Department of Mechanics and Maritime Sciences CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2019 Railway wheel tread damage: Detection and consequences of wheel-rail impact loading Master’s thesis in Mechanics and Maritime Sciences DAVIDE DELLA VALLE MASTER’S THESIS IN MECHANICS AND MARITIME SCIENCES Railway wheel tread damage: Detection and consequences of wheel-rail impact loading DAVIDE DELLA VALLE Department of Mechanics and Maritime Sciences Division of Dynamics CHARMEC CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2019 Railway wheel tread damage: Detection and consequences of wheel-rail impact loading DAVIDE DELLA VALLE © DAVIDE DELLA VALLE, 2019-06-11 Supervisors: Tore Vernersson, Michele Maglio Examiner: Jens Nielsen Master’s Thesis 2019:34 Department of Mechanics and Maritime Sciences Division of Dynamics CHARMEC Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: + 46 (0)31-772 1000 Cover: Snap shot of the modelled wheelset in Abaqus CAD interface. Department of Mechanics and Maritime Sciences Göteborg, Sweden 2019-06-11 I Railway wheel tread damage: Detection and consequences of wheel-rail impact loading Master’s thesis in Master’s Mechanical Engineering DAVIDE DELLA VALLE Department of Mechanics and Maritime Sciences Division of Dynamics CHARMEC Chalmers University of Technology Abstract Swedish transport authorities are investing large amounts of capital in technologies aimed at detecting railway wheel tread damage to reduce the costs for maintenance and repair of wheelsets and railway infrastructure. Wheel impact load detectors (WILDs) based on load cells and accelerometers is one such type of technology currently in use by Trafikverket (the Swedish Transport Administration). This thesis consists of two related parts. The first part is a statistical analysis of data collected by WILDs to assess accuracy and increase confidence in their performance. The statistical tool used to fit the measured data is a multiple linear regression model. For a few selected wheels with evolving rolling contact fatigue damage, data collected over a one-year period have been analysed. At the early stages of wheel tread degradation, it is shown that the measured dynamic loads are considerably influenced by the train speed. For wheels with severe tread damage, the loads are also significantly influenced by the time since the previous wheel maintenance or replacement (probably related to the increasing wheel tread degradation rate over time). It is observed that the accuracy of the detectors cannot be firmly checked since the data are shown to be influenced by train speed and the time of measurement, and because of lack of measured data within a short time window where the tread damage is close to constant. However, a few observations indicating a need for detector calibration are given by comparing the results from different detectors when based on data registered for the same set of wheels. In the second part of the thesis, a Python script for the Abaqus software has been written to automatically generate a parameterized wheelset model. The Python script allows to easily alter the geometrical features of the wheel design, such as the rim thickness. A non-powered wheelset model is used to evaluate the fatigue resistance of the hollow wheelset axle. A case of warning alarm values registered by a WILD is used to define the periodic loads acting on the rolling contact circle of the wheels. It is shown that the Sines’ criterion is the most suitable to calculate the equivalent stress of the most stressed section. For the given set of applied loads, the calculated stress state in the hollow axle does not induce a fatigue damage to the axle. Key words: wheel impact load detectors, multiple linear regression model, parameterized wheelset model, dynamic analysis. II III Acknowledgements I would like to thank my supervisors Jens Nielsen, Tore Vernersson and Michele Maglio for all the support and help they have given to me during these months. I have deeply appreciated their careful revision of all my work. A huge gratitude goes to all my friends who have always encouraged and cheered me up in all the moments of my studies. I would like to thank my girlfriend Luisa because she never made me feel completely far from her despite the distance. I would like to thank my grandmother Maria whose ‘malocchi’ were necessary to pass every single exam. I would like to thank my brothers and my sweet sister who represented my role model. I am the way I am because of them. I would like to thank my parents who have always made me feel their love. Last but not the least, a special thought goes to my grandfather Enrico, my uncle Giancarlo, my aunts Teresa and Nunziata who passed away during my university studies, but they would be happy for this important moment. DAVIDE DELLA VALLE IV Contents Abstract ................................................................................................................................................... I Acknowledgements ........................................................................................................................ III Contents ............................................................................................................................................... IV Preface .................................................................................................................................................. VI Nomenclature .................................................................................................................................. VII 1 Introduction .................................................................................................................................... 1 2 Wheel tread damage ................................................................................................................... 3 2.1 Rolling contact fatigue of railway wheels................................................................. 3 2.2 Wheel flat ................................................................................................................................. 5 2.3 Wear of railway wheels ..................................................................................................... 5 3 Statistical analysis of wheel impact load detector data ............................................. 7 3.1 Wheel impact load detectors .......................................................................................... 7 3.2 Preface to the analysis ....................................................................................................... 8 3.2.1 Introductory notes ..................................................................................................... 8 3.2.2 Aims ................................................................................................................................... 8 3.2.3 Theoretical background to the statistical analysis ..................................... 8 3.3 Influence of train speed on mean wheel load ...................................................... 15 3.4 Investigation on dynamic load.................................................................................... 22 3.4.1 Influence of train speed on dynamic load: low wheel tread damage ... ........................................................................................................................................... 23 3.4.2 Influence of train speed on dynamic loads: severe wheel tread damage ........................................................................................................................................... 33 3.4.3 Concluding comments ........................................................................................... 36 4 Parameterized finite element model ............................................................................... 38 4.1 Implementation phase .................................................................................................... 38 4.2 Wheelset model ................................................................................................................. 38 4.2.1 Material properties ................................................................................................. 40 4.2.2 Python code and modal analysis ...................................................................... 41 5 Fatigue analysis: non-powered wheelset ...................................................................... 45 5.1 Aims and background ..................................................................................................... 45 5.2 Dynamic analysis ............................................................................................................... 46 5.2.1 Loads and boundary conditions ....................................................................... 47 5.2.2 Mesh ............................................................................................................................... 48 V 5.3 Results .................................................................................................................................... 52 6 Conclusions and future work ............................................................................................... 57 6.1 Summary ............................................................................................................................... 57 6.2 Future work ......................................................................................................................... 58 References............................................................................................................................................... 59 Appendix A ................................................................................................................................................ 1 A1 General description of the Matlab Code .................................................................... 1 A2 Matlab Code ............................................................................................................................ 2 Appendix B................................................................................................................................................. 1 B1 Complementary information to Section 3.3 ............................................................ 1 B2 Complementary information to Section 3.4 ......................................................... 