SHIP VIBRATION

B

F. H. TODD, B.Sc., Ph.D., Associate Member,and W. J. MARWOOD

A Paper read before the North East Coast Institution ofEngineers and Shiphuilders in Newcastle upon Tyneon the 20th February, 1948, with the discussion andcorrespondence upon it, and the Authors' reply thereto.(Excerpt from the Institution Transactions, VoI. 64).

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SHIP VIBRATION

By F. H. Toon, B.Sc., Pn.D., Associate Memberand W. J. MARWOOD.

SYNOPSISThe paper gives the results of research on the hull vibration ofpassenger vessels, the contribution made by the big superstructures in such shipsbeing one of the snore doubtful items in the calculation of hull frequency.

1. Introduction

1 HIP vibration has been one of the approved subjects of researchat the Ship Division of the National Physical Laboratory since1928. The principal object of the work has been to develop

methods whereby the natural frequencies of a vessel's hull can becalculated from the drawings while she is still in the design stage. Atfirst this was done by the use of a simple formula involving only theprincipal dimensions of the ship and an empirical coefficient derivedfrom a similar shipW. This method is never very exact, because thereare always some differences between the prototype and the new vessel,and the frequency depends on the condition of loading, the shape ofthe under-water hull and many other features which cannot be takeninto account in a simple formula of this type.

A ship forms an elastic girder of varying cross-section, with a varying loadalong the length, due to the uneven distribution of weight and of the entrainedwater in which she floats. It is possible, if the distribution of all these factorsis known, to calculate the natural frequencies of such a beam from first principles,without the use of any empirical coefficient. Such a method was developed in1933, using results for the inertia of the surrounding fluid obtained mathemati-cally by Lewis in America(2). The acid test of any such calculation is, of course,to see whether or not it gives the correct answer. The method was thereforeapplied to thirteen ships for which the two-node vertical frequency had beenobserved at sea. Except for two cases, where special conditions were knownto exist, the agreement was remarkably good, the calculated and observedvalues for the other eleven ships being generally within about 3O per cent.(3)

This was a great step forward, for it was now possible to calculate the two-node vertical frequency for any ship from the drawings, without the necessityof having measured frequencies on a similar ship, and for any distribution ofcargo, shape of under-water form or distribution of strength in the cross-section.

Two things should be noted about these results: the modulus of elasticity ofthe steel was taken at its test-piece value of 30 x 106 lb/sq. inch and most ofthe ships were tankers (8 out of 13) while the cargo ships had in general onlyshort erectionsthe longest erection covered 36 per cent. of the length, andfor this ship the calculated frequency, giving the long bridge its full value inthe moment of inertia distribution, was 5 per cent. higher than the observedfigure, which suggested that some allowance should be made by tapering offthe inertia at each end of the bridge.

Y

() See Bibliography for references.

194 SHIP ViBR\TI0N

Further calculations made at the same time indicated that the distributionof inertia for the main girder towards the ends was of minor importance, thedifference between using the correct distribution and assuming it to be constantthroughout the length and equal to that at midships being in a typical case onlysome two per cent., whereas the correct distribution of weight and of entrainedwater was essential in order to obtain accurate results.

The three aspects of the problem which remained unsolved by this work werethe frequencies of the higher modes of vertical vibration, the frequencies ofhorizontal vibration, and the effect of superstructures of different lengthsupon the calculated frequencies.

The research has been pursued as and when opportunity occurred, andfurther results have been published from time to time()(). The present papergives the measured and calculated frequencies for a number of ships on whichexperiments have been carried out either just before the outbreak of war in1939, or in the last year or so, during which it has been possible to resume thisparticular branch of our work. While the research has been directed in generaltowards vessels with long superstructures, the opportunity has not been lostof doing similar work upon other ships when facilities were offered.

2. Experimental ProcedureThe vibration has in most cases been measured by means of the Cambridge

Low Period Vibrograph, described in an earlier paper(1. lt records eithervertical or horizontal vibration, the trace being obtained on celluloid, which isvery durable and weatherproof. Simultaneously, records are also marked orithe filin, of time and engine revolutions, the latter signal being obtained from acontact on the propeller shaft. The record is subsequently projected on ascreen and the amplitude, frequency and shaft revolutions measured. Theinstrument has a lowest frequency of about 28 per minute, and so can be usedto measure frequencies as low as 60 per minute without excessive dynamicmagnification. lt has been accurately calibrated, and the necessary correctiondue to frequency is applied to the records before any plotting is made. Whenthe ship is moving in a seaway the instrument fails because of the large move-ments of the hull, and under such conditions a Cambridge Accelerometer isused. This records both horizontal and vertical acceleration and time on acelluloid strip. While it overcomes the pitching and rolling interference, itis not of much use for low frequency vibration, since the accelerations involvedare so small, but it is a useful adjunct to the larger instrument. The idealvibrograph for use on board ship has yet to be designed, although experimentalones involving electronic control or recording are in use.

On any particular ship, records are first taken at a fixed positionpreferablyon the sternwhile the engine revolutions are slowly increased by small steps.If any resonance is observed, the engines are then kept at the requisite speedwhile records are taken along the deck in order to obtain the vibration profileand so to determine the mode of vibration. lt may happen, of course, that thenatural frequencies lie outside the range of engine revolutions, or that theunbalanced forces in the engines are insufficient to excite them. In such casesit is sometimes possible to measure the natural frequency of the two-nodevertical vibration during anchor trials.

3. Description of ShipsThe two-node vertical criticals have been measured on thirteen ships, the

principal particulars of which are shown in Table I. They are numbered13 to 25 in succession to those ships described in earlier papers. Nos. 13 to16 have been briefly referred to previously in paper 5. All but four arepassenger and cargo ships with substantially long erections. The mouldeddepths at side to the uppermost continuous deck are shown, and also to each

TABLE

Principal Particulars of Ships on which Vibration Tests have been carried out

MOULDED DIMENSIONS Length

of Displace- TWO-NODE VERTICAL Super- Draught ment 7 of Observed Observed Type

FREQUENCY PER MINUTE Ship Type structure mid- in full 2-node 3-node of No. Length Beans as ships tons load V.I.P. B/Cl horizontal vertical engines

B.P. mid. Depth feet 7 of L.B.P. d Calculated Observed

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (Ii) (12) (13) (14) (15) (16)

13 T.S. pass. 445' 61' 355 to Upper Deck 100V. 16 -64 8.365 60% 2396 366 ¿o Upper Dk. 79'7 102 Steam and cargo 440 to Br. Deck 43% to Br. Deck 110

525 to Boat Deck 35'5°/

14 T.S. pass. 450' 65' 445 to B Deck 100% 1767 7.776 74% 2-255 3 68 to B Deck 1105 124 Diesel and cargo 53'O to A Deck 59 % to A DeCk 126-5

61 -5 to Sports Deck 45% I

- - 15 T.S. pass. 460' 66' 35-5 to Shel. Deck 100% 19-42 10,300 7947 2235 340 - ad cargo 43-5 to Br. Deck 75% to Br. Deck 956 112 269 Diesel 51 5 to Prom. Deck 62% to Prom. DecklOO -3

605 to Boat Deck 48% to Boat Deck 118-5

12 T.S. pass. and cargo

585' 76' 455to C Deck 100% 24-46 19,918 85% 2'251 3 tI 81 108 Steam Turbine

54-5 toBDeck 81-5 or toBDeck 69-2 60%

625 to A Deck 74-1 or . to A Deck 78-5

48-7% 710 to Boat Deck 48-7%

- 17 Q.S. pass. 650' 84-5' 485 to B Deck 100% 24-37 24,867 100% 2249 3394 79 108 Diesel and cargo 56-5 to A Deck 63 '5%

64-5 to Prom. Deck 63 -5 / to Prom. Dk. 78 -3

725 to Lounge Dk. 47 2°,. to Lounge Dk. 87-8 82-5 to Games Dk. 36-3%

- IS T.S. cargo 495-5' 68-0' 41 -5 to Upper Deck 100% 22-42 14,500 80% 2-162 3-07 to Upper Dk. 80-4 89

I

122 225 Diesel 49 75 to Br.Deck 46-8% to Br. Deck 95-5

27-04 18,130 100% 2-51 114 209 16-91 10,390 57% 401 102 128 243

19 S.S. pass. 375' 54-5' 27-0 to Upper Deck 100°/ 20-0 7,683 89-5% 2-033 2-72 Diesel and cargo 34-75 to Br. Deck 84°2 to Br. Deck 100-5 1175

42-50 to Prom. Dk. 40% to Prom. Dk. 118-0 5050 to Boat Deck 40%

14-79 5,376 626% 2 305 3-69 to Br. Deck 110'6 to Prom. Dk. 129-4

133

20 S.S. pass. 425' 58-0' 34-5 to Upper Deck 100% 12-71 6,070 2-66 465 to Upper Deck 99-5 1055 Diesel and cargo 440 to Br. Deck 364% to Br. Deck 120-8

21 S.S. cargo coaster

180' 32-83' 20-5 to Upper Deck 100% 823 837 234 399 to Upper Dk. 261 I

243 Diesel

22 S.S. cargo 257' 39-33' 22-5 to R.Q. Deck 70% 16-75 3,371 95.5% 1 -91 2-35 including 147 140 196 320 Single collier 1875 to Main Deck 30% hatch

coainings screw

driven by Twin

Diesels

23 S.S. coaster 210' 36-5' 21 -66 to Upper Dk. 100% 14-375 2.178 100% 2-012 254 - 157+ Steam 1408 to 2nd Deck 100% Recip-

rocatIng

24 S.S. cargo 4695' 64-5' 42-0 to Shel. Deck 100% 24-17 14,454 2-66 90 Turbine 500 to Br. Deck 24'2%

58 -0 to Boat Deck 24-2%

25 T. Steam 382' 62-5' 1875 H br. Deck. 9-87 5,159 57-5% 633 78 Steam tanker 27'OO Trunk Deck 1637 8,985 100% 3-81 69 Recipro-

cating

SHIP VIBRATION 195

of the superstructure decks, the percentage of length covered by each of theselatter being also given. The moment of inertia of the midship section has beencalculated to each of these decks in turn. Outline profiles and sections areshown in Fig. I(a) and (b).

4. Method of Calculating the Two-node Vertical Natural FrequencyThe method of calculating the two-node vertical natural frequency has been

described in some detail in the appendix to paper 3, to which reference maybe made. It is sufficient to point out here where some departures have beenmade in the present calculations.

The added virtual mass curve is obtained in the manner described in thatpaper, and added to the weight curve to give the total load curve. The vibrationprofile is then assumed to be the same as that for a uniform, free-free beam,and on the assumption that the vibration is of the simple harmonic, isochronoustype, this profile also represents the acceleration to some scale. Thus theproduct of the ordinates of the total weight curve and the acceleration curveat any station represents the dynamic load at that point, and in this way adynamic-load curve for the whole ship is obtained. Integration of this curvegives the shear force curve. This does not, in general, close, and the base ofthe acceleration curve must be moved until it does, thus ensuring the necessarycondition that the centre of gravity of the ship remains at rest during thevibration. A second integration, of the s.f. curve, gives the bending-momentcurve, and again, in general, this will not close. To correct this means goingback to the assumed profile or acceleration curve and rotating it about thecentroid of the curve of total mass (i.e., including the virtual mass of the water).This, in turn, upsets the s.f. curve again, and in the past a number of calculationshad to be done until the residual bending moment was quite small, when thecurve of bm. was closed by drawing a new base line, and two more integrationsthen gave the derived profile. From this and the assumed acceleration curvethe frequency can be determined.

