PROBLEMS AND EXERCISES PROBLEMS

Transcription

1 64 Fundamentals of Kinematics and Dynamics of Machines and Mechanisms PROBLEMS AND EXERCISES PROBLEMS 1. In Figure 1.14c an inverted slider-crank mechanism is shown. b. If the input is the displacement of the cylinder, what are the unknowns? 2. In Figure 1.14d an inverted slider-crank mechanism is shown. b. If the input is the displacement of the cylinder, what are the unknowns? 3. In Figure P2.1 an inverted slider-crank mechanism is shown. b. If the input is the crank angle, what are the unknowns? d. Express the position of point P in terms of the input angle. FIGURE P In Figure P.2.2 a five-bar linkage is shown, in which AB is parallel and equal to O 2 C. b. If the input is the crank angle, what are the unknowns? d. Express the position of point P in terms of the input angle. 5. In Figure P2.3 a slider-crank mechanism is shown. b. If the input is the crank angle, what are the unknowns? d. Express the position of point P in terms of the input angle.

2 Kinematic Analysis of Mechanisms 65 FIGURE P2.2 FIGURE P In Figure P2.4 a four-bar linkage mechanism is shown. b. If the input is the crank angle, what are the unknowns? d. Express the positions of points P 1 and P 2 in terms of the input angle. FIGURE P For the mechanism in Figure P2.4, a. Formulate the requirement for dimensions in order for the mechanism to be a crank-rocker. b. If the follower is to rock within angle γ, is there a unique set of dimensions to meet this requirement? Outline the solution procedure.

3 66 Fundamentals of Kinematics and Dynamics of Machines and Mechanisms 8. In Figure 2.4 find the maximum angle of rocking of the connecting rod, if r 2 and r 3 are given. 9. For a variable-stroke drive mechanism in Figure P1.7, a. Identify two loops. b. For a coordinate system such that axis x is directed from the center of the driving shaft to the center of the output shaft, outline the solution procedure using solutions in the book for a four-bar linkage. 10. For the double-toggle mechanism in Figure P1.6, a. Identify two loops. b. For a coordinate system such that the x-axis passes through the center of the crankshaft and is directed to the right, outline the solution procedure using solutions in the book for a four-bar linkage and a slider-crank mechanism. 11. For the constant velocity mechanism shown in Figure P1.5, a. Identify two loops. b. For a coordinate system such that the x-axis is directed along the cylinder to the right, outline the solution procedure using solutions in the book for a four-bar linkage and a slider-crank mechanism. FIGURE P A quick-return mechanism, shown in Figure P2.5, is used in machine tools. For a constant angular velocity of the driving crank, it produces slow velocity during the cutting phase and then fast return. a. Identify two loops. b. Define a coordinate system.

4 Kinematic Analysis of Mechanisms 67 c. For a coordinate system such that the y-axis is going through the points O 1 and O 2 upwards, outline the solution procedure using solutions in the book for a slider-crank mechanism. d. Define the time ratio (time of cutting stroke to the time of return stroke) in terms of the distances O 1 O 2 and O 2 A. 13. For the mechanism in Figure P2.1, find a. the velocities. b. the accelerations of point P. Assume that the position, velocity, and acceleration analyses of the skeleton have been done. 14. For the mechanism in Figure P2.2, find a. the velocities. b. the accelerations of point P. Assume that the position, velocity, and acceleration analyses of the skeleton have been done. 15. For the mechanism in Figure P2.3, find a. the velocities. b. the accelerations of point P. Assume that the position, velocity, and acceleration analyses of the skeleton have been done. 16. Assume that dimensions of all links in Figure P1.4 are known. a. What would be the configuration of the links when the pliers are closest? b. Assume that the coupler link is parallel to the frame link, and that the frame link is 10 cm, the coupler link is 7 cm, the link connecting the two jaws is 2 cm, and the fourth link is 3 cm. What is the angle by which the coupler link rotates from the initial to the extreme position? 17. Consider the double-rocker mechanism in Figure P2.6. a. What should the relationship between the links dimensions be in order for the connecting link to make a complete rotation while the arms are rocking? b. Assume 2a = 1 cm, h = 0.5 cm, 2d = 10 cm, and that the arms have equal length 7 cm. What is the angle of arm 1 rocking? 18. For an eight-slot Geneva mechanism a. Find the distance between the centers of rotation, r 1, given the radius of the crank r 2 (distance from the center of the driving plate to the center of the pin) and assuming that the pin enters and leaves the slot smoothly. b. Find the length of the slot.

