Derived Responses”
5.1 Introduction. The aspects of ship response to rough seas that are generally of greatest importance
to the evaluation of seakeeping performance will now be considered. These are the responses that can in principle be derived from the basic six modes of motion.
They include:
Vertical and/or lateral motions, velocities and accelerations at specific points, i.e. local motions.
Relative motions between a location in the ship (as the F.P.) and the encountered waves.
Non-linear effects arise in connection with:
Shipping water and slamming.
Yawing and broaching.
Added resistance and powering in waves.
Wave bending moments and loads on hull and equipment.
All of these linear and non-linear responses will be considered in this section, first in regular and then in irregular seas.
5.2 local and Relative Motions.
(a) General case of six degrees of freedom. If am- plitudes of motion in regular waves in all six degrees
of freedom, with phase angles, have been calculated by methods discussed in Section 3.3 to 3.5, it is possible to compute the resultant local motions, velocities and accelerations a t any position in the ship (Hamlin, 1979).
Such calculations are facilitated by the use of complex number notation, however.
The motions of any point (E, 3, Z,) on the body may be defined by three translations and three rotations.
Since the ship is assumed to be a rigid body, the ro- tational motions are the same at all points on the body.
Thus, the complex amplitudes T4, T5, T6 define the rotational motions at every point. In vector notation we may write
- 0 = vector of rotational motion at any point
= ( 7 7 4 , T5, T6)
The three translational motions a t any point, re- sulting from the combined effect of rotation of the body and translation of its center of gravity, are de- fined in vector notation by
where
kl, Fz, k3
are the complex local amplitudes at2, 3, 2 in Surge, EWay and heave, respectively.
To determine [ we must combine the motion at
l1 Section 5 written by Robert F. Beck, John F. Dalzell and the Editor.
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110 PRINCIPLES OF NAVAL ARCHITECTURE
( E ,
y,
Z ) due to the translations and rotations of the body. From basic kinematics it can be shown that, for small motions,where
2 is the vector of translational motions at the origin
= (771,772,773)
= (Z,
y,
X )- r is the position vector of point Z, 7j, Z
The individual component amplitudes of
2
are given by -L$ = q + z - 5 - - - 6
I ; 2 = 7 2 + - - x 7 7 6 - - - x 7 7 4 (208) I; = q + g - 4 - - - *
77 Y77
- -
77 x77,
The velocity and acceleration a t any point are found by differentiating the motion with respect to time. Re- call that the time dependent motion a t any point is written as: ( ( t ) = L$ eiWet
Thus, the complex amplitudes of the velocities and accelerations are given by
-
4;
the complex amplitude of velocity(209) vector = io, - I;, and
vector = -0: I;
.-.
L$ the complex amplitude of acceleration-
To find the translatory motions, velocities or accel- erations at any point on the basis of the preceding equations it is simply necessary to add several complex numbers and multiply by i w , or -0," as necessary.
Since these equations are in complex number format, they give both the absolute, or scalar, magnitudes and the phase angles for a ship in regular waves. Note that these complex amplitudes have been normalized by dividing by the exciting wave amplitude, so that they represent response per unit wave amplitude, as in Section 4.
To proceed to irregular seas, response spectra can be obtained from wave spectra and appropriate RAOs, using the techniques discussed in Section 4. In this case we are concerned only with the absolute or scalar amplitudes of response, since the phase angles of the component waves of the sea are assumed to be random.
Hence, the RAOs are simply
&, &I, ITji,
where the sidebars signify that only the absolute values of the complex amplitudes have been taken. From the re- sulting response spectra the variance and other sta- tistical quantities can be calculated as needed.If the motions are sep- arated into the longitudinal and transverse modes, as in the case of a symmetrical ship, where there is no
(b) Longitudinal motions.
!
q3Fig. 82 Notation for absolute and relative motions in head seas, as scalar functions at time (i = 0 )
coupling between the longitudinal and transverse mo- tions, the problem is simpler. I t is further simplified if head seas only are considered, or if the point is located on the ship's center line. Insofar as the eval- uation of specific ship or platform designs is concerned the responses that may be derived from longitudinal plane motions are often of more importance than the heave or pitch motions themselves. Specifically, the responses of particular importance are vertical: the vertical motion at any point along the length of the ship (local heave); the vertical velocities at any point along the length; the vertical accelerations at any point; and the vertical motion and velocity a t any point relative to the sea surface, Fig. 82.