In 1932 a method of determining the vertical shift and the rotation of the baseline of the assumed profile, to ensure that both the s.f. and b.m. curves wouldclose, was described by Schladofsky, and a translation of this work has recentlybeen made available to the Authors by Captain H. E. Saunders, U.S.N., untilrecently Director of the David Taylor Model Basin, Washington, for whichcourtesy they wish to extend to him their thanks(s). This method is brieflydescribed in Appendix i of the present paper, and has been used in the calcula-tions for the more recent ships, and results in a great saving of time and labour.

In all the calculations, the moment of inertia has been assumed to be constantalong the whole length of the ship and equal to the value amidships, treatingthe deck as being complete right across the ship, and thus ignoring hatchopenings, and coamings. Thin engine-room casing and deckhouse sides havebeen omitted. The only exception to this is vessel 22, which had very widehatches, the coamings of which were continuous and formed a substantial partof the ship's structure.

In every case, the calculated frequency has been corrected to take accountof the deflection due to shear by using the approximate method describedbefore this Institution by Lockwood Taylor in l927.()

5. Comparison between Calculated and Measured FrequenciesA complete detailed calculation has been made of the two-node, vertical

natural frequencies for eleven of the ships in the condition in which the actualfrequencies were measured. in making these, the moment of inertia has ingeneral been calculated to the uppermost continuous deck, and then to eachsuperstructure deck in turn. These calculated frequencies are shown in column12 of Table 1, and may be compared with those observed on the ships, as shownin column 13.

Considering the complexity of structure in a ship and the difficulties ofallowing for all the discontinuities of decks in way of hatches, engine-roomcasings, etc., the agreement is considered to be good. The greatest differencesof 5 per cent. or so occur generally with the smaller vessels.

The vessels with superstructures fall into two classes: those with long upperworks covering 60 per cent. or more of the length, and those with short bridges.There are three of this latter class in the present selection of shipsNos. J 3,18 and 20. In all these cases the side shell is carried up to the bridge deckthroughout its length, and the latter is therefore of substantial construction.On the other hand, the bridge only covers 43, 46 and 36 per cent. of the lengthrespectively in the three cases, and therefore does not cover the nodes. Aconiparision of calculated and observed frequencies is shown in Table 3.

TABLE 3.Calculated and Observed Values of Two-node Vertical Frequencies forVessels wit/i Short Bridges

In ships 18 and 20, there is no further superstructure deck above the bridgedeck, and it is obvious that the bridge is playing an important part in thestiffness of the girder against vibration, but that this effect decreases withdecrease in length of the bridge. This is to be expected, since when the bridgeis very short, as in tankers, it ceases to act as part of the hull girder and becomespractically speaking only a concentrated load.

Ship No.Frequency per minute Percentage Difference

(+for calculatedabove observed)

Referenceletter ofship inCalculated Observed

1 74-2 785 50 B2 1075 1055 +2-o D3 1025 1045 18 H4 1210 1150 H-50 K5 9F3 905 +08 M6 958 985 28 08 8l5 800 +18 S

10 1122 1090 +29 G12 77.5 785 1221 261 2430 +72 --22 147 1400 +50 -

ShipNo.

Frequency per minutePercentageof lengthcovered by

bridge

Percentage difference(+ for calculated above

observed)Calculatedto upper-most con-

tinuousdeck

Calculatedto Bridge

deckObserved to

deckto Bridge

deck

131820

797804995

110955

1208

10289

l055

434636

208106- 57+ 8-5+ 72+145

196 SHIP VIBRATION

In two of the ships (Nos. 21 and 22) there were no appreciable superstructuresabove the uppermost continuous deck. Results for a number of other vesselsof this kind have already been published(3) and are reproduced for referencein Table 2.

TABLE 2.Calculated and Observed Values of Tii'o-node Vertical Frequencies ji'rVessels with no Substantial Superstructures

On the abo e tasis, the agreement between calculated and observed frequenciesis very good for the first four vessels. Nos. 14, 15 and 16 each had one furtherdeck above that included in the calculations, which was in general a light boator sports deck covering about 47 per cent, of the ship's length. No. 17 had alounge deck covering 47 per cent. and above this a very light games deckcovering 36 per cent., which would, it is thought, have little if any effect onthe natural frequency. No. 19 had a promenade deck and above it a boatdeck each covering 40 per cent. of the length, and these have both been ignoredin the calculation because they did not cover 60 per cent, of the ship's length.1f we assume that the presence of these two decks has virtually the effect oflengthening the promenade deck and we include it in the inertia calculation,then the calculated frequency becomes 118, in good agreement with the observedfigure.

6. Approximate FormulaeWhile it is believed that to take account of all the factors in a new ship it is

necessary to make a complete calculation such as that described above, theAuthors recognize the great convenience to the naval architect of having asimple formula which will give the natural frequency with a minimum ofcalculation from data which are available in the early design stages.

The first such formula was given by Schlick some sixty years ago

N = (1)

where N = frequency per minute of two-node vertical vibration;¡ moment of inertia of midship section in inch 1 feet 2 units;

= displacement in tons,and L = length b.p. in feet.

ç was an empirical coefficient to be derived from actual observations on ships.

ShipNo.

Frequency per minute Percentage oflength covered

highest deckused in

calculation

Percentagedifference in

frequency (+ forcalculated above

observed)

Calculated tohighest deck

covering 60% ormore of vessel's

length

Observed

14 1265 124 60 ± 20IS 1093 112 6216 785 81 74 - 3017 783 79 63 - 0819 1005 1175 84 l45

smp VIBRATION 197

No 13, on the other hand, had a boat deck, covering 35 per cent. of thelength of the ship, above the bridge deck, and this makes the comparison on abasis of the percentage covered by the bridge rather misleading. The figuresfor calculated and observed frequencies suggest that this boat deck is providingsome stiffness, and that in consequence the length of bridge should be virtuallyincreased in this ship for comparative purposes.

The remaining ships were all of the passenger or passenger and cargo typewith long superstructure decks.

An examination of the observed frequencies with those calculated to differentdecks suggests that any deck covering 60 per cent. or more of the length ofship, and therefore covering also the nodes, is fully effective as far as the stressesin vibration are concerned. The data for these ships are shown in Table 4.

TABLE 4.Calculated and observed Values of Two-node Vertical Frequencies forVessels with Long Superstructures

198 SHIP V1BRAT1O

Two major difficulties arise in using this formula. First, it ignores theeffect of the virtual mass due to the surrounding water, and secondly in vesselswith superstructures above the topmost continuous deck there is always con-siderable doubt as to what material should be included in the calculation of I.In 1935, Burril suggested a similar formula, but incorporating two factors totake account of the surrounding water and the shear correction respectively(s).

The results given in the present paper have been analysed, and plotted in

Fig. 3, on the basis of a modified parameter ', where is the dis-

placement including the added virtual mass due1 to the surrounding water.The latter can be calculated from the shape of the underwater hull, making

certain assumptions, in the manner described in paper (4). The ratio -fis called the "virtual inertia factor" and its values for the present series ofships are given in column 10 of Table I.

The moment of inertia I has been calculated to different decks in turn,according to the general arrangement of the particular ship in question, and thespots in Fig. 3 have been arranged to show the effect of including the super-structures of varying length.

1f this diagram is examined carefully, the Authors believe it will be agreed,that the line drawn there is a reasonably good mean of the spots derived fromthose ships having no appreciable superstructures, i.e., tankers and cargoships with only poop, very short bridge and forecastle. Examining the spotsfor the other vessels and their relationship to this line, it appears that in generalmost satisfactory results for ships with long superstructures will be obtainedby including in the moment of inertia all decks covering 60 per cent. or moreof the ship's length, while in certain cases, depending on the particular arrange-ment of the ship, decks covering between 40 and 60 per cent, of the lengthmust also be included in the calculation of I. In other cases, marked on Fig. 3,there were in addition light decks such as sports and boat decks which havenot been included in the calculation. In th case of vessel 20, which had onlyone superstructure deck above the top continuous deck, a bridge covering36 per cent, of the ship's length and with side shell carried up in way of thebridge, it is obviously necessary to allow for this to some extent, to obtainreasonable agreement with the suggested average line.

In all, there are results for twenty-two ships plotted on Fig. 3, and apartfrom five exceptions, the observed frequency is within 5 per cent. of that givenby the drawn line, and in most cases the difference is very much less. Forsome of the exceptions, no explanation can be given, but No. 22 was a vesselof peculiar construction, and in No. 23 the actual frequency was not quitereached because it lay just above the maximum permissible engine revolutions-the real frequency is somewhat above that plotted, as indicated by an arrow inthe figure.

The Schlick formula, as modified above, involves both the calculation ofthe moment of inertia of the cross-section amidships and also of the virtualinertia due to the surrounding fluid, which necessitates a knowledge of theactual lines of the vessel. Both these calculations take time, and it wouldbe a great convenience to naval architects and engineers ifa simpler approximateformula could be evolved which would give results of comparable accuracyand yet avoid the necessity for these calculations. Such a formula was, infact, proposed by one of the present Authors in 1931 ('), the value of I beingassumed proportional to BD3, and for tankers and cargo ships having nosubstantial erections, and in which, therefore, there was no question of theappropriate value of D to be used, it gave very promising results.

(5)

SHIP VIBRATION 199

As originally stated, it was of the form

N = fi (2)

where B -= the breadth moulded in feetand D = the depth moulded at side, in feet, to topmost continuous deck.The other symbols are as previously defined. Values of fi have been given fora number of ships in papers (I) and (4), and it was concluded that empiricalformulae of this type were only useful for comparing vessels of closely similartype in the same general condition of loading.

When we come to consider vessels with long superstructures, we meet theadded difficulty of knowing the correct depth D to use in such a formula inorder to make some allowance for their various lengths. Several methodswere tried to find an equivalent depth for the ship to allow for the4ifferentlengths and heights of superstructures. Finally, that first proposed by Ldndbergin l932() was found to be the most effective, and it has been developed to takeaccount of more than one tier of upper works.

If we have a vessel of length L with, say, two superstructure decks of lengthL1 and L. respectively, the depths to the topmost continuous deck and to thesuperstructure decks being D, D1 and D2, respectively (see Fig. 4), then theequivalent depth of ship has been expressed as

DE = \/D8(lx1) + D3 (x1x2) ± D23x2 (3)

L1 L2where x1 = and x2

This can obviously be extended to more decks as necessary.

Short forecastles and poops and bridges such as those in oil tankers havebeen neglected as being too short to influence the stiffness of the girder.

To avoid the second calculation, that of the virtual mass of the surroundingwater, all the calculated inertia factors have been plotted in Fig. 2 to a baseof beam to draught ratio B'd

The virtual inertia factor is the ratio of the total displacement, includingthe entrained water, to the actual ship displacement, and in the notation used

1.above, is equal to the ratio

It will be seen from Fig. 2 that the expression

= L G- - + 1.2) (4)

gives a very good approximation to the values calculated by the detailed method.

Replacing D and , in equation (2) by the modified values given in (3) and(4), we haveN=ß/E

The values of the observed frequency N have been plotted on this basis inFig. 5.

The available data seem to indicate that on this basis of plotting, tankerswith a longitudinal system of construction must be treated separately fromcargo and passenger ships. Two mean lines have been drawn for these two

classes of ship, and these indicate that for the same value of DE theL3

tan kers are stiffer and give frequencies about 10 per cent. higher than the cargo

For the vessels in reasonably loaded condition, the average value of theratio of the horizontal to vertical two-node frequencies is about 1 37.

In addition to the above figures, the two frequencies were also measuredin a lighter condition (57 per cent. of load displacement instead of 80 per cent.)and the ratio was then found to be l25, the frequencies being 102 and 128per minute. This appears to be the opposite of what would be expected sincedue to the virtual inertia, the horizontal frequency would be expected to increasemore rapidly than the vertical with decrease in draught.

