Mechanism Synthesis, Graphical

Ken Youssefi Mechanical & Aerospace Engineering Dept. SJSU 1

Mechanism Design Graphical Method


• Dimensional Synthesis

Mechanism SynthesisDesign a mechanism to obtain a specified motion or force.

– How many links should themechanism have? How many degrees of freedom are desired?

• Number Synthesis

– given the required performance, what type of mechanism is suitable? Linkages, gears, cam and follower, belt and pulley and chain and sprocket.

• Type Synthesis

– deals with determining the length of all links, gear diameter, cam profile.


Mechanism SynthesisType SynthesisThe Associated Linkage Concept

It is desired to derive various types of mechanisms for driving a slider with a linear translation along a fixed path in a machine. Also, assume that the slider must move with a reciprocating motion.

4-Bar


Mechanism SynthesisType Synthesis - The Associated Linkage Concept (6-Bar)

6-Bar


Limiting Conditions – 4 Bar Mechanism

Toggle positions of a crank-rocker mechanism. Links 2 and 3 become collinear.


Transmission Angle – 4 Bar Mechanism

The angle between link 3 and link 4 is defined as the transmission angle

T4 = F34sin(µ) x (O4D)


Minimum Transmission Angle – 4 Bar Mechanism

Minimum transmission angle occurs when link 2 (crank) becomes collinear with link 1 (ground link)

The minimum transmission angle should be greater than 40o to avoid locking or jamming the mechanism

µ

Min. transmission angle

Max. transmission angle


Mechanical Advantage – 4 Bar Mechanism


Mechanical Advantage – 4 Bar Mechanism

O4B = 2(O2A)

rin = rout

µ = 60O, v = 5O

M.A. = 20

µ

A

B


Mechanism SynthesisDimensional Synthesis

Graphical Methods – provide the designer with a quick straightforward method but parameters cannot easily be manipulated to create new solutions.

– this approach is suitable for automatic computation. Once a mechanism is modeled and coded for computer, parameters are easily manipulated to create new designs.

Analytical Methods


O2

O44. Select two fixed pivot points, O2 and O4, anywhere on the two midnormals.

Graphical Synthesis – Motion Generation MechanismTwo positions, coupler as the output

A1A2

B1

B2

1. Draw the link AB in its two desired positions, A1B1 and A2B2

5. Measure the length of all links,

O2A = link 2, AB = link 3,

O4B = link 4 and O2 O4 = link 1

2. Connect A1 to A2 and B1 to B2.

3. Draw two lines perpendicular to A1 A2 and B1B2 at the midpoint (midnormals).


O4O2

Graphical Synthesis – Motion Generation Mechanism

Three positions, coupler as the output

A1

A2

A3

B1

B2

B3

Same procedure as for two positions.

1. Draw the link AB in three desired positions.

2. Draw the midnormals to A1A2 and A2A3, the intersection locates the fixed pivot point O2. Same for point B to obtain second pivot point O4.

3. Check the accuracy of the mechanism, Grashof condition and the transmission

angle.

4. Change the second position of link AB to vary the locations of the fixed points


O6

4. Select any location on this line for third fixed pivot, O6.

D2

5. Draw a circle with radius C1C2 / 2. The radius is the length of the sixth link.

Graphical Synthesis – Motion Generation MechanismAdding a Dyad to a non-Grashof mechanism.

A1A2

B1

B2

O2

O4

2

3

4

1. Draw the four bar in both positions

C1 C2

2. Select any point C on link 2.

3. Connect C1 to C2 and extend.

56

6. Measure O6D = link 6, DC = link 5



A1A

B1

O4

O6

CDO2

B32 4

5

6

6-Bar Grashof mechanism


Three Position, 6-Bar Grashof ,Motion Generation Mechanism


Three Position, 6-Bar Grashof ,Motion Generation Mechanism



Two positions Grashof 4-Bar mechanism with rocker as the output

D1

C1C2

D2

O2

5. Connect B1 to B2 and extend. Select any location on this line for fixed pivot point O2.

O2A = B1B2 / 2

7. Measure the length of all links, O2A = link 2, AB = link 3, O4CD = link 4 and O2 O4 = link 1

1. Draw the link CD in its two desired positions, C1D1 and C2D2

2. Connect C1 to C2 and D1 to D2 and draw two midnormals to C1C2 and D1D2

O4

3. The intersection of the two midnormals is the fixed pivot point O4.

B1 B2

4. Select point B1 anywhere on link O4C1 and locate B2 so O4B1= O4B2

A2

6. Draw a circle with radius B1 B2 / 2, point A is the intersection of the circle with the B1 B2 extension.



Two positions Grashof 4-Bar mechanism

with rocker as the output

D1

C1C2

A2

O4

O2

B2D2


Two Position, 4-Bar Grashof Motion Generation Mechanism


Graphical Synthesis – Motion Generation MechanismThree positions with specified fixed pivot points,coupler as the output

C1

D1

C2

C3

D2

D3

O4

O2

1. Draw the link CD in its three desired positions, C1D1, C2D2 and C3D3 and locate the fixed pivot points O2 and O4.

2. Draw an arc from C1 with radius O2C2 and another arc from D1 with radius O2D2. Locate the intersection, O’2.

3. Draw an arc from C1 with radius O4C2 and another arc from D1 with radius O4D2. Locate the intersection, O’4.

O’4O’

