DIN 2093 Vibration Transmissibility: Isolation and Resonance Amplification

1. Definition of Vibration Transmissibility

When disc springs are used to support equipment or isolate vibrations, the force transmitted through the spring to the other side changes relative to the excitation force (or base motion). Vibration transmissibility $T$ is defined as the ratio of the transmitted force amplitude to the excitation force amplitude (or displacement transmissibility, which is identical in a single-degree-of-freedom system):

• $T < 1$: The spring provides vibration isolation; the transmitted force is less than the excitation force.
• $T = 1$: Direct transmission, no amplification or attenuation.
• $T > 1$: The system experiences resonance amplification; the transmitted force exceeds the excitation force.

2. Transmissibility Formula for a Single-Degree-of-Freedom System

A disc spring can be simplified as a single-degree-of-freedom system with stiffness $k$ and damping coefficient $c$. Under base displacement excitation (or force excitation), the transmissibility $T$ as a function of the frequency ratio $r = f_{exc}/f_n$ and damping ratio $\zeta = c/(2\sqrt{km})$ is:

Where: - $f_{exc}$ — Excitation frequency (Hz) - $f_n = \frac{1}{2\pi} \sqrt{k/m}$ — Natural frequency of the system (Hz) - $r = f_{exc}/f_n$ — Frequency ratio - $\zeta$ — Damping ratio (typical range for disc spring stacks: 0.02 ~ 0.15, originating from inter-leaf dry friction)

3. Boundary Between the Resonance Region and the Isolation Region

The transmissibility curve exhibits distinctly different characteristics in different frequency ratio ranges, with the key boundary point at $r = \sqrt{2} \approx 1.414$.

3.1 Amplification Region ($r < 1.414$)

• When the excitation frequency approaches the system's natural frequency ($r \approx 1$), the transmissibility increases sharply, causing resonance.
• The magnitude of the resonance peak depends on the damping ratio $\zeta$:
  $$T_{max} \approx \frac{1}{2\zeta}$$

For example, with $\zeta = 0.05$, $T_{max} \approx 10$, meaning a 10-fold force amplification. - Disc spring combinations, due to inter-leaf friction, possess a certain damping capacity ($\zeta \approx 0.05 \sim 0.10$), which can significantly suppress the resonance peak, but prolonged operation in the resonance region should still be avoided.

3.2 Isolation Region ($r > 1.414$)

• When the frequency ratio $r > \sqrt{2}$, the transmissibility $T < 1$, and the system enters the isolation region.
• Here, the excitation force is attenuated during transmission, and the larger the $r$, the better the isolation effect (approximately 12 dB reduction per octave).
• Damping has a slightly adverse effect in the isolation region: higher damping leads to slightly higher $T$ values at high frequencies. However, for disc springs, where damping primarily comes from friction, its impact on high-frequency isolation is acceptable provided resonance attenuation is ensured.

4. Dual Role of Damping

A trade-off is required in design: for equipment that must pass through the resonance region (e.g., frequent start-stop cycles), sufficient damping must be ensured; for equipment operating steadily in the isolation region, friction can be appropriately reduced (e.g., using phosphating lubrication to lower $\mu_f$) to achieve better isolation efficiency.

5. Damping Characteristics of Disc Springs

The material damping of a single disc spring is very low ($\zeta \approx 0.01$), but parallel stacking introduces significant additional damping through inter-leaf dry friction, increasing the total damping ratio to 0.03 ~ 0.15. This characteristic allows disc spring stacks to possess a certain resonance suppression capability without the need for external dampers, outperforming helical springs.

If higher damping is required, additional friction plates or damping coatings can be used.

6. Design Guidelines

To avoid resonance and ensure vibration isolation, the design of a disc spring isolation system should satisfy:

That is, the system's natural frequency should be significantly lower than the minimum excitation frequency, achieving a frequency ratio $r \ge 2.5 \sim 3$ for efficient isolation with $T \le 0.2$ (transmissibility below 20%). Additionally, during equipment startup and shutdown, the system will sweep through the resonance region, relying on the disc spring stack's inherent damping to suppress transient resonance.

7. Calculation Example

Given: - Natural frequency of disc spring assembly $f_n = 20\ \text{Hz}$ (e.g., achieved by stacking multiple springs in series to lower the frequency) - Operating speed 1500 rpm → fundamental frequency 25 Hz, minimum excitation frequency $f_{exc,min} = 25\ \text{Hz}$ - Damping ratio $\zeta = 0.08$

Frequency ratio $r = 25/20 = 1.25 < 1.414$, which lies in the amplification region and does not meet isolation requirements! The natural frequency must be lowered, or the operating speed increased.

If a softer disc spring combination is used to achieve $f_n = 8\ \text{Hz}$, then $r = 25/8 \approx 3.13 > 1.414$, entering the isolation region. The transmissibility is then:

Thus, the transmitted force is only 12.7% of the excitation force, indicating good isolation performance.

Summary: The vibration transmissibility of disc springs follows the classical single-degree-of-freedom system law, with the frequency ratio $r = \sqrt{2}$ marking the boundary between amplification and isolation. In design, the system's natural frequency should be significantly lower than the excitation frequency ($r \ge 2.5$), and the moderate damping provided by inter-leaf friction in disc spring stacks should be utilized to suppress resonance during start-stop transients. Through proper selection and arrangement, disc springs can serve as both elastic supports and vibration isolation elements.