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:
where

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

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

\(\mathrm {\alpha _{2i}\quad }\) Global stiffnessproportional damping factor for frequency \(\mathrm {i}\)
This damping model is the stiffnessproportional term of the general
Rayleigh damping model (see Clough and Penzien (1975)), which in general also
includes a massproportional term. The massproportional term is not
used in VIVANA
as the specified damping level can be obtained for each
response frequency using the stiffnessproportional term alone.
The stiffnessproportional damping factor is found for each response frequency as
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
stiffnessproportional crosssectional 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 stiffnessproportional crosssectional damping
In the model, stiffnessproportional crosssectional 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 stiffnessproportional 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:
where

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

\(\mathrm {\alpha _{2i}\quad }\) Global stiffnessproportional damping factor for frequency \(\mathrm {i}\), given by the relative damping level
RELDAM
specified inVIVANA
. 
\(\mathrm {\boldsymbol{c}_{ij}\quad }\) 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:
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 stiffnessproportional damping factor for frequency \(\mathrm {i}\), given by the relative damping level
RELDAM
specified inVIVANA
. 
\(\mathrm {k_k\quad }\) 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 stiffnessproportional damping factor for all response frequencies. As the global stiffnessproportional 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 stiffnessproportional 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:
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:
where

\(\mathrm {d\quad }\) Damping (energy loss) per cycle per length element
A corresponding stiffnessproportional damping factor
where

\(\mathrm {\omega _i\quad }\) Response frequency \(\mathrm {i}\) in rad/s
The contribution to the system damping matrix for element k is then
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.:
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 * \(\mathrm {n\quad }\) Userspecified 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 stiffnessproportional damping as described above.
1.4. Slip damping
The slip damping is energy loss caused by the hysteresis in a nonlinear momentcurvature curve where the loading and unloading do not follow the same curve. The area enclosed by the momentcurvature 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 momentcurvature relationship is defined using the RambergOsgood model. The four RambergOsgood parameters \(\mathrm {C_Y}\), \(\mathrm {M_Y}\), \(\mathrm {\eta }\) and \(\mathrm {\gamma }\) define the possible momentcurvature 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
The curvature amplitude \(\mathrm {\kappa_0}\) corresponding to a given bending moment amplitude \(\mathrm {M_0}\) can found from the relationship
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:
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 userdefined initial slip curvature \(\mathrm {\kappa_t}\). Above this level the damping will have a smooth transition to the calculated RambergOsgood energy loss.
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 RambergOsgood 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 stiffnessproportional 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
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:
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 inline 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 builtin 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
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 pressuredominated 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 nondimensional frequency is higher than the upper limit for the actual excitation range. The damping force coefficient is given by:
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 nondimensional frequency is lower than the lower limit of the excitation range. The damping force coefficient is given by:
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 builtin 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
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:
\(F_\mathrm {still}\) 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:

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

The damping factor \(R_{sw}\) can be found from the damping coefficient from
\[R_{sw}\dot {x}L=C\dot {x}\quad \mathrm {or}\quad R_{sw}=\frac{C}{L}\] 
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 inline excitation force coefficient curve defined from 3 points, pure IL response in Excitation Force Model), the damping contribution is calculated from
where \(\mathrm {C_e}\) is the actual (negative) CF or IL excitation force coefficient.