Structural and Hydrodynamic Damping

1. Structural damping

1.1. Global relative damping

The relative structural damping level is specified by the user. This gives the global damping level relative to the critical damping level. This damping is intended to be used to describe internal damping in the structure due to friction and local strains. The damping will be applied for each response frequency as a damping matrix proportional to the system stiffness matrix:

\[\boldsymbol{C}_i=\alpha _{2i}\boldsymbol{K}\]

where

  • \(\mathrm {\boldsymbol{C}_i\quad }\) Damping matrix for frequency \(\mathrm {i}\)

  • \(\mathrm {\boldsymbol{K}\quad }\) system Stiffness matrix

  • \(\mathrm {\alpha _{2i}\quad }\) Global stiffness-proportional damping factor for frequency \(\mathrm {i}\)

This damping model is the stiffness-proportional term of the general Rayleigh damping model (see Clough and Penzien (1975)), which in general also includes a mass-proportional term. The mass-proportional term is not used in VIVANA as the specified damping level can be obtained for each response frequency using the stiffness-proportional term alone.

The stiffness-proportional damping factor is found for each response frequency as

\[\alpha _{2i}=\frac{2\xi }{\omega _i}\]

where

  • \(\mathrm {\xi \quad }\) User defined structural damping ratio

  • \(\mathrm {\omega _i\quad }\) Response frequency \(\mathrm {i}\) in rad/s

This gives the same contribution to the relative damping level for all response frequencies.

1.2. Additional damping contributions specified in the model

Additional damping contributions from global seafloor contact and stiffness-proportional cross-sectional damping will be included if they are specified in the model and the damping is larger than the damping level specified in VIVANA. No change is made if the damping is smaller than the damping level specified in VIVANA, i.e. the damping is never reduced below the relative damping level specified in VIVANA.

1.2.1. Additional stiffness-proportional cross-sectional damping

In the model, stiffness-proportional cross-sectional damping may be specified with different factors for the axial, bending and torsional stiffnesses. In the present version of VIVANA, however, these must be identical as the stiffness-proportional damping factor is currently applied to the total element stiffness matrix; i.e. the sum of the axial, bending and torsional material stiffnesses and the geometric stiffness due to the effective tension of the element. This is gives the additional damping contribution:

\[\begin{array}{l} \text{for }\alpha _{2j}>\alpha _{2i}:\quad \boldsymbol{c}_{ij}=(\alpha _{2j}-\alpha _{2i})\boldsymbol{k}_j \\ \text{for }\alpha _{2j}\le\alpha _{2i}:\quad \boldsymbol{c}_{ij}=\boldsymbol{0} \end{array}\]

where

  • \(\mathrm {\alpha _{2j}\quad }\) stiffness-proportional damping factor for element \(\mathrm {j}\), part of the cross section input given in INPMOD

  • \(\mathrm {\alpha _{2i}\quad }\) Global stiffness-proportional damping factor for frequency \(\mathrm {i}\), given by the relative damping level RELDAM specified in VIVANA.

  • \(\mathrm {\boldsymbol{c}_{ij}\quad }\) Additional damping for frequency \(\mathrm {i}\) from element \(\mathrm {j}\)

  • \(\mathrm {\boldsymbol{k}_j\quad }\) Total element stiffness matrix for element \(\mathrm {j}\)

1.2.2. Additional seafloor damping

Specified seafloor damping may give additional damping contributions if the seafloor contact is specified on a global level. Damping specified using seafloor contact components will currently not give additional damping.

Seafloor contact specified on a global level consists of stiffness and damping per element length in the local normal, axial and lateral directions. The friction coefficients in the local axial and lateral directions are not used in VIVANA. The additional damping contribution at each end of an element with contact is found as:

\[\begin{array}{l} \text{for }d_k>\alpha _{2i} k_k:\quad c_{ijk}=(d_k-\alpha _{2i} k_k)l_j \\ \text{for }d_k\le\alpha _{2i} k_k:\quad c_{ijk}=0 \end{array}\]

where

  • \(\mathrm {d_k\quad }\) Seafloor damping per unit length for local direction \(\mathrm {k}\), part of the seafloor contact input given in INPMOD

  • \(\mathrm {\alpha _{2i}\quad }\) Global stiffness-proportional damping factor for frequency \(\mathrm {i}\), given by the relative damping level RELDAM specified in VIVANA.

