A simplified model

There is an obvious parallel between an oscillating boat and the classical dynamic problem of the forced oscillation of a simple damped spring mass system. The latter may be extended to two or more degrees of freedom, just as a boat is. But if we make the initial assumption that coupling effects do not dominate the motion (or even that the essential qualitative features of the motion are exhibited by the equivalent single degree of freedom system) we can keep the model as simple as possible, and therefore gain maximum insight, by investigating the single spring mass analogy. The following discussion will therefore be restricted initially to heave motion.

Even with such a simple analogy the correspondence is evident. Typical graphs in any vibration textbook show strong similarities with seakeeping results. Graphs of amplitude ratio for a spring mass system are analogous to the response of a vessel to a unit wave force. (It will be argued below that variation with frequency of added mass and damping does not significantly affect the response function.) However the seakeeping problem differs significantly from the simple spring mass system in that the wave forcing is a strong function of frequency, and simple equations can also be derived to represent approximately this aspect of the behaviour. Net response is then obtained as the product of thewave force and theresponse to a unit force.

Other differences include the speed dependence of the coefficients that describe the force and response, multiple degrees of freedom, and non-linearities. In terms of the objective of developing a simple model these vary in importance, and each will be discussed below in light of this objective. The main governing parameters will then be identified, typical values determined, and some results will be presented to highlight the implications for the preliminary design of unconventional ships and extrapolation of existing designs.

A total assessment of seakeeping must also include consideration of the seaway in which the vessel is intended to operate, and the criteria for judging performance. In terms of human comfort the latter may be expressed in terms of MSI (discussed briefly in the previous chapter),

but more generally many factors must be considered, as outlined for example by Mandel [65]. It should be stressed that the following discussion of the simplified model aims only at the broadest assessment of only motion response of the vessel. It is intended to shed light on the main trends and is not intended to provide a detailed prediction of motions.

Response to a unit force

The response of the simple spring mass system can be expressed (as derived for example by Rao [81]) as H/2 δst = 1 q (1−r2₎2_{+ (2ζr)}2 (6.1)

where, in terms of the vessel’s hydrodynamic coefficients for a simplified representation of heave response, r = ωe ωn = frequency ratio ωn = s C33 M¡1 + A33 M

¢ = undamped natural frequency

cc = 2 s M µ 1 + A33 M ¶

C33= critical damping coefficient

ζ = B33

cc = damping ratio δst = 1

C33 = static deflection (heave for a unit force of zero frequency)

H = peak to peak heave (6.2)

Application of equation (6.1) requires only the specification of the natural frequency,ωn, and the damping ratio,ζ. These depend in turn on added mass, damping and hydrostatic stiffness. Most of the difficulty in estimating values is associated with the first two of these, and these will be discussed briefly before estimating typical values forωn and ζ.

Added mass Typically the added mass is of similar magnitude to the actual mass (A33 M '1). In the absence of free surface waves (i.e. in the high frequency limit) the heave added mass for a semi-circular section is identical to the actual mass, and figure 3-12 shows that, except at very low frequencies, it does not vary considerably from this value when free surface effects are modelled. For deep narrow sections it is expected to be lower, and for wide shallow sections higher, but, purely for practical purposes, one would not expect typical ship sections to vary too considerably from the aspect ratio of the semi-circle. Added mass affects mainly the natural frequency of the system. Even if it varies considerably, the influence on the natural frequency is approximately halved by dilution by the actual mass in the term¡1 +A33_M ¢, and halved again by the square root. Thus for example a±50% variation fromA33=M would give approximately

Damping The damping ratio influences the magnitude of the resonant peak in the motion response spectrum. This peak varies considerably from one set of experimental results to an- other, and it would be natural to assume that this variation is mainly due to variation in the damping ratio. As such it would be difficult to propose a simple model that would be universally applicable without considerable allowance for variation of empirical coefficients.

