BIZOMAT MAKEOS · Mechanics
Fourbar Crank-Rocker Quick-Return Time Ratio
For a crank-rocker driven at constant speed, what are the crank angles for the forward and return strokes and the time ratio, or what angles does a chosen time ratio need when I design one?
Quick-return time ratio of a Grashof crank-rocker by Norton’s definitions: the crank sweeps α and β between the two toggle positions (crank and coupler colinear), with TR = α ÷ β, α + β = 360° and the construction angle δ = |180° − α| = |180° − β|; for a given linkage the two sweeps and the rocker swing follow from the cosine law in the extended (a + b) and overlapped (b − a) colinear positions, and for design they follow from the time ratio.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Calculate | choice | Whether to analyse a linkage or design for a time ratio. |
| Ground link d | mm | Distance between the fixed pivots. |
| Crank a | mm | Length of the crank, the shortest link. |
| Coupler b | mm | Length of the coupler. |
| Rocker c | mm | Length of the rocker. |
| Time ratio wanted (slower stroke ÷ faster stroke) | Ratio of the longer crank sweep to the shorter, e.g. 1.25 for 1:1.25. |
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: Calculate
- What it is
- Whether to analyse an existing crank-rocker or design for a time ratio.
- Units and format
- Choose one of 2 options from the list: Time ratio of a crank-rocker from its links; Crank angles for a chosen time ratio.
- Example
- The form opens with “Time ratio of a crank-rocker from its links” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses which inputs are used.
- Get it right
- Design first, then analyse the linkage you draw to check it.
What this means: Ground link d
- 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: Crank a
- What it is
- Length of the crank, the shortest link, from 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 30 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
- The crank must be the shortest link of a Grashof linkage.
What this means: Coupler b
- 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 90 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: Rocker c
- What it is
- Length of the rocker 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 70 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: Time ratio wanted (slower stroke ÷ faster strok…
- What it is
- Ratio of the slower stroke time to the faster stroke time.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.25 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the two crank sweeps and the construction angle.
- Get it right
- Enter 1.25 for a time ratio of 1:1.25.
Method and formulas
- TR = α ÷ β with α + β = 360°; δ = |180° − α| (eqs. 3.1 and 3.2).
- Design: β = 360° ÷ (1 + 1/TR) for the slower stroke, α = 360° − β.
- Analysis: the crank angles at the toggle positions follow from the cosine law with O2B = a + b (extended) and b − a (overlapped): cos φ = [O2B² + d² − c²] ÷ (2 d O2B); δ = |φ_overlapped − φ_extended|, sweeps 180° ± δ.
- Rocker swing = difference of the rocker angles at O4 in the two positions, cos γ = [c² + d² − O2B²] ÷ 2cd.
- Example: the crank sweeps 187.87° and 172.13°, a time ratio of 1:1.0915.
Limitations
- The time ratio assumes the crank turns at constant speed.
- The linkage must be a Grashof crank-rocker with the crank as the shortest link.
- Norton notes that fourbar quick-returns work well to about 1:1.5; beyond that the transmission angles become poor.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the time ratio of an existing crank-rocker
- Set the crank angles for a quick-return design
- Find the rocker swing of a crank-rocker
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Crank sweeps α and β
- Time ratio
- Construction angle δ
- Rocker swing angle
Interpreting the result
- Equal sweeps of 180° (δ = 0) give no quick return, as when the crank pivot lies on the rocker chord extended.
- Running the crank the other way turns a quick-return into a quick-forward.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 97–99. Topic reference; no source tables, figures or examples are reproduced.
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Open the Fourbar Crank-Rocker Quick-Return Time Ratio calculator