flowchart TD C2["<b>Ch 2 · Single degree of freedom</b><br/>ωₙ, ζ, RAO H(ω), resonance"] C4["<b>Ch 4 · Kinematics</b><br/>pose η, velocity ν, η̇ = J ν"] C9a["<b>Ch 9 · Random processes</b><br/>stationary, ergodic, Gaussian"] C3["<b>Ch 3 · Fourier transform</b><br/>x̂ = ĥ F̂ / k, h(t), convolution"] C5["<b>Ch 5 · Dynamics</b><br/>M<sub>RB</sub> ν̇ + C<sub>RB</sub> ν = τ<sub>RB</sub>"] C9b["<b>Ch 9 · Wave spectra</b><br/>S<sub>ηη</sub>, PM, JONSWAP, spreading"] C6["<b>Ch 6 · Forces in calm water</b><br/>C, A(ω), B(ω)"] C7["<b>Ch 7 · Forces in waves</b><br/>ωₑ, Froude–Krylov, scattering"] C8["<b>Ch 8 · Motion in regular waves</b><br/>6 × 6 equations, RAOs ĥⱼ(ωₑ)"] R["<b>Ch 9 · Response spectrum</b><br/>S<sub>ξξ</sub> = |ĥⱼ|² S<sub>ηη</sub>"] ST["<b>Ch 9 · Short-term statistics</b><br/>σ, significant amplitude, extremes"] P["<b>Project</b><br/>motion sickness incidence"] C2 --> C3 C4 --> C5 C9a --> C9b C5 --> C6 C6 --> C7 C3 --> C8 C6 --> C8 C7 --> C8 C2 -.->|"same form,<br/>with matrices"| C8 C8 --> R C9b --> R R --> ST ST --> P classDef tools fill:#ebf4ff,stroke:#2b6cb0,color:#1a202c; classDef body fill:#fffaf0,stroke:#c05621,color:#1a202c; classDef sea fill:#f0fff4,stroke:#2f855a,color:#1a202c; classDef result fill:#faf5ff,stroke:#6b46c1,color:#1a202c; class C2,C3 tools; class C4,C5,C6,C7 body; class C9a,C9b sea; class C8,R,ST,P result;
10 Summary of the Course
Learning Objectives
After reading this chapter, you should be able to:
- Trace the path from the response of a single degree of freedom system to the statistics of the motion of a ship in an irregular sea, and name the chapter in which each step was taken
- Identify, for each quantity of the single degree of freedom system, the corresponding quantity of a floating body
- List the steps required to estimate the motion of a given vessel in a given sea state, together with the inputs and the outputs of each step
- State the assumptions made along the way, and recognize the situations in which they no longer hold
10.1 Motivation
At the start of this course we asked a simple question: how will a ship or an offshore platform move when it operates in the ocean? We saw in Chapter 1 that this question became important once ships could travel in any direction relative to the wind and waves, and slamming and green water started to damage hulls and superstructures. Answering the question took us eight chapters. Each chapter added one piece, and the pieces were put together only towards the end. This chapter goes back over the whole course and shows how these pieces fit together. It does not introduce any new material. It is meant to be read at the end of the course, with the earlier chapters at hand for the details.
10.2 The Course at a Glance
The course is shown as a flowchart in Figure 10.1. Each box is a chapter, or a part of a chapter, and the main result that it passes on to the following chapters is written inside the box.
There are three strands in the course:
- The mathematical tools. In Chapter 2 and Chapter 3 we learnt how a linear system responds to a sinusoidal force, and how any force can be written as a sum of sinusoids.
- The floating body. In Chapter 4 and Chapter 5 we described the motion of a rigid body and the equations that govern it. In Chapter 6 and Chapter 7 we found the forces and moments that the water exerts on the body.
- The sea. In Chapter 9 we described the irregular sea as a random process, characterized by its spectrum.
