Numerical calculations of vertical hull vibrations of ships by discretizing the vibration system

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

Netherlands' Research Centre T.N.O. for Shipbuilding and Navigation

SHIPBUILDING DEPARTMENT MEKELWEG 2, DELFT

*

NUMERICAL CALCULATION OF VERTICAL HULL

VIBRATIONS OF SHIPS BY DISCRETIZING

THE VIBRATION SYSTEM

(NUMERIEKE BEREKENING VAN VERTICALE SCHEEPSTRILLINGEN DOOR DISCRETISERING VAN HET TRILLINGSSYSTEEM)

by

J. DE VRIES

Mathematician Institute T.N.O. for Mechanical Constructions

Issued by the Council

This report is not to be published unless verbatim and unabridged

page

Summary 3

i Introduction 3

2 Discretizing 3

3 Mathematical formulation of the vibration problem 6

4 Approximative correction for the influence of rotary inertia 8

5 Approximative correction for the influence of small

differ-ences in mass distribution 8

6 Test problems 9 7 Conclusion 14 8 Cost of calculations 14 References 14 Appendix I 15 Appendix II 16

NUMERICAL CALCULATION OF VERTICAL HULL VIBRATIONS OF SHIPS

BY "DISCRETIZING" THE VIBRATION SYSTEM

by

J. DE \TRIES

Summary

When the ship may be considered as an elementary beam, many methods exist for calculating the natural frequencies from a known mass- and stiffness distribution. One of these is elucidated in this paper. The method is well-suited for calculating more than one natural frequency of the ship. Shear stiffness and rotary enertia can be taken into account.

The presentation in this paper is entirely tuned to the use of an electronic digital computer with a modern programme library. The cost of a calculation with such a machine is given; it proves to be remarkably low.

The accuracy of the method is very satisfactory. It is illustrated in three test problems. Part of the mathematical treatment may be new.

i Introduction

The discrctizing of continuous systems in order to simplify numerical calculations is not new, see for instance [1] and [7]. In combination with the

itera-tion method (same references), it proved to be a

successful method for calculating hull vibrations of

ships by means of desk calculators. However, if more than one natural frequency had to be

cal-culated, the work involved was rather tedious.

Because of its versatility and the satisfactory

results obtained with this method, it was

consid-ered worth-while to program it on an electronic

digital computer. For reasons of time consumption,

the iteration method, however, seemed not well suited to such a machine. Therefore, it was

re-placed by the versatile, economical and accurate

Givens-Housholder method for symmetric matrices.

With free-free vibration systems equilibrium

equations for the system as a whole had to be

introduced. These gave rise to more complicated

matrices than those which go with rigidly

sup-ported systems. However, these complications

could gracefully be solved by the application of

simple products (dyade products).

For the application of matrix-methods, the com-putation of influence numbers is necessary. These computations, which are tedious but not difficult, can be programmed easily.

In deriving the discretized items, a number of

integrals must be computed. These computations

can be performed by known numerical methods

that can be easily programmed.

It is supposed throughout the paper that mass and stiffness distribution of the ship are known,

shear stiffness and rotary inertia included.

2 Discretizing

The ship is supposed to behave as an elementary

beam. This beam is divided into an arbitrary

number of parts of arbitrary lengths. These parts

will be called segments.

The continuous beam will be discretized in two respects: mass and stiffness. [I]

Firstly each of the segments is replaced by one with

discretized mass distribution but with unchanged

continuous stiffness distribution. The deformations

of a new and an old segment will be equal if

func-tions D and M are equal over both segments.

These functions themselves depend on the inertia load on the segment, i.e. on functions myw2 and

t&co2 (see figures 1 and 2). Value w is the circular

frequency of the vibrations.

D c

y

mass segment with length i

= rotation of a cross section D = shear force y = displacement of a cross section M = bending moment

Accordingly, strict equivalence of a new and an

old segment exists if, independently of functions

y and q, the inertia loads on the new and the old

segment give rise to equal functions ZD

and M

for equal functions y and p.

The advantage of simplicity of the discretized system must be paid for with a less strict

equiv-alence. The first price is that equality of¿D and

Mwill only be required at the end of the segment.

Therefore, equivalence is considered to exist i1

independently of functions y and q, equations (1) are satisfied (see also figure 2).

f my dx =

o

/ myx dx = mjyjxj

Old mass segment New mass segment

m1 beam

wthout

m2.t2 mass

segment

m = mass per unit length of the beam

T = rotary inertia per unit length of the beam

rn = discretized mass

T, discretized rotary inertia

Figure 2. Mass-segment

The configuration of m1 and ri introduced

figure 2 is convenient, but not essential.

The second price is that the segment is to be sup-posed so short that with sufficient accuracy may be written:

y = ao+aiX

and

_=ßo

Because equations (1) must be satisfied independ-ently of numbers ao, a1 and ßo, it follows that:

(1) 13

fm dx = fli

¡mx dx = m2 +m31 fmx2dx = 12 m312

dx =

One is free to choose two of the quantities ri ar-bitrarily. A suitable choice, which reduces the

number of degrees of freedom of the segment by

two, is

(2b)

In order to save calculating-time on the computer, the above mentioned expressions may be inverted in formula.

The discretizing of the stiffness runs much the

same as that of the mass. If one replaces a segment

with continuous stiffness by one with discretized

stiffness without changing its mass distribution,

strict equivalence exists if, independently of

func-tionsD and M, the new and the old segment have equal functions \y and Lp (see figure 3) for equal

functions D and M.

stiffness segment

Figure 3. Part of stifIhesssegment

If, however, the mass has been discretized first

and the stiffness segments are chosen exactly

be-tween two successive discretized masses, according to figure 4, no inertia loads will occur in a stiffness

segment, so that D and M have the form:

D =

M = vo+l/ix

Figure 4. Proposed choice of stiffness-segments

(2a) .4 stiffness segment

I

stiffness segment mass segment

In that case strict equivalence exists, if,

independ-ently of numbers yo and yi, the new and the old

stiffness segment have equal values of y and ¿q at the end of the segment. From this follows (see also figure 5): 112 I. i

1dx =

1/2 1/2

/dx

= ql+q3

From these equations, quantities K1 and qi can be calculated. One is free to choose one of the quan-tities qi arbitrarily, e.g.

qi = q3 (3h) Ki+K2+K3

K2-+

K3--mass. rotary inertia O bending joint 1 shear joint

Figure 6. Discretied system

stiffness segment

deformation by bending and shear

Old stiffness segment

stiffness segment

New stiffness segment

nondeformed

beam

Here also the configuration of K1 and q, introduced

in figure 5, is convenient but not essential.

