September 20, Chapter 5. Simple Mechanisms. Mohammad Suliman Abuhaiba, Ph.D., PE
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1 Chapter 5 Simple Mechanisms 1 Mohammad Suliman Abuhaiba, Ph.D., PE
2 2 Assignment #1 All questions at the end of chapter 1 st Exam: Saturday 29/9/2018
3 3 Kinematic Link or Element kinematic link (link) or element: Each part of a machine, which moves relative to some other part
4 4 Kinematic Link or Element A resistant body: capable of transmitting required forces with negligible deformation A link has two characteristics: 1. It should have relative motion 2. It must be a resistant body
5 5 Types of Links 1. Rigid link: does not undergo any deformation while transmitting motion 2. Flexible link: partly deformed in a manner not to affect the transmission of motion Example: belts, ropes, chains, and wires 3. Fluid link: formed by having a fluid in a container and motion is transmitted through the fluid pressure Example: hydraulic presses, jacks, and brakes
6 6 Structure and a Machine Structure: assembly of a number of resistant bodies having no relative motion among them and meant for carrying loads Examples: A railway bridge, a roof truss, machine frames
7 7 Structure vs. a Machine Machine Parts move relative to one another Transforms available energy into useful work Links may transmit both power & motion Structure Parts do not move relative to one another No energy is transformed into useful work Parts of a structure transmit forces only
8 8 Kinematic Pair A pair: two links of a machine, in contact with each other kinematic pair: the relative motion between the links is completely or successfully constrained
9 9 Types of Constrained Motions
10 10 Types of Constrained Motions Completely constrained motion Motion between a pair is limited to a definite direction irrespective of direction of applied force
11 11 Types of Constrained Motions Incompletely constrained motion Motion between a pair can take place in more than one direction.
12 12 Types of Constrained Motions Successfully constrained motion
13 13 Types of Constrained Motions Successfully constrained motion shaft may rotate in a bearing or it may move upwards: a case of incompletely constrained motion If load is placed on shaft to prevent axial upward movement of shaft, then motion is successfully constrained.
14 14 Classification of Kinematic Pairs According to type of relative motion between elements Sliding pair Turning pair Rolling pair Screw pair Spherical pair According to type of contact between elements Lower pair Higher pair According to type of closure Self closed pair Force - closed pair
15 15 Classification of Kinematic Pairs According to type of relative motion between elements Sliding pair one pair can only slide relative to other completely constrained motion Examples: Piston & cylinder Tail stock on lathe bed
16 16 Classification of Kinematic Pairs According to type of relative motion between elements Turning pair One pair can only revolve about a fixed axis of another link Completely constrained motion Examples: Crankshaft in a journal bearing Lathe spindle supported in head stock
17 17 Classification of Kinematic Pairs According to type of relative motion between elements Rolling pair One pair rolls over another fixed link Example: Ball and roller bearings
18 18 Classification of Kinematic Pairs According to type of relative motion between elements Screw pair One element can turn about the other by screw threads Example: Lead screw of a lathe
19 19 Classification of Kinematic Pairs According to type of relative motion between elements Spherical pair One element with spherical shape swivels about the other fixed element Examples: Ball & socket joint Attachment of a car mirror Pen stand
20 20 Classification of Kinematic Pairs According to type of contact between the elements Lower pair Two elements of pair have surface contact when relative motion takes place Surface of one element slides over surface of other Examples: Sliding pairs, turning pairs and screw pairs
21 21 Classification of Kinematic Pairs According to type of contact between the elements Higher pair Two elements of a pair have a line or point contact when relative motion takes place Motion between the two elements is partly turning and partly sliding Examples: A pair of friction discs Toothed gearing Belt & rope drives Ball & roller bearings Cam & follower
