Spur and Helical Gear Geometry: Method

Spur and Helical Gear Geometry: Involute Geometry, Measurements, Contact Ratio, Interference and Gear Trains

Scope and conventions

The tool computes the involute geometry of one external or internal gear pair with parallel axes, spur or helical, and the ratios, speeds and torques of gear trains. Member 1 is always the external pinion; member 2 is the gear, or the ring gear of an internal pair. Both members share the normal module m_n (or the normal diametral pitch P_nd = 25.4/m_n teeth per inch), the normal pressure angle α_n, the helix angle β (zero for spur gears) and the basic rack with the addendum coefficient h_a* and the dedendum coefficient h_f*; the profile shift coefficients x₁ and x₂ differ. A positive profile shift withdraws the rack from the blank by x m_n, which thickens the tooth at the reference circle by 2 x m_n tan α_n, raises the tip by x m_n and shortens the dedendum by the same amount. For the ring gear of an internal pair a positive x₂ widens the tooth space by 2 x₂ m_n tan α_n and moves its bore (tip circle) and root circle outward; with that convention the ring's working pressure angle equation carries x₂ − x₁, as in the usual internal-gear formulas. Stress and life are the business of the Gear Stress tool; this one draws no conclusion about load.

Basic geometry

Transverse pressure angle tan α_t = tan α_n/cos β; transverse module m_t = m_n/cos β (NASA RP-1152 eqs. 13 to 17). Reference diameter d = m_t z, base diameter d_b = d cos α_t, base helix angle sin β_b = sin β cos α_n (equivalently tan β_b = tan β cos α_t). Addendum h_a = (h_a* + x) m_n, dedendum h_f = (h_f* − x) m_n, tip diameter d + 2 h_a, root diameter d − 2 h_f; for the ring d_a2 = d₂ − 2 (h_a* − x₂) m_n and d_f2 = d₂ + 2 (h_f* + x₂) m_n. Pitches: transverse p_t = π m_t, normal p_n = π m_n, base pitches p_bt = p_t cos α_t and p_bn = p_n cos α_n, axial pitch p_x = p_n/sin β, lead π d/tan β. The virtual (equivalent) spur gear of a helical gear has z_v = z/cos³β teeth of module m_n (RP-1152 eq. 21). Transverse reference tooth thickness s_t = m_t (π/2 + 2 x tan α_n), normal s_n = m_n (π/2 + 2 x tan α_n). No tip shortening is applied; the tip-shortening coefficient that would restore the standard clearance at a modified centre distance is reported, with the actual bottom clearances.

Involute function, centre distance and backlash

The involute function inv α = tan α − α is the polar angle of the involute point whose pressure angle is α (Buckingham 1928, eq. 36). Tooth thickness at any radius r follows from the thickness at the reference circle (Buckingham eq. 40): s_y = 2 r (s_t/d + inv α_t − inv α_y) with cos α_y = d_b/(2 r); for a ring tooth the sign of the involute terms reverses because the tooth widens outward. The tooth is pointed where inv α_p = s_t/d + inv α_t (eq. 41); the normal tip thickness s_an = s_at cos β_a, tan β_a = tan β d_a/d, is compared with the entered minimum s_amin* m_n (0.25 m_n is a common lower bound for through-hardened teeth, 0.4 m_n for surface-hardened ones; the value is an input).

Two mating teeth fill one pitch at the working pitch circles when the backlash is zero. Writing the thickness equation for both members at their working radii and using d_w = d cos α_t/cos α_wt gives Buckingham's eq. 43: inv α_wt = inv α_t + 2 (x₁ + x₂) tan α_n/(z₁ + z₂) for an external pair and inv α_wt = inv α_t + 2 (x₂ − x₁) tan α_n/(z₂ − z₁) for an internal pair. The tool solves it for α_wt by Newton's method on tan α − α (starting from the cube-root estimate inv α ≈ α³/3) and then a_w = a cos α_t/cos α_wt (eq. 44) with a = m_t (z₁ ± z₂)/2. When a centre distance is entered instead, the working pressure angle is cos α_wt = a cos α_t/a_w and the transverse backlash is computed exactly as the working pitch less the two working thicknesses, j_t = p_wt − s_w1 − s_w2 (ring: space width less pinion thickness), which reduces to j_t = 2 a_w (inv α_wt − inv α_wt,0) for a pair whose thicknesses fill the pitch at a_w,0; the normal backlash is j_n = j_t cos α_wt cos β_b. The linear estimate of NASA RP-1152 eq. 7, Δj_t = 2 Δa tan α, is printed beside the exact value; as RP-1152 notes, for an internal pair an increase of centre distance reduces the backlash, which the exact calculation reproduces.

