MENG 371 Chapter 3 Notes Graphical Linkage Synthesis. 3.3 Limiting Conditions (Toggle) 3.3 Limiting Conditions

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1 MENG 371 Chapter 3 Notes Graphical Linkage Synthesis Dr. Keith Hekman The American University in Cairo Mechanical Engineering Department All figures taken from Design of Machinery, 3 rd ed. Robert Norton Function, Path, and Motion Generation Function Generation correlation of an input motion with an output monition a mechanism Path Generation control of a point in the plane such that it follows some prescribed path Motion Generation the control of a line in the plane such that it assumes some prescribed set of sequential positions Planar vs. Spatial Mechanisms many spatial mechanisms duplicate planar mechanisms Limiting Conditions (Toggle) 3.3 Limiting Conditions Toggle a point where the link cannot rotate anymore. Determined by the colinearity of two moving links. Need to check when making a design (either by making a cardboard model or Transmission angle (µ) the absolute value of the acute angle of the pair of angles at the intersection of the two links. Want the force in link 3 to rotate link 4 Optimum value of 90 degrees Try to keep the minimum value above 40 degrees working model. 3 4

2 3.4 Dimensional Synthesis Dimensional Synthesis the determination of the proportions (lengths) of the links necessary to accomplish the desired motions. Types of synthesis -Rocker output (pure rotation) (function generation) and coupler output (complex motion) (motion generation) Rocker Output -Two Positions with Angular Displacement 5 6 Rocker Output Rocker Output Draw O 4 B in two extreme positions Draw chord B 1 B 2 in either direction Select point O 2 Bisect B 1 B 2 and draw circle of that radius at O 2 Crank-O 2 A, Coupler AB, Rocker O 4 B, Ground O 2 O 4 7 8

3 Rocker Output Rocker Output Two positions with Complex Displacement. Want to move from C 1 D 1 to C 2 D 2 Construct perpendicular bisectors C 1 C 2 and D 1 D 2 Intersection of the bisectors is the rotopole (the ground location) The output link is shown in its two positions 9 10 Rocker Output Two positions with Complex Displacement. You can add a dyad by picking point B on the output link 11 Coupler Output Two Positions with Complex Displacement. Want to move from C 1 D 1 to C 2 D 2 Construct bisectors of C 1 C 2 and D 1 D 2. Any point of bisector of C 1 C 2 can be O 2 and any point on bisector of D 1 D 2 can be O 4 Links are O 2 C 1, C 1 D 1, D 1 O 4, and ground O 2 O 4 Pick Pick 12

4 Driving a non-grashof linkage with a dyad The dyad does not have to be along the O 2 C 1 line. Three Position Motion Synthesis Want the coupler to go from C 1 D 1 to C 2 D 2 to C 3 D 3 This allows a choice of many places for O 6 C 1 D 1 D 2 C 2 D 3 B 1 B 1 C Three Position Motion Synthesis Construct bisector of C 1 C 2 and C 2 C 3. Where they intersect is O 2. Construct bisector of D 1 D 2 and D 2 D 3. Where they intersect is O 4. Links are O 2 C 1, C 1 D 1, and D 1 O 4, and ground is O 2 O 4 15 Three position synthesis with alternate attachment points The given points do not have to be used as the attachment points Draw points E and F relative to C and D at each position Solve to move from E 1 F 1 to E 2 F 2 to E 3 F 3 Can add a driver dyad D 1 C 1 C 2 D D 3 2 C 3 16

5 Three position motion with specified Three position motion with specified Inversion Problem. Move the ground while holding the link fixed Transfer the relative position of C 2 D 2 O 2 O 4 to C 1 D 1 O 2 O 4 O 2 O Three position motion with specified Transfer the relative position of C 3 D 3 O 2 O 4 to C 1 D 1 O 2 O 4 Three position motion with specified Now we have the three ground positions relative to the first link Label them E 1 F 1, E 2 F 2, E 3 F 3. O 2 O 4 E 3 F 2 E 2 O 4 O 2 O 2 O 2 E 1 O 4 F 3 O 4 F

6 Three position motion with specified Solve the problem assuming you want to move E 1 F 1 to E 2 F 2 to E 3 F 3 finding ground positions G and H Three position motion with specified The completed fourbar linkage which moves E 1 F 1 to E 2 F 2 to E 3 F 3 G and H become the attachment points for the original linkage Three position motion with specified The completed linkage Quick Return Fourbar Mechanism Quick return goes quicker in one direction (α) then the other (β) Time Ratio T R =α/β α+β=360 β=360/(1+t R ) Max TR of 1:1.5 β α 23 24

