DIN 2093 Natural Frequency: Vibration Characteristics of Disc Springs

1. Single-Degree-of-Freedom Natural Frequency Formula

The axial vibration of a disc spring can be simplified as a single-degree-of-freedom spring-mass system, with its undamped natural frequency given by:

• $f_n$ — Natural frequency (Hz)
• $k$ — Tangent stiffness (N/m) of the disc spring at the operating point (given preload deflection $s$)
• $m_{eff}$ — Effective mass (kg) of the vibration system, typically including the effective mass of the disc spring itself and any additional load mass

2. Calculation of Tangent Stiffness $k$

The force-deflection relationship of a disc spring is nonlinear, with stiffness varying with compression. The tangent stiffness at the operating point $s$ is derived from the Almen-Laszlo theory:

Symbol meanings are as defined in previous sections ($E$ elastic modulus, $t$ thickness, $h_0$ free cone height, $D_e$ outer diameter, $K_1$ shape factor).

For commonly used low cone height ratio disc springs ($\eta = h_0/t \le 0.9$), stiffness typically monotonically increases (or slowly decreases) during compression, so the natural frequency rises with increasing preload.

3. Effective Mass $m_{eff}$

The effective mass during disc spring vibration consists of two parts:

1. Effective mass of the disc spring itself: Not all mass of a distributed-mass spring participates in vibration; typically 1/3 of its mass is taken as the equivalent mass:
   $$m_{spring} \approx \frac{1}{3} \cdot \rho \cdot V$$

where $\rho$ is material density (steel ≈ 7850 kg/m³), and $V$ is the disc spring volume. The mass of a single disc spring is very small (typically a few grams to tens of grams), having much less influence on natural frequency than stiffness.

1. External load mass: If a concentrated mass $M$ (e.g., a clamped flange or workpiece) is supported on the disc spring, this mass should be fully included:
   $$m_{eff} = M + m_{spring}$$

When analyzing only the dynamic characteristics of the disc spring itself, external mass can be neglected, considering only the spring's effective mass.

4. Increase in Natural Frequency with Preload

As disc spring compression $s$ increases: - Tangent stiffness $k$ increases (especially for low $\eta$ disc springs). - Effective mass remains unchanged. - Natural frequency $f_n$ increases proportionally to $\sqrt{k}$.

Example: For a certain disc spring, stiffness near the free state is about 5 N/μm, with a natural frequency of about 1 kHz; when compressed to 0.75 $h_0$, stiffness increases to 12 N/μm, and natural frequency rises to about 1.5 kHz.

5. Typical Range of Natural Frequencies

The natural frequency of a single standard steel disc spring typically ranges from 1 kHz to 10 kHz, much higher than the excitation frequencies of general mechanical equipment (rotating machinery vibration < 500 Hz, reciprocating machinery typically < 200 Hz). Therefore, in conventional applications, disc springs rarely experience resonance, and their dynamic response can be considered quasi-static.

When using disc spring combinations: - Parallel (stacked in same direction): Total stiffness increases by $n$ times, total mass also increases by $n$ times, natural frequency remains essentially unchanged (theoretically identical). - Series (stacked in opposite directions): Total stiffness decreases to $1/i$, total mass increases by $i$ times, natural frequency decreases significantly. - Mixed: Parallel increases load capacity, series decreases frequency; natural frequency can be adjusted accordingly during design.

6. Calculation Example

Disc spring specifications: $D_e = 40\ \text{mm}$, $D_i = 20.4\ \text{mm}$, $t = 2.0\ \text{mm}$, $h_0 = 1.2\ \text{mm}$ ($\eta=0.6$), steel material.

1. Calculate operating point stiffness (preload to $s = 0.6\ \text{mm}$, $\delta = 0.3$) From the above parameters, $K_1 \approx 0.68$, $C_F \approx 6.5 \times 10^4$ N, thus operating point stiffness $k \approx 8\,000\ \text{N/mm} = 8 \times 10^6\ \text{N/m}$.

2. Calculate effective mass Disc spring volume $V \approx \frac{\pi}{4}(D_e^2 - D_i^2) \cdot t \approx 1.5 \times 10^{-6}\ \text{m}^3$, mass $m = 7850 \times 1.5 \times 10^{-6} \approx 0.0118\ \text{kg}$.
Spring effective mass $m_{eff} \approx m/3 \approx 0.0039\ \text{kg}$ (assuming no additional load).

3. Natural frequency

If an additional load mass of 0.1 kg is added, $m_{eff} \approx 0.104\ \text{kg}$, then $f_n \approx 1.4\ \text{kHz}$, still much higher than common mechanical vibration frequencies.

7. Key Points for Engineering Design

• Avoid resonance: When disc springs are installed in equipment that may generate high-frequency excitation (e.g., ultrasonic devices, high-speed presses), the natural frequency should be checked to ensure it is offset from the excitation frequency.
• Preload adjustment: Increasing the initial compression can raise the natural frequency, serving as a simple and effective frequency tuning method.
• Guidance effects: Guidance clearance and friction introduce damping, slightly reducing the natural frequency, but with minimal impact on the frequency magnitude.
• Combination use: To lower the natural frequency, series combinations can be used; to increase stiffness without changing frequency, parallel combinations are employed.

Summary: The natural frequency of a disc spring is determined by tangent stiffness and effective mass, with typical values of 1–10 kHz. Preload significantly increases frequency by increasing stiffness, while the disc spring's own mass is very light, making external additional mass the key factor determining the lower frequency limit. The high natural frequency allows disc springs to be treated as rigid elements in conventional mechanical applications, with fast and stable dynamic response.