Das Kentern von Schiffen in unregelmässiger längslaufender See

Complete English translation of the original German-language document (143 pages).

[page 1: title page]

INSTITUTE FOR SHIPBUILDING, UNIVERSITY OF HAMBURG — Report No. 249

The Capsizing of Ships in Irregular Head and Following Seas

by Sigismund Kastner

Hamburg, December 1968

[page 2]

The present work is the unabridged version of a dissertation approved by the Faculty of Mechanical Engineering of the Technical University of Hanover.

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Errata

Pages 38, 40, 49: All F are replaced by Fn.

Page 43 (bottom): 1 Fn = −2Kf₀ (instead of F₀ − f₀)

Page 48: In formula (5.3.8): Σ

Pages 47, 48: n − Γ

Page 29: The square-root sign is missing in (4.1.12):

ξ = Hₙ(t) = (N / √(3nπ)) · Σ fₚ · sin(ωₚt + εₚ)

[pages 4–5: table of contents]

TABLE OF CONTENTS

Overview — p. 1
The Influence of Sea State on the Capsizing Safety of Ships — p. 4
- 2.1 Practical Significance — p. 5
- 2.2 Previous Methods — p. 5
- 2.3 Task — p. 7
- 2.4 What Is Capsizing? — p. 6
- 2.5 Capsizing Safety — p. 10
Computational Approach for the Capsizing Ship — p. 14
- 3.1 Systems Theory and Stochastics — p. 17
- 3.2 Spectral Representation of Stochastic Quantities — p. 19
- 3.3 Equation of Motion — p. 22
Analysis of the Equation of Motion — p. 29
- 4.1 Righting Levers — p. 36
- 4.2 Stochastic Rolling Motion — p. 44
Influence of Ship Speed — p. 46
- 5.1 Transformed Lever Spectra — p. 51
- 5.2 Transformation of Bandwidth — p. 53
- 5.3 Discrete Representation of the Spectra — p. 57
Numerical Solutions — p. 58
- 6.1 Why Analog Computing? — p. 51
- 6.2 Normalization for the Analog Computer — p. 53
- 6.3 Control Circuit on the Analog Computer for Statistical Solutions — p. 57
- 6.4 Accuracy — p. 58
Results — p. 61
- 7.1 Overview of the Computed Cases — p. 65
- 7.2 Characteristic Capsizing Processes — p. 68
- 7.3 Statistical Evaluation — p. 73
- 7.4 Mean Capsizing Transit Time for Various Parameters — p. 78
- 7.5 Relative Frequency of Roll Angles — p. 80
Summary — p. 80
Notation — p. 85
References — p. 88
Appendix — p. 95
List of Figures and Figures — p. 96

1. Overview

The problem of capsizing safety in ships has occupied the attention of naval architects for more than a hundred years. Since statistical methods for describing the irregularity of sea states have opened new avenues for treating the seakeeping behavior of ships, the capsizing of a ship under the influence of waves has also come to be seen in a new light. Waves running along the length of a ship produce changes in the ship’s righting moments, which are often associated with a risk of capsizing. For irregular sea states, investigations of this type that incorporate the nonlinear shape of a ship’s righting-arm curve as a function of heel angle have hitherto been conducted only experimentally. In recent years, calls have repeatedly been made to provide a stronger theoretical foundation for the measurements known as the “Plöner capsizing experiments.” The present work introduces a method for the calculation and practically feasible execution of statistical computations of rolling behavior at large heel angles. The intuitive picture of the capsizing process acquired during the capsizing experiments has been a valuable aid. The investigations are confined to a ship’s heading along or directly against the direction of wave propagation.

Chapter 2 gives a brief account of the methods previously employed to account for the influence of sea state on ship stability. After a discussion of the concept of “capsizing,” capsizing safety is introduced as a probabilistic quantity.

The introduction of the concepts of stochastics and mathematical statistics in Chapter 3 forms the basis for a mathematical treatment of processes that are subject to chance. Representing the ship in a seaway as a system with an input signal and an output signal (or “system response”) corresponds to the modern statistical viewpoint. A subdivision of the ship-in-seaway system into two subsystems serves to clearly separate the problem into (a) the calculation of righting-lever variations in an irregular sea state, and (b) the calculation of the resulting roll angles of the ship.

In Chapter 4, the equation of motion for one degree of freedom is examined more closely. A mean normalized spectrum is introduced for the time-irregular fluctuations of the righting moment in a seaway. The dependence on the heel angle is reduced to the magnitude of the power content of the lever spectrum. A log-normal distribution is assumed for the shape of the spectrum.

* Assumption: the waves are unaffected by the ship. Computation uses the Froude–Krylov hypothesis. The equation of motion incorporates a superlinear dependence of damping on roll amplitude.

The transformation of the sea-state spectra and the righting-lever spectra for the moving ship is examined in Chapter 5, with reference to the work of St. Denis and Pierson. To synthesize stochastic time functions for the righting-moment lever variations, the spectra are approximated by discrete harmonics.

For the numerical solution of the equation of motion, a program and a control circuit are developed for the analog computer; these are described in Chapter 6. In parallel, a digital program has been elaborated — extending a program developed by Abicht and Zunker for capsizing in regular waves — to cover the irregular-sea approach. The digital program is used for the computation of individual cases, for verification runs on the analog computer, and for an uninterrupted computation over a longer time period. The analog computer has the advantage of time compression, i.e., a reduction of computation time relative to the real time of the rolling ship.

Chapter 7 summarizes the numerical results. In addition to examining individual characteristic capsizing events, the statistical evaluation of the computed capsizing transit times is described. The mean capsizing transit time, as the parameter of the exponential distribution, is discussed for various influencing quantities. An example of the calculation of the relative frequency of roll angles is presented.

The summary in Chapter 8 contains a concise presentation of the essential points that emerged during the execution of this work and that are important for further investigation.

2. The Influence of Sea State on the Capsizing Safety of Ships

2.1 Practical Significance

It is the task of the design engineer to dimension a ship so that it satisfies the demands placed upon it during its service life. Part of this task is to make the ship seaworthy and safe against capsizing. It is the task of research to develop suitable calculation bases and methods for this purpose.

For the problem of capsizing safety in ocean-going ships, the approach proposed by Wendel [74] has been followed: to compute all righting and heeling moments acting on a ship underway individually and to compare them in a balance. Reference is made here to the comprehensive lecture delivered by Wendel before the Schiffbautechnische Gesellschaft (Society of Naval Architects) in 1965 [79].

In this balance, particular importance attaches to the variations in righting moments in longitudinal (head/following) sea states. It is well known that moment fluctuations in a seaway can lead to large roll amplitudes or to capsizing of a ship. For irregular sea states, capsizing processes have previously not been amenable to calculation. However, in order to make statements about capsizing safety and to develop criteria, knowledge of the expected motions of a ship at large roll amplitudes is required. Especially for the multitude of new ship types that are being and will certainly continue to be developed, it is first desirable to have available methods of calculation. The following work is intended to make a contribution to that end.

2.2 Previous Methods

In historical sequence, the following classification can be applied to the assessment of ship stability:

Seaworthy and capsizing-safe ships were built solely from experience: the “trial and error” method.
A specified vertical position of the center of gravity was observed.
Freeboard and the righting-arm curve for calm water were taken into account (“Captain” and “Monarch,” 1870).
Stability criteria based on the righting-arm curve for calm water, in which the influence of sea state is included in a global manner without detailed knowledge. This includes various attempts to capture dynamic effects through the integral of the righting levers over the roll angle; these are today part of various national stability regulations or recommendations [45]. Representative of this method is the Rahola criterion of 1939 [62].
Moment (lever) balance from righting and heeling moments: Wendel, 1958 [72, 73]. In this approach, the maximum lever fluctuations and a mean lever curve in longitudinal regular waves are used as criteria. In subsequent years, meaningful requirements for these criteria were established by comparison with capsizing experiments in natural irregular model waves on an inland lake [48, 78]. For most practical cases, this method appears to be adequate. At least, the diversity of ship forms and ship sizes is well captured by it. However, further measurements and calculations to refine these criteria are desirable.
Explicit representation of the expected ship motions in a seaway, in particular rolling under the influence of externally applied heeling moments and the righting-lever fluctuations of the righting moment in a seaway. This method is currently still confined to research areas or to individual cases. The first solutions of rolling motion up to capsizing in an irregular sea state have been obtained only experimentally.

These “Plöner capsizing experiments” were conducted with free-running models in the natural waves generated by wind on a lake, using a continuous spectrum similar to that of ocean seas. They yielded, for defined model conditions, various transit times of a model in the seaway until capsizing occurred. A statistical evaluation of these experimental capsizing transit times provided statements on capsizing safety. These experiments were first reported in 1962 at the Symposium on Ship Theory at the Institute for Shipbuilding, University of Hamburg [46, 63]. Since then, these experiments have been continued and the measurement method further developed [48, 49].

To account for the irregularity of the sea state with respect to its influence on stability, Grim in 1961 [42] proposed an approach using a so-called “effective wave.” In this approach, the irregular surface-elevation fluctuations along the ship’s hull are approximated by a standing wave of regular form but with a time-irregular amplitude. From this effective amplitude and its statistical parameters, indications for critical ship speeds and natural periods could be derived.

In 1962, Kräppinger [55] published a “New Capsizing Criterion,” in which, using Grim’s effective wave, he makes a probabilistic approach to capsizing safety. The basis is a calculation of the frequency of occurrence of dangerous limiting heel angles, carried out by solving an equation of motion in regular waves over at most one encounter period with various initial conditions. Abicht [29] reported on the execution of these computations.

2.3 Task

An approach is to be introduced here that allows the excitation of a ship to large roll amplitudes by a longitudinal irregular sea state to be calculated. This means solving an equation of motion for the rolling ship over a longer transit time of the ship through a succession of waves of varying height and length. Particular emphasis is placed on the occurrence of extreme roll angles that can lead to capsizing. The approach makes it possible to carry out comparative calculations against experimental results from capsizing experiments in irregular seas. On the other hand, it is intended to provide guidance on the selection of the characteristic cases to be investigated in model experiments, since an experimental scanning of all parameter ranges with the available means would be too costly. Furthermore, the relationship of capsizing safety to important influencing quantities of the ship — such as the vertical position of the center of gravity, ship form, freeboard, natural period, damping, speed, etc. — should be calculable.

The statistical methods to be applied and the way of thinking are no longer new in naval architecture either. The statistical treatment of ship motions in irregular sea states was introduced into ship theory by St. Denis and Pierson in their paper before the SNAME in 1953, building on the foundations developed by Rice [66]. Statistical methods are already finding application in other design problems as well. Thus, Wendel in 1960 employed a probabilistic approach to assess the sinking safety of ships by including the chance-determined (i.e., unforeseeable) flooding accidents [76, 77].

2.4 What Is Capsizing?

A definition was given by Wendel in 1958 in his work “Safety Against Capsizing” [74]:

“Capsizing designates the property of a floating body to pass into another floating equilibrium without further expenditure of force upon reaching a certain angle of inclination. It corresponds to the tipping of rigid bodies on a solid support.”

This definition is to be made somewhat more specific: under “capsizing” is to be understood the assumption of a new stable floating attitude by the ship without further expenditure of force at heel angles so large that loss of the ship through incapacitation or flooding is to be expected.

Capsizing thus denotes a specific sequence of motions of the ship. Whether a ship capsizes or not is a question of its moment equilibrium. For equilibrium, the general condition is that the sum of all moments acting on the ship equals zero:

ΣM = 0 …(2.4.1)

or, in the usual representation using levers as the moments normalized to the displacement:

h + k = 0 …(2.4.2)

where:

h = righting lever
k = sum of the heeling levers

The equilibrium is:

stable if dh/dφ + dk/dφ > 0 …(2.4.3)
indifferent if dh/dφ + dk/dφ = 0 …(2.4.4)
unstable if dh/dφ + dk/dφ < 0 …(2.4.5)

The definition given applies to a static consideration. For the ship as a dynamic system that performs rolling oscillations, the analogous statement is:

If a ship is excited to rolling oscillations of such large amplitude that it passes into another stable floating attitude at which incapacitation or flooding is to be expected, this process is called capsizing.

It is readily understood that in capsizing due to irregular, non-deterministic fluctuations of the righting moment, the transit time of the ship until a capsizing event occurs will also be non-deterministic.

Figure 1 shows three examples of righting-arm curves. The heeling levers are set to zero for simplicity.

If the heel angles become greater than the angles of unstable equilibrium (without considering here how the ship reaches this position), the ship capsizes, provided that the righting-arm curve remains constant for the duration of the capsizing process. An exception is the unstable equilibrium in the third case of Figure 1 at zero degrees. The two stable equilibrium positions at approximately 10° lie within the ship’s operating range; the ship is not yet considered to have capsized.

By contrast, the coastal motor vessel “Lohengrin” met with an accident in 1961 — the second case in Figure 1 — by reaching the equilibrium state at approximately 45° in following seas. It could no longer right itself [41]. The ship is considered to have capsized, even though high hatch coamings as well as superstructures and bridge prevented it from turning over completely. The “Lohengrin” sank in the subsequent hours.

2.5 Capsizing Safety

Through the application of probability theory, the concept of safety has become a defined and numerically specifiable quantity. Reference is made to the extensive literature on this subject. An overview was given by Kräppinger in 1961 before the Schiffbautechnische Gesellschaft [56].

Capsizing safety is understood to mean the probability that a ship will not capsize during a given period of time. To determine capsizing safety, it is more practical to determine the probability of a capsizing event occurring within this time interval.

Since safety and failure probability are complementary, capsizing safety is thereby also given:

P(no capsize) = 1 − FK(t) …(2.5.1)

It can also be stated that the duration of time that a ship requires from the beginning of its voyage until capsizing is a random variable. The distribution FK of the capsizing transit time tK must now be determined. For this purpose, one starts from the so-called intensity function, or more specifically, the “failure rate.”

The expression μK(t) dt gives the conditional probability that the event — in this case the capsizing of the ship — occurs in the time interval (t, t + dt), given that it has not yet occurred up to time t:

μK(t) = fK(t) / [1 − FK(t)] …(2.5.2) and (2.5.3)

where fK(t) dt is the probability that a ship which begins its voyage at time t = 0 will capsize between times t and t + dt:

fK(t) = dFK(t)/dt …(2.5.4)

The denominator in equation (2.5.2) gives the probability that capsizing does not occur in the time interval (0, t):

1 − FK(t) …(2.5.5)

where FK(t) is the probability that capsizing occurs in the time interval (0, t):

FK(t) = ∫₀ᵗ fK(u) du …(2.5.6)

For a given intensity function, the probability FK(t) can then be calculated (after Gumbel [12]) from:

FK(t) = 1 − {1 − FK(t₁)} · exp[−∫ₜ₁ᵗ μK(u) du] …(2.5.7)

where t₁ is an arbitrary value of t.

For t₁ = 0, FK(t₁) = 0:

FK(t) = 1 − exp[−∫₀ᵗ μK(u) du] …(2.5.8)

Under the assumption that in every time interval dt during the voyage of a ship the probability of capsizing is constant, i.e., for a constant failure rate:

μK(t) ≈ const …(2.5.9)

equation (2.5.8) yields the distribution function for capsizing:

FK(t) = 1 − exp(−μK · t) …(2.5.10)

with the probability density function:

fK(t) = μK · exp(−μK · t) …(2.5.11)

Here:

μK = 1/TK …(2.5.12)

In the case of radioactive decay, the expression “half-life” is customary for TK. Here, TK is the mean capsizing transit time. As the parameter of the exponential distribution for the transit time of the ship until capsizing, it equals the first-order moment of the probability density fK(t):

TK = ∫₀^∞ t · fK(t) dt …(2.5.13)

Given a sample of n values tKi, the mean capsizing transit time is estimated by:

T̄K = (1/M) · Σᵢ₌₁ᴹ tKi …(2.5.14)

Observation and experiment show that for the transit time of a ship until capsizing there is a minimum time required — the actual capsizing time of the ship — which cannot be undercut. This time t₀ may be called the “incubation time” [38]. This incubation time gives the duration of the actual capsizing event itself, and thus corresponds to a capsizing time, as distinct from the longer “capsizing transit time” tK. It can be assumed that the capsizing time t₀ is normally distributed as N(t̄₀, σ²t₀).

The distribution of the transit times t′ of the ship until the beginning of the incubation time t₀ — which is set to a mean of t̄₀ — is then, with t′ = t − t₀ and T′K = TK − t₀:

fK′(t′) = (1/(TK − t₀)) · exp(−t′/(TK − t₀)) …(2.5.16)

3. Calculation Approach for the Capsizing Ship

3.1 Systems Theory and Stochastics

The rolling ship is treated as a system with an input signal and an output signal (Fig. 3). The external conditions to which the ship is exposed constitute the input signals. The corresponding ship behavior is the system response to those input signals. If nothing is known about the internal structure of the system, or if it is mathematically difficult to represent, knowledge of the input and output alone may suffice to describe the system. This approach is called the black-box method. For the capsizing model tests in irregular seas carried out at Plön, this method was applied in practice.

For a deeper understanding and calculation of system behavior, an analysis of the structure of the system and of the governing parameters for ship behavior is required. This analytical path is supplemented by the synthetic method, in which an attempt is made, starting from a simple approach and successively incorporating additional influencing quantities, to match the theoretical model ever more closely to actual events.

The capsizing model tests have the advantage that all influencing quantities are simultaneously active. Among these may be counted:

Passing frequency of waves relative to ship
Disturbances from lateral waves and wind forces
Magnitude of righting-lever fluctuations
Hydrodynamic effects at the ship hull

In the present work, an analytical approach will be introduced that captures the essential influencing quantities, particularly the irregularity of the righting-lever fluctuations.

Fig. 4 shows the subdivision of a ship rolling in longitudinal seas into two sub-systems. The first system

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describes the hydrostatic determination of the righting-moment levers in the seaway. The second system describes the rolling ship as a dynamic system. Because of the irregular nature of the seaway, the treatment of both systems proves to be demanding. For the mathematical description of the systems, the nature of the input signal and the “distortion” of the input signal within the system play a decisive role.