7 Quick Return Fourbar Mechanism Sixbar Quick-Return Draw initial angle θ 4 Extend line through B 1 crossing B 2 O 4 Draw line through B 2 at angle δ δ= β 180 = 180 α δ Larger time ratios of 1:2 can be obtained Based on a Grashof fourbar crank-crank mechanism Intersection is O 2 Extend arc from B 1 to find twice driver length Return is α, going is β β α Sixbar Quick-Return Draw line of centers X-X at convenient location Generate line Y-Y at convenient location Draw circle of radius O 2 A at O 2 Draw α symmetric about quadrant 1 Find points A 1 and A 2 (90 α)/2 A 1 A 2 α Sixbar Quick-Return Pick radius for coupler CA such that it will cross X-X twice. Find C 1 and C 2 Bisect C 1 C 2 to find O 4 Points B 1 and B 2 are the same distance apart as C 1 and C 2 Draw a line at an angle (180-γ)/2 from B 1 and B 2 to find O 6 (180-γ)/2 O 6 B 1 B 2 C 1 A 1 O 4 A 2 α C

8 Same base fourbar linkage (O 2 ACO 4 ) can be used for a slider output Sixbar Quick-Return Crank Shaper Quick Return Can be used for larger time ratios Has disadvantage of a slider joint Crank Shaper Quick Return Locate ground on vertical line. Draw a line at angle α/2. Pick length for link 2. Draw line to first at slider. Where this line intersects vertical line is the ground Length of output motion can be chosen by moving attachment point up or down same length α/2 Path of a point on the coupler Closed path, even for non- Grashof linkages Coupler Curves Capable of generating approximate straight lines and circular arcs

9 Coupler Curves Categorized by shape Cusp instantaneous zero velocity Crunode multiple loop point Coupler Curves Hrones and Nelson has atlas of coupler curves Each dash represents 5 degrees of rotation Coupler Curves (Examples) Film advance mechanism in camera is used to pause between frames Suspension is used to make the point of tire contact move vertically 3.7 Cognates Roberts-Chebyschev theorem Three different planar, pin-jointed fourbar linkages will trace identical coupler curves 35 36

10 Cognate Construction Start with linkage with desired coupler curve Align links 2 and 4 with AB Create similar triangles to obtain other links geometry. Note the parallel links of equal length Cognate Construction Cont. Restore links 2 and 4 to the ground position. O C will move to its new position. Can also create from parallel lines and similar triangles from this position All cognates can move together Final Cognates Together Cognates of a Straight Line Linkage Similar cognates can be found, even if the coupler is not triangular 39 40

11 Cognate creation For the 1 st cognate, point P is between A and B For the 2 nd cognate, point B is between A and P For the 3 rd cognate, point B is between P and A Parallel Motion Cognates can be used to create parallel motion. Watt s sixbar based on two eightbar cognates O C is between O A and O B B 1 A 1 B 2 B3 A 2 A Straight-Line Mechanisms Straight-Line Mechanisms A common application of coupler curves is in the generation of straight lines 43 44

12 Optimum Straight- Line Linkages Optimum Straight-Line Linkages Based on table Based on Hoekens straight-line linkage Optimized for straightness or constant velocity Basic geometry given in figure. Calculated from a table. 45 β (deg) Range of Motion θ start (deg) % of cycle 5.6% 11.1% 16.7% 22.2% 27.8% 33.3% 38.9% 44.4% 50.0% Optimized for Straightness Maximum c y % % % % 0.001% 0.004% 0.010% 0.023% 0.047% 0.096% V % 0.38% 1.53% 3.48% 6.27% 9.90% 14.68% 20.48% 27.15% 35.31% _V x _ L 2 ω Link Ratios L 1 /L 2 L 3 /L 2 x/l Optimum Straight-Line Linkages Based on table β (deg) Range of Motion θ start (deg) % of cycle Optimized for Constant Velocity Maximum c y % V % _V x _ L 2 ω 2 L 1 /L 2 Link Ratios L 3 /L 2 x/l 2 Single-Dwell Linkages Find a coupler curve with a circular arc Add a dyad with one extreme position at the center of the arc % 11.1% 16.7% 22.2% 27.8% 33.3% 38.9% 44.4% 50.0% 0.006% 0.038% 0.106% 0.340% 0.910% 1.885% 3.327% 5.878% 9.299% 0.137% 0.274% 0.387% 0.503% 0.640% 0.752% 0.888% 1.067% 1.446%

13 Double Dwell Sixbar Linkage Find a coupler curve with two straight line segments Use a slider pivoted at the intersection of the straight lines 3.10 Other Useful Linkages Washing machine agitator. See book for other examples 49 50