  • \(\mathrm {k_k\quad }\) Seafloor stiffness per unit length for local direction \(\mathrm {k}\), part of the seafloor contact input given in INPMOD

  • \(\mathrm {c_{ijk}\quad }\) Additional damping for frequency \(\mathrm {i}\) from node \(\mathrm {j}\) in local direction \(\mathrm {k}\)

  • \(\mathrm {l_j\quad }\) Half element length at node \(\mathrm {j}\)

The seafloor damping corresponds to a constant stiffness-proportional damping factor for all response frequencies. As the global stiffness-proportional damping factor varies between response frequencies, additional seafloor damping may be added for some response frequencies and not for others.

1.3. Detailed structural damping specified in VIVANA

In addition to the structural damping described above, additional structural damping can be specified for part or all of the structure. This damping can include both material and slip damping contributions and will in general be dependent on the response. In the present version, the additional structural damping is assumed to be only due to the dynamic curvatures. As the damping is dependent on the response, the system damping matrix must be modified during the response iterations.

The element contributions to the system damping matrix are proportional to the element’s total stiffness matrix; i.e. axial, bending, torsional and geometric stiffnesses are all included.

At the start of a response iteration the following is done for each element with additional structural damping: - calculate the average curvature amplitude for the element - find the additional material and slip damping contributions - calculate the elastic energy from the initial bending stiffness given in INPMOD and the curvature amplitude - estimate relative damping from the damping and the elastic energy - calculate the stiffness-proportional damping factor, \(\mathrm {\alpha _{2j}}\), for this relative damping and response frequency - add the element damping contribution as \(\mathrm {\alpha _{2j}\boldsymbol{k_j}}\), where \(\mathrm {\boldsymbol{k_j}}\) is the total stiffness matrix for element j.

The elastic energy per element length corresponding to this curvature amplitude is calculated as:

\[e_{elast}=0.5\quad EI\quad (\Delta \kappa)^2\]

where

  • \(\mathrm {e_{elast}\quad }\) Elastic energy per element length

  • \(\mathrm {EI\quad }\) Bending stiffness of the element

  • \(\mathrm {\Delta \kappa\quad }\) Curvature amplitude

The relative damping level \(\mathrm {\xi }\) for this element is estimated as:

\[\xi =\frac{d}{4\pi \quad e_{elast}}\]

where

  • \(\mathrm {d\quad }\) Damping (energy loss) per cycle per length element

A corresponding stiffness-proportional damping factor

\[\alpha _2}] is then found as latexmath:[\mathrm {\alpha _2=\frac{2\xi }{\omega _i}\]

where

  • \(\mathrm {\omega _i\quad }\) Response frequency \(\mathrm {i}\) in rad/s

The contribution to the system damping matrix for element k is then

\[\boldsymbol{c}_k=\alpha _2\boldsymbol{k}_k\]

where

  • \(\mathrm {\boldsymbol{c}_k\quad }\) Additional damping contribution from element l

  • \(\mathrm {\boldsymbol{k}_k\quad }\) Total element stiffness matrix for element k

1.3.1. Material damping

The material damping is given as a table of energy loss per cycle for a matrix of static effective tension and curvature amplitude values. The specified exponent, n, is used in the interpolation between the given values. n is often a material property. The input table of damping values are normalized by dividing by the corresponding curvature values to the nth power.:

\[d_{norm}=\frac{d}{(\Delta \kappa)^n}\]

where * \(\mathrm {d_{norm}\quad }\) Normalized damping per cycle per length element * \(\mathrm {d\quad }\) Damping per cycle per length element * \(\mathrm {\Delta \kappa\quad }\) Curvature amplitude * \(\mathrm {n\quad }\) User-specified exponent for interpolation

Damping at intermediate tension / curvature values are found using linear interpolation on the normalized damping. Flat extrapolation is used outside of the ranges of effective tension and curvature amplitude values. The resulting material damping is then found by multiplying the normalized value found by interpolation / extrapolation by the curvature to the nth power.

The resulting material damping may then scaled to account for frequency and / or temperature effects.

The resulting material damping is then included in the system damping matrix as stiffness-proportional damping as described above.

1.4. Slip damping

The slip damping is energy loss caused by the hysteresis in a nonlinear moment-curvature curve where the loading and unloading do not follow the same curve. The area enclosed by the moment-curvature lop corresponds to the dissipated energy during a cycle, and is thus the energy loss per element length and cycle for the given curvature amplitude.