The first two strands meet in Chapter 8, where the equations of motion are solved for the response amplitude operators (RAOs). The RAOs and the wave spectrum then meet in Chapter 9, where they give the response spectrum and from it the statistics of the motion. The project (Chapter 11) applies the whole chain to a design question.
10.3 Step by Step Through the Course
10.3.1 A Single Degree of Freedom System
In Chapter 2 we started with the spring mass damper system:
\[\begin{align} m\ddot{x} + b\dot{x} + kx = F(t) \end{align}\]
with the natural frequency and the damping ratio
\[\begin{align} \omega_n &= \sqrt{\frac{k}{m}} \nonumber \\ \zeta &= \frac{b}{2\sqrt{mk}} \end{align}\]
For a sinusoidal force \(F(t) = F_0\cos(\omega t)\), the steady response is a sinusoid at the same frequency. Its amplitude relative to the static displacement \(F_0/k\), together with its phase, is given by the RAO:
\[\begin{align} H(\omega) = \frac{1}{\left(1 - \frac{\omega^2}{\omega_n^2}\right) + i 2 \zeta \frac{\omega}{\omega_n}} \end{align}\]
Three regions of the RAO are worth remembering. Well below \(\omega_n\) the RAO tends to one and the response is quasi-static: the spring alone resists the force. Near \(\omega_n\) the response is amplified, and the amount of amplification is limited only by damping. Well above \(\omega_n\) the RAO tends to zero, because the mass is too heavy to follow the force. Since the equation is linear, the principle of superposition holds: the response to a sum of forces is the sum of the responses to each force. Most of what we later learnt about a ship is this same idea written with matrices in place of numbers.
10.3.2 From a Sinusoidal Force to Any Force
In Chapter 3 we wrote a periodic force as a Fourier series, and a general force as a Fourier transform. Taking the Fourier transform of the equation of motion turns the differential equation into an algebraic one:
\[\begin{align} \hat{x}(\omega) = \frac{\hat{h}(\omega)\hat{F}(\omega)}{k} \end{align}\]
where \(\hat{h}(\omega) \equiv H(\omega)\) is the RAO of Chapter 2.
The inverse Fourier transform of the RAO is the impulse response function \(h(t)\), and the response to any force is then given by the convolution integral:
\[\begin{align} x(t) = \frac{1}{k}\int_{-\infty}^{t} F(\tau) h(t - \tau)\, d\tau \end{align}\]
The upper limit \(t\) expresses causality: a force can only affect the response at the same time or later. Thus we have two equivalent ways of computing the response of a linear system. In the frequency domain we multiply by the RAO, and in the time domain we convolve with the impulse response function. Both are used again for the ship in Chapter 8.
10.3.3 Describing the Motion of a Rigid Body
In Chapter 4 we defined the global coordinate system (GCS), fixed to the Earth, and the body coordinate system (BCS), fixed to the vessel with its origin at a geometric point such as the intersection of the midship section, the centerline and the waterline. The pose of the body is described by the position of the BCS origin and the Euler angles, and its velocities are expressed in the BCS:
\[\begin{align} \{\eta\} &= \begin{bmatrix} x_0 & y_0 & z_0 & \phi & \theta & \psi\end{bmatrix}^T \nonumber \\ \{\nu\} &= \begin{bmatrix} u & v & w & p & q & r \end{bmatrix}^T \end{align}\]
The rotation matrix \(\boldsymbol{R}\) for the ZYX sequence of Euler angles transforms vectors from the BCS to the GCS. The Euler angles have a singularity (gimbal lock) at \(\theta = \pm 90^\circ\), which a ship does not reach in normal operation. The kinematic equations relate the rate of change of the pose to the velocities:
\[\begin{align} \{\dot{\eta}\} = \boldsymbol{J}\left(\{\eta_2\}\right) \{\nu\} \end{align}\]
For small motions they reduce to \(\{\dot{\eta}\} = \{\nu\}\).
