The twist

Unrolling the pitch cylinder onto a plane turns every tooth of a helical gear into a straight band inclined at \(\beta\) to the axis, as in Figure 9.3.1. Moving a distance \(z\) along the axis shifts the tooth by \(z\tan\beta\) along the pitch circle, which is a rotation of the transverse section through

\[ \theta(z) = \frac{z\tan\beta}{r_p} = \frac{2z\tan\beta}{d} \tag{9.3.1}\]

where \(d = mN\) is the pitch diameter. The rotation is linear in \(z\), so the tooth trace is a helix, and one full turn takes the axial distance \(L = \pi d/\tan\beta\), the lead. A herringbone gear of face width \(b\) consists of two halves of width \(b/2\), and each turns its section through

\[ \theta_h = \frac{b\tan\beta}{d} \tag{9.3.2}\]

between the mid-plane and its end face, the two halves in opposite senses. The twist depends on \(b\) and \(d\) as well as on \(\beta\). A twist typed into SolidWorks as a fixed number therefore gives a different helix angle as soon as \(N\), \(m\) or the face width changes, and it has to be an equation.

The twisted section also gives the exact flank. Every transverse section of the gear is the transverse profile turned through \(\theta(z)\), so the flank is swept by a planar involute rotating at a constant rate as it advances along the axis. That surface is the involute helicoid, the flank of every involute helical gear [2], and its sections perpendicular to the axis are true involutes of the base circle. A twisted sweep of the spur profile is therefore not an approximation of the helical flank but the flank itself. The sections that are not involutes are those taken normal to the tooth.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patheffects as pe

m, N, b, beta = 1, 20, 10, np.radians(30)
p = np.pi * m                     # circular pitch: one tooth and one space
h = b / 2
halo = [pe.withStroke(linewidth=3, foreground='white')]

fig, ax = plt.subplots(figsize=(7, 4.2))
z = np.array([-h, 0, h])
shift = np.abs(z) * np.tan(beta)  # pitch-circle arc length the tooth has moved, per eq-hb-theta
for k in range(0, 5):
    left = k * p - p / 4 + shift
    right = k * p + p / 4 + shift
    ax.fill(np.r_[left, right[::-1]], np.r_[z, z[::-1]],
            facecolor='#bcd4e3', edgecolor='#236b8e', lw=1)

ax.axhline(0, color='0.35', lw=0.8, ls='--')
ax.plot([0, 0], [0, h], color='0.35', lw=0.8)
ax.plot([0, h * np.tan(beta)], [0, h], color='0.35', lw=0.8, ls='--')  # tooth centreline

# helix angle between the axis direction and the tooth centreline
arc = np.linspace(np.pi / 2 - beta, np.pi / 2, 30)
ax.plot(0.6 * h * np.cos(arc), 0.6 * h * np.sin(arc), color='0.35', lw=0.8)
ax.text(0.17 * h, 0.66 * h, r'$\beta$', fontsize=13, path_effects=halo)

# the arc length the section turns through over one half
ax.annotate('', xy=(h * np.tan(beta), h + 0.6), xytext=(0, h + 0.6),
            arrowprops=dict(arrowstyle='<->', color='0.35', lw=0.8))
ax.text(h * np.tan(beta) / 2, h + 0.9, r'$r_p\theta_h = \frac{b}{2}\tan\beta$',
        ha='center', fontsize=12, path_effects=halo)
ax.annotate('', xy=(-1.4, h), xytext=(-1.4, 0),
            arrowprops=dict(arrowstyle='<->', color='0.35', lw=0.8))
ax.text(-1.6, h / 2, r'$b/2$', ha='right', va='center', fontsize=12, path_effects=halo)
ax.text(-1.6, -0.6, 'mid-plane', ha='right', va='center', fontsize=10, color='0.35')

ax.set_xlim(-4.5, 5 * p)
ax.set_ylim(-h - 0.5, h + 2)
ax.set_aspect('equal')
ax.axis('off')
plt.show()

The helix also sets how many teeth share the load. The axial distance between neighbouring teeth is the axial pitch \(p_x = \pi m/\tan\beta\), and the face width measured in axial pitches is the overlap ratio

\[ \varepsilon_\beta = \frac{b\tan\beta}{\pi m} \tag{9.3.3}\]

