VLCC deckhouse vibration, calculations compared with measurements

77

ARCHIEF

tab

_f

v.

_i

_i

Schéepsbouwkunk

_I

_II.

_I

I- i.

uecnniscne nogescnooi

Delfi

MONOGRAPH PUBLISHED BY THE NETHERLANDS MARITIME INSTITUtE

VLCC deckhouse

vibration

Calai lations compared

with measurements

S. Hylarides

R. van de Graaf

M6

July 1976

i'

VLCC deckhouse

vibration

Calai lations compared

with measurements

Dr. Ir. S. Hylarides

CONTENTS

page

Summary 5

I Introduction 6

2 Particulars of ship and exciter tests 6

3 Calculation method 6

3.1 The mathematical model 6

3.2 Normal mode method and damping 6

4 Theresults 7

4.1 Presentation of the results 7

4.2 Discussion of the results 7

S Conclusions 12

References 13

LIST OF SYMBOLS A indices B indices Q damping matrix E modulus of elasticity stiffness matth

K* _{condensed stiffness matrix}

L mass matrix

M* _{condensed mass matrix}

N matrix of eigen vectors T kinetic energy

Y potential energy

aj j-th generalized normal coordinate

a vector of normal coordinates

d damping value of dashpot

j

force vector

Ç the force applying in the i-th degree of freedom

f (t) time function

h value of hysteretiá damping

j mode numbering

k stiffness value of spring m mass value of body

normalized mass of the j-th mode

n number of considered degrees of freedom

t

time

¿ vector of participation of the excitation over all the normal modes

(3 dimensionless damping coefficient, defined as a fraction of the critical damping

6 variation operator or displacement function displacement vector

K ratio between damping and stiffness matrix

exponent

¡.L ratio between damping and mass matrix

n j-th normal mode

displacement of the i-th degree of freedom in the j-th normal mode

angular frequency

w eigen angular frequency of the j-th mode phase lag of the j-th normal mode

i

Introduction

In present-day ships, more and more attention is being paid to vibrations in the accommodation. The reasons are twofold:

due to the high excitations, generated by the highly

loaded propeller, and due to the response characteristics

of the ship structure, the vibration amplitudes may reach

high values;

due to the introduction of vibration criteria in interna-tional standards.

Ilierefore, owners are specifying permissible levels of

vibra-tion levels in their quotavibra-tions. Consequently there is a growing need for reliable tools to predict the vibration level at the design stage. In the development of the pre-diction technique, the following items concerning

struc-turai response have to be investigated:

concerning structural response have to be investigated:

- the correlationbetween the calculated and full scale

re-suits (including the superstructure), using a detailed mathe-matical model of the ship structure.

- the effect of the hull-deckhouse connection. - the influence of added mass of the hull girderand in tanks.

* Netherlands Ship Model Basin, Wageningen

** Verolme United Shipyard

VLCC DECKHOUSE VIBRATION

CALCULATIONS COMPARED WITH MEASUREMENTS

by

Dr. Ir. S. HYLARIDES* and h. R. VAN DE GRAAF**

Summasy

Vibrations in the deckhouse of a VLCC, generated by an exciter located on the steering gear platform, have been measured.

the generated vibrations have also been calculated, using a three-dimensional, finite element model, accounting for damping, and compared with the measured results.

The calculated results lie within the repeatability of the measurements.

Suggestions are given for improvement of the measurements and calculations.

In this report, calculations of the natural frequencies and forced responses to an exciting force are compared with the results of full scale exciter tests, performed on board a VLCC. The calculations are based on a detailed three-di-mensional fmite element model of the afterbody (Figs. i and 2).

Fig. 2 Finite element mesh - perspex model

I

;----

_J s

Fig. 1 The ship investigated and its finite element breakdown

2 Partkiulars of ship and exciter tests

The exciter tests, were performed on a 228,000 DWT tan-ker, of which the overall dimensions are given in Table I.

The objective of these tests was to obtain correlation data for the calculations with regard to the natural frequencies and corresponding modes, especiälly for the deckhouse, situated on the aft ship as shown in Fig. 1. The frequency range had to rim from 4 to 16 Hz. Therefore, two rotary out-of-balance exciters were uséd, 'a heavy one running from 4 to 10 cps and a lighter one covering the frequéncy range from8 to 16 cps. They wére installed in the steering gear flat at the centre line and were used for generating

the' exciting force.