Measurements

Chordal thickness and height (gear-tooth vernier caliper): the chord subtending the tooth at the reference circle and the distance from the tip to that chord, s̄ = d sin(s/d) and h̄ = h_a + (d/2)(1 − cos(s/d)) (Buckingham eqs. 115 and 116; NAVEDTRA 12204 chapter 15 works the 24-tooth, 8-pitch example that the test reproduces). A helical tooth is measured in the normal plane as the tooth of its virtual spur gear, so d is replaced by d_v = m_n z_v and s by s_n; this is the usual shop approximation and is stated as such. For a ring tooth the height is measured from the tip inward, h̄ = h_a − (d/2)(1 − cos(s/d)).

Span measurement over k teeth (Wildhaber's base-tangent method): a vernier caliper spanning k teeth touches two involute flanks along a line tangent to the base circle, so the reading equals the base-circle arc between the two profiles and does not depend on where on the flanks the jaws touch: W_k = m_n cos α_n [π (k − 0.5) + z inv α_t] + 2 x m_n sin α_n, that is k − 1 base pitches plus the base thickness of one tooth, projected on the normal plane for a helical gear (W_n = W_t cos β_b). Buckingham prints this as eq. 126, M = r cos α (T/r + πS/N + 2 inv α) with S the number of spaces between the measured profiles; the space term should read 2πS/N (one base pitch per space is 2π r cos α/N), and his worked example (6 d.p., 20°, 30 teeth, S = 3, printed 1.0524 in) follows the misprinted form — the correct reading across four teeth is 1.7904 in, and 1.0524 in lies between the two- and three-tooth spans, where no caliper could rest. The automatic k is the one that brings the contact points nearest the reference circle, k ≈ [d_b tan α_t cos β_b − 2 x m_n sin α_n]/(π m_n cos α_n) − z inv α_t/π + 0.5 rounded; the contact-point diameter d_M = √(d_b² + (W_k/cos β_b)²) is checked against the form and tip diameters, and a helical gear needs a face width of at least W_k sin β_b for the jaws. Ring gears are not measured this way.

Measurement over pins or balls (Buckingham, Problem 13 and Chapter IX): the centre of a pin of diameter d_p resting in a tooth space lies on an involute of the same base circle, offset from the flank involute by d_p/2 along the base tangent, so with the space centre line as reference inv α_M = s_t/d + inv α_t + d_p/d_b − π/z (eq. 53), which with the reference thickness becomes inv α_M = inv α_t + d_p/(m_n z cos α_n) − π/(2z) + 2 x tan α_n/z; the term d_p/(m_n z cos α_n) equals d_p/(d_b cos β_b) and so holds for balls on helical gears. Buckingham calls the offset approximate; it is exact, because parallel involutes of one base circle are a constant distance apart along their common normal, and the numerical construction in the tool's test (a circle pressed into the space between two flank point clouds) agrees with the formula to a micrometre. For a ring the signs of the pin and thickness terms reverse. The pin centres lie on d_c = d_b/cos α_M; over two pins of an even-toothed gear M = d_c + d_p, for an odd number of teeth M = d_c cos(90°/z) + d_p (eqs. 120 and 121), and between the pins of a ring the pin diameter is subtracted. The default pin, 1.728 m_n, is a common size that touches the flanks near the reference circle of an unshifted gear; any pin may be entered. The test reproduces Buckingham's roll examples (24 teeth, 6 d.p., 0.150-in roll radius; 25 teeth, 5 d.p., 0.375-in rolls, M = 5.5652 in).

