ME317

1. ME317 Chapter 2 Kinematics Fundamentals

2. Definition • A mechanical system’s mobility (M) is classified according to the number of degrees of freedom (DOF) that it has. • The system’s DOF is equal to the number of independent parameters necessary to uniquely define its position in space at any given time. • It is defined relative to a selected frame of reference

3. DOF Example • A system of the pencil in a plane has three degrees of freedom to define its position at any instant of time. • Three parameters are required to define the position of the pencil.

4. DOF Example • A system of the pencil in three dimensional space has six degrees of freedom. • One possible set of parameters • Three length (x, y, z) and three angles (θ, φ, ρ)

5. Types of Plane Motion • Pure translation motion • All points on the body move along parallel straight or curved lines (rectilinear or curvilinear) • Pure rotational motion • The body posses one point that has no motion, all other points on the body move along circular paths relative to this point. • General (complex) motion • A combination of translation and rotation

6. Links, Joints and Kinematics Chains • A link is a body (rigid) with at least two nodes used for attachment to other links.

7. Links, Joints and Kinematics Chains • A joint or pair is a connection between two or more links at their nods which allows some motion, or potential motion, between the connected links. • Joints are classified by the: • type of contact between the elements, line, point, or surface. • Lower pair • Higher pair • number of DOF allowed at the joint. • type of physical closure of the joint: either force or form closed. • number of links joined (order of the joint).

8. Lower Pair Joints • Lower pair describes joints with surface contacts • Examples:, • A door joint to a frame with hinges (revolute (R) pair) • A sash window (prismatic (P) pair) (a pin surrounded by a hole) • If there is a clearance between a pin and a hole (to allow for motion), the so-called surface contact becomes line contact.

9. Higher Pair Joints • Higher pair (describes joints with point or line contacts) • Examples: • Gear teeth, cam and follower, two curved surfaces • At a macroscopic level, a block sliding on a surface level, has contacts at discrete points, the top of the surface asperities’ or surface roughness

10. Advantage of Lower Pairs Joints • The lubricant traps between their enveloping surfaces. • Results in low wear and long life

11. Lower Pair Joints The revolute (R) and Prismatic (P) pairs are the only lower pair joint used in planer mechanisms. The screw (H), Cylindric (C), Spherical (S), and Flat (F) lower pairs are combinations of revolute/prismatic pairs and are used in 3D mechanisms.

12. Examples of Planar Lower Pair Joints The pin joint allows one rotational degrees of freedom between the joint links. The slider joint allows one translational degrees of freedom between the joint links.

13. Examples of Planar Lower Pair Joints The pin joint is a special case of screw and nut joint. If the helix angle is zero, the nut rotates without advancing. The slider joint is a special case of screw and nut joint. If the helix angle is 90 degrees, the nut translates along the axis of screw.

14. The Six Lower Pairs Joints The revolute (R) and prismatic (P) are also called full joints (i.e. full=1 DOF)

15. Examples of Higher Pair (2-freedom) Joints Two-degrees of freedom joints (higher pairs) allows two simultaneous independent relative motions, translation and rotation, between the joined links. This 2 DOF joint is also referred to half joint ( with its 2 freedom placed in the denominator) or roll-slide joint.

16. Example of Higher Pair (3-freedom) Joints Spherical, or ball-and socket joint, allows three independent angular motions between the two links joined. Examples include, joystick or ball joint in automobile suspension system.

17. Higher Pair Joints Depending on friction the wheel, may roll (planar pure-roll), may slide (planer pure slide) or roll-slide.

18. Higher Pair Joints A force-closed joint requires some external force to keep it together. (a slider on a surface) A form-closed joint is kept together or closed by its geometry. (a pin in a hole or a slider in a two sided slide)

19. Joint Order Note: it takes two links to make a single joint Joint order = number of links joint -1

20. Definitions • A kinematic chain: • An assemblage of links and joints, interconnected in a way to provide a controlled output motion in response to a supplied input motion • A mechanism: • A kinematic chain in which at least one link has been “grounded”, or attached to the frame of reference which may be in motion. • A machine: • A combination of resistant bodies arranged to compel the mechanical forces of nature to do work accompanied by determinate motions. • A collection of mechanisms arranged to transmit forces and do work.

21. Definitions • A crank: • A link that makes a complete revolution and is pivoted to ground. http://www.flying-pig.co.uk/mechanisms/pages/crank.html • A rocker: • A link that has oscillatory (back and forth) rotation and is pivoted to ground. • A coupler or connecting rod: • A link that has complex motion and is not pivoted to ground. • Ground: • Any link or links that are fixed with respect to the reference frame.

22. Drawing Kinematic Diagrams • Draw a schematic kinematic diagram • A kinematic link, or link edge is a line between joints that allows relative motion between adjacent links. • Possible joint motion must be clear from the kinematic diagram. • The diagram must clearly indicate which joints or links are grounded and which can move.

23. Schematic Notation For Kinematic Diagrams

24. DOF or Mobility • Degrees of Freedom Definition • The number of inputs that need to be provided in order to be able to create a predictable output for a system.

25. DOF or Mobility DOF definition: The number of independent coordinates required to define the systems’ position. In this example, the position of both links can be described using only four parameters  DOF = 4.

