Strain-wave ring generated for a prescribed deformation

RatioLab generates the rigid circular spline from the complete swept outline of the matching flexible tooth reference. The deformation remains a prescribed pitch-arc/ellipse map. It is a geometric model, not a prediction of elastic shape, stresses, fatigue, loaded contact or manufacturing tolerances.

Reference and coordinate conventions

Yu and Ting, Application of Finite Element Analysis for the Strain Wave Gear Tooth Surfaces Design and Modification, Gear Solutions, 15 September 2019, Kinematic Model, printed p. 37 (unnumbered speed-ratio equation): select tooth motion, transform a complete flexible tooth profile through that motion, then find the envelope of the trajectories. Their Compliant Cup Deformation and Loaded Tooth Contact Pattern sections separately address elastic deformation and longitudinal modification. RatioLab uses their envelope construction principle with its own prescribed map and reference tooth; it does not reproduce their double-circular tooth profile, finite element results, contact ratio or prototype performance. Original article.

All rotations are positive about world +Z. Internal equations use radians; lengths and normal offsets use millimetres. With zf flex teeth and zc=zf+2 ring teeth, the nominal member equation is:

2*omega_wave + zf*omega_flex - zc*omega_ring = 0

Fixed ring gives omega_flex=-2/zf*omega_wave. Fixed flex gives omega_ring=2/zc*omega_wave. Subtracting the ring rotation reduces either configuration to the same relative tooth motion, so both output choices use the same generated ring. A negative phase reverses that relative motion.

Prescribed pitch-arc/normal map

The reference pitch radius is rp=m*zf/2. The chosen amplitude sets ellipse semiaxis a=rp+amplitude; b is solved so its perimeter is 2*pi*rp. Uniform material angle u means pitch arc length divided by rp, not ellipse polar angle. Eight Fourier modes integrate the ellipse speed; CPU and GLSL use the same coefficients and Newton inversion:

xi(t) = t + sum(c[k]*sin(2*k*t), k=1..8)
xi = u + flex_angle - wave_angle
E(t,d) = (a*cos(t),b*sin(t)) + d*unit(cos(t)/a,sin(t)/b)
P = rotate_Z(wave_angle)*E(t,d)

d is the reference radial offset from the undeformed pitch circle. The tooth band uses full deformation. The cup wall blends continuously from a circular base to that band using an immutable per-vertex weight; cup, flat cup and hat retain their continuous body, diaphragm or flange and independent output bore.

Complete outer swept envelope

Let g=zc/zf. In the fixed-ring frame, xi=u-g*psi, where psi is the wave angle. Eliminating psi gives every trajectory point as:

psi = (u-xi(t))/g
ring_angle = u/g + arg(E(t,d)) - xi(t)/g
radius = length(E(t,d))

A material tooth shift u+2*pi/zf shifts the ring angle by 2*pi/zc. Likewise, t+pi shifts it by one ring pitch. Thus one complete reference tooth and a half-turn of ellipse parameter cover all ring teeth and phases.

The solver retains all trajectory branches, reference edges and corners. It samples the two-parameter reference/trajectory grid, including both edge directions and patch diagonals, and takes the maximum radial intersection at each fixed ring angle. Keeping only the largest-radius branch for each reference point is incorrect and is not used. Adaptive doubled-grid comparison must converge within a measured 0.002 mm radial change, subject to a bounded grid size; unresolved cases fail explicitly.

The ring has 128 angular cells per tooth, starts at -pi/zc, and contains exactly 128*zc cyclic hole vertices without a duplicated endpoint. Its outer wall leaves the chosen rim thickness outside the largest cavity radius.

Numerical enclosure and physical backlash

The actual flexible reference tooth is thinned by half the specified pair backlash. The generating reference is instead thickened by that half, using the same rack-generated tooth form. This is a reference tooth-thickness allowance; the application does not equate it to loaded rotational lost motion of an elastic reducer.

Numerical enclosure is calculated separately. For a cell with angle step h and endpoint radii Ra,Rb, a chord's inverse radial distance at angle theta is:

A = sin(theta_b-theta)/sin(h)
B = sin(theta-theta_a)/sin(h)
1/R(theta) = A/Ra + B/Rb

Each interior sampled grid vertex (theta,r) requires cell scale max(1,r*(A/Ra+B/Rb)). Each endpoint receives the larger adjacent cell scale. The resulting chord half-plane contains every clipped sampled segment because its boundary crossings are already bounded and its interior endpoints satisfy the same half-plane inequality. This preserves branches and corners without adding an arbitrary clearance or changing the wave amplitude.

The maximum additional radial relief and measured refinement change are reported separately in wave_mapping.ring_envelope and calculation reports. This encloses the sampled sweep patches; it is not a mathematical global bound for the continuous nonlinear deformation between all grid points. Independent off-grid checks retain that distinction. Small sub-threshold penetrations can remain between the different sampled polygons; they are subject to the interference check's stated geometric tolerance.

Moving interference checks and validation

The checker uses each station's immutable material coordinates, validates that they reconstruct the nominal shared vertices within 1e-7 mm, and evaluates the same CPU map afresh at each pose. Triangle connectivity and triangle count are retained, including cap and bore vertices. Unknown/malformed deformation, stale references, unsupported warps and invalid posed surfaces remain explicit failures. Cancellation, timing, phase coverage and triangle limits apply to the full moving assembly; no expected interface suppresses a collision pair.

Regression tests independently move literal tooth-polygon edges against the actual ring polygon at unaligned positive and reverse phases, including both output choices, several feasible tooth counts/amplitudes and all body variants. Zero-backlash tests require useful flank proximity with a nonzero normal torque arm; tip or root tangency alone is insufficient. A deliberately wrong ring phase must create an interior witness. These are sampled geometric checks, not an elastic solution or a continuous clearance certificate. Real bearings, supports, fatigue fillets, tooling, lubrication and production acceptance still need their own engineering checks.