The Rail Technology Unit
Optimisation of Railway Wheel Profiles using a Genetic Algorithm Persson I., Iwnicki S.D.
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OPTIMISATION OF RAILWAY WHEEL PROFILES USING A GENETIC ALGORITHM PERSSON I.1 AND IWNICKI S.D.2 SUMMARY This paper presents the procedures and preliminary results of a novel method for designing wheel profiles for railway vehicles using a genetic algorithm. Two existing wheel profiles are chosen as parents and genes are formed to represent these profiles. These genes are mated to produce offspring genes and then reconstructed into profiles that have random combinations of the properties of the parents. Each of the offspring profiles are evaluated by running a computer simulation of the behaviour of a vehicle fitted with these wheel profiles and calculating a penalty index. The inverted penalty index is used as the fitness value in the genetic algorithm. The method has been used to produce optimised wheel profiles for two variants of a typical vehicle, one with a relatively soft primary suspension and the other with a relatively stiff primary suspension.
1. BACKGROUND The selection of railway wheel and rail profiles is a challenge that has faced engineers since the dawn of the railway age. From the first cylindrical wheels running on flat plates, wheels were made conical to give better guidance and flanges were added for safety. Modern wheels often have complex profiles based on the shape of worn wheels in an attempt to improve their life. Rails also now have a complex profile with different radii on the rail head where the wheel tread contacts and on the corner where the flange contacts. A high level of conicity will allow good curving behaviour even in the tightest curve without flange contact. This can however, lead to a relatively low critical speed and possibly dangerous hunting instability. A low level of conicity on the other hand will allow stable operation at high speeds but the flangeway clearance will quickly be used up in curves, resulting in flange contact and possible flange climb derailment. Flange angle and root radius are also variables that can have a significant effect on the possibility of derailment. In addition to the vehicle behaviour engineers must consider the stresses on the wheel and on the rail. These have a major influence on the development of rolling contact fatigue which can have expensive and sometimes dangerous consequences. The methods used to design these profiles have been mainly based on the experience of railway operators using various rules of thumb regarding conicity, flange angle, flange root radius etc [1],[2]. 1
DESolver, Optand 2876, S-831 Ostersund, Sweden Manchester Metropolitan University, Chester Street, Manchester M1 5GD, UK
2
In nature these complex design problems are often faced and solutions found, sometimes of astonishing complexity, effectiveness and efficiency. In this work the same evolutionary principles are used to obtain profiles optimised for performance on typical vehicles and track. Genetic algorithms have been used in other engineering applications including optimisation of suspension components and have given useful results [3]. 2. THE GENES The cross sectional profile of the wheel is initially described with a series of x,y coordinates and these are converted into a binary sequence. Several methods have been tried for producing the binary sequence including direct conversion of the cartesian coordinates of the profile into a series of binary numbers and a fourier transform of the profile description into a series of harmonic components but the best results so far are from a series of consecutive derivatives of the coordinates. From the nominal running circle to the inside of the wheel the 4th order derivative of the profile, calculated every 0.5mm, is used. On the flange side of the nominal running circle the 1st order derivatives are used. These numbers are converted into binary and joined together to form the gene. The profiles are digitised at 0.5mm spacing but in reforming the profiles spline interpolation is used to increase the number of data points to the 0.1mm used in Gensys. The genes for the two parent profiles are mated by taking random sections from each to make a child. The children will represent different profiles to the parent but will share similar characteristics. Mutations are also made by randomly changing the genes to introduce occasional larger variations and are used to avoid local minima in the optimisation process. 3. THE SIMULATION The selected wheel profile is incorporated into a GENSYS [4] simulation model of a simple motored bogie vehicle with an axle load of 20 tonnes. The vehicle bodies are assumed to be rigid and the main primary and secondary suspension stiffness is linear. The vehicle has vertical primary dampers as well as secondary lateral, vertical and yaw dampers. Traction rods and anti-roll bars are included in the model and the yaw dampers have blow off valves and include series stiffness. The nominal wheel diameter is 1m. For these tests two versions of the vehicle were set up, one with soft primary suspension and the other with a stiffer primary suspension and no yaw damper. The main vehicle parameters are shown in table 1. The track selected for the tests was a section of Swedish main line (class K1). The vehicle was run at 145 km/h on straight track for 275m then into a 120m linear transition onto a curve of 800m radius and cant of 150mm (130mm cant deficiency). The rails are inclined at 1:40 and the track standard according to CEN/TC 256WG at 145 km/h is QN2. Measured track irregularities are included and the average gauge
is 1430.76. After running for 227m around the curve the simulation was stopped and the results used to calculate the penalty index. 4. THE PENALTY INDEX In order to evaluate the effectiveness of each profile a penalty index has been introduced and is calculated after each simulation run. The aim of the penalty index is to provide an assessment of the vehicle behaviour in a single value. In the current work factors have been included in the penalty index to reflect the maximum contact stress; the maximum lateral force on the track; the maximum derailment quotient (lateral force divided by vertical force at the contact point); total wear and the total ride index. The penalty index is the sum of the individual penalty factors. Each of the factors that make up the penalty index can be weighted to reflect their importance to the designer or the particular service type. Other factors could also be included in the penalty function if they were felt to be important. 4.1.