In this case Equation (208) for vertical local motion reduces to
-
(210) L$ = q - - -
where
i3
is the complex amplitude of vertical motion- at point Z, 0, P, with p = 180 deg, assuming that both I; and
q3
have been normalized by dividing by wave amplitude,<
. For calculating responses to long-crested irregular head seas it is only necessary to knowE3j
and
E5),
the amplitudes of heave and pitch. The RAO is then7 7 5
where
1 3 1
represents the absolute magnitude of the complex amplitude. (Note that phase information em- bodied inT3
and T5 was needed to calculatek3).
Fig.282 defines the notation for simple problems in head seas on which the preceding equations are based.
The asymptotic properties of the local vertical mo- tion RAO are the same as for heave (Section 3.2), that is, as wave frequency approaches zero (long waves) pitch approaches wave slope (which approaches zero) and the heave RAO approaches unity, so that the local vertical motion RAO also approaches unity. At very high frequencies both pitch and heave RAO's tend to zero, so that the local vertical motion RAO also does.
The qualitative differences between the heave RAO and the local vertical motion RAO are slight. When the point of interest is far removed from the origin,
MOTIONS IN WAVES 1 1 1
the pitching terms usually have the effect of producing a much larger apparent resonance peak of vertical motion than for heave alone near midship. Because of the phasing between heave and pitch, the amplitude of vertical motion is usually much larger a t the bow than at the stern. The longitudinal location of the cen- ter of flotation is also a factor.
In general, if the amplitude of heave-or of vertical motion at any point-is known in regular waves, the corresponding velocity and acceleration amplitudes can be determined from
Hence, the response amplitude operators (squared) can be transformed from motion to velocity by multiplying successive points by we and to acceleration by multi- plying by w,2. In evaluating ship performance the ver- tical acceleration at critical locations is of particular importance.
Also of interest is the relative vertical motion be- tween a point in the ship and the surface of the en- countered wave. This relative motion in regular wave is found by subtracting the free-surface motion from the vertical ship motion a t the desired point, taking account of their phase relationship. The free-surface motion is composed of the incident wave, the diffracted
wave, the radiated wave (Section 3) and the wave due to the ship’s steady forward speed. The traditional assumption is that the principal component is the in- cident wave and that the other components tend to cancel each other; i.e., the incident wave is not distorted by the presence of the ship. This may or may not be true and will be discussed further in the next subsec- tion.
Assuming that the wave is not distorted by the pres- ence of the ship and B = 0, the amplitude of the relative vertical motion in general is given by
k
R= k
3 - J - - = T Z Y 3 - - - xr/5- - -
(213)
+
V % -t??
= - 713 -
zq, +
gq4- -
ye-&(”
eos p + sin +)where
rzYg
the complex amplitude of the wave at x= Z, y = y relative to the body coordinate system.
In the case of pitch and heave in head seas this reduces to
where
tz
is the complex wave amplitude-at x-= iE relative to the body coordinate system, or <-; = J eikz.Then the RAO which requires only the scalar or absolute amplitude is
6 I 1
I
F,
-
0.14 1
I I I I
0 1 2 3
I
I I ILJ L
1 2 3
0
Fig. 83 Measured magnitude of relative motion RAOs in regular head waves (O‘Dea, 1983)
112 PRINCIPLES OF NAVAL ARCHITECTURE
I t is important to note that the relative motion RAO has entirely different asymptotic properties than heave. These properties are illustrated in Fig. 83 (O’Dea, 1983), which shows a plot of the total relative motion divided by the incident wave amplitude at sta- tions 0-3 (on a 20-station ship) for a high-speed cargo ship (SL-7) model. At very long wavelengths the rel- ative motion tends to zero because the ship contours the waves. For very short wavelengths both heave and pitch RAO’s go to zero, and the relative motion is due only to the wave motion, yielding a value of around 1.0. In between the two extremes the relative motion peaks, typically around a wavelength to ship-length ratio of 1 to 1.5. In general the relative motion is greatest at the bow. However, as seen in the figure for Froude number 0.3, there are cases where the relative motion is greater abaft the bow. O’Dea also gives curves for non-head seas. For this case, the char- acter of the relative motion curves is the same as Fig.
(a) WAVE SPECTRUM
5m SIGNIFICANT WAVE HEIGHT (m, = 1.56m2)
WAVE FREQUENCY, w
L
15001000 500 300 2 0 0 150 100 75 50 40 WAVE LENGTH, L,m
0) RESPONSE OPERATOR 150m (500-FT)
SHIP AT 11.3 KNOTS
0 0.2 0.4 0.6 0.8 1 .o 1.2
m, RADBEC.
0 $! 5 0 - 40- -
-
b 3- ,30
20 10
v)
-
-
0 0.2 0.4 0.6 0.8 1 .o 1.2
w, RADEEC.
Fig. 84 Relative bow motion in an irregular head sea
83, except that the magnitude is increased on the weather side and decreased on the lee side.
