Homework5bsol.spring11

# Homework5bsol.spring11 - ME 352 Machine Design I Spring...

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2 Solution to Homework Set 5. Part A. A suitable set of vectors for the sliding-block linkage are shown in Figure 2. Figure 2. The vectors for the sliding-block linkage. The vector loop equation (VLE) for the linkage is ?? 0 123 4 I RRR √√ + −= (1) The X and Y components of Equation (1) are 11 2 2 3 4 3 4 cos cos cos 0 RR R θ θθ + (2a) and 3 4 sin sin sin 0 R + (2b) Differentiating Equations (2) with respect to the input angle θ 2 gives 2 2 34 34 34 34 34 sin sin cos 0 R −+ = (3a) and 2 2 34 34 34 34 34 cos cos sin 0 R (3b)
3 Equations (3) can be written in matrix form as 34 34 34 34 22 34 34 34 34 sin θ cos θθ sin θ cos θ sin θ R cos θ R R R R ⎡⎤ = ⎢⎥ −− ⎣⎦ (4) The determinant of the coefficient matrix is 34 34 34 34 34 34 34 34 34 34 34 sin θ cos θ sin θ cos θ cos θ sin θ R DET R R R R == = (5) Using Cramer’s rule, the first-order kinematic coefficient for link 3 can be written as 3 4 3 4 2 2 34 2 2 34 34 34 sin cos cos sin sin sin cos cos R R RR DET R θ (6a) which can be written as 3 4 34 34 cos ( ) R R ′ = (6b) The first-order kinematic coefficient for links 3 and 4 can be written as 34 34 2 2 34 34 2 2 2 34 2 34 2 34 2 34 34 34 sin sin cos cos cos sin sin cos R DET R −+ (7a) which can be written as 23 4

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Homework5bsol.spring11 - ME 352 Machine Design I Spring...

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