Springing. Wave induced ship vibrations

SPRINGING.

WAVE I N D U C E D SHIP VIBRATIONS

b y Ir. F . F . v a n G u n s t e r e n •)

Summary.

A theory of wave induced main hull vibrations in regular and irregular waves is developed. The vibrating hull is represented by a beam-element-model. The wave exciting forces are determined with a strip theory f o r oblique waves.

Numerical results for a 500. 000 tons tanker are obtained and discussed.

The given solution proves to be an efficient procedure for the analysis of springing phenomena.

1. Introduction.

The theory of vibrations is known for a long time [ 1 , 2]. Ship vibrations are mainly excited by the propeller and the machinery of the ship.

But recently, measurements on board ships have shown the influence of wave loads on main hull vibrations [3, 4].

The periodic deflection of the main hull of large tankers could be visually observed.

Untill now the calculation of wave induced hull vibration stresses has drawn little attention [4, 5, 6].

Since the vibrating ship can be considered as a multi-spring-mass system, the phenomenon of wave induced ship vibrations is called springing . If the superposition principle is valid, the ship vibrations in irregular waves can be determined, when the frequency characteristics and the energy spectrum of the waves are known. The springing stresses in regular waves appear to be linear with the amplitudes of the waves. The springing stresses in irregular waves can therefore be obtained by superposition of the stresses result-ing f r o m the regular wave components, that constitute the irregular seaway [7].

It w i l l be shown in section 2 that, as a consequence of orthogonality of the normal modes of vibration, the springing stress frequency characteristics can be determined by super-position of the characteristics of all modes of vibration. The ship w i l l be excited by the wave forces in all modes of vibration, which all contribute to the total vibration stresses. The frequency response function of the springing stresses f o r one mode of vibration results f r o m *) Sea Transport Engineering N. V . , Amsterdam, The Netlierlands.

the resonance curve for unit excitation amplitude and f r o m the frequency characteristic of the ex-citation amplitudes f o r the considered mode of vibration.

This is illustrated in Figures l a , l b and I c . Generally, the two-node vibration w i l l dominate, because, due to the nature of a wave k spectrum, the component which excitates the two-node vibration far exceeds the other components of excitation.

The relation between the amplitudes and the frequencies of the wave components of a seaway is determined by the wave-spectrum.

In normal conditions, the frequency of encounter of the top of the wave spectrum is less then the two-node natural frequency. The spring-ing stresses w i l l be relevant, if there is sufficient wave energy, i . e. if the top of the wave spectrum is close to the top of the frequency response function. Consequently, the following ship types are sensitive f o r springing:

- large tankers

- fast, long container ships - destroyers.

Principally, the theory of springing is valid for vertical as well as horizontal vibrations. The vertical vibrations are not affected by the horizontal and torsional vibrations for ships having symmetrical cross sections.

For vertical vibrations the amplitudes of excitation are mostly larger and the natural frequencies lower than the corresponding ones for horizontal and torsional vibrations. The analysis of springing can therefore be confined to vei'tical vibrations.

F i g u r e l a . Amplitudes of f o r c e d v i b r a t i o n f o r unit excitation amplitude.

F i g u r e l b . Amplitudes of e x c i t a t i o n f o r unit wave energy.

F i g u r e I c . Amplitude characteristic of springing.

F i g u r e l e . Springing spectrum.

F i g u r e . 1. I l l u s t r a t i o n of the calculation of s p r i n g i n g .

2. The ship structure in forced vibration with damping.

Mathematical model.

The calculation of the vibrations of the complex three dimensional ship structure can nowadays be accomplished by finite element techniques [8]. However, the classical beam theory is applied for the calculation of springing, since:

- springing is dominated by the two-node vibration.

- for lower modes of vibration the finite element technique only gives l i t t l e more accurate results, but requires considerable computing time f o r the numerical solution. - the finite element method needs three

dimensional information about stiffness, wave loads and added mass, which is difficult to obtain.

- the probability of springing can be calculated in the preliminary design stage.