I t should be noted that the model test data given in Fig. 83 include certain dynamic effects to be discussed subsequently. So far our discussion of relative bow motion is based on the simplifying assumption that the ship and its motions do not affect the encountered wave, giving a so-called kinematic solution.
If the RAO’s for absolute or relative motions have been calculated and the system is assumed to be linear, then the corresponding responses to irregular seas can be determined by the methods of Section 4. The cal- culation of relative bow motion in an irregular sea is of particular interest. Fig. 84 is an example of the prediction of relative bow motion for the 150-m (500- ft) ship of Fig. 72 (Section 4). In this example the influence of p i t a iKrelatively strong and results in peak values of
I E R / C I
in excess of 4.0 a t o = 0.6 rad per sec. This is not an untypical magnitude for a ship at speed in head seas. Depending upon speed and full- ness the peak relative motion RAO for ships can vary between 2 and 5. In the example of Fig. 84 the effect of unity high-frequency relative motion RAO is slight.It becomes of more concern as the RAO peak induced by pitch becomes smaller, or as the wave spectrum peak shifts to higher frequency.
The significance of the relative motion response is that the moments of the spectrum provide probability measures related to anticipated deck wetness and bow emergence-the latter affecting the likelihood of slam- ming-as discussed in the following sub-sections.
However, in general we can expect that comparative calculations of relative bow motion will give useful qualitative information on the seakeeping performance of alternative ship designs. It should be emphasized that so far we have considered only the simplest pos- sible, or kinematic, treatment of relative motion. Dy- namic effects will be discussed under shipping water in the next sub-section.
5.3 Shipping Water Forward. An important aspect of relative bow motion is the probability of bow sub- mergence and hence of shipping water on deck, par- ticularly in head seas, since this greatly affects attainable speed in service. Predicting the shipping of water involves the comparison of the relative bow mo- tion previously discussed with the available bow free- board. One approach is to compute the probable fraction (or percentage) of time that the foredeck is awash or that the deck edge or bulwark is submerged, a t a specific longitudinal location. This can be done on the basis of the underlying assumption that relative motion can be considered to be a Gaussian process, so that values taken at equal intervals of time follow a normal or Gaussian density function. The probability of relative bow motion exceeding a particular value of freeboard F , , is 1 minus the cumulative normal dis- tribution, which is readily obtainable from probability tables.
M O T I O N S IN WAVES
Fig. 85 Required bow freeboard ratio for constant probability of wetness, P. Average of all headings
Typical values are given in the abbreviated Table 18, where
&
is the standard deviation and m , is the area under the response spectrum. For example, if the bow freeboard is twice6
a t the bow F , /& is 2.0.Then the probable fraction of time that the foredeck is awash would be 0.0228.
Often we are interested in the probability that an event such as shipping of water will occur in any par- ticular cycle of motion; i.e., the probability that a peak response will exceed the freeboard F,. Here we are concerned with the visible peaks (or maxima) of the record rather than equally spaced points in time. In most cases it may be assumed that the relative low motion spectrum is narrow band and that a Rayleigh density function will apply.
In Section 4 it is stated that, on the basis of a Ray- leigh distribution, the highest expected amplitude in 1000 oscillations is 3.85
&
for any ship response, such as relative bow motion, where m o is the area under the response variance spectrum. For a larger number of cycles, N, than 1000, the highest expected value isJm.
Hence, if we place the bow free- board F , equal to 4-N and solve for N we can say that the water will be expected to reach the weather deck once in NL cycles,113
>
NL = e F l 2 / 2 m 0 (216)
Alternatively, the cumulative Rayleigh distribution, whose values are the reciprocal of the right-hand side of Equation (216), can tell us directly the percentage of cycles (or maxima) in which it is to be expected that the deck will be wet by the sea. These are given by the probabilities of exceeding F , ,
(217) p = 1 - e - F 1 2 / 2 m 0
Fl
By calculating the average apparent period T , from Equation (52) Section 2, or assuming T , = T n 5 , the result can be expressed in terms of number of times per hour,
N , = 3600 P J T , (218)
Table 18-Probability of Exceeding a Particular Value of a Gaussian (or Normal) Function
Fl/& Probability
0 . . ... .0.5000 0.5 ... .0.3085 1.0 ... 0.1587 1.5 ... .0.0668 2.0 ... .0.0228 3 . 0 . . ... .0.0013
114 PRINCIPLES OF NAVAL ARCHITECTURE
The water can be expected to reach the deck on the average N , times per hour, so long as conditions re- main unchanged and the total number of cycles N , is large.
