10.4 Design of Spur, Helical, Bevel & Worm Gears

Design Versus Kinematics

The Theory of Machines treatment of gearing answers what motion results: velocity ratio, interference, path of contact, contact ratio. Gear design answers a different question — will the tooth break, and will the surface wear out? The CIL Mechanical syllabus lists gears under both headings, and the exam draws on both.

A gear tooth can fail in two independent ways, and a competent design checks both:

Tangential Tooth Load

The starting point of every gear calculation is the tangential force transmitted at the pitch circle:

$P_t = \frac{2T}{d} = \frac{60\times10^{6}\,(\text{kW})}{\pi\,d\,n} \;\text{N}$

with $d$ in mm and $n$ in rev/min. A service factor $C_s$ accounts for shock from the prime mover and driven machine, so the design tangential load is

$P_{\text{design}} = \frac{P_t\,C_s}{C_v}$

where $C_v$ is the velocity factor, discussed below. For a coal-handling drive with a heavy-shock crusher, $C_s$ may reach 1.75 or more; for a steady electric-motor-driven fan it is near 1.0.

The Lewis Equation: Beam Strength

Wilfred Lewis modelled the tooth as a cantilever of uniform strength, fixed at the root and loaded at the tip. Inscribing a parabola of uniform strength inside the tooth profile and applying simple bending theory gives

$\boxed{S_b = \sigma_b\,b\,m\,Y}$

where $\sigma_b$ is the allowable bending stress, $b$ the face width, $m$ the module and $Y$ the Lewis form factor.

The form factor depends on the number of teeth $z$ and the pressure angle, and for a 20-degree full-depth involute is approximated by

$Y = \pi\left(0.154 - \frac{0.912}{z}\right)$

$Y$ increases with the number of teeth, so a small pinion has the weaker tooth. This is the reason the pinion is usually made of the stronger material.

Which member is weaker?

Since beam strength is proportional to $\sigma_b Y$ and $b$ and $m$ are common to both gears in mesh, the weaker member is the one with the smaller product $\sigma_b Y$. Design for that member.

A typical case: a steel pinion with $\sigma_b = 200$ MPa and $Y_p = 0.308$ gives $\sigma_b Y = 61.6$; a cast-iron gear with $\sigma_b = 55$ MPa and $Y_g = 0.408$ gives $22.4$. The cast-iron gear is weaker despite having more teeth, and it governs.

Face width

Face width is conventionally taken as a multiple of module:

$b = (9 \text{ to } 15)\,m$

with 10$m$ a common starting value. Too narrow a face wastes capacity; too wide risks uneven load distribution from shaft misalignment.

Dynamic Load

Real gears carry more than the static tangential load because of tooth errors, elastic deflection and inertia of the accelerating masses. Two approaches are standard.

Velocity factor (approximate)

$P_d = \frac{P_t}{C_v}$

Buckingham's equation (accurate)

$P_d = P_t + \frac{21v\left(b\,C + P_t\right)}{21v + \sqrt{b\,C + P_t}}$

where $v$ is pitch line velocity in m/s, $b$ the face width in mm and $C$ the deformation factor depending on tooth error and material combination. The incremental dynamic load is the second term.

Wear Strength

Surface fatigue produces pitting long before the tooth breaks in many drives. Buckingham's wear strength is

$S_w = d_p\,b\,Q\,K$

where $d_p$ is the pinion pitch diameter, and the ratio factor is

$Q = \frac{2\,z_g}{z_g + z_p} = \frac{2i}{i+1}$

The load-stress factor $K$ derives from the Hertzian contact stress $\sigma_{es}$:

$K = \frac{\sigma_{es}^2\sin\phi}{1.4}\left(\frac{1}{E_p} + \frac{1}{E_g}\right)$

Note that $K$ depends on the square of the allowable surface endurance stress, which in turn correlates with surface hardness. Doubling hardness therefore roughly quadruples wear capacity, which is why case-hardening and nitriding are the standard responses to a pitting problem.

The Design Criteria

A gear pair is satisfactory when both conditions hold:

$S_b \geq P_d \quad \text{(no tooth breakage)}$

$S_w \geq P_d \quad \text{(no surface pitting)}$

The factor of safety is $S_b/P_d$ in bending and $S_w/P_d$ in wear, and the lower of the two describes the drive.

If beam strength is deficient, increase the module or the face width, or use a stronger material. If wear strength is deficient, increase the surface hardness or the pinion diameter — increasing module alone helps far less, because $S_w$ does not contain $m$ explicitly.

Worked Example

A spur pinion of 20 teeth, module 5 mm, face width 50 mm transmits power at 900 rev/min to a 60-tooth gear. Take $\sigma_b = 140$ MPa for the pinion and $Y = 0.32$.

Pitch diameter: $d = mz = 5\times20 = 100$ mm.

Pitch line velocity:

$v = \frac{\pi d n}{60000} = \frac{\pi\times100\times900}{60000} = 4.71 \text{ m/s}$

Beam strength:

$S_b = \sigma_b\,b\,m\,Y = 140\times50\times5\times0.32 = 11200 \text{ N}$

Using the ordinary-cut velocity factor $C_v = 3/(3+4.71) = 0.389$, the permissible tangential load is

$P_t = S_b \times C_v = 11200\times0.389 = 4357 \text{ N}$

and the power transmitted is

$P = P_t\,v = 4357\times4.71 = 20.5 \text{ kW}$

Other Gear Types

Helical gears

Teeth are cut on a helix at angle $\alpha$, so engagement is gradual and running is quieter with a higher contact ratio. Two moduli must be distinguished:

$m_n = m_t\cos\alpha$

The Lewis equation uses the normal module and a form factor based on the virtual (formative) number of teeth:

$z_v = \frac{z}{\cos^3\alpha}$

The penalty is an axial thrust $P_a = P_t\tan\alpha$, which the bearings must carry. Double helical (herringbone) gears cancel this thrust by using two opposite helices.

Bevel gears

Used for intersecting shafts, normally at 90 degrees. The tooth section varies along its length, so the Lewis equation carries a correction:

$S_b = \sigma_b\,b\,m\,Y\left(1 - \frac{b}{L}\right)$

where $L$ is the cone distance. The form factor uses the virtual number of teeth $z_v = z/\cos\gamma$, with $\gamma$ the pitch cone angle. Face width is limited to about $L/3$.

Worm gears

Used for non-intersecting perpendicular shafts and very high reduction ratios in a single stage — commonly 20:1 to 100:1. The action is predominantly sliding, so efficiency is comparatively low and heat dissipation usually governs the design rather than tooth strength. Efficiency is

$\eta = \frac{\tan\lambda\,(1 - \mu\tan\lambda)}{\tan\lambda + \mu}$

where $\lambda$ is the lead angle. When the lead angle is small enough that $\tan\lambda < \mu$, the drive becomes self-locking — the worm can drive the wheel but the wheel cannot back-drive the worm. That property is exploited deliberately in hoists and lifting gear, where it prevents a load from running away.