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Rayleigh damping: Guidance |
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Rayleigh damping: Guidance |
Classical Rayleigh damping is viscous damping which is proportional to a linear combination of mass and stiffness. The damping matrix $\mat{C}$ is given by $\mat{C} = \mu\mat{M} + \lambda\mat{K}$, where $\mat{M}$ and $\mat{K}$ are the mass and stiffness matrices respectively and $\mu$ and $\lambda$ are constants of proportionality.
Rayleigh damping does afford certain mathematical conveniences and is widely used to model internal structural damping. One of the less attractive features of Rayleigh damping however is that the achieved damping ratio $\xi$ varies with response frequency. The stiffness proportional term contributes damping that is linearly proportional to response frequency and the mass proportional term contributes damping that is inversely proportional to response frequency.
This graph illustrates the way in which the mass and stiffness damping terms contribute to the overall damping ratio:
| Figure: | Variation of damping ratio with frequency |
Consider a system which has two primary responses: one at the wave frequency and the other at a much lower frequency, for example due to vessel drift. Clearly Rayleigh damping constants must be chosen carefully to avoid the mass proportional term resulting in over-damping of the low frequency response.
It is common practice to do this by using the stiffness proportional term only. For example the DNV dynamic riser code DNV OS F201 (appendix A, K103) makes the following recommendation: "it should also be observed that the mass proportional damping would give damping due to rigid body motions. The mass proportional damping is therefore normally neglected for compliant structures undergoing large rigid body motions." in other words the recommendation for such systems is to use stiffness proportional damping (the red curve above).
This is a good argument. However, an analogous argument can also be made about any high frequency response. If only stiffness proportional damping is used then any high frequency response will be over-damped. It is quite common for systems to have responses at frequencies higher than the wave frequency and, since high frequency responses are often damaging to a system it is important to model them accurately. Stiffness proportional damping is however very effective at artificially removing high frequency responses from an analysis: this is a danger that must be avoided.
We recommend the following procedure for applying Rayleigh damping:
This technique will apply the specified damping ratio at responses with frequency $f_\textrm{min}$ and $f_\textrm{max}$. For frequencies in between $f_\textrm{min}$ and $f_\textrm{max}$, the damping ratio will be less than the specified damping ratio which ensures that artificial over-damping is avoided.
| Note: | While this approach avoids over-damping, it could be argued that the response will instead be under-damped. Generally, this conservative under-damping is however far less important than the non-conservative over-damping that we are trying to avoid. |
If your system response is concentrated at the wave frequency for all critical design cases, rather than across a range of frequencies, then it is safe to apply stiffness proportional damping.
The procedure described here will help you avoid non-conservative over-damping of high or low frequency responses in your system. However, it is quite laborious and time consuming. Can we adopt a simpler approach?
For a great many of the systems that are analysed by OrcaFlex it turns out that structural damping has little or no effect on system performance. For subsea lines the structural damping is usually negligible in comparison with the damping due to hydrodynamic drag, and so Rayleigh damping can usually be ignored for analysis of subsea lines. If you have such a system which does show significantly different responses when Rayleigh damping is applied, you should check that this is not in fact due to the over-damping issues described above.
For in-air lines (e.g. jumper hoses) the situation is different. These lines have no hydrodynamic damping and so the structural damping can be significant. For such lines it is very easy for resonant responses to be excited, and if no damping is included then these responses do not decay. In this situation, Rayleigh damping can be very useful – it is, of course, important to apply it carefully, as described above.
Turbines blades undergo rigid body motion as the rotor turns. As discussed above, mass proportional damping acts to damp out this type motion and so it is probably inappropriate. For these models, it is recommended that the mass coefficient is set to zero.
One common phenomenon that is sometimes modelled with Rayleigh damping is the damping due to internal friction between layers of a pressurised, unbonded flexible riser. This effect is strongly amplitude dependent and is poorly represented by Rayleigh damping. The problem is that the damping ratio depends on the amplitude of response, making it very difficult to set and load-case dependent. Instead we recommend that you use a hysteretic bend stiffness which gives a more accurate model of the riser in this situation.