Comparison between theoretically and experimentally determined natural frequencies and modes of vibration of ships

AR

August 1954

Lab.

_{y. Scheepsbouwkunje}

Technische Hor5 Tó

Deift

by

RT. McGoJrick

NAVY

DEPARTMENT

ThE DAVID W. TAYLOR

MODEL BASIN

WASHINGTON 7.

D.0

COMPARISON

BETWEEN

dEORETLCALLY

AND EXPERt

MENTALLY

DETEM1NED

NATURAL

FREQUENCIES AND MODES

OF

VIBRATtON OF SHiPS

Report 906

ABSTRACT

rhe results of iitxution-generator tests and theoretical calculations of

natural frequencies and normal modos of ibrat-ion on eight

vessels of wiuely different types arcdiscùssed in this piogress report. The tests

ere made

with the 1MB mouiun vibration generator producing driving

forces up to 30,000 lb single amplitude. The calculations were made by means of the TMB

elecu4cal network analyzer. The erocs in the calculated

frequencies aro

tabu-lated. fly using correction factors for the arious modes based on

the

acéurnu-lated experimental data, fiore reliable estimates should be

possible in thofuture.

DaLa on thehorizontal modes are also given, butcoupling between horizontal ana

torsional motionsrequires further investigation before the reliability of

predic-tions of horizontal modes can be forecast.

INTRODUCTION

The treatment of a ship together with the water mo%ing with it as a

free-free beam

having numerous natural modes of Itexural and torsionaL

vibation covers ar extensive

litera-Lure whichit is not intended to review in this report. The

David Taylor Model Basin attempt..

ed to establish a set of equationsapplicabi to the dynamics

of this system in a report

en-titled "RecentDevelopments in the Theory of Ship

Vibration."1 This repolt is conterned chiefly with the expemental verification of the theory given thorein.

lt does not seem to have ueen sufficiently emphasized so far that the treatment of the huit und the surrounding water as a single vibrating mass is highly artificial;

There re

many limitations to thistheoretical treatment among which aro the uncertainties

in evaluating the parameters representing bending rigidity, shearing

rigidity, and effective masswhich enter into the calculations themselves. As the frequency

increases, a stage is reached at

- which alt semblance to ordinary beam vibration

disappears, and the equations can no longer be considered even approximately valid. :lthough this procedure has proved Its utility in

the calculation of natural frequencaes and normal modes

of vibration of hulls, it must be recognized that a beam vibrating on the surface of a dense fluid presents a dynamical

sytem

very different from that of a free-free beam in ompt. space.

The use of different values of virtual mass for different modes of vibration will

increase the accuracy obtainable n the frequency calculations.

't formula fur estimatIng the variation

in virtual mass for vertical modes has been proposed

b EJI. Kennard,2 but it is clear that

the general dynamical problem cannot bu treated

by the equations given n Ueferene i if the

mass must be considered tßvary with the fruc-ncy.

AS horizontal and torional ntions of the hull may be coupled, the measuring tech-nique must permit une to distinguish between these two motions. Such a techtech-nique requires measurement of phase as well as amplitude.

The TMB medium vibration genetor can be adjusted to produce a pure couple by re-oìng two eccentrics on the same side of the tnathine and rcplacing them afic-r rotating them ISO deg. However, the coupe thus podi.red is much lesq than the moment about i.be center of mass that is developed when the machine is adjusted for an athwartship sinusoidal (orce

and installed on the main deck, since the Lever attn is so large in the latter case. Up to the time of riting this report, experience ¡ri investigating the torsional modes of hulls by using the TMB medium vibration generator when set for a pure couple is lacking. However, in the experiments scheduled to be conducted on a dry cargo essel of the !larinor class under the auspice. of Panel S-6 of the Society of Xaal Architects and Marine Engineers, it is planeed

to run suc) a test.

-Since the vibration g,neracor perriiits the determination of the amplitud.e prodiced by a known exciting force, it may also be used for the detori.ination of the effective exciting forces due to propeller action. However, details of this phase of the ship vibration rese.rch program will not be discussed in this report.

Since at resonance the amplitude of the hull is linited only by damping, it is theoreti-cally possible to establish damping salues frrri the resonance curves obtained daring

vibra-tion generator tests. As shoi.n in this report th eamping "constants" thus determined are

not. actually constant but va-: with the frequency.

