BIZOMAT MAKEOS · Mechanics
Fourbar Linkage Grashof Condition, Type, Transmission and Toggle Angles
From the four link lengths of my fourbar linkage, is it Grashof, what kind of motion does this inversion give (crank-rocker, double-crank, double-rocker or triple-rocker), and what are its extreme transmission angles or toggle positions?
Fourbar linkage behaviour by Norton’s Grashof condition S + L ≤ P + Q with the shortest S, longest L and other links P and Q: Class I (S + L < P + Q) gives a crank-rocker when a link next to the shortest is grounded, a double-crank when the shortest is grounded and a Grashof double-rocker when the link opposite the shortest is grounded; Class II (S + L > P + Q) gives triple-rockers; Class III (S + L = P + Q) has change points. It also gives the extreme transmission angles of a crank-rocker, μ = arccos{[b² + c² − (d ± a)²] ÷ 2bc}, and for a non-Grashof linkage the input toggle angle θ2 = arccos[(a² + d² − b² − c²) ÷ 2ad ± bc ÷ ad] (equation 4.33) with the transmission angle at the toggle where the input and coupler are in line.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Ground link d (link 1) | mm | Distance between the two fixed pivots O2 and O4. |
| Input link a (link 2) | mm | Length of the input link, from O2 to the coupler joint A. |
| Coupler b (link 3) | mm | Length of the coupler, from A to B. |
| Output link c (link 4) | mm | Length of the output link, from O4 to B. |
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: Ground link d (link 1)
- What it is
- Distance between the two fixed pivots O2 and O4 of the fourbar.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the motion and the angles of the whole linkage.
- Get it right
- Measure centre to centre of the frame pivots.
What this means: Input link a (link 2)
- What it is
- Length of the input link from its fixed pivot O2 to joint A.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the motion and the angles of the whole linkage.
- Get it right
- Measure pin centre to pin centre.
What this means: Coupler b (link 3)
- What it is
- Length of the coupler from joint A to joint B.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 120 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the motion and the angles of the whole linkage.
- Get it right
- Measure pin centre to pin centre, not the overall link length.
What this means: Output link c (link 4)
- What it is
- Length of the output link from its fixed pivot O4 to joint B.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the motion and the angles of the whole linkage.
- Get it right
- Measure pin centre to pin centre.
Method and formulas
- S, L, P, Q are the shortest, longest and remaining links; Class I if S + L < P + Q, Class II if S + L > P + Q, Class III (special-case Grashof, change points) if equal.
- Class I inversions: shortest link adjacent to ground → crank-rocker; shortest grounded → double-crank; shortest is the coupler → Grashof double-rocker. Class II: triple-rocker.
- Crank-rocker with the input as crank: μ1 = arccos{[b² + c² − (d + a)²] ÷ 2bc}, μ2 = arccos{[b² + c² − (d − a)²] ÷ 2bc} (eq. 4.29).
- Non-Grashof: θ2,toggle = arccos{[a² + d² − b² − c²] ÷ 2ad ± bc ÷ ad}, taking the sign that gives an argument within ±1 (eq. 4.33); the toggle positions are ± that angle. With input and coupler colinear, μ = arccos{[(a + b)² + c² − d²] ÷ 2c(a + b)} (eq. 4.30).
- Example: S + L = 160 against P + Q = 180.
Limitations
- The condition applies to pin-jointed fourbar linkages; whether the linkage assembles at all needs the longest link shorter than the sum of the others.
- Transmission angles below about 40° give poor force transmission in practice; check your own limit.
- Class III linkages need their motion limited or a carry-through link at the change points.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check whether a fourbar crank will turn fully
- Classify a fourbar inversion before analysing it
- Check the transmission angle range of a crank-rocker
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Shortest, longest and the other two links
- Grashof class and the motion of this inversion
- Extreme transmission angles of a crank-rocker
- Toggle angles of the input for a non-Grashof linkage
Interpreting the result
- The Grashof condition does not depend on the order in which the links are assembled; the inversion (which link is grounded) decides the motion.
- The transmission angle is the angle between the coupler and the output link; 90° is ideal.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 46–49, 169–172. Topic reference; no source tables, figures or examples are reproduced.
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These BIZOMAT Library readings explain the method this tool uses, with worked examples.
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