An elliptical crank can be illustrated as a perpendicular double-slider trammel: a rigid bar joins two guided pins, and a point on that bar follows an ellipse. This calculator shows the point between the pins and keeps its coordinates, speed and tracing path tied to the same geometry.
Elliptical Crank Interactive Calculator
Adjust the ellipse dimensions, initial phase and cycle time. Follow a rigid two-slider trammel while its tracing point, coordinates and speed update together.
Equation Used
- Rigid bar with P between its slider pins.
- Perpendicular ideal guides.
- VP=a, PH=b and HV=a+b.
- Uniform prescribed phase rate.
Horizontal and vertical semi-axes may be in either size order. Uniform prescribed phase; ideal guide geometry.
Two sliders constrain one rigid bar
Pin H runs in the horizontal guide and pin V in the vertical guide. The gold bar joins them. Point P lies between the pins, a distance a from V and b from H. Their fixed separation is therefore a+b.
As the prescribed phase advances, the horizontal pin moves through ((a+b) cos θ, 0), while the vertical pin moves through (0, (a+b) sin θ). Those constraints keep the bar length constant and put P at (a cos θ, b sin θ).
The drawing is a plan-view kinematic model. The green ellipse is the point’s projected trajectory. Guide clearances, motor hardware and the out-of-plane construction are omitted. The mechanism itself stays visible throughout the cycle, including when the bar lines up with a guide.
Explore elliptical motion
A trammel is a useful way to see how sliding and pivoting joints produce an exact geometric curve. Historical ellipsographs use related arrangements to carry a drawing point. The cited Smithsonian instrument places its drawing bracket beyond its sliders; the arrangement shown here deliberately puts P between them so both positive semi-axis inputs remain valid independently.
The semi-axis controls set the ellipse dimensions. The phase control sets the initial angle, and animation advances from that setting at the entered cycle time. The live coordinates and speed refer to the point shown at that instant.
Position and speed from the constrained geometry
With L=a+b, H=(L cos θ,0), V=(0,L sin θ), and P=(a cos θ,b sin θ), the distances HV=L, VP=a and PH=b are constant. Eliminating θ gives x²/a²+y²/b²=1.
For a uniform phase rate ω=2π/T, the velocity components are vx=−aω sin θ and vy=bω cos θ. Their magnitude is ω√(a² sin²θ+b² cos²θ). The mechanism completes 60/T cycles per minute.
The horizontal and vertical semi-axes are named by direction. Either can be larger. When a=b, the path is a circle and point speed is constant. Otherwise the point speeds up and slows down even though phase advances uniformly.
Default point on the ellipse
At a=120 mm, b=60 mm, initial phase 45° and T=4 s, the point starts at approximately (84.85,42.43) mm. Its initial speed is about 149.02 mm/s, and the mechanism rate is 15 cycles per minute.
The slider pins stay 180 mm apart. P stays 120 mm from the vertical pin and 60 mm from the horizontal pin. At phase 0°, P is (120,0) mm and travels at about 94.25 mm/s; at 90°, it is (0,60) mm and travels at about 188.50 mm/s.
Geometry model and construction details
This calculator does not infer forces, drive torque, guide wear or manufacturing tolerances. Clearance, wear and perpendicularity tolerances require the actual instrument dimensions and service requirements.
Nonperpendicular guides change the coordinate relationship but do not justify calling every resulting curve an egg-shaped error. The displayed model assumes exactly perpendicular guides and a rigid bar of constant length.
Elliptical crank questions
Why can the vertical semi-axis be larger?
The inputs specify horizontal and vertical dimensions, not a fixed major/minor ordering.
What happens when the two semi-axes are equal?
P becomes the bar midpoint and follows a circle at constant speed for the prescribed phase rate.
Is the displayed cycle rate a motor specification?
No. It is the imposed mechanism cycle rate. The motor and transmission are not modeled.
Why does the point speed change during a cycle?
The horizontal and vertical velocity components have different amplitudes when a and b differ.
References
- Caltech ME115: Notes on the Elliptical Trammel — two sliding joints, a rigid body and ellipse geometry.
- Smithsonian: Elliptic Trammel — a surviving drawing instrument with adjustable sliders.
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