- The tests were performed for three different loading

con-ditions of thé ship: 100% 50% and 40% of the maximum dèadweight /1/.

table II Particulars of the exciter tests

3 Calculation method. 3.1 The máthematical model

Calculations were established using the finite element

me-thod. A detailed model was c nstructed, representing the afterbody, including the engine room, superstructure and aft peak. Forward of the deckhoùse, a more simple repre-sentation by means of a "Timoshenkä" beam was thought to be acceptable.

The computer costs of this model would have been

extre-mely high. In order to reduce the number of degrees of

freedOm, symmetric behaviour only was considered, as was

the case in the exciter tests and the condensation techm que was used to obtain a second reduction of the number

of dègrees of freedom.

From the remaining set of degrees of freedom, the

eigen-value problem was formulated, leading to natural

frequen-6

cies and corrésponding modes.

For more information reference is made to Appendic I. 3.2. Normal mode method and damping

In order to calculate the structural response, the normal

mode method was used. This technique replaces the

origi-nal set tif coupled differential equations describing the structure, by an equivalent, uncoupled set. Stated simpl, the structural response is now composed by the sum of the individual modal responses /2!.

This method is treated in some detail m Appendix II 1he denvation of the mobility of each point is given, leading to the formula:

n W!fljj cos (wt

--=

f.

j=1''

m.

Table I Overäll ship pâUcu1ar

+ 2f3w ±w2)2w2

= the velOcity vector

= the force applying in the i-th dégrée

of freedom

= the circular frequency of the excita-tion force

= the j-th normal mode

= the displacement of the i-th degree of freedom in the j-th normal mode

m = normalized mass of the j-th mode

w. natural circular frequency of the j-th mode

ji = viscous damping, proportional to the mass distribution

-= viscous damping, proportional to the stiffness distribution

(3 viscous damping, defmed as a

frac-tion of the critical damping, reläted to the relevant mode

= phase lag of the j th normal mode

In this expression,(3is separately defined for each mode as

the fraction of the critical damping for that mode.

Cri-tical damping revers to the lowest value of the viscous

dam-ping forces for which the highest response to any excitation system is obtained at zero frequency. In the present

analy-sis, the' same valúe for (3 was used for all modes, so that (3,

needs not to be distinguished by means of the suffix j. The valùefor (3 was chosen at 006, or 6% of the critical damping. The value for k was set at 0.002. The efféct of

has not been considered because it has been shown /4/ that the effect at low frequency is too strong and, at higher fre4uency, too weak.

In order to clarify the effects of the various damping sy-stems, the formulation fôr the modal response, as given above, must be studied. The total effect of modal damping

is given by:

(Kw2 + 2 ¡3w. + ji)2

Using only (3 the form becomes (2(3w)2

Using only K the modal damping is expressed by (kw2)2.

Using only

ji

the form becomes (ii)2.

Length 314.120 m

Breadth

48.680 m

Depth 25.600 m

Deadweight (maximum) 228,000 ton

(metric) Installed shaft power 32,000 BHP

at 82 RPM

Propeller diàmeter 9.40 rn

Number of blades 4

Maximum ship speed (full load) 16 knots

Heavy exciter:

frequency range 4- 105 cps

maximum output (only vertical) 3000 kgf

Light exciter:

frequency range 8 - 16 cps maLximurnoutput (only vertical) 2500 kgf

Ship speed during tests

l-3 knots

Weátherconditions rnoaerate throughout Deadweight 100, 50 and 40% of maximum deadweight inwhich f flu

(D z Q-z 4 -J 4 o o z z w -J o w I I I I I I 10 11 12 13 14 15 16 FREQUENCY [HZ]

Fig. 3 The effect of the several approaches of damping on the equivalent modal damping as a function of the frequency

These three forms of the modal damping were plotted as a function of the frequency in Fig. 3. For the chosen values

of ß and , it follows that around 10 Hz theeffect is simi-lar. Below 10 Hz jI has stronger effects and above 10 Hz K

is more effective. Also the effect of u is shown.

4 The results

4.1 Presentation of the results

For a restricted number of nodal points the response is shown and compared with the results of the full scale

measurements viz.:

- the aft peak - the bridge deck - the funnel top.

The calculations were performed for the 50% deadweight

Table Ill Description of the most important mode shapes for the longitudinal vibration of the deckhouse due to vertical excitation in the steering gear room (these modes

have been selected on base Fig. 4).

Most important modes

Nat.

Freq.

Hz No

Description of mode shape

deckhouse out of phase with stern

deckhouse in phase with longitudinal vibration hull

deckhouse in phase with longitudinal vibration hull

deckhouse in phase with stern

deckhouse out of phase with engine casing deckhouse out of phase with engine casing

deckhouse out of phase with longitudinal hull vibration

deckhouse out of phase with engine casing

condition. These results are directly comparable with the full scale results at 50% deadweight, but additional

informa-tion with regard to the 40% deadweight is thought to be pos-sible.

