hmewrk_03_sol - Solution Homework #3: Metal Cutting 2.008...

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Solution Homework #3: Metal Cutting 2.008 Design and Manufacturing II Spring 2004 Problem 1: (a) Consider the Merchant’s Cutting Force diagram in Figure 1 and the chip-workpiece interface during orthogonal cutting in Fig. 2. α φ R F t F c F n F s F N α β β−α Fig 1. Merchant’s diagram Fig 2. Cutting FBD Merchant’s hypothesis is that the shear plane is located to minimize the cutting force, or where the shear stress is maximum. Derive the Merchant’s relationship between shear angle, rake angle, and friction angle as below from the diagram above. o α β φ = 45 + 2 2 (b) Consider the directions of the cutting force and the thrust force. Fc, cutting force, is always positive, since the material is removed. Is the thrust force, Ft, also positive at all times? If not, explain why. Also explain how you can make Ft = 0 for a given friction coefficient between the tool and the work piece. Solution (a) From the Force diagram (we discussed it in class so you know how each of the elements came in), we know that Fs = Fc*cos φ - Ft*sin φ (1) Fn = Fc*sin φ + Ft*cos φ (2) Where Fc = net horizontal cutting force Ft = net vertical thrust force
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Fs = shear force along shear plane Fn = shear force normal to shear plane There are two ways to derive Merchant´s Relationship: find the minimum of the cutting force or the maximum of the shear stress. We will have a look at both. i) Find the minimum of the cutting force Fc A t 0 i w (3) F i A s s s s s sin ϕ sin ϕ F t = tan( β−α ) (4) F c With σ s as the shear stress and A s the shear area; t 0 is the thickness of the uncut chip, called the depth of cut, w is the width of the work. (3) and (4) in (2): tw i σ 0 = Fcos ϕ− F s in ϕ tan ( ) s c c sin ϕ ii σ 1 F = 0 s i (5) c sin ϕ cos ϕ sin ϕ tan( β α ) In order for the cutting force to be minimum at angle ϕ , the first derivative has to be zero: F c = 0 ∂ϕ Substituting equation (5) and solving: !
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hmewrk_03_sol - Solution Homework #3: Metal Cutting 2.008...

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