A proposal for a criterion for multiply frequency vibration acceptability aboard ships, Paper for ISO/TC108/SC2/WG2

TECHNISCHE HOGESCHOOL DELFT

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE

DELFT UNIVERSITY OF TECHNOLOGY

Department of Shipbuilding and Shipping

Ship. Structures Laboratory

Report No.

_{SSL 231a}

Paper prepared J'or the ISO/TC1O8/SC2/WG2

A PROPOSAL FOR A CRITERION FOR MULTIPLE FREQUENCY

VIBRATION ACCEPTABILITY ABOARD SHIPS

by

R. Wereldsrna

A PROPOSAL FOR A CRITERION FOR MULTIPLE FRESUENCY VIBRATIONACCEPTAB1ILITY ABOARD SHIPS

by R. Wereldsma

Introduction

-1-It is very difficult in nature to find a pure harmonic time dependent phenomenon. Most of the vibrations encountered are of random nature.

As a matter of fact the pure harmonic function is man-made and is a good

help in understanding the physical problems encountered. The randomness of ship vibrations asks for a spectral analysis, where less approximations

are made with .respect to the time function.

Local resonance phenomena may influence the randomness in such a manner

that a harmonic function is closely approximated. This condition is.

ex-ploited when considerations are made concerning the 'maximum repetitive amplitude', a definition that generates problems when interpretations are

to be made. For that reason a more sophisticated approach is necessary to

avoid problems of interpretation.. This approach is found iñ the spectral

analysis of the signais encountered /1/.

Further interpretation of the power density spectrum of narrow band

rdom signals

In the appendix of this paper an outline. isgiven of _{the interpretation} of power density spectra and the approximations by d:iscrete frequency signals, to which is referred.

For a narrow band random signai having one dominant f_{req.uency (to which} the ISO/TC1O8/SC2/WG2 has restricted itself for the time being) the fol-lowing interpretation can be made (see Fig. I).

Since the original signal has one dominant frequency the total power

distribution over the frequency range is not interesting and a simple

'root mean square' instrument can read the amplitude of the harmonic function of the single frequency approximation and the frequency itself

is determined by the character of the signal (half the number of zero

crossings pe.r time unit). This 'root mean square' value has to be

con-verted to the maximum repetitive amplitude by means of a 'cres.t fac.tor'

and an evaluation of the amplitude against the developed single frequency criterion can take place. For pure sinusoidal signals the crest factor

amounts .to /2.

Since we refer to the maximum repetitive amplitude a .higher crest factor (e.. g.. 2 or 3 times /2) may be applied (a value which we have to agree. upon).

Multiple f requency criterion

For signals having more dominant frequencies a more complex analysis is necessary, see Fig. 2.

In the first place it is necessary to determine _{the shape of the power}

density spectrum.

This can be done by a .tracking filter having a sufficient small bandwidth

(see appendix). In the case of Fig. 2 this bandwidth eq:uals half the

dis-tance between. two adjacent dominant frequencies. If w2 and w3 happen

to be blade freq.uencies and their multiples (e.g 10, 20 and 30 cps) the

required filter characteristic shows a bandwidth of say 5-8 cps, which is a strong requirement, not so easy to be. obtained with passive filtering.

Having this filter, the hatched area in Fig. 2 can be sensed by, a real

'mean square' instrument (Watt-meter), giving the square of the amplitude A

of signal S2 in Fig. 2.

r( \2 (s2\

ç)

tc)

< I

M=

overn

n

If there should be no frequency dependency in the single frequency criterion

a more simple rule could be developed,. because M1, M2 and M3 should be equal.

in that case the signais S2 and 53 can first be summed before evaluated, which means that also the original signals can be added together without

filtering.

Our criterion can then be replaced by a somewhat stronger criterion but more simple to handle. If we assume the single frequency criterion to be constant valued, with maximum allowable single fréquency amplitude M, the 3-component signai, measured with a broad band 'root mean square' meter and factorized with the crest factor can simply be evaluated against M. This procedure is

proposed to do in the first place. When the acceptability is discussable a more advanced method needs to be applied for a precize analysis.

