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Linkage Design: Function Generation, Rocker Amplitude and Path Synthesis

Linkage Design: Function Generation, Rocker Amplitude and Path Synthesis. Chapter 6, Continuing Dr. R. Lindeke, ME 3230. 4-Bar Linkage Function Generation.

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Linkage Design: Function Generation, Rocker Amplitude and Path Synthesis

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  1. Linkage Design: Function Generation, Rocker Amplitude and Path Synthesis Chapter 6, Continuing Dr. R. Lindeke, ME 3230

  2. 4-Bar Linkage Function Generation • The goal is to design a linkage who’s coupler or cranks follow the motion defined as a function between an input variable and a desired output result • A typical solution method is to find “Precision Points” – these are specific solutions of the functional equation controlling what we desire: • f(, , a1, a2, a3, a4) as an input relationship where ai’s are design variables defining the linkage • and ,  are used to solve the output function g(, ) to give some number of “precision points” [if we have 4 ai’s then it is 4 (, ) precision points!] ME 3230

  3. Lets try Pr. 6.23: • Desired Input-Output relationship is: •  = 2a1 + a2Sin • Function we wish to Generate: y = 2x2 (angles in radians) • Here  corresponds to ‘y’ and  corresponds to ‘x’ • Solution to be held over the range: 0<  < /2 • So, with only 2 unknowns (a1 & a2) we need only 2 precision points (x1[1], y1[1]) and (x2[2], y2[2]) • We could guess that they are at 1/3 and 2/3 of the  range (0.5235 and 1.0472 radians) but let’s use a precise way to determine them by using the Chebyshev Polynomial -- a tool to reduce approximation error (N = # of precision Points, i = 1 to N) : ME 3230

  4. Chebyshev Polynomial continued: • Here N = 2 thus we will find x1 [1] & x2 [2] ME 3230

  5. Finding x’s [’s] and y’s [’s]: ME 3230

  6. Solving for a1 & a2 ME 3230

  7. Determining model error at  = 45 ME 3230

  8. The actual Goal is Linkage Design • For 4 Bar design we will use Freudenstein’s equation • We start with 3 values for  and  related to the desired function • Then we will develop “X-, Y- Component Models” for the links (with a unit base) ME 3230

  9. Active Models: ME 3230

  10. Our Text (rightly) suggests a change of variable as: This last equation can be “easily” solved for the Zi’s and then ri’s using CoeffINV and Mathematica or Matlab ME 3230

  11. Solving for a Linkage and Pr 6.24 (generating function:  = 2 between.5 and 1 radian) • Get 3 Precision Points using Chebyshev’s Polynomial: NOTE: we never use end points as precision points but the middle is used if an odd number of precision points is needed ME 3230

  12. Resorting to Mathematica: ME 3230

  13. Design Parameters: • r2 = 1/z2 = 1/-.00188 = -531.91 • r4 = 1/z3 = 1/.1011 = 9.89 • r3 = (1+r2^2 + r4^2 - 2r2r4z1) = 521.44 • Scaling back to Base size of 2 (not 1!) • r2 = 1063.8 • r3 = 1042.9 • r4 = 19.8 • r1 = 2 (given in problem) • Looking at these values: since they are so different (link ratios are large) this is likely not a very good design (structurally unsound) • Additionally, it is a type 2 Grashof design ME 3230

  14. Generalizing: • In the previous slides we have explored the techniques using precision points • While the results we observed here were not satisfying, the technique was straight forward and easy to follow • Usually, however, we are not given output and input angles directly so we must relate input as change lengths to inputs as angles and similarly for outputs • Then we typically choose change in input and change in output or:  and  for the design along with starting poses for  and  ME 3230

  15. Given a Function: y = mx + b • Y from 0 to 1.0” while X is .5 to 2.3” • Therefore: b = -0.179 and m = 0.357 • Y = 0.357x -0.179 • We want  to change by -45, (110 to 65) • Rad: -0.78539, (1.9199, 1.1345) • While  changes by -25, (135 to 110) • Lets use: • linear model relationships between X/ and Y/ • 3 Precision Points ME 3230

