A Polynomial Chaos Approach for Stochastic Modeling of Dynamic Wheel-Rail Friction
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Accurate estimation of the coefficient of friction (CoF) is essential to accurately modeling railroad dynamics, reducing maintenance costs, and increasing safety factors in rail operations. The assumption of a constant CoF is popularly used in simulation studies for ease of implementation, however many evidences demonstrated that CoF depends on various dynamic parameters and instantaneous conditions. In the real world, accurately estimating the CoF is difficult due to effects of various uncertain parameters, such as wheel and rail materials, rail roughness, contact patch, and so on. In this study, the newly developed 3-D nonlinear CoF model for the dry rail condition is introduced and the CoF variation is tested using this model with dynamic parameters estimated from the wheel-rail simulation model. In order to account for uncertain parameters, a stochastic analysis using the polynomial chaos (poly-chaos) theory is performed using the CoF and wheel-rail dynamics models. The wheel-rail system at a right traction wheel is modeled as a mass-spring-damper system to simulate the basic wheel-rail dynamics and the CoF variation. The wheel-rail model accounts for wheel-rail contact, creepage effect, and creep force, among others. Simulations are performed at train speed of 20 m/s for 4 sec using rail roughness as a unique excitation source. The dynamic simulation has been performed for the deterministic model and for the stochastic model. The dynamics results of the deterministic model provide the starting point for the uncertainty analysis. Six uncertain parameters have been studied with an assumption of 50% uncertainty, intentionally imposed for testing extreme conditions. These parameters are: the maximum amplitude of rail roughness (MARR), the wheel lateral displacement, the track stiffness and damping coefficient, the sleeper distance, and semi-elliptical contact lengths. A symmetric beta distribution is assumed for these six uncertain parameters. The PDF of the CoF has been obtained for each uncertain parameter study, for combinations of two different uncertain parameters, and also for combinations of three different uncertain parameters. The results from the deterministic model show acceptable vibration results for the body, the wheel, and the rail. The introduced CoF model demonstrates the nonlinear variation of the total CoF, the stick component, and the slip component. In addition, it captures the maximum CoF value (initial peak) successfully. The stochastic analysis results show that the total CoF PDF before 1 sec is dominantly affected by the stick phenomenon, while the slip dominantly influences the total CoF PDF after 1 sec. Although a symmetric distribution has been used for the uncertain parameters considered, the uncertainty in the response obtained displayed a skewed distribution for some of the situations investigated. The CoF PDFs obtained from simulations with combinations of two and three uncertain parameters have wider PDF ranges than those obtained for only one uncertain parameter. FFT analysis using the rail displacement has been performed for the qualitative validation of the stochastic simulation result due to the absence of the experimental data. The FFT analysis of the deterministic rail displacement and of the stochastic rail displacement with uncertainties demonstrates consistent trends commensurate with loss of tractive efficiency, such as the bandwidth broadening, peak frequency shifts, and side band occurrence. Thus, the FFT analysis validates qualitatively that the stochastic modeling with various uncertainties is well executed and is reflecting observable, real-world results. In conclusions, the development of an effective model which helps to understand the nonlinear nature of wheel-rail friction is critical to the progress of railroad component technology and rail safety. In the real world, accurate estimation of the CoF at the wheel-rail interface is very difficult since it is influenced by several uncertain parameters as illustrated in this study. Using the deterministic CoF value can cause underestimation or overestimation of CoF values leading to inaccurate decisions in the design of the wheel-rail system. Thus, the possible PDF ranges of the CoF according to key uncertain parameters must be considered in the design of the wheel-rail system.
- Doctoral Dissertations