Meriam, Kraige & Bolton — Analysis of Structures: Problem 6_103

⚡ Mecademy AIENG정역학 · ch6  Problem Statement Replace the flat belt and pulley of Fig. 6/11 by a V-belt and matching grooved pulley as indicated by the cross-sectional view accompanying this problem. Derive the relation among the belt tensions, the angle of contact, and the coefficient of friction for the V-belt when slipping impends. A V-belt design with would be equivalent to increasing the coefficient of friction for a flat belt of the same material by what factor ? Problem 6/103 (a) Derive the relation among the belt tensions, angle of contact, and coefficient of friction for the V-belt 1. Formula: Consider a differential element of the belt of length subtending an angle . For a V-belt, the normal force that would act on a flat belt is replaced by two normal forces acting on the side walls of the groove. The equilibrium in the radial direction is: The equilibrium in the tangential direction is: where the total friction force is . 2. Substitution: From the radial equilibrium equation, express in terms of tension and the groove angle : Substitute this expression into the tangential equilibrium equation ( ): α=35 ∘ n rdθ dθdN dN ′ F = ∑ r 0: 2d N sin − ′ 2 α Tdθ=0 F = ∑ t 0:(T+dT)−T−dF=0 dF=2μdN ′ 2dN ′ Tα 2dN= ′ sin(α/2) Tdθ dT=dF= 2μdN ′ dT= μ ( sin(α/2) Tdθ ) 3. Calculation: Rearrange the terms to separate variables and integrate over the total contact angle : 4. Result: Exponentiate both sides to find the final tension ratio: ● Final Conclusion: The derived relation for