4LIN! 1977
.ARCKIEE
Lab.
v.
Scheepsbouwkunde
Technische HogeschbaL
Delft
1..________.
MONOGRAPH PUBLISHED BY THE NETHERLANDS MARITIME INSTITUTE
Finite element
ship hull vibration
analysis
compared with
full scale
measurements
T. H. Oei
M7
July 1976
Finite element
ship hull vibration
analysis
compared with
full scale
measurements
PREFACE
The investigation reported in the present monograph was
sponsored by the Netherlands Ship Research Centre TNO.
The Netherlands Maritime Institute has been entrusted with the publication.
CONTENTS page List of symbols 6 Summary 7 1 Introduction 7 2 Principles of calculation 7
3 Particulars about the applied finite element
pro-graimne 8
3.1 Substructuring and condensation 8
3.2 Finite elements. 8
4 Particulars about the ship and its finite
ele-ment representation 9
4.1 The ship 9
4.2 The finite element model 9
5 Details about the mass representation 12
5.1 Structural mass 12
5.2 Added mass 12
6 Comparison of the results of the calculations
and measurements 15 6.1 General information 15 6.2 Vertical vibrations 15 6.3 Horizontal vibrations 18 6.4 Torsional vibrations 22 7 Conclusions 24 - References 25
LIST OF SYMBOLS
ai, bi constants
f
Driving force vectorf Frequency
fn Natural frequency
F Statical load vector
K Stiffness matrix
M Mass matrix
m Mass
meg. Equivalent mass
8 Vector of displacements
u, v Displacement functions in x and y directions
x, y Cartesian co-ordinates
w Circular frequency
FINITE ELEMENT SHIP HULL VIBRATION ANALYSIS COMPARED WITH FULL
SCALE MEASUREMENTS
by
Dipl. Ing. T. H. Oei
Netherlands Ship Model Basin Wakeningen
Summary
The results of a complete ship hull vibration analysis, for which substructuring and the finite element technique have been used; are presented and compared with the results of full scale measurements:
It is shown that lower natural frequencies of the main hull and natural frequencies of local structural details can be determined accurately. Correct prediction of the actual ship Vibration level at propeller blade excitation frequency also seems to be possible.
Essentials about the finite element analysis are described.
1 Introduction
In 1967, hull vibration measurements were carried out on the cargo liner Koudekerk" /1/.
In order to check te adequacy of a finite element ap-proach compared with that of the classical beam method, a preliminary analysis with a coarse finite element model of the ship who performed in 1968 /2/.
From this study, it was learned that, in spite of the
coarseness Of that first model (see Fig. 1), the finite element
technique had at least the same accuracy as the classical beam method in determining the lower natural frequencies of the ship.
One of the striking results of the full scale measurements, however, two 4-noded modes occurring at two different natural frequencies, could not be simulated correctly in the
preliminary investigations.
The main purpose of the present analysis therefore was to
determine whether a considerable refinement of the grid of the model would remove this discrepancy.
Although at first not actually pursued, it became evident almost at the end of the analysis that a more important matter should and could be elucidated; namely the
ques-tion, whether a costly finite element analysis is justified as a
common means to determine not only natural frequencies, but also vibration levels throughout the ship for the higher
excitation frequencies at service speed.
2 Principles of calculation
Assuming harmonic excitation due to driving forces with
constant circular frequency w and neglecting damping, the following set of linear equations written in matrix notation
Fig. 1 Finite element model of the cargo-liner `Koudekerle, used for the rust vibration analysis of 1969
characterize the steady state vibrations of the finite element
model:
[K co2 M] 5 = f
where,
K = stiffness matrix expressing all force-deformation rela-tionships between the nodal points.
= circular frequency of the harmonic excitation forces. M = diagonal matrix representing simple lumped ship
mas-ses and added masmas-ses at the nodal points.
column vector with all unknown nodal displacements.
column vector with all amplitudes of the known ex-citation forces.
Spring elements as a representation of the hydraulic resto-ring forces acting on the real ship were not attached to the totally unconstrained model.
