Chosen problems and their final solutions of Chap. 2 (Waldron)- Par 1

Transcription

1 Chosen problems and their final solutions of Chap. 2 (Waldron)- Par 1 1. In the mechanism shown below, link 2 is rotating CCW at the rate of 2 rad/s (constant). In the position shown, link 2 is horizontal and link 4 is vertical. Write the appropriate vector equations, solve them using vector polygons, and b) determine a C4, α 3, and α 4. Link lengths: AB = 75 mm, CD = 100 mm. Sol: (a) v C4 = 43 mm/ s ω 3 = 0.86 rad / s CW, ω 4 = 0.43 rad / s CW. (b) a C4 = 435 mm/ s 2, α 3 = rad/s 2 CCW, α 4 = rad/s 2 CCW. 2. In the mechanism shown below, link 2 is rotating CCW at the rate of 500 rad/s (constant). In the position shown, link 2 is vertical. Write the appropriate vector equations, solve them using vector polygons, and b) determine a C4, α 3, and α 4. Link lengths: AB = 1.2 in, BC = 2.42 in, CD = 2 in Sol: (a) v C4 = 858 in/ s ω 3 = rad / s CCW, ω 4 = 429 rad / s CCW. (b) a C4 = in/ s 2, α 3 = rad/s 2 CW, α 4 = rad/s 2 CW. 3. In the mechanism shown below, link 2 is rotating CW at the rate of 10 rad/s (constant). In the position shown, link 4 is vertical. Write the appropriate vector equations, solve them using vector polygons, and b) determine a C4, α 3, and 4. Link lengths: AB = 100 mm, BC = 260 mm, CD = 180 mm Sol: (a) v C4 = 990 mm/ s ω 3 = 0.12 rad / s CCW, ω 4 = 5.5 rad / s CW. (b) a C4 = 5700 mm/ s 2, α 3 = 18.4 rad/s 2 CCW, α 4 = 9.88 rad/s 2 CW.

2 4. In the mechanism shown below, link 2 is rotating CW at the rate of 4 rad/s (constant). In the position shown, θ is 53. Write the appropriate vector equations, solve them using vector polygons, and link lengths: AB = 100 mm, BC = 160 mm, CD = 200 mm, b) determine a C4, α 3, and α 4. Sol: (a) v C4 = 300 mm/ s ω 3 = rad / s CCW, ω 4 = 1.5 rad / s CW. (b) a C4 = 3250 mm/ s 2, α 3 = 3.87 rad/s 2 CCW, α 4 = 16.1 rad/s 2 CW. 5. In the mechanism shown below, link 2 is rotating CCW at the rate of 4 rad/s (constant). In the position shown, link 2 is horizontal. Write the appropriate vector equations, solve them using vector polygons, and link lengths: AB = 1.25 in, BC = 2.5 in, CD = 2.5 in, a) Determine v C4, ω 3, and ω 4. b) Determine a C4, α 3, and α 4. Sol: (a) v C4 = 3.75 in/ s ω 3 = 2.5 rad / s CCW, ω 4 = 1.5 rad / s CW. (b) a C4 = 7.32 in/ s 2, α 3 = 1.87 rad/s 2 CCW, α 4 = 1.87 rad/s 2 CW. 6. In the mechanism shown below, link 2 is rotating CW at the rate of 100 rad/s (constant). In the position shown, link 2 is horizontal. Write the appropriate vector equations, solve them using vector polygons, and link lengths: AB = 60 mm, BC = 200 mm. a) Determine v C4 and ω 3. b) Determine a C4 and α 3. Sol: (a) v C4 = 4500 mm/ s ω 3 = 0.12 rad / s CW, (b) a C4 = mm/ s 2, α 3 = 1060 rad/s 2 CCW,

3 7. In the mechanism shown below, link 4 is moving to the left at the rate of 4 ft/s (constant). Link lengths: AB = 10 ft, BC = 20 ft. Write the appropriate vector equations, solve them using vector polygons, and a) Determine ω 3 and ω 4. b) Determine α 3 and α 4. Sol: ω 3 =.115 rad / s CW, ω 4 =0 rad / s; α 3 = rad/s 2 CW, α 4.= 0 rad/s In the mechanism shown below, link 4 is moving to the right at the rate of 20 in/s (constant). Link lengths: AB = 5 in, BC = 5 in. Write the appropriate vector equations, solve them using vector polygons, and a) Determine ω 3 and ω 4. b) Determine α 3 and α 4. Sol: ω 3 = 2.82 rad / s CW, ω 4 = 0 rad / s; α 3 = 7.76 rad/s 2 CCW, α 4.= 0 rad/s In the mechanism shown below, link 4 is moving to the left at the rate of 0.6 ft/s (constant). Write the appropriate vector equations, solve them using vector polygons, and determine the velocity and acceleration of point A 3. Link lengths: AB = 5 in, BC = 5 in. Sol: v A3 = 1.34 ft/s, a A3 = 4.93 ft/s 2.