Ship2-node vertical

frequencyper minute

2-node horizontalfrequencyper minute

Ratio ofhorizontal frequency

vertical frequency

16171822

817989

140

108108122196

134l37137l-40

2DO SHIP VIBRATION

and passenger ships. Results for some 9 tankers are shown in Fig. 5, andfor 8 of these the departure from the mean line never exceeds 3 per cent. Theexception is No. 6, which does not plot on Fig. 3 either, although the detailedcalculation gave a result within 3 per cent, of the observed figure. There is noobvious explanation of this difference.

Tankers Nos. 2, 3, 4 and 5 were small and of trunk-deck type used for feederservices in the West Indies. They all had narrow harbour decks, and the depthused in the above equation has been measured to the top of trunk. No. 25was ari ocean-going tanker and had a very wide harbour deck along each side(12 feet). For this vessel DE has been calculated to make allowance for thissection.

In drawing the mean line for the cargo and passenger ships, primary con-sideration was given to those vessels with no appreciable superstructures, viz:10, 11, 12, 18, 21 and 22. For vessels with long superstructures, the effect ofincluding or neglecting the topmost decks when these are of light construction,and therefore of using different equivalent depths DE, is clearly shown in Fig. 5.

Considering the variation in types of ships and in the extent and arrangementof their superstructures, the results have plotted extraordinarily well. TheAuthors believe that the intelligent use of this diagram in association with theprofiles shown in Figs. I(a) and (b) will enable designers to make a very closeestimate of the two-node vertical frequency for a new ship before the informationrequired for a detailed calculation is available.

7. Eflèct of Change in Displacement

For three ships results have been measured for two different draughts. Twoof them were cargo-passenger types, Nos. 18 and 19, and one a tanker, No. 25.

Calculating ¿, from the same approximate formula (4), these can be plottedand are shown in Fig. 6 together with the mean lines already drawn on Fig. 5.lt will be seen that with decreasing displacement there is a tendency for thefrequency to increase rather more rapidly than would be expected from theslope of the mean lines.

8. Horizontal VibrationThe two-node horizontal natural frequency has been observed on four ships,

and the results are listed in Table 5.

TABLE 5

For a uniform beam the ratio is 2 76, while for one of uniform depth butconsisting of two wedges in plan, it is 2'26. The above ratios therefore appearto be reasonable. For very fine ships such as destroyers there is evidencethat the ratio approaches 20.

10. Conclusions

It is believed that the results given in this paper represent a further steptowards assessing the strength value of superstructures in vibration problems.The detailed method of calculation remains the only one possible for a newdesign where measured results for similar ships are not available. In using it,discretion must be used in making allowance for the superstructure decks, butthe results given here would suggest that any deck covering more than 60 percent. of the vessel's length may be taken as fully effective. The inclusion of anyhigher decks of shorter length will depend very much on the arrangement ofthe individual ship, and some guidance on this point can be obtained fromthe profiles in Fig. 1 and the remarks in the text and tables.

The use of the method of correcting the vibration profile base line due tSchladofsky has proved very useful in saving time and reducing the labourinvolved in the detailed calculation.

A further refinement of this calculation would be to use a moment of inertiacurve showing the actual distribution along the length rather than a uniformvalue equal to that amidships. This, however, would involve a great increasein the work, since it would mean calculating I for perhaps 10 or 12 sectionsalong the length, and there has not been time to carry this out even if all therelevant information were available. In any case, calculations to find the effectof such a process have shown that it is of a secondary character (paper 3).

Two approximate formulae have also been developed. The first of theseinvolves the detail calculation both of the moment of inertia of the midshipsection and of the amount and distribution of the entrained water. In thesecond, the moment of inertia has been assumed to be proportional to BDE3,where DE S an equivalent depth designed to allow for the varying lengths ofsuperstructure, while the amount of entrained water has been assumed todepend on the ratio of beam to draught. The approximation to the inertiaof midship section will only be expected to apply to vessels built to a commonstrength standard, such as, for example, the classification societies rules, andspecial care would have to be taken where owners ask for additional scantlings,the vessel is strengthened for ice, or similar cases.

Either formula appears to form a good basis for approximate estimates orthe two-node vertical frequency. The first, which includes I, involves the useof a certain amount of judgment because it does not intrinsically take anyaccount of the lengths of the superstructure decks. In the second, this is allowed

z

Ship2-node vertical

frequencyper minute

3-node verticalfrequencyper minute

Ratio3-node frequency2-node frequency

1518

22

11289 (80% load)

102 (57% load)140

269225243320

24O253238228

SHIP VIBRATION 201

9. Three-node Vertical FrequenciesThe three-node vertical natural frequency was measured on three ships

(Table 6):

TABLE 6

I. F.H.Todd.

F. M. Lewis.

F.H.Todd.

BibliographySome Measurements of Ship Vibration." Trans. N.E.C.

Inst., 1931/2, Vol. 48.

The Inertia of the Water Surrounding a VibratingShip." Trans. Amer. Soc. N. & Mar., 1929, Vol. 37.

"Ship VibrationA Comparison of Measured withCalculated Frequencies." Trans. N.E.C. Inst. 1932/3,Vol. 49.

F. H. Todd. "Vibration in Ships." Gothenburg Society ¡f Engineers,1935.

F. H. Todd. "Vibration in Ships." British Association, Section G.,Cambridge Meeting. 1938.

Dr. Ing. E. Schadlofsky. "The Calculation and Measurement of Elastic NaturalFrequencies of Ships' Hulls." Schiffbautechnische Gesell-schaft, 1932, Vol. 33.

J. L. Taylor. "Ship Vibration." Trans. N.E.C. Inst. 1927/8, Vol. 44.

X. L. C. Burrill. "Ship Vibration: Simple Methods of Estimating Critical,Frequencies." Trans. N.E.C. inst., 1934/5, Vol. 51.

9. S. Lündberg. "Vibrationföreteelser." Gothenburg Society of Engineers.1932.

202 SHIP VIBRATION

for on an empirical basis. It would appear from the second plotting that thecross-sections of tankers are somewhat stiffer than those of cargo and passengerships for the same beam and equivalent depth, because it is necessary to dis-criminate between these two classes and draw separate lines for them. Thisdifference exists even between tankers and cargo ships when the latter alsohave no substantial superstructures and is presumably a real difference betweenlongitudinally and transversely framed ships.

The measurement of higher frequencies is much less easy. The amplitudesare very small and it is very often extremely difficult to decide from the dis-tribution along the deck just what is the proper vibration profile, i.e., whetherit has three or four or more nodes. The figures given in the paper for higherfrequencies are believed to be correct for the modes of vibration, stated, althoughit was not possible to measure the profiles. A large number of measuredfrequencies of these higher types are required in order either to compare themon an empirical basis or with detail calculations, and it is hoped that withthe return of normal trial procedures more opportunities will occur to continuethis work.

11. Acknowledgments

The work described above has been carried out as part of the researchprogramme of the National Physical Laboratory and this paper is publishedby permission of the Director of the Laboratory.

The Authors wish to express their thanks to the shipbuilders, engine buildersand shipowners concerned, not only for permission to carry out the tests ontheir ships, but also for the very willing and generous help that has alwaysbeen given during the fitting up of the apparatus and the conduct of theexperiments.

They would also like to acknowledge the assistance rendered by Mr. F.Gridley of the Ship Division Staff, who has attended most of the trials and hassuggested, and made, many improvements to the recording instruments.

SIm' VIBRATION 203

APPENDIX I

In a paper given by Todd in 1933 (3) a method was detailed for calculating thetwo-node vertical frequency when all the necessary information was available. Thismethod has been adhered to in all the calculations made for this publication. It was,however, soon discovered that in spite of the correction for closing the dynamic-shear-force curves, the integration of this s.f. curve invariably failed to close and in manycases left a residual bending moment at the fore perpendicular which was far toogreat to correct by simply joining the ends of the bending-moment curve as shownin Fig. 7.

Fig. 7

The following method was adopted by the Authors to ensure that both the s.f.diagram and bending-moment diagram closed, with a minimum of calculation. Themethod is due to E. Schadlofsky and was first published in 1932 () but the Authorsbelieve it has not yet been published in this country.

By using two simple formulae it is possible to predict from a single performanceof the standard calculation, combined with a calculation of the longitudinal momentof inertia of the total mass curve about its own axis, the vertical parallel movementof the vibration profile base in order to close the shear-force curve, and the rotationof the base required to close the bending-moment diagram.

(I) The parallel shift of the baseYs = Residual Dynamic Shear Force at Fore Perpendicular ¡Total Mass

where the residual dynamic shear force is the value of the s.f. ordinate at the f.p. andthe total mass is the sum of the ship's weight and that of the entrained water.

(2) The rotation of the base at the f.p. about the centre of gravity of the total masscurve

F. p. = J (y F.P._) [ RMF. L2RSF.P. ]where L2 is the distance in feet from the f.p. to the centre of gravity of the total

mass curve.

j is the longitudinal moment of inertia of the total mass curve about its ownaxis.

YIF.P. is the original ordinate at the f.p. of the assumed vibration profile(which is ¡ 0).

R1F.F'. is the residual bending moment at the fore perpendicular (i.e. theamount by which the b.m. diagram fails to close).

RSF.P. 'is the residual dynamic shear force at the f.p. (i.e. the amount by whichthe s.f. diagram fails to close).

After the two corrections have been applied to the original vibration profile theordinate of this curve at the fore perpendicular is equated to unity. The wholevibration calculation is repeated and both the dynamic s.f. and the dynamic bendingmoment curves should close. If there are any discrepancies they are usually so smallas to be negligible.

204 SHIP VIBRATION

Several points must be borne in mind regarding signs, and the following rules mustbe observed:

The area below the base is negative.

The shifting of the base y is given the negative sign when it is downwards.

The residual dynamic s.f. is to be made negative when the lower parts of thedynamic load curve, m.y., are greater than the upper.

The residual dynamic bending moment will be negative when the dynamic massmoment curve ends below the base assuming that its ordinates at the afterperpendicular take an upward course.The rotation of the base must be anti-clockwise at the forward perpendicularwhen the residual moment is negative.

These facts can be verified easily by inspection of the conditions applying to eachindividual case.

The ends of the tilted vibration profile are joined by a straight line, and the maximumordinate between this line and the profile, measured perpendicular to the original base,is the maximum deflection. The end deflection can then be obtained by simpleproportion and the frequency then calculated as described in detail in the appendixto an earlier paper (3)

Several calculations have been done by the Authors using this method and it hasbeen found to be very satisfactory.

011

2A

19375 54-5 270 70 UPPER DECK -

M[ASUR[D 2 ( VERTICAL PROFILE CC3LCULATED 2 rc VERTICAL PROFILE

Fig.1 AArrange.nents of Decks and Superstructures-4easured and calculated vibration profi1e.

j

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SHIP VIBRATION 205

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2578 3933 225 TO R. a: DECKPROFILE TOO SMALL TO MEASURE

210'8 363 21:67 TO UPPER DELl<

3R2' 62-5 IB75 TO HARBOUR DECK

j-',Fig. lBArrangements of Decks and Superstructures

Measured and calculated vibration profilesMeasured 2-node vertical profilesCalculated 2-node vertical proffles

Io

CDITIOr4 AS FOR 2 NODE VERT$CAL

H-AMRJTUDL -:

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Fig. 1 CMeasured Vibration Profiks2-node horizontal

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208 SHIP VIBRATION

'0004 '0006 '0008 0040 0012 '00)4 0046 00)8 0020 '0022

Fig. 3- Plot of Measured 2-Node Vertical Frequencies to Base of IVL3

Includes effect of surrounding water (see section 6 of paper)I Is calculated to different decks as follows :-i Including material only up to topmost continuous (100%) deck

Including all decks covering 60% or more of length of ship+ ,, ., 40%

25

24

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22

210

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SHIP VIBRATION 209

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Fig. 4

For Fig. 5 seo p. 210

00 Q2

Fig. 6Plot Shoiving Effect of Change of Load oi 2-Node Vertical Frequency

L3

Fig. 5Plot of Measured 2-Node Vertical Frequencies to Base of IB DVL3Where DE is the equivalent depth (see equation 3 in text)

= includes effect of surrounding water (see section 6 of paper)B = breadthL = length bp.