26 VI Preface The present Master’s thesis was carried out between December 2018 and June 2019 at the Division of Dynamics at the Department of Mechanics and Maritime Sciences, Chalmers University of Technology, Sweden. The work is part of the EU project In2Track2, which is one of the many ongoing projects within the National Center of Excellence in Railway Mechanics, CHARMEC. Further, the project was carried out in collaboration with Trafikverket and Green Cargo AB. The accomplishment of this project would not have been possible without the help of the following people: Mr Lars Fehrlund, from Green Cargo AB, who handed over the data registered by the wheel impact load detectors. Dr Matthias Asplund, from Trafikverket, who shared information regarding the wheel impact load detectors (WILDs) and provided feedback on the first part of this project regarding the analysis of WILDs data. I want to particularly thank my supervisors, Prof. Jens Nielsen and Dr Tore Vernersson, whose insights and guidance were fundamental for the completion of my thesis. A special thanks to my youngest supervisor, Mr Michele Maglio, who was my firm reference point during the development of my work. Finally, it is worth mentioning that this MSc thesis was produced within the Erasmus+ bilateral agreement between Chalmers University of Technology and Politecnico di Torino. DAVIDE DELLA VALLE VII Nomenclature Abbreviations FE Finite element PRESS Predicted error sum of squares RCF Rolling contact fatigue WILD Wheel impact load detector Symbols 𝐹𝑑𝑦𝑛 Dynamic wheel impact load 𝐹𝑙 Mean wheel load 𝐹𝑃 Peak wheel impact load 𝐼𝑚 First invariant of the mean stress tensor 𝑀 Mean stress sensitivity 𝑀𝑆𝑟𝑒𝑠 Mean sum of residual squares 𝑅2 Coefficient of determination 𝑆𝑆𝐸 Sum of squares explained 𝑆𝑆𝑅 Sum of the residuals squared 𝑆𝑆𝑇 Sum of the total squares 𝜎𝑎,𝑖 Alternate component of the ith principal stress 𝜎𝑖 ith principal stress 𝜎𝑚,𝑖 Mean component of the ith principal stress CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 1 1 Introduction For the dynamics of a railway vehicle, the interaction between wheel and rail is by far the most important aspect. Keeping the wheels and vehicles in an acceptable condition is a major concern for both railway operators and infrastructure owners [1]. Since damage induced on the wheel tread might be detrimental for both track and vehicle components, infrastructure owners have adopted several countermeasures in order to reduce costs for repair and maintenance. To this aim, criteria for determining when a railway wheel should be replaced or reprofiled have been debated and updated over time. For instance, in the last decades Swedish criteria were based on identifying only a critical length of the wheelflat, whereas recent studies have shown that the depth of the flat may have a more considerable influence than the length [2]. Severe wheel-rail impact loads are also generated by other discrete wheel tread defects, such as clusters of rolling contact fatigue (RCF) damage leading to pieces of material breaking out from the tread. Wheel flats and severe RCF damage, apart from having a large impact on noise generation, might also cause safety problems. Moreover, they induce large impact forces, see Figure 1.1. The wheel-rail impact forces occur whenever a wheel flat on the wheel tread meets the rail. The wheel flat results in a transient and periodic loading due to an impact for each revolution of the wheel. During this phase, the wheel moves downwards to compensate for the missing wheel material and at the same time the rail moves upwards. Since the wheel and rail cannot utterly compensate for the irregularity due to their inertia, there is a reduction in the contact force. After passing the centre of the flat, the wheel continues downwards because of its higher inertia. This results in a peak in the contact force, which is followed by a damped transient response [3]. These impact forces aggravate damage on the wheel tread. Damage on the railway wheels occur due to mainly two phenomena, fatigue and wear, and their synergetic effects. Before introducing what countermeasures have been adopted by infrastructure owners to reduce costs for repair and maintenance of track infrastructure, it is worthwhile to give a brief introduction regarding the most common types of damage of railway wheels. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 2 Figure 1.1: Theoretical description of an impact force acting on the wheel of a railway vehicle [4]. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 3 2 Wheel tread damage 2.1 Rolling contact fatigue of railway wheels The development of rolling contact fatigue damage (RCF) can in general be divided into four phases. The first phase involves the crack initiation. Then follows a second phase during which an early crack propagation occurs. The third phase refers instead to an extended crack propagation, whereas the last phase is the one responsible for the final fracture [5]. Surface-initiated RCF cracks occur when a railway component in rolling contact is subjected to cyclic tangential wheel-rail contact loads of high magnitude due to braking and curving. Under these conditions, the contact surface deforms plastically. Consequently, the accumulated plastic strains lead to crack formation as the fracture strain is exceeded [6]. RCF cracks are commonly classified based on their appearance and location on the wheel. As a matter of fact, the lateral and longitudinal (tangential) forces that occur within the wheel tread may lead to damage in four specific zones. The most common form of RCF cracks are located on the field side of the tread (Zone 1, see Figure 2.2). These cracks are mainly due to the interaction between the wheel and the lower inner rail during curving and their orientation depends on the loading. The cracks are commonly oriented perpendicularly to the direction of the resulting tangential contact force during curving, which is usually around 30°-45° with respect to the wheel axis [7]. The same mechanism and forces, but acting during the interaction between the wheel and the higher outer rail during curving are responsible for damage occurrence in Zone 2. Cracks affecting Zone 2 are typically oriented by 30°-60° with respect to the wheel axis, see picture on the left in Figure 2.2. Zone 3 RCF occurs far less frequently than in Zone 1 or Zone 2. Fatigue cracks occurring in Zone 3 are located in the centre of the tread and they arise from repeated application of high longitudinal tractive forces which in turn result from an increase in the longitudinal creep. On the contrary, cracks occurring in Zone 4 have a significant longitudinal orientation of around 60°-80° to the wheel axis, which indicates that they have been induced by a resultant lateral creep force [8]. Lastly, RCF clusters arising from mainly braking operations appear at the centre of tread. They are in general oriented parallel to the wheel axis, see Figure 2.2. Figure 2.1: Schematic sketch regarding plastic deformation of the surface material in a railway wheel due to tangential loading. The dashed lines indicate material planes before and after deformation [6]. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 4 Figure 2.2: RCF damage on a locomotive wheel. [Picture courtesy Michele Maria Maglio]. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 5 2.2 Wheel flat Wheel flats are the most common form of wheel tread damage. Wheel flats are caused by the sliding of the wheel along the rail that occurs as a consequence of a blockage or partial blockage of the wheelset. The reason for the sliding may be either a fault in the brake system or an excessive applied braking force [8, 9]. Consequently, the dissipated friction energy turns into heat and makes the tread surface locally flat. This local flat defect is designated as wheel flat, see Figure 2.3. However, under the significantly high temperatures (e.g. 800˚) reached during the generation of the flat, the pearlitic microstructure of the wheel steel transforms into martensite. Martensite is a very hard and brittle form of steel microstructure. As a consequence of further cyclic loads arising from the wheel-rail contact, cracks can develop in the brittle material and grow considerably until the brittle surface starts to spall out from the wheel tread [8, 9]. The cavities on the tread surface, such as RCF clusters and wheel flats with spalling, produce a local deviation from the nominal wheel radius. This deviation may generate impact loads in the wheel-rail contact that further aggravate the irregularity [9]. Figure 2.3: Wheel flat on the tread surface. [Picture courtesy Michele Maria Maglio]. 2.3 Wear of railway wheels The broadest definition of “wheel wear” includes any kind of damage occurring on the running surface that involves loss of material. Wheel wear can be categorized into three main classes, deeply described in [10]: • Flange wear, involving the reduction of the flange thickness that leads to a reduction in the strength and a worsening in wheel-rail contact, see Figure 2.4; • Wheel tread wear, involving an increase in the flange height and an increase in the flange thickness which in turn might cause severe problems in turnouts or crossings; • Out-of-round wheel is often caused by the presence of a mixed microstructure within the wheel tread as a result of heat treatments issues during wheel manufacturing [8]. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 6 All of these phenomena lead to a significant change in the shape of the wheel profile, which in turn could severely affect the dynamics of the vehicle and the safety against derailment. These modifications may be recovered by reprofiling the wheel, i.e. turning it to its original shape. However, a limited number of reprofiling operations may be carried out before the height and thickness of the rim do not comply anymore with the mechanical resistance of the wheel. As this limit is reached, the wheel must be replaced [12]. Figure 2.4: Typical railway wheel profile. The picture shows also flange and tread wear phenomena [11]. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 7 3 Statistical analysis of wheel impact load detector data The first goal of the present thesis project has been to analyse data detected by wheel impact load detectors located in Sweden in order to draw some conclusions from the arising trends and to increase the confidence in the performance of the detectors. 3.1 Wheel impact load detectors A significant amount of capital is being invested by the Swedish transport authorities in technologies aimed at detecting railway wheel damage on time to reduce costs deriving from maintenance and repair and to meet noise legislation. One countermeasure that has been adopted is represented by wheel impact load detectors (WILDs). A WILD system consists of sensors set along the sleepers connecting the two rails. The system makes use of a combination of vertical load receptors, lateral receptors and accelerometers. These sensors collectively provide coverage for approximatively five meters of tangent track, which becomes the detection zone as shown in Figure 3.1. This system is equipped with an automatic vehicle identification (AVI) system that detects the unique vehicle number characterizing each passing wagon, regardless of the travel direction, through the detection zone [14]. The WILD system provides the following information: • Train speed; • Direction of travel; • Mean wheel load (𝐹𝑙) on left and right rails; • Peak wheel impact load (𝐹𝑃) on left and right rails; • Dynamic wheel impact load (𝐹𝑑𝑦𝑛 = 𝐹𝑃 − 𝐹𝑙) on left and right rails; In addition, the WILDs generate alarms when any of the above-mentioned loads exceeds a threshold value configured by the operators or set by the regulations. These output data are then saved in a file, which is stored on disk and is available to the operators. These results can be used for statistical analyses as has been done in the present thesis work. Figure 3.1: Wheel impact load detector (WILD) installed in track [13]. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 8 3.2 Preface to the analysis 3.2.1 Introductory notes The analysed data have been provided in different Microsoft Excel files by Mr Lars Fehrlund, vehicle engineer at Green Cargo AB. More specifically, each Excel file refers to a specific axle of one specific locomotive. Each file contains information about the dates and time instants at which the data were registered, the name of the detector station, the travel direction of the train, the train speed, and the mean and peak impact loads registered for both wheels. Based on additional information provided by Lars Fehrlund, data detected at Mellansjö USP MJ MDS and Mellansjö NSP HJ MDS stations have been excluded from the analysis because these detectors have not yet been calibrated. In addition, data detected for low train speeds (lower than 40 km/h) have been excluded because of a relatively high risk that the WILD is not able to identify the locomotive correctly. 3.2.2 Aims As far as the aims are concerned, the first goal has been to create a unique Matlab code that reads any Excel file having the same format as the ones provided by Lars Fehrlund. Its functionalities are explained in Appendix A. Then, several analyses have been carried out using statistical tools. More specifically, the influence of the train speed on the mean wheel loads has been investigated first, and then the dependence on dynamic impact loads of the train speed and the time instants at which data have been registered has been studied. Lastly, the performance of different detectors has been evaluated by comparing measured dynamic loads when subjected to the same wheel tread defect. More specifically, the analysis focused on sequences of data where, according to detector data, the degradation rate of the RCF defects led to a slow increase in dynamic load over time. 3.2.3 Theoretical background to the statistical analysis 3.2.3.1 Simple linear regression model The statistical technique that has been used for investigating different relationships between the data registered by the wheel impact load detectors is the regression analysis. Reference [15] has been taken into account to provide a theoretical background allowing for carrying out a simple but solid statistical analysis. Regardless of the selected pair of variables to be analysed, it is possible to display any couple of data registered by WILDs in a scatter diagram. A scatter plot, as the one shown in Figure 3.2, might suggest a possible relationship between the two variables. Based on a first impression given by the scatter plot in Figure 3.2, the two variables can be related by a linear relationship. The straight line obtained by linear interpolation can be described by Eq. (3.1): 𝑦 = 𝛽0 + 𝛽1𝑥 (3.1) CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 9 where 𝑦 represents the dependent variable, 𝑥 represents the independent variable, 𝛽0 is the intercept and 𝛽1 accounts for the slope. These two latter coefficients are also known as regression coefficients and are statistically computed based on the sample data. However, since the data points do not lie exactly on the straight line, the difference between the observed data (𝑦) and the straight line (𝛽0 + 𝛽1𝑥), namely the statistical error ( 𝑒 ) or residual, should be considered. Hence, a more plausible model is represented by the following equation. 𝑦 = 𝛽0 + 𝛽1𝑥 + 𝑒 (3.2) Equation (3.2) is called a linear regression model. The independent variable 𝑥 is usually denoted as the predictor or regressor variable, whereas the dependent variable 𝑦 is the response variable. The linear regression model described by Eq. (3.2) is referred to as a simple linear regression model because it involves only a single regressor variable. Moreover, the mean value of the errors is assumed to be zero, their variance is unknown and they are supposed to be uncorrelated, that means that the value of one error does not depend on the value of any other error. (a) (b) Figure 3.2: (a) Example of a typical scatter plot. (b) Linear interpolation between independent and dependent variable. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 10 A simple linear regression model has been used to study the influence of the train speed and the time instants at which data have been registered on the mean loads measured by the detectors. Simple linear regression models have been generated in Matlab using the function fitlm [16], see Appendix A. This function saves any statistical parameters of interest in a nested structure. These parameters include the regression coefficients, which in turn are estimated through the method of least squares, meaning that the sum of the squares of the differences between the observations 𝑦i and the straight line has a minimum. The sum of squared errors S is written as: 𝑆(𝛽0, 𝛽1) = ∑ (𝑦𝑖 − 𝛽0 − 𝛽1𝑥𝑖) 2 𝑛 𝑖=1 (3.3) Therefore, the least-squares estimators of 𝛽0 and 𝛽1 must satisfy 𝜕𝑆 𝜕𝛽1 = −2 ∑ (𝑦𝑖 − 𝛽0 − 𝛽1𝑥𝑖)𝑥𝑖 𝑛 𝑖=1 = 0 (3.4) and 𝜕𝑆 𝜕𝛽0 = −2 ∑ (𝑦𝑖 − 𝛽0 − 𝛽1𝑥𝑖) 𝑛 𝑖=1 = 0 (3.5) By simplifying Eq. (3.4) and Eq. (3.5) and solving for 𝛽0 and 𝛽1, it is possible to deduce the equation representing the simple linear regression model. Before applying the fitted model, the significance of the regression model has to be checked along with the accuracy of the basic assumptions. One statistical tool used to check if the data agree with certain predictions is the hypothesis test. This procedure requires the additional assumption that the errors 𝑒𝑖 are normally distributed. It considers two hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis (𝐻0) is the hypothesis that is directly tested, for instance the assumption that no regression relationship exists between the response ( 𝑦) and the predictor variable (𝑥). As opposite to the null hypothesis, the alternative hypothesis (𝐻1) will consequently indicate that there is instead a regression relationship between the response and the predictor variable. The P-value approach is usually used to check the validity of the null hypothesis. After having assumed the null hypothesis as true, the hypothesis testing procedure involves subsequently the calculation of t-statistics. The t-statistics is defined as the ratio between the estimated value of a parameter (β ̂) from its hypothesized value (β0) and its standard error [15] . 𝑡𝛽 ̂ = 𝛽 ̂− 𝛽0 𝑠𝑒(𝛽)̂ (3.6) According to the aim of the statistical analysis, the estimated parameter involved in Eq. (3.6) might be either the intercept or the slope of the regression model. Both these parameters are determined through the method of least squares. Moreover, the t- statistics follows a t-distribution with n-1 degrees of freedom, where n accounts for the number of observed values. Hence, using the known distribution of the t-statistics, CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 11 it is possible to compute the P-value. The latter then has to be compared to a significance level α, which in this study has been set to 0.05 and assumed as default in the regression models generated by the fitlm function in Matlab. If the P-value is less than α, the null hypothesis will be rejected in favour of the alternative. On the contrary, if the P-value is larger than α, the null hypothesis should not be rejected [17]. Another way to assess the goodness of the fit is through the evaluation of the coefficient of determination 𝑅2 . The coefficient of determination, customarily also denoted as R-squared, is defined according to Eq. (3.7): 𝑅2 = 𝑆𝑆𝐸 𝑆𝑆𝑇 = 1 − 𝑆𝑆𝑅 𝑆𝑆𝑇 (3.7) where: • 𝑆𝑆𝑇 is the sum of the quadratic deviations of the observed values (𝑦𝑖) from their average; • 𝑆𝑆𝐸 is the sum of the quadratic deviations of the response values from their average; • 𝑆𝑆𝑅 is the sum of the squares of the residuals deduced from the fitted model. Since 𝑆𝑆𝑇 is a measure of the variability in 𝑦 without considering the effect of the regressor variable 𝑥, and 𝑆𝑆𝑅 is a measure of the variability in 𝑦 remaining after 𝑥 has been considered, R-squared accounts for the proportion of variation in 𝑦 explained by the regressor 𝑥. It ranges between 0 and 1 and the larger its value is, the better the variability in 𝑦 is explained by the adopted regression model [18]. 3.2.3.2 Multiple linear regression model Multiple linear regression models have been used to analyse the correlation between the measured dynamic impact loads, the time at which the data have been registered and the train speed. A multiple linear regression model relates the response variable (𝑦) to k regressor variables. The model is described by Eq. (3.8): 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + ⋯ + 𝛽𝑘𝑥𝑘 + 𝑒 (3.8) or equivalently in matrix form by Eq. (3.9): 𝒚 = 𝜷𝑿 + 𝒆 (3.9) where each regression coefficient 𝛽𝑘 indicates the expected change in the response variable per unit of change in xk when all the other regressor variables are held constant. It is worth underlining that the term linear refers to a response variable expressed as a linear function of the unknown regression coefficients and not of the regressor variables. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 12 As already explained in Section 3.2.3.1, the first step is to calculate the regression coefficients through the method of least squares. For the sake of brevity, the mathematical passages used to compute the regression coefficients that minimize the sum of squared errors are here omitted but their final expression is reported in Eq. (3.10). 𝜷 = (𝑿′𝑿)−𝟏𝑿′𝒚 (3.10) Consequently, the vector of the fitted values 𝒚̂ turns out to be: 𝒚̂ = 𝑿𝜷 = 𝑿(𝑿′𝑿)−𝟏𝑿′𝒚 = 𝑯𝒚 (3.11) The 𝑛 𝑥 𝑛 matrix 𝐇 = 𝐗(𝐗′𝐗)−𝟏𝐗′is known as “hat matrix” and allows to correlate the vector of observed values to the vector of fitted values. This matrix has particular properties and plays a relevant role in multiple regression analyses [15]. As far as the test for significance of the adopted regression model is concerned, a very close procedure to what has been presented in Section 3.2.3.1 is usually performed. This test still involves the definition of two hyphotheses: the null hypothesis that implies that no regressor variables influence the response variable (𝐻0: 𝛽0 = 𝛽1 = ⋯ = 𝛽𝑘 = 0), and the alternative hypothesis that implies that at least one regressor variable affects significantly the response variable (𝐻1: 𝛽𝑘 ≠ 0 for at least one k- variable). The rejection of the null hypothesis can be checked by looking at F- statistics or the P-value. Once having created a multiple regression model in Matlab through the fitlm function, all these parameters of interest may be accessed through the anova function [16], see Appendix A. After having verified that the response variable is significantly influenced by at least one regressor variable, it is worth figuring out which multiple regression model best fits the observed data. This cannot be achieved by looking at the R-squared value because its value cannot decrease by adding regressor variables to the model [17]. Instead, this may be achieved by looking at the adjusted R-squared that is defined as: 𝑅𝐴𝑑𝑗 2 = 1 − [ 𝑆𝑆𝑅 𝑛−𝑝−1 𝑛−1 𝑆𝑆𝑇 ] (3.12) where n accounts for the number of observed data and p is the number of regressor variables used in the regression model. Since 𝑆𝑆𝑇 (𝑛 − 1)⁄ is constant regardless of how many variables have been introduced in the model and 𝑆𝑆𝑅 (𝑛 − 𝑝 − 1⁄ ) is the residual mean square, 𝑅𝐴𝑑𝑗 2 will only increase if the addition of the regressor variable reduces the residual mean square [18]. This statistical tool allows therefore to evaluate and compare regression models. Another useful statistical tool in comparing regression models is the PRESS statistics. Reference [15] defines the predicted residual sum of squares (PRESS) as the difference between the ith observed value and the predicted value of the ith observed response by fitting a model that excludes the ith observation. This definition of the PRESS residual or predicted error is outlined by Eq. (3.13) CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 13 𝑒(𝑖) = 𝑦𝑖 − 𝑦(𝑖)̂ (3.13) According to the definition given above, it would seem that to calculate all PRESS residuals requires to fit n regression models. However, it is possible to deduce the PRESS residuals from the results of the least squares fit. As a matter of fact, the ith PRESS residual may be calculated according to Eq. (3.14): 𝑒(𝑖) = 𝑒𝑖 1−ℎ𝑖𝑖 (3.14) Eq. (3.13) defines the PRESS residual as the ordinary residual, weighted according to the diagonal elements of the hat matrix ℎ𝑖𝑖. Consequently, from the definition of the PRESS residual it is possible to define the predicted error sum of squares, commonly known as PRESS statistics, that is defined according to Eq. (3.15): 𝑃𝑅𝐸𝑆𝑆 = ∑ (𝑦𝑖 − 𝑦(𝑖)̂)2 𝑛 𝑖=1 = ∑ [ 𝑒𝑖 1−ℎ𝑖𝑖 ] 2 𝑛 𝑖=1 (3.15) Both the adjusted R-squared and PRESS statistics approaches suggest the model that best fits the data. Generally, the PRESS statistics tend to recommend smaller models than the adjusted R-squared. However, the regression model that does not exhibit large gaps in the PRESS statistics and the adjusted R-squared values with respect to higher order polynomials is commonly the one that is chosen. 