- the simplicity of the beam theory makes the description of the phenomena more lucid. The present theory f o r the description of the forced damped vibration of a beam is based on the work of Koch [1], (1929).

The inverse method, using mass - and stiffness elements, is preferred to other methods such as the Integral-, Holzer- or Stodola method, because:

- the numerical solution of the free vibration.

- the forced vibrations can simply be calculated f r o m the frqe vibration analysis.

The mass- and stiffness discretization is described in the references [1] and[9], which also provides the numerical solution of the free vibration.

The influence of rotatory inertia is neglected. For prismatic bars, this influence on the natural frequency is a quarter of the influence of shear, whichisalready of small influence in comparison with bending.

D i s c r e t i z i n g the ship s t r u c t u r e .

The mass and stiffnesses of the beam, with known mass- and stiffness distributions, are

discretized.

The beam is divided into a numer of sections as indicated in Figure 3.

The continuous mass of each section is substituted by three concentrated masses. These three discretized masses are calculated f r o m the conditions, that the continuous and discretized system have equal mass, linear and quadratic

c o n t i n u o u s beann s e g m e n t ma s s s e g m e n t s t i f f n e s s s e g m e n t '2

'(1

JkAG q, =q. •J m= m a s s / u n i t length • m. : d i s c r e t i z e d m a s s E l = b e n d i n g s t i f f n e s s kAG= s h e a r s t i f f n e s s o K;= b e n d i n g J o i n t |Z|q;= s h e a r j o i n t mdx = m. +m.. +m. m x d x = m i» m i '2 ' mx''dx=m4.*m 6-«7 > e

f r o m the conditions of equal deflection and slope at the section ends, i f equal loads are assumed. When the number of sections increases, the

approximation of the continuous system by the discretized system is improved.

Free vibration a n a l y s i s .

The elasticity of multi-mass-systems can conveniently be described by influence numbers.

The influence number a _ is defined by: deflection in point i , caused by a unit force in point j . These influences numbers are calculated f r o m the stiffnesses of any system, discretized or not. The equations of motions can then simply

be written as linear deflection equations, instead

of second order equations of motion, which is called the inverse method. Consider now the ship structure as a long beam, discretized in n masses, m. (i = 1 , . . . . n), which can move in the plane through one of the two principal axes, perpendicular to each other. The moving system can then be substituted by a static system, which is loaded in the points, x., by the inertia forces:

m.

in which the deflection, y . , in point x. is given by: ^ d y

Y = 2 m — — a.,

' 3 3 , ^ 2 13 (1)

This is a set of n (i = 1 , . . . n) linear second order differential equations in t , corresponding with n masses.

According to the theory of vibrations, i t is assumed that the masses m. p e r f o r m harmonic motions with the same frequency and different amplitudes. y. = r|. sin CO t so: (2)

4

dt^ CO ri. sin CO t

n r|. = 2 m, M 1. «.. J 1 i = l J J I J (3) 2 m r| . n,. = O i K i l i (7) i = l or: 2 (m^ a.. n. = O J 1] J

These n equations (4) are homogeneous in n. and possess only a non-zero solution if the n-th order determinant of the coefficients is zero:

("^1 " l l - -2> CO m a 2 12 .m a n I n "^1 "21 ("^2 ' ^ 2 2 - ^ ) - m a„ n 2n m a 1 n l (m a n nn

for which proof is referred to reference [1]. Normal modes n . satisfying (6) and (7) are called

K l

(4) orthonormal. As to the support, the ship can be considered as a beam with free ends: f r e e f r e e -beam.

The influence of the elastic support by the surrounding water is neglected, but can be taken into account by application of the law of Southwell [10]:

2 2 2

M = CO + 0 )

s 1 2 =0 in which:

= natural frequency caused by effect 1 i . e. free-free-beam.

CO = natural frequency caused by effect 2 i . e.

r i g i d body on elastic support.

