Stocker’s rotary engine has two sector-shaped pistons turning about the same axis. Their speeds vary through each revolution, so the spaces between them alternately grow and shrink. Slotted arms on their concentric shafts connect through sliding blocks to a crank on a separate, offset axis. Use the two synchronized views below to follow the pistons and their external coupling.
Stocker Rotary Engine Interactive Calculator
Set chamber radius, shaft offset ratio and width to compare the two chamber spaces. The angle slider sets the position used by the result tiles; the animation continues from that position. Pressure gives an ideal work comparison for one chamber expansion, not engine power. Dimensions and proportions are illustrative.
Equation Used
The model uses two 90-degree sectors, hub radius h=0.12R and crank radius r=0.6R. The shaft offset is e=kr. Opposite crank pins are (±r cos θ, e±r sin θ); their directions from the piston axis set the slotted-arm angles. δ is the positive angle from the blue arm to the gold arm. Volumes use the open annular spaces between the sectors, excluding passages and clearances. Over a cycle, δ ranges from π−2 atan(k) to π+2 atan(k). The pressure input is an assumed constant pressure difference acting through one expansion. Port events, steam expansion, losses and net cycle work are not calculated. Playback runs at one fifth of the entered rpm.
Two pistons, one chamber axis
The blue and gold sectors are separate pistons mounted on concentric shafts. Each shaft carries a slotted arm outside the working chamber. A block slides along each arm and pivots on one end of the offset crank. The two crank pins are opposite one another.
Because the crank axis is displaced from the piston axis, equal crank-angle increments produce unequal piston-angle increments. One sector gains on the other and then falls behind it, changing the chamber spaces. The left view shows the sectors inside the casing; the right view shows their external coupling, with the same colors identifying connected parts. The two circles at the common centre indicate the concentric shafts.
Hiscox’s entry 343 and Reuleaux’s Plate XXI, figure 4 show this two-sector arrangement. The illustration uses ideal flat sections and separates the chamber from its coupling for clarity. Shafts and crank links occupy different axial layers. Ports are shown as openings; their admission and exhaust sequence is not simulated.
Find chamber volume from the sector angles
Let O=(0,0) be the piston axis and E=(0,e) the crank axis. The two crank pins are P₁=(r cos θ,e+r sin θ) and P₂=(−r cos θ,e−r sin θ). Each pin stays within its radial slot, so the arm directions are α₁=atan2(P₁y,P₁x) and α₂=atan2(P₂y,P₂x). Each piston is opposite its slotted arm.
For two 90-degree sectors, the open chamber angles are δ−π/2 and 3π/2−δ, where δ is α₂−α₁ wrapped into 0 to 2π. Multiplying either angle in radians by ½(R²−h²)b gives its volume. Here h=0.12R, r=0.6R and e=kr. The entered ratio k is limited to 0–0.6, which keeps the sectors separated throughout the cycle.
The extreme chamber angles are π/2−2 atan(k) and π/2+2 atan(k). Their volume difference is the expansion swept by one chamber. Multiplying that difference by an assumed constant pressure difference gives ideal expansion work. With volume in cm³ and pressure in kPa, divide their product by 1,000 to obtain joules.
Compare zero offset with an offset crank
With R=100 mm, b=40 mm and k=0, each chamber holds about 309.64 cm³ throughout the rotation. Both pistons turn together at constant angular separation. There is no chamber expansion, so the calculated expansion work is zero.
With the same radius and width but k=0.3, chamber volume ranges from about 194.73 to 424.54 cm³. One expansion sweeps about 229.81 cm³. An assumed constant pressure difference of 100 kPa gives 22.98 J over that expansion. This is not net work per engine revolution: actual port timing and the pressure in both chambers are also required.
A historical differential-piston engine
The arrangement converts pressure acting between sector pistons into rotation through the external linkage. Reuleaux discusses the same geometry as a pair of turning-block mechanisms and explains how the sector width can make the intervening space nearly disappear at one position.
This example retains clearance between the sectors. Increasing the offset ratio makes their relative speed variation and chamber-volume excursion greater. Increasing radius scales chamber volumes with R²; increasing width scales them directly with b.
Geometry is only part of an engine calculation
The model solves rigid linkage positions and geometric volumes. It does not determine steam admission, exhaust, cut-off, condensation, leakage, seal friction or bearing forces. Its pressure-work comparison assumes a constant pressure difference during one chamber expansion and does not include the return portion of a thermodynamic cycle.
Crank speed controls playback but does not change the geometric volumes or assumed work per expansion. Dimensions, sector width and hub size are explanatory choices, not manufacturing dimensions or operating settings for a surviving engine.
Frequently asked questions
Why do the two pistons change speed?
Their radial slots rotate about one axis while the crank pins rotate about another. The offset makes the slot angles advance non-uniformly even when the crank turns steadily.
Are the two pistons fixed to the same shaft?
They have separate concentric shafts. Each shaft connects its piston to one external slotted arm; the two pistons can change their angular separation.
What happens when the offset ratio is zero?
The axes coincide and the pistons rotate at constant separation. The chamber volumes stay constant, so this model has no expansion volume or expansion work.
Is the work result the output of the engine?
No. It is assumed constant pressure difference multiplied by the volume increase of one chamber. Net engine work needs the pressure history and valve events for the complete cycle.
Why does the animated volume differ from the selected volume tile?
The tile reports the crank angle chosen with the angle slider. The drawing continues moving from that angle and displays its current chamber volume. Pause and scrub let you inspect other positions.
Does rpm change chamber volume?
No. It changes playback speed. Geometry fixes the chamber volume at each crank angle; steam dynamics and speed-dependent losses are not included.
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