Ship motions memo no. 1. Notes on the analysis of graphical ship motion records

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SHiP MOTIONS MEMO NO3.

NOT ON E AIWIB

of

GBAIHIQIJ SHIP MOTION BECB

A. he Ship Motion Record..

In the analysis of ship-wave motions the basic init

of reference is thó individual recoz, not the individual

of cycles, and should motion taken on one course and

speed only. More specifically, tbe individual ship motion

record should satisfy the following x'equtements.

Record Lenbh: For larger vessels, and for iaotions

such, as roll, pitch, surge, and way,. each record should. be about 20-30 minutes long. ilternativély, the record

should contain 30-100 cycles of the lowest important

frequency component. Records considerably shorbe than

this can be used for amplitude analysis, but wUl not satisfactorily define the frequency characteristics of

the motion.

Time Scale: V/here frequency characteristics are

important, it is necessary to' use a. fairly high paper

speed. For typical records, the time base should be such

that it is possible to resolve 0.5 seconds or less. In

time inteivale in the whole record..

irlitude Scale: The amplitude scale should be zilch

-2-between the highest peak in the record and. zero. This

means ± 10 ° _± 20 levels. Obviously, both amplitude and

time should have rather finely-divided coordinate grids. VJhere the original curves are small-sized, _{as in the case}

of Brush oscillograph records, it is helpful to use some kind of a projector to enlarge the record. The A!icro-Caxd

reader can easily be converted to this application. For

curves as large as the Esterline-.Angus oscillographs,

such a reader is not necessary.

B. _{Analysis of Amplitude Characteristics.}

s shown in Figure 1, we can distinguish two types.

of amplitude variables in a given record: (1) the

ins1an-taneous amplitudes of motion-here called x; and (2) the peak, or envelope, amplitudes of motion-hare called . The characteristics of these tvo amplitude variables are closely related, as will be seen below, In addition, we can define a third amplitude vathb1e, z, which is the

collection of y's over all records.

Thstantaneous Amplitude: Instantaneous amplitudes should be measured at intervals sufficiently close

together to give about 1000 ordinates in a r20 minute

record. The time increments can be spaced uniformly or

at random; however, they should not be much _{closer together}

than one-quarter the shortest important period in the record. The amplitudes can be measured from _{any reference}

level. Negative amplitudes aheuld be distinguished from positive amplitudes.

Twenty to forbi levels in all should be defined an& amplitudes should be ccurmilated in. these levels to provid

an amplitude distribution, or histogram. The distribution should be normalized nd then converted to a cumulative

distribution, as indicated in Figure 2.

ExDerience has shown that the distribution of

instan-taneous amplitudes is clo'sely Gaussian. The cumulative

distribution should therefore be plotted on standard

probability paper, where the cumulative Gaussian

distri-button appears as a. straight line. Having this plot,, it..,

is. a very simple matter to pick off the mean . amplitude

C zero reference) and the RMS amplitude (or standard

deviation from zero reference). These .two paxméters

completely characterize the distribution. The RLS

amplitude should be noted and retained. The mean amplitude

is

Peak Amplitude.: If the frequency spectrum of the

motion appears to be fairly narrow-band (i.e. more or less

sinusoidal motion) nothing.furthèr heeds to be done,

since ithasbeen found in thiscase that peak amplitudes

in a given record have a Rayleigh distribution with a

mode equal to the RM$. amplitude, established above. That

is to say, if the probability density of peak amplitude

is g (.y), then x

I.

f(x) _=.

_e

and

where

= BMS instantaneous amplitude = most likely

peak amplitude. Prom equation (2) it follows that the peak amplitudes for a given record (with narrowband spectrum) are completely characterized by the RMS

instantaneous amplitude, which also happens to be the most likely peak amplitude.

T3here the spectrum is fairly bread, the distribution

of peak amplitudes will differ from the Rayleigh

distri-bution, but not markedly. Where it appears desirable to

detenine this distribution directly, the procedure is

However, in this case it is necessary to have a correct

zero line. No distinction should be made between positive and negative peaks. The distribution can be foried by

simply counting peaks or by picking points off an envelope. In either case there will be about 100-200 stgnifLcant

peak ordinates in the typical record, as defined above.

Long-Tei _{Peak 4mplitudes: Each record has its own}

RMS amplitude, and hence its own Rayleigh distribution.

It has been found that the summation of thase distributions

over a period of time approaches a limiting foin that is

very close to a log-normal distribution. This two-parameter distribution then characterizes the I,ong-term peak amplitud behavior of the motion under consideration.

As each individual record is analyzed, the number of

peaks at specified levels should either be deteiined

increment between levels. These numbers should be accuinu-lated. over all -ecorde to provide the long-term

distribu-tion of peak amplitude, or distribudistribu-tion of z. The

disti-bution should be normalized and put in cumulative form.

