# How to use Rayleigh damping

Ashes offers the possibility to use Rayleigh damping to dampen out the response of the structure. The Rayleigh damping characteristics can be changed in the Damping section of the Analysis tab of the Analysis parameters window.
Rayleigh damping is implemented following the equations given in the Damping section. This produces a damping ratio equal to

$$\xi = 0.5\left(\frac{\mu}{2\pi f}+\lambda2\pi f\right)$$
where
$$\xi$$
is the damping ratio,
$$f$$
is the frequency of the response and
$$\mu$$
and
$$\lambda$$
are the mass and stiffness damping coefficients, respectively. This method produces a damping ratio that is dependent on the frequency of the response. Given that there are 2 coefficients that can be input by the user, it is possible to select those coefficients so that 2 frequencies correspond to 2 damping ratios selected by the user. It is common to specify the damping ratios of the first two eigenmodes of the structure, which yields a system of two equations where the unknowns are
$$\mu$$
and
$$\lambda$$
, as follows:
$$\xi_1=0.5\left(\frac{\mu}{2\pi f_1} + \lambda 2\pi f_1\right)$$

$$\xi_2=0.5\left(\frac{\mu}{2\pi f_2}+\lambda 2\pi f_2\right)$$

where
$$\xi_1$$
and
$$\xi_2$$
are the damping ratios of the 1st and 2nd mode shapes, respectively and
$$f_1$$
and
$$f_2$$
are the 1st and 2nd eigenfrequencies, respectvely. For example, within the OC5 project, Robertson et al. (2016b) reports the following characteristics for the support structure of the NREL 5 MW turbine:

 Eigenfrequency [Hz] Damping ratio [-] 1st mode 0.28 0.017 2nd mode 2.0 0.027

In order to achieve these characteristics, the damping ratios and eigenfrequencies are input into the system of two equations. The mass and stiffness coefficients that produce these characteristics are
$$\mu=0.047$$
and
$$\lambda=0.004$$
. This is illustrated in the figure below: Note that with this method, it is not possible to specify the damping ratios of more that 2 frequencies. For example, in this case, the damping ratio at 3 Hz will be close to 4%, and it is not possible to modify that without changing the damping ratios of the first two modes.

Mass damping will dampen out rigid body motions. For rotating blades for example, which undergo rigid body motions, this means that using mass proportional damping will artificially dampen out the motion of the blades, leading to lower power output than expected. For these cases, it is recommended not to use mass proportional damping. This is discussed in section 103 of DNV-OS-F201 (2010)

Ashes offers different options to define your damping characteristics based on the input you have. These are defined by the Structural damping mode and the Damping input parameters in the Damping section of the Analysis tab in the Analysis parameters window.

## 1 Rayleigh damping defined by the structural characteristics

If you know the eigenfrequencies and the targeted damping ratios of your structure, you can input them so that Ashes solves the system of equations and determines the corresponding damping coefficients. This is obtained by setting the Structural damping mode parameter to Rayleigh and the Damping input parameter to Damping ratios.

Note: the mass and stiffness damping coefficients thus obtained can be found in the Model Information dialog

## 2 Rayleigh damping defined by the mass and stiffness damping coefficients

If you know the mass and stiffness damping coefficients, that is
$$\mu$$
and
$$\lambda$$
, you can input them directly into Ashes. This is obtained by setting the Structural damping mode parameter to Rayleigh and the Damping input parameter to Explicit coefficient(s).

Note: in this case, you can still input periods. These will not be used to compute the damping, i.e. they will not influence the simulation in any way. They are only used to show you relevant damping information in the Model Information dialog.

## 3 Stiffness proportional only

This mode can be used when the mass proportional part of rayleigh damping should be neglected. This can be obtained by setting the Structural damping mode parameter to Stiffness proportional, and is equivalent to having Rayleigh damping with the mass damping coefficient
$$\mu=0$$
. As for Rayleigh damping, it is possible to explicitely set the stiffness coefficient or to set the requested damping ratio at a given period.

Note: in this case, you can still input a period. This will not be used to compute the damping, i.e. it will not infuence the simulation in any way. It is only used to show you relevant damping information in the Model Information dialog.

## 4 Mass proportional only

This mode can be used when the stiffness proportional part of rayleigh damping should be neglected. This can be obtained by setting the Structural damping mode parameter to Mass proportional, and is equivalent to having Rayleigh damping with the stiffness damping coefficient
$$\lambda=0$$
. As for Rayleigh damping, it is possible to explicitely set the mass coefficient or to set the requested damping ratio at a given period.

Note: in this case, you can still input a period. This will not be used to compute the damping, i.e. it will not influence the simulation in any way. It is only used to show you relevant damping information in the Model Information dialog.