3.2.3.3 Model adequacy checking As mentioned in the previous sections, the regression models and the significance tests are based on some assumptions. In fact, it has been assumed that the error term 𝑒 has constant variance and is uncorrelated. Additionally, the hypothesis testing procedure requires an additional assumption, namely that the errors are normally distributed. All these assumptions must be checked before the fitted model can be applied. The detection of any violations of these assumptions, along with the detection of any model inadequacy, is generally carried out through the study of residual plots. These plots usually involve scaled residuals. The first category refers to standardized residuals that use the approximate average variance of residuals, that in turns is represented by the mean sum of squares (𝑀𝑆𝑟𝑒𝑠), as scaling factors. The definition of a standardized residual is herein reported as 𝑑𝑖 = 𝑒𝑖 √𝑀𝑆𝑟𝑒𝑠 (3.16) However, using 𝑀𝑆𝑟𝑒𝑠 as the variance of the ith residual is just an approximation. This approximation may be improved by using as scaling factor the exact standard deviation of the ith residual. This is considered in the definition of studentized residuals, whose expression is proposed by Eq. (3.14): 𝑟𝑖 = 𝑒𝑖 √𝑀𝑆𝑟𝑒𝑠(1−ℎ𝑖𝑖) , 𝑖 = 1,2, … , 𝑛 (3.17) CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 14 Studentized residuals are usually preferred in the graphical analysis [15]. The graphical analysis involves the study of different residual plots. The first one is the plot of residuals versus the corresponding fitted values. This plot allows to detect model inadequacies and, for the sake of clarity, a typical one is proposed in Figure 3.3. If the residuals can be contained within two horizontal bands, the model has no obvious deficiencies. Patterns that deviate from this one might instead indicate a non- constant variance or nonlinearity and therefore the models should be corrected by other statistical procedures [15]. Especially if the hypothesis testing that is based on the normal distribution assumption of residuals has been carried out, the normality must be checked. This property is usually checked through a normal probability plot of residuals. This graph is constructed in such a way that the cumulative of the normal distribution will appear as a straight line. Consequently, if the residuals, commonly the studentized residuals, are plotted against the cumulative probability on the normal probability plot, they should lie approximatively on the straight line as shown in Figure 3.4 for the case in which the normal distribution assumption holds. Any eventual strong deviation from the straight line will therefore indicate that the distribution is not normal and consequently the hypothesis testing procedure cannot be considered as reliable. Figure 3.3: Typical plot of residuals versus corresponding fitted values. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 15 Figure 3.4: Typical normal probability plot of studentized residuals. 3.3 Influence of train speed on mean wheel load In the following section, the influence of train speed on the mean wheel load has been investigated for several detector stations. The mean wheel loads measured by the WILDs should correspond to the static wheel loads that are merely depending on the weight of the vehicle and any possible external static load acting on it (for example snow and ice). Hence, the registered mean loads are expected to be independent of train speed. Further, for a given wheel on a given vehicle, it is expected that the different detectors should measure the same mean load. Nonetheless, the mean loads on the left and right wheels become quasistatic and slightly dependent on the train speed during curving because of several factors, such as curve radius and cant deficiency. This may lead to a re-distribution of wheel loads between the left and right rails [10]. However, because of lack of information regarding the exact track geometry at the WILDs, it has not been possible to consider this relevant aspect in the statistical analysis. Simple linear regression models, described by Eq. (3.2), involving data registered by each detector station have been determined to investigate the dependence of the train speed (regressor variable) on the mean wheel load (response variable). The analysis has been carried out for loads measured on both wheels of the first axle of three different freight locomotives, each of them identified by a vehicle number. These vehicle numbers are 917400014341, 917400014234 and 917400014325. The analysis involved data collected between March 18th 2018 and January 29th 2019 during which maintenance interventions took place. The same analysis was carried out also by considering restricted time intervals in order to reduce the amount of scattered data. First the period from May 1st to October 18th was considered, and secondly October 18th to December 18th. For these cases, similar results were achieved that led to the same conclusions. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 16 Detector stations at which less than thirty passages were registered during this time interval were excluded from the analysis. An example of results arising from this analysis is reported in Figure 3.5. The fitted linear model suggests that the mean loads do not depend on train speed (because the slope of the regression line given by the regression coefficient 𝛽1 is close to zero). Along with the linear model, the plot of residuals against fitted values must be analysed to prove that the model does not suffer any significant deficiency, see Figure 3.6. Figure 3.5: Simple linear regression model between train speed and mean wheel loads detected for the left wheel of the first axle of locomotive 917400014325 between March 18th 2018 and January 29th 2019 at Skorped detector station. Red dashed lines account for the confidence bounds. Figure 3.6: Plot of studentized residuals versus fitted values derived from the simple linear regression model between train speed and mean wheel loads detected for the left wheel of the first axle of locomotive 917400014325 between March 18th 2018 and January 29th 2019 at Skorped detector station. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 17 Since Figure 3.6 shows that the studentized residuals are randomly distributed around zero, it is possible to state that the adopted linear regression model has no obvious deficiency. Graphs showing linear regression models for other detector stations and for all the considered locomotives are reported in Appendix B. However, for some models the plot of the studentized residuals against the fitted values shows a double- row pattern indicating a non-constant variance. In those cases, reference [15] suggests to perform a transformation to either the response variable or regression variable in order to stabilize the variance. However, in order to prove that the mean loads may be considered as independent of train speed at a given confidence level, hypothesis testing has been performed according to the procedure described in Section 3.2.3.1. In such a case, the null hypothesis states that no linear relationship exists between the regressor and the response variable (𝐻0: 𝛽1 = 0 ). Failing to reject the null hypothesis by the P-value approach will therefore imply that there is no linear relationship between train speed and mean loads. The results arising from the hypothesis testing with a 95 % confidence interval care listed in the following tables. Each table provides information regarding the estimate of the regression coefficients, the number of observed data n, the P-value and the coefficient of determination. Furthermore, the statement concerning whether the null hypothesis is rejected or not is included in the rightmost cell of the table. By looking at the results listed in the tables, it is observed that the null hypothesis stating no linear relationship exists between mean loads and train speed has not been rejected for all models. Therefore, based on the data measured by the detectors, the results do not completely rule out that the mean wheel loads may be seen as independent of the train speed. On the other hand, a further insight helping in stating that the mean loads do not depend on the train speed may be obtained by looking at the R-squared values. Due to their low values in all models, it is possible to confirm that the used linear regression model does not indicate that the variance in the response variable is expressed to a large extent by the regressor variable. Therefore, the train speed does not exhibit a high influence on the mean loads. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 18 Table 3.1: Results arising from the hypothesis testing for each detector station. Regression analysis involved mean loads registered by WILDs for the left wheel of the first axle of locomotive 917400014314 between March 18th 2018 and January 29th 2019. Locomotive 1 RFID: 917400014341 Detector Station Estimate n P-value R2 Rejected Null hypothesis 𝛽0 𝛽1 𝛽1 Bodsjön 90.47 0.040 118 0.070 0.028 No Degerbäcken 92.03 -0.057 94 0.018 0.059 Yes Jörn 97.88 -0.089 120 0.001 0.089 Yes Koler 104.17 -0.136 96 0.000 0.192 Yes Skorped 94.78 0.022 107 0.338 0.008 No Sunderbyns Sjukhus 95.75 -0.019 121 0.465 0.004 No Table 3.2: Results arising from the hypothesis testing for each detector station. Regression analysis involved mean loads registered by WILDs for the right wheel of the first axle of locomotive 917400014314 between March 18th 2018 and January 29th 2019. Locomotive 1 RFID: 917400014341 Detector Station Estimate n P-value R2 Rejected Null hypothesis 𝛽0 𝛽1 𝛽1 Bodsjön 91.56 0.100 118 0.000 0.111 Yes Degerbäcken 93.45 0.012 94 0.637 0.002 No Jörn 105.35 -0.132 120 0.000 0.126 Yes Koler 94.247 0.051 96 0.039 0.044 Yes Skorped 98.676 0.001 107 0.780 0.000 No Sunderbyns Sjukhus 94.39 0.040 121 0.077 0.026 No CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 19 Table 3.3: Results arising from the hypothesis testing for each detector station. Regression analysis involved mean loads registered by WILDs for the left wheel of the first axle of locomotive 917400014234 between March 18th 2018 and January 29th 2019. Locomotive 2 RFID: 917400014234 Detector Station Estimate n P-value R2 Rejected Null hypothesis 𝛽0 𝛽1 𝛽1 Bodsjön 90.07 0.070 176 0.000 0.079 Yes Degerbäcken 90.17 -0.004 90 0.806 0.001 No Jörn 93.69 -0.010 184 0.655 0.001 No Koler 93.19 0.002 158 0.884 0.000 No Skorped 90.04 0.082 170 0.000 0.078 Yes Sunderbyns Sjukhus 97.21 -0.001 179 0.877 0.001 No Table 3.4: Results arising from the hypothesis testing for each detector station. Regression analysis involved mean loads registered by WILDs for the right wheel of the first axle of locomotive 917400014234 between March 18th 2018 and January 29th 2019. Locomotive 2 RFID: 917400014234 Detector Station Estimate n P-value R2 Rejected Null hypothesis 𝛽0 𝛽1 𝛽1 Bodsjön 86.92 0.095 176 0.000 0.094 Yes Degerbäcken 92.88 -0.009 90 0.726 0.001 No Jörn 93.67 -0.039 184 0.124 0.013 No Koler 99.66 -0.058 158 0.016 0.036 Yes Skorped 97.27 -0.024 170 0.172 0.011 No Sunderbyns Sjukhus 94.03 0.008 179 0.829 0.001 No CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 20 Table 3.5: Results arising from the hypothesis testing for each detector station. Regression analysis involved mean loads registered by WILDs for the left wheel of the first axle of locomotive 917400014325 between March 18th 2018 and January 29th 2019. Locomotive 3 RFID: 917400014325 Detector Station Estimate n P-value R2 Rejected Null hypothesis 𝛽0 𝛽1 𝛽1 Bodsjön 90.40 0.061 175 0.001 0.061 Yes Degerbäcken 87.73 0.005 59 0.847 0.001 No Hållsta 100.13 -0.040 39 0.123 0.063 No Jörn 95.06 -0.034 174 0.163 0.011 No Koler 88.20 0.048 157 0.023 0.033 Yes Skorped 96.24 0.010 164 0.643 0.001 No Sunderbyns Sjukhus 101.22 -0.055 166 0.055 0.022 No Table 3.6: Results arising from the hypothesis testing for each detector station. Regression analysis involved mean impact loads registered by WILDs for the right wheel of the first axle of locomotive 917400014325 between March 18th 2018 and January 29th 2019. Locomotive 3 RFID: 917400014325 Detector Station Estimate n P-value R2 Rejected Null hypothesis 𝛽0 𝛽1 𝛽1 Bodsjön 91.91 0.053 175 0.007 0.041 Yes Degerbäcken 94.53 -0.028 59 0.470 0.009 No Hållsta 82.33 0.217 39 0.000 0.452 Yes Jörn 96.02 -0.061 174 0.006 0.042 Yes Koler 95.54 -0.009 157 0.614 0.002 No Skorped 94.55 0.012 164 0.527 0.002 No Sunderbyns Sjukhus 91.71 0.038 166 0.224 0.009 No For those models where the null hypothesis was rejected, the normal distribution assumption has been checked via a graphical approach and reported in Appendix B, see Section B1. According to the graphs, the normal distribution assumption is conserved for all models. As a matter of fact, the graphs do not show substantial deviations from the straight line. Nonetheless, some of them show a typical defect, that is the occurrence of a few residuals exhibiting low offset from the straight line. This phenomenon is usually an indication that the corresponding observations are outliers [15]. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 21 Lastly, a brief comparison between the detectors is proposed in terms of the average value of the mean loads registered over time. The comparison was made by taking into account data registered during the summer time to exclude any influence of snow and ice packed in the bogies. In addition, because of limited amount of data registered during this time period for the locomotive identified by vehicle number 917400014341, the analysis focused on data measured for the first axle of the locomotive identified by vehicle number 917400014234. The average values of the mean loads measured for both wheels at each detector station in the period of time May 1st to November 1st 2018 is shown in Figure 3.7 and Figure 3.8. Figure 3.7: 2D-Bar plot showing the average value of mean loads registered for the left wheel of the first axle of locomotive 917400014234 at each detector station in the period of time May 1st 2018 to November 1st 2019 . The vertical axis shows the average of the mean load. The thin vertical line for each bar allows to identify the upper and lower bounds for the sample data standard deviation. Figure 3.8: 2D-Bar plot showing the average value of mean loads registered for the right wheel of the first axle of locomotive 917400014234 at each detector station in the period of time May 1st 2018 to November 1st 2019 . The vertical axis shows the average of the mean load. The thin vertical line for each bar allows to identify the upper and lower bounds for the sample data standard deviation. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 22 By looking at the Figure 3.7 and Figure 3.8, it is observed that not all the wheel impact load detectors exhibit the same performance in registering the mean loads. This aspect may be evidenced by looking at the average values of the mean loads registered at each detector station. The average value (estimated around 97 kN) of mean loads registered for the left wheel is rather similar at the Bodsjön, Skorped and Sunderbyns Sjukhus detector stations, but evident differences from these values can be observed at the Degerbäcken, Jörn and Koler detector stations. The same differences, but slightly lower, may be observed for the loads registered for the right wheel. As far as the standard deviation is concerned, it generally ranges between 2.5 kN and 3.5 kN. However, a larger value is derived for data registered at the Sunderbyns Sjukhus detectors station which is estimated 7.7 kN. Since the standard deviation accounts for the variability in the measured data, it is possible to state that there are some other hidden factors related to the registered mean load that make the statistical results not completely reliable. 3.4 Investigation on dynamic load The dynamic loads due to wheel-rail impact caused by severe wheel tread damage are of major interest because they are responsible for wheelset and track damage occurrence and growth. Their values in turn depend on the extent of the damage on the wheel tread; larger defects on the wheel tread are responsible for the occurrence of higher dynamic loads [11]. In addition, for a specific type of damage on the wheel tread, a linear increase in dynamic load with increasing speed was shown in [13]. Based on these expectations, the analysis proposed in this section aims at finding any possible correlation between dynamic load, train speed and the time since initiation of the wheel tread defect in order to increase the confidence in WILDs detection ability. Among the three locomotives considered in the analysis proposed in the previous section, Lars Fehrlund also provided additional information regarding the maintenance of one of the locomotives, the one identified by the vehicle number 917400014341. According to this information, the wheelset corresponding to the first axle has undergone maintenance more than once during the time period in which data have been measured. More specifically, on April 9th and May 23rd both wheels were replaced, whereas on October 18th 2018 the wheels were reprofiled. The match of this information may be found by looking at the variation in dynamic loads over time for the right wheel that is shown in Figure 3.9. The corresponding impact load distribution for the left wheel may be found in Appendix B, see Figure B1.45. As is possible to notice from Figure 3.9, the dynamic loads show increasing trends corresponding to a deteriorating state of the wheel tread. Moreover, the sudden reduction in 𝐹𝑑𝑦𝑛 values occurs at specific time instants which perfectly match the dates provided by Lars Fehrlund at which the wheelset was maintained. Another decrease in 𝐹𝑑𝑦𝑛 occurs in the middle of February. Another maintenance action may be expected to have taken place in this period, but no information about this has been received. Thus, having confirmed that the registered dynamic impact loads depend on the time since the initiation of the wheel damage, the analysis was focused on restricted time intervals. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 23 More specifically, the influence of the train speed on the dynamic impact loads over a time period with low wheel tread damage has been investigated first. Secondly, the same investigation has been repeated during a time interval where the damage on the wheel tread is more significant. Figure 3.9: Scattered data of dynamic impact loads generated by the right wheel of the first axle of locomotive 917400014341 and registered by all WILDs over one year period. The data have been plotted for three different train speed ranges. 3.4.1 Influence of train speed on dynamic load: low wheel tread damage In this section, the influence of train speed on the dynamic impact loads measured shortly after the wheelset was maintained on October 18th is investigated. More specifically, the time period between October 18th and December 18th has been considered. To this aim, an insight helping for the understanding of what can be expected from the influence of train speed on dynamic loads is provided by a 3D-bar plot illustrating the mean (average value) of the dynamic loads registered during different time intervals and for different speed ranges. A typical trend can be seen in Figure 3.10. By looking at the trends shown in Figure 3.10, it is possible to notice that the dynamic impact loads tend to increase with increasing train speed. As expected, the trend shows also an increase of the dynamic impact loads over time. This is an expected result because of continuous growth of the RCF damage. Hence, not only the time instants at which data were registered but also the train speed influence somehow the dynamic loads. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 24 Figure 3.10: 3D-Bar plot showing the mean of dynamic loads registered for the right wheel of the first axle of locomotive 917400014341 at the Sunderbyns Sjukhus detector station during specific time intervals and for specific train speed ranges. The vertical axis shows the dynamic load values. The Y- axis shows the time intervals at which data have been registered, while the X-axis shows the considered speed ranges. 3.4.1.1 Results from a multiple regression analysis The next step has been to determine the polynomial that best fits the measured data. To this aim, multiple linear regression models described by Eq. (3.8) have been generated. Data subject of the following statistical analysis refer to dynamic impact loads on both wheels of the first axle of the locomotives identified respectively by vehicle numbers 917400014341and 917400014234 and registered between October 18th and December 18th 2018 by the detector stations located at Sunderbyns Sjukhus and Bodsjön. Similar results were achieved for all the other detectors. However for the sake of brevity, only results related to data detected at Sunderbyns Sjukhus detector station are presented here. Results regarding the data measured at the Bodsjön detector station are presented in Appendix B, see Section B2. Several regression models have been generated. First, only one regressor variable was included, either time or train speed. Then models including a combination of both variables and including higher power exponents have been generated. Then, the models have been compared by looking at PRESS statistics, adjusted R-squared and 𝑀𝑆𝑟𝑒𝑠, whose details are given in Section 3.2.3.2. Lastly, the model characterized by the best results in terms of PRESS statistics and 𝑅𝐴𝑑𝑗 2 has been chosen as the one that best fits the measured dynamic impact loads. The results are listed in the following tables. Each table provides information regarding the number and type of regressor variables used in the fitted model. