REPORT No. 144 S December 1970 (S3/143)
NEDERLANDS SCHEEPSSTUDIECENTRUM TNO
NETHERLANDS SHIP RESEARCH CENTRE TNOSHIPBUILDING DEPARTMENT LEEGHWATERSTRAAT 5, DELFT
*
CRITICAL CONSIDERATION OF PRESENT HULL
VIBRATION ANALYSIS
(KRITISCHE BESCHOIJWING VAND E HUIDIGE ANALYSÈMEÏH0DEÑ
VOOR SCHEEPSTRILLTNGEN)by
IR. S. HYLARIDES
Netherlands Ship Model Basin
Evenals het in 1965 versehenen rapport no. 75 S ,,Scheepstril-lingen van het vracht en passagiersschip m s Oranje Nassau door van Horssen, geeft ook dit rapport een vergelijking tüssen gemeten en berekende eigenfrekwenties en trilvormen van een bestaaiid schip. Ook hièr zijn de berekeningen uitgevoerd door het schip als een Timoshenko balk te beschouwen en ook hier blijken de afwijkingen tussen berekende en gemeten waarden ñièt onaanzienlijk.
Weliswaar is het mogelijk de balkmethode te verbeteren, maar het feit blijft bestaan dat het balkmodel, zeker voor de trillingen van hogere orde fundamenteel onjuist is.
De auteur van dit rapport heeft zieh reeds enige tijd bezig gehouden met de toepassing van de eindige elernentenmethode voor trillingsanalyse, het pakket computerprogramma's hiërvoor is in principe gereed;
De eindige elementenmethode, die weliswaar een ved gede-tailleerder analyse van de scheepsconstructie nodig maakt, lijkt aanzienlijk betere perspectieven te bieden. Binnenkort zal een rapport hierover gepubliceerd worden als vervolg op het reeds eerder verschenen rapport 107 S.
HET NEDERLANDS SCHEEPSSTUDIECENTRUM TNO
Like report no. 75 S "Hull vibrations of the cargo-passenger motor ship 'Oranje Nassau", by van Horssen, issued in 1965, this report gives a comparison between measured and calculated natural frequencies and vibratioñal modes of an existing ship.
Here too, for the calculationS, the ship has been considered as a Timòshenko beam and here too the differences between the calculated and the measured valúes appear to be not inconsider-able.
Indeed it is possible to improve the beam method, but the fact remains that the beam model is fundamentally incorrect especi ally for the higher modes of vibration.
The author of this report has already for Some time been en-gaged on the application of the finite element method to vibra-tional analysis, the computer software for this in principle is available.
The finite element method, although it requires a much more détailed analyses of the ship structure, appears to offer better perspectives. A report on this subject will be published shortly as a sequel to report 107 S.
THE NETHERLANDS SHEP RESEARCH CENTRE TNO
CONTENTS
page List of symbols . 6 Summary. .. . . 7 i Introduction 7 2 Investigated ship 73 Measurement of the hull vibrations 9
4 Calculation of the hull vibrations 9
4.1 Transverse vibrations 9
4.2 Torsional vibrations 10
5 Discussion of the measured and calculated resuI1 of the hull vibrations 11
6 Double bottom vibration 13
7 The finite element technique in ship vibration analysis 14
8 Conclusions 14
References 15
Appendix Ï Virtually added mass of water 16
Appendix Il Bending- and shear stiffness 17
LIST OF SYMBOLS
b breadth on waterline
d draught
f
frequencyf, natural frequency of sprung mass g acceleration due to gravity
k empirical constant
¡ length
m virtually added mass of water m1 effective mass
meq çquivalent mass n natural frequency
n calculated natural frequency measured natural frequency
s shear stiffness t thickness of plating
A sectional area
B ship's breadth
CH virtual inertia coefficient for horizontal vibrations
C, virtual inertia coefficient for vertical vibrations D ship's depth
G modulus of rigidity
I
polar moment of inertiaJ
reduction factor for added mass L ship's length (over al!)M mass V shear force cc angle , wave length densityof water r shear stress A ship's displacement
CRITICAL CONSIDERATION OF PRESENT HULL VIBRATION ANALYSIS
by
Ir., S. HYLARIDES
Summary
Calculations of the natural hull frequencies and vibrational môdes have ben carried out, the ship being considered as a Tithoshenko beam Results are compared with values obtamed experimentally It is concluded that to improve the accuracy of the calculations it is necessary to brmg mto account the influence of local vibrating parts of the hull the reduction of the bending stiffness with increasing number of nodes and the coupling between the horizontal and t rsional hull vibration.
In fact these shortcomings of the beam method are caused by neclecting the three dimensional character of the ship structure hence a higher accuracy may be expected wh6n taking the spatial cOmposition of the hull into account.
As an effective means to do this, the finite element method issuggested.
i
InfroductknIn recent years ship's dimensions have grown rapidly. As a consequence the natural frequencies of the hull vibrations became lower. This means that. the higher natural frequencies of these large ships come within the range of excitation frequencies of the propulsive machinery and propeller.
To avoid vibration problems it is necessary to have t one's disposal an accurate method to calculate in the design stage the higher natural frequencies of the hull vibrations. The usual method is to consider the ship as a Timoshenko beam for which the natural fre-quencies and vibration modes can be calculàted. This method shows a diminishing accuracy when the. num-ber of nodes of the natural vibrations increases. The general trend is that the calculated values of the
natu-ral - frequencies increase faster than the measured
values.
To improve the accuracy one may try, either to develop methods which are adapted better to the prob lem, or to refine the available methods.
By means of measurements and calculations of the natural hull vibrations of the fast cargo liner "Koude-kerk" it was expected that the several shortcomings of the beam method could be analysed. In this way it should be possible to indicate whether the conventional method could be refined or had to be. abandoned.