The resulting discretized system has the appearance

shown in figure 6. It will be clear that where two successive mass segments touch, the discretized masses may be united.

mass segment

1/2 (3a) EI = bending stiffness -o- K1 = bending joint

1x2

Jdx =

12

K2-+

K3--12 GS =shear stiffness q1 = shear joint

Fig. 5 Stiffness-segment

coinciding coinciding

at one point at one point

It may happen that the described discretizing

method gives small negative values of mt at the

ends of the ship. This is closely connected with the

shape of the mass curve of an extreme segment.

A concave curve yields negative values. In that case one must apply at the ends a less sophisticated

dis-cretizing method.

The discretized system is conceivable from the viewpoint of mechanics. Therefore all the laws of mechanics will hold for it.

One can easily think of other methods for dis-cretizing a continuous system, which are not

nec-essarily less suitable. In comparing different

meth-ods, the criterion must be the overall cost of the calculations necessary to obtain a given number

of natural frequencies with a given accuracy.

How-ever, no attempt has been made for such a com-parison. Firstly because with the described dis-cretizing method fair allowance has been made for the pertaining properties of the continuous

system. Accordingly, it proved to give satisfactory

results in the test problems. Secondly because one may doubt whether the gain in cost of calculations

mass se.ment mass se.merit

Figure 7. Indices when the beam is simply supported at the ends

L X2

-xl

xn+1xi

m3 m4

s''

xn

For ease of calculation, the lengths of the mass segments are preferably chosen equal.

By Maxwell's theorem of reciprocity:

a5 = ait

The equation that governs the vibrations of the

free-free beam now reads (see also figure 9)

Xj

i

.Yn+1 =_>m1w2,jatI xn+1 (4a)

I!

I

s

Xr,.1Xi Xi xn1 0 + Ynl

No

mr..2

s

will be sufficient to warrant the expense of testing cheaper discretizing methods.

3 Mathematical formulation of the vibration

problem

It is well-known that the influence of rotary inertia

is small. In order to save calculating-time on the

computor, the natural frequencies are therefore

calculated without considering the rotary inertia. Instead, a correction for its influence is introduced afterwards. The indices introduced when

discretiz-ing are now obsolete. Instead, new indices are

introduced here as indicated in figure 7.

First of all, influence numbers are calculated for the case that the beam is simply supported at

masses mo and m+i. Supports at the ends give a convenient notation. In connection with the

ac-curacy of the numerical calculations, however, it

is recommended not to take the supports at the ends because of the often very small stiffness of

these ends. The derived formulas will hold, if one takes the indices as indicated in figure 8.

m1 w2y

m2

m,1 m1 m

s

s

s

Figure 8. Indices when the beam is simply supported at one mass-segment from the ends

yn+l

Xn

j=1

( = 1, 2, 3, ... n)

Figure 9. Vibration of a free-free beam, with discretized mass distribution mn,l m

s

x X4

with the equilibrium conditions: n-l-mjwy5 = O i = m5w2y5x5 = O i = t) From (4b): yo = _L.jmo

_X-j

¡=1 Yn+1 =

Substitution in (4a) gives:

n ç

(x)+x) (x+ix5)

+

m moX2+i

+

5=1 XiX 'I

Y=

n V m5

Lm+i

xn+1 Xi mjw2yiaij (i = 1, 2, 3,

These are n homogeneous linear equations in y with n natural circular frequencies w = wk (k =

1, 2, 3, ... n). The trivial solutions where the non-deformed beam rotates or translates with w 0,

have disappeared from these equations. One can

check this by insertion. The reason of the dis-appearance lies in the division by w2, when

de-ducing (5a) from (4b). After substitution of: l'i

=

equation (5h) obtains symmetrical matrixes:

[A_9B]F=0

(6)

where

A = [aij \/mim5]

B = 1+

{mimi ç (xnixi) (x+ixj)

_mx2

+

XjX5

+

_m11+x2+

10

J = 0

1. Y1 Y2 (5h)

with

C = B-'AB

and

=

the equations can suitably be solved by the method

of Givens-Householder [3], [5]. Another possible shape is:

B1A_

11

7=0

(F)2

which, however, has the drawback that matrix

B_lA is not symmetrical.

Shapes

(w21_A_1B)? = O and

(w2I_ABAF)A V

= O

are not well-suited for those numerical methods

which give the higher eigenvalues with the highest accuracy.

For ease of calculation it is remarked that

A = DA0D with A0 = [ai5]

and D =

O \/rn2 L i/rn11 and that

B = I+UU'+WW'

ßi32+2yiv13+arii2+2ii =

a132+2y723+ß22+22 = 1

Yi32+ (2 + wji + ßj2) /3+ i)72 = where

ß= W,W

= U,W= W'U.

-with m - Xi mi X12 m Xn+i m+1 Xn+i X2/7fl2 X m2

Xn+i mO

_{and W=}

Xn+i m+

The author did not have a fast programme avail-able for solving equations (6). In the shape:

(c_-i)

=o

(7)

xn+i Xn x m11

Xflfi

m_

_Xn+i m11+1_

U' and W' are the transposeds of U and W.

