Cycloidal & Planetary Reducer: Stress, Stiffness and Backlash

The equations are implemented here from geometry, equilibrium and elastic strain energy. Blagojević et al. (2011), §3.2, motivates clearance-dependent pin participation and moment balance; its FEM von Mises stresses are not Hertz pressures. Wu, Guo and Li (2024), Eq. (14), requires moment equilibrium for each disc. Gorla et al. (2008) provides force-distribution context for a different external-ring architecture; this tool models the conventional disc inside a pin ring.

For $N$ ring pins and $z=N-1$ lobes, the roller-centre locus in disc coordinates is $C(t)=[R\cos t-e\cos Nt,\ R\sin t-e\sin Nt]$. The disc flank is its inward normal offset by the generating roller radius. Apply $R+\Delta R$ and $r+\Delta r$ while retaining the actual pin positions and radii. Newton closest-point iterations find each normal gap $g_j$, moment arm $l_j$ and signed flank curvature. Each disc has its own elastic rotation $\delta_d$, with ring forces $F_j=k_j\max(l_j\delta_d-g_j,0)$.

The local indentation stiffness follows Wu et al. (2024), Eq. (9): $k_H=\pi E^*b/4$, where $1/E^*=(1-\nu_1^2)/E_1+(1-\nu_2^2)/E_2$. Pin cantilever compliance $L^3/(3EI)$ is added in series at each contact. The Hertz kernel is explicit: $a=\sqrt{4FR^*/(\pi bE^*)}$ and $p_0=\sqrt{FE^*/(\pi bR^*)}$. Signed curvature includes concave disc regions and internal pin/hole contact.

For output pin $j$ and disc $d$, the contact distance is $s_{dj}=|c_d+Q(\delta_d)h_j-Q(\theta)p_j|$, with fixed eccentric centre $c_d$, rotation matrix $Q$, and common carrier angle $\theta$. The hole force is $k_o\max(s_{dj}-e-c_o,0)$. Distance derivatives give the actual moment arms about the disc centre and carrier axis. Newton iteration with an energy line search enforces every disc's ring-plus-hole moment balance and the total output torque simultaneously, in both load directions. No equal torque split is imposed. Unloaded backlash is the first complete ring–disc–carrier contact path in each direction, found from the circle-intersection boundaries.

The full tangent matrix includes geometric contact derivatives. Condensing the individual disc rotations gives $K_r=H_{cc}-\sum_d H_{cd}^2/H_{dd}$; the mounting stiffness is then added in series. Inactive discs contribute no load or stiffness. Identical-disc assembly requires both $z\Delta\beta\equiv\pi$ and $n_o\Delta\beta\equiv0$ modulo $2\pi$. The tool selects 180° body clocking when compatible, otherwise another compatible hole-pattern increment, and rejects combinations with no solution.

Planetary kinematics uses $i=1+z_r/z_s$ and $z_r=z_s+2z_p$. Local involute roll radii determine external/internal Hertz curvature. Tooth compliance integrates bending, shear and axial strain energy over the varying tooth section. Concurrent tooth pairs act in parallel; the two meshes and carrier pin bending act in series. Equally spaced planet phasing includes opposite-flank indexing. A unilateral equilibrium solve distributes torque between branches with the specified seating gaps.

Pressure margin is permissible pressure divided by computed peak pressure, not a torque multiplier. Nominal bending margins use the respective entered pin and unnotched tooth-section stress limits.

Quasi-static, frictionless, aligned line contacts with uniform face loading and small elastic strain. Materials are isotropic. Eccentric centres are rigidly located. Ring compatibility is linearized separately about each disc's nominal pose; output-hole distances retain the separate finite disc rotations and common carrier rotation. Opposed discs are identical and use the computed compatible body clocking. Their load shares are solved rather than prescribed. Support-pin bending uses the entered effective cantilever arm. Shared-disc pin forces are summed in magnitude for a conservative nominal stress; detailed axial coupling of bending along shared pins is not represented.

The Wu et al. stiffness convention describes local indentation rather than the full displacement of finite contacting bodies. Disc-web, root-foundation, rim, bearing and input-restraint compliance must be represented by a suitable additional output stiffness or a detailed structural model. Planet teeth are unshifted, full-depth spur involutes with a clamped radial root section; the root fillet is not represented. The planet seating spread is a specified symmetric cosine gap pattern, not a measured manufacturing-error distribution. No preload, frictional hysteresis, efficiency loss or wear growth is modeled. The schematic shows one cycloidal disc; planetary tooth symbols are schematic.

The load verdict screens Hertz pressure and nominal support-pin bending; planetary mode also screens nominal tooth-section bending. It does not rate disc webs, root fillets, bearing life, case depth, contact fatigue, scuffing, micropitting or dynamic impact. Supply life- and material-specific permissible stresses. For production involute gear ratings use ISO 6336-2:2019 (Edition 3, corrected 2020) and the applicable bending parts, or ANSI/AGMA 2101-E25; check against the standard in force. This tool implements Hertz contact and an elastic beam model, not the complete ISO/AGMA rating factors.

The cycloidal envelope solves the assembled equations at 21 ring-phase positions across one pin pitch and nine output-hole pattern phases across one hole pitch, in both torque directions. This covers the possible common hole-to-lobe indexing: its extrema are conservative across indexing choices, rather than a trace of one specified part's indexing. Every sampled configuration satisfies individual disc equilibrium and total carrier torque. The reversal curve and load bars use the single assembled phase pair that governs primary pressure. A 1441-point geometric locus is interpolated with PCHIP for the contact positions; the pressure plot uses PCHIP between the 21 solved envelope positions. The coarser phase grid keeps the coupled solve near three seconds at defaults. Narrow seating transitions and peaks between phase nodes can be missed; interpolation smooths the picture without adding solved extrema. Confirm critical geometries with a refined contact analysis.

Planetary mode uses 81 mesh-phase points, 48 tooth-section integration stations and a 181-point compliance lookup with PCHIP interpolation. The 125-point torque trace includes the exact unloaded angular interval and both loaded branches. Numerical equilibrium tolerances apply to every disc and to the carrier, not just their aggregate.

Local Hertz theory needs a patch much smaller than either curvature radius; a condition flags a half-width/radius ratio above 0.1. Profile cusps, pin interference, incompatible identical-disc clocking, undercut, insufficient contact ratio, planet overlap and incompatible planet indexing are rejected. Keep sizing rotations below 0.05 rad. Tooth entry/exit can produce abrupt load transfers because no tip relief is specified. Detailed 3D compliance and face misalignment can materially change pressure and stiffness.