In the previous note we reviewed the derivation of capstan equation that solves the tension developed in a rope (web, belt) undergoing kinetic friction on a capstan (roller, pulley). We set up a problem in which a 1-D belt is tensioned, constrained statically, and in contact for a wrap angle of 90° on a cylinder that is rotating at a constant angular velocity. So the belt is in equilibrium and the belt-cylinder contact interface is in complete slip condition. This setup avoids the complication that would be introduced by slip-stick condition. However, the slip-stick condition universally exists in similar contact problems. In this note we will review the slip-stick condition, its mechanism, and its location at the contact interface in 1-D belt-roller contact problems with various boundary conditions. We will use \(\{\)rope, belt, web, string \(\}\) and \(\{\)capstan, cylinder, roller, pulley \(\}\) interchangeable, respectively, since in 1-D cases elements in each set are equivalent to others in the physical sense. All vector quantities are treated as scalars in computation with their direction visually shown.
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This note reviews the derivation of capstan equation. A capstan is a cylinder-like device that turns to wind a rope or cable around it to lift or haul things. Ropes and cables are flexible enough to be wound and work as a medium to transfer force to other bodies. The capstan’s cylindrical geometry couples rope’s tangential and radial loads as the rope is in equilibrium on the capstan. This coupling transfers a portion of the rope’s tangential tensile load into its radial contact pressure with the capstan, thus, enlarges the frictional effect. This behavior results in a distinct feature of capstan devices—the ratio of tensile forces at two ends of the rope wound on a capstan is an exponential function of the product of kinetic friction coefficient and wrap angle (slip angle to be precise) between them. Looping multiple threads of rope on the capstan would significantly increase the contact area where the friction can take place on, thus, significantly alter the ratio of two end forces of the rope, which sometimes would be nice to take advantage of. The same mechanics universally governs the problems with similar geometry and loading members, for instances, a belt transporting on pulleys or a web transporting on rollers. The capstan equation solves the tensile force developed in the rope (belt, web) undergoes kinetic friction on capstans (pulleys, rollers).
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