To give the beam method a fair chance, great atten-tion had to be paid to the determinaatten-tion of the distri-bution of mass and stiffness.
2 Investigated ship
In table I the main characteristics of the ship are
presented.
The consttuction of thê ship satisfies the require-ments of Lloyd's Register of Shipping, classffication
Draught forward düring measurements at the beginning 6.60 m at the end 6.76 m Draught after during measurement at the beginnirg 7.56 in at the end 7.16 m Deadweight as an open shelter decker ± 9940 metric tons Deadweight as a closed shelter decker ± 12200 metric tons
Maximum propulsive power 14.200 BHP
at1l7rprn
Service speed 20 knots
Diameter of the propeller 6.00m
Number of blades 4
Fig. 1. Stern arrangement of m.v. "Koudekerk". Table L Main particulars of m.v. "koudekerk'.
Length overall 164.95 m
Length between perpendiclilars 152.40 m
Breadth moulded 21.03 in
Depth to upper deck 189m
Summer draught as an open shelter decker 8.00 m Summer draught as a clOsed shelter decker 8.91 m
meas uriñ gi
point
exciter on strengthened tweendeck measuring point 40 at the exciter
A ...A
3.31. C/seç,4noded vertical, vibratioh mode
L.15/sec, 1. noded vertical vibration mode
10 11
trame no
Fig. 2. Location of the meas'urihg points and the exciter during the measurementOf the hull vibrations.
17
399 /,3riode horizontal vibration mode
cdupled viith thé i ridded tdFsional
vibration mode
3838 37a37
97 108 117 127
-
- --
margin plateside girder-
_
'centre, girderengine foundation 29 28 31 34
26 29 32 39
27 30 33 36
measuring pOint doibLè
bottom hoLd.
i 59 178 192 205
2.46 C/sec,3 nOded verticaL vibration mode
1.96 C/sec,2noded horizontal vibration mode
Fig. 4. Measured horizoñtai and torsional vibration modes. Fig. 3. Mèasured vertical vibration rnodes
8
trame no
loo A 1 (longitudinal framing at bottom and
upper-deck).
It is a single-screw motor cargo ship of the
con-vertible open/closed shelter deck type.
The ship's stern has been constructed with a long boss and a spade rudder (figure 1). In this way large
screw clearances have been realised, resulting in a low vibration level of the vessel.
3 Measurement of the hull vibrations
Extensive measurements of the natural hull vibrations of the m.v. "Koudekerk" have been carried out by the Institute TNO for Mechanical Constructions [1]. For this an out of balance rotary exciter has been installed at the reinforced tweendeck in the after peak (figure 2). Thus with a constant frequency a vertical or horizontal harmonic force could be exerted to the ship. At several points along the hull vibration pick-ups were installed (figure 2). By this arrangement the several natural fre-quencies and natural modes have been measured (figures 3 and 4). 27 246 C, 3.34 isec C, 4.15 'see
Fig. 5. Measured vertical vibration patterns of the inner bottom of hold III at resonance conditions of the vertical hull vibrations. The numbering of the measuring points is given in figure 2. There exists no relation between the drawn amplitudes at the various frequencies.
3g
During excitation at the natural frequencies of the vertical hull vibrations also the vibration pattern of the inner bottom of hold III and of the foundation of the main engine have been determined (figures 5 and 6).
It has been found that a four-noded vertical vibra-tion mode sets in at two different frequencies, viz, at 3.34 and 4.15 c/sec. The location of nodes and anti-nodes is slightly different at these frequencies (figure 3). Also a small difference in the pattern is observed.
The vibration pattern of the inner bottom of hold III shows an amplification at these frequencies (figure 5), suggesting a natural frequency in the range of these two natural hull frequencies.
The measurements of the horizontal vibrations (figure 4) revealed that the three noded horizontal mode was coupled with the one-noded torisonal mode.
4 Calculation of the hull vibrations
4.1 Transverse vibrations
The ship has been represented by a slender beam with corresponding mass and stiffness distribution [2, 3, 4]. To calculate its natural frequencies the slender beam
dec k 2.46 C/sec engine foundation dec k 334 engine foundation dec k 4.15 Cj1 engine foundation
Fig. 6. Vibration patterns of main engine foundation at various natural frequencies of the vertical hull vibrations. The numbering of the measuring points is given in figure 2. There exists no relation between the drawn amplitudes of the various frequencies.
lo
u
t,am, noÍ. 50 - i_. 1.0-0.5 mass kgf sec cm2J In J1 ¿f
s-1
j
mass öf the ship
mass of the cargo
Fig.7. Mass distributiön of ship and cargo.
1
Fig. 8. Added mass distribution for vertical hull vibratiOns.
is represented by a discretied system of mass lumps, bending- and shearjoints. For this problem a computer program has been developed by the Institute TNO for Mechanical Constructions [5]. By means of this calcu-lation technique an approximation of the ñatural frequencies of the hull has been obtainéd.
For the calculations the distribution of stiffness and mass along the hull is needed. The added mass of water has to be taken into account. The mass of the ship has
been obtained from tables and drawings prepared during the construction of the ship. The distribution of the mass of the cargo has been obtained from the load-ing plan. The results are presented in figure 7. It must be noted that the 'cargo' includes fuel oil, stores etc..
To obtain an accurate distribution of the added mass the current calculations refer to 20 uniformly
spaced sections of the hull. Reduction of the added mass with the decrease in wave length of the hull vibra-tion has been taken into account. A detailed descrip-tion of these calculadescrip-tions has been given in Appendix I. The results have been represented in the figures 8 and 9. The distribution of the bending and shear stiffness has been given in figure 10 and has also been based on 20 uniformly spaççd sections. More details of these calcUlations are given in Appendix II.
In the figures 11 and 12 the calculated as well as the measured modes of the vertical and horizontal vibra-tions have been represent.ed.
