Ship vibration evaluation through narrow band spectral analysis, Discussion of ship vibration evaluation in ISO/TC, London, 1980

TECHNISCHE HOGESCHOOL DELFT

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKWÑDE

DELFT UNIVERSITY OF TECHNOLOGY

Department of Shipbuilding and Shipping

Ship Structures Laboratory

Report No. 7L

_{SSL 231}

SHIP VIBRATION EVALUATION

THROUGH NARROW BAND SPECTRAL ANALYSIS

by

R. Wereldsma

Contribution to the discussion

of

ship vibration

evaluation in ISO/TC 208/Sc 2/wa 2 - London-September 1980.

H,

:1.

SHIP VIBRATION EVALUATION TH:ROUCF NARROW BAND SPECTRAL ANALYSIS

by R. Wereldsma

Introduction

The ship vibration phenomenon is nowadays of great concern for the shipbuilder and the shipowner. Requirements, norms and standards for acceptable vibration levels are now discussed and issued for international adaptation, in order to quantisize the habitability aboard ships.

For that purpose studies have been undertaken by a.o. the I.S.O.-organisation (TC 108 and its sUbcommittees) to design standards of acceptability.

These standards are based on the human response on a harmonic vibratory envi-ronment and it is necessary to interrelate this deterministic information with the more or less irregular, randcm information _{as observed aboard ships.}

1. Nature of ship vibrations a. Frequencies

The steady ship vibration, of importance for the habitability, _{stems from} the propeller and engine. Other vib±atory phenomena such as: whipping,

slamming and springing, wifl not be considered, because they have an

inter-ruptive nature and can be avoided by relatively simple _means.

The frequencies of structure excitation of the ship are strongly related to the propeller rpm. Engine forces will have frequencies _{equal to the rpm}

and itS lower multiples.

Propeller shaft forces will have- frequencies. equal to the blade. rate and its first three multiples.

Propeller hull forces (or pressures) will have frequencies begiiming with blade rate up to a relatively high number of multiples. The important contribution of higher harmonics is caused by the impulsive repetition of

the pressure fluctuations dUe to the growth and collapse of cavities from the propeller. Besides this deterministic portion there will be a part

which can be characterized as random, having a larger frequency range than the deterministic part. The higher frequencies of this part of the

excitation will not be considered., because they fall in the range of sòund

frequencies nt bing the subject of our investigation.

For propellers having different blade numbers. the excitation frequencies

will change accordingly. Also a frequency modulation occurs due to the rpm

fluctuations.

Generally speaking the excitation ranges f.r regular _{cases from 2 c.p.s.}

to 'appr. 100 c.:p.s..

The mentioned excitation forces are transferred through a number of systems, before they are sensed by the crew and passengers as vibratory displacements.

These dynamic systems can be characterized as:

- local 'wet details' (hull plating in the vicinity of the propeller); - global hull dynamics (main hull structure);

- local 'dry details' (mass-elastic substructures inside the. ship).

The nearby-propelier-soiid-boundary_pressure_fluctuatjons _{are sensed by}

the' local plating in the vicinity of the propeller and convert these

pressures into plate vibrations and accompanying forces on the supporting edges. This mechanism behaves in general as a multiple mass-spring system

and the force transfer will' be amplified in the range of the resonant fre quencies. Various different resonant frequencies may be distinguished.,

due to the plating and its supporting structure, which means that prefer-ence for a certain frequency can not be established.

excita-

-2-- tion. Its global characteristics are determined by the first 8-10 normal modes, natural frequencies of which may coincide with.the lower excitation frequencies, in this way giving rise to resonant amplification.

Much of the higher frequency excitation will be strongly attenuated by the overcritical excitation. In this respect it can be stated that preference

is given to the, lower frequencies.

The 'dry details', of importance for the habitability 'of the crew are base

excited by the main hull of the ship. Similarly as for the wet details no frequency preference can be determined, although for some details,. (cabin

floor, or working desk) a strong amplification may exist in the natural

frequency of the detail and the more or, less. attenuated excitation.