For the treatment of ship capsizing, a large range of roll angles must be captured, over which even an approximate linearization is not possible. An equivalent to the linear-theory representation does not unfortunately exist for nonlinear systems. For nonlinear systems with a stochastic(1) input, all individual cases must therefore each be computed separately. For more general statements on a statistical basis, the output signals can again be used.

For Sub-system I defined above, the righting levers can be calculated quasi-statically at equidistant time intervals for irregular seas as well, based on the ship’s form. However, the numerical effort increases considerably compared to regular waves. It appears feasible, after computing a number of hull forms, to use reliable approximations in this context. Further details are given in Section 4.1.

Even more demanding is the treatment of System II. The large number of capsizing processes to be computed from the solution of the equation of motion, together with the required long-duration statistical calculations, and combined with variation of ship parameters such as speed or damping, entail extensive

(1) A stochastic quantity is a random variable that cannot be expressed analytically. Stochastic is used synonymously with random or irregular (English: “random”). The expression “aleatory” is also used in part (from Latin aleos — the die).

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numerical calculations.

Searching for parameter regions that represent a high capsizing hazard for the ship is, with many parameters and some stochastic quantities, very laborious even with the use of modern computational tools. It seems sensible to restrict the cases to be computed to the important regions and, through knowledge of the tendency of parameter influences, to limit the computational effort. To an even greater degree this applies to statistical capsizing model tests. Once a calculation method such as the one proposed here is available, they should be restricted to characteristic individual cases, so that the theoretical approach can be tested and improved. For this purpose, deterministic cases must also be measured, in which the initial conditions and the wave sequence at the ship are recorded by instrumentation. Under “individual cases,” however, statistical test series are also understood here — ones that supply statistical data on motion quantities and experimental capsizing-time distributions. Nevertheless, the effort for conducting and evaluating such tests remains considerable.

Solution methods making use of a random process are today often termed the Monte Carlo method. Originally, after von Neumann [20], it referred only to the solution of a simple stochastic problem, whose experimental solution approximates arbitrarily closely the solution to the original problem — which can be formulated but is analytically too difficult to solve.

Covering the entire important parameter range in experiments in the sense of a Monte Carlo method would be too costly. In that respect, the approach described in the following can be regarded as a necessary complement to the capsizing model tests.

If the numerical calculations are not carried out in purely digital form but instead an analogue representation of the formulated problem is used — for example by means of an electrical oscillating circuit on which sampling experiments are performed — this is called a simulation. A simulation can facilitate the calculation.

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3.2 Spectral Representation of Stochastic Quantities

In irregular seas one cannot specify what ordinate ζ the ocean surface will have at a given point at a given instant. The theory of stochastic processes serves to describe such events. Using probability calculus, statements can be made about characteristic properties of such processes. Numerical values are obtained via statistical means through integration over sufficiently large distances or time periods.

On the assumption that irregular seas constitute an ergodic and stationary stochastic process, one can work with the temporal ordinate fluctuations at a single point on the water surface [50].

It has proved expedient, for the mathematical description of stochastic processes, to introduce a relative power measure, namely the time-averaged central moment of second order:

$$m_2(\zeta) = \overline{\zeta^2(t)} = \zeta^2_\text{eff} = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} \zeta^2(t), dt \tag{3.2.1}$$

The spectrum then represents the distribution density of the power over frequency:

$$S_\zeta(f) = \frac{d,\overline{\zeta^2(t)}}{df} \tag{3.2.2}$$

where S = sea spectrum, f = frequency.

Equation (3.2.2) in principle contains the so-called Parseval’s theorem.

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The representation in terms of f is preferred here. The following relations hold:

$$\omega = 2\pi f, \quad d\omega = 2\pi, df, \quad S(f) = 2\pi S(\omega) \tag{3.2.3}$$

Sometimes the spectrum is represented only over positive frequencies. In that case it is to be noted that the spectral values are doubled:

$$S^+(f) = 2, S(f) \tag{3.2.4}$$

So that:

$$\overline{\zeta^2} = \int_{-\infty}^{+\infty} S(f), df = 2\int_{0}^{\infty} S(f), df = \int_{0}^{\infty} S^+(f), df$$

The spectrum is obtained numerically from the Fourier transform F of the time record ζ(t), or alternatively from the autocorrelation function R:

$$S_n(f) \sim \left|\frac{1}{T} F[\zeta(t)]\right|^2 \tag{3.2.5}$$

$$S_n(f) \sim \mathcal{F}[R(\tau)] \tag{3.2.6}$$

The component waves of the seaway have an irregular phase distribution. The probability density of the phase angles ε is given by a rectangular (uniform) distribution:

$$w(\varepsilon) = \frac{1}{2\pi}, \quad 0 \le \varepsilon \le 2\pi \tag{3.2.7}$$

It is therefore not possible to compute an amplitude spectrum. However, time-averaging yields, via the power spectrum, a statistically determined function for characterizing the stochastic process.

As stated in Section [3.1], no transfer function can be given for nonlinear systems. To carry out individual calculations, a time function must be developed from the spectrum:

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$$\zeta(t) = \int_0^{\infty} C(f), \sin{2\pi ft + \varepsilon(f)}, df \tag{3.2.8}$$

$$C(f) = \sqrt{2, S(f’)} \tag{3.2.9}$$

The quantity C(f) here denotes an amplitude density. For continuous spectra, a discrete frequency therefore has only an infinitesimal amplitude. The amplitude density derived from the spectrum no longer describes the stochastic process completely, but only in conjunction with the random phases ε(f) in equation (3.2.8). Equation (3.2.8) can thereby yield a temporal excerpt T from the sea function ζ(t).

It is possible to approximate the continuous spectrum by discrete partial components through a piecewise integration of the spectral density. From (3.2.8) the following summation expression then follows:

$$\zeta(t) = \sum_{p=1}^{N} \sqrt{2, S(f_p), \Delta f_p}; \sin(2\pi f_p t + \varepsilon_p) \tag{3.2.10}$$

In practice, random draws from the uniformly distributed population must be substituted for ε_p.

3.3 Equation of Motion

The ship is treated as a rigid body. Every body in motion follows its trajectory in such a way that, between two instants of time, the difference between kinetic and potential energy is minimized (Hamilton’s principle). The necessary condition for this is, according to Euler, for one degree of freedom:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{\varphi}} - \frac{\partial L}{\partial \varphi} = 0 \tag{3.3.1}$$

$$L = T - U \tag{3.3.2}$$

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where:

L = kinetic potential (Lagrangian function)
T = kinetic energy
U = potential energy
φ = coordinate

In the rolling ship, the potential energy U is formed by the Earth’s gravitational field and the instantaneous buoyancy vector at the hull. If the buoyancy vector depends on the angle of inclination φ and on time t, one can write:

$$U = -\Delta \cdot g \cdot e(\varphi, t) \tag{3.3.3}$$

where:

Δ = displacement of the ship
e = stability path (righting lever)

The displacement of the ship is assumed to be constant.

The kinetic energy can be expressed in the form:

$$T = \frac{1}{2}, I_{\varphi\varphi}, \dot{\varphi}^2$$

Thus equation (3.3.1) becomes:

$$I_{\varphi\varphi}, \ddot{\varphi} + \Delta \cdot g \cdot h(\varphi, t) = 0 \tag{3.3.5}$$

with:

$$h(\varphi, t) = e(\varphi, t) \tag{3.3.6}$$

In addition there are the moments that act independently of the potential. These include the dissipation moment M_D and external exciting moments M_k(t). The general form of the equation of motion for one degree of freedom is then obtained:

[eq. 3.3.7 — see following text]

A ship in longitudinal seas can experience such changes in its righting levers h that it capsizes. Roll motions with very large amplitudes are thus produced by temporal changes in the characteristic curve of the ship system,

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where the characteristic curve is to be understood as the graphical representation of the righting lever h as a function of roll angle φ — familiar in naval architecture as the GZ curve (lever-arm curve).

A system whose properties change with time is called “rheonomous” following Klotter [15]. Depending on the form of the system’s characteristic curves, one then speaks of “rheolinear” or “rheo-nonlinear” oscillations. The equivalent designations quasi-harmonic and pseudo-harmonic have also been used in part, though these are not employed here. In electrical engineering the same type of oscillation has established the concepts of “parametric excitation” or “parametric de-damping.”

For a ship in irregular seas, the temporal changes in the system characteristic curves are stochastic functions. The ship system is then characterized, in accordance with the above definitions, as a “stochastically rheo-nonlinear system.” For the type of excitation one can speak of “stochastic parametric excitation.”

For linear systems with harmonic parametric excitation, the solutions of the Mathieu equation are known [17].

Grim [40] investigated the ship with periodic parametric excitation, both theoretically and experimentally, treating specifically the equation:

$$I_{\varphi\varphi}, \ddot{\varphi} + N_{\varphi\varphi}, \dot{\varphi} + \Delta, g \left(a_0 + a_1, \tan^2\varphi + \Delta\text{GR}, \sin\nu t\right)\sin\varphi = 0 \tag{3.3.8}$$

with initial conditions φ_0 and φ̇_0.

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4. Analysis of the Equation of Motion

4.1 Righting Levers

The time-varying righting lever can be represented by its time-averaged mean value and a fluctuation function about that mean:

$$h(\varphi, t) = \bar{h}(\varphi) + \Delta h(\varphi, t) \tag{4.1.1}$$

Equation (4.1.1) is intended to hold for a constant angle of inclination φ. For a regular wave, these curves are shown in Fig. 5 for various values of φ for ship No. 4212 W of the Series 60. The determination of lever values in the seaway corresponds, by the definition in Section 3, to the treatment of System I. For the practically common hull forms, the lever function h(φ, t) must be computed point by point numerically from the periodic sea function. Nevertheless, the results — here the levers for regular waves — show the fundamental period of the individual wave for all angle ranges. A deviation from the sinusoidal form can be interpreted as a distortion of the input signal by the hull form.

The graphical representation of levers over one encounter period, given in Fig. 5 with φ as parameter, cannot satisfy more general statements about this distortion. Other plotting methods cannot change this either. The course of the levers according to Fig. 5 is certainly connected with characteristic properties of the hull form. It is already common practice today to calculate characteristic points of these lever curves — for example, for the position of the ship in the wave trough and at the wave crest. In this way the possible fluctuation range of the levers has been captured for the usual hull forms. In addition to the main proportions of the ship, special characteristic properties of the hull form — such as a full midship section, a cruiser stern, or a transom stern — enter into the result.

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In order to capture the periodic lever variation according to Fig. 5, knowledge of the levers at intermediate positions of the wave–ship encounter period is required. At a minimum, one point between crest and trough must be calculated in each case, to gain an indication of the deviation from the sinusoidal form. These points are designated the first and second intermediate positions. The first intermediate position corresponds to the location of the wave crest at the midship section of the forward half of the ship, and the second intermediate position corresponds to the location of the wave crest at the midship section of the after half. It is natural to regard these two intermediate positions as characteristic for the influence of the forward and after hull form on the seaway levers, respectively. It is therefore proposed that these phase positions always be determined alongside the seaway lever calculations as a matter of principle.

Since a continuous course of the h_φ(t₀) curves in regular waves can always be assumed for the usual hull forms — if only because of the smoothing produced by integration — these four points already suffice to characterize the typical course. The extent to which this also holds quantitatively is shown in Fig. 6. There, the deviations of the mean seaway lever from the calm-water lever are plotted against the angle of inclination. The mean seaway lever was averaged from two phase positions — crest and trough; from four phase positions — crest, trough, and the two intermediate positions; and from eight phase positions. In addition, the mean seaway lever was determined by integration over all phase positions. It can be seen that, for this hull form, the step from two to four phase positions is considerable, whereas adding still more phase positions changes little further. To capture the characteristic properties of the distortion, one should therefore calculate the seaway levers for four phase positions of a ship in a regular wave.

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To obtain more quantitative results that go beyond a purely visual comparison of the levers of different ships, a harmonic analysis of the lever fluctuations for various φ is proposed. The distortion will then be expressed in terms of the amplitudes of the harmonics, and particular features for different ship types may be identified. Figs. 7 and 8 show the result of such analyses for the standard ship of Series 60. They contain the mean lever in the seaway, the fundamental period of the lever fluctuation — which equals the seaway period — as well as the harmonics. The harmonic series can be truncated after the third order, since the higher orders are usually very small and barely modify the levers any further.

Since the additional computational effort can, as described above, be considerably reduced, such spectra could soon be available for many ships. It is conceivable that in this way a systematic correlation between seaway levers and hull forms can be established.

As Fig. 6 makes clear, the mean curves of the levers in regular waves deviate strongly from the calm-water curve. Especially for large angles of inclination, smaller mean levers are usually found. The resulting hazard to ships is well known, and particularly in the last decade there have been intensive efforts to incorporate this into the practical assessment of capsizing safety. The reason is that the lever reductions at the wave crest, compared to calm water, are greater than the lever increase in the trough. In investigations in the seaway, the calm-water curve must therefore be abandoned and cannot serve as a reference line. Since it is, however, standard practice to calculate GM for calm water — if only because the inclining experiment is conducted in calm water — this value is used as an equivalent datum for the height of the center of gravity above the keel (KG) in all following investigations in irregular seas as well. The specification of GM_CW is therefore intended solely to characterize the position of the center of

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gravity, since KM_CW is a defined geometric property of the ship:

$$KG = KM_{CW} - GM_{CW} \tag{4.1.2}$$

In irregular seas, the righting lever will now also vary irregularly over time. This stochastic lever can be represented statistically for φ = const by spectra and distribution functions in the same way that is customary for irregular seas. For the accurate numerical calculation of the stochastic levers for a standard hull of Series 60 — to whose lines a new model for the Plön capsizing tests was also built — a research project is underway with support from the Deutsche Forschungsgemeinschaft. Stochastic levers can in principle be calculated in the same manner as for regular seas under the assumption of the Froude–Krylov hypothesis. The ship’s lines and displacement are given. The ship’s underwater volume is bounded above by an irregular surface, and this boundary contour changes over time. With the aid of digital programs it is today possible, at least for fundamental investigations, to carry out numerical calculations of such scope. Regarding the method and first results, reference is made to the available interim report [37]. Since this project is not yet concluded, plausible assumptions are made in the present work for the stochastic levers used — assumptions that can be modified or varied in later calculations.

Let the lever function h_φ(t) for a constant angle of inclination φ be a stationary stochastic process. For it, the following expression exists:

$$\bar{h} = \lim_{T \to \infty} \frac{1}{T} \int_0^T h_\varphi(t), dt \tag{4.1.3}$$

(The overbar always denotes the mean value of the quantity over which it is drawn, except in the case of segment designations by two endpoints, e.g., KM.)

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Expression (4.1.3) represents the linear mean value of the stochastic levers. If h̄ is determined for all φ in the range −φ_K ≤ φ ≤ φ_K, with φ_K = 90°, the mean curve of the righting levers in the seaway, h̄(φ), is obtained. For initial investigations it is accepted as permissible to apply the mean curve h̄(φ) derived from regular seas also to irregular seas.

The temporal fluctuations of the stochastic levers about the mean curve are then, for a constant angle φ:

$$\Delta h_\varphi(t) = h_\varphi(t) - \bar{h}_\varphi \tag{4.1.4}$$

It can further be assumed that the ordinates of the function Δh are normally distributed. For the angle φ = 30°, Fig. 9 shows the ordinate distribution of the stochastic levers calculated for the chosen Series 60 hull form.

The variance σ²(Δh) of the lever distribution is now a function of the angle of inclination:

$$\sigma^2_{\Delta h} = \sigma^2_{\Delta h}(\varphi) \tag{4.1.5}$$

This means that, depending on the angle of inclination, a different range Δh is covered by the lever fluctuations. The fluctuation range of the levers is to be defined as the lever change in the seaway that is exceeded only with small probability. If it is assumed that the maximum lever fluctuations occur in one ship-length of wave, a design wave height can be chosen from statistics of the associated wave heights.

Fig. 10 shows the wave height plotted against relative frequency for 100 m waves in the North Atlantic, computed from statistical data by Roll [65]. The fluctuation range of the levers was calculated for a wave height of 6.7 m, which for 100 m waves is exceeded by only 2‰ of all waves.

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Because wave crest and wave trough, owing to the influence of the intermediate positions on the mean lever h̄, yield different deviations from the mean value in regular waves, the mean of the crest deviation and the trough deviation is here taken as the fluctuation range for calculation purposes (see Fig. 11):

$$\Delta h(\varphi) = \frac{1}{2}(h_T - h_B) \tag{4.1.6}$$

If it is now assumed that this fluctuation range so determined represents the 3σ_Δh range of the normally distributed lever fluctuations, a representation of the probability density of the levers as a function of angle of inclination results, approximately as shown in Fig. 12.

On the analog computer, the mean seaway lever curve h̄(φ) and the fluctuation width Δh(φ) can be reproduced with function generators.

For the representation of the temporal fluctuations of the levers within the range so defined, an irregular time function must now be introduced. The time function of the lever fluctuations is given, for regular seas, by the curves in Fig. 5, or equivalently by superposition of the Fourier components according to Fig. 8. For irregular seas, continuous lever spectra are to be used analogously. However, it should be noted that a direct correlation exists among the individual time functions with φ as parameter. Therefore, for a given roll computation, different realizations h_φ(t) must not be taken for different values of φ = const; rather, the previously computed lever values must be retained in their correct temporal assignment. The spectra here serve only as statements about statistical mean values.

A brief excerpt from the temporal fluctuations of righting levers in irregular seas for several constant inclination angles φ_i is shown in Fig. 13. As already indicated, the various curves are directly

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coupled to each other in time. Phase draws may only be made for the exciting sea state, to which a coherent family of lever curves is then assigned.

It is now plausible from Fig. 13 that the temporal lever fluctuations, referred to the fluctuation range, will differ only insignificantly from one φ_i to another. In order to incorporate the irregularity into the calculation in a fundamental way, the approximate approach using a single mean normalized spectrum for the entire range of inclination angles therefore appears justified. It makes it possible to calculate for all inclination angles using only one normalized fluctuation function H_n(t). This is particularly important for a solution on the analog computer.

