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Gear Stress: Method

Open the Gear Stress tool

Gear Stress: Lewis and AGMA Spur-Gear Rating

Scope

The tool rates one spur-gear mesh of standard full-depth involute teeth (20° or 25° pressure angle, addendum 1/P) in bending and in surface durability. It follows Chapter 14 of Shigley's Mechanical Engineering Design, 10th edition, which presents the ANSI/AGMA 2001-D04 method with the factors as the standard tabulates them, and it reproduces the book's Example 14-4 with the default inputs. Helical, bevel and worm gears, internal gears, non-standard addenda, scoring and lubrication are outside its scope.

Geometry and load

Pitch diameters d = N/P (P the diametral pitch in teeth per inch; a module m in millimetres is converted by P = 25.4/m), speed ratio mG = NG/NP, pitch-line velocity V = π dP nP/12 ft/min and transmitted tangential load Wt = 33,000 H/V with H in horsepower, or Wt = 2TP/dP from the pinion torque. Circular pitch p = π/P, base pitch pb = p cos φ.

Lewis equation

Lewis (1893) treated the tooth as a cantilever of constant strength: a parabola with its vertex at the point where the load line meets the tooth centreline and tangent to the root fillet defines the critical section of thickness t at a distance l below the vertex, and the bending stress there is σ = 6 Wt l/(F t²). With the dimensionless form factor Y = 2 x P/3 (x = t²/4l) this is Shigley's Eq. 14-7, σ = Wt P/(F Y), and Eq. 14-8 adds the Barth velocity factor Kv for the dynamic increment of the load: (600 + V)/600 for cast-profile cast-iron teeth, (1200 + V)/1200 for cut or milled teeth, (50 + √V)/50 for hobbed or shaped teeth and √((78 + √V)/78) for shaved or ground teeth, V in ft/min. The form factors of Table 14-2 are for the load at the tip of a 20° full-depth tooth and are interpolated linearly in the tooth number. The Lewis stress ignores the radial component of the load, stress concentration, load sharing and life; it is the preliminary check the AGMA method refines.

AGMA bending stress and strength

Bending stress number, Eq. 14-15: σ = Wt Ko Kv Ks (Pd/F) (Km KB/J). Allowable bending stress, Eq. 14-17: σall = St YN/(KT KR). The AGMA bending factor of safety is SF = σall/σ (Eq. 14-41).

  • Ko, overload factor: entered from the service (Fig. 14-17; 1.00 for a uniform power source and uniform driven machine).
  • Kv, dynamic factor, Eqs. 14-27 and 14-28: B = 0.25 (12 − Qv)2/3, A = 50 + 56 (1 − B), Kv = [(A + √V)/A]B with V in ft/min, valid up to Vmax = [A + (Qv − 3)]² (Eq. 14-29). Qv is the AGMA transmission accuracy level, 3 to 7 for most commercial gears, 8 to 12 for precision gears.
  • Ks, size factor, Section 14-10 Eq. (a): Ks = 1.192 (F √Y/P)0.0535 in inch units, Y the Lewis form factor; taken as 1 when the fit gives less.
  • Km, load-distribution factor, Eq. 14-30: Km = 1 + Cmc (Cpf Cpm + Cma Ce). Cmc = 1 for uncrowned teeth, 0.8 crowned (Eq. 14-31). Cpf, the pinion proportion factor (Eq. 14-32): F/(10d) − 0.025 for F ≤ 1 in, F/(10d) − 0.0375 + 0.0125 F for 1 < F ≤ 17 in, F/(10d) − 0.1109 + 0.0207 F − 0.000228 F² for 17 < F ≤ 40 in, with F/(10d) taken as 0.05 when smaller, d the pinion pitch diameter. Cpm = 1 for a straddle-mounted pinion with S1/S < 0.175, 1.1 otherwise (Eq. 14-33). Cma = A + B F + C F² (Eq. 14-34) with Table 14-9: open gearing 0.247, 0.0167, −0.765 × 10⁻⁴; commercial enclosed 0.127, 0.0158, −0.930 × 10⁻⁴; precision enclosed 0.0675, 0.0128, −0.926 × 10⁻⁴; extraprecision enclosed 0.00360, 0.0102, −0.822 × 10⁻⁴. Ce = 0.8 when the gearing is adjusted at assembly or lapped, 1 otherwise (Eq. 14-35). Because Cpf uses the pinion diameter, Km is the same for both members.
  • KB, rim-thickness factor, Eq. 14-40: KB = 1.6 ln(2.242/mB) for a backup ratio mB = tR/ht below 1.2, otherwise 1.
  • J, bending geometry factor (Section 14-6): J = Y Kψ/(Kf mN) with the form factor Y of the tooth loaded at the highest point of single-tooth contact (HPSTC), the Dolan-Broghamer stress correction factor Kf and the load-sharing ratio mN = 1 for spur gears. Shigley's Fig. 14-6 (the AGMA 908 chart) gives J against the tooth number for several mating tooth numbers; the tool takes an entered reading or computes J as described below, and always shows both.
  • YN, bending stress-cycle factor (Fig. 14-14): above 3 × 10⁶ cycles the figure gives a band, upper line YN = 1.3558 N−0.0178 (the line Shigley's examples use) and lower line 1.6831 N−0.0323 (critical service); below it the hardness lines 9.4518 N−0.148 (400 HB), 6.1514 N−0.1192 (250 HB), 4.9404 N−0.1045 (160 HB), 3.517 N−0.0817 (case carburized) and 2.3194 N−0.0538 (nitrided). The gear sees NP/NG of the pinion cycles.
  • KT, temperature factor: 1 up to 250 °F; KR, reliability factor, Table 14-10: 0.70, 0.85, 1.00, 1.25 and 1.50 for reliabilities 0.50, 0.90, 0.99, 0.999 and 0.9999 (Eq. 14-38 between them).