In overall terms no frequency preference is expected, for important

de-tails in the contrary this may be the case, i.. e. in local areas the

vi-bratory behaviour of the structure and the living environment may have a strong preference for a certain frequency that falls in the range of the

natural frequency of the detail: under consideration. Because habitability

indeed is bound to the response of details it can be stated that signals of interest for evaluation will be amplified through a resonator and will therefore have a narrow band character. From place to place the centre frequency of this narrow band may differ due to the difference in natural freuency of the detail governirg the vibratory behaviour of that place. b. Amplitude and frequency modulation

For ships in operation various causes may give rise to instationary phenomena..

Propeller, load variations for a ship in a seaway will cause rpm

fluctua-tions., i.e. frequency fluctuations.

For twin screw vessels a small difference in 'rpm of the two propellers will, cause beating phenomena. The variable added mass, its variability

caused by the shipmotions and wave pattern, will continuously change the natural frequencies of the global hull, and as a consequence the transfer factor of the global hull dynamics (amplilude modulation).

The variable stress conditions of the plating of the hull, due to bending

and, submergence will also cause variable dynamic'characteris.tics of the wet details and dry details.. Therefore a variation in the transfer factor

of these details may also be expected.

From these considerations it can be concluded that the vibration phenomena

aboard ships are a.o. characterized' by a non-constant frequency and

variable amplitude of the harmonic components of the total phenomenon. Even these harmonic components will individually have an ever changing

amplitude.

e. Conclusion

Having noticed the preference for a certain frequency range at a specific place and the non-stationary character of amplitude, phase and frequency

of the harmonic components, it can be stated that the vibration phenomenon' aboard ships can be described as a narrow band random yhenomenon to be

dealt with by statistical- and spectral-analysis.. .

The treatment of 'random signals as dealt with in this report remains

re-.stricted to one single variable. For more complex situations, where more independent or correlated signals are to be considered simultaneously,

(convolution, cross correlation, cross .spectra and transfer functions),

reference is made to the available text books and literature (see refer-ence list.).

Statistical analysis

b. Time domain

E(x2)

3

Probability and its distributions, although of importance for random signal analysis will not be considered in this repor in detail. Attention will be

focussed on the spectral analysis and its reiation with functions in the time domain. As an example in Fig. i the probability density function p(x) of two time functions is given. It can be concluded that when the random function is only represented by its probability density function the in-formation about the frequency content of the signal is lost.

In Figs. 2 and 3 respectively a sine wave and the probability density func-tion (PDF) of a sine wave is shown.. This PDF-funcfunc-tion is not dependent on the frequency of the original sine wave.

In Fig. 4 the PDF of four different types of signals are given, beginning with a harmonic single frequency signal and followed by signals of increas-ing randomness. The shape of the PDF-function characterizes _{the original} signal. The signal is said to be gaussian if its PDF has the shape of the

Gauss-curve given in Fig. 5. This shape is controllediby two _parameters, i.e. the mean value m and the value of the variance a. In Fig. 6 the change of shape of the curve as a function of a is shown. For _{a large o the s'ignal}

shows large excursions from the mean value. m.

For the statistical description of a phenomenon or signal it is nècessary to collect as much information as possible about the phenomenon (suffici-ently large population or number of samples). For continuous signals, _as in our case, this means that information needs _{to be available over a long} period. In the fundamental theory this period will _{last from - to}

+

in

time.

Another important property for our signais is assumed, _{it is ergodicity.} A signal is said to be ergodic if all the statistical parameters as

obtain-ed by integration in the time domain are equal to similar parameters ob-tamed by integration of an ensemble obtained from many reaiisations at one time instant. See Fig. 7.

The most important and simple statistical value of a signal f(t) is its mean value m:

+T/2

mlim

.

f

f(t)dt

(also called expectation E(x)).