The anchor-dropping tochniqur .as poved extremely ueful in finding the frequency of the fundamental vertical mode of hulls, although on occasion it may excite a higher mode. lt consists simply in letting die anchor fall a 1e feet and suddenly erresting it. The result-ing impulse usually sots the hull into %ertical vibration. A continuous recoraresult-ing of this vibra-tian will give not only the natural frequency but the logarithmic decrement (rom which a value of the damping constant may also be dori'.ed.

CALCULATED AND EXPERIENTAL NATURAL FREQUENCIES

AND DAMP$4G FACTORS

:though a nuibor of other %ibration-generator testS have been run on naval and

mor-cht vessols, the test data assembled Fee are limited to those obtaino1 on vessels tested

in a sufficient depth of ater to eliminate s!ialtow-ater and side effects. Test results on

eight vessels of widely arying types are available for study at this time.

AU those vessels were tesia&d wji the T3 ,nedium vibation generator cescribed in Reference 3; the tests were conducted b c-ther of the methûds previously described.

A'h

p.incip. data on 'he:' esseis are gken in Table I.

Profiles and midship

TnLr: i

Principal tJata on \essols Discussed in This leport u Tp* tsn 0rt ßt.Iit liii auo liti t' -Waiø r,, r 4? $lf* V41I. Ss41

f

64tfli Fii Lt Fui ¡444 Dil.ce.eu! uii I') S2c?Ion. j 544V lieu. 4I (Dj b-i' L?i 44f u.tisl P4rt#d-4i's. L

r-_f--4 -Piupeau: i It ft-rn fi-ii Itni i ft it fl-se Yiii1i 4441114u1$ NOiiiOfiiII

i---

1 -'

r----tUWIAGMA T,,.ie,t 4GO 504 -15-O $740 5500 j»?l1 310 I I70 Sfl;-4.70 -10.4 lS? 0* ¡R 4 '52

4- 4.--+ --+ ---i--i ---e - - - --- -. - -IllSii*7.R.SAR6st4?,f 313 4G-44 14.0 3400 iio ¡3O zv 507 liPS 1.4$ 257 ijÇ) ifl St 4 14GO - .4---- .4- e

--

' -SSE.J.3Lt OriCn*r SR -O $-Ilt '3OO IIS ¡*0* 32-0 24*0 -4?) Iii lii I.I1561 flO 4 -III

t - -iS0Ç.*.PAIh. Oi.aCvritt 520 544 21.41* I6 14200 '34 310 101 jOn 4.37 46.3 ¡64 374 -102 113-4 n; 340 5$-Q 5*00 S3 ¡47 nt * 21H ORO -23G * (3 ¡U 2.4 .21 ¡10 1 IÖO A0IJTTE 21 t .

-1

t -

1

--. --t 51O.00flLO*Y 1OtrCuip 523 1$-0 ¡I-4 -23000 11400 74-0 44-4 18?I ¡tC 443 ¡413 11.* Lu 1.55 ¡05 4 420

I.

---- o

-t - 4-4 -. U5 O*T4IWT-)iiCi,usuc 044 70-3 24-0 11100 I 3$ I 24G 52-4 30 20 0 *03 ¿T.I 17.511 1.33 340 4 3310

I--.---: t

f

-

--t I t6.555T:T01 jicesciiotr 0 43-4 1_ 450*

!i*p j

e* 4' tic j__$*_j $.rSIU 313} 110-3 1 320 Short tons. (-n

r

ri.

ITHT--

:

iT 4*_=4

2' 200 90 'ß'.

. r

Froma 200

,Qt.ot Gairolof Loc?ej

at Fro 200

USS NLAGARA (APA 87)

JJSS CL-AR LES R. WARE (DO 85) 6 SS E.). KULAS As 0 .20 nO O 90 AO 7C. 60 8e0 ---flD,,u P'oV"a

1'igure - Profiles tid cetùns of

Vcs'eb Tested

!'ro1e .d SS C.\. PML. was flot avuttable.

----I---.--

-Y 0 .-2?.,.. 0? u*1 01 LA.I Uo OscA F,)t PtO Or?' S.cor4 Ptøtfon,i

!.&a6sd 8as LAI

From. '72

-. O.c

-j, e..