  16. Modeling: ME 3230

  17. To angles for the linkage: With these angular inputs, we enter the “Matrix Design” Models of earlier. Solve for Zi’s convert to r2, r3, r4 (with unit base length) ME 3230

  18. “Matrixifying”: (in Mathematica) ME 3230

  19. If r1 = 1 then: • r2 = 1/z2 = 1/3.77042 = 0.2652 • r4 = 1/z3 = 1/.68066 = 1.4692 • r3 = (1+r2^2 + r4^2 – 2r2r4z1) =1.9143 • This is a Type I Grashof unit (s+l < p+q) and r2 is the crank in a crank rocker type of linkage ME 3230

  20. Synthesis of Crank-Rockers for specified Rocker Amplitude • These are often used a replacements for CAM linkages! • lower contact forces • No Retaining Springs • Closer Joint tolerance • Design looks at the limits of Crank-Coupler • Often we need to control the “Time Ratio” as well • These designs must be Grashof Type I’s (of course)! ME 3230

  21. Solution Ideas: • The text lays out some very good graphical methods for solving the designs • One based purely of oscillation magnitude and Time ratio • Goal is to design an appropriate Transmission angle • Graphical Procedure based on Specified Base Fixed Pivots • This second can also be approached Analytically (we shall explore it!) ME 3230

  22. Analytical modeling with Base Length Given (O2 – O4) • Our concerns are with Q (Time Ratio) and  (input Crank + coupler oscillation to output link -- in degrees) • The Center of the B2 solution circle (the position of the rocker at end of return): (xg , yg) • Finally, the allowable range for  which limits the solution space of the Locus of B2 (see below) ME 3230

  23. From the Givens, The B2 Circle has the center and radius (case 1: 0<<): ME 3230

  24. Limits of B2 Circle • 1st Limit is O2 • Second limit is xm, ym: •  angles are given by ME 3230

  25. See Text for models as  changes • Case 2: (/2 - /2) <  < 0 • ym = 0; xm = 2rBCos((/2)- ) • Case 3:  =  • B2 becomes a line thru O2, a slider is suggested for the coupler and the mechanism is an Inverted Slider-Crank • Case 4: (/2 - /2) >  >  • This changes everything from circle center, radius (xm, ym) and  equation (see text pg 306-307) ME 3230

  26. For Case 1, 2, and 4 with O2O4 scaled to 1: • We select any reasonable value of  in range • Using this angle then: ME 3230

  27. Continuing: B1 & B2 represent the limits of the rocker locations r2 = (O2B1 – O2B2)/2 r3 = (O2B1 + O2B2)/2 ME 3230

  28. Optimizing Design (controlled by Transmission Angles and Link length ratios): • When we wish to minimize the motor power needed to operate the crank for a given output torque • Optimizing Equation (to seek a minimum): • U = U’ + WU’’ • W is 1 to 5 as a length ratio weighting factor Optimization of U is a function of the  angle – to find the optimum value, vary the chosen  and resolve U to a minimum ME 3230

  29. Designing for Path Generation • Path generation differs from function generation since here we are trying to control the trajectory (path and speed) of a single point on a linkage • This allows the point of interest to track the desired shape at varying speeds along the path • The method usually starts by finding an appropriate “Coupler Curve” and then refining the linkage by moving the point slightly or choosing slightly different link lengths ME 3230

  30. Uses: • The method is used to build complex linkages where output segments are driven by paths generated by 4-bar crank-rockers • Please see Text Sections 6.6 for the techniques – they follow a straight forward modeling study based on coupler curves that match the desired path trajectory • We then “Tweak” the designs to obtain the desired trajectory • Try problem 6.43 as you work on this idea! ME 3230

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