se pojavljajo nezvezne geometrijske oblike, kot so zareze, luknje, stopnice in krivine. Nenadna sprememba prereza, nezveznosti in potek sil povzročajo koncentracijo napetosti v napetostnem območju. Koncentracije napetosti v materialu lahko nastanejo zaradi dimenzijskih sprememb ali sprememb smeri delovanja sil. Taka koncentracija napetosti se obravnava kot zarezni učinek v materialu. Zarezni učinek lahko povzroči porušitev ali izkrivljenje materiala. V tej študiji je predstavljen matematični model umetne nevronske mreže (ANN), ki lahko obravnava faktor zareznega učinka okrogle palice z utorom, okrogle gredi s prečno odprtino, obremenjeno s torzijo, in okrogle gredi z zaokroženim prehodom. Model določa faktor zareznega učinka z uporabo dimenzij, navora in radija zaokrožitev. Razvit ANN-model omogoča hitrejše in bolj zanesljivo določanje faktorja zareznega učinka, ki ga sicer dobimo iz katalogov z zapletenimi analitičnimi preračunavanji. V tem modelu so odpravljene nezanesljivosti, ki se pojavljajo pri analitskem preračunavanju, odpravljene so računske napake in prihranjeno nam je dolgotrajno preračunavanje. Pregledane so bile zmogljivosti ANN-modela, razvitega za torzijo okrogle palice z utorom, torzijo okrogle gredi s prečno odprtino in okrogle gredi z zaobljenim prehodom. V študiji je bila izvedena tudi multipla regresijska analiza podatkov, vendar ni bilo mogoče izluščiti ugotovitve, ki bi prispevala k oceni podatkov. Ključne besede: gred, faktor zareznega učinka, torzija, umetna nevronska mreža, statistična analiza 1 INTRODUCTION Breaks and deformations are observed on almost all machine parts used for a power and force transmission. In order not to have these undesired effects, the notch factor is considered in the calculations. Thus, a formation of such effects is minimized or eliminated. Theoretical notch factors used in the calculations according to the change in the calculations or type of strain affecting the shafts vary. For each different type of strain, there are many table values available, but it is an inconvenient procedure to obtain the values required for the design from such tables. Mechanical damages formed as a result of fatigue have been a subject of engineering studies for many years. One of the first studies on this subject was made by W. A. J. Albert who tested metal chains lifted up under cyclic loadings in Germany in 1828. The term "fatigue" was first used in 1839 by J. V. Poncelet.1 During the studies he made in 1850s in Germany, August Wohler started to develop design strategies in order to avoid fatigue damage, testing iron, steel and other metals under torsion, bending and axial loadings. With his studies, Wohler proved that fatigue was affected by the average stress as well as by cyclic stresses.2 McClintock made the first theoretical research related to the ductile damage, taking place as void growth.3 In this research, it was concluded that the rate of void growth definitely depends on three axial stress regions as well as on the rate of hydrostatic equivalent stress. As a result of his experiments, McClintock concluded that different samples do not always have the same unit deformation in crack formation. The study made by Rice and Tracey took McClin-tock's study to a higher level. With this study, it was concluded that the rate of void growth definitely depends on three axial stress regions as well as on the rate of hydrostatic equivalent stress.4 With their experimental study, Hancock and Mackenzie supported the idea that the orientability of ductility for construction materials could be three-axial, and revealed that the material damage had been caused by high-degree hydrostatic pressure.5 By using the results of their experimental study, Bridgman, Hancock and Mackenzie revealed damage-unit-deformation and representation parameters of triaxiality in a closed damage curve.6 Hancock and Brown examined stress-unit deformation spaces on a notched sample.7 In the study, damage was reviewed at the centre point of the minimum cross-section where triaxiality is the highest on a cylindrical notched sample. Ozkan made a study about the notch-sensitivity determination of shafts. He used an ANN model.8 Ozkan et al. made a study about determining the notch factor on the shafts under tensile stress. They also used an ANN model.9 Recorded literature studies have revealed that notched tensile tests are commonly applied experiments. They show that notched tensile experiments include a large number of notch types. Therefore, it is obvious that modelling the data obtained from standard-experiment results and notched tensile experiments will provide an increase in the number of variables in experimental studies. The notch-factor selection and the calculations made afterwards require long and inconvenient procedures and, consequently, a significant amount of time and labour. It is necessary to