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 25 Table 3.7: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. The analysis involved dynamic impact loads measured for the right wheel of the first axle of locomotive 917400014341 at Sunderbyns Sjukhus detector station between October 18th and December 18th. LOCOMOTIVE 917400014341 Sunderbyns Sjukhus N° of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 54.52 -0.007 26.99 x 1 54.97 0.601 10.69 x 2 57.38 0.608 10.50 x x 2 54.97 0.601 10.69 x x 2 54.00 0.622 10.13 x x 3 57.31 0.623 10.11 x x x 4 61.11 0.614 10.33 x x x x Table 3.8: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. The analysis involved dynamic impact loads measured for the left wheel of the first axle of locomotive 917400014341 at Sunderbyns Sjukhus detector station between October 18th and December 18th. LOCOMOTIVE 917400014341 Sunderbyns Sjukhus N° of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 56.65 0.057 25.62 x 1 57.22 0.593 11.05 x 2 62.27 0.666 9.07 x x 2 57.22 0.593 11.05 x x 2 60.96 0.637 9.84 x x 3 70.31 0.694 8.31 x x x 4 82.81 0.690 8.41 x x x x CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 26 Table 3.9: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. The analysis involved dynamic impact loads measured for the left wheel of the first axle of locomotive 917400014234 at Sunderbyns Sjukhus detector station between October 18th and December 18th. LOCOMOTIVE 917400014234 Sunderbyns Sjukhus N° of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 41.75 0.175 42.35 x 1 44.14 0.599 20.58 x 2 51.82 0.737 13.52 x x 2 44.14 0.599 20.58 x x 2 55.17 0.612 19.92 x x 3 68.99 0.766 11.99 x x x 4 83.45 0.775 11.55 x x x x Table 3.10: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. The analysis involved dynamic impact loads measured for the right wheel of the first axle of locomotive 917400014234 at Sunderbyns Sjukhus detector station between October 18th and December 18th. LOCOMOTIVE 917400014234 Sunderbyns Sjukhus Number of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 45.98 0.089 52.90 x 1 47.74 0.514 28.24 x 2 45.78 0.579 24.44 x x 2 47.73 0.514 28.24 x x 2 44.91 0.585 24.09 x x 3 55.3 0.67 19.11 x x x 4 252.72 0.661 19.66 x x x x By comparing the results reported in the tables above, it is possible to notice that time is not the most relevant regressor variable in the regression model. This can be seen by looking at the adjusted R-squared and the mean of the sum of residual squares values. As a matter of fact, the adjusted R-squared for the regression model involving merely the time as regressor variable turns out to have a very low value. This means that the variability in the dynamic loads expressed by the time variable accounts for a small fraction. In reality this indicates that over a two-month period, namely between October 18th and December 18th 2018, the damage on the wheel treads did not grow enough to lead to a drastic increase in the dynamic load values. The opposite conclusion may be derived for the train speed. In fact, due to the large adjusted R- squared values, the train speed is confirmed to have a high influence on the dynamic impact loads. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 27 Additionally, the regression model that best fits the measured data turns out to be the one expressing the dynamic impact load values as a linear function of time and including both linear and quadratic terms of train speed. This can be deduced because the regression model leads to a slight improvement in the R-squared value with respect to the ones derived for lower-order polynomials and at the same time it does not show huge differences in the PRESS-statistics value. The outcome slightly contradicts the experimental results demonstrated in [13], where the dynamic loads exhibited an linear increase with the train speed for the same wheel tread damage. The surface fitting of the observed data by using a multiple regression model expressing the dynamic impact loads as a linear function of time and both linear and quadratic terms of the train speed is presented in Figure 3.11 and Figure 3.12. According to these figures, it is observed that the dynamic impact loads show an increase with train speed and also a slight increase over time. Figure 3.11: Surface fitting of dynamic impact loads expressed as a linear function of time and including both linear and quadratic terms of the train speed. Data have been registered by the Sunderbyns Sjukhus detector station and referring to dynamic impact loads measured for the right wheel of the first axle of locomotive 917400014341 between October 18th and December 18th 2018. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 28 Figure 3.12: Complementary view to Figure 3.11. 3.4.1.2 Low wheel tread damage: comparison between wheel impact load detectors In the previous section, it was shown that the dynamic loads registered when the wheel tread damage is not severe depend mainly on the train speed. The next step was to use simple linear regression models involving merely the train speed as regressor variable in order to make some comparisons on the data registered by each wheel impact load detector. Figure 3.13 illustrates observed data and corresponding fitted linear regression models for each detector station over the period October 18th to December 18th. Complementary information to Figure 3.13 are presented in Table 3.11. The table highlights the values of the slope corresponding to each regression model reported in Figure 3.13. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 29 Figure 3.13: Simple linear regression models between train speed and dynamic loads detected for the right wheel of the first axle of locomotive 917400014341 between October 18th 2018 and December 18th for each detector station. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 30 Table 3.11: Regression coefficient (𝛽1) values corresponding to the regression models reported in Figure 3.13 Detector station Regression coefficient 𝜷𝟏 Bodsjön 0.494 Degerbäcken 0.353 Jörn 0.137 Koler 0.361 Skorped -0.011 Sunderbyns Sjukhus 0.255 By looking at Figure 3.13 it is possible to notice that the linear models derived based on data registered at Bodsjön, Degerbäcken, Koler and Sunderbyns Sjukhus detector stations show a sharp increase in the dynamic loads with increasing train speed. A less steep trend arises from data registered at the Jörn HJ detector station. The slope of the fitted models varies between the detectors. However, for the detectors at Degerbäcken, Koler and Sunderbyns Sjukhus, the slope (𝛽1) is rather similar and in the order of 0.3 kN/(km/h). On the contrary, the fitted linear model derived for data collected at the Skorped detector station does not exhibit an increase in dynamic loads with increasing speed. This is also confirmed by the acceptance of the null hypothesis stating that no linear relationship exists between regressor and response variable. A reason for this might be the lack of data measured at higher train speeds which have a high influence in the results for all the other detectors. It could also indicate that the vehicle dynamics at Skorped is different compared to at the other detectors (leading to a different rolling band that may not interact with the position of the tread damage) or that the performance of this detector needs to be calibrated. Figure 3.13 also illustrates a significant scatter in the data measured for similar train speeds. In particular, this is evident for the Degerbäcken detector station. Based on this observation, the accuracy of each detector in measured dynamic loads in a restricted train speed range is reported in Figure 3.14. Figure 3.14 shows dynamic loads registered over a two-month period and corresponding to a measured train speed ranging between 95 km/h and 105 km/h. Despite the restricted train speed range, the dynamic loads turn out to be still very scattered. This implies that despite a period of time with low wheel tread damage was considered, a small growth of the damage still occurred. Therefore, based on this consideration, the time interval was reduced even further. Figure 3.15 illustrates the updated information by considering only the dynamic loads measured between October 18th and October 30th and for train speeds ranging between 95 km/h and 105 km/h. Results derived from other time intervals and a more restricted train speed range may be found in the Appendix B, Figure B1.46 and Figure B1.47. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 31 Figure 3.14: Target diagrams involving dynamic impact loads registered at each detector station and referring to the right wheel of the first axle of the locomotive identified by the vehicle number 917400014341. Data were measured between October 18th and December 18th 2018 and correspond to measured train speed ranging between 95 km/h and 105 km/h. The green dashed line indicates the mean value of the dynamic loads. The blue solid lines represent an accuracy of 5%, the red solid lines an accuracy of 10% whereas the black solid lines an accuracy of 20% computed with respect to the mean value. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 32 Figure 3.15: Target diagrams involving dynamic impact loads registered at each detector station and referring to the right wheel of the first axle of the locomotive identified by the vehicle number 917400014341. Data were measured between October 18th and October 30th 2018 and correspond to a measured train speed ranging between 95 km/h and 105 km/h. The green dashed line indicates the mean value of dynamic loads. The blue solid lines represent an accuracy of 5%, the red solid lines an accuracy of 10% whereas the black solid lines an accuracy of 20% computed with respect to the mean value. By analyzing the information in Figure 3.15, Figure B1.46 and Figure B1.47 the following conclusions can be drawn. Data registered at Dergebäcken detector station are very scattered, which may indicate that this station has a poor accuracy and needs to be calibrated. A further assessment of the performance of this detector station is recommended. All the other detectors exhibit a better performance in measuring data because the dynamic loads magnitudes measured for the same wheel over a short period of time and for a limited train speed range are close to each other. The accuracy of each detector may be estimated around 20%. However, an accuracy of 20% does not match the one prescribed by the authorities which is 5%. It is clear that the poor accuracy CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 33 can partly be explained by the fact that the data have been measured on different occasions and at slightly different speeds. Therefore, it is not possible to assess the exact accuracy of each detectors because of the lack of data registered in a very short time interval (implying that the wheel tread damage would be constant) and for the exact same train speed. Further, since the RCF damage has a local appearance on the wheel tread, it is not possible to exclude the possibility that the damage is only partly, or perhaps not at all, interacting with the rolling band when the damaged wheel is passing a given detector. 