Question

A stationary ship has longitudinal symmetry. The surge, sway and heave motions are represented by indices 1-2-3, respectively and roll, pitch and yaw motions are represented by indices 4-5-6, respectively. Which of the following are TRUE about the added mass (\(A_{ij}\))?

• A. \( A_{35} = A_{53} \)
• B. \( A_{62} = A_{26} \)
• C. \( A_{46} = A_{64} \)
• D. \( A_{33} = A_{55} \)

Hint:

The added mass matrix (and also the damping matrix) is always symmetric, i.e., \(A_{ij} = A_{ji}\). This is a powerful property to remember. Any option that follows this format for valid coupling terms is likely to be correct. Be wary of options that equate different diagonal terms, as they represent physically different quantities.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The added mass matrix \(A_{ij}\) is a 6x6 matrix used in seakeeping analysis. The term \(A_{ij}\) represents the hydrodynamic force (or moment) in the i-th direction due to a unit acceleration in the j-th direction. The question asks about the symmetries of this matrix for a ship with longitudinal symmetry (i.e., symmetric about the x-z plane or port-starboard symmetry).
Step 2: Key Properties of the Added Mass Matrix:
1. General Symmetry: For any shape of body in an ideal fluid, the added mass matrix is symmetric, meaning \(A_{ij} = A_{ji}\). This is a fundamental property derived from potential theory (Green's theorem).
2. Geometric Symmetries: If the body has geometric symmetries, many of the off-diagonal terms become zero. For a ship that is symmetric about the x-z plane (port-starboard symmetry), the motions can be decoupled into two groups:
- Symmetric motions: Surge (1), Heave (3), Pitch (5)
- Asymmetric motions: Sway (2), Roll (4), Yaw (6)
The coupling terms \(A_{ij}\) between a symmetric mode and an asymmetric mode are zero. For example, \(A_{12} = A_{21} = 0\), \(A_{34} = A_{43} = 0\).
Step 3: Detailed Analysis of Options:
- (A) \( A_{35} = A_{53} \): This relates heave (3) and pitch (5). Both are symmetric motions, so coupling between them is expected. The statement reflects the general symmetry property \(A_{ij} = A_{ji}\). Therefore, (A) is TRUE.
- (B) \( A_{62} = A_{26} \): This relates sway (2) and yaw (6). Both are asymmetric motions, so coupling between them is expected. The statement reflects the general symmetry property \(A_{ij} = A_{ji}\). Therefore, (B) is TRUE.
- (C) \( A_{46} = A_{64} \): This relates roll (4) and yaw (6). Both are asymmetric motions, so coupling between them is expected. The statement reflects the general symmetry property \(A_{ij} = A_{ji}\). Therefore, (C) is TRUE.
- (D) \( A_{33} = A_{55} \): This equates the heave added mass (\(A_{33}\)) with the pitch added moment of inertia (\(A_{55}\)). These are diagonal terms and represent the direct force/moment due to acceleration in the same mode. While both are non-zero, there is no physical reason for them to be equal. \(A_{33}\) has units of mass, while \(A_{55}\) has units of moment of inertia (mass \(\times\) length\(^2\)). They cannot be equal. Therefore, (D) is FALSE.
Step 4: Why This is Correct:
Options (A), (B), and (C) are all direct consequences of the fundamental symmetry of the added mass matrix (\(A_{ij} = A_{ji}\)). This property holds for any underwater body in an ideal fluid, regardless of its shape. Option (D) incorrectly equates two different physical quantities.