utilise computer programs to solve such problems. The aim of this study was to develop a mathematical model that can provide for the best notch factors on a grooved round bar in torsion, a round shaft with a transverse hole in torsion and a round shaft with a shoulder fillet in torsion by considering the formal characteristics of the material affecting the notch factor and the torsio-nal-stress effect influencing the shaft. The mathematical model was developed using a multilayer feedforward ln|"" Weights Addition Operation Activity of Operation Output * f( Activity ) Threshold Figure 1: Artificial neural network Slika 1: Umetna nevronska mreža artificial neural network (MLP). In the study, a multiple regression analysis of the data was made. Multiple-regression and ANOVA analyses were also made, but since their results did not help us interpret the data, the study was focused on ANN. The artificial-neural-network model developed within the study consisted of three inputs for the round shaft with a shoulder fillet in torsion and the grooved round bar in torsion, two inputs for the round shaft with a transverse hole in torsion, one hidden layer and one output. 2 ARTIFICIAL NEURAL NETWORKS The concept of artificial neural networks first appeared as the idea of simulating the principle operation of the brain on digital computers. An artificial neural network is a mathematical model inspired by the functional structure of a biological neural network.10 Artificial neural networks consist of many operation elements connected to each other. Operation elements in artificial neural networks (nodes) function like simple nerves. An artificial neural network consists of many nodes connected to each other. The main unit of an artificial neural network is an artificial nerve. An artificial nerve is much simpler than a biological nerve. In Figure 1, an artificial neural element is shown. All the artificial neural networks are derived from this main structure. Differences in this structure allow different classifications of artificial neural networks. An ANN model consists of two main steps: the training and the test. The meaning of learning in artificial neural networks is to allow a neural network to produce correct outputs by establishing the right connections between the input and the output data relating to the problem. This procedure continues until the difference between the estimated output and the desired output decreases to a certain value. Artificial neural networks learn with experience just like humans. For that purpose, an experimental group is divided into two parts: the training group and the test group. During the training period, the network uses an inductive training model to train the training group.11 The training process continues in the network until the desired output value is ob-tained.12 When certain amounts of the input are entered in the network during learning, the network makes changes to itself to be able to give similar responses. Here, the error in question is the difference between the estimated output and the generated output. After training, the network is tested to find whether ANN has actually learned, instead of just memorizing, the data. In the test section, the data not used during the training is used. The performance of a developed ANN model is determined using different error-analysis methods. In general, such methods can be ranked as the absolute fraction of variance (R2), the root-mean-square error (RMSE) and the mean absolute-percentage error (MAPE). The best performance of an ANN model is at the highest value of R2 and at the lowest values of RMSE and MAPE.13 Such parameters are defined with the following equations: R2 =1- £(MRexp, -mrann t)2 I (mrann )2 (1) 3.1 Stress-concentration factor (K) and notch-sensitivity factor (q) The stress-concentration factor (Kt) is defined as the ratio of the biggest stress generated at bottom of the notch to the nominal stress:1819 K =- (4) RMSE = ,j N £ (mrann ,, - MRev ,,)2 (2) MRann - MReXp MAPE =-—-- 100 MR.m, (3) In the calculation of torsional stress, the relations in equations 5 and 6 are used: M (5) T n W p M T n = KW (6) 3 STRENGTH-REDUCTION FACTORS Resistance diagrams are obtained using standardexperiment test-bar surfaces that have been polished. Dimensional and surface features of the actual machine elements are different from the test bars. Therefore, the values taken from a permanent-resistance diagram cannot be used without considering the resistance-reduction factors.14 The resistance limits of materials are affected by the factors such as notch, surface roughness, dimension, manufacture method, heat treatment, environmental effect, etc.15 In some cases, the results obtained for machine elements with experiments show the existence of the stresses much bigger than the normal stresses. The reason for that is the geometrical difference