Excitation Force Model
1. Basic assumptions
The excitation force in CF and IL direction at a given position on the structure is defined as the component of the hydrodynamic force that is in phase with the response velocity at the same position and for the CF and IL direction respectively. The force on an element with length \(\mathrm {\Delta L}\) is given by
The frequency of the force in CF direction follows the vortex shedding frequency, while the IL force acts at the double frequency.
The excitation force coefficient \(\mathrm {C_{e,CF/IL}}\) is a function of the nondimensional amplitude \(\mathrm {A_{CF/IL}/D_H}\) and the corrected nondimensional frequency \(\mathrm {\hat{f}_{c,CF/IL}}\). Note that for the same response condition \(\mathrm {\hat{f}_{c,IL}}\) will be identical to \(\mathrm {\hat{f}_{c,CF}}\) even if the IL response takes place at twice the CF frequency.
It is well known that VIV is a selflimiting process, which is easy to understand by realizing that \(\mathrm {C_e}\) becomes negative if the response amplitude exceeds a certain value. \(\mathrm {F_e}\) will hence change from being an excitation force to become a damping force.
A complete excitation force model consists of

data for the excitation force coefficient, which makes it possible to find vales of \(\mathrm {C_{e,CF/IL}}\) for combinations of response amplitudes and frequencies.

the bandwidth for the nondimensional frequency where the excitation force coefficients should be applied. Outside this range an empirical damping model will be used to find the force in phase with the velocity, and this force will always give damping.
The bandwidth is referred to as "the excitation range", and the part of the structure where the nondimensional frequency falls within this range is referred to as "the excitation zone", see also Excitation zone. Figure 1 illustrates this concept for a vertical riser that has a buoyancy zone with increased hydrodynamic diameter. The excitation zones for the lowest response frequency are indicated. Note that damping may still take place within the excitation zone if the response amplitude becomes so large that the excitation force coefficient becomes negative.
Excitation force coefficients, excitation range and damping coefficients are defined as default values in the program, but these are valid for a smooth circular cylinder. The user is therefore allowed to specify her own data, which in particular is important to account for helical strakes or cross sections with satellite pipes as found on drilling risers. The following rules apply:

Use of builtin values of \(\mathrm {C_{e,CF/IL}}\) and default values for the excitation range:

Builtin or user specified damping coefficients will be used outside the excitation range


User specified values for the excitation range:

The user specified range should not exceed the valid range for builtin or given coefficient data. If a larger range is wanted, the program will redefine the range and apply the actual damping model (default or user specified) outside the modified range. A warning will be printed.


User specified excitation force coefficients:

The program will check that data for coefficients (user specified or builtin) are given for the desired range. If not, the range will be redefined to the maximum range of actual data, and the damping model will be applied outside this range. A warning will be printed.

Note that the program allows use of different concepts and sets of curves for each segment along the structure.
2. Default excitation coefficient model, CF response
The default excitation coefficient model for crossflow VIV in VIVANA is based on the coefficients found by Gopalkrishnan (1993). However, some modifications to these curves were made in order to include the experience from flexible beam tests. Gopalkrishnan’s map of the excitation coefficient in a nondimensional amplitude and frequency plane is shown on Plot of contour curves for CF excitation coefficient in an amplitude / frequency map, from Gopalkrishnan (1993).
Instead of defining the excitation force coefficients as a twoparameter function of amplitude and frequency, VIVANA’s builtin model applies a set of parameters that defines the coefficient as a function of the amplitude. The parameters are in turn given as functions of the frequency. This is done in order to make it easy to modify the model on the basis of new information without being able to define the complete set of data as shown on Plot of contour curves for CF excitation coefficient in an amplitude / frequency map, from Gopalkrishnan (1993).
Figure 2 shows how the CF excitation coefficient is defined as a function of the nondimensional response amplitude for a given nondimensional response frequency. The curve is assumed to have a maximum value (horizontal tangent) at \(\mathrm {B}\), meaning that \(\mathrm {AB}\) and \(\mathrm {BC}\) can be given as two second order polynomials when the three points \(\mathrm {A}\), \(\mathrm {B}\) and \(\mathrm {C}\) are defined.

Point \(\mathrm {A}\) gives the excitation coefficient value for zero response amplitude, \(\mathrm {C_{e,CF,A/D=0}}\).

Point \(\mathrm {B}\) is defined by the maximum excitation coefficient, \(\mathrm {C_{e,CF=max}}\), and the response amplitude level, \(\mathrm {A/D_{C_e,CF=max}}\), that gives maximum excitation force.

