Symbol	Name	Unit
mu_bite	mu_bite	—
mu_flat	mu_flat	—
preload_N	preload_N	N
size_key	size_key	—

NFE 25-511 Locking Torque Calculation

1. Definition and Physical Mechanism

The locking capability of the NFE 25-511 single-sided toothed conical spring washer originates from the mechanical interlocking effect produced by radial tooth tips biting into the mating surface. When the bolt is subjected to an external loosening rotational force, the tooth engagement between the washer, the connected part, and the bolt head/nut generates a tangential resistance far higher than ordinary flat friction, thereby resisting rotational loosening.

The locking torque $M_{lock}$ refers to the maximum static friction torque that the washer can provide under the preload force $$F_M$$ , i.e., the critical torque that must be overcome for the bolt to begin loosening rotation.

2. Core Formula

Locking torque formula based on the equivalent tooth friction coefficient $\mu_{eff}$ :

\boxed{M_{lock} = F_M \cdot \mu_{eff} \cdot \frac{D_{km}}{2}}

Where: - $M_{lock}$ : Locking torque (N·mm), the torque resisting bolt loosening rotation - $$F_M$$ : Current preload force (N), typically the minimum residual preload $F_{Mmin}$ is used for conservative evaluation - $\mu_{eff}$ : Equivalent tooth friction coefficient (dimensionless), incorporating the amplification effect of tooth tip mechanical engagement and flat friction - $D_{km}$ : Equivalent friction diameter of the washer bearing surface (mm), taken as the washer mean diameter $$D_m = (D_e + D_i)/2$$

3. Equivalent Tooth Friction Coefficient $\mu_{eff}$

The $\mu_{eff}$ of a single-sided toothed conical spring washer is significantly higher than the flat surface friction coefficient because it includes additional resistances from tooth plowing, cold welding, and mechanical interlocking. Its calculation formula is:

\boxed{\mu_{eff} = \mu_{base} + \Delta\mu_{dent}}

\Delta\mu_{dent} \approx \frac{\tau_{joint}}{p_{nom}} \cdot \frac{\sum A_{dent}}{A_{nom}}

Where: - $\mu_{base}$ : Friction coefficient of a smooth flat surface under the same material/lubrication conditions (0.08~0.25) - $\Delta\mu_{dent}$ : Additional tooth term, contributed by the mechanical engagement effect of tooth tips - $\tau_{joint}$ : Shear strength at the tooth engagement points (MPa), taken as the shear strength of the softer material (≈ 0.6  $$R_m$$ ) - $p_{nom} = F_M / A_{nom}$ : Nominal bearing surface pressure (MPa) - $A_{nom} = \frac{\pi}{4}(D_e^2 - D_i^2)$ : Nominal annular area of the washer (mm²) - $\sum A_{dent}$ : Sum of effective bearing areas of all tooth tips (mm²)

Typical Reference Values (Steel washer + Steel connected part):

Lubrication Condition	Smooth Flat $\mu_{base}$	Toothed Washer $\mu_{eff}$
Dry, oil-free	0.18 – 0.25	0.35 – 0.50
Light oil film	0.12 – 0.16	0.30 – 0.45
Well lubricated	0.08 – 0.12	0.25 – 0.35

If measurement is not possible, it is recommended to use $\mu_{eff} \approx 0.35 \sim 0.40$ for conservative design.

4. Locking Safety Factor

Compare the locking torque $M_{lock}$ provided by the washer with the external loosening disturbance torque $M_{disturb}$ to define the locking safety factor:

\boxed{S_{lock} = \frac{M_{lock}}{M_{disturb}} \ge 1.5}

$M_{disturb}$ : Loosening disturbance torque from external vibration or impact (N·mm), typically taken as the thread back-off torque $M_{thread\_backoff}$ or measured vibration torque

For connections subjected to transverse vibration, $M_{disturb}$ can be determined by Junker testing or estimated using the following formula:

M_{disturb} = F_M \cdot \frac{d_2}{2} \cdot \left( \frac{\mu_G}{\cos\beta} - \tan\varphi \right)

Where $\mu_G$ is the thread friction coefficient under vibration (can be as low as 0.03~0.05), $\beta$ is the half flank angle (ISO thread 30°), and $\varphi$ is the thread helix angle.