The matrix B" can be calculated as follows. If

one writes

B' = I+1UU'+2WW'+213 [UW'+WU'],

it follows from B 2B'12 = B,

that numbers i, and d3 must satisfy the equa-tions:

i ₍₈₎

(4b)

As will be derived in Appendices I and II, a suit-able solution of (8) reads:

fi 1/I

=

+

+

VÇ2«ß)

a

+

y--a fi

+

Vç2aaß)2

_y2aß

_y2aß

i

)73= (yi+V)

with

- 2

(a+fi+2)_21/(1+a)(1+fi)_y2

y

_2+(aß)2

J If one writes: = I+61UU'+62W1V"+83[UW'+WU'] it follows from: B' 1B'12 J

that numbers El, 62 and 63 must satisfy the

equa-tions:

= O = O

(1 +ß2+y3)E2+(rì2+aa)r3+ì2 = O

(ßa+yr1l)E2+(1+y13+a)l)e3+ì]3 = O

It is readily verified that these equations are

satisfied by:

y(ì1i2i32)

1/3

63

- 1 +ai +ß2±2yi3+ (afiy2) (r1v22r/32)

rIl+ (yr/l+fi13)E3

1=

_l+a1+p)3

72+(yr,2+aÌ3)e3 62

₌

+ß112+r/3 & = CL) fi o

y2aß

+

y2aß

a O

+

(9) (IO)

4 Approximative correction for the influence

of rotary inertia

If it is supposed that the vibration profiles are not

influenced by the rotary inertia, a correction for

the influence on the natural frequencies can easily be calculated by Rayleigh's energy method. If w is a natural circular frequency of the beam

with-out rotary inertia and y the vibration profile be-longing to it, the corrected natural circular

fre-quency & follows from:

where 1j i rotation of rj in this vibration profile.

With a given natural circular frequency w the

values of j can be calculated from the known

vibration profile as follows (see also figure 10).

Figure 10. Shear-forces and bending-moments in a

discretized system

The shear force between m and mj+i is:

D = (L)2 E mjy1 (13)

i.0

Bending moment M, at m, can then be calculated with the recurrence formula:

= M1+D111/2 (14)

and

M0 = O

From figure 11 is easily derived:

yí+i i + 2D,q1

=

- MK1(M1+D11/4)

K2 . (l5a) Alternatively, when qo has been calculated with this expression, one can find the other values of q, using the recurrence formula:

=

+ (M + D11/4) K2 + M+1K13 . (1 5b)

DL q

Figure 11. Rotation in a discretized system

)

5 Approximative correction for the influence

of small differences in mass distríbution

n+I

Often it is necessary to know the approximative

influence of small differences in the

mass-distribut-i

/

(12)

ion. Formula (12) can easily be extended to the

computation of such corrections:

where:

i

=w

// n± I i=O ¡-=0

/ E ñbj2+ E r2+D+i,E

'n+l n+1 = (17a) / _{E mjyj2} i = o W = (L) n+l

E ñkyt+ E

1=0 ¿=0 n -f- t Em1y12

The difference between (16) and (17) will usually

be negligible.

6 Test problems

Three test problems have been solved in order to

ñyt

TÇ9t

myi2

where:

= different discretized mass.

A more correct approximation is obtained with the vibration profile:

= i +1' + I'X1,

=

ï + V,

where ¡i and r are calculated from the equilibrium

conditions: n1

Eñ=O and

i- o --1 n±l

+ E

= O. ¡=0 ¿=0 It follows that:

CDAE

and v=

== = A B - C2

As it turns out, the improved correction formula

reads:

i

(16)

(17b)

study the accuracy and time-consumption of the proposed calculation method.

The first and second test problem concerned a prismatic beam, with equal concentrated masses m added at the ends, as shown in figure 12.

m m

L

= mass per unit length EI -= bending stiffness

Figure 12. First and second test problem

Continuous system

If shear deformations are disregarded, the exact

natural frequencies are derived from:

tg + tgh +

₎

(_cotg + coth +

= O

where:

m

/AL/60 = dimensionless end-mass

= W

J/L4

_M

= dimensionless natural frequency.

The beam was divided in 10 mass-segments. After discretizing in accordance with Section 2,

the distribution of the discretized masses and

bend-ing joints was as shown in figure 13.

I

1.,6 2 f,4 1.5

The discretized mass-distribution of a ship will

often show very small masses at the ends. Because

the disappearance of such a mass lowers the num-ber of degrees of freedom of the system, it is con-ceivable that very small masses might impair the

numerical accuracy. In order to study this, the

first test problem was 6 = 0.999. The second

test problem was the plain prismatic bar, b = 0.

The natural frequencies of both discretized

systems were calculated with all four

sign-com-binations of Appendix I. The supports were chosen

0000000

ffn 0

000000

1424242

2 424241

L

Figure 13. First and second test problem

Discretized system

.

4 2 4 4

2_J

A = E mi

¡=0 = n+l

B= E ñixj2+ E Tj

¿=0 1=0

C= Eñ1x1

i=0 n+1_

f

E=

iyx+ E T(J'i

¡=0 i=0

CEBD

¡t =

A B - C2 = n+1 =

TABLE I. Results of first test problem (ò = -0.999)

Dimensionless frequency

TABLE Il. Results of second test problem ( = 0)

Dimensionless frequency x

at one mass segment from each end, as shown in

figure 8. The results of the lower 12 natural fre-quencies are given in tables I and II.

It appears that in both test problems the mutual

differences between the sign-combinations are far smaller than those between the exact solution on

the one hand and that of the sign-combinations on the other hand. From the numerical point of

view, the four sign-combinations must be

consid-ered as entirely independent, except for the

cal-culation of matrix A, vectors U and W and

num-bers a,

fi and y. It was easily checked that no

numerical fault was introduced in calculating the influence numbers, i.e. matrix A0. The remaining manipulations for calculating A, U, W, a, fi and y

are so simple that a fault of any importance can

2-node 3-node 4-node 5-node 6-node 7-node

exact 24.0812646 66.6732543 131.301476 218.079768 327.350964 459.396393 E 1 24.0810279 66.6682401 131.261887 217.882353 326.574761 456.688269 2 24.0810277 66.6682401 131.261884 217.882356 326.574743 456.688243 3 24.0810277 66.6682401 131.261887 217.882350 326.574750 456.688286 . c 4 24.0810277 66.6682401 131.261882 217.882362 326.574710 456.688226 - .

