Helical, bevel and worm gears and gear trains

The spur version of the GR-5 works, but the utility's specification limits noise in its pump rooms, and spur gears at nearly 5 m/s are noisy. Farah asks Owen to compare a helical version. The same week, two other jobs arrive: a mixer drive that turns the motor's power through a right angle, and a valve actuator that must hold its position when the motor stops. Each needs a different kind of gear, and the sludge pump itself, it turns out, should run even slower than 417 rpm. This lesson covers helical gears, bevel gears, worm gears and the trains that combine gears to reach large ratios.

Helical gears

Helical gears have teeth cut at an angle, the helix angle \(\psi\), to the axis of the gear. Contact between mating teeth starts at one end of a tooth and spreads across the face as the gears turn, instead of along the whole face at once as in spur gears. Load is picked up and released gradually, and more teeth share the load at any moment, so helical gears run more quietly and smoothly and carry more load for the same size. The price is an axial thrust force that the bearings must carry, and slightly higher sliding losses. Helix angles are typically 15° to 30°. Mating helical gears on parallel shafts have equal helix angles of opposite hand.

Helical geometry is described in two planes. The normal plane is perpendicular to the teeth; the transverse plane is perpendicular to the axis. The cutting tool works in the normal plane, so the normal module \(m_n\) and normal pressure angle \(\phi_n\) are the standard values. In the transverse plane:

The face contact ratio, or overlap ratio, \(m_F = F\tan\psi/(\pi m_n)\), measures how far a tooth's contact advances across the face; a value of at least 1 is recommended for smooth running. Helical gear forces resolve into three components:

Double helical (herringbone) gears cancel the thrust by using two opposite-hand helices side by side; they are used in large turbine and mill drives. Crossed helical gears on non-parallel shafts make point contact and carry only light loads.

Bevel gears

Bevel gears transmit power between intersecting shafts, usually at 90°. Their pitch surfaces are cones that roll on each other, with the cone apexes meeting at the intersection of the shaft axes. The pitch angle of the pinion is \(\gamma = \arctan(N_P/N_G)\) for a 90° pair. Straight bevel gears have straight teeth tapering toward the apex; spiral bevel gears have curved, oblique teeth that run more smoothly and are used where speed or load is high; hypoid gears, as in vehicle axles, have offset axes. Forces are calculated at the mean radius of the face, \(r_{av} = d_P/2 - (F/2)\sin\gamma\), and the radial and axial components on the pinion are

For the mixer drive, a 20-tooth pinion of module 4 mm drives a 40-tooth gear, and the pinion carries 120 N·m. The program gives a pitch angle of 26.6°, a mean radius of 34.4 mm and a tangential load of 3,487 N, with 1,135 N radial and 568 N axial on the pinion. Bevel pinions are often overhung, and their bearings must locate them accurately along the axis, because the mesh is sensitive to axial position.

Worm gears

A worm gear set has a screw-like worm meshing with a worm wheel on a shaft at 90° to it. One revolution of a single-start worm advances the wheel by one tooth, so ratios of 10 to 100 are easy in a single stage. The lead angle \(\lambda\) of the worm is the angle of its thread relative to a plane perpendicular to its axis. Contact is mostly sliding, so friction is significant and efficiency depends strongly on the lead angle and the coefficient of friction f:

Small lead angles give low efficiency but can make the drive self-locking: the wheel cannot drive the worm backwards if the static coefficient of friction satisfies \(f \ge \cos\phi_n\tan\lambda\). Self-locking is useful for hoists and valve actuators, where the load must hold when the motor stops, although vibration can reduce friction and cause creep, so a separate brake is still used where safety depends on it. Worm wheels are usually bronze, running against a hardened steel worm, and heat generation often limits the rating rather than strength.

Gear trains

A gear train combines several meshes. Its train value e is the ratio of output speed to input speed, the product of the driving tooth numbers divided by the product of the driven tooth numbers:

with the sign positive if input and output turn the same way. In a simple train, each shaft carries one gear and idlers change only the direction. In a compound train, some shafts carry two gears, so large ratios fit in a compact space. A single stage is usually limited to a ratio of about 6 to 8 for spur and helical gears, because the pinion becomes too small or the gear too large.

The sludge pump's process engineer now asks for about 105 rpm, a ratio of about 16.7, too much for one stage. A two-stage compound train with 17 and 71 teeth in the first stage and 19 and 76 in the second gives e = (17 × 19)/(71 × 76) = 0.05986, an output of 105.4 rpm. Choosing tooth numbers for a required ratio is a small search problem: the program shows the result, and a few lines of code can test all combinations within limits on tooth numbers.

