Considerations on the application of stochastic processes to the response of ship on the ocean, Part 2

CH1EF

Considerations on the Application of Stochastic Processes to

the Response of Ship on the Ocean

Part 2

Yasufumi YAMANOUCHI Ship Research Institute

Presented to the 2nd meeting of the Japan-U.S. Joint Seminar on

Applied Stochastics Held at Washington, D.C. Sept. 19-25, 1968 Lab. _{y. Scheepsbouwkunde} Technische Hogeschool Deift

r

Considerations on the Application of Stochastic Processes to the Response of Ship on the Ocean

Part 2

Yasufumi YAMANOUCHI Ship Research Institute

1. Introduction

The purpose of the application of the stochastic process analysis technique to the analysis of the dynamic system is to find out its dynamic behaviour in its real operating conditions, which is usually in the random

environments. This is also the case for the analysis of the response of

the ship on the sea, and we naval architects wish to find out the behaviour of the ship, namely for example, the response character of the ship

oscil-lations, stresses induced on the hull structure, the increase of thrust and

torque on screw propeller shaft of the ship that is working heavily on the irregular sea waves, and also their mutual interactions, in order to make

clear the mechanism of the behaviour and to find out the way to improve the

ship performance in the waves. In this case, the ship is one of the dynamic systems working under the stochastic or random inputs. When the mecha-nism of the response is rather clear and simple, and one output responses to one input, the response character of the system, for example in the

shape of the frequency response characters can be obtained from the spectrum analysis as were shown by many authors, - 3)

This author also showed 4) a result of a trial to get the impulse response

-1-function directly from auto and cross correlations, and insisted that the time domain analysis could also be practical.

On the other hand, this author has intended to use the coherency function as an index to show the degree of linearity of the response, and for this purpose, the apparent loss of coherency caused by the spectral window was found to be an annoyance. This was examined, and the method to remove this difficulty was shown first practically and then theoreti-cally.3)

However, it was found that the coherency was also low when some of

the inputs actually affecting this system were omrnitted or failed to be

counted. In order to make the coherency really an effective index to show the feasibility of the linear explanation for some definite system, usually the multi-inputs analysis, which was already reported by many authors6)

-9) _{was found to be necessary.}

It was also realized that multi-inputs analysis was applicable to detect the character of non-linear response even when the input was single.

Here using the case of the analysis of stress, induced on the transverse

member of the ship' s hull, the above mentioned facts will be shown. The description will be made following the order of the steps actually taken, by

try and error method, in the process of analysis.

2. Data used as example

Simultaneous records of ship motions of 1 3 test runs taken by our group on board a certain cargo liner in 1 964, io) sailing across the Pacific Ocean west-wards from Panama Strait to Yokohama were used as examples

to be analysed here. Among the measured items, rolling, pitching

oscil-lations, relative wave height at midship, the transverse stress induced

on the hull structure and sometimes the virtical accelalations were

digitized and punched on tapes during the runs, for a series of tests,

per-formed generally once a day during the vogage. Here the relative wave height means the variation of water level relative to the ship side, measured

from the deck side virtically downwards by supersonic probe. The

trans-verse stress was measured on the web frame near the midship section. The rolling and pitching are the angular oscillations around the horizontal and virtical axis, measured by a free gyro with automatic horizontal setting apparatus. The general aspect of the environment of the ship at test runs and the particulars of analysis are shown in Table i

3. The process of analysis and the considerations on the results 3.1 Basic analysis and averaging

At the beginning, as examples are shown in Fig. i the ordinary spectrum of the measured items in these test runs were computed by the

standard process, as are used in our oup at present. The total number of the sample M, the maximum lag number m, the sampling time

interval t are shown in Table 1 . M/m was usually i O with a few

excep-tions. As the spectrum window, W2, proposed by Akaike with the weight-ing factors a0=O.6398, a1 =a1 = 0.2401, a2= a2 = -0.0600, was used. Many informations can be obtained of course even from these plain spectra

for us naval architects.

As a trial, a kind of average were taken expecting that through

--averaging the spectra of these processes, the effects of the spectrum density of the sea waves, the input for these cases, which varies quite much each other by the test runs must be averaged down, and the averaged spectrum will approach to the response to the input with quite uniform spectrum density. In ideal case, then the averaged shape of the response spectrum must show the shape of the squared response function themselves, as the response to the input with white spectrum must show just the shape of the squared frequency response function.

This average was performed in normalized correlation coefficients, and after averaging in time domain, the averaged normalized correlogram was Fourier transformed. Namely the averaged normalized correlation of response is

I

rr()/rr(°)J

(V)/2

(i)

Its Fourier transform is

function H (w), then

&S (w) = Ç H (w)

Accordingly

= f H

- L -yr

r

(w)/2j

(2)

On the other hand the spectrum S1Çw)of' response r(t) is expressed by the frequency response function H(tO) of response r(t) to wave (t), and the

spectrum of' waves S (w) as

yy()

) Hr(W)

'S

()

(3)

Here suppose that the spectrum of the waves S (w) is the response of the

sea to some imaginary random and white process 42 with a response

(4)

2 ,4y _A2

=

Hy()i

(A/

)JH4()!2

()

and the term with average sign has the tendency to become less variant with the frequency, thus results P (&) to have the shape very much like that of in ideal case.

