DIN 2093 Stack Configuration Tool: Automatic Determination of Parallel and Series Disc Counts

1. Purpose of the Configuration Tool

In disc spring design, a single disc spring often cannot simultaneously meet the requirements for force and deflection.
The Stack Configuration Tool is a logical calculation process that, based on design targets (total load $F_{req}$, total deflection $s_{req}$) and installation space constraints, uses the force-deflection relationship of standard disc springs to automatically calculate the reasonable number of parallel discs $n$ and series groups $i$, providing a feasible stacked combination $n \times i$, and verifying strength and stability.

2. Basic Formulas: Single Disc Characteristics

A selected single disc spring (according to the DIN 2093 standard series) has the following known parameters:

• Outer diameter $D_e$, inner diameter $D_i$, thickness $t$, free cone height $h_0$
• Material elastic modulus $E$, Poisson's ratio $\nu$ (0.3 for steel)
• Shape factor $K_1 = \dfrac{1}{\pi} \cdot \dfrac{[(c-1)/c]^2}{(c+1)/(c-1) - 2/\ln c}$, $c = D_e/D_i$

Single disc force-deflection formula (Almen‑Laszlo):

• Flat force $F_{flat} = F_{single}(h_0)$;

• Single disc allowable elastic force (considering the 0.75 h₀ deflection limit and safety factor $S_F \ge 1.3$):

  $$F_{adm} = \frac{F_{single}(0.75\,h_0)}{S_F} \quad \text{or conservatively } \frac{F_{flat}}{1.5}$$

• Single disc effective elastic deflection: $\Delta s_{single} \le 0.75\,h_0$.

3. Mechanical Superposition Rules for Series and Parallel Configurations

Therefore, for a target total force $F_{req}$ and target total deflection $s_{req}$:

4. Disc Count Calculation Formulas

Solving for the disc counts from the above constraints:

If $n_{min} > 1$ and $i_{min} > 1$, a mixed configuration is required, with total disc count $N = n_{min} \times i_{min}$.

Limiting Conditions:

• The number of parallel discs should not exceed 4 (DIN 2093 recommendation, considering load uniformity and friction);
• The number of series groups has no hard limit, but stability requirements must be met (free length $L_0$ to outer diameter ratio $\lambda = L_0/D_e \le 3$ without guidance, or $\le 6$ with guidance); otherwise, segmental guidance is required.
• The final selected $n$ and $i$ must be integers, and the actual force and deflection should be back-calculated from them.

5. Complete Configuration Process

1. Input Target Values: Total working load $F_{req}$, total required elastic deflection $s_{req}$, installation space constraints (maximum outer diameter $D_{e,max}$, maximum total height $L_{max}$).
2. Select Single Disc Specification from Standard Series: Must satisfy $D_e \le D_{e,max}$, inner diameter suitable for shaft/hole fit. If no suitable single disc is found, select a larger specification.
3. Calculate Single Disc Characteristic Values: $F_{flat}$, $F_{adm}$, $\Delta s_{single}$.
4. Determine $n$ and $i$:
   $$n_{min} = \lceil F_{req} / F_{adm} \rceil, \quad i_{min} = \lceil s_{req} / \Delta s_{single} \rceil$$

If $n_{min} > 4$, the disc spring specification must be increased, or parallel groups must be added in series (but the total force is still borne by the parallel group; n>4 is not feasible, requiring a thicker disc spring). 5. Verify Combined Total Force and Total Deflection: - Total flat force $= n \cdot F_{flat}$, requirement: $F_{req} \le n \cdot F_{flat} / S_F$; - Total deflection $= i \cdot \Delta s_{single}$, actual working deflection should be $\le i \cdot 0.75h_0$. 6. Stability Check: Calculate total free height $L_0 = i \cdot h_0 + (N-1) \cdot t$ (approximate). If $L_0/D_e > 3$, a guidance device is mandatory, and the ratio must not exceed 6. 7. Friction Correction: Considering friction between parallel discs, actual loading force ≈ theoretical force × (1.05~1.15), unloading force is slightly reduced. 8. Stress Check: The stress level of a single disc is determined by the compression amount $s = s_{total}/i$. Check stresses at the OM and I points. 9. Output the recommended stack configuration: $n \times i$ combination, number of each disc type, total height, guidance requirements.

6. Calculation Example

Requirements: Load $F_{req} = 15\,000\ \text{N}$, required elastic compensation deflection $s_{req} = 2.2\ \text{mm}$.
Initial Disc Spring Selection: DIN 2093 Series A, $D_e=40\ \text{mm}, D_i=20.4\ \text{mm}, t=2.0\ \text{mm}, h_0=1.2\ \text{mm}$.

• Calculate flat force $F_{flat} \approx 12\,800\ \text{N}$;
• Single disc allowable elastic force $F_{adm} \approx F_{flat}/1.3 \approx 9\,850\ \text{N}$ (using a safety factor of 1.3);
• Single disc effective elastic deflection $\Delta s_{single} = 0.75 \times 1.2 = 0.9\ \text{mm}$.

Disc Count Estimation:

Configuration Scheme: 2 parallel × 3 series, total disc count 6.
Verification:

• Total allowable force $2 \times 9\,850 = 19\,700\ \text{N} > 15\,000\ \text{N}$, satisfied.
• Total effective deflection $3 \times 0.9 = 2.7\ \text{mm} > 2.2\ \text{mm}$, satisfied.
• Total free height $L_0 \approx 3 \times 1.2 + (6-1) \times 2.0 = 3.6 + 10 = 13.6\ \text{mm}$; $\lambda = 13.6/40 \approx 0.34 \le 3$, no guidance required (but light guidance is still recommended to prevent misalignment).
• Actual single disc compression $s_{single} = s_{total}/i$, when $s_{total} = 2.2\ \text{mm}$, $s_{single} \approx 0.733\ \text{mm}$, less than $0.75h_0=0.9\ \text{mm}$, stress is safe.

Conclusion: Using a 2×3 mixed configuration meets the requirements.

Stack Configuration Tool Summary: By back-calculating the required number of parallel discs $n = \lceil F_{req}/F_{adm} \rceil$ and series groups $i = \lceil s_{req}/\Delta s_{single} \rceil$ from the target force and deflection, a stack configuration scheme is quickly generated, and strength, stability, and guidance requirements are automatically verified, enabling the automated design of disc spring combinations.