3.4.2 Influence of train speed on dynamic loads: severe wheel tread damage Contrary to the analysis proposed in the previous section, the following analysis focuses on a time interval where the damage on the wheel tread is more significant leading to higher dynamic loads. Such a time interval may be deduced from the variation in dynamic loads over time, as the one proposed in Figure 3.9. As a matter of fact, a sharp increase in the dynamic impact load values over time marks the presence and evolution of severe RCF damage on the wheel treads. The same type of analysis as in the previous section has been repeated for the different time interval. The analysis whose results are presented below is based on data collected by the detector stations at Sunderbyns Sjukhus and Bodsjön and referring to both wheels of the first axle of the locomotive identified by vehicle number 917400014341. Particularly, dynamic impact loads registered from January 14th to February 14th 2018 have been considered because of their abnormal trend shown in Figure 3.9 and Figure B1.45, available in Appendix B. As it has been proposed in the previous section, an insight helping for the understanding of the expected trends may be obtained by looking at Figure 3.16. According to the figure, the dynamic loads exhibit a sharp increase over time. Figure 3.16: 3D-Bar plot showing the mean of dynamic loads registered for the right wheel of the first axle of locomotive 917400014341 at the Sunderbyns Sjukhus detector station during specific time intervals and for specific train speed ranges. The vertical axis shows the dynamic load values. The Y- axis shows the time intervals at which data have been registered, while the X-axis shows the considered speed ranges. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 34 Table 3.12: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. Analysis involved dynamic impact loads measured for the left wheel of the first axle of locomotive 917400014341 at Sunderbyns Sjukhus detector station between January 14th and February 14th. LOCOMOTIVE 917400014341 Sunderbyns Sjukhus N° of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 27.41 0.637 45.92 x 1 29.44 -0.025 129.56 x 2 35.07 0.762 30.02 x x 2 29.44 -0.025 129.56 x x 2 45.40 0.008 125.34 x x 3 56.83 0.796 25.65 x x x 4 150.87 0.795 25.89 x x x x Table 3.13: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. Analysis involved dynamic impact loads measured for the right wheel of the first axle of locomotive 917400014341 at Sunderbyns Sjukhus detector station between January 14th and February 14th. LOCOMOTIVE 917400014341 Sunderbyns Sjukhus N° of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 32.46 0.781 44.17 x 1 28.34 -0.051 212.32 x 2 40.28 0.849 30.50 x x 2 28.34 -0.051 212.32 x x 2 40.84 -0.092 220.51 x x 3 47.48 0.842 31.90 x x x 4 93.73 0.835 33.28 x x x x CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 35 Table 3.14: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. Analysis involved dynamic impact loads measured for the left wheel of the first axle of locomotive 917400014341 at Bodsjön detector station between January 14th and February 14th. LOCOMOTIVE 917400014341 Bodsjön N° of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 29.45 0.695 42.25 x 1 32.65 -0.046 144.87 x 2 34.19 0.683 43.89 x x 2 32.65 -0.046 144.87 x x 2 37.13 -0.103 152.79 x x 3 36.13 0.684 43.73 x x x 4 41.28 0.681 44.13 x x x x Table 3.15: Results in terms of PRESS statistics, 𝑅𝐴𝑑𝑗 2 and 𝑀𝑆𝑟𝑒𝑠 derived by the adoption of different regression models involving the regressor variables indicated in the rightmost cells of the table. Analysis involved dynamic impact loads measured for the right wheel of the first axle of locomotive 917400014341 at Bodsjön detector station between January 14th and February 14th. LOCOMOTIVE 917400014341 Bodsjön N° of regressor variables PRESS 𝑅𝐴𝑑𝑗 2 𝑀𝑆𝑟𝑒𝑠 𝑡 𝑣 𝑡2 𝑣2 𝑣3 1 80.21 0.468 856.58 x 1 61.87 0.074 1491.3 x 2 143.74 0.689 501.71 x x 2 61.87 0.074 1491.3 x x 2 67.43 0.028 1564.7 x x 3 163.97 0.676 521.34 x x x 4 176.81 0.696 488.97 x x x x By looking at the results listed in the tables above, it is possible to notice that the train speed is no longer the most influencing regressor variable. This is evidenced by the results arising from the adoption of multiple linear regression models involving merely the train speed as variable. Instead, the regression models involving the presence of time as a regressor variable provides a better fit to the observed data. A typical surface fitting arising from the adoption of a multiple linear regression model involving both train speed and time as regressor variables is shown in Figure 3.17. As it is possible to notice from Figure 3.17, in this time interval the dynamic loads exhibit a (in relation to the time dependence) low increase with train speed but instead a sharp increase over time. These results might be related to a fast wheel tread damage growth over time. This implies that the contribution to the increase in the dynamic loads registered by the detector stations due to the fast deterioration of the wheel tread is much more CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 36 significant than the increase induced by higher train speeds. This conjecture will therefore result in regression models that suffer the train speed dependence. Further, it is even more difficult to compare the performance of the different detectors since the measured dynamic loads are not stable over time. Figure 3.17: Surface fitting of dynamic impact loads expressed as a linear function of time and train speed. Data have been registered at the Sunderbyns Sjukhus detector station and referring to dynamic impact loads detected for the right wheel of the first axle of locomotive 917400014341 between January 14th and February 14th 2018. 3.4.3 Concluding comments In conclusion, from the results achieved in Chapter 3 the following key points may be highlighted: • The mean loads registered by the WILDs cannot be considered as completely independent of train speed. The linear simple regression model between mean load and train speed does not allow to properly correlate these two variables. • The dynamic loads registered by WILDs exhibit an increase with increasing train speed and over time. • For low wheel tread damage, the measured dynamic loads depend mainly on the train speed. This confirms the results presented in [13], where a linear increase in the dynamic loads with increasing train speed for the same wheel tread damage (rolling contact fatigue) was shown. • For severe wheel tread damage, the progressive growth over time of the wheel tread damage leads to an increase in the loads acting on the wheels that is much more significant than the increase induced by higher train speed. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 37 • For both above-mentioned cases, the multiple linear regression model between dynamic load, train speed and time instants at which data were measured provides a good fit to the registered data. • It is not possible to firmly assess the accuracy of the wheel impact load detectors because of the lack of data registered in a very short time and for the same train speed. However, indications for necessary calibration of a wheel impact load detector may be given by comparing data registered by each device for a given wheel in a very short time period and for a restricted range of train speeds. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 38 4 Parameterized finite element model 4.1 Implementation phase The second part of the present thesis project aimed at generating a parameterized three-dimensional wheelset model that will be used for further analyses in the EU project In2Track2. For this purpose, Python scripts have been used to generate FE models in Abaqus. Abaqus is a commercial software package for finite element analysis developed by Dassault Systèmes Simulia Corp [19]. The Abaqus scripting procedure is summarized in Figure 4.1. The Abaqus/CAE graphical user interface (GUI) was used to create the wheelset model. The GUI generates all command functions accounting for each performed operation and saves them in a journal-file whose extention is .jnl [20]. The journal-file is written in object- oriented programming language and therefore it can be saved as a Python-file. Subsequently, the Python script may be edited to add parameterization of some geometrical features. In addition, it allows to easily alter the model properties such as materials, mesh size and element type. The Python code may then be run in Abaqus/CAE. The Abaqus kernel interprets the commands to generate the modified model [21]. Figure 4.1: Block diagram summarizing the implementation procedure in Abaqus. 4.2 Wheelset model A conventional wheelset consists of two coned wheels rigidly connected to the axle so that both wheels exhibit the same angular velocity and the distance between the wheels is preserved. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 39 Figure 4.2:Schematic of a conventional wheelset. Measures in mm. [22] The surface of the wheel where contact with the rail occurs is characterized by two parts: the flange and the wheel tread. The flange is located at the inner side of the wheel and aims to provide lateral guidance especially in curves. The wheel tread is the conical running surface that meets the rail [22]. Additionally, the wheelset involves the presence of brake discs that are mounted on the wheels (or on the axle). A powered wheelset also includes a traction gear box which is mounted on the axle of the wheelset. The modelled wheelset refers to a powered wheelset of a freight locomotive whose drawings were provided by the Bombardier Transportation company. The drawings included a detailed sketch of the wheel and the assembly drawing of the whole wheelset. However, because of the lack of detailed sketches regarding the gear box housing, gear wheel and brake discs, some simplifications in their modelling were made. Reference [23] was used as a guideline to perform these simplifications. As a matter of fact, in reference [23], the FE model of a powered railway wheelset is characterized by similar geometry and characteristics as the one analysed in this thesis project. Hence, components for which detailed information were missing have been modelled in such a way that their characteristics were similar to the corresponding ones reported in [23]. The main assumptions and simplifications adopted in the model are: • The wheels are rigidly connected to the wheelset axle. • The brake discs are rigidly connected to the wheels. • The gear wheel has been modelled as a hollow cylinder rigidly connected to the axle. • The gear housing is rigidly connected to the gear wheel. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 40 • The roller bearings ensuring the interface between axle and gear box were modelled by springs applied in lateral and radial directions. More specifically, a 500 MN/m spring stiffness was assumed in the longitudinal (𝑘𝐿) and radial vertical directions (𝑘𝑅𝑉 ), whereas a 1 MN/m spring stiffness was used to model the interaction in the radial horizontal direction ( 𝑘𝑅𝐻 ), [23]. An illustration showing the mentioned interaction is presented in Figure 4.3. The CAD model of the wheelset is shown in Figure 4.4. Figure 4.3: Schematic illustrating the modelled interaction between axle and gear box. Figure 4.4: Frontal view of the wheelset model in Abaqus CAD interface. 