Vibration Analysis of Superstructure by Mode Synthesis Method
1. Introduction
Recently an international standárd has been established about the allowable limits of hull vibrations following a
mounting demand for the reduction ofvibrations and noise. It is hence very important to estimate the vibration characteris-tiÇs of hull structures precisely and design taking antivibration measures into consideration.
The Vibrations of ships are extremely complicated as the vibrations of a local structure such as the superstructure are coupled with the vibrations of the main hull structure and the various forces of excitation include the main engine excitatioh force, propeller excitation force and wave excitation force1. In particular, to quantitativély estimate the vibration response of the superstructure and other structures, it is not sufficient to analyze the vibrations of local structures and the main hull
individually, but it is necessary to estimate the coupled
vibration characteristics by comprehensively considering these vibration characteristics A comprehensive estimation of the vibration characteristics is also important for structural efficiency and optimization from the aspect of vibration control plan, and to understand the complicated vibration characteristics of the complex structure, it is necessary to distinguish the principal vibrating bodies in the coupled
vibration mode, and to understañd the effects of constituent structures on each other, it is required to calculate vibrations repeatedly in various conditions.
To analyze such complicated coupled vibrations of a ship's
structure iìi higher modes, along with recent progress in
computer technology as shown in Fig. 1, it has become possible to apply the finite element method to vibration analysis of the entire hull including local structures23> To apply the finite element method to every ship, however, as it would requirean enormous length of time in preparation and editing the input and output data, and it would be too expensive, désign tasks must be improved. At MHI, accordingly, as a quick, precise
and practical method of estimation by using the results f the finite element method, a technique called MAST based on the mode synthesis method has been developed and is put into actual use"1.
This report, referring first to the outline of these analytical techniques discusses the validity of computing techmques by
MTB179February1988
Koji Kágawa * KazunobU FUjita * Kazuhisa Yanagi
A mode synthesis method was developed for estimating the dynamic characteristics
of
ship structures, in which the coupled Vibrationofthe main hull with substructures such as a superstructure the main engine propulsion shaft etc were taken into consideration It is expected that the practical simplificationof the junctioñ points between the main structure and the substructures will make the method convenient for use in the design stage.
The application of the method to longitudinal vibrations of the superstructure is shown with examples of an LNGcamer and a
container ship The method is also used for parametrical studies of the vibration of the superstructure excited by shaftaxial vibrations Comparisons of the calculated results with the measured values indicate that the method is valid for vibration studies at different design stages from early planning to detailed design.
Calculated 207.6 prn Measured 2O8cpm
Calculated 228 cpa Measured 222 cpm
Measured vibrations of the ships bottom structuÑ
Fis. i Example of calculated mode by 3-dimensional FEM
In vibration analysis of a ship which is a complicated structure a
large scale three dimensional FEM detailed calculation is applied and the satisfactory agreement of calcúlated and measured válùes is shown.
applying them to large LNG carriers and contäiner ships, and introduces the results of sensitivity anlysis by applying this mode synthesis method to the response of superstruçtures induced by shaft vibrations.
2. Outline of mode synthesis method
As the method of obtaining the vibration characteristics of a compound structure composed of several constituent struc-tures, the coupled vibration mode calculation method has beén recently reported as an efficient method. This method divides the compound structure into its constituent structures, expres-ses the vibration characteristics of these constituent structures in modal parameters and synthesize these component struc ture characteristics to obtain the vibration áharacteristics of the entire compound structure. This method requires thé development of analytical techniques suiting the object struc ture.
The method of synthesizing the constituent structures for an actuál ship (hereinafter called the mode synthesis method) must take the features of the hull vibratiOns ho.rn in Tables i and 2 into consideratiôn sufficiéntly. That is, the vibration characteristics of individual structures of a ship are evaluated in the design stage. Furthermore, it is often required to study the coupled vibration characteristics with mutually related structures.
Table i Characteristics of ship vibration There are large and small local structures, and the construction is complicated.
Because of steel structure, the vibratory damping factor is small, and vibratiòns are likely to be transmitted. The vibration source and the living quarters are close to each other, and vibration problems tend to occur.
In each local structure, rational boundary conditions are set, and design takes into consideration the avoidance of resonance. For this purpose, it is necessary to predict and evaluate the effect of coupling from other structures, in each local structure.
To compare the vibration response with the allowable limit, complicated, large-scale vibration calculationsare needed, including the vibration transmission characteristics from the vibration source to the response point.
