A Tusi couple is the special two-cusped hypocycloid formed when a circle rolls inside a fixed circle of exactly twice its diameter. A point on the rolling circle traces a diameter of the fixed circle.
Tusi Couple Interactive Calculator
Enter fixed and rolling-circle diameters. Straight-line stroke and amplitude are returned only when the fixed diameter equals exactly twice the rolling diameter.
Equation Used
- The rolling circle remains internally tangent to the fixed circle without slip.
- The traced point lies on the rolling-circle circumference at the initial tangency point.
- Fixed diameter is exactly twice rolling diameter for valid results.
- Circles are ideal, coplanar and rigid.
- Clearance, gear teeth, backlash, friction, speed, acceleration, force, stress and manufacturability are omitted.
One Circle Rolling Inside Another
The Tusi couple uses a rolling circle whose diameter is exactly half the fixed circle's diameter. With no slip and a traced point on the smaller circumference, two circular motions combine into back-and-forth motion on one diameter.
What This Calculator Represents
The tool checks the defining two-to-one diameter condition and, only when it holds, reports the fixed-circle diameter as ideal straight-line stroke, half that value as centered amplitude, the diameter ratio and zero ideal side deviation.
Exact Tusi-Couple Geometry
rolling radius r = Dr/2 = R/2
traced point x = R cos(theta)
traced point y = 0
stroke = 2R = Df
amplitude = R
When Df is not exactly 2Dr, the curve is not the straight-line Tusi couple and the dependent results are withheld.
Worked Geometry Example
For fixed diameter 100 mm and rolling diameter 50 mm, R=50 mm and r=25 mm. The rim point moves from x=+50 mm to x=-50 mm with y=0, giving 100 mm stroke and 50 mm centered amplitude.
Model Boundary
- Calculated: exact ideal two-to-one geometry, stroke and amplitude.
- Not calculated: off-ratio hypocycloid dimensions, clearance, gears, backlash, friction, velocity, acceleration, force, torque, stress, wear or design limits.
Frequently Asked Questions
Why must the diameter ratio be exactly 2:1?
That equality makes the transverse components cancel and produces a line.
Is the stroke twice the rolling diameter?
For the valid 2:1 case, yes; it also equals the fixed diameter.
What happens off ratio?
The traced curve is a different hypocycloid, so this Tusi-couple calculator withholds the straight-line results.
Does zero side deviation include clearance?
No; it is ideal geometry.
Are velocity and acceleration calculated?
No.
Can this size a physical mechanism?
No.
References
- University of California, Riverside, Rolling Circles and Balls, Part 3: proofs of the twice-as-large fixed circle and straight-line trace.
- Mathematical Association of America, Hands On History: Tusi couple from a small circle rolling inside a circle of twice the radius.
- H. Nussbaumer, Sun, Stars and Planets: historical description of the Tusi couple and linear oscillation from circular motions.
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