Point \(\mathrm {C}\) defines the \(\mathrm {A/D}\) value, \(\mathrm {A/D_{C_e,CF=0}}\), that gives zero excitation force for the actual nondimensional frequency.
The numerical values for \(\mathrm {A}\), \(\mathrm {B}\) and \(\mathrm {C}\) coordinates as functions of the nondimensional frequency are defined in Figure 3 for crossflow VIV. Appendix B gives the same values in a table.
Note that the excitation force coefficient will define damping for larger response amplitudes than \(\mathrm {A/D_{C_e,CF=0}}\) within the excitation zone. The curves give a reasonably good approximation to Gopalkrishnan’s data also for negative coefficients.
Examples of coefficients used in the program are shown on Figure 4. Each curve on the figure represents the CF excitation coefficient as function of the response amplitude for a specific value of the nondimensional response frequency.
3. Default excitation coefficient model, pure IL response
Gopalkrishnan (1993) has published results from forced harmonic inline motions in uniform flow. An excerpt of these results are presented in terms of contour plots for the IL excitation coefficient, \(\mathrm {C_{e,IL}}\), as functions of nondimensional frequency and amplitude ratio, see Figure 5. The \(\mathrm {C_{e,IL}=0}\) contour line defines the ideal IL lockin condition simply because zero excitation force means that there is no energy transmission between the cylinder and the fluid  which also is the case for an undamped cylinder under steady state lockin.
The excitation range for IL response is defined as \(\mathrm {0.2<\hat{f}<0.9}\) and is indicated on Figure 5. The full range of available data from Aronsen’s experiments is included, even if damping is seen to take place for all amplitude values at low and high \(\mathrm {\hat{f}}\) values. The reason for this decision is that direct use of Aronsen’s data will give a better damping model for inline vibrations than use of the Gopalkrishnan model  which is based on forced CF motion tests. Damping at lower \(\mathrm {\hat{f}}\) values than \(\mathrm {0.2}\) is not likely to take place since the response must be expected to shift to a higher eigenfrequency. At the other extreme, \(\mathrm {\hat{f}>0.9}\), we have response at very low flow velocity, which can be modelled as still water damping. Hence, IL and CF vibrations will have the same damping for such cases.
Figure 6 shows the parametric IL excitation coefficient curve. This is slightly different from the CF case since the coefficient is assumed to be zero for zero amplitude. Parameters that define the position of the points \(\mathrm {A}\), \(\mathrm {B}\) and \(\mathrm {C}\) on Figure 6 are given as functions of the nondimensional frequency on Figure 7. The curves that are defined from these parameters will be close to what would be found directly from the contour curves on Figure 5. Note that these curves are valid for pure IL response and must not be used for cases with simultaneous response in both directions. Numerical values for all points on Figure 7 are found in Appendix B.
4. Default excitation coefficient model, IL in combination with CF
IL response is known to be significantly larger in combination with CF than for the pure IL case. Hence, the coefficients that were found from pure IL experiments (Section 3) can not be used for combined IL/CF cases. Gopalkrishnan (1993) proposed to describe the IL amplitude as function of dominating mode order. However, the review of existing data presented by Gopalkrishnan (1993) could not verify this assumption.
As a preliminary solution for describing the IL excitation coefficient for combined response, we have decided to use the results from the PhD work of Gopalkrishnan (1993). The amount of data is far less than what is needed to cover all cases of practical interest, and no true interaction between IL and CF is accounted for. The excitation coefficient curve has the same shape as for CF (Figure 2), and the values of the 3 points that define the curve for a given value of nondimensional frequency is given in Figure 72. Numerical values for these points are found in Appendix B.
Note that these data must be considered as preliminary. The curves might be changed as soon as new data becomes available.
5. General excitation coefficient model
A general excitation coefficient model in VIVANA allows for easy input of model test results or results from CFD calculations. Such data can be applied by use of two different formats:

Give a table with sets of parameters that define amplitude dependent curves for given nondimensional frequency in the same way as shown for CF and IL coefficients in the previous sections.

Give coordinates for discrete points along curves that define the excitation force coefficient as function of amplitude for a set of nondimensional frequencies.
An example of the second alternative is illustrated on Figure 9. These curves are valid for a specific design of helical strakes, and are seen to give damping (negative excitation coefficients) for all amplitudes and frequencies. The coefficient for a specific combination of frequency and amplitude is found by interpolation.
Use of curves as shown on Figure 9 can adequately handle cases with up to approximately \(\mathrm {75~\%}\) coverage of VIV suppression devices. For these cases the bare pipe controls the VIV behaviour (frequency, mode etc.), and the section with strakes will provide a significant damping instead of excitation on a bare riser.
If strakes are attached to a larger part of the pipe, this section will take control of the VIV behaviour. Model tests for such cases indicate a different physical behaviour than for cases controlled by the bare pipe. This behaviour seems to be strongly dependent on pitch and height of the strakes. The responding frequencies and modes are generally lower than expected. VIV analyses by VIVANA of such cases are therefore not recommended.