To meet the varied dpmands of shipowners and shortening the planning and building period, adequate estimation evalution is necessary.
Since the structure is complicated, the vibration estimation method is limited to each structure, and it ishard to establish an estimating technique for the entire system Feedback of actual ship data and experience into servicing and plänning often depends on the learning effect of casestudies by individual designers.
In both exciting force and structural vibration characteristics, it is necessary to understand the factors having influence and feed them back for antivibration design, clarifying the degree of any effects.
Table 2 Requirements Of mode synthesis for ship vibration
(1) So it can be applied simply and quickly to various vibrations depending on the progress stage of planning and design, such as simplified calculation, datailed' cal.ilstion and design modification
To be able to analyze the entire system since the effects of the fore part of the main hull may not be always ignored although vibration problems often occur n the stern. To analyze the entire system, in applied calculatiön, from various vibration sources to responsepoints.
(4) If the vibration mode of a local structure is the lower mode, it may beone of higher modes when considered for the entire system. Hence, lowering the precisioiof ahigher mode calculation may bave relatively little influence, and the determination of resonant frequencies of the local structures and countermeasures should be easy. (5) Since many points are unknown about excitng forces, the evaluation of structural calculation models is not yet established. Therefore, calculation models should be
evaluated frequently.
(6) So that it is easy to establish measures against trouble, it should make use of intermediate results, etc., while considering the changes m frequency and mode, plysical orooerties and local structures, and the effècta of coupling of structures, etc.
(7) Conventional calcúlating methods, design concepts. andactual data should be directly and efficiently utilized. Especially at the time of initial planning, simple data should be usable, and coupling calculations of new structures from former structure should be easy. Also, to be suitable for the accumulation of know.how; it should systematically
accumulate data.
(SiSince the dasnoine factor la generally small, it may be approximated in calculations, but the damping factor should be considered for each local structure.
(9) Because of varied and diverse exciting forces, the calculation model of parts subjected to exciting forces should be easy to investigate with respect to the effective exciting force and other elements depending on the properties of exciting forces and characteristics of local structures.
In the evaluation of the -vibration characteristics of local structures, the independent vibration characteris1ic is usually evaluated iñ the boundary conditions including past data and design concepts. Therefore, as a techhiqúe for computing these coupled vibrations, as stated above, it is preferable to analyze the coupled vibration characteristic by efficiently combining the constituent structure models representing the independent vibration characteristics determined in the boundary condi-tions intrinsic to each structure and evaluating the coupling effect among structures.
From such a concept, to divide the compound structure in the mode synthesis method, it is classified into the main structure. and súbstructures so the vibration characteristics ofindividual structures can be evaluated independently. As the boundary condition at the connecting point, the vibration characteristic in the unconstrained mode is determind for the main structure with the connecting point set free, while the vibration charac-teristic in the constrained mode is determined for substructures with the connecting points fixed. By synthesizing the
charac-teristics of constituent structures obtained in this way, the
ship's vibration characteristic and response characteristic as
the compound structure are determined, and in order to
facilitate applicatiôn to actual ship calculations, a simplified method considering the features of ship structures is introduced concerning constituent structures. By synthesizing the damping of constituent structures, the damping of coupled
vibrations is determined.
-Details of the theory of the calculation are given in
References (4) and (5), and only an outline is shown here.Assuming the structure to be composed of a main struclure (System 1) and substructure (system 2), the vibration character-istics are obtained separately, and are then synthesized andthe coupled vibrations of the entire structure is calculated. The vibration mode of system i is expressed by the overlapping of vibration modes with the part connecting to system 2 set fiee, and the vibration mode of system 2 is expressed by the sum of the Vibration mode of the cystem with the connecting part ith system i fixed and a term due to forced displacement from
system 1.
Kinetic energies T1, T2 of system 1, system 2, and strain energies U1, U2 are given as follows.
