Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| De | De | mm |
| Di | Di | mm |
| h0 | h0 | mm |
| t | t | mm |
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DIN 2093 Sensitivity Analysis: Influence of Design Parameters on Load
1. Purpose of Sensitivity Analysis
The load $F$ of a disc spring is influenced by multiple geometric parameters (thickness $t$, free cone height $h_0$, outer diameter $D_e$, inner diameter $D_i$) and material parameters (elastic modulus $E$). Manufacturing tolerances, material batch variations, and operational wear can cause these parameters to deviate from their nominal values, leading to load fluctuations.
Sensitivity analysis quantitatively calculates the relative degree of influence of each parameter on the load, helping designers to: - Identify critical parameters: Parameters with a large influence on load require tight tolerance control; - Relax non-critical parameters: For parameters with little influence, precision requirements can be appropriately reduced to save costs; - Guide dimensional adjustments: When measured load deviates from the design, quickly determine which parameter should be modified first.
2. Mathematical Definition of Sensitivity
Let the disc spring load $F$ be a function of the parameter vector $\mathbf{x} = (t, h_0, D_e, D_i, E)$. The relative sensitivity of parameter $x_i$ is defined as: the percentage change in load when this parameter changes by 1%. Its mathematical expression is the dimensionless form of the partial derivative:
- $S_i$ — Sensitivity coefficient of parameter $x_i$ (dimensionless)
- $\partial F/\partial x_i$ — Partial derivative of load with respect to this parameter
- $x_i/F$ — Normalization factor, eliminating dimensions
Physical meaning: If $S_t = 3.0$, it means that for every 1% increase in thickness $t$, the load $F$ increases by approximately 3%.
3. Sensitivity Derivation Based on the Almen‑Laszlo Formula
The load formula is:
where $K_1$ depends only on the outer-to-inner diameter ratio $c = D_e/D_i$. Taking partial derivatives with respect to each parameter yields analytical sensitivity expressions.
3.1 Sensitivity of Thickness $t$
Thickness appears in the $t^4$ term in the numerator, in terms independent of $K_1$ in the denominator, and in the dimensionless quantity $\delta = s/t$. After differentiation, for commonly used low-cone disc springs ($\eta = h_0/t \le 0.9$), the thickness sensitivity in the working range ($s/h_0 \le 0.75$) is approximately:
Typical range: $S_t \approx 1.5 \sim 2.8$, i.e., a 1% change in thickness results in a 1.5% to 2.8% change in load. Thickness is one of the highest sensitivity parameters.
3.2 Sensitivity of Free Cone Height $h_0$
Cone height mainly affects the degree of nonlinearity of the disc spring effect. The sensitivity is:
Typical range: $S_{h0} \approx 0.6 \sim 1.2$, with higher sensitivity for high-cone disc springs. $h_0$ is a highly sensitive parameter.
3.3 Sensitivity of Outer Diameter $D_e$
The outer diameter appears mainly in the $D_e^2$ term in the denominator and in $K_1(c)$. Since $K_1$ varies relatively slowly with $c$, the main contribution comes from $D_e^{-2}$:
Typical range: $S_{D_e} \approx -1.8 \sim -2.2$, the absolute value is also large, indicating that the influence of the outer diameter is equally significant. However, if the outer diameter itself is large, its relative percentage change is small, so the absolute tolerance band can be slightly wider.
3.4 Sensitivity of Inner Diameter $D_i$
The inner diameter affects the load only through the shape factor $K_1$, and $K_1$ has low sensitivity to $D_i$, especially when the outer-to-inner diameter ratio $c$ is large.
Typical range: $S_{D_i} \approx 0.1 \sim 0.4$, which is a low sensitivity parameter.
3.5 Sensitivity of Elastic Modulus $E$
The load is proportional to $E$ (ignoring Poisson effects), therefore:
That is, a 1% fluctuation in elastic modulus between material batches results in a corresponding 1% fluctuation in load. This is a medium sensitivity level and must be controlled through material standards.
4. Sensitivity Levels and Parameter Control Recommendations
Based on engineering experience, sensitivity is divided into three levels:
| Absolute Sensitivity $\lvert S_i\rvert$ | Level | Control Recommendation |
|---|---|---|
| > 0.5 | High Sensitivity | Tolerances must be strictly controlled (e.g., $t$, $h_0$, $D_e$), prioritize precision manufacturing processes, and focus on outgoing inspection |
| 0.1 \~ 0.5 | Medium Sensitivity | Control with standard tolerances (e.g., general outer diameter tolerances), perform sampling inspection when necessary |
| \< 0.1 | Low Sensitivity | Tolerances can be appropriately relaxed (e.g., inner diameter $D_i$ can be relaxed as long as assembly requirements are met) to reduce costs |
For disc springs, thickness $t$ and cone height $h_0$ are always highly sensitive parameters, outer diameter $D_e$ is next, and inner diameter $D_i$ is usually a low sensitivity parameter. This also explains why disc spring standards impose the strictest tolerance requirements on thickness and cone height.
5. Example of Sensitivity Calculation
Conditions: Disc spring specification $D_e=40\ \text{mm}, D_i=20.4\ \text{mm}, t=2.0\ \text{mm}, h_0=0.9\ \text{mm}$, working deflection $s=0.5\ \text{mm}$, material spring steel.
Using the load formula, perturb each parameter by ±1% and calculate the relative change in load to obtain the sensitivity coefficients:
| Parameter | Sensitivity $S_i$ | Sensitivity Level |
|---|---|---|
| $t$ | +2.4 | High |
| $h_0$ | +0.9 | High |
| $D_e$ | -2.0 | High |
| $D_i$ | +0.15 | Medium-Low |
| $E$ | +1.0 | Medium |
Interpretation: - If the thickness exceeds tolerance by +0.02 mm (+1%), the load will be approximately 2.4% higher than expected (approx. +140 N). - If the inner diameter exceeds tolerance by +0.1 mm (+0.5%), the load change is only about 0.075%, which is almost negligible. Therefore, the inner diameter tolerance can be appropriately relaxed. - The sensitivity of the outer diameter is negative, meaning that an increase in outer diameter reduces the load (due to the dominant $D_e^2$ term in the denominator).
6. Application of Sensitivity Analysis in Tolerance Design
Given the sensitivity coefficients, the statistical fluctuation of the load can be approximated as:
If the tolerances of each parameter follow a normal distribution, this formula can be used to estimate the ±3σ distribution range of the load, guiding preload design and tightening torque settings.
Example: Assuming tolerances for $t$, $h_0$, and $D_e$ are all ±1%, the relative load fluctuation amplitude is approximately:
That is, the load scatter is approximately ±3.25%, which has a significant impact on preload and must be controlled through strict process control.
7. Design Recommendations
- Critical Dimension Control: Classify $t$, $h_0$, and $D_e$ as key quality characteristics and clearly define tolerance requirements with suppliers.
- Compensation Design: If manufacturing tolerances are difficult to reduce, the working deflection $s$ can be adjusted to accommodate load deviations (leveraging the elastic compensation characteristics of disc springs).
- Robustness Optimization: Based on multi-objective Pareto optimization, select parameter combinations with lower sensitivity to improve batch consistency.
Summary: Sensitivity analysis provides a quantitative basis for the geometric tolerance design of disc springs. By calculating the partial derivatives of each parameter with respect to load, high, medium, and low sensitivity parameters can be clearly distinguished, allowing for a rational allocation of manufacturing precision and achieving a balance between performance and cost.