Associated with the mean normalized lever fluctuation H_n(t) is the normalized lever spectrum S_n(f). For these investigations it was assumed to follow a logarithmic normal distribution. First numerical results for lever spectra of the Series 60 hull form were able to confirm the justification for this approach. Fig. 14 shows a comparison of a normalized sea spectrum according to Pierson and Moskowitz [61] with the normalized lever spectrum for a heel angle of 30°.

If it is assumed that the fluctuation range Δh(φ) corresponds to the 3σ_Δh range of the lever fluctuations distributed as N(0, σ²_Δh), then the associated standard deviation σ_Δh can easily be calculated:

$$\sigma_{\Delta h}(\varphi) = \frac{1}{3}, \Delta h(\varphi) = t_h \cdot \Delta h(\varphi) \tag{4.1.7}$$

In general, let t_h = σ_Δh / Δh be the ratio of standard deviation to fluctuation range.

Using Parseval’s theorem, which states that the area under the spectrum equals the variance of the time record:

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$$\int_0^{\infty} S_{\Delta h}(f), df = \overline{(\Delta h)^2} = \sigma^2_{\Delta h} \tag{4.1.8}$$

the spectrum is also known when a logarithmic normal type spectrum is assumed (M = log e = 0.4343):

$$S_{\Delta h}(f) = \frac{\sigma_{\Delta h}}{\sqrt{2\pi}, \Delta h, f} \exp!\left[-\frac{(\log f - \log f_\text{med})^2}{2, \Delta h^2}\right] \tag{4.1.9}$$

The normalized spectrum S_n is the lever spectrum referred to its effective value:

$$S_n(f) = \frac{S_{\Delta h}(f)}{\sigma^2_{\Delta h}} \tag{4.1.10}$$

It then follows that:

$$\sqrt{2, S_n(f), \Delta f} = t_h \cdot \Delta h(\varphi) \cdot \sqrt{2, S_n(f), \Delta f} \tag{4.1.11}$$

With this, the following expression is obtained for the stochastic lever as a function of roll angle φ and time:

$$h(\varphi, t) = \bar{h}(\varphi) + t_h \cdot \Delta h(\varphi) \cdot H_n(t)$$

$$\text{with } H_n(t) = \sum_{p=1}^{N} \sqrt{S_{np}, \Delta f_p}; \sin(\omega_p t + \varepsilon_p) \tag{4.1.12}$$

4.2 Stochastic Rolling Motion

The ship is excited into rolling oscillations by external disturbances. If the characteristic curves of the ship — or, in other terms, the coefficients of the equation of motion — change with time, the rolling motion is influenced additionally. Even at relatively modest external disturbances, the change in the system characteristic curves alone, depending on the amplitude, frequency, and phase of these fluctuations, can lead to capsizing. Since the present work is restricted to the investigation of the influence of longitudinal seas,

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no external heeling moments from oblique seas or from wind pressure — which would represent an increase in capsizing hazard — are assumed here.

The rolling motion of the ship — designated in Fig. 4 as the output signal of equivalent System II — is a dynamic process. In Section 3.3 the general form of the equation of motion for the rolling ship with one degree of freedom was derived (see equation 3.3.7):

$$I_{\varphi\varphi}, \ddot{\varphi} + N_{\varphi\varphi}, \dot{\varphi} + \Delta, g, h(\varphi, t) = M_k(\varphi, t)$$

In particular, the influence of heave oscillations of the ship on the rolling motion is not taken into account.

It is known that, for the oscillating ship in the seaway, in addition to the hydrostatically determined fluctuations of the righting levers according to the wave contour at the ship — investigated more closely in Section 4.1 — hydrodynamic effects play a role. One of these is the so-called “Smith effect,” which accounts for the dynamic processes in a wave without the ship being present. It characterizes the dependence of the pressure fluctuations in a wave on depth below the water surface. The Smith effect generally produces a reduction in the hydrostatically determined righting levers. Furthermore, the wave pattern is disturbed by the presence of the ship. The influence of the ship hull causes a deformation of the waves [28, 43]. This deformation generally leads to a smaller effective wave height compared to the undisturbed wave pattern. For regular waves in the linear roll angle range, these effects have already been captured both by measurement and by potential theory. In principle, it appears possible to account, at least approximately, for these hydrodynamic effects also for the ship in the nonlinear range under irregular seas. However, in the present work the validity of the so-called Froude–Krylov hypothesis is assumed,

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which presupposes that the ship does not influence the waves. The Smith effect is also not taken into account in the calculation of the righting levers. Since both effects act in opposite directions, it may be assumed that these simplifications do not accumulate into large errors.

In addition to the hydrodynamic influences on the hydrostatically determined righting moments of a ship in the seaway, the dynamic and hydrodynamic quantities of the ship participate in the oscillating system. They are contained in equation (3.3.7) in the mass moment of inertia I_φφ and in the damping quantity N_φφ. It is certain that the hydrodynamic quantities will likewise experience stochastic changes when the flow around the hull changes stochastically. Thus, the time-irregularly changing form of the submerged hull volume — particularly at large roll angles — will produce changes both in the hydrodynamic mass moment of inertia

$$I_{\varphi\varphi,\text{hydro}}$$

and in the damping quantity N_φφ. Nothing is, however, known about the detailed relationships and the magnitude of these changes in irregular seas at large roll amplitudes. For the numerical calculation, therefore, strong simplifications are made for the time being, for which it can be expected that they do not distort the results.

The resultant mass moment of inertia, composed of the ship’s own mass moment of inertia and the hydrodynamic added mass moment of inertia:

$$I_{\varphi\varphi} = I_{\varphi\varphi,\text{ship}} + I_{\varphi\varphi,\text{hydro}} \tag{4.2.1}$$

can now be related to the ship’s mass:

$$i_{\varphi\varphi} = I_{\varphi\varphi} / \Delta \tag{4.2.2}$$

The radius of gyration i_tr is here set proportional to the ship’s beam B, as is also customary in simplified approaches:

$$i_{tr} = 0.4, B \tag{4.2.3}$$

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The restriction here is to the case of the ship traveling in the direction of the waves — following sea — and to the ship traveling against the wave direction — head sea. For following sea the course angle is X* = 0, for head sea X* = π.

The contour of a single wave traveling in the direction of ship travel, in the space-fixed system, can then be described by the expression:

$$\zeta_p(x, t) = \zeta_{ap} \cos!\left(\frac{2\pi}{\lambda_p} x - \omega_p t - \varepsilon_p\right) \tag{5.1.4}$$

for X = 0.

Transforming to the ship-fixed system, using (5.1.1), one obtains:

$$\zeta_p(x^, t) = \zeta_{ap} \cos!\left(\frac{2\pi}{\lambda_p} x^ - \left(\omega_p - \frac{2\pi}{\lambda_p} v\right) t - \varepsilon_p\right) \tag{5.1.5}$$

The encounter frequency between waves and ship is now:

$$\omega^* = \omega - \frac{2\pi}{\lambda} v \tag{5.1.6}$$

Using the relationship for wave speed:

$$c = \frac{\lambda}{T} = \frac{\omega \lambda}{2\pi} \tag{5.1.7}$$

one obtains:

$$\omega^* = \omega!\left(1 - \frac{v}{c}\right) = \omega(1 - \alpha) \tag{5.1.8}$$

$$\alpha = \frac{v}{c} \tag{5.1.9}$$

which represents the ratio of ship speed to wave speed.

The derived relationships apply to the case of pure following sea. For head sea, with X = π:

$$\omega^* = \omega!\left(1 + \frac{v}{c}\right) \tag{5.1.10}$$

In practice, the formulas for following sea are always used. If the ship reverses direction by 180°, one simply inserts the ship speed v as negative.

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For the ship without forward speed, the space-fixed and ship-fixed systems coincide. The ship-fixed system for the ship without speed will hereafter be called the original system.

The encounter frequency according to equation (5.1.8) will now be examined more closely. For head sea with sign(v) = −1, the encounter frequencies increase with ship speed. In following sea with sign(v) = +1, on the other hand, the encounter frequencies decrease. Depending on the magnitude of v, however, certain ranges arise:

a) v < c, α < 1 — The ship is slower than the wave and is overtaken by it.

b) v = c, α = 1 — The ship travels at wave speed.

c) v > c, α > 1 — The ship is faster than the wave and overtakes it. From the ship, it appears as though the wave is coming from ahead. From equation (5.1.8) negative encounter frequencies are obtained in this case. The negative sign has a tangible meaning: it states that waves, relative to the ship, apparently come from the opposite direction to what is actually the case.

The following reformulation of equation (5.1.8) is introduced using:

$$F = \frac{v}{\sqrt{gL}} \quad \text{Froude number} \tag{5.1.12}$$

$$\Lambda = \frac{L}{\lambda} = \frac{L}{L_w} \quad \text{ratio of ship length to characteristic wavelength} \tag{5.1.13}$$

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$$K = \frac{2\pi}{g \Lambda L_w^{1/2}} \quad \text{[sec]} \quad \text{wave number} \tag{5.1.14}$$

$$\omega = 2\pi f \tag{5.1.15}$$

This yields the following relationship for the encounter frequency:

$$f^* = f(1 - KFf) \tag{5.1.16}$$

Here, F must be inserted as positive for following sea and negative for head sea.

The shift quantity is:

$$\Delta f = Df = KFf \tag{5.1.17}$$

From relationship (5.1.16), the corresponding f* can be computed from the input quantities L_w, Λ, F, and f.

For case b) in following sea, when the ship travels at wave speed, v = c, α = 1, the corresponding wave frequency can readily be stated as a function of ship length and Froude number.

From (5.1.16), setting f* = 0:

$$f_v = \frac{1}{D} \tag{5.1.18}$$

Attention is now turned to the lever-arm spectra of the underway ship. For the ship in the original system (ship without speed, F = 0), let the lever-arm spectrum for a constant inclination angle in longitudinal seas be given by S_hh(f).

Applying the transformation relationships for the sea state to the lever-arm spectrum of the underway ship, the transformed lever-arm spectrum can be written as [66, 36]:

$$S_{hh}^(f^) = \frac{S_{hh}(f)}{|1 - 2\alpha|} \tag{5.1.19}$$

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where α* is defined analogously to equation (5.1.17) as:

$$\alpha^* = \frac{v}{c^} = KFf^ \tag{5.1.20}$$

Inserting (5.1.16) into (5.1.19) gives:

$$\alpha^* = \alpha - \alpha^2 = KFf - (KFf)^2 \tag{5.1.21}$$

Using (5.1.16) and (5.1.21), the transformed spectral value in following sea at frequency f_v according to (5.1.18) follows immediately from (5.1.19) as:

$$S_{hh}^(f_v^) = S_{hh}(f_v) \tag{5.1.22}$$

because f_v = 0 and α*(f_v) = 0. This means that the spectral density at frequency f_v does not change its magnitude in the transformed domain — an intuitively expected result.

In the transformed spectrum a discontinuity can arise in following sea. When the denominator in (5.1.19) becomes zero, S*_hh grows without bound. From:

$$1 - 2\alpha^* = 0 \implies \alpha^*_\infty = \frac{1}{4} \tag{5.1.23}$$

The corresponding frequency is, from (5.1.20):

$$f^*_\infty = \frac{1}{4D} \tag{5.1.24}$$

In the untransformed domain, the corresponding frequency f_∞ follows from (5.1.21):

$$f_\infty = \frac{1}{2D} \tag{5.1.25}$$

with:

$$\alpha_\infty = \frac{1}{2} = 2\alpha^*_\infty \tag{5.1.26}$$

The point of discontinuity is simultaneously the maximum frequency in the transformed domain, as is immediately seen from differentiation of

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(5.1.16) with respect to f:

$$\frac{df^*}{df} = 1 - 2KFf = 0$$

$$f_{\max} = f_\infty = \frac{1}{2KF} \tag{5.1.27}$$

$$\omega_{\max} = \omega_\infty, \quad \alpha_\infty = \frac{1}{2} \tag{5.1.28}$$

The two distinguished points on the frequency axis — f_v and f_∞ in the original system, respectively f* = 0 and f*_∞ in the transformed system — divide the spectrum into three regions that also admit a straightforward physical interpretation.

Figure 17 shows these regions in an untransformed spectrum.

On both sides of the frequency f_∞, the transformed frequencies f* decrease again. At frequencies f = 0 and f = f_v, the transformed frequency f* becomes zero. Of practical importance of the two is only the point f_v. For frequencies f > f_v, the f* values become negative.

In accordance with St. Denis/Pierson, the regions are numbered with Roman numerals in ascending frequency. They are therefore:

Region I: $$0 < f < f_\infty, \quad 0 < f^* < f^_\infty, \quad 0 < \alpha < \tfrac{1}{2}, \quad 0 < \alpha^ < \tfrac{1}{4}$$

Region II: $$f_\infty < f < f_v, \quad f^_\infty > f^ > 0, \quad \tfrac{1}{2} < \alpha < 1, \quad \alpha^* > \tfrac{1}{4} > 0$$

Region III: $$f > f_v, \quad f^* < 0, \quad 1 < \alpha < \infty, \quad 0 > \alpha^* > -\infty$$

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The boundary between Region I and II is defined by:

$$\alpha_\infty = \frac{1}{2}, \quad \alpha^*_\infty = \frac{1}{4}$$

$$2f^_\infty \quad \text{with} \quad f^_\infty = \frac{1}{4KF}$$

Because the spectral regions above and below f_∞ in the original system overlap below f*∞ in the transformed system, the term “fold frequency” or “folding frequency” is introduced for this boundary f∞.

The boundary between Regions II and III is defined by:

$$\alpha_v = \alpha^*_v = 1$$

For f > f_v, f* < 0 (Region III), the spectral components appear from the ship to come from ahead (X* ~ π), even though they travel in the same direction as the ship (X ~ 0). The ship is therefore faster than these components.

In practice, when measuring aboard the ship or observing its effects on the ship, this directional distinction cannot be perceived. Rather, this portion of the spectrum on the negative axis will superimpose with the spectrum on the positive axis at frequencies of equal absolute value.

The location of frequency f_v in the original spectrum is, according to (5.1.18), inversely proportional to the Froude number F and the square root of the ship length L ≡ Λ · L_w. For a given original spectrum, let L_w be a reference length that, according to the classical wave formula:

$$L_w = \frac{g}{2\pi f^2}$$

corresponds to an encounter frequency of the spectrum. The median f_med of the original spectrum was chosen as reference frequency, with f_med ≈ 0.125 Hz and L_w = L_med ≈ 100 m.

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This form of representation is meaningful because, for practical investigations, a sea-state spectrum of the operating area or the accident location must always be known. To this spectrum, the ship is then assigned with its data — here essentially its length and its speed — when following sea is involved.

Depending on ship length L, for a constant Froude number one obtains different boundary values f_v. For ship lengths L between 50 and 200 m, corresponding to Λ values between 0.5 and 2, the f_v values are plotted versus F with Λ as parameter in Figure 18. A log–log representation yields straight lines. It can be seen that with increasing Froude number, Region III of the spectrum becomes ever larger.

Frequency Region III — the region with negative transformed frequencies in following sea — grows only with the square root of ship length. On the other hand, Region III vanishes to zero when no component in the original spectrum travels slower than the ship, c > v, i.e., when the boundary frequency is greater than the largest upper frequency still to be considered in the original spectrum: f_v > f_0.

For further numerical treatment, the special case is restricted to:

$$\Lambda = 1, \quad L = L_w = L_{\text{med}} \tag{5.1.29}$$

The corresponding transformed spectra are shown graphically in Figure 20 for various Froude numbers. If the folding frequency f*_∞ is greater than f_0, then even in following sea only Region I occurs. For Λ = 1 this is the case up to a Froude number of F = 1/(KF_0). For F > 1/(KF_0), Region II is added to Region I (here F = 0.0775 with K = 20 s and f_0 = 0.3225 Hz).

Region III first appears for F > 1/(2KF_0)

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(here F = 0.155 with K = 20 s and f_0 = 0.3225 Hz).

Figure 19 shows the transformed spectrum in following sea for a Froude number of 0.2. It also contains the resulting transformed spectrum S*_d plotted over the absolute value of frequency.

5.2 Transformation of the Bandwidth

It is to be investigated how the width of the spectrum transforms — i.e., what frequency range Δf* in the transformed domain the frequency range Δf = f_0 − f_u of the original domain covers. This question has practical significance in that when the total energy of the spectrum is distributed over a narrower band, the intensity within that band must increase, and preferred excitation frequencies supply higher energy.

Dangerous resonance of the ship to irregular excitation can more readily occur the narrower the transformed spectrum of the underway ship is. Of additional importance, however, will be whether the maximum possible power density for the given original spectrum also concentrates in this narrowest band. That is certainly the case when the band center frequency f_m = f_∞ lies in the region of greatest spectral density. In that case, approximately at f_med or f_mod (median or modal value), the peak of the original spectrum will map onto the narrowest band Δf_min.

For Λ = 1, L_w = 100 m, and f_med = 0.125 Hz, a critical Froude number of F_min = 0.2 is obtained from (5.2.5).

If the ship is longer than the waves of maximum spectral density, the critical speed shifts to a smaller Froude number; if the ship is shorter, to a larger Froude number.

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This is entirely obvious, because the absolute ship speed v must maintain the same ratio to the wave speed c, and with larger L, F = v/√(gL) becomes smaller and vice versa.

A band of constant width in the transformed domain with f_∞ as upper boundary, e.g., Δf_c, captures different areas of the original spectrum depending on the Froude number. Within the transformed band Δf* or Δf_c, the transformed spectral density Sd is not uniformly distributed, but for certain Froude numbers is strongly asymmetrically concentrated toward the upper band boundary f*∞. The greatest asymmetry in the band Δf*_c occurs at Froude number F_min. A clear picture is given by the transformed spectra in Figure 20 for various Froude numbers.

