Damped hull vibrations of a cargo vessel, calculations and measurements

16. NOV. 197

ARCHIEF

Lab.

y.

Scheepsbouwkund

Technische HogesnI

h Iii

weIR

-MONOGRAPH PUBLISHED BY THE NETHERLANDS MARITIME INSTITUTE

Damped hull

vibrations

of

a cargo

vessel,

calculations

and

measurements

S. Hylarides

M5

May 1976

Damped hull

vibrations of

a

cargo

_vessel,

calculations

and

measurements

Dr. Ir. S. Hylarides

CONTENTS

Page

List of symbols 4

Summary 5

Introduction 5

Particulars of ship and the measurements S

CalcUlation of thehull vibrations 6

Presentation of the results 7

Critica! considerations 7

Conclusións 12

4 LIST OF SYMBOLS D damping matrix K stiffness matrix M mass matrix

f

force vector i imaginary unit gravity acceleration displacement vctor

ratio between damping and stiffness matrix

.1. Introduction

Present day ships often suffer from excessive vibrations, which make operation at reduced speed necessary. The

cause is twofold:

- large excitations, generated by the high powered

propel-ler;

- local resonances and hull girder response.

Resonances in the hull girder response can be left out of consideration because the main source of exciting forces is the propeller, generating forces,at frequencies in the range

of resonance frequencies of the complex hull girder vibration

modes. The amplification at resonance in this frequency range is moderate to small and can not longer be considered as a fundamental cause of unacceptable vibrations.

In order to obtain tools to control the vibrations aboard

a ship, investigations are being performed with regard to the.

excitations and the structural response. In the latter the three-dimensional representation of the structure and the formulation of damping are essential.

The full scale (exciter generated) hull vibrations of a motor cargo ship have been compared with the results of calculations based on the finite element computer program DASH /1/. These calculations did not account for damping.

With a slender beam representation of this ship the

effects of various formulations of damping were studied /2/:

- viscous damping (damping proportional with velocity) in which the damping matrix was put proportional to the

mass or stiffness distribution

hysteretic damping (damping proportional to the

dis-placement) with a distribution proportional to the

stiffness distribution. This latter assumption leads to a

complex expression for the stiffness matrix or to a

com-plex modulus of elasticity.

From this parameter investigation it was concluded that a viscous damping with the damping matrix proportional to the stiffñess matrix gives best results in comparison with the full scale measurements over the frequency range of interest.

This so-called "stiffness damping" was further also applied in a three-dimensional finite element representation

of the ship, showing also .a good agreement between

calcUla-DAMPED HULL VIBRATIONS OF A CARGO VESSEL, CALCULATIONS AND MEASUREMENTS

by

Dr. Ir. S. HYLARIDES

Netherlànds Ship Model Basin Wageniñgen

Summary

For the motor cargo vessel "Koudekerk" a detailed, three-dimensional finite element model /1/ and the full scale response to an exciter

/3/ were available.

By means of the principle of stiffness.damping /2/ the effect of damping on the vertical excited hull vibrations has been calculated and compared with results of the measurements.

Taking account of the effect of noise on the measurements the agreement between measurements and calculations is good

Due to the unknown effect of the noise no quantitative consideration about the found differences can be made. For this problem further investigationsare required. Care of this phenomenon has to be taken at further exciter experiments.

tions and measurements./2/.

The above theoretical study only considered the response of the point at which the exciter was mounted on board the

ship.

In the present report the investigation into

the.usefull-ness of the applied theory has been extended to other points of the same ship.

2. Particulars of the ship and the measurements.

The main particulars of the investigated motor cargo ship "Koudekerk" are given in table 1. Her main outline and the location of the exciter are shown.in figure 1. In figure 2 the brealc-down of the ship into finite elemeñts is shown /1/. The calculations refer to the numbered grid points.

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r Sr-i- I- r i- I I I I I I

Fig. 1. Bird's view and profile.of the M.V. "Koudekerk".

5

JRy

The full scale measurements refer to vibrations generated

by an out-of-balance rotary exciter, which was installed at the specially reinforced tweendeck in the aft ship /3/.

Along the hull several pick-ups were installed to determine

the vibration modes at the resonance frequencies. These

modes and resonance frequencies confirm the calculated

re-suits/l/.