The results are shown in Fig. 2, the ordinate of the spectrum being in arbitrary unit. The shape of the averaged rolling and pitching spectrum shows a reasonable shape as the response of roll and pitch respectively, the position of the peak corresponding to the natural frequency of each mode of oscillations. The shape of the frequency response character of the transverse stress to the wave has never been shown by the structure group of naval architects, because the transverse strength has been considered by them to be less important for the strength of the ship than the longitudinal stresses that are induced by the longitudinal bending moment which has been treated as one of the serious concern to them.

Recently, however, with the increase of the size of the ship, the trans-verse strength of the hull has come to be noticed as one of the important

items that can not be neglected by the designer. The statistical characters

of the transverse stress will be examined here, though the character of the

transverse stress itself is not our main concern here. This was ta.ken just

as an example of the response of ship to the waves, and also the motions. As the starting point, however, we will use our engineering sense on

the transverse stress which we know from our experience. That is closely

related to rolling motion as the longitudinal stress is to the pitching motion, and are also believed to be correlated to the relative water level.

Investigating the shape of the averaged spectrum of the stress, _{we will}

find that the position of the peak is just about at the frequency two times of that of the peak of rolling.

As the reason of this fact, the following possibilities were considered. By sorre mechanism, the transverse stress might vary with the double frequency of that of the rolling.

The response of transverse stress might strongly be under the

non-linear, or square effect of rolling.

The first one should be checked from engineering point of view, however so far, the proper explanation was not obtained along this line. The natural frequency of the transverse stress is of course far high than the frequency of the range which we are interested in now. At this moment, the 2nd model seemed to be considered. Here this will be examined.

3.2 Investigation of non-linear responses

3.2.1 Considerations on weak non-linearity

Bef ore to go into the discussion of the non-hlLearity of response, here

the stress, the non-linearity of motions, here assumed as inputs will be

discussed first.

An approximate method to get the effect of non-linear component that

exist in the response character of a dynamic system on the spectrum was previously derived 4) by this author.

That was say for the case with weak linearity where the non-linear damping, or the velocity square damping was assumed to exist in a very simple oscillation system, that can be expressed by the second order differential equation of motion, as for the simplifiEd ship' s rolling oscillation.

linear term

[-i, (w)]

=

is the variance of the angular velocity as is

o

fcw)

(12)

and, (.o) being the double convolution of the spectrum5 to itself

as

The same kind of perturbation method was applied to another typical non-linear system, the simple oscillation with non-linear _{restoring term,} as we have sometimes also in simplified rolling.

t

=

(14)

-7-(io) (13)

I

W) e ₍₆₎ or

±tScic!

CL)05

= fll(t)

(7)

The result of a perturbation method, starting from the zero order approxi-mation of linear response 0(t), gave us as the spectrum expression

(a)) of the first approximation çb (t), as the modification of linear spectrum

SjuJ) by additive terms as

2t of e0

6. S

(o) t

S.

)

(8)

here

(w) = w1Sw)

(9)

and H,,,1 (w) is the linear frequency response function of 4(t) to the

non-24

t

-

(15) By the effect of the term k33 , the spectrum S

() of the first

approxi-mati on (t) vas given as the modification of the linear spectrum S

(te) as Ac.

=

Sui)-Hn(w) ?

6S(u)J

(16) where

Hui)

f H(w)j

W2u1,2 _+1d (w- wz..)

(Q)Z)Z4ZZ

f (18)

and again (w) is the double convolution of the spectrum

Sw)

=j J$aU

₍₁₉₎

These both equation (i 3) and (1 6) show the spectrum of the first

approxi-mation of non-linear system, and the 2nd and the 3rd term are the modifi-cation of linear spectrum by the existence of non-linear damping _¡34i141

or non-linear restoring term As were already discussed, when

S or S _{is narrow banded as is in the case of ship' s rolling, the}

double convolution of or _{to itself has sharp peak at its}

original peak position LO _{, and also at its 3 times of the frequency,}

3 LO0

Although the effect of this double convolution _{at 3 &. might be very} much reduced by the small value of JH4.iw)J2multiplied to _. (ù), on the

3rd term of eq. (13) and (16), especially for the case with very small

spectrum, there is still a possibility to have small peak at 34 , besides

the original main peak, that is also modified by the existence of the non-linear term, by non-non-linear damping or non-non-linear restoring term

differ-ently.

Here investigating carefully the spectrum of rolling, plotted by the logartithmic ordinate as is shown in Fig. 9, this might be a possible ex-planation of small peak that can be detected always at around 3 .

However these peak are so low in energy, and accordingly it is not safe to guess too definitely from this point the non-linearity of rolling.

Accordingly, here, the non-linearity of rolling is assumed not to be so strong. The rest of the inputs, pitching and the relative wave height can be assumed more possibly be linear.