Finally, it is suggested, when the multiple frequency criterion developed on the basis of spectral analysis becomes feasible, to transform the single. frequency criterion based on 'maximum repetitive amplitudes' to a 'mean

square' criteriOn, which is more adap.ted for spectral evaluation, and avoids the problematic 'crest factor'.

References

Similarly the strength of S3 and S3 can be obtained, so that the discrete frequency approximation can be made and an evaluation against the developed criterion is possible.

The strength of the three signals S1, S2 and S3 has to be converted to.

'maximum repetitive amlitudes' by means of a crest factor (see page 1). in this way we obtain SI, S2 and S3.

Now we have to design a method to evaluate the 3 simultaneously occurring signals against the single frequency criterion as developed.

Under the assumption that squared linear superposition is applicable (which has to be verified; in the first place it is assumed that no cross correla-.

tion exists between the various frequencies of the signal, in the second place the human sensitivity needs to allow this superposition /2/), the

followingrule can be developed (see Fig. 3).

For each of the components a single frequency evaluation can be made.

If the strength of one of the components happens to be the maximum allowable value, the other components are not acceptable (the total 'volume of

accept-ance' is consumed)

The 'volume of acceptance' determines the maximum contribution of each of the components relative to each other, which results in the following rule:

or in generai:

/1/ Wereldsmna, R. 'Ship vibration evaluation through narrow band spectral

analysis'. SSL-report No. 231, Sept. 1980.

/2/ Fothergill, L.C. and Griffin, M.J. 'The evaluation of discomfort pro-duced by multiple frequency whole-body vibration',

PO.S. of narrow

band random

signal with one

dominant

frequency

_Wd

idealized I.PO.S.

A

single frequency

representation

having the same

"mean square value"

as the original

narrow band

random signal,

w

Fi:g..1 Narrow band random signal with

one dominant frequency

apprdximated by a singLe

s

s

Li w V)

PD.S. of multiple

narrow band random

signal having dominant

frequencies w

_I

- w- w

_{2 5}

idealized I.PO.S.

w.

discrete frequency

approximation

of the

PD.S.

Fig.2 Multiple narrow band random

signal approximation

SI

single frequency criterion for

maximum repetitive amplitudes.

maximum acceptabLe repetitive

amplitudes for single frequency

signaLs

criterion for acceptance

w3

w-maximum repetitive amplitudes

of the three composing discrete

sinusoidal signal components.

2 2

M1

M3

=

al

-Appendix

THE POWER DENSITY SPECTRUM AND THE BEAM LOAD ANALOGY

The power density spectrum reflects the distribution of the 'power' located in an infinitely small frequency range.

. . r (amplitude)2

The dimension of the. spectrum equals i .

. [Unit of frequency

A similar presentation can be found in the distributed _{loading of a}

canti-levered beam. .

r

-I

The load of the beam is expressed _{as I .} orce _{and can be interpreted}

.

L'"

of leng.thj

as a load density along the beam length (normally. named 'distributed loadt).

In Fig. AI.a a powe.r density spectrum _{(P.D.S.) and beam load distribution}

q(x) are shown. To obtain the shearing force _{in the beam the distributed} load has to be integrated along the beam axis.

The shearforce in the beam equals the integrated _{distributed load from the} free end to the point of observation.

Similarly the power density spectrum can be integrated along the frequency axis and results in the 'power spectrum'. (This word may lead to

misunder-standing becaus,e in the professional language 'power spectrum' often means

'power density spectrum'. Therefore the wording 'integrated power density spectrum' is used in order to avoid misunderstanding (I.P.D.S.)).

This function represents the total 'power' A2 _{which is located in the} fre-quency range from O Hz through the frefre-quency of observation. In Fig.