T+ -T/2

This. value is not of interest for _{our studies (see Introduction),}

To have some information about the vibratory character _{of the signal f(t)} we consider the signal to have no mean value (i.e. _{m O) and look for the}

'power dissipated in a l resistor', as the electrical engineer says,

i.e. mean square value E(x2):

i

+T/2

um

_T

f2(t) dt.

T+ -T/2

This value tells us something about the strength of the vibratory charac-ter. (Often this value is compared with a similar value of _{a single} har-monic function. The amplitude of a harhar-monic function having the same E(x2) as the original signal f(t) is called the r.m.s.-vaiue of signal).

Fôr Gaussian ergodic signais it can be shown that E(x2) equals the square of the variance (02) of the signai.

From this figure however nothing can be said about _{the 'Time scaling',} i.e. how many events occur per time unit. _{Theréfore the mean square}

anal-ysis has been extended to the autocorrela-tion function:

f(t):x=A(sinwtp) B

Fig. 1. _{Probability density functions (PDF) of random} signal x f(t).

P(x)

Fig. 2. Marmonic wave. Fig. 3. Probability density function of harmonic

X=f(t)

harmonic

AÁAAA £AAAA

''''VV,,'''

harmonic plus noise

-5-wideband random

Fig. . Different type of signals and .their probability density functions.

rn

m-a- m+

Fig. 5. Gauss-curve of the probability

density function.

X

P(x)

Fig. 6. Different Gauss-curves with increasing o.

t2

-6

Ensembte at t1

/

EnsembLe at t

Fig. 7. Illustration of ergodicity.

-7-+T/2

R (T)

um

!

_j f(t) .

f(ti-

T) dt.

XX

TT_T/2

For T O we have the mean square value of the signal again. For T O

the

number

of events per time unit and their real time separation isre-flected in the correlation function R _{(T) on the T-scaling. The ACF is} symmetric about the T O axis. XX

In Fig. 8 a square wave function and its autocorrelation function (ACF)

is shown.

in Fig. 9 a square wave having a mean value and its ACF with its mean value is shown.

in Fig. 10 a sine wave and its ACF, being a cosine function. Figure 11 gives another example of a periodic function.

Figure 12 shows a non-periodic function and its ACF. Now the periodicity of the ACF is not present.

Figure 13 shows the ACF of a random signal with a periodic compönent. When T is large enough the random part of the signal does not correlate anymore and the periodicity of the.ACF is due to the periodic component

alone.

Figure lIt shows the character of the ACF and its modifications when the

ratio of the periodicity and randomness of the signal changes.

Remarks:

1. The mean square and the autocorrelation function as stated have a

dimen-sion being the square of the dimendimen-sion of the original signal devided by the time dimension often referred to as power.

2. For non-continuous signals or transients, sini1ar functions and quan-tities may be derived, but since the integral over the time will have

a finite value, division by the integration time is not carried out,, so

that the dimension equals the square of the dimension of the original

signal or the, energy.

c. Frequency dómain

Instead of working in the original physical time coordinate we _{may transfer} this coordinate to a frequency coordinate. In general this is done by

re-placing the time series by a composite of harmonic signals each runtüng in time from - to toe and each having a distinct amplitude. This means that

the original time coordinate 'and magnitude is eliminated and replaced by a frequency coordinate and an amplitude. Analysis in this frequency

coordin-ate is referred, to as ectrai analysis.

-The Fourier integral plays an important role in the conversion from the time domain to the frequency dömain and reads:

F(ici)

Um

$

f

f(t)

e1t

dt

T+

-T/2

where F(iw) is the amplitude spectrum.

- For the case f(t) i's a deterministic periodic function the integration

time can be limited to one period and the result is a line spectrum representing the amplitudes of the composing harmonics having discrete frequencies (amplitude line spectrum).

In Fig. 15 an arbitrary periodic function is shown and the decomposition

in its harmonic components. Sine and cosine components are to be distiri-guished.

