Optimization and validation of a finite element methodology for thermo-structural analysis of polyhuretane wheels for roller coaster application

Polymeric materials are widely used in several engineering sectors. Among these, a particularly critical sector for this type of material is that of roller coasters. The wheels are indeed made with an aluminum hub and a compact polyurethane coating, which, being in contact with the track, is subject to high speeds dynamic loads. Due to the viscoelastic behavior typical of these materials, such loads induce overheating of the coating and therefore a rapid degradation of the wheel. This results in machine downtime and, consequently, significant waste of time and money. In this context, the authors have developed a methodology for finite element thermo-structural analysis capable of quickly evaluating the temperature reached during work cycles and proving very useful in selecting the type of wheels to use. In this work, this methodology was firstly computationally developed and then validated by comparing the analysis results with data obtained from experimental tests conducted by the manifacturer. The comparison demonstrated the effectiveness of the proposed method, highlighting, however, a constant error in terms of maximum temperature reached attributable to a non-exact material characterization.


Introduction
Roller coasters are a engineering sector that is continuously evolving, particularly in terms of track length, speed (exceeding 100 km/h), and forces.Roller coasters are essentially trains composed of carriages (typically 10) that move along a track consisting of two or three steel tubes.The connection between the carriages and the track is achieved through a system known as the 'three-wheel locking system'.This system involves the use of at least 4 'main wheels' for each axle, which support the weight of the carriage and maintain the train's movement on the track, 4 'lateral wheels' that keep the train centered on the track to prevent derailment, and 4 'stop wheels' that prevent the train from detaching from the track during maneuvers.Overall, there are approximately 240 wheels for each roller coaster (considering an average of two axles per carriage for an average of 10 carriages).These wheels, although of different sizes depending on their function, are made with a steel or aluminum hub and a polyurethane coating.The hysteresis behavior typical of these materials [1][2][3], while useful for absorbing shocks and vibrations, dissipates energy and thus generates heat, which can lead to overheating of the material [4][5].Such overheating is one of the main causes of damage to the polyurethane coating 1306 (2024) 012015 IOP Publishing doi:10.1088/1757-899X/1306/1/012015 2 (other types of damage are due to mechanical fatigue and photo-degradation) [6][7].These damages can take various forms, such as surface wear, crumbling, cracking and/or fissuring of the coating, and the formation of bubbles.Due to the high speeds and loads to which the polyurethane coating of the wheels is subjected, especially with the latest developments in roller coasters, these damage phenomena occur on average three times a year for standard applications and approximately every 30 days for particularly demanding applications.Considering that the average cost to replace the coating of a single wheel is approximately €500, the maintenance cost of such installations solely due to wheel damage is very high.For this reason, the purpose of this work is to develop and experimentally test a simulation methodology capable of assessing the temperature reached inside the wheel coating [8].This way, the company, during the design phase of the installation, can choose the appropriate type of wheel to maximize its lifespan, significantly reducing maintenance costs.The most immediate approach to simulate this phenomenon would be to perform a coupled transient thermo-structural analysis using finite elements [9].However, this type of simulation, due to the large deformations (caused by the viscoelastic behavior of polyurethane), wheeltrack contact, and dynamic phenomena, results in very long computation times and is therefore impractical [10].For this reason, the authors have developed an analysis methodology that, although based on the decoupling of thermal and structural aspects, has been shown to assess the temperature distribution in the wheel during actual operating conditions with relatively short computation times [8].This methodology is based on the assumption that the material's behavior remains unchanged with varying temperatures and is divided into two phases.The first phase involves conducting structural transient analyses to evaluate the energy dissipated by the material and the heat per unit volume.The results of the structural analyses are then used as input for a transient thermal analysis (phase two) performed on a finite element model identical to the one used in the structural analyses, but aimed at obtaining the temperature during the machine's operating cycles.The assumption of decoupling structural phenomena from thermal ones, although it may seem limiting, is not actually so when considering the use of this method in the design phase, aimed at selecting the most suitable wheel to maximize its lifespan.However, in this study, it was decided to test the method through a numerical-experimental comparison of two different types of wheels, tested using a twin-disc machine capable of replicating the real load and speed timehistory.The numerical-experimental comparison has shown that the proposed method yields results in line with experimental data.What is evident is an overestimation of the temperature, which can, however, be attributed to the material parameters used, which were derived from literature data of a similar material.Nonetheless, the results obtained are encouraging and allow the company to choose the most appropriate wheel.However, it is necessary to conduct material characterization in order to define a material model that can reasonably describe the real behavior.