This frequency equation of the n-th order in 1 / (5)

2 has n positive real roots (k = 1,.. .n). The homogeneous linear equations (4) have non-zero solutions f o r n., i f CO = K . The n values for

1 K

«2 are arranged, such that: 2 2 2 2

CO < M < a . . . < w

1 2 3 n

The system of r|. values corresponding with w

1 K

is r| . A point i in this system is indicated with

K

Tne values co are called natural frequencies.

M = natural frequency caused by effects 1 and

2 i . e . elastic beam on elastic support. The influence numbers can only be calculated for the beam supported in two points. Then the equations (3) are only valid, i f the forces of support are taken into account. The set of equations (3) is transformed then into:

r i . = 2 m u a r| 1 j = l ] l j j X V i - ^ i X . 1 n + n _{o X , n+1} n+1 n+1 (8) ( 1 = 1 , . . . n)

The displacements of the semi-supports and r i ^ ^ ^ , see Figure 4, can be expressed in 1.» n„ - • • n by the condition of equilibrium of the

1 z n The corresponding configurations of vibration inertia forces: amplitudes, n ^, are called normal modes of ^ vibration.

The normal modes n . differ f r o m each other by

K l

a proportionality factor only. This factor is chosen, such that;

2 m M (x - x ) r | = 0 j=0 j n+1 j j n 2 2 m n = 1 1=1 i K l (6) n+1 , 2 m M 1=1 j X . n. (9) (10)

A very important feature of the normal modes is their orthogonality:

The n equations, resulting f r o m inserting (9) and (10) in (8) have non-zero roots for r|. if their determinant is zero.

m. rrij ' m m m -/ h—^ I i I m. u V a i - ' f t * F i g u r e 4. D i s c r e t i z e d mass system.

The normal modes of vibration and the natural frequencies of this corrected set of equations (8) can be calculated analogue to the solution of (4).

A numerical procedure is developed in [9], which:

discretizes mass and stiffnesses, calculates influence numbers,

calculates natural frequencies corresponding modes of vibration.

and

The deflection can be considered to be constituted by superposition of certain contributions of each normal mode: n y. = 2 ei. . n . 1 K K l K = l in which: n a = 2 m. y. n . K 1 1 K l 1=1 {i = l , . . . n ) (13) ( K = l , . . . n ) (14)

Each normal mode corresponds with a system offerees P ., acting in x., i f M = co K l 1 I P . follows f r o m : K l 2 P . = m. w n . K l 1 K K l (15) Therefore, n normal modes correspond with n 'normal loads'. The system of external periodical forces Q. can be resolved into the normal loads inherent to the beam:

n

Q = 2 b P . i K K l

K = l

(i = l , . . . n ) (16)

Because of the orthogonality of the normal modes given by (7) the coefficients b can be written as:

Mechanical vibrations excited by periodical forces.

The periodical forces Q., (i = 1, n), and the viscous dampingforces are assumedtoact on the discretized masses in the points x- ( i = 1 , . . . n). The equations of motion- (1) can then be written as:

(11)

n dy. d y.

2 IQ - c — - - m. — - I

' r-iV

^ d t

Ut'l

The viscous damping coefficient is assumed to be proportional to the mass:

c = c m (12) j J in which c = constant ~ 100 , n Q. P . 1 1 K l , b = —- 2 ( K = 1 , . . .n) k 4 m. w 1=1 1 K Substitution of (15) gives: 1 b = — 2 Q. n . (K = l , . . . n ) (17) K 2 . , 1 K l K> 1=1 K

Substitution of (13) and (16) in (11) gives: n n n / da 2 a n . = 2: 2 b P . c.—^ n , -^ K K l . , 1 \ K K j 3 dt K3 K = l 3=1 K = l \ d \ ^ \ - m . a . . ( i = 1 , . . .n) ^ dt^

n _{" / 2} K = l j = l K = l \ dt (i = 1 , . . .n) Inserting (3) in (18): n S a n , K K l K = l b - c K K m T] a j i j 2 \ da d a \ 1 ^ K K ' 1 ^ dt 2 i 1 2 dt _{d t /} co K (i = l , . , . n ) (19) Now, each equation of (19) is multiplied by

m. n • and the resulting equations are added.

