BIZOMAT MAKEOS · Mechanics
Linkage Degree of Freedom (Gruebler and Kutzbach)
How many degrees of freedom does my planar (or spatial) linkage have, from its numbers of links and joints, and is it a mechanism, a structure or a preloaded structure?
Mobility of a linkage by the equations Norton gives: Kutzbach’s form of Gruebler’s equation for planar mechanisms M = 3(L − 1) − 2J1 − J2 with full (one-freedom) and half (two-freedom) joints, multiple joints counting as one less than the number of links they join, and the Kutzbach equation for spatial linkages M = 6(L − 1) − 5J1 − 4J2 − 3J3 − 2J4 − J5, with the classification into a mechanism (M > 0), a structure (M = 0) or a preloaded structure (M < 0).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Linkage | choice | Whether the links move in one plane or in three dimensions. |
| Number of links L (including the ground) | links | Number of links, counting the ground (frame) as one link. |
| One-freedom (full) joints J1 | joints | Full joints: pin and slider joints that allow one freedom; a multiple joint counts as one less than the links it joins. |
| Two-freedom (half) joints J2 | joints | Half joints: roll-slide joints such as cam or gear contacts that allow two freedoms. |
| Three-freedom joints J3 (spatial) | joints | Spatial joints with three freedoms, such as a ball joint. |
| Four-freedom joints J4 (spatial) | joints | Spatial joints with four freedoms. |
| Five-freedom joints J5 (spatial) | joints | Spatial joints with five freedoms. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Linkage
- What it is
- Whether the links all move in one plane or in three dimensions.
- Units and format
- Choose one of 2 options from the list: Planar (Gruebler–Kutzbach); Spatial (Kutzbach).
- Example
- The form opens with “Planar (Gruebler–Kutzbach)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the planar or spatial mobility equation.
- Get it right
- Most machine linkages are planar; use spatial for ball-jointed or skewed linkages.
What this means: Number of links L (including the ground)
- What it is
- Number of links in the linkage, counting the frame as one link.
- Units and format
- Enter a number in links: links. The form accepts 1 links to 1000 links. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 links as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each link adds three (planar) or six (spatial) freedoms.
- Get it right
- Count every rigid body, including the ground or frame once.
What this means: One-freedom (full) joints J1
- What it is
- Number of joints that allow one relative freedom, such as pins and sliders.
- Units and format
- Enter a number in joints: joints. The form accepts 0 joints to 1000 joints. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 joints as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each removes two (planar) or five (spatial) freedoms.
- Get it right
- Count a joint shared by n links as n − 1 full joints.
What this means: Two-freedom (half) joints J2
- What it is
- Number of joints that allow two freedoms, such as cam or roll-slide contacts.
- Units and format
- Enter a number in joints: joints. The form accepts 0 joints to 1000 joints. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 joints as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each removes one planar freedom.
- Get it right
- Gear tooth and cam follower contacts are usually half joints.
What this means: Three-freedom joints J3 (spatial)
- What it is
- Number of spatial joints that allow three freedoms, such as ball joints.
- Units and format
- Enter a number in joints: joints. The form accepts 0 joints to 1000 joints. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 joints as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each removes three spatial freedoms.
- Get it right
- Only used for spatial linkages.
What this means: Four-freedom joints J4 (spatial)
- What it is
- Number of spatial joints that allow four relative freedoms.
- Units and format
- Enter a number in joints: joints. The form accepts 0 joints to 1000 joints. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 joints as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each removes two spatial freedoms.
- Get it right
- Only used for spatial linkages.
What this means: Five-freedom joints J5 (spatial)
- What it is
- Number of spatial joints that allow five relative freedoms.
- Units and format
- Enter a number in joints: joints. The form accepts 0 joints to 1000 joints. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 joints as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each removes one spatial freedom.
- Get it right
- Only used for spatial linkages.
Method and formulas
- Planar: M = 3(L − 1) − 2J1 − J2 (Kutzbach’s modification of Gruebler’s equation); a multiple joint joining n links counts as n − 1 full joints.
- Spatial: M = 6(L − 1) − 5J1 − 4J2 − 3J3 − 2J4 − J5.
- M > 0: a mechanism with M independent inputs; M = 0: a structure; M < 0: a preloaded structure.
- Example: 4 links with 4 full joints give M = 1.
Limitations
- The equations count links and joints only; special link proportions (such as parallel links) can give motion the count does not predict.
- Count each multiple joint as one less than the number of links joined at it.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a proposed linkage has one degree of freedom before designing it
- Count the joints of a sketch to see whether it will move
- Find the mobility of a spatial linkage
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Degree of freedom (mobility) M
- Whether it is a mechanism, a structure or a preloaded structure
Interpreting the result
- A fourbar linkage (4 links, 4 full joints) has one degree of freedom and needs one input.
- Three links joined by three full joints form a structure; two links joined by two full joints are preloaded.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 29–33. Topic reference; no source tables, figures or examples are reproduced.
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Open the Linkage Degree of Freedom (Gruebler and Kutzbach) calculator