Purpose of the webpage is to document the results of my exploration of optimizing the classical 4-bar linkage as schematically shown in Figure 1 to convert the rotational motion of point 1 to a nearly linear, vertical motion of point 4 or vice versa. Interest in this linkage was sparked by a suggestions of Rick Topf, a member of the SESUSA ( Stirling Engine Society of the USA ) user group, to use , what he calls , a bowtie mechanism to convert the linear motion of a piston into rotational motion with the objective of reducing the friction between piston an cylinder walls.
The essential portion of this mechanism is known in the kinematics/dynamics community as a 4-bar linkage and is sketched out in Figure 1. By choosing suitable values for the dimensions r,a,b,c,d,e, and f the motion of point 4 approaches that of a vertical ( y-direction ) straight line as point 1 moves around on a circle centered at point 0.
Principally, it is not possible for such a linkage to achieve an exactly linear motion of point 4 and there does not exist a single optimal configuration.
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| 4 = connecting point, to be moved along a vertical line as much as possible 01 = crank, length r, rotating about fixed point 0, the crank 23 = swing bar, length c, pivoting about fixed point 2 1354 = bellcrank = rigid body, pin/hole connection at points 1 and 3 dimensions : 13=d , 35=f , 54=e f can be positive ( point 5 below point 3 ) , zero , or negative ( point 5 above point 3 ) line 1-3 is perpendicular to line 54 |
As the angle α is increased from 0 to 360° point 4 moves around on a close-looped orbit which, depending on the values of the paramters r,a,c,d,e,f, and η, has different shapes ranging from asymmetric ovals, ovals with cusps, ovals with vanishing waist, and bowtie-type orbits. Notice that in these examples only the lengths c and d have been varied by less than 2 mm ( r=10 mm) while the difference between the largest and smallest x-value for the orbit changes by a factor of 10.
As there does not exists a single optimum configuration - in fact there exists infinitely many - three other criteria have to enter when deciding upon the values for the parameters r through f.
Table 1 below shows a subset of optimal configuations. As the crank turns once around the x_location of point 4 differs by an amount given in the column labelled x4_diff. Optimal means that if you change any of the parameters a,b,c,d,e, and f by just a tiny bit (like 10-6mm) the value of x4_diff will increase.
Two of the three design decision which were made to generate Table 1 are :
Lastly, as the third design decision, we have to choose which of the shown configuration to build, there is clearly a trade-off between the desire to minimize x4_diff and the size of the rectangular box surrounding the entire mechanism. Please follow the links in the 1. column to get more information about the motion of the connecting point 4.
The vertical motion of connecting point 4 is equivalent to the stroke of a piston's engine and will therefore differ from design to design.
Still, Table 1 can be used rather easily assuming that the designer has made a decision on the stroke at hand and either the size of the surrounding box or the value of x4_diff to be achieved.
Example : y4_diff=58mm and x4_diff ≤ 0.3 mm is desired
Step 1 : Scale back x4_diff to 0.3*50/58=0.2586mm
In Table 1 the 7'th row contains in its 1.column x4_diff=0.250039, slightly better than desired.
Choose the values in this 7'th row as template, that is take the displayed values for the parameters and scale them up by a factor of 58/50 :
Your configuration is :
| Your optimal Configuration, y4_diff=58mm | ||||||||
|---|---|---|---|---|---|---|---|---|
| x4_diff | box-width | box-height | r | a=e | b | c | d | f |
| 0.290045 | 203.050479 | 131.038827 | 20.406263 | 102.031316 | 102.033892 | 103.052930 | 102.546626 | -0.512733 |
and it is exactly as optimal as the configuration shown in row 7 of Table 1.