8-node 9-node 10-node Il-node 2-node 13-node

exact 614.397481 792.379510 993.178012 1216.44744 1461.71496 1728.45941

i 605.259777 759.214953 1215.65216 1344.39314 1565.98807 1819.64169

2 605.259644 759.214931 1215.65445 1344.38920 1565.99058 1819.64028

u 3 605.259462 759.215129 1215.65117 1344.39192 1565.98968 1819.64907

4 605.259777 759.214810 1215.65441 1344.39647 1565.99191 1819.64081

2-node 3-node 4-node 5-node 6-node 7-r ode

exact 22.3732854 61.6728226 120.903391 199.859448 298.555533 416.990784

1 22.3731759 61.6706866 120.887622 199.785526 298.284733 416.135365

2 22.3731757 61.6706866 120.887619 199.785521 298.284726 416.135378

3 22.3731758 61.6706863 120.887619 199.785521 298.284723 416.135427

c 4 22.3731758 61.6706863 120.887617 199.785515 298.284716 416.135378

8-nodn 9-node 10-node Il-node 12-node 13-node

exact 555.165247 713.078915 890.731797 1088.12388 1305.25517 1542.12569

.9 1 552.690849 706.020145 918.345344 964.291983 1328.86495 1539.32080

2 552.690826 706.019938 918.345968 964.292219 1328.86484 1539.32089

3 552.690798 706.020326 918.345538 964.292101 1328.86443 1539.32229

scarcely be introduced. Therefore, the mutual dif-ferences between the results of the four sign-com-binations may be considered as a measure for the numerical fault introduced by the restricted word-length of the computer in the entire computation.

The computer had a wordiength of 39 bits. O'

these, 9 bits were used to represent the exponent

in floating point arithmetic, and i to represent

the sign of the fraction. The fraction itself there-fore had an accuracy of 29 bits, that is just under

9 significant decimal digits.

Shear deformations were disregarded. They

would only have affected the influence-numbers.

With given influence-numbers, the entire

math-ematical course is independent of

deforma-tions. It is therefore to be expected that

shear-TABLE III. Physical faults introduced by the discretizing

Physical faults are expressed in percentages of exactly calculated natural frequencies. A positive sign means that the natural frequency of the discretized system is the higher one.

Table III contains the physical fault in percentages

of the exact natural frequency. It appears that this fault is often very small, even if the displacements of the continuous system are by far not linear over the length of a mass-segment, as was supposed in

Section 2. See for instance the 7-node vibration

/

/fr

o--.'

first test problem (ô-O.ÇÇ1)

- - second test problem (ôO) Figure 14. 10-node vibration

deformations would not have influenced the

nu-merical fault considerably.

The difference between the exact solution on

the one hand and that of the sign-combinations on the other hand must have been introduced by the

discretizing. This difference may be called the

physical fault, because it is connected with the fact

that instead of the natural frequencies of' the con-tinuous system the natural frequencies of another,

though related, system were calculated.

The physical fault is introduced entirely during

the discretizing of the mass. During the discretizing

of the stiffness no additional physical fault is

in-troduced. Accordingly it is to be expected that

shear deformations would not have influenced the

physical fault considerably.

in table III. Such an unexpected good accuracy must be credited to a, more or less accidental,

lucky placing of the discretized masses with respect

to the nodes and crests of the vibration profile.

The 10-node vibration profiles offer a good

illustra-tion of this aspect, as demonstrated in figure 14. In the first test problem the heavy masses appear

to be placed close to the nodes, and the light masses

close to the crests. Therefore the 10-node natural

frequency of the discretized system of the first test problem is much higher than that of the continuous

system. By contrast, in the second test problem, part of the heavy masses are placed nearly as far from the crests as the light masses. Therefore the natural frequency of the discretized system is only slightly higher than that of the continuous system. If the number of nodes is about equal to the num-ber of mass-segments, an unlucky placing of the masses of one mass-segment will tend to repeat at the other segments, which leads to accumulation of faults, i.e. to serious faults. See also the 9-node

vibration (figure 15) and the 11-node vibration

(figure 16).

2-node 3-node 4-node 5-node 6-node 7-node

o = 0.999 0.00098 0.0075 0.0302 0.0905 0.237 0.590

o = 0 0.00049 0.0035 0.0130 0.0370 0.0907 0.205

0-node 9-node l0-nodc I-node 12-node 13-node

o = -0.999 1.487

4.19

+22.4(1

10.5

7.13 +5.28

//

/

_r

first test problem(Ör-Oggg)

- -

second test problem (ÓO)

Figure 15. 9-node vibration

\

/

\

first test problem (ô=-O.9) - - second test problem (OO)

Figure 16. 11-node vibration

The conclusions must be that the physical fault

has a considerable random aspect and that one

should be careful to choose a sufficient number of mass-segments. In general with 10 mass-segments it seems justified to expect an accuracy of 1% or better up till the 7-node vibration.

The third test problem was based on an

unpublish-ed report [6]. In this report the lower three natural frequencies of the vertical vibrations of a ship in half-loaded condition were calculated, both with

and without rotary inertia. Firstly the vibration

system had been discretized. The natural frequen-cies of the discretized system were calculated on

a desk calculator with the iteration method [1]. In applying this method it is necessary to calculate

the deformations produced by a given load on the system. This was noi done by using the influence

numbers, but by calculating from equilibrium

con-ditions the bending moments and the shear forces at the elastic joints. From these the deformations

of the joints, and therefore of the whole system,

could be found easily. In consequence, the natural

frequencies of the discretized system vere

cal-culated entirely independently of the method

proposed in this paper. Moreover, the influence of

the rotary inertia of the discretized system was

com-puted correctly, and not by means of an approxi-mative correction afterwards. Thus, the third test

problem consisted of:

A comparison of the results for zero rotary inertia from [6] with those calculated by the

proposed method applied to the same

discret-ized system.

A comparison of the correct influence of the rotary inertia of the discretized system from

[6] with that calculated by the approximative method described in this paper.

In [6] the mass- and stiffness distribution had been

taken from a real ship in half-loaded condition. The rotary inertia of this ship, however, was not

known, and was therefore supposed to be equal to 5 &/E times the bending stiffness EI of the ship.

E and are Young's modulus and specific mass of steel respectively. Factor 5 stands for those parts

of the ship that contribute to the rotary inertia, but not to the bending stiffness. This va]ue was chosen on the strength of the supposition that it

should be about equal to the ratio of the total mass

of the ship in the half-loaded condition (without

the virtual mass of the water) to that of those parts that contribute to the bending stiffness.

The discretized system as used in [6] is given in table IV. For the third test-problem it is irrelevant that the method for discretizing the mass had been

different from the one described in this paper. However, the present author had a programme for an uneven number of masses at his disposal.