10.3.4 What Causes the Motion
In Chapter 5 we applied the conservation of linear and angular momentum to the rigid body and obtained the Newton–Euler equations of motion in the BCS:
\[\begin{align} \boldsymbol{M}_{RB} \{\dot{\nu}\} + \boldsymbol{C}_{RB}\left(\{\nu\}\right)\{\nu\} = \{\tau_{RB}\} \end{align}\]
Writing them in the BCS keeps the mass and the inertia tensor in \(\boldsymbol{M}_{RB}\) constant in time. For small motions the Coriolis term, which is a product of velocities, can be neglected and the equations become \(\boldsymbol{M}_{RB}\{\dot{\nu}\} = \{\tau_{RB}\}\). The rest of the course was about finding \(\{\tau_{RB}\}\), the forces and moments that the water exerts on the body.
10.3.5 Forces on a Floating Body in Calm Water
In Chapter 6 we described the motion by the small displacements \(\{\xi\}\) from the equilibrium position, so that \(\boldsymbol{M}_{RB}\{\ddot{\xi}\} = \{\tau_{RB}\}\), and split the forces into hydrostatic, radiation and wave forces.
- The hydrostatic restoring force is \(-\boldsymbol{C}\{\xi\}\). The restoring matrix \(\boldsymbol{C}\) has non-zero entries only in heave, roll and pitch, with \(C_{33} = \rho g A_{WP}\), \(C_{44} = \rho g \nabla GM_T\) and \(C_{55} = \rho g \nabla GM_L\) as the main terms. Surge, sway and yaw have no restoring force.
- The radiation force is the force due to the waves that the body itself generates as it moves. Since the viscous forces on an oscillating hull are small compared to the pressure forces, the flow can be assumed to be inviscid, incompressible and irrotational. A velocity potential then exists and satisfies the Laplace equation. The free surface and body boundary conditions are linearized and applied at the mean free surface and the mean wetted surface \(S_{B0}\), and the problem is solved one mode and one frequency at a time, usually by a panel method. The pressure integrated over the hull gives the added mass \(\boldsymbol{A}(\omega)\) and the radiation damping \(\boldsymbol{B}(\omega)\). The radiation damping represents the energy carried away by the radiated waves.
Since \(\boldsymbol{A}\) and \(\boldsymbol{B}\) depend on frequency, the equations of motion in the time domain contain a convolution with the retardation function \(\boldsymbol{K}(\tau)\). These are the Cummins equations:
\[\begin{align} &\left(\boldsymbol{M}_{RB} + \boldsymbol{A}(\infty)\right)\left\{\ddot{\xi}\right\} + \boldsymbol{B}(\infty) \left\{\dot{\xi}\right\} \nonumber \\ &+ \int_{-\infty}^{t} \boldsymbol{K}(t-\tau) \left\{\dot{\xi}(\tau)\right\} d\tau + \boldsymbol{C} \left\{\xi\right\} = \left\{\tau_{wav}\right\} \end{align}\]
10.3.6 Forces on a Floating Body in Waves
In Chapter 7 we added the waves. A regular incident wave of frequency \(\omega\) has the wave number \(k\) given by the dispersion relation \(\omega^2 = gk\tanh(kh)\), which becomes \(\omega^2 = gk\) in deep water. A ship moving at speed \(U\) at a heading \(\beta\) relative to the waves meets them at the encounter frequency
\[\begin{align} \omega_e = \omega - kU\cos\beta \end{align}\]
The wave force is made of two parts. The Froude–Krylov force is obtained by integrating the pressure of the undisturbed incident wave over \(S_{B0}\). The scattering force is due to the wave scattered by the body, and is found from a boundary value problem of the same kind as the radiation problem. Together they make up the diffraction force \(\{\tau_D\} = \{\tau_I\} + \{\tau_S\}\). In long waves the wave forces tend to the hydrostatic limit, and in short waves the pressure cancels along the hull and the forces become small.
