Kinematic Synthesis of Mechanisms

Kinematic Synthesis: To design or create a mechanism to yield prescribed motion characteristics (kinematic parameters) satisfying the various constraints and under specified input motions. The general steps in kinematic synthesis are: Type Synthesis refers to the kind of mechanism selected (e.g., linkage, cams, gear trains and so on. Number Synthesis deals with the number of links or joints (pairs) to obtain certain mobility. Dimensional Synthesis is the detailed design process, in which the dimensions (say length) of individual links are decided. Note: Some authors combine Type and Number Synthesis and call it as Type Synthesis, since Number synthesis is not a generally applicable step for all kinds of mechanisms.

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Kinematic Synthesis - Requirements: In general the motion requirements specified for a synthesis problem could be classified into: Function Generation: There should be a specified relation [like y = log(x)], between the input motion (say, the rotation of input crank) and output motion (say, the rotation of the follower) Path Generation: A point on the mechanism (usually a point, on the coupler link, called the coupler point) should generate a prescribed curve (coupler curve). If the coupler point has to reach specific positions on the path at prescribed time (or rotation of input link), the synthesis problem is called Path Generation with prescribed timing. Motion Generation (Body Guidance): The requirement is to move a rigid body from a position to another (the orientation of the rigid body at each position and in between points will be significant).

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Function Generation: The usual requirement in function generation is to design the mechanism for 2-5 precision points. Precision Point: In context of function generation, the precision points can be defined as the points at which the specified function and the function generated by the mechanism are exactly matching. Similar definitions can be given in path and motion generation cases also! If the precision points are not prescribed in the synthesis problem, they could be chosen to minimize the structural error, which is the difference between the specified function and the function generated by the mechanism. A first estimate for precision points is obtained by using the Chebychev Spacing (roots of Chebychev polynomial), which could be further optimized to reduce the structural error.

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Function Generation – Two Precision Points: Problem:

A1

A2

B2 2 1

O2

B1 2 1

O4

Kinematic Requirements: For the function generation, the precision points are given by ( 1, 1) & ( 2, 2), where the first coordinate is the input rotation and second coordinate is the output rotation.

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Function Generation – Two Precision Points: A2’

Solution:

A1

A2

O2’

2

B2

2 1

O2

B1

2 1

12

O4

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Function Generation – Two Precision Points: A2’

Solution:

A1

A2

O2’

2

B2

2 1

O2

B1

2 1

12

O4

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Function Generation – Three Precision Points: Problem:

A3

A2

A1

B3

B2 23

3

2

B1 12

??

1

O2

O4

Kinematic Requirements: For the function generation, the precision points are given by ( 1, 1), ( 2, 2) & ( 3, 3), where the first coordinate is the input rotation and second coordinate is the output rotation.

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Function Generation – Three Precision Points: Solution:

A3” A2’

O2” A3

A2

3

B2 B3

2

B1

O2’ A1 3

23 13

2 1

O2

12

12

O4

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Function Generation – Three Precision Points: Solution:

A3” A2’

O2” A3

A2

3

B2 B3

2

B1

O2’ A1 3

23 13

2 1

O2

12

12

O4

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Function Generation – The Overlay Method: Problem: Four or five precision points

A4

A2

A3

A1

B4

B3

B2 B1

O2

O4

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Function Generation – The Overlay Method contd.… B4 B3 B2 A4

A2

A3

A1

B1 O2

Figure on Transparent sheet 12

Function Generation – The Overlay Method contd.…

B3

B2

B4

B1

O4

Figure on another sheet of paper 13

Function Generation – The Overlay Method contd.…

A4

B4

A2

A3

A1

B3 B2

B1

O2 O4

Overlay both the figures to get a suitable mechanism. 14

Function Generation – The Overlay Method contd.…

A4

B4

A2

A3

A1

B3 B2

B1

O2 O4

Overlay both the figures to get a suitable mechanism. 15

Path Generation: The coupler point has to pass through a fixed number of points on a curve (usually 3 – 6 points). The task becomes more difficult in case of path generation with prescribed timing. A graphical method similar to the function generation can be used, but the inversion to be taken is by fixing the coupler.

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Path Generation – Three Prescribed Positions: Problem: P2

P3

P1

A2

A3

A1 O2

O4

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Path Generation – Three Prescribed Positions: Problem: P2

P3

P1

A2 O2”

B1 A3

A1

O4

O2 O2’ O4’ O4”

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Path Generation – Three Prescribed Positions: Problem: P2

P3

P1

A2 O2”

B1 A3

A1

O4

O2 O2’ O4’ O4”

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Path Generation with Prescribed Timing: Problem: Three prescribed positions P3

P2 P1

A2 A1

12

13

O2

A3 O4

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Path Generation with Prescribed Timing: Solution: P3

P2

P2’ P1

P3’ A1

13

12

12 13

O2

O4

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Motion Generation: Assume that the coupler link has to pass through a number of positions (rigid body motion – translation & rotation). Example for two prescribed positions.

B A’

B’

A

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Motion Generation: Example for two prescribed positions.

B A’

B’

A

O2

O4

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Motion Generation: Example for three prescribed positions. B’ B

A’ A”

A B”

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Motion Generation: Example for three prescribed positions. B’ B

A’ A”

A B” O4

O2 25

Motion Generation: Example for three prescribed positions. B’ B

A’ A”

A B” O4

O2 26