Multi Frequency Response

1. Excitation zone

For a given response frequency candidate, \(f_{\mathrm {osc},i}\), one can calculate the non-dimensional frequency along the structure by

\[\hat{f}_i(s)=\frac{f_{\mathrm {osc},i}D(s)}{U_N(s)}\]

where \(\mathrm {s}\) is a coordinate along the structure and \(\mathrm {U_N}\) is the flow velocity perpendicular to the structure. From the tests by Gopalkrishnan (1993) with forced CF motions of a rigid cylinder we can see that a positive excitation coefficient is found in the interval \(\mathrm {0.125-0.3}\) for the non-dimensional frequency, see Figure 1. Note that this default range for excitation is larger than what was applied in the earlier versions of VIVANA; version 3.6 and lower had the interval \(\mathrm {0.125-0.2}\) as default.

tm excitation contour Gopalkrishnan CF
Figure 1. lot of contour curves for CF excitation coefficient in an amplitude / frequency map, from Gopalkrishnan (1993)

This interval is applied to define an excitation zone along the structure for each response frequency candidate. [tm_overlapping_excitation_zones] shows how such zones will be defined for a vertical riser with uniform cross section in sheared current. Each frequency will have its own unique zone, and these zones may overlap.

Overlapping zones imply that the vortex shedding process may simultaneously act at more than one frequency at a specific position along the structure. This is in contrast to what normally is observed for VIV of slender structures: The response is almost always dominated by one frequency even if there are many response frequency candidates. From experience it is therefore reasonable to assume that one frequency among the set of possible response frequencies will dominate, and hence occupy its entire excitation zone. However, even if one frequency will dominate, other frequencies may control vortex shedding outside the excitation zone for the dominating frequency. Consequently there is a need for ranking of the response frequency candidates and defining how their excitation zones should be determined in order to avoid overlaps.

Overlapping excitation zones on riser with uniform cross section and exposed to sheared current.

In order to define the priority among all possibly acting frequencies, an excitation parameter based on energy considerations has been defined:

\[E_i=\int_{L_{e,i}}U_N^3(s)D_H^2(s)(\frac{A}{D})_{C_e=0}\mathrm {d}s\]

The integral for the excitation parameter is taken over the excitation zone for each frequency, \(\mathrm {L_{e,i}}\). The parameter \(\mathrm {\displaystyle (\frac{A}{D})_{C_e=0}}\) is the non-dimensional amplitude where the excitation coefficient shifts from positive to negative value, see The CF excitation force coefficient curve defined from three points. This parameter is needed for ranking cases with uniform current since all actual frequencies will have the same normal velocity \(\mathrm {U_N}\) and hydrodynamic diameter \(\mathrm {D_H}\). The result of this calculation is that the response frequency candidates can be ranked according to the magnitude of the excitation parameter. The frequency with the largest \(\mathrm {E_i}\) is referred to as ``the dominating response frequency''.

The interpretation and use of the excitation zones \(\mathrm {L_{e,i}}\) and parameters \(\mathrm {E_i}\) has two different options in VIVANA depending on the assumption for multiple frequency response. The two assumptions are:

  • simultaneously acting frequencies

  • time sharing between frequencies

Use of the parameters will be described in the following sections.

2. Simultaneously acting frequencies

The vortex shedding process at a specific position on the structure can not take place at several frequencies at the same time. Hence, if excitation zones for various frequencies overlap, there is need for a rule to decide the frequency that actually will become active.

Observations from laboratory experiments with flexible beams indicate that even if several response frequencies could become active, only one frequency will be seen (Gopalkrishnan (1993)). This is the background for allowing the dominating frequency to take its total excitation zone, while other frequencies will take their zones from segments along the structure that have not been taken by frequencies with higher priority. This leads to the following steps in the procedure for identification of frequencies and excitation zones for the ``simultaneously acting frequency'' method:

  1. Calculation of the excitation parameter for all possible response frequencies.

  2. The frequency with the largest excitation parameter becomes the primary response frequency and its total excitation zone will be allocated to this frequency.

  3. Other frequencies will have their zones reduced by sections that have been occupied by frequencies with higher excitation parameters.

  4. Response is calculated for all frequencies without any consideration of possible interaction effects.

  5. Response frequencies with a maximum response amplitude less than \(\mathrm {1~\%}\) of the minimum diameter will be omitted in the fatigue analysis.

  6. Fatigue damage is calculated by rainflow counting of time histories for stresses generated by linear superposition of the contributions from each active frequency. Interaction between frequencies is hence disregarded. See also Fatigue Analysis.

Figure 2 illustrates how the excitation zones for frequency 1, 2 and 3 will be defined when this method is applied.

tm excitation zones concurrent
Figure 2. Excitation zones for simultaneous response frequencies.

The user may overrule this way of ranking frequencies by specifying which frequency that should be selected as the primary frequency for a specific case. This should normally not be done in a design analysis. The option is present in order to perform a sensitivity study for fatigue calculation.

3. Time sharing

Time sharing is assumed to appear as a sequence of single frequency responses as illustrated in Figure 3. The relative duration of these responses is found by using the same excitation parameters as used to rank the response frequencies when the "simultaneously acting frequencies" approach is applied. The process is illustrated in Figure 3.

tm time shearing process
Figure 3. Illustration of the time shearing process.

Formally the excitation zones will be allowed to overlap, but they are not simultaneously active. Within the time period a specific frequency keeps control, this frequency will have its entire excitation zone. The zones will hence follow the definition shown in Figure 1.

The total time \(\mathrm {T_i}\) dedicated to response frequency \(\mathrm {i}\) during a period \(\mathrm {T}\) will hence be given by

\[T_i=T\frac{E_i}{\displaystyle \sum_{n=1}^kE_n}\]

where \(\mathrm {E_n}\) is the excitation parameter for frequency \(\mathrm {n}\) given by Equation (2), and \(\mathrm {k}\) is the total number of response frequencies.

Note that this method does not describe how often the response will shift from one frequency to another, or why the shift takes place. It is, however, possible to calculate the fatigue damage from this type of response by using the relative duration as described. The relevance of this way of defining time shearing has so far not been verified by analysis of measured response. This method may hence be modified in the future.