Because the computer program does not take into account the coupling between the horizontal and tor-siònal vibrations, this effèct has been neglected.
4.2 Torsional vibrations
For the torsiònal hull vibrations no detailed calculation technique is available at present. In order to obtain an estimate of the lowest naftiral torsional frequency use has been made of an approximative formula deducted by Horn [6]. Details of this techniques are represented in Appendix IL frame no 0.5 0.4 0.3 0.2 0.1 50 1_00 mass[kfsec 2 noded 3 noded L
Fig. 9. Added mass distribution for horizontal hull vibrations.
to-.
0.5-frame no o lo
/
¡
/
ri
_._ o-. El L1kf cm2] o " hcrizonta bending stiffness/
/
__x7
vertucffbending 0kf] 1.0 ÌAG0.5-verticaL shear stiffness Fig. lo Distribution of bending and shear stiffness.
-' .' I -'
-I I\334%)
i 3.95%e 4rioded - I\.
_,
I 4noded \L.l5%ec) -H-I 5node.w
-S / enoineroem hold = 50 I- f5 1 15s calcuL L . I I - - measurd 1.24 %e 2 noded5 Discussion of the measured and calculated results of the hull vibrations
From several investigations it follows that the ratio between the calculated and measured natural freqUen-cies of the vertical hull vibrations will generally surpass unity considerably with the increasê of nodes in the vibration mode [3, 4, 7, 8, 9, 10, il] (see table II). Up Table IL Review Of the ratio nc/nm of the calculated and
measured values of the natural frequencies from several ships [3, 4, 7, 8, 9, 10, 11].
number of nodes of vertical vibration
-calculated -...measured references 2 3 4 5 6 7 [7] integral 1.200 0.865 1.011 1.114 1.230 method 0.894 1.006 1.071 1.113 0.739 1.118 1.256 Holzer- 1.111 1.296 1.433 Myklestad method 1.045 1.128 1.415 [8) 1.037 1.116 1.309 [3] 0.912 0.968 1.111 1.242 1.356 1.345 0.884 0.963 1.025 1.183 1.156 1.036 [4] 0.955 1.011 1.078 1.233 1.215 1.115 [9] 0.960 0.978 1.004 1.022 Eloi 0.891 0.955 0.983 1.056 1.036 [11] average of 14 ships 0.959 0.950 0.963 0.970
Fig. 11. Çalculated and measured vertical, vibration modes. Fig. 12. Calculated and measured hOrizontal vibration modes. Exciter frequencies between brackets. Exciter frequencies between brackets.
12
to the five-noded mode thesè results have been approx= imated linearly and thus reported in figure 13.
The same trend also holds for the natural frequencies of the vertical hull vibration modes of the
"Koude-kerk", indicated in figure 13 and by their valùes in table III, together with the results of the horizontal vibratiöns.
Table III. Calculated and measured natural frequencies of the
hull vibrations of the mx. "Koudekerk". Due to
difficulties at the measurements the queried values are not quite certain.
The pósitive gradient of the ratio between the calcu lated and measured natural frequencies as a function of the number of nodes of the vertical vibration modes
D]
number of nodes
Fig. 13. Ratio of the calculated and measured natural hull fre-quencies 77c/utm as a function of the number of nodes. Up to the 5 noded modes this relation may be approx-imated linearly.
The pints+correspond to the ratios c/m of the m.v. "Koudekerk".
Figures in square brackets refer to the references.
is probably caused by the fact that shear lag has not been taken into account in the calculations [12]. Due
to shear lag in the horizontal members of the hull
structure the bending stiffness against vertical deflec-tion will decrease in the neighbourhood of the nodes. In the vibration calculations this stiffness is supposed to be constant. Hence, at the higher noded modes the hull stiffness is overestimated so that the calculated natural frequencies will also be overestimated.
Up to now only Csupor [13] has taken shearlag into accoUnt in the calculation of vertical hull vibrations
[1 1 ]. This result is the average of 14 ships, whose in-dividual results are vei-y scattered. Further, the calcula-tions have been based on the same added mass for the several modes, thus leading to a smaller gradient of the ratio between the calculated and meastired natural hull frequencies.
To calculate the effect of shear lag it is necessary to decompose the bending moment into its harmonic components. This is a very complicated and time eon-sumirìg process, for which the wave lengths of the
vibrations have to be known beforehand. Therefore, it can be supposed that Csupor did not get the accurate reduction in bending stiffness, which explains the scattered results and the small gradient of the ratio
between the calculated and measured natural frequen-cies.
Nevertheless, it can be stated that by the results of Csupor the importance of shear lag is clearly indicated.
Another remarkable fact is the measurement of a
four-noded vertical hull vibration mode at two dif-ferent frequencies (table III and figure 3), whereas the calculations give only a four noded mode at one fre-quency. Qualitatively this can be explained by con-sidering the lowest natural frequency of the bottom in hold IlL The calculation of this natural frequency and the discussion of its influence is given at length in the following chapter.
The estimate of the lowest natural frequency of the torsional hull vibrations gives a value of 3.25 c/sec. The measurements indicate a natural frequency
be-tween 3.52 c/sec and 3.99 c/sec (figure 4). For the
three-noded horizontal hull vibration the calculated naturel frequency is 3.95 c/sec and the measurements give 3.99 c/sec. Although this result of the calculations seems very good, it is doubtful whether this may be stated, bearing in mind the measured strong coupling between horizontal and torsional vibrations (figure 4), whereas it was assumed to be negligible in the calcu-lations.
