Bevel differential

RatioLab constructs straight and logarithmic spiral bevels from exact spherical involute working flanks. The analytic flanks satisfy the common-normal meshing condition; preview and exports sample those surfaces into facets. Nonworking root relief is a separate construction. Loaded contact, strength, backdrivability, bearing capacity and continuous collision clearance require additional engineering validation.

Wrist coordinates and speed equations

Lengths are millimetres and rates are rpm. A and B are the two independent input speeds about the same world +X direction, even though the shaft ends face opposite sides. Their side gears have equal tooth counts Zs. The output gear has Zo teeth and points along the carrier's +Z axis at the nominal pose. C is carrier rotation about world +X; S is output spin relative to its moving carrier. Rigid gears and ideal rolling without slip are assumed.

The two carrier-relative rolling constraints and their solution are:

Zs (A - C) + Zo S = 0
Zs (B - C) - Zo S = 0
C = (A + B) / 2
S = (Zs / Zo) (B - A) / 2
A = C - (Zo / Zs) S
B = C + (Zo / Zs) S

All tooth ratios are dimensionless. Equal inputs produce carrier rotation; opposite inputs produce relative output spin. For A=12, B=-12, Zs=24 and Zo=48, C=0 and S=-6 rpm; the inverse reconstructs both inputs. With A=0 and B=12, the same gearing gives C=6 and S=3 rpm. The original PR2 hardware manual, Wrist Drivetrains and PR2 Coordinate System, describes differential flex/roll actuation; its motor-facing sign conventions must be transformed before comparing with these world-axis definitions.

The output pose is Rx(carrier angle) Rz(relative spin angle), so the output spin axis follows the carrier. Speed changes preserve the four physical angles of A, B, C and S in the preview. Inspection and exports use the nominal model. No single reduction ratio, combined power balance or output torque is inferred from two prescribed input speeds.

Classic configuration: direct carrier input and two side outputs

The Classic configuration uses the same cone axes and equal side tooth counts, but the carrier is the physical input and A/B are the two side output speeds. The perpendicular bevel is an internal spider, freely turning on a carrier-fixed pin. There is no external perpendicular output shaft or flange.

The primary reference is MathWorks Simscape Driveline, Differential, Equations / Ideal Fundamental Constraints. Its first fundamental constraint couples both side gears to the carrier. With equal side/spider ratios, its non-divided form is A+B=2C; this form also covers equal speeds and stationary members. RatioLab drives C directly and does not include the reference block's additional longitudinal crown/pinion drive.

Carrier speed alone does not determine both output speeds. For visualization, the user prescribes Delta=B-A as a separate speed boundary condition:

A = C - Delta/2
B = C + Delta/2
S = (Zs / Zo) Delta/2
C = (A+B)/2
Delta = B-A

C, A, B and Delta use rpm about world +X. S is the internal spider's spin relative to its carrier's moving +Z axis; Zo is now the spider tooth count. All tooth ratios remain dimensionless. C=12/Delta=0 produces A=B=12 and S=0. C=12/Delta=24 with equal tooth counts produces A=0, B=24 and S=12. C=0/Delta=24 produces A=-12, B=12 and S=12. C=-5/Delta=-6 gives A=-2, B=-8 and S=-3. Both rolling constraints from the wrist equations remain satisfied.

Delta is not a second physical motor input or a load-dependent speed split. A prescribed stationary side is not a differential lock. No traction, friction, limited-slip behavior, output torque or power balance is inferred. Changing Zo changes internal spider spin, while the equal-side output constraint remains the same. Preview time follows the fastest derived member so zero carrier speed with a nonzero prescribed difference still animates.

Both side outputs can have solid shafts, hollow shafts or drilled flanges. A direct coaxial annular carrier mount clears output A. The side extension is measured beyond its outer carrier interface, including the full input-mount thickness on A. The internal spider bore equals the pin diameter plus twice the selected positive radial running clearance. Nominal axial retainers remain separated from the spider by the assembly clearance. These are geometric interfaces; complete pin retention, bearing/journal design, fits and lubrication in CAD. Hollow side passages do not establish an unobstructed through-route across the spider and its carrier-fixed pin.

Pitch cones and interfaces

With outer module m, the reference radii are rs = m Zs / 2 and ro = m Zo / 2. The common cone distance is R_outer = hypot(rs, ro); the perpendicular shaft cone angles are delta_s = atan(Zs/Zo) and delta_o = atan(Zo/Zs), summing to 90 degrees. These are the pitch-cone relations in KHK's dimension reference, section 4.4. Face width b is measured along the cone generator and limited to b <= R_outer/3; R_inner = R_outer - b must remain positive. Shaft radii are checked against the small-end tooth roots.

The end faces are planes at local Z = R_outer cos(delta) and Z = (R_outer-b) cos(delta). Module and face width refer to the pitch cone. At another polar angle theta, a vertex's actual apex distance is R = Z/cos(theta), which can extend beyond the pitch-cone reference interval. The mating surfaces engage only where their two axial face intervals overlap; there is no fictitious clamping of every vertex to R_outer. Cylindrical bores remain circular with the same radius at every axial section.

