ME317

1. ME317 Chapter 3 Graphical Linkage Synthesis

2. Definition • Function Generation: Is the correlation of an input motion with an output motion in a mechanism • Given an input produce a predictable output • Cam-follower is an example of mechanical function generator • Today, function generators are electromechanical devices • Cheaper, and programmable

3. Definition • Path Generation: Is the control of a point in a plane such that it follows some prescribed path. • A subset of motion generation

4. Definition • Motion Generation: Is the control of a line in a plane such that it assumes some prescribed set of sequential positions.

5. Limiting Conditions • The linkage must be checked to reach all the specified design positions without encountering a limiting position. • C1D1 position crank and coupler are overlapped • C2D2 positions crank and coupler are aligned Grashof crank-rocker toggle positions

6. Limiting Conditions • Toggle positions: • Undesirable, if it prevents the linkage from getting from one position to a desired position . • Desirable, it creates a self-locking mechanism if it is moved slightly beyond the toggle position and against a fixed stop.

7. Limiting Conditions

8. Transmission Angle (μ) • The acute angle between the output link and the coupler • It is a measure of the quality of force and velocity transmission at the joint

9. Transmission Angle (μ)

10. Dimensional Synthesis • Dimensional synthesis of a linkage is the determination of the proportions (length) of the links necessary to accomplish the desired motion. • Methods used: • Graphical • Analytical • Numerical • Graphical method is the simplest and quickest method for a fourbar linkage. • It works well for up to three design positions.

11. Dimensional Synthesis • Two position synthesis • Link moves through 2 positions as rocker • Rocker output • The output function is defined as two discrete angular positions of a rocker • Coupler output • Two positions of a line in a plane are defined as the output.

12. Dimensional Synthesis • Example 3-1 • Two position synthesis • Design a fourbar Grashof crank-rocker to give 45˚ of rocker rotation with equal time forward and back, from a constant speed motor.

13. Dimensional Synthesis • Two position synthesis • Rocker output • The output function is defined as two discrete angular positions of a rocker

14. Two position Motion Synthesis with Rocker Output Link 4 is represented by a line, complex motion, reduced to problem of pure rotation, as a rocker output of a fourbar linkage

15. Two position Motion Synthesis with Coupler Output Additional links are necessary to limit the motion to desired positions.

16. Two position Motion Synthesis with Coupler Output Additional links are necessary to limit the motion to desired positions.

17. Two position Motion Synthesis with Coupler Output(Adding a dyad (Twobar chain) to example 3.3 Select a point on link 2, not on line O2C1, label it B1 Draw an arc about center O2 through B1 , label it B2 The cord B1B2 gives us example problem 3-1.

18. Two position Motion Synthesis with Coupler Output(Adding a dyad (Twobar chain) to example 3.3)

19. Two position Motion Synthesis with Coupler Output(Adding a dyad (Twobar chain) to example 3.3)

20. Three Position Motion Synthesis with Specified Moving Pivots

21. Three Position Motion Synthesis with Specified Moving Pivots Note: Links 3 and 4 are in toggle in position 1, and links 2 and 3 are in toggle position 3. Drive link 3 with a driver dyad. Pivot locations of O2 and O4 may not be desirable

22. Three Position Motion Synthesis with Alternate Attachment Point for Moving Pivots

23. Three Position Motion Synthesis with Alternate Attachment Point for Moving Pivots

24. Three Position Motion Synthesis with Alternate Attachment Point for Moving Pivots

25. Three Position Motion Synthesis with Fixed Pivots • The fixed pivot positions are defined. • Three positions of the moving link is defined. • Use inversion principal

26. Three Position Motion Synthesis with Fixed Pivots • Problem: • Invert a fourbar linkage which moves the link CD shown from position C1D1 to C2D2 and then to C3D3 . Use specified fixed pivots O2 and O4.

27. Three Position Motion Synthesis with Fixed Pivots Step 1: Step 2:

28. Three Position Motion Synthesis with Fixed Pivots Step 3: Transfer this relationship back to the first coupler position C1D1 The new ground position has the same relation to C1D1 as O2O4 had withC2D2.

29. Three Position Motion Synthesis with Fixed Pivots Step 4: Transfer this relationship back to the first coupler position C1D1 The new ground position has the same relation to C1D1 as O2O4 had with C3D3.

30. Three Position Motion Synthesis with Fixed Pivots Three inverted positions of the ground plane corresponding to the original coupler position

31. Three Position Motion Synthesis with Fixed Pivots • The inversion process swaps the roles of coupler and ground plane. • The problem now is similar to example 3-5.

32. Finding the Moving Pivots for Three Position and Specified Fixed Pivots • Problem: • Design a fourbar linkage to move the link CD from position C1D1 to C2D2 and then to C3D3 . Find the required moving pivot locations on the coupler by inversion.

33. Finding the Moving Pivots for Three Position and Specified Fixed Pivots Finding rotopoles G and H Correct inversion of the desired linkage

34. Finding the Moving Pivots for Three Position and Specified Fixed Pivots Comparing the original problem with the solution Reinvert to obtain the result: Line E1F1 O2O4 Line GH  coupler link 2 represents line CD

35. Finding the Moving Pivots for Three Position and Specified Fixed Pivots Reinvert Reinvert to obtain the result: Line E1F1 O2O4 Line GH  coupler

36. Finding the Moving Pivots for Three Position and Specified Fixed Pivots Replace CD on link 3

37. Non-Quick Return Mechanism

38. Quick Return Mechanism Note: with non-quick return mechanism the pivot O2 would be along the dotted blue line

39. Fourbar Quick Return Mechanism • Advantages • Simple • Compact • Disadvantage: • It works for small time ratio <1.5 • It will run into bad transmission angle and rough running

40. Quick Return Mechanism Time ratio TR

41. Quick Return Mechanism • Redesign Example 3-1 to provide a time ratio of 1.25 with 45˚ rocker output. α=160˚ β =200˚

42. Quick Return Mechanism Measure O2B1 and O2B2 to calculate crank and coupler’s length Link 2 and 3 are collinear Coupler + Crank= O2B1 Link 2 and 3 are collinear (Overlapped)Coupler - Crank= O2B2 α=160˚ β =200˚

43. Quick Return Mechanism Calculate Grashof condition. If non-Grashof move O2 further away from O4 This method works well for time ratios down to 1.5.

44. Sixbar Quick Return Mechanism • Advantages • Large time ratio up to 2 to 1. • Smooth action • Time ratio can be as high as 3 to 1, but may not be as smooth action as with 2 to 1 time ratio • Disadvantage: • More links

45. Sixbar Quick Return Mechanism • Time ratios of up to about 1:2 can be obtained. • Design a fourbar drag link mechanism with the desired time ratio • Add a dyad (twobar) output stage driven by the dragged crank. • It can have either a rocker or translating slider as the output link.

46. Sixbar Quick Return Mechanism • Example: Provide a time ratio of 1.4 with 90˚ rocker motion. Show video, Start time =28 min. α=150˚ β =210˚