1 K l

F r o m orthogonality it follows that:

a = b K K. 1 1 K 1 d a K 2 ^ 2 2 CO dt d t K K or: d t da dt K 2 — + K> a K K 2 CO b ( K = 1 , . . . n' K K (20) The solution of this second order non-homo-geneous linear differential equation for small damping is:

- I '

a = e [C sin qt + C cos qt] K 1 2 particular integral (k = 1 , . . . n) in which: 2 c

"

-u

C andC are calculatedfromthe initial condi-1 2

tions.

After some time the solution w i l l be independent of the initial conditions. Then, the particular integral determines a . The particular solutionis

K

obtainedby considering the right side of (20), in which b is resolved in series of Fourier.

K b = b + 2 b sin ( p c o t + 9 ) K K O K P e K P p = l ( K = l , . . n ) (21)

The particular solution which satisfies the equa-tions (20) when (21) is inserted, is:

a = b + 2 K K O b w sin ( p w t + 9 + E ) K p K e K p K p p=i

y.

' • •

( M - p o i ) + C p M K e e in which: C CO p

" " ° ^ ^ I T T

^ 2

\ P M - CO e K • (22) (23)

According to (13) the deflection curve of the beam is obtained by superposition of the deflec-tion curves of each mode of vibradeflec-tion, multiplied by a .

K

The solution for the forced vibration, given by (13) and (22) is determined by the b values, which are

K

determined by Q. (17) when the free vibration analysis is known. So, the maximum y. value is determined by Q., which corresponds with

physical interpretations.

The maximum of the deflection curve y,-.^^^, 1 IIlclX given by insertion of (22) in (13) can be determined.

The fictitious static forces p., acting in x., and corresponding with the maximum dynamic deflection curve, can be calculated with the p r e -calculated influence numbers, since they determine the relation between deflections and loads.

The effect of she a r t s all owed for at the computa-tion of the influence numbers.

If the deflections y and y, in respectively x and

a b a x^, which were the points of support f o r the

determination of the influence numbers, are zero, the influence numbers can be used to calculate the

static loads f r o m the deflections.

Therefore the deflection curve is transformed to: (^b - \ ^

(X - X )

I V 1 m a x a ( ^ " -^g.) ^ ^

( i = l , . . . n )

The fictitious static forces corresponding with the deflection curve y. is given by: _{1 n i t i x}

p.. n

= 2 y. / « . . J=l

( i = l , . . . n ) (25)

The maximum springing stress in the beam can easily be calculated f r o m the static loads and the section modulus of the beam, so the springing stresses resulting f r o m damped vibration excited by periodical forces can be determined.

Finally, the calculation is reviewed in the flow chart, given in Figure 5.

M a s s d i s t r i b u t i o n S t i f f n e s s d i s t r i b u t i o n s A d d e d m a s s d i s t r i b u t i o n D i s c r e t i z e d m a s s I n f l u e n c e n u m b e r s N a t u r a l f r e q u e n c y N o r m a l m o d e P e r i o d i c a l e x t e r n a l f o r c e s Max. d e f l e c t i o n c u r v e of f o r c e d v i b r a t i o n M a x i m u m m o m e n t and c o r r e s p . s t r e s s F i g u r e 5. F l o w c h a r t of the f o r c e d v i b r a t i o n s t r e s s calculation.

3. P e r i o d i c a l forces in regular waves.

The strip theory [11] is assumed to give adequate results as to the exciting forces.

As both the beamelementmodel and the s t r i p -theory are two dimensional, their joint applica-tion seems to be the logical approach. For head waves and for the range of wave length ratios and forward speeds of practical interest the s t r i p -theory has been provedto agree with experiments [11]. For oblique and very short waves the s t r i p -theory needs experimental investigation.

But, in view of the errors in the stiffness - and mass distributions, the striptheory is expected to be sufficiently accurate f o r the calculation of

/ / / / / /

F i g u r e 6. D e f i n i t i o n of wave and m o t i o n s .

springing stresses with the beam-element-model.

A review of the calculation of the strip forces in regular oblique waves is given now.