Contact ratio and path of contact

Contact runs along the line of action, the common tangent of the base circles through the pitch point, from A, where the mate's tip circle cuts it, to E, where the pinion tip circle cuts it. With the tangency points T₁ and T₂ of the base circles a distance a_w sin α_wt apart, the length of the path is g_α = √(r_a1² − r_b1²) + √(r_a2² − r_b2²) − a_w sin α_wt for an external pair (Buckingham eq. 58, RP-1152 eq. 8) and g_α = √(r_a1² − r_b1²) − √(r_a2² − r_b2²) + a_w sin α_wt for an internal pair, where both tangency points lie on the same side of the pitch point. The transverse contact ratio is ε_α = g_α/p_bt (RP-1152 eq. 11), the overlap ratio of a helical pair ε_β = b sin β/(π m_n) = b/p_x, and the total ε_γ = ε_α + ε_β. The approach (A to P) and recess (P to E) lengths and the diameters at which the active profiles start on each member are reported. The test reproduces Buckingham's Problem 14 geometry (30 and 40 teeth, 7-in centres, 3.2- and 4.2-in outside radii; the book prints 1.70 teeth where its own numbers give 1.68) and the contact ratios 1.81 and 1.80 of the two gear sets in Table I of NASA TM 83725 (50 teeth at 10 d.p. with addendum ratio 1.1, 100 teeth at 20 d.p. with 1.04, 22°).

Specific sliding

At a contact point a distance t from the pitch point along the line of action the radii of curvature of the two involutes are their distances from the tangency points, ρ₁ and ρ₂ (RP-1152 eqs. 9 and 10). Pure rolling occurs only at the pitch point; elsewhere the profiles slide, and Buckingham's specific sliding (eqs. 23 and 24), the sliding per unit length of profile, is ζ₁ = 1 − ρ₂ z₁/(ρ₁ z₂) on the pinion and ζ₂ = 1 − ρ₁ z₂/(ρ₂ z₁) on the gear; each becomes unbounded where its profile reaches the base circle, which is why a pinion with few teeth wears at the root. The sliding velocity is |t| (ω₁ + ω₂) for an external pair and |t| (ω₁ − ω₂) for an internal one, reported as a fraction of the pitch-line velocity. Figure 2 draws ζ₁, ζ₂ and the velocity ratio from A to E. The test confirms the formulas by the kinematics of the two contact points (their velocities projected on the common tangent).

Undercut and interference

A rack-type cutter with a sharp corner of addendum (h_a* − x) m_n undercuts the tooth when the corner passes inside the point where the line of action touches the base circle, a distance r sin²α_t below the pitch line (Buckingham eq. 31, A = r cos²α for the radius of that point); hence z ≥ z_min = 2 (h_a* − x) cos β/sin²α_t, or equivalently x ≥ x_min = h_a* − z sin²α_t/(2 cos β). For h_a* = 1 and spur teeth z_min is 17.10 at 20°, 31.90 at 14.5° and 11.20 at 25°, so the smallest whole numbers without any undercut are 18, 32 and 12 (the familiar 17 and 32 come from rounding 17.1 and 31.9). Without undercut the involute extends down to the form diameter d_Ff = √(d_b² + (d sin α_t − 2 (h_a* − x) m_n/sin α_t)²), the radius at which the rack corner crosses the line of action; the start of active profile must lie above it. A fillet on the cutter corner lets it go deeper by the fillet height (Buckingham eq. 60), which the tool does not credit. The test checks the undercut decision by following the corner's trochoid in the gear frame and detecting whether it crosses the involute above the base circle.

