Analytical Synthesis

1
Analytical Synthesis
2
Complex Numbers and Polar Notation
x2 1 0, x2 x 1 0, x ?
3
Analytical Synthesis Rotational Operator
Stretch Ratio
y
P1
x
4
Analytical Synthesis Standard Dyad Form
4 Bar mechanism
P
B
3
A
4
2
Design the left side of the 4 bar ? r2 r'3
Design the right side of the 4 bar ? r4 r?3
5
Analytical Synthesis Standard Dyad Form
Apply the same procedure to obtain the Dyad
equation for the right side of the four bar
mechanism.
P
r?3
B
r4
4
Rotation of link 4, g
6
Analytical Synthesis Two Position Motion Path
Generation Mechanisms
Left side of the mechanism
P2
d2
a2
r'3 eia2
Motion generation mechanism, the orientation of
link 3 is important (angle alpha)
Parallel
A2

• Draw the two desired positions accurately.

ß2
r2 eiß2

• Measure the angle a from the drawing, a2

• Measure the length and angle of vector d2

7
Analytical Synthesis Two Position Motion Path
Generation Mechanisms
Apply the same procedure for the right side of
the 4-bar mechanism
8
Analytical Synthesis Three Position Motion
Path Generation Mechanisms
9
Analytical Synthesis Three Position Motion
Path Generation Mechanisms
Three position motion gen. mech.
2 free choices
10
Analytical Synthesis
Four position motion generation mechanism
Given a2, a3, a4 d2, d3 and d4
Four position motion gen. Mech.
Solve for r2 and r'3
Select ß2 or ß3 or ß4
One set of infinite solution
11
Analytical Synthesis
Five position motion generation mechanism
r2 (eiß2 1) r'3 (eia2 1) d2
Non-linear equations
r2 (eiß3 1) r'3 (eia3 1) d3
Dyad equations
r2 (eiß4 1) r'3 (eia4 1) d4
r2 (eiß5 1) r'3 (eia5 1) d5
8 unknowns r2 , r'3 , ß2 , ß3 , ß4 and ß5
0 free choice
Given a2, a3, a4, a5, d2, d3, d4, and d5
Four position motion gen. Mech.
Solve for r2 and r'3
Select 0 choice
Unique solution, not desirable