Since the distribution is closely log-normal it should..

appear as a straight line on probability paper when

cumulative probability is plotted aainst the logarithm

of peak amplitude. The long-term distribution can now be

defined in terms of average log-amplitude and spread. These

parameters tend to limiting values quite rapidly as the

number of records increases _{- fore-given ship operating in}

a given route or area.

Analysis of Preguency Characteristics,

Power _{pectruin Analysis: flhere it is necessazy to imow} the frequency characteristics of the motion in detail a

power spectrum analysis must be made. in the ease of

graphical records, this is done by picking about 2000 poin of instantaneous aI!rplitude off a given record,- auto-correla-ting these, and then compuauto-correla-ting the Pourier transform of the

auto-correlation.functjon. This last is the poWer spectrum, or the beat estimate thereof, assuming proper smoothing of

the. data.

The analrsis is quite straight-forward but involves

so much mathematical computation that it is almost precluded

where automatic digital computers are not available.

Usually, in the case of graphical records, one will simply undertake to analyze .the distributiOn of periods. While

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this treatment is imich cruder, it also involves much less

work0

Period Distribution nalys is: As shown in Figure 3,

we can distinguish. at least two types of periods (or half-j

periods): C].) per1ods0---or the time between: alternate zero crossings, and. (2) per'ioda1- - or the time between alternatc zero crossings of the Tflrst derivative of the motion. There

js obiousiy a hierarchy of

here., but these will do for most. practical purpQses. n.

accurate. knowledge of the zero level is required for the..

determthatiOn of periods0., but. not for the determination

of periods.

A distribution can 'be formed fOr either type of period by accumzlating periods in specified levels as waa 'dOne for. amplitudes. The period dIstributions' will not in general

have forms as simple as the.. amplitude distributions..

However, the garmna distribution has been found 'to: give a

pretty good .f it. when the pOwer spectrun is narrow-band,.

The medIan (or 5O' cumulative 'probability) period will be designated T0. In'the case of periOds0 and in the case

of periods1 WI].]. further define N0 as the number Of

zero crossings of the motion per unit time and N1 as the number of zero crossings of the. derivative _{per unit time,}

For a' narrow-band spectrum it will be found that the

two distributions are closely similar, that T0.is almost

e qua]. to .T1, and that' R0 is almost equal to N. As the spectrum becomes wider, the shapes of the distributions

-7.-smaflar than N1. The ratio N0/N1 gives a pretty good

tdt.i-cation of the spectitim bandwidth. UsuaUy it is sufficient

to calculate the distribution of periods1 and the ratio.

$inrpltfied Period Ant; An even siaple.r ana1yei3

of frequeny content can be mad by calculating N, and

N1, and the ratio L1JN1e Note that 1/N0 is approximately the average period0. and 1/N1 is appro*lwiteiy the average period1.

Lon-Texm Period Distribution: As was done for

anip1i-tudes, one can determine a long-term distribution for

periods. Little is

of such a distribution, if any.

D, Final Results of na]ysis.

. As demonstrated in the above paragraphs, the essential

information in tyitca1 ship motion records can be summarized

in terms of a hand.fi1 of paetY4.. The most important of

these part

outlined for finding their values in the case of any given record. It is convenient to divide these parameters into two groups: first-order parameters, and second-order ara-meters.

First-Order Parameters: The greater part of the information

in the record, particularly if it is reasonably narrow-band, can be defined in terms of three qualities power; center

frequency; and frequency bandwidth. In the notation

-8--= RS instantaneous amplitude-for power; T1 median period1-fo.r center frequency; and = zero crossing

ratio-for bandwidth. The center frequency and bandwidth

could be described.by slightly different parameters but the results would not be essenti3ly different. For example we could replace P1 by P0 or by l/N3, etc. Basically, we. have 0 , P1, and N0/N10

- Second-Order Paraneters: Where a more detailed anaiysi

is required, and particularly where the narrow-band hypothesis does not held, a few more parameters are

equired. These parameters add. more detail to the shape of the power spectrum.

Specifically, we would add. to the above: two more

para-meters. to define the shape of the period distribution

(spread. and skew), plus P0 and. two parameters to define

the shape of the period0 distribution. For resolution

beyond this, one ought to go to the complete power

spectrum analysis. In many cases, however, even these

second-order parameters may be superfluous.

Long-Term Distributions: The above discussion refers

to the. characteristics of individual records. In ad.d.it ion to characterizing the individual records one should. also accuxthlate the overall or "long-terxa" distribution of:

peak amplitudes, and possib1y of periods. The long-term

peak amplitude distribution, in. particular, is kaown to assume alimiting form, characteristic of the vessel and

itS. service.

J.H. CILtflWICK/n Mch 12, 1956.

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