20

-ao 30 20 0 Ö'c.'.

AP 200 190 tSO .10 '60 50 40 *0 .20 i0A &0F .00 90 80

70 60 0 40 .Q .20 *0 tP

A,,

; .i;-;r 'G 5 3 Fjo ' fo

l., r....

-

o as Sg.r...

.4 J j t. I I. 'r SS PLRE tPIRQUT1E 21

r:

SS tiLO COL'INY W¼RUER

tSS NORTHP'0 (CLCD

SS STAT SLMW

T.U3LK Cornpaison of Experirnt?ntid a.-i Utdculated Frequencies of Vert.caI Modos Vessel NlAGA CHARLES R. VARE E. J. 'UJLAS C. A. PAUL PERE MARÇ'UTTf 21 (OLD CCLOIIY I NORTHAMPTCN STATP 1SLAID Fiequenc:e incpn ist e*p 110 79 45 !12 82 68 1280 28?iSlO 1592 720936 _i

ot

tuiv

er.tftcd *' ihi mod.'.

T.1LE :1 -

('opaon

of Fxperureneal ;Lnu ('aluuhtteii Frer1uercíes of Hotzor*al Modes

ssel Fie erp:s n cpm Ist 2nd orie !ode 3r P'rde 4th node 5th ¡.Ode exp ci ca

t'

.ah exp cal exp cal

NIAtAR .190 1J7 40? 365 5.5 53 CHARLES . AR( 132 101 746 207 E. J. $UL.S . si :;51s9 3C' 306 375 444 C. A. PAUL 183 148 3CC 243 PERE t.AQUETTE 21 220 201 393 423 CL COLOY MARINER 118 107 ?0 234 35 369 435 495 tORTHATON 103 83 j 13 166 26 277 327t 405 392 530 STA1TN LAND 40 375

Lxperia1dr'ier"in..tion f nu'ter of no.ies no: -4e, totu1ation mt to

yield bcs xjeenei* with cetluIto1 '.aiueS.

t3 wh.uwr this is a fl.1ural or ¿ì -aor.,1 mode.

TABI.E 4 - One-Nued Tor'ionaL

Frequency in C11T

kidt'-. Found h Vibration Ui.nerator Tests

Vesse! Vessel Fceency n cpíi

:IAI'(LtS :. i.

lLA

A. f'AU. wME 322 262' 1(FERE ARCETT[ 21

'Li CLV

AER

THA'í TO' IATEN !YD 346'

tte nti oie !.'uce 4th Mode 5th !,odc 6th io 7th L'ode SLhLote

r

cil expkl

4Il exp

c.'

x

r.jI exp c.! exp cal ez cal

92CU 190 297 26 355 3 448 462 771 ff I19 _{2Cl ?/C 3C0 38E} 31 7 74 15C 126 200 18' 246 237 285 Z4 34 348 360 396 3S li6 % 16. 367 210 24i 312 315 354 33 432 459 113 224 346 335 512 524g 73 155 14 227 233 270 31S 64 13 130 207 2& 2S3 359 357 431 434 500 500

The vertical r.'odes aro the most important as far as verification of present theory is concerned, and the results obtained for those modes willtherefore be presented (1mL Table 2 gives acomparison boteen experimentaland calculated frequencies of vertical modes.

As has been pointed out elsewhere, pure horizontal and pure torsional modes may not be found on certain vessels, and modes aro possible in which both horizontal flexure and torsion occur simultaneously in various proportions. Those are the so-called torsion-bending

modes. The present stage of accumulation ofexperimental data is not sul(1ciently advanced

to permit tabulation of definite tctsion- bendingmodes in this report. Therefore only modes that have been identified either as horizontal floxural or totsional are tabulated. This does

not imply, however, that the measuring tethniçue employed was alvays adequate to establish

that the mode was, purely of one or the other typo. Table 3 gives a comparison between

cal-culated and expeimental

v1s of horizontal

modes.

At this ritiag no reliablo calculations are available relative to pute torsional modes

foc the vessels under c-onsidoration. Thec are given therefore in Table 4 only the 8cant

experimental data on suhmodes.

lthough tleexpacimontaldeterminatice of propellerexciting rorces is not discussed

here, some coErelations between driving forces and amplitudes wilL be atLepted and it

should be noted that at resonance the azpLitudes are determined by the damping as well as the driving force..

