Hi All
I am currently using Rayleigh damping to calculate the damping matrix of the wind turbine tower and plan to use this matrix for whole wind turbine analysis. However, I have some doubts regarding the selection of damping coefficients.
Rayleigh damping is calculated using the following formula:
Taking the 5MW-baseline wind turbine as an example, the structural-damping ratio is 1%
For the isolated tower, I use the finite element method for simulation, where the tower was discretized into 20 beam elements. The undamped modal analysis yielded the first four natural frequencies of vibration for the tower as follows:
1st Tower Fore-After/Side-to-Side: 0.8876 Hz
2st Tower Fore-After/Side-to-Side: 4.3531 Hz
Therefore, according to the yellow-highlighted portion in the above figure, a0 and a1 should be calculated using this formula at this point:
a0,1 = 2 * 0.01 * (2 * pi * 0.8876 * 2 * pi * 4.3531)/(2 * pi * 0.8876 + 2 * pi * 4.3531) = 0.0926
a1,1 = 2 * 0.01/(2 * pi * 0.8876 + 2 * pi * 4.3531) = 6.0739e-04
However, in the modal analysis of the entire wind turbine, the modal results of the tower structure differ from those of a isolated tower due to the rotor and nacelle mounted at its top. For the 5MW-baseline wind turbine, the first four natural frequencies are:
If these frequencies are used as the natural frequencies for calculations, then:
a0,2 = 2 * 0.01 * (2 * pi * 0.32 * 2 * pi * 2.9)/(2 * pi * 0.32+2 * pi * 2.9) = 0.0362
a1,2 = 2 * 0.01/(2 * pi * 0.32+2 * pi * 2.9) = 9.8854e-04
If a0,1 and a1,1 are adopted as the coefficients for Rayleigh damping calculation, then the first-order frequency of 0.32Hz during the vibration of the entire wind turbine does not fall between the natural frequencies of 0.8876Hz and 4.3531Hz of the isolated tower. Moreover, the damping ratio of Rayleigh damping increases rapidly outside the selected frequency range, as shown in the figure below.
However, if a0,2 and a1,2 are used as the coefficients for Rayleigh damping calculation, the damping ratio of the vibration mode of the isolated tower would no longer be 0.01, which contradicts the provided data.
Which set of coefficients is more suitable?
Please forgive me for any errors in my English writing. I hope someone can clarify my confusion, point out errors in my calculations, or identify any logical or knowledge gaps in my understanding. Thank you!
Best regards!
Yangyudong.Liu
[1] Jonkman, J. M. , Butterfield, S. , Musial, W. , & Scott, G. . (2009). Definition of a 5mw reference wind turbine for offshore system development. office of scientific & technical information.
[2] Dynamics of structures, Ray W. Clough and Joseph Penzien