Equation 2.1. in this analysis therefore describes the vibration of a completely free - free" beamlike 3-dimen-sional model and it will yield a stable solution, except in the case of w being zero (rigid body modes) or w being one of the natural frequencies of the model. In this study, unit excitation forces were used for the driving force vector f
After solving Equation 2.1., the resulting responses 5
can be compared directly with the results of the measure-ments /1/, which were presented as mobility values or res-ponses per unit excitation force.
When the displacements 5, obtained for various frequencies
w, are plotted against these frequencies, the resulting
response curves clearly indicate with their peaks where the
natural frequencies of the system are.
This procedure is necessary, because Equation 2.1. was solved directly in this analysis, without first solving the eigenvalue problem.
The above approach completely neglects damping. Therefore, resonance peaks will also be found at high
fre-quencies. It is known, however, that for real ship vibrations,
damping becomes increasingly stronger at the higher
fre-quencies, so that resonance vibrations are smoothed down
to a kind of mean level.
3 Particulars of the applied finite element program 3.1 Substructuring and condensation
The applied program DASH (Dynamic Analysis of Ship Hulls) has been developed at the N.S.M.B., especially for
use in ship vibration problems /3/.
Due to the limited core capacity of the CDC 3300 computer used (32 K words), from which only a part can be used for matrix handling, a substructuring technique has become a major feature of the above program.
The philosophy of substructuring is based on the fact that, without loss of accuracy; in vibration problems inertia
may be described much coarser than elastic behaviour.
Therefore, substructures with a dense network of nodal points, needed to properly describe the elasticity, are con-densed to substructures with a considerably reduced num-ber of points.
These are coupled to bigger combinations, which are again
reduced and coupled in an arbitrary repetitive manner.
5=
f=
(2.1.)
8
Reduction or condensation of nodal points refers to the algebraical elimination of the unknown displacements of these points in the static deformation equations, K S = F. By doing so, the influence of the finite elements con-nected to nodal points, which have to be reduced, will be
transmitted correctly to the remaining coefficients of the reduced stiffness matrix, see /2/.
However, nodal points located on sections of sub-structures where these will intersect in a next coupling stage, should not be reduced after the above so-called exact reduction, unless boundary gaps are to be taken for granted at such locations after coupling.
In that case, another kind of reduction has been prac-tised in the current analysis, based on prescribed con-straints, and chosen in such a way that the displacements of the reduced points become linearly dependent on the dis-placements of the remaining non-reduced point in between which they lie on such boundary locations.
Although this linear reduction method saves conside-rable computer time because it enables a higher reduction rate in early stages of substructuring, it has been abandoned in following analyses due to its undefined stiffening effect. A visual check of the vast amount of finite element input
data is necessary. This is realized by a thorough
examina-tion of the plots of the substructures.
After the reduction of stiffness matrices of the
sub-structures and after their coupling, such checks are no longer
possible because no input data for plotting exist for the
new situations.
Therefore, a subroutine is used to examine the row terms of all initially evaluated or just coupled matrices for possible nulls. The stiffness matrix would be singular if a row is
found, which is completely filled with zeros. Such a row indicates that a certain nodal point (known from the row
number) is allowed to move freely in one of its degrees of
freedom.
Resequencing of the numbers of the nodal points, which
is needed in the various stages of coupling, easily leads to serious errors, which may endanger all previous efforts.
An adequate check against these kinds of errors can
only be made by means of clear schematic drawings, which
allow a spatial view of the coupled substructures in their
various coupling stages.
3.2 Finite elements
Except for a beam element type, having 6 effective degrees of freedom at each nodal point for stiffness, 3 rotational and 3 translatory, all other elements used with DASH are
simple membrane and bar elements, which consider
stiff-ness only on the in-plane or axial translatory degrees of freedom at their nodal points.
Bending elements very swiftly increase the number of unknowns due to their additional rotational degrees of
freedom.
Their use, can be limited however to such places in the model where a correct description of bending is required, such as for the shafting arrangement, rudder horn or points where propeller induced moments must be applied.