The 1:1 preset derives Zo=Zs. Spin reduction 2:1 derives Zo=2Zs; it changes the relative output/spider S while retaining the carrier mean equation. Custom mode preserves both supplied counts. Output connection, carrier construction and tooth variant are independent saved choices.

The hollow shaft has a real coaxial passage from the output gear toe to the shaft end, with at least 1 mm radial shaft wall. This does not establish a full cable route through the mechanism or its tilt range. Split carrier members have open bearing-seat halves and overlapping bridge tabs in separate axial layers, with matching joining holes. Nominal layer/ligament and clearance checks do not establish clamping strength. Bearings are envelopes; fits, retention, fasteners and mounting require completion in CAD.

Spherical involute flanks and nonworking roots

The primary surface reference is Dong et al., Spatial analysis of contact deformation, stiffness, and flash temperature in involute gear pairs, DOI 10.3389/fmech.2025.1643228, section 2, equation (1). Its Cartesian spherical involute can be expressed in polar form about each gear's own shaft as follows:

sin(delta_b) = sin(delta) cos(alpha_t)
t = acos(cos(theta)/cos(delta_b))
inv_s(theta) = t/sin(delta_b) - atan2(sin(t), sin(delta_b) cos(t))
half(theta) = pi/(2 Z_teeth) - j/(4 r_pitch)
              + inv_s(delta) - inv_s(theta)
phi = tooth_phase + trace(R) ± half(theta)
P = R (sin(theta) cos(phi), sin(theta) sin(phi), cos(theta))

Angles inside these formulas are radians; UI pressure and spiral angles are degrees. Here theta is polar angle from the gear axis, delta is pitch-cone half-angle, delta_b is base-cone half-angle, t is the generating involute parameter, j is the outer transverse backlash field in mm, and r_pitch = m Z_teeth/2. Each member receives half the requested tangential tooth-thickness reduction, so a mating pair has the requested nominal backlash at the outer pitch radius. The pressure field is transverse/spherical alpha_t, preserving the existing stored parameter convention. For the logarithmic spiral's pitch angle beta, normal pressure satisfies tan(alpha_n) = tan(alpha_t) cos(beta); the same numerical value in the transverse and normal fields would describe different spiral gears.

The meshing reference is Wang et al., The Equation of Meshing of Spiral Bevel Gears Manufactured by Generating-Line Method, DOI 10.2174/1874155X01105010051, section 4, equation (22): n dot ((omega_1-omega_2) cross P) = 0. Both shaft axes intersect at the common apex, so no translational term occurs. Tests solve this equation using independently finite-differenced runtime surfaces and verify mate positions, parallel normals, and zero relative normal velocity on both flanks, both hands, equal/unequal and odd tooth counts. They also compare the angular involute against Dong's Cartesian equation (1). Zero backlash is used for the nominal coincident-flank calculation; prescribed positive backlash separates the two tooth surfaces until a flank takes up play.

The historical heel tip radius r_pitch+m is retained. Its polar angle is theta_tip = atan((Z_teeth+2)/Z_mate). The initial root angle is atan((Z_teeth-2.5)/Z_mate). Where necessary it is deepened to pi/2-theta_tip_mate-0.25 m/R_outer, ensuring a positive angular clearance between the root blank and the mate tip cone. Shaft-wall checks use this actual root rather than the old nominal root radius. Below the base cone, constant half-angle continuation joins the root; it is a nonworking sharp construction, not a spherical involute or a generated manufacturing fillet. No reversed involute branch is used. A mate addendum requiring below-base contact is refused using the base-cusp limit theta_tip_mate < acos(sin(delta_b)^2/(cos(delta_b) tan(delta))). Pointed/overlapping teeth and exhausted sampling budgets are also refused.

This is an explicit custom planar blank, with m defining the outer pitch module. Its heel addendum measured normal to the pitch cone is m cos(delta). It is not the full-depth standard blank in KHK table 4.17, whose linked dimensions include addendum h_a=m, tip radius r_pitch+h_a cos(delta) and crown plane Z=R_outer cos(delta)-h_a sin(delta). Replacing just the cone angle while retaining the current pitch-height plane would not implement those dimensions. RatioLab retains its existing mounting/end planes and checks the resulting actual contact domain; some ordinary-looking high-pressure or narrow parameter sets therefore require different dimensions, as detailed below.

