DIN 2093 Tolerance Analysis: RSS Method for Load Scatter Estimation

1. Purpose of Tolerance Analysis

The final load $F$ of a disc spring is affected by fluctuations in multiple geometric parameters (thickness $t$, free cone height $h_0$, outer diameter $D_e$, inner diameter $D_i$) and the material elastic modulus $E$. In mass production, these parameters cannot be perfectly consistent, leading to load scatter.
The purpose of tolerance analysis is to quantify this scatter, to confirm whether the load variation is acceptable within the given tolerance range, and to identify the critical dimensions that contribute most to load scatter, thereby enabling rational tolerance allocation.

2. RSS Estimation Model for Load Scatter

Using the classic Root Sum Square (RSS) method, assuming all parameters are independent and follow a normal distribution, the relative standard deviation of the load can be obtained by taking the square root of the sum of squares of the product of each parameter's sensitivity and its relative tolerance:

Where: - $\delta_{F,RSS}$ — Relative standard deviation of the load (%), i.e., $\frac{\sigma_F}{F} \times 100\%$, characterizing the magnitude of load scatter. - $S_i$ — Relative sensitivity of parameter $x_i$, $S_i = \frac{\partial F}{\partial x_i} \cdot \frac{x_i}{F}$ (dimensionless), see "Sensitivity Analysis" section. - $\frac{\Delta x_i}{x_i}$ — Relative tolerance of parameter $x_i$ (typically half the tolerance band, i.e., 1/2 tolerance, used as the approximate standard deviation $\pm\Delta x_i$). - $n$ — Number of parameters included in the analysis.

Criterion: - $\delta_{F,RSS} < 10\%$: Tolerance design is reasonable, load consistency is good, meeting most engineering requirements. - $\delta_{F,RSS} \ge 10\%$: Load scatter is excessive, which may lead to insufficient or excessive preload. Tightening tolerances on critical parameters or changing the design (e.g., adjusting nominal dimensions to reduce sensitivity) is required.

3. Sensitivity and Tolerance Contribution of Each Parameter

Based on sensitivity analysis, typical sensitivity values for the main parameters of a disc spring are: - Thickness $t$: $S_t \approx 2.0 \sim 2.8$ - Free cone height $h_0$: $S_{h0} \approx 0.8 \sim 1.2$ - Outer diameter $D_e$: $S_{D_e} \approx -1.8 \sim -2.2$ (negative sign indicates inverse proportionality) - Inner diameter $D_i$: $S_{D_i} \approx 0.1 \sim 0.3$ - Elastic modulus $E$: $S_E = 1.0$

The relative tolerance $\frac{\Delta x_i}{x_i}$ for each parameter is given by manufacturing standards and drawings. For example, thickness $t = 2.0 \pm 0.015\ \text{mm}$, then relative tolerance $\frac{\Delta t}{t} \approx \frac{0.015}{2.0} = 0.0075 = 0.75\%$.

The contribution of each parameter can be defined as:

The parameter with the largest contribution is the one requiring the strictest tolerance control.

Rule: Due to the very high $S_t$ and the fact that thickness tolerances are typically on the order of ±0.01~0.02 mm, its contribution often accounts for over 60% of the total deviation, making it the absolute core of tolerance control.

4. Calculation Example

Disc spring specification: $D_e = 40\ \text{mm}, D_i = 20.4\ \text{mm}, t = 2.0\ \text{mm}, h_0 = 0.9\ \text{mm}$
Tolerances (per DIN 2093 or company standard): - $t = 2.0 \pm 0.015\ \text{mm}$ → $\Delta t/t = 0.75\%$ - $h_0 = 0.9 \pm 0.05\ \text{mm}$ → $\Delta h_0/h_0 = 5.56\%$ (cone height tolerance is typically larger) - $D_e = 40 \pm 0.1\ \text{mm}$ → $\Delta D_e/D_e = 0.25\%$ - $D_i = 20.4 \pm 0.1\ \text{mm}$ → $\Delta D_i/D_i = 0.49\%$ - $E$ taken as ±3% variation → $\Delta E/E = 3\%$

Calculated sensitivities (as before): - $S_t = 2.4$ - $S_{h0} = 0.9$ - $S_{D_e} = -2.0$ - $S_{D_i} = 0.15$ - $S_E = 1.0$

Variance component calculation:

Sum:

Relative standard deviation:

Conclusion: - $\delta_{F,RSS} = 6.12\% < 10\%$, tolerance design is reasonable, load scatter is acceptable. - Largest contributor: $h_0$ variance $25.0\times10^{-4}$, accounting for approximately 66.7%; followed by $E$ at about 24%, and $t$ at about 8.6%. This indicates that, under the tolerance settings of this example, the wider cone height tolerance makes it the primary source of deviation. If the cone height tolerance can be tightened in actual manufacturing, the total scatter will decrease further.

Note: Typically, if the cone height tolerance is well controlled (e.g., ±0.02 mm instead of ±0.05 mm), the contribution of thickness will become dominant. Common experience suggests that thickness tolerance accounts for over 60% of the total deviation.

5. Tolerance Optimization and Design Recommendations

• Prioritize tightening parameters with high sensitivity and high contribution: such as thickness and cone height. If $\delta_{F,RSS}$ exceeds the limit, first reduce the tolerance bands for $t$ and $h_0$.
• Relax tolerances on low-sensitivity parameters: Even doubling the tolerance for inner diameter $D_i$ has a negligible effect on total scatter, which can reduce manufacturing costs.
• Utilize statistical tolerancing: If the production process is stable, tighter statistical tolerances (e.g., Cpk ≥ 1.33) can be applied to further reduce scatter.
• Preload compensation: If load scatter is unavoidable, it can be compensated during assembly through group selection or by adjusting the compression amount (changing shim thickness), though this increases process complexity.

Summary: RSS tolerance analysis predicts the statistical scatter of disc spring load by taking the square root of the sum of squares of the product of sensitivity coefficients and parameter tolerances. The design target is $\delta_{F,RSS} < 10\%$, with a focus on controlling tolerances for high-sensitivity parameters such as thickness and cone height to ensure consistent performance in mass production.