Chosen problems and their final solutions of Chap. 2 (Waldron)(Waldron)- Par 2
19. Draw the velocity polygon to determine the velocity of link 6. Points A, C , and E have the same vertical coordinate. Sol: v E 6 = 8 in/ s.
20. Link 2 of the linkage shown in the figure has an angular velocity of 10 rad/s CCW. Find the angular velocity of link 6 and the velocities of points B, C , and D. Sol: v B4 = 3.29 in/ s, vC 3 = 6.789 in/ s, = 4.52 rad / s CCW. v D6 = 6.78 in/ s, ω6
21. The linkage shown is used to raise the fabric roof on convertible automobiles. automobiles. The dimensions at given as shown. Link 2 is driven by a DC motor through a gear reduction. If the angular velocity, ω2 = 2 rad/s, CCW, determine the linear velocity of point J , which is the point where the linkage connects to the automobile near the windshield.
Sol: v J 8 = 73.6 in/ s.
22. In the mechanism shown, determine the sliding velocity of link 6 and the angular velocities of links 3 and 5.
Sol: v F 6 = 43.33 in/ s = 1.187 rad / s CCW, ω3 = 0.862 rad / s CW. ω5
23. In the mechanism shown, vA2 = 15 m/s. Draw the velocity polygon, and determine the velocity of point D on link 6 and the angular velocity of link 5.
Sol: v D6 = 12.31 m/ s ω5 = 4.488 rad / s CCW.
24. In the mechanism shown below, points E and B have the same vertical coordinate. Find the velocities of points B, C , and D of the double-slider mechanism shown in the figure if Crank 2 rotates at 42 rad/s CCW.
Sol: v B4 = 17.76 in/ s,
vC 3
= 18.615 in/ s, v D6 = 5.63 in/ s.
25. Given vA4 = 1.0 ft/s to the left, find
.
vB6
Sol: v B6 = 1.23 ft/ s.
26. If vA2 = 10 cm/s as shown, find
.
vG6
Sol: vG6 = 3.65 cm/ s.
27. If vA2 = 10 in/s as shown, find the angular velocity of link 6.
Sol: ω 6 = 1.375 rad/s CCW. 28. The angular velocity of link 2 of the mechanism shown is 20 rad/s, and the angular 2 acceleration is 100 rad/s at the instant being considered. Determine the linear velocity and acceleration of point F 6. Sol: v F 6 = 3.79 in/s, a F 6 = 152.7 in/s2.
29. In the drag-link mechanism shown, link 2 is turning CW at the rate of 130 rpm. Construct the velocity and acceleration polygons and compute the following: aE5, aF6, and the angular acceleration of link 5.
2
Sol: aE5 = 308.0 ft/s , 2 aF6 = 83.4 ft/s , 2 α 5 = 29.3 rad / s CCW.
30. The figure shows the mechanism used in two-cylinder 60-degree V-engine consisting, in part, of an articulated connecting rod. Crank 2 rotates at 2000 rpm CW. Find the velocities and acceleration of points B, C , and D and the angular acceleration of links 3 and 5. Sol: (a) vB4 = 212.7 in/s, vC3 = 243.3 in/s, vD6 = 189.2 in/s. 2 (b) aB4 = 14710 in/ s , 2 aC3 = 39300 in/ s , 2 aD6 = 24000 in/ s , 2 α 3 = 12820 rad/s CCW, 2 α 5 = 10300 rad/s CW.
31. In the mechanism shown, ω2 = 4 rad/s CCW (constant). Write the appropriate vector equations, solve them using vector polygons, and a) Determine vE3, vE4, and ω3. b) Determine aE3, aE4, and α3. Also find the point in link 3 that has zero acceleration for the position given. Sol: (a) vE3 = 2.522 m/s, vE4 = 0.797 m/s, = 3.0 rad / s CW. ω3 (b) aE3 = 15.84 m / s 2, 2 aE4 = 32.19 m / s , 2 α 3 = 20.16 rad/s CCW.
32. In the mechanism shown, point A lies on the X axis. Draw the basic velocity and acceleration polygons and use the image technique to determine the velocity and acceleration of point D4. 2
2
Sol: vD4 = 6.77 in/s , aD4 = 54.0 in/s .
33. In the mechanism shown below, the velocity of A2 is 10 in/s to the right and is constant. Draw the velocity and acceleration polygons for the mechanism, and record values for angular velocity and acceleration of link 6. Use the image technique to determine the velocity of points D3, and E 3, and locate the point in link 3 that has zero velocity. Sol: v D3 = 10.2 in/s, v E 3 = 12.7 in/s, ω 6 = 3.97 rad/s CW, 2 α 6 = 26.4 rad/s CCW.
34. If the accelerations of Points A and B are as given in the rigid body shown below, find the Point C in that link at which the acceleration is zero.
Sol:
35. The following are given for the mechanism shown in the figure: ω2 =6.5 rad/ s (CCW); 2 Draw the velocity polygon, and locate the velocity of Point α2 =40 rad/ s (CCW). E using the image technique. Sol: v E = 5.4042 in/s
36. In the mechanism shown, find image.
ω6
and
. Also, determine the acceleration of D3 by
α3
2
2
Sol: ω 6 = 4.229 rad/s CW, α 3 = 22.856 rad/s CW, aD3 = 91.39 in/s . 2
37. In the mechanism shown, ω2 = 1 rad/s (CCW) and α2 = 0 rad/s . Find ω5, α5, vE6, aE6 for the position given. Also find the point in link 5 that has zero acceleration for the position given. Sol: ω 5 = 0.706 rad/s CCW, 2 α 5 = 0.626 rad/s CW, 2 v E 6 =0.441 m/s, a E 6 = 0.042 m/s .
O’ of link 5 has zero acceleration.
38. Part of an eight-link mechanism is shown in the figure. Links 7 and 8 are drawn to scale, and the velocity and acceleration of point D7 are given. Find ω7 and α7 for the position given. Also find the velocity of G7 by image. 2
Sol: ω 7 = 2.27 rad/s CCW, α 7 = 28.4 rad/s CCW, vG7 = 6.016 in/s.
39. In the mechanism shown below, link 2 is rotating CW at the rate of 3 rad/s (constant). In the position shown, link 2 is horizontal. Link lengths: AB = 3 in, BC = BE = CE = 5 in, CD = 3 in . Write the appropriate vector equations, solve them using vector polygons, and a) Determine vC4, vE3, ω3, and ω4. b) Determine aC4, aE4, α3, and α4. Sol: (a) vC4 = 6.57 in/s, vE3 = 13.5 in/s, ω3 = 2.232 rad/s CW, and ω4 = 2.19 rad/s CCW. 2 2 2 (b) aC4 = 48.87 in/s , aE3 = 34.29 in/s , α3 = 0 rad/s , 2 and α4 = 15.57 rad/s CCW.
40. Part of a 10-link mechanism is shown in the figure. Links 7 and 8 are drawn to scale, and the velocity and acceleration of points D7 and F 8 are given. Find ω8 and α7 for the position given. Also find the velocity of G7 by image. 2
Sol: ω 8 = 0 rad/s, α 7 = 0 rad/s , vG7 = 9.1 in/s.
