A crossed-slider chain uses two perpendicular guides and a rigid bar pinned to a slider in each guide. A point fixed on the bar traces an ellipse. This calculator places that tracing point between the two slider pins.
Related actuator design tools
The trammel calculator shows geometry at selected bar angles. A separate actuator-driven version requires analysis of the chosen drive point, changing leverage and end positions; the illustrated cycle does not calculate actuator speed or force.
With leverage, force is traded against travel. A bracket change that improves force can also increase required stroke, slow the output motion, or move the actuator into a poor mounting angle.
The mechanism may also change speed as the linkage angle changes. A constant actuator extension speed does not always create constant output speed, so timing should be checked at the actual output point.
- Map actuator stroke to the actual output travel before choosing a model.
- Check for binding, changing leverage, and peak load near the ends of travel.
- Decide whether timed motion is acceptable or position feedback is required.
For a FIRGELLI design review, calculate the load case with the FIRGELLI actuator force calculator and plan the switching or controller arrangement with the linear actuator wiring diagram generator. If an actuator or cylinder is already installed, the linear actuator replacement finder helps keep the comparison tied to real dimensions.
Crossed-slider Chain Interactive Calculator
Change the two distances from tracing point P to the slider pins. See the rigid bar, perpendicular guides and ellipse update together.
Equation Used
- Straight perpendicular guides and rigid pin-connected bar.
- P lies between A and B, with AP=b and PB=a.
- Guide travel is pin-center travel; block length and end clearance require additional space.
- Plan view with bar above slider blocks; thickness schematic.
- No forces, drive speed or tolerance prediction.
Width and height describe the tracing point. Each slider pin travels 2(a+b).
Two slots, two sliders, one bar
Pin A moves horizontally and pin B moves vertically. Their fixed separation is L=a+b. Point P divides the bar so that AP=b and PB=a. These distances stay fixed while both sliders move.
The illustration shows separate slider blocks in the cross-shaped opening of a common frame. Each block is longer than the slot crossing so it retains guidance while crossing the center. The bar sits above the guides. P is a geometric tracing marker; a practical drawing instrument needs a holder and an unobstructed drawing surface.
Tracing an ellipse
An elliptic trammel is a physical way to generate an elliptical path. Different tracing positions along a rigid bar produce different ellipses. This model retains a point between the pins; other instruments place the stylus beyond them and use a different pin-spacing relation.
Tracing-point motion and slider motion
Let A=(L cosθ,0) and B=(0,L sinθ), with L=a+b. Then AB=L for every angle. Point P=(a cosθ,b sinθ) lies on the bar with AP=b and PB=a.
Substituting gives x²/a²+y²/b²=1. The traced ellipse is 2a wide and 2b high, and its area is πab. These widths are the tracing point’s coordinate ranges. They are not the slider strokes: each slider moves from −L to +L and therefore requires 2(a+b) of pin travel.
Physical guides also need room for the block length and end clearance. Those manufacturing dimensions are not calculated here.
Default dimensions
For a=100 mm and b=60 mm, the pin span is 160 mm. P traces a 200 mm by 120 mm ellipse with area approximately 18,849.6 mm². Each slider pin needs 320 mm of end-to-end guide travel.
Setting a and b equal puts P at the midpoint. Its path becomes a circle of that radius, while each slider still travels twice the full pin spacing.
Geometry and practical limits
The equations assume straight, perpendicular guides and a rigid bar. They do not predict errors from clearance, bending, bearing friction or misalignment. The model does not prescribe a universal machining tolerance, material section or service life.
A single slider’s position does not by itself select every assembly direction over the entire cycle. The animation uses bar orientation as its phase coordinate. It is not a force, actuator-drive or constant-speed-slider calculation.
Crossed-slider questions
Why are the slider travels larger than the ellipse?
Each slider moves through twice AB. The point between them moves through smaller coordinate ranges set by AP and PB.
Which distance is a?
a is PB, the distance from tracing point P to vertical-slider pin B. b is AP, the distance from P to horizontal-slider pin A.
Can this calculator place the stylus beyond the pins?
No. It models the between-pin configuration. An external stylus requires a different pin-spacing relation.
Are the apparent bar and slot crossings collisions?
The bar lies above the sliders and fixed frame. The drawing is a plan view, with thickness and vertical separation schematic.
References
Jodlbauer and Kovács: Ellipsograph of Archimedes as a simple LEGO construction, for a built perpendicular-guide trammel and the importance of slider clearance. That construction uses an external stylus; this calculator places the tracing point between the pins.
Matemáticas Visuales: Ellipsograph or Trammel of Archimedes, for coordinate geometry and the between-pin tracing case.
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