Four Different Positions (Inversions) of Model C01

David W. Henderson and Daina Taimina

Inversions

In this picture is the 4-bar linkage of model C01 with the link of length 100 held fixed.. Reuleaux designed a stand that made it possible to hold fixed any of the other three links -- the results are different motions.  These different positions of the C01 linkage are called inversions.  Below we explore the geometry of these four inversions of C01.

 

First Inversion

See 4-bar linkages for a discussion of the eight main types of 4-bar linkages. Using the notation from 4-bar linkages, we have  g = 100, a = 30, h = 94, and b = 39. Thus,

b+h = 133 > 130 = a+g  and  |a-g| = 70 > 55 = |b-h|

a+h = 124 < 139 = b+g  and  |b-g| = 61 < 64 = |a-h|.

From the chart in 4-bar linkages we have that this is a crank-rocker, which means that the input link (on the left) can rotate completely around but the output link on the right will rock back and forth between

 

and

.

Second Inversion.

 

In this inversion fix the link of length 30 as the fixed link g and consider the link of length 94 as the input link a. (The crank handle on the mechanism will no longer function as a crank in this inversion.)  We then have   g = 30, a = 94, h = 39, and b = 100. Thus,

b+h = 139 > 124 = a+g  and  |a-g| = 64 > 61 = |b-h|

a+h = 133 > 130 = b+g  and  |b-g| = 70 > 55 = |a-h|.

From the chart in 4-bar linkages we have that this is a double crank, which means that both the  input link (length 94) and the output link (length 100) can rotate completely around.

Third Inversion.

 

In this inversion fix the link of length 94 as the fixed link g and consider the link of length 39 as the input link a.  We then have   g = 94, a = 39, h = 100, and b = 30. Thus,

b+h = 130 < 133 = a+g  and  |a-g| = 55 < 70 = |b-h|

a+h = 139 > 124 = b+g  and  |b-g| = 64 > 61 = |a-h|.

From the chart in 4-bar linkages we have that this is a rocker-crank, which means that the output link (length 30) can rotate completely around but the input link (length 39) will rock back and forth between

 

and

.

Fourth Inversion.

In this inversion fix the link of length 39 as the fixed link g and consider the link of length 100 as the input link a.  We then have   g = 39, a = 100, h = 30, and b = 94. Thus,

b+h = 124 < 139 = a+g  and  |a-g| = 61 < 64 = |b-h|

a+h = 130 < 133 = b+g  and  |b-g| = 55 < 70 = |a-h|.

From the chart in 4-bar linkages we have that this is a double crank, which means that both the  input link (length 94) and the output link (length 100) can rotate completely around.