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Linkage Design: Function Generation, Rocker Amplitude and Path Synthesis

This chapter focuses on designing linkages that achieve specific motion outputs based on input variables. It introduces the concept of "Precision Points," crucial for defining a linkage's operational range through a functional equation. Utilizing techniques such as the Chebyshev Polynomial for approximation, the chapter outlines methods to determine unknown parameters in linkage design. The discussion also covers Grashof conditions and the significance of Crank-Rocker types in linkage synthesis, emphasizing graphical and analytical design approaches for optimizing transmission angles and minimizing error.

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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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