507 Ways to Move Part 4: Spur & Internal Gears

External Spur Gears

Internal Spur Gears

Gear Ratio & Speed Calculations

Interference & Undercutting

Friction Gearing

Spur Gear Design Procedure

Follow these steps for a systematic spur gear design:

1. Define requirements: Input power, input speed, output speed (or ratio), service life, space constraints, noise limits
2. Calculate gear ratio: i = n_input / n_output. Determine if single-stage (i < 6) or multi-stage is needed
3. Select number of teeth: Choose pinion teeth (minimum 17 for 20-degree PA, more for quieter operation). Gear teeth = pinion teeth * ratio
4. Select module: Based on tooth bending strength (Lewis equation) and surface contact stress (Hertz equation). Use AGMA 2001 or ISO 6336 rating methods
5. Calculate geometry: Pitch diameters, center distance, addendum/dedendum, face width
6. Check contact ratio: Must be > 1.2 (preferably > 1.4)
7. Check for interference: Verify no undercutting; apply profile shift if needed
8. Verify stresses: Bending stress and contact stress must be below allowable limits for chosen material and heat treatment
9. Specify tolerances: Select AGMA/ISO quality grade based on speed and noise requirements
10. Select material and heat treatment: Through-hardened, case-carburized, nitrided, or induction-hardened based on stress levels and application

Case Studies

Mechanical clocks use trains of spur gears to step down the high-speed escapement (4-5 beats per second) to the minute and hour hands. A typical clock gear train uses 4-5 stages of reduction, with ratios carefully chosen so that the minute hand shaft rotates once per hour and the hour hand once per 12 hours. The gear teeth are very fine (module 0.3-0.5), typically made from brass, and operate at negligible loads — accuracy of the gear ratio is the primary design criterion, not load capacity.

A conveyor belt requires 50 RPM from a 1750 RPM motor. The total ratio needed is 35:1, far too high for a single spur gear stage. Solution: a 3-stage spur gear reducer with ratios of approximately 3.3:1 per stage (3.3^3 = 35.9). Each stage uses a 20-tooth pinion and 66-tooth gear (ratio 3.3:1), module 3, 20-degree pressure angle. The total gearbox is compact, efficient (0.98^3 = 94.1%), and straightforward to manufacture.

An internal gear arrangement is used in the final stage of a conveyor drive where coaxial input and output shafts are required. The motor drives a sun pinion (24 teeth), which meshes with an internal ring gear (72 teeth), giving a 3:1 reduction in a compact, enclosed package. The same-direction rotation of the internal gear arrangement eliminates the need for an idler gear. This configuration forms the basis of simple planetary reducers used extensively in conveyor and winch drives.

Python Calculations

import math

class SpurGearPair:
    """Complete spur gear pair design calculator."""

    def __init__(self, module, pinion_teeth, gear_teeth,
                 pressure_angle_deg=20.0, face_width_factor=10):
        self.m = module
        self.Np = pinion_teeth
        self.Ng = gear_teeth
        self.phi = math.radians(pressure_angle_deg)
        self.phi_deg = pressure_angle_deg
        self.face_width = face_width_factor * module

    # --- Gear Ratio ---
    @property
    def ratio(self):
        return self.Ng / self.Np

    # --- Pitch Diameters ---
    @property
    def dp(self):
        """Pinion pitch diameter (mm)."""
        return self.m * self.Np

    @property
    def dg(self):
        """Gear pitch diameter (mm)."""
        return self.m * self.Ng

    # --- Center Distance ---
    @property
    def center_distance(self):
        """Center distance for external gear pair (mm)."""
        return (self.dp + self.dg) / 2

    @property
    def center_distance_internal(self):
        """Center distance for internal gear pair (mm)."""
        return (self.dg - self.dp) / 2

    # --- Speed & Torque ---
    def output_speed(self, input_rpm):
        """Calculate output RPM given input RPM."""
        return input_rpm / self.ratio

    def output_torque(self, input_torque, efficiency=0.98):
        """Calculate output torque given input torque (Nm)."""
        return input_torque * self.ratio * efficiency

    # --- Interference Check ---
    @property
    def min_pinion_teeth(self):
        """Minimum pinion teeth to avoid undercutting."""
        return math.ceil(2 / (math.sin(self.phi) ** 2))