In order to be able to use it, a zero-mass was added

at the end of table IV which thus gives 10 mass-frictionless nondeirmc±e

joints massless _X

coinciding

at one point = discretized mass

= discrezed rotary nera dx

t

= rotary inertia per unit length -Q- = bending joint

= shear joint

TABLE IV. Discretized system of third test problem

segments. In addition, faked elastic joints had to be introduced too. This zero mass proved to give no difficulties in the calculations. Neither is it of much importance that in [6] a different (and more refined) model had been used with respect to the discretized rotary inertia. This model is illustrated

in figure 17.

* as used in [6] E = 2.1 10 kg/cm2

* * in accordance with the proposed method

G = 0.817 10 kg/cm2

O faked

Distance between two successive masses: 656 cm

Discretizing of the rotary inertia as proposed in

this paper gave the fifth column in table IV. This

column can be derived from the fourth column

by adding together every two successive discretized

rotary inertias. It was used for calculating the in-fluence of the rotary inertia by the approximative method described in this paper.

mi(kgsec2\cm ) / rad KkgCflI) q,[cm/kg] T,* [kgcrnsec] r ** [kgcrnsec2] 44.6 _{0.49695 1/G} 3040.34 10-9/E 5.40 10 0.49695 171 54 157.0 251:19 25.44 10 0.21635 580.67 20.04 0.2 1635 -309 109.91 0.2089 399.56 29.01 0.2089 I4 () J 86.838 1 81.071 ((iP 0. 17465 196.909 61.45 0.17465 - 904 o J 28.208 1 32.199 0. 1207 119.829 03.2! 0. 1207 1 99083 1123.0 28:074 201.27 0.1139 107.611 106.06 0.1139 - 1)560 26.686_26.399 0. 11675 110.142 104.62 0. 11675 1176.0 { 220.04 0.1154 98.733 ¡15.42 0.1154 - 1)60 0 f 24.400 1 24.540 0. 1154 98.017 117.14 0.1154 1238.0 233.12 0.1132 98.755 115.98 0.1132 11580 j 25.368 27.967 0.1103 125.504 92.97 0.1103 /kgsec2\ rn. j cm J / rad KI_\kgcm q [cm!kg r.* [kgcmsec2] [kgcrnsec] 1200.0 Ç 10-9/E 170.99 106 0.11525 11G 150.126 78.02 10 0.11525 11380 35.413 k 32.852 ((.1213 130.517 85.56 1)1213 72.0 143.53 0. 1378 198.673 57.97 0.1378 r 55.621 100. 0 1 56.061 ft i -136 253.730 46.01 t). I 1: ft3 848.0 f 106.02 (1. 1362 187.228 60.01 0. 1362 6700 -. 47.268 1. 45.975 (1.12945 207.930 55.22 (1.12945 332.0

f

96.82 (3.11955 275.515 41.60 0.11955 -- - j 85.201 I.L).() I -0.055 (.2 155 1284.667 15.69 0.2155 2:.o = 15.69

The results of the third test problem are given

in table V. The supports were taken as illustrated in figure 8, the faked mass being considered as a

real one.

TABLE V. Natural frequencies of the discretized system of the third test-problem as calculated by proposed method and as given by 161

Natural Icquencies in cycles per second.

7 Conclusion

The conclusion from the three test problems must

be that the proposed method gives accurate and

reliable results, also when the influence of shear

and rotary inertia are involved. By far the most

serious fault is the physical fault, which is

intro-duced by discretizing the mass. In general, with

10 mass-segments it seems justified to expect an accuracy of 1%, or better, uptill the 7-node vibra-tion.

The numerical fault in the calculations is excep-tionally small.

8 Cost of calculations

The calculations were performed in autocode on

an Elliott 803A machine. This is a machine of moderate capacity and operating speed. The ca-pacity of its magnetic ferrite core store is 4092

words. It has no floating point unit. Therefore the

floating point arithmetic was performed by

sub-routines. The pertaining operating speed is 0.070 sec for multiplication as well as for division,

addi-tion and subtracaddi-tion.

Calculating time on this machine is charged at

Dfls. lOO./hour.

The calculating times of a system with 10

mass-segments on this machine were about:

Reading

Discretizing

Calculating matrix B1'

Calculating matrix A

Calculating matrix B» AB-1' Tridiagonalisation

Calculating one natural frequency

with pertaining vibration profile

2 minutes 3 minutes 4 minutes 10 minutes 10 minutes 8 minutes 8 minutes Total (37+8n) minutes

where n = number of natural frequencies wanted.

On an Elliott 803B with floating point unit, the

total calculating time will be about (5+n) minutes.

This machine is charged at Dfls. 200./hour. On an Elliott 503 the calculating time will be

about (5+n)/l00 minutes + 1 minute output. This

machine will be charged at about Dfls. ]000./

hour.

The amounts at which the machines are charged

may differ from machine to machine. However,

the amounts mentioned give a good impression of the cost of these calculations.

For the calculations specified in the break-down above, it was supposed that data of stiffness and

mass distribution where available on punched

paper tape.

References

KOCH, J. J.: Enige toepassingen van de leer der eigen-functies op vraagstukken uit de Toegepaste Mecha-nica (Some applications of the theory of eigenfunc-tions in applied mechanics). Diss. Deift, 1929. BODEWIG, E.: Matrix calculus. 2nd ed. North-Holland

Pub!. Cy. Amsterdam, 1959.

WILKINSON, J. T-1.: Householder's method for the solution

of the Algebraic Eigenproblem. The computer Jour-nal, Vol. 3 p. 23, 1960.

Roy, S. N., and A. E. SARHAN: Biometrika, 43 (1956)

227-31.

D. WILKINSON, J. H.: Householder's method for symmetric

matrices. Numerische Mathematik 4. Band 4, Heft

S 354-361 (1962).

DORT, D. VAN: Berekening van de eigenfrequenties van

verticale scheepstrillingen (Calculation of natural frequencies of vertical hull vibrations). Unpublished report of the Institute T.N.O. for Mechanical

Con-structions.

BIEZENO, C. B., and R. GRAMMEL: Technische Dynamik. Berlin, 1953.