In Fig. 16 the complex spectral presentation of a harmonic function is shown. The phasing of the signal is reflected in the real and imaginary parts of the complex line spectrum.

f(t)

III

₂₂₂

Fig. 8. Square wave time function and autocorrelation function

_R(T).

Fig. 9. _{Square wave time function and autocorrelation function}

/1/

R(t)

Fig. 10. _{Harmonic wave function and autocorrelation function.}

a

t

/

R(t) of the

harmonic signal.

-

10

-Random square wave time function

o

Autocorretatiori function Rt

Fig. 12. Random square wave and autocorrelation function.

R(t)of total signal

R(T)of the noise

A

A

V V

'vl

V V

w w

Fig. 13. Autocorrelation function of harmonic signal

11

-R(t)

o

t

Fig. ]L _{Autocorrelation function of:}

Harmonic time function.

Harmonic time fUnction with random noise. © Narrow band random noise.

® ® Random noise with increasing bandwidth.

f(t)

T'

sine components

f(t)=coswt

-t.

f (t)=cos(wt+)

AIA

,t

,

f(t)=sínwt

AIA

y''

f (t)=s, n(wt+)

AAAA.t

-

12

-time function

amplitude tine spectra

B

w(k1,2,3--)

î

T

cosine components

Fig. 15. Periodic signal f(t) decomposed in sine and cosine components..

Real (cosine)

mag i nary (sine)

e.

Im.

Re.

imaginary ampI.

tine spectrum

W.

_{(sine components)}

Fig. 16. Complex spectral presentation.

real amplitude

tine spectrum

(cosine components)

comple:x ampI.

tine spectrum

real arnpL

Line spectrum

(cosine components)

We have:

which could be referred to as the amplitude density spectrum, but equals zero, for the case we have pure noise because the mean value of this noise signal weighted with

eWt

equals O in the limit.

If we assume a harmonic component to be present in the random signal f(t), the randomness may be represented by a variable phasing of this component. If we try to establish the amplitude by a simple Fourier transform, the mean value of the amplitude will show to be _{zero because}

of selfcancella-tion, (see Fig. 16). From this fact can be concluded that

a random signal can not be decomposed in harmonic components other than by an infinitely large number of harmonic components having an infinite-ly small amp1itude.

The spectral functions up till now are amplitude functions

F(iw).

For given amplitude functions a reconstruction of thé original time function S: _{possible through the reverse transform:}

f(t)

f A(iw)

et

_dw

which in fact is a synthesis of the time function through its spectral

content.

d.. Other uantities in the frequency domain

The mean sguare value of the signal is important for probabilistic estima-tions. Having a spectral presentation of the signal, relations exist

be 13 be

-In Fig. 17 is shown how the harmonic components add _{up to the original} periodic signal.

Figure 18 shows examples of periodic functions broken down into harmonic components, represented by the modulus of the composing sine and cosine components and the phase.

Figure 19 shows how a number of cosine functions adds _{up to a time} signal. More components result in a function having a more impulsive nature. In the limit (infinité number of cosine functions) the time function equals a repeated impulse function.

- For transient signals of finite duration the signal is assumed to be periodic with infinite period and premultiplication with lIT results in

F(iw) = 0.

Therefore: F(iw)

L

f(t)

eWt

dt.

In this case no discrete frequencies canbe distinguished and the func-tion F(iw) is a continuous funcfunc-tion (amplitude density spectrum).

Figure 20 shows an aperiodic function of time with its sine and cosine

amplitude density spectra.

In Fig. 21 a symmetrical rectangular impulse and its continuous cosine

spectrum.

Figure 22 shows how the characteristics of the cosine amplitude density spectrum are modified when the time function changes from a unit impulse function through rectangular function with increasing length to a con-stant continuous value.

