The document discusses limiting positions, displacement diagrams, and coupler curves in mechanisms. It defines limiting positions as the extreme locations of an output link's motion path. A stroke is the displacement between two limiting positions. Displacement diagrams graphically show the motion of components over a full cycle. Coupler curves represent the path generated by a point on a linkage coupler as it moves, which can be used to generate complex motion. Examples of determining slider stroke, throw, and generating coupler curves are provided.

1. Mechanics of Machines
MDB3043
Lecture 06:
Limiting Positions, Displacement
Diagrams & Coupler Curves
Azman Zainuddin

2. Outline of Lecture 06
 Limiting Positions
 Displacement Diagram
 Coupler Curves
Copyright © Azman Zainuddin 2004,2006

3. Limiting Positions

4. Limiting Positions
 The output link of a mechanism is at a
limiting position when it is placed at an
extreme location along its motion path.
 There are normally two limiting positions in
a linkage.
 The displacement between two limiting
positions is called a stroke.
 The displacement of the stroke can be
linear or angular.

5. Engine Cylinder
Piston Stroke
STROKE

6. Example of a
Limiting Position
limiting
position

7. Example of a
Limiting Position
limiting
position

8. Example of a
Limiting Position
limiting
position
limiting
position

9. Limiting Positions in
In-line Slider-Crank
limiting
position 2
STROKE
limiting
position 1

10. In-line Slider-Crank Stroke
L1 L2
L1 + L2
L2 – L1 2 L1
Stroke = 2 x crank radius

11. In-line Slider-Crank Stroke
L1 L2
L1 + L2
L2 – L1 2 L1
Stroke = 2 x crank radius

12. Limiting Positions of a 4-bar Linkage
crank
rocker
coupler

14. limiting position 1

15. limiting position 1

16. limiting position 1

17. limiting position 1

18. limiting position 1

19. limiting position 1

20. limiting position 1

21. limiting position 1

22. limiting position 2
limiting position 1

23. limiting position 1
limiting position 2

24. limiting position 2
limiting position 1

25. STROKE
limiting position 2
limiting position 1
Note that the limiting positions of the rocker are achieved when the
crank and the coupler are in line.

26. Offset Slider-Crank
12
43
22
What is the stroke of the slider driven by the 12-mm crank through a
43-mm link. The slider has an offset of 22 mm from the crank centre

27. Offset Slider-Crank
43-12
22
What is the stroke of the slider driven by the 12-mm crank through a
43-mm link. The slider has an offset of 22 mm from the crank centre.
Lmin
The extreme left
position of the slider

28. Offset Slider-Crank
43+12
22
What is the stroke of the slider driven by the 12-mm crank through a
43-mm link. The slider has an offset of 22 mm from the crank centre.
Lmax
The extreme right
position of the slider

29. Offset Slider-Crank
55
22
What is the stroke of the slider driven by the 12-mm crank
through a 43-mm link. The slider has an offset of 22 mm from the
crank centre.
31
Stroke = Lmax - Lmin
= √(552-222) - √(312-222) = 50.4 - 21.8 = 28.6 mm
Lmin
Lmax

30. Throw Determination by Analytical
Method
Analytically calculate the maximum angular
displacement (‘throw’) of the ram for the crushing
mechanism.
180
420
360
60
ram
crank

31. Throw Determination by Analytical Method
 Analytically calculate the maximum angular
displacement (‘throw’) of the ram for the crushing
mechanism. The angular limits are achieved when the
coupler and the crank are in line.
180
420
360
60
throw

32. Throw Determination by Analytical Method
The throw of the ram is the difference between
α1 and α2.
180
360
180
360
α1
α2
α1 = cos-1(1802+3602-4802)
2(180)(360)
= 121.9˚
α2 = cos-1(1802+3602-3602)
2(180)(360)
= 75.5˚
α = α1 – α2
= 121.9˚-75.5˚ = 46.4˚

33. Displacement Diagram for
a Complete Cycle
 Motion of a mechanism is normally cyclical especially when the
input motion actuator is a motor that rotate in a single direction.
 All components of the mechanism move in phase to assume
different configurations. The mechanism is said to have completed
a cycle when it assumes its original configuration.
 More practical to perform complete cycle analysis by using
computerized analytical method rather than graphical because the
tasks are repetitive and too tedious to be performed in the latter.

34. Displacement Diagram
 This diagram can be drawn after performing position
analysis for a complete cycle of a mechanism.
 The information from the diagram is more reliable if the
analysis is performed at greater number of points, so
analytical method is more practical for this purpose
compared to graphical.
Slider
displacement
diagram for a full
revolution of the
crank.
Crank angle (deg)
Slider
position
(mm)

35. Coupler Curves
 Curve generated by a point on a link, normally the coupler of a
4-bar linkage, as the linkage moves
 Simple solution to generate complex motion path
 The points on a curve might represent the path taken by a
cutter, a spray painting nozzle or an adhesive applicator.
crank
rocker
coupler

36. Coupler Curve Generation

37. Coupler Curve Generation

38. Coupler Curve Generation

39. Coupler Curve Generation

40. Coupler Curve Generation

41. Coupler Curve Generation

42. Coupler Curve Generation

43. Coupler Curve Generation

44. Coupler Curve Generation

45. Coupler Curve Generation

46. Coupler Curve Generation

47. Coupler Curve Generation

48. Coupler Curve Generation

49. Coupler Curve Generation

50. Coupler Curve Generation

51. Coupler Curve Generation

52. Coupler Curve Generation

53. Coupler Curve Generation

54. Coupler Curve Generation

55. Coupler Curve Generation

56. Examples of Coupler Curves

57. Bring these to the Tutorial!

58. Tutorial Week 3
4-6 & 4-56:
Compressor Linkage
4-20, 4-47 & 4-60:
Aircraft Nose Wheel Actuator