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210 SHIP VIBRATION

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DISCUSSION ON "SHIP VIBRATION *

Sir WESTCOTT ABELL, K.B.E., Fellow:I am afraid I am rather out-of-date and

have no intimate knowledge of the problemof vibrations such as are described in Dr.Todd and Mr. Marwood's paper. I haverather a simple idea about this problem; itis to me rather like a simple pendulum:that is to say the frequency is inversely asthe square root of deflection! I ask myself,is there anything in the deflection of theship under the loading system which Iought to look into first?

I may tell you one story from which Iderived, I suppose by intuition, a con-siderable amount of kudos. We receiveda message in Lloyd's office that a ship ontrial was shaking herself to bits. Theyasked what they should do about it, andwhat were Lloyd's doing in this matter.This was in St. Nazaire, if I rememberrightly. We only got this warning thatthe ship was shaking herself to pieces.We looked at the dimensions of the ship;there was nothing in the problem thatseemed unusual, and it occurred to me itwas the loading of the ship. When I gotthe reply it seemed the builders were undercontract to put the ship on trial at her loaddisplacement, and they thought the cheapestpossible way of loading a ship to displace-ment was to get sand from the river andpour it into the hold amidships. I imme-diately jumped to the conclusion that thedeflection was enormous and brought downher frequency to something that wouldagree more or less with reciprocating-engine revolutions and I sent back to tellthem to take out the load and distributeit more evenly over the whole ship. Therewas no more trouble.

A French naval architect arrived at thetime they were re-trying the ship and hetried to persuade me that the ship waswrong. I said all that was wrong wasthat some way or another they had increasedthe deflection beyond what might reasonablybe expected since the load was not evenlydistributed.

On the question of stiffness with oiltankers, I think i am right in saying thereis increased stiffness and less deflection in atanker with its longitudinal bulkheads thanwith the ordinary ship. Therefore, if onecould estimate in some way the difference ofdeflection between the systems of construc-tion, a direct numerical correction betweenthe ordinary type and the one with longi-tudinal strengthening, could be made.

* Paper by F. H. Todd, B.Sc., Ph.D., AssociateMember and W. J. Marwood. Seep. 193, ante.

21

I had a lot of trouble with a certain shipand went through the usual process. Wecrept up the revolutions of the engines oneat a time, and finally the clapper on theship's bell on the forecastle head began toring violently. lt came time to go tolunch and i said we must leave it. Afterlunch we returned, started below therevolutions and went up; the ship's bellrang again. It was a question of balancingcertain weights and when balanced Out, atthe next trial, we stepped up the revolutionsand the ship's bell did not ring so we knewwe were much better. I forgot to add thatI found someone had muffled the clapper

Mr. HARRY HUNTER, O.B.E., Fellow:I would agree with the Authors as to the

danger of relying on simple empiricalformulaewhile it is true that in themajority of cases the observed naturalfrequency is in line with such calculationsyet exceptions can and do arise withdifficulties for all concerned and consequenttrouble is not helped by the fact that it isan exception. Surely the calculations re-lating to vibration are equal, or nearlyequal, in importance to those relating tostability and propulsion and therefore thenaval architect should accept the necessityfor proper investigation.

On the matter of ship vibration, broadlythere are two prime factors involved: theship which may vibrate and somethingwhich may cause it to vibratein fact it isthe very common case of the hull being a" bell" which will certainly " ring" ifstruck by a suitable" clapper." Some bell-clapper problems are best solved by goingafter the bell and others by going after theclapper and others again demand simul-taneous action in respect of both sides;in my view ship vibration belongs to thislatter class just as does torsional vibration ofshafting and for the same reason that aviolin (" bell ") gives its best performancewhen it and the bow (" clapper ") are underone control.

Dr. T. W. F. Brown in his 1939 paperbefore this Institution clearly Sets out thebenefits and some principles of "paralleltreatment" in the section "The Ship andher Main Engines ". This view of paralleltreatment has also been adopted by theBritish Shipbuilding Research Associationin Setting up the Ships' Vibration Committeeand one of the first actions of the committeewas to order an electrically driven vibration

f" Vibration Problems from the Marine EngineeringPoint of View," Vol. 55.

exciter for the purpose of investigating shipvibration generally and including the effectof varying magnitude and location ofexcitation. This machine has been in usein several ships and has given valuableinformation.

The present paper, as with the vastmajority of previous papers on the samesubject, deals entirely with the bell"side of the problem and the only referenceto the "clapper" is on p. 194, "It mayhappen . . . that the unbalanced forces inthe engines are insufficient to excite them ",(i.e. hull vibration). That one sentenceputs my main point in a nutshell, namely,that we engineers are most anxious to knowwhat magnitude, type and location ofexcitation is acceptable.

Incidentally, the Authors rather implythat the main engines usually provide theexcitation, yet two or three of his ships,(16, 24, and 13 (?)) are turbine driven and,therefore, presumably some other "clap-per" is at work. Have the Authors anyinformation on this point?

Any reciprocating engine, steam orDiesel, or propeller rotating in the disturbedwake, sets up various forces some of whichare under ready control in the design stagebut much more difficult to deal with whenthe vessel and machinery are completed.Unfortunately very little information isavailable on this "clapper" end and Iwould propose to put the sort of require-ments in the form of a proforma request tothe Authors for further information whilerealizing that the information requested isfar more than can be expected in the replyto a discussion.

In case of each vessel, can the Authorsstate the designed propeller r.p.m. at servicespeed; also in case of reciprocating enginesthe number of cylinders and whether theDiesels are 2-stroke or 4-stroke? From thepaper one assumes that in each case whenvibration occurred the propeller r.p.m.coincided with the frequency of vibration;was this so?

The location of main engine in relationto the nodes may well he of importancesince at the node one might expect a coupleto be a more effective "clapper" than aforce, and vice versa il engine is at an anti-node a force should be avoided. Since inengine balance one can, in the design stages,often ring the changes between forces andcouples, some definite information as totheir relative importance would be mosthelpful. Further, the magnitude of anyengine forces or couples at the criticalspeed would be most valuable information.

Apart from the main engines there are,of course, other possible "clappers ", forinstance, those arising from the propellerworking in the disturbed wake. This maywell set up horizontal and vertical forcesacting at the stern-tube bushperhaps agood position for setting up vertical

or horizontal vibration. Also there will bea varying thrust acting, well away from theneutral axis and capable of exciting, onethinks, vertical and also possibly in a twin-screw ship, horizontal vibration; and so itgoes on.

While it is not expected that the Authorscan answer all of the foregoing requests itis suggested that future papers on thissubject of ship vibration should includesome reference to the "clapper" on thelines indicated.

Prof. L. C. BURRILL, Member of council:From the naval architect's point of view,

there are four main aspects of the ship-vibration problem. First of all, there isthis question of carrying out a long calcu-lation which is worth while, and we haveheard to-night that it takes nearly a fort-night to work Out such a calculation. Itwas not stated that it may take two or threeweeks to assemble the information to startthe calculation, to prepare a proper loadingdiagram for a given condition and to makea detailed moment-of-inertia calculation.It is obvious, therefore, that before under-taking such a calculation in a shipyard it isnecessary to be sure that a satisfactoryanswer will be obtained.

I think we can now say that a satisfactoryresult can be obtained, if sufficient care istaken and a sufficient length of time is spenton the calculation. The principle ofbalancing the strain energy in the shipstructure in its extreme bent position to thekinetic energy when the ship is passingthrough the neutral position has provedvery satisfactory, and I think it is a matterof congratulation that most of the develop-ments of this method are due to studentsof King's College, and not least to the workof Dr. Todd who has pursued this subjectfor many years.

In the second place, there is the problemof establishing a short calculation methodwhich will give a reasonably accurate answerat an early stage in the design. Theprinciple of this method goes back toSchlick who said in effect that the frequencyof vibration of a free-free rod is given by

where i is the moment ofWV

inertia of the cross-section, W is the weight,L is the length and C is a constant, and thattherefore for ships of a given type it shouldbe possible to substitute another value of Cwhich takes into account the variationsfrom a uniform beam. With this in mind,he made some tests on various shipstowards the end of last century, and foundthat constant was in fact nearly the sameas for a uniform beam having a momentof inertia equal to that of the midshipsection and the same total weight. As aresult he published three constants, for usewith different types of vessels, but it was

D114 SHIP VIBRATION

later found by experience that it was some-hat difficult to choose a suitable value of

the constant.This Schlick method has been developed

by various observers, and it now takes twoprincipal forms, namely:

F=C/J and F=C2/3In the first of these expressions the

constant C1 takes into account the effectof entrained water, the effect of sheer-strainenergy and the effect of distribution of massand moment of inertia along the length,and in the second the item W1 is the sum ofWand the entrained water. This inclusionof the effect of entrained water was, I think,an important step forward, and the resultsobtained from such a formula should becorrect to within five to seven per cent.The use of BD3 in place of I can, however,be very misleading, and I would join withMr. Hunter in stressing this point. Theaccuracy of this formula using BD'1 dependsentirely upon the assumption that I is aconstant times BD1, and I think that thediagram at the end of the paper shows thatthe variation in the useful range is suchthat there is quite a wide range of choicein putting a mean line through the diagram.There was one instance about two years agoin which the shipbuilders used a formula ofthis type and estimated a frequency ofabout 117 whereas the actual frequency ontrial was about 77 per minute and therewas considerable vibration as this was nearthe working revolutions. Other similarinstances have occurred and I wouldaccordingly recommend to shipbuilders thatthey use this forni of the expression withgreat caution.

The third problem is the question ofcollecting as many ad hoc records fromactual ships concerning their frequencies ofvibration in the lower nodes, for comparisonwith the calculated values. It is in thisdirection, I think, that the Authors are tobe congratulated on being able to placebefore us a good deal of new informationwhich represents the work of about twelveyears or so. lt is a long time now since wehad a paper giving us new data concerningthe actual frequencies measured on shipsat sea. In my view, there is room for manyinvestigators in this field, and any navalarchitect who can interest himself in thecollection and publication of reliablefrequencies for actual ships will be helpingtowards the final solution of this vibrationproblem. It would, in fact, be highlydesirable that we should reach a stage inwhich we could plot the frequencies fordifferent classes of vessels to a base oflength or displacement, as this would enablea very rapid decision to be made concerningthe engine revolutions or type of enginewhich might give trouble in any particularinstance.

SHIP VIBRATION Dl 15

On p. 195 Dr. Todd has drawn attentionto the base line corrections suggested bySchladlofsky in 1932 and it is suggested inthe paper that these corrections wereunknown in this country. I think I shouldcorrect this suggestion, in that quite anumber of investigators who have beeninterested in the vibration problem notonly knew of Schladlofsky's method ofdealing with this matter but have alsoapplied it in carrying out such calculations.The method is, for example, referred to inmy 1935 paper on ship vibration.*

It is true that Schladlofsky's paper hasnot appeared in English and that it is notour habit to translate many of the valuablepapers published on the Continent ontechnical subjects of this kind, but I amglad to note that through The Shipbuilderand Marine Engineer translations of currentforeign papers of interest are beginning tobe available for readers in this country. Inparticular I would refer to the very valuablepaper entitled "The Vertical Vibration ofShips ", which was read before the Associa-tion Technique Maritime et Aéronautiquelast year by Professor Prohaska of Copen-hagen University, which appeared in TheShipbuilder not long ago. This paperrepresented a very important advance inconnexion with the estimation of ship-vibration frequencies, particularly in con-nexion with the corrections for entrainedwater.