| Table 1a : Optimal Configurations, e/a=1.0 ; y4_diff=50mm | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| x4_diff | box-width | box-height | r | a | e | b | c | d | f |
| 1.002325 | 85.105144 | 68.441794 | 17.357154 | 43.392885 | 43.392885 | 43.410969 | 45.138005 | 44.296907 | -0.885938 |
| 0.858824 | 92.388670 | 72.017463 | 17.399357 | 46.978264 | 46.978264 | 46.992569 | 48.596714 | 47.812397 | -0.819828 |
| 0.695247 | 103.271165 | 77.375399 | 17.448829 | 52.346488 | 52.346488 | 52.356876 | 53.805896 | 53.094296 | -0.737421 |
| 0.510529 | 121.314806 | 86.291416 | 17.506557 | 61.272951 | 61.272951 | 61.279464 | 62.526742 | 61.911211 | -0.631747 |
| 0.390773 | 139.277980 | 95.192915 | 17.545106 | 70.180422 | 70.180422 | 70.184774 | 71.279201 | 70.737410 | -0.552636 |
| 0.308716 | 157.181767 | 104.083090 | 17.572057 | 79.074254 | 79.074254 | 79.077305 | 80.052024 | 79.568468 | -0.491163 |
| 0.250039 | 175.043516 | 112.964506 | 17.591606 | 87.958031 | 87.958031 | 87.960252 | 88.838733 | 88.402264 | -0.442011 |
| 0.173624 | 210.677664 | 130.708323 | 17.617425 | 105.704550 | 105.704550 | 105.705833 | 106.439255 | 106.074146 | -0.368313 |
| 0.127556 | 246.236360 | 148.434728 | 17.633192 | 123.432342 | 123.432342 | 123.433150 | 124.062505 | 123.748835 | -0.315686 |
| 0.097659 | 281.746440 | 166.149661 | 17.643507 | 141.148058 | 141.148058 | 141.148598 | 141.699689 | 141.424819 | -0.276220 |
| 0.077162 | 317.216309 | 183.856686 | 17.650617 | 158.855557 | 158.855557 | 158.855937 | 159.346043 | 159.101464 | -0.245527 |
| 0.062501 | 352.673899 | 201.558050 | 17.655723 | 176.557226 | 176.557226 | 176.557503 | 176.998758 | 176.778476 | -0.220973 |
| 0.051653 | 388.105392 | 219.255232 | 17.659510 | 194.254613 | 194.254613 | 194.254821 | 194.656069 | 194.455705 | -0.200884 |
| 0.043403 | 423.525661 | 236.949242 | 17.662397 | 211.948764 | 211.948764 | 211.948924 | 212.316811 | 212.133067 | -0.184143 |
| 0.036982 | 458.941091 | 254.640789 | 17.664647 | 229.640413 | 229.640413 | 229.640539 | 229.980181 | 229.810517 | -0.169978 |
Table 1 can be extended indefinitly to values of x4_diff above, below, or anywhere inbetween those already listed. Furthermore, the design condition of e=a can be dropped as well, providing additional optimal configurations.
Surprisingly, dropping the design decision of a=e has little influence on the compromise one must make between lowering x4_diff and increasing the horizontal extent, Hbox, of the box surrounding the mechanism. Over the range investigated, the ratio e/a has very little affect on the relationship between the two, as shown in Figure 2a.
The vertical size, Vbox, of the mechanism experiences the same fate, there is pretty much a one to one relationship between x4_diff and Vbox which is almost independent of the ratio, e/a, click here for graphs of Vbox as function of x4_diff and Vbox as function of Hbox.
Therefore the ratio e/a is still a free design parameter. It is known though (link will follow later at this spot), that if the 4-bar linkage is used in the classical bowtie configuration for beta-type Stirling engines, the ratio e/a influences strongly the ratio between the maximum size of the compression to expansion volume which in turn is determined by thermodynamic considerations.
Click here for tables like Table 1a but for e/a ranging from 0.5 to 2.0.
| Figure 2a : Horizontal size, Hbox, of box surrounding
mechanism versus horizontal size of box surrounding orbit of point 4 for various ratios e/a at y4_diff=50mm |
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