Figure 21 shows the result of a moment calculation of the transformed spectra in following sea presented in Figure 20, with respect to the discontinuity point f_∞, as a function of Froude number F with Λ as parameter. The quantity M_1 here represents the centroid of the area under the spectrum relative to f_∞. For Λ = 1 — i.e., ship length equal to characteristic wavelength — the minimum occurs at F_min = 0.19.

Changes in Λ produce shifts of F_min. Since a ship will certainly frequently encounter sea-state spectra in this Λ range, a very narrow encounter spectrum and thus resonance is possible from case to case within quite a wide range of Froude numbers. The calculation of the second-order moment with respect to f*_∞ yields the same tendency of the curves as in Figure 21.

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5.3 Discrete Representation of the Transformed Spectra

For the computational representation and generation of time functions from the transformed spectra, it is expedient to approximate them by discrete sinusoidal components:

$$H_r(t) = \sum_{p=1}^{N} h_{ap} \cos(2\pi f^*_p t + \varepsilon_p) \tag{5.3.1}$$

For this purpose, the corresponding spectral densities are plotted over the absolute value of frequency:

$$S_d^(|f^|) = S_{hh}^(f^) + S_{hh}^(-f^) \tag{5.3.2}$$

This corresponds to the physical reality that the ship experiences only this superposition and makes no distinction based on the sign of the frequency. For the discrete representation this approach has the advantage that with the same number of components a frequency range twice as wide (for + and −) can be captured. This is especially important for representation on the analog computer due to the limited number of computing components.

The S*_d values are thus determined from the sum of the three partial regions:

$$S_d^(|f^|) = S_I^* + S_{II}^* + S_{III}^* \tag{5.3.3}$$

In the original domain, a normalized spectrum with log-normal distribution was taken as the basis — see Figure 17 — in the form:

$$S_n(f) = \frac{1}{f} \varphi(u), \quad \int_0^\infty S_n(f), d(\log f) = 1$$

$$u = \frac{\log f - \log f_{\text{med}}}{\sigma}, \quad f_{\text{med}} \text{ median}, \quad \sigma \text{ standard deviation}$$

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$$\varphi(u) = \frac{1}{\sqrt{2\pi}} \exp!\left(-\frac{u^2}{2}\right)$$

$$H = \log e = 0.4343$$

The following numerical values were chosen:

$$\log f_{\text{med}} = -0.90309 \quad \text{for } f_{\text{med}} = 0.125 \text{ Hz and } L_{\text{med}} = 100 \text{ m}$$

$$\sigma = 0.1144$$

$$f_u = 0.0484 \text{ Hz} \quad \text{for } u = -3.6$$

$$f_0 = 0.3225 \text{ Hz} \quad \text{for } u = +3.6$$

The discrete representation is carried out as follows: the frequency range Δf* to be captured is subdivided into n sub-intervals Δf*_i. A meaningful criterion for the type of subdivision is the condition:

$$\frac{1}{n} = \int_{\Delta f_i} S_d^(|f^|), df \approx \text{const} \quad \text{for } i = 1, 2, \ldots, n \tag{5.3.5}$$

This means in practice that at locations of higher power density the frequency axis is subdivided more finely — i.e., a more precise discrete capture of spectral regions of higher density. If the spectra S_d are not given in closed analytical form, exact adherence to this condition is only possible through iterative calculation. A simple method was therefore chosen that approximately satisfies condition (5.3.5). Three different fixed algorithms for domain subdivision were programmed. Depending on the type of spectrum S_d, one of these subdivisions is used that best satisfies the chosen condition.

These are:

1. Logarithmic subdivision with the generating law:

$$\Delta f^*_p = \Delta f_m \cdot u^{p-m}, \quad m = \text{ENTIER}!\left(\frac{n}{2}\right) \tag{5.3.6}$$

Page 48

Suitable for spectra having the character of a log-normal distribution, primarily for transformation during ship travel against the waves and for following sea up to Froude numbers of 0.1. The reduction factor u together with the number of components n determines the frequency range captured.

2. Arithmetic subdivision with the generating law:

$$\Delta f^_p = \frac{n - p + 1}{n} \cdot f^_\infty \cdot \Delta p$$

Suitable for transformed spectra in the range of Froude numbers 0.15 ≤ F ≤ 0.30 in following sea. Figure 22 shows the arithmetic subdivision into seven components for F = 0.25.

The arithmetic subdivision yields an increasingly fine subdivision with increasing frequency f. The narrowest frequency interval is at the discontinuity point f*_∞.

3. Linear subdivision:

$$\Delta f^_p = \frac{1}{n} f^ \quad \text{for all } p$$

Suitable for all transformed spectra S*_d with regions of approximately constant density, for Froude numbers 0.3 < F ≤ 0.4 in following sea.

For all subdivisions, f_∞ was chosen as the upper boundary of a frequency interval. The program allows specification of how many components are to lie above frequency f∞ and how many below. Power density concentrates numerically above f*∞. With increasing Froude number in following sea, however, the tail of the transformed spectrum toward negative frequencies grows, increasingly exceeding the magnitude of f*_∞ — see also Figure 20.

Page 49

After the subdivision is established, the integrations of the spectrum S*_d over the frequency intervals are not carried out in the transformed domain.

Using the relationship:

$$\int_{\Delta f^} S^, df^* = \int_{\Delta f} S, df$$

the back-transformation from f* to f is:

$$f = f(F, K) = \frac{1 \mp \sqrt{1 - 4KFf^}}{2KF} = \frac{1}{2\alpha^} \tag{5.3.10}$$

Both the negative and the positive frequencies f must be inserted, and for each, because of the square root, two solutions are obtained. For positive frequencies f*, the negative root gives Region I and the positive root gives Region II.

For negative frequencies f*, the positive root gives Region III.

The fourth region, which formally results from the negative root with f* < 0, is practically without significance. It maps the negative frequencies in the original domain: f_IV < 0.

A region integral in the transformed domain is therefore composed of three partial integrals in the original domain:

$$\int_{\Delta f_i} S^, df^ = \sum_{l=\text{I}}^{\text{III}} \int_{\Delta f_l} S, df \tag{5.3.11}$$

It must further be mentioned that the spectra presented in the figures

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Damping Term

The quantities in the damping term of equation (6.2.3) are defined as follows:

$$d_{\varphi n}^{*} = 2,\delta_{\varphi n} = \frac{2}{k_o},D_0,\overline{\omega}_0$$ (6.2.11)

normalized damping factor
damping fraction
decay constant
mean natural frequency of the system in irregular seas
damping measure for the linearized system

$$D_B \quad \text{= reference damping measure}$$ (6.2.12)

Here $\widetilde{D}{\varphi n}$ is a normalized nonlinear damping function that is set on a function generator. The normalized quantities $d{c,n}$ and $d_{p,n}$ are coefficients set on potentiometers.

$\widetilde{D}_{\varphi n}$ was assumed proportional to the magnitude of the cube of the heel angle:

$$\widetilde{D}_{\varphi n} = 1{,}5,|\varphi|^3$$ (6.2.13)

The individual quantities in the lever term of equation (6.2.3) are:

$$c_n = \frac{c \cdot \varphi_{nB}}{1} = \frac{1}{2} \cdot \frac{\overline{\omega}0^2 \cdot \varphi_s}{\varphi{nB}} \cdot \frac{h_B}{\varphi_{nB}} \cdot \frac{1}{k_o^2}$$ (6.2.14)

with $h_B$ as the reference quantity for normalizing the mean lever $\bar{h}$. The coefficient $c_n$ is set on a potentiometer. It is a measure of the mean stiffness of the system and is

(page 56 continued)

quadratically dependent on the time scale $k_o$. The coefficient

$$\alpha_n = \frac{\Delta h_B}{h_B}$$ (6.2.15)

gives the ratio of the reference quantity for the lever variation to the reference quantity for the mean lever.

The quantities $\alpha_h$ and $\alpha_A$ are coefficients that determine the linear fraction in the mean lever $\bar{h}$ and in the lever variation $\Delta h$. The levers marked with a tilde denote the nonlinear fractions:

$$\bar{h}(\varphi) = \alpha_h,\varphi + \widetilde{h}(\varphi)$$ (6.2.16)

$$\Delta h(\varphi) = \alpha_A,\varphi + \widetilde{\Delta h}(\varphi)$$ (6.2.17)

where

$$\widetilde{h}_n(\varphi_n) = \frac{\widetilde{U}_n(\varphi_n)}{h_B}$$ (6.2.18)

$$\widetilde{\Delta h}_n(\varphi_n) = \frac{\widetilde{\Delta h}(\varphi)}{\Delta h_B}$$ (6.2.19)

The two functions $\widetilde{h}_n$ and $\widetilde{\Delta h}_n$ are set on function generators. Figures 24 and 25 show the course of these curves for the various computation cases in normalized form.

In accordance with the time transformation (6.2.4) from real time to computer time, computation is performed with the time function $\xi(\tau)$ instead of the exciting time function $\xi(t)$.

The irregular time function is formed, according to Section 5.3, from a superposition of harmonic components:

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$$\xi(\tau) = \sum_{p=1}^{N} c_p^,\sin!\left[\Omega_p^,\tau + \varepsilon_p^*\right]$$ (6.2.20)

$$\overline{\xi}(\tau) = \xi(\tau) / \hat{\xi}$$ (6.2.21)

The normalized frequencies for computer time are then

$$\Omega_{p,n}^* = \frac{1}{k_o},\Omega_p^*$$ (6.2.22)

The quantity $k_o$ is chosen so that the potentiometers are well driven for as many computation cases as possible. For $k_o = 1$ and $\frac{1}{\varphi} = 3$, the computations run three times faster than the actual process aboard ship. By switching to $k_o = 10$, computation proceeds ten times faster. A time scale factor of 30 is then achieved for the statistical calculations.

With regard to the computer-side aspects of normalization and the significance of the amplifier characteristic $k_o$, reference is made to the book by G i l o i and L a u b e r [9], which was the primary reference used.

6.3 Control Circuitry on the Analog Computer for Statistical Solutions

Three desktop computers RAT 700 by Telefunken, with a total of 45 amplifiers, were available for the calculations. One computer (abbreviated AR 1) was set up with the equation of motion to be investigated; the other two computers (AR 2 and AR 3) provided the excitation for the equation of motion. Computers 2 and 3 are connected in parallel by a cable; computer 2 takes over control of computer 3.

When a preset threshold value of $\varphi$ in the equation of motion is exceeded ($0.9$ for $\varphi \leq 90°$), a comparator on AR 1 reconnects the p-contact of AR 1 to ground.

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AR 1 thereby switches back to pause (indicated by the white pause button lighting up).

All values at AR 1 are thereby erased; one run of the equation of motion is complete. AR 2 and AR 3 remain switched on and continue computing.

The threshold exceedance of $\varphi$ causes, in addition to putting AR 1 on pause, the activation of the timing circuit in AR 2 and AR 3, and the cycle begins again from the beginning as described above: after a set pause duration $\tau_p$ is reached, AR 1 is switched on and starts computing. At the beginning of computation, the timing circuit is discharged and can therefore start from zero again at the next end of a computation run on AR 1 (when the threshold value of $\varphi$ is exceeded). It is possible, during the computation period, to preselect a different pause duration for the following pause by adjusting a potentiometer.

The pause time and computation time thus alternate completely automatically, with the timing circuit on AR 2 and AR 3 switching on the equation-of-motion circuit on AR 1 and switching itself off and resetting, while AR 1 via its equation-of-motion circuit switches on the timing circuit and switches itself off and resets. In this way, statistical long-term calculations with a large number of individual runs can be carried out automatically.

The analog value for the number of runs is fed to the Y-input of an XY/t recorder, which plots the duration of each computation run against its sequence number — see the histograms in Section 7.4.

6.4 Accuracy

As mentioned in Section 6.1 on the advantages and disadvantages of analog computation, thorough accuracy checks of the analog circuit are necessary. Since the results depend not only on the calculus but also on the accuracy of the computing elements involved, the accuracy considerations must be more extensive than for a purely digital program. If, as in the present case, only a computer of a certain accuracy class is available, the first question is whether this computer can yield useful numerical solutions at all. If this question is answered in the affirmative, the problem must be programmed in a computer-compatible manner — above all, the computing elements must be driven as close to full scale as possible, since only then can the available accuracy class be fully exploited. Various test methods, which check the circuit for systematic errors via known properties of the programmed system, provide the necessary confidence for reliable execution of the calculations.

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A desktop computer of the so-called $10^{-3}$ accuracy class was used. With computers of this class, value ranges of $10^3$ can be captured. For the roll angle range of $100°$, this means a maximum resolution of $0.1°$.

The following analog computing elements are involved in the circuits: amplifiers connected as integrators or summers, multipliers, potentiometers, function generators, relays, and diodes.

For the calculations performed, three sub-circuits can be distinguished: the computation circuit for the equation of motion, the circuit for generating the excitation, and the control circuit for the automatic execution of the statistical computations.

Apart from all test procedures, the clearest proof of accuracy is a comparison with solutions obtained numerically by a more accurate method, such as a digital computer. If one wishes to carry out this comparison for all the computed variations, the effort involved would negate the actual advantages of analog computation. It therefore seems appropriate, after performing various tests and comparing with easily surveyed system parameters, to proceed with the actual calculations. Relatively fast results are then obtained for the solution of the nonlinear equation of motion with irregular parametric excitation.

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For the computation of individual cases, a digital program was developed. A program for computing roll motions in regular seas, developed by A b i c h t and Z u n k e r [29, 30] at the Chair for Ship Design at the Institute for Shipbuilding of the University of Hamburg, was available and could be extended to the irregular approach. The step-by-step computation of the motion sequence is based on the Runge-Kutta method [27]. A prescribed accuracy is maintained by incorporating the Nyström criterion.

Figure 26 shows results of comparison calculations. Since all three analog computers were only simultaneously available for a short time, these calculations were limited to a single harmonic component of the excitation and to linear damping. Good agreement between the analog- and digital-computed roll angles can be reported. Figure 26 also clearly illustrates the nonlinear distortion of the harmonic input signal in substitute system II. Furthermore, a case was chosen that demonstrates the resonance effect particularly clearly. At negative roll angles, a wave crest passes that reduces the effective restoring levers and causes greater heeling. At positive roll angles, a wave trough passes, whose larger levers reduce the roll amplitudes. Since the sequence of crests and troughs coincides exactly with the sequence of roll amplitudes, the ship builds up oscillation to large negative roll amplitudes that lead to capsizing.

7. Results

7.1 Overview of the Computation Cases

Using the described analog computer circuit, distributions of capsizing voyage times were determined for various parameters. In addition, a digital program is now available that carries out the same computational approach and was used for verification computations and long-term calculations. Several individual cases were computed digitally and the relative frequency of roll angles was determined — see Section 7.5.

As described in Section 2.5, it is expedient for determining the capsizing safety of a ship to determine the frequency of capsizing events. To limit the computational effort, cases that lead to capsizing relatively quickly are computed through. This method was also applied in the Plön capsizing experiments. Although on an inland lake with model-like sea spectra the measurement runs are longer than in a tank, the duration of the test runs and the extent of the statistically determined measurement data still play a role. In the experiment, cases are therefore preferably investigated in which capsizing occurs with a frequency sufficient for statistical evaluation. The same principle applies to a computational approach with regard to computation time. An additional consideration when using the analog computer is that long computation times were not possible due to the error influences of its computing components. This applies particularly to the damping of the excitation amplitudes $c_p^*$ on the analog computer as a function of the number of periods elapsed and the frequency. Furthermore, drift errors of the computational amplifiers can have less effect the shorter the computation times on the analog computer. Due to the dynamic errors of the computing components, the frequency must, however, remain below 100 Hz with time compression. For all calculations on the

(page 62)

analog computer, a uniform time scale of $\lambda = 30,\mu$ was introduced.

On the analog computer, low mean levers and very large variation ranges of the restoring levers were investigated. Calculations were performed with significantly larger lever variations than those resulting from the standard calculation wave of height $H_W$ for the hull form of Series 60 — specifically with $\hat{h} = 1$ in equation (4.1.11). The values of $c_p^*$ are given in Table (6.2.6). The amplitudes decrease over computation time. By grouping amplitudes into ranges, a classification is obtained. The statistical calculations were usually not continued until the excitation amplitudes had decreased to the value $\Delta h_{M2}$, since under these initial conditions and mean levers hardly any capsizing occurred. The pre-heel was restricted to the smallest value practicable on the analog computer, $0.05$. The initial values $\varphi_0$ and $\dot{\varphi}_0$ were not imposed and were therefore set to zero. Buildup of rolling motion was achieved on the analog computer through the change in restoring lever with pre-heel $f_v$. The ship speed, linear damping, and height of the ship’s center of gravity were varied. A total of 3,000 capsizing cases were computed.

For practical reasons, the new phase drawing of the excitation for the next computation case was carried out using uniformly distributed draws of a pause time $t_p$, as described in Chapter 6, with the original time function being retained. This yields, in accordance with the associated relationship

$$\xi(t) = \sum_{p=1}^{N} c_p^,\sin!\left[\omega_p^,(t + T_0) + \varepsilon_p^*\right]$$ (7.1.1)

for various $T_0$, various new excerpts from the sea state function — or in other words, new phase draws, as can easily be seen from reformulating the argument:

(page 63)

$$\xi(t) = \sum_{p=1}^{N} c_p^,\sin!\left[\omega_p^,t + \varepsilon_p^{*,\text{new}}\right]$$ (7.1.2)

with

$$\varepsilon_p^{,\text{new}} = \omega_p^,T_0 + \varepsilon_p^*$$ (7.1.3)

By sequentially computing capsizing cases over the selection of $T_0$, short-term computation is obtained for the equation of motion while long-term computation is obtained for the excitation circuit.

The times $T_0$ are then composed of the sum of the preceding capsizing voyage times $t_{K}$ and pause times $t_p$:

$$T_{0,i} = t_{p,i} + \sum_{j=1}^{i-1}\left(t_{K,j} + t_{p,j}\right)$$ (7.1.4)

In the digital program, the stochastic time function $H(t)$ according to equation 7.1.1 was programmed so generally that other time excerpts can be selected by computing uniformly distributed phase draws for $\varepsilon_p^$, but also by inputting various times $T_0$. By calling a procedure that generates uniformly distributed random phases, even for a very large number of spectral components, frequent re-drawing of the values $\varepsilon_p^$ is hardly more effort. Inputting $T_0$, however, also allows a new computation to be performed using the same numbers from the pseudo-random procedure (which repeat when the program is reloaded), while still selecting a different time excerpt.