In this analysis no bending elements were applied, only membrane and bar elements for which the following
for the rectangular plate element:
u= ao +ao + a2y + a3
(vx2+y2) 2b3 yv = bo +bix + b2y+b3 (x2+vy2) 2a3xy
(the normal stresses deduced from this displacement field are a linear function of the co-ordinates and the shear stresses are constant over the plate area).
for the triangular plate element: U = ao + al x +a2y
V =bo +b1 x+b2 v
-(all the stresses related to this displacement field, are constant over the plate area).
for the bar element (which only has axial translatory
degrees of freedom)
u=a +a x
o(this gives a constant normal stress over the length of
the bar).
In the above functions, u and v are the displacements in
respectivelY, the x- and y-direction of the local co-ordinate
systems of the elements; al and b1 are coefficients.and v is
the Poisson's ratio.
A detailed description is given in /4/ as to how the coefficients of the stiffness matrices of the various elements are determined with the above assumed deformation
pat-terns.
Since four points usually do not lie in one plane, the program defines a mean plane" in between the four points of each rectangular element in which the calculated
stiff-ness is supposed to be effective. In the case of modelling,
the shell plating it is therefore not needed to choose
cauti-ously planes between the curves of the frames in which
rectangular elements are to be described.
4 Particulars of the ship and its finite element repre-sentation
41 The .ship
The main particulars of the cargo liner Koudekerk" are
given in the following Table I.
Table 1 Main particulars of cargo liner Koudekerk".
4.2 The finite element model.
The finite elethentmodel represents the ship as a whole, reaching from port to starboard. This was done to enable the incorporation of asymmetry of the ship's masses and construction.
For a proper introduction of exciter forces into the main ship's hull, the aft body finite element mesh has been much more finely chosen, compared with the grids of the forward
substructures.
The average fineness of the grid aft, is about 3 frame
spa.-cings (1.81 m) in longitudinal direction and in vertical and
transverse direction the grid is laid in the decks, floors and main longitudinals. In the forward parts, the average fineness is about 7 frame spacings (5.6 m).
Parts of the ship construction lying in between the
chosen grid planes are simply lumped and added to neigh-bouring elements of the grid.
The aft body itself consists of 4 initial spatial substruc-tures which are shown in Figs. 2 and 3, with their particular numbering sequences needed before and after coupling and
after the first reduction of nodal points.
These two figures not only illustrate the fineness of the
grid, but also show how manual resequencing of the num-bering should and can be visualized to avoid errors.
The other more forward substructures, generally con-sist of 3-dimensional double bottom and plane deck sub-structures between the main bulkheads. Examples of these
substructures are given in Figs. 4 and 5.
After reduction of several nodal points, as illustrated in Figs. 4 and 5, all substructures were coupled together to the complete finite element model, which then had a total of
146 nodal points, see Fig. 6.
In order to reduce the solution time even more, a fur-ther reduction of points was made; the final finite element
model with 102 nodal points is shown in Fig. 7.
Triangular plate elements have only been used for the
representation of the shell plating in the aft body. All other plate fields are represented by rectangular plate elements.
Bar elements are used for the frames and deckbeams. The sizes of the initial substriictures were relatively small in this analysis, the biggest only had 66 grid points.
A great number of relatively small substructures need
many matrix manipulations with rows and columns of the concerning matrices, which are more or less invisibly" done by the computer.
Therefore, visualization of fewer but much bigger
sub-structures offers a better guarantee against errors.
Modelling the whole ship by only one complete finite
element model is not practicable either, the earlier men-tioned advantage would be lost due to a confusing plot of
the model crowded with overlapping elements.
Therefore, substructures with nodal point numbers up to a maximum around 200 were chosen in further analyses. Input data of such substructures can be completed fast
enough by separate teams, computatiOn time is still
reaso-nable and, more importantly, restarts due to unforeseen constructional changes or late detected errors can be done
very easily and considerably cheaper.
