This Hoeken four-bar example turns crank rotation into an approximately straight portion of a tracer path. The tracer lies on an extension of the coupler, and the return portion curves back around the mechanism. Scale the defined geometry and compare the travel and sideways error over the 90°–270° crank interval.
Straight-line Mechanism Interactive Calculator
Scale the explicitly defined Hoeken four-bar teaching example and enter its physical crank speed. The calculator measures the tracer locus over the disclosed 90°–270° crank interval rather than assuming a straight-zone length.
Equation Used
- The same circle-intersection configuration is used throughout the cycle; a position without geometric closure has no motion result.
- All link lengths retain the displayed 1:2.5:2.5:2 proportions as the scale changes.
- Endpoint travel is the projected span along the line through the two interval endpoints; maximum deviation is the largest absolute normal distance from that line.
- Links and pin joints are rigid and planar. Clearance, tolerance, elasticity, force, stress and speed variation are not predicted. Hoeken motion is approximate, not exact.
The orange path marks the evaluated 90°–270° interval; the grey curve shows the full cycle. Travel and normal deviation are measured relative to the line joining the interval endpoints.
How the tracer path is formed
The input crank O2A turns about fixed pivot O2. Coupler AB connects it to follower BO4, which turns about fixed pivot O4. Tracer P lies on the extension of AB beyond B.
In this defined example, crank length is u, ground spacing O2O4 is 2u, and both AB and BO4 are 2.5u. P is placed so that AP = 5u. Its position is P = A + 2(B − A), putting B halfway between A and P.
One segment of the resulting closed curve is approximately straight. The calculator evaluates crank angles from 90° to 270°; the remainder of the curve is the return interval. Neither segment is forced onto an artificial straight guide.
Reading the results
- Tracer distance AP: distance from crank pin A to the tracing point along the coupler extension.
- Endpoint travel: the range of projected tracer positions along the line joining the interval endpoints.
- Maximum normal deviation: the largest sampled perpendicular distance from that line.
- Crank period: time for one revolution at the entered physical rpm.
What this example demonstrates
A coupler curve can provide a nearly straight working interval and a curved return without a sliding guide. The visible full path helps distinguish that motion from a point constrained to remain on a straight rail.
This model is useful for comparing geometric scale and timing. It does not simulate a supported load, contact with a surface, a walking machine’s stability or a drawing tool’s engagement. Real accuracy also depends on joint clearance and structural deflection.
Geometry and path measurement
O2O4 = 2u; O2A = u; AB = BO4 = 2.5u; AP = 5u. Joint B is found from the intersection of the two equal-radius circles about A and O4, keeping the same configuration throughout the cycle. The tracer position is P = 2B − A.
The calculation samples 1,201 angles from 90° through 270°. It defines a line through the first and last tracer positions, projects all samples along and normal to that line, and reports projected travel and maximum absolute normal distance.
Crank period = 60/N seconds for N in rpm. Playback is slowed for inspection and does not run at the entered physical speed. Increasing rpm changes the period result, not the calculated geometric curve.
Worked example: scale and speed
Set u = 50 mm and N = 6 rpm. The ground spacing is 100 mm, the coupler and follower are each 125 mm, and AP = 250 mm.
- The current geometry calculation gives 200 mm projected travel over the 90°–270° interval.
- Its maximum sampled normal deviation from the endpoint line is about 0.488 mm.
- Crank period = 60/6 = 10 seconds.
At u = 100 mm, all geometric distances double: AP = 500 mm, travel = 400 mm and sampled deviation ≈ 0.975 mm. If speed is also raised to 12 rpm, period becomes 5 seconds. These are ideal geometric results; changing scale does not establish the strength or accuracy of a physical build.
Approximate paths and exact constraints
The four-bar example provides a working interval that is close to a line, followed by a curved return. An ideal exact straight-line linkage uses a different geometric constraint to produce a line, while a linear guide constrains a carriage along the guide.
The choice of geometry does not remove physical errors. Joint clearance, dimensions, mounting and compliance matter even for an exact mathematical linkage. This calculator reports only the ideal path of its stated proportions and tracer location.
Frequently asked questions
Is the working interval perfectly straight?
No. It has a sampled normal deviation from the line through its endpoints. The drawing and results describe an approximation.
Where exactly is the tracer point?
For this example AP = 5u, with P = A + 2(B − A). It lies on the straight extension of coupler AB beyond B; B is halfway between A and P.
Why does scale change the absolute error?
Every link dimension and tracer coordinate scales with u. Doubling u doubles travel and absolute normal deviation for the same angular interval.
Why does rpm not change the path error?
The model is geometric. Rpm sets the crank period but does not change link dimensions or tracer positions. Dynamic deformation is not calculated.
Does the animation play at the entered rpm?
No. Playback is slowed for viewing. The crank-period output gives the physical time per revolution implied by the entered rpm.
How is the straight reference line chosen?
It joins the tracer positions at 90° and 270°. Maximum deviation is the largest absolute normal distance among 1,201 samples over that interval.
Can this predict precision or joint life in a real mechanism?
No. It has rigid links and ideal planar joints, with no clearance, elasticity, forces, stress or wear model.
References
- Beroz, Awtar and Hart, ASME DETC2013-12582, Figure 5 and page 7: Hoekens-based four-bar definitions and the cited 2.5 length ratio. The calculator separately discloses its chosen ground length.
- Robert L. Norton, Design of Machinery, Hoekens linkage geometry and structural-error discussion.
- Application-oriented classification and performance analysis of precision linear feed mechanisms, Hoekens linear mechanism section.
- Michio Innami, design of an approximate straight-line mechanism using optimization.
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