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

**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

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Mechanical : Kinematics of Machinery : Basics of Mechanisms : Kutzbach criterion, Grashoff's law Kutzbach criterion |

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