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Design of Planer Devices

Design of Planer Devices. Dr. R. Lindeke, PhD ME 3230 – Kinematics and Mechatronics. Some Random Design Thoughts:. Amp  re defined Kinematics as “the study of motion of mechanisms AND methods to create them [that is the mechanisms]

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Design of Planer Devices

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  1. Design of Planer Devices Dr. R. Lindeke, PhD ME 3230 – Kinematics and Mechatronics ME 3230

  2. Some Random Design Thoughts: • Ampre defined Kinematics as “the study of motion of mechanisms AND methods to create them [that is the mechanisms] • We are about to begin the study of the means and method to design (create) them! ME 3230

  3. Some Design Thoughts: • Binary Link: 1 length • Ternary Link: 3 lengths, 2 lengths & 1 angle; 2 Angles & 1 length • Bell Crank (same as ternary link) • Cam & Follower: CL distance, Follower Length and radial length, Ref. direction with an infinite number of cam radii relative to it • Gear Offset and tooth ratio • Eccentric “Cam” a single eccentric offset – binary! ME 3230

  4. Some Design Thoughts: • Starting Configurations: • (a) crank starting position • (b) Slider Starting position • 4-bar linkage start requires 2 crank angles – or – with only one we solve to either (c) or (d)! ME 3230

  5. Focus is on 4-Bar Mechanisms • They are extremely popular solutions for generating irregular motions! • As a designer we typically try to make the motion achieve a certain number (2, 3 or perhaps 4 or 5) “Precision Points” along a desired path • And then hope that all other positions are close to what we seek! • A second approach is to select a large number of points and then minimize the sum of “Squared Error” in actual and desired paths positions • Using the design methods we are looking for the link lengths ME 3230

  6. Design Goals: • Motion Generation – coupler as a whole follows a desired trajectory • Function Generation (like sine, straight line, or log based motion!) for points along the coupler path • Rocker Amplitude – oscillation control systems • Point path for a single location on the coupler, typically to interact with a separate device or mechanism ME 3230

  7. 2-Position Design – for a Double Rocker • Start with a known base length (O2 to O4) and the length of the Output Link (O4B) • We desire that the mechanisms 2 rockers be in precisely two location (at some time) – they are separated by the angle  with respect to link 2 &  wrt Link 4 • The problem can be solved Graphically – we make use of an inversion approach (sitting on Link 2) see figure (b) bottom  ME 3230

  8. 2-Position Design – for a Double Rocker Givens: Each Rocker Angle, Base Length & Output Crank Length • To solve, lay out 2 points (O2—O4) • From O2 strike two line separated by  • From O4 strike Output Link (O4B) at two positions separated by  • Then strike a line from O2 to B2 about O2 by - (the inversion!) • Connect B1 and B2’ • Form a perpendicular bisector to this line • Where it strikes O2 – ‘A1’ line locates A1 • Sets Link 2 Length anda line from A1 to B1 establishes Coupler length ME 3230

  9. Also (Simply) by Analytical Means:  Of course r2 is an unknown when we start! ME 3230

  10. After Manipulation: ME 3230

  11. Dealing with the 2-Position Coupler Problem: • Here we see 3 positions of a coupler as it “flows along” a motion path • Each point A, B, C traces a unique trace • With only 3 positions it is possible to find a single center will allow the motion to be accomplished ME 3230

  12. Dealing with the 2-Position Coupler Problem: • Using a 4 bar mechanism: • Construct perpendicular bisectors for each (or any 2) pairs of points on the coupler • Crank Pivots can be placed anywhere along them – one along A’s perpendicular Bisector and one along B’s (C’s) perpendicular Bisector • Notice, If the Perpendicular Bisectors are extended far enough they will intersect at the single pivot position – the Displacement pole P12 (we could build a 3-bar solution!) ME 3230

  13. Moving to the 3-Position Problem: • We consider Lamella AB or other points (or lines) as the coupler (it is a moving plane) • End points (or any point) on the chosen line-segment are called Moving Pivots • A “Fixed Pivot” is located at the center of the circle on which all three equivalent points on the moving coupler line lie – A1, A2, A3 which are the same moving center at the 3 locations • We find these points at the intersection of the Perpendicular Bisectors of each moving pivot taken in pairs • The Crank(s) are drawn for the Fixed Pivot to the 1st pose of the moving Pivot ME 3230

  14. Leads to this design “A Crank”: Note: Shows the locating of Fixed Pivot A (using 3-pt circle and P.B. Pairs) ME 3230

  15. And 4-Bar after Adding “B Crank” We should activate the design to check for motion clearance. See if all positions can be reach without reconstructing the Cranks! ME 3230

  16. After Design: • Check Grashof Criteria for type • s =1.131 (Base) (Check actual values) • l = 3.708 (Crank 1) • P = 1.75 (coupler) • Q = 2.785 (Crank 2) • This is a Type II device not capable of full rotation (p+q<s+l) • This might not be the solution we seek – but it could solve my “Box-Up” problem where the input is an intermittent lift-lower function! • Perhaps we need to explore a design from a Fixed Pivot location? ME 3230

