ken youssefi mechanical & aerospace engineering dept, sjsu

Ken Youssefi Mechanical & Aerospace Engineering Dept, SJSU


Complex Numbers and Polar Notation

x2 + 1 = 0, x2 + x + 1 = 0, x = ? Euler (1777), i = √ -1

i2 = -1

O AA′

OA′ = - OA

OA′ = i2 OA

i2 represents 180o rotation of a vector

i represents 90o 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 eiθ

eiθ = cos(θ) + i sin(θ),

Euler’s Formula

e-iθ = cos(θ) - i sin(θ)


Analytical Synthesis Rotational Operator & Stretch Ratio

j

P1

θ1

r1

θjrj

x

y

Pj

r1 = r1 eiθ1

rj = rj eiθj = r1 e

iθ1 (rj

r1

) eij = r1 r1

) eij rj

(

r1 Original vector

eij Rotational operator

rj

r1

( ) Stretch ratio, = 1 if length of the link is constant

r1

r1

ei(j + 1)rj=


Analytical Synthesis – Standard Dyad Form

4 Bar mechanism

A

B

O2O4

2

3

4

P

Left side Right side

r2r4

r3

r′3r″3

Design the left side of the 4 bar → r2 & r′3Design the right side of the 4 bar → r4 & r″3

αj

βj

δj

Closed loop vector equation – complex polar notation

r2 + r′3 + δj = r2 eiβj + r′3 e

iαj

Left side of the mechanism

A1

O2

2

P1

r2

r′3Aj

Pj

r2 eiβj

r′3 eiαj

r2 (eiβj – 1) + r′3 (e

iαj – 1) = δj

Standard Dyad form

Parallel


Analytical Synthesis – Standard Dyad Form

Apply the same procedure to obtain the Dyad equation for the right side of the four bar mechanism.

B

O4

4

P

r4

r″3

Rotation of link 4

r2 (eiβj – 1) + r′3 (e

iαj – 1) = δj

Standard Dyad form for the left side of the mechanism

α → rotation of link 3β → rotation of link 2

r4 (eij – 1) + r″3 (e

iαj – 1) = δj

Standard Dyad form for the right side of the mechanism

→ rotation of link 4 α → rotation of link 3


Analytical Synthesis Two Position Motion & Path Generation Mechanisms

α2

β2

δ2

Left side of the mechanism

A1

O2

2

P1

r2

r′3A2

P2

r2 eiβ2

r′3 eiα2

Parallel

r2 (eiβ2 – 1) + r′3 (e

iα2 – 1) = δ2

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, α2

3. Measure the length and angle of vector δ2

There are 5 unknowns; r2, r′3 and angle β2 and only two equations (Dyad).

Select three unknowns and solve the equations for the other two unknowns

Given; α2 and δ2

Select; β2 and r′3Solve for r2

Two position motion gen. Mech.

Three sets of infinite solution


Analytical Synthesis Two Position Motion & Path Generation Mechanisms

Apply the same procedure for the right side of the 4-bar mechanism

r4 (eij – 1) + r″3 (e

iαj – 1) = δj

Given; α2 and δ2

Select;, 2 , r″3

Solve for r4Two position motion gen. Mech.

Given; β2 and δ2

Select; α2 and r′3Solve for r2

Two position path gen. Mech.

Three sets of infinite solution

Path Generation Mechanism (left side of the mechanism)


Analytical Synthesis Three Position Motion & Path Generation Mechanisms

δ2

A1

O2

2

P1

r2

r′3

α2

β2

Parallel

P3

A3

α3

β3

δ3

A2

P2

r2 eiβ2

r′3 eiα2

r2 eiβ3

r′3 eiα3


Analytical Synthesis Three Position Motion & Path Generation Mechanisms

Three position motion gen. mech.

Given; α2, α3, δ2, and δ3

Select; β2 and β3 Solve for r2 and r′3

Three position motion gen. Mech.

Two sets of infinite solution

2 free choices

4 scalar equations

Dyad equationsr2 (e

iβ2 – 1) + r′3 (eiα2 – 1) = δ2

r2 (eiβ3 – 1) + r′3 (e

iα3 – 1) = δ3

6 unknowns; r2 , r′3 , β2 and β3


Analytical Synthesis

Four position motion generation mechanism

Given; α2, α3, α4 δ2, δ3 and δ4

Select; β2 or β3 or β4 Solve for r2 and r′3

Four position motion gen. Mech.

One set of infinite solution

Dyad equations

r2 (eiβ2 – 1) + r′3 (e

iα2 – 1) = δ2

r2 (eiβ3 – 1) + r′3 (e

iα3 – 1) = δ3

r2 (eiβ4 – 1) + r′3 (e

iα4 – 1) = δ4

Non-linear equations

7 unknowns; r2 , r′3 , β2 , β3 and β4

1 free choices

6 scalar equations


Analytical Synthesis

Dyad equations

r2 (eiβ2 – 1) + r′3 (e

iα2 – 1) = δ2

r2 (eiβ3 – 1) + r′3 (e

iα3 – 1) = δ3

r2 (eiβ4 – 1) + r′3 (e

iα4 – 1) = δ4

Non-linear equations

Five position motion generation mechanism

r2 (eiβ5 – 1) + r′3 (e

iα5 – 1) = δ5

8 unknowns; r2 , r′3 , β2 , β3 , β4 and β5

0 free choice

8 scalar equations

Given; α2, α3, α4, α5, δ2, δ3, δ4, and δ5

Select; 0 choice

Four position motion gen. Mech.

Unique solution, not desirable

Solve for r2 and r′3

ken youssefi mechanical & aerospace engineering dept, sjsu

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