Force components on a helical gear tooth

Geometry of the tooth force

In a helical gear the tooth force $W_n$ (normal to the tooth surface) acts in the normal plane, which is inclined to the transverse plane. $\phi_n$ is the normal pressure angle, $\psi$ the helix angle.

 Normal plane:        Pitch plane:
      W_n             axial ^   /
      /|                    |  / helix
     / | W_r                | /  (psi)
    /phi_n                  |/
   +-----               ----+------> tangential
   W_n cos(phi_n)

Step 1: Components in the normal plane

• Radial component: $W_r = W_n\sin\phi_n$.
• Component in the pitch plane, perpendicular to the tooth (helix) direction: $W_n\cos\phi_n$.

Step 2: Resolve the pitch-plane component

The helix line makes angle $\psi$ with the axis. Along the transverse tangent (tangential) and the axis:

• Tangential component: $W_t = W_n\cos\phi_n\cos\psi$
• Axial component: $W_a = W_n\cos\phi_n\sin\psi$

Results

$$W_n = \frac{W_t}{\cos\phi_n\cos\psi}$$ $$W_r = W_n\sin\phi_n = \frac{W_t\tan\phi_n}{\cos\psi} = W_t\tan\phi_t$$ $$W_a = W_n\cos\phi_n\sin\psi = W_t\tan\psi$$

where $W_t = 2T/d$ is found from the torque, and $\tan\phi_t = \tan\phi_n/\cos\psi$.

Axial thrust

$W_a = W_t\tan\psi$ increases with helix angle. For $\psi = 15^\circ$, $W_a = 0.268W_t$; for $\psi = 30^\circ$, $W_a = 0.577W_t$. Thrust bearings (angular contact or taper roller) must take it. The direction of the thrust depends on the hand of the helix and direction of rotation, and the gear and pinion have equal and opposite axial forces.

Double helical (herringbone) gears

A double helical gear is two helical gears of opposite hand cut side by side on one blank. Each half produces an axial thrust of $W_t\tan\psi/2$, and the two thrusts are equal and opposite, so the net axial force on the shaft is zero. This allows larger helix angles (up to 45 degrees), giving smoother and more powerful drives with no thrust bearing needed. Applications: heavy reduction gearboxes, turbines, rolling mills.