Kutzbach criterion, Grashoff's law Kutzbach criterion

Kutzbach criterion, Grashoff's law Kutzbach criterion:

·        Fundamental Equation for 2-D Mechanisms: M = 3(L – 1) – 2J1 – J2

Can we intuitively derive Kutzbach’s modification of Grubler’s equation? Consider a rigid link constrained to move in a plane. How many degrees of freedom does the link have? (3: translation in x and y directions, rotation about z-axis)

·        If you pin one end of the link to the plane, how many degrees of freedom does it now have?

·        Add a second link to the picture so that you have one link pinned to the plane and one free to move in the plane. How many degrees of freedom exist between the two links? (4 is the correct answer)

·        Pin the second link to the free end of the first link. How many degrees of freedom do you now have?

·        How many degrees of freedom do you have each time you introduce a moving link? How many degrees of freedom do you take away when you add a simple joint? How many degrees of freedom would you take away by adding a half joint? Do the different terms in equation make sense in light of this knowledge?

Grashoff's law:

·        Grashoff 4-bar linkage: A linkage that contains one or more links capable of undergoing a full rotation. A linkage is Grashoff if: S + L < P + Q (where: S = shortest link length, L = longest, P, Q = intermediate length links). Both joints of the shortest link are capable of 360 degrees of rotation in a Grashoff linkages. This gives us 4 possible linkages: crank-rocker (input rotates 360), rocker-crank-rocker (coupler rotates 360), rocker-crank (follower); double crank (all links rotate 360). Note that these mechanisms are simply the possible inversions (section 2.11, Figure 2-16) of a Grashoff mechanism.

·        Non Grashoff 4 bar: No link can rotate 360 if: S + L > P + Q

Let’s examine why the Grashoff condition works:

·        Consider a linkage with the shortest and longest sides joined together. Examine the linkage when the shortest side is parallel to the longest side (2 positions possible, folded over on the long side and extended away from the long side). How long do P and Q have to be to allow the linkage to achieve these positions?

·        Consider a linkage where the long and short sides are not joined. Can you figure out the required lengths for P and Q in this type of mechanism