The dynamical characteristics of some dry hulls

Journal of Sound and Vibration (1977) 54(l),

THE DYNAMICAL

29-38

CHARACTERISTICS

OF SOME DRY HULLS

R. E. D. BISHOP,W. G. PRICE AND P. K. Y. TAM Department of Mechanical Engineering, University College London, London WC 1E 7JE, England (Received 10 January 1977)

With a destroyer, a 250 000 DWT tanker in ballast and the same tanker fully laden used as examples, data are quoted for the dynamical characteristics of dry hulls. By use of the principal co-ordinates of the dry hulls, the properties quoted are converted to generalized form in the Lagrangian sense. 1. INTRODUCTION

A ship presumably responds to wave excitation in “modes” and the prospect of analysis in terms of wet modes was pointed out some time ago [l-3]. But since the vessel in water is essentially non-conservative, it is by no means obvious what the form and properties of those modes are. In a number of recent papers, two of the writers have discussed the responses of ships to waves-both bodily motions and distortions-in terms of the mechanical properties of the dry hull [4-61. By using these properties it is possible to avoid many of the mathematical difficulties that are introduced by hydrodynamic actions. Unfortunately the mechanical characteristics of the dry ship cannot be measured directly and they are therefore unfamiliar. The writers know of no literature in which they are cited in any systematic fashion. The modes and frequencies of a dry free-free hull can be calculated with fair ease and precision, however, and one can at least make a shot at estimating damping properties [7]. Accordingly this paper seeks to make good the deficiency in the literature. With a destroyer and a large tanker, the latter both in ballast and fully loaded, taken as examples, results are given for natural frequencies, principal modes and modal damping factors. Further, with a convenient modal scaling, the generalized stiffnesses, generalized damping forces and generalized masses are quoted, corresponding to the principal coordinates. Although the mathematical apparatus now exists [8, 91 at least to make a start on antisymmetric responses, the necessary data do not appear to be available. (Indeed some difficult4 was experienced in finding suitable data for the symmetric case.) This paper therefore deals only with the characteristics of symmetric responses. The results are quoted for modes up to and including that of order 7. This is admitted11 neither strictly justified nor necessary. If one were concerned only with response to waves., the 2-node mode would be of overwhelming importance, while anything above the 4- or 5-node mode is likely to be of limited interest. If, on the other hand, propeller excitation is of the essence, the higher modes would be important; but they might well demand a more sophisticated representation of the hull than that of a straightforward Timoshenko beam. Furthermore, the destroyer computations were carried out by using a 20-mass ProhlMyklestad representation so that a 7-node mode places much too great a strain on the idealization employed. (Fifty masses were used for the tanker.) Nevertheless it is instructive to quote results for the range O-7 of modal index, if only to illustrate the way things can go with the higher order modes. 29

30

R. E. D. BISHOP, W. G. PRICE AND P. K. Y. TAM

2. THE SHIPS Figure 1 refers to a destroyer. The outline of the hull is shown in Figure l(a), the weight distribution &x)g is shown in Figure l(b) and Figure l(c) gives the variation of second moment of area, Z(x). Figure l(d) represents the shear area U(x) and Figure l(e) gives the rotary inertia per unit length, Z,,(x). The vessel is of 111.25 m length overall, displacement = 25.1 MN, breadth moulded = 5.79 m and mean draught 3.52 m.

0

I

I

/

I

25

50

75

100

I25

x(m)

Figure 1. (a) Outline of destroyer hull; (b) variation of weight/unit length p(x)g; (c) second moment of area Z(x); (d) shear area kA(x); (e) rotary inertia/unit length Z,(x).

Figure 2 relates to a 250 000 DWT tanker in ballast. The various curves correspond to those of Figure 1, though that of Figure 2(e) had to be estimated by the rule of thumb Z,,(x) + pZ(x), where p, the density of steel, has been taken as 7850 kg/m3. For this vessel, the length overall is 348 metres and the displacement in ballast is 764 MN. Figure 3 refers to the same tanker, fully loaded. Again the rotary inertia/unit length has been estimated in Figure 3(e). Before proceeding it is profitable to consider the outlines of the hulls in Figures l(a), 2(a) and 3(a). They are prismatic, moderately “thin” structures, and it is by no means unreasonable to idealize them as Timoshenko beams. Figure 4 shows a number of types of vessel, drawn to different scales so as to have the same length in the diagram. It will readily be seen that the large tanker of cathedral-like dimensions is, paradoxically, the thinnest beam.