From [6] Proposed method

2 NV 3 NV 4NV 2.0921 c/s 3.8551 c/s 5.5718c/s 2.0922 c/s 3.8552 c/s 5.57l7c/s Without rotary inertia

From [6] (correct) Proposed method (approximate) 2 NV 2.072 1 c/s 2.0721 c/s 3 NV 3.8150 c/s 3.8151 c/s 4NV 5.5221 c/s 5.5209 c/s

Appendix L Insertion of (A15) and (A16) in (A14) gives: For the numerical calculations one needs only one

suitable solution of the equations (8). Solutions

can be produced as follows.

ßj32+2y1)l3+al2+2ìl1 = i

. . . (Al) 132+2y1)21)3+ß1)22+21)2 = i . . . (A2) ?1/32+(2+1)1+ß1)2)1)3+V1)11)2 = O . (A3) From (A3) follows:

2+yris+ß1)2

1)3 ar/3+yr/2

and

2+3+a1)1

ris - ßris+

From (Al) follows:

2+'îs+a1)1

ßi3+vI1

-and from (A2):

2+y3+ß/2

a1)3 + '1J2

From (A4) and (A7):

1/11)2 i

1)3

- /

a1)3+y172

and from (A5) and (A6):

7)11)2 i

1)3 -

ßri3+yl/1

From (A8) and (A9):

y (iii 172)

1/3 =

aß

From (Alo) and (Al) can be derived:

(a1)jß1)2)2 =

(a_ß)2

j(v2_aß)1)12_2ß1)i+ß} Similarly from (A 10) and (A2):

(a1)iß1)2)2 =

=

(a_ß)2{(2ß)2+}

With

O = (y2aß)1)12-2ßiji+ß =

= (v2aß)1/222a1/2+a

both (All) and (Al2) read:

(a1)iß1)z)2

(a_ß)2

o

/o2òoo

. (A15) a aß Say: and where

± 26V1

(a+ß)+aß+O(ß+aaß

02 0 02 à\a ¡9 0

= 20+ (a+ß) +

0 (aß'\2

aò (A17)

àt\

) 0f

Squaring gives:

4ò2-46(a+ß)

+4aß+4O(6

+

aà a ß) +402=

a ¡9 (M)

_{= 462-40(a+ß)+(a+ß)2}

_{+[_46+2(a+ß) +}

(Alo) (All) (Al2) or 401a2+ß2 _{(a+ß)I + 402 =} 1y2_aß I 4aß

+ 2(aHß)

-=

y2aß

oI(ß)2+2jJ

(aß)2}

or

(ß)2{l22+"

y2

+

024y2+(aß)2}7/4 from which

2(a+ß+2) ± 2v'(1+a)(l+ß)_y2

4y2H (aß)2

Because it may happen that

zß

it is preferable not to calculate 173 from (Alo), but

e.g. from (Al) which reads with (A13):

1)3 = -

iii +

Vß_2ß1)i+1)12(y2_aß) =

=(y17i± \/

(Al9)

Insertion of numerical solutions shows that not all of them satisfy equations (A2) and (A3). In order to obtain valid solutions the following combina-tions of signs should be adhered to:

=0

(A18)

The best choice of these combinations will be

discussed in Appendix II.

O (aß'\2

aO ¡90

0 (aß'\2

czò ßà

+6k)

ß

aOky!

ß à

sign in sign in sign in sign in

combi- (Al8) (A15) (A16) (A19)

nation O _{172 173}

+

+

+

2

+

+

3

+

+

+

4

1)2+

à

i/6

0 00 (A16)

0

aß

- aß

IABLE A5.

TABLE A2.

TABLE A3.

TABLE A4.

Prismatic beam Ship

u, 0.795495130 0.00000000 u, 1.50260191 1.12562483 u, 1.32582522 1.59099026 u4 0.875000000 1.08293922 u5 1.14904852 1.88272429 u4 0.750000000 1.44769086 u, 0.972271826 1.63612425 u4 0.625000000 1.112724449 u9 0.795495130 1.41528714 u10 0.500000000 0.806590501 u11 0.618718434 1.12445861 u12 0.375000000 0.757899575 u1, 0.441941739 0.852184154 u14 0.250000000 0.492144168 u15 0.265165043 0.458332777 u16 0.125000000 0.200642351 u17 0.0883883478 0.126316596 u14 -0.0883883478 -0.0697619377 u19 -0.0883883478 0.0201099917

Prismatic beani Ship

WI 0.0883883478 0.00000000 W2 -0.0883883478 -0.0604441284 W, w. 0.0883883478 0.125000000 0.0968245836 0.141226247 w5 0.265165043 0.396619446 w6 0.250000000 0.440518300 w7 0.441941739 0.678895576 W4 0.375000000 0.617417237 W9 0.618718434 1.00486904 W15 0.500000000 0.736313019 Wi1 0.795495130 1.31976717 w12 0.625000000 1.15310748 w, w34 0.972271826 0.750000000 1.71145511 1.34779231 w15 1.14904852 1.81306089 W16 0.875000000 1.28212398 w17 1.32582522 1.72966122 Wis 1.50260191 1.08262196 w', 0.795495130 1.65220441

Prismatic beam Ship

mi i M 0.000 M 4 0.954 m0 2 0.850 m3 4 2.448 rn 2 1.302 m5 4 4.564 m 2 3.167 ns, 4 2 4.814 2.765 rn 4 5.381 rn,0 2 2.2 12 rn,1 4 5.615 2 3.472 4 6.32 1 mil 2 3.294 mis 4 5.079 2 2.190 4 3.472 m2 2 1.020 m15 4 1.059 m19 1 0.022 Total 60 M 60 M

Prismatic beam Ship

(7 10.4062500 20.2915027

10.4062500 19.7175513

y 3.59375001 11.1538438

Prismatic beam Ship

Sign

combi-nation

lB1Is IBI-V {(1+a)(1+ß)_y2}_h/a B-'/i {(1+a)(1±ß)

1 2 3 4 -0.0923760403 -0.0923760445 0.0923760408 0.092376049 1 +0.0923760431 ±0.0923760431 -0.0561922307 -0.0561922302 0.0561922290 Q.0561922346 +0.0561922296 +0.0561922296 Appendix II

In order to find the combination of signs that gives

the most accurate numerical results, a number of computations have been performed. These com-putations were made on a binary digital computer with a floating point fraction of 29 bits. These 29

bits correspond with a decimal word-length of

2910 1g2 = 8.73 digits.