The nonlinear moment-curvature relationship is defined using the Ramberg-Osgood model. The four Ramberg-Osgood parameters \(\mathrm {C_Y}\), \(\mathrm {M_Y}\), \(\mathrm {\eta }\) and \(\mathrm {\gamma }\) define the possible moment-curvature loops for given conditions. The parameters can be given for different statuc effective tension values as well as varying between segments in the model.

The curvature, \(\mathrm {\kappa}\), and bending moment, \(\mathrm {M}\), follow the initial loading curve

\[\frac{\kappa}{C_y}=\frac{M}{M_y}[1+\eta (\frac{M}{M_y})^{(\gamma -1)}]\]

The curvature amplitude \(\mathrm {\kappa_0}\) corresponding to a given bending moment amplitude \(\mathrm {M_0}\) can found from the relationship

\[\frac{\kappa_0}{C_y}=\frac{M_0}{M_y}[1+\eta (\frac{M_0}{M_y})^{(\gamma -1})]\]

To go in the opposite direction; i.e. from the curvature amplitude \(\mathrm {\kappa_0}\) to the bending moment amplitude \(\mathrm {M_0}\); either iterations or interpolation between precalculated values may be used. Interpolation is used in the in the present implementation.

The energy loss per element length for a given curvature amplitude can then be calculated as:

\begin{eqnarray} d(\kappa_0,M_0)=4\quad M_0\quad \kappa_0(1-\lambda)\frac{\gamma -1}{\gamma +1} \end{eqnarray}

where \(\mathrm {\lambda=\frac{M_0\quad C_y}{M_y\quad \kappa_0}}\)

This equation will give slip damping contributions at all curvature levels, even at very small levels. In order to have an initial stick region, the value is corrected to be zero up to the user-defined initial slip curvature \(\mathrm {\kappa_t}\). Above this level the damping will have a smooth transition to the calculated Ramberg-Osgood energy loss.

\begin{eqnarray} d_{slip}(\kappa_0,M_0)=0\quad \quad for\quad \kappa_0\le\kappa_t\\ d_{slip}(\kappa_0,M_0)=d(\kappa_0,M_0)-d(\kappa_t,M_t)\frac{\kappa_t}{\kappa_0}\quad \quad for\quad \kappa_0>\kappa_t \end{eqnarray}

where

  • \(\mathrm {d_{slip}(\kappa_0,M_0)\quad }\) Damping per cycle per length element for for curvature amplitude \(\mathrm {\kappa_0}\) and moment amplitude \(\mathrm {M_0}\)

  • \(\mathrm {\kappa_t\quad }\) Slip transition curvature

  • \(\mathrm {d(\kappa_0,M_0)\quad }\) Energy loss per cycle per length element from [tm_eq_10_12].

For static effective tension between the ones the Ramberg-Osgood parameters are specified for, the damping is found by interpolation between the values found for the given curvature amplitude at the closest tension levels.

The resulting slip damping is then included in the system damping matrix as stiffness-proportional damping as described previously.

2. Hydrodynamic damping outside the excitation zone

Hydrodynamic damping is always defined by use of a distributed damping coefficient \(\mathrm {R}\) with unit \(\mathrm {kg\,m^{-1}\,s^{-1}}\) (or \(\mathrm {Nm^{-2}s}\)). Outside the excitation regions, the damping force \(\mathrm {F^D}\) on an element with length \(\mathrm {\Delta L}\) is found by

\[F^D=R\dot {u}\Delta L\]

where \(\mathrm {\dot {u}}\) is the response velocity in CF or IL direction

The damping coefficient for translation degrees of freedom in the damping matrix is found from standard FEM theory:

\[c_{ij}=\int_LR(x)N_i(x)N_j(x)\mathrm {d}x\]

where \(\mathrm {N}\) denotes the shape functions for the element and indexes \(\mathrm {i}\) and \(\mathrm {j}\) refers to local degrees of freedom in the element damping matrix.

The damping coefficient \(\mathrm {R}\) can be modelled in two different ways in VIVANA. One option is to use Venugopal’s model for low and high flow velocity and still water, see Clough and Penzien (1975). This model is strictly valid for circular smooth cross section only. The other possibility is to define data for the excitation force coefficient outside the excitation zone in combination with a general model for still water damping. It should also be noted that hydrodynamic damping may take place within the excitation zone if the response amplitude becomes larger than \(\mathrm {(A/D)_{C_e=0}}\), see The CF excitation force coefficient curve defined from three points and The in-line excitation force coefficient curve defined from 3 points, pure IL response in Excitation Force Model. The three damping models will be briefly described in the following.

