This paper did a theoretical study on the Nadal's L/V ratio. The analysis is based on a mechanical model of an object sliding on an incline (or slope), which is widely used in college physics. The key is that the direction of frictional forces is always opposite to the direction of the motion of the sliding object. Therefore, there are two directions (upward or downward) for the frictional forces between the object and incline depending on the states of motion of the object. Thus, there must be two L/V ratios for the object sliding on the incline for the same reason. The theoretical demonstration shows that Nadal's L/V is the same with the L/V which governs the downward motion of the object on the incline, because the direction of frictional force between the object and the incline is set to be upwards in the derivation of the Nadal's L/V. Thus, Nadal's L/V is for the object going down the incline. A detail examination was performed on the Nadal's L/V for some typical configurations, such as the critical angle; the zero and 90 degrees angles, further proving that the Nadal's L/V is not for an object going up on the incline, thus cannot be used as the criterion for wheel climb. A new L/V ratio was created by setting the direction of frictional force downwards to simulate the object going up on the incline, and was named as Huang's L/V. Wheel flange/rail contact produces frictional forces between them to consume the pulling power, like a braking to slowdown wheel rotation. Thus, wheel climb is only 1/3 of the whole story of wheel flange/rail contact. The other two are 1). A retarder derailment mode is created by the braking and 2). A braking, large enough, will cause a wheel locked. Therefore, there are two derailment modes with wheel/flange rail contact, wheel climb modes and retarder mode. A method to determine which mode was initiated was demonstrated in the paper. Angle of Attack (AoA) introduces a complicated scenario for wheel climb calculations. It is almost impossible to determine a correct L/V ratio under AoA.
This paper deals with the problem of static and dynamic (or kinetic) friction, namely the coefficients of friction for the two states. The coefficient of static friction is well known, and its theory and practice are commonly accepted by the academia and the industry. The coefficient of kinetic friction, however, has not fully been understood. The popular theory for the kinetic friction is that the coefficient of dynamic friction is smaller than the coefficient of static friction, by comparison of the forces applied in the two states. After studying the characteristics of the coefficient of friction, it is found that the comparison is not appropriate, because the inertial force was excluded. The new discovery in the paper is that coefficients of static friction and dynamic friction are identical. Wheel “locked” in wheel braking is further used to prove the conclusion. The key to cause confusions between the two coefficients of friction is the inertial force. In the measurement of the coefficient of static friction, the inertial force is initiated as soon as the testing object starts to move. Therefore, there are two forces acting against the movement of the object, the frictional force and the inertial force. But in the measurement of the coefficient of kinetic friction, no inertial force is involved because velocity must be kept constant.
Wheelset hunting is a motion with two degrees of freedom. The second degree of freedom of hunting is investigated in this paper. Rolling radius difference is commonly understood as the root cause of wheelset hunting. Normally, hunting will begin as soon as the truck begins to move. Rolling radius difference will initiate the hunting and can only be used to determine the configuration of the first quarter of a hunting cycle. Then a mechanism will come into exist to determine the configuration of the second quarter of the hunting cycle, which is exactly the same as the one of first quarter but in the opposite direction. The second half of the hunting cycle is a mirror of the first half. The information from wheelset hunting is too large and too complicated for people to understand and analyze. However, the radius of curvature of wheelset circular motion can be calculated exactly, especially at some special configurations, even though the radius of curvature of wheelset circular motion keeps changing in both magnitude and direction. Furthermore, comparison of the radii of curvature between hunting curve and a cosine curve shows that the curve for the wheelset in hunting motion is sinusoidal. This is proven mathematically in the paper. The limits in Klingel's results were discussed. Due to the fact that Klingel's results are unable to provide a strong theoretical base for further understanding the complicated dynamic characteristics caused by hunting, misleading in hunting is overwhelmed. After understanding the principles of wheelset hunting, three critical speeds in hunting and dynamic loading for truck design in three directions were calculated; wheelset hunting curve while curving was generated; wheelset dynamic interaction was analyzed and the question "Why trains stay on tracks" was answered correctly, just to name a few. Thus, a total solution to the hunting problems can be expected and was discussed in the paper.
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