Two different discretized mass distributions were chosen, one of a prismatic beam, one of a loaded

ship including vibrating water. Each mass

distribu-tion consisted of 21 equidistant masses. The

sup-ports were chosen at the third mass from each

end, as shown in figure 8. For the masses, pertain-ing vectors U and iV and numbers a, ß and y, see

tables Al, A2, A3 and A4. These tables contain the masses in the correct physical order. For reasons of

convenience the total mass of the prismatic beam

was taken 60 M. For comparison the total mass

of the ship too has been taken equal to 60 M.

With these numbers a, ß and , numbers e,, e

and e3 were calculated for all four sign-combina-tions of numbers ì.

Now the determinant value of matrix B was computed on the one hand from its elements which

themselves were calculated from numbers e and vectors U and 1V and, on the other hand, from the

determinant value of matrix B, also computed

from its elements. The results are mentioned in

table A5.

Moreover it can be proved that:

= (l-[-a)(l+ß)-y2.

The proof will be omitted here.

Next numbers ô of matrix:

B-' = I+ò,UU'+ò2V'+53[UW'+íVU']

were calculated in two different ways. On the one hand matrix:

B =

i+UU'+WlV'

TABLE A6.

was inverted in analogy of the inversion of matrix

B'

by means of equation (10). On the other hand matrix:

B-12 = I+e UU'+e2 WW'+e3[UW'+ WU']

was squared:

B-i = B-'12B-2 =

= I+(aEi2+ße32+2e1+2ye1E3) UU' + (ße22+ae32+2c2±2ye2ea) WJ-V'

+ (ye32+2e3+yele2+aelea+ße2e3) [UW'H- WU']

The results are presented in table A6.

It appears that all sign-combinations do very well,

though combination 4 seems slightly inferior to the others and combination 3 seems best on the whole.

The test problems showed that the numerical

fault was very small and offered no grounds for a

choice.

Because a choice had to be made, combination

3 was chosen.

Sign

com-bination

Prismatic beam Ship

From [B'/2]2 From [B] From [B'l] From [B]

ò 2 -0.0973333332 -0.0973333332 6, 62 -0.0654170495 -0.0672293392 2 63 6, 62 0.0306666660 -0.0973333339 -0.0973333339 6, 6, 62 0.0352190054 -0.0654170497 -0.0672293396 63 0.0306666676 61 = -0.0973333334 63 0.0352190055 ò - -0.0654170494 62 = -0.0973333334 62 - -0.0672293393 :3 6 62 -0.0973333335 -0.0973333335 63 = 0.0306666668 6 62 -0.0654170495 -0.0672293392 6, = 0.0352190054 63 0.0306666666 63 0.0352 190054 1 6 62 -0.0973333330 -0.0973333330 6, 62 -0.065417090 -0.0672293387 ò3 0.0306666665 0.0352190048

Reports

No. i S The determination ofthe natural frequencies ofship vibrations (Dutch).

By prof. ir H. E. Jaeger. May i 950.

No. 3 S Practical possibilities of constructional applications of aluminium alloys to ship construction.

By prof. ir H. E. Jaeger. March 1951.

No. 4 S Corrugation of bottom shell plating in ships with all-welded or partially welded bottoms (Dutch).

By prof. ir H. E. Jaeger and ir H. A. Verbeek. November 1951.

No. 5 S Standard.recommendations fcr measured mile and endurance trials ofsea-going ships (Dutch).

By prof. ir J. W. Bonebakker, dr ir W. J. Muller and ir E. J. Diehi. February 1952.

No. 6 S Some tests on stayed and unstayed masts and a comparison ofexperimental results and calculated stresses (Dutch.

By ir A. Verduin and ir B. Burghgraef. June 1952. No. 7 M Cylinder wear in marine diesel engines (Dutch).

By ir H. Visser. December 1952.

No. 8 M Analysis and testing of lubricating oils (Dutch).

By ir R. N. M. A. Malotaux and irj. G. Smit.July 1953.

No. 9 5 Stability experiments on models of Dutch and French standardized lifeboats.

By prof. ir H. E. Jaeger, prof. ir J. W. Bonebakker andj. Pereboom, in collaboration with A. Audigé. October 1952.

No. 10 S On collecting ship service performance data and their analysis. Byprof. irj. W. Bonebakker.January 1953.

No. 1 1 M The use of three-phase current for auxiliary purposes (Dutch). Bv irj. C. G. van WjIc. May 1953.

No. 12 M Noise and noise abatement in marine engine rooms (Dutch). By " Technisch-Physische Dienst T. N. O.- T.H. ' ' April 1953.

No. 1 3 M Investigation of cylinder wear in diesel engines by means of laboratory machines (Dutch).

By ir H. Visser. December 1954.

No. 14 M The purification of heavy fuel oil for diesel engines (Dutch).

By A. Bremer. August 1953.

No. 15 S Investigation of the stress distribution in corrugated bulkheads with vertical troughs.

By prof. ir H. E. Jaeger, ir B. Burghgraef and I. van der Ham. September 1954.

No. 16 M Analysis and testing of lubricating oils II (Dutch).

By ir R. N. M. A. Malolaux and drs J. B. Zabel. March 1956.

No. 17 M The application of new physical methods in the examination of lubricating oils.

By ir R. N. M. A. Malotaux and dr F. van Zeggeren. March 1957.

No. 18 M Considerations on the application of three phase current on board ships for auxiliary purposes especially with regard to fault protection, with a survey of winch drives recently applied on board of these ships and their in-fluence on the generating capacity (Dutch).

By ir J. C. G. van Wjk. February 1957. No. 19 M Crankcase explosions (Dutch).

By ir J. H. Minkhorst. April 1957.

No. 20 S An analysis of the application of aluminium alloys in ships' structures.

Suggestions about the riveting between steel and aluminium alloy ships' structures. By prof. ir H. E. Jaeger. January 1955.

No. 21 S On stress calculations in helicoidal shells and propeller blades.

By dr ir J. W. Cohen. July 1955.

No. 22 S Some notes on the calculation of pitching and heaving in longitudinal waves. By ir J. Gerritsma. December 1955.

No. 23 S Second series of stability experiments on models of lifeboats. By ir B. Burghgraef. September 1956.

