Law of Gearing Tooth Profiles

Definition

The Law of Gearing states that for two gear wheels to transmit motion with a constant angular velocity ratio, the common normal at the point of contact between the two tooth profiles must always pass through a fixed point on the line of centers, known as the pitch point.

Main Content

1. The Pitch Point

• The pitch point is the point where the two pitch circles of the meshing gears make contact.
• According to the law of gearing, this point divides the line joining the centers of the two gears in the inverse ratio of their angular velocities.

2. Constant Velocity Ratio

• To achieve smooth power transmission, the ratio of the angular velocity of the driving gear to the driven gear must remain constant at every instant of contact.
• If the velocity ratio fluctuates, it leads to vibration, noise, and potential failure of the gear teeth.

3. Conjugate Action

• Conjugate action describes the behavior of tooth profiles that satisfy the law of gearing.
• When the motion transmitted by the teeth is equivalent to the motion transmitted by two cylinders rolling against each other without slipping, the profiles are said to be "conjugate."

       Gear 1 (Driver)
          / \
         /   \
  O1----P-----O2 (Line of Centers)
        |
      Pitch Point (P)
        |
       Gear 2 (Driven)

Visual representation: P is the fixed point on the line connecting centers O1 and O2.

Working / Process

1. Establishing the Normal

• At any point of contact (Q) between two tooth profiles, draw a line perpendicular to the surfaces (the common normal).
• This line must intersect the line of centers (O1-O2) at a specific point (P).

2. Maintaining the Velocity Ratio

• The gear geometry must ensure that the point of intersection (P) remains fixed throughout the engagement.
• If the point P shifts while the gears rotate, the velocity ratio will change, causing a "jerky" motion.

3. Ensuring Smooth Engagement

• As the teeth move, the point of contact changes position.
• The tooth profile must be mathematically shaped (usually an involute curve) so that the common normal always passes through the pitch point P, ensuring constant velocity throughout the tooth mesh.

Advantages / Applications

• Constant Speed: Essential for precision machinery like clocks, watches, and CNC machine tools.
• Reduced Wear: By maintaining a constant velocity ratio, impact forces are minimized, leading to longer gear life.
• Power Efficiency: Involute gear profiles are used in automotive gearboxes and heavy industrial transmissions to ensure high torque and efficient power transfer.

Summary

The Law of Gearing is the fundamental principle ensuring that two meshing gears maintain a constant angular velocity ratio by ensuring their common normal at the contact point passes through a fixed pitch point.

• Pitch Point: The constant intersection point on the line of centers.
• Velocity Ratio: The fixed proportion of rotational speeds between the driver and driven gear.
• Conjugate Profiles: Tooth shapes designed to obey the law of gearing.
• Involute Curve: The most common geometric profile used to satisfy the law of gearing.