10.3.7 Motion of a Floating Body in Regular Waves
In Chapter 8 we put all the forces into the equations of motion. Taking the Fourier transform gives the equations of motion in the frequency domain:
\[\begin{align} &\left(-\omega_e^2\left(\boldsymbol{M}_{RB} + \boldsymbol{A}(\omega_e)\right) + i\omega_e \boldsymbol{B}(\omega_e) + \boldsymbol{C} \right) \left\{\hat{\xi}(\omega_e)\right\} \nonumber \\ &= \left\{\hat{\tau}_{D}(\omega_e)\right\} \end{align}\]
This \(6 \times 6\) complex system is solved at each frequency, heading and speed. The response per unit wave amplitude, \(\hat{h}_j = \hat{\xi}_j/A\), is the RAO of mode \(j\). Its magnitude gives the amplitude of the motion, and its phase tells us how far the motion leads or lags the wave crest at the BCS origin. The rotational RAOs are usually given per unit wave slope, \(\hat{\xi}_j/kA\). In long waves the vessel follows the water surface, and in short waves it hardly responds. Roll needs special care. Its radiation damping is very small, so the roll RAO depends mainly on viscous damping. A panel method cannot compute the viscous damping, and it must be added separately. The inverse Fourier transform of each RAO gives its impulse response function \(h_j(t)\), and the response to any wave record \(\eta(t)\) is again a convolution.
10.3.8 Irregular Waves and the Statistics of the Response
In Chapter 9 we treated the wave elevation at a point as a random process. Over a period of a few hours the sea can be taken to be stationary, ergodic and Gaussian, so that one long record represents the sea state and the whole process is described by its spectrum. The spectral moments give the main statistics:
\[\begin{align} m_n &= \int_0^\infty \omega^n S_{\eta\eta}(\omega)\, d\omega \nonumber \\ \sigma_\eta^2 &= m_0 \nonumber \\ H_s &= 4\sqrt{m_0} \nonumber \\ T_z &= 2\pi\sqrt{\frac{m_0}{m_2}} \end{align}\]
Rice’s results give the rates of upcrossings and peaks, and for a narrow banded process the amplitudes follow the Rayleigh distribution. A sea state is described by the Pierson–Moskowitz or the JONSWAP spectrum with its \(H_s\), \(T_p\) and \(\gamma\), and, for a short crested sea, by a directional spreading function.
For a linear system the response spectrum is the input spectrum multiplied by the square of the RAO. For a floating body,
\[\begin{align} S_{\xi_j\xi_j}(\omega) = |\hat{h}_j(\omega)|^2 S_{\eta\eta}(\omega) \end{align}\]
and at forward speed the same relation holds with the encounter spectrum. The response of a linear system to a Gaussian sea is itself Gaussian, so the formulas derived for the waves apply to the response with the moments of the response spectrum. For a narrow banded response, the standard deviation is \(\sigma_\xi = \sqrt{m_0^{(\xi)}}\), the significant amplitude is \(2\sqrt{m_0^{(\xi)}}\), and the level exceeded on average once in \(N\) oscillations is \(\xi_N = \sqrt{2m_0^{(\xi)}\ln N}\).
10.3.9 The Project
In the project (Chapter 11) we used the whole chain for a design question: which hull gives the crew the lowest motion sickness incidence. The RAOs of the ship are combined into the vertical acceleration at a cabin, its spectrum is computed in the design sea state, and the motion sickness incidence follows from its standard deviation and mean frequency.