9
length overall 164.95 m
Length between perpendiculars 152.40 m
'Breadth moulded 21.03 m Depth to upper deck 11.89 m Summer draught
as an open shelter decker 8.00 m
as a closed shelter decker 8.91 m
Deadweight
as an open shelter decker 9,940 metric tons
as a closed shelter decker 12,200 metric tons
Service speed 20 knots
Delivered power at 117.5 RPM 14,200 BHP
Number of propeller blades 4
Propeller diameter 6.00 m
Frame spacings in aft and fore peak 0.61m
Fig. 4 10 ISO framerdonbarg 107 DECK SUBSTRUCTURES J. 'IS
0 MAIN NODAL POINT
NODAL POINT FOR EXACT REDUCTION 0 NODAL POINT FOR LINEAR REDUCTION
T211
Fig. 2 Aft body substructure, consisting of 4 initial substructures, Fig. 3 Aft substructure, with resequenced numbering of points A, B, C en D, with the nodal point numbering sequences before after first coupling and reduction of points
Fig. 5
105
100
I WAS nUMber-11---40 Cr:C1(- AND DOUBLE BOTTOM SUBSTRuCTURE,
115
MAIN NODAL POINT
NODAL POINT FOR EXACT REDUCTION 0 NODAL POINT FOR UNEAR REDUCTION
..ilz-Namrds1.-,e1 --arr
TAZIANL
7.16*Allagn7411Mnfigi
4111illasiw r AM-740131 koraessima... $10Warirjaile
WASIE1041/
Iiiinligil Libit gniNatiostO
phATROVIlit, '1174 -0411075ONO
*3.'07
iroe
Joe
kffIKFig. 6 General plan of the finite element substructures of the cargo-liner `Koudekerle, with 146 nodal points before the last reduction
Fig. 7 Finite element model of the cargo-liner `Koudekerk% with 102 final nodal points after last reduction
Table II gives a comparison between the first
investiga-ted finite element model in 1969 and the finer meshed one
which was studied in the 1972 analysis.
Table II Comparison of the finite element models of the
kOudekerk"
5 Details about the mass representation
5.1 Structural mass
The mass matrix was evaluated by means of a simple mass lumping program, which maintains the original mass distri-bution as close as possible.
In other finite element programs use is made of the
so-called consistent mass matrix, which also maintains the
original kinetic energy per finite element. This is done by using the same deformation fields to express the kinetic
energies as have been used to express the strain energies of the finite elements, the derivatives of Which lead to the stiffness matrices.
The disadvantage of this way of determining the mass
matrix for all initial nodal pdints is that it depends on the frequency. Consequently, for each frequency considered the mass condensation up to the final number of grid points must be repeated.
Therefore, in this analysis the masses were luniped to
the final nodal points from the start. This lumping procedure is based on the conservation of the following static demand.s
for the masses: same centre of gravity, same moment and moments of inertia.
The lumped Mass matrix is a so-called diagonal matrix, whereas the consistent 'Matrix also has non-zero off diago-nal terms.
FOr a proper distribution of the lumped masses, all the
masses of the structure, stores, cargo and other items must be known with their centres of gravity.
The structural mass data, can of course, be evaltiated from the vast amount of finite eletnent input data. But at
the time of this analysis, such a program was not available.
In this analysis therefore detailed information provided by the shipyard about all ships masses, including those of the construction was used for the lumping procedure.
A computer program first of all distributes each given
mass to the two nearest aft and forward vertical grid planes, then further in these planes to the nearest lower and upper
grid levels and, finally, transversely over the three nearest points on these particular levels.
12
This distribution is done in such a way that in all main co-ordinate directions the centre of gravity is kept the
same.
For the transverse distribution the moment of inertia has been maintained too.
Figs. 8 and 9 give the longitudinal mass distribution of steel Furthermore special nodal points have been chosen exactly on those locations in the ship where heavy mass concentrations are found, in order to avoid unrealistic mass distributions.
Such mass concentrations are caused by heavy machinery,
anchor chain, etc.
and cargo of the original ship and the coarse redistribution
of the according lumped mass system used in this analysis. Liquids in the various tanks were regarded as solid masses
and treated accordingly.
