Freedom and Constraints

A body that is free in space can move in three, independent perpendicular directions and rotation in three ways about those directions. It is said to have 6 degrees of freedom (DoF).

The number of degrees of freedom are the number of components of motion that are required in order to generate the motion.

If a joint is constrained to move along a line, it has 1 degree of freedom. If a joint is constrained to a plane with no rotation then it has two degrees of freedom.

Gruebler's Equation of Degrees of Freedom

M = 3L - 2J - 3G

where M is degrees of freedom, L is number of links, J is number of joints, and G is number of grounded links.

if DoF > 0, it is a mechanism with relative motions
if DoF = 0, it is a structure without motion
if DoF < 0, it is a preloaded (or overly constrained) structure without motion but possibly additional stress

Grashof Condition

Four bar linkage is the simplest possible pin-jointed mechanism.

The Grashof condition is a simple relationship that predicts the linkage behaviour based only on the lengths of the links. A linkage is a Grashof linkage if at least one link is capable of making a full revolution and the following equation holds true:

S + L < P + Q

where

S is the length of shortest link
L is the length of longest link
P is the length of one remaining link
Q is the length of other remaining link

Triple Rockers

If S + L > P + Q then it is not a Grashof linkage and is known as a triple rocker.

Double Cranks / Crank Rockers

S + L = P + Q

Limit Conditions

Toggle is an important test to check if the linkage can reach all specified positions without encountering a limit or toggle position, also called a station configuration.

Toggle positions may be determined by collinearity of two links

Transmission Angle

Transmission angle, \mu, is the angle between the output link and the coupler.

It is a useful test to check the quality of the force transmission in a linkage.

Ideally \mu = 90 deg but at minimum \mu \ge 40 deg.

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