Ride comfort penalty factor
The ride comfort penalty factor consists of the sum of the filtered RMS accelerations weighted according to ERRI Question B153 at the floor of the vehicle over both bogies. A lateral and a vertical value of the ride comfort penalty factor is calculated. 4.2.
Lateral track-shifting force penalty factor
The lateral track force at each wheel is first filtered with a sliding 2m window and then the maximum value for all axles is taken. A non linear weighting is applied to this force to establish the penalty factor. If the maximum track-shift force is less than 40% of the permissible value (the Prud’Homme limit) the penalty factor will be 0. If the maximum track-shift force is between 40-90% of the permissible value, the penalty factor will increase linearly to 2.0, if the maximum track-shift force is over 90% of the permissible value the penalty factor increases to 100 at 100%. 4.3.
Maximum derailment quotient penalty factor
The likelihood of derailment is linked to the widely used derailment quotient and this has been selected as one of the factors in the penalty index. The lateral force is divided by the vertical force at each contact point and this is filtered in a sliding 50ms window. The largest maximum value for the simulation is stored and is weighted in exactly the same way as the maximum track shifting force with the maximum permissible limit being taken as 0.8.
4.4.
Wear penalty factor
The level of wear at the wheel and rail surface is indicated by the energy dissipated in the contact patch and this is calculated by taking the product of creepage and creep force (including spin). A high value of energy here is seen as indicating a high rate of wear of the wheel profile and to be avoided. Some wear is however seen as beneficial as it will probably remove small cracks which develop through rolling contact. A very low level of wear is also undesirable as it probably indicates that wear is taking place over a very limited section of the profile and the shape of the profile will not then be stable. For these reasons the weighting on the wear energy dissipation at values between 80 and 120 Nm/m is set to 0 and not included in the wear penalty factor. Outside this range the value of the wear penalty factor increases linearly to 0.5 at 0. and at 200. 4.5.
Maximum contact stress penalty factor
High levels of contact stress are likely to result in rolling contact fatigue or other rail or wheel damage. The evaluation of the maximum contact stress penalty factor takes the 99.85th percentile of the contact stress at all wheels (taken as the normal force divided by the contact area) and retains the maximum value. This value is divided by 1e9 N/m2 to produce the normalised penalty value. 5. SELECTION OF THE FITTEST PROFILES Once the penalty index has been calculated for all the profiles the ‘parents’ for the next generation are selected. A random selection weighted by the penalty index is used to select the parents for the next generation and mating as described above is carried out. Four profiles are also retained into the next generation (using the same weighted random selection). Simulations are carried out for the ‘child’ profiles and the process is repeated. In this way a chance mating which results in poorer performance than the parents will not damage the population excessively. 6. RESULTS 6.1.
Profiles
The genetic algorithm and assessment simulation was run until no further improvement was seen (110 generations for the vehicle with the soft bogie and 121 generations for the stiff bogie vehicle). The profiles produced are shown in figures 1 and 2.