26. DOF or Mobility • A system constrained by a revolute pair has one degrees of freedom

27. DOF or Mobility • A system constrained by a Prismatic pair has one degrees of freedom

28. DOF or Mobility • A system constrained by a higher pair has two degrees of freedom 1: Translating along the curved surface 2: Rotating about the instantaneous contact point

29. Kinematic Chains or Mechanism • Open • An open mechanism of more than one link will always have more than one DOF. • Requires as many actuators (motors) as DOF. • Example: Robot • Closed • No open attachment points or nodes • May have one or more DOF.

30. Kinematic Chains or Mechanism • An open kinematic chain of two binary links and one joint is called dyad.

31. Effect of Joints on DOF (Mobility • Joints remove DOF • Any link in a plane has 3 DOF. • A system of L unconnected links in the same plane has 3L DOF.

32. DOF (Mobility) in Planar Mechanism • Gruebler condition: • Any link in a plane has 3 DOF. • A system of L unconnected links in the same plane has 3L DOF.

33. DOF (M) in Planar Mechanism • When connecting links are by a full joint, Δy1 and Δy2 are combined as Δyand Δx1 and Δx2 are combined as Δx. • This reduces DOF by 2, resulting in 4 DOF.

34. DOF (M) in Planar Mechanism • When connecting links by a half joint, Δy1 and Δy2 are combined as Δy. • This reduces DOF by 1, resulting in 5 DOF.

35. DOF (M) in Planar Mechanism Gruebler condition: • Any link in a plane has 3 DOF. • A system of L unconnected links in the same plane has 3L DOF. • When links are connected by a full joint, its DOF is reduced by two. • When system of links are connected by a half joint, its DOF is reduced by one. • When any link is grounded or attached to a reference frame, all three of its DOF will be removed. Gruebler’s equation: M=3L -2J-3G (2.1a) M= DOF or mobility L=number of links J= number of joints G= number of grounded links

36. DOF (M) in Planar Mechanism • In real mechanism, because there can be only one ground plane , even if more than one link of the kinematic chain is grounded, the net effect is to create one larger, higher order ground link. • Thus G is always equal to one. Gruebler’s equation: M=3L -2J-3G (2.1a) M=3(L -1) -2J (2.1b) Derivation: Each of the L links have 3 DOF except for the fixed link which has none M=3L – 3=3(L – 1). Each of the J single DOF pairs remove 2 DOF:  M=3(L-1) – 2J.

37. DOF (M) in Planar Mechanism • Since half joints only remove one DOF, the modified Gruebler’s equation, called Kutzback’s equation is easier to work with. Kutzback’s equation: M=3(L -1) -2J1-J2 (2.1c) M= DOF or mobility L=number of links J1= number of one DOF (full) joints J2= number of two DOF (half) joints Note: Multiple joints count as one less than the number of links joined at that joint.

38. Examples of Finding DOF (M) in Planar Mechanism L=8 J1= 10 J2= 0 Kutzback’s equation: M=3(L -1) -2J1-J2 M=3(8-1) -2(10) -0 =1

39. Examples of Finding DOF (M) in Planar Mechanism L=6 J1= 7 J2= 1 Kutzback’s equation: M=3(L -1) -2J1-J2 M=3(6-1) -2(7) -1 =0

40. Mechanisms and Structures • DOF of an assembly completely defines the character of an assembly of links. • Positive DOF  assembly is a mechanism  links will have relative motion • Zero DOF  assembly is a structure  no motion is possible • Negative DOF  assembly is a preloaded structure  no motion is possible ad some stress may be present at the assembly time

41. Examples of Finding DOF (M) in Planar Mechanism Kutzback’s equation: M=3(L -1) -2J1-J2 M=3(3-1) -2(2) -1=1 Kutzback’s equation: M=3(L -1) -2J1-J2 M=3(3-1) -2(3) -0=0

42. Grubler’s Formula Summary • Steps to calculate mobility: • Count number of elements/links: n • Count number of single DOF pairs: J1 • Count number of two DOF pairs: J2 M = 3(L – 1) – 2 J1 – J2 Caution • Must count ground link since its effect is included in the formula (the L – 1 term) • Any joint at which k links meet must count as k–1 joints. • Cases of slip and no-slip must be considered for rolling pairs.

43. Grubler’s Formula Summary • Grubler’s formula is a theoretical result formula does not take into account the geometry (size and shape) of the mechanism, therefore, it can give misleading results. • The actual mobility of a mechanism can only be calculated by inspection • Grubler’s formula is applicable to a wide variety of mechanisms commonly encountered in engineering applications.

44. Number Synthesis • Number synthesis means determination of the number and order of the links and joints necessary to produce motion of a particular DOF. • Here order refers to the number of nodes per link, i.e. binary, ternary, etc. • It gives the designer all possible link combinations to solve a motion control problem.

45. Number Synthesis • Hypothesis: • If all joints are full joints, an odd number of DOF requires an even number of link and vice versa. • Proof: See lecture note

46. Paradoxes A B Grubler’s equation: M=3(L -1) -2J1-J2 M=3(5-1) -2(6) =0

47. Paradoxes C Grubler’s equation: M=3(L -1) -2J1-J2 M=3(3-1) -2(3) =0 This linkage does move, DOF=1

48. Isomers • Isomer means having equal parts • In Chemistry:

49. Isomers • Linkage isomers are obtained by connecting the links together using various joints. • The assembly will have different motion properties.

50. Isomers Invalid isomer