DIN 2093 Free Length and Stability

1. Definition of Free Length $L_0$

In a disc spring stack, the free length $L_0$ is the total axial height of all disc springs in a no-load, free state. Whether for a single disc spring or a parallel, series, or mixed stack, $L_0$ is a fundamental design parameter that directly affects installation space and stability verification.

For a single free disc spring, the free height is determined by the thickness $t$ and the free cone height $h_0$. When multiple springs are stacked, the calculation method depends on the stacking arrangement.

2. Geometry of a Single Disc Spring

Schematic cross-section of a disc spring in the free state:

Outer diameter De
┌─────────────┐
│ ╲ ╱ │
│ ╲ t ╱ │
│ ╲ ╱ │
│ ╲ ╱ │
│ ╲ ╱ │
│ ╳ h0 │
│ ╱ ╲ │
│ ╱ ╲ │
└─────────────┘
Inner diameter Di

In the free state:

• The nominal axial height of a single disc spring ≈ $t + h_0$ (when the outer edge rests on a flat surface, the inner edge lift height is $h_0$, and the total height is the thickness plus the cone height).
• More precisely, when the disc spring is placed on a flat surface, its total free height $l_0 = t + h_0$, where $h_0$ is the vertical distance from the inner edge to the surface.

When measured in a fixture, the free height is typically given as $l_0 \approx t + h_0$.

3. Free Length of Stacked Springs

3.1 Parallel Stacking (Same Direction)

In a parallel stack, $n$ disc springs are stacked closely in the same direction, with surface contact between adjacent springs and the cone surfaces fitting together. The total free height is:

Because all springs have the same cone direction, only the top surface of the uppermost spring exhibits the cone height $h_0$.

3.2 Series Stacking (Opposing Direction)

In a series stack, $i$ disc springs are arranged alternately in opposite directions, with adjacent springs contacting only at the inner and outer edges. Each spring's cone height contributes to the total axial height. If the single spring free height is $l_0$, then:

Because each spring occupies its full free height.

3.3 Mixed Stacking (Parallel × Series)

If the stack consists of $n$ springs in parallel forming one group, with $i$ groups in series, the height of each parallel group follows the parallel formula, and the series groups are added:

It can also be estimated using the total number of springs $N = n \times i$ and the cone structure:

In engineering, $L_0 = i \cdot h_0 + N \cdot t$ is often used for quick evaluation.

4. Stability Parameter $\lambda$

When a disc spring stack has a large free length, it behaves like a slender column under axial load and may experience lateral instability (buckling). DIN 2093 uses the dimensionless parameter $\lambda$ to assess stability:

• $L_0$ — Total free length of the spring stack (mm)
• $D_e$ — Outer diameter of the disc spring (mm)

5. $\lambda$ Criterion and Design Rules

Based on extensive experiments and practical experience, DIN 2093 provides the following stability criteria:

Guidance Design Points:

• Guide Rod: The outer diameter of the rod should be slightly smaller than the inner diameter $D_i$ of the disc spring. Recommended radial clearance: 0.1–0.5 mm (smaller clearance for smaller disc springs).
• Guide Sleeve: The inner diameter of the sleeve should be slightly larger than the outer diameter $D_e$ of the disc spring. Recommended radial clearance: 0.2–0.8 mm.
• Too small a clearance increases hysteresis due to friction; too large a clearance reduces guidance effectiveness.
• Guide surfaces should be hardened and lubricated to reduce wear and friction.

6. Why $\lambda$ is Limited

When $L_0/D_e$ is large, the disc spring column is susceptible to bending instability under slight eccentricity during compression, leading to:

• Some disc springs jumping out of their normal contact positions
• Local stress spikes causing early fatigue failure
• The force‑deflection curve deviating from theoretical superposition

Therefore, $\lambda$ must be checked when designing long-stroke springs, and guidance or segmentation should be introduced when necessary.

7. Calculation Example

Stack: 3 disc springs in series. Single spring specifications: $D_e = 40\ \text{mm}$, $t = 2.0\ \text{mm}$, $h_0 = 1.5\ \text{mm}$.

Single spring free height $l_0 = t + h_0 = 2.0 + 1.5 = 3.5\ \text{mm}$
Total series free height $L_0 = 3 \times 3.5 = 10.5\ \text{mm}$

Stability parameter $\lambda = 10.5 / 40 = 0.2625 \le 3$, no guidance needed.

If 20 springs in series: $L_0 = 20 \times 3.5 = 70\ \text{mm}$, $\lambda = 70 / 40 = 1.75 \le 3$, still no guidance needed.

If the outer diameter is reduced to 15 mm, 20 springs in series give $\lambda = 70/15 \approx 4.67$, requiring guidance.

8. Summary

• Free length is calculated based on the stacking arrangement using single spring geometry and is fundamental for space and stability.
• Stability parameter $\lambda = L_0/D_e$ is a simple and effective buckling criterion.
• $\lambda \le 3$: free installation; $3 < \lambda \le 6$: guidance required; $\lambda > 6$: avoid or segment.
• Proper verification of free length and stability ensures safe, reliable, and accurate force output from the disc spring stack.