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chap85 - Belt Friction Let's look at a flat belt passing...

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Belt Friction Let's look at a flat belt passing over a drum Let's take a look at a differential element Motion is assumed to be impending. dN dF s μ = The normal force is a differential force because it acts on a differential element of area. 0 2 cos ) ( 2 cos 0 = + + - - = Σ θ μ θ d dT T dN d T F s x 0 2 sin ) ( 2 sin 0 = + + - - = Σ dN d dT T d T F y θ θ For small angles cos( d θ ) = 1 sin( d θ ) = d θ 0 = + + - - Σ dT T dN T F s x μ dN dT s μ = T 1 P P' β θ d θ P 1 P 2 O dN d θ μ s dN T+dT T d θ/2 d θ/2 x y r
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0 2 ) ( 2 = + + - - Σ dN d dT T d T F y θ θ 0 2 2 2 = + - - - dN d dT d T d T θ θ θ 0 2 = θ d dT (2 small numbers squared = 0) Thus: θ d T dN = Substituting: θ μ θ μ d T dT d T dT s s = = Integrating: θ μ β d T dT O s T T = 2 1 B T T S μ = - 1 2 ln ln Thus: B T T s μ = 1 2
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