1. 1 All figures taken from Design of Machinery, 3 rd ed. Robert Norton 2003 MENG 372 Chapter 5 Analytical Position Synthesis

2. 2 5.1 Types of Kinematic Synthesis Function Generation: correlation of an input function with an output function in a mechanism Path Generation: control of a point in the plane such that it follows some prescribed path Motion Generation: control of a line in the plane such that it assumes some sequential set of prescribed positions

3. 3 The points, positions prescribed for successive locations of the output (coupler or rocker) link in the plane. In graphical synthesis: move from C 1 D 1 to C 2 D 2 In analytical synthesis: move from P 1 to P 2 while rotating coupler  2 (note: angles are measured anticlockwise) 5.2 Precision Points P1P1 22 P2P2

4. 4 22 Can define vectors Z and S from the attachment points E and F to P Note: the coupler is not triangular, but 3 points are defined on the coupler Points E and F are called A and B Precision Points P1P1 P2P2 Z1Z1 S1S1 A B

5. 5 5.3 Two Position Synthesis Want to move from P 1 to P 2 while coupler rotates  2 Given P 21,  2 and  2 Design each half separately Write vector loop equation(s) to include given values, find free choices to make problem easy to solve.

6. 6 1.Choose any coordinate system X-Y 2.Draw vector P 21 inclined at d 2 3.Define position vectors R 1 and R 2 4.Draw an arbitrary vector Z 1. Then form vector Z 2 with same magnitude but angle a 2 with Z 1. 5.Draw vectors W 1 and W 2 to meet at O 2. 6.Write vector loop equation. Problem Statement Design a 4-bar linkage which will move P 1 to P 2 while coupler rotates thru a 2. P lies on coupler. Find the lengths and angles of all links. X Y R2R2 R1R1 Z1Z1 Z2Z2 W1W1 W2W2 d2d2 P 21 a2a2 P1P1 P2P2 O2O2

7. 7 Two Position Synthesis Vector loop equation W 2 + Z 2 - P 21 - Z 1 - W 1 = 0 Write complex vectors Expand exponents Combine terms

8. 8 Two Position Synthesis Variables w, ,  2, z, ,  2, P 21,  2 = 8 Given P 21,  2,  2 =-3 Complex equations: 1 can solve for 2 unknowns =-2 Free Choices=3

9. 9 Two Position Synthesis Choose ( ,  2,  ) Gives 2 simultaneous eqns.

10. 10 Two Position Synthesis Choose (  2, z,  ) from which the magnitude and angle can be calculated w=abs(Q),  =angle(Q) The other side can be calculated similarly

11. 11 Two Position Synthesis Once both sides have been solved, the coupler and ground can be calculated using v=abs(V 1 ) g=abs(G 1 )

12. 12 Two Position Synthesis Comparison For graphical, position of attachment points A and B relative to P in x and y directions (4) and points of O 2 and O 4 along the perpendicular bisectors (2) gives 6 total For analytical, 3 free choices each side * 2 sides=6 total

13. 13 5.6 Three Position Synthesis Want to move from P 1 to P 2 while coupler rotates  2 and from P 1 to P 3 while coupler rotates  3 Given P 21,  2, P 31,  3,  2 and  3.

14. 14 Three Position Synthesis Vector loop equations W 2 + Z 2 - P 21 - Z 1 - W 1 = 0 W 3 + Z 3 - P 31 - Z 1 - W 1 = 0 Write complex vectors Combine terms.

15. 15 Three Position Synthesis Variables w, ,  2,  3,z, ,  2,  3,P 21, P 31,  2,  3 = 12 Given P 21,P 31,  2,  3,  2,  3 =-6 Complex equations *2 2*2=-4 Free Choices=2

16. 16 Three Position Synthesis Choose (  2,  3 ) Two linear equations Gives solution w=abs(W),  =angle(W) z=abs(Z),  =angle(Z).

17. 17 Eliminate W to get: Then solve for W: (USE MATLAB) Solution

18. 18 Choose (  2,  3 ) Two linear equations REPEAT FOR RIGHT-HAND SIDE OF LINKAGE (USE MATLAB)

19. 19 Three Position Synthesis Comparison For graphical, position of attachment points A and B relative to P in x and y directions (4) For analytical, 2 free choices each side * 2 sides=4 total

20. 20 Example Design a 4-bar linkage to move A 1 P 1 to A 2 P 2 to A 3 P 3

21. 21

22. 22 3 Position Synthesis with Specified Fixed Pivots. Want to move from P 1 to P 2 while coupler rotates  2 and from P 1 to P 3 while coupler rotates  3 and attach to ground at O 2 and O 4 Given R 1,R 2,R 3,  1,  2,  3,  2 and  3 Note: if R 1 and R 2 are satisfied, P 21 is satisfied, and R 1 and R 3 give P 31 

23. 23 3 Position Synthesis with Specified Fixed Pivots. Vector loop equations W 1 +Z 1 =R 1 W 2 +Z 2 =R 2 W 3 +Z 3 =R 3 Use relationships to get 

24. 24 3 Position Synthesis with Specified Fixed Pivots. Variables w, ,  2,  3,z, ,  2,  3,R,  1,  2,  3 = 12 Given R,  1,  2,  3,  2,  3 =-6 Complex equations *2 3eqn*2=-6 Free Choices (Sub)=0 This makes the problem hard 

25. 25 Use this to eliminate Z 1 Divide 2 eq’ns to eliminate W 1 Cross Multiply 3 Position Synthesis with Specified Fixed Pivots. From 1 st equation:

26. 26 Arrange into form where using s and t: gives 3 Position Synthesis with Specified Fixed Pivots. 0 3 2    i i eeCBA

27. 27 Taking conjugate Since s and t represent angles Multiplying by st From (a) Substituting into (b) gives a quadratic function of only t (a) (b) 3 Position Synthesis with Specified Fixed Pivots.

28. 28 where Solving gives Only one of the t will be valid. s can be solved using Any 2 of the first eqns can be used to solve for W 1 and Z 1 3 Position Synthesis with Specified Fixed Pivots.

29. 29 Summary of calculations (for MATLAB implementation) w=abs(W 1 ),  =angle(W 1 ), z=abs(Z 1 ),  =angle(Z 1 ) 3 Position Synthesis with Specified Fixed Pivots.

30. 30 Example Problem Move from C 1 D 1 to C 2 D 2 to C 3 D 3 using attachment points O 2 and O 3 Call point C, P   