The theory of forced vibration given in Reference i makes use of a distributed viscous

damping constant c which is the damping force per unit vek .y per unit. leagth and is also

assumed to be proporUoial to the mass per unit length z. Also defined in Reference i are

the "effective mass" 'f, and the "effective damping constant" c, for eachnormal mode. It

follows from these definitions that C'/4f cija where the large leaers represent the effective

values fr the ith

normal mode and the small letters represent the values per unitlength.

If the damping were actually of the type discussod in Reference 1, c would be

inde-pendent of bDLh amplitude and frequency, and it might be expected that the values of c/gz

v.ojld fall' within areasonably narrow rango foc ail classes of ships and foc all modes of vibration. The existence of.a universal value of this ratio could be extremely useful in

súmaLing hull amplitudes for resonant conditions. Since in reality the damping appears

w increase with frequency, it seems more likely that the ratio c' would remain constant (« being the circular frequency of the mode). In a vibration generator test

the ener' input

per cycle at resonance W is given by the relation

li' = 17F0 y,,

where F0 is the amplitude of the driving foree and y0 is the dilacement amplited.e o! the hull at the location of the vibration onerato ("the driving point").

10

Under the assumptions stated in elerence I, the enerr absorbed por cycle is given

y the equation

Jicy2drS_I

o L

j0

ty2dz

since c"t is assumed constant.

By equating these two ener expressions thero is obtained the formula

c=

O0

LA 1L py2(Lr

0

from which c.i may be evaluated if the 1riving the amplitude at. alt points along the

hull, cnd the mass per unit length including virtual mass are given.

In Table 5 are given values of cig' and c/LA c obtained from data taken for vertical

modes during vibration-generator tests on the vessels under discussion by the use of this

equation. The average ofall aluesof c,lii given in the last ctunn of Table 5 is 0.034.

An estimate of datping crinalso be made by observing the rate of ec1y of free

vibra-tion which yields the logarithmic Qecremerit. 1f the damping wore viscous and proportional

to mass, the logarithmic decrem.nc would be related to the ratio c/; by the equation

s-f i

cúgA

where is the natural circular frequency of the rnod in question. It would also follow that the Logarithmic decrement would vary inversely with the frequency ofthe mode.

To date, the few decrementobservations reported were made durirg anchor-drop tests,

a method that usually excites appreciably only the fundamental mode. Almost identical values

of S were found for the 1unciaen tal vertical mode on two destroyers of considerably different

displacement. 01 those the value for the ChARLES l. ARE was 0.022. As the frequency

as 79 epm, thi corresponds to a c/ of 0U5S and to a value of c/uu of 0.007.

In deriving lampiag values for the hriontal moues it is to be notd that the value of

will be di1fezeuL since the virtual mass values are differentfor the two cases. In Tab 6

are given the damping factors derived for those modos. The averageof all values of cIicu

11

TABLE 5

Damping Derived fr3m Vbration Generator Tests under Resonance Conditions

(Vertical Mode9 Only)

Vessel Mode ,

rad. sec 1. sec

-C/poi oriving Fosce tons iv..g Poiot Single Amplitude ist 11.5 0.49 0.043 0.51 0.0011 2nd 20.9 0.41 0.019 1.68 0.0021 3rd 30.5 0.83 0.027 3.57 0.0014 4th 31.1 2.6 0.067 5.27 . 0.005$ 5th 46.8 2.20 0.047 8.44 0.0049 CHARLES R. WARE Ist B.? 0.17 0.021W 0.30 0.0050

2nd 12.2 0.17 0.010 1.32 0.0070 3rd 27.3 0.31 0.014 3.32 0.0031 4th 31.6 1.3 0.035 6.29 0.0092 I _{E. J. KOLAS} 5th 29.8 0.80 0.021 2.77 0.0006 C. A. PAUL ist 4.71 0.029 0.006 0.16 0.0079 md 11.1 0.144 0.010 0.76 0.0064