The restriction to seven components on the analog computer without compensation for amplitude damping was determined by the capabilities of the available desktop computer. With seven components, the resulting period of the time function $H(\tau)$ is already so large that one can speak of irregular excitation — particularly for the discrete representation of spectra that become narrower under transformation.

The larger the number of discrete harmonics, the better the approximation to the continuous spectrum. For the long-term digital calculations, 41 components were used.

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The width of the mean normalized lever spectrum — Figure 17 — was initially chosen heuristically in reference to a sea state spectrum measured on Lake Plön [46]. According to already available calculations and analyses of stochastic levers — Figure 14 — generally even narrower spectra will occur. In the digital program for computing the discrete components, the desired spectral width can easily be changed by inputting a $t_R$ value. This value specifies the magnitude of the scatter of the log-normal distribution:

$$\sigma_f = \frac{\log f_o - \log f_u}{2,t_R}$$ (7.1.5)

An underlying $t_R$ of $3.6$ applies to the analog calculations. For the digital long-term calculations in Section 7.5, $t_R$ was doubled — that is, computation was performed with half as large a spectral scatter.

The statistical results determined in this manner can already demonstrate some relationships quantitatively; they are not sufficient for a thorough understanding of the behavior of this nonlinear oscillation system. Important parameters to add are the influence of various damping moments and various initial conditions. Since the superposition principle for different input signals:

$$h(t) = h_1(t) + h_2(t) \neq \text{const}$$ (7.1.6)

is not valid for nonlinear systems, all occurring cases must be investigated separately.

To develop the approach and compute initial solutions, the available analog and digital computer resources were exploited. There is no doubt that, with the rapid further development of computing technology, the problems described here will soon present no further difficulties.

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Hybrid computing technology, combining the advantages of analog and digital computation, can be particularly useful for this task.

7.2 Characteristic Capsizing Processes

From the statistical multitude of capsizing processes investigated on the analog computer, several individual cases were selected and the various motion quantities were recorded over time.

Capsizing generally occurs through a large wave crest that strikes the ship at a large roll angle. If the roll angle and lever reduction coincide such that the ship is on the steeply descending branch of the lever curve — which then acts to increase inclination rather than to restore it — capsizing occurs very rapidly. The fact that a wave crest coincides with this roll angle is subject to chance. An individual computation case is indeed reproducible, since specific phase draws of the excitation and specific initial conditions for the equation of motion were chosen, and the overall time sequence was kept short. For a ship that has been in following seas for an extended period, it would be only one possible individual case among many other possible capsizing situations — statistically speaking, one draw from the population of capsizing events.

A distinction must therefore be made: an individual case is examined on the one hand to investigate the relationships of the motion process and make it accessible to calculation, while statements about the expected behavior of the ship and its capsizing safety can only be probabilistic statements.

From the examination of individual cases, it can be intuitively recognized that the hydrostatically determined restoring levers in irregular seas should be considered together with the other hydrodynamic quantities when ensuring or improving capsizing safety. Of primary importance is the existence of sufficient restoring levers. In irregular seas, these are defined by mean value and variation range. If the mean levers are too small and the lever variations are very large, measures via damping have hardly any effect. On the other hand, damping — if the levers are large enough that capsizing is only rarely to be expected — can reduce dangerous roll amplitudes that could lead to capsizing upon coincidence with a wave crest. The phase curves illustrate this well: with greater damping, they describe smaller circles.

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Figure 27 shows a typical resonance case for mean excitation period equal to mean natural period at a Froude number of 0.2. The circles of the phase curve grow continuously larger without significant irregularity until the angle of 80° is reached, from which the ship can no longer right itself but capsizes completely.

By contrast, at the same Froude number of 0.2 but a lower mean lever $\bar{h}$, the ship exhibits detuning, and capsizing then no longer occurs through such a regular buildup of oscillation. Such behavior was to be expected based on the considerations in Section 5.2.

The fact that the frequent occurrence of resonance averaged over time also affects the mean capsizing voyage time will be seen in the statistical evaluation. An increase in damping, however, produces an unambiguous elimination of the capsizing risk for this resonance case. Capsizing occurred only rarely for the doubled damping measure — $D_0 = 0.05$ — while for a damping measure of 0.10 and 0.15 no capsizing was recorded at all.

Figures 28 to 30 show the motion sequence for a capsizing case in following seas at the high Froude number of 0.4. The individual wave crests in irregular succession strike the ship at different roll angles, causing different roll deflections. The large roll angle of approximately 40° in this case then decays rapidly due to the following wave trough and damping; however, the next large crest strikes the ship just when it still has a large amplitude of approximately 20° and brings it to capsizing. Figure 29 shows the corresponding angular acceleration plotted against roll angle $\ddot{\varphi} = f(\varphi)$; Figure 30 shows the corresponding phase curve $\dot{\varphi} = f(\varphi)$.

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The time-domain representation in Figure 28 also clearly shows the time-varying natural period of the ship. For a ship rolling about the angle of heel, the natural period at small amplitudes is

$$T_1 = \frac{2\pi,i_\varphi}{\sqrt{h - k}}$$ (7.2.1)

where:

$h$ = restoring lever
$k$ = resultant heeling lever
$i_\varphi$ = radius of gyration
$t_s$ = heel — intersection of $h$ and $k$, hydrostatic equilibrium

The temporal variations of the restoring levers cause a change in the natural period of the oscillating system — for small roll deflections in the linear range:

$$T(t) = \frac{2\pi,i_\varphi}{\sqrt{\left.\frac{d}{d\varphi}\left[h(t) + k\right]\right|_{\varphi = t_s}}}$$ (7.2.2)

In addition, at large amplitudes the influence of the nonlinearity of the levers is superimposed, so that the roll period becomes a function of both time and angle.

7.3 Statistical Evaluation (continued)

If the constant heeling lever produces a large list angle, the period variation as a function of angle becomes dependent on the direction of motion for large roll amplitudes. With a pre-existing heel to starboard, the sub-linear shape of the lever curve causes the half-period for the starboard excursion to be greater than for the port excursion.

The half-periods of a rolling ship in following seas again possess a probability density. It is particularly important to note that the mean natural period of the ship in seaway T_m only rarely coincides with the free-roll natural period of the ship in calm water. This is already evident from the differing lever curves of the ship in calm water and in a seaway. It is therefore of little value to search for resonance points of the rolling ship by comparison with the calm-water roll period, just as the calm-water lever curve must be abandoned when operating in a seaway.

7.3 Statistical Evaluation

From the set of capsize calculations performed with variations of various parameters, groups of data for time-to-capsize can now be assembled. Statistical evaluation of these groups yields a statement about the frequency of occurrence of capsize events within a given time interval, or the expected mean time-to-capsize together with its associated distribution function.

In total, more than 3,000 capsize cases can be evaluated statistically from the computed results. More cases were computed, but those exhibiting very long times-to-capsize are excluded from the statistics because determining a larger sample for such cases would require excessive computation time.

Sample Sizes

Wrongly accepting a hypothetical distribution function (d.f.) on the basis of a sample is called, following the theory of Neyman and Pearson, a type-II error. The probability of error for a type-II error is denoted by beta, and in general one may write:

(7.3.1)

The probability of a type-II error decreases as the sample size increases. The larger the sample, therefore, the smaller the risk of determining an incorrect distribution function; in other words, the larger the sample, the more confidently one can conclude that the sample genuinely reproduces the essential parts of the population and permits a reliable statistical statement. Excessively large samples, however, unnecessarily increase the effort if they can no longer provide additional information.

More than one hundred capsize cases were computed on the analog computer per configuration. Smaller samples with sizes below 20, which arise from classification of the excitation amplitudes, were no longer evaluated statistically. Sizes above 130 were not pursued; computations were terminated at that point.

If the type of a distribution is known from theoretical considerations, then only the parameters of the distribution need to be fitted to the respective sample. This task is simplified here because only one parameter must be determined. The goal is to find the best possible estimate of the correct parameter value from the sample.

By the maximum-likelihood method, the mean value of the available data is the most probable value. The mean is therefore computed from the available t_K values as the estimate:

M = (1/n) * sum_{i=1}^{n} t_{K,i} (7.3.2)

If the so-called incubation time t_0 is taken into account, one must write:

M = (1/n) * sum_{i=1}^{n} (t_{K,i} - t_0)

Once the type of distribution has been established from theoretical considerations and the respective parameter determined from the computed samples, the theoretically expected distribution for a given sample is at hand. A hypothesis test can then be used to examine the extent to which this expected distribution function agrees with the actually observed sample.

First, the hypothesis is stated that the present sample originates from a population possessing a specific theoretical distribution function with the computed parameters. This hypothesis is the null hypothesis; let the associated d.f. be denoted F_0. Let the distribution function derived from the sample be F_1. The test procedures for the null hypothesis are based on applying criteria for the deviations between the two distribution functions. It is then possible to render a judgment as to whether the null hypothesis must be rejected on the basis of the present sample. However, if the null hypothesis cannot be rejected, this does not yet mean that it has been confirmed.

(By way of illustration: otherwise one might, for example, with only two values of the sample distribution function that happen to coincide with the hypothetical distribution function, and without knowledge of further values, conclude that the null hypothesis has been proved — which is clearly inadmissible.) It can only be stated that a test result does not speak against the null hypothesis. If, however, the hypothesis is rejected even though it is correct, this is called a type-I error. The associated error probability is denoted alpha, and one may write:

P{reject | F_1 = F_0} = alpha (7.3.4)

For the comparison of the distribution function with the sample, a difficulty arises. The theoretical distribution function is continuous, whereas the sample consists of discrete individual values. If the discrete sample values are grouped into intervals and the values per interval are counted, then, normalized to the sample size, a relative frequency within the chosen interval is obtained, which can be compared with the theoretical probability for that interval. This grouping into intervals is called classification; the respective intervals are the classes. For each class, the number of sample values falling into it is counted.

No universally valid rules can be established regarding the number of classes to be chosen. It is immediately apparent, however, that the sample must not be too small for classification. Most authors regard an absolute frequency of five values per class as the minimum. Since the shape of the distribution function is to be determined, a number of points on the curve — corresponding to a number of classes — must be established. A relationship between the feasible or required number of classes and the sample size is therefore plausible. The number of classes can only be fixed once the sample size is known. Various empirically founded recommendations exist for the class

number. For the digital classification program used for evaluation, the following relationship was chosen:

m = [function of M] (7.3.5)

where M is the sample size.

The largest value of the sample, t_{K,max}, is then determined and divided by m. This gives the class width:

H = t_{K,max} / m (7.3.6)

If this partitioning results in fewer than five sample values falling into individual classes, those classes are merged, starting from the top, with the adjacent lower class.

Through classification, points of an experimental distribution function are obtained from the discrete sample; these can be compared with the corresponding points of the expected theoretical distribution function. For this comparison, the so-called chi-squared method (chi^2 test) is available. It permits testing when the type of distribution is known but the parameters must be estimated, as is the case with the present time-to-capsize distributions. The Smirnov–Kolmogorov test is additionally parameter-free, i.e., it requires no information about theoretical values. For the ordered sample, a confidence band is specified within which any curve is admissible.

7.4 Mean Time-to-Capsize for Various Parameters

The following parameters were varied:

Ship speed
Spread of the righting levers
Linear damping coefficient D_a
Height of the center of gravity (expressed by the calm-water initial stability GM_GW)

Variations of the heeling moments and of the initial conditions were not evaluated statistically on the analog computer.

Figure 32 shows, as an example, the histogram from a reproducibility test carried out before each computation cycle. The initial conditions are constant, as are the pause times between a capsize and the start of a new computation. A repetition shows the identical computation sequence.

Figures 33 and 34 show various histograms as recorded by the Y-t plotter for different computation cases. For the statistics, the pause time was varied according to a random table between 0 and 10 seconds.

In addition, for some cases different segments of the excitation function were selected by inputting different draws of uniformly distributed initial phases. These cases are designated P0 and P1. It is found that differences in the results are still noticeable for these two draws. Since both time functions are draws from the stationary stochastic process, a stationary value must also be obtained for the distribution of the ship’s time-to-capsize. In principle, a single draw of the excitation function would suffice if the time duration were chosen long enough. This is not possible with the circuit used on the analog computer, as described in Chapter 6, because of the damping of the excitation amplitudes. With this circuit, several such phase draws would have to be computed, and it is not possible to limit oneself to the phase draw described in Section 7.1 by choosing the pause times. Since this would mean resetting the potentiometers for the initial phases of the excitation components, this was foregone when computing the remaining parameter range. The essential trends can also be identified with a single drawn time function. For improving the accuracy of the results, it is considered more expedient to employ correspondingly better computing equipment.

In Figure 35, the mean values for both phase draws P0 and P1 are combined. The case under consideration is a ship in following seas with a Froude number of 0.2 and a linear damping coefficient of 0.025. It is clearly recognizable that a lowering of the ship’s center of gravity by 150 mm here produces an increase in capsizing danger — entirely contrary to the intuitive expectation. For comparison, Figure 35 also shows the result for a Froude number of 0.4. For this case, a larger GM on average leads to less frequent capsizing. How is this difference to be explained?

The cause of this different behavior of the ship in following seas at different speeds can only be sought in resonance. This is evidently a case in which, even in irregular seas, the periodically possible resonant oscillations are sufficiently frequent that they occur not only in relatively short time intervals but also over longer periods, and thus become dominant for the ship’s behavior even in the statistical mean. A first indication and a mathematical derivation of this effect in irregular following seas was given by Grim in 1961 [42].

It is known that for Mathieu-type excitation, resonance occurs for the following ratios of excitation frequency to natural frequency of the system:

f_err / f_eig = 1/2, 1, … or equivalently f_eig = 2, 1, … * f_err

Taking f_00* as the characteristic frequency of the excitation, the natural period of the ship for resonance must be:

f_eig = f_00* = approximately 62.5 mHz at F = 0.2

T_eig = T_err = 16 [sec] (T* = 7 [dimensionless])

or:

f_eig = (1/2) * f_00* = approximately 31.25 mHz

T_eig = 2 * T_err = 32 [sec] (T* = 10 [dimensionless])

If, therefore, the mean natural frequency of the ship lies near f_00* (or slightly below it), and the ship’s speed concentrates the energy content of the sea spectrum very close to that frequency, then a case approaching regular excitation is possible even in irregular seas with a continuous spectrum. The regularity promotes a build-up of rolling motion that can extend over several periods without being interrupted and damped by an irregularity in the excitation. This is the case here for a Froude number of 0.2 and a mean natural period of T_m = 16 s (corresponding to a dimensionless characteristic number T_m* = 5 for L = 100 m). The corresponding calm-water GM as the characteristic number for the height of the center of gravity is, for the computed ship, GM_GW = 25 cm.

The resonance phenomenon just described is also expressed in the reduction of the mean time-to-capsize shown in Figure 35 for GM_GW = 25 cm compared to GM_GW = 10 cm.

The comparatively regular build-up of rolling until capsizing is clearly visible in the phase curve in Figure 27. In contrast, for the higher Froude number of 0.4 no resonance phenomenon is discernible; instead, a larger GM reduces the capsizing danger. Comparing Figure 35 at F = 0.4 with the case F = 0.2, it is striking that despite twice the damping, the mean time-to-capsize is shorter. This finds its explanation in the fact that at high forward speed in following seas, large reductions in the righting lever can persist for relatively long periods. This case is sufficiently well known for regular seas. It had not previously been possible, however, to make any statement about the possible duration of such reductions in irregular seas or their quantitative effect on rolling behavior. Grim [42] provided indications of the expected behavior in irregular seas at high Froude numbers of 0.3 to 0.4. Grim calculated the mean amplitude of an effective wave at the ship within a chosen time interval. For equal time intervals, the mean value increases with increasing Froude number.

This tendency also corresponds to the intuitive picture that at higher Froude numbers in following seas, the ship can remain longer in a dangerous wave crest. The statistical mean value given by Grim shows that, in a hydrostatic treatment, the ship is also more endangered in the statistical mean. The capsize computations carried out confirm this and enable quantitative statements about the mean time of travel of the ship until capsizing. The situation here is similar to the resonance considered above: while the same processes do not occur as with a regular wave travelling with the ship, for Froude numbers of 0.35 to 0.4 the case of irregular following seas most closely approaches the regular case, since the transformed spectrum then concentrates toward zero frequency. Figure 40 shows, for comparison, the transformed spectra for F_n = 0.2 and 0.4.

If the required stability of a ship is determined for the case of high Froude numbers of approximately 0.4, one is therefore on the safe side with respect to capsizing danger compared with lower speeds. Nevertheless, at the lower Froude numbers actually used in practice, ranges with unpleasantly large roll motions can occur. The present method now opens up the possibility of determining, for practically occurring cases, the roll amplitudes and the associated capsizing danger. A further step has been taken from the mere identification of resonance points toward the associated calculation of rolling behavior.

Systematically computing all possible parameter variations proves too laborious for initial investigations. The parameter variation computations performed with different center-of-gravity heights KG and different damping coefficients D_0 at several Froude numbers are shown in Figures 35 to 39. These values alone are based on 3,000 capsize computations.

Figure 36 shows the mean times-to-capsize for a Froude number of 0.25 at two different heights of the center of gravity. Here, a larger GM is also associated with a longer mean time-to-capsize, since at GM_GW = 25 cm for F_n = 0.25 the resonance that occurs at F_n = 0.2 is no longer present. The parameter in Figure 36 is the linear damping coefficient D_0. An increase in damping does exhibit the expected tendency that the ship becomes more capsize-resistant, but this effect is very small, particularly for GM_GW = 8 cm, even at high damping values.