No. 24 M Outside corrosion of and slagformation on tubes in oil-fired boilers (Dutch). Bydr W.J. Taat. April 1957.

No. 25 5 Experimental determination of damping, added mass and added mass moment of inertia of a shipmodel.

By ir J. Gerritsma. October 1957.

No. 26 M Noise measurements and noise reduction in ships.

By ir G. J. van Os and B. van Steenbrugge. May 1957.

No. 27 S Initial metacentric height of small seagoing ships and the inaccuracy and unreliability of calculated curves of

righting levers.

By prof. ir J. W. Bonebakker. December 1957.

No. 28 M Influence of piston temperature on piston fouling and piston-ring wear in diesel engines using residual fuels.

By ir H. Visser. June 1959.

No. 29 M The influence of hysteresis on the value of the modulus of rigidity of steel.

By ir A. Hoppe and ir A. M. Hens. December 1959.

No. 30 S An experimental analysis of shipmotions in longitudinal regular waves.

By ir J. Gerritsma. December 1958.

No. 31 M Model tests concerning damping coefficients and the increase in the moments of inertia due to entrained water

on ship's propellers.

By N. J. Visser. October 1959.

No. 32 S The effect of a keel on the rolling characteristics of a ship. By ir J. Gerritsma. July 1959.

No. 33 M The application of new physical methods in the examination of lubricating oils. (Continuation of report No. 17 M.)

By ir j. Gerritsma. February 1960.

No. 36 S Experimental determination ofbending moments for three models ofdifferent fullness in regular waves.

By ir J. Ch. De Does. April 1960.

No. 37 M Propeller excited vibratory forces in the shaft of a single screw tanker.

By dr ir J. D. van Manen and ir R. Wereldsma. June 1960. No. 38 S Beamknees and other bracketed connections.

By prof ir H. E. Jaeger and ir J. J. W. Nibbering. January 1961.

No. 39 M Crankshaft coupled free torsional-axial vibrations of a ship's propulsion system. By ir D. van Dort and N. J. Visser. September 1963.

No. 40 S On the longitudinal reduction factor for the added mass of vibrating ships with rectangular cross-section. By ir W. P. A. Joosen and dr J. A. Sparenberg. April 1961.

No. 41 S Stresses in flat propeller blade models determined by the moiré-method. By ir F. K. Ligtenberg. June 1962.

No. 42 S Application of modern digital computers in naval-architecture.

By ir H. J. Zunderdorp. June 1962.

No. 43 C Raft trials and ships' trials with some underwater paint systems.

By drs P. de Wolf and A. M. van Londen. July 1962.

No. 44 S Some acoustical properties of ships with respect to noise-control. Part I. By ir J. H. Janssen. August 1962.

No. 45 S Some acoustical properties of ships with respect to noise-control. Part II. By ir J. H. Janssen. August 1962.

No. 46 C An investigation into the influence of the method of application on the behaviour of anti-corrosive paint systems

in Seawater.

By A. M. van Londen. August 1962.

No. 47 C Results oían inquiry into the condition of ships' hulls in relation to fouling and corrosion.

By ir H. C. Ekarna, A. M. van Landen and drs. P. de Wolf. December 1962.

No. 48 C Investigations into the use of the wheel-abrator for removing rust and millscale from shipbuilding steel (Dutch) Interim report.

By ir J. Rernrnelts arid L. D. B. van den Burg. December 1962.

No. 49 S Distribution ç,f damping and added mass along the length of a shipnìodel.

By prof, ir J. Gerriisma and W. Beulcelnian. March 1963.

No. 50 S The influence of a bulbous bow on the motions and th propulsion in longitudinal waves.

By prof. ir J. Gerritsrna and W. Beukelman. April 1963.

No. 52 C Comparative investigations on the surface preparation of shipbuilding steel by using wheel-abrators and the application of shop-coats.

By ir H. C. Ekaina, A. M. van Londen and ir J. Rernrneltc. July 1963.

No. 53 S The braking of large vessels.

By prof ir H. E. Jaeger. August 1963.

No. 54 C A study of ship bottom paints in particular pertaining to the behaviour and action of anti-fouling paints.

By A. M. van Landen. September 1963.

No. 55 S Fatigue of ship structures.

By ir J. J. W. Nibbering. September 1963.

No. 56 C The possibilities of exposure of anti-fouling paints in Curaçao, Dutch Lesser Antilles.

By drs P. de Wolf and Mrs M. Meuter-Schriel. November 1963.

No. 58 S Numerical calculation of vertical hull vibrations of ships by discretizing the vibration system. By J. de Vries. April 1964.

No. 57 M Determination of the dynamic properties and propeller excited vibrations of a special ship stern arrangement.

By ir R. Werelds,na. Maart 1969.

Communications

No. I M Report on the use of heavy fuel oil in the tanker "Auricula" of the Anglo-Saxon Petroleum Company (Dutch).

August 1950.

No. 2 S Ship speeds over the measured mile (Dutch).

By ir W. H. C. E. Rösingh. February 1951.

No. 3 S On voyage logs of sea-going ships and their analysis (Dutch).

By prof ir J. W. Banebakker and ir J. Gerriisma. November 1952.

No. 4 S Analysis of model experiments, trial and service performance data of a single-screw tanker. By prof ir J. W. Bonebakker. October 1954.

No. 5 S Determination of the dimensions of panels subjected to water pressure only or to a combination of water pressure and edge compression (Dutch).

By prof ir H. E. Jaeger. November 1954.

No. 6 S Approximative calculation of the effect of free surfaces on transverse stability (Dutch). By ir L. P. Herfsl. April 1956.

No. 7 S On the calculation of stresses in a stayed mast.

By ir B. Burghgraef August 1956.

No. 8 S Simply supported rectangular plates subjected to the combined action ol a uniformly distributed lateral load and compressive forces in the middle plane.

By ir B. Burghgraef. February 1958.

No. 9 C Review of the investigations into the prevention of corrosion and ftuling of ships' hulls (Dutch). By ir H. C. Ekama. October 1962.

No. 10 S/M Condensed report of a design study for a 53,000 dwt-class nuclear powered tanker.

By tite Dutch International Team (D.I. T.) directed by ir A. M. Fabery de Jonge. October 1963.

M = engineering department S = shipbuilding department