Consider a right hand cartesian coordinate system Ox^y^z^, fixed i n space. See Figure 6. The velocity potential O of an ideal f l u i d with regular waves of small wave slope follows f r o m the Laplace equation and the free surface condi-tions : K Z o CD ( X Q , Y Q , Z Q , t) = ^e sin (kx^ cos M + + ky^ sin u - mt) (1) in-which; \ = wave amplitude CO = wave frequency k = wave number = 2 T T / A

u = angle between x^-axis and direction of travel of the waves.

The pressure is given by the Bernoulli equa-tion:

90

P (^o' yo' ^ o ' t ) = - p y t " p (2) The velocity potential with respect to a right hand coordinate system Oxyz, which travels with the ship's speed V relative to systemx, y, z along the positiveXQ-axis, is given by the transforma-tion:

X Q = V t + X

yo = y z^ = z

f r o m which follows:

kx^ cos M - cot = k x cos n - u t Wg = w - k V cos n 2 V M = CO - M — COS n e p-(3) thus; , = circular frequency encounter kz of O (x, y , z, t) =i -e sin (kx cos n + ky sin ^ - M g t ) ₍₄₎ Substitution of (3) in (2) gives the pressure with respect to the x y z system:

kz

p ( X, y, z, t) = p g e cos (k x cos n +

(, = l,^ cos OD^t ₍₉₎

The equations for heave and pitch in waves are given by:

pVz =

ƒ

F ' dx^ (heave) (10)

pVë ƒ F ' dx^ (pitch) (11)

L

in which: F ' is the cross sectional force f o r a distance x^ f r o m G.

yy longitudinal radius of inertia of the ship. Consider the coordinate systemx y z , which

b b b is fixed on the ship as indicated in Figure 6. According to the strip theory the cross sectional value of the vertical force i n x^ i s :

F ' = F ' + F ' + F ' 1 2 3 F ' = - 2 p g y (z - X e) + F ' 1 w b 5 F^ = - N ' (z - X j ^ é + v e ) + F ! F ' = - m ' ( Z - X S + 2 v è ) + 3 b d m ' . • + V - — (z - X , e + Ve) + F -d x / \ ^ in which: (12) + k y sin n - M t) - p g z (5)

The profile of the wave follows f r o m : 1 30

g at z=o (6)

y = the half width of the cross-section at the waterline

m ' = the sectional added mass

N ' = the sectional damping coefficient. and:

I = i cos (kx cos p + k y sin p - t) (7)

a e If surge is neglected, the vertical ship motions

are defined by:

e = e cos ( M t + £ J a ^ e e^' Z = Z cos (K) t + E ) a e zl' (pitch) >(8)

inwhiche^ andz^ ax-ethe amplitudes and e „ ^ and

a a o E r are the fase angles with respect to the wave,

Zb defined by: , F ^ = 2 p g y ^ r F ^ = N ' r F : = m ' r - v ^ r b (13) (heave) i n which: k T

^* = ^ e cos (kx cos n + k y sin p - M t)

a e

\ W -draught

The sectional wave forces , F- and F-^. are obtained by integration of the pressure (5) over the draught and based on the assumption that the water pressure is not influenced by the presence of the ship (Froude-Kriloff hypothesis). The stationary displacement force in s t i l l water is not included in F'because by integration over the length of the shipthis force is equal to the weight of the ship and has the opposite direction. Besides, the vibration displacement is assumed very small in comparison with the wave height.

The sectional added mass m ' , the slope of the d m '

added mass distribution and the sectional d X b

damping coefficient can be calculated by conformal transformation of the solution of Ursell for the circular cylinder.

The calculation of the sectional wave forces (13) can be pei-formed according to [12].

The frequency response functions f o r pitch and heave are found by solving the equations of motion (10) and (11), in which (12) and (13) are substituted. The strip forces (12) in regular waves can then be calculated for a certain ship speed, wave length and wave direction. These strip forces are the periodical forces which excitate ship vibration in regular waves.