_ALb

both integrated functions are presented. _{The shearing force is the analogy}

for the power.

The integration over the full beam length gives _{us the total shearing force} felt by the support of the beam. The integration of the total frequency range gives us thetotal 'power' of the signal as dissipated in a 1Q

re-sistor.

It is this integrated power density spectrum that gives us the possibility to make interpretations with deterministic _{pure harmonic signals, similarly} as the integrated load distribution of the beam gives _{us the possibility to} make interpretations with deterministic forces.

The best single force approximation for the beam loading has a magnitude

that generates the same shea-ring force _{at the beam support, and is located}

in the centre of gravity of the,distributed beam load.

The best single frequency approximation for the signal under consideration

has an amplitude gene-rating the _{same power in a IQ resistor and has a f}

re-quency that is located in the 'centre of- gravity' of the power density

spec-trum, see Fig. Al.c. The amplitude of the single frequency _{approximation is} known as the 'root-mean-square-value' of the _{total signal.. Going back to the} presentation in densities we obtain the _{patterns as shown in Fig. AI.d.}

In the beam load analogy the load distribution of the single force approxima-tion results in an impulse of infinite height _{having an area equal to the} force F, being the integration of the original _{distributed load q(x).} In the spectral presentation the single frequency approximation results in an impulsive spectrum, having infinite height and infinitely _{small bandwidth} having an area equal to the total _{power of the.harmonic signal.}

It can be concluded that not very much resemblance exists between the original power density spectrum and the approximated single frequency _{power density}

spectrum. A better approximation _{an be obtained by 3 harmonic deterministic}

signals.

I

In Fig. A2.a the resulting integrated power densit spectrum and the power density spectrum are shown as well _{as for the vibration signal as well as fo-r}

the beam analogy. Comparison with the original _{power density spectrum and} load distribution respectively is not very satisfactory either.

a2

-Amuitiple harmonic approximation is shown in Fig. A2.b. It can be concluded that the approximation is improved. For an infinite number of harmonic sig-nais having an infinitely small amplitude the apprOximation of the integrated power density spectrum and the power density spectrum itself is accurate and

in the limit exact.

An alternative for the approximation of a broad banded random signal is the

breakdown into a number of narrow band random signals (N..B.R.S.).

In Fig. A3.a an illustration is given of this approach and it may be concluded that a closer approximation is possible than for a multiple pure harmonic

ap-proximation. S

In the limit again an infinite large number of infinite small banded random signals having an infinite small power gives an exact representation of the original broad banded random signal, (see Fig. A3.b). .

Another element in the analogy is found in the determination of the power density spectrum and its sensitivity for the bandwidth of the filters applied. Accoring Fig. 30 and Fig. 31 of report No.. 231 of the Ship Structures

Labor-atory ) there exists a strong sensitivity for the applied bandwidth of the

equipint that measures the power density spectrum.

In the beam analogy this effect can be illustrated easily and is shown in Fig. A4 where for the original beam load distribution 3 types of sensors are used to measure this distribution. It can be noticed that a smoothing effect occurs when the length of the sensor is increased.

The 'characteristic length' of the beam loading and the length of the sensor are related to each other for an accurate measurement. The length of the sensor determines the solvability of the instrument. Strong gradients in the

loading are only properly sensed by 'short' sensors, Similar effects occur when the P.D.S. of a signal is measured with a filter having a variable

band-width.

Therefore, before measuring a power density spectrum it is necessary from a point of view of accuracy to know the character of the spectrum (which is unknown!) and efforts should be paid, in a specific field of applicàtion as we are concerned with, to learn by advanced instrumentation the character of the spectra, generally encountered, so that the required bandwidth for more

.simple equipment can be determined in connection with the requ.ired

solvabil-ity. The bandwidth of the filters applied should be at maximum one half of the

bandwidth of the narrow banded signals,, a subject to be standardized. Finally, the beam analogy will be applied for a 'multiple frequency' signal

composed of e.g. three narrow band random signals. In Fig. A5 the discrete frequency approximation of the original signal is shown together with the beam load analogy.