between the parts. The notch is the generally defined dimensional difference.16 In design of machine elements, geometric figure differences or discontinuities such as notches, holes, steps or various groove roundings and keyways can occur for certain reasons. Sudden section changes and discontinuities cause concentrations in the force flow, particularly in the stress area. Such stress concentrations cause a notch effect on the material.17 In the machine elements, the stress that is times the calculated nominal stress is generated at the bottoms of geometric figures. If the material is brittle, the notched material is broken due to the static stress that is times lower than the nominal stress. For example, if there is a notch with a concentration factor Kt = 3 on a machine element made of hardened steel, such an element is three times more fragile than the unnotched one:1719 Kc =1+q(Kt -1) (7) The stress-concentration factor (Kt) is a value depending on geometry. The fatigue-strength-reduction factor indicating an active reduction in the material strength is Kc. The notch factor depends on the geometrical shape of the notch and the sensitivity of the material to the notch. If the effect of the notch's geometrical shape is represented with the theoretical notch factor Kt , and the sensitivity of the material to the notch is represented with the notch-sensitivity factor q, the notch factor is calculated using the relation given in equation 7. 4 RESULTS AND DISCUSSION The data in this study was obtained by examining the graphics relating to the notch factor from Peterson's book "Design Factors for Stress Concentration".20,21 The Table 1: Numbers of trainings and tests for the shafts under torsional stress Tabela 1: Število usposabljanj in podatki za gred, izpostavljeni torzijski obremenitvi Notch-factor values for the shafts under torsional stress Training Data Test Data Total Data Round shaft with a shoulder fillet in torsion Round shaft with a transverse hole in torsion Grooved round bar in torsion Torsion Torsion Torsion 590 130 450 100 30 110 690 160 560 max n graphics were transformed to digital values, obtaining the data for the ANN learning and testing. In the notch charts, there are three basic figures for the shafts under the torsional-stress effect. These are a round shaft with a shoulder fillet in torsion, a round shaft with a transverse hole in torsion and a grooved round bar in torsion. In Table 1, there are the numbers of trainings and tests used for determining the notch factors for the three basic figures. In Table 1, the classification and the numbers for the shafts under the torsional-stress effect are presented. The input data used in ANN includes the maximum shaft diameter (D), the minimum shaft diameter (d) and the chamfer radius (r), while the output data is the notch factor (Table 2). Table 2: Input and output values for the notch factor of the shafts Tabela 2: Vhodne in dobljene vrednosti za faktor zareze gredi Determination input/output parameters for the shafts under torsional stress Symbol Name Input/output D Maximum diameter of the shaft Input d Minimum diameter of the shaft Input r Chamfer radius Input K, Stress-concentration factor Output As ANN has been generated, not all the experiment data is used in the training. After the ANN system has been established and the training procedure finished, 10 % of the experiment data is hidden from the system to check whether ANN has given correct results. In the scope of the study, 690 pieces of data for the round shaft with a shoulder fillet in torsion, 160 for the round shaft with a transverse hole in torsion and 560 for the grooved round bar in torsion have been obtained with theoretical calculations (Table 3). Out of such data, 590 pieces for the round shaft with a shoulder fillet in torsion, 130 for the round shaft with a transverse hole in torsion and 450 pieces for the grooved round bar in torsion were used for the training purposes. The other data was saved for the test purposes. The test data is used to find the error rate of the ANN system estimations. In the study, a feedforward, multiple-layer neural-learning mechanism was used as the learning mechanism. For the learning model, the Levenberg-Marquardt algorithm (LMA) was used. During the determination of the learning criteria in ANN, different network structures were tried and the network structure with the minimum error and maximum learning rate was selected. According to that, the best learning for the round shaft with a shoulder fillet in torsion took place within a 3-4-1 network structure, for the grooved round bar in torsion within a 3-3-1 structure and for the round shaft with a transverse hole in torsion it took place within a 2-3-1 network structure (Figure 2). In this study, a single output layer and a single hidden layer were selected for Table 3: Input and output samples used