3.2.2

Considerations on general non-linear response

Considering the rolling as the input, the stress has very little re-sponse to the rolling even at its peak frequency. This fact is also

reflect-ed to the low value of coherency as an example is shown in Fig. 3 obtainreflect-ed

by cross-spectrum analysis of stress to the rolling, that has been

perform-ed following our standard process.1 i Accordingly if we think over the possibility of non-linear response of stress, the considerations related in

preceding section, that dealt with the system with rather weak

non-linearity can not be applied. Here at this situation, we have to think over

the possibility of strong non-linearity.

As a technique to find out the non-linearity of a system operating

under a Gaussian input, the higher order spectra or higher order

12)-14)

spectra has been proposed, by a few authors. Assuming the

sea waves as Gaussian, if we could compute the bispectra (u),, &)2)

of rolling 4) (t), and also TB1- (4 ¿L)2) of response r (t). we will be

able to know the extent of the non- linearity by computing the skewness

p=

(20)

where

j:

The skewness for stress must be much larger than that of the rolling, if

the hypothesis ii) in 3.1 holds, namely the stress is a strongly

non-linear response of rolling. However, at this moment it needs a great

efforts to compute the bispectra, and so, leaving this as a future work, here a simpler technique was adopted.

At first, the square of the rolling records was computed, namely the

rolling squared process was artificially made. Using this process as the

input, the coherency with the stress was calculated and compared with the

co-herency of the original rolling to the stress as was already shown in Fig. 3.

This is a special case of cross bispectrum analysis or the 'mixed

spectrum' analysis proposed by Akaike. 14) Namely, in the expression by Akaike as

fl7í-r,o) = EJ

'(1r)-j (u3,1 o)du3,cJoJ N (o,

O)<3>

₍₂₂₎

(23)

J(w)dc = Rô) =

₍₂₁₎

(26) B

_(f

)

ft)

(t,

) d (24) is put to be zero,

t) = E

[

±-_&*tri}J

2( _{) (25)}

z(f)

I J-o.0

There were computed. This ß1z (f) correspond directly to the

represen-tation

f().

(-t)t

(27)

Using this (f) and and the response function analysis was performed. From the results, just an example of coherency is shown

in Fig. 4, This can be compared with the coherency of the original rolling

to the stress, shown in Fig. 3.

Results are not necessarily good, and the improvement was not so noticeable except around zero frequency, as was expected. As the reason

of this poor results, the following possibility was considered. The new input is the squared process of rolling, accordingly as are noticed by its spectrum in Fig. 6 orSR2ß2 (eu) in Fig. B or by the shape of the process itself shown in Fig. 5, that has very low frequency component with rather high energy that comes from the difference of the component frequencies, besides the oscillating variations that oscillate with the sum of the

frequencies.

This could be anticipated easily, as the spectrum of the square of the Gaussian process which has narrow banded spectrum centered at i4 , has an

impulsive peak at zero frequency, and because of the convolution of the spectrum of the original process to itself, it has also a peak at the

frequency 2 u) _{twice of the frequency of the original peak, with the} band width of about twice of that for the original peak.

This fact again seems to give us another evidence to believe the model ii). Looking through the spectra of each process of all test runs, we find that the spectra of stress have almost always a clear peak at zero

frequen-cy, and spectra of other process do not. This peak can be assumed as the

response to the peak of the spectra of squared rolling process at zero

frequency. The band width of the averaged spectrum of the stress at 2)

twice of that frequency of the rolling frequency is about twice of that of the averaged spectrum of rolling, thus makes us form conjecture that the spectrum of stress might be the response to the spectrum of squared rolling.

However, here we have a rather poor results in coherency against our expectation. In fact, the energy of the low frequency component in squared process is quite high as are shown in its spectrum, and the effect

of high frequency component might possibly be masked away. So as the second trial, the low frequency component was cut off by a numerical filter. That was performed just by taking the difference of succeeding samples as

Xt) -

(Xir-ì)

(28)

The series (t) is the rolling squared process in this case. The shape of

this numerical filter is just a simple cosine function at frequency domain,

and cut off completely zero frequency component _{as is shown by an}

-

(

1-wt

_z

)XX()

13

-(29) The filtered process (roll squared) was used as the new input and the response analysis was performed again. The example of coherency is

shown in Fig. 4. The coherency shows almost the same value and

improve-ment was not noticed even at double frequency of the peak of the roll spectrum where the new process shows a prominent peak.

Considering these results, the necessity of multiple input analysis was fully realized. Namely in thinking of some definite component of responses,

the response characters might be very much contaminated by the effect of

other co-existing responses. This is especially the case as is treated here,

when the component concern has rather small energy compared with

co-existing other components.

Hereupon, the multiple input analysis were performed as will be report-ed in the next section.

3.3 Multiple input analysis

In these records of 13 test runs, rolling was always included, and pitching and transverse stress was included in 1 2 records, and relative

wave height was taken just in 8 test runs. Thinking of the stress as the

final output, it is clear that the only one source of input comes from the sea waves. However, to our regret, the encountering sea waves could not be

measured. Here, just the change of the relative water level was measured

at ship side, besides two modes of ship' s bodily oscillations. The bodily motions of the ship' s hull induced by the waves which are correlated each

other will also have effects on the stress secondarily. There are 6

degrees of freedom on the ships motion, and among them, rolling, pitch-ing and heavpitch-ing are the motions which were considered to be most closely

related to the transverse stress. Namely the response pattern was

assumed as is shown in Fig. 7.

