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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# Category Archives: Mechanics

# Mass Conservation of a Moving Web in Steady State

Posted in Mechanics on .The longitudinal behavior of a moving web in steady state is like a 1-D steady state fluid. The law of the mass conservation on the steady state web relates the spatial velocity of web to its longitudinal stain. This note reviews the derivation of the relation between the longitudinal strain and spatial velocity in a continuous web traveling in steady state in space.

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# Capstan Equation

Posted in Mechanics and tagged Contact Mechanics on .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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# Extremum Problems of Functionals and Calculus of Variations

Posted in Mechanics and tagged Math on .Last note, Functionals, Euler’s Equation, and Equilibrium, reviewed the concept of functionals and the method that solves the extremum problems of the simplest integral form of functionals—the Euler’s equation. The derivation adopted in that post is a directional derivatives like method. This treatment was simply to bypass the review of calculus of variations at that time. Calculus of variations is the subject of mathematics directly concerns the extremum problems of functionals, which underlies quite a lot of mechanics laws. The notation \(\delta\) frequently appears in many principles of mechanics and numerical formulations of complex mechanics problems. This memento reviews the concept of calculus of variations.

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# Functionals, Euler’s Equation, and Equilibrium

Posted in Mechanics and tagged Math on .This note reviews the concepts of functionals, Euler’s equation, equilibrium, and the equivalence between the equilibrium equations and the principle of minimum potential energy.