Two tooth profiles, two operating characteristics
In spur gearing, the teeth run parallel to the gear axis. Mesh engagement begins and ends abruptly across the entire tooth width—this generates periodic impacts and noise and limits the load-carrying capacity to the profile overlap alone.
In helical gearing, the teeth are inclined relative to the axis by the helix angle β (typically 8–20°). As a result, the contact moves smoothly across the tooth width: the force is gently applied and dissipated, which makes the operation significantly smoother. In addition to the profile overlap ε_α, there is also the jump overlap ε_β—the total overlap ε_γ = ε_α + ε_β increases, and on average, more teeth are in contact at the same time.
Forces on Helical Gearing
The inclined tooth line decomposes the tooth force into three components. The torque M and the pitch diameter d give rise to circumferential, axial, and radial forces:
F_t = 2 · M / d
F_a = F_t · tan β
F_r = F_t · tanα_n/cos β
F_t = Circumferential force (N) · F_a = Axial force (N) · F_r = Radial force (N) · M = Torque (Nmm) · d = Pitch diameter (mm) · β = Helix angle · α_n = Normal pressure angle (usually 20°)
The shaft bearing must reliably absorb this axial force—often, a simple deep groove ball bearing is no longer sufficient for this purpose, and angular contact ball bearings or tapered roller bearings are required.
Direct Comparison
| Characteristic | Spur gearing | Helical gearing |
|---|---|---|
| Smooth Operation / Noise | loud (sudden engagement) | significantly quieter (sliding engagement) |
| Load capacity | Standard | bis ca. +30 % |
| Axial force | keine | vorhanden |
| Efficiency | very high | slightly lower |
| Manufacturing / Costs | simple, affordable | more complex |
The special type known as arrow-tooth gearing (double helical gearing) combines two mirror-image halves on a single gear—the axial forces cancel each other out, resulting in smooth operation and high load-carrying capacity without axial bearing load, although the manufacturing process is more complex.
Mnemonic
Worked example
Given
A helical spur gear transmits M = 150 Nm with a pitch circle diameter d = 80 mm, helix angle β = 12°, and normal mesh angle α_n = 20°.
Calculation
- Circumferential force: F_t = 2 · M / d = 2 · 150,000 Nmm / 80 mm = 3,750 N
- Axial force: F_a = F_t · tan 12° = 3,750 · 0.2126 ≈ 797 N
- Radial force: F_r = F_t · tan 20° / cos 12° = 3,750 · 0.3640 / 0.9781 ≈ 1,395 N
The axial force here corresponds to approximately 21% of the circumferential force—a value that must be taken into account from the very beginning when selecting bearings.
Description and values of the figure
Three force components act at the point of tooth engagement: the circumferential force F_t= 2M/d, the axial force F_a=F_t·tan β (resulting from the helix angle β), and the radial force F_r=F_t·tan αn/cos β. For the worked example with M = 150 Nm, d = 80 mm, β = 12°, and αn = 20°, the results are F_t=3,750 N, F_a≈ 797 N (approximately 21% of F_t), and F_r≈1,395 N (approximately 37% of F_t).
| Force | Formula | Value |
|---|---|---|
| Umfangskraft F_t | 2 · M/d | 3,750 N |
| Axial force F_a | F_t · tan β | ≈ 797 N |
| Radial force F_r | F_t · tan αn / cos β | ≈ 1,395 N |
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