The flow chart of the calculation of the periodical exciting forces in regular waves is shown in Figure 7, left part.

4 . S p r i n g i n g in r e g u l a r wases.

According to the strip theory, as reviewed in the preceding section, the periodical exciting forces Q., ( i = l , . . .n), are sinusoidal in regular waves.

Then, the exciting force on a strip can be written as:

Q. = F. cos (w^t + E . ) ( i = l , . . . n ) (1) in which:

F. = amplitude of the strip force CO = circular frequency (of encounter) £. = phase angle between wave and force. Q can be written as:

1

Q. = F. cos E cos u t - F sin e sin co t (2)

1 1 1 e i i e

Substitution of (2) in equation (17) of section 2 gives: Oj b = cos CO t 2 (F cos E , r | ) + K K e . ^ i i K i 1=1 sin w t 2 (- F. sin E , r | ) (3) e .^^ 1 1 Ki

The right side of this equation (13) can be written as:

CO b = F sin ( o j t + 9 ) ( K = l , . . . n ) (4)

K K K e K

in which:

2 2

( 2 F. cos E..I1 .) +( 2 -F sin E , n )

i = l ' ' i=l ' ' (5) n 2 F cos E , r | . , i 1 K 1 1=1 n \^ - F sin E . n / . i i K i 1=1 arctg (6)

The second order linear differential equation, which describes the forced damped vibration of a multi-mass-element-beam is given by equation (20) in section 2: d a da K K 2 2 c + OJ a = CO b ( K = 1 , . . . n) (7) , 2 K K K K dt dt Inserting (4) in (7) gives; d a da K K 2 — + c + CO a = F sin ( o j 1 + 9 ) K K K e K dt dt ( K = l , . . . n ) (8) A particular integral, which satisfies this differential equation (8), is given by:

a = F sin ( w t + 9 + E ) K e K K r~2 2^2 2 2~ ( M - Oj ) + c Oj K e e ( 9 )

in which: E = arctg c w 2 2 .a - CO e K ' ( K = l , . . . n ) (10)

According to (14) of section 2, the deflection of the multi-mass-element-beam is given by:

n

y. = 2 a • n .

1 ^ K K 1

K = l

(11)

So, the deflection of the beam and the c o r -responding stresses can be calculated by substitution of (10) in (11).

As is already stated in section 1, two-node vibration w i l l dominate, for ship speeds and wave lengths of practical interest.

The maximum deflection of the 2-node vibration is given by:

own amplitude, length and direction of travel, is accepted [7], for the calculation of springing stresses using the beam analogy.

The frequency springing stress response func-tions can be obtained f r o m calculafunc-tions accord-ing to the precedaccord-ing sections. The spraccord-ingaccord-ing stress appears to be linear with the wave height, within the range where Hooke's law, the strip theory and beam analogy are valid.

For practical applications i t may be assumed that the springing stress amplitudes follow the Rayleigh distribution law f o r a narrow band

springing stress spectrum.

Now, the prediction of springing stresses i n irregular waves is possible by superposition of the stresses caused by the individual regular wave components.

For a given wave spectrurri S^^, the variance of the springing stresses is given by:

y = ( y ) = (3- ) n _{i m a x i max K = 1 max K = l , i} (1=1,... n) (12) H (o. ) [ a e 8^^(00^) d o . ^ (1) (a ) = + , n K max - - . / ^ ^ 2 2 ^ (13) ( M - CO ) + C Oj ' K e e

Thus the maximum deflection curve of the 2-node vibration is given by;

y. 1 max K = l 2 2 2 2 w ) + C Oj e e l . i (14) (1=1,...n) in which H (oj ) ai ^ e^ a (2)

is the frequency springing stress response function and Sf-^(oj ) is determined by the distribution of the square wave height as a function of circular frequency of encounter:

1 2 S (oj ) d&j = - r (oj )

[ ^ e' e 2 ^a ^ e' (3)

The fictitious static load and resulting stresses, which correspond with this deflection curve y can be calculated with the method outlined

1 max

in section 2.