T14±1TM1±I, T2=-±2TM2.2
U1 = --Z1TK1Z1, U2 = --Z2TK2Z2 (i)
Z1=1q1, Z2= Tøq1+ø2q
where,
M1, M2 : mass matrix of system i, system 2 K1, K2 : stiffness matrix of system i, system 2
Z1, Z2 : displacement vector of system 1, system 2
01, 0 : mode shape matrix of system 1, system 2
q1, q2 modal coordinate of system 1, system 2 T : conversion matrix for obtaining the term due
to forced displacement from system 1. Simultaneous equations for the free vibration of the couple1d
systems are expressed as follows according to Lagranges
formula, assuming the modal coordinates to be qij; q2 as generalizing coordinates:
rGll Gu1rj10
LG21 G22]L2J
The frequency determining
detjGI=O where, 1= q2 Lq2.]T(i=1_._N2) G11=["i11 (cvii2_22),J _22P1+S1 G12 = - 22p2T +S2T G21 =.À2P2 +52
G=fl2i(co2i2_A2)J
À circular frequencyN1, N2 : number of modes of system 1, system 2
rn2 effective mass of i-th order mode of system 1,
System 2
circular frequency of i-th order mode of system 1, system 2
The foráed vibration of a coupled system is calcúlated by overlapping coupled modes, but the damping factor of the coupled system must be determinecL Generally, the value of the damping factor is determined on the basis of measured data, and damping factor data is often collected from individ-ual constituent structures. If each constituent structure has a different damping factor, it is necessary to consider how thé damping of each structure would contribute to the damping fáctor of the coupled system. The damping factor of the coupled system synthesizing the damping factors of the
constituent structure is determined from the following, equa-tion.
.(u)
(i!)12 2ô2J(L)(L)2.2
(5)where,
effective mass of ith order mode of coupled system logarithmic decrement of ith order mode of cou-pled system
,: logarithmic decrement of i-th order mode of system 1, system 2
3. ActUal ship calculations using modé synthesis method
3.1, Vibration calculation for LNG carrier
The superstructure of a ship is always occupied by the crew, and is the command station fòr steering the ship and cargo handling, but since its location is close to the main sources of vibrations Such as the propeller and thàin engine, its antivibra-tion design i the most important item in antivibratio' design of the hull. However, the structure is very complicated, and there are too many errors when the simplified estimation of vibrations sed and calculation for the entire ship by the finite element method is too time-consuming to be applied in routine
design work. On the other hand, in the estimation method
usiñg the mode synthesis method, first the vibrations of a single
structure are calculated as a primary estimate, and as the
design progresses, the modé synthesis method is repeatedly applied, and the entire structure is estimated.
Using this method, it is possible to estimate wjth increasing
precision as the design progresses, and necessary measures can be taken at each stage. In order to apply the mode synthesis
method to the analysis of actual ship's structures, it is
necessary to use a practical calculation model suiting the
actual Structure, that is, to divide the ship structure into proper component structures and to modelize connected parts. In the vibrations of a superstructure, since relative displacement occurs in the verticâl direction with respect to the main hull, the stiffness of local elastic deformation of the main hull undér
the Superstucttùe is assumed to be the base spring of the
superstructure, as a part of the superstructure, or a substruc-ture (system 2), separating from the main strucsubstruc-ture (system 1) which is the main hull.
In coupled vibrations with the main hull, the superstructure and the main hull are connected at the front and aft ends of the superstructure for the sake of simplicity, and the basespring of the superstructure issupported by thelower end of the. straight line which links these two points. Since the bouhdary condition of the top of the superstructure is free, when the connecting
points are simplified in this way, the forced displacement
component of the superstructure due to movement of the main
hïll, that
is, the forced displacement compOnent of the substructure (system 2) becomes a rigid body motion, and inthis case, the coupled vibration calculation is extremely
simplified, which is practical for actual ship calculatiOns. In this paragraph, hereinafter, an example of application of this méthod to an LNG carriér is shown.
The LNG carrier has two superstructures in the stern, the accommodation and funnel. As shown in Fig. 2, this vessel is divided into the main hull and two superstructures, with the main hull regarded as the main structure and the two super-structures as subsuper-structures.