The fourth and final aspect of the ship-vibration problem is, in fact, this questionof the effect of entrained water. At thepresent time we have to rely mainly on thetheoretical work of Professor Lewis, to-gether with a partial verification byexperimental means obtained by Messrs.Moulliri and Brown. So far as their workon the vibration of free-free beams ofvarious cross-sections is concerned, it wouldappear that the practical values of theentrained mass effect are about 90 per cent.of the calculated values. There is room fora great deal of further careful investigationon this aspect of the problem, and experi-ments are at present being carried out atKing's College towards this end, underthe auspices of the British ShipbuildingResearch Association.

The present paper by Dr. Toddcarries the general investigation a con-siderable stage forward in that it dealsmainly with the effect of erections on thefrequencies of vibration in the fundamentalmode. The present position can, I think,be summed up as follows. For shipshaving no erections, a reasonably satisfac-tory answer can be obtained by applyingexisting methods, and for ships having afairly long set of erections, the same applies;but there is a transition region between

" Ship Vibration : Simple Methods of EstimatingCritical Frequencies," Vol. 51.

these two types where at present it isdifficult to obtain a really satisfactoryanswer.

The new information and the methodssuggested by Dr. Todd in this paper willcertainly help the designer in estimatingwhere a proposed new design lies in thistransition region. There is no doubt thissubject must be pursued further and weshall welcome a further paper on the saniesubject in due course.

The Authors, on p. 194, state that theyusually take their records at the stern ofthe vessel, whereas I have always thoughtit most convenient to take such records atthe forward end. There is no doubt thatthe two free ends give the best records, butI would say that the local influence ofengines and propellers is liable to be greaterat the after end than at the extreme forwardend of the ship.

Mr. H. G. YATES, Member:Have any measurements been made at

speeds near resonance sufficiently accurateto determine the degree of damping presentin the system?

As Mr. Hunter says, what we want to

CORRESPONDENCE

Mr. N. CARTER, Member:This paper gives a very simple method for

a preliminary assessment of the two-nodevertical frequency. The following list givesthe results of a few ships on which thefrequency was observed and which agreereasonably with the curves given in thepaper:-

/ /B.D3 ObservedType of Length A

The two cargo ships are complete super-structure types with midship house about20 per cent. long, the figures being givento the uppermost continuous deck.

It is significant that the ships mentionedin the paper without erections, and thosewith long erections, give consistent results,while the ships with erections between 30per cent, and 60 per cent. are not so good.The Authors state that discretion must beused in making allowance for superstructuredecks. Is it logical to assume a meandepth or mean ¡ to cover these shortererections? The effect of these erectionswill be to stiffen up the girder amidshipsand thereby change the shape of the

know is the exciting force which the shipcan stand without dangerous or unpleasantvibration. That can only be determined bya knowledge of its reaction to frequenciesabove and below resonant frequency.

With reference to Mr. Hunter's query onthe point of damping in the region wherethe exciting force is acting, I think one cansay with reasonable certainty that it doesnot matter where the damping is for agiven mode of vibration, it has the sameeffect whether it acts near the excitingforce or somewhere else. The excitingforce may come from the engine amidshipsor aft or from the screw, butin all cases itwill be possible to make a reasonabledetermination of the exciting force, andonly the damping is necessary to give theresulting velocity. The amplitude can bedetermined as soon as the frequency isknown.

VOTE OF THANKSOn the motion of the PRESIDENT

(Mr. H. B. Robin Rowell, A.F.C., D.L.) avote of thanks was accorded to Dr. Toddand Mr. Marwood for their paper.

5050= 331

Then,/'' _54.5 X 34.753='0020

M L3 l5600x 33lNo. 17 ship should not be included in

Fig. 5 which uses depth as a basis, as anexpansion joint was fitted in the gamesdeck also the decks from the promenadeupwards were overhanging the normalbreadth of ship.

Dr. J. F. C. CONN, Member:Thanks to Dr. Todd's earlier work,

calculations of the two-node verticalfrequency give reliable results for ordinaryvessels. The present paper deals with theeffects of long erections and the highermodes of vibration.

Ship b.p. V V jfrequencyTanker 490 000391 00085 77Tanker 483 000402 P00088 72Tanker 460 000405 00088 73Tanker 420 000451 00096 83Tanker 500 '000365 '000787 69Tanker 460 '000406 000865 76Cargo 433 000625 00158 104Cargo 412 000522 00134 88

D2 = 50'50 L = 375

D1 = 34'75 L = 375

ol 16 SHIP VIBRATION

deflection curve and this suggests that amodification should be made to the lengthrather than to the depth. The spots canbe brought more into line by using adepth D1 in conjunction with a length L.

whereL = l.b.p. X 3\/D2

D2 = depth to top erection 30 percent to 60 per cent.

D1 = depth to highest deck over 60per cent.

For example No. 19 ship in Table 1.

So far as structural rigidity is concerned,it is already clear that the two-nodefrequency depends mainly on the inertiavalue about amidships, but with highermodes this will not remain true.

Further refinements in frequency calcu-lation must depend upon the experimentalresults available, and the Authors havegiven useful additions to the amount ofpublished information. lt appears to mcthat only extensive series of tests on vesselsof different types, where vertical andhorizontal vibration of fundamental andhigher modes are produced by means of anexciter, can provide the required data.

The Authors will probably agree that thecorrection for shear in ship vibrationcalculations is an appreciable, if not a large,one. lt is extraordinary that while Dr.Lockwood Taylor's shear correction iscommonly applied in vibration calculations,the same shear corrections as applied tostress and deflection are not yet commonlyaccepted in structural strength calculations.

Mr. R. W. L. GAWN, O.B.E., Member:The title of the paper while commendably

brief appears too comprehensive and would,it is suggested, be more representative ifexpanded to include "The Natural Fre-quency and Amplitude of Main HullVibration ". The particular emphasis inthe paper is rather on the degree of exacti-tude of prediction of these characteristics ofvibration from design drawings of ships.This is indeed a sufficiently wide andimportant subject. The complexities ofelasticity and hydrodynamics involved arereflected in the many excellent contributionsto the subject that have been made in thepast in which company the present paperfinds a very good place indeed.

The Authors rightly draw attention to theimmense labour involved in ship calculationsof this type from design drawings. Theirsolution which reduces the labour effortand gives predictions of primaty naturalfrequency ranging from +Th per cent. to5 per cent. of the measured frequencywill be welcomed as an important achieve-ment by all interested in the detail ofvibration problems.

The accuracy of prediction of amplitudeby calculation brought out in Fig. lB whilenot so close on the whole as for thefrequency can nevertheless be regarded asextremely satisfactory in view of thecomplexity of the problem. It is, however,disconcerting to find that amplitudes aslarge as about 300 thousandths of an inchare recorded for ship 25. This largevibration may possibly be explained by thereciprocating Diesel machinery but evenwith the two turbine-driven ships consideredin the paper, the amplitude is 90 thousandthsfor ship 24 and 60 thousandths for ship 16.The amplitude is large for other ships andin fact the only record of a small vibration

SHIP VIBRATION o117

is for ship 22, the amplitude then being 8thousandths. This is a shorter ship, thelength being 257 ft., but even the smallership 21 has an amplitude aft of about 50thousandths of an inch associated naturallywith a high frequency.

If these large vibrations occur within theoperational range of speed of the shipsconcerned they must be unpleasant. Theconsequential reactions on the efficiency ofa radar set and possibly wireless or otherequipment might be serious if local reso-nances occur. It would be of interest ifthe Authors could give an explanation ofthe excessive amplitudes and remark orithe operational aspect including speed atwhich the large vibrations occur. It wouldalso be appreciated if particulars could begiven of the engine and shaft revolutions,ship speed, hull clearance and number ofpropeller blades, to permit of furtherconsideration of this important aspect.There would appear to be scope forimprovement, possibly by modification ofpropellers.

The first report on vibration of H.M.ships completed at Admiralty ExperimentWorks, Haslar, is dated 1889. A vibro-graph was specially designed and made forthe trials. The vibration was recorded assatisfactory, the amplitude of movement ofthe deck of Gleaner being 150 thousandthsof an inch at 20 knots and 375 thousandthsfor Medusa. These trials were primarilyto ascertain whether vibration prejudicedstructural strength in a seaway and it is inthat sense that the vibration was recordedas unimportant. The vibration measuredon these ships sixty years ago is as satisfac-tory as many of the ships dealt with in thepaper. The present-day requirements forvibration of H.M. ships are governed bymany considerations other than structuralstrength and very refined standards arerequired.

It is necessary to evaluate the momentof inertia for strength estimates so that theobjection raised by the Authors to Schlick'sformula that the moment of inertia isunknown will surely not apply to ships forwhich longitudinal strength calculationshave been made. Vibration formulae canaccordingly he readily applied to thesecases. Fig. 3 of the paper can be comparedwith Schlick's formula. The frequencylocus as drawn does not, however, passthrough the origin, and it would be interest-ing if the Authors could remark as to anyexplanation of this. The locus appears tohave been partly governed by the resultsfor ships 21 and 23. Ship 21 is a single-screw cargo coaster of 180 ft. length andship 23 a single-screw coaster of 210 ft.length. The majority of the ships con-sidered are of cargo and passenger type oflength ranging from 375 to 650 ft. I havedrawn a new frequency line through theorigin, averaging approximately the mea-

sured frequencies excluding the two shortships referred to. This line gives afrequency of about 150,000 times theabscissa parameter. The coefficients quotedoriginally by Schlick are 156,850 fordestroyers, 143,500 for liners and 127,900for full-cargo ships but Schlick's parameterdid not allow for entrained water. If theAuthors' parameter is modified to dis-regard the effect of entrained water thenthe coefficient of 150,000 is reduced toabout 100,000. The extent to which thisfalls below Schlick's coefficient is worthyof some consideration because Schlickstated that his formula gave generallyreliable values and this claim is substan-tiated for certain classes of warship. If inworking out the coefficient from Fig. 3regard is paid to the parameter for whichthe moment of inertia included materialonly up to the topmost continuous deckthe coefficient is increased to about 127,000.This approximates to that quoted bySchlick. The implication appears to bethat Schlick's formula could be applied togive a fairly close estimate of the primaryfrequency of ships of the type dealt within the paper with the exception of shortships 21 and 23 which will require somereduction in coefficient. The Authors'remarks on this point would be muchappreciated.

The paper brings out the advantage ofcombining fundamental theory with experi-ment and the considerable effort involvedin ship vibration problems. The effort isgreatly intensified by the numerous recordsrequired of local vibration to supplementthe hull vibration dealt with in the paper.Avoidance of resonance by a good marginis the one method of obviating or reducingthe vibration of ships. The Authors' workmakes an important contribution to thisaim in ship design and thanks are due tothem for this important advance.

Mr. F. MeALISTER, Associate Member:The great value of such papers as this

is the undoubted contribution to the storeof data available on the subject. Thepapers on vibration listed by Dr. Toddform in themselves a large and extensivestore of authentic data without equal inour records.

Although no doubt it is valuable toexamine in post-mortem fashion such casesof vibration as are investigated in detailin the paper, the supreme test of the dataso analysed must lie in their applicationto new designs and for that purpose littletime is available at that stage for determin-ing the critical revolutions by involvedinvestigation.

The design of any ship revolves roundweights and to determine machinery weightsthe r.p.m. must be known. This is at avery early stage when even the erectionsare not fully determined. The preliminary

investigation for critical r.p.m. should thenbe determined by an approximate applica-tion of known data, and one useful formulais L1Primary Critical r.p.m.