22 22 Classification of Kinematic Pairs According to type of closure Self closed pair Two elements of a pair are connected together mechanically in such a way that only required kind of relative motion occurs. Lower pairs are self closed pair
23 23 Classification of Kinematic Pairs According to type of closure Force - closed pair Two elements of a pair are not connected mechanically but are kept in contact by action of external forces Example: cam & follower, kept in contact by forces exerted by spring & gravity
24 24 Kinematic Chain (KC) A kinematic chain: A combination of kinematic pairs, joined in such a way that each link forms a part of two pairs Relative motion between links or elements is completely or successfully constrained
25 25 Kinematic Chain (KC) Example: Crankshaft of engine forms a kinematic pair with the bearings which are fixed in a pair ConRod with crank forms a 2 nd pair Piston with ConRod forms a 3 rd pair Piston with cylinder forms a 4 th pair Total combination of these links is a KC
26 26 Kinematic Chain (KC) If each link is assumed to form two pairs with two adjacent links, the relation between # of pairs (p) forming a KC & # of links (L) is: L = 2 p 4
27 27 Kinematic Chain (KC) Relation between # of links (L) & # of joints (j) which constitute a KC is given by: j = 1.5 L - 2
28 28 Kinematic Chain (KC) Above Equations are applicable only to KC, in which lower pairs are used These equations may be applied to KC, in which higher pairs are used: each higher pair = two lower pairs + a link
29 29 Kinematic Chain (KC) Examples Arrangement of three links with pin joints It is not a KC: no relative motion is possible
30 30 Kinematic Chain Examples constrained kinematic chain of one DOF
31 31 Kinematic Chain Examples It is not a KC Unconstrained chain
32 32 Types of Joints in a Chain Binary joint Two links joined at same connection 4 links & 4 binary joins Use A.W. Klein equation to determine whether chain is: locked chain (structure) kinematic chain unconstrained chain j = #of binary joints h = # of higher pairs, l = # of links
33 33 Types of Joints in a Chain Ternary joint Three links joined at same connection Example: 6 links 3 binary joints at A, B & D 2 ternary joints at C & E One ternary joint two binary joints Equivalent binary joints = = 7
34 34 Types of Joints in a Chain Quaternary joint Four links joined at same connection Equivalent to three binary joints When L number of links are joined at same connection, joint is equivalent to (L 1) binary joints
35 35 Types of Joints in a Chain Quaternary joint one binary joint at D 4 ternary joints at A, B, E and F 2 quaternary joints at C & G # of binary joints in chain are Chain is not a KC locked chain: structure
36 36 Types of Joints in a Chain Quaternary joint Total number of binary joints = = 13 LHS = RHS, chain is a KC or constrained chain
37 37 Mechanism Mechanism: one of the links of a KC is fixed Used for transmitting or transforming motion Example: engine indicators, typewriter
38 38 Mechanism Simple mechanism: 4 links Compound mechanism: > 4 links Machine: When a mechanism is required to transmit power or to do some particular type of work
39 39 Number of DOF for Plane Mechanisms DOF: number of input parameters which must be independently controlled in order to bring the mechanism into a useful engineering purpose
40 40 Number of DOF for Plane Mechanisms One variable to define relative positions of all links # of DOF = 1
41 41 Number of DOF for Plane Mechanisms Two variables q 1 & q 2 to completely define relative positions of all links # of DOF = 2
42 42 Number of DOF for Plane Mechanisms Two links AB & CD in a plane motion Each link has 3 DOF before it is connected to any other link when CD is connected to AB by a turning pair at A, position of link CD is now determined by a single variable & thus has one DOF
43 43 Number of DOF for Plane Mechanisms When a link is connected to a fixed link by a turning pair (lower pair), two DOF are destroyed
44 44 Number of DOF for Plane Mechanisms In a mechanism, one of the links is to be fixed, therefore: number of movable links = (L 1) total number of DOF = 3 (L 1) before they are connected to any other link
45 45 Number of DOF for Plane Mechanisms # of DOF of a mechanism with L number of links connected by j number of binary joints or lower pairs (single DOF pairs) and h number of higher pairs (two DOF pairs) is given by: Kutzbach criterion for movability of a mechanism having plane motion
