Question

The heave natural frequencies of a Jacket structure, FPSO and a semi-submersible are \(\omega_J\), \(\omega_F\) and \(\omega_S\) respectively. Each one of them has a pay load capacity of 10000 tonnes. Which of the following is TRUE?

• A. \(\omega_J<\omega_F<\omega_S\)
• B. \(\omega_J>\omega_F>\omega_S\)
• C. \(\omega_J<\omega_S<\omega_F\)
• D. \(\omega_J>\omega_S>\omega_F\)

Hint:

For offshore structures, remember the design drivers: Semi-submersibles are designed to be "transparent" to waves by having small waterplane areas, leading to long natural periods (low frequencies). Ship-shaped vessels (FPSOs) are the opposite, with large waterplane areas and shorter periods (higher frequencies). Fixed platforms like Jackets are fundamentally different and have very high natural frequencies.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The natural frequency (\(\omega_n\)) of a system in simple harmonic motion (like heave motion for a floating body) is determined by its stiffness (\(k\)) and mass (\(m\)). The relationship is \(\omega_n = \sqrt{k/m}\). We need to compare the effective stiffness and mass characteristics for heave motion of the three different offshore structures.
Step 2: Key Formula or Approach:
For a floating structure, the heave stiffness is the hydrostatic restoring force per unit displacement, given by \(k = \rho g A_{wp}\), where \(A_{wp}\) is the waterplane area. The mass includes the structure's mass and the added mass of the water that moves with it. The structure with higher stiffness and/or lower mass will have a higher natural frequency.
Step 3: Detailed Explanation or Calculation:
Let's analyze each structure:
1. Jacket structure: This is a fixed structure, piled into the seabed. It does not float. Therefore, it does not have a "heave" natural frequency in the same sense as a floating body. Its vertical stiffness is determined by the axial stiffness of its foundation piles, which is extremely high. Any vertical vibration would occur at a very high frequency. Thus, \(\omega_J\) is very large compared to floating structures.
2. FPSO (Floating Production Storage and Offloading unit): This is typically a large, ship-shaped vessel. Ship shapes have a very large waterplane area (\(A_{wp}\)) for their displacement, which gives them a high hydrostatic heave stiffness (\(k = \rho g A_{wp}\)). This results in a relatively high heave natural frequency (or a short natural period).
3. Semi-submersible: This type of platform is designed specifically for low motion response in waves. It achieves this by having most of its buoyancy provided by large submerged pontoons, while the waterplane area is minimized by using slender vertical columns to connect the pontoons to the deck. This small \(A_{wp}\) results in a very low hydrostatic heave stiffness. The combination of low stiffness and large mass (including a significant added mass from the submerged pontoons) gives the semi-submersible a very low heave natural frequency (a long natural period, typically over 20 seconds). This is done intentionally to move the natural frequency away from the range of common wave frequencies, thus avoiding resonance.
Comparison:
- Stiffness: \(k_{Jacket}\) (structural) \(> k_{FPSO}\) (hydrostatic) \(> k_{Semi-sub}\) (hydrostatic).
- Natural Frequency: Since \(\omega_n\) is proportional to \(\sqrt{k}\), the order of frequencies will follow the order of stiffness.
Therefore, we have \(\omega_J>\omega_F>\omega_S\).
Step 4: Final Answer:
The correct ordering of heave natural frequencies is \(\omega_J>\omega_F>\omega_S\).
Step 5: Why This is Correct:
The physical characteristics and design philosophies of the three structures lead to this order. Jackets are extremely stiff fixed structures (high \(\omega\)). FPSOs have large waterplane areas (intermediate \(\omega\)). Semi-submersibles are designed with small waterplane areas for stability in rough seas, resulting in a low natural frequency (low \(\omega\)). This matches option (B).