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I'm having trouble understanding how to apply the Kutzbach Gruebler equation to all but the simplest of mechanisms.

As I understand it, the DOFs of a mechanism is a sum of the DOFs of all the moving links minus the sum of the all the DOFs constrained by the joints. For a planar four bar linkage with one link grounded, there are 3 moving links (so total DOFs = 3 links * 3 DOF = 9) and 4 rotary joints (constrained DOFs = 4 joints * 2 DOF = 8). The total DOF = 9 - 8 = 1.

Now consider the pantograph mechanism shown in the attached picture. I can see that there are a total of 7 links (including the base), 6 of which are moving (total DOFs = 6 links * 3 DOF = 18). However, I can only identify 7 joints (5 rotary, 2 prismatic), which would constrain a total of 7 * 2 = 14 DOF, giving the total DOFs as 18 - 14 = 4, which I know is incorrect. According to the book I got the picture out of, there are actually 8 joints, which gives the correct answer of 2 DOF.

Likewise, I also have a problem with the mechanism on the left hand side of the following image:

http://www.cs.cmu.edu/~rapidproto/mechanisms/figures/dcacu.gif

I can see 5 moving links (15 DOF) but only 6 joints (5 rotary and 1 prismatic: 12 DOF) instead of 7, giving a total DOF = 3, whereas the correct answer (according to the site) is 1.

So, for some reason, there always seems to be an extra joint that I'm unable to identify. What am I missing?

Thanks

Amr

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# Applying the Kutzbach Gruebler criterion to a pantograph

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