10.4 From One Degree of Freedom to Six
Looking back, the motion of a floating body was computed in the same way as the response of the spring mass damper system of Chapter 2. Table 10.1 lists the corresponding quantities side by side. There are three main differences. The numbers become \(6 \times 6\) matrices, so that the modes are coupled. The added mass and the damping depend on frequency. The input is the wave elevation instead of a force, so that the force per unit wave amplitude is contained in the RAO. The random response explorer of Chapter 9 and the KCS response explorer show the same calculation for the two cases.
| Quantity | Single degree of freedom system | Floating body |
|---|---|---|
| Displacement | \(x(t)\) | \(\{\xi(t)\}\), six modes |
| Inertia | \(m\) | \(\boldsymbol{M}_{RB} + \boldsymbol{A}(\omega)\) |
| Damping | \(b\) | \(\boldsymbol{B}(\omega)\), with viscous damping added in roll |
| Stiffness | \(k\) | \(\boldsymbol{C}\), zero in surge, sway and yaw |
| Excitation | \(F(t)\) | \(\{\tau_D(t)\} = \{\tau_I(t)\} + \{\tau_S(t)\}\) |
| Natural frequency | \(\omega_n = \sqrt{k/m}\) | \(\omega_n \approx \sqrt{C_{jj}/(M_{jj} + A_{jj})}\) for heave, roll and pitch |
| Frequency response | \(\hat{h}(\omega) = \hat{x}/(\hat{F}/k)\) | RAO \(\hat{h}_j(\omega_e) = \hat{\xi}_j/A\) |
| Impulse response | \(h(t)\) | \(h_j(t)\) |
| Response to any input | \(x(t) = \frac{1}{k}\int F(\tau)h(t-\tau)\,d\tau\) | \(\xi_j(t) = \int \eta(\tau)h_j(t-\tau)\,d\tau\) |
| Random input | \(S_{FF}(\omega)\) | \(S_{\eta\eta}(\omega)\), or the encounter spectrum |
| Response spectrum | \(S_{XX} = \lvert\hat{h}\rvert^2 S_{FF}/k^2\) | \(S_{\xi_j\xi_j} = \lvert\hat{h}_j\rvert^2 S_{\eta\eta}\) |
| Variance of the response | \(\sigma_x^2 = \int_0^\infty S_{XX}\,d\omega\) | \(\sigma_{\xi_j}^2 = m_0^{(\xi)}\) |
10.5 Computing the Motion of a Vessel in a Sea State
The procedure for estimating the motion of a given vessel in a given sea state is shown in Figure 10.2. It is the procedure that we followed in the project.
flowchart TD
V["<b>1 · Vessel</b><br/>hull form and loading"]
W["<b>6 · Site or route</b><br/>scatter diagram: H<sub>s</sub>, period"]
H["<b>2 · Hydrostatics and mass</b><br/>C, M<sub>RB</sub>"]
PM["<b>3 · Panel method</b><br/>A(ω), B(ω), τ<sub>D</sub> at each ω, β, U"]
SP["<b>7 · Wave spectrum</b><br/>PM or JONSWAP, spreading"]
EQ["<b>4 · Equations of motion</b><br/>add viscous roll damping,<br/>solve 6 × 6 at each ωₑ"]
EN["<b>8 · Encounter spectrum</b><br/>for speed U and heading β"]
RAO["<b>5 · RAOs</b><br/>modes, or a point of interest"]
RS["<b>9 · Response spectrum</b><br/>|ĥ|² S<sub>ηη</sub>, × ωₑ² or × ωₑ⁴"]
MO["<b>10 · Spectral moments</b><br/>m₀, m₂, m₄"]
ST["<b>11 · Statistics</b><br/>σ, significant amplitude, T<sub>z</sub>,<br/>exceedance, largest value"]
CR{"<b>12 · Criteria<br/>met?</b>"}
OK["Accept the design<br/>or the operation"]
CH["Change the speed or heading,<br/>or the hull or loading"]
V --> H
V --> PM
H --> EQ
PM --> EQ
EQ --> RAO
W --> SP
SP --> EN
RAO --> RS
EN --> RS
RS --> MO
MO --> ST
ST --> CR
CR -->|"yes"| OK
CR -->|"no"| CH
CH -.-> V
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classDef sea fill:#f0fff4,stroke:#2f855a,color:#1a202c;
classDef result fill:#faf5ff,stroke:#6b46c1,color:#1a202c;
class V,H,PM,EQ,RAO body;
class W,SP,EN sea;
class RS,MO,ST,CR,OK,CH result;
The steps, with the chapters in which they were introduced, are as follows:
- Describe the vessel. The hull form and the loading condition fix the displaced volume, the position of the centre of gravity and the radii of gyration (Chapter 5).
