DIN 2093 Damping Estimation: Interleaf Friction Energy Dissipation Model

1. Source of Damping

When disc springs are arranged in parallel stacks (same direction stacking) or mixed stacking, contact pressure exists between the conical surfaces of adjacent discs. Under dynamic loading (vibration, impact), micro-slip occurs between the discs, and dry friction work converts mechanical energy into heat, thereby generating damping. Friction from the guide mandrel/sleeve also contributes a small amount of damping, but interleaf friction usually dominates.

Compared with coil springs, this structural damping of disc spring stacks does not require additional dampers, which is one of its outstanding advantages.

2. Measurement of Damping

The equivalent viscous damping ratio $\zeta$ is commonly used in engineering to measure damping level:

• $\Delta W$ — Area of the hysteresis loop in one complete loading-unloading cycle, i.e., energy dissipated by friction (N·mm)
• $W_{max}$ — Maximum elastic stored energy in that cycle (N·mm), usually approximated as $W_{max} \approx \frac{1}{2} F_{max} s_{max}$ (linear approximation)

The logarithmic decrement $\delta = 2\pi\zeta$ can also be used.

3. Damping Ratio Estimation Formula (Based on Interleaf Friction)

For a combination of $n$ discs stacked in parallel in the same direction, the damping ratio generated by interleaf friction can be estimated using the following simplified formula:

Where: - $\mu_f$ — Interleaf friction coefficient, typically 0.03 ~ 0.08 (phosphated + oil can be ≤ 0.03) - $n$ — Number of discs in parallel ($n \ge 2$ for interleaf friction to occur)

Derivation idea: - Total interleaf friction force $F_f \approx \mu_f (n-1) F$, where $F$ is the theoretical force of a single disc - The hysteresis loop is approximated as a parallelogram, with loop area $\Delta W \approx 2 F_f \cdot \Delta s$ ($\Delta s$ is the stroke) - Maximum stored energy $W_{max} \approx \frac{1}{2} F \Delta s$ - Substituting into the definition of $\zeta$ yields the above result.

This formula shows that: the damping ratio is proportional to the friction coefficient and the number of discs in parallel, and has little relation to the stiffness, stroke, or external load of a single disc spring. For most industrial disc spring stacks ($n=2\sim4$, $\mu_f=0.05$), $\zeta$ is approximately in the range 0.03 ~ 0.10.

If guide friction (mandrel/sleeve) exists, an equivalent friction coefficient $\mu_{eff} = \mu_f + \mu_g \cdot \frac{D_m}{D_e}$ can be introduced, where $D_m$ is the mean diameter of the disc spring and $\mu_g$ is the guide friction coefficient (0.05~0.10). In this case, simply replace $\mu_f$ in the formula with $\mu_{eff}$.

4. Typical Range of Damping Ratio

Rule: The more discs and the higher the friction, the greater the damping. However, excessive friction can cause severe heating and wear, so a trade-off is required in design.

5. Nonlinear Characteristics of Damping

The dry friction damping of disc spring stacks exhibits significant amplitude dependence:

• At small amplitudes: The relative slip between discs is extremely small, possibly not reaching full sliding, resulting in lower damping.
• At large amplitudes: Full slip occurs between discs, damping increases and tends to saturate.

Therefore, the damping ratio $\zeta$ is not constant but increases with amplitude, exhibiting typical nonlinear damping characteristics. In engineering, equivalent linearized damping is often used, i.e., measuring the equivalent damping ratio at the expected operating amplitude.

6. Determination of Accurate Damping Values

Theoretical estimation formulas can only provide approximate values. Accurate damping parameters must be obtained through dynamic testing:

1. Hysteresis loop method: Apply sinusoidal displacement excitation to the disc spring stack on an electro-hydraulic servo testing machine, simultaneously record the force-displacement curve, directly measure the hysteresis loop area $\Delta W$ and maximum stored energy $W_{max}$, and calculate $\zeta$.
2. Free decay method: Give the spring stack an initial displacement, record the free decay vibration curve, and determine damping from the logarithmic decrement.
3. Frequency response method: Perform a frequency sweep test and determine damping at the half-power bandwidth.

It is recommended to perform at least one set of validation tests when designing critical vibration damping systems.

7. Design Recommendations

• When high damping is required: Increase the number of discs in parallel, use dry friction or sandblasted surfaces (high $\mu_f$), but be aware of wear and heat buildup.
• When low damping is required: Use phosphating + oil lubrication to reduce interleaf friction; or reduce the number of discs in parallel.
• Guide system: Avoid introducing additional uncontrollable friction due to overly tight guiding, as this can cause excessive damping and lead to sticking.
• Long-term stability: The friction coefficient may change over long-term operation due to wear or lubricant aging. A conservative damping value should be used in design, and checks should be performed during maintenance.

Summary: The damping ratio of DIN 2093 disc spring stacks mainly originates from interleaf dry friction and can be quickly estimated using $\zeta \approx \frac{2}{\pi} \mu_f (n-1)$, with a typical range of 0.02~0.15. Accurate values must be obtained through dynamic testing. Reasonable use of dry friction damping can effectively suppress resonance, but friction variation and wear must be controlled.