AGMA contact stress and strength

Contact stress number, Eq. 14-16: σc = Cp √[Wt Ko Kv Ks Km/(dP F) · Cf/I]; the same for both members except for Ks. Allowable contact stress, Eq. 14-18: σc,all = Sc ZN CH/(KT KR), and SH = σc,all/σc (Eq. 14-42).

  • Cp, elastic coefficient, Eq. 14-13: Cp = √{1/[π ((1 − νP²)/EP + (1 − νG²)/EG)]} in √psi; Table 14-8 rounds it (steel on steel 2300, steel on cast iron 2100, steel on tin bronze 1900 …), and the book's examples use the table values.
  • I, pitting geometry factor, Eq. 14-23 for external spur gears: I = (cos φ sin φ/2) · mG/(mG + 1).
  • Cf, surface condition factor: 1 (no standard values exist).
  • ZN, pitting stress-cycle factor (Fig. 14-15): 2.466 N−0.056 below 10⁷ cycles; above, the upper line 1.4488 N−0.023 or the lower line 2.466 N−0.056; nitrided steels 1.249 N−0.0138.
  • CH, hardness-ratio factor for the gear only (the pinion takes 1). Eq. 14-36 for through-hardened members: CH = 1 + A′ (mG − 1), A′ = 8.98 × 10⁻³ (HBP/HBG) − 8.29 × 10⁻³ for a hardness ratio between 1.2 and 1.7, A′ = 0 below 1.2 and 0.00698 above 1.7. Eq. 14-37 for a surface-hardened pinion (above 48 HRC) with a through-hardened gear (180 to 400 HB): CH = 1 + B′ (450 − HBG), B′ = 0.00075 exp(−0.0112 fP), fP the pinion surface finish in µin rms.