For the higher natural frequencies the measurements do not show dominant resonance phenomena, hence,
no finn conclusions can be drawn regarding this
aspect.
horizontal 2 2.00 1.96 1.02
3 3.95 3.99 0.99
4 £94
5 8.15
direction number calculated measured of vibratiOn of nodes n (c/sac) n (c/sec cIm
vertical 2 1.24 1.4 ?) 0.885 (?) 3 2.54 2.46 1.03 4 3.95 3.34 1.18 4.15 0.95 5 5.17 6 6.56
6 Double bottom vibration
The measurements of the vertical hull vibration of the m.v. "Koudekerk" revealed a fOur-noded vibration mode at two different natural frequencies (figure 3). At these frequencies also the vertical vibration pattern of the double bottom of hold III has been measured (figure 5). The results of these measurements suggest a natural frequency of this bottôm in this range. This is caused by the fact that the cargo is connected elastically to the hull by the double bottom on which it is stored. It is a realistic approximation to consider cargo and double bottom as a single degree of freedom mass-spring system, with its individual natural frequency. This frequency is determined by the effective stiffness of the double bottom and by the effective mass of the cargo, the double bottom and the added mass of water As has been indicated in [3] and [14] the effective mass, mef, can replaced by an equivalent mass, meq, rigidly attached to the hull. In the absence of damping the relation between these quantities is
meq
m1
1f 2/f,
wheref is the frequency of the vibrating hull andf the natural frequency of the sprung mass.
Obviously the local vibrating structure can have a considerable influence if f and f have almost the same value, thus if the natural frequency of the bottom is in
the neighbourhood of the vertical hull resonance frequency!
By means of a sandwich theory [15] an estimate can be made of the lowest naturaÌ frequency of a double bottom and the cargo stored on it, taking into account the added mass of water. This has been done fnrthe'
engine room and for hold III of the m.v. "Koude-kerk". In [15] a detailed description of this calculation has been given for the engine room. Similar considera-tiOns hold for the double bottom of hold III. It has
been found that the natural frequency of the double
bottom of the engine room equals 5.9 c/sec and of hold III 3.6 c/sec.
The vibration patterns of the main engine founda-tion have been measured at 2.46 c/sec, 3.34 c/sec and 4.15 c/sec. In comparison with the hull vibratiOns the results of these measurements suggest a natural fre-quency above 4.15 c/sec, thus more or less verifying the theoretically obtaiñed value of 5.9 c/sec (figure 6). The measurements of the vibrations of the inner bottom of hold III have been carried out with accelera-tion pick-ups with a rather low sensitivity. Therefore it was not- possible to detect the correct phase relative to the hull vibrations. The experiments thus only indicate a natural frequency in the vicinity of 3.34
c/sec (figure 5). The theoretically obtained value of 3.6 c/sec is in good agreement with this result and is assumed to be correct in the following analysis. For an explanation of the cecurrence of a four-noded vibration mode at two different naturál hull frequencies the following consideration can be given.
With the calculated natural frequency of the bottom of hold III of 3.6 c/sec, the-equivalent mass of this hold will be at 3.34 c/sec mer
me =
q = 7.15nzî 1_(3.34/3.6)2 and at 4 15 c/sec merme =
q = 3.O3mCf 1_(4.15/3.6)2The total cargo mass on the bottom of hold III is roughly one tenth of the total ship mass, excluding
the added mass of water. This means that due to the amplification of the mass in hold III the total vibrating mass is considerably increased at 3.34 c/sec and reduced at 4.15 c/sec. Thus at the two four-noded vibration modes notably different mass distributions occur,
giving a lower natural frequency for the larger total mass and a higher value for the reduced mass distribu-tion. In this way the occurrence of the two four noded modes can be understood.
Furthermore this consideration also verifies that the calculated natural frequency of the bottom of hold III has to be situated between the two four-noded natural hull frequencies, as has been stated above.
Another explanation of the phenonena observed
has been given by Den Hartog [14]. In the chapter dealing with the vibrations of systems with two degrees of freedom, a mass-spring-system provided with a second, smaller mass, attached elastically to the larger mass, has been considered. Between these two masses also a damper has bees located (figure 14).If damping is taken into account the resonance
dia-gram of the larger mass shows a maximum at two
different frequencies, thus two natural frequencies. For an undamped absorber the dotted lines of figure
14 hold.
Considering only the larger mass it seems that it possesses two nätural frequencies with the same vibration mode. A similar behaviour can be concluded from the results of the hull vibration measurements solely (figure 3). A plausible explanation pf this phe-nomenon is thus given by the example mentioned: the
larger mass representing the hull of the ship, the
14 C e a u a withoUt j damping II with I damping I / frequency
Fig. 14. Resonance diagram for the larger thass of the indicated system (m < M).
This diagram has been taken from reference [14] in which its derivation is fully described.
Although both considerations are only indicative,
they show clearly the influence of the double bottom elasticity on the hull vibrations. FOr a more precise prediction of hull vibrations it is thus essential to take this effect into account. The coupling with adjacent double bottoms, not considered here, has also to be taken into account.
In order to study the influence of the whole com-plex of holds containing cargo it seems acceptable to represent this system by a flat plate with bending stiff-ness (the double bottom) and a given mass distributiOn (cargo, bottom and added mass).
The boundary conditions are given by the ship
structure. It can be stated that the bottom is simply supported on the sides of the ship and on the bUlk-heads.
For this system the natural frequencies have to be calculated. Then the equivalefft masses of the cargo in the several holds can be incorporated in the vibration calculations of the hull.
To solve this problem the finite element technique will be a fruitful method.
7 The finite element technique in ship vibratiOn anaJ'sis
In the finite element technique a complex structure
like a ship's hull is considered as a composition of
elements for which the stress-strain relations can be deduced easily. Hence, the three-dimensional
proper-ties of the hull are maintained [16, 17]. Due to the fact that in the finite element technique the three-dimen-sional character of the hull structure is maintained the imperfections of the beam method, essentially caused
by considering the hull as a one dimensionally
vi-brating beam, are eliminated.