DYNAMICAL

0

PROPERTIES

50

100

OF SOME DRY HULLS

200

150

250

300

31

350

xim) Figure

2. Data for the tanker in ballast.

3. NATURAL

FREQUENCIES

If a hull is represented as a Timoshenko beam, there are four levels of accuracy that can be adopted in calculations. They are set out as Cases I to IV in Table 1. Case I is that of the Bernoulli-Euler beam and we shall discover that it is somewhat crude and should only be used in practice when lack of information makes this inescapable. We shall find that Case III has little to commend it. There is very little difference between the results obtained in Cases II and IV and, since the former requires much less data than the latter, Case II is the most useful in practice. There are many ways in which natural frequencies may be calculated [lo], but perhaps the simplest is that of the Prohl-Myklestad technique [l 1, 121. Table 2 shows the natural frequencies of the destroyer, up to and including that of the 7-node mode. The “best” results are those of Case IV and, since solid data were available to permit them to be made (i.e., I,(x) did not have to be estimated-though it does seem to be open to question), this is the Case we shall adopt henceforth when referring to the destroyer. For obvious reasons Cases I and III may be discarded as too inaccurate. We shall therefore refer only to Cases II and IV for the tanker; as we shall discover, there is little difference between them and inaccuracies in the estimation of rotary inertia are really neither here nor there. Table 3 gives the appropriate results for the tanker and, roughly speaking, they can be summed up (along with those of the destroyer) as in Table 4.

R. E. D. BISHOP, W. G. PRICE AND P. K. Y. TAM

0

50

100

150

200

250

300

350

x(m) Figure 3. Data for the tanker loaded.

4. PRINCIPAL

MODES

Almost inevitably the computation of natural frequency carries with it the calculation of a modal shape. There will be small differences between the modes found for each of the Cases in Table 1 but, generally speaking they are not large. The scale of a modal deflection may be fixed arbitrarily, and we have consistently used the convention that unit positive (upward) deflection (i.e., 1 metre) shall occur at the stern, where x = 0. Once a scale has been assigned to the mode, it becomes possible to determine corresponding curves of bending moment and shearing force. If the unit modal deflection in mode r is TABLE

1

Possible assumptions in the idealization of a hull as a beam

Case

Correction made for shear deflections

Correction made for rotary inertia

I II III IV

No Yes No Yes

No No Yes Yes

DYNAMICAL

PROPERTIES

33

OF SOME DRY HULLS

0

(a)

(bl

0

0 Cd;

I-J I

n

0 L--c72

I

I

“‘\

TT

u

L,

Figure 4. Outlines of ships. (a) Tanker (LOA 326 m); (b) bulk cargo carrier (LOA 201 m);(c) frigate (LOA 130 m); (d) tanker (LOA 224 m); (e) cargo ship (LOA 156 m); (f) pilot vessel (LOA 55 m). It will be seen that the largest vessel is the most closely represented by a thin beam.

/

___200 150 100 St I

50

T 3

0 -50 -100 -150 -200

20

40

60

80

100

I20

x(m)

Figure 5. Curves of principal modes w,(x), modal bending moment M,(x) and modal shearing force v+) for the destroyer hull (r = 2,3,4).

34

R. E. D. BISHOP, W. G. PRICE AND P.

K. Y.

TAM

TABLE 2 Calculated natural frequencies (in rad/s) for symmetric principal modes of the warship hull

Modal index r

w, (rad/s) in Case number *

I I

2

14.78 33.76 57.25 96.82 140.62 184.99

3 4 5 6 7

\

II

III

IV

13.65 27.43 40.32 58.38 74.63 86.99

14.66 33.25 56.05 93.55 133.63 173.58

13.57 27.22 40.05 58.05 7417 86.45

TABLE 3 Calculated natural frequencies (in rad/s) for symmetric principal modes of a dry tanker hull

Modal index r

Loaded condition wr (rad/s) in Case number A \ /IV (I,” 0) (Z, estimated)

Ballast condition w, (rad/s) in Case number (-A, (I,”

2 3 4 5 6 7

0)

(Z, estilated)

6.73 15.83 29.90 44.42 57.82 69.54

4.01 10.05 16.7’ 24.10 33.11 42.02

6.68 15.69 29.55 43.76 57.02 68.84

4.00 10.01 16.63 23.98 32.95 41.78

TABLE 4 Approximate

w,(x), them. The M,(x) results ballast

naturalfrequencies

in Hz

Modal index, r

2

3

4

5

6

7

Destroyer hull Tanker in ballast Tanker loaded

2.2 1.1 0.6

4.3 2.5 1.6

6.4 4.7 2.7

9.2 7.0 3.8

11.8 9.1 5.3

138 11.0 6.7

then these are M,(x) and V,(x), respectively,

and scales are automatically

assigned

to

curves shown in Figure 5 are a sequence of results for the destroyer. They show w,(x), and V,(x) for r = 2, 3 and 4. These curves are for Case IV. Figure 6 shows comparable for the tanker; the full line curves are for the loaded hull and the broken line is for the condition. 5. DAMPING

Damping cannot be estimated with much confidence. Assuming nevertheless, that crossdamping between the principal modes is negligible and that hydrodynamic damping of distortion modes can also be ignored we may “guess” the values of the damping factors (i.e.,

DYNAMICAL

35

PROPERTIES OF SOME DRY HULLS

/’ Im

I 3

2000 1500 1000 f I

500

_,

0

I:

-500 -1000 -1500 I 103

-2000'

1 200

I 300

I

x(m) Figure 6. Curves like those of Figure 5 for the tanker of Figures 2 and 3. The full lines refer to the loaded vessel and the broken line to the vessel in ballast.

