1. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU

2. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Complex Numbers and Polar Notation x 2 + 1 = 0, x 2 + x + 1 = 0, x = ? Euler (1777), i = √ -1 i 2 = -1 O A A′ OA′ = - OA OA′ = i 2 OA i 2 represents 180 o rotation of a vector i represents 90 o rotation of a vector Real axis Imaginary axis r P x iy Argand Diagram θ r = x + iy x = rcos (θ) y = rsin (θ) r = rcos (θ) + i rsin (θ) real part imaginary part r = r e iθ e iθ = cos (θ) + i sin (θ), Euler’s Formula e -iθ = cos (θ) - i sin (θ)

3. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis Rotational Operator & Stretch Ratio jj P1P1 θ1θ1 r1r1 θjθj rjrj x y PjPj r 1 = r 1 e iθ 1 r j = r j e iθ j = r 1 e iθ 1 ( rjrj r1r1 ) e i  j = r 1 r1r1 ) e i  j rjrj ( r 1 Original vector eijeij Rotational operator rjrj r1r1 ( ) Stretch ratio, = 1 if length of the link is constant r1r1 r1r1 ei(jei(j +  1 ) rjrj =

4. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis – Standard Dyad Form 4 Bar mechanism A B O2O2 O4O4 2 3 4 P Left sideRight side r2r2 r4r4 r3r3 r′3r′3 r″ 3 Design the left side of the 4 bar → r 2 & r ′ 3 Design the right side of the 4 bar → r 4 & r ″ 3 αjαj βjβj δjδj Closed loop vector equation – complex polar notation r 2 + r ′ 3 + δ j = r 2 e iβ j + r′ 3 e iα j Left side of the mechanism A1A1 O2O2 2 P1P1 r2r2 r′3r′3 AjAj PjPj r 2 e iβ j r′ 3 e iα j r 2 (e i β j – 1 ) + r ′ 3 (e i α j – 1 ) = δ j Standard Dyad form Parallel

5. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis – Standard Dyad Form Apply the same procedure to obtain the Dyad equation for the right side of the four bar mechanism. B O4O4 4 P r4r4 r″ 3 Rotation of link 4   r 2 (e i β j – 1 ) + r ′ 3 (e i α j – 1 ) = δ j Standard Dyad form for the left side of the mechanism α → rotation of link 3 β → rotation of link 2 r 4 (e i  j – 1 ) + r ″ 3 (e i α j – 1 ) = δ j Standard Dyad form for the right side of the mechanism  → rotation of link 4 α → rotation of link 3

6. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis Two Position Motion & Path Generation Mechanisms α2α2 β2β2 δ2δ2 Left side of the mechanism A1A1 O2O2 2 P1P1 r2r2 r′3r′3 A2A2 P2P2 r 2 e iβ 2 r′ 3 e iα 2 Parallel r 2 (e i β 2 – 1 ) + r ′ 3 (e i α 2 – 1 ) = δ 2 Dyad equation for the left side of the mechanism. One vector equation or two scalar equations 1.Draw the two desired positions accurately. Motion generation mechanism, the orientation of link 3 is important (angle alpha) 2.Measure the angle α from the drawing, α 2 3.Measure the length and angle of vector δ 2 There are 5 unknowns; r 2, r ′ 3 and angle β 2 and only two equations (Dyad). Select three unknowns and solve the equations for the other two unknowns Given; α 2 and δ 2 Select; β 2 and r ′ 3 Solve for r 2 Two position motion gen. Mech. Three sets of infinite solution

7. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis Two Position Motion & Path Generation Mechanisms Apply the same procedure for the right side of the 4-bar mechanism r 4 (e i  j – 1 ) + r ″ 3 (e i α j – 1 ) = δ j Given; α 2 and δ 2 Select;,  2, r ″ 3 Solve for r 4 Two position motion gen. Mech. Given; β 2 and δ 2 Select; α 2 and r ′ 3 Solve for r 2 Two position path gen. Mech. Three sets of infinite solution Path Generation Mechanism (left side of the mechanism)

8. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis Three Position Motion & Path Generation Mechanisms δ2δ2 A1A1 O2O2 2 P1P1 r2r2 r′3r′3 α2α2 β2β2 Parallel P3P3 A3A3 α3α3 β3β3 δ3δ3 A2A2 P2P2 r 2 e iβ 2 r′ 3 e iα 2 r 2 e iβ 3 r′ 3 e iα 3

9. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis Three Position Motion & Path Generation Mechanisms Three position motion gen. mech. Given; α 2, α 3, δ 2, and δ 3 Select; β 2 and β 3 Solve for r 2 and r ′ 3 Three position motion gen. Mech. Two sets of infinite solution 2 free choices 4 scalar equations Dyad equations r 2 (e i β 2 – 1 ) + r ′ 3 (e i α 2 – 1 ) = δ 2 r 2 (e i β 3 – 1 ) + r ′ 3 (e i α 3 – 1 ) = δ 3 6 unknowns; r 2, r ′ 3, β 2 and β 3

10. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis Four position motion generation mechanism Given; α 2, α 3, α 4 δ 2, δ 3 and δ 4 Select; β 2 or β 3 or β 4 Solve for r 2 and r ′ 3 Four position motion gen. Mech. One set of infinite solution Dyad equations r 2 (e i β 2 – 1 ) + r ′ 3 (e i α 2 – 1 ) = δ 2 r 2 (e i β 3 – 1 ) + r ′ 3 (e i α 3 – 1 ) = δ 3 r 2 (e i β 4 – 1 ) + r ′ 3 (e i α 4 – 1 ) = δ 4 Non-linear equations 7 unknowns; r 2, r ′ 3, β 2, β 3 and β 4 1 free choices 6 scalar equations

11. Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU Analytical Synthesis Dyad equations r 2 (e i β 2 – 1 ) + r ′ 3 (e i α 2 – 1 ) = δ 2 r 2 (e i β 3 – 1 ) + r ′ 3 (e i α 3 – 1 ) = δ 3 r 2 (e i β 4 – 1 ) + r ′ 3 (e i α 4 – 1 ) = δ 4 Non-linear equations Five position motion generation mechanism r 2 (e i β 5 – 1 ) + r ′ 3 (e i α 5 – 1 ) = δ 5 8 unknowns; r 2, r ′ 3, β 2, β 3, β 4 and β 5 0 free choice 8 scalar equations Given; α 2, α 3, α 4, α 5, δ 2, δ 3, δ 4, and δ 5 Select; 0 choice Four position motion gen. Mech. Unique solution, not desirable Solve for r 2 and r ′ 3