Figure 37 shows the same values plotted against the damping coefficient with GM as parameter. This tendency is reliable insofar as the analog computer settings were not altered for this computation case; only the damping was increased via a potentiometer and a check via the free-oscillation test was performed. Possible scatter in the setting of the coefficients of the equation of motion is therefore excluded. For GM_GW = 8 cm, quadrupling the damping compared with D_0 = 0.025 only approximately doubles the time-to-capsize. The large damping coefficient 0.15 does cause an increase in the curve (T_K - t_0) in Figure 37, but capsizing still occurs relatively frequently.

In contrast, Figure 38 shows a marked increase in time-to-capsize with damping. For D_0 = 0.15, capsize events occurred so rarely that no capsize statistics were computed any longer. The underlying Froude number here is a smaller value of 0.15. At the same time, however, GM_GW = 10 cm is somewhat larger and the spread of the levers is somewhat smaller than in Figure 37, so that the influence of only one of these parameters cannot be uniquely identified. Nevertheless, the comparison permits the conclusion that, owing to large lever reductions in severely endangered ships, even an extremely high damping coefficient cannot prevent capsizing. If, however, certain minimum hydrostatic values are present — such as the mean value and the wave-crest curve of the righting levers, which follow from the mean value and the range of variation — then improvement in capsize safety can be achieved through damping. Damping alone can accomplish very little; equally, its influence should not simply be ignored. Only a judicious weighting of all quantities involved yields a more capsize-resistant ship.

Figure 39 shows the mean time-to-capsize as a function of the height of the center of gravity for a Froude number of 0.4. An increase in damping has no significant influence here, while for GM_GW greater than or equal to 50 cm no capsizing occurs at all.

7.5 Relative Frequency of Roll Angles

For computing roll-angle distributions, the analog voltage of the equation of motion can be evaluated in a classification device — known in nuclear physics as a pulse-height analyzer. An analog-to-digital conversion of the computing voltage followed by evaluation of the digitally stored values is also possible. Both methods require the appropriate electronic equipment. Since such equipment was not available, a number of long-duration computations of the rolling motion were carried out using the developed digital program. All step-by-step computed motion quantities were stored on tape. Since the time steps are of varying size for reasons of accuracy, the time steps delta_t_i were also stored on tape. Relative frequencies were then determined by classifying values interpolated to equidistant time steps. These are shown graphically in Figure 41 for the roll angle.

The normalized time function of the lever fluctuations was formed from 41 sine components for various Froude numbers in following seas. A constant heeling lever of 5 cm was used for all heel angles. The representation in Figure 41 shows that quite large roll angles occur in some cases, and one histogram also includes capsizing.

8. Summary

With the investigations described, an approach and a computational method are now available for determining numerical solutions of the equation of motion of a ship in parallel with the capsize experiments conducted on the Grosser Ploener See in irregular seas. The computations performed here are based on the lines according to which a new free-running model has been built [49]. In designing this free-running model, the experience gained with earlier models was taken into account.

A comparison of a measured and a computed statistics of times-to-capsize cannot yet be presented. The investigation carried out here does, however, provide important guidance for the selection of experiments to be conducted in irregular seas. A step has thereby been taken beyond the purely experimental stage in the treatment of capsizing of ships in irregular seas. Since the capsizing of a ship depends on a very large number of influencing variables, a comprehensive representation is very difficult. Two essential characteristics of this problem are taken into account in the present work: the nonlinearity of the righting moment as a function of heel angle, and the irregular changes in the righting moment in longitudinal seas. Distributions of time-to-capsize computed with this approach exhibit the theoretically expected and experimentally observed form of an exponential distribution (Figure 31). A minimum travel time is subtracted from the total time of the ship’s passage from the start of travel until the end of the capsize event; this minimum time is present on average for each capsize event and is never undercut. This actual capsize time can descriptively be called the “incubation time.”

The influence of the ship’s forward speed on the spectra of the hydrostatic righting levers in longitudinal seas was examined more closely. In general, it can be stated that in following seas the motion spectrum most closely approaches that of a regular sea when the discontinuity arising in the transformed spectrum is assigned to the region of the spectral maximum for the ship at rest:

(8.1)

Resonances in the Mathieu sense can occur depending on the mean natural frequency of the ship in the seaway.

The computations are based on the Froude-Krylov hypothesis. Following analyses of lever fluctuations of the righting moment in regular (Figures 7 and 8) and irregular seas (Figure 14), a mean normalized lever spectrum S_n was introduced for all heel angles phi. This approach facilitates numerical treatment and makes it possible to use an analog computer for solving the equation of motion. In place of the 1-permille computer used, a more accurate computer of the 0.1-permille accuracy class with hybrid extension would be more appropriate. A computation program was developed that combines the advantages of analog and digital computing aids.

The main reason for employing an analog computer is the possibility of time compression. Due to the scope of computation required for statistical solutions, a large time scale factor makes more extensive computations possible in the first place. Unfortunately, the full complement of the equipment used was available for only half a year. Since the program had to be developed during that period, this time is not comparable with the inherently short analog computing time. A complete hybrid computing installation or a faster digital computer would be advantageous here. For comparison computations and

important individual cases — for example, in design projects or for the clarification of maritime accidents — a corresponding digital program is now available.

For the large range of roll angles investigated, it is no longer adequate to compute with linear damping alone. A nonlinear damping function could easily be incorporated into the computation on the analog computer. The capsize computations performed with various linear damping coefficients give reason to recommend that this hydrodynamic quantity also be taken into account in practical stability assessments. As a first feasible measure, the linear damping coefficient in the upright position should be determined and specified for those new vessels for which a capsizing danger due to seaway is to be expected. The dependence of damping on the ship’s forward speed should also be taken into account in this context.

External heeling moments of the ship must be regarded as important for the solution of the equation of motion. These are always present to a small degree even for a ship traveling in longitudinal seas. Small disturbances cause only small roll amplitudes. The irregular Mathieu excitation can only enlarge the roll angles up to the point of capsizing. In addition to the resonance effect, the magnitude of the maximum lever fluctuations in the seaway, together with the mean righting lever, is of significance. The greater the range of fluctuation and the lower the mean value, the more frequently negative resultant levers can occur — levers that cause the ship to heel over.

An extension of the present approach to stochastic external disturbances — which can arise in longitudinal seas from obliquely running waves, lateral wind components, or rudder forces — should be carried out next. In the same direction lies the admittedly more difficult treatment of capsizing due to obliquely following irregular seas. The author hopes that his approach constitutes a further step also toward the solution of these problems.

Acknowledgments

The author wishes to express particular thanks to Professor Wendel for the opportunity to have deepened knowledge in the field of stability and capsize safety of seagoing ships over several years of work on research projects and expert assessments. The approaches of the present work grew out of engagement with the problems encountered in that work. Thanks are due to Professor Grim for the stimulation received from his publications and occasional verbal suggestions. The author also wishes to thank Professor Weinblum for the far-sighted judgment that led him, with the assistance of the German Research Foundation, to acquire an analog computer for the Institute of Shipbuilding in Hamburg. To this the author wishes to attach the wish that it may continue to be possible in the future, given the rapid advance of technology, to procure equipment commensurate with the state of development.

Thanks are owed to Professor Lerbs for permission to carry out the analog computations at the Hamburg Ship Model Basin using its two analog computers there, and to Dr. Schwanecke for the support rendered in that connection.

In setting up the control circuit for the analog computer and in carrying out the computations, the electronic expertise of colleague H. Poehlsen was of inestimable help, and the author wishes to express thanks to him here as well. The numerical computations of the hydrostatic levers for the Series 60 hull were carried out by Dipl.-Ing. N. Heinecke at the Technical University Hannover in cooperation with Dipl.-Ing. F. Behn using the digital program “Archimedes.” Dipl.-Ing. W. Abicht is thanked for making available the digital program for computing rolling motions in regular seas, whose Runge-Kutta-Nystrom iteration procedure was adopted for the irregular-sea capsize program. Thanks go to the German Research Foundation for supporting the computation of righting levers in irregular seas through a research project, thereby enabling the numerical foundation for the approach introduced here to be broadened.

9. Notation

The most important symbols are compiled below. The meaning of symbols not listed, or of symbols used with multiple meanings, follows directly from the text or from the respective context.

Symbol	Meaning
x, y, z	Space-fixed coordinates
x, y, z*	Body-fixed (ship-fixed) coordinates
zeta	Sea surface ordinate
phi	Roll angle
phi_0	Reference angle for normalization
phi_k	Normalized roll angle
t	Real time
t_R	Computing time on the analog computer
k_t	Time scale factor
rho	Density of seawater
g	Acceleration due to gravity
Delta	Displacement of the ship
I	Mass moment of inertia of the ship about its longitudinal axis
i	Roll radius of gyration
D	Damping coefficient
c	Spring constant (restoring coefficient)
l	Lever of the righting moment
l_m	Mean righting lever in the seaway
delta_l_max	Maximum lever reduction in the seaway
v_l	Ratio of lever scatter to maximum lever change in regular seas
l_k	Lever of the sum of heeling moments

Symbols and Notation (continued)

Symbol	Meaning
c_f	decay constant
D_o	linear damping measure
D_φ	nonlinear damping measure as a function of the roll angle
ω_m	mean natural frequency of the ship in a seaway
T_m	mean natural period of the ship in a seaway
v	ship speed
F_n	Froude number
K	wave-number
Λ	ratio of ship length to characteristic wavelength
L	ship length
L_w	characteristic wavelength
c	wave celerity (phase speed)
c_p	amplitudes of harmonic partial components of the spectrum
f_p	frequency of harmonic spectral components
ω_tp	circular frequency of harmonic spectral components
φ_N	phase of harmonic spectral components
N	number of discrete spectral components
H_w, H_n(t), H_n(t_2…)	computed height of a regular wave; normalised fluctuation function of the righting levers
S	spectrum
S_d*	transformed spectrum plotted over the magnitude of the encounter frequency in following seas
w	probability density
W, f_K, f_UK	probability / probability density; relative frequency of voyage times for capsizing and non-capsizing, respectively

Symbol	Meaning
F_K, F_NK	distribution function of the ship’s voyage times for capsizing and non-capsizing, respectively
T̄_K	mean capsizing voyage time T_K
α	significance level for type-I error
β	significance level for type-II error
m	sample size
K	number of classes
Δ	class width

Subscripts:

* as index: for all normalised quantities
~ as index: for all quantities transformed into the ship’s reference frame
K as index: for capsizing

Notation:

E[…] — expected value of …

Abbreviations:

Abbreviation	German	English
AR	Analogrechner	analog computer
DR	Digitalrechner	digital computer
KO	Kathodenstrahloszillograph	cathode-ray oscilloscope
FG	Funktionsgeber	function generator
P	Potentiometer	potentiometer
V	Verstärker	amplifier
M	Multiplizierer	multiplier
Vf.	Verteilungsfunktion	distribution function

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Roll, H. U.: Höhe, Länge und Steilheit der Meereswellen im Nordatlantik. [Height, length, and steepness of ocean waves in the North Atlantic.] Individual publication of the Deutschen Seewetteramtes, No. 1, Hamburg, 1953.
St. Denis, M. and W. J. Pierson: On the Motion of Ships in Confused Seas. The Society of Naval Architects and Marine Engineers, 1953, pp. 280–332.
St. Denis, M.: On a Problem in the Theory of Non-linear Oscillations of Ships. Schiffstechnik, Vol. 14, 1967, Issue 70, pp. 11–14.
Stefun, G. P.: Comparative Seakeeping Tests at the David Taylor Model Basin, the Netherlands Ship Model Basin, and the Admiralty Experiment Works. David Taylor Model Basin, Report 1309, May 1960.
Todd, F. H.: Some Further Experiments on Single-Screw Merchant Ship Forms – Series 60. Transactions of SNAME, Vol. 61, 1953, pp. 516–589.
Vossers, G., W. A. Swaan, and H. Rijken: Experiments with Series 60 Models in Waves. International Shipbuilding Progress, May 1961, pp. 201–232.
Weinblum, G. and M. St. Denis: On the Motion of Ships at Sea. Transactions, The Society of Naval Architects and Marine Engineers, Vol. 58, 1950.
Weinblum, G.: Kräfte bei Bewegungen von Körpern in oder nahe der freien Oberfläche. [Forces on bodies moving in or near the free surface.] Schiffstechnik, Vol. 1, 1952, Issue 1, pp. 21–31.
Wendel, K.: Stabilitätseinbußen im Seegang und durch Koksdeckslast. [Stability losses in a seaway and due to deck cargo of coke.] Hansa, Vol. 91, 1954, pp. 2000–2022. Investigations on a medium-sized ocean-going vessel (“Irene Oldendorff”).
Wendel, K.: Sicherheit gegen Kentern. [Safety against capsizing.] VDI-Zeitschrift, Vol. 100, 1958, No. 32, pp. 1523–1533.
Wendel, K.: Safety from Capsizing. Fishing Boats of the World: 2, London, 1960.
Wendel, K.: Die Wahrscheinlichkeit des Überstehens von Verletzungen. [The probability of surviving damage.] Schiffstechnik, Vol. 7, 1960, Issue 36, pp. 47–61.
Wendel, K.: Die Bewertung von Unterteilungen. [The evaluation of subdivision.] Jahrbuch der Schiffbautechnischen Gesellschaft, Vol. 55, 1961, pp. 190–208.
Wendel, K., S. Roden, and S. Kastner: Studium der Bewegungen der Seeschiffe unter dem Einfluss von Seegang und Wind. [Study of the motions of ocean-going ships under the influence of seaway and wind.] German contribution to Theme 2, Section II — Division for Ocean Shipping — for the XXI International Navigation Congress, Stockholm, 1965.
Wendel, K., B. Arndt, K. Boie, and F. Seefisch: Vortragsgruppe “Schiffsstabilität”. [Lecture group “Ship Stability”.] Jahrbuch der STG, 1965.
Zunker, H.: Aufstellen eines Rechenprogramms zur Berechnung der Rollbewegungen von Schiffen im Seegang. [Compilation of a computation program for calculating the rolling motions of ships in a seaway.] Chair for Ship Design, Universität Hamburg, 1966. Unpublished.

11. Appendix

The appendix is not included with this work but may be consulted on request. It contains the associated digital programs in Algol and their descriptions, as well as printouts of the results.

A — Transformation of continuous spectra to the moving ship
B — Discretisation of transformed spectra
C — Classification and χ²-test for exponentially distributed quantities
D — Classification and test for normal distribution
E — Fourier analysis for regular variations of righting levers
F — Capsizing under stochastic parametric excitation
G — Stochastic phases for the settings on the analog computer

A more detailed version of this work also exists, in which the analog computing and control circuit is described in depth.

12. List of Figures

Figure No.	Title
1	Various types of righting-lever curves
2	Lever ranges in a seaway
3	Block diagram: ship as a system
4	Substitute systems for the rolling ship in longitudinal seas
5	Righting levers; wave–ship passing period for regular waves with λ as parameter
6	Deviations of the mean seaway lever from the calm-water lever for various phase positions in a regular wave
7	Fourier analysis of the regular lever variation for 45° heel
8	Fourier components of the righting levers in a regular wave
9	Ordinate distribution of stochastic levers at 30° heel
10	Relative frequency of characteristic wave heights for 100-m waves in the North Atlantic according to the statistics of Roll [65]
11	Lever differences as the variation range in irregular seas
12	Probability density of the lever ordinates in irregular, longitudinally running seas
13	Stochastic levers in longitudinal seas, plotted against time t with the heel angle φ as parameter
14	Comparison of the normalised spectra of irregular seas and the corresponding levers at 30° heel in longitudinal seas
15	Influence of initial conditions on the time to capsizing
16	Block diagram for transformed levers
17	Division of the original spectrum into three regions in following seas for F_n = 0.25
18	Boundary frequency f_v of region III plotted against the Froude number with λ/L as parameter
19	The three regions of a transformed spectrum
20	Transformed spectra in head and following seas for various Froude numbers — overview
21	Centre of gravity of the transformed spectra with respect to the fold frequency
22	Discretised spectrum in following seas for a Froude number of 0.25 with arithmetic spacing
23	Computing circuit on the analog computer for the equation of motion
24	Nonlinear fractions η_n of the mean seaway lever curve, normalised for various λ/L
25	Normalised nonlinear fraction Δη of the variation range of the levers
26	Comparison of analog and digital computation
27	Phase-plane curve and f(φ) for resonance F_n = 0.2; T_m·√F = 5
28	Motion quantities over time for capsizing with F_n = 0.4
29	Angular acceleration φ̈ over roll angle φ for capsizing with F_n = 0.4
30	Phase-plane curve for capsizing with F_n = 0.4
31	Distribution of capsizing voyage times with confidence interval — example
32	Reproducibility test on the analog computer — histogram
33 and 34	Histograms of statistically determined capsizing voyage times on the analog computer
35–39	Mean capsizing voyage time (T̄_K·t_0) for various parameters
40	Comparison of the transformed spectra S_d* for F_n = 0.2 and F_n = 0.4
41	Probability density of roll angles at various Froude numbers in following seas

Table 1 — Normalised Frequencies and Amplitudes of the Excitation Components at the Analog Computer

Time scale: — A₁ =

n	F = 0.15		F = 0.20		F = 0.25		F = 0.40
	ω*_pn [sec⁻¹]	c*_p [–]	ω*_pn [sec⁻¹]	c*_p [–]	ω*_pn [sec⁻¹]	c*_p [–]	ω*_pn [sec⁻¹]	c*_p [–]
1	0.7156	0.045	0.1473	0.08	0.1178	0.20	0.0421	0.33
2	0.9425	0.126	0.4208	0.10	0.3366	0.22	0.1262	0.33
3	1.1345	0.282	0.6522	0.14	0.5217	0.27	0.2104	0.33
4	1.2915	0.415	0.8415	0.21	0.6732	0.32	0.2945	0.33
5	1.4137	0.477	0.9888	0.34	0.7910	0.38	0.3787	0.33
6	1.5010	0.478	1.0940	0.50	0.8751	0.45	0.4628	0.33
7	1.5533	0.510	1.1571	0.71	0.9257	0.66	0.5470	0.36

[page 98: Figure 1 (2.4.1) — Various types of righting-lever curves h(φ) for calm water. S = stable equilibrium, L = unstable equilibrium. Curves shown for Series 60 No. 4212W and the coastal motor vessel “Lohengrin”.]