=where K is a constant from similar shipsand L the length in feet. The constant, ofcourse, varies with the type of ship,displacement, disposition of structure andmany other factors, yet in similar typesof ship this formula gives a very good guidein the initial stages of the design.

For example, for oil tankers K is 75 andfor trunk-type tankers, coasters, etc., Kis about 54 to 58, K rises 110-115 in inter-mediate passenger vessels and up to 130or more in large vessels.

At a later stage in the design detailedcalculations can be made to confirm thatthe fundamentals of the design are sound.

Still searching on this point of theapproximate solution of the problem Dr.Todd's allowance for superstructures is inthe right direction and is, in fact, muchthe same equivalent depth as that used bynaval architects in estimating their K.G's.Most of my own weight data and centreof gravity data are based on a rather moreelaborate form of equivalent depth and Iendorse Dr. Todd's formula (3) as a usefulbasis to adopt.

I do think, however, that instead ofapproximating the critical two-node verticalfrequency and comparing it with theobserved critical, it might be more powerfulto analyse the actual critical back to theequivalent depth and plot the excess depthabove that to the uppermost continuousdeck on a basis of percentage of erectionsor some other suitable parameter.

Mr. R. G. MANLEY, Associate Member:First, it is noted that the experimental

results were obtained by using the mainengines of the ships as exciters. Has ityet been possible to use artificial vibratorsto excite the vibrations? Admittedly theships' engines and the propellers are theprincipal sources of vibrations, but theremight be circumstances in which it wouldbe desirable to see the effect of excitingfrom some other part of the structure, andagain it might be more convenient at timesto carry out vibration tests without callingupon the engine-room for co-operation.

Is it a practical proposition to performvibration tests on structural models?There are several obvious reasons why sucha procedure would be difficult to employusefully, but it would be interesting tolearn whether the Authors know of anysuccessful work on these lines.

Amongst the vibration records obtainedduring a typical test, what proportion arereasonably pure sine-waves? What is theAuthors' preferred method for analysingthose that are not?

D118 SHIP VIBRATION

It is noted that in Fig. IC on p. 206 themeasured vibration profile for ship No. 16turns back towards the zero line at the sternvery much more markedly than is the casewith the other profiles shown. Is thereany known reason for this? Unless thereis a very great concentration of mass atthe stern the appearance is rather odd.

The Authors use the term "moment ofinertia ", and occasionally simply" mertia",for the quantity proportional to BD3 whichdetermines the stiffness of the structure.This seems to me to be unfortunate, as itleads to statements such as that at the topof p. 194: " ... the distribution of inertiafor the main girder towards the ends wasof minor importance . . whereas thecorrect distribution of weight . wasessential." The vibration characteristics ofa beam depend upon the distribution ofmass and stiffness. The engineer wouldprefer to talk of weight or of inertia insteadof mass, but it is important to note thatthe inertia which affects the issue isassociated with the mass. The BD3 quan-tity, which is properly if horridly termed"second moment of area ", is a purelygeometric quantity which is independent ofthe density and the physical properties ofthe material, and it is misleading to referto it as " inertia ". lt is, after all, merelycoincidence that the mathematics of thesituation makes the second moment ofarea of a cross-section something similarin form to a moment of inertia. Myobjection could perhaps be justifiablytermed a mere quibble were it not for thefact that true inertia operates in a mannerquite opposite to the so-called " moment ofinertia ", for an increase in mass inertiadecreases the natural frequencies whereasan increase in the area moment increasesthem. I should like to see the term"moment of inertia" deleted from all thetextbooks and papers except where itmeans what it says, namely, the rotationalanalogue of mass in equations of motion.

Mr. J. M. MURRAY:This paper continues the records of ship

vibration which Dr. Todd has given to theInstitution and is of value both on accountof the specific cases which he furnishes andthe general formula which he has evolved.The approximations which he gives areexceedingly useful, but it must be borne inmind that, as the Authors point out, theyare only approximations and in applyingthem a certain amount of judgment mustbe used. The results given in Figs. 3 and5 have been applied to several cases whichhave come within my experience with verygood results. lt is of interest to quote twoof these cases(l) a tanker 460 feet longand (2) a passenger ship 570 feet long.

For the tanker very complete records ofvibration in three conditions were available

SHTP VIBRATION D1l9

and the frequency at load draught coincidedwith that given for an oil tanker in Fig. 5;at lighter draughts the divergence from thecurve noted in Fig. 6 was confirmed. Theratio of the three-node vibration to two-node vibration was 247, which is inaccordance with the results given in Table6. Since tankers are more or less of astandard design and do not differ muchin proportions from ship to ship, it is notsurprising that such close agreement wasobtained with Figs. 5 and 6.

The case is altered considerably when wecome to passenger vessels, for here theremay be a wide variation in the characteris-tics of different ships. Nevertheless, in thecase of the 570-foot passenger ship, withtwo tiers of erections, there was alsocoincidence with the results derived fromFig. 5. At the same time, it is not certainthat such a favourable result would beobtained in every case for I have notfound that the inertia is proportional toBD3 irrespective of the proportions of theship. Dr. Todd, in his 1931/32 paper tothe Institution,* has given a diagram whichseems to demonstrate that the inertiacoefficient does not vary with L/D. I havenot had this experience and have found that,in general, there is a very definite variationwith the proportions of the ship. Someconfirmation of this point of view may beobtained from a comparison between Fig. 3and Fig. 5. 1f the frequency is plottedwith respect to inertia, as is done in Fig. 3,it will be observed that tankers, passengerand cargo ships fall on the same line.When the inertia is related to BDE as inFig. 3, two curves are necessary, and it issuggested that this is due as much to thedifference in proportions of the two classesof vessels under consideration as todifferences in type or construction of ship.

lt is considered that the method of indi-cating the characteristics of the ships inTable 1 and Fig. lA and B is extremelyuseful and should be of considerableassistance in determination of the naturalfrequency of a ship in the initial stages ofdesign.

Arising from this paper, though notdirectly concerned with it, there is one pointto which reference might be made, and thatis that general or even local vibration is nota sign of weakness in a ship as it is some-times considered to be, and that it is flotoften that the introduction of additionalmaterial can have any sensible effect.

Prof. C. W. PROHASKA, Member.The aim of the paper has obviously been

to find the effect of superstructures onfrequency, and although Fig. 3 gives someidea of this effect, it seems difficult from thisdiagram to draw definite conclusions.

"Some Measurements of Ship Vibration," VoL48.

I think it should be possible to obtaincloser agreement between calculated andobserved data than that generally borne outby the Authors' investigation.

The reasons for the rather confusingdifferences present in some cases, are, inmy opinion, mainly due to the followingcauses:

Moment of inertia has been taken asconstant,Full moment of inertia has been usedfor top decks, not decks to the hull,Shear correction is too approximate, andVirtual inertia factor is probably over-estimated, if observed frequencies cor-respond to deep-water condition.

But apart from that, it is good to remember,that an uncertainty of one or two per cent.is quite normal in observed frequencies.

Ori p. 201 it is stated that the effect ofvariation of moment of inertia along thelength of the vessel is of secondary charac-ter. In this ¡ disagree, and T should liketo refer to a paper read before the Associa-tion Technique Maritime and Aeronautiquein Paris* last year, in which it is shownthat the necessary correction in some casesamounts to 20 per cent. or more. In myopinion, therefore, it is absolutely essentialto make this correction for practically allvessels, except tankers. The correctionfactor to be used is greater or smaller thanunity according to the I value introducedin the calculation.

For top decks of light scantlings andsupported at the sides of the ship bystanchions only, the wing parts of the deckare less effective than those rigidly supportedby deck-house sides, and will carry but apart of their theoretical load. It thereforeseems logical to introduce a reductionfactor for the moment of inertia to suchdecks.

The shear correction has been esti-mated in accordance with Dr. Taylor'sapproximate method. This method isbased on theoretical deductions for abox-vessel with uniform thickness of plating,hut the curvature for pure bending isrelated to the corresponding deflection byan empirical and rather arbitrary constant.

Dr. Taylor gives: , withNB i/l+r

D2 3a3+9a2+6a+l23a+l

a = - . If the proportion of deck and

bottom thickness to that of the side shell

= 3,5

*C W. Prohaska 'Vibrations Verticales du Navire."Bulletin de l'ATMA, Paris 1947. English extract inThe Shipbuilder and Marine Engine-Builder, Oct..1947. (See also Prohaska: Lodrette Skibssving-ninger med 2 Knuder, Copenhagen, 1941.)

is A, this expression can be generalizedinto:

Dt 3a3A+9a2A2+6aÀ+l2r=35xj X 3aA+lWhen we deal with ships with severaldecks and a double bottom, it is obviousthat a value of A > I must be introduced,which results in an increase in r, andconsequently in the shear correction onfrequency. For the box girder of non-uniform thickness, it can be shown (loe.cit.), that:

p2 J I 3aA+lAXDSF2X aA+l ,and:

C = (3a5À+9a2.k+6aA+ I 2)

where: C=Ox = r2dA,St2

a's

(A = cross section area of material, Qshear force, G = modulus of rigidity,

O=shear angle, r=shear stress, St=staticalstatic moment about neutral axis of materialoutside the point of the section considered,& = thickness of plating, and s = girthco-ordinate). Introducing c and in Dr.Taylor's modified r value, the followinggeneral expression is obtained:

'D' ZNS+B =1 ±l+42Cx (r)- r2Nß

The variation of r is graphically given inFig. 8. As the constant 42 is arbitrary,these curves cannot be expected to give thetrue value of r5. Schadlofsky dealt withthe question on similar lines, but usedinstead of Dr. Taylor's arbitrary constant,the figure 286, which is correct for theuniform free-free beam. Curves showingSchadlofsky's correction are also givenin Fig. 8.

Solving Timoshenko's complete differen-tial equation for the vibration of ahomogeneous bar and adjusting the resultsto be fit for the non-uniform ship-girder, Lhave produced the diagram in Fig. 9 (loe.cit.), which shows corrections somewhatintermediate between those found from thetwo sets of curves in Fig. 8. Taylor'scorrection, therefore, might be acceptable,if the proper values of c were used.The Authors, I suppose, used Taylor'soriginal data corresponding to A = 1, whichgive too low a correction. This may beseen from Fig. 10, in which the variationof c with a = BID and A is shown, forrectangular sections of non-uniform thick-ness. The four points marked ADcorrespond to Schadlofsky's calculationsfor four ship sections alI with doublebottom, but with one, two, three and fivedecks respectively. As Schadlofsky's sec-tions have uniform thickness of plating, thevalues indicated are somewhat higher than

D 120 SHIP VIBRATION

those found for actual ships. I thereforerecalculated ship B with correct Lloyd'sscantlings and found the cf value markedB1, which is much closer to the line forA= 1, but still gives a 2 per cent, higherreduction on the frequency than accordingto Taylor. More calculations of exact cfvalues will have to be carried out, beforeit is advisable to use this graph for ships,but it gives some indication of the variationof c.

lo 12100

.95

.90

.85

.80

- TAYLOR-SCHADLOFS

r;Fig. 8Shear Correction Factor:r2 derived from Lockwood Taylor'sand Schadlofsky's Investigations

SHIP VIBRATION D 121

3a'À9a'Á+ 6'aA+l.212 3a+l--P-----

10

a-B

4. The virtual inertia of entrained waterhas been assessed according to the methoddeveloped by Lewis. In my opinion, thecorrection for three-dimensional flow shouldbe performed in accordance with the resultsfound by Lockwood Taylor and not bymeans of Lewis's more approximateformula. But it is still more important totake into consideration the depth of water,as the virtual inertia factor increases rapidlywhen depth decreases (bc. cit.). Noreference is made in the paper to depth of

water, but it is suggested that the Authorsinclude such data in their reply to thediscussion, to make the paper still morevaluable for future exploration.