    @property
    def has_undercutting(self):
        return self.Np < self.min_pinion_teeth

    # --- Contact Ratio ---
    @property
    def contact_ratio(self):
        """Profile contact ratio for external spur gear pair."""
        rp = self.dp / 2
        rg = self.dg / 2
        rap = rp + self.m  # addendum circle radius
        rag = rg + self.m
        rbp = rp * math.cos(self.phi)
        rbg = rg * math.cos(self.phi)
        C = self.center_distance
        base_pitch = math.pi * self.m * math.cos(self.phi)

        path_length = (math.sqrt(rap**2 - rbp**2) +
                       math.sqrt(rag**2 - rbg**2) -
                       C * math.sin(self.phi))
        return path_length / base_pitch

    # --- Pitch Line Velocity ---
    def pitch_velocity(self, pinion_rpm):
        """Pitch line velocity (m/s)."""
        return math.pi * self.dp * pinion_rpm / 60000

    # --- Lewis Bending Stress Estimate ---
    def lewis_stress(self, tangential_force_N, lewis_form_factor=0.32):
        """
        Estimate bending stress using Lewis equation (simplified).
        Args:
            tangential_force_N: Tangential tooth force in Newtons
            lewis_form_factor: Y factor (0.32 typical for 20T, 20deg PA)
        Returns:
            Bending stress in MPa
        """
        return tangential_force_N / (self.face_width * self.m * lewis_form_factor)

    def summary(self, input_rpm=1750):
        print(f"=== Spur Gear Pair Design ===")
        print(f"Module: {self.m} mm | PA: {self.phi_deg} deg")
        print(f"Pinion: {self.Np} teeth, d={self.dp:.1f} mm")
        print(f"Gear:   {self.Ng} teeth, d={self.dg:.1f} mm")
        print(f"Ratio:  {self.ratio:.3f}:1")
        print(f"Center distance (ext): {self.center_distance:.2f} mm")
        print(f"Center distance (int): {self.center_distance_internal:.2f} mm")
        print(f"Face width: {self.face_width:.1f} mm")
        print(f"Contact ratio: {self.contact_ratio:.3f}")
        print(f"Min teeth (no UC): {self.min_pinion_teeth}")
        print(f"Undercut risk: {'YES' if self.has_undercutting else 'No'}")
        print(f"\nAt {input_rpm} RPM input:")
        print(f"  Output speed: {self.output_speed(input_rpm):.1f} RPM")
        print(f"  Pitch velocity: {self.pitch_velocity(input_rpm):.2f} m/s")
        if input_rpm > 0:
            power_W = 5000  # 5 kW example
            Ft = power_W / (self.pitch_velocity(input_rpm))
            print(f"\n  For 5 kW power transmission:")
            print(f"  Tangential force: {Ft:.1f} N")
            print(f"  Est. bending stress: {self.lewis_stress(Ft):.1f} MPa")


# Example: Design a 4:1 reduction
pair = SpurGearPair(module=3, pinion_teeth=20, gear_teeth=80)
pair.summary(input_rpm=1750)

# Example: Internal gear arrangement
print("\n--- Internal Gear (same teeth) ---")
print(f"Center distance (internal): {pair.center_distance_internal:.2f} mm")
print(f"Package OD: {pair.dg + 2*pair.m:.1f} mm (very compact!)")
print(f"Tooth difference: {pair.Ng - pair.Np} (min recommended: 12)")

Exercises & Self-Assessment

Conclusion & Next Steps

You now have a comprehensive understanding of spur and internal gears. Here are the key takeaways from Part 4:

• External spur gears are the simplest, most efficient (98-99%), and most widely manufactured gear type — the default choice for parallel-shaft power transmission
• Internal gears provide same-direction rotation, compact coaxial design, higher contact ratio, and form the basis of planetary gear systems
• Gear ratio = N_driven/N_driver — when speed decreases, torque increases proportionally (conservation of power)
• Undercutting occurs below 17 teeth (at 20-degree PA) and weakens the tooth at its most critical point — profile shift correction can prevent it
• Friction gearing offers zero backlash and silent operation for light loads, with modern applications in CVTs and precision instruments
• Systematic design procedure from requirements through geometry, stress analysis, and material selection ensures reliable gear performance