### Summary

**Project Title:**

**Prinicipal Investigators:**

**Sponsor:**

**Completion Date:**

**Final Report:**

**Scope of Work:**

Recently it has become common to use time domain procedures to predict the responses of compliant structures to wave and current loading. In extreme loading cases Morison’s equation with modifications has been utilized. An alternative is to compute first- and second-order diffraction forces, usually in the frequency domain, and then to solve the dynamic equations of motion in the time domain, considering other possible nonlinearities.

In this project the nonlinear response of spar platforms to wave loading is explored in the frequency domain. The structure is considered linear at first- and second-order (i.e. its restoring stiffness from mooring is linear). First, the linear response alone is considered. Next, second-order nonlinearities are considered in the wave forcing and in the fluid damping (structural damping is assumed to equal zero in all cases). Four types of quadratic forcing are considered in the form of force quadratic transfer functions (force QTF’s). They are:

- QTFfree (t) the free-body force QTF’s due to the total (first- plus second-order) velocity potential,
- QTFfree (1) the free-body force QTF’s due to the first-order total velocity potential, only,
- QTFfix (t) the fixed-body force QTF’s due to the total velocity potential, and,
- OTFD/S (1) as presented by Donley and Spanos (1990). Three alternatives are examined:
- Wave elevation drift force (moment)
- Velocity head drift force (moment)
- Velocity head drift force (moment)

Here QTF is used to denote collectively the separate surge, pitch and heave force QTF’s. The four damping types considered are:

- radiation damping only as computed by potential theory (damping type, dt0),
- addition of drag damping from the linearized drag term of Morison’s equation (dt1),
- addition of wave drift damping to the B11 entry of the damping matrix (dt2), and
- radiation plus wave drift damping (dt3).

The relative importance of these nonlinear forces and damping contributions are investigated. Response predictions (in surge, pitch and heave) are systematically compared to experimental results. Two types of spar platform models were tested: a large spar and a tethered small spar. Three types of loading were considered: monochromatic, bichromatic, and irregular waves. The irregular waves were to simulate operational loading conditions in one case and the 100-year survival condition in the other. All response predictions are computed in the frequency domain except for damping cases dt2 and dt3 were wave drift damping is involved; here it is necessary to compute the response in the time domain.

An important objective is to determine the adequacy of steady-state second-order frequency domain response solutions. The majority of the responses presented here resulted from taking the actual input wave’s complex coefficients (or equivalently modulus and phase) and substituting them into the steady-state force LTF’s and QTF’s the equations of motion were then solved for the unknown responses.

The major computational tasks performed in this work were:

- convergence studies for the first- and second-order forces and radiation coefficients
- computation of first-order forces and radiation coefficients for ten cylinder aspect ratios; simplified formulae as a function of normalized frequency were also fitted to the results
- mean drift forces were computed for two cylinder aspect ratios and for two cases:
- for a fixed cylinder and
- for a free cylinder; simplified formulae were also obtained for the surge and pitch forces for both cases

- complete second-order forces for surge, pitch and heave (in the form of force QTF’s) were computer for two cylinder aspect ratios and for the four QTF cases outlined above; the forces are presented as surface plotted versus two normalized incident wave frequencies
- linearized drag forces and drag damping coefficients were computed from the relative velocity form of Morison’s equation considering center of rotation effects; the drag coefficient was determined from free decay test simulations
- wave drift damping for the B11 matrix entry was computed using the modified gradient drift method
- first- and second-order responses were computed in the frequency domain using various combinations of the second-order forces and damping contributions mentioned above
- numerically predicted responses were compared to model test results of two spar platform concepts subjected to monochromatic, bichromatic, and irregular wave loading
- for irregular loading cases numerical ensemble statistics were computed to characterize the behavior of the two spar concepts to realistic wave loading conditions.

**Related Publications:**

Weggel, D.C. and Roesset, J.M., “The Behavior of Spar Platforms,” Proceedings, Coastal Ocean Space Utilization Conference, Buenos Aires, Argentina, December 1996.

Weggel, D.C. and Roesset, J.M., “Second Order Dynamic Response of a Large Spar Platform: Numerical Predictions versus Experimental Results,” Proceedings, OMAE 96 Conference, Florence, Italy, June 1996.

Mekha, B.B., Weggel, D.C., Johnson, C.P. and Roesset, J.M. “Effects of Second Order Diffraction Forces on the Global Response of Spars,” Proceedings, 6th ISOPE Conference, Los Angeles, California, May 1996.