On the other hand if the added mass does not vary considerably from one hull form to another it is also reasonable to postulate that damping does not either. An exception perhaps is as a result of submergence, in which reduced proximity to the free surface would result in smaller free surface waves, hence a reduced mechanism for the energy dissipation that must be associated with damping. It will be shown that in fact a damping ratio that does not vary much from one boat to another could be entirely consistent with the observed considerable variation of the resonant peak magnitude, the effect being able to be explained in terms of a variation in the forcing function.

Frequency dependence Suppose that good estimates are known of the added mass and damping at the resonant frequency. We note that added mass and damping are generally weak functions of frequency, so motion should be reasonably well predicted anywhere in the vicinity of the resonant frequency. But at frequencies significantly differing from resonance we also note that the motion response is only weakly affected by the choice of resonant frequency or damping ratio, particularly where the latter is small and resonance effects are confined to a narrow frequency band. This is because the normalised response will always tend to unity in the zero frequency limit, and to zero in the high frequency limit, although the manner in which these limits are approached may vary (for example some of the low frequency pitch responses of SWATH forms presented later in this chapter (section 6.2.1) pass through a zone of low amplitude due to cancellation effects from heave coupling). Therefore, even if the estimates applicable for the resonant frequency are not the appropriate ones at other frequencies, the effect on motion predictions will be small. In other words, treating added mass and damping as constants for a given vessel (invariant with frequency) is quite adequate for the purposes of a broad simplified model.

Speed dependence With the exception of coupling (due to hydrodynamic asymmetry, as suggested by Gerritsma [32], and described in section 4.3), forward speed effects are not as dra- matic as generally thought, are difficult to quantify, and recalling from section 4.3 that Newman [74] questions the use of forward speed corrections altogether in conventional strip theory, it is debatable whether they are accurately predicted by standard strip theories. The simple model gives reasonable predictions without allowing for speed effects onωn andζ. Certainly the effect is expected to be of a higher order than those that are predicted by the proposed simplified model.

the observed speed dependent effects. This will be discussed below.

Typical coefficients In terms of length,L, beam,B, and draught,T, block coefficient,CB, and water-plane area coefficient,CW P, the hull stiffness in heave can be expressed asρgCW PLB, and the total effective mass as ρCBLBT¡1 + A33

M ¢

. Thus the dimensionless natural frequency in heave (assuming one degree of freedom) is approximately

ω∗ n=ωn s L g =α r L T (6.3) whereαis given by α= s CW P CB¡1 +A33_M ¢

and is a characteristic dimensionless parameter of a particular hull form, varying only slightly with speed through theA33 term. For a given hull shape CW P and CB can be quite precisely

quantified, and as was shown above the exact value ofA33was not extremely critical.

For a general hull, if we assume 0.4 < CB <0.7, 0.6< CW P <0.8, and 0.5 < A33 M <1.5, then we obtain 0.6< α <1.2. Even this is not a considerable variation given the possible range of hulls represented by the chosen coefficient ranges.

The main point to make here is that the dimensionless natural frequency of a boat is primarily a function of itsL

T ratio, or, almost equivalently, the water-plane area to displacement ratio1. The latter ratio is the more relevant one for the description of SWATHs since these can have values of CB (orCW P depending on the definition ofB used) significantly outside the ranges suggested. However one could refer to a notional effective draught, Te, as a measure of ‘SWATHness’, chosen to give the correct natural frequency using the same value of α as for an equivalent conventional hull2_{. In the discussion of the wave force it will be shown that it is both convenient}

and appropriate to do this. An L

T ratio of 20 is fairly typical of a conventional boat,and values as low as TL = 4 have been chosen to represent a notional semi-SWATH in the following discussion. This corresponds to an 80% reduction in water-plane area compared to the conventional hull. A value ofα= 1 has also been used.

