Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| nominal_dia | nominal_dia | — |
| reuse_cycles | reuse_cycles | — |
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DIN 9250 Reuse Determination: Tooth Height Wear Model and Maximum Allowable Cycles
1. Formula Definition and Physical Background
During each installation and removal cycle of DIN 9250 toothed lock washers, the tooth tips undergo severe plastic deformation and frictional wear against the mating surface, causing a gradual reduction in tooth height. This reduction directly compromises the engagement depth and locking torque. To quantitatively control the number of reuse cycles, the following tooth height wear model is adopted:
- $h_{tooth}(0)$ — Nominal tooth height of a new washer (mm), i.e., the height from tooth tip to root
- $h_{tooth}(k)$ — Remaining tooth height after $k$ assembly/disassembly cycles (mm)
- $w$ — Relative tooth height wear rate per assembly/disassembly cycle (dimensionless, 0 < w < 1)
- $k$ — Number of assembly/disassembly cycles (1 complete tightening + 1 complete loosening counts as 1 cycle)
This model assumes that the wear per cycle is proportional to the current tooth height, exhibiting an exponential decay pattern.
2. Maximum Allowable Number of Cycles $k_{max}$
When the tooth height drops below a certain threshold, the washer no longer provides sufficient engagement depth and locking torque and must be discarded. Typically, a tooth height allowable loss rate of 15% is used as the criterion, requiring:
Substituting the wear model yields the maximum allowable number of cycles:
Taking the natural logarithm:
Since $w$ is small, $\ln(1-w) \approx -w$, leading to the simplified form:
The engineering approximation is:
Meaning: The larger the single-cycle wear rate $w$, the fewer the allowable reuse cycles. For example, if $w = 0.03$ (3%), then $k_{max} \approx 5$ cycles.
3. Determining the Wear Rate $w$
The value of $w$ depends on several factors:
| Influencing Factor | Trend |
|---|---|
| Hardness ratio $HV_{washer}/HV_{mating}$ | Higher ratio (softer mating surface) reduces washer tooth wear, decreasing $w$ |
| Tooth geometry (tip angle, fillet radius) | Blunter angles and larger fillet radii improve wear resistance, reducing $w$ |
| Coating/lubrication | Dry friction increases wear; good lubrication reduces $w$ |
| Preload magnitude | Excessive preload intensifies single-cycle plastic deformation, increasing $w$ |
| Disassembly method | Forced disassembly significantly increases $w$ |
Typical reference values (steel washer + steel mating surface, normal assembly/disassembly):
| Condition | Single-cycle wear rate $w$ |
|---|---|
| Good lubrication, low to medium preload | 0.01 – 0.02 |
| Normal condition (light oil film, standard preload) | 0.02 – 0.04 |
| Dry friction, high preload, or repeated impact | 0.05 – 0.10 |
Method for obtaining accurate $w$: Measure the tooth height of washers after the 1st, 5th, and 10th assembly/disassembly cycles, then fit the $(1-w)^k$ curve to back-calculate $w$.
4. Determination Procedure
- Measure the new washer tooth height $h_{tooth}(0)$ (using a profilometer or micrometer to measure the tooth tip height).
- Determine the wear rate $w$ (from tables or experimental calibration).
- Calculate the current number of cycles $k$ and compute the remaining tooth height $h_{tooth}(k)$ using the formula.
- If $k > k_{max}$ (i.e., $k > 0.15/w$), or if the measured tooth height has decreased by more than 15% compared to the new washer, the washer must be discarded.
- Combine with locking torque verification: Even if the tooth height is acceptable, the washer should be replaced if the locking torque has dropped below the required value.
5. Calculation Example
Given: New washer tooth height $h_{tooth}(0) = 0.50$ mm, experimentally determined single-cycle wear rate $w = 0.03$.
Maximum allowable cycles:
Remaining tooth height after the 3rd use:
Tooth height loss is approximately 8.7%, still within the 15% allowable range, so it can continue to be used.
After the 6th use (exceeding $k_{max}$):
Loss is 16.7%, exceeding the 15% threshold, must be discarded.
6. Application Notes
- This model assumes a constant wear rate per cycle. In practice, the wear rate may change as the tooth tips become blunter; periodic inspection is recommended.
- Distinction from surface indentation depth: This formula focuses on the change in the washer's own tooth height, not the indentation depth on the mating surface.
- Safety margin: For critical connections, a stricter tooth height loss limit (e.g., 10%) may be applied, corresponding to $k_{max} = 0.10/w$.
- Record keeping: Reused washers should be marked with the number of cycles to avoid exceeding the limit.
Summary:
The reusability of DIN 9250 washers is governed by the exponential tooth height wear model $h_{tooth}(k) = h_{tooth}(0) (1-w)^k$. When the cumulative wear exceeds 15% or the number of cycles reaches $k_{max} \approx 0.15/w$, the washer must be discarded. Correctly evaluating the wear rate $w$ and strictly implementing the replacement criteria are fundamental to ensuring the long-term locking reliability of toothed washers.