Three-dimensional contact surface modeling and stress analysis of interference fit based on cylindricity error

Abstract

The main failure mode of a wheel–axle assembly, which is an important moving part of rail vehicles, in the interference fit zone is partial fatigue. Given that neither the plane analytical method nor the axisymmetric finite element method can reveal the local and stochastic contact stress concentration at the end of the wheel seat, this study analyzed various factors influencing a machined mating surface and found that the cylindricity error is a key factor in transforming the plane contact into a 3D contact. In addition, the experimental data of the pressing force applied to 60 groups of wheel–axle assemblies were analyzed, and the circumferential contour of the mating surface was found to conform to a beta distribution. Accordingly, a method for obtaining random interpolation points on the circumferential contour was proposed. To ensure the continuity of the mating surface, the circumferential Hermite interpolation method and the axial cubic interpolation method were employed, and the contact surface with random characteristics was modeled. Based on the random ergodicity of the mating surface, a small-sample 3D finite element model of a wheel–axle assembly was established using the random cylindricity error, and a calculation method for the 3D contact stress was proposed. Taking the RE_2B axle of railway locomotive as an example, compared with the axisymmetric method, the inhomogeneity analysis of the contact stress of the small-sample model proposed in this paper could help explain the problem of local stress concentration at the end of the wheel seat. The analysis results showed that the maximum stress of the 3D model based on the cylindricity is greater than that of the axisymmetric model, enabling strength evaluation in the design of wheel–axle assemblies.

Appendix 1: Introduction of beta distribution

The beta distribution refers to a set of continuous probability distributions defined in the interval (0,1), and its probability density function is:

$$f\left( {x;\alpha ,\beta } \right) = \left\{ {\begin{array}{*{20}l} {\frac{{x^{\alpha - 1} \left( {1 - x} \right)^{\beta - 1} }}{{\int_{0}^{1} {u^{\alpha - 1} } \left( {1 - u} \right)^{\beta - 1} du}} = \frac{{\Gamma \left( {\alpha + \beta } \right)}}{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}x^{\alpha - 1} \left( {1 - x} \right)^{\beta - 1} ,} \hfill & {0 < x < 1} \hfill \\ {0,} \hfill & {{\text{other}}} \hfill \\ \end{array} } \right.$$

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where $\alpha \ge 0, \, \beta > 0$.

Mathematical expectation and variance are expressed as follows:

$$\mu = E\left( x \right) = \frac{\alpha }{\alpha + \beta }$$

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$${\text{Var}}\left( X \right) = \frac{\alpha \beta }{{\left( {\alpha + \beta } \right)^{2} \left( {\alpha + \beta + 1} \right)}}$$

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$\Gamma ( \cdot )$ represents the $\Gamma$ distribution:

$$f\left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\lambda ^{e} }}{{\Gamma \left( \alpha \right)}}x^{{a - 1}} e^{{ - \lambda x}} ,} \hfill & {x \ge 0} \hfill \\ {0,} \hfill & {x < 0} \hfill \\ \end{array} } \right.$$

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