In Figs. 23, 24 and 25 other examples of transformation are given. - Similarly for non-periodic continuous signals we may assume a

periodic-ity with infinite period time. Again premultiplication with 1/T (T integration time) is necessary to have finite values for F(iw).

+T/2

F(iw)lim

_f

f(t)e1tdt=O

f(t)

f(t)=sinwt

14

-i harmon-ic component

2 harmoni.c components.

fit)

many harmonic components

amplitude Une

spectrum

(imaginary)

ci)

3W 5W

1W

Fig. 17. _{Periodic time functions and line spectra.}

fR)

ta

o

f(t )

-a

fR)

t

a T T

2Çt

n

II

2

2a

_ampi.

t

Line spectrum

'T

ial

2a

T

lai

a

n

-

15

-O2.32i

T T

ampLitude

IL phase Line

2

_spectrum

2n

2n

T 3r

ampt.

Line spectrum

ampt.

Une spectrum

4

w

phase Line

spectrum

234

I I

phase tine

sp e ct rum

I

i

I

phase Line

spectrum

f(t)

T

2T

impuLs train

-

16

-t..

cosIne

tine spectrum

i

harmonic

compone nt

cosine

tine spectrum

3 harmonic

components

cosine

tine spectrum

5 harmonic

compone nt s

cosine

tine spectrum

oo harm oflic

components

Fig. 19. Synthesis of tim functions from special amplitude

O=f(t)= e_0tt

t)O

time function

f (t)

-

17

-cosine ampi.

density spectrum

i

4noc

w

Fig. 20. Transient function and amplitude density spectrum.

b

sine ampi.

density spectrum

4itoc

Fig. 21. ing1e square wave function and amplitude density spectrum.

ampi.

density spectrum

w

i

2toç

Re. F(iw)

m.F(iw)

a

f(t)

: ¡F(iw)I

f

-

f(t)=short

.aj

rectangular

t

f(t)

IFtiw)I a

f (t)

a

T/2

f(t)=ö.irnpuls

t

t

fit)= medium

rectan guiar

f(t)= wide

rectangular

Lt

2T

-

18

-f(t)=constant value

t

i

w

(j)

Fig. 22. Effect of time duration o rectangular time function on the shape of the amplitude density spectrum.

time function

ampi. density spectrum

f(t)

JF(iw)

o t0

time function

Fig. 23. Effect of shifted impuls on amplitude density spectrum.

f (t)

t

i:mpuls

function ö(tt0)

t

o

I

ampi. density spectrum

1(t)

IF (

t

1g

sine

spectrum

_1

w

o

i

F(iw)

TtÖ

cosine spectrum

sine spectrum

i

w

sine spectrum

2

w

f (t)

i

Ve0(t2

IC IC

t

Gaussian

éxponential

20

-j:F (I w)I a2 a2+w2 o Gauss i an

IF( w)l

Lorentzi'an

cosine ampt.

density spectrum

e 4a

w

cosine ampt.

density spectrum

w

21

-tween the mean square value and the spectral presentation.

Since these quantities refer to the square of the signal, i.e. the energy or power, it is necessary to define energy and power spectra.

For deterministic periodic functions the total power geneted in a one-Ohm resistor equals A2 (A is the amplitude of the n harmonic

component) and for each discrete frequency this power equals A2 (simply the square of the amplitude). So the distribution of the power over the frequencies equals the square of the distribution of the amplitudes and is referred to as the power line spectrum.

- For transient signals the power generated over a long period will be zero, therefore in this case we refer to energy (it is the integration of the power, we do not refer to the mean square value but amply to the squared value of the signal integrated over the time, i.e.

_f

f(t)2dt). Having the continuous amplitude density spectrum IF(w)'I we tain the total energy by the integration of this spectrum, i.e.: 2ir

_f

_IF(w)12dw.

The energy density spectrum equals

_27r1F(w)12,

being in esseie the square

of the amplitude density spectrum., see Fig. 26, where the energy density

spectrum of a rectangular time function is shown. This spectrum is the square of the spectrum of the rectangular time function of Fig. 21. Note that the time shift does not effect the energy density spectrum.

