I am trying to model the damping ratios that I've got from experimental modal analysis of a Titanium blisk. I am trying to fit the Rayleigh damping model to my experimental data. I am getting negative values of alpha and beta in some measurements.
Thanks for you response Filippo Piccini . I was wondering if there are any standardized values of alpha and beta.
Mathematically alpha and beta can have negative values. For a certain pair of values for alpha and beta there corresponds a curve between the frequency of vibration and the damping ratio.
According to the expressions of alpha and beta as shown attached; if the first frequency is 0.3 Hz and the selected damping ratio is 10% and if the second frequency is 0.7 Hz and the corresponding damping ratio is 3%, then the value of alpha is 0.06405 Hz and that of beta is -0.045 1/Hz.
Of course physically the values of the frequency and the damping ratio should be positive and within a reasonable range.
As far as I know, the Rayleigh damping has no physical justification, so I am not surprised that you encounter issues by trying to match it with experiments.
I see two reasons why the Rayleigh damping model is popular. First, it is mathematically simple and easy to implement. More importantly, it guarantees that, for an appropriate parameter selection, in the model the damping is always underestimated for a given frequency interval. The latter property is of a great value if you want your analysis to be conservative, i.e. overestimating the amplitude of the dynamic response.
Therefore, I think that fitting Rayleigh parameters from experiments does not make any sense.
However, if you can establish from experiments an overall critical damping in the frequency range of interest, than you can select Rayleigh parameters such that your numerical damping will be underestimated and the analyses conservative.
Hello everyone!
I agree with my colleague Dr. Markovic. Even if Rayleigh damping was designed for mathematical practical reasons to introduce viscous damping forces directly expressed as a function of velocities in the equations of motion and proportional the two matrices of stiffness and mass (so as to guarantee a projection on a modal basis which takes advantage of the orthogonality of the natural modes), it is possible to justify this modeling from a viscoelastic constitutive model, expressed in terms of strain rates. We can cite on this subject: Christensen R.: Theory of Viscoelasticity. An Introduction, 2nd edition. Academic Press, 1982; see also: https://www.researchgate.net/publication/334430043_Modelisation_de_l'absorption_dans_les_structures_en_beton_arme_sous_seisme?_sg%5B0%5D=yG307DOYi6KkukYiuWyUjav9JiR2-NvoU3XgGUkEeOWDj31qfxx5aEgsOCmYozFmUGAr RmaO7PDrBen7hEIo9mo-nlxUtBbcjflrcxLK.IAEd8dV_2ryMnO3KXXuDvgTdfbL7sF3nYyX51VkdygyVgUIEqj4seX2b_uLUHV97TxH6mnMv9JeO7yXlCJAg3g&_tp=eyJjb250ZXh0Ijp7 ImZpcnN0UGFnZSI6InByb2ZpbGUiLCJwYWdlIjoicHJvZmlsZSIsInByZXZpb3VzUGFnZSI6InByb2ZpbGUiLCJwb3NpdGlvbiI6InBhZ2VDb250ZW50In19: at the section 3.1 we can see that once integrated by finite elements, we obtain the global viscous Rayleigh damping forces (Pay attention to the naming in this paper: alpha and beta are reversed compared to your convention). In particular, we can, thanks to this constitutive model, ensure the objectivity of the viscous damping forces, therefore independency with respect to the chosen frame of reference. The two alpha and beta coefficients, calibrated to ensure relevant viscous damping in the frequency range of the dynamical transient response to be calculated, must be positive so that the energy dissipation rate is always positive. It is recommended to choose a strictly positive value of beta (stiffness proportional damping) in order to ensure a dissipation of spurious high frequency numerical disturbances of the transient response to be calculated, due to the time integration scheme.
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