An autonomous underwater vehicle is made of a long cylinder with a semi-ellipsoid at the forward end and a hemisphere at the aft end as shown in the figure. The origin of the reference frame is located at the centroid of the cylinder. The positive x, y and z axes, respectively, are pointing towards forward, port and upward directions. The surge, sway and heave motions are represented by indices 1-2-3 and roll, pitch and yaw motions are represented by indices 4-5-6, respectively. If A = [A\(_{ij}\)] is the added mass matrix, then which of the following are NOT zero?
Show Hint
In rigid body dynamics of underwater vehicles, non-zero values in the added mass matrix often reflect the geometric and hydrodynamic couplings between translational and rotational motions.
Step 1: Understand the added mass matrix.
In the context of underwater vehicles, the added mass matrix represents the inertia forces that resist motion in various directions. The non-zero values of this matrix are determined by the relative motions between the vehicle's axes.
Step 2: Evaluate the options.
- Option (A): A\(_{15}\) is expected to be zero because it represents the coupling between surge (1) and pitch (5), which typically does not contribute in a rigid vehicle model.
- Option (B): A\(_{35}\) is non-zero, representing the coupling between heave (3) and pitch (5), which is present due to the vehicle’s geometry.
- Option (C): A\(_{46}\) is typically zero, as roll (4) and yaw (6) do not have direct coupling in this setup.
- Option (D): A\(_{26}\) is non-zero, representing the coupling between sway (2) and yaw (6), which exists because of the vehicle's shape and symmetry.
Thus, the correct answers are (B) and (D).