46 46 Application of Kutzbach Criterion to Plane Mechanisms a structure and no relative motion between the links is possible redundant constraints in the chain and it forms a statically indeterminate structure
47 47 Application of Kutzbach Criterion to Plane Mechanisms
48 48 Application of Kutzbach Criterion to Plane Mechanisms 3 links, 2 binary joints and one higher pair L = 3, j = 2 & h = 1
49 49 Application of Kutzbach Criterion to Plane Mechanisms 4 links, 3 binary joints & one higher pair L = 4, j = 3 & h = 1
50 50 Grubler s Criterion (GC) for Plane Mechanisms GC applies to mechanisms with only single DOF joints Substituting n = 1 and h = 0 in Kutzbach equation, 1 = 3 (L 1) 2 j simplifies to 3L 2j 4 = 0
51 51 Grubler s Criterion (GC) for Plane Mechanisms A plane mechanism with a movability of 1 & only single DOF joints can not have odd number of links. Simplest possible mechanisms: four bar slider-crank
52 52 Types of Kinematic Chains Three types of kinematic chains with four lower pairs are considered: 1. Four bar 2. Single slider crank 3. Double slider crank
53 53 Four Bar Chain Four links, turning pairs at A, B, C & D Grashof s law for a four bar mechanism: For a continuous relative motion between links 2 & 4, S shortest & longest link lengths S remaining two link lengths
54 54 Four Bar Chain Crank or driver: will make a complete revolution relative to the other three links, if it satisfies Grashof s law Follower: Link BC (2) ConRod: Link CD (3) Frame of mechanism: link 1
55 55 Single Slider Crank Chain Modification of basic four bar One sliding pair and three turning pairs Converts rotary motion into reciprocating motion & vice versa
56 56 Inversion of Mechanism Inversion of mechanism: method of obtaining different mechanisms by fixing different links in a kinematic chain Relative motions between various links is not changed, but their absolute motions may be changed drastically
57 57 Inversions of Single Slider Crank Chain Hand Pump Inversion #4
58 58 Inversions of Single Slider Crank Chain Crank and slotted lever quick return motion mechanism - Inversion #3 Used in shaping machines, slotting machines and in rotary internal combustion engines.
59 59 Inversions of Single Slider Crank Chain Crank & slotted lever quick return motion mechanism - Inversion #3 Forward stroke: position CB 1 to CB 2 (b) cw Return stroke: position CB 2 to CB 1 (a) cw Crank has uniform angular speed,
60 60 Inversions of Single Slider Crank Chain Whitworth quick return motion mechanism Inversion #2 Used in shaping & slotting machines
61 61 Inversions of Single Slider Crank Chain Whitworth quick return motion mechanism Time of cutting stroke = time to drive crank from CA 1 to CA 2 Time of return stroke = time to drive crank from CA 2 to CA 1 Crank link CA rotates at uniform angular velocity
62 62 Example 5.3 The driving crank BC is 30 mm long & time ratio of the working stroke to the return stroke is to be 1.7. If the length of the working stroke of R is 120 mm, determine the dimensions of AC & AP.
63 63 Example 5.3
64 Example In a Whitworth quick return motion mechanism, distance between the fixed centers is 50 mm & length of driving crank is 75 mm. The length of slotted lever is 150 mm and length of ConRod is 135 mm. Find the ratio of time of cutting stroke to time of return stroke and also the effective stroke.
65 65 Example 5.4
66 66 Double Slider Crank Two turning pairs & two sliding pairs
67 67 Inversions of Double Slider Crank Chain Elliptical trammels When links 1 & 3 slide along their respective grooves, any point on link 2 such as P traces out an ellipse on the surface of link 4. AP & BP are semi-major axis & minor axis of the ellipse.
68 68 Inversions of Double Slider Crank Chain Elliptical trammels Coordinates of point P on link BA will be
69 69 Inversions of Double Slider Crank Chain Scotch yoke mechanism Convert rotary motion into a reciprocating motion. Inversion is obtained by fixing either link 1 or link 3 When Link 2 (crank) rotates, link 4 (frame) reciprocates Fixed link 1 guides the frame
70 70 Inversions of Double Slider Crank Chain Oldham s coupling Used for connecting two parallel shafts whose axes are at a small distance apart
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