The proposed approach
The direct approach to assess wheel temperature involves a coupled finite element thermostructural analysis.However, this method is hindered by the long computational time due to nonlinearity and transient nature of the analysis, frequency sampling (about 1000 Hz) required for accuracy and the need for fine meshing to assess the exact deformation.This makes such kind of simulation nearly impractical, especially when considering various wheel types and multiple track circuits.To address these challenges, the authors devised a faster finite element technique for temperature estimation [8].This method decouples thermal and structural analyses, assuming material behavior remains unchanged with temperature changes.The proposed method comprises three  As shown in Figure 1 the track is pressed against the wheel and then it is rotated around the wheel with an angular velocity proportional to the carriage's forward velocity.Introducing frictional contact between the wheel and track, the wheel experiences both rolling and sliding conditions.Structural simulations are used to determine the dissipated power, calculated as the product of the nodal forces vector and nodal velocities vector, for each element within a slice of elements located at the center of the considered segment (Figure 1).To significantly reduce computational effort and account for the effect of deformation in elements before or after those in the considered slice, it has been observed that, in the case of a regular mesh, each element positioned before or after the slice experiences the same time histories of force and velocity and, consequently, dissipated power but with a time phase shift of δt = β/ω (where β is the angular width of the element slice, and ω is the angular velocity of the wheel).Thus, it is possible to calculate the energy dissipated in the time interval ∆t (the temporal span during which the radial row of considered elements influences energy dissipation) for each ring k of elements, as indicated in equation 1, also considering the influence of n F sections upstream and downstream.
In equation 1, j represents the index for the n F considered sections.To account for the influence of deformation in sections upstream and downstream of the slice of elements being considered, it's sufficient to shift forward and backward in time, with increments of δt, the power curves obtained and verify whether they still fall within the ∆t interval.Once the dissipated energy for each ring of elements is known, it's possible to calculate the power dissipated in the time interval ∆t, as described in equation 2.
By repeating this process for each pair of force and carriage velocity, it is possible to obtain power dissipation surfaces for each element of the slice, as a function of the normal force at the contact point and the carriage's forward velocity (Figure 3).These surfaces can be interpolated instant by instant to obtain a time-history of dissipated power W (t).This result can be used to perform a transient thermal analysis [11][12].In fact, at the core of the thermal analysis, the equation to be solved for each element of the model is the one shown in equation 3.
where ρ is the density, C is the specific heat, T is the temperature, t is the time vector, {L} is the is the derivative operator, {V } is the relative velocity vector for heat exchange mass transport, {q(t)} is the thermal flux vector between the wall and the fluid and {q * (t)} represents the internally generated heat per unit volume.This last quantity represents the input to be provided for a transient thermal analysis and can be calculated as shown in equation 4.
where W (t) is the time history of dissipated power, calculated by interpolating the surface, element by element, with the instantaneous values of the normal force at the contact point and the forward velocity, and V k is the volume of the row of elements associated with the i−th element of the considered slice.Figure 2 depicts a flowchart of the procedure used.
The illustrated method allows for the evaluation of the temperature reached within the coating for various types of tracks (provided that the forces and velocities fall within the initially It is essential to note that the assumption that the material behavior remains unchanged with temperature variation must hold.To further speed-up the computation time, given the simple geometry of the wheel, symmetry has been exploited to greatly reduce the number of degrees of freedom [13].Symmetry has been utilized both in structural and thermal analysis.In this manner, the computation time required to conduct a complete analysis is approximately 4 days, making it compatible with the company's design schedules.Another crucial aspect to highlight is that the power dissipation surfaces can be reused to assess the temperature reached by the same wheel on different types of tracks without the need to repeat the entire characterization process aimed at obtaining the surfaces of dissipated power through transient structural analyses.

Validation of the proposed method with experimental data
The method presented in the previous section has been employed solely in the design phase until now, i.e., for selecting the type of wheel capable of achieving lower temperatures.However, this methodology has never been compared with experimental data.For this reason, in this study, the proposed method was tested by comparing the simulation results with experimental data conducted by the wheels manufactures.

Experimental setup
To carry out the experimental tests required for the evaluation of the proposed method, the experimental setup shown in Figure 3a was utilized.
As shown in Figure 3a, the wheel is pressed against a flywheel using a hydraulic cylinder.This cylinder replicates a force load history at the contact point provided by the roller coaster manufacturer, while the flywheel is driven by an electric motor that imposes an angular velocity IOP Publishing doi:10.1088/1757-899X/1306/1/0120156 proportional to the forward speed of the roller coaster.The time histories are obtained from a multibody simulation and those used during the tests are depicted in figure 3b and c.The force and velocity time histories provided by the company were replicated multiple times, thus simulating multiple track circuits.Two types of wheels were tested: one with an outer diameter of 516 mm and the other with an outer diameter of 570 mm.The coating width is 75 mm and 85 mm respectively, while the coating thickness is the same for both wheels at 15 mm.Each wheel was tested at different load percentages: 70%, 85%, and 100% of the load provided by the company, while the velocity remained unchanged.The tests were conducted at room temperature (22 °C), and the temperature was measured on the outer surface of the wheel after a sufficient number of cycles to ensure temperature stabilization.

