On the varying brachistochrone shape with allowance for chute loading limitation

In this paper, the problem of the effect of stress limit on the chute shape is analyzed for the first time. Here, the dynamic equations for the motion of the material body rolling down the chute are formulated neglecting the friction forces. It is shown that if the stress limit for the chute material is taken into account, the shape of the chute varies greatly as a function of the parameter. Four possible cases are analyzed when the parameter is: equal to zero, more than unity, less than unity, and equal to unity. It is found that if the parameter is more than unity, the chute shape represents almost horizontal and vertical segments of a trajectory, which is clear from a physical point of view, since for this type of the trajectory the chute is least affected by the body moving along. If the parameter is equal to unity, the chute takes a specific loop-like shape. If the parameter is equal to zero, the system of equations describes a classical brachistochrone. The solution to the problem is applicable in practice for predicting the shape of the chute withstanding high loads when the stress limit for the material is known.