Question

A ship with controls fixed is modeled as a two degrees of freedom system. For the linear maneuvering equations of motion for coupled sway and yaw, if the derived eigenvalues are real and negative, then the ship must possess:

• A. Positional motion stability
• B. Directional stability
• C. Straight line stability
• D. Both directional and positional motion stabilities

Hint:

For stability in linear systems, real and negative eigenvalues indicate asymptotic stability, which ensures the system's return to equilibrium after disturbances.

Answer 1

The Correct Option is C

Step 1: Understand the significance of eigenvalues. 
In a two degrees of freedom system for coupled sway and yaw, the eigenvalues of the system matrix determine the stability of the ship's motion: - Real and negative eigenvalues imply that any deviation from equilibrium decays over time, resulting in a stable response. 

Step 2: Relate eigenvalues to straight-line stability. 
Straight-line stability refers to the ship's ability to maintain a straight trajectory or return to a straight course after being disturbed. In this context: - Negative eigenvalues indicate the damping effect in both sway and yaw motions, ensuring that the ship naturally returns to a straight-line motion over time. 

Step 3: Analyze other options. 
Positional motion stability (Option A): This refers to the ship's ability to maintain its position, which is not directly determined by sway and yaw dynamics.
Directional stability (Option B): While it involves yaw stability, it does not fully describe the straight-line motion stability as required in this question.
Both directional and positional motion stabilities (Option D): This is broader than what the eigenvalues of sway and yaw specifically describe.

Conclusion: If the eigenvalues of the coupled sway and yaw system are real and negative, the ship possesses straight-line stability.