Mechanisms Design MECN 4110

1. Mechanisms Design MECN 4110 Professor: Dr. Omar E. Meza Castillo omeza@bayamon.inter.edu http://www.bc.inter.edu/facultad/omeza Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus

2. Tentative Lectures Schedule

3. One thing you learn in science is that there is no perfect answer, no perfect measure. A. O. Beckman Topic 2: Mechanism and Kinematics Kinematics Fundamentals

4. Course Objectives • Up on completion of this chapter, the student will be able to • Explain the need for kinematic analysis of mechanism. • Define the basic components that comprise a mechanism. • Draw the kinematic diagram from a view of a complex mechanism. • Compute the number of degrees of freedom of a mechanism. • Identify a four bar mechanism and classify it according to its possible motion. • Identify a slider crank mechanism.

5. 2.1 NUMBER SYSTHESIS The term number synthesis has been coined to mean the determination of the number and order of links and joints necessary to produce motion of a particular DOF. Order in this context refers to the number of nodes per link, i.e., binary, ternary, quaternary, etc. The value of number synthesis is to allow the exhaustive determination of all possible combinations of links which will yield any chosen DOF. This then equips the designer with a definitive catalog of potential linkages to solve a variety of motion control problems.

6. 2.1 NUMBER SYSTHESIS • Hypothesis: If all joints are full joints, an odd number of DOF requires an even number of links and vice versa. • Proof: Given: All even integers can be denoted by 2m or by 2n, and all odd integers can be denoted by 2m - 1 or by 2n - 1, where n and m are any positive integers. The number of joints must be a positive integer. • Let: L = number of links, J = number of joints, and M = DOF = 2m (i.e., all even numbers) • Then: rewriting Gruebler's equation (Equation 2.1b) to solve for J,

7. 2.1 NUMBER SYSTHESIS • Try: Substituting M=2m, and L=2n (i.e., both any even numbers) • This cannot result in J being a positive integer as required • Try: M=2m-1, and L=2n-1 (i.e., both any odd numbers) • This also cannot result in J being a positive integer as required

8. 2.1 NUMBER SYSTHESIS • Try: M=2m-1, and L=2n (i.e., odd-even) • This is a positive integer for m>=1 and n>=2. • Try: M=2m, and L=2n-1 (i.e., even-odd) • This is a positive integer for m>=1 and n>2 • So, for our example of one-DOF mechanism we can only consider combinations of 2, 4, 6, 8 … links

9. 2.1 NUMBER SYSTHESIS • Letting the order of the links be represented by: • The total number of links in any mechanism will be: • Since two link nodes are needed to make one joint • and nodes

10. 2.1 NUMBER SYSTHESIS • Then • Substitute J and L in the Gruebler’s equation • The DOF is independent of number of ternary link in the mechanism. But because each ternary link has three nodes, it can only create or remove 3/2 joints

11. 2.1 NUMBER SYSTHESIS

12. 2.2 PARADOXES • The Gruebler’s criterion pays no attention to link sizes or shapes, it can give misleading results in the face of unique geometric configurations.

13. 2.3 ISOMERS • The word isomer is from the Greek and means having equal parts. • Linkage isomers are analogous to chemical compounds in that the links (like atoms) have various nodes (electrons) available to connect to other link’s nodes.

14. 2.4 LINKAGE TRANSFORMATION • There are several transformation techniques or rules that we can apply to planar kinematic chains: • Revolute joints in any loop can be replaced by prismatic joints with no change in DOF of the mechanism, provided that at least two revolute joints remain in the loop. • Any full joint can be replaced by half joint, but this will increase the DOF by one. • Removal of a link will reduce the DOF by one. • The combination of rules 2 and 3 above will keep the original DOF unchanged.

15. 2.4 LINKAGE TRANSFORMATION • Any ternary or higher-order link can be partially “shrunk” to a lower-order link by coalescing nodes. This will create a multiple joint but will no change the DOF of the mechanism. • Complete shrinkage of a higher-order link is equivalent to its removal. A multiple joint will be created, and the DOF will be reduced.

16. 2.4 LINKAGE TRANSFORMATION • A fourbar crank-rocker linkage transformed into the fourbar slider-crank by the application of rule #1.

17. 2.4 LINKAGE TRANSFORMATION • A fourbar slider-crank transformed via rule #4 by the substitution of a half joint for the coupler.

18. 2.4 LINKAGE TRANSFORMATION • A fourbar linkage transformed into a earn-follower linkage by the application of rule #4. Link 3 has been removed and a half joint substituted for a full joint between links 2 and 4.

19. 2.4 LINKAGE TRANSFORMATION (a) shows the Stephenson's sixbar chain transformed by partial shrinkage of a ternary link (rule #5) to create a multiple joint. It is still a one-DOF Stephenson's sixbar.

20. 2.4 LINKAGE TRANSFORMATION (b) shows the Watt's sixbar chain from with one ternary link completely shrunk to create a multiple joint.

21. 2.5 INTERMITENT MOTION • Is a sequence of motions and dwells. Dwell; is a period in which the output link remains stationary while the input link continues to move.

22. 2.5 INTERMITENT MOTION

23. 2.6 INVERSION • An inversion is created by grounding a different link in the kinematic chain. Thus there are as many inversions of a given linkage as it has links.