Tooth engagement and the shared face interval

Collision-free geometry alone does not ensure a tooth is available throughout the mesh cycle. The implementation also checks complete ideal engagement on both flanks, including both planar end caps. Parameterize the unit-sphere action great circle by s, measured from the pitch point:

X = cos(delta) cos(s) - sin(alpha_t) sin(delta) sin(s)
Y = -cos(alpha_t) sin(s)
Z = sin(delta) cos(s) + sin(alpha_t) cos(delta) sin(s)
R_low  = max(R_inner cos(delta_A)/X, R_inner cos(delta_B)/Z)
R_high = min(R_outer cos(delta_A)/X, R_outer cos(delta_B)/Z)
gear_phase = s/sin(delta_b_A) - k_A ln(R/R_outer) + constant

Addendum-cone limits and R_low <= R_high bound a connected contact domain. Each flank's unwrapped gear-phase interval must span more than one tooth pitch 2 pi/Z_A; reflecting the flank reverses the sign of k in this check. Its boundary consists of R = constant/X or constant/Z, switching at s=0. On each smooth branch, phase extrema occur at interval ends or tan(s+offset) = 1/(k sin(delta_b_A)). Evaluating these analytic stationary points and the branch junction gives the complete ideal phase span, rather than inferring engagement from a finite number of collision-free poses. The reported contact_phase_span_teeth is the smaller of the two flank spans.

For m=2 mm, 20-degree transverse pressure and zero spiral, the 24/24 pair with 8 mm face width has span 1.19202469688 tooth pitches. A 24/48 pair with 4 mm width gives 1.14572350061. A formerly accepted 16/60 pair with 4 mm width has only 0.76980163019 in the straight construction and 0.59766453066 on its limiting 35-degree spiral flank: these narrow cases are now rejected because the planar caps remove part of the required contact interval. With 8 mm width, that spiral pair gives 1.14273421889. Increasing width, reducing module or changing tooth counts can restore complete engagement when cap trimming is the limiting factor. High transverse pressure can instead make the addenda themselves too short: 24/24 at 30 degrees has a straight-flank span of only 0.94630958702 even when the caps do not trim contact. Reducing pressure, changing tooth counts or a supported spiral overlap can restore engagement; the generator does not silently extend teeth. The outer face-width limit b <= R_outer/3 still applies. These are unloaded geometric engagement spans, not load distribution or strength calculations.

The conjugacy_verified metadata flag is explicitly scoped to these analytic working flanks. It does not certify the sampled mesh, nonworking root, bearings, manufacturing process or an entire continuous motion cycle.

Spiral trace and local handedness

The spiral variant retains the same ideal speed equations. Its pitch-cone trace uses Cumbicus et al., Obtaining the geometry and 5-axis machining paths for the manufacture of spiral bevel gear, DOI 10.1007/s00170-025-16738-7, section 3.2, equations (8)-(9), also available in the author preprint.

theta(R) = h tan(beta) / sin(delta) * ln(R / R_outer)
R sin(delta) dtheta/dR = h tan(beta)

R is distance along the cone generator from the common apex, delta is the pitch-cone half-angle, beta is an unsigned spiral angle, and h is local hand (-1 or +1). The UI stores beta in degrees; internal trigonometry and theta use radians. Beta here corresponds to the paper's spiral angle lambda, not its involute parameter beta. The logarithm argument is dimensionless and positive. The heel phase is zero; zero beta uses exactly the straight geometry.

The side gear frames are right handed, with local +Z along world +X for A and world -X for B. Both side traces use the same local hand h; the output or internal spider trace uses -h. Because Zs/sin(delta_s) = Zo/sin(delta_o), both contact lines satisfy Zs theta_s(R) + Zo theta_o(R) = 0 at a shared cone distance. Together with the spherical involutes, this preserves conjugate indexing along the face. Assigning the same local hand to all three gears or using a twist linear in axial Z would not implement this construction. The heel pitch point has zero trace; other heel-plane vertices have the nonzero correction trace(R_outer cos(delta)/cos(theta)). Omitting this angle-dependent correction would produce incorrect spiral flanks despite correct-looking tooth traces.

All gears use matching cone-distance stations. A conservative second-derivative bound controls the chord deviation of each sampled outer-vertex trace to 0.003 mm. For f(R)=a R exp(i k ln(R/R_outer)), |f''| = a |k| sqrt(1+k²)/R; the maximum is bounded at R_inner and the largest profile radius. Uniform cone-distance intervals satisfy max|f''| deltaR²/8 <= 0.003 mm. Bores remain fixed circular sections. This bound applies to corresponding sampled vertices along the axial loft; it is separate from the section profile's quarter/midpoint refinement threshold of 0.003 mm. Refinement compares analytic points with their chord and refuses budget exhaustion. The profile threshold is a numerical refinement criterion, not a proved global bound on the entire surface or an industrial flank tolerance. Triangles between stations additionally approximate a curved two-parameter surface. At zero or very small backlash, discretization can affect apparent contact, so mesh intersection results must be interpreted at their stated sampling precision rather than substituted for an analytic contact calculation.

Preview, STEP and STL consume the same section surfaces in the nominal pose. SVG/DXF retain individual sections and placement metadata; they are not cutting templates. Axial caps are planar, avoiding nonplanar polygon caps in faceted STEP. Regression tests exercise actual shared meshes at independent tooth-cycle poses for wrist/classic arrangements and both mesh sides, and include an intentional tooth-on-tooth phase error as a collision negative control. These finite tests establish those sampled cases; they are not complete continuous clearance or load-capacity certification.