Figure 1. The profile produced by the genetic algorithm for the stiff bogie
Figure 2. The profile produced by the genetic algorithm for the soft bogie
Figure 3. shows the conicity produced by each of the profiles. The P8 profile has been developed for rails with a 1:20 inclination and is steeper in the tread area than the S1002 profile, which is usually used on 1:40 or 1:30 rails. For the stiff bogie, the genetic algorithm has produced a wheel profile with a lower conicity than either the P8 or the S1002 profiles. The soft bogie has, in contrast, resulted in a wheel profile with a high conicity.
mm Figure 3. Conicity of the original and new profiles against lateral displacement of the wheelset
6.2.
Wear
Figure 4. shows the calculated wear index from the Gensys simulation as the vehicle runs over the virtual test track. The upper two plots show the results for the vehicle with the stiff primary suspension and the lower two are for the vehicle with the soft primary suspension. The first and third plots show the results with the S1002 wheel profiles and the second and fourth profiles are with the best profile from the genetic algorithm for the specific vehicle as presented in section 5.4. above. The solid line shows the results for the outer leading wheel on bogie 1 and the dashed line shows the results for the outer leading wheel on bogie 2. Table 2 summarises the wear penalty factors for the different cases. Table 2. Wear penalty factor
Wear penalty factor Wheel profile S1002 Best wheel profile from Genetic Algorithm
Stiff bogie 3.68 3.24
Soft bogie 0.509 0.306
It can be seen that the profiles produced by the genetic algorithm give reduced wear for both vehicles. The soft bogie already has better wear performance but one significant peak has been eliminated and the reduction in the wear penalty index is still significant. 6.3.
Contact stress
Figure 5. shows the simulated contact stress for the s1002 wheel profile and the best profile from the Genetic Algorithm for both vehicles and table 3. shows the penalty factors. The solid line on the plot is the right wheel and the dashed line is the left wheel, both on the leading axle. For the vehicle with the stiff bogie there is an overall reduction in the contact stress in the curved section of the track with a corresponding 10% reduction in the penalty index. For the vehicle with the soft bogie some clear peak stress values have also been eliminated. Table 3. Contact stress penalty factor
Contact stress penalty factor Wheel profile S1002 Best wheel profile from Genetic Algorithm
Stiff bogie 1.28 1.16
Soft bogie 1.3 1.03
Figure 4. Simulation results for wear index
6.4.
Derailment quotient
The derailment quotient penalty factor values are also shown in table 4. For the stiff vehicle with the S1002 profiles the derailment quotient is always below this limit and with the soft bogie there is only one small excursion above this lower limit. The Genetic Algorithm has eliminated the 1 excursion for the soft vehicle but has actually increased the penalty factor for the stiff vehicle to bring the overall level of the derailment quotient up towards the lower limit. Table 4. Derailment quotient penalty factor
Derailment quotient penalty factor Wheel profile S1002 Best wheel profile from Genetic Algorithm
Stiff bogie 0 0.1
Soft bogie 0.23 0.
Figure 5. Simulation results for contact stress
6.5.
Track shift forces
The results for the Lateral Track-Shifting force Penalty are shown in table 5. Table 5. Lateral track-shifting force penalty factor
Lateral track-shifting force penalty factor Wheel profile S1002 Best wheel profile from Genetic Algorithm
Stiff bogie 0.989 0.730
Soft bogie 0. 0.
For the stiff bogie the lateral forces have been reduced by using the new wheel profile and the maximum lateral track-shifting force is lower. The soft bogie does not generate lateral forces higher than 40% of the Prud’Homme limit. 6.6.
Passenger comfort
The results for the ride comfort penalty are shown in table 6.