in the ANN model Tabela 3: Vhodni in izhodni vzorci, uporabljeni v ANN-modelu Round shaft with a shoulder fillet in torsion Grooved round bar in torsion Round shaft with a transverse hole in torsion D d r Kt D d r Kt D d Kt 2 2.04 0.024 2.29 1 1.02 0.025 2.082 2 153.846 3.643 3 3.06 0.051 2.12 2 2.04 0.068 1.928 3 120.000 3.430 4 4.08 0.084 2 3 3.06 0.12 1.835 4 117.647 3.300 5 5.1 0.125 1.926 4 4.08 0.2 1.767 5 100.000 3.160 13 13.26 1.95 1.33 11 11.22 1.375 1.464 6 96.774 3.080 14 14.28 2.45 1.297 12 12.24 1.644 1.44 7 93.333 3.000 15 15.3 3 1.264 13 13.26 1.95 1.414 8 80.000 2.910 16 16.32 3.6 1.242 14 14.28 2.45 1.386 9 72.000 2.840 35 36.75 8.75 1.286 15 15.3 3 1.36 12 60.000 2.708 36 37.8 9.9 1.264 16 16.32 3.6 1.325 13 57.778 2.680 37 38.85 11.1 1.242 17 17.34 4.25 1.3 14 56.000 2.650 38 41.8 0.456 2.7 18 18.36 4.95 1.276 15 54.545 2.640 39 42.9 0.663 2.5 35 36.75 7.875 1.425 16 53.333 2.630 62 93 3.844 1.925 36 37.8 9 1.4 63 94.5 4.725 1.8 37 38.85 10.175 1.364 64 96 5.568 1.728 38 39.9 11.4 1.338 65 97.5 6.5 1.66 45 67.5 3.375 2.144 66 99 8.25 1.584 46 69 3.956 2.04 67 100.5 10.05 1.51 47 70.5 4.7 1.94 87 261 17.4 1.457 48 72 5.376 1.872 88 264 19.8 1.41 53 79.5 10.6 1.574 89 267 22.25 1.374 54 81 12.15 1.53 90 270 24.75 1.34 55 82.5 13.75 1.486 91 273 27.3 1.32 56 84 15.4 1.44 min - x =- nor / r _Y ^ max An ) (8) Table 4: Determination of the appropriate network design Tabela 4: Določanje oblikovanja primerne mreže Round shaft with a shoulder fillet in torsion Grooved round bar in torsion Round shaft with a transverse hole in torsion MLP 3-15-1 MLP 3-22-1 MLP 2-3-1 MLP 3-13-1 MLP 3-19-1 MLP 2-11-1 MLP 3-23-1 MLP 3-30-1 RBF 2-9-1 MLP 3-13-1 MLP 3-5-1 MLP 2-7-1 MLP 3-20-1 MLP 3-8-1 MLP 2-7-1 RBF 3-24-1 MLP 3-32-1 RBF 2-2-1 MLP 3-30-1 RBF 3-7-1 MLP 2-5-1 MLP 3-30-1 RBF 3-18-1 MLP 2-8-1 MLP 3-20-1 RBF 3-15-1 MLP 2-4-1 MLP 3-21-1 RBF 3-22-1 RBF 2-10-1 MLP 3-47-1 RBF 3-30-1 RBF 2-5-1 Here, Xr represents the actual input value, Xmn is the minimum input value and Xmax is the maximum input value. The values used for the normalization are given in Table 5. Table 5: Values used for normalization Tabela 5: Vrednosti, uporabljene za normalizacijo Figure 2: Suitable network structures for the notch factors of the shafts for: a) round shaft with a shoulder fillet in torsion, b) grooved round bar in torsion, c) round shaft with a transverse hole in torsion Slika 2: Primerne strukture mreže za faktor zareznega učinka na gredi pri torziji: a) okrogla gred z zaokroženim prehodom, b) okrogla palica z utorom, c) okrogla gred s prečno odprtino each type of the shafts. As a result of the experimental-data training, it was observed that optimum outputs were the models having eight neurons for the round shaft with a shoulder fillet in torsion, seven neurons for the grooved round bar in torsion and six for the round shaft with a transverse hole in torsion. For all of these experimental-data trainings, determination of the network structure and its optimization, the Pythia software was used. In the software, for each different notch situation (the grooved round bar in torsion, the round shaft with a transverse hole in torsion and the round shaft with a shoulder fillet), the ANN model with the highest performance was determined. For this purpose, different variations were tried to determine the notch factor of the shafts under the torsional effect, and the model with the highest performance was selected as the ANN model (Table 4). In order to test the network structure of ANN, a normalization of the inputs was implemented at first. The normalization of the inputs and outputs was made within the ranges of (-1, +1) or (0, -1). The normalization of the input (Xnor) is made with equation 8: r - Shafts under torsion Parameters xmax xmin Round shaft with a shoulder fillet in torsion D (maximum diameter of shaft) 91 2 d (minimum diameter of shaft) 273 2.04 r (chamfer radius) 27.3 0.024 Grooved round bar in torsion D (maximum diameter of shaft) 57 1 d (minimum diameter of shaft) 85.5 1.02 r (chamfer radius) 17.1 0.025 Round shaft with a transverse hole in torsion D (maximum diameter of shaft) 16 2 d (minimum diameter of shaft) 153.846 53.333 Formulization of neurons was made with the Fermi-transfer function that is widely used in the ANN training (equation 9). The Fermi-transfer function is a commonly preferred function in the studies conducted in different areas: 1 (9) F = , -4(T xnor-w, -0.5) 1+e ^ Here, Xnor represents the normalized value of the input as (I = 1, 2, 3, ..., n) and represents its weight value. The weights obtained in the ANN model are given in Table 6. The Fermi functions created for each shaft type considered in the study are given in equations 10, 11 and 12: F 1 F erfilet(1 -4 ) d(1 - 3) 1+ e Dnor -W21 + dnor -w21 + rnor -w21 -°.5) 1+ e -4(T D nor -W21 + dnor - w21 + rnor - w21 -°.5) F 'Transverse hole (1 -3) 1+ e -4