To take the three-dimensionality of the hull rigour ously into ccount a fine mesh of elements has to be chosen, resulting in a large number of linear equations with an equal number of unknowns. To solve these problems fast computers with a large memory-store are needed. As a further consequence it is necessary to supply the computer with a very detailed information of the structure.
The advantage is that no separate calculâtions of the distribution over the shiP's length of the bending and shear stiffnesses are needed, thus eliminating the intro-ductory calculations.
Many times already the finite element technique has shown its abilities. Especially in the strength
calcula-tions of statically loaded structures this method is widely used, leading to satisfactory results. Also, how-ever, for dynamical loadings good results have been
obtained (for example flutter phenomena of an aero-plane wing).
The application of the finite element technique in ship vibration analysis seems therefore very fruitful, especially if elements are used that are adapted to the construction of the ship. That means that also plate elements have to be taken into account, that can resist inplane stresses and normal stresses, whereas up to nowin the finite element technique the construction has generally been supposed to be composed of bars.
Sometimes also plates, which resist only shear, are taken into account.
8 COnclusions
The lower natural frequencies of the vertical hull
vibrations calculated with the beam method show the generally obtained difference with the experimentally obtained natural frequencies, in spite of the fact that
the bending stiffness has been presented in a very
detailed way. This has obviously been caused by the neglection of shear lag.
Once again the coupling between horizontal and torsional vibrations has been clearly illustrated.
The most striking result of this investigation is the discovery of the influence of the cargo, being elastically attached to the hull by way of the flexible double bottom. For the vessel being investigated this influence results in a four-noded vertical hull vibration at two different frequencies at which the hull is in resonance. Thus in the calculation of hull vibrations this
phenom-enon is significant. Probably this effect is partly also responsible for the general trend in the difference be-tween the calculated and measured natural frequencies of the vertical hull vibrations.
In the analysis of hull vibrations it is thus needed to take into accoint:
- shear lag,
- coupling between horizontal and torsional vibra-tions,
- local vibrating masses.
To iñcorporate shear lag and local vibrations use has to be made of estimates of the location of the nodes and the natural frequencies of the several hull vibra
tions. These estimates have to be checked with the
results of the calculations and, if needed, with better estimates a further calculation of the hull vibrations has to be performed, thus leading to an iterative Way of caiculätions.
The finite element technique takes these effects and that of coupling between the several vibration modes implicitly into account, thus leading to a direct way of calculation.
References
'T HAJtT, H. H., Hull vibrations of the cargo-liner "Koudò-kerk". Neth. Ship Research Centre TNO, Report No. 143 S,
October 1970.
TODD, F. H., Ship hull vibration. London, 1961
MCGOLDRICK, R. T. and V. L. Russo, Hüll vibration In-vestigation on SS "Gopher Mariner". SNAME, Vol. 63,
1955.
LEiBowiTz, R. C. and E. H. KENNARD, Theory of freely vi-brating non-uniform beams; including methods of solution and application to ships D.T.M.B. Report 1317, 1961. Viuzs, J. DE, Numerical calculation of vertical bull vibra. tions of ships by discretizing the vibration system. Neth.-Ship Res. Centre TNO, Reportno. 58 S, April 1964. HORN, F. Horizontal und Torsiònsschiffsschwingungen auf Frachtschiffen Werft, Reederei, Hafen, 1925, no. 18 and 19.
VREUODENHIL, C. B., Natural frequencies of fEce vertical ship vibrations. Neth. Ship Research Centre TNO, Report no. 60 5, August 1964. Also: I.S.P. Vol. 11, no. 122, 1964. HORSSEN, W. VAN, Hull vibrations of the cargo-passenger motor ship "Orañjè Nassau". Neth. Ship Res. Centre TNO, Report no. 75 S, August 1965.
KATSUo, O., T. Fwio, H. FUKIJDEN and O. MASATOÑO, A study of vertical vibration of ships Journ Zosen Kiokai Vol. 119, June 1966.
10.. ABAMSOM, H. N., A guide for the analysis of ship structures. U.S. Department of Commerce, 1960.
11. Committee Nò. 9: Vibration. Third thtemational Ship
Structures Congress. Vol. II, Oslo 1967.
12 SCHNADEL, G, Die Mittragende Breite in Kastenträgern und, Doppelboden. Werft, Reederei, Hafen, 1928, no. 5.
CSIJPOR, D., MethOden zur Berechnung der freien Schwin-gungen des Schiffskörpers..Jahrbuch ST.G., 50. Band, 1956. HARTOG, J. P. DEN, Mechriicàl vibrations. McGriw-Hill,
1956.
HYLARIDÈS, S., Estimation of the natural frequencies Of a ship's double bottom by means of a sandwich theory. Neth. Ship Res. Centre TNO, Report no. 89 S, April 1967. HYLARIDES, S., Ship vibration analysis by finite element techniqûe Part I: General reviev and application to simple structures, statically loaded. Neth. Ship Res. Centre TNO, Report no. 107 S, Decëiiiber 1967.
HYLARIDES, S, Méthode des éléments finis pour le calcul des vibrations de la poutre-navire. A.T.M.A., Session 1968. LS.P. Vol. 15, No. 169, September 1968.
JoosEN, W. P. A. and J. A. SPARENBERG, On the.longitudinal feduction factor fòr the added mass of vibrating ships 'with rectangular crOss-section. Neth Ship Res. Centre TNO, Report no40 S, 1961.
19 LEwis, F. M., The inertia of the water suftounding a vibrat-ing ship. SNAME, Vol. 37, 1929.
TAYLOR, J. L., Some hydrodynamical inertia coefficients. Phil. Mag., Vol. 9, 7th.series 1930.
LANDWEBER, L. and M. C. DE MACAGNO, Added mass of two-dimensional foñus escillating in a free surface. Journ. Of Ship Research, November 1967.
