Gear Stress

Gear Stress

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 m_G = N_G/N_P, pitch-line velocity V = π d_P n_P/12 ft/min and transmitted tangential load W_t = 33,000 H/V with H in horsepower, or W_t = 2T_P/d_P from the pinion torque. Circular pitch p = π/P, base pitch p_b = 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 W_t l/(F t²). With the dimensionless form factor Y = 2 x P/3 (x = t²/4l) this is Shigley's Eq. 14-7, σ = W_t P/(F Y), and Eq. 14-8 adds the Barth velocity factor K_v 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 = W_t K_o K_v K_s (P_d/F) (K_m K_B/J). Allowable bending stress, Eq. 14-17: σ_all = S_t Y_N/(K_T K_R). The AGMA bending factor of safety is S_F = σ_all/σ (Eq. 14-41).

• K_o, overload factor: entered from the service (Fig. 14-17; 1.00 for a uniform power source and uniform driven machine).
• K_v, dynamic factor, Eqs. 14-27 and 14-28: B = 0.25 (12 − Q_v)^2/3, A = 50 + 56 (1 − B), K_v = [(A + √V)/A]^B with V in ft/min, valid up to V_max = [A + (Q_v − 3)]² (Eq. 14-29). Q_v is the AGMA transmission accuracy level, 3 to 7 for most commercial gears, 8 to 12 for precision gears.
• K_s, size factor, Section 14-10 Eq. (a): K_s = 1.192 (F √Y/P)^0.0535 in inch units, Y the Lewis form factor; taken as 1 when the fit gives less.
• K_m, load-distribution factor, Eq. 14-30: K_m = 1 + C_mc (C_pf C_pm + C_ma C_e). C_mc = 1 for uncrowned teeth, 0.8 crowned (Eq. 14-31). C_pf, 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. C_pm = 1 for a straddle-mounted pinion with S₁/S < 0.175, 1.1 otherwise (Eq. 14-33). C_ma = 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. C_e = 0.8 when the gearing is adjusted at assembly or lapped, 1 otherwise (Eq. 14-35). Because C_pf uses the pinion diameter, K_m is the same for both members.
• K_B, rim-thickness factor, Eq. 14-40: K_B = 1.6 ln(2.242/m_B) for a backup ratio m_B = t_R/h_t below 1.2, otherwise 1.
• J, bending geometry factor (Section 14-6): J = Y K_ψ/(K_f m_N) with the form factor Y of the tooth loaded at the highest point of single-tooth contact (HPSTC), the Dolan-Broghamer stress correction factor K_f and the load-sharing ratio m_N = 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.
• Y_N, bending stress-cycle factor (Fig. 14-14): above 3 × 10⁶ cycles the figure gives a band, upper line Y_N = 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 N_P/N_G of the pinion cycles.
• K_T, temperature factor: 1 up to 250 °F; K_R, 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: σ_c = C_p √[W_t K_o K_v K_s K_m/(d_P F) · C_f/I]; the same for both members except for K_s. Allowable contact stress, Eq. 14-18: σ_c,all = S_c Z_N C_H/(K_T K_R), and S_H = σ_c,all/σ_c (Eq. 14-42).

• C_p, elastic coefficient, Eq. 14-13: C_p = √{1/[π ((1 − ν_P²)/E_P + (1 − ν_G²)/E_G)]} 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) · m_G/(m_G + 1).
• C_f, surface condition factor: 1 (no standard values exist).
• Z_N, 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.
• C_H, hardness-ratio factor for the gear only (the pinion takes 1). Eq. 14-36 for through-hardened members: C_H = 1 + A′ (m_G − 1), A′ = 8.98 × 10⁻³ (H_BP/H_BG) − 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): C_H = 1 + B′ (450 − H_BG), B′ = 0.00075 exp(−0.0112 f_P), f_P the pinion surface finish in µin rms.

Allowable stress numbers

S_t and S_c 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 S_t = 77.3 H_B + 12,800 lb/in² and S_c = 322 H_B + 29,100 lb/in²; Grade 2 S_t = 102 H_B + 16,400 and S_c = 349 H_B + 34,300 (valid for 180 to 400 HB). Nitrided through-hardened steels take S_t from Fig. 14-3 (Grade 1: 82.3 H_B + 12,150; Grade 2: 108.6 H_B + 15,890, core hardness) and nitriding steels from Fig. 14-4 (Nitralloy 135M Grade 1: 86.2 H_B + 12,730, Grade 2: 113.8 H_B + 16,650; 2.5 % chrome Grades 1 to 3: 105.2 H_B + 9,280 / 22,280 / 29,280), with S_c 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): S_t 45,000 / 55,000, S_c 170,000 / 190,000 for Grades 1 and 2. Carburized and hardened steel: S_t 55,000 / 65,000 / 75,000 and S_c 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 S_t 5,000 / 8,500 / 13,000 and S_c 50,000 / 65,000 / 75,000; ductile iron A536 60-40-18, 80-55-06, 100-70-03 and 120-90-02 S_t 22,000 / 22,000 / 27,000 / 31,000 and S_c 77,000 / 77,000 / 92,000 / 103,000. Bronze: sand cast S_t 5,700, S_c 30,000; heat treated 23,600 and 65,000. For any other material choose "Entered" and type S_t and S_c.

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 − √(r_o,mate² − r_b,mate²) with C₆ = (r + r_mate) sin φ, and the highest point of single-tooth contact is a base pitch further, C₄ = C₁ + p_b (contact ratio m_p = (C₅ − C₁)/p_b with C₅ = √(r_o² − r_b²)). The load angle follows from tan φ_nW = C₄/r_b as φ_nL = tan φ_nW − tan φ + φ − π/(2N) and the load line meets the tooth centreline at the radius r_nL = r_b/cos φ_nL, 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 h_F tan β = ξ of the standard's iteration (its Eq. 5.63). With the critical thickness s_F = 2ξ and height h_F, Y = 1/[(cos φ_nL/cos φ)(6 h_F/s_F² − tan φ_nL/s_F)] (Eq. 5.78, the second term being the radial compression), K_f = H + (s_F/ρ_F)^L (s_F/h_F)^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 ρ_F = r_T + (b − r_T)²/(r + b − r_T) at the root (Eq. 5.68 for a rack), and J = Y/K_f. 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 S_F with S_H² 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 S_F,P, S_F,G, S_H,P² and S_H,G², because none of the factors depends on the load. The margins of safety are reported on the same load basis, S_F − 1 in bending and S_H² − 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.