This is well understandable,, because the total energy will not change

due to a time shift. The squaring process necessary to obtain the energy density spectrum is realised by complex conjugate multiplication, in this way eliminating the time shift sensitive phase of the components.

- For random continuous signals we have to refer to power again and the power of the signal as a function of the frequency is found in a similar

way as in the earlier cases:

hm

1

+T/2

If(t) e_tI2dt

T- -T/2

This function is referred. to as the power density spectrum. in Fig. 27 examples are given for power density spectra ('PDS).

A pure harmonic signal is represented by a s-function. The surface under this s-function equals the power of the original harmonic function.

Since we refer to power density, i.e. the pöwer per frequency rane, this power density function has an infinite value for an infinite small frequency range. When more randomness is introduced in the time

function.,

the PDS becomes smoother, (see Fig. 27).

The nature of the PDS is a refléction of the nature of the time signal,

as is shown in Fig. 28. To avoid s-functions in the PDS sometimes the integrated PDS is used, where the -functións are presented by a finite

jump, (Fig,. 29). .

The total power of the random signal equals:

-I-T/2

him

_f

f(t)2dt

_f

P(w) dw

T+°°

-T/2

-which is the integration of the power density spectrum.

For the experimental determination of PDS, the signal is fed into a filter, having a certain bandwidth. This bandwidth may have a pronounced influence on the shape of the PDS when strong harmonic components are present in the signal, see Figs. 30 and 31.

f (t)

-

22

Fig. 26. _{Energy spectrum of single square wave function.}

®

P(u)

I

P(u)

P()

A P(w)

Fig. 27. _{Time functions and correspondinc power spectra:}

Harmonic wave. Narrow band signal.

Harmonic wave with noise. Broadbanded noise.

f (t)

f (t)

f (t)

Fig. 28. _{Various time functions and corresponding power spectra.}

w

f (t)

P(w)

f (t)

- 2'4

-P(w)

-3w -2w1 -w1 O w1 2w1 3w1

power density spectrum of rándorn wave and

periodic component, having a periodicity T=

J

P(w)dw

-3w1-2w1 -Wi O Wi 2w1 3w1

integrated power density spectrum of

random wave with a periodic component

Fig. 29. Spectral densities of random wave and

P(w)

P(w)

A

25

-

27rw-

2nw'-bandwidth 50Hz

bandwidth 10 Hz

bandwidth 2 Hz

loo

100

Fig. 3Ó. _{Experimentally determined power density spectrum} showing the effect of bandwidth.

2nw

26

-smaLl

bandwidth

medium bandwidth

broad

bandwidth

27

-' e. Relations between the quantities in the time- and frequency domain

The autocorrlation function of a continuous non-periodic signal gives information about 'repeating effects' in the signai, Which _{as we may expect} is closely related to the frequency content of the signai, or the power density spectrum.

- The important relation between the autocorrelation function and the power density spectrum is known as the Wiener-Khinchine-reiation and reads:

1-T/2

P(w)

um

f R(T)

e1Wt

dt

T-,

-T/2

R

(T)

autocorrelation function.

Amutual.Fourier transformation P(w)

R(T)

is applicable.

- The mean square value of the signai is related to the power density spec-trum by simple integration:

1-T/2 -foe,

um

_f

f(t)2dt

f

_P(w.) dw

T--oe

-T/2

-and is known as. the Parseval theorem..

- Similar relations between energy and power of the signal, although of a more simple nature, may be found for deterministic and transient signals. - For the case the probability distribution of the phenomenon under cn_

sideration has a Gaussian distribution, the integrated power (either from the spectral presentation or from the time function itself) equals the

(variance)2 of the Gaussian distribution (a2).