Criterion: When $S_{lock} \ge 1.5$ , the connection is considered not to loosen under vibration conditions.

5. Calculation Example

Given: - M10 bolt, preload force $F_{Mmin} = 15,000\ \text{N}$ - NFE 25-511 washer: $D_e = 20\ \text{mm}$ , $D_i = 10.5\ \text{mm}$ → $D_{km} \approx 15.25\ \text{mm}$ - Equivalent tooth friction coefficient $\mu_{eff} = 0.40$ (dry, oil-free) - Thread parameters: $d_2 = 9.026\ \text{mm}$ , $\varphi \approx 3.03^\circ$ , vibration $\mu_G = 0.05$ , $\beta = 30^\circ$

Step 1: Calculate locking torque

M_{lock} = 15,000 \times 0.40 \times \frac{15.25}{2} = 15,000 \times 0.40 \times 7.625 = 45,750\ \text{N·mm} = 45.75\ \text{N·m}

Step 2: Calculate thread back-off torque under vibration

M_{disturb} = 15,000 \times \frac{9.026}{2} \times \left( \frac{0.05}{\cos30^\circ} - \tan3.03^\circ \right)

= 67,695 \times \left( \frac{0.05}{0.866} - 0.053 \right) \approx 67,695 \times (0.0577 - 0.053) \approx 67,695 \times 0.0047 \approx 318\ \text{N·mm} = 0.32\ \text{N·m}

Step 3: Calculate safety factor

S_{lock} = \frac{45.75}{0.32} \approx 143 \gg 1.5

Conclusion: The locking capability of this washer under these conditions is extremely ample. Even under severe vibration where the thread friction coefficient drops to 0.05, it can reliably prevent loosening.

6. Key Factors Influencing Locking Torque

Factor	Influence Law	Design Recommendation
Preload Force $$F_M$$	Proportional; higher preload yields higher locking torque	Use minimum residual preload for conservative evaluation
Tooth Profile and Count	More teeth, sharper teeth → higher $\mu_{eff}$	Select per NFE 25-511 standard
Material Hardness Matching	Washer hardness ≥ 1.2 × connected part hardness	Ensure teeth can effectively bite in
Surface Coating	Thin coating has no effect; thick coating hinders biting	Avoid thick coatings like hot-dip galvanizing
Reuse	Tooth tip wear significantly reduces $\mu_{eff}$	Inspect tooth profile after disassembly; replace if necessary

7. Integration with VDI 2230 and DIN 25201

VDI 2230 R13: In torque-preload calculations, use $\mu_{eff}$ to replace the conventional $\mu_K$ for the bearing surface friction term
DIN 25201 Comparison: Wedge-locking washers achieve locking through wedge geometry ( $M_{lock} = F_M \tan(\alpha+\rho) D_{cam}/2$ ), where locking torque is independent of friction, making them more suitable for extreme vibration; NFE 25-511 combines elastic compensation and tooth-face locking, suitable for moderate vibration and applications requiring relaxation compensation
Test Verification: Final locking capability should be confirmed via Junker transverse vibration testing (ISO 16130); a residual preload ratio ≥ 70% is considered acceptable

Summary: The core formula for the locking torque of NFE 25-511 washers is $M_{lock} = F_M \cdot \mu_{eff} \cdot D_{km}/2$ , which quantifies the mechanical engagement effect through the equivalent tooth friction coefficient $\mu_{eff}$ . During design, use the minimum preload force and the thread back-off torque under vibration for safety factor verification, ensuring $S_{lock} \ge 1.5$ .

Anti-Loosening Torque

Formula Expression

Parameters

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