# Helical, bevel and worm gear forces, and gear train values
import math
P, n1 = 7500.0, 1760
T1 = P / (2 * math.pi * n1 / 60)
# Helical replacement for the GR-5 stage: same ratio, normal module 3 mm, helix angle chosen for a near-equal spacing
mn, phin, psi = 3.0, math.radians(20), math.radians(15)
Np, Ng = 18, 76
mt = mn / math.cos(psi)
dp = mt * Np; dg = mt * Ng
phit = math.atan(math.tan(phin) / math.cos(psi))
Wt = 2 * T1 / (dp / 1000)
print(f"helical: transverse module {mt:.3f} mm, pitch diameters {dp:.1f} / {dg:.1f} mm, shaft spacing {(dp+dg)/2:.1f} mm")
print(f"transverse pressure angle {math.degrees(phit):.2f} deg; Wt = {Wt:.0f} N, Wr = {Wt*math.tan(phit):.0f} N, thrust Wa = {Wt*math.tan(psi):.0f} N")
F = 36.0
print(f"axial contact ratio for a {F:.0f} mm face: {F*math.tan(psi)/(math.pi*mn):.2f}")

# Straight bevel pair turning the drive through 90 degrees (mixer drive), forces at the mean radius
Npb, Ngb, mb, Fb = 20, 40, 4.0, 25.0
gam = math.atan(Npb / Ngb)
dpb = mb * Npb; rav = dpb / 2 - Fb / 2 * math.sin(gam)
Wtb = 120.0 / (rav / 1000)
print(f"\nbevel pinion pitch angle {math.degrees(gam):.1f} deg, mean radius {rav:.1f} mm, Wt = {Wtb:.0f} N")
print(f"pinion radial load {Wtb*math.tan(math.radians(20))*math.cos(gam):.0f} N, pinion thrust {Wtb*math.tan(math.radians(20))*math.sin(gam):.0f} N")

# Worm drive for a valve actuator: running efficiency and self-locking at rest
phin_w, f_run, f_static = math.radians(20), 0.05, 0.10
for lam_deg in (3.5, 6.0, 12.0):
    lam = math.radians(lam_deg)
    eff = (math.cos(phin_w) - f_run * math.tan(lam)) / (math.cos(phin_w) + f_run / math.tan(lam))
    locks = f_static >= math.cos(phin_w) * math.tan(lam)
    print(f"lead angle {lam_deg:4.1f} deg: running efficiency {eff:.1%}, {'self-locking at rest' if locks else 'can back-drive'}")

# Compound train for the PC-150 sludge pump: target 1760 -> about 105 rpm
pairs = [(17, 71), (19, 76)]
e = 1.0
for a, b in pairs: e *= a / b
print(f"\ncompound train value e = {e:.5f}, output {n1*e:.1f} rpm, overall ratio {1/e:.2f}")
# Planetary stage (sun driven, ring fixed, carrier output): Willis equation
Ns, Nr = 21, 63
print(f"planetary: sun {Ns}, ring {Nr}, planets {(Nr-Ns)//2}; ratio sun/carrier = {1 + Nr/Ns:.2f}")

helical: transverse module 3.106 mm, pitch diameters 55.9 / 236.0 mm, shaft spacing 146.0 mm
transverse pressure angle 20.65 deg; Wt = 1456 N, Wr = 549 N, thrust Wa = 390 N
axial contact ratio for a 36 mm face: 1.02

bevel pinion pitch angle 26.6 deg, mean radius 34.4 mm, Wt = 3487 N
pinion radial load 1135 N, pinion thrust 568 N
lead angle  3.5 deg: running efficiency 53.3%, self-locking at rest
lead angle  6.0 deg: running efficiency 66.0%, self-locking at rest
lead angle 12.0 deg: running efficiency 79.1%, can back-drive

compound train value e = 0.05986, output 105.4 rpm, overall ratio 16.71
planetary: sun 21, ring 63, planets 21; ratio sun/carrier = 4.00

Planetary trains

In a planetary (epicyclic) train, one or more planet gears mesh with a central sun gear and an internal ring gear, and the planets are carried on an arm called the carrier. With one member held, the train gives a fixed ratio; with all three free, it can add or split speeds, as in a vehicle differential. Analysis uses the relative motion of the gears with respect to the carrier, often written as the Willis equation:

where e is the train value with the carrier held. With the ring fixed and the sun driving, the ratio is \(1 + N_R/N_S\). For a sun of 21 teeth and a ring of 63, the ratio is 4.0, and the planets have (63 − 21)/2 = 21 teeth. Because the load is shared among several planets, planetary gearboxes are compact and carry high torques; they are common in mixers, wind turbines and the large sludge pumps Tidewater buys from specialist makers.

In US units, the helical GR-5 has a normal diametral pitch near 8.5 (the nearest standard is 8), a thrust of about 88 lbf and a tangential load of about 327 lbf. US gear reducer makers rate their products to AGMA 6013 and related application standards, which set service factors for driven machinery; a progressive cavity sludge pump typically needs a service factor of about 1.25 to 1.5.