BIEZENO, C. B., J. J. KOCH and J. G. LEiuluceiucna, Onder. zoek naar de trillingsmogelijkheden van een schip 1956 (un published report).
'T HART, H. H., Vervòlg van de trillingsmçtingen met be-huip van de 5-tons scheepsexcitator aan boord van het m.s.
,,Naess Falcoñ" IWECOTNO Deift, Report no. 3082,
16
Appendix I
Virtually added mass of water
Since the inflúence ofthe water is hot the same for the vertical and horizontal vibrations the added mass is
different for both types of vibration.
The câlciatons tefer to 20 unifornily spaced
sections of the hull.
Added mass at vertical vibration
For the calculation of the added thass of water at the vertical vibrations use has been made of the theory described in reference [18] (figure 15) This theory has been based on a rectangular, infinitely long flexible cylinder, vibrating with a wave Ïength 2.
For the application of this theory an estimate of the wave length has to be made. From previous experi-ments these estimates have been chosen to be
= 110m
13= 83m
14= 66m
15= 55m
'6
46mFroth the calculated modes it föllöws that
22=130.4m
23= 94.7m
14 72.5m
25 58.5m
49.6m
For the midship section this theans that for the first vibration niode the calculated added mass was 6.9% too low.
If this value is used to cortect the Whole added mass at the vertical vibration, it may be stated that the total
j
2.0 C 1.8 1.4 1.2 1.0 0.8 0.6 Ö.4 0.8 1.2. 1.6 0.2 2.0 2.4 2.8 3.2 0.1 bearn/draft cy1ndsi -tiat stdp a sy mp t ot io a I behaviourFig. 15. Relation between the added mass of water per unit of length and the wave length of a rectangular crôss-section in vertical vibration.
mass incorporated was almost f x 6.9% too low, be-cause, roughly, the mass of ship and its loading is the
same as that of the added mass of water. Then the calculated natural frequency, which is roughly propor-tional to the inverse of the square root of the total
mass, has to be reduced by 1.7%
This way of correction results in a reduction of 1.4%
0.5 1.0 1.5 2.0
Fig. 16. Virtual inertia coefficient C, for verticalihull vibrations..
4.4 3.6
and 1.0% for the calculated natural frequencies of the three- and four-noded vertical hull vibration. This re-duction of the öorrection fâctor is in agreement with
figure 15.
From these simplified considerations it may be con-cluded that the trend of the relation. between the ratio lc/lm and the number of nodes in figure 13 for the. "Koudekerk" is somewhat too weak.
The theory to determine the added mass of water was based on a rectangular cylinder (figúre 15), which, in general, differs from a ship section. Lewis [19] has 'shown in 1929 that a relation exiSts between the
sec-tional area coefficient and the added mass of water
for the most current sections. Todd [2] has fixed this relation graphically (figure 16). From this figure the relation between the added mass of a ship section and
of its circumscribed rectangle (b xd) can be deter-mined.
Thus from the theory of Joosen and Sparenberg,
corrected with Todd s diagram, the added mass of the various sections has been calculated (figure 8).
Added mass at horizòñtál vibration
For the calculation of the added mass at the horizontal hull vibration use has been made of the classic calcula-tion methods developed by Taylor, Lewis,
Land-weber and Macagno [19, 20, 21] and represented by figure 17 [2h. The reduction factor from two to three
dimensional motion has been taken from reference [.19] (figure 18). By means of extrapolation the reduc-tion factors J for the fòur- and five-noded mode have been obtained. 0.55 0.50 0.45 .0.40 J I n 0.5 1.0 1.5 2.0 2.5 3.0 bearr/draf t
Fig. 17. Virtual inertia coefficient Cif for horizontal vibrations.
The calculations of the added mass for 20 equally spaced sections have been based on the equation [2]
mad = fCHJQIVd2
in which
CH = the virtual ihertia coefficieñt (figure 17) J = the reduction factor (figure 18)
Q the density of the surrounding water
d = draught of the section under consideration The results have been presented in figure 9.
1.0- 2 nded - noded
-
4,odd 0.9 Snoded O .8 12 14 1 18 20 length /dr ftFig. 18. Correction factor from 2 to 3 dimensioñál, hon.-zontal vibratidns.
For 4 and 5 noded vibrations the factor J has been obtained by extrapolätion.
Appendix II
Bendiflg- and shear stiffness
To represent the stiffness of the bull against transverse deformations only those parts of the construction have been considered from which it can be expected that they will be fully loaded due to these deformations. This means that all the longitudinal structural members up to the uppermost continuous deck have been
con-sïdered. This includes shell and deck plating, keel,
innerbottom and continuous longitudinal bulkheads
and striñgers which do not terminate at or in the
vicinity of the section in questioñ [2, 3, 4].
The deckhouses have not been taken into account. To have a good representation of the stiffness distri-bution, the bending and shear stiffness of 20 equally spaced sections along the length of the ship have been calculated. It is assumed that the curve through these points (figure 10) will represent satisfactorily the distri-bution of the hull stiffness.
Bending stWness
The bending stiffness is determined by the moment of inertia I of a section with respect to the neutral axis of that section. This moment bas been multiplied by the
mcx.ulus of elasticity (E .= 2.05 x 106 kgf/cm2) to obtain the bending stiffness EI. In figure 10 the
distri-18
bution of the bending stiffness for the horizontal and vertical vibrations is represented.
In the calculation use is made of the expression 1/EL To avoid difficulties in the calculations (division by
small quantities) it is advisable to overestimate the
bending stiffness at the ends of the ship This has been done by taking EI at the ends constant over one tenth
of the length.
These considerätions also hold for the shear
stiff-ness.
The determination of the bending stiffness is very laborious. Therefôre very often the calculations are restricted to a reduced number of sections.
To investigate the effect of the accuräcy, for the
current vessel fòur sections have been chosen located in the neighbourhood of frame nos. 40, 80, 120 and 1 6Ó. At the fOre- and afterpeak the stiffness is zero!