4.2.1 Material properties Three different materials were used to assign the mechanical properties to each component of the wheelset. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 41 Common steel was used to assign the mechanical properties to the brake discs, gear wheel and gear housing. A slightly lower density steel was assigned to the wheels to match the wheel mass prescribed by the drawing provided by Bombardier Transportation. Lastly, EA1N steel was assigned to the axle [24]. The mechanical properties defined for each component of the wheelset and the corresponding mass are presented in the following tables. Table 4.1: Mechanical properties for each modelled component Component Material Young Modulus [𝑮𝑷𝒂] Density [ 𝒌𝒈 𝒎𝟑 ] Poisson coefficient [-] Axle (hollow) EA1N 210 7500 0.26 Wheel Steel 210 7700 0.3 Brake disc Steel 210 7800 0.3 Gear wheel Steel 210 7800 0.3 Gear housing Steel 210 7800 0.3 Table 4.2: Mass of each component of the wheelset and overall mass of the wheelset. Component Quantity Total Mass [𝒌𝒈] Axle 1 430 Wheel 2 1222 Brake disc 4 528 Gear wheel 1 241 Gear housing 1 642 ∑ 3063 4.2.2 Python code and modal analysis The Python script allows to create railway wheelset models that differ in terms of geometrical features of the wheel, such as the rim thickness. The goal was to create a parameterized wheelset model that enables to generate automatically different models characterized by railway wheels with varying external diameter, thus considering the wear of the tread. The use of the Python script is to easily make available different wheelset models to reduce the time demanded to create each of them directly in the Abaqus GUI. The following section presents a description of the Python script and results arising from a modal analysis of the wheelset for two different thicknesses of the wheel rim. Finally, a few suggestions for further analyses are proposed. 4.2.2.1 Description of the Python code As mentioned in Section 4.1, the modelling process has been carried out in the Abaqus graphical user interface (GUI). By saving the model and converting the journal-file into a Python file it was possible to get access to information regarding all the operations executed. However, this information appears in the form of numbered CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 42 entities, thus not allowing to modify the code. Therefore, the first action was to type the instruction “session.journaloptions.setvalues(replaygeometry = coordinate, recovergeometry = coordinate)” into the command line interface [20]. By applying this instruction, the information available in the Python code appears as function of the coordinates of each selected entity. Consequently, it was possible to parameterize each entity belonging to the rim of the wheel as a function of the rim thickness. The Python script takes only the external diameter of the wheel as input. The modelled wheel has a nominal external diameter of 1.25 m. Nevertheless, the rim thickness of the wheel diminishes gradually due to wear of the tread until it reaches a critical value. For this wheel type, the prescribed threshold diameter beyond which the wheel must be replaced is 1.17 m. An idea about the maximum allowable range of wheels due to the reduction of the rim is given in Figure 4.5. Figure 4.5: Cross section view of the modelled railway wheel: the wheel having nominal external diameter is shown on the left , whereas critical prescribed external diameter thereof is shown on the right. 4.2.2.2 Modal analysis of free wheelset This section proposes the numerical results arising from a modal analysis of the wheelset for two different cases: the first case was derived by the adoption of the wheelset model characterized by the wheel having nominal external diameter, whereas the second one considered a wheel having the limit prescribed external diameter. The analysis may be seen as starting point of further analyses aimed at studying the dynamic response of the wheelset for different wear conditions of the railway wheel. The analysis was carried out in Abaqus CAE/Standard. The elastic modes of the wheelset in the frequency range 0 - 1000 Hz were calculated. Solid 8-node linear CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 43 hexahedral elements were used to mesh all the components of the wheelset except the gear housing. The latter was instead meshed with 10-node quadratic tetrahedral elements. A convergence study revealed that a 50 mm element size was appropriate. Finally, free boundary conditions were assumed and the Lanczos solver was adopted to run the analysis. Table 4.3 shows some of the significant modes and the corresponding eigenfrequencies for both analysed cases. The table also provides the percentage of the difference between the eigenfrequencies derived for the two mentioned cases. Figure 4.6: Elastic modes of the wheelset: (a) Undeformed wheelset. (b) Mode 3. (c) Mode 4. (d) Mode 7. (e) Mode 11. (f) Mode 17. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 44 Table 4.3: Eigenmodes and corresponding eigenfrequencies of the wheelset for two different cases: wheelset characterized by a wheel with nominal dimensions and wheelset characterized by a wheel having critical external diameter. Percentage increase in terms of eigenfrequencies is shown in the rightmost column. Frequency [Hz] Mode Type Nominal Limit Limit - nominal difference [%] 1 - 2 Rigid body 0 0 - 3 First torsion mode 42.7 51.5 17.0 4 First bending mode 53.3 60.3 11.6 7 Second bending mode 96.2 113.7 15.4 11 First umbrella mode 194.9 227.3 14.2 17 Second umbrella mode 363.9 399.3 8.9 By comparing the results listed in Table 4.3, an increase in the eigenfrequency values with reduced external diameter of the wheel is observed. This is an expected and reasonable result. As a matter of fact, a reduction in the rim thickness results in a decrease in the overall mass of the system that leads consequently to an increase in the eigenfrequency values since the eigenfrequencies are inversely proportional to the mass. Further, it is possible to observe that the percentage increase in the eigenfrequency values due to the reduction of the rim thickness of the wheel tends to decrease for higher modes of the wheelset. The parameterized wheelset model might be used for innumerable purposes. Many current researches in the railway sector focus on the wheel–rail interaction. For instance in [25], a three dimensional wheel–rail model was object of study to predict the occurrence of defects on both rail and wheel surfaces. In this context, a useful means of the parameterized wheelset model might be to go into more specific issues and investigate how the reduction of the rim thickness due to the wear may affect the interaction between wheel and rail trying to predict how the damage occurrence might be influenced. Additionally, the study of the wheelset flexibility might be another interesting lead to be undertaken. For this purpose, the deformation of the axle and wheels through the receptance method might be investigated for different wheel geometries and for different train speeds [26]. In conclusion, the Python scripting for finite element analysis (FEA) softwares is a suggested tool to create parameterized models because it allows to generate different models automatically, and moreover it opens up many avenues for further analyses. CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 45 5 Fatigue analysis: non-powered wheelset 5.1 Aims and background In the last chapter, a dynamic analysis aimed at checking the fatigue resistance of the axle of a non-powered wheelset while subject to periodic sinusoidal loads acting on the wheels is performed. The intent of this analysis is to increase the confidence in the alarm limits currently set for the wheel impact load detectors, but at the same time also to encourage other studies and research for improved wheel design and for defining a precise timeframe within which the maintenance of the wheels must be carried out to prevent any consequent and serious damage to the railway axles. The fatigue phenomenon is a well understood type of damage which is triggered by variable loading conditions on a mechanical component that might lead to a sudden failure thereof. Railway axles are designed for infinite life which means that the time- varying stress state is below the limit (endurance limit) that causes crack initiation. This cautionary condition is ensured by generous safety factors applied to the material fatigue properties [27]. Railway axles undergo a complex and three-dimensional state of stress. Therefore, multiaxial fatigue criteria must be used to determine a scalar quantity (equivalent stress) that equivalently describes the stress state of the component. Subsequently, the equivalent stress must be compared to the fatigue limit prescribed by the regulations. Multiaxial fatigue criteria can be categorised in several typologies, see [28]. The use of the most suitable criterion depends on the case being analysed. The main distinction between different multiaxial fatigue criteria entails the type of loading. Multiaxial loadings can be classified into two categories: proportional and non- proportional loading. The difference refers to the direction of the principal stresses over time. For the first category, the direction of the principal stresses does not change over time, whereas multiaxial non-proportional loadings lead to the change in principal stresses directions instant by instant [29]. In section 5.3 a proportional loading condition is proven to reflect the stress state in the wheelset axle caused by first order sinusoidal loads applied on the rolling circle of the wheels. Based on this outcome, the multiaxial fatigue criterion that best suits the proportional loading condition turned out to be the Sines’ criterion. The Sines’ criterion is an invariant based criterion; the term invariant is used because the first invariant of the mean stress tensor appears in the formulation of the equivalent stress [30]. The definition of the equivalent stress is described by Eq. (5.1): 𝜎𝑎,𝑒𝑞 = 𝜎𝑣𝑀,𝑎 + 𝑀 ⋅ 𝐼𝑚 (5.1) where: • 𝜎𝑣𝑀,𝑎 is the von Mises equivalent alternating stress which is described by Eq. (5.2): CHALMERS, Mechanics and Maritime Sciences, Master’s Thesis 2019:34 46 𝜎𝑣𝑀,𝑎 = 1 √ 2 √(𝜎𝑎,1 − 𝜎𝑎,2) 2 + (𝜎𝑎,2 − 𝜎𝑎,3) 2 + (𝜎𝑎,3 − 𝜎𝑎,1) 2 (5.2) where 𝜎𝑎,𝑖 accounts for the alternate component of the ith principal stress; • 𝐼𝑚 is the first invariant of the mean stress tensor, as described by Eq. (5.3): 𝐼𝑚 = 𝜎𝑚,1 + 𝜎𝑚,2 + 𝜎𝑚,3 (5.3) where 𝜎𝑚,𝑖 accounts for the mean component of the ith principal stress; • 𝑀 is the mean stress sensitivity and depends on the material properties. FKM-Richtline provides an estimate of the mean stress sensitivity based on experimental results [31]. This factor can be calculated according to Eq. (5.4): 𝑀 = 𝑎 ⋅ 10−3𝑅𝑚 + 𝑏 (5.4) where 𝑅𝑚 (expressed in MPa) is the ultimate strength of the material, whereas 𝑎 and 𝑏 are coefficients depending on the material. The coefficients are estimated to 0.35 and -0.1 for different steel alloys. According to Sines’ criterion, the equivalent stress described by Eq. (5.1) must be lower than the fatigue limit prescribed by the technical standards to prevent any initiation and propagation of cracks. 5.2 Dynamic analysis The