As an example the calculated longitudinal response of the bridge deck is treated.

In Fig. 4, the separate responses of a number of modes with a natural frequency in the range from 4 to 16 Hz are' given as a function of the frequency. These responses only refer to the longitudinal mobility of the bridge deck to a vertical force at the exciter location in the steering gear

room.

Fig. 4 demonstrates that around 10 Hz only a few modes are in resonance, whereas out of this said number only

some lead to a high mobility value in the frequency range. These have been summarized in Table III. A rough descrip-tion of their shape is also given.

Summation of all modal mobilities of Fig. 4 is given in

Fig. 5, together with the full scale measured mobility.

For the aft peak these same results are given in Fig. 6 and 7 and Table IV.

The results of the funnel top mobility are shown in Fig. 8 No information of the modal mobiities has been presented. 4.2 Discussion of the results

Considering the calculated and measured mobility curves from Figs. 5, 7 and 8, some comment is called for.

First, the non-coincidence of the response to both exci-ters in the overlapping frequency range calls for attention.

This indicates the problem encountered in experimental

de-termination of the structural response:

- the effect of noise.

Noise, generated by onboard machinery, sea state and the like, can be expected to be of the same order of magnitude as the vibrations generated by the exciter. This can cause a good deal of inaccuracy in the evaluation of the signals.

- consequences of the exciter method.

The measurements were made by gradually increasing the exciter RPM. Experience in other cases shows that a

different rate of increase of RPM can lead to different

vibrations levels and resonance frequencies. Coupling

effects between the exciter behaviour and the vibrating structure are to be expected, especially around resonan-ce frequencies. The actual rotary speed of the exciter,

7 20 6,67 25 8,29 29 9,10 31 9,42 33 10,36 34 10,45 35 11,03 46 14,60

4

wa)4.

i

d!A(4PLid41t4

#'

.----A

1PW4'

A

R.

gR

,iii

FREQUENCY Hz r r i CRITICAL DAMPING (3 0.06

r

u IA

z

E I--j o

I

10 10

Fig. 4 Contribution of the several modes in the longitudinal response of bridge deck

8

as well as the amplitude ofthe exciting force, can thus be affected.

Secondly, regarding the calculated results and accoun-ting for the above problems in the measurements, the conclusion may be drawn that there is a fair rate of

Table 1V Description of the most important mode shapes for the vertical vibration of the stern due to vertical excitation in the steering gear room (these modes have been selected on base of Fig. 6)

Most important modes

Nat.

Freq.

No. _Hz

Description of mode shape

stern in phase with deckhouse

stern and rudder out of phase with deckhouse stern vibration

stem out of phase with deckhouse

longitudinal bulkhead in vertical vibration deckhouse out of phase with engine casing

longitudinal hull vibration out of phase with deckhouse

deckhouse out of phase with engine casing

accuracy. Peaks and levels have been calculated within the repeatability ofthe measurements.

With regard to the calculations, the following points merit close consideration:

- though no general information can yetbe given, it is

9

'o

o

4

Fig S Mobility curves for:

light excster 3. A. byO.O6 B Mensuring point

/

IO

A

I"

5 6 7 8 9FREQUENCY -Hz 11 12 13 14 15 16 12 4,54 13 4,78 20 6,67 23 7,95 30 9,42 34 10,45 35 11,03 46 14,63

lo

u U) z E -J o lo-lo Io

£

_,441L1h___

_{__WA}

'

r,

iA

A 46

WA

WDi WI

IL VI

V

____

-VA

4'

AW

-VÄ

rl

20 FREQUENCY 4 5 6 7 8 9 10

il

12 13 14 15 16

11 u, z o

-O

Fig. 7 Mobility curves for: 1. Measured value

heavy exciter 2. Measured value

3. calated values with damping

re-4.alculated'is presented by K w .002 A Exintatson station B Measuring point ¡

\

___

,

o

O4,

IWAW1IV

_{______________}

.j

-ir

-4 5 G 7 8 9FREQIJENCY Hz 11 12 13 14 15 1G (D 4 A

A1Ì%

Fig 8 Mobility curves for

4.EVS0.06

presented byK=.002

/

______w

-

I

-k'-,

I I I

t,

i.