  17. Doing The Invert Think – another way to design and check Cranks: • Working with nominated Fixed Pivots (rather than moving pivots as before) • Method suggests that the Coupler (moving Lamina) become the reference frame • Again it relies on “Apparent” rather than “true” positions that lie at vertices of Congruent triangles with the desired lamina positions for the fixed pivot of design interest. • Once the series of apparent “fixed” pivot positions are found as above, the position of the moving pivot on the 1st coupler position is set at the center of the circle that include all of the apparent “fixed” pivot locations • For the second crank we need to perform a similar relationship with a different nominated fixed Pivot ME 3230

  18. Starts Here: We will prepare Apparent Positions C*2 and C*3 by making “Congruent Triangles” upon A1-B1 working with triangles A2B2C* and A3B3C* The true position of the crank starting out is C* to C1 where C1 is the center of the circle containing C* -- C2* -- C3* developed in the earlier steps ME 3230

  19. Getting C2* -- used intersecting Arcs equal in length to A2-C* and B2-C* ME 3230

  20. Now After the Position of C1 determined: The Crank is C*-C1 Found by making the 2 congruent triangles and then drawing a 3-point circle thru C*, C2* & C3* -- true position C1 is the center of this circle ME 3230

  21. Similarly if a 2nd Fixed Pivot is desired! • Make Congruent triangles to find apparent positions of the Fixed Pivot • True Position of first instance of the moving pivot is the center of circle containing D* - D2* - D3* • Then we would check for Grashof type and ultimately ability to solve our problem. ME 3230

  22. Slider Crank Design: 3 Coupler Positions • Note the Coupler Positions • The “Circle of Sliders” is formed by finding poles of positional pairs 1-2, 1-3 & 2-3 and “Image Pole” P23’ – note this image pole is a reflection of the “True” pole P23 about a line connecting poles 1-2 and 1-3 • We need to define a fixed pivot (A* or B*) and connect the crankfrom the “*” fixed pivot and appropriate end of the coupler (the one used to find fixed pivot) ME 3230

  23. Generating P12: Similar work for other two poles ME 3230

  24. After Generating all three poles: ME 3230

  25. With Image Pole P23’ ME 3230

  26. After Adding the Slider Circle & Slider line The Triangle of Coupler position to points on the slider line are all congruent – here the lowest vertex is “C” marking the Slider Line – the Circle is 3-pt including I12, I13 & I23’ ME 3230

  27. A Solution – But Will it work? • We must determine if the linkage can be assembled and travel without being disassembled and reassembled in the other “Solution Branch” • For 4-bar linkages, we can define a angle  -- an angle between the coupler and the Driven (longer) crank in all positions • If the angle changes “sign” or direction, we are looking at an impractical solution – one that can’t be operated ME 3230

  28. Change of Branch? It looks like there is no change of Branch! Afix – Ai are all CCW; ME 3230

  29. What about a Slider Crank? • We need to check order of points • The individual “Ci’s” should be traversed in the positional order – else the design fails as sketched (like the one seen earlier as it turns out!) ME 3230

  30. Approaching the Solution – Analytically! • Modeling is based on “Coordinate Transformations” – we will consider them is Homogeneous ones – both ‘ends’ of vector space share SO geometry (orthogonal spaces is planes or 3-space) • We then speak of rotational transformations (R’s) as matrices and translational vectors relating origins of coor. systems (see above) • Relating Coupler (AP) to Frame (O) is seen by vectors: ME 3230

  31. Continuing with this Transform Idea: • Coordinate Transformation from Coupler to Frame is: ME 3230

  32. Doing the Inverse: • Transforming Frame to Coupler: ME 3230

  33. What We do with these models: • Typically we would know positions of the coupler in ‘Frame Space’ (ax, ay) • We can compute the Angle: • Circle Points are given wrt Coupler coordinates – these must be converted to Frame space using the [A] based model • Center points are given in Frame Coordinates. To use CP’s to generate circles they must be converted to coupler space using the [R] based models • Crank lengths can be determined when both circle and center points are defined in a common coordinate system ME 3230

  34. Analytical Solutions: • Finding poles (Pij) – where a pole is a point from which two lines can be repositioned by a simple rotation – see EQN 6.16 it contains the solution • point locations assumed known in Frame space • We must be aware of matrix singularity issues • Find Circle Centers on which 3 points lie • Treat Bi in Eqn 6.16 as Aj (the middle point of the 3) ME 3230

  35. Treating the Design Problem using Analytical Models • Crank Design using “Circle Points” – moving points on coupler • Requires circle point coordinates be Transformed to Frame Reference From (X,Y) to (x,y) • Find “Poles of the 3 positions” to define Fixed Pivot (x*, y*) • Crank is (x*, y*) to (x1, y1) from transformed circle point 1 • Locate 2nd Crank similarly and build the linkage ME 3230

  36. Treating the Design Problem using Analytical Models • Given “Fixed Pivot” or Center Point (coor. in Frame space) • Transform coordinates Of (x*, y*) To Coupler Space (for each coupler position) using A matrices • Find Coupler Pole of the 3 “*’ed” transformed position – it becomes (X,Y), now transform this pole to Frame Coordinate For positioni (chosen one! And use its R matrix) • Crank in posi is located by (x*, y*) to (xi, yi) ME 3230

  37. Treating the Design Problem using Analytical Models • Design of Slider: • Follow the steps on page 282 – 283 • We use all the tools: • Transforming coor. Between frame and coupler • Finding poles • Finding image poles • Finding Circles (center and radius) • Defining slider line points and ultimately the slider line – parametrically • Determining if slider order is correct ME 3230

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