Correcting and clarifying the basic locomotive skidding equation

Determining whether a railroad locomotive will skid under varying conditions is a fairly routine, pervasive problem. However, some of the equations provided are wrong, others are at best murky. All require correction or clarification. Skidding during heavy braking is a crucial safety problem, possibly resulting in loss of control and collision. It can also be an economic detriment, ruining wheels and tracks and, by inhibiting acceleration, causing reduced full-speed times. While skidding is less likely for rubber-tired equipment, a comparable problem exists for tractor-trailers operating on poor surfaces. The primary reference is to the SME Mining Engineering Handbook (pp. 14-11, Eqs. 7 and 8), which is taken as authoritative and normative (Brantner, 1973). Fundamentally, the physical weight of the locomotive can never be negative (-). However, based on this reference it can be negative on adverse grades, during braking (Eq. 7), and on favorable grades, whether accelerating or braking (Eq. 8). Additionally, it should be noted that the term rW should be Rw. To avoid having the locomotive weight turn out negative, "absolute value" brackets must enclose the right-hand expression of both equations. Also, the weight of locomotive can be greater than the right-hand side value introducing the common "greater than or equal to" symbol (>_). While these alterations are easy to make, the use of the algebraic sign associated with each term is still confusing, as