A compound epicyclic drive can create non-sinusoidal reciprocation by adding two rotating offsets. An input carrier turns a planet around an equal-sized fixed sun gear. The planet spins through twice the carrier angle. A pin on that planet runs in a vertically slotted yoke, so the yoke follows the pin’s horizontal position.
Complex Alternating Reciprocal Motion Interactive Calculator
A fixed sun and equal orbiting planet constrain the two-frequency motion. The front-plane pin drives a vertical slot in a horizontally guided yoke. Result cards use your reference angle; the animation shows the full cycle.
Equation Used
- Fixed sun; equal external gear pitch radii and 24 teeth.
- Rigid carrier center distance R1 and planet pin offset R2.
- Vertical slot in a horizontally translating yoke.
- Ideal no-slip gearing and zero-clearance pin/slot.
Equal 24-tooth gears; pitch radii R1/2. Tooth bodies schematic. Pin crank and yoke in front planes. No force or durability rating.
How the gears constrain the two-frequency motion
The fixed gold sun and orbiting blue planet have equal pitch radii and equal 24-tooth counts. Their centers remain R1 apart. The sun stays fixed, the carrier turns through θ, and external rolling contact makes the planet’s absolute angle 2θ.
A crank pin lies R2 from the planet center. Its position is x=R1 cosθ+R2 cos2θ and y=R1 sinθ+R2 sin2θ. The pin slides vertically in the yoke’s slot; the yoke is guided horizontally. That slot is essential: an ordinary finite connecting rod would not give the same simple projection formula.
The crank arm and yoke are in planes in front of the gears. An overhung pin may extend beyond the planet pitch circle. The illustrated epicyclic arrangement uses equal gears and a fixed sun. Its proportions describe an ideal mechanism, not the dimensions of a particular production machine.
Explore waveform shape rather than unequal travel
Use R1 to set the carrier’s center distance and R2 to change the secondary harmonic. R2=0 gives a simple sinusoidal displacement. Larger R2 flattens and reshapes one end of the motion; if 4R2 exceeds R1, extra reversals appear around the far end.
The reference-angle control selects the position used by the result cards. The autoplay demonstration then shows a complete cycle from that angle. Its live position and graph cursor follow the same pin geometry.
Calculate the true extrema
Position x=R1 cosθ+R2 cos2θ. Its angular derivative is dx/dθ=−R1 sinθ−2R2 sin2θ. Multiply that derivative by carrier speed in radians per second to obtain instantaneous slider velocity.
The maximum is R1+R2. If R1≥4R2, the minimum is −R1+R2. If R1<4R2, the minimum instead occurs at cosθ=−R1/(4R2) and equals −R2−R1²/(8R2). Full positional span is maximum minus minimum.
Positions measured from the origin are different from distances traveled. A slider returning between the same endpoints covers the same distance in both directions. With additional reversals, total travel during a cycle can exceed twice the positional span.
Example: R1=75 mm, R2=16 mm
The maximum is +91 mm and minimum is −59 mm, giving a 150 mm span. At 90°, position is −16 mm. Because 75 exceeds 4×16, there are no extra reversals between the two principal extremes.
For R1=50 mm and R2=40 mm, maximum is +90 mm and minimum is −47.8125 mm. The 137.8125 mm span includes additional turning points; the minimum is no longer the position at 180°.
Model assumptions and limits
The gears set the two-to-one angular relationship; the slot sets the horizontal projection. The drawing uses simplified tooth profiles and omits bearing and axial stack detail. Backlash, pin contact stress, slot wear, inertia and drive torque are not calculated.
Changing R1 rescales both equal gear pitch radii while retaining 24 teeth. That represents a different module, not an adjustment possible on an already manufactured pair. The calculation does not determine tolerances, fatigue strength or service life.
Compound reciprocation questions
Why does the planet spin twice per carrier turn?
Its center moves around a fixed equal-size sun. Relative to the carrier, the external gear mesh contributes one additional planet turn, giving two absolute turns.
Why use a slotted yoke?
The slot lets the pin move vertically while imposing its horizontal coordinate on the output. A finite connecting rod would require a different displacement equation.
Are forward and return distances different?
Travel between the same two endpoints is equal in both directions. The velocity profile and intermediate reversals can differ.
What happens with zero secondary offset?
The pin sits at the planet center and the yoke follows R1 cosθ with span 2R1.
Epicyclic gearing reference
Brown’s sun-and-planet movement 39 illustrates equal external gears linked at a fixed center distance. Its steam-engine connection differs from this page’s fixed-sun input-carrier arrangement. The model here explicitly fixes the sun and derives planet motion from no-slip pitch-circle contact.
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