Involute interference in the mesh occurs when a tip circle reaches beyond the tangency point of the mate's base circle, so that contact would have to take place below that base circle where there is no involute. The largest permissible tip radius of each member is √(r_b² + (a_w sin α_wt)²) (RP-1152 eq. 12); for the internal pair the condition on the ring tip is the usual z₁/z₂ ≥ 1 − tan α_a2/tan α_wt written in the same geometry. For the internal pair the tool also follows the two tip corners of a pinion tooth through one ring pitch of relative motion in the ring's frame (the ring turning z₁/z₂ of the pinion angle in the same sense, so one relative pitch takes 2π/(z₂ − z₁) of pinion rotation) and reports their least clearance from the ring tooth material. The ring tooth is modelled with involute flanks down to its root circle, so by conjugate action the clearance is zero exactly where the corner is the point of contact at the end of recess, and it turns negative only when a corner passes the ring root circle (the radial clearance c₁ has vanished) or enters a ring tooth. The trochoid interference proper — the pinion tip against the fillet of a ring cut by a pinion-type cutter — depends on the cutter's tooth number and is not modelled; a ring with few more teeth than its pinion should be checked against its cutter. Radial assembly of the pinion into the ring is not checked either.

Gear trains

Series (compound) trains: the overall ratio is the product of the driven tooth numbers over the product of the driver tooth numbers (RP-1152 Section 9.1; NAVEDTRA 14037 chapter 6 works the 10/40 then 20/10 train with a speed ratio of ½ and the anchor winch 22/88 then 30/90 with 12 to 1); the output speed is n_in/i, the output torque T_in i η^N with the mesh efficiency applied once per stage, and each external mesh reverses the direction.

Epicyclic trains by the tabular method: with every member locked the train turns one revolution; with the carrier held the sun turns one revolution through the ordinary train, whose ratio is the basic ratio i₀ = (ω_r − ω_c)/(ω_s − ω_c) = −z_s/z_r for a simple train and −z_s z_pr/(z_p z_r) for a compound planet (RP-1152 Tables 21 and 22, arm fixed). Every motion is a combination of the two rows (Willis' relation), which gives the RP-1152 ratios with one member fixed — ring/arm = 1 + z_s/z_r with the sun fixed, sun/arm = 1 + z_r/z_s with the ring fixed — and the speeds of all members for any fixed and input member. Torques follow from the power balance with the efficiency on the output and from the condition that the three torques sum to zero (the fixed member carries the reaction). Assembly with q equally spaced planets needs (z_s + z_r)/q whole for a simple train and (z_r z_p − z_s z_pr)/q for a compound one (RP-1152 Section 9.3), equal modules need z_r = z_s + 2 z_p (or z_s + z_p + z_pr), and adjacent planets clear each other when the chord between their centres, 2 a_sp sin(π/q), exceeds the planet tip diameter m (z_p + 2); the largest planet count is reported. In the limit of many teeth that clearance condition gives the maximum simple-train ratios RP-1152 quotes, about 6.8 with four planets and 4.8 with five.

Figure 1

The drawing is generated point by point from the involute equations in the transverse plane: three teeth of each member with their rack-generated form radius, the working pitch circles (dashed), the base circles (dotted) with their tangency points T₁ and T₂, and the line of action through the pitch point P at the working pressure angle with the path of contact A–E on it. The pinion tooth above P has its lower flank through P; the mating tooth below P has its upper flank through P. Root fillets are drawn as radial lines and tooth tips as arcs; the tooth thickness at the tip, the pointing diameter and the form diameter are the computed ones.

Sources and checks

The method is Buckingham's 1928 Spur Gears (public domain), NASA RP-1152 for the helical relations, the backlash and interference equations and the epicyclic tables, Wildhaber's base-tangent method, NAVEDTRA 12204 for the caliper example and NAVEDTRA 14037 for the train examples; AGMA 908 / 2001, Buckingham's 1949 Analytical Mechanics of Gears, Dudley and Machinery's Handbook are cited only. The test script reproduces every worked problem of Buckingham's Chapters III and IX used here to the printed digits (thickness at another radius, pointed radius, tight-mesh centre distance of a pair and of similar gears, roll position, roll measurement, chordal values, undercut radius), the NAVEDTRA example, the TM 83725 contact ratios, and checks every formula against an independent construction from the involute curve itself: flank point clouds for the span (support widths), a circle pressed into the space for the pins, bisection on the thickness equation for the centre distance, line-circle intersections in coordinates for the path of contact, contact-point kinematics for the sliding, the corner trochoid for the undercut and a tip-corner scan for the internal pair, on the default and on twenty seeded pairs.