Superstructure i : accommodation (living quarters and
wheelhouse)
Superstructùre 2 : funnel (stack and engine casing) In ships without longitudinal bulkhead such as cargo ships, container ships and LNG carriers, since the frequency of the ship's bottom structure vibration, is low, the Ship's bottom vibrations and vértical vibrations of main hull 'are coupled,
JJ-i- -J
Funnel structure ....V' Aommodationstructure
L xExDx d=269.0X 44.5 X25.0 Xl1.5rii
MTB179February 1988
rl
n-H
r'i
r'
dMain huH
Fig. 2 Subdivision of LNG carrier for mode synthesis method
The structure was subdivided for application of the mode
synthesis method. formula is q11= i 1Ni) q21= 21eiAt= lNz) P1=Ø1T7'TJf27'Ø1 "2=02TM2T01 = Ø1T7'TJ( S2=ø2K2Tø of coupled system
and, as a result, the frequency and vibration mode are
considerably different from the vibrations estimated for asingle beam(6x7). Therefore, as for the main hull of the LNG carrier, ignoring the two superstructures, the boundary condi-tion at the connecting part of each superstructure is free, and
coupling with the ship's bottom vibration
is taken into
cnsideration in calculation.As the superstructure, the accommodation and funnel structures are taken into consideration. To calculate the
individual free vibration characteristics, it is necessary to set up a calculation model considering the stiffness of members in the main hull of each structural base as the base spring, but instead of setting up such individual models for the sake of convenience, a three-dimensional finite element method calcu-lation model is assumed by considering the engine room and
the afterold, mainly relàting to the stern superstructure of
the LNG carrier. In this calculation modél, the number of nodes is 2 283, the number of beam elements is 1 527, and the number of plate elements is 3 704. In the calculation of the accommodation structure, in the condition of constraining the funnel structure, the stiffness and mass of the accommodation
structure are taken into consideration, the free vibration characteristic is calculated assuming the stiffness of the members in the main hull as the base spring. On the other hand, in the calculation of funnel structure, the vibration
characteristic is similarly calculated by constraining the
accommodation structure.
The independent frequency of each superstructure in shown in Table 3. In this table, the longitudinal primary vibration mode of the accommodation structure is expressed by S-1, and that of the funeLstructure by F-1. The longitudinal vibration modes ofthe accommodation and funnel are shown in Fig. 3(a) and (b). The vibrations of the accommodation structure
consist of the rotation of the base spring and the shear
deformation of the accommodation structure itself. The vibration of the funnel structure is almost due to the rotationof the funnel, as compared with that of the engine casing.
To analyze the coupled vibration by the mode synthesis
method described in Chapter 2, first the coupled vibration is calculated taking the main hull as the main structure and the
Table 3 Coupled frequency of LNG carrier
=ii.
ii -:-
--' L_--+ .i r -- -r---.-'- o'-'-
-
--, ---- u_-,___ ,_._,._.__co._-u_-,___ 4. __,_ _.__ ...ri'r
T-4-(a) Longitudinal vibration mode of accommodation otructure
vI!,1, LI
-:----'
s-=:-hwa :1r
I
it
r-'r-r---- _____
-
::4-___,:_, T- -'- -w-
--(b) Longitudinal vibration mode of funnel
Fig. 3 Vibration mode of accommodation and funnel of LNG carrier by FEM model
Vibration modes of accommodation structure and funnel struc-ture alone obtained by three-dimensional FEM calculation.
accommodation structute as the substnicture. The. cóupled
system formed in this way is assumed to be a new main
structure, and the funnel structure is regarded as a substiuc-ture, and the coupled vibration is similarly calculated, and the coupled vibration characteristic
of the entire system
is obtained. To determine the logarithmic decrement of thecoupled vibrations, the logarithmic decrement of the main hull is determined by reference to Taylor's paper8 relating to the
vibrations of a ship, and those of the superstructure, by
reference to thé measufed data fòt exisiting ships. The resiilt of calculation of the coupled frequency is shown in Table 3, and the coupled vibration mode is given in Fig. 4.
Using the results ofthese calculations, response calculations were performed. When a unit exciting force was applied to the stern end inthe vertical direction, the acceleration at the top of
the superstructure was calculated, the results of which are
shown iñ Table 4. For comparison with these- results, the vibration of the LNG carrier was measured by utilizing sa
trial test condition. First, by analyzing the vibrations in an anchor test, the frequency in the lower mode was obtained. Then, by vibrating the stern in the vertical direction with a vibration generator mounted on the stern, the resonant fre-quency of the longitudinal vibration of the superstructure was measured. The measured frequency is shown in Table 3, the measured acceleration in Table 4, and the resonance curve of accommodatiòn and -fÚnel in Fig. 5. From these results,, th following findings wire -obtained.
The frequency calculation results by this method have ai error range of 5 to 10%, from the lower mode to the higher mode, in comparison with the measured frequency.
The vibration response results calculated by this method have an errOr range of 20 to 25% of the measured values.