_i

5 G 7 8 9 FREQUENCY Hz 11 12 13 14 15 16

clear that the formulation of mass and stiffness matrices require the highest attention. In particular, the applied condensation technique can probably be too coarse. - no direct conclusions can be made with regard to the

mode pattern in correlation with the full scale measured

patterns. As shown by Fig. 4 and 6 the measured patterns are substantially affected by several modes, so the fmal

results can differ appreciably from the mode pattern which is sought for. This is illustrated by Fig. 9 and 10. 5 Conclusions

- Concerning measurements on board ships, great care

must be taken to obtain a good signal-noise ratio. To

inves-tigate the resonances of a part of a given structure, much can be gained by locating the exciter in a position where optimal excitation of the expected thodes of interest

oc-curs.

12

Calculation

- The instability and the rate of increase of exciter RPM merits further investigation to quantify the effect and to offer a remedy.

- The use of the normal mode method in the way demen.

strated serves well to give the analyst a clearer insight into

the complex structural behaviour by breaking it down into the sum of a set simply understandable modal _responses. Concerning the use of the damping coefficient and(3,

the comment can be made that the difference inresponse

calculated for the two cases is less than the deviation of

the measured results themselves. Which case to use seems arbitrary; good values for (3 and K can be given as .06 and

.002, respectively. In summary, results have been submitted demonstrating a fair degree of accuracy. As yet, noanswer has been given concerning the questions raised with re&pect

to hull deckhouse connection, nor concerning the added mass influences. The general validity of the finite element method has once again been demonstrated, although a number of refinements merit further study.

Experiment

Fig. 10 Comparison of 33-rd normal mode with closest resonance vibration pattern

Calculation Experiment

References

Schulze,R.A.PJ.: S.T.T. "British Progress". deckhouse mechani-cal vibration measurements

INSTITUTE for mechanical constructions - TNO report No 10840. April 1974.

Hurty. W.C.. M.F. Rubinstein: Dynamics of structures.

Prentice-Hall. Inc., May 1965.

Den Haxtog,J.P.: Mechanical vibrations.

McGraw-Hill, 1956.

Hylarides, S.: Damping in propeller-generated ship vibrations; Netherlands Ship Model Basin, publication No. 468. October

1974:

Appendix I

Equations used to formulate the eigenvalue problem. For free vibrations, the following matrix equationmust be

solved.

MS+K6=O

(1)

where M is the mass matrix K is the stiffness matrix

S the nodal point displacement vector.

When the system is "condensed", the fòUowing operation is carrièd out; let . A denote the degrees of freedom defining

the dynamic behaviour, and let denote degrees of freedom bound to A by the Stiffness relation only:

B=KBBAA

(2)

Rewriting I in.partitioned form:

ÍMAA B1

Í1+ [

KAB [A1

_{= o}

[!BA MBBJ [nj LKa

BBJ [B]

-

(3)

and substituting 2 equatiOn 3 becomes

M*+KQ

(4)

where

-

AB K8MBA MAB KBA +

+KABKBMBBKBKBA (5)

and

írp* - r,iAA Z'

_ABBBBA

¡P-1 Z' (6)

The system described by (4) assumes that the _A degrees

of freedom can accurately describe the dynamic behaviour. A further Simplification can be carried out by assuming the mass forces, which in (4) are distributed over all the nodal points, may be represented by a lumped set in the A degrees of freedom. ThenMBB 'MAB MBA =12th eq. (3) and equation (5) becomes:

M* =M (5)

In this case Mp.. differs from the above.

This condensation technique was used in the analysis under

discussion. A justification of this technique can be found by regarding the governing equatiOn for the structure /2/:

(T+ V) = O (7)

here T, kinetic energy, is given by

T=½f ni2dv

(8)

vol

and V, potential energy, can be expressed as

V=½fEe2dv

(9)

vOI

in which E is the modulus of elasticity.

The strains e are derivatives of the displacement functions.

Thus the kinetic energy (depending on the displacement

14

functions themselves) is approximated a higher degree of accuracy than the potential enetgy

This discrepancy can be balanced by using a finer gridwòrk

for the description of the potential energy than for the ki. netic energy. This implies that for the equations of motión the stiffness matrix requires a finer network than themass

matrix. Appendix II

Normal mode method.

The matrix equations for fórced vibrations can be thown

as:

M6+c+K6=f

¶1)

The eigenvalue equations M + K = Q

as given in Appendix I do not account for damping. In thé first place, because in lightly damped structures (damping ranging from O to 10 percent of critical damping), natral

frequencies and mode shapes are. practically uftaffected by damping forces; in the Second place ,because the uncertainty

as to the content of the damping matrix Çin equation (1)

is very high.