- Having a given power density spectrum of a signal it is not possible to synthesize the original signal again, because the phase reiatioñ of the composing components is lost by the calculation of the power density

spectrum. .A new realization, of the signal might have the same power

spec-trum and total power, but will differ from the original signal. Only the statistical quantities are equal, (frequency distribution, total power and variance).

3. Digital determination of PDS

In modern instruments equipped with sampling and analog to digital conversion a number of effects and inaccuracies are introduced.

The well-known aliasing problem is illustrated in Fig. 32 and 33, where is shown how high frequency phenomena may be interpreted,as low frequency

coin-ponents (folding).

The effect on the spectrum is shown in Figs. 3L1, 35 and 36.

Avoiding of this effect is possible by introducing a low pass filter with cut-off frequency e.g. 1/3 of the sample frequency (in theory of the

sample frequency, Nyquist frequency). Another inaccuracy is shown in Fig. 37.

When the autocorrelation function and power spectrum is calculated by mul-tiplication and integration of the original signal and the time shifted

signal having a limited time duration, errors will be introduced 'at the.

beginning and end of the recording. These errors are identified as 'wrap

28

-Ncr. 32 Illustration of aliasing

due to sampling.

and _{different time-signals,} same sampled output.

f(t)

original

time function

o

sampling

impulses

1f I

4

f

-w

o

sampled

time function

P(w)

power spectrum of

original

time functio,n

w.

O

power spectrum of

sampling impulses

t r t

t L

power spectrum

of sampled

time function

Fig. 3't. Effect of sampling on harmonic content of

Fig. 33. _{Different time functions,} power spectrum.

29

-P(w)

1/true

spectrum

w

càtcutated spectrum

true spectrum

_{alias spectra}

,,

"I

\

I

_/

/

I--

--->----half the sample frequency

Fig. 35. _{Error in calculated spectrum when the signal frequency} exceeds half the sampling frequency (folding).

ali ased

spectrum

r

uua.

[errone.ous effects due to

"wrap around"

Fig. 37. Effect of "wrap around" on the calculation OF the autocorrelation function when time-limited signals

are used.

Fig. 36. Aliased power spectrum due to folding.

30

-L. Narrow Band Random Analysis

From the preceding paragraph follows that the power density spectrum and its surface integral are the main tools for a statistical evaluation of a random

signal.

Narrow band random signals are characterized by a small range of frequencies where the power of the signal is situated (Figs. 38 and 39).

These types of signals have specific properties, under the assumptions that the process is stationary, Gaussian and ergodic.

They are:

The narrow band process resembles a harmonic wave of varying amplitude

and phase.

There is only one maximum value for each zero crossing.

The probability for the value of a maximum peak between two zero crossings is according the Rayleigh distribution, having its maximum value at the standard deviation 0 of the signal (02 equals the surface integral of the power density spectrum, or the mean square value of the signal). (See

Figs. '40 and '41), /6/, /8/.

Application

For the definition of the 'maximum repetitive amplitude' of the signal we may

define that level that is only by 1% of all maximum values exceeded.

For our Rayleigh distribution this means a value of the magnitude of 3o, so

that three times the r.m.s. -value is equal to the 'maximum repetitive

am-plitude'.

This also means that a 'crest factor' of 3/2 is incorporated in this approach when we refer to the r.m.s.-value of the vibration.

Multiple frequency evaluation

-From the propeller and engine a number of harmonic components is generated in shaft force- and hull pressure signals, that may result in 2 or more narrow

band signals. Although these signals originally are strongly correlated, their correlation will be diminished when they have arrived at the point of

observa-tion, i.e. in the living quarters.

Under this condition we may assume that the two narrow band random signals do not have a finite cross correlation, so that the autocorrelation integrand can be developed as follows:

[f1t

f2(t)j[f1(t

+ T) + f2(t + T)]

f1(t) . f(t 1- T) f1(t) . f2(t + T) + f2(t) . f(t + t)

. .f(t + T)

f1(t) . f1(t 1- T) + f2(t). f2(t+ T)

and the autocorrelation function equals:

R(T)

um

_f

f1(t) . f1(t + T) dt +

T+ -T/2

+T/ 2

11m

_f

f2(t) . f 2(t + T) dt

T+ -T/2

See Figs. I2 and '43.