3.3311 no 4 33434 SI 0 IT/ 71
I
,
Fig. 8.Mass distribution of ship and cargo
rI H
t.--,MASS DISTRIBUTION OF SNIP AND CARGO
... REDISTRIBUTION FROM LUMPED MASS MATRIX
DISTRIBUTION GIVEN BY SHIPYARD mass oi the cargo
.liT
so as 113 1117 lste analysis Kouclekerk (1969) 2nd analysis Koudekerk (1972) Number of nodal points 59 102 -Number of bar andplate elements 110 2451 Reduction stages none -Initial number of nodal points before reduction
Fig. 9 Added mass distribution for vertical hull vibrations
2-MODED
3- MOOED
4-110000
Fig. 10 Added mass distribution for horizontal hull vibrations
5.2 Added mass
Added masses of the surrounding water were calculated
separately and added to the final model mass matrix. These
added masses were derived in the same way as for the preli-minary analysis, described in /1/.
The vertical added mass for example was determined
after the theory described in /5/, which gives the added mass as a function of the draught and breadth of a section and also of the particular wavelength of the vibration mode
which is being investigated. Afterward, a correction was
made by means of the well known Lewis reduction factors /6/ for the various sectional area coefficients. The virtual inertia coefficients needed for the calculation of the hori-zontal added mass have been derived from /7/ and the cor-rection factors for 3-dimensional flow, which depend on the vibration mode, have been taken from /8/.
A more detailed explanation of the above procedure is given in the appendices of /9/.
The resulting added mass distributions are given in Figs. 9 and 10.
Because originally only vertical vibrations up to the
4-noded modes had to be investigated, no further added
masses for the higher modes were evaluated. For the same
reason, the horizontal added mass of the rudder has not
been included in the analysis. Longitudinal added masses were also neglected.
The above added mass distributions have been lumped to
the main vertical grid planes and added to the lumped
struc-tural masses on grid points below the water surface in these planes.
The change to a higher vibration mode must be judged from the resulting vertical and horizontal responses during the calculations, in order to enable consideration of the appro-priate added masses in the mass matrix for further calcula-tion.
The vertical and horizontal added masses, elialuated for the 4-noded vibration mode, have also been used for the
calcu-lation of forced vibrations at frequencies where actually
higher noded modes occur.
For the calculations at 2 Hz, where the vertical added
masses of the 2-noded as well as of the 3-noded modes were used, no big differences were found for the calculated levels of the vibrations responses.
Only in the aft body for point nos. 1 to 7, two or three times smaller responses were found when calculated with
the added masses of the 3-noded mode, but this local effect is apparently caused by a shift of the vibration node aft. In general, however, the choice of a more or less correct added mass seems to have little effect on the calculated
vibration levels.
6 Comparison of the results of the calculations and
measurements
6.1 General information
The results of the vibration measurements in /1/ are presen-ted as velocities per unit force excitation, also known as
mobilities. (cm. s-1.kgf-1).
The measurements were carried out on board the ship during good weather conditions. Ship motions therefore
were very moderate.
1 -T 0 MOBILITY so,
111111.1111111111 1111111111111111=11111
111111M1111111111111111111111111111111
III
IIIIII
. 1111111111 IIIEWINOMMIINIII
II I li
-I -I
1111111111111111
11
III
---irrnikl=111111111 1111
II
II
0___.---;--'''
-111111111IIIIII
MMORININI11111111111111111
11 Mtit:Wbft-- -IN= 3=111111
11 111WwiTIEMEMEINIIIM1111
111111111111111_ 11111111M11111
11 11111011111111100115^15=11111
n
el omminaliiiiimmui
II 1192=1111111111111111111111
II MEC _:::muum" """t"."9121111111111
a. PAING11111111111111111111111111111
1 111111:11,111111111111111111111111111
11 1111111111M1111111111111111=11111111
11 IIIIMINCM111111111=1111111
11 IIIII1111gli,7
-PROPELLER SERVICE J1--3 FRE°UF NCI I
1111.11111t
it
MIIIIVIMM
11111111111111111111Samw21111111.I
1111
NMI
I II 111111111=
1111
IMI
MI
1111111111141 II
MI=11M
MI
- .