[page 98: Figure 3 (3.1.1) — Block diagram: Ship as a System. Input signal x(t) / X(ω) → System → Output signal y(t) / Y(ω); Excitation → Ship → Ship behaviour.]

[page 98: Figure 4 — Substitute systems for the rolling ship in longitudinal seas. Block diagrams showing the various analog-computer sub-circuits corresponding to the equation-of-motion components (damping, restoring moment, parametric excitation, stochastic input), with integrators and summer blocks indicated by standard analog-computer symbology.]

[page 109: figure only]

Fig. 5 (4.1.3) — Righting-lever differences between sea-state conditions

(Rotated diagram; axis labels and legend text only partially legible at this resolution. The figure shows curves of righting-lever differences Δh [cm] plotted against heel angle φ [°], comparing multiple sea conditions. Legend entries include: Glattwasser (calm water), and entries for various Trochoidenwelle (trochoidal wave) parameters. Ship Series 60 No. 4212 W. Parameter values listed: L, T_m, Δh_M.)

[page 110: figure only]

Fig. 6 (4.1.3) — Righting-lever differences between sea-state conditions

(Two side-by-side rotated graphs showing Δh [cm] vs. φ [°] for different wave conditions. Legend identifies curves for: Glattwasser (calm water), and trochoidal wave cases at multiple inclinations. Labels indicate ship parameters and wave parameters. Axes: horizontal — Δh [cm]; vertical — φ [°].)

Fig. 7 (4.1.4a) — Fourier Analysis of the Regular Righting-Lever Variation in a Trochoidal Wave at 45° Inclination

The graph plots righting-lever amplitude h [cm] against normalized time t/T* (0 to 1) for one complete wave period.

Curves shown:

Δh₄₅°(t) — total righting-lever variation at 45° wave inclination
h̄ (GM̄_GW = 0) — mean value of the righting lever for GM_GW = 0 (approximately 10 cm)
c₁ sin(ω*t + ε₁) — first Fourier harmonic
c₂ sin(2ω*t + ε₂) — second Fourier harmonic
c₃ sin(3ω*t + ε₃) — third Fourier harmonic

The total variation Δh₄₅°(t) reaches a maximum of approximately +30 cm and a minimum of approximately −30 cm over one period.

Fig. 8 (4.1.4b) — Fourier Components c_p and Phases ε_p for Regular Righting-Lever Variation in a Trochoidal Wave

The figure consists of a bar chart and an accompanying table.

Governing equations:

h_y(t) = h̄_y + Δh_y(t) for GM̄_GW = 0

Δh_y(t) = Σ c_p · ε_pn · sin(pω*t + ε_p)

h̄_y = a₀ = ½ h̄

Table of Fourier coefficients (normalized amplitudes c_pn and phases ε_pn) as a function of wave inclination angle γ [°]:

	γ = 30°	γ = 45°	γ = 60°	γ = 75°
h̄ [cm]	14.6	8.2	−17.8	−535
c₁ [cm]	27.4	33	26.9	19.3
c₁n	0.764	0.761	0.762	0.769
ε₂n	0.81	0.794	0.792	0.744
c₃n	0.56	0.695	0.763	0.80
ε₄n	0.375	0.270	0.48	0.009
c₅n	0.509	0.514	0.49	0.598
c₆n	0.676	0.614	0.594	0.815

The bar chart shows the relative magnitudes of the normalized Fourier coefficients c_pn versus harmonic order number p (1 through 6), for each of the four wave inclination angles (30°, 45°, 60°, 75°). The bars are grouped by order number along the horizontal axis; the vertical axis shows c_pn values from 0 to approximately 1.0 in steps of 0.1.

Fig. 9 (4.1.10) — Relative Frequency of Righting Levers in Head Seas for φ = 30°, Pierson–Moskowitz Sea Spectrum, GM̄_GW = 0

Parameters of the sample:

M = 1000 (sample size)
Δt = 1 sec (time step)
T = M · Δt = 16.65 min (total duration)
S̄² = 1.1 m² (mean-square wave elevation)
s_Δh = 6.9 cm (standard deviation of righting-lever variation)
σ_Δh = E[s_Δh] (expected value of s_Δh)

Graph description:

The histogram shows the empirical relative frequency distribution n_i/M of the righting lever h₃₀°(t_i) for the 1000-sample simulation. The horizontal axis is h [cm], running from approximately −10 to +40 cm, with reference marks at h̄ − 3s_Δh, h̄ − 2s_Δh, h̄ − s_Δh, h̄ + s_Δh, h̄ + 2s_Δh, h̄ + 3s_Δh. The vertical axis has dual scales:

Left (inner): 10³ · w(h) [cm⁻¹], ranging 0–5
Left (outer): n_i/M [−], ranging 0–0.22

A superimposed smooth curve shows the theoretical probability density of a normally distributed population N(h̄; S²_Δh).

Reference values:

h̄ = 15.1 cm (sample mean)
Trochoidal wave value: h = 15 cm
Calm-water value: h = 17 cm

Fig. 10 (4.1.8) — Relative Frequency of Characteristic Wave Heights for 100 m Waves according to Roll Statistics

Graph description:

A single curve plotted on log–log axes:

Horizontal axis: p [‰] (permille probability), ranging from 0.01 to approximately 10
Vertical axis: (H_W/H̄_W) · 100 [%], ranging from 0 to approximately 10

The curve is monotonically increasing. A dashed reference line marks p = 2‰, at which point the characteristic wave height H_W = 6.7 m is indicated.

The graph allows determination of the wave height exceeded with a given probability per unit time, expressed as a fraction of the mean wave height H̄_W.

Fig. 11 (4.1.9) — Righting-Lever Differences for Ship Series 60 No. 4212 W

Graph description:

Three curves plotted as righting-lever difference Δh [cm] (horizontal axis, 0 to 50 cm) vs. wave inclination angle φ [°] (vertical axis, 0° to 90°):

Δh_M2 = |h_B − h_M2| = |h_T − h_M2| — righting-lever difference based on second-moment approximation
Δh_BM4 = h_B − h_M4 — righting-lever difference based on fourth-moment approximation
Δh_TM4 = h_T − h_M4 — righting-lever difference based on fourth-moment approximation

Definitions:

h_M2 = ½(h_B + h_T) — mean of broadside and trough levers (second-moment)
h_M4 = Mittel aus vier Phasenlagen in längslaufender Trochoidenwelle = Mean of four phase positions in a head-running trochoidal wave
H_W/L_W = 0.0667

The curves show that all three difference measures increase monotonically with wave inclination angle from 0° to 90°, with Δh_TM4 and Δh_BM4 diverging slightly at higher angles while Δh_M2 remains between them.

[page 116: figure only]

Fig. 12 (4.1.11) — Three-dimensional representation of righting-lever variations

(Rotated 3-D surface diagram; text annotation identifies it as Fig. 12 (4.1.11), depicting righting-lever behavior (Hebelkurven) for Ship Series 60 No. 4212 W. Additional annotation text, partially legible: “Aufrichtender Hebel” (righting lever), “Längslaufende Trochoidenwelle” (head-running trochoidal wave), “HW/LW = 0.0667”. Axes not fully legible at this resolution.)

Fig. 13 (4.1.12) — Stochastic Righting Lever in Head Seas, GM̄_GW = 0

Graph description:

Two-panel time-history plot:

Upper panel — Righting lever h [cm] (vertical axis, scale approximately −2 to +10 cm) vs. time t [sec] (horizontal axis, 0 to approximately 40 sec). Three time traces are overlaid:

h₃₀°(t) — righting-lever variation at 30° heel angle (solid line)
h₄₅°(t) — righting-lever variation at 45° heel angle (dashed line)

Labels indicate reference points “Wellenberg” (wave crest) and “Wellental” (wave trough).

Lower panel — Wave surface elevation S_w(t) [m] (vertical axis, scale approximately −1 to +1 m) vs. time t [sec]. Shows the stochastic wave record used to drive the computations.

Additionally, a separate trace h₆₀°(t) appears in the lower portion, showing righting-lever variation at 60° heel.

Fig. 14 (4.1.14) — Comparison of Normalized Spectra of Sea State and Righting-Lever Variation, Ship without Forward Speed

Graph description:

Normalized spectral density S_n [sec] (vertical axis, 0 to approximately 20 sec) vs. frequency f [Hz] (horizontal axis, 0.10 to 0.40 Hz).

Two spectra are shown:

S_hhn(30°) — normalized spectrum of the righting-lever variation at 30° heel angle. Narrow, tall peak near f ≈ 0.13 Hz reaching S_n ≈ 20 sec.
S_ssn — normalized spectrum of the sea-state surface elevation. Broader peak near f ≈ 0.20 Hz, reaching S_n ≈ 12 sec.

Normalization definitions:

S_n = S_± / (2R)
R = ∫₀^∞ S(f) df (total spectral energy)

Reference values:

S̄² = 0.406 m²
Δh²₃₀° = 19.5 cm²

The righting-lever spectrum is noticeably sharper and shifted to lower frequencies relative to the wave spectrum, consistent with parametric excitation at half the wave frequency.

Fig. 15 (4.2.2) — Time to Capsizing for Various φ₀ at φ̇₀ = 0

Graph description:

Normalized time to capsize t_K/T_m (vertical axis, 0 to approximately 20) vs. initial roll angle φ₀ [°] (horizontal axis, 1 to 20°, logarithmic scale).

System parameters:

GM̄_GW = 0.25 m
T_m = 15.5 sec (mean natural roll period)
D₀ = 0.025 (linear damping coefficient)
|φ_K| = 80° (capsizing angle)

Parametric excitation with two components:

H(t) = q₀ · Σ_{p=1}^{2} c_p* · sin[2π(f_p* · t* + ε_pn)]

F = 0.2

Component table:

p	c_p*	f_p* [Hz]	ε_pn
1	0.413	0.0580	0.2709
2	0.587	0.0614	0.1862

Two curves are plotted, parametrized by excitation amplitude q₀:

q₀ = 1: time to capsize decreases steeply from approximately 20 T_m at φ₀ ≈ 1° to approximately 4 T_m at φ₀ = 10°, with a local maximum (⊕ symbol) near φ₀ = 5° indicating a region of increased capsize resistance.
q₀ = 2: time to capsize is much shorter throughout, dropping below 2 T_m at small initial angles.

Critical values of φ_max are indicated on the vertical axis:

φ_max at −6°, −4° (labeled ⊕ ⊕) — stable equilibria
φ_max at −17.5° (labeled ⊕) — unstable equilibrium or boundary

[page 120: figure only]

Fig. 16 — Block Diagram for Transformed Righting Lever

Two parallel processing chains are shown:

Left chain:

Input: S_ss (sea-state spectrum in the stationary frame)
Block: Schiffsform (ship-form transformation)
Output: S*_ss (transformed sea-state spectrum in the ship frame)
Block: Geschwindigkeitstransformation (speed transformation)
Output: S*_hh (transformed righting-lever spectrum)

Right chain:

Input: S_ss
Block: Geschwindigkeitstransformation (speed transformation)
Output: S_hh (righting-lever spectrum in intermediate frame)
Block: Schiffsform (ship-form transformation)
Output: S*_hh

Both chains yield the same final result S*_hh, confirming that the speed transformation and ship-form transformation commute.

Fig. 17 (5.1.2) — Transformed Sea-State Spectrum in Following Seas

Parameters:

F_n = 0.25 (Froude number)
Λ = 1 (ship-length-to-wavelength ratio)
L = L_W = 100 m

Graph description:

Normalized spectral density S_n*(t) [—] (vertical axis, 0 to 10) vs. frequency f [mHz] (horizontal axis, 0 to 250 mHz).

The spectrum is divided into three regions:

Region I (high frequency, right): −∞ < α < ½ — the waves overtake the ship; 0 < f < f_∞. Shaded (hatched) region. Order numbers of 7 discrete components indicated.
Region II (central): ½ < α < 1 — the waves overtake the ship; 0 < f < f_∞. Solid curves for α* = 0 and α* = 1.
Region III (left, including negative frequencies): 1 < α < ∞ — the ship overtakes the waves; −∞ < f < 0.

Reference lines:

α* = 0 (zero-speed condition)
α* = 1 (ship speed equals wave phase speed)

The falling frequency f_∞ is indicated: f_∞ = 2f*_∞ is the Froude-number-dependent fold frequency.

Annotation: “Ordnungszahlen der Bereiche bei 7 diskreten Komponenten” (Order numbers of the regions for 7 discrete components); numbers 1–7 are marked on the right-hand axis.

Fig. 18 (5.1.3) — Fold Frequency f_V as a Function of Froude Number F_n

Governing relations:

f_V = 1/(KF_n) = 2f_∞° = 4f_∞*

K = |2π/g · √(g · Λ · L_W)|

L_W = 100 m

Λ = L/L_W

Graph description:

Log–log plot of f_V [Hz] (vertical axis, 0.02 to 10 Hz) vs. F_n [−] (horizontal axis, 0.01 to 1).

Five curves are shown, parametrized by Λ:

Λ = 0.5
Λ = 0.75
Λ = 1
Λ = 1.5
Λ = 2

All curves are parallel straight lines with negative slope (−1) on the log–log plot, displaced vertically according to Λ. As Froude number increases, the fold frequency decreases; as Λ increases, f_V also increases.

Fig. 19 (5.1.9) — Transformed Spectral Density S, S_d vs. Encounter Frequency f*, Following Seas

Parameters:

F_n = 0.25
Following seas (achterliche See)
Λ = 1
L = L_W = 100 m

Graph description:

Spectral density S*, S_d* [sec] (vertical axis, 0 to approximately 40 sec) vs. encounter frequency f* [mHz] (horizontal axis, −50 to +50 mHz).

Two curves are plotted:

S(f)** — transformed one-sided spectrum. Rises steeply toward the fold frequency f_∞ (marked on the right-hand side of the hatched zone). The curve is defined only for f < f*_∞.
S_d(|f|)** — doubled (two-sided folded) spectral density, defined as S_d* = S*(f*) + S*(−f*). Shown for both positive and negative encounter frequencies.

Three spectral regions (Bereich I, II, III) are indicated with hatching and labels corresponding to the regions defined in Fig. 17.

The singular behavior as f* → f*_∞ reflects the mathematical fold (encounter frequency approaching zero relative group velocity); the spectral density diverges at this fold point.

Fig. 20 (5.1.5) — Transformed Spectra in Head Seas and Following Seas, F_n as Parameter

Graph description:

(Rotated diagram.) Transformed spectral density S_n* [sec] (horizontal axis, −5 to approximately 40 sec) vs. encounter frequency f* [mHz] (vertical axis, −50 to approximately 350 mHz).

Multiple curves are shown with Froude number F_n as the parameter:

F_n = 0, −0.10, −0.15, −0.20, −0.25 (negative values indicating head seas / opposing course) F_n = 0.1, 0.15, 0.20, 0.25, 0.30, 0.35, 0.375 (positive values indicating following seas)

Dashed curves: F_n = −0.10, −0.15, −0.20, −0.25 (head-sea cases) Solid curves: F_n = 0 through +0.375 (following-sea cases)

The figure demonstrates how the spectral shape is compressed (following seas) or stretched (head seas) as a function of ship speed. In following seas at high F_n, the spectrum develops a pronounced peak and the fold singularity becomes prominent at low encounter frequencies.

Fig. 21 — First Spectral Moment M₁ of the Transformed Spectrum vs. Froude Number F_n

Governing equations:

M₁ = Σ_{i=1}^{N} (f_i* − f_∞) · S_di · Δf_i*

S_d* = S*(f*) + S*(−f*)

N = 41

Λ = L/L_W

Graph description:

M₁ [mHz] (vertical axis, 0 to −30 mHz; values are negative) vs. Froude number F_n (horizontal axis, 0.10 to 0.40).

Three curves are shown, parametrized by ship-length-to-wavelength ratio Λ:

Λ = 0.75 (leftmost minimum)
Λ = 1 (intermediate minimum)
Λ = 1.5 (rightmost minimum)

Solid lines represent one phase of the computation; dashed lines represent a complementary computation. Each curve shows a minimum (most negative M₁) at an intermediate F_n, then rises back toward zero at higher F_n. The minima shift to higher F_n as Λ increases.

The first spectral moment M₁ characterizes the mean encounter frequency offset from the fold frequency f_∞* and is used to assess the degree of spectral redistribution caused by ship speed in following seas.

Fig. 22 (5.3.1) — Discretized Transformed Spectral Density S_d* for Arithmetic Frequency Division

Parameters:

F_n = 0.25
Following seas (achterliche See)
Λ = 1
L = L_W = 100 m
Arithmetic (uniform) frequency spacing
N = N_U = 7 (number of discrete frequency components)

Graph description:

S_d* [sec] (vertical axis, 0 to approximately 40 sec) vs. |f*| [mHz] (horizontal axis, 0 to approximately 70 mHz).

A smooth continuous curve of S_d*(|f*|) rises steeply as |f*| approaches f_∞ (approximately 50 mHz), exhibiting the fold singularity. Superimposed are seven vertical hatched bars representing the discretized spectral components at frequencies f₁ through f₇*, with the spacing Δf* uniform between 0 and f_∞. The bars at f₆ and f₇* (closest to the singularity) are tallest, reflecting the strong concentration of spectral energy near the fold frequency.

The reference frequency f*_V (the lowest discrete component) is labeled on the horizontal axis.

This figure illustrates the consequence of arithmetic (equal-interval) discretization: because the spectral density diverges near f*_∞, uniform spacing assigns a disproportionately large weight to the last few components, motivating the use of non-uniform (e.g., logarithmic or energy-equal) discretization schemes instead.

Figure 23 (§ 8.2.1): Computing Circuit for the Equations of Motion on the Analog Computer

[page 127: figure only — schematic diagram of the analog computer circuit used for the equations of motion. The circuit is drawn as a block diagram with operational amplifiers, integrators, multipliers, summers, and function generators interconnected. Labels at left refer to the parametric excitation input H_p(t); internal blocks are labeled with the corresponding computation steps for righting-lever variation, roll angle, and roll velocity. No body text appears on this page.]