I should have liked to try my suggestedcorrection on all the ships given in the paper,but it is only for ship No. 15 that it ispossible from the data given to calculatethe moments of inertia corresponding toall the superstructure decks. I thereforehave to content myself with this single

8 10 1214L,6

Fig. 9Shear Correction Factor:r2 (Prohaska, ¡oc. cit.)

calculation, which is to be considered moreas an example than as a proof of mymethod:

From an enlarged photograph of Fig. 3,

I read off the values of\/ 'L ,and thus

for ship No. 15 found the following valuesof I for the three top-decks: l'27 , 10G,

105 . 106 and '73 10'. ¡for the shelter-deck is not given, but is, due to the extentof the superstructures, without great im-

1- 1.5 20Fig. 10AD indicates Schadlofsky's ShipSections of Uniform Thickness, B, ship B

re-calculated with Correct Scantlings

portance, and can be estimated to about-3 to 4 10. Fig. il depicts theapproximate variation of I. The dottedlines indicate an arbitrary reduction for theupper decks due to the above-mentionedcauses. Calculating now the reductionfactor r1 by means of a method I havedescribed previously (toc. cit.), I found:r1=l 26 when using the bridge-deck

Boat Dk. Is12 710' ç1.2010'

Prom. DkIaWS.10'

&Dk ! .731

_Li. Dk. I-..' -36t0

I J

moment of inertia in the calculation of thefrequency. Similarly was found: r1 = 1 065and r1=0985 for the promenade andboat-decks respectively. With c= 60

and -4-- to boat deck=76, r2= 85 was

lifted from Fig. 9. With a further correctionfor transverse stresses (loe. cit.): ra= 98,I get for the three cases considered, a totalcorrection of:¡ to bridge deck:

R=l26 x85x98=l-050I to prom. deck:

R=l065x85x98= 887I to boat deck:

R= 985x85x98 82l

The Authors' corresponding values of

amounts to: 901 887 and 870.

Thus their calculated frequencies of: 95109-3 and 118-5, multiplied by R/irand by the square root of the proportionbetween the reduced and the actual valuesof moment of inertia give:

1050 ¡.73Bridge deck: 956x

91ÒX =111

Prom. deck: l093x- x\/=108Boat deck: ll85x X4,/- =109

These values are in good accordancemutually, but fall a bit short of the observedfrequency: 112. This might be due to thev.i.f., which according to the Authors wastaken as 2235. Had Lockwood Taylor'sreductions been used instead of Lewis's,v.i.f. would have been 207, changing theabove calculated figure of 109 into 113.No correction for depth of water was used.

Fig. 11-Approxiinat e Variation of ¡for Ship No. 15Dotted lines indicate suggested reduction due toineffective connexion of wing parts of decks

to the hull (compare Fig. I)

'S

ER

I should like the Authors to state whetherthe frequency 112 was observed in deepwater. 1f that was not the case, theagreement might be less perfect. As faras I can see from this paper and fromDr. Todd's earlier papers, the vibrationprofile used in the calculations was in allcases taken as that of the uniform bar. Ifthe calculations were not repeated to giveclose accordance between the curve usedfor the determination of the accelerationforces and the resulting deflection curve,errors to the amount of several per cent.might be present in some of the calculatedresults.

The use of the correction factors, such asr1 and r2, is, of course, unnecessary whena complete calculation is performed. Theamount of extra work involved, to includecorrectly the effect of variation of momentof inertia and shear deflection, is not pro-hibitive (loe. cit.).

I agree, however, with the Authors thatapproximate methods are of value at thedesign stage, but I do not think the modifiedLundberg formula fit to take accountcorrectly of variation in moment of inertia.In Fig. 12 curves of r1 derived from thisformula, for the case of one step only inthe moment of inertia curve, are comparedwith r1 curves found by direct calculation(loe. cit.) for a uniform distribution ofload. It is seen that the formula under-estimates the correction, except, of course,at zero length of bridge. For usual ship-distributions the differences become evengreater.

In the Appendix the Authors have dealt

Dl 22 SHiP VIBRATION

at length with the necessary shift of zero-linefor the vibration profile, and attributed themethod to Schadlofsky. This is not quitecorrect. The method dates back at leastto Poisson, and has been described inEnglish several times before. (Barfoed:"The Natural Vibration of Ships ", TheMotor Ship, Feb., 1926; Lewis: "Vibrationand Engine Balance in Diesel Ships ",Trans. Soc.N.A. & ME. 1927, The Ship-builder and Marine Engine-Builder, Oct.,1947, p. 543-544).

Finally, I would draw attention to thecalculated figures for ship No. 14, whichI think are not in mutual accordance, ornot in accordance with the correspondingIIdata of Fig. 3, as 'V A1'L3 which ought

N/lrto be the same for the two decks, differsabout 15 per cent.

.ko. '1.0JaR

6

.4

o

Mr. S. B. RALSTONSome of the earlier papers dealing with

vibration have tended to include factorswhich cannot be predicted before the shipis actually constructed, so that the con-clusions were of little assistance to theharassed naval architect. The Authors ofthis paper are to he congratulated on theirpractical approach to the problem andappear to be justified in their belief thattheir work will enable designers to makevery close estimates at an early stage beforefull information is available.

The Authors have examined the effect ofsuperstructures, and Figs 3 and 5 showpairs of spots for each ship with andwithout the superstructure. The mean lineis drawn as a straight line, but must thisline be straight? I suggest that a curve

SHIP VIBRATION ol23

Fig. 12

drawn through the mean of each pair ofspots will cover more ships than anystraight line, and such a curve (to work inthe reverse as it were) may lead the wayto a more generally correct approach to theeffect of superstructures.

lt is noted that the mean straight line ismade to coincide with the 180 ft. coasterNo. 21, while the small ships No. 22 and 23do not conform to the line. Small shipsare, in the nature of things, more liableto differences of type, and it is suggestedthat the mean line or curve need notnecessarily agree with No. 21.

Dr. J. LOCKWOOD TAYLORThe general conclusions as to the

possibility of calculating from first prin-ciples the two-mode vertical vibration seemto tally with those of my 1930 l.N.A.paper.* I believe, however, that the three-dimensional correction to the virtual inertia,

0 20 40 60 80PER CENT BRIDGE

which I based on mathematical results foran ellipsoid, should be applied. I was ablealso to get agreement for the three-mode,between observation and calculation.

The apparent discrepancy for the two-mode horizontal seems to require investiga-tion. Further calculations of virtual inertiafor different sections allowing for the free-water surface and of the effect of addingmass at an offset from the shear axis of thestructure suggest themselves as subjectsworth looking into. As I pointed out inan earlier papert, there is always sometorsion combined with lateral vibration.

*" Vibration of Ships," Vol. 72.

t" Ship Vibration Periods" Vol. 44, N.E.C. Inst.

----.

LUNDBERG-FORMULA.EXACT CALCULATIOM

Prof. G. VEDELER, Member:lt is usually instructive to compare

calculated or measured values with knownresults for simple bodies. To start withFig. 2 it may be of interest to note that thevirtual inertia factor for a rectangular boxwith fairly good approximation within theusual range of Bld may be given by(B/d + 1.2) which lies well above thestraight line for actual ships.

For a homogeneous rod the coefficientç in Schlick's formula (1) is 140,000. Thisshould be unaltered when the rod is partlyimmersed when replacing by , atleast for fairly long rods for which it shouldbe sufficiently correct to assume the addedvirtual mass to be evenly distributed overover the length. Now the equation ofthe straight line drawn in Fig. 3 isN = I l0,000x + 29, when x is the abscissa.The line does not pass through the originand will be cut by the line for the homo-geneous rod at about N = 132.

Also in this figure it seems natural todraw a separate line for the tankers, e.g.through a point midway between 2 and 3and another point midway between 7and 8. This line will lie entirely above andparallel to the line for the homogeneousrod N = 140,000x.

If the solid rod is of rectangular shape,it has I = BD2/12, which means that it hasa coefficient of equation (5) of = 140,000/-V'12 = 40,400. The corresponding line inFig. 5 lles a good distance to the right ofall the points from actual ships. Thestraight lines drawn by Dr. Todd seem tohave the equations : N= 46,500x + 27for passenger and cargo ships, N = 52,200x+ 288 for tankers, where x is now theabscissa of Fig. 5.

Oil tankers having a moment of inertiaof the midship section divided by LBDtsome 10 to 15% larger than other shipswithout superstructures and some 20 to25 °/ larger than ships with super-structures*), it is quite natural that theyshould fall outside the line suitable forother ships in Fig. 5. Supposing that theclassification societies require the samemoment of inertia for transverse framingas for longitudinal framing, the type offraming should have no influence unlessthe vessel is so heavily loaded that somepart of the structure has been stressedbeyond the elastic limit.

The critical stress for deck plates incompression is undoubtedly lower fortransversely framed ships than for longi-tudinally framed ones. But it would riotbe reasonable to expect that the critical

aSee Prohaska: "Vibralions verticales du navires."Ass. Techn. Mar. et Aeronaurique, 1947, Fig. 12.

stress under normal circumstances hasbeen reached in any part of the platingwhen the natural frequency is beingmeasured in port.

There is, however, a fundamentalsimilarity between the calculation ofnatural frequencies of vibration and thecalculation of critical stress in elasticstability. In both problems similar sets ofhomogeneous differential equations appear.In such a complex structure as a ship thehigher modes of vibration must have someinfluence on the two-mode vibration justas in the elastic stability problem termswhich describe a higher number of half-waves influence also the magnitude of thelower critical load through the requirementthat the factor determinant must be zero.The influence may not be large, buttheoretically it should be there. This leadsto the conclusion that the number of bulk-heads may have some influence on thefrequency, in a similar way to the influenceof lattices on the critical load of a latticedstrut. The influence of the bulkhead ismuch less, however, because the hull properis rather strong with only few bulkheads.Nevertheless the large number of bulkheadsin oil tankers may have some influence ontheir frequency, say the difference noted inFig. 3 between tankers and other ships.

It is astonishing that the straight linesin Fig. 5 fit as well as they do. Theabscissa of this figure has been arrived atby substituting I in the abscissa of Fig. 3by aBD3, which may be correct for a solidrectangle, but not for a ship. The momentof inertia of the midship section should besubstituted by the value used by the classi-fication societies. This should be nearlyproportional to the bending moment,which again is proportional to the dis-placement by the length. If the distancefrom the neutral axis to the upper deck isconsidered to be proportional to D Itmight be worth while trying the abscissa±1 DL 'V l2+B/drection due to the permissible stressincreasing with the length of ship.

The curves given by professor Prohaska(loe. cit.) show that the midship momentof inertia I divided by LBD2 has a minimumat a ship's length between 350 and 400 feetand rises considerably for smaller lengths.Ships with lengths below 300 feet built toLloyd's rules must, therefore, be relativelymuch stronger than ships with lengths over400 feet. lt would, therefore, not hesurprising if the frequency of small shipsshould fall outside the curve of frequenciesfor larger ones when tried against theabscissa suggested above.

with an eventual cor-

D124 SHIP VIBRATION

Reply to Sir Wescolt Abel!Sir Wescott Abell in his remarks has

illustrated the very great effect whichloading can have on the natural frequencyof a vessel's hull. Calculations done on atanker and published in an earlier paper3have shown that assuming the load to beequally distributed along the length thereis a difference in calculated frequency ofover 20 per cent. from that calculated usingthe correct curve of load distribution. ltis interesting to hear that Sir Wescott'spractical experience bears out the calcu-lations in this respect.

Reply to Mr. HunterThe Authors are glad to find that Mr.

