507 Ways to Move Part 5: Helical, Herringbone & Crossed Gears

Helical Gears

Normal vs Transverse Planes

Axial Thrust Forces

Herringbone (Double Helical) Gears

Crossed Helical Gears

Spur vs Helical Comparison

Manufacturing Helical Gears

Helical gears are manufactured using the same fundamental processes as spur gears, with one critical difference: the workpiece and/or the cutting tool must be tilted or differentially rotated to produce the helix angle.

Hobbing remains the primary production method. The gear blank is tilted on the hobbing machine's work table by the helix angle. The machine's differential gearing coordinates the hob rotation, blank rotation, and hob feed to generate the correct helix. Modern CNC hobbing machines handle this automatically from the gear data input.

Key manufacturing considerations for helical gears:

• The hob must have the correct normal module and normal pressure angle — the same hob that cuts spur gears of a given module can cut helical gears of the same normal module at any helix angle (by adjusting the machine setup)
• Right-hand and left-hand helical gears require different machine setups (reversed differential)
• Face width must be sufficient for at least one complete helix wrap: minimum face width = pi * m_n / sin(psi) for a face contact ratio of 1.0; practical face widths provide face contact ratios of 1.5-3.0
• Grinding of helical gears requires threaded grinding wheels that match the helix angle, or profile grinding with CNC path control

History: From Spur to Helical

The transition from spur to helical gears in automotive transmissions tells a compelling engineering story. Early automobiles (1900s-1920s) used straight-cut (spur) gears in their transmissions, producing the characteristic "whine" that classic car enthusiasts recognize — and that modern drivers would find unacceptable.

As automotive speeds increased and passenger comfort expectations rose, the need for quieter transmissions drove the adoption of helical gears. By the 1930s-1940s, most passenger car transmissions had switched to helical gears for the forward ratios. However, the reverse gear in most manual transmissions remained (and still remains) a spur gear — which is why reverse gear produces a distinctive whining noise. The brief engagement of reverse makes the spur gear's lower cost and simpler engagement mechanism preferable to helical's smoothness.

Modern automatic transmissions use helical gears exclusively, with helix angles of 25-35 degrees providing the NVH (Noise, Vibration, Harshness) performance that luxury vehicle buyers demand. The latest electric vehicle transmissions use specially optimized helical gear profiles with micro-geometry modifications (tip relief, lead crowning) to further reduce gear noise, since there is no engine noise to mask gear whine in an EV.

Case Studies

A modern 6-speed manual transmission uses helical gears for all forward ratios. The 3rd gear pair typically has: module 2.0-2.5 mm, pinion 22-26 teeth, gear 35-40 teeth, helix angle 28-32 degrees, face width 18-24 mm. The gears are made from 20MnCr5 or 16MnCr5 case-carburizing steel, carburized to 0.8-1.0 mm case depth, with surface hardness of 58-62 HRC. After carburizing, the gears are hard-finished by either generating grinding or honing to achieve AGMA quality 11-12 with surface roughness Ra < 0.3 micrometers. Total gear whine at 3000 RPM in 3rd gear is typically below 50 dBA at the driver's ear — nearly inaudible against road noise.

A naval frigate's propulsion system uses herringbone reduction gears to step down a 22,000 RPM gas turbine to a 150 RPM propeller shaft. The gearbox transmits 25 MW (33,500 HP) through a two-stage herringbone arrangement. The first stage (high-speed) has a helix angle of 35 degrees, module 8, and AGMA quality 13. The second stage (low-speed) has module 14 gears weighing several tons each. Total gearbox efficiency exceeds 98%. The herringbone design eliminates thrust bearings entirely, and the zero-thrust characteristic is critical because any axial force on a ship's propeller shaft would affect the stern tube seals and the ship's propulsion alignment.

The Tesla Model 3 uses a single-speed helical gear reducer to connect the permanent magnet motor (17,000 RPM max) to the wheels (~1,000 RPM max). The approximately 9:1 reduction uses a two-stage helical gear train with opposing helix hands to minimize thrust. Gear noise is a critical design parameter since there is no engine noise to mask it. Tesla employs advanced micro-geometry optimization: the tooth flanks are intentionally modified with 5-15 micrometers of tip relief and 3-8 micrometers of lead crowning to minimize transmission error under loaded conditions, achieving gear whine levels below 35 dBA at highway speeds.

Python Calculations

import math

class HelicalGear:
    """Complete helical gear calculator with force analysis."""

    def __init__(self, normal_module, num_teeth, helix_angle_deg,
                 normal_pressure_angle_deg=20.0):
        self.mn = normal_module
        self.N = num_teeth
        self.psi = math.radians(helix_angle_deg)
        self.psi_deg = helix_angle_deg
        self.phi_n = math.radians(normal_pressure_angle_deg)
        self.phi_n_deg = normal_pressure_angle_deg

    # --- Module conversions ---
    @property
    def transverse_module(self):
        """Transverse module (mm)."""
        return self.mn / math.cos(self.psi)

    # --- Pressure angle conversions ---
    @property
    def transverse_pressure_angle(self):
        """Transverse pressure angle (radians)."""
        return math.atan(math.tan(self.phi_n) / math.cos(self.psi))

