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A Brief Description of Rack Over-Pin Dimension and Backlash

To measure the tooth thickness tolerance of a rack, an over-pin measurement method is used. Two pins are inserted between the rack teeth, and the distance between the top of the pins to the bottom of the rack is measured, as illustrated in Figure 1. This distance is defined as the over-pin dimension (M).

Formula for Over-Pin Dimension

The over-pin dimension can be calculated by the following formula:

M = A₁ + (D_M / 2) × (1 / sin α_n + 1) - (m_n / 4tan α_n)

Normal pitch: P_n = m_n × π

Relation between Tooth Thickness Tolerance of Rack and Rack-Pinion Backlash

When the rack and pinion mesh, the theoretical line of contact is called the pitch line. When the pitch line is at its ideal position, the tooth thickness equals P_n / 2. To ensure smooth meshing between the rack and pinion, the tooth thickness must be maintained within a specific range, defined as the tooth thickness tolerance.

According to theoretical calculations, for each reduction in tooth thickness by c / 2, the corresponding decrease in tooth height (d) can be calculated using the following formula:

d = (c / 2) × tan α_n

c	d	Unit
0.01	0.0137	mm
0.02	0.0275	mm

This relationship also allows engineers to estimate the variation in gear backlash when tooth thickness changes.

Over-Pin Measurement Instructions for YYC Racks

The table below lists the standard pin diameters and corresponding over-pin dimensions for different module sizes in YYC racks.

Module (M)	Pin Diameter ØDM	Over-Pin Dimension A0	Pin No.
1.5	ø3	18.149	17
2	ø4	25.532	24
2.5	ø5	25.915	29
3	ø6	31.298	29
4	ø8	42.064	39
5	ø10	42.830	39
6	ø12	52.396	49
8	ø16	85.128	79
10	ø20	106.659	99

Figure 1 illustrates the over-pin measurement principle, while Figures 2 and 3 demonstrate the variation in tooth thickness and backlash. This measurement method provides a practical way to assess rack manufacturing precision and serves as an important reference for adjusting backlash during rack-pinion assembly.