ch03 - Solutions to Chapter 3 Exercise Problems Problem 3.1...

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- 33 - Solutions to Chapter 3 Exercise Problems Problem 3.1 In the figure below, points A and C have the same horizontal coordinate, and ω 3 = 30 rad/s. Draw and dimension the velocity polygon. Identify the sliding velocity between the block and the slide, and find the angular velocity of link 2. 4 2 B 3 , B 4 A ω 3 3 AC = 1 in BC = 3 in r = 2.8 in C 45˚ Position Analysis: Draw the linkage to scale. AC = 1 in BC = 3 in 30 in/sec Velocity Polygon O v b 3 b 4 b 2 , A B C 2 in 3 2 45.0° Velocity Analysis: 1 v B 3 = 1 v B 3 / A 3 = 1 ω 3 × r B 3 / A 3 1 v B 3 = 1 ω 3 r B 3 /A 3 = 30(2.2084) = 66.252 in/sec 1 v B 4 = 1 v B 2 = 1 v B 3 + 1 v B 4 / B 3 (1)
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- 34 - 1 v B 2 = 1 v B 2 / C 2 = 1 ω 2 × r B 2 / C 2 Now, 1 v B 3 = 66.252in/sec in the direction of r B /A 1 v B 2 = 1 ω 2 × r B/C ( to r B/ C ) 1 v B 4 / B 3 is on the line of AB Solve Eq. (1) graphically with a velocity polygon. From the polygon, 1 v B 4 / B 3 = 15.63in/sec Also, 1 ω 2 = 1 v B 2 / C 2 r B/ C = 68.829 3 = 22.943 rad /sec From the directions given in the position and velocity polygons 1 ω 2 = 22.943 rad/sec CW
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- 35 - Problem 3.2 If ω 2 = 10 rad/s CCW, find the velocity of point B 3 . 1 ω 2 Α Β 2 3 4 E C D 45˚ 18˚ 110˚ CA = 1.5" DE = 2.5" CD = 4.0" AB = 1.6" Position Analysis Draw the linkage to scale.
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- 36 - Velocity Analysis 2 2 2 2 2 1 1 1 1 1 1 / 2 / 2 / 0 10(1.5) 15 in/s A C A C A C A A C ω ω = + = + × = = = v v v r v r (1) 3 3 3 3 2 3 3 1 1 1 1 1 / / E A E A A E A = + = + v v v v v 3 4 3 4 1 1 1 / E E E E = + v v v 4 4 4 4 1 1 1 1 / 4 / 0 E D E D E D ω = + = + × v v v r Now, 2 1 15 /s A in = v / ( to ) A C r 3 3 1 1 / 3 / / ( to ) E A E A E A ω = × v r r 4 4 1 1 / 4 / / ( ) E D E D E D to ω = × v r r and 3 4 ω ω = , need to get 3 ω to find 3 1 B v . Define the point F where AF DF in position polygon. 3 3 3 3 1 1 1 / F A F A = + v v v 3 4 3 4 1 1 1 / F F F F = + v v v 4 3 4 3 1 1 1 / F F F F = + v v v 4 4 4 1 1 / F F D = v v Solve Eq. (1) graphically with a velocity polygon.
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- 37 - After finding point “f 3 ”, construct the velocity image to find the point “b 3 a line to AB through the point “a” a line to BF through the point “f 3 fine the point “b 3 From the polygon, 3 1 9.4 in/s B = v
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- 38 - Problem 3.3 If ω 2 = 100 rad/s CCW, find v B 4 . A B 2 3 4 C D 4 95˚ G E F AD = 1.8" CD = 0.75" AE = 0.7" CF = 0.45" FG = 1.75" CB = 1.0" DB = 1.65" 125˚ Position Analysis Draw the linkage to scale.
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- 39 - Velocity Analysis 2 2 2 2 1 1 1 1 / 2 / 0 G A G A G A ω = + = + × v v v r (1) 3 3 3 3 4 1 1 1 1 1 / 3 / G C G C C G C ω = + = + × v v v v r 3 2 3 2 1 1 1 / G G G G = + v v v 4 3 4 4 4 1 1 1 1 1 / 4 / 0 C C D C D C D ω = = + = + × v v v v r Now, 2 1 1 2 / / 100(3.44) 344 in/s ( ) G G A G A to ω = = = v r r 1 1 3 2 ω ω = 3 3 1 1 / 3 / / 100(2.65) 265 in/s ( ) G C G C G C to ω = = = v r r 3 2 1 / G G v is on the line of EG 4 4 1 1 / 4 / / ( ) C D C D C D to ω = × v r r Solve Eq. (1) graphically with a velocity polygon.
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- 40 - To find the point “ 4 b use velocity polygon 4 4 1 / 612.14 / C D in s = v
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- 41 - Problem 3.4 If ω 2 = 50 rad/s CCW, find v D 4 . 2 3 4 A B C D 50˚ 150˚ BC = CD BD = 3.06" Position Analysis Draw the linkage to scale.
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- 42 - Velocity Analysis 2 2 2 2 1 1 1 1 / 2 / 0 B A B A B A ω = + = + × v v v r (1) 3 3 3 3 4 1 1 1 1 1 / 3 / B D B D D B D ω = + = + × v v v v r 3 2 3 2 1 1 1 / B B B B = + v v v 4 3 1 1 D D = v v Now, 2 1 1 2 / / 50(2.39) 119.5 in/s ( ) B B A B A to ω = = = v r r 1 1 3 2 ω ω = 3 3 1 1 / 3 / / 50(3.06) 153 in/s ( ) B D B D B D to ω = = = v
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