Useful zero friction simulations for assessing MBS codes Pascal's formula giving wheelsets frequency for zero wheel-rail friction

Abstract

Present study provides a simple analytical formula, the “Klingel-like formula” or “Pascal's Formula” that can be used as a reference to test some results of existing railway codes and specifically those using rigid contact. It develops properly the 3D Newton-Euler equations governing the 6 degrees of freedom (DoF) of unsuspended loaded wheelsets in case of zero wheel-rail friction and constant conicity. Thus, by solving numerically these equations, we got pendulum like harmonic oscillations of which the calculated angular frequency is used for assessing the accuracy of the proposed formula so that it can in turn be used as a fast practical target for testing multi-body system (MBS) railway codes. Due to the harmonic property of these pendulum-like oscillations, the square ω2 of their angular frequency can be made in the form of a ratio K/M where K depends on the wheelset geometry and load and M on its inertia. Information on K and M are useful to understand wheelsets behavior. The analytical formula is derived from the first order writing of full trigonometric Newton-Euler equations by setting zero elastic wheel-rail penetration and by assuming small displacements. Full trigonometric equations are numerically solved to assess that the formula provides ω2 inside a 1% accuracy for usual wheelsets dimensions. By decreasing the conicity down to 1 × 10−4 rad, the relative formula accuracy is under 3 × 10−5. In order to test the formula reliability for rigid contact formulations, the stiffness of elastic contacts can be increased up to practical rigidity (Hertz stiffness × 1000).