 Gear helix angle

The gear helix angle $\beta$ measures a cylindrical gear tooth's inclination to the rotation axis, normally on the reference cylinder. A straight spur gear has $\beta =0$; a helical gear has a nonzero inclination and a specified hand. The cylinder and reference direction matter because another measurement plane or cylinder radius can give another angle.^[ref]

Let $L$ be the positive gear lead, the axial advance of one tooth helix in a complete turn, and let the reference diameter be $d$. Unwrapping that cylinder produces a right triangle with circumferential side $\pi d$ and axial side $L$:For $d=60\mathrm{mm}$ and $|\beta |=30{}^{\circ }$, $L\approx 326.484\mathrm{mm}$. Doubling the lead on that same cylinder gives angle approximately $16.102{}^{\circ }$, rather than15degrees: the relation involves a tangent, not a linear angle scaling.

With $N$ equally spaced teeth, gear axial pitch is $p_x=L/N=p_t/\tan |\beta |$, where $p_t$ is gear transverse pitch. For nonzero $\beta$, this also gives $p_x=\pi m_n/\sin |\beta |$. The corresponding axial-pitch quantity diverges in the straight-tooth limit; it is not a finite replacement for the transverse tooth spacing of a spur gear.

Larger helix angle changes reference diameter at fixed gear normal module and tooth count, increases axial thrust for a fixed tangential load, and increases face-overlap geometry at fixed face width. These consequences must be evaluated together with gear contact ratio and support loads. A magnitude such as30degrees does not by itself specify right versus left hand, nor the thrust direction under a particular applied torque.