-Vibration mode Uncoupled frequency (cpm) Coupled freqienc' (cpm) Ratio Calculated -Measured N., N0/N0 NOB/NO N,,1/N,, N0 - N0 N0, V-1 45.9 45.5 45.2 46 0.991 0.985 0.989 V-2 97.6 96.8 96.0 90 0.992 0.984 1.067 V-3 145.1 144.0 143.0 - 135 0.992 0.986 1.009 V-4 186.7 185.6 184.9 172 0.994 0.990 1.075 V-5 222.6 221.8 221.2 213 0.996 0.994 1.038 V 6 260 8 260 0 259 2 241 0 997 0 994 1 076 V-7 282.2 281.8 281.3 271 0.999 0.997 1.038 V-8 283.8 .283.7 283.7 293 1.000 1.000 0.968 S 1 310 5 289 5 289 8 303 0 932 0 931 0 954 F-1 351 7 - 360 3 352 - 1 024 1 024
V-3: 143.Ocprn V-4:184.gcpm /
r
)!'
V-5:221.2cpm V-6:259.2cpm V-12:39 1.lcpniJ'27>
V-13:403.2cpm V-14:410.5cpm 7 .' ..- î; V-15:414.6cpm 7____.___ i",,L-i:329.3cpm Ship's side Ship's bottom
Fig. 4 Coupled vibration mode of LNG carrier calculated by present method
Coupled vibration mode calculation result of main hull, accommodation, and funnel by mode synthesis method.
By this method, it is possible to caicülate with high
precision over a wide frequency ränge, from the low mode mainly composed of nodal vibrations of the: main hull to the high mode of superstructure vibrations. It is also possible to easily understand the frequency before and after coupling the superstructure to the main hull.
The longitudinal frequency of the stern superstructure of a large LNG carrier is lower than blade rate frequency in normal propeller revolution (number of rev. X number of propeller blades), in both the accommodation and funnel. The frequency change due to coupling with the main hull is 7% for the accommodation, and 2% for the funnel.
3.2 Vibration calculation foi container ship
In the theory proposed in Chapter 2, the coupled vibrations are calculated by using modal parameters, such as the natural frequency, effective mass, damping factor and vibration mode of individual constituent structures, but as a result of simplifi-cation of the connecting method of the maiñ structure and substructure mentioned in 3.1, the coupling calculation in this method does not require -a stiffness matrix for each constituent structure, and only the mass distribution of the substructures is
MTB179 February 1988 - 25 50 Frequency (cpm) 0 200 300 400 500 600 700 Frequency (cpm)
Fig. 5 Measured response curves of accommodatiòn and funnel of LNG carrier
Response curve of accommodation and funnel whén stern is excited.
Superstructure frequency is 303 cpm, and funnel frequency is 352 cpm.
L X B X D Xd =213D X30$X165 X95m
H
Fig. 6 Rough arrañgement of container ship
The above shows roughly figured general arrangementsofcontainer ship.
necessary. Therefore, if detailed structural införmation is not available as needed in the finite element method, it is possible to calculate coupling by this method with only modal parame-ters obtained separately by súitable methods.
In the initial state of design, the structural information
necessary for analysis of details is often not available, and in such a case the modal parameters of the constituent structures are determined on the basis of the results in previous shipbuild-ing experience. This paragraph therefore refers to a simplified estimating method of a ship's vibration characteristic using such modal prameters. In this case, the target of calculation is a large container ship. A rough arrangement of the ship is shown in Fig. 6. It is divided, as in 3.1, into the main hull as its main structure and superstructure.
The modal parameters of the main hull àrid superstructure are determined on the basis of previous shipbuilding data. Various simplified calculation formulae have been proposed for the natural frequency of the main hull, and by finding the
- Vibration
Calculated
-Measured Ratio(calculated/
freq. acc. freq. acc.
Superstructure Direction
- mode
measur
(cpm) (gal/t) (cpm) (.gal/t) freq. acc. Accommodation -
-structure Longitudinal S-1 289.8 10.8 303.0 14.5 0.954 0.745 Funnel structure Longitudinal F-1 360.3 34.1 352.0 29.0 1.024 1.176 V-1:45.2 V-11 :352.tcpn, Table 4 Response of accommodation and funnel to unit exciting force
of LN& carrier V-7:281.3cpm V-16:425.5cpm
\' y
A-
--V-8:283.7cpm V-17 : 453 .4 cpm r,r1 .-. -f-'
S-1:289.2cpm V-18:478.lcpm -V-9:289.8cpm V-19:544.Ocpm '--V- 10: 311.lcprn rìfl V-2:96.Ocpm F-U360.3cpmTable 5 Comparison between calculated andmeasured results of coitainer ship in frequency and response
correction coefficiént from these experimental values, the natural frequency was determined by the method reported in Reference (9). The effective mass of the máin hull was
determined on the basis of caiculatiöns developed from
previous shipbuilding experience, as was the vibration mode of
the main hull. As for the longitudinal frequency of the
superstructure, too, several simplified estimatiOn methods have been proposedt9. In this paragraph, the frequency was determined by the method shown in Reference (10). The effective mass of the superstructure was determined from past
shipbuilding records and also the vibratiOn mode. The determination of damping factor was the saine as in 3.1.