P(w)

W0

wo

P(xI

__1"

\density function

E(x)

Fig.. 39.. Idealized characteristics of a narrow band signal.

f(t).

da

a

-

31

-Fig. 38. _{Illustration of the generation of a narrow band process.}

power spectrum

P(w) . R (t)

xx

autocorretation function

J

_L

Fig. 40. Determination of number of peaks in the range a through a + da of a narrow band processa

broad banded

transfer-f(t)

noise

function

narrow banded

P(a)

-

32

-a2

2G2

_{ayteigh distribution)}

Low density for small

and large peaks

Fig. 41. _{Probability density function for maximum}

values

f1(t)

-

33

-FIg. 1-t2a. Narrow band random signal, centre frequency w.

/J

Fig. 42b. Narrow band random signal, centre frequency w2 2w1.

f1(t)+f2(t)

ri

34 -R

(t)of f(t)

xx

I

-2w1

-w1 O W R

(t)of f(t)

XX 2

(t) of f (t)+f (t)

Xx 1 2

Fig. 143 Autocorrelation function of two non-correlated narrow frequency band

random signais f 1(t) and f

[1

W22w1

W

Fig. 44. Power density spectrum of 2 narrow frequency

mean square value 2 of the sum of both signals is both individual mean square values and

The r.m.s.-value of the total signai

T equals distribution is accordingly modified.

For the distribution of maximum values of the combined made to /111.

Note.:

35

-equal to the sum of

and the probability

signal reference is

The names power density spectrum and energy density spectrum., although in common use., may lead to misunderstanding in connection with future

-36--List of symbols

A _{amplitude, amplitude of sine component}

a magnitude

B amplitude of cosine component

B constant value

b = time function

E(x) expectation or mean value

E(ü) energy density spectrum

F(iw) amplitude density spectrum

f(t) time function

H(iw)

transfer function

k = integer

in mean value

P(x) probability of x

P(w)

= power density spectrum

R _{autocorrelation function} xx

T time of period

t time coordinate

X _{value of time function}

dirac impuls function

variance of Gaussian time signal T = time shift -.

37

-/1/ Blackman, R.B. and Tukey., J.W.: 'The measurement of power spectra'.

Dover Publication Inc., New York, 1958. /2/ Bendat,, J.S. and Piersol, A.C.:

'Random Data: analysis and mèasurement procedures'.

Wiley-Interscience, 1971.

/3/ Lee, Y.W.:

'Statistical theory of communication'. John Wiley and Sons Inc., 1960,.

l'-i-I Lange, P.1-I.:

'Korrelat ions Elektronik'.

VEB Verlag Technik, Berlin, 1959. /5/ Wiener, N.:

'Extrapolation, interpolation and smoothing of stationary time series'.

The M.I.T.-Press, Cambridgç Mass., 19'49.

/6/ Crandall, S.H. and Mark,, W.D.:

'Random vibrations in mechanical systems'. Academic Press, New York, 1963.

/7/ Robson, J.D.:

'An introduction to random vibrations'.

Eseviers Publishing Company, 1970. ¡8/ Newland, D.E.:

'An introduction to random vibrations and spectral analysis'.

Longman Croup Ltd., 1975.

/9/ _{Crandall, S.H., Chandiramani, K.L. and Cook, R.C.:}

'Some first-passage problems in random vibration'.

Transactions ASME, Sept. 1966. /10/ Crandall, S.H.:

'First-crossing probabilities of the linear oscillator'.

J. Sound Vib. (1970), 12(3). /11/ Crandall, S.H.:

'Distribution of maxima in the response of an oscillator to random

excitation'. .