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Analytical Synthesis

Analytical Synthesis. Euler (1777), i = √ -1. i 2 = -1. O. A. A′. OA′ = - OA. i 2 represents 180 o rotation of a vector. OA′ = i 2 OA. i represents 90 o rotation of a vector. Argand Diagram. x = rcos ( θ ) y = rsin ( θ ). r = rcos ( θ ) + i rsin ( θ ). iy.

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Analytical Synthesis

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  1. Analytical Synthesis Mechanical & Aerospace Engineering Dept, SJSU

  2. Euler (1777), i = √ -1 i2 = -1 O A A′ OA′ = - OA i2 represents 180o rotation of a vector OA′ = i2 OA i represents 90o rotation of a vector Argand Diagram x = rcos(θ) y = rsin(θ) r = rcos(θ)+i rsin(θ) iy imaginary part real part r = x + iy Imaginary axis P Euler’s Formula r eiθ =cos(θ) + i sin(θ), e-iθ =cos(θ) - i sin(θ) θ x r = r eiθ Real axis Complex Numbers and Polar Notation x2+1 = 0, x2+x+1 = 0, x= ? Mechanical & Aerospace Engineering Dept, SJSU

  3. j Pj θj rj r1 θ1 r1= r1 eiθ1 r1 rj rj + 1) rj ei(j rj= rj eiθj ) eij ) eij = r1 eiθ1( = r1 ( = r1 r1 r1 Original vector rj ( ) r1 Stretch ratio, = 1 if length of the link is constant r1 eij Rotational operator Analytical Synthesis Rotational Operator & Stretch Ratio y P1 x Mechanical & Aerospace Engineering Dept, SJSU

  4. Left side of the mechanism r″3 Pj r′3 δj P1 αj r′3 eiαj r3 Parallel r′3 r4 r2 A1 Aj O2 O2 2 βj r2 eiβj O4 r2 Left side Right side Closed loop vector equation – complex polar notation r2 +r′3+δj= r2 eiβj+ r′3 eiαj Standard Dyad form r2(eiβj – 1) + r′3(eiαj – 1) = δj 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 Mechanical & Aerospace Engineering Dept, SJSU

  5. Standard Dyad form for the right side of the mechanism r4(eigj – 1) + r″3(eiαj – 1) = δj O4 g→ rotation of link 4 α → rotation of link 3 Standard Dyad form for the left side of the mechanism r2(eiβj – 1) + r′3(eiαj – 1) = δj β → rotation of link 2 α → rotation of link 3 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 Mechanical & Aerospace Engineering Dept, SJSU

  6. Dyad equation for the left side of the mechanism. One vector equation or two scalar equations P1 r′3 r2(eiβ2 – 1) + r′3(eiα2 – 1) = δ2 A1 O2 2 r2 Select three unknowns and solve the equations for the other two unknowns There are 5 unknowns; r2, r′3andangleβ2and only two equations (Dyad). Three sets of infinite solution Given;α2andδ2 Two position motion gen. Mech. Solve for r2 Select; β2and r′3 Analytical Synthesis Two Position Motion & Path Generation Mechanisms Left side of the mechanism P2 δ2 α2 r′3 eiα2 Motion generation mechanism, the orientation of link 3 is important (angle alpha) Parallel A2 • Draw the two desired positions accurately. β2 r2 eiβ2 • Measure the angle αfrom the drawing, α2 • Measure the length and angle of vector δ2 Mechanical & Aerospace Engineering Dept, SJSU

  7. Given;α2 andδ2 Two position motion gen. Mech. Solve for r4 Select;, g2, r″3 Path Generation Mechanism (left side of the mechanism) Three sets of infinite solution Given;β2andδ2 Two position path gen. Mech. r4(eigj – 1) + r″3(eiαj – 1) = δj Solve for r2 Select; α2and r′3 Analytical Synthesis Two Position Motion & Path Generation Mechanisms Apply the same procedure for the right side of the 4-bar mechanism Mechanical & Aerospace Engineering Dept, SJSU

  8. P1 P2 r′3 δ2 A1 r′3 eiα3 r′3 eiα2 O2 α2 2 Parallel δ3 r2 A2 P3 α3 β2 r2 eiβ3 r2 eiβ2 β3 A3 Analytical Synthesis Three Position Motion & Path Generation Mechanisms Mechanical & Aerospace Engineering Dept, SJSU

  9. r2(eiβ2 – 1) + r′3(eiα2 – 1) = δ2 r2(eiβ3 – 1) + r′3(eiα3 – 1) = δ3 Dyad equations 4 scalar equations 6 unknowns; r2 , r′3 , β2and β3 Given;α2, α3, δ2, andδ3 Three position motion gen. Mech. Solve for r2andr′3 Select; β2and β3 Two sets of infinite solution Analytical Synthesis Three Position Motion & Path Generation Mechanisms Three position motion gen. mech. 2 free choices Mechanical & Aerospace Engineering Dept, SJSU

  10. r2(eiβ2 – 1) + r′3(eiα2 – 1) = δ2 Non-linear equations r2(eiβ3 – 1) + r′3(eiα3 – 1) = δ3 Dyad equations r2(eiβ4 – 1) + r′3(eiα4 – 1) = δ4 6 scalar equations 7 unknowns; r2 , r′3 , β2, β3and β4 1 free choices Analytical Synthesis Four position motion generation mechanism Given;α2, α3, α4δ2, δ3 and δ4 Four position motion gen. Mech. Solve for r2andr′3 Select; β2or β3 or β4 One set of infinite solution Mechanical & Aerospace Engineering Dept, SJSU

  11. 8 scalar equations Analytical Synthesis Five position motion generation mechanism r2(eiβ2 – 1) + r′3(eiα2 – 1) = δ2 Non-linear equations r2(eiβ3 – 1) + r′3(eiα3 – 1) = δ3 Dyad equations r2(eiβ4 – 1) + r′3(eiα4 – 1) = δ4 r2(eiβ5 – 1) + r′3(eiα5 – 1) = δ5 8 unknowns; r2 , r′3 , β2, β3, β4and β5 0 free choice Given;α2, α3, α4, α5, δ2, δ3,δ4, andδ5 Four position motion gen. Mech. Solve for r2andr′3 Select; 0 choice Uniquesolution, not desirable Mechanical & Aerospace Engineering Dept, SJSU

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