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Fourbar Three-Position Motion Synthesis (Analytical Dyad Method)
I need a fourbar whose coupler carries a line through three given positions: from the moves of the coupler point and the coupler rotations between the positions, and my free choices of the crank and rocker rotations, what are the link lengths, the fixed pivot locations, the coupler point and the linkage angles at each position?
Three-position motion generation by Norton’s analytical method: the coupler point moves P21 at angle δ2 and P31 at angle δ3 from its first position while the coupler turns α2 and α3. Choosing the rotations β2 and β3 of the input link and γ2 and γ3 of the output link as free choices makes each dyad a set of four linear equations (eqs. 5.24–5.27 and 5.30–5.31), solved here for the vectors W and Z of the input dyad and U and S of the output dyad. The coupler V = Z − S and the ground G = W + V − U (eqs. 5.2), the fixed pivots follow from the first position, and the tool gives the crank, coupler and rocker angles, the transmission angle and the assembly circuit at each position, with the Grashof class.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Move of the coupler point from position 1 to 2, p21 | mm | Straight-line distance the coupler point moves from the first to the second position. |
| Direction of that move δ2 | ° | Angle of that move measured anticlockwise from the X axis. |
| Move from position 1 to 3, p31 | mm | Straight-line distance the coupler point moves from the first to the third position. |
| Direction of that move δ3 | ° | Angle of the move from the first to the third position, anticlockwise from X. |
| Coupler rotation from position 1 to 2, α2 | ° | Rotation of the coupler between the first and second positions, anticlockwise positive. |
| Coupler rotation from position 1 to 3, α3 | ° | Rotation of the coupler between the first and third positions, anticlockwise positive. |
| Chosen input link rotation β2 | ° | Chosen rotation of the input link between positions 1 and 2. |
| Chosen input link rotation β3 | ° | Chosen rotation of the input link between positions 1 and 3. |
| Chosen output link rotation γ2 | ° | Chosen rotation of the output link between positions 1 and 2. |
| Chosen output link rotation γ3 | ° | Chosen rotation of the output link between positions 1 and 3. |
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: Move of the coupler point from position 1 to 2,…
- What it is
- Straight-line distance the coupler point moves from the first to the second position.
- 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 71.07 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales the linkage with the size of the move.
- Get it right
- Draw the three positions to scale and measure P1 to P2.
What this means: Direction of that move δ2
- What it is
- Direction of the move from P1 to P2, anticlockwise from the X axis.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -31.19° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Turns the second position about P1.
- Get it right
- Measure from a horizontal line through P1.
What this means: Move from position 1 to 3, p31
- What it is
- Straight-line distance the coupler point moves from the first to the third position.
- 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 99.54 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales the linkage with the size of the move.
- Get it right
- Measure P1 to P3 on the same drawing.
What this means: Direction of that move δ3
- What it is
- Direction of the move from P1 to P3, anticlockwise from the X axis.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -16.34° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Turns the third position about P1.
- Get it right
- Measure from the same horizontal line through P1.
What this means: Coupler rotation from position 1 to 2, α2
- What it is
- Rotation of the coupler line between the first and second positions.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -45° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes both dyads and so every link.
- Get it right
- Anticlockwise is positive; measure the change in the line’s angle.
What this means: Coupler rotation from position 1 to 3, α3
- What it is
- Rotation of the coupler line between the first and third positions.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 9.3° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes both dyads and so every link.
- Get it right
- Anticlockwise is positive; measure from the first position.
What this means: Chosen input link rotation β2
- What it is
- Chosen rotation of the input link between position 1 and position 2.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 342.3° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the fixed pivot and changes the link lengths.
- Get it right
- Pick a value, solve, and change it until the pivots and links suit the space.
What this means: Chosen input link rotation β3
- What it is
- Chosen rotation of the input link between position 1 and position 3.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 324.8° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the fixed pivot and changes the link lengths.
- Get it right
- Pick a value, solve, and change it until the pivots and links suit the space.
What this means: Chosen output link rotation γ2
- What it is
- Chosen rotation of the output link between position 1 and position 2.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30.9° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the fixed pivot and changes the link lengths.
- Get it right
- Pick a value, solve, and change it until the pivots and links suit the space.
What this means: Chosen output link rotation γ3
- What it is
- Chosen rotation of the output link between position 1 and position 3.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -360° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80.6° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the fixed pivot and changes the link lengths.
- Get it right
- Pick a value, solve, and change it until the pivots and links suit the space.
Method and formulas
- Input dyad: A = cos β2 − 1, B = sin β2, C = cos α2 − 1, D = sin α2, E = p21 cos δ2, F = cos β3 − 1, G = sin β3, H = cos α3 − 1, K = sin α3, L = p31 cos δ3, M = p21 sin δ2, N = p31 sin δ3 (eq. 5.25); solve the 4 × 4 system of eq. 5.27 for W1x, W1y, Z1x, Z1y.
- Output dyad: the same with γ2 and γ3 in place of β2 and β3 (eqs. 5.30, 5.31) for U1 and S1.
- Coupler V = Z − S; ground G = W + V − U (eqs. 5.2); O2 = −Z − W and O4 = −S − U from the first precision point.
- At each position the angles are taken from the ground line O2O4; the circuit is found by comparing the rocker angle with the open and crossed fourbar solutions (eq. 4.10).
- Example: the links are 229.35, 173.53, 35.542 and 81.27 mm.
Limitations
- Each set of free choices gives a different linkage; many will be impractical, with pivots too far away or circuit, branch or order defects.
- The linkage reaches the three positions exactly but its motion between them is not controlled; run it in a fourbar analysis to check it.
- Angles are measured anticlockwise; all data are relative to the first position, so the global origin is at the first precision point.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Design a fourbar to carry a part through three positions
- Try free choices of the link rotations until the linkage is practical
- Find the fixed pivot positions and link lengths for a three-position task
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Vectors W, Z, U and S of the two dyads
- Link lengths and fixed pivot positions measured from the first precision point
- Coupler point location on the coupler
- Crank, coupler and rocker angles, transmission angle and circuit at each position, and the Grashof class
Interpreting the result
- If the circuit differs between positions the linkage would have to be taken apart to reach all three; change the free choices.
- Keep the transmission angle above about 40° at every position.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 155–157, 191, 201–209, 258. Topic reference; no source tables, figures or examples are reproduced.
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