Synthesis of Six-bars

The six-bar linkage is the next simplest 1-DOF planar mechanism after the four-bar. However, designing these mechanisms presents a challenging set of synthesis equations. For example, a four-bar motion generator is determined by a polynomial system that has 4 isolated roots (of which one is the origin), where each root represents a potential design. On the other hand, a six-bar function generator is determined by a polynomial system that could have as many as 4.13×108 roots. Chances are the actual number of roots of this system is much smaller (by 2 or 3 magnitudes), but that number is unknown. It is much harder to find complete solution sets for general six-bar design problems than it is for four-bars.

From a practical point of view, six-bars have 2 more moving parts than four-bars. With the increased complexity comes advantages:

1. More design options. As mentioned above, each root above represents a design candidate, and surely it is better to have thousands of design candidates than 3. Especially because most of these design candidates will suffer from linkage defects rendering them useless.

2. Six-bars do more complex stuff. Four-bars can move through 5, 5, and 9 positions for function, motion, and path generation, respectively. On the other hand, six-bars can move through 11, 8, and 15 positions. The maximum degree of the curve traced by a point on the coupler link of a four-bar versus a six-bar is 6 to 18. Six-bars have even been designed that can perform 2:1 gear ratios.

3. You can put things where they need to be. That is there is much more design freedom shaping links and choosing the locations of pivots. This is useful if your linkage needs to mount to specific hard points or fit within a certain envelope. Four and six-bar motion generators provide a good example. There will be a maximum of 6 four-bars that move through 5 task positions with no pivots specified. Meanwhile, there will be a maximum of 5,735 six-bars that move through 6 task positions with both ground pivots specified!

Here is a bit on my research with six-bars. Six-bars are divided into 5 kinematic inversions. For simplicity, we’ll just call these types. There are three common motion requirement tasks. The table below provides a breakdown of the size of the synthesis problem for each linkage type over the three motion tasks.

This table shows six-bar info

Function Motion Path
Watt I Max positions: 5
Degree: 4
Max positions: 8
Degree: 3.43×1010
Max positions: 15
Degree: 2.28×1046
Watt II Max positions: 11
Degree: 5.50×107
Max positions: 5
Degree: 4
Max positions: 9
Degree: 268,720
Stephenson I Max positions: 5
Degree: 4
Max positions: 5
Degree: 4
Max positions: 15
Degree: ?
Stephenson II Max positions: 11
Degree: 4.13×108
Max positions: 5
Degree: 4
Max positions: 15
Degree: ?
Stephenson III Max positions: 11
Degree: 4.13×108
Max positions: 5
Degree: 4
Max positions: 15
Degree: ?

 

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