Figure 24 (§ 6.2.5): Non-Linear Component of the Mean Lever Curve

Normalized for Various Height Positions of the Centre of Gravity; Set on the Function Generator

[page 128: figure only — graph of the non-linear component of the mean righting-lever curve. The horizontal axis is the normalized lever variable x = Δl_E / (Δl_E)_B, ranging from −1 to +1. The vertical axis shows the normalized amplitude φ / φ_B, ranging from 0 to 1 (marked at 0.06, 0.09, 0.1, etc. up to 1). Four curves are plotted as parameter the value of GM_GW [cm] combined with the corresponding centre-of-gravity height h_B [cm]:

GM_GW [cm]	0	10	25	50	45
h_B [cm]	96	84	70	—	—

The curves diverge progressively from the origin, with higher GM_GW values producing higher amplitudes at given x.]

Figure 25 (§ 6.2.6): Non-Linear Component of the Righting Lever in the Oscillation Range

Normalized Representation for Function Generator

[page 129: figure only — graph showing the non-linear component Δh̃ of the righting lever plotted against the normalized variable Δh_n / h_B on the horizontal axis (range approximately −0.2 to +0.2). The vertical axis shows the normalized variable s / s_B (range 0 to approximately 1.0). A single curve with specific annotated values is shown:

At Δh_n / h_B = 0.012, the corresponding ordinate value is noted
Further annotated points: 0.065, 0.058, 0.175, 0.55, 0.94 The caption notes the parameters: Δh̃ = Δh − α_A · φ; h_B = 33.5 cm; φ_B = 100°.]

Figure 26 (§ 6.5.14): Comparison of Analog and Digital Computation

[page 130: figure only — time-history chart comparing analog computer output (continuous trace) with digitally computed points (marked with × symbols). Two plots are shown on the same time base:

Upper trace: parametric excitation H_p(t) = sin[2π(0.058 t + 0.2709 t)]
Lower trace: roll angle φ in degrees, showing large-amplitude oscillations leading toward a capsize event

A “Berg” (mountain/crest) annotation marks the peak of the excitation. The record spans approximately 60 time steps. Parameters: GM_GW = 25 cm; D_0 = 0.025. The comparison demonstrates good agreement between the two computation methods.]

Figure 27 (§ 7.2.5): Phase Curve

[page 131: figure only — phase-plane portrait (φ̇ vs. φ) for a capsize event in following seas. The spiral trajectory begins in a region of stable rolling near the origin and progressively diverges outward, eventually crossing into the capsize region at large roll angles (beyond approximately ±60°). Axes: horizontal = roll angle φ [degrees], range approximately −60° to +80°; vertical = roll velocity φ̇ [rad/s], range approximately −0.15 to +0.15. Parameters: F_n = 0.20; GM_GW = 25 cm; T_m^{1/2} = 5; D_0 = 0.025. Note in chart: “Due to the pen speed only qualitative values — dated 21.11.1967.”]

Figure 28 (§ 7.2.7): Capsizing in Following Seas — Froude Number F_n = 0.4

[page 132: figure only — time-history recorder traces for a capsize run in following seas. Two traces are shown with a time offset between them:

Upper trace: wave excitation H_p(t), showing irregular oscillation
Lower trace: roll angle φ, showing growing oscillations culminating in capsizing

Parameters: GM_GW = 10 cm; T_m = 18.7 sec; D_0 = 0.05. Caption notes: “Same case as Sheets 9 and 10. The traces have a time offset.” Time scale: 6 sec per division, time-scale factor = 3.]

Figure 29 (§ 7.2.8): Angular Acceleration vs. Roll Angle During Capsizing in Following Seas

[page 133: figure only — phase-plane plot with roll angle φ [degrees] on the horizontal axis (range approximately −60° to +80°) and angular acceleration φ̈ [rad/s²] on the vertical axis (range approximately −0.1 to +0.1). The trajectory spirals outward from a central stable region, with multiple loops visible near the origin, then breaks away to a large-amplitude capsize trajectory. Parameters: F_n = 0.4; GM_GW = 10 cm; D_0 = 0.05. Sheet 2/40/10, dated 13.12.67.]

Figure 31 (§ 7.3.1): Capsizing Time Distribution

[page 134: figure only — two-panel statistical chart.

Upper panel — Probability Density:

Horizontal axis: t’_K = t_K − ⟨t_0⟩ [sec], range 0 to ~400 sec
Vertical axis: n_k / n (normalised frequency density f_K [sec⁻¹])
Bar histogram shows the empirical probability density from classification of the sample
Smooth curve shows the normalized theoretical probability density (exponential distribution)
Annotated: T’_K = 80 sec

Lower panel — Cumulative Distribution Function:

Horizontal axis: grouped classes U (1, 2, 3, 4, 5, 6≤2H, 7≤4H)
Vertical axis: F_K [−], range 0 to 1.0
Step function = empirical cumulative distribution
Smooth curve = theoretical exponential distribution
Dashed band = 99% and 95% confidence interval

Test statistics:

Sample size M = 125
Class width H = 36.8 sec
χ² = 13 < χ²_{99%,5} = 15.1 → fit accepted at 99% level

Parameters: GM_GW = 10 cm; D_0 = 0.05; F_n = 0.4; ⟨t_0⟩ = 30 sec]

Figure 32 (§ 7.4.1): Reproducibility Test

[page 135: figure only — bar chart of capsize times from repeated analog computer runs used to assess reproducibility of the simulation. Vertical axis: capsize time t [sec], scale 0 to approximately 6 sec (real-time equivalent; full bar length represents the ship’s voyage time). Horizontal axis: run numbers n = 1 through 12, grouped in two sets of runs (labeled 5 and 6 at bottom). Each bar represents the capsize time for one simulation run. The spread of bar heights indicates the variability due to stochastic excitation.

Parameters: F_n = 0.4; GM_GW = 25 cm; D_0 = 0.05. Sheet 2/40/15 from 13.12.67. Pause time T_p = 10 sec; real time t = 30 T (scale factor 30). Note: “The full bar length corresponds to the ship’s voyage time.”]

Figure 33 (§ 7.4.4): Excitation Amplitudes

[page 136: figure only — bar chart displaying the excitation amplitudes ζ’ (in units of seconds, vertical axis 0 to 2.5 sec) for successive simulation runs. Horizontal axis: sub-groups a and b within each of six main groups (Untergruppe Nr. 1 through 6). Within each main group a pair of bars (a and b) represents two runs with the same nominal parameters. The amplitudes vary considerably from run to run, reflecting the stochastic nature of the irregular-sea excitation. No body text on this page.]

Figure 35 (§ 7.4.8): Mean Capsizing Time t̄_K − t̄_0 — Excitation Phases P0 and P1 Combined

[page 137: figure only — line graph.

Horizontal axis: GM_GW [cm], range 0 to 25 cm
Vertical axis: t̄_K − t̄_0 [min] (left scale, 0 to 10 min) with corresponding [sec] scale on right (0 to 500 sec)
Two parameter sets are shown, each with two curves (labeled a and b):
- F_n = 0.20, D_0 = 0.025: curves a and b both show mean capsizing time increasing with GM_GW; curve b lies above curve a; values reach approximately 7–8 min at GM_GW = 25 cm
- F_n = 0.40, D_0 = 0.05: curves a and b lie much lower; they intersect and converge near GM_GW = 10–15 cm; values remain below ~3 min across the range
Note: Results are averaged over computing groups No. 1 and 2 (Rechengruppe Nr. 1 u. 2 gemittelt)]

Figure 36 (§ 7.4.9): Mean Capsizing Time — F_n = 0.25; D_0 as Parameter

[page 138: figure only — line graph.

Horizontal axis: GM_GW [cm], range 0 to 25 cm
Vertical axis: t̄_K − t̄_0 [min] / [sec] (dual scale, 0 to 10 min / 500 sec)
Multiple curves are plotted, each identified by a run-group label (e.g., 9a+b, 9c+d, 11c+d, 12a+b, 12c+d, 13a+b, 14a b, 15a b, 15, 15c d) and corresponding D_0 values
D_0 values shown as parameter: 0.025, 0.05, 0.10, 0.15
General trend: mean capsizing time increases with GM_GW for all D_0 values; higher damping (larger D_0) produces longer mean capsizing times; the upper cluster of points at D_0 = 0.05 reaches ~5–7 min at GM_GW = 25 cm]

Figure 39 (§ 7.4.12): Mean Capsizing Time — F_n = 0.40; D_0 as Parameter

[page 139: figure only — line graph.

Horizontal axis: GM_GW [cm], range 0 to 25 cm
Vertical axis: t̄_K − t̄_0 [min] / [sec] (dual scale, 0 to 10 min / 500 sec)
Symbols: circles (○) for D_0 = 0.05; squares (□) for D_0 = 0.10
Annotations on curves: run-group identifiers 16a, 16b, 16, 17a, 17b, 17, 18a, 18b, 18, 19a, 19b, 19c, 19, 20a, 20b, 21a, 21, 22a, 22b, 22, 23a, 23
Notes:
- For D_0 = 0.15: no change compared to D_0 = 0.10
- For GM_GW = 50 cm: no capsizing occurs (kein Kentern mehr)
The F_n = 0.40 case shows considerably shorter mean capsizing times and weaker dependence on GM_GW compared to lower Froude numbers]

Figure 40 (§ 7.4.13): Comparison of Transformed Spectra S_d* for F_n = 0.2 and F_n = 0.4

[page 140: figure only — spectral plot.

Horizontal axis: |f*| [mHz], range 0 to ~70 mHz; vertical lines mark the encounter-frequency singularities f_v* (at ~0 mHz), f_∞* (at ~30 mHz for F_n = 0.4), and f_∞* (at ~62 mHz for F_n = 0.2)
Vertical axis: S_d* (normalized transformed spectrum density), range 0 to approximately 40
Two curves:
- Labeled 0.4 (F_n = 0.4): broad, relatively flat curve with a mild peak around 20–25 mHz, reaching a maximum of ~25–27; drops sharply at f_∞* ≈ 30 mHz
- Labeled 0.2 (F_n = 0.2): rises steeply from near zero at low frequencies; crosses the 0.4 curve at approximately 35 mHz and rises steeply to a sharp singularity at f_∞* ≈ 62 mHz
The plot illustrates how increasing ship speed (higher F_n) shifts and broadens the transformed excitation spectrum relative to the singularity frequency]

Figure 41 (§ 7.5.1): Digitally Computed Roll-Angle Distributions in Following Seas

[page 141: figure only — four-panel histogram figure comparing roll-angle probability density distributions for a Series 60 hull form under varying conditions.

Common parameters for all panels: φ_0 = 10°; Δφ = 0°; Δt = 0.5 sec

Panel 1 (upper left):

GM_m = 30 cm; T_m = 21 sec; k = 5 cm; F_n = 0.20; D_0 = 0.025; T = 10 min
Histogram of roll angle φ [degrees] from −20° to +30°; distribution approximately centred near 0°, slight positive skew

Panel 2 (upper right):

GM_m = 55 cm; T_m = 15.5 sec; k = 5 cm; F_n = 0.20; D_0 = 0.025; T = 7 min
Distribution narrower and more symmetric; range −20° to +20°

Panel 3 (lower left):

GM_m = 30 cm; T_m = 21 sec; k = 5 cm; F_n = 0.25; D_0 = 0.025; T = 10 min
Broader distribution; extends to approximately +50°; strong positive tail indicating approach to capsizing

Panel 4 (lower right):

GM_m = 55 cm; T_m = 15.5 sec; k = 5 cm; F_n = 0.25; D_0 = 0.025; T = 7 min
Distribution moderately broad; extends to approximately +30°

Note at bottom: “Das längsströmende See” (following sea with wave propagation in the ship’s direction of travel)]

Abstract (German) — Zur Kenterung von Schiffen in unregelm äßigem längsaufendem See

[page 142: text page]

Thesis details:

Scope: 98 pages, 41 figures
Published in: Schiffstechnik
Author: Sigismund Kastner
Date of promotion: 19 June 1969
First referee: Prof. Dr.-Ing. K. Wendel
Second referee: Prof. Dr.-Ing. O. Grim
Department: Schiffstechnik (Naval Architecture)

Abstract (German):

A method is presented for carrying out statistical computations of the rolling behaviour of ships at large angles of inclination in irregular seas running in the longitudinal direction. For the time-irregular fluctuations of the righting moment, a normalized spectrum averaged over the angle of inclination is introduced. Its dependence on the angle of inclination is reduced to the magnitude of the power content of the lever spectrum.

The transformation of the spectra of the seaway and of the righting levers for a ship underway is examined, taking into account the work of St. Denis and Pierson. For the synthesis of stochastic time functions for the lever variations of the righting moment, the spectra are approximated by discrete harmonics.

In addition to the examination of individual characteristic capsizing events, the statistical evaluation of the computed capsizing voyage times is described. The mean capsizing voyage time, as a parameter of the exponential distribution, is discussed for various influencing quantities. An example of computing the relative frequency of roll angles is given.

The new approach enables the systematic computation of lever spectra of the righting moment and of capsizing cases. It can be used to select the test cases for capsizing experiments with free-running ship models and can provide an overview of the influence of parameter variations on capsizing safety.

Summary (English)

[page 143: text page — English-language abstract]

A method is introduced for the statistical determination of the rolling behaviour of a ship in a longitudinal irregular seaway for large rolling angles. The time-varied irregularity in the variation of the righting moment is represented by a normalized spectrum (mean over angle of inclination). Its dependence on the angle of inclination is reduced to the power content of the lever spectrum.

The transformation of the spectra of the seaway and the righting levers is analysed with due regard to the paper by St. Denis and Pierson. For the establishment of stochastic time functions for the variation of the righting moment levers the spectra are approximated by discrete harmonics.

In addition to the discussion of certain characteristic capsizing events the statistical evaluation of the computed time periods until capsizing is described. The mean period until capsizing as a parameter of the exponential distribution is discussed for different factors of influence. Finally an example is given for the computing of the relative frequency of rolling angles.

The new approach allows the systematic computation of lever spectra of the righting moments as well as of capsizing events. This computation can be used when selecting the test samples for capsizing tests with automotive ship’s models and may provide a survey of the effect of parameter variation on safety from capsizing.

Das Kentern von Schiffen in unregelmässiger längslaufender See

1. Overview

2. The Influence of Sea State on the Capsizing Safety of Ships

2.1 Practical Significance

2.2 Previous Methods

2.3 Task

2.4 What Is Capsizing?

2.5 Capsizing Safety

3. Calculation Approach for the Capsizing Ship

3.1 Systems Theory and Stochastics

3.2 Spectral Representation of Stochastic Quantities

3.3 Equation of Motion

4. Analysis of the Equation of Motion

4.1 Righting Levers

4.2 Stochastic Rolling Motion

Page 37

Page 38

Page 39

Page 40

Page 41

Page 42

Page 43

Page 44

5.2 Transformation of the Bandwidth

Page 45

Page 46

5.3 Discrete Representation of the Transformed Spectra

Page 47

Page 48

Page 49

Pages 50–54

Damping Term

(page 56 continued)

(page 57)

6.3 Control Circuitry on the Analog Computer for Statistical Solutions

(page 58)

6.4 Accuracy

(page 59)

(page 60)

7. Results

7.1 Overview of the Computation Cases

(page 62)

(page 63)

(page 64)

(page 65)

7.2 Characteristic Capsizing Processes

(page 66)

(page 67)

7.3 Statistical Evaluation (continued)

7.3 Statistical Evaluation

Sample Sizes

7.4 Mean Time-to-Capsize for Various Parameters

7.5 Relative Frequency of Roll Angles

8. Summary

Acknowledgments

9. Notation

Symbols and Notation (continued)

10. Bibliography

Books

Articles

11. Appendix

12. List of Figures

Table 1 — Normalised Frequencies and Amplitudes of the Excitation Components at the Analog Computer

Fig. 7 (4.1.4a) — Fourier Analysis of the Regular Righting-Lever Variation in a Trochoidal Wave at 45° Inclination

Fig. 8 (4.1.4b) — Fourier Components c_p and Phases ε_p for Regular Righting-Lever Variation in a Trochoidal Wave

Fig. 9 (4.1.10) — Relative Frequency of Righting Levers in Head Seas for φ = 30°, Pierson–Moskowitz Sea Spectrum, GM̄_GW = 0

Fig. 10 (4.1.8) — Relative Frequency of Characteristic Wave Heights for 100 m Waves according to Roll Statistics

Fig. 11 (4.1.9) — Righting-Lever Differences for Ship Series 60 No. 4212 W

Fig. 13 (4.1.12) — Stochastic Righting Lever in Head Seas, GM̄_GW = 0

Fig. 14 (4.1.14) — Comparison of Normalized Spectra of Sea State and Righting-Lever Variation, Ship without Forward Speed

Fig. 15 (4.2.2) — Time to Capsizing for Various φ₀ at φ̇₀ = 0

Fig. 17 (5.1.2) — Transformed Sea-State Spectrum in Following Seas

Fig. 18 (5.1.3) — Fold Frequency f_V as a Function of Froude Number F_n

Fig. 19 (5.1.9) — Transformed Spectral Density S*, S_d* vs. Encounter Frequency f*, Following Seas

Fig. 20 (5.1.5) — Transformed Spectra in Head Seas and Following Seas, F_n as Parameter

Fig. 21 — First Spectral Moment M₁ of the Transformed Spectrum vs. Froude Number F_n

Fig. 22 (5.3.1) — Discretized Transformed Spectral Density S_d* for Arithmetic Frequency Division

Figure 23 (§ 8.2.1): Computing Circuit for the Equations of Motion on the Analog Computer

Figure 24 (§ 6.2.5): Non-Linear Component of the Mean Lever Curve

Figure 25 (§ 6.2.6): Non-Linear Component of the Righting Lever in the Oscillation Range

Fig. 19 (5.1.9) — Transformed Spectral Density S, S_d vs. Encounter Frequency f*, Following Seas