The damping ratio is probably most easily estimated empirically. All boats are significantly underdamped (ζ ¿ 1). This is not necessarily evident from typical response curves, and the reason will be made apparent below. However one only needs to visualise the free oscillation of a hull that has been given an initial heave displacement to verify this. In all the simulations illustrated below the valueζ= 0.18 has been used, giving realistic results for the extremes of hull form and speed considered. It is reasonable to assume that a value in this vicinity is generally applicable for a simplified model of ship motions.

1_{The absolute natural frequency is mainly a function of draught.}

A consequence of the low damping ratio (ζ¿1) is that the damped natural frequency, ωd=ωnp1−ζ2_{, and the frequency of maximum response,ωn}p_1−2ζ2_{, will be for the present}

purposes almost the same as the natural frequency,ωn.

Wave force

The incident wave is generally defined in terms of the surface elevation. Thus two aspects of the wave force need consideration.

First is the question of the actual incident wave definition, generally in terms of one of several standard spectra. This is highly dependent on the proposed route for which a boat is designed, and (when presenting results in terms of dimensionless quantities) also dependent on the boat length. This deals with the application of the hull, which is beyond the scope of the present work (being mainly concerned with inherent hydrodynamic properties of the hull itself). Since the towing tank results that follow are nondimensionalised to represent a constant wave height that is what will be assumed here.

The second aspect, and the one that will be presently considered, is the question of wave force for a unit wave height, as a function of frequency (or equivalently wavelength).

Total wave force is made up of the hydrostatic and Froude-Krylov forces, which are easily obtained by integrating the incident wave pressure field over the hull surface, and the diffraction force, which is far more difficult to evaluate. Both are functions primarily of wave frequency (as opposed to encounter frequency, the primary variable influencing response). Fortunately the diffraction force is small compared with the hydrostatic and Froude-Krylov forces, as is shown consistently in the many sets of results contained in Magee and Beck [64]. Therefore for our simplified model it is sufficient to ignore it.

An approximate expression for the hydrostatic and Froude-Krylov force can be obtained by assuming the hull to be a rectangular prism. Given that the pressure field underneath a wave in head seas can be expressed in the form p

ρg =H2weky+i(kx−ω0t), the magnitude of the total heave

force on such a hull would be

|F|=ρgBHw 2 ¯ ¯ ¯ ¯ ¯ Z _L 0 e−kT+ikxdx ¯ ¯ ¯ ¯ ¯. If we takeρgBLHw

2 as the unit force (the heave force at zero frequency), then, after integrating,

the dimensionless force magnitude,F∗_{, would be}

F∗ _{= e}−kT s 2|1−cos (kL)| (kL)2 = e¡−_kLkT 2 ¢ ¯ ¯ ¯ ¯sin µ kL 2 ¶¯ ¯ ¯ ¯. (6.4)

This is a function of wave frequency since (for deep water)k= ω20

g . It should further be noted that, in the notation of equation (6.1), with the above definition of a unit force,δst = Hw

2 , the

The force on a real hull can be estimated in terms of the force on a rectangular prism by considering an effective length and draught. (We note that in dimensionless terms the force is independent of beam.) The effective length and draught will both be slightly less than the actual ones.

Examples of equation (6.4) can be seen in figures 6-4, 6-5, and 6-6. Thee−kT _{term represents} the Froude-Krylov correction resulting from the decay with depth of the non-hydrostatic com- ponent of pressure, while the remaining term accounts for the cancellation of force along the hull due to the changing phase in the longitudinal direction. At long wavelengths (low frequency) the force is hydrostatic (F∗ _{= 1), and reduces to zero when the wavelength and boat length are}

equal (ω∗

0 =

√

2π) because of exact cancellation of phase along the hull. (The discontinuity in slope represents a phase reversal; the sign, and not the magnitude, of the slope changes because the absolute value is taken.) At higher frequencies the force is generally small, and is also zero when the ratio of boat length to wavelength is an integer.