2.1. Venugopal Damping Model

VIVANA uses the damping model proposed by Clough and Penzien (1975) as the default model. This model was verified by Clough and Penzien (1975). The model defines three coefficients, valid for still water, low velocity and high velocity zones, respectively. VIVANA has built-in values for these parameters, but the user is allowed to define her own values if such information is available for the actual cross sections.

2.1.1. Damping in still water

This damping model is used if the normal velocity component is zero. The damping force coefficient is defined by

\[R_{sw}=\frac{\omega \pi \rho D_H^2}{2}[\frac{2\sqrt{2}}{\sqrt{\mathrm {Re}_\omega }}+k(\frac{x_0}{D_H})^2]\]

where \(\mathrm {Re}_\omega =\omega D_H^2/v\).

The first part corresponds to skin friction according to Stoke’s law. The second part is the pressure-dominated force. The factor \(\mathrm {k}\) is found from curve fitting of empirical data to be \(\mathrm {0.25}\).

2.1.2. Damping in low velocity regions

This damping model is used outside the excitation zone where the non-dimensional frequency is higher than the upper limit for the actual excitation range. The damping force coefficient is given by:

\[R_{lv}=R_{sw}+\frac{1}{2}\rho D_HU_NC_{vl}\]

The damping is increasing linearly with respect to the incident flow velocity. The coefficient \(\mathrm {C_{vl}}\) is found to be \(\mathrm {0.36}\) based on measurements.

2.1.3. Damping in high reduced velocity regions

This damping model is used outside the excitation zone where the non-dimensional frequency is lower than the lower limit of the excitation range. The damping force coefficient is given by:

\[R_{hv}=\frac{1}{2}\rho \frac{U_N^2}{\omega }C_{vh}\]

This model is independent of the amplitude ratio. The coefficient \(\mathrm {C_{vh}}\) is found to be \(\mathrm {0.4}\) based on measurements.

2.2. Damping combined with user specified excitation coefficients

All built-in coefficients in VIVANA are valid for circular cross sections only. If structures with helical strakes or fairings should be analysed, hydrodynamic coefficients for the actual cross section geometries must be specified. The most convenient way to define damping, is to refer directly to the data for user defined excitation coefficients. The data should include negative values for amplitude and frequency combinations that can cover ordinary VIV situations. Typical excitation coefficient curves for helical strakes, general model in Excitation Force Model illustrates how such data may look for helical strakes that will give damping for almost all oscillation conditions.

The distributed damping coefficient in Equation (10) can be calculated from the (negative) excitation coefficient by

\[R_{cl}=-\frac{\rho D_HU_N^2C_e}{2\omega A}\]

where \(\mathrm {C_e}\) is the excitation force coefficient.

Damping in still water can not be found from Equation (9) since \(\mathrm {U_N}\) will be zero. A general formulation for \(\mathrm {R_{sw}}\) is therefore applied:

\[R_{sw}=\frac{\omega \pi \rho D_H^2}{2}(1+(\frac{A}{D_H})^2)F_\mathrm {still}\]

\(F_\mathrm {still}\) is a scaling factor that needs to be specified as input. If a standard decay test is carried out for a cylinder with length \(\mathrm {L}\), data can be used as follows:

  1. The damping coefficient \(C\) can be calculated directly from the measurements by using the logarithmic decrement and standard equations for damped free oscillations

  2. The damping factor \(R_{sw}\) can be found from the damping coefficient from

    \[R_{sw}\dot {x}L=C\dot {x}\quad \mathrm {or}\quad R_{sw}=\frac{C}{L}\]

  3. The input scaling factor \(F_\mathrm {still}\) can be calculated form

    \[F_\mathrm {still}=\frac{R_{sw}}{\displaystyle \frac{\omega \pi \rho D^2}{2}(1+(\frac{A}{D})^2)}\]

For further details about this damping model, see Clough and Penzien (1975).

3. Hydrodynamic damping in the excitation zone

3.1. Damping for large response amplitudes

If an element is within the excitation zone and the excitation coefficient is negative (normally because of large amplitude, confer The CF excitation force coefficient curve defined from three points and The in-line excitation force coefficient curve defined from 3 points, pure IL response in Excitation Force Model), the damping contribution is calculated from

\[R_{C_e}=-\frac{\rho D_HU_N^2C_e}{2\omega A}\]

where \(\mathrm {C_e}\) is the actual (negative) CF or IL excitation force coefficient.