For these narrow band random signals the approximation by the three discrete frequencies results in an acceptable reproduction of the power density

spec-trum and can be handled simultaneously in a similar way as the single fre-quency approach by the 'maximum repetitive amplitude' method.

R. Wereldsma: 'Ship vibration evaluation through narrow band spectral

fi.

LI w U)

fig. Ala

o

"centre of frequency"

Po.s.

power density

spectrum

I.PD.s..

integrated power

density spectrum

best single harmonic

approximation

for equal total

power

w

power density

spectrum of

best single

harmonic

approximation

fig.Alc

fig. Aid

o

of Length q(x)

load

per unit

411111

X

fqdx

fig. Alb

F

best single force

approximation

FJq Cx) dx

over beam length

Fig. Al Simple single frequency approximation of continuoUs spectrum

and its beam

analogy.

shearing

force

X

-best single force

approximation

for equal

shearing force

4

Original Pos;

I

_{power density}

spectrum

tIu,

I I I

i

I i i I I I I I .1 I I

i

I. I I I I I

I

I

1,

A;

original

I. ROS.

and 3 step appr.

I

i'

e

1,;/J\3I3m0

app

mulliple harmonic

appr. of

J, PO.S.

A2..a

ifli

i i i

IItI

T I I

liii

i I

multiple

:

i

harmonic

l'li

i i

1approf

I

i'RO.S.

'i

I I A2. b F

shearing force

and 3 step appc

I I I

ii.

I

JI

-

j

3 force appr.

/

i 'i

of origiruL

,

_{N_i}

'¼

toad

s. s. s.

multiple step

appr. of

shearing force

I'

I I I I i i

III

I I .1 I I I i

::

i I I i I I i i i i

niuttip(e

I i

t_i

_i i

lit

i

i force appr.

I

I'

ti l I

i of original

/

'i

i

! beam Load

Fig.A2 Improved multiple frequency approximation

_{of continuous spectrum}

and its beam analogy.

e.

--

originaL

/'

integrated

appr. beam Loading

i i i

yand

(shearing force)

its

i I _(.°

-

I I X

-

1

!n!

i

____oppr.

P.DS i

'I

___originaL

POS.

loading

L

_original

_beam

Aillili

X-

X

power

density

spectrum

of

composing

_narrow

band

random

signais.

Fig.

A3a

Power

density

spectrum

approximated

by

three

_narrow

bond

random

signals

and

its

beam

load

analogy.

i

'u

appr.

distribution

of

beam

loading

j

N.

/

'

original

beam Load _in

w

u,

power

density

spectrum

approximate

representation

.by

narrow

band

random

signaLs

:havin:g

bandwidth

and

power

equaL

to

hatched

_area

_e.g.

Fig.

A3

b MuLtipLe

_narrow

band

random

signal

approximation

of

broadband

random

signaL

beam Load

analogy

X

-f

x.

approximation

of

beam Load

by

constant

distributed

Loadings

ranging

_oVer

_x11-x1

and

having

an

integrated

weight

equal

to

originaL

beam load

distribution

X -'

1''

dimensions of 3 types

of sensors measuring

2 '

_{the beam loading}

(comparable to pressure

3 '

pick-ups in the

2-dimensional. case)

sensor moving along the beam axis

and sensing the hatched area,

devided by the sensor Length.

X

-measured beam load,

distorted by sensor length

Fig. A4 Effect of filter bandwidth

A

A2 3

"2

appr. LPD.S.

approximate P.O.S. impulsive of

nature,due to discrete frequency

approximations.

i

I appr. shear

force function

shear force

function

distributed

beam loading

X

-approximate beam loading,

impulsive of nature, due to

-

discrete force approximations.

Fig.A5 Narrow band random signai having three dominant frequencies

and its approximation.