Hunter is in agreement with them on theimportance of making as correct a calcu-lation as possible of natural hull frequenciesand on the dangers of the use of empiricalformulae. It is, of course, true that anyhull vibration must in the first instance beexcited by some out-of-balance force, eitherin main engines, propellers, or auxiliarymachinery. From our experience the maincause of a ship's excitation has generallybeen the main engine. In the two turbine-driven ships to which Mr. Hunter callsspecial attention, the frequencies measuredindicate that the out-of-balance force is inthe main propelling machinery and ofone-per-revolution frequency. This may,of course, be due to an out-of-balance forceeither in the main engines, shafting. orpropellers. In the latter event it meansthat either the centre of gravity of thepropeller is not on the axis, or one of theblades is badly out of pitch. In the caseof vessel No. 16, which was turbo-electric,most exhaustive tests were done in dockwith each auxiliary machine at a time, andit was proved beyond doubt that the excitingforce was either in the main motors,shafting or propellers. These forces canonly be dealt with from the marine engineer-ing side by keeping them to the veryminimum possible, but there remains thosehydrodynamic forces resulting from therotation of the propeller in the disturbedwake. Some disturbing force must beaccepted here, but every endeavour shouldbe made to reduce its magnitude by carefulattention to such matters as clearancebetween propeller blade and hull or fin andto the shaping of shaft webs, bossings,A-brackets and the endings of the water-lines at the after end, in single-screw ships.

Repl.v to Prof BurrillProfessor Burrill has given a very lucid

summary of the present position of ourknowledge regarding ship vibration. As

one of the Authors* was responsible forintroducing the use of a formula involvingthe BD3 to which Professor Burnill takessome exception, we should like to make itquite clear that we did not intend that itshould be preferred to the use of oneinvolving the correct value of the momentof inertia 1. It was more on account toprovide the practitioner with an approxi-mate formula which he could use when theinformation necessary for the more detailedcalculation of J was not available to him.We ourselves have found it quite useful onmany occasions and cannot believe that thedifference between 77 and 117 per minutequoted by Professor Burrill can have beendue to the reasonable use of this formula.We are quite sure some other factor musthave been present to account for this bigdifference.

It is very gratifying to note that experi-ments are to be carried out at King'sCollege to investigate the effect of entrainedwater, and we shall await the results of theB.S.R.A. work with great interest.

It has been the Authors' custom to takerecords at the stern rather than at the bowor midships. At midships it is quitepossible to miss certain of the criticalfrequencies since, for example, a three-nodevibration would give no amplitude at thatpoint. Admittedly the stern is subject tocertain local vibration due to propeller andpossibly auxiliary machinery, but it has theadvantage of being in general easier ofaccess to the engine-room for communica-tion purposes, and it has also been found onmore than one occasion that the amplitudeat the fore end of the ship is less than wouldbe expected from the general trend of thevibration profile.

Reply lo Mr. YatesCurves showing the relation between

amplitude and frequency over the resonantrange have been published for ships fromtime to time, and a sample may be foundin the literature on the subject, in particularNos. i and 5 of the Bibliography. Theseshould enable Mr. Yates to make someestimates of the degree of damping.

Reply to Mr. CarterMr. Carter's contribution to the dis-

cussion is extremely useful and gratifying,since he gives the results of another eightships which can be added to the dataalready given in the paper. It will be found

Dr. Todd

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AUTHORS' REPLY

L) 126 SHIP VIBRATION

on comparing these with Fig. 5 that theobserved frequencies fall very closely alongthe lines there drawn. There are, of course,many ways of attempting to allow for thedifferent length of superstructure by someempirical factor, and an obvious alternativeto assuming an equivalent depth is to alterthe length of the erections, since theirstrength effect will taper off gradually intothe main strength of the hull. TheAuthors will examine Mr. Carter's suggestedmethods of using different length Lc and atthe same time they are grateful to him forpointing Out that ship No. 17 had anexpansion joint fitted into the depth, whichmeans in effect that the spot in Fig. 5which shows the results of including thegames deck should not have been included,and it will be seen that taking the equivalentdepth only to the lounge deck it is in factin very good agreement with the main line.

Reply to Dr. ConnUndoubtedly the most satisfactory way

of obtaining really reliable vibration datais to use an exciter on the ship, as Dr.Conn suggests. This is, however, a costlyand a time-consuming process, and theAuthors feel that while this may be donein certain chosen cases with a resultantsteady accumulation of information, therewill still for a considerable time be a placefor those observers who are able to attendship trials with recording instruments withthe hope of being able to pick up criticalfrequencies from time to time. We wouldagree with Dr. Conn on the importance ofthe shear correction in all vibration orstrength calculations. In the former itsinclusion in the calculations may alter thefrequency by as much as 10 per cent, andit would certainly seem advisable that itshould be included also in strength anddeflection calculations.

Reply to Mr. GawnMr. Gawn has drawn attention to what

at first sight appears to be satisfactoryagreement between measured and predictedamplitudes. The Authors must, however,disclaim any such credit and apologize toMr. Gawn if he has been misled by theresults shown in Fig. IB. The measuredprofiles shown there are, of course, correctand are measured on actual ships. Inmaking the calculations, however, anarbitrary scale is assumed for the accelera-tion curve, and no attempt has been madeto calculate the actual amplitude. In orderto compare the shape of calculated andobserved profiles, however, the calculatedprofile has been superposed upon themeasured one in such a way as to makeamplitudes amidships agree with that

actually measured. The comparison shownin the figure, therefore, extends no furtherthan to the shape of the curve. Theamplitudes measured on these ships arequite normal, and on no less than threeoccasions amplitudes of over one inch havebeen measured'. The Authors' experiencesuggests that an amplitude of '05 inchesor more is sufficient to attract generalattention on a ship, but to anyone interestedin this question of vibration an amplitudeof '02 can quite definitely be felt withoutthe use of any instruments.

Reply to Mr. MeA listerThe Authors are very glad to have Mr.

McAlister's endorsement of their suggesteduse of an equivalent depth in the empiricalformula which they have suggested in thepaper. They also look forward to applyinghis own formula to the results given in thepaper in order to determine the appropriatevalues of K. This formula is, of course,an extremely useful one, since it involvesnothing more than the length of the shipand a single empirical factor varying withtype of ship and which can in fact be easilymemorized.

Reply to Mr. ManleyFrom time to time vibration exciters

have been used on board ship in order todetermine critical frequencies. Such ex-citers, however, are very heavy and needelaborate electrical control gear in order toensure not only a fine graduation in speedbut also absolute constancy at any givenspeed. Their use necessitates devoting theship to vibration measurements for somedays while the exciter is fitted and after-wards removed from the ship. This iscostly and needs the resources of some largeorganization behind it.

Vibration tests have been carried Out onstructural modelsas long ago as 1908,Professor J. B. Henderson carried out suchtests with models of the Lusitania andPathfinder. These models consisted of flatbars of steel of constant thickness, thewidth being proportional to the moment ofinertìa of the cross-section of the ship atany point. The beam was loaded withstrips of lead to represent the distributionof weight without contributing to the stiff-ness of the bow. In these experiments,Henderson neglected the effect of thesurrounding water, and concluded from hisexperiments that the value of E for thesteel of the ship's structure must be takenas 10,000 tons per square inch rather thanthe test-piece value of 13,000. Morerecent tests have been carried out on actualmetal models of ships at the David W.Taylor Basin in Washington.

In analysing the records the envelopemethod has been generally used and was infact illustrated in paper' as long ago as1931. The reduction in amplitude at theextreme end of a ship to which Mr. Manleydraws attention has been observed severaltimes and is obvious not only in No. 16of this paper but also in No. 25.

While one might in theory agree withMr. Manley's objections to the use of thewords "moment of inertia ", this hasbecome so much a part of the literature ofnaval architecture that we doubt whether itwould be wise to change it now.

Reply fo Mr. MurrayIt is interesting to have Mr. Murray's

confirmation of the usefulness of Fig. 5 asshown by the agreement between the resultsfor the two ships which he has stated. Insearching for a suitable parameter on whichto plot the results of these vibration tests,the Authors had in mind the possibility ofvariation in moment of inertia with LID,and indeed it was one of the methods ofplotting which was tried among many othersbefore adopting that used in Fig. 5. Withmore knowledge of the subject, it may bepossible in the future to take account ofthis variation by adding another term tothe formula in equation (5).

Mr. Murray's statement that vibration isnot a sign of weakness in a ship drawsattention to an aspect of the subject whichis still rather widespread, particularlyamong owners. To alter the frequency ofa ship's hull by say 5 per cent. to remove acondition of resonance would mean ineffect adding some 10 per cent. to themoment of inertia of the cross-section,which is almost impossible in a completedship, and, therefore, this method is not onewhich is generally available in dealing withsuch problems.

Reply to Prof. ProhaskaProfessor Prohaska's remarks form an

exceedingly valuable addition to the paper.The Authors were aware of his paperbefore the Association Technique Maritimeand Aeronautique but when this came intotheir hands the calculations for the secondpaper were all complete and it was notpossible in the time available to make anycomparison with those which might havebeen found using Professor Prohaska'smethod. Detailed calculations involvingthe use of an inertia curve along the lengthof the hull would probably give betteragreement if the actual inertia distributionwere correctly known rather than by theuse of a constant value equal to that

amidships, but it is in fact not possible touse a correct inertia curve because of theeffect of the end of the superstructuredecks. It is obvious in fact that the inertiaof the decks must be graded up into theinertia of the main hull at their ends, butany such grading would have to be anapproximation made without true know-ledge. On the other hand we have foundthat using the amidship inertia gives resultsvery close to those found from the ship,and certainly have no knowledge ofdifferences as large as the 20 per cent.quoted by Professor Prohaska.

The same reasoning applies to the secondpoint raised in this discussion, namely theeffect of the top decks in vibration. In theAuthors' experience even quite light decksplay a considerable part in the strength ofthe hull structure and certainly we believethat insufficient is known at present toenable any reduction of strength to be madeother than by the use of some overallempirical factor such as is included inequation (5).

Professor Prohaska's remarks on theshear correction are most interesting andthe Authors hope to make use of hiswork in future papers on vibration. He isright in his assumption that we usedTaylor's original data in making thecalculations of the present paper.

As regards the allowance for entrainedwater, the ship results were always obtainedin relatively deep water, and no allowancewas considered necessary for any deptheffect.

Professor Prohaska states that the virtualinertia factor increases rapidly when depthdecreases, and this is what one wouldexpect. On the other hand, certain in-formation which we have obtained recentlyon a large ship in varying depths of waterin a river estuary have not shown any sucheffect, although the tests were designedspecifically to find it. This result was verysurprising and the experiments have beenrepeated on a sister vessel with much thesame result.

The frequency of 112 per minute forvessel 15 was obtained in deep water.

In making the calculations, the originalassumed vibration profile was always thatfor a uniform beam, but the calculationswere repeated as necessary, in each caseusing the derived profile as the new assumedprofile until very close agreement wasobtained between the two of them.

Reply to Mr. RalstonThere is no fundamental reason why the

mean lines through the spot need bestraight since the parameter on which they

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nl 28 SHIP VIBRATION

are plotted cannot be expected to takeaccount of every factor which influencesthe hull frequency. On the other hand,there is perhaps insufficient evidence inFig. 5 to justify more than a straight lineat present. As more data become availableit may well be that some curves will befound to fit them more closely than thelines at present drawn.

Reply fo Dr. TaylorThe Authors agree that further work is

required on the effect of a free watersurface upon the virtual mass before the

accuracy of calculation for horizontalvibration can approach that for vertical.

Reply ro Dr. VedelerIt was very interesting to have Dr.

Vedeler's comparisons between our resultsand those for simple prismatic bodies.These seem in a general way to bear outwhat we have found for the different typesof ship.

His suggestion that small ships arerelatively much stronger than large ones isin fact borne out by experience withvibration and the small ship is a notoriouslydifficult one to deal with.