Having obtained the natural frequencies (w1, w2, ...) and mode shapes (m, fl2, ...), the following co-ordinate trans-formation is used:

ajj

(2)

where aj is the thnormal coordinate and Nïs a matrix

whiôh columns are the eigen modes ni.

By substitution of this expression fot. and pre.multl. plication by the original' set of equations of motion (i)

now becomes:

NTMNa±NTCNa+NTK!Ta=NTf (3)

FOr the undamped system

NT

ö+NT

T

follows, making use of the orthogonality properties /2/:

= mfori=j

= O fori*j

(4) and

= wfori=j

= O

fori*j

(5) that the matricesNTMIYand NTK are diagonal matrices. Then the j-th normal mode equation can be considered separatély from the others. This j-th equation is given by:

m a, + wmaj =

Tj.

in which the expression n Tf is called the participation of the excitation fin the j-th normal mode equation. This expression or articipation factor is generally replaced by X which, togetherwith the values of the other normal modes, is grouped in the vector F, thus:

The j-th normal mode equation of motion then becomes:

mäj + ci. maj = X (6)

sub b: In undamped case, the j-th normal mode equation of motion (6) is in fact the equation of motion of a

single mass-spring system. As in that case, this

equation is completed with a viscous damping c:

mäj + ci +

=

k. in which

J m

In the investigation described, ¡3 has been assumed constant

For a single.mass-spring system it is known that for for all modes, so that its index j has been omitted.

c = 2m o the maximum response is obtained for frequency zero.

This is called the critical damping.

15

In the setofequations for the damped forced response (3), the mass matrix.M and stiffness matrix K are dearly

defined. For the formulatiön of the damping matrix C no information is available. Two different approaches are

generally used:

assume the damping matrix to be a linear combination of the mass and stiffness matrix:

C=LM+K

(7) assume the damping such that it results into a viscous damping per normal mode equation, comparable with

the dashpot damping of a single mass-spring system

(8)

subá: C=jtM+cK

That means that from equation (3) NTCN = jT + K]vTICN

Also in the current analysis the concept of critical c is defined as the rate of this critical value

damping dcritj = 2 m1w is used. The actual value of

ci

according to

-Ccrit.

The normal mode equation of motion then becomes:

mä + 2!3wmãj +cma = X

(10) Comparing now the results of the 2 approximations of damping, equations (9) and (10), show their concurrence. Therefore, they are combined into one expression, containing both damping effécts:

mäj + (p +

+ icw)maj + wmaj =

The solution of this set of equations is:

= m...J(p,2

-

w2)2+ (p + +

22

In the investigation described, the ship structure was

excited by an exciter which generated a single force

(vertically oriented). In normal mode technique, this means that only in the i-th degree of freedom the excitation

vector f differs from zero. Then X =nf=n1f1.

Further, the measured and calculated values are given in

mobility terms, i.e., velocity divided by the magnitude of the force amplitude. Using equation (2) for the total response the mobility vector becomes:

n

m\/(c

-

w2)2+ (p +

_{2ßw +Kc)2}

2

I)

fj

ji

(p + +gw

accounting for the phase = arctg

2 2

Using the orthogonality properties (4) and (5),

NTCN becomes a diagonal matrix, so that for this case of damping the mentioned manipulation with the mode matrix_Nthe set of equations (3) is uncoupled and the j-th normal mode equation

becomes:

PUBLICATIONS OF ThE NETHERLANDS MARITIME INSTITUTE

Monographs

M I Fleetsimulation with conventional ships and seagoing tug!

barge combinations, Robert W. Bos, 1976.

M 2

Ship vibration analysis by rrnite element technique. Part lii:

Damping in ship hull vibrations, S. Hylarides, 1976.

M 3

The impact of Comecon maritime policy on western shipping, Jac. de Jong, ¡976.

M 4

liifluence of hull inclination and hull-duct clearance on perfor-.

mance, cavitation and hull excitation of a ducted propeller, Part I, W. van Gent and J. van der Kooij, 1976.

M 5 Damped hull vibrations of a cargo vessel, calculations and

measurements, S. Hylarides, 1976.

M 6 VLCCdeckhouse vibration, calculations compared with measurements, S. Hylarides, 1976.

M 7 Finite elements ship hull vibration analysis compared with full scale measurements, T. H. Oei, 1976.

M 8 Investigations about nôise abatement measures in way of ship's accommodation by means of two laboratory facilities,

J. Buiten and H. Aartsen, 1976.

M 9 The Rhine-Main-Danube connection and its economical implications for Europe, Jac. de Jong, 1976.