MENG 372 Chapter 5 Analytical Position Synthesis All
- Slides: 30
MENG 372 Chapter 5 Analytical Position Synthesis All figures taken from Design of Machinery, 3 rd ed. Robert Norton 2003 1
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 2
5. 2 Precision Points • The points, positions prescribed for successive locations of the output (coupler or rocker) link in the plane. • In graphical synthesis: P 1 move from C 1 D 1 to C 2 D 2 a 2 • In analytical synthesis: P 2 move from P 1 to P 2 while rotating coupler a 2 (note: angles are measured anticlockwise) 3
Precision Points • 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 P 1 a 2 Z 1 S 1 A P 2 B 4
5. 3 Two Position Synthesis • Want to move from P 1 to P 2 while coupler rotates a 2 • Given P 21, d 2 and a 2 • Design each half separately • Write vector loop equation(s) to include given values, find free choices to make problem easy to solve. 5
Problem Statement Y 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. R 2 R 1 d 2 1. Choose any coordinate system X-Y P 2 P P 1 2. Draw vector P 21 inclined at d 2 21 3. Define position vectors R 1 and R 2 Z 1 a 2 Z 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. W 1 6. Write vector loop equation. W 2 O 2 X 6
Two Position Synthesis • Vector loop equation W 2 + Z 2 - P 21 - Z 1 - W 1 = 0 • Write complex vectors • Expand exponents • Combine terms 7
Two Position Synthesis • Variables w, q, b 2, z, f, a 2, P 21, d 2 = 8 • Given P 21, d 2, a 2 =-3 • Complex equations: 1 can solve for 2 unknowns =-2 • Free Choices =3 8
Two Position Synthesis • Choose (q, b 2, f) Gives 2 simultaneous eqns. 9
Two Position Synthesis • Choose (b 2, z, f) from which the magnitude and angle can be calculated w=abs(Q), q=angle(Q) • The other side can be calculated similarly 10
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) 11
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 12
5. 6 Three Position Synthesis • Want to move from P 1 to P 2 while coupler rotates a 2 and from P 1 to P 3 while coupler rotates a 3 • Given P 21, d 2, P 31, d 3, a 2 and a 3. 13
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 14
Three Position Synthesis • Variables w, q, b 2, b 3, z, f, a 2, a 3, P 21, P 31, d 2 , d 3 = 12 • Given P 21, P 31, d 2, d 3, a 2, a 3 =-6 • Complex equations *2 2*2 =-4 • Free Choices =2 15
Three Position Synthesis • Choose (b 2, b 3 ) • Two linear equations. • Gives solution w=abs(W), q=angle(W) z=abs(Z), f=angle(Z) 16
Solution Eliminate W to get: Then solve for W: (USE MATLAB) 17
REPEAT FOR RIGHT-HAND SIDE OF LINKAGE Choose (g 2, g 3 ) Two linear equations (USE MATLAB) 18
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 19
Example Design a 4 -bar linkage to move A 1 P 1 to A 2 P 2 to A 3 P 3 20
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3 Position Synthesis with Specified Fixed Pivots. • Want to move from P 1 to P 2 while coupler rotates a 2 and from P 1 to P 3 while coupler rotates a 3 and attach to ground at O 2 and O 4 • Given R 1, R 2, R 3, z 1, z 2, z 3, a 2 and a 3 • Note: if R 1 and R 2 are satisfied, P 21 is satisfied, and R 1 and R 3 give P 31 f 22
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 f to get 23
3 Position Synthesis with Specified Fixed Pivots. • Variables w, q, b 2, b 3, z, f, a 2, a 3 , R, z 1, z 2, z 3 = 12 • Given R, z 1, z 2, z 3, a 2, a 3 =-6 • Complex equations *2 3 eqn*2 =-6 • Free Choices (Sub) =0 f This makes the problem hard 24
3 Position Synthesis with Specified Fixed Pivots. From 1 st equation: Use this to eliminate Z 1 Divide 2 eq’ns to eliminate W 1 Cross Multiply 25
3 Position Synthesis with Specified Fixed Pivots. Arrange into form A + Be where ib 2 + Ce ib = 0 3 using s and t: gives 26
3 Position Synthesis with Specified Fixed Pivots. Taking conjugate (a) Since s and t represent angles Multiplying by st (b) From (a) • Substituting into (b) gives a quadratic function of only t 27
3 Position Synthesis with Specified Fixed Pivots. 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 28
3 Position Synthesis with Specified Fixed Pivots Summary of calculations (for MATLAB implementation). w=abs(W 1), q=angle(W 1), z=abs(Z 1), f=angle(Z 1) 29
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 f 3 f 2 f 1 30