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Mathematics of Four-Bar Linkages

 
Planar and Spherical

by David W. Henderson and Daina Taimina

 

The four bar linkage is a mechanism that lies in a plane (see C01) (or spherical surface, C03) and consists of four bars connected by joints that allow rotation only in the plane (or sphere) of the mechanism. We will add this red color to indicate the slight changes that are necessary when considering spherical linkages instead of planar linkages.

 

In normal practice one of the links is fixed so that it does not move. In the linkage to the right we assume that the link OC is fixed and investigate the possibilities of motion for the other three links. We suggest that the reader play with the simulation by changing the length of links h and b and experiencing the motions that are possible. 

We call the link OA the input crank and link CB the output crank. Similarly, we call the angle θ the input angle and angle  the output angle.

 

On a sphere we measure the lengths of the edges by the angle (in radian measure) that the side subtends from the center of the sphere. We assume here that all links have lengths less than  = 180 degrees (that is, we assume that links have lengths less than half a great circle) This is not a serious restriction because a link of length  l >  can always be replaced by a link of length   - l  that together with the original link completes a great circle. However, below we sometimes will need to consider what happens when two links are collinear (for example, when A-O-C are in a line) then the sum (in the example, a+g) of the lengths of the two links may be l >  and then the distance between the end points (in the example, A and C) will not be l but will rather be the length of the shorter great circle arc, l- . [We suggest that you try this out by drawing great circle arcs on an orange or tennis ball.] For this reason, when we write |l|_S we will mean the distance between the end points.

• On the plane |l|_S = |l|, the usual absolute value.
• On the sphere, |l|_S = |l|, if  |l|
  
  , and |l|_S =
  
   - |l|, if  |l|
  
  .

 

The reader may experiment with the above crank-crank simulation and discover that the input crank will be able to swing opposite C (when the input angle is π = 180 degrees) only if the distance between C and A is allowed to be |a+g|_S and this will only be allowed by the mechanism if |a+g|_S < b+h. See also movie of a crank-crank linkage. There will be a maximum input angle if  |a+g|_S > b+h. (If  |a+g|_S = b+h  then the link folds, that is, it has a configuration in which all four links lie in the same straight line.) Experience this in the rocker-crank simulation and the movie.

 

By applying the Law of Cosines to this configuration (when A-B-C is a straight line) we see that this maximum angle satisfies:

.

Or using the spherical Law of Cosines:

.

 

Similarly, there will be a minimum input angle whenever  a+|b-h| > g (if  a<g)  or  g+|b-h| > a (if  a>g)  which are equivalent to  |b-h| > |g-a|.  The minimum angle is actualized when either B-A-C is straight or A-C-B is straight, as in the figure.

 

In both of these cases we can use the Law of Cosines to conclude:

.

.

Thus we have four types of input cranks:

• A crank if the link OA can freely rotate completely around O. 
  In this case,  b+h > |a+g|_S  and  |a-g| > |b-h|.
• A 0-rocker if there is a maximum input angle but the link OA can rotate freely thru θ = 0.  In this case,  b+h < |a+g|_S  and  |a-g| > |b-h|.
• A π-rocker if there is a minimum input angle but the link OA can rotate freely thru θ = π.  In this case,  b+h > |a+g|_S  and  |a-g| < |b-h|.
• A rocker if there is a maximum input angle and minimum input angle.
  In this case,  b+h < |a+g|_S  and  |a-g| < |b-h|.

 

The analysis of the output crank is exactly symmetric to the above with the lengths a and b interchanged. In particular:

 

If  |b+g|_S > a+h, then there is a maximum output angle _max and, by applying the Law of Cosines to this configuration (when B-A-O is a straight line) we see that this maximum angle satisfies:

 

.

 

 

 

 

 

If  |a-h| > |g-b|, there is a minimum output angle _min and, by applying the Law of Cosinges (when either A-B-O is straight or B-O-A is straight) we see that this minimum angle satisfies:

           

 

 

 

 

.

 

 

Thus we have four types of output cranks:

• A crank if the link CB can freely rotate completely around C. 
  In this case,  a+h > |b+g|_S  and  |b-g| > |a-h|.
• A 0-rocker if there is a maximum output angle but the link CB can rotate freely thru
  
   = 0.  In this case,  a+h < |b+g|_S  and  |b-g| > |a-h|.
• A π-rocker if there is a minimum output angle but the link CB can rotate freely thru
  
   = π.  In this case,  a+h > |b+g|_S  and  |b-g| < |a-h|.
• A rocker if there is a maximum output angle and minimum output angle.
  In this case,  a+h < |b+g|_S  and  |b-g| < |a-h|.

 

Putting these together we get eight types of 4-bar linkages:

1. A double crank in which both the input and output links are cranks. 
   b+h > |a+g|_S,  |a-g| > |b-h|,  a+h > |b+g|_S,  and  |b-g| > |a-h|. See crank-crank simulation and movie.
   
2. A crank-rocker if the input link is a crank and the output link is a rocker.
   b+h > |a+g|_S,  |a-g| > |b-h|,  a+h < |b+g|_S,  and  |b-g| < |a-h|. See crank-rocker simulation and movie.
   
3. A rocker-crank if the input link is a rocker and the output link is a crank.
   b+h < |a+g|_S,  |a-g| < |b-h|,  a+h > |b+g|_S,  and  |b-g| > |a-h|. See rocker-crank simulation and movie.
   
4. A rocker-rocker if both the input link and the output link are rockers.
   b+h < |a+g|_S,  |a-g| < |b-h|,  a+h < |b+g|_S,  and  |b-g| < |a-h|. See rocker-rocker simulation and movie.
   
5. A 00 double rocker if the input and output angles both have maximums but no minimums and thus both cranks move freely across the fixed link OC. b+h < |a+g|_S,  |a-g| > |b-h|,  a+h < |b+g|_S,  and  |b-g| > |a-h|. See 00 double rocker simulation.
6. A 0π double rocker if the input angle has a maximum and no minimum but the output angle has a minimum but no maximum. Thus the input crank moves freely across OC while the output crank moves freely on the side of C opposite O.
   b+h < |a+g|_S,  |a-g| > |b-h|,  a+h > |b+g|_S,  and  |b-g| < |a-h|. See 0pi double rocker simulation.
7. A π0 double rocker if the input angle has a minimum and no maximum but the output angle has a maximum but no minimum. Thus the output crank moves freely across OC while the input crank moves freely on the side of O opposite C.
   b+h > |a+g|_S,  |a-g| < |b-h|,  a+h < |b+g|_S,  and  |b-g| > |a-h|.  See pi0 double rocker simulation.
8. A ππ double rocker if the input and output angles both have minimums but no maximums and thus neither cranks move freely across the fixed link OC, but both move freely on the sides opposite OC.
   b+h > |a+g|_S,  |a-g| < |b-h|,  a+h > |b+g|_S,  and  |b-g| < |a-h|.  See pipi double rocker simulation.

 

You can check that the other eight combinations (a 0 or π rocker combined with a crank or rocker) are not possible. This can be done either analytically (by showing that the resulting inequalities have no solutions), or geometrically (by noting that a 0 or π rocker has a motion that is symmetric by mirror symmetry across OC whereas the crank and rocker do not have this symmetry).

 

All the other 4-bar linkages are in the case when one or more of the inequalities become equalities, in each of these cases the linkage can be folded. That is, the linkage has a configuration in which all the links line up with OC. Some 4-bar linkages can be folded in more than one way; for example, the linkage with a=h=b=g can be folded in three different ways (try it!).