Multi-Hertzian Methods are used in Railway codes to calculate elastically normal forces of Wheel/Rail Contacts. Such methods may be properly applied to calculate forces of several elliptical simultaneous contacts. They may also be applied as a sum of Hertzian contacts to approximate normal forces of non elliptical patches such as presented in the 3D drawing. Solving such cases necessitates to choose the location, the ellipticity and the maximum indentation of elementary ellipses. In order to solve this problem it is usual to consider the maxima of the function of un-deformed profiles indentation in the symmetry plane YOZ. However these methods cannot solve cases when this function has only one maximum and has no symmetry. The “Rigid-Multi-Hertzian Method” solves these difficult cases on a topology basis: at first by searching the locations of all potential contacts in the rigid situation during a pure lateral translation (OY) of the wheel across the rail. During this translation, contact locations can be discontinuous and forbidden areas are identified as “Gutters” of which the Edges are stored. The second phase uses the indentation function to calculate indentations at gutter edges where secondary ellipses are assumed to be located; the main ellipse is located as usual. This method allows to solve conformal cases and develops smooth continuous contact forces during dynamical simulations. This paper presents the method in more details using the example of a quasi-conformal pair of profiles (S1002 and UIC60). Taking advantage of the analytical definition of these profiles, it is proposed as a benchmark to calculate the resultant normal force (amplitude and direction in upper figure) for one case of which all the surface details are disclosed: numerical data of both surfaces can be re-produced using the attached software. Contact Forces could be compared either using this method (results are given) or using numerical data as inputs to FEM calculations of commercial codes available to researchers who would like to assess the accuracy of this Rigid-Multi-Hertzian Method with respect to more sophisticated tools. One FEM calculation, using ANSYS, is presented and results are in good agreement with this method. Note: This method was first mentioned in 1991 & 1993 in papers [3], [4] but it could not be applied by third parties because implementation details were not disclosed. However it has been used since by the author to produce lookup tables of equivalent mono-patches as input data of Vocodym successful “rigid” software. It has now been developed to be used on line in the dynamical elastic software Ocrec.

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