Determining the d.o.f. count

Determining the d.o.f. count#

Grübler’s formula for planar mechanisms#

It is a systematic method to find out the number of d.o.f. of a mechanism.

Formula
\[G=3(N-1) -2 p_{I} - p_{II}\]

With:

  • \(G\): Overall d.o.f. (the result we are looking for!).

  • \(N\): Number of links, including the ground.

  • \(p_{I}, p_{II}\): Number of degree I and degree II joints.

Formula discussion:

  • Each free planar body has 3 d.o.f.; hence, the \(N-1\) moving links have \(3(N-1)\) d.o.f. (the ground has none).

  • Each degree I joint (such as R or P) connecting two links removes 2 d.o.f.

  • Each degree II joint (such as a cam) connecting two links removes 1 d.o.f.

Kutzbach’s general formula#

Grübler’s formula is a special case of Kutzbach’s formula, which is more general and applies to both planar and spatial mechanisms.

For planar mechanisms: It is identical to Grübler’s.

\[G = 3(N-1) - 2p_I - p_{II}\]

For spatial mechanisms (3D):

\[G = 6(N-1) - \sum_{i=1}^{5} (6-i)p_i = 6(N-1) - 5p_1 - 4p_2 - 3p_3 - 2p_4 - 1p_5\]

where a free body in space has 6 d.o.f., and \(p_i\) is the number of joints allowing \(i\) degrees of freedom. For example, a revolute joint (\(p_1\)) removes 5 d.o.f.

Practical application of the formula#

Follow these steps

  • Number all links: Ground is #1, etc.

  • Identify the kinematic pairs: Tell degree I from degree II joints.

  • Apply the formula: \(G=3(N-1) -2 p_{I} - p_{II}\)

Numbering options#

It is common to find more than one way to number links and joints, but they will eventually be equivalent. For example:

gruebler

The slider block is modeled as an additional link, with two degree I joints.

gruebler

The block is not counted as a link, but one degree II joint is counted.

Exceptions and problematic cases#

The Grübler/Kutzbach formula can fail in certain cases:

  • Redundant constraints: Sometimes a mechanism has more constraints than strictly needed to produce a determinate motion. The formula may predict 0 or fewer d.o.f., yet the mechanism still moves because of a very specific geometry (e.g., a five-bar parallelogram mechanism). These extra constraints are not independent.

  • Passive or local degrees of freedom: Some mechanisms have local motions (such as a ball rolling in a track) that do not affect the mechanism’s overall motion. The formula may count these d.o.f., leading to an incorrect result for the main mobility.

Numerical methods criterion#

When using multibody formulations for machine and mechanism analysis, there exists a more advanced alternative to Grübler’s method:

  • Based on the Jacobian matrix of the constraints.

  • It works for both 2D and 3D mechanisms.

  • It detects singular configurations (Grübler fails!).

This method will not be studied in the present introductory course.