111111
F: E 11111mom
1Ho
!hull
oloommol
mumudi
Li 3 11111111111111111 MUM
T111111111 1111111111 I NI
2=111111111111111111111
1111(
111111111
1111111
I El MOBILITY ISEC.IIGF1111.1111111111111111
.11
11111M11111
In
11111111111
1111mai
II
111111111111uI
PROPELLER SERVICE1111.ADE2iiECIUENCY1111111 1 In
4 41)=Gel II
1111
MOBILITY reClil lIT MOBILITY11111
ii
111iuii
II
MN IN 1111
lull
III
111I
III
LL PROPE1
NUMMI illam
R S R.VICEBL AOLFREOUENC YA mechanical exciter with out-of-balance masses was .
placed in the afterbody on the stiffened tweendeck above the steering gear compartment.
The exciter point on the model correspond with nodal point number 5, which lies in the plane of the tweendedk, see Fig. 7. In fact, the horizontal exciter force had to be applied at a location somewhat above that tweendeck, due to the particular exciter construction, this circumstance has been ignored here:
An electronic filter with a' bandwidth of 2 Hz was used to separate noise from the recorded signals.
The vibration measurements were carried out with out-of-balance masses fittedlt different radii, for a number
of frequency ranges.
OVerlapping measured vibration data, due to the
change of position of the out-of-balance masses often differ
considerably, causing the somewhat peculiar vertical shifts
of the mobility curves in the figure, which are presented
in the following sections.
6.2 Vertical vibrations.
Calculated and measured vertical mobility curves are given
in Figs. 11 to 14. Indications in the Figures about the
oc-currence of the modes in the various frequency ranges, were
made after a study of the calculated responses.
All measured mobility curves in the Figures were derived
from data evaluated by means of the narrow band filter
techniquell/.
Only in Fig. 11, a measured mobility curve is given, which was evaluated by means of a sampling technique.
Results of the first analysis in 1969 with the coarse finite
element model have also been included in Fig. 11. In judging Fig. 11, it can be stated first Of all that the
re-finement of the finite element model has resulted in a fair simulation of the two 4-noded vibration modes.
The calculated and measured mode shapes are shown in Fig. 15.
Nothing definite can be said about the possibility of also getting a better agreement between the calculated and
measured natural frequencies of the 4-noded mode by means of a further refinement of the initial grid mesh. The discrepancy may have another reason. We know, for example, from /1/, that the double bottom has caused
the peculiar 4-noded mode, because it acts as a local mass
spring attached to the hull. In /9/it has been shown how
strong ship vibrations are affected by the mass resting on
the double bottom.
The actual double bottom mass (m) may be replaced by an equivalent mass (rne.) rigidly attached to the hull; meg. can be calculated with the following equation from /9/:
,where f is the frequency of the
yibrating hull andf the natural
frequency of the double bottom.
It is shownin/9/ using the above equation, that the
equi-valent mass meg of the double bottom of hold III of the Koudekerk", Will have a value of about seven times and minus three times the actual mass m at the two natural frequencies of the 4-noded mode.
16
A small increase or decrease of the double bottom mass, which is roughly one-tenth of the total ship mass, will
therefore have a considerable influence on the concerned natural frequencies themselves. Considering this, it seems more likely that the mass distribution used for the analysis (one single nodal point in the middle of the hold) was not accurate enough for the particular matter of the 4-noded
mode frequencies.
The natural frequencies, which can be derived from Fig. 11 are summarized and compared with the other
re-sults in Table III.
Table HI Calculated and measured vertical natural frequen-cies of the hull.
More important than a reasonable agreement between the natural frequencies is the agreement in level between the
calculated and measured responses.
In the lower frequency range, the measured vibrations indicate almost no damping and the measured and calcula-ted curves show a similar tendency with regard to the natu-ral frequencies and with regard to the level.
In the higher frequency range, the measured vibrations
show strong damping, leading to certain vibration levels
around which the undamped calculated values are situated. Only the calculated mobility values of the double bot-tom point No. 66, see Fig. 13, are found fluctuating around
vibration levels considerably lower than measured. No ex-planation could be given for this behaviour.
If local natural frequencies are of interest, these can be found in a rather simple way, by drawing the resonance
Table IV. Calculated and measured local double .bottom na-tural. frequencies.