2

O’2

O’4



C1

D1

C2

C3

D2

D3

O4

O2

O’2

O’4

Three positions with specified fixed pivot points,coupler as the output

4. Draw an arc from C1 with radius O2C3 and another arc from D1 with radius O2D3. Locate the intersection, O”2.

5. Draw an arc from C1 with radius O4C3 and another arc from D1 with radius O4D3. Locate the intersection, O”4.

O”2

O”4

O”2

O”4


C1

D1

C2

C3

D2

O4

O2

O”2

O”4

O’2

O’4

G

H

Graphical Synthesis – Motion Generation MechanismThree positions with specified fixed pivot points,coupler as the output

D3

6. Connect O2 to O’2 and O’2 to O”2 . Draw two midnormals and locate the intersection, G.

7. Connect O4 to O”4 and O”4 to O’4 . Draw two midnormals and locate the intersection, H.

8. O2G is link 2 and O4H is link 4.

9. Construct a link (3) containing GH and CD.

10. Verify the solution by constructing the mechanism in three position



C1

D1

C2

C3

D2

O4

O2

G

H

D3


Graphical Synthesis – Motion Generation MechanismThree positions with specified fixed pivot points, coupler as the output.


O4O2

2. Select the location of the fixed pivot points, O2 and O4.

Graphical Synthesis – Path Generation MechanismThree prescribed points.

5. Measure angles α1 (O2A1P1), α2 and α3.

α1α2 α3

P1 P2

P3

1. Draw the three desired points, P1, P2, and P3.

A13. Select the length of the crank O2A

and the coupler side AP. A3

A2

4. With A1P1 established, locate A2 and A3, A1P1 = A2P2 = A3P3.

Design a 4-Bar in such a way that a point on the coupler passes thru three specified points


Graphical Synthesis – Path Generation MechanismThree prescribed points.Locate moving pivot B by means of kinematic inversion. Fix coupler AP in position 1 and rotate O2O4.

O4O2

P1 P2

P3

A1

11. Verify the mechanism.

B

O”4

O’2

6. Rotate A1O2 about A1 by (α2 – α1) to O’2 .

O’4

7. Draw an arc from O’2 with radius O2O4 , draw another arc from P1

with radius P2O4 , locate the intersection, O’4 .

O”2

8. Rotate A1O2 about A1 by (α3 – α1) to O”2 .

9. Draw an arc from O”2 with radius O2O4 , draw another arc from P1

with radius P3O4 , locate the intersection, O”4 .

10. Connect O4 to O’4 and O’4 to O”4 and draw the midnormals. Locate the intersection, B.


O2

1. Select location of the fixed pivot point O2.

Graphical Synthesis – Path Generation Mechanismwith Prescribed Timing

Three prescribed pointsTiming requirements: input crank rotation α, mechanism moves from P1 to P2 input crank rotation β, mechanism moves from P1 to P3

P1 P2

P3

6. Follow the same procedure as before , for without timing, to locate the moving pivot point B.

A

Note: timing takes away the free choices of the crank length and coupler length AP.

P’2

α

2. Rotate O2P2 , in the opposite direction

of motion, through angle α, P’2.P’3

β

3. Rotate O2P3 ,in the opposite direction of motion, through angle β, P’3.

4. Draw midnormals to P1P’2 and P1P’3.and locate the intersection A.

5. Measure O2A = link 2 and AP.


Graphical Synthesis; Quick – Return Mechanism

Q = time of advance stroke / time of return stroke

Q > 1 quick-return mechanism

Advance stroke – mechanism operates under the load.

Return stroke – mechanism operates under no load.

4-Bar crank-Rocker mechanism


Quick – Return Mechanism

Consider the two toggle positions of a crank-rocker mechanism.

O4O2

B1

2

3 4

A1

B2

A2

C

Locate point C to satisfy the following two conditions;

1) C is on extension of line A2B2.

2) O2C = O2B1 = r2 + r3B2C = r2 +r3 - (r3 – r2) = 2r2

r 3 – r 2


Quick – Return Mechanism

O4O2

B1

2

3 4

A1

B2

A2

C

α

180 – α, Return stroke

Q = advance / Return = (180 + α) / (180 – α), Time Ratio


Synthesis of a Quick – Return MechanismKnown or selected; Rocker angle, φ Rocker length, r4

Time ratio, Q

Determine; r1, r2, r3

O4

1. Select the location for the fixed pivot point, O4.

O2

6. The intersection of XX’ and YY’ is the other fixed pivot, O2

X4. Construct an arbitrary line XX’

through point B1.

X’

5. Construct the line YY’ through point B2 making an angle α with XX’.

Y

Y’

α

2. Draw the two toggle positions, knowing r4 and φ.

B1

B2

φ

3. Calculate the angle α from known time ratio Q = (180 + α) / (180 – α)


Synthesis of a Quick – Return Mechanism

O2

X

Y’

O4

X’

Y

B1

B2

7. Locate point C on YY’ so O2C = O2 B1.

C

9. Calculate the length of link 3, AB = r3 = O2 B1 – r2

8. Measure length B2 C, Link 2 = r2 = (B2 C) /22r2

A1

r2

A2

A

O4O2

B

10. Verify the motion of the mechanism and check the minimum transmission angle.

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Mechanism Synthesis, Graphical