- Hydrostatics and mass. From these we compute the restoring matrix \(\boldsymbol{C}\) and the rigid body mass matrix \(\boldsymbol{M}_{RB}\) (Chapter 6, Chapter 5).
- Hydrodynamic coefficients and wave forces. A panel method solves the radiation and scattering problems and gives \(\boldsymbol{A}(\omega)\), \(\boldsymbol{B}(\omega)\) and \(\{\hat{\tau}_D\}\) for each frequency, heading and speed (Chapter 6, Chapter 7).
- Equations of motion. We add the damping that potential flow cannot give, mainly the viscous damping in roll, and solve the \(6 \times 6\) system at each encounter frequency (Chapter 8).
- RAOs. The solution gives the RAO of each mode. If the motion at a particular point is needed, such as the vertical acceleration at a cabin in the project, the RAOs of the modes are combined into the RAO of that motion.
- Site or route. The wave scatter diagram of the area of operation gives the significant wave height and a period for each sea state.
- Wave spectrum. We choose a spectrum and its parameters, converting the period of the scatter diagram into \(T_p\), and a directional spreading function if the sea is short crested (Section 9.13).
- Encounter spectrum. For a ship with forward speed the wave spectrum is transformed to the encounter frequency (Section 9.15).
- Response spectrum. The squared RAO is multiplied by the encounter spectrum. The spectra of the velocity and the acceleration are obtained by multiplying by \(\omega_e^2\) and \(\omega_e^4\).
- Spectral moments. We integrate the response spectrum to obtain \(m_0\), \(m_2\) and \(m_4\).
- Statistics. From the moments we compute the standard deviation, the significant amplitude, the mean zero upcrossing period, the rate at which a level is exceeded and the largest value expected in the sea state (Section 9.15).
- Check the criteria. The results are compared with the criteria for the vessel, such as limits on roll, vertical acceleration or slamming. If they are not met, we can change the speed or the heading, which is a decision for the operation of the vessel, or the hull and the loading, which is a decision for its design.
10.6 Assumptions and Their Limits
Several assumptions were made along the way. Each one made the problem easier to solve, and each one has a range outside which it no longer holds. Table 10.2 collects them.
| Assumption | Introduced in | What it allows us to do | When it does not hold |
|---|---|---|---|
| The vessel is a rigid body | Chapter 4 | Describe the motion by six degrees of freedom | Long and flexible hulls, where the hull girder vibrates (springing and whipping) |
| The motions are small | Chapter 4, Chapter 5, Chapter 6 | Linearize the equations, use superposition and the frequency domain | Large amplitude roll and capsizing, slamming, green water |
| The flow is inviscid, incompressible and irrotational | Chapter 6 | Use a velocity potential, the Laplace equation and panel methods | Roll damping, which is mainly viscous; flow separation at bilge keels and sharp edges |
| The waves are of small steepness | Chapter 6, Chapter 7 | Linearize the free surface and body boundary conditions | Steep and breaking waves; second order effects such as the mean and slowly varying drift forces |
| Linear theory at forward speed | Chapter 7, Chapter 8 | Compute the loads and the RAOs at the encounter frequency | Following seas where \(\omega_e \rightarrow 0\), and high speeds |
| The sea is stationary, ergodic and Gaussian | Chapter 9 | Represent a sea state by its spectrum and obtain all its statistics from the spectral moments | Periods longer than a few hours, over which the sea state changes; very steep seas with nonlinear crests |
Note that these assumptions limit the conditions in which the results of the course can be used. In most operating conditions of ships and offshore platforms, linear seakeeping theory gives good estimates of the motions, and it is the basis of most design calculations. For the conditions in the last column, the results of linear theory are a starting point that has to be corrected with model tests, with nonlinear simulations or with empirical methods.
