DIN 2093 Disc Spring Energy Storage Calculation

1. Definition and Purpose of Energy Storage

During axial loading of a disc spring, the work done by the external force is converted into elastic strain energy stored within the spring body; upon unloading, this energy is released and performs external work. The stored energy is numerically equal to the area under the force‑deflection curve, i.e.:

Energy storage is a key indicator for evaluating the vibration absorption, cushioning, and energy absorption capacity of disc springs. Under the same external dimensions, the larger the relative cone height $h_0/t$ (the stronger the disc effect), the higher the energy storage per unit volume.

2. Energy Storage Formula Based on Almen‑Laszlo Theory

The load formula for a disc spring (dimensionless form) is:

where:

• $\delta = s/t$, $\eta = h_0/t$
• $C_F = \dfrac{4E}{1-\nu^2} \cdot \dfrac{t^4}{K_1 D_e^2}$
• $K_1 = \dfrac{1}{\pi} \cdot \dfrac{\left(\dfrac{c-1}{c}\right)^2}{\dfrac{c+1}{c-1} - \dfrac{2}{\ln c}}$, $c = D_e/D_i$

Substituting $F(\delta)$ into the integral and using $ds = t\,d\delta$, the energy storage at any compression $s$ (corresponding to $\delta$) is:

Or expressed in dimensional form:

3. Energy Storage at Flat Deflection

When the disc spring is compressed to flat deflection ($s = h_0$, $\delta = \eta$), the energy storage reaches its maximum value, denoted as $U_{flat}$:

Substituting $C_F$ yields:

Low cone height approximation: When $h_0/t \le 0.8$, the $\eta^4/8$ term is typically less than 10% of the $\eta^2/2$ term and can be neglected. The energy storage then approximates the triangular area under the force‑deflection curve:

where $F_{flat} = C_F\,\eta$ is the flat deflection force. This approximation is simple and intuitive, suitable for quick evaluation.

4. Increased Energy Storage with Higher $h_0/t$ Ratio

From the energy storage formula, it can be seen that $U_{flat}$ is proportional to $h_0^2$ and also depends on the $\eta^4$ term. The larger the relative cone height $h_0/t$, the stronger the nonlinearity of the force‑deflection curve, and the area enclosed by the curve increases significantly. Therefore, when designing energy-absorbing and cushioning components, disc springs with high $h_0/t$ are preferred.

However, it should be noted that a high $h_0/t$ also increases the stress at the OM point, and strength verification must be performed simultaneously.

5. Calculation Example

Disc spring specification (DIN 2093 Series A, outer diameter 20 mm, inner diameter 10.2 mm):

• $D_e = 20\ \text{mm}$, $D_i = 10.2\ \text{mm}$, $t = 1.5\ \text{mm}$, $h_0 = 1.0\ \text{mm}$
• Material spring steel: $E = 206\,000\ \text{MPa}$, $\nu = 0.3$

Step 1: Calculate parameters

,

Step 2: Flat deflection energy storage

Exact value:

Triangular approximation:

Error approximately 10%, acceptable for engineering purposes.

6. Applications of Energy Storage Calculation

• Vibration absorption and cushioning: Compare the $U_{flat}$ of different disc springs and select one with sufficiently large energy storage.
• Dynamic response: Energy storage and stiffness together determine the natural frequency and energy dissipation.
• Stacked disc springs: For parallel stacking, the energy storage adds ($U_{total} = n \cdot U_{single}$); for series stacking, the energy storage also adds, but the force remains unchanged while the deflection increases.

Summary: The energy storage of a disc spring is obtained by integrating the Almen‑Laszlo force‑deflection equation. The flat deflection energy storage formula is $U_{flat} = C_F t (\eta^2/2 + \eta^4/8)$. High cone height disc springs have greater energy storage per unit volume and are preferred for energy-absorbing elements. Design should be combined with strength verification to prevent local stress exceeding limits.