Gruebler’s and Kutzbach’s Equations

Kinematic chains

An organized arrangement of interconnected links which are rigid bodies, that are either firmly interconnected or in direct contact with one another.

If a single one of these chains is immobilized and the displacement of a different link causes the link to move to a predictable position, the system is then a constrained kinematic chain

If not, the system is an unconstrained kinematic chain

Joints
A full joint has 1 DOF and do not allow relative movement between connected members

A half joint has 2 DOF and is a semi-rigid connection which allows rotational movement between connected members.

Degrees of Freedom

Degrees of Freedom

The number of independent parameters needed to uniquely identify a systems position

You calculate DOF / Mobility using…
Greubler’s Equation
$M=3L-2J-3G$
Where:

• M = degrees of freedom or mobility
• L = number of links
• J = number of joints
• G = number of grounded links

Deception

Greublers Equation can sometimes be misleading since it doesn’t consider link geometry…

Kutzbach’s Modification of Greubler’s Equation
$M=3\left(L-1\right)-2J_1-J_2$
Where:

• M = degrees of freedom or mobility
• L = number of links
• $J_1$ = number of full joints (1 DOF)
• $J_2$ = number of half joints (2 DOF)

This modification is independent of link size or shape and for multiple joints, count one less than the number of links joined and add to the $J_1$ variable (full joints.

Counting Links and Joints

Sliders count as an extra link and an extra full joint and not a half joint

Interpreting DOF Results

If:

• DOF > 0 links have relative motion ⇒ Mechanism
• DOF = 0 no motion is possible ⇒ Structure
• DOF < 0 no motion is possible and stresses might be present ⇒ Preloaded Structure