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Rayleigh damping |
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Rayleigh damping |
Rayleigh damping is used to model structural damping for lines and turbine blades. It is only available under the implicit integration scheme.
| Warning: | Rayleigh damping can easily be misused – please refer to our guidance to avoid this. |
Classical Rayleigh damping uses a system damping matrix $\C$ defined as \begin{equation} \C = \mu \M + \l{} \K{} \end{equation} where
$\mu$ is the mass proportional Rayleigh damping coefficient
$\l{}$ is the stiffness proportional Rayleigh damping coefficient
$\M$ is the system structural mass matrix
$\K{}$ is the system structural stiffness matrix
With this formulation the damping ratio is the same for axial, bending and torsional response.
Classical Rayleigh damping results in different damping ratios for different response frequencies, according to the equation \begin{equation} \xi = \frac12 \left(\frac{\mu}{\omega} + \l{} \omega \right) \end{equation} where
$\xi$ is the damping ratio (a value of 1 corresponds to critical damping)
$\omega$ is the response frequency in rad/s.
It can be seen from this that the mass proportional term gives a damping ratio inversely proportional to response frequency and the stiffness proportional term gives a damping ratio linearly proportional to response frequency.
In addition to classical Rayleigh damping, OrcaFlex offers a separated Rayleigh damping model in which $\C$ is defined as \begin{equation} \C = \mu \M + \l{a} \K{a} + \l{b} \K{b} + \l{t} \K{t} \end{equation} where
$\l{a}$ is the stiffness proportional Rayleigh damping coefficient for axial deformation
$\l{b}$ is the stiffness proportional Rayleigh damping coefficient for bending deformation
$\l{t}$ is the stiffness proportional Rayleigh damping coefficient for torsional deformation
$\K{a}$ is the system structural stiffness matrix corresponding to axial deformation
$\K{b}$ is the system structural stiffness matrix corresponding to bending deformation
$\K{t}$ is the system structural stiffness matrix corresponding to torsional deformation
Note that $\K{} = \K{a} + \K{b} + \K{t}$.
The separated model allows for different damping ratio in axial, bending and torsional response.
| Note: | Rayleigh damping is not applied to any line segment whose unstretched length is actively varying because of line feeding. This is because the Rayleigh damping tends to work against the feeding and can cause the simulation to become unstable. Rayleigh damping will be applied to all line segments once they are fully grown to their reference unstretched lengths. |
Each line type and turbine has structural damping properties determined by a named Rayleigh damping coefficient data set. Multiple such data sets can be defined, each with the following data.
OrcaFlex offers four distinct methods for specifying the Rayleigh damping coefficients:
These two modes allow you to define structural damping in terms of % critical damping ratio. In addition you need to give response periods at which the damping ratio will be achieved. For stiffness proportional damping only one response period can be specified.
If response period 1 is '~', then OrcaFlex will choose response periods automatically, based on the wave period (for regular waves) or on peak period $\Tp$ (for irregular waves). This method is generally recommended; it is particularly useful if you are running a batch of cases with varying wave conditions.
In these two modes, OrcaFlex uses the classical Rayleigh damping model and reports the values of the mass and stiffness proportional coefficients $\mu$ and $\l{}$.
In these modes, you specify the damping coefficients (as defined above for the two different modes) directly.
This controls how the geometric stiffness for bending deformation is handled. If it is false, the stiffness matrix $\K{}$ (for classical) and $\K{b}$ (for separated) is calculated excluding the geometric stiffness terms.
This option caters both for scenarios in which it is desirable to apply damping coefficients to geometric stiffness terms, and for others where that is not the case.
The damping ratio graph plots the damping ratio that will be achieved for a range of response frequencies. You may choose between period and frequency scales for the graph's $x$-axis.