A spur gear has \(\varepsilon_\beta = 0\). Once it exceeds one, some part of a tooth is in contact at every instant across the face, independently of the transverse contact ratio. For the gear of Figure 9.3.2, with \(N = 20\), \(m = 1\) mm, \(b = 10\) mm and \(\beta = 30°\), 9.3.2, the lead and 9.3.3 give

import sympy as sp
from mechanicskit import ltx

m_, N_, b_, beta_ = 1, 20, 10, sp.rad(30)
d_ = m_ * N_
theta_h = b_ * sp.tan(beta_) / d_
L_ = sp.pi * d_ / sp.tan(beta_)
eps_beta = b_ * sp.tan(beta_) / (sp.pi * m_)

ltx(r"\theta_h =", sp.deg(theta_h).evalf(4), r"^\circ,\quad L =", L_.evalf(4),
    r"~\text{mm},\quad \varepsilon_\beta =", eps_beta.evalf(3))

so the twist each half needs is modest, a little under one angular pitch of \(18°\), and nearly two teeth overlap across the face.

from mpl_toolkits.mplot3d.art3d import Poly3DCollection
from gear_geometry import gear_profile_points

P = gear_profile_points(N_teeth=20, module=1)[::2, :2]
z = np.linspace(-h, h, 41)
theta = 2 * np.abs(z) * np.tan(beta) / (m * N)      # eq-hb-theta, mirrored at the mid-plane
X = np.cos(theta)[:, None] * P[:, 0] - np.sin(theta)[:, None] * P[:, 1]
Y = np.sin(theta)[:, None] * P[:, 0] + np.cos(theta)[:, None] * P[:, 1]
Z = np.repeat(z[:, None], len(P), axis=1)

fig = plt.figure(figsize=(5.5, 5.5))
ax = fig.add_subplot(projection='3d')
ax.plot_surface(X, Y, Z, color='#bcd4e3', edgecolor='none', shade=True,
                rstride=1, cstride=1, antialiased=False)
ax.add_collection3d(Poly3DCollection([np.column_stack([X[-1], Y[-1], Z[-1]])],
                                     facecolor='#bcd4e3', edgecolor='#236b8e', lw=0.6))
ax.set_box_aspect((1, 1, b / (m * (N + 2))))
ax.view_init(elev=25, azim=-60)
ax.set_axis_off()
plt.show()

Building the gear in SolidWorks

We start from the finished spur gear part of the spur gear procedure. Its solid supplies the transverse profile, and the herringbone is built as a second body beside it.

Add two global variables to the equations,

where Twist is 9.3.2 converted to degrees, with FaceWidth as the full width \(b\) of the herringbone.

Open a sketch on the Top Plane, select the face of the spur gear that lies in that plane, and use Convert Entities. The sketch now holds the complete closed outline of the gear, flanks, root fillets and tip arcs, and it follows the solid through every change of \(N\) (Figure 9.3.3). This is the one reference to a model face in the construction, which the spur gear chapter warns against, so it is the first thing to check when a rebuild fails. Then open a second sketch on the Front Plane and draw a line from the origin along the gear axis, normal to the Top Plane, of length "FaceWidth" / 2.

Sweep the outline along the line with Insert, Boss/Base, Sweep. Under Options set Profile Twist to Specify Twist Value, Twist Control to Degrees, and enter = "Twist" in the angle field. Clear Merge result, so the sweep becomes a body of its own instead of fusing with the spur solid it overlaps (Figure 9.3.4).

The profile turns about the path, which is why the path starts at the origin and runs along the axis; a path anywhere else would swing the gear off centre as it twisted. Then remove the spur solid with Insert, Features, Delete/Keep Body, keeping the sweep (Figure 9.3.5). The converted sketch still resolves, since it comes before the deletion in the feature tree.

Finally mirror the sweep with Insert, Pattern/Mirror, Mirror, taking the Top Plane as the mirror plane, the sweep under Bodies to Mirror, and Merge solids checked. A reflection in a plane perpendicular to the axis reverses the hand of a helix, so the mirrored half twists the other way and the two halves meet in the V at the Top Plane, now the mid-plane of the gear. Both halves come from one sweep, so they agree by construction.

Two meshing herringbone gears have opposite hands, the V of one pointing into the V of the other. The mating gear is a copy of the part with its own N and the angle field set to = -"Twist", the sign of the twist setting the hand.