A matched pair of ideal equal elliptical pitch curves rotating about corresponding foci can produce a cyclic angular-speed ratio. This calculator evaluates one explicit phase map and then projects the follower angle onto a simple cosine slider coordinate.
Variable Rotary Motion via Elliptical Gear and Slotted Bar Interactive Calculator
Enter semi-major axis, eccentricity, input phase and speed, plus a projection radius. Results are ideal pitch-curve kinematics—not tooth geometry, a closed slotted-bar linkage, force or machine capacity.
Equation Used
- The two ideal pitch curves are equal ellipses with common eccentricity e and center distance 2a.
- Shafts lie at corresponding foci and the phase convention is exactly the stated atan2 mapping.
- Input angular speed is constant and output speed is the magnitude derivative of follower phase.
- Slider coordinate is defined independently as x=R cos(phi).
- Gear teeth, local normal geometry, backlash, contact ratio, torque, efficiency, force, stress, clearances, dynamics, wear and safety are omitted.
Ideal Equal-Ellipse Phase Map
NASA mechanism material presents an elliptical-gear formula and relative-speed graph, while modern noncircular-gear patents emphasize that conjugate pitch curves must be designed for the desired velocity profile. This calculator uses the explicit equal-ellipse phase map below and does not generate teeth.
What This Calculator Represents
The tool converts input phase theta to follower phase phi, differentiates that map for instantaneous speed-ratio magnitude, applies input rpm and reports a separate cosine projection x=R cos(phi). Semi-major axis sets only the ideal 2a center distance.
Ideal Phase, Speed and Projection Calculations
phi=2 atan2(k sin(theta/2), cos(theta/2))
dphi/dtheta=k/[cos²(theta/2)+k²sin²(theta/2)]
Nout=Nin(dphi/dtheta)
C=2a
x=R cos(phi)
External contact reverses physical direction; output speed is reported as a magnitude.
Worked Idealized Example
For a=50 mm, e=0.30, theta=0 degrees, Nin=120 rpm and R=40 mm, k and the instantaneous ratio are about 0.5385. Ideal center distance is 100 mm, output-speed magnitude is about 64.6 rpm and the projection coordinate is 40 mm.
Model Boundary
- Calculated: one ideal follower-phase map, its instantaneous ratio, output-speed magnitude, 2a center distance and cosine projection.
- Illustrated: matched equal elliptical pitch curves and an illustrative slotted bar; the cosine projection is reported separately.
- Not calculated: tooth profiles, a closed slotted-bar linkage, slider guide loads, torque, efficiency, contact stress, backlash, wear, capacity or safety.
Frequently Asked Questions
Why does the formula differ from some ellipse formulas?
Phase origin and which member is treated as input change the sign and reciprocal convention; this page states its exact map.
Does the calculator generate conjugate teeth?
No.
Is x a complete slotted-bar solution?
No; it is only R cos(phi).
Does semi-major axis change the speed ratio?
No; in this normalized equal-ellipse model it sets center distance only.
Are forces or stresses included?
No.
Can this size a noncircular gear pair?
No.
References
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