As the sea waves was not measured, we will limit our discussions to the response of stress to the ship motions and relative wave high. Here, the record of heaving was not taken, however, its effect was considered to be included in relative wave height. Of course there might be other paths of inputs besides these, figured out in Fig. 7. All of these effects will be counted as the noise to the response.

The concept and the procedure of analysis of multiple input process has been made clear by Tick, 6) Goodman Akaike 8) and Enochson,

but rather few results of application have been published. Here for the convenience of discussions, the general scheme of procedure of this tech-nique will be summarised. This can be considered as the expansion of the single input single output linear case into a vector input to output relations, and all steps of analysis can be expressed by the same form as have been used in single input analysis.

The output (t) is the sum of the elemental responses (t),

k

WLt) ₌

,

f)

(30)

each elemental response (t) being assumed to be linear to each input which is also a stationary random process.

Accordingly the correlation function is =

(1) =

=

(32) (34) (40)

and the spectrum is expressed by

/S(f) - z

f-(f) H(1) Scf)

(33)

H*(f) showing the conjugate function of response character H(f).

Similarly

here RTy1), &a(1) are

= E

íx(t)'4(i]

(35)

(f)

() =

ja()e2

î

₍₃₆₎

= E Í«)

(&)J

=

o (37)

Namely thinking the input, frequency response function or the cross spectra as p-dimensional,

,(&) (&),

- J (38)

the spectrum of output and the cross-spectrum can be written as

=

H(f)S'(fY

(i)

(41)

H* _{(f) showing the complex coniugate transpose matrix, namely}

(1 = 1H(1 HL(f) H(f)JS11 (f)

()

,2()

2cÑ -SU) H)

1)

k(f) S'*k(f) and also

=

H'()

(43) namely or

-

&()

/S'/LJÇ)

(f)

_ASk2(f) HJf' (42) (44)

From eq. (43) the frequency response function of N-dimension can be

solved as

=

1) 4' (5)

(45)

H2(f) I _A3y2(f)

All these relations are analogous to simple input-output relations, as

H()(i) R1)

(47)

5i

if)

Sxz (f) (48)

=

(49)

=

cf)

5x(1)

(50)

In simple input-output relations, the coherency functions were defined as the ratio of squared frequency response functions obtained by two ways as from eq. (48) and eq. (50) namely

eyec

p2(f)

,L()

(si)

In ideal case, when the inputs are all Gaussian and the response

characters are all linear and noise is included neither in inputs nor in

output measurement, the frequency response function obtained through both way must be equal. This will results to give the value of the co-herency to be i . Actually, however none of the above conditions are ful-filled. Here, the inputs are assumed to be ideally Gaussian, and no noise is included in its measurement, thought in output, the noise n(t) is consider-ed to be inevitably includconsider-ed. Usually this noise can be assumconsider-ed as uncor-related to the input x(t), namely

=0

then

=

-t-- it&) ₍₅₂₎

&(f) &q(f) t5(f)

_H2(!(5

(53)

-17-(f) =

=

(fi.

5(f)

(54) Therefore Coherency H2()

H)5,(f)

.

S,,1c-Ç)

_-

_-1

E -

-

(f) (55)

This shows, the value of Coherency ¿1'2(f) is always less than unity

° _cf)

i

and that the extent of the contamination of output by the noise is shown in

Coherency. Heie the word "noise" should be interpreted in wide sense, and includes all other part of the output y(t) that is not explained by the linear response to the input concerned, and accordingly is called as error

Accordingly the value of coherency (f) could be used as the index

of linearity of this response to certain inputs now concerned. In rather early days of the application of the stochastic process analysis, this author wished to estimate the extent of non-linearity of responses from the value of coherency for single input-output case, and found that because of the finite length of record, or the effect of the lag window on cross spectrum computations, the cross correlation are apt to be cut down, and thus

results to show low coherency than it is. It is quite annoyable to have this effect, and one practical method to remove this difficulty was proposed.

That was to shift the time axis of cross correlation, until the origin of lag comes to the position of the most highly correlated lag point. This method

. . 3)

From eq. (51)

(H(f)JS(f)

/

Coherency

_(f)

_x(f)H(f) (56)

In multiple input analysis, the coherency can also be obtained as

2

Coherency

S'(f) Sj8(f)

Sxx(f)

SCt)

Ç f /

Sxx(f)

(f)

-Therefore writing in matrix form

Multiple Coherency

f2--- *-

'

1H,(f )-12(J - -.-_'ljk(f)) k51(f)

up

Using the concept of the conditional spectra expressions by Tick, the

partial coherency is expressed as

Partial Coherency

19

Here,

_{ì?z-Î---k(f)}

show the conditional spectra and conditional cross spectra respectively, as Tick, Akaike or Enochson defined.

Using the diffinition and the form of expression of the conditional spectra and cross-spectra, the frequency response function H(f) can

be expressed as

)-C1)

H,J..k()

Just as for the case of simple input output analysis, Akaike derived the confidence region as A (.f) - Hd (fi j

j

/ (f) j j = where D (L

i

i ( î -' N-k (60) (61)

- i.