The current report refers only to some points for which the responce over the complete investigated frequency

range has been measured /3/.

Table 1. Main particulars cargo-liner "Koudekerk"

3. Calculation of the hull vibrations

From the total finite element break-down of the ship, as shown in figure 2, large parts were taken to calculate their own stiffness matrix. The stiffness matrices of these sub. structures, roughly the size of one hold, were then reduced, resulting in a much smaller number of grid points. Next,

the stiffness matrices were assembled to obtain the complete system, after which many coupling points at the connections were eliminated. The successively reduction or condensation

of nodal points finally resulted in the 102 grid points as

shown in figure 2. This condensation method is generally

applied in finite element calculations /4, 5/to keep the computer time within reasonable bounds without loss of much accuracy /1/.

6

The mass matrix has also been restricted to these 102 nodal points. The lumped mass distribution was obtained such that the centre of gravity of the various parts of the structure and of the equipment remained unaltered. Ap-proximated representations for the moments of inertia /1/ had to be accepted.

The added masses have been calculated for each of the

vibration directions. separately (vertical and athwartships) and added to the constructional mass in the nodal points of interest, keeping the centre of gravity unchanged / 1/.

For the damping distribution no information exists. As mentioned in the introduction, the conclusion of a study /2/ where various formulations for the distribution were investigated was that a viscous damping matrix, proportio-nal to the stiffness matrix, yields best results. Based on the

results of that previous study this so-called stiffness damping

has been studied in more detail, by applying it in the cal-culations of the exciter generated hull vibrations of the "Koudekerk", for which full scale measurements were

available.

In matrix notation the set of equations for the vibrations

of the ship is given by:

+ D + K = f

in whìchM is the mass matrix, P is the damping matrix, L is the stiffness matrix,

is the displacement vector, leading to velocity and acceleration vectors ô and 6,

f

is the force vector.

Putting the damping matrix proportional to the stiffness

gives = ¡<K, so that for harmonic vibrations the following

set öf equations holds:

[-w2M +(1+iw<)K]6 f

That means that the complex matrix [-w2 M + (1 + iwK)K]

had to be inverted for each value of w to obtain the solution for the given right hand member f. This latter is given by

the excitation, which consists of a vertical force, applying at the point where the exciter had been situated.

This complex matrix refers to 102 grid points, each with 3 degrees of freedom in general, so that the order of the

Length overall 164.95 m

Length between perpendiculars 152.40 m

Breadth moulded 21.03 m

Depth to upper deck

ll.98m

Summer draught

as an open sheiterdeck 8.00 m

as a closed shelterdeck 8.91 m

Deadweight

as an open sheiterdeck 9940 metric tons

as a closed shelterdeck 12200 metric tons

Service speed 20 knots

Delivered power at 117 RPM 14,200 BHP

Number of propeller blades 4

Propeller diameter 6.00 m

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5. Critical considerations.

In evaluating the differences between calculations and

measurements it has to be realized that in a vibration

inves-tigation four different phenoniena which have been brought

together are essential:

-

the ship structure and its mass distribution,

- the mathematical model of the ship,

- the way of determination of the full scale vibrations, - the way of calculation of the vibrations.

In this report only vibrations have been considered which are generated by an exciter. 0f this exciter its location is known and the generated force is well defined in amplitude and direction.

The force is in fact a centrifugal force, generated by two out-of-balance rotating masses of which the eccentricity is known. Furtlermore,the force applied to the ship structure has been measured by mounting the exciter by means of

force pick-ups to the deck of the ship. The direction

(ver-tical or horizontal athwartships) is given by the phase adjustment of the two out-of-balance wheels. Therefore, in first instance the magnitude of the excitation force itself

is omitted in discussing the differences between the calcu-lated and measured results, because it can be considered as

rather accurate.

7

FuH scale measured response:

Fig. 3. Vertical response of exciter foundation to vertical exciter forces.

complex matrix amounts to 102 x 3 x 2 = 612.

4. Presentation of the results.

For a few points the response to the exciter has been measured over the complete frequency range, running from 1.5. up to 10Hz /3/. These points are:

- exciter foundation - after peak

- fore peak

- center of double bottom of hold 3.

The calculations of the undamped system take account of a vertical and horizontal (athwartships) excitation.

