Why should cross-rod/ball distance and common normal be measured at the circle of graduation?

Everyone knows that when parts like gears are controlled to determine whether the processing is in place or whether the final parts are qualified, it is usually necessary to measure the cross-rod/ball distance or the common normal line. When measuring, it is necessary to pay attention to the fact that the measuring column or probe needs to contact the pitch circle of the tooth profile. Have you ever thought about the reason behind this?

First, let's take the standard gear as an example, and then explain some special cases later.

1. To ensure that it is near the pitch circle, the first thing is to ensure the consistency of the geometric datum. The pitch circle is an important datum for gear design. The parameters such as tooth thickness and pressure angle here are all theoretical design values. The calculation formulas of M value and W value (such as invα_M, cross-tooth number k, etc.) are all based on the pitch circle parameters. Whether it is the cross-rod distance or the common normal line, the essence is to indirectly reflect the tooth thickness through measurement. The tooth thickness and tooth width at the pitch circle of the standard gear are symmetrical. The measurement results can directly reflect the deviation between the gear processing accuracy and the theoretical value, ensuring that the measurement datum and the design datum are unified. Otherwise, when there is a deviation, the data needs to be corrected according to the actual pressure angle, etc.

2. The tooth profile at the pitch circle is a pure involute, without the interference of transition curves. When measuring, the measuring tool contacts the tooth surface in this area to avoid the measurement error caused by the root fillet or the tooth top modification. In addition, the tooth top and root area are also easily affected by tool wear and heat treatment deformation. Measuring at the pitch circle can ensure the stability of the data.

3. The contact point near the pitch circle can also better reflect the actual meshing of the gear. When the gear is driven, the tooth surface contact mainly occurs near the pitch circle, so the measurement here can better represent the working state. If the measuring point deviates, the tooth thickness may not be correctly evaluated, resulting in incorrect calculation of the side clearance, affecting the transmission accuracy and noise.

4. From the perspective of the measuring tool, the tooth surface curvature radius near the pitch circle is larger, the measuring tool contact is more stable, and the wear of the measuring tool can be reduced. The contact of the measuring rod or probe is also more stable. The sudden decrease in the curvature radius of the tooth top area is easy to cause poor contact, which reduces the measurement repeatability by 30%-50%.

5. For small module gears (m<0.5mm) or plastic gears, measurement near the pitch circle can also avoid the influence of material elastic deformation as much as possible. For example, when plastic gears are measured by spanning rods, pitch circle contact can reduce the tooth surface indentation error caused by the measuring force. The above is for standard non-displaced spur gears. Let's look at other situations.

1. Displacement gear processing

For gears with displacement, the pitch circle tooth thickness changes to s=m(π/2+2*x*tanα). At this time, the common normal needs to be corrected according to the formula W′=W+2x*m*sinα, and invαM needs to be corrected by 2*x*tanα/z to adjust the measurement distance M to ensure that the contact point is still equivalent to the position near the pitch circle.

2. Helical gear measurement compensation

The normal pitch circle of helical gears is mn=mt*cosβ. The module and pressure angle in the calculation formula of the measuring rod diameter need to be replaced with normal parameters, and the axial contact position should be adjusted according to the β angle to avoid the projection error caused by the helix angle.

3. Segmented measurement of large-module gears

When the module m>10, it is recommended to measure 3 points within the range of 15% tooth height above and below the pitch circle, and eliminate the influence of tooth concave according to the weighted average. The formula is: Mavg=0.3Mtop+0.4Mmid+0.3Mbot.