Gear helix angle
= Gear helix angle
{title2=$\beta$}
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.<KHK Gear Nomenclature>[Technical reference §4.3, Figure4.7, reference-cylinder helix and lead; Basic Guide B §2, p.19]{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 $L$:
$$
\tan|\beta|=\frac{\pi d}{L}.
$$
For $d=60\,\mathrm{mm}$ and $|\beta|=30\,{}^{\circ}$, $L\approx326.484\,\mathrm{mm}$. 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.