F (2,2 (Ñ-k S) (62)

N is the intecrer nearest to frl

2

'X JQ/

M, m, and ct,being the number of sample and the maximum lag number for

one input or output, and shows the weights 1 to be used at the smoothing of spectrum. Here we used the values for window W2 as was already mentioned in 3. 1

The program of computaticn for multi-input analysis was made f ollow-ing thoroughly Akaike' s procedure 01 analysis 8), and under his instruc-tion. The main part of the computation wa carried out by CI(Itoh Chu)

Service Centre, by CDC-3600 computor, a fEw complementary computations being performed by NEAC-2206 and FACOM-230-10 computers at Ship

Research Institute. The whole procedure of computation are shown for example in Figs. 8 & 9.

Originally, half of the punched records (T. No. 214- 219) _{have 3}

channels of rolling, pitching and stress, and the rest 8 runs (T.No.

204-209, 211 and 220) have 4 channels of rolling, pitching, relative wave height and stress with few exeptions. 2 inputs (rolling & pitching) analysis,

were carried out for all of the runs to compare the runs on the same basis, and 3 inputs analisis with adding the relative _{wave height to the inputs} were perfomed for 8 runs.

Comparing these results in the examples shown in Figs. 10 and 11, we

find that the multi coherency becomes higher by taking account of the

relative wave height as an additional input. We find also that by taking account of rolling pitching and relative _{wave height as inputs, the stress is} fairly nicely explained by a linear model, especially at the frequency domain, where the energy of stress keeps considerable level, the multi

coherency keeping very high value around _{0.9. Around this region, the}

pitching and relative wave height contribute _{to the stress response more} than the rolling. However the following fact should be taken into account

in the above statement. That is, the _{contribution of inputs, reflected in}

the partial coherency is high at the frequency range where that keeps high level of energy, and the frequency range where the rolling keeps considerable energy is far low than that for

_{the stress. The rolling}

con-tributes to the stress _{response at lower frequency range where the rolling} spectrum shows a peak. At any rate, _{coherency keeps considerably high}

values through whole range of the frequency where the analysis was

-performed, that we have never experienced in single input frequency re-sponse analysis.

By the way, as an example of very high coherency, the results of Run No. 211 is shown ir. Fig. 12, although this digress a little from the subject,

as in this case, the virtical acceleration at the deck side of the ship, where

the relative wave height was measured, was taken as the output instead of

the stress, which is now our concern.

The result of Run No. 220 in Fig. 13, where the pitching was replaced by the virtical acceleration, al o shows us that the pitching contributes to

the stress more than the acceleration.

Then the same analysis were performed adding the roll squared process as the 2nd additional input. This is to find the non-linear re-sponse, namely here the contribution of the square of the rolling process,

under the condition of the existence of other inputs such as pitching and relative wave height, as well as the linear rolling. The example of the

results are shown in Figs. 10 & 11 , where the inprovement of rrulti-coherency and high value of partial rrulti-coherency for this squared process are noticed near zero frequency and also around the frequency twice of that for the peak of rolling spectrum. This shows the usefulness of multi-input analysis in the detection of non-linear response characters.

The frequency response characters of all inputs, in the form of the frequency amplitude gain arid also the phase shift were obtained for all test

runs. The results for some cases are shown in Figs. 8 & 9. Amplitude

gain is expressed in the conditional form, namely _HSRPWR2 shows

pitch-ing (p), relative wave height (w) and squared rollpitch-ing (R2) were taken into

account.

Including the single input analysis for relative wave height to rolling, the following analysis were tried, for example for Test No. 207.

The channel numbers were used for the expression of spectrum and also for coherencies to identify the input and output processes.

From these examples, the usefullness and applicability of this multi-input analysis was fully realized. These can be expressed in many ways as follows.

In order to know the feasibility of the adoption of a linear model with many inputs, as the response character of a system, the multi-coherency obtained by multi-input analysis gives us a good basis for judgernent. In other words, to find the extent of linearity of a system, well designed model taking account of all inputs that effect to the system is necessary to be chosen and examined. For this purpose, the multi input analysis _{may be an}

effective and promising approach.

By increasing the number of inputs counted in the analysis, the

multi 23 multi

-No. of

Inputes

Inputs

Channel No. _{(Channel No.)}Output Resultshown

in

1 2 3 4

1 Roll Rel.W. Height (2) Fig. 3

1 Rel.W.H.

Stress

(2) Fig. 3

1 Roll

Stress

(2) Fig. 3

(Roll)2 --filtered

Stress

(2)

Stress

(2)

Fig. 4

Fig. 4

(Roll)2

2 Roll Pitch

Stress

(3) Fig.10

3 Roll Pitch Rel.W.H.

Stress

(4) Fig.10

coherency increases, when that was really related to the

response. The

increase of coherency depends upon the relatedness of the present input to the output that is also reflected to the value of the partial coherency of that definite input to the output.

In order to find out the extent of contributions made by some definite

inpute to the output, under the existence of other inputs, the multi-input analysis is effective to eliminate its spurious response that makes the ap-parent correlations higher or sometimes lower than it is, by the contribu-tions through the other inputs correlated closely to the _{output and also to} the concerning input. _{Partial coherency, or the conditional coherency} calculated under the condition that all of the related inputs were taken into considerations in multi-inputs analysis,is necessary to be investigated for this purpose.