Allowable stress numbers

St and Sc are the AGMA 2001-D04 allowable stress numbers for 10⁷ cycles at 99 % reliability, as Shigley prints them. Through-hardened steel follows the lines of Figs. 14-2 and 14-5: Grade 1 St = 77.3 HB + 12,800 lb/in² and Sc = 322 HB + 29,100 lb/in²; Grade 2 St = 102 HB + 16,400 and Sc = 349 HB + 34,300 (valid for 180 to 400 HB). Nitrided through-hardened steels take St from Fig. 14-3 (Grade 1: 82.3 HB + 12,150; Grade 2: 108.6 HB + 15,890, core hardness) and nitriding steels from Fig. 14-4 (Nitralloy 135M Grade 1: 86.2 HB + 12,730, Grade 2: 113.8 HB + 16,650; 2.5 % chrome Grades 1 to 3: 105.2 HB + 9,280 / 22,280 / 29,280), with Sc from Table 14-6 at 83.5 HR15N (150,000 / 163,000) for the nitrided through-hardened steels, 90 HR15N for Nitralloy (170,000 / 183,000) and 87.5 HR15N for the chrome steels (155,000 / 172,000 / 189,000). Flame or induction hardened steel (50 HRC, Type A): St 45,000 / 55,000, Sc 170,000 / 190,000 for Grades 1 and 2. Carburized and hardened steel: St 55,000 / 65,000 / 75,000 and Sc 180,000 / 225,000 / 275,000 for Grades 1 to 3 (Tables 14-3 and 14-6). Cast irons (Tables 14-4 and 14-7, the lower value of each printed range): gray iron A48 class 20 / 30 / 40 St 5,000 / 8,500 / 13,000 and Sc 50,000 / 65,000 / 75,000; ductile iron A536 60-40-18, 80-55-06, 100-70-03 and 120-90-02 St 22,000 / 22,000 / 27,000 / 31,000 and Sc 77,000 / 77,000 / 92,000 / 103,000. Bronze: sand cast St 5,700, Sc 30,000; heat treated 23,600 and 65,000. For any other material choose "Entered" and type St and Sc.

Computed geometry factor J

When no chart reading is at hand the tool lays out the tooth by the procedure of AGMA 908-B89 Section 5 and the drawing shows the result. The pinion and gear outside radii fix the line of action: the start of the active profile on the member analysed is C1 = C6 − √(ro,mate² − rb,mate²) with C6 = (r + rmate) sin φ, and the highest point of single-tooth contact is a base pitch further, C4 = C1 + pb (contact ratio mp = (C5 − C1)/pb with C5 = √(ro² − rb²)). The load angle follows from tan φnW = C4/rb as φnL = tan φnW − tan φ + φ − π/(2N) and the load line meets the tooth centreline at the radius rnL = rb/cos φnL, the vertex of the Lewis parabola. The fillet is the trochoid swept by the tip radius of the generating rack (addendum 1/P, whole depth 2.4/P and tip radius 0.35/P, the rack of the AGMA chart); the point where the inscribed parabola is tangent to the fillet is found by minimising ξ²/h along the fillet, which is the tangency condition 2 hF tan β = ξ of the standard's iteration (its Eq. 5.63). With the critical thickness sF = 2ξ and height hF, Y = 1/[(cos φnL/cos φ)(6 hF/sF² − tan φnL/sF)] (Eq. 5.78, the second term being the radial compression), Kf = H + (sF/ρF)L (sF/hF)M with H = 0.331 − 0.436 φ, L = 0.324 − 0.492 φ, M = 0.261 + 0.545 φ (φ in radians, Eqs. 5.72 to 5.76) and the minimum fillet radius ρF = rT + (b − rT)²/(r + b − rT) at the root (Eq. 5.68 for a rack), and J = Y/Kf. The computed values agree with the chart readings within about 0.02 (17-tooth pinion with a 52-tooth gear: 0.317 computed against 0.30 read; the 52-tooth gear 0.405 against 0.40). With the same layout, the tip load and no radial term the procedure reproduces the Lewis form factors of Table 14-2 within a few percent (rack 0.479 against 0.485), which is the check on the geometry. A pinion of fewer than about 18 teeth is undercut by this rack; the tool says so and its computed J is then an approximation.

Comparison and rating

Bending stress is proportional to the load while contact stress grows with its square root, so Shigley compares SF with SH² to see which failure mode governs each member (Section 14-19). The power rating at unit AGMA safety factor is the transmitted power times the least of SF,P, SF,G, SH,P² and SH,G², because none of the factors depends on the load. The margins of safety are reported on the same load basis, SF − 1 in bending and SH² − 1 in wear; the AGMA factors already contain the life, reliability and temperature effects.

Units

Lengths, forces, torques, stresses, power and speed convert with the unit switch; the diametral pitch and module, the hardness, the factors and the elastic coefficient (√psi, as the AGMA tables print it) are plain numbers. The AGMA fits are evaluated in inch units internally whatever the display system.