    @property
    def transverse_pressure_angle_deg(self):
        return math.degrees(self.transverse_pressure_angle)

    # --- Diameters ---
    @property
    def pitch_diameter(self):
        """Pitch diameter (mm)."""
        return self.transverse_module * self.N

    @property
    def base_diameter(self):
        return self.pitch_diameter * math.cos(self.transverse_pressure_angle)

    @property
    def outside_diameter(self):
        return self.pitch_diameter + 2 * self.mn

    @property
    def root_diameter(self):
        return self.pitch_diameter - 2.5 * self.mn

    # --- Virtual (equivalent) teeth ---
    @property
    def virtual_teeth(self):
        """Equivalent number of teeth in normal plane."""
        return self.N / (math.cos(self.psi) ** 3)

    @property
    def min_teeth_no_undercut(self):
        """Minimum actual teeth to avoid undercutting."""
        n_min_spur = 2 / (math.sin(self.phi_n) ** 2)
        return math.ceil(n_min_spur * (math.cos(self.psi) ** 3))

    # --- Force Analysis ---
    def forces(self, torque_Nm):
        """
        Calculate all tooth force components.
        Args:
            torque_Nm: Torque on this gear in Newton-meters
        Returns:
            dict with tangential, radial, axial, and normal forces (N)
        """
        d = self.pitch_diameter / 1000  # convert to meters
        Ft = 2 * torque_Nm / d
        Fr = Ft * math.tan(self.transverse_pressure_angle)
        Fa = Ft * math.tan(self.psi)
        Fn = Ft / (math.cos(self.phi_n) * math.cos(self.psi))

        return {
            'tangential_N': Ft,
            'radial_N': Fr,
            'axial_N': Fa,
            'normal_N': Fn,
            'resultant_N': math.sqrt(Ft**2 + Fr**2 + Fa**2)
        }

    # --- Overlap ratio ---
    def face_contact_ratio(self, face_width_mm):
        """Face contact ratio (overlap ratio)."""
        axial_pitch = self.mn * math.pi / math.sin(self.psi)
        return face_width_mm / axial_pitch

    def summary(self, torque_Nm=100, face_width_mm=None):
        if face_width_mm is None:
            face_width_mm = 10 * self.mn

        print(f"=== Helical Gear Analysis ===")
        print(f"Normal module: {self.mn} mm")
        print(f"Transverse module: {self.transverse_module:.3f} mm")
        print(f"Teeth: {self.N} (virtual: {self.virtual_teeth:.1f})")
        print(f"Helix angle: {self.psi_deg} deg")
        print(f"Normal PA: {self.phi_n_deg} deg")
        print(f"Transverse PA: {self.transverse_pressure_angle_deg:.2f} deg")
        print(f"Pitch diameter: {self.pitch_diameter:.2f} mm")
        print(f"Outside diameter: {self.outside_diameter:.2f} mm")
        print(f"Min teeth (no UC): {self.min_teeth_no_undercut}")
        print(f"Undercut: {'YES' if self.N < self.min_teeth_no_undercut else 'No'}")
        print(f"Face contact ratio: {self.face_contact_ratio(face_width_mm):.2f}")

        forces = self.forces(torque_Nm)
        print(f"\nForces at {torque_Nm} Nm torque:")
        print(f"  Tangential: {forces['tangential_N']:.1f} N")
        print(f"  Radial:     {forces['radial_N']:.1f} N")
        print(f"  Axial:      {forces['axial_N']:.1f} N")
        print(f"  Normal:     {forces['normal_N']:.1f} N")
        print(f"  Axial/Tang: {forces['axial_N']/forces['tangential_N']*100:.1f}%")


# Example: Automotive 3rd gear
pinion = HelicalGear(normal_module=2.5, num_teeth=24, helix_angle_deg=30)
gear = HelicalGear(normal_module=2.5, num_teeth=37, helix_angle_deg=30)

print("--- Pinion ---")
pinion.summary(torque_Nm=150, face_width_mm=22)

print("\n--- Gear ---")
gear.summary(torque_Nm=150 * 37/24, face_width_mm=22)

ratio = gear.N / pinion.N
print(f"\nGear ratio: {ratio:.3f}:1")
print(f"Center distance: {(pinion.pitch_diameter + gear.pitch_diameter)/2:.2f} mm")

Exercises & Self-Assessment

Conclusion & Next Steps

You now understand helical, herringbone, and crossed helical gears. Here are the key takeaways from Part 5:

• Helical gears achieve 50-80% noise reduction over spur gears through gradual tooth engagement — making them the standard for automotive and industrial applications where NVH matters
• The helix angle trade-off: higher angles mean smoother operation but more axial thrust force (F_a = F_t * tan psi) and slightly lower efficiency
• Normal vs transverse planes are essential concepts — cutting tools work in the normal plane, gear ratios are calculated in the transverse plane, and the virtual number of teeth (N_v = N/cos^3 psi) determines undercutting limits
• Herringbone gears cancel thrust forces entirely through opposing helix hands, enabling very high helix angles and maximum load capacity for power generation and marine applications
• Crossed helical gears handle non-parallel, non-intersecting shafts with simple construction, but point contact limits them to light-load applications
• Modern EV transmissions push helical gear design to new extremes of NVH performance through micro-geometry optimization and advanced manufacturing