4 10. In the mechanism shown below, link 4 moves to the right with a constant velocity of 75 ft/s. Link lengths: AB = 4.8 in, BC =16.0 in, BG = 6.0 in. Write the appropriate vector equations, solve them using vector polygons, and a) Determine v B2, v G3, ω 2, and ω 3. b) Determine a B2, a G3, α 2, and α 3. Sol: (a) v B2 = 91.2 ft/s, v G3 = 79 ft/s, ω 3 = 52 rad / s CCW, ω 2 = 228 rad / s CW. (b) a B2 = ft/ s 2, a G3 = ft/ s 2, α 3 = rad/s 2 CW, α 2 = rad/s 2 CW 11. For the four-bar linkage, assume that ω 2 = 50 rad/s CW and α 2 = 1600 rad/s 2 CW. Write the appropriate vector equations, solve them using vector polygons, and a) Determine v B2, v E3, ω 3, and ω 4. b) Determine a E3, α 3, and α 4. Sol: (a) v B2 = 87.5 in/s, v E3 = in/s, ω 3 = 12.7 rad / s CW, ω 4 = 33.7 rad / s CW. (b) a E3 = 5958 in/ s 2, α 3 = 303 rad/s 2 CW, α 4 = rad/s 2 CW. 12. In the mechanism shown below, link 2 is rotating CW at the rate of 180 rad/s. Link lengths: AB = 4.6 in, BC = 12.0 in, AD = 15.2 in, CD = 9.2 in, EB = 8.0 in, CE = 5.48 in. Write the appropriate vector equations, solve them using vector polygons, and a) Determine v E3, ω 3, and ω 4. b) Determine a C3, a E3, α 3, and α 4. Sol: (a) v E3 = 695 in/s, ω 3 = 48.6 rad / s CCW, ω 4 = 51.6 rad / s CW. (b) a C3 = in/ s 2, a E3 = in/ s 2, α 3 = 8073 rad/s 2 CW, α 4 = rad/s 2 CW.

5 13. The accelerations of points A and B in the coupler below are as given. Determine the acceleration of the center of mass G and the angular acceleration of the body. Draw the vector representing a G from G. Sol: α = 1122 rad/s 2 CW, a G = 6980 in/s Crank 2 of the push-link mechanism shown in the figure is driven at a constant angular velocity ω 2 = 60 rad/s (CW). Find the velocity and acceleration of point F and the angular velocity and acceleration of links 3 and 4. Sol: v F3 = 4.94 m/s, ω 3 = rad/s CW, ω 4 = rad/s CW, α 3 = 484 rad/s 2 CW, α 4 = 136 rad/s 2 CCW, a F3 = 256 m/s For the straight-line mechanism shown in the figure, ω 2 = 20 rad/s (CW) and α 2 = 140 rad/s 2 (CW). Determine the velocity and acceleration of point B and the angular acceleration of link 3. Sol: v B3 = 77.3 in/s, ω 3 = 20 rad/s CCW, α 3 = 140 rad/s 2 CCW, a B3 = 955 in/s2.

6 16. For the data given in the figure below, find the velocity and acceleration of points B and C. Assume v A = 20 ft/s, a A = 400 ft/s 2, ω 2 = 24 rad/s (CW), and α 2 = 160 rad/s 2 (CCW). Sol: v B = 11.9 ft/s, v C = ft/s, a B = ft/s 2, a C = ft/s In the mechanism shown below, link 2 is turning CCW at the rate of 10 rad/s (constant). Draw the velocity and acceleration polygons for the mechanism, and determine a G3 and α 4. Sol: α 4 = rad/s 2 CW, a G3 = 116 in/s If ω 2 = 100 rad/s CCW (constant) find the velocity and acceleration of point E. Sol: v E =28.4 in/s, a E = 4600 in/s 2.