The distributioù thus obtained is in good agreement with the more detailed calculated curve (figuré 10). It may be expected that the loss in accuracy is acceptable. Shear stiffness
the sectional area represents to a certain degree the stiffness against vertical and horizontal shear defor-mations In principle the shear stiffness, s in a cross-section is given by the formula
T r2dA =
where
r = shear stress
V = shear force
A = sectional area of lougitudinal structural members
G modulus of rigidity. (8.01 x iø kgf/cm2)
Calculations of the stiffness of several sections [221 have indicáted that for the vertical shear Stiffness the calculations can be restricted to the summation
A1cos2 where
= area of member i
= the angle of that member with the veitical
(figure 19)
n = the number of structural members of interest
Multiplication of this summation by 0.8G gives a satis-factory approximation of the shear stiffness. The factor 0.8 has been based on the results of detailed calcula-tions which are compared with the described method
j+i
V
X
Fig. 19. Structural member i in the calculation of the shear stiffness.
[22]. Another result is that this summation can be
restricted to the outer plating of the hull and the con-tinuous longitudinal bulkhead and deck plating.
Similarly the horizontal shear stiffness is given by 0.8GA sin2 cc
the resúlts of these calculations are given in gure 10.
Appendix ifi
Calculation of the torsional vibrations
Ïn order to obtain an estimate of the lowest natural
torsional frequency Only an approximative equation, developed by Horn [6], is available
gGI =
¿(B2 + D2)L in which
n natural torsional frequency in c/sec = acceleration due to gravity
G = modulus of rigidity (801 x 1ø kgf/cm2) I,, = polar moment of inertia of midship section
= displacemeñt B breadth D = depth
L = length over all k = empirical constant
The polar moment of inertia I is approximated by [6] 4B2D2
IP
in which
= length along the outer plating of the midship section
The summation comprises the entire contour of the midship section.
Detailed calculations of the polar moment of inertia of a section indicate a small difference with the approx-imative equation used here In view of the accuracy of the torsional vibration calculation the approximated value of i,, will be satisfactory [6].
The emperical constant k is based on measurements of the one noded torsional vibration. It has been found
that for the
Wasgenwald[6] k = 1 58, for the
"Gopher Mariner" [3] k = 1.64 añd for the "NassFalcon" [23] k = 2.05.
The first two vessels have large hatches similar to
the "Koudekerk". This is nOt the case with the
"Naess Falcon". For the "Koudekerk" k 1.61 has been chosen being the mean value of the two com-parable vessels.FrOm the drawing of the midship section and the
main ship dimensions it follows that I,,= 70.2 m4. From the loading plan and the weight of the ship (figure 7) it has been found that A = 14073 tonf. Then the first natural torsional frequency equals 3.25 c/sec.
Measurements of the horizontal hull vibrations
(figure 4) indicate that at 3.99 c/sec a strong coupling between the horizontal three-noded and the torsional
one-noded vibration mode exists. At 3.52 c/sec a
strong vibration has been measured [1]. The signals, however, are very difficult to interpret and are, there-fore, omitted in this report. It is acceptable to expect that the naturäl frequency for the one-noded torsional hull vibration is highet than the calculated value, heñce the välue 1.61 for k is somewhat too low for this ship. In order to obtain an approximation of the natural frequency of the torsional vibration, the equation
PUBLICATIONS OF THE NETHERLANDS SHIP RESEARCH CENTRE TNO PUBLISHED AFTER 1963 (LIST OF EARLIER PUBLICATIONS AVAILABLE ON REQUEST)
PRICE PER COPY DFL.
IO,-M = engineering department S = shipbuilding department C = corrosion and antifouling dpartment
Reports
57 M Determination of the dynamic properties and propeller excited vibrations of a special ship stern arrangement. R. Wereldsma,
1964.
58 S Numerica] calculation of vertical hull vibrations of ships by discretizing the vibration system, J. de Vries, 1964.
59 M Controllable pitch propellers, their suitability and economy for large sea-going ships propelled by conventional, directly coupled engines. C. Kapsenberg, 1964.
60 S Natural frequencies of free vertical ship vibrations. C. B. Vreug-denhil, 1964.
61 S The distribution of the hydrodynamic forces on a heaving and pitching shipmodel in still water. J. Gerritsma and W. Beukel-man, 1964.
62 C The mode of action of anti-fouling paints : Interaction between anti-fouling paints and sea water. A. M. van Londen, 1964. 63 M Corrosion in exhaust driven turbochargers on marine diesel
engines using heavy fuels. R. W. Stuart Mitchell and V. A. Ogale,
1965.
64 C Barnacle fouling on aged anti-fouling paints ; a survey of pertinent literature and some recent observations. P. de Wolf, 1964. 65 S The lateral damping and added mass of a horizontally oscillating
shipmodel. G. van Leeuwen, 1964.
66 S Investigations into the strength of ships' derricks. Part I. F. X. P. Soejadi, 1965.
67 S Heat-transfer in cargotanks of a 50,000 DWT tanker. D. J. van der Heeden and L. L. Mulder, 1965.
68 M Guide to the application of method for calculation of cylinder liner temperatures in diesel engines. H. W. van Tijen, 1965. 69 M Stress measurements on a propeller model for a 42,000 DWT
tanker. R. Wereldsma, 1965.
70 M Experiments on vibrating propeller models. R. Wereldsma, 1965. 71 S Research on bulbous bow ships. Part H. A. Still water perfor-mance of a 24,000 DWT bulkcarrier with a large bulbous bow. W. P. A. van Lammeren and J. J. Muntjewerf, 1965.
72 S Research on bulbous bow ships. Part II. B. Behaviour of a 24,000 DWT bulkcarrier with a large bulbous bow in a seaway. W. P. A. van Lammeren and F. V. A. Pangalila, 1965.
