The Rayleigh damping coefficient of the tower structure of the wind turbine

[Yangyudong.Liu]

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

[Jason.Jonkman]

Dear @Yangyudong.Liu,

I haven’t gone through all of your arguments, but the tower damping ratio inputs to ElastoDyn (TwrFADmp, TwrSSDmp) are explained in the following forum topic: Natural frequency and damping ratio calculation. To summarize, ElastoDyn uses stiffness proportional damping for the isolated tower with a fixed base and no tower-top mass/inertia.

Best regards,

[Yangyudong.Liu]

Dear @Jason.Jonkman

Thank you for your answer and explanation. It is very helpful!

Best regards!

Yangyudong.Liu

[Yingxin.Lv]

Dear @Jason.Jonkman
In the INPUT FILE for ELASTODYN TOWER and blade, items such as:

1 TwrFADmp(1) - Tower 1st fore-aft mode structural damping ratio (%)
1 TwrFADmp(2) - Tower 2nd fore-aft mode structural damping ratio (%)
1 TwrSSDmp(1) - Tower 1st side-to-side mode structural damping ratio (%)
1 TwrSSDmp(2) - Tower 2nd side-to-side mode structural damping ratio (%)

How do these items affect the structural damping matrix? My understanding is that the structural model may have more than 20 degrees of freedom. Assuming the Tower 1st fore-aft, Tower 2nd fore-aft, Tower 1st side-to-side, and Tower 2nd side-to-side modes correspond to the 1st, 2nd, 3rd, and 4th degrees of freedom, respectively, the input values for TwrFADmp and TwrSSDmp will modify the overall structural damping matrix. For instance, the updated term C(1,1) would be calculated as the existing C(1,1) + 2 * TwrFADmp(1) * 0.01 * sqrt(K_fa1) * sqrt(M_fa1), where K_fa1 and M_fa1 represent the modal stiffness and modal mass of the tower’s first fore-aft mode, respectively. The same logic applies to the second fore-aft mode and the side-to-side modes. Could you point out if my understanding is correct?
In addition to this, do the tower structural damping ratios affect off-diagonal elements of the damping matrix, other than just C(1,1), C(2,2), C(3,3), and C(4,4)? Furthermore, regarding the calculation of modal stiffness: for the tower and blades, should the influence of axial load be considered? And for the blades, should the effect of Twist also be taken into account?
Thank you again for your reply and help.
Best regards,

[Jason.Jonkman]

Dear @Yingxin.Lv,

The structural damping matrix of the tower as used by ElastoDyn is explained in the forum topic I linked to above: Natural frequency and damping ratio calculation. That is, if K_ij is the tower elastic stiffness matrix, the tower structural damping matrix C_ij is:

C_ij = K_ij * beta_j

where

beta_j = zeta_j / ( pi * f_j )

with

f_j = 1 / ( 2 * pi ) * SQRT( K_jj / M_jj )

Carrying out the math to eliminate beta_j and f_j separately for the tower-fore aft (TFA) and tower side-side (TSS) components:

CTFA_ij = KTFA_ij * 0.01 * TwrFADmp(j) * 2 * SQRT( MTFA_jj / KTFA_jj )
CTSS_ij = KTSS_ij * 0.01 * TwrSSDmp(j) * 2 * SQRT( MTSS_jj / KTSS_jj )

So, your math, is close, but not exact. Just a few clarifying points:

• The tower in ElastoDyn has 4 DOFs, not 20, the 1st and 2nd TFA and 1st and 2nd TSS modes.
• If the tower elastic stiffness matrices have off-diagonal terms (nonzero terms for i /= j), so, will the tower structural damping matrices, as implied by the math above.
• I’m not sure I understand your point about axial load.

For the blades, the process is similar, but the effect of structural pretwist must be accounted for, which coupled in-plane and output-of-plane bending, as discussed in other forum topics.

Best regards,

[Yingxin.Lv]

Dear @ Jason.Jonkman,
I would like to thank you for your response.

1. Regarding axial load

CTFA_ij = KTFA_ij * 0.01 * TwrFADmp (j) * 2 * SQRT( MTFA_jj / KTFA_jj )
CTSS_ij = KTSS_ij * 0.01 * TwrSSDmp (j) * 2 * SQRT( MTSS_jj / KTSS_jj )

In the formula above, KTFA_ij represents the modal mass of the tower.

KTFA_ij = ∫ E I(h) (d²φ(h)/dh²)² dh

However, if we consider the influence of gravity/loads stemming from the upper RNA and the tower’s self-weight, the calculation formula for KTFA_ij should be:

KTFA_ij = ∫ EI(h) (d²φ(h)/dh²)² dh - ∫ N(h) (dφ(h)/dh)² dh

Regarding the blade, if the axial load is considered, the calculation of modal stiffness should also be similar to the expression above.Therefore, when calculating the damping coefficient, which calculation formula should be adopted for KTFA_ij?
2. Regarding the number of DOFs, I apologize for not making it clear earlier. I am referring to all the structural degrees of freedom, including the platform’s 6 DOFs, the tower’s 4 DOFs, the blades’ 9 DOFs, and other DOFs, totaling over 20 DOFs. Taking the tower as an example, assuming the Tower 1st fore-aft, Tower 2nd fore-aft, Tower 1st side-to-side, and Tower 2nd side-to-side modes correspond to the 1st, 2nd, 3rd, and 4th degrees of freedom, I now understand that the coefficients contributed by the structural damping ratio to C(1,1), C(2,2), C(3,3), and C(4,4) in the structural damping matrix can be calculated using the formula you provided.

However, I am unclear whether the structural damping ratio has any effect on the off-diagonal elements of the damping matrix. If it does, according to the above formula, for instance, the corresponding coefficient for an off-diagonal element like C(1,2) would be

CTFA_12=KTFA_12 * 0.01 * TwrFADmp(2) * 2 * SQRT( MTFA_22 / KTFA_22 ),

where KTFA_12 should be calculated using the following expression.

KTFA_12 = ∫ E I(h) (d²φ_mode1(h)/dh²)* (d²φ_mode2(h)/dh²) dh

Could you please indicate whether there are any errors in the calculation process mentioned above?
However, The structural damping ratios set for the tower and blades will not contribute a damping coefficient to the elements in the structural damping matrix that correspond to coupling terms, such as those between the tower and blades or between the tower/blades and the platform.Is my understanding correct?
I would appreciate any thoughts or insights.
Thank you and best regards,

[Jason.Jonkman]

Dear @Yingxin.Lv,

Regarding (1), thanks for clarifying regarding the axial load. I agree with your augmented expression for the tower stiffness, which can be applied to the gravitational de-stiffening of the tower. In ElastoDyn, only the elastic stiffness (the first term in your augmented expression of KTFA_ij) is used to compute the tower structural damping. While the gravitational de-stiffening is calculated and implemented within ElastoDyn separately, this stiffness does not directly influence the CTFA_ij and CTSS_ij matrices in ElastoDyn.

Regarding (2), ElastoDyn does not construct a full damping matrix for all DOFs as part of its solve (that said, the effective damping matrix can be extracted through the linearization process). Rather; the structural damping contribution gets added along with other forces on the right-hand side of the equations of motion (that is, M * qdd = f(q,qd)). Beyond direct structural damping of the tower and blades, there can be off-diagonal terms in this effective damping matrix associated with the system kinematics. Regardless, I agree with your expression for KTFA_12.

Best regards,