Table 6. Ride comfort penalty
Ride Comfort Penalty Wheel profile S1002 Best wheel profile from Genetic Algorithm
Stiff bogie 3.66 3.18
Soft bogie 3.62 3.29
It can be seen that the best profile from the Genetic Algorithm gives an improvement in the vehicle ride in both cases. The improvement is greater for the stiff bogie. 7. CONCLUSIONS This work has shown that a genetic algorithm can be used to optimise the wheel profiles for a typical railway vehicle running on main line track and that different profiles are produced for different vehicles. The method can be tuned to reflect the importance of various factors such as contact stress or ride comfort etc. The method could also be used to optimise rail profiles and the wheel/rail profile combination. Further investigation is required to test the profiles produced by the genetic algorithm method. In particular the effect of wear of the wheel and the change this will cause to the profile must be understood. A possible initial alternative to practical testing may be to use the methods developed by Jendel and Berg [5] where ‘load collectives’ are used to model typical vehicle traffic including traction braking, curving etc on wear. REFERENCES 1. Esveld C. ‘One procedure for optimal design of wheel profile’ Proc. 1st Wheel-Rail Interface management conference, Amsterdam 2002, IQPC London 2 Shen G., Ayasse J.B., Chollet H. and Pratt I. ‘A unique design method for wheel profiles by considering the contact angle function’ Proc. Instn. Mech. Engrs. Vol.217 Part F. 2003 3. Mei T.X. and Goodall R. M. ‘use of multiobjective genetic algorithms to optimise inter-vehicle active suspensions’, Proc. Instn. Mech. Engrs. Part F 2002, 216, pp53-63 4. Persson I. ‘Using GENSYS 0203’ ISBN 91-631-3110-2 DEsolver 2002 5. Jendel T. and Berg M. ‘Prediction of wheel profile wear methodology and verification’ Proc. 17th IAVSD Symposium, Copenhagen 2001, ISBN 0042-3114
Figure 6. Simulation results for derailment quotient
Parameter PRIMARY SUSPENSION Lateral semi-spacing to primary suspension Semi-wheelbase Longitudinal primary stiffness Vertical primary stiffness Viscous damping in primary stiffness Vertical damping Vertical bump-stop clearance Vertical bump-stop stiffness SECONDARY SUSPENSION Lateral spacing to secondary suspension Bogie centre spacing Vertical secondary stiffness Lateral secondary stiffness Longitudinal secondary stiffness Lateral bump stop clearance Lateral bump stop stiffness Lateral damping Vertical damping Yaw damping INERTIA PROPERTIES Bogie mass Centre of gravity above track Bogie roll moment of inertia Bogie pitch moment of inertia Bogie yaw moment of inertia Wheelset mass Centre of gravity above track Wheelset roll moment of inertia Wheelset pitch moment of inertia Wheelset yaw moment of inertia Body mass Centre of gravity above track Body roll moment of inertia Body pitch moment of inertia Body yaw moment of inertia
Stiff bogie
Soft bogie
Units
1 1.5 20e6 1200e3 2e3 30e3 0.05* 20e6*
1 1.5 4e6 1200e3 2e3 30e3 0.05* 20e6*
m m N/m N/m Ns/m Ns/m m N/m
1 16 1000e3 600e3 600e3 0.09* 22e6* 40e3 40e3 0
1 16 1000e3 600e3 600e3 0.09* 22e6* 40e3 40e3 500e3*
m m N/m N/m N/m m N/m Ns/m Ns/m Ns/m
10000 0.7 3000 10000 15000 2000 0.5 1200 200 1200 52e3 2 80e3 1800e3 1800e3
10000 0.7 3000 10000 15000 2000 0.5 1200 200 1200 52e3 2 80e3 1800e3 1800e3
kg m kgm2 kgm2 kgm2 kg m kgm2 kgm2 kgm2 kg M kgm2 kgm2 kgm2
(* non-linear component, simplified linear properties given)
The Rail Technology Unit The Rail Technology Unit based at Manchester Metropolitan University carries out research and consultancy into the dynamic behaviour of railway vehicles and their interaction with the track. We use state of the art simulation tools to model the interaction of conventional and novel vehicles with the track and to predict track damage, passenger comfort and derailment. Our simulation models are backed up by validation tests on vehicles and supported by tests on individual components in our test laboratory. We are developing methods to investigate the detailed interaction between the wheel and rail. January, 2004 Simon Iwnicki (RTU Manager)
Core expertise:
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Wheel/Rail Interface Engineering
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•Railway vehicle suspension dynamics –Computer simulation –Lab and field testing •Wheel-rail interface modelling –Profile design –Wear –Rolling contact fatigue
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