73 S Stress and strain distribution in a vertically corrugated bulkhead. H. E. Jaeger and P. A. van Katwijk, 1965.
74S Research on bulbous bow ships. Part I. A. Still water investiga-tions into bulbous bow forms for a fast cargo liner. W. P. A. van Lammeren and R. Wahab, 1965.
75 S Hull vibrations of the cargo-passenger motor ship "Oranje Nassau", W. van Horssen, 1965.
76 S Research on bulbous bow ships. Part I. B. Thebehaviour of a fast cargo liner with a convéntional and with a bulbous bow in a sea-way. R. Wahab, 1965.
77 M Comparative shipboard measurements of surface temperatures and surface corrosion in air cooled and water cooled turbine outlet casings of exhaust driven marine diesel engine turbo-chargers. R. W. Stuart Mitchell and V. A. Ogale, 1965. 78 M Stern tube vibration measurements of a cargo ship with special
afterbody. R. Wereldsma, 1965.
79 C The pre-treatment of ship plates: A comparative investigation
on some pre-treatment methods in use in the shipbuilding
industry. A. M. van Londen, 1965.
80 C The pre-treatment of ship plates: A practical investigation into the influence of different working procedures in over-coating zinc rich epoxy-resin based pre-construction primers. A. M. van Londen and W. Mulder, 1965.
81 S The performance of U.tanks as a passive anti-rolling device. C. Stigter, 1966.
82 S Low-cycle fatigue of steel structures. J.J. W. Nibbering and J. van Lint, 1966.
83 S Roll damping by free surface tanks. J. J. van den Bosch and J. H. Vugts, 1966.
84 5 Behaviour of a ship in a seaway. J. Gerritsma, 1966.
85 S Brittle fracture of full scale structures damaged by fatigue. J. J. W. Nibbering, J. van Lint and R. T. van Leeuwen, 1966. 86 M Theoretical evaluation of heat transfer in dry cargo ship's tanks
using thermal oil as a heat transfer medium. D. J. van der
Heeden, 1966.
87 5 Model experiments on sound transmission from engineroom to accommodation in motorships. J. H. Janssen, 1966.
88 5 Pitch and heave with fixedand controlled bow fins. J. H. Vugts,
1966.
89 5 Estimation of the natural frequencies of a ship's double bottom by means of a sandwich theory. S. Hylarides, 1967.
90 S Computation of pitch and heave motions for arbitrary ship forms. W. E. Smith, 1967.
91 M Corrosion in exhaust driven turbochargers on marine diesel engines using heavyfuels. R. W. Stuart Mitchell, A. J. M. S. van Mòntfoort and V. A. Ogale, 1967.
92 M Residual fuel treatment on board ship. Part II. Comparative cylinder wear measurements on a laboratory diesel engine using filtered or centrifuged residual fueL A. de Mooy, M. Verwoest and G. G. van der Meulen, 1967.
93 C Cost relations of the treatments of ship hulls and the fuel con-sumption of ships. H. J. Lageveen-van Kuijk, 1967.
94 C Optimum conditions for blast cleaning of steel plate. J. Rem-melts, 1967.
95 M Residual fuel treatment on board ship. Part I. The effect of cen-trifuging, filtering and homogenizing on the unsolubles in residual fuel. M. Verwoest and F. J. Colon, 1967.
96 S Analysis of the modified strip theory for the calculation of ship motions and wave bending moments. J. Gerritsma and W. Beú-kelman, 1967.
97 S On the efficacy of two different roll-damping tanks. J. Bootsma and.J. J. van den Bosch, 1967.
98 S Equation of motion coefficients for a pitching and heaving des-troyer model. W. E. Smith, 1967.
99 S The manoeuvrabiity of ships on a straight course. J. P. Hooft, 1967.
100 S Amidships forces and moments on a C. = 0.80 "Series 60" model in waves from various directions. R. Wahab, 1967. 101 C Optimum conditions for blast cleaning of steel plate. Conclusion.
J. Remmelts, 1967.
102 M The axial stiffness of marine diesel engine crankshafts. Part I. Comparison between the results of full scale measurements and those of calculations according to published formulae. N. J. Visser, 1967.
103 M The axial stiffness of marine diesel engine crankshafts. Part II. Theory and results of scale model measurements and comparison with published formulae. C. A. M. van der Linden, 1967. 104 M Marine diesel engine exhaust noise. Part I. A mathematical model.
J. H. Janssen, 1967.
105 M Marine diesel engine exhaust noise. Part II. Scale models of exhaust systems. J. Buiten and J. H. Janssen, 1968.
106 M Marine diesel engine exhaust noise. Part ifi. Exhaust sound criteria for bridge wings. J. H. Janssen en J. Buiten, 1967. 107 S Ship vibration analysis by finite element technique. Part I.
General review and application to simple structures, statically loaded. S. Hylarides, 1967.
108 M Marine refrigeration engineering. Part I. Testing of a decentraI-ised refrigerating installation. J. A. Knobbout and R. W. J. Kouffeld, 1967.
109 S A comparative study on four different passive roll damping tanks. Part I. J. H. Vugts, 1968.
110 S Strain, stress and flexure of two corrugated and one plane bulk-head subjected to a lateral, distributed load. H. E. Jaeger and
P. A. van Katwijk, 1968.
111 M Experimental evaluation of heat transfer in a dry-cargo ships' tank, using thermal oil as a heat transfer medium. D. J. van der Heeden, 1968.
112 S The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. J. H. Vugts, 1968.
113 M Marine refrigeration engineering. Part H. Some results of testing a decentralised marine refrigerating unit with R 502. J. A. Knob-bout and C. B. Colenbrander, 1968.
114 S The steering of a ship during the stopping manoeuvre. J. P. Hooft, 1969.