For the damped system only the response to a vertical excitation has been calculated. For these damped responses

two values ofK,the proportionality of the damping matrix

with regard to the stiffness matrix, have been considered: 0.001 and 0.002.

The calculated and measured response for the above

locations are given in figures 3 to 6.

In figures 7 to 9 the calculated response of the undamped

system to a horizontal excitation are shown and compared

with the full scale results.

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Il

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1 2 3 4 5 6 7

8

9 10

frequency Hz

( undamped:

Calculated response

effect of viscous damping, X: 0.001: x

(

effect of viscous damping,

: 0.002: o

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By application of the finite element technique it is possi-ble to realize an accurate mathematical model of the ship.

However, due to restricted computer capacity, due to

restrictions in funds and time, such a model becomes

gene-rally rather coarse, see figure 2. A further reduction in cost

and time was still needed and has been realized by reducing

the network of figure 2 further down to the mesh of which the grid points are shown and numbered in this figure 2. Therefore, with the final coarse network no information of very local aspects can be expected. Only with regard to the overall behaviour the correlation between calculations and measurements can be investigated. In the higher

frequency range, shifts in resonance locations will occur, so

that differences in the shape of a resonance curve and

different values in the vibration-level at a given frequency

have to be accepted. The aim of this report is to show to what degree the calculations fit with the measurements for this particular ship and to draw some conclusions for the

general case.

The measured vibrations have been evaluated by means

of a narrow band filter technique. The bandwidth was 2 Hz,

which is rather large in the frequency range running from 1.5 up to lO Hz. Only the vertical response of the

founda-tion of the exciter has been evaluated by a sampling

tech-nique, which cross-correlates the registrations of the

mea-sured vibration to the shaft rotations of the exciter /6/. In this case the bandwidth is reduced to almost zero, so that participation of all signals with frequencies different from

9 I I

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2 3 4 5 6 8 9 10 frequency Hz undamped:

Calculated response

effect of viscous damping )t: 0.001: x

(

effect of viscous damping X: 0.002: o

Full scale measured response:

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Calculated uridamped torsional response of exciter transverse.

4 3

After peak with location of

exciter (tween deck) and

vibration pick-ups 4 and 3

at top deck.

Fig. 9. Torsional response of transverse section at exciter location to horizontal exciter forces.

i j I i i

il

8 9 10

frequency Hz

11

After peak with location

of

exciter (tween deck) and

vibration pick-ups 3 and 3a

at top deck.

measured torsional response of exciter Full scale L30 transverse:

II

I,uiuuuu..uuaìuuiuui

w

1IÏiiiI1Iii1Íiiii1j!

il

t i 7 6 5 4 3 2 8 9 i frequency Hz

the exciter frequency or its multiples can be eliminated

practically completely. The level of the signal with regard to the noise level (signal-noise ratio) determines the number

of periods of.the signal required for proper evaluation of

the signal. For weak vibration-signals, disturbed by strong noise signals, a very large number of periods has to be taken. Generally, resonance vibrations will suffer less from nOise than out-of-resonance vibrations.

Noise is caused by the random vibrations of the ship due to many circumstances, in thecurrent, case mainly due to the propeller generated vibrations, because the

measure-ments were necessarily carried out on a sailing ship.

Pre-cautions have been made by running the ship at propeller speeds,. generating blade frequencies different from the exciter frequency. The propeller generated vibrations, however, may be assumed to be of the order of 0.01 g, which agrees with the general experience on board ships

having a low vibratiön level. For frequencies around 5 Hz

0.01 g corresponds to a displacemènt amplitUde of

10 m 100 jim. From the calculated and measured

response curves follows an average response of 0.2 x 10 m/kgf, which gives fòr án average exciter force of 3000 kgf

an overall displacement amplitude of 0.6 x 10 m =

60 jim, hence about half the propeller generated vibrations.

This relative vibratiOn level is further confirmed by the

fact that from model tests of this ship it followed that the level of the hydrodynamic shaft excitation with blade fre-quency runs from 2 to 12 tonf. Thus, on a average, twice as large as the exciter force. These hydrodynamic excita-tions, mutually combined and further combined with the hull pressure forces, will certainly lead to a total excita-tion that is considerably larger than the exciter force.

