Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_98

⚡ Mecademy AIENG정역학 · ch6  Problem Statement The endless belt of an escalator passes around idler drum and is driven by a torque applied to drum . Belt tension is adjusted by a turnbuckle at , which produces an initial tension of in each side of the belt when the escalator is unloaded. For the design of this system, calculate the minimum coefficient of friction between drum and the belt to prevent slipping if the escalator handles people uniformly distributed along the belt and averaging each. (Note: It can be shown that the increase in belt tension on the upper side of drum and the decrease in belt tension at the lower drum are each equal to half the component of the combined passenger weight along the incline.) Problem 6/98 (a) Minimum coefficient of friction 1. Formula: First, we determine the geometry of the incline and the total load. The angle of inclination is found from the rise and run: The total weight of the passengers is . The component of this weight along the incline is: According to the provided note, the tensions on the tight side () and slack side () at the driving drum are: The minimum coefficient of friction is determined by the capstan equation for belt friction: where is the wrap angle in radians. For an endless belt turn-around, ( ). AM BC 1000 lb μB 30 150 lb BA μ θ θ= tan −1 ( L H ) W=n⋅W avg W = x Wsinθ T 1 T 2 B T = 1 T + 0 W , T = 2 1 x2 T − 0 W 2 1 x μ = T 2 T 1 e ⟹ μβ μ= ln β 1 ( T 2 T 1 ) ββ=π rad 180 ∘ 2. Substitution: Given values: , , , , . 3