Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_76

⚡ Mecademy AIENG정역학 · ch6  Problem Statement An end of the thin board is being sanded by the disk sander under application of the force . If the effective coefficient of kinetic friction is and if the pressure is essentially constant over the board end, determine the moment which must be applied by the motor in order to rotate the disk at a constant angular speed. The board end is centered along the radius of the disk. Problem 6/76 (a) Determination of the required motor moment 1. Formula: For the disk to rotate at a constant angular speed, the motor must supply a moment that exactly balances the resisting moment due to friction . The friction moment is found by integrating the differential moments created by the friction force across the contact area: where is the contact pressure, is the coefficient of kinetic friction, and is an infinitesimal element of the contact area. 2. Substitution: According to the diagram, the board end is a rectangle centered along the radius. Its radial length is , and it is positioned between an inner radius and an outer radius . Let the board have a small thickness . Since the pressure is essentially constant, the total applied force is: From this, we can express the constant pressure-thickness product as: 3. Calculation: Consider a differential radial element of width at radius . Its area is . The differential friction force acting on this element is: P μ M M MM f M=M = f r d F = ∫ f r (μpdA) ∫ pμdA L=R/2 r = 1 R/4r = 2 R/4+R/2=3R/4 tpP P= pdA