In order to find the non-linear response character, of _{some system}

not through bispectra the multi-inputs analysis is helpful to proceed the analysis subtracting the effect of linear response, which usually plays a considerable role even in the response of non-linear _{system. In these} analysis, the 'mixed spectrum' technique proposed by _Akaike 14) _combined

with the multiple input analysis might afford _{an appropriate way of approach.}

4. Conclusions and remarks on future work

The process of the analysis on the stress, induced on the transverse

member of the ship' s hull on the _{waves was examined. On doing so, the} results of trials, made to detect the non-linear response characters, by

The multi-input analysis technique was found to be a necessary way of access to get the actual frequency response character of some input to the response, under the usual condition where the co-existence of other inputs correlated to the response should be premised. This technique was also found to be effective to detect the non-linear response character of response to input extracting the non-linearity which is usually covered by rather large linear effects of that input and of others.

This multi-input analysis technique will afford a promising approach in the analysis of ship' s behavior on waves in future. This will give us the way to find out the leading parameter or the leading inputs from many inputs affecting to a definite response. For example this technique will give us a

clue to find out which mode of motions among rolling, pitching, yawing

surging swaying and heaving has the greatest influence on the increase of

the thrust of screw propeller in waves. This will alse help us to find the

extent of the non-linearity of this increase of thrust and torque of the pro-peller in the waves, and will help us to predict the increase of the pro-.

pulsive power in rough sea.

To promote the availability of this technique, the concept of 'mixed spectrum' proposed by Akaike seems to give us a good approach to be taken in future.

The practical application technique of bispectra, cross bispectra or

more general higher order spectra should be developed more actively and its applicability should be checked practically.

-In closing this report, the author wishes to express his deep thanks to Dr. H. Akaike who made him possible to pursue these analysis using the program of multi-input analysis he developed, and also gave him instructive suggestions through many discussions. He also thanks to Miss A. Shimura

who assisted him preparing the figures and typing the manuscript.

Reference

i) Blackman, R.B., and Tukey, J.W.; "The measurement of power

spectra from the point of view of communications engineering" Part I and II, Bell System Tech. Jour., Vol. XXXVII, No. 1, Jan. 1958

and No. 2 March, 1958

Goodman, N.R.; "On the joint estimation of the spectra, cospectrum and quadrature spectrum of a two dimensional stationary _Gaussian

processes", Scientific Paper No. 10 Engineering Statistics

Laboratory, New York Univ., March, 1 957

Akaike H. and Yarnanouchi Y.; "On the statistical estimation of

frequency response function", Annals of the Institute of _Statistical

Mathematics, Vol. XIV, No. 1 1962

Yamanouchi, Y.; "Sorne Remarks on the Statistical _{Estimation of} Response Functions of a Ship", Fifth Symposium on Naval

Hydro-dynamics, Sept., 1964, Bergen Norway, Proc. ACR 112, Office of

Naval Research, Washington, D.C., pp 97- 126

Yamanouchi, Y.; "On the analysis of the Ship _{Oscillations among}

waves _{Part il", Jour, of the Soc. of Naval Architects of Japan,}

Vol. 110, Dec. 1961 (in Japanese)

Tick, L. J.; "Conditional Spectra, Linear Systems,

and

Coherency", Proc. of the Symposium on Time Series Analysis,

John Wiley & Sons, New York, London, 1963, _{pp 197-203}

Goodman, N.R.; "Spectral Analysis of Multiple _{time Series",}

Proc. of the Symposium on Time Series _{Analysis, John Wiley & Sons,}

Akaike, H; "On the Statistical Estimation of the Frequency Response Function on a System Having Multiple Input", Annals of the

Institute of Statistical Mathematics, Vol. 17, No. 2, 1965

Enochson, L. D.; "Frequency Response Functions and Coherency Functions for Multiple Input Linear Systems", Measurement Analysis Corporation, National Aeronautics and Space Administration NASA

OR-32

io) Takaishi Y., Ando S. and Kadoi H.; "Test on Service Performance of M.S. Yamataka-Maru at North Pacific Route", Report of Ship Research Institute, Vol. 2, March 1965 (in Japanese)

ii)

Akaike, H., "Yamanouchi, Y., Kawashima, R. and others;

"Studies on the statistical estimation of frequency response function",

Annals of the Institute of Statistical Mathematics, Supplement 111,1964

12) Tick, L.J.; "The estimation of 'Transfer Function' of quadratic

systems", Technometrics Vol. 3, Nov. 1 961

1 3) Hasselmann, K., Munk, W. and Mac Donald, G.; "Bispectra of

Ocean Waves", Proc. of the Symp. on Time Series Analysis, John

Wiley Sons, New York, London, _{1963, pp 125 - 139}

14) _{Akaike, H.; "Note on Higher Order Spectra", Annals of the}

Institute of Statistical Mathematics, Vol. 18, No. 1, 1966

)

H -,

ZQA

1

/ LA-L /0, N5 I,

i-:-

(? _{6' shA}I'? &

-Table - i Environments & Particulars of Analysis for Test Runs

Run

NO

Date

Time

Wind Sea State Waves Digitized Channel Contents

Sample

Dir. (dey)