Number
of
nodes
Natural frequencies of the vertical hull vibration (Hz)
Measured
Finite element calculations
1969 1972 about 1.4 1.2 1.3 2.46 2.4 2.45 4 __ _3.34 4.15 3.75 3.48 4.5 .
Natural freq of the double bottom (Hz)
Measured After sandwich
theory /10/ 'Finite element calculation . 1969 1972 -Double bottom
of hold III near to3.34
-3.14 Double bottom of hold IV
_
395
_ Double bottom of engine room 9.." 5.9 5.23 meg. = 1._f2/f2curves of relative movements between such local points and
nearby points on the main hull girder. This is shown for two double bottom points and the engine foundation in
Fig. 16.
Natural frequencies of these local parts are clearly
found in this way and are given in Table IV, compared
with values estimated from the measurements. Results of calculations on the same double bottoms, but isolated from the main hull girder, by means of the sandwich theory /10/
are also included.
6.3 Horizontal vibrations
Calculated and measured horizontal mobility curves are given in Figs. 17 and 18.
For the higher frequencies, a somewhat better
agree-ment of the mean vibration level between the scattered
calculated values and the measured vibrations is observed as
compared with the vertical vibrations.
For the lower frequency range ( 2 Hz), a considerably lower calculated vibration level is found.
No explanation could be found for this discrepancy. The change to a different vibration mode could not be determined so clearly and unmistakeably from the complex horizontal responses of the nodal points as was possible for
the vertical vibrations.
So an appropriate choice of the correct added mass for the calculations becomes rather doubtful.
It does not seem particulary useful therefore to evalu-ate different horizontal added masses for a great number of modes for the analysis, as their differences are small.
The lower natural frequencies which can be derived
from Fig. 18 and those resulting from the measurements are
summarized in the following Table V.
Table V Calculated and measured horizontal natural fre-quencies of the hull.
The calculated and measured horizontal vibration modes
are given in Fig. 19.
6.4 Torsional vibrations
Interpretation of torsional vibrations from the calculated
response data is difficult, since no purely torsional
excita-tion moments were applied to the model.
17
Number
of
nodes
Natural frequencies of the horizontal
hull vibration (Hz)
Measured
Finite element calculation
1969 1972
2 1.96 2.2
3 3.99 4.05
Fig. 15 Calculated and measured vertical vibration modes. Fig. 16 Magnification curves of double bottom in hold II.
Exciter frequencies between brackets engine room and hold II
05 90 las al tat 12 22 2 38 nodal point no 4 23 79 37 57 03 98 n ea 01 50 84 0 78 VI 92
1111. 00 MOBILJTY {UAW] % X MOBILITY [34R14 0 0 0 PRoVELLE E 2 uI
INI-
MI
INN
1111MEM
BLADE FREOLENC 0I 1222 27
n2d0Ip512n2 IS 22 20
The calculated and measured torsional vibrations modes are given in Fig. 21.
35 53 so 7 al 65 77 03 91
37 WI 51 59 DI 70 711 el
--341402122141ASURED
CALCULAIZD
Fig. 19 Calculated and measured horizontal vibration modes. Exciter frequencies between brackets
irg Ica
Le
However, the horizontal excitation force in point 5, while not applying exactly in the shear centre, will cause torsional
movements, which will cause vertical responses of off- to-4
center-line nodal points.
Studying these vertical responses due to the horizontal excitation force, it was found that maximum values are reached between 34 and 3.6 Hz and also between 4.0 and 4.2 Hz. But it is not clearly known by what particular reso-nance vibration they are caused. Therefore, Fig. 20 was
drawn, giving the vertical and horizontal responses of the
portside nodal point number 23 due to a unit horizontal excitation force in 5. It can be seen that the horizontal vibration dominates for the most part, except for the fre-quency range from 2.3 to 3.6 Hz. In that range the vertical response amplitudes are defmitely higher, implying that a strong resonance vibration occurs, which can only be a torsional one.
The natural frequency that can be derived from the
above mentioned figure is given in the Table V with a simi-larly determined full scale value.
Table VI Calculated and measured torsional natural fre-quency of the hull.