5 68 Fundamentals of Kinematics and Dynamics of Machines and Mechanisms FIGURE P2.6 FIGURE P In Figure P2.7 a linear intermittent motion mechanism is shown. Given the angular velocity of the crank and its radius, r 2, how would velocity and acceleration of the slider change during the cycle of the crank? EXERCISES (PROJECTS) WITH MATHEMATICA 1. A motor drives a film-advancing mechanism with constant velocity ω (Figure P2.8) Link 2 is the driver. The path of point C should be as indicated, so that during the engagement with the film point C moves along a straight line. a. Find (by trial and error) such dimensions of the mechanism that the needed trajectory of point C is achieved. c. Plot the velocity and acceleration of point C over the cycle. Find the velocities and accelerations during engagement and disengagement with the film. Does the velocity remain constant during the engagement?

6 Kinematic Analysis of Mechanisms 69 FIGURE P A motor drives a carrier mechanism with constant velocity ω (see Figure 2.26). The path of point A should be such that at the lowest position the line AB is parallel to the horizontal conveyor, and at the highest position the line AC should be parallel to the inclined conveyor (so that the load can be transferred from one conveyor to another). a. Find (by trial and error) such dimensions of the mechanism (for the assumed positions of the conveyor belts) that the needed trajectory of point A is achieved. (Hint: At extreme positions the motion of point A is reversed, which means that at these positions the links DC and CB are collinear.) c. Plot the velocity and acceleration of point A over the cycle. What are the velocities and accelerations at the points of load transfer? 3. For the dimensions of the double-rocker given in problem 17: a. Animate the motion. b. Plot the trajectory, velocity, and acceleration of the tracing point over the cycle. 4. The oscillating drive-arm in Figure P2.9 has the maximum operating angle π/6. For a relatively short guideway, the reciprocating output stroke is large and it follows a straight line. a. Find (by trial and error) such dimensions of the mechanism that the needed trajectory of the tracing point is achieved. c. Plot the velocity and acceleration of point P over the cycle. What are the velocities and accelerations at the end points of the stroke?

7 70 Fundamentals of Kinematics and Dynamics of Machines and Mechanisms FIGURE P For a complex mechanism operating a dump truck in Figure 1.6 assume that the bed moves from a horizontal to a π/3 position. a. Find (by trial and error) such dimensions of the mechanism that the needed trajectory of the bed is achieved. 6. In Figure P1.6 a double-toggle puncher is shown. When the drive crank rotates clockwise, the second toggle begins to straighten to create a strong punching force. a. Find link dimensions such that the desired motion is achieved. c. Plot velocities and accelerations of the point P over the cycle. 7. In Figure P1.7 a variable-stroke drive, which is a combination of two four-bar linkages, is shown. The driving member rotates the eccentric, which, through the linkage, causes the output link to rotate a fixed amount. The ratchet on the output shaft transfers motion in one direction only. Thus, on the return stroke, the output link overrides the output shaft. As a result, a pulsating motion is transmitted to the output shaft, which is needed in many applications, such as feeders and mixers. A smoother drive can be produced by mounting on the same shaft the same device but with some shift in phase with respect to the first one. A continuously variable drive can be designed by mounting a few of such devices on the same shaft and using the control link to change the position of the adjustable pivot. a. Perform a complete kinematic analysis for one position of the adjustable pivot. Plot velocities and accelerations of the output link over the cycle.

8 Kinematic Analysis of Mechanisms In Figure P1.3 an adjustable stroke mechanism is shown. The output link slides along the horizontal line, and the stroke is controlled by the position of the pivot point. a. Perform complete kinematic analysis for one position of the adjustable pivot. Plot velocities and accelerations of the output link over the cycle. 9. For the loader shown in Figure 2.3, assume that rods 5 and 6 move during the same time interval (synchronized motion), but their velocities may be different. a. Find by trial and error such dimensions that the bucket rotates over the π/2 range from the lowest to the highest position. b. Perform a complete kinematic analysis. Plot angular velocity and acceleration of the bucket when it moves from the lowest to the highest position. c. Animate the motion.