Ve1.(m/s),Swefl\

Sketch Swell/Sea Total

1'ax. Ht.Period C) "Sea -, Sec. 1 2 3 4 Lag 204 Jan.15 -70 4.8 8.5 R P S W 630 1335 1345 1.2 5.3 205 Jan.16 -127 2.2 9.0 R W 600 0910 0920 \ 6 0.7 4.0 60 206 Jan.17 -80 _3Ç 3.5 11.3 R P S W 600 0900 0910 8 0.7 60

I

Jan.18 0900 0910 -7O f'_i 2.9 9.7 R P S W 1.5 600 11.5 1.0 6.2 60 Jan.19 ₂ \ ° 3.0 10.6 R P S W 600 0900 0910 .5 0.5 3.5 1.5 60 209 Jan.20 0900 091 2 1.5

8.7RPSW

1.5 1.0 5.8 211 Jan.22 6 ! 44' 3.9

9.3RPAW

0900 0910 12.5

T_

1.5 5.6 1.5 214 Jan.23 -10

-A-_

8 7.8 13.5 R p s / 800 0805 0820 12

U-

3.2 6.0 1125 215 Jan.23 1400 1415 6 6.4 11. R P S / 1125 2.1 6.2 216 Jan.25 6.2 11.0 R P S / 1317 1330 8 1.6 55 1125 / 218 Jan.26 8.7 R P S / 800 1330 1350 - 8 20 6.8 1125 219 Jan.26 1530 1545 35 6.2 8.5 R P S / 1125 800 17 1.8 220 Jan.27 2 3 1.5

9.:RASW

800 0900 0918 1.5 8.0 1.5 60

R: Rolling S: Stress (A): Verte Acc. P: Pitching W: Rel. Wave Ht.

r

- ¡:,_2)S LO - 0.80. 214 0oo / 112 - O.,15 (k._/.2)2 -,-.-/4F0 I 0 20 40 60 80 I I I I UA00B/L o2 3.990 0.82 I I 0 20 0.- STRESS 1.0 PITCH M . 000 00 At 1.125 nO / L0 2 F 2.80? o.g2 F 0/ At 40 60 00 00 30 40 50 tO 70 80 00

A-

J0 20 7? 40 50 60 70 300 50 > 143 30

Fig. i Examples of spectra of each process

Fig. 2 Averaged correlograms and spectra

29 02000111 10.0. 0.0.1 0 0.8 1.0 14 0.0 Oq :1, ¿ o., r., oo 2.0 PF T OH TREE S - AO

CIRCULER FRESOENCY 141 SEC 0 L ' Ç° - 0.2115

1.0 0.0 000 1.0 20 oS

r

20 EO'HEREÑC/ E5 TEST NU 207 30 L i 1) OHERENCIES E5T N00207 L.L_' 30 0 50 100 15 °: RELOViOH,/ROLL

lilt

105 j L__J Y -J_ 20 DME5R 60

: ;1JÌ: \k1J.,

CDHERENIES

TES7 NO,2O7 3W STRESS/REL.W. H.

S 01 2 IO COHEHENL TEST NO1 207 i i I _\

Nil

/

¡

L' \i 31 --±0

t.

50 - 60

Fig. 4 _{Coherency of squared rolling process and filtered-squared}

rolling process to the stress

50 -J.-2O 0MG 60 r Ic' I \ C J

j

_'i

%1 l5 20 0ME64 COHEPENC I ES TEST Nl 207 3R.fj].t Stress/CR

Other

Effect

o

dog2 (I5ol1)2 _filtered

Pit ching

Rel. Wave ¿o

Fig. 5 Roll, (Roll)2,

-Other Motions Rolling

1.0 1.0

Fig. 6 Spectra of Roll, (Roll)2, (Roll)2 filtered

si0 Roll)2n(Roll)2n-i-1 } Process (t) x3' Height-- h3(t) y2 ) y3 y Stress

LOI I 05,2 so / / - L,,, .. :7 60 I 60 a * 1.5 1021 k 6.10. 207 2, (916201 Fig. 8 Example of 4

Inputs Analysis -(I)

(a)

Correlations and Spectra

0 l2-6 9 . 600 62 01.0717 6.02.. r r -.4 i r J I r 40 .-'-2.170012/2.... -652 1 /6565 2. '',.-.

'-4-r 01.1552 91_4 '6 2 22 S65 6;) S I r .L i i i_, 0.05210 91J...22.»

