Nadal’s Formula and High Speed Rail Derailments

Several railroad vehicle derailment criteria are based on the $(L/LVV)$ ratio where $L$ is the lateral force and $V$ is the vertical force acting on the wheelset. Derailment is assumed to occur if this ratio exceeds a certain limit. The $(L/LVV)$ ratio has its roots in Nadal’s formula which was introduced more than a century ago. When a planar analysis that corresponds to zero angle of attack is used, Nadal’s formula can be derived using a geometric approach, while a Nadal-like formula can be derived using a kinetic approach. In the geometric approach, a coordinate transformation is used to define the normal and friction forces in another coordinate system. In this case, the lateral and vertical forces are interpreted as the components of a vector that defines the normal and friction forces in another coordinate system without consideration of other external forces that are applied to the wheelset. Because the geometric approach does not account for other forces, it should not be used as the basis for derailment studies. In the kinetic approach, on the other hand, the $L$ and $V$ forces are interpreted as the resultant of the forces excluding the normal and friction forces at the contact point. That is, in both approaches discussed in this paper, the $L$ and $V$ forces cannot be interpreted as the resultant forces acting on the wheelset. The condition for the wheel climb using the kinetic approach is obtained and examined. It is shown that formulas obtained in this paper are based on assumptions that do not capture the gyroscopic moments, and therefore, these formulas should not be used as the basis for general high speed rail derailment criteria. It is also shown that in the case of zero angle of attack and using single degree of freedom planar kinetic model assumptions, an increase in the lateral force $L$ can reduce the tendency for wheel climb, while reducing $L$ can increase the wheel climb risk. Furthermore, the single degree of freedom assumptions used to obtain the wheel climb formula presented in this paper do not allow for wheel lift, and as a consequence, wheel lift derailment scenarios that can be the result of large moment should not be investigated using Nadal’s formula or one of its derivatives that employ the same assumptions. Furthermore, since the analysis presented in this paper assumes zero angle of attack, the conclusions obtained in this investigation do not apply to the wheel climb scenarios with nonzero angle of attack. It is also important to point out that this paper is not intended as a discussion on how Nadal’s formula is interpreted by researchers and engineers; instead, the paper is mainly focused on examining the roots of this formula and the problems that can arise from the assumptions used in its derivation.