Using these independent modal parameters simplified
estimating results for the main hull and superstructure, the
cupled vibration was estimated in a simplified manner..
Furthermore, by using coupled vibration calculating results, the respOnse of longitudinally exciting the top of the super.
structure by unit exciting force was calculated. These calculation results are shown in Table 5. On the other hand, just before an ocean trial of the container ship, the vibration test of superstrùcture was carrièd out and the vibration of the superstructure was measured. By analyzing the vibration in anhor test, the frequency of main hull was obtained. These
measured results are shown in Table 5, and the resonance
curve of the superstructure is given in Fig. 7. By comparing the result of these calculations and the results of measurement,
it has 'been confirmed that this. calculation is sufficiently
practicable as a verification of the initial design.
4. Sensitivity analysis by mode synthesis method
By making use of this calculation method, the sensitivity of the coupled frequency characteristic with changes in the local structure characteristics was analyzed. That is, coupled
vibrations composed Of four elements longitudinal vibration of the propeller shaft system, verticál vibrations and longitudi-nal vibrations of the main hull, and longitudilongitudi-nal vibrations of the superstructure - were estimated in a simplified manner, and the closeness of the shafting frequency and superstructure
frequency and relation with the superstructure vibration
response were obtained, and the vibration excitation
phenome-non of the superstructure due to shafting vibrations was studied.
4.1 Shafting, hull and superstructure-coupled vibrations
In a low speed diesel vessel, when the propeller shaft
resonates with longitudinal vibratiOns, it has been reported
10
Shafting eA Q
--f
Main hull
FIg. 8 Calculation, model of superstructure fOr shafting excited vibration
Coupled vibration calculation model of shaft, hull, su'pr-strutüre using, this method. In this calculation model, the superstructure is induced by shaft ecitation.
that excessive vibrations may be induced in the superstruc-ture111, as a result of vibrations of the main hull through\the thrust bearing induced by the shafting vibration inertial force in the longitudinal direction. In this paragraph, relating this longitudinal vibrations of the superstructure modal pararne ters to express the vibration characteristics of the main hill, shafting and superstucture are set by reference to the measued data, and the tendency of past results is investigated by series calculation of the coupled vibration characteristic.
As the vibration calculation modél of. the superstrÙct1re induced by the resonance of' the pr9peller Shaft systém to longitudinal vibrätions, a structural system composed of the main hull, shafting and superstructure is assumed as shäwn! in Fig. 8. In the calculating procedure of coupled vibrations, first the coupled vibration is calculated by supposing that thè main hull is the main structure and the shafting is the substructue, and coupled system obtained in this way is newly' regarded as the maiñ structure while the superstructure is next considered as the. substructuré to calculate the coupled vibration similarly, and then the coupled vibratiOn characteristic of three elemeits
consisting of the main hull, shafting and superstructüre 'is
determined.
The vibration characteristic of the main hull is considered to be composed of vértical vibrations and longitudinal vibration. In order to take the effects of the ship type into considérátioi, oÍl tankers of 30 000,, 50 000, 70 000, and 100 000 tons displace-ment are used as examples. The vibration characteristic df the main hiíllwas determined by reference to previous shipbuilding
records.
As the vibration characteristics of the shafting, the
longitudi-nal vibration of the shafting presents the vibration mode
elastically supported in the longitudinal direction at the thrust
bearing stand, and the system from the propeller to the
crankShaft front end is the 0-nodé vibration mode vibrating iñ
Vibration mode
Calculated Measured Ratio(calculated/mesured)
freq. (cpm) acc. (galït) ft-eq. (cm) acc.
(gal/t) freq. acc.
V-1 54.1 56 0.966 V-2 108.2 ' ' 110 0.985 V-3 162.3 160 1.014 V-4 216.1 214 1.010 V-5 257.9 ' ' 256 1.007 S-i 491.6 16.3 514 15.5 0.956 1.052 0" 500 100o Frequency (cpm)
Fig. 7 Measured response curves of superstruc ture of container ship
Response curve of superstructure when superstructure is excited. Superstructure frequency is 514 cpm.
