Question 1 of 6: Definitions in Ship Dynamics

Question 1: Definitions in Ship Dynamics (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Four kinds of motion stability in ship maneuvering. Whether a ship "settles down" after a disturbance to its heading/track — and what it settles into — is classified into four distinct notions of stability, each a strictly weaker requirement than the last:

1. Straight-line (dynamic) stability. With the rudder fixed amidships, a ship disturbed from a straight course (e.g. by a wave or a brief yaw) returns asymptotically to travelling in a straight line, parallel to but not necessarily coincident with the original track, and ends on the original heading.
2. Directional stability. The ship returns to its original heading after the disturbance, but the final track may be laterally offset from the original line (a permanent sideways drift is tolerated as long as the heading recovers).
3. Positional motion stability. The ship returns to travel along the original line itself, but not necessarily on the original heading (it may cross the original track at a small residual angle rather than run parallel to it). This is a different (and generally harder to satisfy) requirement than directional stability, since neither implies the other.
4. Stability on a turn (steady-turning stability). With the rudder held at a fixed angle, the ship settles into a steady circular turn of constant radius and rate — the disturbed motion converges to a steady-state turning circle rather than to a straight path. This is the condition exploited in Question 6(b).

(b) Froude-Krylov force. The Froude-Krylov force is the wave-exciting force (or moment) obtained by integrating the pressure of the undisturbed incident wave (i.e. the pressure field that would exist if the ship's hull were not there) over the ship's wetted surface. It ignores the diffraction of the incident wave by the presence of the hull and the radiation waves generated by the ship's own motions — both of which are added separately in a full linear seakeeping (strip-theory or panel-method) solution as the diffraction force and the radiation (added-mass/damping) forces. Froude-Krylov theory is exact only in the limit of a body that is small or transparent to the wave field, but it is the leading-order, and often dominant, term of the total wave-exciting force for slender hulls; Question 5 is a direct Froude-Krylov derivation.

(c) Wave slope. The wave slope is the local inclination of the free-surface profile, $\alpha(x,t) = \partial \eta/\partial x$, where $\eta(x,t)$ is the surface elevation. For a regular wave $\eta = \eta_a\sin(kx-\omega t)$, the slope is itself sinusoidal with amplitude $\alpha_a = k\eta_a$ (a wave of amplitude $\eta_a$ and wave number $k=2\pi/\lambda$ has a maximum surface slope of $k\eta_a$ radians). Wave slope is the governing excitation for roll: a wave-borne ship tends to align its deck locally parallel to the instantaneous sea surface, so the wave-slope amplitude — not the wave height directly — is what is used (via the magnification factor) to estimate roll amplitude in Question 3(b).

(d) Added mass. When a body accelerates in a fluid, it must also accelerate some of the surrounding fluid with it; the added mass (or added moment of inertia) is the fictitious extra mass that, if it moved rigidly with the body, would produce the same inertial reaction force as this entrained fluid motion. It appears as an in-phase-with-acceleration hydrodynamic reaction force (as opposed to damping, which is in phase with velocity) and must be added to the body's structural mass/inertia before writing Newton's second law for the motion — exactly the role $0.20\,I$ plays in Question 3's roll equation of motion, and $m'$ plays (implicitly, through the acceleration terms dropped at steady state) in Question 6(b).

(e) Hydroelasticity. Hydroelasticity is the coupled interaction between a ship or ocean structure's own structural flexibility (elastic deformation, e.g. hull girder bending/torsion, or local plate/stiffener flexing) and the surrounding fluid's hydrodynamic loading, in which neither can be solved independently of the other — the structure's deformation changes the wetted-surface pressure field, which in turn changes the load driving the deformation. It governs phenomena such as springing and whipping of large ships and flexible offshore structures. Two standard analysis methods:

1. Two-dimensional hydroelastic beam theory (segmented-model / Timoshenko-beam strip approach). The hull is idealized as a non-uniform elastic beam (Timoshenko or Euler beam, including shear and rotary inertia) whose distributed mass and stiffness are obtained from the ship's structural section properties, coupled to a strip-theory hydrodynamic model (2-D added mass, damping and Froude-Krylov/diffraction force per station) through the beam's modal equations of motion.
2. Three-dimensional hydroelasticity (FEM + 3-D potential-flow panel/boundary-element coupling). The structure is modelled with a full 3-D finite-element model (giving its dry natural modes and generalized mass/stiffness), and the fluid loading (added mass, damping, wave excitation) on the wetted hull is computed with a 3-D radiation/diffraction panel method; the two are coupled modally to solve for the hydroelastic response, which is necessary when the structure is not slender enough for a beam idealization (e.g. multi-hulls, very-large floating structures).