Numerical -experimental comparison
In order to validate the proposed approach, the experimental tests were simulated using finite element analysis (FEA) following the method introduced in the previous section.For this purpose, FEA models were created for the two wheels experimentally tested.To correctly model the viscoelastic behavior of the polyurethane coating in finite element analysis [14][15][16], the generalized Maxwell model was used.The elasticity was modelled using the Mooney-Rivlin hyperelastic material model [17][18][19] with two parameters, while the viscosity was modelled using the Prony series with 5 coefficients [20][21].The parameters used for the viscoelastic material model are presented in Table 1 and Table 2 The coefficients of the Mooney-Rivlin hyperelastic material model and the coefficients of the Prony series shown in Table 1 and in table 2 were obtained from quasi-static compression tests or DMA tests conducted by the authors in the past on a similar material.The hub of the wheel was modeled using aluminum with the following properties: Young's modulus E = 2 × 10 11 Pa, Poisson's ratio ν = 0.34, and density ρ = 2700 kg/m³.For the track, steel with the following properties was considered: Young's modulus E = 7.2 × 10 10 Pa, Poisson's ratio ν = 0.3, and density ρ = 7850 kg/m³.To determine the thermal power, the wheels were numerically mapped considering the load and velocity time history used in the experimental tests shown in Figure 3.A friction coeffient equal to 0.1 was used between the track and the wheel.The ranges considered for obtaining the thermal power surfaces as a function of vertical force at the contact point and carriage advancement velocity ranged from 0 N to 30000 N with a step of 2000 N for force and from 0 m/s² to 30 m/s² with a step of 5 m/s² for acceleration.In this way, it was necessary to conduct 112 transient structural analyses to obtain the information required for thermal power calculation.By interpolating these surfaces instant by instant with the values of vertical force at the contact point and the roller coaster forward velocity, following the procedure shown in Figure 2, it was possible to obtain the time histories of dissipated power for each element and thus the heat internally generated per unit volume {q * (t)} to be applied in a transient thermal analysis.Exploiting the symmetry it was possible to simulate multiple laps.For this the time histories of dissipated power were replicated to simulate 7 consecutive laps of the track.The thermal analysis were conducted considering the entire wheel model, consisting of both the aluminum hub and the coating.However, the track was not considered, and thus, neither the thermal conduction between the covering and the track.Nonetheless, convective heat transfer with the external air was taken into account, assigning a convection coefficient of 60 W/(m² K) at 293 K.The material parameters used in the thermal analysis are listed in Table 3.The conducted thermal analyses allowed evaluating the temperatures reached by the wheel.Figure 4 shows a comparison between the temperature maps for the two models at the instant when the maximum temperature is reached (T = 1342 s).
As shown in Figure 4, the maximum temperature is reached at the core of the coating.Indeed, due to its low thermal conductivity, polyurethane retains heat inside, and there is no heat dissipation either to the outside or to the wheel hub.However, what is particularly interesting is the temperature evolution over time.Indeed, after a certain number of track circuits, the temperature stabilizes at a constant value and no longer increases.This phenomenon has also been experimentally observed in the results obtained by the wheel manufacturer.The results obtained from these simulations (shown in Figure 4) were then compared with the experimental results obtained by the company.To this aim, Table 4 shows a comparison between the numerical and experimental results for all test conditions for both wheels.The comparison of Table 4 shows that the proposed allows obtaining temperature values very close to the experimental data, although there is an overestimation of the reached temperatures.However, this difference could be attributed to the parameters used in the material model that being obtained from similar materials (from tests conducted by the authors or from bibliographic data) may not accurately describe the actual behavior of the wheel coating.Table 4 also reports the ratio (normalized to 293K) between the numerical values and those obtained by linearly interpolating the experimental results for each load percentage.As evident, the slope of the line interpolating the data is not equal in both tests.This is because when no load is applied (0% of the imposed load), the obtained temperature must tend to ambient temperature (293 K).However, as previously introduced, this may be caused by the material model and a correct characterization of the actual material will be the topic of future research.

Conclusion
This work has introduced and experimentally validated a method for the thermo-structural analysis of wheels with polyurethane coatings.The proposed method involves decoupling the structural and thermal simulation, assuming that the material's behavior remains constant with temperature variations.This calculation methodology allows for transient analyses to evaluate the temperatures reached in the wheels during operation, within timeframes compatible with the company's design phase.This result would not be achievable with a coupled thermostructural analysis due to the various nonlinearities present in the model (contact, material, large deformations).The proposed method was validated by comparing the results obtained from simulations with experimental data provided by the wheels manufactures.Tests were conducted on two different types of wheels with three different load percentages.The numerical results show a good correspondence with experimental data, although there are still differences in terms of the maximum temperature reached.However, this can be attributed to the parameters used to describe the viscoelastic behavior of polyurethane.These parameters, obtained from tests on a similar material, represent a significant source of error, and it is therefore necessary to characterize the material used in order to define a material model capable of representing its real behavior.

Figure 1 .
Figure 1.Speed and vertical load application in transient structural analysis

Figure 2 .
Figure 2. Flowchart of the procedure used for the thermo-structural simulation of polyurethane wheels.

Table 1 .
. Adopted parameters for elastic potential of Mooney-Rivlin

Table 2 .
Adopted parameters for the Prony series

Table 3 .
Material parameters for transient thermal analysis

Table 4 .
Numerical-experimental comparison of the temperatures reached on the external surface of the wheels Load % Experiment [k] Experiment mean [k] Simulation [k] Ratio