Superstructure
diIuIIl ,uuIuupp. 8th 16th Superucture longitudinal vibration Hull vertical and longitudinal vibration ___v' Shafting longitudinal ,'' vibration / 22th 19th
LU
Fig 9 Coupled vibration mode calculated by present method Result of calculation of coupled vibration mode of shaft, hull and
superstructure by mode synthesis method.
phase in the longitudinal direction. The shafting is coupled with the main hull at the base of the thrust bearing stand. The frequency of the shaft longitudinal vibration is calculated in series by varying in a. range of 500 to 700 cpm from the
frequency results in actual ships. The vibration modé and effective mass of the shafting were determined by reference to actual data. In order to see the effects of the longitudinal
frequency of the superstructure and the frequency of the
shafting approaching each other, the frequency of the super-structure was varied in a range of 300 to 800 cpm for series calculation. The vibration mode and effective mass of the superstructure were determined by reference to actual data. An example of calculated coupled vibration mode is shown in Fig. 9.
4.2 Vibration response of superstructure
Using actual ship calculation results from the previous paragraph, the relation between the closeness of shafting frequency and superstructure longitudinal frequency and
superstructure vibration response, the frequency difference necessary to avoid resonance and effects of displacement are studied below. In summing up the results of series calculation of the coupled vibration, the followiñg parameters are used. First, for simplicity's sake, aurning a system with i degree of freedom, only the resonance mode, the longitudinal accelera-tion for resonance of the superstructure is given as follows.
or
a,--. -
U m4 where,a5 : longitudinal accéleration of superstructure top (gal) Q shaft longitudinal vibration exciting force
(equi-valent exciting force at crankshaft front end) (t) amplitude of superstructure top based on crank-shaft front end in coupled vibration mode
circular constant
ô' logarithmic décrement of coupled vibration mA effective mass of coupled vibration based on shaft
front end (t/gal) Supposing, 0.3 0.2 0.1 a = 10 a=20 MTB179February 1988 a=30 I
-0.5 1.0 1.5 (b) N*NA N,INAFig 10 Response of superstructure obtained by par-ametrical study
To show the relation between the c1osenes of superstructure frequency N5 and shafting frequency NA and Z to indicate the
magnitude of the superstucture résponse. As N, -and NA approach each other, the superstructure response increases.
a = a5/Qo
(N\2
Qc=Qoyj)
Çb5 mAg whére,Q0 value of Q0 at shaft resonance point (t)
NA : shafting longitudinal resonant-frequency (frequency of shafting main body after coupling) (cpm) N, : superstructure longitudinal resonant frequency
(frequency of superstructure main body after coupling) (cpm)
N : exciting frequency (NA or N,) (cpm)
WA ':shafting weight (t)
g : acceleration of gravity (gal)
a is the parameter to express a8, and X is the parameter to express ,/HíA; from equation (6) the following equation is obtained. or (8) (7) 4(t) Cal. Measured 30000 D 50000 Ô 70000
0
100000 'es'*
o*
o OL
23thNow, when X is obtained from the series calculation result of coupled vibratÏons, the value of X becomes large at N N8and N NA. Therefore, when NNa and NNA, respectively, X
may be graphically expressed as shown in Fig. 10, with
NS/NA plotted on the horizontal axis. The solid line shows the envélope of the calculated value. From this diagram and equation (8), the upper limit of a is estimated.. Here, assuming c and WA are. mean values, when the relation between X and a (gal/t) is determined from equation (8), the result becomes as shown by the broken line in Fig. 10, from which the following facts are known.
When NSNA, it is predicted that the response of the
superstructure may be increased. This agrees with the
tendency of actual ship results".
When X is calculated in the reverse way and graphically found from the measured superstructure top response data, the value settles within the envelope as shown in Fig. 10, and
a large value is indicated in the vicinity of N3
NA. Therefore, from Fig. 10 and equation (8), the maximum value of response of the superstructure top may be approxi-mately estimated. Incidentally, in both actual ship's records and calculated results, the value, of X is sometimes small even when close to N3 NA. This is because the stimulation excitation to the superstructure is small depending on the mode when, for example, the position of the superstructure is at the loop of the vibrations of the main hull.In both superstructure resonant frequency N3 and shafting resonant frequency NA, the response of the superstructure becomes large, and the largest response occurs when N3 NA. As N3 and NA get further apart from each other, the amplitude of the superstructure decreases. The rate of
amplitude decrease suddenly drops as NA becomes further from N3 when N NA, but the rate of amplitude decreasé as N3 departs from NA is small. That is, at the resonant point of the súperstructure, large vibrations do not occur when the shafting vibration is somewhat far from the frequency
of the superstructure, but at the resonant point of the
shafting, the response of the superstructure is great if the frequency of the superstructure is somewhat far fröm the frequency of the shafting. Therefore, for the control of vibrations of the superstructure, it is necessary not only to avoid resonance with the propeller excitation frequency, but also to consider the effect of excitation by the shafting.