MEASURED CASOULAIED
Fig. 21 Calculated and measured torsional vibration modes due to horizontal excitation.
Exciter frequencies between brackets
96 CO 102
100
19
Natural frequencies of the torsional
Number
of
hull vibration (Hz)
estimated
from Finite element calculation
nodes 1969 1972 measurements between 3.52 and 3.99 3.49 10 AO pgsmig___
Fig. 20 Horizontal and vertical response curves of point 23 (see fig. 7) due to horizontal excitation.
7 Conclusions
The general conclusions from this finite element analysis are:
Lower natural frequencies of the ship can be determined as accurately as with the classical beam method, with the advantage that corrections due to the effective breadth concept can be omitted.
Natural frequencies at which, equally noded modes occur, due to the introduction of an extra mode by a local
spring-mass system, can be determined.
Natural frequencies of large local spring-mass systems, as
for example the double bottom, can be determined fairly
well.
20
At higher frequencies, for example around propeller blade frequency at service RPM, the calculated, undamped
responses fluctuate around values which correspond well with the full scale vibration levels, in which resonance and
anti-resonance responses are already smoothed out by
dam-ping.
For the description of the elastic behaviour of the model
it seems generally sufficient to choose an initial grid
fineness of about 5 frame spacings (3 to 4 m).
The condensation of the initial nodal points should not
be continued so far as was done in this analysis. A
some-what finer final model simulates the influence of local
spring-mass systems in a better way and it also provides a
better description of the complicated defoimation patterns
References
't Hart, H.H., "Hull vibrations of the cargo-liner "Koudekerk". Netherlands Ship Research Centre, T.N.O., Report No. 143 S. October 1970.
Hylarides, S., "Ship vibration analysis by finite element tech-nique", part II. Netherlands Ship Research Centre T.N.O., Report No. 153 S. May 1971 or N.S.M.B. Report No.
69-164-AS.
Hylarides, S., "DASH, Computer program for Dynamic Ana-lysis of Ship Hulls". Netherlands Ship Research Centre T.N.O., Report No. 159 S. September 1971.
Hylarides, S., "Ship vibration analysis by finite element tech-nique", part I, General review and the application to simple structures, statically loaded. Netherlands Ship Research Centre T.N.O., Report No. 107 S or N.S.M.B. report No. 69-088-AS. Joosen, W.P.A. and J.A. Sparenberg, "On the longitudinal reduction factor for the added mass of vibrating ships with
rectangular cross-section." Netherlands Ship Research Centre T.N.O., Report No. 40 S. 1961.
Todd, F.H., "Ship hull vibration", Arnold Ltd., London 1961.
Landweber, L. and M.C. de Macagno. "Added mass of
two-dimensional forms oscillating in a free surface", Journal of
Ship Research, November 1967.
Lewis, F.M., "The inertia of the water surrounding a vibrating ship", SNAME. Vol. 37, 1929.
Hylarides, S., "Critical consideration of present hull vibration analysis". Netherlands Ship Research Centre T.N.O., Report
No. 144 S. December 1970 or N.S.M.B. report No. 68-127-AS. Hylarides, S., "Estimation of the natural frequencies of a ship's double bottom by means of a sandwich theory". Nether-lands Ship Research Centre T.N.O., Report No. 89 S. April
1967.
PUBLICATIONS OF THE NETHERLANDS MARITIME INSTITUTE
Monographs
M 1 Fleetsimulation with conventional ships and seagoing tug/
barge combinations, Robert W. Bos, 1976.
M 2
Ship vibration analysis by finite element technique. Part III:Damping in ship hull vibrations, S. Hylarides, 1976.
M 3
The impact of Comecon, maritime policy on western shipping,Jac. de Jong, 1976.
M 4
Influence of hull inclination and hull-duct clearance onperfor-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 VLCC-deckhouse vibration, calculations compared with
measurements, S. Hylarides, 1976.
M 7
Finite elements ship hull vibration analysis compared with fullscale measurements, T. H. Oei, 1976.
8 Investigations about noise abatement measures in way of ship's accommodation by means of two laboratory facilities, J. Buiten and H. Aartsen, 1976.