10.7 Beyond This Course
The tools of this course lead on to several topics that we did not cover:
- Long-term statistics. We computed the statistics of the response in one sea state. Over the life of a vessel it meets many sea states, each with the probability given by the scatter diagram. Combining the short-term statistics of all of them gives the long-term distribution of the response, which is used for design extreme values and for fatigue.
- Nonlinear time-domain simulation. The Cummins equations of Chapter 6 and Chapter 8 can be solved in the time domain with nonlinear forces added, such as the nonlinear restoring force at large roll angles or the forces from mooring lines.
- Second order wave forces. The mean and slowly varying drift forces are small compared to the first order forces, but act at very low frequencies. They decide the design of moorings and dynamic positioning systems.
- Maneuvering. The kinematics and dynamics of Chapter 4 and Chapter 5 are also the starting point for the maneuvering of ships in calm water, where the nonlinear equations are kept and the hydrodynamic forces are expressed in terms of the velocities of the ship.
- Model tests and full scale measurements. These are needed to check the predictions of the theory and to supply what it cannot, such as the viscous roll damping.
10.8 Summary
- The course built up the motion of a floating body in an irregular sea from three strands: the response of a linear system (Chapter 2, Chapter 3), the motion of a rigid body and the forces on it (Chapter 4 to Chapter 7), and the description of the sea as a random process (Chapter 9).
- The equations of motion of a floating body have the same form as those of the spring mass damper system, with \(\boldsymbol{M}_{RB} + \boldsymbol{A}(\omega)\), \(\boldsymbol{B}(\omega)\) and \(\boldsymbol{C}\) in place of \(m\), \(b\) and \(k\), and the diffraction force in place of the applied force.
- The RAO gives the response to one regular wave. The response spectrum \(|\hat{h}_j|^2 S_{\eta\eta}\) gives the response to an irregular sea, and its area is the variance of the response.
- Since the response of a linear system to a Gaussian sea is Gaussian, all the statistics of the response follow from the moments of the response spectrum, in the same way as the statistics of the waves follow from the moments of the wave spectrum.
- The results rest on the assumptions of a rigid body, small motions, potential flow, small wave steepness and a stationary Gaussian sea. Viscous roll damping has to be added to the potential flow results in every case.
10.9 Questions for Revision
These questions go across the chapters. Each one can be answered in a few lines, with the chapters indicated as the place to look.
- In the spring mass damper system, the response well below resonance is controlled by the stiffness, near resonance by the damping and well above resonance by the mass. Identify the corresponding quantities for the heave motion of a ship, and explain what the heave RAO looks like in each of the three regions (Chapter 2, Chapter 8).
- Explain why the added mass and radiation damping of a floating body depend on frequency, while the mass and damping of the spring mass damper system do not. What does this dependence change in the equations of motion in the time domain (Chapter 6)?
- Surge, sway and yaw have no hydrostatic restoring force. What does this mean for their natural frequencies, and for their RAOs in long waves (Chapter 6, Chapter 8)?
- Why is the roll RAO computed by a panel method not reliable near resonance, and how is this corrected (Chapter 8)?
- The same sea is met by a ship in head seas and in following seas at the same speed. Explain how the encounter spectrum differs in the two cases, and why its area does not change (Chapter 7, Chapter 9).
- Explain why the response of a linear system to a Gaussian sea is Gaussian, and why this allows the statistics of the response to be computed from its spectrum alone (Chapter 9).
- A designer has the RAOs of a vessel and the scatter diagram of its route. List the steps needed to find the largest roll angle expected in a three hour storm, and state the assumptions made at each step.
- Pick one assumption from Table 10.2 and describe a situation in which it fails. What would you use in place of linear theory in that situation?