Chebychev–Grübler–Kutzbach criterion (simple closed chain)

Description

The Chebychev–Grübler–Kutzbach criterion determines the degree of freedom of a kinematic chain, that is, a coupling of rigid bodies by means of mechanical constraints.These devices are also called linkages.The Kutzbach criterion is also called the mobility formula, because it computes the number of parameters that define the configuration of a linkage from the number of links and joints and the degree of freedom at each joint. The mobility formula counts the number of parameters that define the positions of a set of rigid bodies and then reduces this number by the constraints that are imposed by joints connecting these bodies. Joints that connect bodies in this system remove degrees of freedom and reduce mobility. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. For a simple closed chain, n moving links are connected end-to-end by n+1 joints such that the two ends are connected to the ground link forming a loop. In this case, we have N=j and the mobility of the chain is given by the Chebychev–Grübler–Kutzbach criterion formula ( with N=j).

Related formulas

Variables

MchainclosedThe mobility of the chain (dimensionless)
jThe final number of joints (dimensionless)
iNumber of joints (1,2,3....j) (dimensionless)
fiThe number of joints' freedom (dimensionless)