* This model and analysis was a collaborative project with Stephen Wilkey, Kevin Healey, and Matthew Turcotte.*

The offshore wind turbine can be described as a single degree of freedom system to allow for proper modeling. The single degree of freedom system involves a mass, spring and damper. When an excitation force is applied to the system, the magnitude and phase of the response is strongly dependent on the frequency of the excitation. There are three main types of steady state response regions. The first, quasi-static, occurs when the excitation frequency is so far below that of the natural frequency of the system that the response does not change very much. The displacement of the system follows the time varying force almost as if it were loaded statically. The second region is when the excitation force is close to that of the natural frequency of the system. In this case, the inertia force and the spring force almost cancel each other out, causing the displacement that has a much larger magnitude then the response would be statically. The amplitude becomes a factor of the damping occurring in the system. When analyzing vibrations, it is important that such vibrations are not too close to the structure’s natural frequencies. You cannot manipulate wind and wave driven vibrations, but you can change structural frequencies.

This structure includes a lumped mass on top, which represents the mass of the turbine. For the sake of the model, the structure is 80 meters tall (Kurian, Ganapathy, 3) and 35 meters of the structure is below the waterline (Kurian, Ganapathy, 3). It is made of A36 Steel, which has a density of 7800 kg/m^3. The cross-section of the structure can be resembled by a pipe geometry, which was used to compute the second moment of Inertia (bending) for the natural frequency calculation. The thickness of the flagpole structure was chosen to have a diameter of 3 meters and a thickness of .1 meters (Kurian, Ganapathy, 3).

When the excitation frequency is close to the natural frequency of the system, resonance can occur and cause severe load cases and even failure. For this reason, any structure in which dynamics are involved it is important to gain knowledge of the expected excitation frequencies and the natural frequencies of the system. The most visible source of excitation in a wind turbine system is the rotor. For this model, we will only be covering the excitation force due to wave loading on the structure.

The above graph is a log plot of the frequency analysis showing the wave excitation frequency, first natural frequency, and non-linear effects from the drag force. The plot shows a peak at the wave input frequency of 0.08Hz in green, and a peak at the natural frequency of the structure, 0.44 Hz, in red. The light blue line depicts a peak which is believed to be from the nonlinear drag coefficient. This value is 3 times the wave forcing or 0.24 Hz. The other peaks are leakage from the wave forcing nonlinearity.

The damping coefficients, alpha and beta, used in Abaqus to obtain the frequency above are found using the equation:

The damping percent is a known value which was 2% or .02 for both of the first two frequencies. The frequencies were then plugged into the equation and using two equations to solve two unknowns, the values of alpha and beta were determined to be, α=0.0152 and β=0.0125.

The response of the structure can be seen above. The displacement is observed for a period of 74 seconds and shows 6 periods of the structure response. The maximum displacement of 0.0125 is earlier in the response at 10 seconds. The damping causes the structure to be displaced less and less until it reaches a steady state situation at about 35 seconds. The average maximum displacement at this time period was found by observation to be 0.0105.

Ultimate limit state analysis uses the largest wave forcing to determine if the stress added to the structure exceeds the allowable stress of the material it is made out of. The allowable stress for our structure was found to be 250 MPa. The stress received from Abaqus shows that the max stress felt by the structure was 3.23 MPa which means that the structure will not fail for the maximum wave loading.

The design of offshore wind turbines relies heavily on the frequency analysis of the wind turbine system and the wave loading. The frequency comparison of the monopile wind turbine to the wave loading and turbine rotor frequencies results in a structure design that avoids amplified damage due to resonance. The monopile design analysis results in a first natural frequency of 0.44 Hz. This is compared to the wave forcing frequency of 0.8 Hz and the rotor 1P and 3P frequencies of 0.2 Hz and 0.6 Hz respectively. From this analysis, along with the parameters associated with the structure location, the result is a soft-stiff wind turbine structure.