A-ì-''

65, J 2 ç I r .. ... I I i I i_05_ I r i I t i i r r r 0.5 i r i r i i 04022. 0.2075 12/6,2.0.62 0 I 0422/65652 - '+.. ,.t d-2.. + I /a0 i t i r -o.,-!-I I i i I I r 5 ç r i i J . i-. ----r -.1. r r r I

liLi

e.... 207 bull k

fl

4 t 4020.f L. o 00 09 O 24l ,, /2-,' II,' - 0.2915 20 092/ 02 .09__._ 40 oo 60

sa.4.i

I It/S,. 157. I 40 O, 910. 1.2.1 H 'o-'j' 2.a.f i, 1' 0.142 6l5 0,1 66/6540, -Ç,42,.2.%491.) a,,, /52 1.910 4912 . ç 2 16 /6565 -'t

j

1j

Iv

1i L 20 40 60 [ 1.40. 201 20 40 60

r

1.00. 207 Fig. 8

Example of 4 - Inputs Analysis -(i)

(b) Amplitude Gain and Phase Shift

1221.22021 2'

!î21 J''\f, /''

1.20. 201 .5 I 9202.0201 \T . a. 20

4060

--.1.2.222.22 .Ifl2.&22 (.2.2 .2 L t -5 \o V Y 20 60

1.0 LO 1.0 40 60 5.996' 0.42

IA AA

V

YVVV

8 800 I 80 At = 1.125 2 I 80 20 40 60 00 Rt3;l1I202/kR2 2 2 147281 0eg4 I I I I I..0.9_. I I I -80 -60 -40 -20 0 20 i i 0.9-4j-i p I 0.9402kg/....d, 1 800 / 884 ..p.._ \.\ -- .

.

.0.5 I I ; I I 0.5 642 3t63 4g/.0.g4 I I = 1.123 k/...8eg : 641; 108 / 6980 80' lash 40 Fig. 9 Example of 3 Inputs Analysis

(a) Correlations and Spectra

t t 0.5 13.74 0843 R; p023 / 6o I I I I I I I_45_ I I t I 60 60 .80 -60 -40 -20 0 20 40 60 00 I I I I I I L lost I I I t I I iOoZ4 fl.52 0 IL. 0,2 42p / 6026? t / 4* 8=000 = 1.125 seo. .4 / 4* 0 20 40 60 80 fo5 p501

its.

140 I ." -t 8 .8 / 8 'lji-044; 093 / 682 I 682 _tO305(60/2)2 I 0.80. 214 0.80. 204 66 3.349 04 I0.9_-t I I I t t t i 621 040 /66 -0.54 4* 1.125 seo. -0. -80 -40 -40 -20 6 /4* ₆₀ 03

Thm -V- i AF -, _L ii J\ iI i Ii It I

'i L

9I Fig. 9

Example of 3 - Inputs Analysis

(b) Amplitude Gain and Phase Shift

-sI

:.

1

f

i a 54 i o i G C C H E P E N C I ES TEST Nl 207 COHEPENC I ES TEST Nl 207 COHEPENC I ES TEST Nl 207

-

37 -CIIqNNEL I CHNNEL 2 MU..TWLE 3RPV-- '' --- r;H4NEL LH4NEL 2 CHANNEL 3 f22'RPW---SR.PWR Y --- CI-iNNEL j DIqNNEL 2 sw. im- - - X --- CHqNNEL 3 SR.RPW - --- CI4'NNEL ! t5s.jjwR ID --- MIJ.TIPLE

Fig. i O Examples of Multi and Partial Coherencies (j)

Sa - 60 215 OMEOR I

:

Y * 4

h

¿ = -; 5554-T1\ _-* : S S\/P :, * _x 1G 2G 3G 4G fl fl F iì. -, . L

/

r

1

_.4 T ? /4 /: r :/: 5

;:

'b:f

c:c::

_v:

¶Íz:.J;

;74t l.a L.5 2.0 OMEGR la 20 30 40 50 60 l O 1.5 2O GMEGR la 20 30 G a 1.5

G _L' COHEPENE ILS TEST Nl 21'-E O H 21'-E 21'-E N 21'-E I 21'-E S TEST NO 21L1 Fig. 11

Examples of Multi arid Partial Coherencies -(ii)

2O t'SR.p__. V LH-LNNE III4NN[L 2 3.PR MTLr1 25 OMEGA SR.PR---y CHNft i CHNEL 2 X CINNEL 3 ai MLtItf'Li L ,

¡

\.

; i \ F

: 'j

::* £

&

70 80 25 OMEGA 60 2 O i O I G 70 80 2 00

10 0Ü a o COHEHEN lES TEST Nl 211 CflHEPENL I ES TEST Nl 211 AR.PW CIIRNN CH HNN CHHNN

Fig. 12 Example of Multi, Partial Coherencies -(III)

Vert. Acc. - Roll, Pitch Rel. W.H.

-

39 --. Y CHqNNEL CHNNEL 2 tA.RY -- MULTIELL 40 50 OMEOR 2 ii

-i''

t ' 4

;/\

_{):[ /;'}

44

:'

:

:

10 20 30 40 50 15 20 0MEGR 20 30 1. ti

i o L OH E P E NC I ES T5T NO0 220 C O H E P E N C I E S TEST Nl 221 CH4IL. 2 MUL1PL Y CH4MEL EI+NEL 2 X CH4L i t'S.P..AW - MLj.JIFL Jo 21ì 0MU;

Fig. 1 3 Example of Multi, Partial Coherencies -(IV)

Stress - Roll, Vert. -Acc. & Rel.W.H.

E

t

lIA A p

t/

/ /í

:\jj 4

:/-4

»

/\.

-i vO'v i / ¡

ii

k

\

* L r :

'

L ,. A .0 1 1) 10 2O OMEGA 10 20 30 40 LO .4 n, u