24. 2.6 INVERSION

25. 2.6 INVERSION – All inversions of the Grashof fourbar linkage

26. 2.7 THE GRASHOF CONDITION 3.- Coupler Link 4.- Follower Link 2.- Input Link 1.- Fixed Link

27. 2.7 THE GRASHOF CONDITION 3.- Coupler Link 4.- Follower Link 2.- Input Link 1.- Fixed Link

28. 2.7 THE GRASHOF CONDITION • The four bar linkage, shown in previous slide, is a basic mechanism which is quite common. Further, the vast majority of planar one degree-of-freedom (DOF) mechanisms have "equivalent" four bar mechanisms. The four bar has two rotating links ("levers") which have fixed pivots, (bodies 2 and 4 above). One of the levers would be an input rotation, while the other would be the output rotation. The two levers have their fixed pivots with the "ground link"(body 1) and are connected by the "coupler link" (body 3).

29. 2.7 THE GRASHOF CONDITION - Definitions • Crank- a ground pivoted link which is continuously rotatable. • Rocker- a ground pivoted link that is only capable of oscillating between two limit positions and cannot rotate continuously.

30. 2.7 THE GRASHOF CONDITION - Definitions • Grashof Condition- is a very simple relationship which predicts the rotation behavior or rotability of a fourbar linkage's inversions based only on the link lengths • Let: • S=length of shortest link • L=length of longest link • P=length of one remaining link • Q=length of other remaining link • Then if: S+L<=P+Q

31. 2.7 THE GRASHOF CONDITION • The linkage is Grashof and at least one link will be capable of making a full revolution with respect to the ground plane. This is called a Class I kinematic chain. • If the inequality is not true, then the linkage is non-Grashof and no link will be capable of a complete revolution relative to any other link. This is a Class II kinematic chain. • The order of the assemble in the kinematic chain in S, L, P, Q, or S, P, L, Q or any other order, will not change the Grashof condition.

32. 2.7 THE GRASHOF CONDITION • The motions possible from a fourbar linkage will depend on both the Grashof condition and the inversion chosen. The inversions will be defined with respect to the shortest link. The motions are: • For the Class I case, S + L < P + Q: • Ground either link adjacent to the shortest and you get a crank-rocker, in which the shortest link will fully rotate and the other link pivoted to ground will oscillate.

33. 2.7 THE GRASHOF CONDITION

34. 2.7 THE GRASHOF CONDITION • Ground the shortest link and you will get a double-crank, in which both links pivoted to ground make complete evolutions as does the coupler. • Ground the link opposite the shortest and you will get a Grashofdouble-rocker, in which both links pivoted to ground oscillate and only the coupler makes a full revolution.

35. 2.7 THE GRASHOF CONDITION

36. 2.7 THE GRASHOF CONDITION • For the Class II case, S + L > P + Q: • All inversions will be triple-rockers in which no link can fully rotate.

37. 2.7 THE GRASHOF CONDITION

38. 2.7 THE GRASHOF CONDITION • For Class III case, S+L = P+Q • All inversion will be either double-cranks, or crank-rocker

39. 2.7 THE GRASHOF CONDITION • For Class III case, S+L = P+Q • All inversion will be either double-cranks, or crank-rocker

40. 2.7 THE GRASHOF CONDITION • For Class III case, Special Grashof Case

41. 2.7 THE GRASHOF CONDITION

42. 2.8 LINKAGES OF MORE THAN FOUR BARS • Geared Fivebar Linkages

43. 2.8 LINKAGES OF MORE THAN FOUR BARS • Sixbar Linkages

44. Application Problems

45. Example • Statement: • Find the Grashof condition and the Baker classification. Solution: • Grashof Condition S + L < P + Q (12 + 32)<(26+30) 44<56 CRANK-ROCKER • Baker Classification Type 2, L2=s=input Class I-2, Baker’s designation Grashof crank-rocker-rocker, Code GCRR, also known as crank-rocker. L2

46. Example– GRASHOF CONDITION - SOLIDWORKS • Examples of Grashof Criterion for Four-Bar Mechanisms • http://www.me.unlv.edu/~mbt/320/Grashof.html • Examples of links, planar joints in SolidWorks • http://wahyu-tjakraningrat.blogspot.com/2009/02/solidwork-planar-joints.html

47. 2.9 ROTABILIDAD • Mecanismo de cuatro barras

48. 2.9 ROTABILIDAD • Nomenclatura • El eslabón 1, MN, cuya longitud es a1, se conoce como bastidor, marco o eslabón fijo. • El eslabón 2, MA, cuya longitud es a2, se supone el motriz y se conoce como manivela, eslabón de entrada, motriz o conductor. • El eslabón 3, AB, cuya longitud es a3, se conoce como eslabón acoplador. • El eslabón 4, NB, cuya longitud es a4, se conoce como seguidor, eslabón de salida o conducido.

49. 2.9 ROTABILIDAD • Dependiendo de la capacidad de rotar de los eslabones motriz y conducido respecto a su eje de rotación, rotabilidad, los mecanismos de cuatro barras se clasifican en: • Doble oscilatorio - doublerocker - cuando ambos eslabones únicamente pueden oscilar, obviamente, el ángulo de oscilacion es menor a 360◦. • Rotatorio oscilatorio - crankrocker - cuando uno de los eslabones motriz o conducido puede rotar, mientras que el otro solamente puede oscilar. • Doble rotatorio - doublecrank - cuando ambos eslabones pueden rotar.

50. 2.9 ROTABILIDAD – POSICIONES CRITICAS • La rotabilidad de los eslabones de entrada y salida de un mecanismo, esta íntimamente ligada a la aparición de ciertas posiciones conocidas como posiciones criticas. Existen dos diferentes tipos de posiciones criticas: • Posición límite: Una posición límite para el eslabón de salida, en un mecanismo de cuatro barras, ocurre cuando el ángulo interior entre el eslabón acoplador y el de entrada es de 180o o 360o; es decir, las uniones M, A y B están en línea, vea la figura.