Probability techniques have been applied to deter- mination of desirable trends in bow freeboard with ship type and size. For example, calculations were made (Band, 1964) beginning with relative bow motion for four lengths of full tankers in five different short- crested head seas. The probability of bow submergence at reasonable speeds in each sea was determined. Fi- nally, considering the frequency of occurrence of each sea condition the combined probability of bow sub- mergence in all head seas was calculated as a function of freeboard. Results are plotted in Fig. 85 in the form of required freeboard/ length ratio versus ship length for different probabilities of bow submergence in both the North Atlantic and on typical tanker routes (Eu- rope or U.S. to Mediterranean). In this case, absolute values are less important than trends, and it is inter- esting to note that all of the curves indicate approxi- mately constant required freeboard for ship lengths above about 180m (600 ft). Hence, if a bow freeboard of 0.05L = 9m (30 ft) has been found satisfactory for 600-ft cargo ships it would appear that 9m (30 ft) should be satisfactory for any longer ships of the same type in head seas.
However, if accurate quantitative predictions of shipping water for specific ships in specific seas are needed, a detailed analysis of the deck wetness prob- lem reveals that the effective freeboard does not equal the nominal freeboard, and the relative motion is al- tered by hydrodynamic effects not accounted for in the simple kinematic approach. The actual effective free- board can be considered as the sum of several com- ponents. The most important is the nominal freeboard, usually defined as the distance from the calm waterline to the deck or top of the bulwark at any longitudinal location. The second is the change in freeboard due to the sinkage and trim caused by the forward speed of the vessel. The forward speed also creates a calm water wave profile which further modifies the free- board. Tasaki (1960) called these two effects static swell-up. Finally, as introduced by Newton (1960), the above-water body shape, freeboard, flare, knuckles, and other special features will alter the necessary rel- ative motion required to produce deck wetness. Al- though this influence is often considered as a change in effective freeboard, it is convenient to consider it here along with the dynamic effects to follow.
Static swell-up can easily be evaluated by model tests in still water, and it can be assumed that there is linear superposition of the ship’s wave and the en- countered wave. The theoretical prediction of static swell-up has been extensively investigated in conjunc- tion with ship wave-resistance theory. Various theo- retical methods are available ranging from simple slender-body theories to “thin-ship” theory, to three-
dimensional source-panel distribution methods. A good summary may be found in Bai and McCarthy (1979) and Noblesse and McCarthy (1983). Lee, et a1 (1982) used a thin-ship theory to determine the steady wave profile and devised an empirical formula for the pre- diction of sinkage and trim of destroyer-type hulls (Bishop and Bales, 1978) and (Tasaki, 1963).
The actual relative motion may differ from that ob- tained by the simple kinematic approach because of the presence of the ship, as mentioned in Section 5.2.
The first effect is the diffraction of the incident wave system and the second is the radiation of waves caused by the motions of the vessel. The change in relative motion due to the diffracted and radiated wave systems is often referred to as dynamic swell-up since it re- sults from the dynamics of ship and wave motions.
Dynamic swell-up can again be determined by model tests. In model tests of a Mariner hull in head seas (C, = 0.61), Hoffman and Maclean (1970) found a dy- namic swell-up factor of 1.12 to 1.15. Experimental trends of swell-up are given by Journee (1976a) for full-load and ballast conditions of a high-speed cargo ship (with bulb). I t is noted that the dynamic swell-up is much greater a t a station 10 percent of length abaft the FP than at the FP. O’Dea (1983) and O’Dea and Jones (1983) also measured the components on a model of an SL-7 high-speed containership.
Since simple general formulas for all types of ship are not yet available, the best solution in a specific case appears to be model experiments for the designs under consideration. However, theoretical and exper- imental research continues and some highlights will be mentioned.
Lee, O’Dea and Meyers (1982) extended the basic ship motion theory, as presented in Section 3, to predict the vertical motions of a point in a ship relative to the free surface, retaining the assumptions of strip theory and linearity. Calculations for a high-speed contain- ership and a typical naval combatant were then com- pared with model tests in head and bow seas. Results showed some discrepancies at the higher speed, which were attributed to inaccuracies in prediction of the phase relations among the incident, diffracted and ra- diated wave components, and may be due in part to neglect of non-linear effects, including the influence of above-water hull form. On the other hand, the com- parisons reveal no conclusive evidence that the theo- retical refinements in calculation of relative motions provide much improvement compared to using only the kinematic solution.
Beck (1982) measured experimentally the radiation and diffraction components in head seas about a math- ematical hull form with parabolic waterlines. The com- ponents were also predicted theoretically using a strip theory for the radiated waves and a slender-body the- ory for the diffracted waves. Researchers have used three-dimensional computations to improve the agree- ment between theory and experiment for the mathe-