Calculation of Response Frequencies
1. Basic assumptions
The purpose of this part of the analysis is to identify all possible response frequencies when taking into account that the flow velocity, diameter, added mass and Strouhal number may vary along the structure. The procedure for identifying the response frequencies applies the nondimensional frequency \(\mathrm {\hat{f}}\) as controlling parameter. Excitation and damping zones related to a specific oscillation frequency can then be defined as shown in Multi Frequency Response.
When a structure becomes excited by vortex shedding in sheared current, added mass will vary along its length, and the response frequency will appear as an eigenfrequency that is influenced by this added mass distribution. However, since added mass depends on the frequency, the correct distribution of added mass can not be found directly. Consistency between the frequency and added mass distribution must therefore be obtained through an iterative process for each response frequency candidate. The starting point for this iteration is a set of eigenfrequencies valid for still water. The information needed to carry out this analysis can be summarised as follows:

\(\mathrm {f_{01},f_{02},\dot sc,f_{0n}\quad }\) Eigenfrequencies in still water

\(\mathrm {U_N(s)\quad }\) Current velocity perpendicular to the structure

\(\mathrm {D_H(s)\quad }\) Hydrodynamic diameter

\(\mathrm {St(s) }\) Strouhal number

\(\mathrm {C_{a0}(s)\quad }\) Added mass coefficient for still water
2. Added mass, CF response
Added mass as a function of the nondimensional frequency must be established. This information is available from forced motion tests of cylinder sections. Such test programs consist of a large set of individual tests. Harmonic motions in CF or IL direction are applied, and the frequency and amplitude are systematically varied. The most elegant way of presenting data from such test programs is to make contour plots in a nondimensional frequency/amplitude plane. The contour lines define combinations of amplitudes and frequencies that have equal value of the added mass coefficient.
The response frequency iteration is based on eigenfrequency calculations, which means that response amplitudes are not taken into account. This is formally incorrect since experiments show that both frequency and amplitude will have an influence on added mass. The approach is still applied in order to simplify the final response calculation, and can be accepted since amplitudes are less important for added mass than frequency. This is illustrated by Contour plot of the CF added mass coefficient based on forced harmonic motions. From Gopalkrishnan (1993) that shows contour lines for CF added mass as presented by Gopalkrishnan (1993). The map shows a plateau to the left with negative added mass, then a steep wall with almost frequency independent transition to another plateau with positive values. This observation makes it possible to eliminate the amplitude as a variable and establish a unique relationship between frequency and added mass that is applied in the frequency iteration.
The CF added mass model in VIVANA is shown on Figure 2. It is found by taking the added mass values from Contour plot of the CF added mass coefficient based on forced harmonic motions. From Gopalkrishnan (1993) at a nondimensional amplitude of \(\mathrm {0.5}\) (confer the horizontal red line on Contour plot of the CF added mass coefficient based on forced harmonic motions. From Gopalkrishnan (1993)). This is a pragmatic decision which may fail for very low and high amplitudes, but is still expected to give reasonable results in most practical cases.
3. Added mass, pure IL response
Figure 3 shows contour lines for IL added mass as presented by Gopalkrishnan (1993). Note that Aronsenâ€™s tests had pure IL motions, which means that these curves are not valid for oscillations with combined IL and CF motions. The curves are seen to have some of the same features as the CF added mass curves in the sense that the contour lines are close to vertical. This means that IL added mass is more sensitive to frequency variations than variations of amplitude, which allows us to apply a simplified added mass model as illustrated in Figure 4. Here added mass is assumed to be defined by the nondimensional frequency, and the actual curve is found in the contour plot for a nondimensional amplitude of \(\mathrm {0.075}\). The red horizontal line on Figure 3 illustrates how this added mass model is established.
On Figure 3, the frequency/amplitude combinations that give \(\mathrm {C_{e,IL}=0}\) are also shown as the broad line.
4. Added mass, IL in combination with CF
It is a common understanding that the dominating IL response frequency always is two times the dominating CF frequency. The CF frequency is found by an iteration that seeks consistency between flow parameters, added mass and the eigenfrequency, see Section 5. Consequently, the IL response frequency should be defined as \(\mathrm {2\omega _{CF}}\), but in general this frequency will not necessarily be an IL eigenfrequency based on still water added mass. Figure 5 illustrates this situation. By adjusting the IL added mass, both \(\mathrm {\omega _{j,IL}}\) and \(\mathrm {\omega _{j+1,IL}}\) can be adjusted to match the required frequency. It is, however, not obvious which of the two candidates that should be selected.
Figure 6 shows added mass in IL and CF direction found from forced motions of a rigid cylinder. The motions are observed trajectories for cross sections on a flexible beam. The experiments are described by Gopalkrishnan (1993). The plot illustrates that the IL and CF added mass are not likely to be reduced from still water values (\(\mathrm {1.0}\)) at the same position. IL added mass is also seen to be larger than \(\mathrm {1.0}\) in most cases. By increasing IL added mass \(\mathrm {\omega _{j+1,IL}}\) will move in the direction of \(\mathrm {2\omega _{i,CF}}\) on Figure 5. This is in fact a conservative approach since stresses from vibrations that are dominated by mode \(\mathrm {j+1}\) will be higher than stresses from mode \(\mathrm {j}\). The general approach for selecting the IL response frequency is therefore to adjust added mass so that \(\mathrm {\omega _{j+1,IL}}\) becomes equal to \(\mathrm {2\omega _{i,CF}}\).
5. Iteration scheme.
The initial situation is illustrated in Figure 7.
For each possibly active eigenfrequency, \(\mathrm {i}\), the following iteration is carried out:

Assume that the response frequency is identical to the still water eigenfrequency:
\[f_{\mathrm {osc},i}^k=f_{0i}^k\]\(\mathrm {k}\) is the iteration step; here \(\mathrm {k=1}\), meaning that \(\mathrm {f_{0i}^1}\) is the initial eigenfrequency \(\mathrm {i}\) based on user defined added mass given in the
xxx_inpmod.inp
file. 
Calculate the corrected nondimensional frequency along the riser (See Appendix A):
\[\displaystyle \hat{f}_{c,i}^k(s)=\frac{f_{\mathrm {osc},i}^kD_H(s)}{U_N(s)}\frac{\mathrm {St}_E}{\mathrm {St}(s)}\] 
Find the added mass coefficient along the riser from builtin or user defined added mass curves (Figure 2 for CF and Figure 4 for IL response), recalculate the added mass matrix and total mass matrix.

Solve the eigenvalue problem with the new mass distribution and identify the wanted eigenfrequency \(\mathrm {f_{0i}^{k+1}}\).

Convergence test:
\[f_{0i}^{k+1}f_{0i}^k\leq\varepsilon\]where \(\varepsilon\) is a builtin criterion for convergence.

If the test fails, go to 1. If the test is satisfied, go to the next step.

Accept \(\mathrm {f_{0i}^{k+1}}\) as a possible response frequency \(f_{\mathrm {osc},i}\).

Calculate the corrected nondimensional frequency along the riser from:
\[\hat{f}_{c,i}(s)=\frac{f_{\mathrm {osc},i}D_H(s)}{U_N(s)}\frac{\mathrm {St}_E}{\mathrm {St}(s)}\]
The end result of these iterations is a set of possible response frequencies, associated added mass distributions and the variation of the corrected nondimensional frequency along the structure for each frequency. This information makes it possible to identify active response frequencies since excitation requires values of \(\mathrm {\hat{f}_{c,i}(s)}\) within an interval that gives positive excitation coefficients. Further details will be given in Multi Frequency Response.