Therefore, it can be stated that the vibrational noise, induced by the running propeller, is appreciably larger than the exciter generated vibration lvel.

Furthermore, noise is induced by the irregularity of ro-tation of the exciter (frequency modulation). This also

induces transient vibrations of the same level as the inves-tigated vibrations.

The effect of noise is an increase of the finally evaluated signal and can be explained by the following. When at the

full scale tests the frequency of the exciter was increased, also the excitation forces increased, being proportional to

the centrifugal forces of the out-of-balance masses, hence

proportional to the frequency squared. Therefore, at cer-tain frequencies the eccentricity of the out-of-balance weights had to be reduced, so that the next frequency range started with a smaller excitation. From the measured

re-sults, as shown in the figures 3 to 9 as well as in the original

report of these measurements /3/, it has been found that after reducing the eccentricity most times the fmal signal was increased*. This is due tot the proportional reduction of exciter amplitude and vibration amplitude, so that the registration did suffer more from noise, which resulted into

higher values of the fInally evaluated signal.

The way of calculating the forced response is based on

determining the particular solution of the set of second order differential equations at the considered frequency.

* these are the discontinuities in the full scale measuredresponse

curves, given ¡n figure 3 to 9. 12

Therefore, the results do not contain any effect of noise or transient vibrations, so that in general their values must be

lower than the measured values.

These considerations are satisfactorily confirmed by the

response curves given in figures 3 to 9. A further

confirma-tion is found by the faët that the difference between

calcu-lated and measured values is smaller when the calculations

are compared with the measured results, evaluated by means of the sampling technique (Figure 3), than with the results, evaluated by means of the narrow band filter tech-nique (the other figures). As already mentioned this is be-cause of the fact that the sampled results will generally be

less affected by noise than the filtered signals. With regard

to the calculations there is, however, one aspect that needs

a further investigation. From the vertical vibrations,

calcula-ted as well as measured, it follows that the response per unit of excitation force decreases with the increase of the frequency. This aspect is also found in the measured response to a horizontal excitation, but to a less degree in the calculated response, especially in the lower frequency range. Probably this is caused by the assumptions which have been made in modelling the ship structure with re-gard to the exciter location, the construction of its foun-dation and its height above deck. In the calculations the horizontal exciter force has been applied at a point in the

deck, whereas on full scale the exciter applied a force and a moment to the deck. The cause of this is that originally the fmite element model had to serve the investigation of

ver-tical vibrations.

6. Conclùsions.

In the lower frequency range the response calculations are in good agreement with the measurement results

concer-fling the magnitudes of the natural frequencies as well as

with regard to the nature of the response curves.

However, for this frequency range the detailed finite ele-ment model as used here is not required. The lowest natural

frequencies can also be found by means of a

one-dimensio-nal slender beam representation of the hull girder, although for the representation of the two four-noded, vertical hull modes a more complex beam representation is needed.

The detailed Imite element model only serves the cal-culation of the ship response in the higher frequency range, especially with regard to the vibration level in the super-structure. This was not foreseen in the original set-up of the hull measurements, so that no conclusions can be made in

this respect.

With regard to the hull girder vibration level it is shown

that the trend and the level of the calculated response are in good agreement with the measurements. To quantify the observed differences it is first required to investigate the

effect of noise on the results of the vibration measurements.

Due to the participation of noise in the measurements the resulting values of the response may be expected to be

References

Oei, t.H.Finite element ship hull vibration analysis compared

with full scale measurements.

Hylandes, S.: Dampingin ship vibrations. Delft Technological

University thesis, 1974. N.S.M.B. pUblication No. 468.

't Hart, H.H.: Hull vibrations òf the cargo-liner "Koudekerk" NSS-TNO report No. 142 S, OctOber 1970.

Anderson, R.G.: Irons, B.M and Zienkiewiez, 0.0.: Vibration añd stability of plates using finite elêments. mt. Journ. of So-lids andStructures, 4, 1031-1055 (1968).

Hylarides, S.: Ship vibration analysis by finite elenfènt technique.

Part II: Vibration analysis. NSS-TNO Report No. 153 S, Mày

1971. NSMB publication No. 386.

Wereldsma, R.: Dynamic stress measurements improved by

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