PERE MARQUETTE 21 -15t

ILl

0.168 0.014 0.89 0.0052 NORTHAMPTON 2nd 13.9 0.298 0.021 1.21 0.0008 3rd 21.4 0.512 0.024 2.86 0.0001 4th 30.2 0.722 0.024 5.11 0.0004 5th 37.6 1.33 0.035 8.84 0.0004 6th 45.8 2.55 0.056 12.95 0.0001 7th 52.4 7.80 0.149 17.19 0.0302

STATEN ISLAND- Ist 29.3 0.976 0.033 2.81 0.0038_4

12

TABLE 6

DampIng Factors Dervod irom a oration Generator Test.s.undor Resonance Conditions

(Horizontal Modos Only)

FORCED RESONANT VIBRATION ESTIMATES BASED ON EXPERiMENTAL DATA

The problem of estiniating the forced vibration of a y ossei in its design stage is one

of considerable importance to the naval architect. heroas much progress is to be expected

in this field in the near future, sorne forecasting is possible even with the meager experime.. tal data available at the present time.

It is assumed in this discussion that the vertical and horizontal components of the exciting (orco have already beed predicted. It is also assumed that the natur*I frequencies and normal medos of vertical and horizontal vibratLon have been calculated by methods pre%a-ousiy discussed. Torsional or torsion-bending modes are not considered at this time.

It is attempted hero to deal only vith the case of forced vibration at a bull resonan in which case the amplitude is limited only by the darr.pirig. The values of C/j.ic.J given in

Tables 5 and 6 should be helpful in making the forecast.

In accordanco with tho equaticns given Oli page 24 of Reference I te amplitude of

resonant forced vibration will be given by the relation

yCr) -¿)2 _{¡4 X2 (z) dz} o - X1(z)

r-

Vessel Mode j rai sec c/p 1/sec C/pL.J dimensionless Otiving Force -Ions Driving Point Single Aphtude ft -NIAGARA ist 19. 0.293 0.015 1.76 0.0030 2nd 42.1 0.943 0.022 6.79 0.0010 3rd 61.3 2.49 0.041 593 00003 CHARLES R. WARE ist 13.6 0.t6S 0.006 0.25 0.0)44

NORTHAMPTON ist 10.3 0.735 0.668 0.71 0.0008

2nd 19.2 0.975 0.051 2.32

3rd 26.9 i.&f 0.055 5.36 0.0004

4th 34.2 2.15 0.063 1.14

0O3

5th 41.1 1.82 0.044 10.71 0.0002 STATEN ISLAND is' 44.0 3.62 0.082 2.81 0.007

I

16

can be corrected by applying factors based on average errors as sh&wn in this report. This

pro-oss should become more reliable as further experimental data aro accumulated.

On the basis of experimentally dei.ormined damping values, forced vibration estimaLes

are osb!e Io;

oant condiuon providedthe exciting forces have previously been

predicted.

Much less accuracy appets attainable lathe caso of horizontal modos than in the

case of ertical modes and the effect. of coupling boteenhorizontal and torsional motions

requires further investigation.

AC K H OW LEDGME NT S

The information given in this report as deried from experimental and theoretical

vork involving most o the porsor;el of the Vibrations Division. In particular the author is

indebted to Mr. N.H. J9sper, Mr. E. Kaploff, Mr. .J.T. Birmingham, Mr. R. Milam, and

Mrs. :.Yt. Matheson. The comments of Dr. E.U. Kecinard in reviewing this report have also been helpful.

REFERENCES

McGoldrick, R.T.. et al, "Recent t velopmc;ts in the Theory of Ship Vibration,"

Taylor Model Basin Report 730 (Feb 1051).

McGoldriek, R.T., "Determination of the Hull Critical Frequencies on the Or Carrier

SS F.J. NULAS by Means of a Vibration Generator," Taylor Model Basin Report 762 (Jun 1951).

;L Robinson, Q.R., "Vibration Machines at the David W. Taylor Model Basin," Taylor Model Basin Report 821 (Jul 1952).

Hardy, V.5., "Vibration Studies of Ship Hulls by Means of Vibration Generators,"

Taylor Model Basin Report C-SO (Nov 1049).

Bordahl, E.O., "Construction nd Operation of the Taylor Model Basin 5000-Pound 'ibration Generator," Taylor Model Basin Report 524 (Apr 1044).