As for a3, when the allowable acceleration is taken, the
tolerance of X is determined from equation (8), and by
utilizing Fig. 10 the tOlerances of the superstructure fre-quency and shafting frefre-quency can be obtained. Their
relation may be utilized in evaluating the sensitivity of
superstructure. response when both frequencies are varied.
5. Conclusions
As a practical estimating method of coupled vibrations in a ship structure, the coupled vibration of the superstructure and main hull was analyzed by the method depending on the mode
synthesis method, and compared with measured rusults.
Furthermore, by performing the sensitivity analysis making
use of the mode synthesis method in the vibratiön control
studies of the superstructure caused by shaft vibration, the
following conclusions were obtained.
In a large LNG carrier, the coupled vibration was
calculated by connecting the stérn superstructure and maihhull by the mode synthesis method, the results were
compared with the measured values, and this method was proved to have a sufficientlypractical precision 'from low to high modes of vibration.As an application of this analytical method, without
obtaining the independent vibration characteristics of the
main structure and substructure by precise calu1aion, a
simplified method to estimate coupled vibration by connect-ing them usconnect-ing previous shipbuildconnect-ing data was presented, and, by using this, the coupled vibrations of the superstruc-ture and main hull of a large container ship utilizing the past shipbuilding data, was calculated and was conared with the measured results, and the validity of this simplified estimating method has been confirmed. As a result, this simplified estimating method has been proved to be effec-tive for the estimation of vibration in the initial planning
and designing stage, the necessity of which has been
previously indicated,
i
In the longitudinal vibration of the superstructure excited by the longitudinal vibrations of the propeller shaft system, this method was applied in the sensitivity analysis bf the
changes in the coupled vibration to changes in local
structural characteristics. As a result, thÏs simplified
estimating method has been found to be effective rn the sensitivity analysis of coupled vibrations which was diffi-cult using conventional methods such as the finite element method because of the problems in labor, time, etc.
References
K. Ohtaka, Method of Study on Higher Mode Vibration of Hull, 4th Sümmer Course of the Society ofNaval Architects of Japan (1978-8)
K. Kagawa, T. Hara, A Kawamura, T. Kobayashi, R. Nagamoto, Correlàtion Study on Higher Mode Vibration of Bulk Carriers (ist Report) (2nd Report), Transactions of the West-Japan Society of Naval Architects No. 69 (1985-3) and
No. 71(1986-3)
K. Kagawa, T. Hara, D. Sakai, M. Sondá, R. Nagamoto, Vibration of After Part Structure of LNG Carrier, jourial of the Society of Naval Architects of Japan No. 160
(1986-11)
K. Kagawa, K. Fujita, K. Ota, Y. Hayashi, Vibration Analysis of Hull Structure by Modal Synthesis Methbd, Mitsubishi Juko Gibo Vol. 17 No. 5 (1980)
K. Kagawa, K. Fujita, A Study on Higher Mode Vibration of Ships (Ist report), JSNA No. 143 (1978-5), (2nd repert) JSNA No. 147 (1980-5)
M. Yamakoshi, S. Ohnuma, On the Coupling of Hull Vibration and Bottom Vibration of Ships, Journal of the Society of Naval Architects of Japan No. 118 (1965-l2)
K. Ohtaka, Vertical Vibration of Bulk Carrier, JSNA West Japan No. 56 (1978-6)
T.L. Taylor, Vibration of Ships, TINA (1930)
Guide to Ship Vibration, Nippon Káiji Kyokai (1981-7) S. Ohnuma, T. Yamamoto, M. Nakano, M. Onoue, Vibration of Superstructurè of Aft-Bridge-Ship, Journal of the Socity of Naval Architects of West-Japan No. 38 (1969-7) A Study ón Vibromotive Force of Engine and Propeller and
Hull Response, Task 112 of the Shipbuilding Research Association of Japan, Material No. 114 (1970-3) and No. 147
