DIN 2093 Critical Speed: Rotational Resonance Calculation

1. Definition of Critical Speed

When a disc spring is used as a rotating component (e.g., an elastic support rotating at high speed with a shaft), resonance occurs when its natural vibration frequency coincides with the rotational frequency or an integer multiple thereof, leading to severe vibration or even failure. The corresponding rotational speed is called the critical speed $n_{crit}$.

Since the axial natural frequency of disc springs is typically above 1 kHz, the corresponding critical speed is extremely high (> 50 000 rpm). General industrial applications (< 10 000 rpm) usually do not reach this range, but verification is mandatory for high-speed spindles, turbines, and centrifuges.

2. Basic Formula for Critical Speed Calculation

In rotating machinery, the most dangerous resonance occurs when the rotational frequency (fundamental frequency) equals the system's natural frequency. Therefore, the critical speed can be estimated as:

Where: - $n_{crit}$ — Critical speed (revolutions per minute, rpm) - $f_n$ — Axial natural frequency (Hz) of the disc spring (or assembly)

If strong harmonic excitation exists (e.g., unbalance force at 1×, blade passing frequency at 2×, 3×, etc.), the critical speed may also be $n_{crit,k} = 60 \cdot f_n / k$, where $k = 1, 2, 3, \dots$. Typically, fundamental frequency resonance ($k = 1$) is the most severe.

3. Calculation of Disc Spring Natural Frequency

The axial natural frequency of a single disc spring is detailed in the "Natural Frequency" section:

• $k$ — Tangential stiffness (N/m) of the disc spring at the operating point (given preload deflection), derived from the Almen‑Laszlo formula:

  $$k = \frac{4E}{1-\nu^2} \cdot \frac{t^3}{K_1 D_e^2} \left[ \left( \frac{h_0}{t} \right)^2 + 1 - 3\frac{h_0}{t}\cdot\frac{s}{t} + \frac{3}{2}\left( \frac{s}{t} \right)^2 \right]$$

• $m_{eff}$ — Effective mass (kg), typically including the effective mass of the disc spring itself (approximately 1/3 of the disc spring mass) and any external added mass. For vibration of the disc spring alone, $m_{eff} \approx \frac{1}{3} \rho V$.

For multi-disc assemblies: - Parallel (stacked in the same direction): Natural frequency $f_{n,stack} = f_{n,single}$ (unchanged). - Series (stacked in opposite directions): Natural frequency $f_{n,stack} \approx f_{n,single} / i$ ($i$ is the number of series groups). - Mixed: Natural frequency follows the series rule, dominated by the number of series groups $i$.

4. Quick Determination of Critical Speed

Substituting the natural frequency into the basic formula yields:

• If $f_n = 2\,000\ \text{Hz}$ → $n_{crit} \approx 120\,000\ \text{rpm}$
• If $f_n = 500\ \text{Hz}$ → $n_{crit} \approx 30\,000\ \text{rpm}$
• If $f_n = 200\ \text{Hz}$ → $n_{crit} \approx 12\,000\ \text{rpm}$

It is evident that the critical speed drops below 10 000 rpm only when the natural frequency is below approximately 160 Hz. This is nearly impossible for a single disc spring (single disc $f_n$ is typically > 1 kHz), but large-stroke series assemblies can significantly reduce the natural frequency and must be carefully considered.

5. Calculation Example

Given: - Single disc spring: $D_e = 40\ \text{mm}$, $D_i = 20.4\ \text{mm}$, $t = 2.0\ \text{mm}$, $h_0 = 1.2\ \text{mm}$ - Operating preload $s = 0.6\ \text{mm}$, calculated tangential stiffness $k \approx 8.0\times10^6\ \text{N/m}$ - Effective mass (disc spring only) $m_{eff} \approx 0.004\ \text{kg}$

Natural frequency:

Critical speed:

Far higher than common machinery speeds; no concern is necessary.

If 6 discs are used in series (to increase stroke), the assembly natural frequency drops to $7100 / 6 \approx 1\,183\ \text{Hz}$, and the critical speed is approximately 71 000 rpm, still above most industrial speeds. However, if the external added mass is large, the natural frequency may drop to a few hundred Hz, requiring detailed verification.

6. Correction of Critical Speed for Centrifugal Force

At high rotational speeds, centrifugal force causes radial expansion of the disc spring, slightly altering its axial stiffness and stress distribution. However, the centrifugal force correction typically affects the axial force magnitude, and its influence on natural frequency is a secondary effect. Near the critical speed, due to resonance amplitude amplification, the centrifugal effect may become significant. Accurate calculation requires rotor dynamics analysis (e.g., considering rotational softening and stress stiffening). For conventional industrial high speeds (< 30 000 rpm), the correction of frequency due to centrifugal force can be neglected, and the above calculation is sufficient for safety verification.

7. Design Guidelines

• General industrial speeds: Only ensure $f_n > 10 \cdot f_{exc}$ (to avoid resonance); the critical speed will naturally be far higher than the operating speed, and no separate verification is needed.
• High-speed applications (> 10 000 rpm):
• Calculate the natural frequency $f_n$ of the disc spring assembly at the operating point.
• Calculate the fundamental critical speed $n_{crit} = 60 f_n$.
• Ensure a sufficient safety margin between $n_{crit}$ and the operating speed range (typically, avoid ±20% of the operating speed).
• If the critical speed falls within the operating speed range, change the disc spring specifications, adjust the stacking arrangement, or increase the number of series discs to lower the frequency (avoidance), or enhance damping to suppress resonance.
• Assembled disc springs: Prioritize attention to the frequency reduction effect of the number of series groups $i$; parallel stacking does not affect the critical speed.

Summary: The critical speed of a disc spring is $n_{crit} = 60 f_n$. Due to its high natural frequency, it is typically far above general industrial speeds. When designing high-speed rotating disc springs, verify the natural frequency of the assembly and ensure the critical speed avoids the operating speed to guarantee safety.