TABLE 5

Representative selection of damping factors Modal index r

I Warship

Value of v, for A

\

Tanker in ballast

Tanker loaded

0.002 0.005 0.008 0.010

2 3 4 5 6

OGO6 0+09 o-014 0.020 0.025

0.004 0.007 O-010 0.015 0.024

7

0.030

0.030

0.012 0.016

the ratios v, = modal damping/critical modal damping) in line with the suggestion in rererence (71. The values given in Table 5 were arrived at in this way. t

contained

t The results for the destroyer are based on the assumption that the hull is all-welded. In actual fact this was not so, the ship concerned being an old one that is now scrapped. The hull was partly welded and partly rivetted.

36

R. E. D. BISHOP,

W. G. PRICE

AND P. K. Y. TAM

6. GENERALIZED

The generalized is given by

MASS

mass coefficient a,, corresponding

to the rth and sth principal

co-ordinates

w, w, + Z, 0, 0,) dx = a,, 6,s. 0

This result is fully explained elsewhere 1131 and we have merely to note that w,(x) is the rth scaled modal deflection and e,(x) is the corresponding slope due to bending. The quantities p(x) and Z,,(x) are respectively the mass/unit length and rotary inertia/unit length. The symbol a,, is the so-called Kronecker delta function which is such that

Orthogonality

of the principal

r

0,

4, =

#s,

r = s.

i 1,

modes ensures that the cross-terms

1

I 0

otw,

w,

+

4

6

&I

dx,

r #

s,

vanish and any reputable program for computing the modes will check that this is so. (These integrals may not be exactly zero by reason of the approximations introduced in the process of integration.) By way of example the computations for the destroyer give the results quoted in Table 6. 6

TABLE

Values of a,, for the warship (in tonne

Mode 0 1 2 3 4

m2)-Case

IV of Table 1

0

1

2

3

4

2559.7 -0.5 I.0 0.1 1.4

-0.5 629.7 0.3 0.0 0.3

I.0 0.3 338.2 -0.2 2.0

0.1 0.0 -0.2 247.6 0.3

1.4 0.3 2.0 0.3 248.7

. . . .

Having ascertained that the ars are all sensibly zero for r # s we may list the constants arr. Notice particularly that the values depend on the scaling of the modes. The results are given in Table 7. TABLE

7

Values of generalized

Modal index r 0 1 2 3 4 5 6 7

mass

Value of urr in tonne-m2 for h

r

Tanker in ballast Warship (4 # 0) 2560 630 338 248 249 518 592 713

r 1, = 0

Zyestimated

77 888 23 621 22 010 7868 3433 4834 2683 2616

77 888 23 621 22 251 8034 3541 4931 2792 2566

fl

Tanker loaded A I, = 0

284 699 53 174 20 091 11 101 8776 7163 6580 6889

\

\ I, estimated 284 699 53 174 20 170 11 191 8873 7254 6675 7003

DYNAMICALPROPERTIESOFSOMEDRY

7. THE GENERALIZED

37

HULLS

STIFFNESS

The generalized stiffness is c,,, where c,, = &urs, and so the cross-terms c,, (r # s) again vanish [ 131. We are left with c,, = o: arr and, when the values quoted in Tables 2, 3 and 7 are used, the values given in Table 8 are found. These too depend on the scaling of the modes. TABLET Values of generalized

I Modal index

Warship

r

(4 # 0)

0 1 2 3 4 5 6 7

0 0 62.2 184 399 1746 3257 5329

st@ness

Value of clr in MN m for h , Tanker in ballast Tanker loaded A \ cph, c Iy = 0 I, estimated IY= 0 Z, estimated 0 0 997 1972 3069 9536 8970 12 650

