Junker Residual Preload Ratio
Junker Residual Preload Ratio
Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| amplitude_mm | amplitude_mm | mm |
| mu_eff | mu_eff | — |
| preload_N | preload_N | N |
| size_key | size_key | — |
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Contact Engineering TeamDetailed Calculation Guide
Calculation of Tooth Tip Penetration Depth
1. Physical Background and Engineering Significance
The radial teeth of single-sided toothed conical spring washers (NFE 25‑511) and wedge-locking washers (DIN 25201) press into the surface of the connected part under preload, creating mechanical interlocking. Penetration depth $h_{pen}$ is a key indicator for assessing the quality of engagement:
- Too shallow penetration → Insufficient slip resistance, potential loosening under vibration.
- Too deep penetration → Surface crushing of the connected part, or plastic deformation of the tooth tip itself.
Therefore, it is necessary to quantitatively evaluate the penetration depth based on preload, tooth geometry, and material hardness to ensure reliable engagement without damaging the mating surface.
2. Single Tooth Force Analysis
Assuming the preload $F_M$ is evenly distributed among $z$ teeth, each tooth bears a normal force:
The tooth tip is simplified as a wedge (included angle $2\theta$, i.e., half-angle $\theta$), pressing into a plastic material surface. Under normal force, the contact surface pressure reaches the plastic flow pressure (hardness) of the connected part, causing plastic flow, and the tooth continues to penetrate until the contact area is sufficient to bear $F_{tooth}$.
3. Theoretical Formula for Penetration Depth
Based on indentation hardness theory (analogous to Brinell/Vickers hardness testing), the tooth tip penetration depth $h_{pen}$ can be expressed as:
or equivalently:
Where: - $F_{tooth}$ — Normal force per tooth (N) - $F_M$ — Preload (N) - $z$ — Number of teeth - $H_{plate}$ — Indentation hardness of the connected part (MPa), can be taken as Vickers hardness HV or $H \approx 3\sigma_y$ ($\sigma_y$ is yield strength) - $2\theta$ — Tooth tip included angle (°), $\theta$ is the half-angle - $C$ — Constraint factor, for a conical indenter (approximately triangular cross-section): - Fully plastic, unconstrained: $C \approx 2.0 \sim 2.5$ - Constrained by surrounding elastic material (groove effect): $C \approx 1.5 \sim 2.0$
Derivation Approach
During indentation, the tooth tip contact area $A_{proj} \propto h_{pen} \times w_{tooth}$, and tooth width $w_{tooth} \propto h_{pen}\tan\theta$ (projected length of the wedge face), hence $A_{proj} \propto h_{pen}^2 \tan\theta$.
From force equilibrium: $F_{tooth} = p_{yield} \times A_{proj} = H_{plate} \times (C_1 h_{pen}^2 \tan\theta)$, solving gives $h_{pen} \propto \sqrt{F_{tooth}/(H_{plate} \tan\theta)}$. Introducing the constraint coefficient $C$ yields the above formula.
4. Parameter Value Guidelines
| Parameter | Recommended Value/Range | Description |
|---|---|---|
| $z$ | Per washer specification, typically 6–18 | More teeth reduce force per tooth |
| $\theta$ | 30°–45° (total tip included angle 60°–90°) | DIN 25201 washer tooth angle approx. 60°, NFE 25-511 similar |
| $H_{plate}$ | Mild steel (S235) ≈ 1200–1400 MPa (HV 120–140) Medium carbon steel (S355) ≈ 1800 MPa Aluminum alloy (6061-T6) ≈ 600–800 MPa |
Can be approximated as $H \approx 3R_{p0.2}$ |
| $C$ | 1.5 (groove constraint case) 2.0 (free surface) |
Teeth adjacent to elastic material, recommend 1.5 |
5. Calculation Example
Given:
- M10 connection, preload $F_M = 18\,000$ N
- Washer teeth count $z = 12$
- Tooth tip half-angle $\theta = 30°$ ($\tan 30° = 0.577$)
- Connected part: S235 steel, hardness $H_{plate} \approx 1300$ MPa
- Take $C = 1.5$
Force per tooth:
Penetration depth:
Assessment:
A penetration depth of approximately 1.16 mm is clearly excessive for ordinary steel (expected to be ≤ 0.3 mm). This indicates that the single-tooth load is too high or the hardness value is too low — in reality, S235 yield strength is ~235 MPa, but the surface may have higher local hardness due to work hardening, oxide layers, etc. Moreover, the tooth tip is not an ideal wedge; the actual contact width increases rapidly with depth, quickly increasing load capacity and limiting depth. Additionally, the tooth is not freely indented under axial load but is constrained by adjacent material, making the $C$ factor closer to 2.5, which reduces depth.
If using $C=2.5$:
Still too large. This shows that with the above parameters, 12 teeth may not meet the requirement of not penetrating too deeply; increasing the number of teeth or reducing preload may be necessary. If the connected part is high-strength steel ($H \approx 3000$ MPa):
Still relatively large. In practice, washer designs optimize tooth geometry to control penetration depth to the order of 0.1–0.3 mm, because tooth width increases rapidly with depth — the above simplification does not account for width variation. For greater accuracy, finite element analysis or experimental calibration is needed. However, the formula clearly reflects the relative influence of each parameter.
6. Engagement Effectiveness Evaluation Criteria
6.1 Minimum Penetration Depth (Ensuring Engagement)
should be greater than the surface roughness of the connected part and the thickness of any contamination layer, ensuring the tooth tip breaks through oxide layers, coatings, etc., to achieve metal contact. General requirements: - Steel connected parts: $h_{min} \approx 0.05 \sim 0.10$ mm - Soft materials like aluminum alloy: $h_{min} \approx 0.10 \sim 0.20$ mm
6.2 Maximum Penetration Depth (Preventing Crushing)
is determined by the material thickness of the connected part, the depth of the surface hardened layer, or the acceptable deformation. Typically: - For thin plates or parts with surface hardened layers: $h_{max} \le$ 50% of the hardened layer thickness - General steel structures: $h_{max} \approx 0.3$ mm
6.3 Ideal Penetration Depth Range
Combining the above, the design target range:
7. Other Factors Affecting Penetration Depth
- Washer Hardness: Washer hardness should be ≥ connected part hardness + 100 HV, otherwise the tooth tip may crush first.
- Surface Coating: Soft coatings like Dacromet, zinc-nickel can be penetrated by teeth, but excessively thick coatings (>20 μm) reduce effective penetration depth.
- Multiple Assembly/Disassembly: Tooth tip wear increases the included angle, effectively increasing $\theta$, reducing penetration capability.
- Dynamic Load: Under impact, local stress may exceed static hardness, potentially deepening penetration.
8. Design Recommendations
- Select Appropriate Tooth Count: Ensure moderate single-tooth force, neither overloaded nor underloaded.
- Check Surface Pressure: While meeting penetration depth requirements, also verify via R10 surface pressure check.
- Experimental Validation: Measure actual indentation depth through cross-sectional metallography or profilometry to calibrate the $C$ factor in the calculation formula.
- Material Matching: For high-hardness connected parts (HV > 300), use specially hardened washers or consider tooth geometry optimization.
$R_z$Summary:
The tooth tip penetration depth $h_{pen} = \sqrt{F_{tooth} / (C \cdot H_{plate} \cdot \tan\theta)}$ integrates the effects of preload, tooth geometry, and material hardness, serving as a core physical quantity for evaluating the engagement effectiveness of serrated washers. Design should aim to control penetration depth within 0.1–0.3 mm to balance anti-loosening and anti-crushing requirements.