These results show remarkable similarity to results given in Magee and Beck [64] for real hull forms (particularly if the former are plotted as a function ofkL, as Magee and Beck do, instead ofω0

q L

g). As an illustrative example, a typical figure of Magee and Beck’s heave exciting force results has been reproduced below in figure 6-1, and this can be compared with figure 6-6. The main difference is the frequencies at which exact cancellation of the forces occurs, being slightly higher for the real hulls, confirming that the effective length is slightly less than the actual length.

Figure 6-1: Reproduction of Magee and Beck [64] figure 310: Magnitude of heave exciting force as a function of kL for SL-7 containership, F r = 0.3. (3D panel method, strip theory and experiment.)

Net response

Net response for a unit wave height is obtained as the product of response for a unit force (equation (6.1)) and force for a unit wave height (equation (6.4)), taking care to recognise that the former is a function of encounter frequency and the latter of wave frequency. The result is summarised as H Hw = 2e−kT kL ¯ ¯_sin¡kL 2 ¢¯ ¯ q (1−r2₎2_{+ (2ζr)}2 (6.5)

where in deep waterr=³ω∗

0+ (ω∗0) 2

F r´

√

T /L

α in head seas, which embodies the Doppler shift between encounter and wave frequency, andk=ω20

g (orkL= (ω∗0) 2

).

Examples are given in figures 6-4, 6-5, and 6-6 illustrating the effects of both SWATHness and speed on total response, and these will be discussed below. Each of these figures shows the forcing for a unit wave height, the response for a unit force, and the net response for a unit wave height, illustrating the importance of both forcing and resonance effects. Also the dimensionless acceleration for a unit wave height is given as a means of evaluating options, as this in the primary stimulus to seasickness. The forcing is based on the rectangular prism analogy, andα andζ are taken respectively as 1.0 and 0.18.

An investigation of the effect of speed Figures 6-4 and 6-5 show the effect of speed respectively for a notional conventional hull (L

T = 20) and a notional SWATH (TLe = 4, corre-

sponding to an 80% water-plane area reduction on the notional conventional hull for the same displacement and length), as predicted by the simple model just described.

The only effect of speed is to change the wavelength at which the resonant frequency is encountered, shown by a left shift in the resonant peak when plotted against the wave frequency. The actual magnitude of the peak of thedimensionless heave amplitude for unit forcegraph does not change. The result is that this resonant peak is brought into coincidence with significant forcing, and the net response (dimensionless heave for unit wave height) is significantly increased. At low speeds the forcing is so small that the resonant peak is not even evident (which could be mistaken for supercritical damping). Because the forcing is asymptotic to the hydrostatic value at infinite wavelength (zero frequency) the net response approaches a maximum value as speed increases.

The SWATH, because of a lower natural frequency, reaches the maximum net response sooner as speed is increased (compare the dimensionless heave for unit wave height in figures 6-4 and 6-5). However again because of a lower natural frequency, the accelerations at high speed are significantly smaller for the SWATH. This must always be the case in the high speed limit because of the ceiling on the response. The maximum possible response will be ³H/_δ 2

st

´

max = 1 2ζ√1−ζ2 ([81], equation (3.33)), or, given that ζ ¿ 1,

³ H/2 δst ´ max ' 1

2ζ. If the assumption that ζ does not vary significantly is correct, and the further assumption made that the simplified model still holds at high speeds, then the conclusion to be made is that for high speed (at which F∗ _{= 1,}

i.e. δst = Hw

2 ) the only significant change the hull shape can make is to decrease the natural

frequency, and with it maximum accelerations.

An investigation of the effect of water-plane area reduction (‘SWATHness’) SWATH- ness, as suggested above, can be expressed in terms of the ratio L

Te. The effect is twofold. First,

there is a reduction of the resonant frequency, embodied in equation (6.3). Second, the force is reduced because of the Froude-Krylov term in equation (6.4).

One could argue that if the water-line beam is reduced while the maximum beam is main-