A straight-line linkage guides a chosen point using pivoted links. The Watt four-bar shown here produces an approximately straight portion of its coupler curve. Enter three link lengths and an input sweep to compare travel with the sampled sideways deviation. The path is not an exact straight line.
Straight-line Linkage Interactive Calculator
Enter the three Watt four-bar link lengths and a symmetric input half-sweep. The calculator solves the positive circle-intersection branch and reports actual sampled travel and lateral error.
Equation Used
- The positive circle-intersection branch containing the defined central configuration is retained continuously.
- A sweep crossing a circle-intersection toggle is invalid and produces no travel/error result.
- Links and pin joints are rigid and planar; clearance, elasticity, force, stress and bearing loads are not predicted.
- Watt motion is approximate, not an exact straight line.
The calculator solves the four-bar geometry at points across the selected sweep. It reports travel and sideways deviation only when that sweep stays within one continuous assembly configuration.
How the Watt four-bar guides a point
The fixed pivots O and C support the left arm OA and right arm CB. Coupler AB joins their moving ends. A tracing point P lies along the coupler; it follows the path shown as the left arm moves through the selected angle.
In the reference configuration O = (0,0), A = (a,0), B = (a,h) and C = (a+b,h). The tracer is placed a distance x = hb/(a+b) from A along the coupler. The calculation holds the same circle-intersection solution throughout the sweep.
The central portion is nearly straight, but curvature grows away from the reference position. A toggle occurs when the circles used to locate the moving joint become tangent. A sweep that reaches that condition is rejected instead of combining two different assembly configurations.
What the outputs describe
- Ground length: distance between fixed pivots O and C.
- Tracer distance: distance from A along AB to P.
- Approximate-line travel: sampled range along the central path direction.
- Maximum lateral error: largest sampled absolute sideways displacement from the reference line.
Exploring an approximately straight path
This model helps compare link proportions and input sweep when a guided point is required to stay near a line. It also illustrates why a path can look straight at one scale while retaining a small geometric error.
Geometric straightness is only one part of a working mechanism. Pin clearance, compliance, mounting error, load and speed can affect an actual point’s motion. Those effects are not included, so the ideal result is not a precision, durability or motor-sizing specification.
Geometry and sampled path error
The fixed-pivot spacing is g = √[(a+b)²+h²], and the tracer distance from A is x = hb/(a+b). For each input angle θ, A = (a cos θ, a sin θ). Joint B is found at the selected intersection of a circle of radius h about A and a circle of radius b about C.
The tracer position is P = A + [b/(a+b)](B − A). Its reference position is (a,x). The sideways error is measured from the line through that point in the central travel direction.
The calculator samples 801 positions over the symmetric sweep and takes the largest absolute lateral displacement. This is a numerical maximum over those samples, not a universal error factor or a mathematical bound between sample points. It separately checks the full sweep for circle tangencies.
Worked example: increasing the input sweep
Set a = b = h = 100 mm. The ground length is √50,000 ≈ 223.607 mm and the tracer lies 50 mm from A.
- At an input half-sweep of 5°, the current geometry calculation gives about 17.431 mm travel and 0.000127 mm maximum sampled lateral error.
- At a half-sweep of 10°, it gives about 34.730 mm travel and 0.004098 mm maximum sampled lateral error.
- The lengths and tracer position are unchanged. Doubling the angular sweep therefore increases path error much faster than a simple constant-times-stroke rule would suggest.
The small numerical errors describe ideal rigid geometry. They do not include real joint play or manufacturing error and should not be read as achievable machine accuracy.
Approximate geometry and physical accuracy
| Approach | What determines the path |
|---|---|
| Approximate straight-line linkage | A selected portion of a coupler curve; error depends on proportions, tracer location and sweep. |
| Exact straight-line linkage | An exact geometric constraint in an ideal model; physical joints still introduce error. |
| Linear guide | A sliding or rolling constraint along the guide; real straightness depends on the guide and installation. |
The calculator evaluates the first approach for the displayed Watt arrangement. It does not supply price, lifespan or a universal tolerance ranking among these alternatives.
Frequently asked questions
Is this an exact straight-line mechanism?
No. The Watt four-bar produces an approximately straight region. The maximum-lateral-error output measures the departure over the selected sweep.
What does input half-sweep mean?
The left arm is evaluated from the negative entered angle to the positive entered angle around its reference position. A 5° half-sweep covers −5° to +5°.
Why is an invalid sweep reported?
The selected angular window reaches a circle-intersection toggle or cannot stay on the selected continuous configuration. Reduce the sweep or change the link lengths to examine a valid interval.
Is the displayed lateral error the guaranteed worst case?
It is the maximum over 801 sampled positions in the ideal geometry. It is not an error bound between samples or a tolerance including real joint clearance and deflection.
Where is the tracing point on the coupler?
It is hb/(a+b) from A along coupler AB. When the two side arms are equal, this point lies at the coupler midpoint.
Can I use the result to choose a motor or pin clearance?
No. The model calculates positions and path deviation. It has no force, inertia, elasticity, stress or joint-clearance calculation.
References
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