0 0 993 1978 3092 9443 9078 12 160

8. GENERALIZED

0 0 323 1121 2450 4160 7213 12 164

0 0 323 1121 2454 4171 7247 12 224

DAMPING

The modal damping factors given in Table 5 are independent of the modal scaling. But if they are converted to specify generalized damping, they do become dependent on the scaling. According to elementary vibration theory [14] the generalized damping coefficient is b,, = 2a,,w,v,. We are thus in a position to calculate these coefficients from the data contained in Tables 2, 3, 5 and 7. The results are given in Table 9. TABLET Values of generalized

damping

Value of b,, in tonne mz/s for Modal index r 0 1 2 3 4 5 6 7

Ip~-

\

Warship 0 0 55 122 279 1203 2195 3698

Tanker in ballast (1, = 0) 0 0 1185 1744 2053 6442 7446 10 915

Tanker loaded (1, = 0) 0 0 322 1116 2346 3453 5229 9263

9. CONCLUSIONS The main purpose of this paper is to give some idea of the orders of magnitude of mechanical constants that are likely to be met in the analysis of dry hulls. The ships for which results are quoted are a destroyer and a large tanker, the latter being in ballast and fully loaded.

38

R. E. D. BISHOP, W. G. PRICE AND P. K. Y. TAM

It is essential to note that the observable characteristics of a ship (i.e., those of a wet hull) may be very different from those of the dry vessel (i.e., those of the ship in vacua without supports). It simply will not do, generally, to regard the wet ship as having approximately the same dynamical characteristics as the dry one. It will be appreciated that the data given in this paper have had to be calculated by means of a computer. The programs that have been used were specially written for the purpose and a substantial effort has been put into making them both versatile and economical. These programs are part ofa comprehensive “suite” which will eventually handle the hydrodynamics side, as well as the structural. This has meant that the needs of the one have had to be borne in mind when a program for the other has been written. In other words, the data presented here represent a small fraction of those that can be quoted, and indeed are needed in the general analysis of ship response. The object has thus been only the limited one of trying to impart a “sense of feel”. It is not suggested that results of the type quoted could be used for excitation of greater frequency than that of importance in wave spectra. Specifically, an accurate analysis of propeller-excited vibration would require more refined techniques than those relating to Timoshenko beams. It is the writers’ belief that the greatest obstacle to progress up the frequency scale lies in the present state of “shear diffusion” (or “shear lag”) theory; it does not appear to be possible to adapt that theory with any confidence to the dynamics of a ship hull.

ACKNOWLEDGMENTS The writers wish to acknowledge the financial support of the MOD(PE) in this work. They would also gratefully mention their indebtedness to members of the Ship Department. of Yarrow Shipbuilders Limited and of BSRA for meeting requests for information. REFERENCES 1. R. E. D. BISHOP 1971 South Afvican Mechanical Engineer 21, 2-17. The John Orr Memorial Lecture: On the strength of large ships in heavy seas. 2. R. E. D. BISHOP, R. EATOCK TAYLORand K. L. JACKSON 1973 Transactions of the Royal Institution of Naval Architects 115,257-274. On the structural dynamics of ship hulls in waves. 3. R. E. D. BISHOP and R. EATOCK TAYLOR 1973 Philosophical Transactions of the Royal Society A275, l-32. On wave-induced stress in a ship executing symmetric motions. 4. R. E. D. BISHOP and W. G. PRICE 1974 Proceedings of the Royal Society London A341, 121-134. On modal analysis of ship strength. 5. R. E. D. BISHOP and W. G. PRICE 1975 (April) The Naval Architect, 61-63. Ship strength as a problem of structural dynamics. 6. R. E. D. BISHOP and W. G. PRICE 1976JournalofSoundand Vibration45,157-164. On therelationship between “dry modes” and “wet modes” in the theory of ship response. 7. C. V. BETTS, R. E. D. BISHOP and W. G. PRICE 1976 Transactions of the Royal Institution of Naval Architects Paper W2. A survey of internal hull damping. 8. R. E. D. BISHOP and W. G. PRICE 1976 Proceedings of the Royal Society London A349, 157-167. Antisymmetric response of a box-like ship. 9. R. E. D. BISHOP and W. G. PRICE 1976 Proceedings of the Royal Society London A349,169-182. On the transverse strength of ships with large deck openings. 10. R. E D BISHOP, G. M. L. GLADWELL and S. MICHAELSON 1965 The Matrix Analysisof Vibration. Cambridge University Press. 11. N. 0. MYKLESTAD 1944JournalofAeronautica~Sciences 11,153-l 62.A new method of calculating natural modes of uncoupled bending vibration of airplane wings and other types of beams. 12. R. E. D. BISHOP 1956 The Engineer 202, 838-840 and 874-875. Myklestad’s method for nonuniform vibrating beams. 13. R. E. D. BISHOP and W. G. PRICE 1976 Journaloj Soundand Vibration 47,303-31 I. Allowance for shear distortion and rotatory inertia of ship hulls. 14. R. E. D. BISHOP and D. C. JOHNSON 1960 The Mechanics of Vibration. Cambridge University Press.