5. Springing in irregular waves.

Assuming that the superposition principle is valid, the springing stresses in irregular waves can be determined when the frequency character-istics and the energy spectrum of the waves are known.

The assumption that the seaway is composed of many regular wave components each having their

The probability that the springing stress

amplitude a exceeds a certain value a is given _{a a}

by:

2 m

P [a >,T *] O a

(4) The prediction of springing in irregular waves is analogue to the prediction of ship motions in irregular waves [13].

6. Numerical results.

A computer program has been developed for the calculation of the springing stresses in regular

B o d y p l a n M a s s d i s t r i b u t i o n S t i f f n e s s d i s t r i b u t i o n s ) B o d y p l a n M a s s d i s t r i b u t i o n S t i f f n e s s d i s t r i b u t i o n s ) T r a n s f o r n n a t i o n c o e t f i e n t s D i s c r e t i z e d m a s s I n f l u e n c e n u m b e r s S h i p s p e e d W a v e l e n g t h W a v e d i r e c t i o n S e c t i o n a l a d d e d m a s s a n d h y d r. d a m p i n g c o e f f . D i s c r e t i z e d a d d e d m a s s 1 1 S e c t i o n a l w a v e f o r c e L o n g i t u d i n a l s h i p m o t i o n s S e c t i o n a l f o r c e s in r e g u l a r w a v e s N a t u r a l f r e q u e n c y N o r m a l m o d e • ' 0 F o r c e d v I b r a t I o n : d e f l e c t i o n c u r v e Ivlax. m o m e n t a n d s t r e s s F r e q u e n c y s p r i n g i n g s t r e s s r e s p o n s e f u n c t i o n W i n d v e l o c i t y W i n d d i r e c t i o n W a v e s p e c t r u m S p r i n g i n g s t r e s s s p e c t r u r P r o b a b i l i t y c a l c u l a t i o n

F i g u r e 7. Flow chart of the calculation of springing.

and irregular long crested waves according to the present theory.

A simplified flow chart of the program is given in Figure 7.

Verification of the theory by comparison of numerical results with experiments is not yet possible. Full scale measurements [3, 4] have been carried out, but it is not clear what part of the total measured bending stresses originates f r o m springing. Model experiments on springing have been reported [5], but information on the test conditions is insufficient f o r making a proper analysis.

The discussion of the results has therefore to be limited to a qualitative analysis.

Numerical results of computation of springing stresses and vertical ship motions are obtained for a 500. 000 tons tanker. The main particulars of the tanlier are given in Table 1.

It appeared f r o m the calculation of the cross-sectional values of the added mass and hydro-dynamical damping coefficient, that the convergence of the series used in the numerical evaluation of the potentials is sufficient f o r frequencies up to w = 2 rad/sec.

For frequencies exceeding this value, the computerprogram could not be used. Therefore the asymptotic behaviour was investigated. The vertical ship motions and hydrodynamical

Table 1. Main particulars of 500. 000 tons tanker.

Length 1338'

Breadth 233'

Draft aft loaded 76'

Draft forward loaded 63. 3'

Draft aft ballast 46.25'

Draft forward ballast 33. l '

Volume loaded 48 95 81 m^

Volume ballast 26 75 54 m^

Midship bending stiffness 5, 5 X 10^ kgcm^

2,61 X 10^° kgf

Midship shear stiffness

5, 5 X 10^ kgcm^

2,61 X 10^° kgf

(^-node) 2,1 rad/sec

«2 (3-node) 4,5 rad/sec

damping coefficient tend to zero, for high frequencies. The cross sectional added mass approaches to a non-zero constant value asymp-totically. This asymptotic value can be calculated with the formula given in the appendix.

The main calculations of the springing stresses in regular and irregular waves at high frequencies is therefore based on the following assumptions:

- the vertical shipmotions are zero

- the hydrodynamical damping coefficient is zero

- the asymptotic value of the added mass is a sufficient approximation f o r the added mass for high frequencies.

The results of the calculations f o r the two frequency ranges are shown in Figure 8.

It shows the behaviour of the ship in regular waves.

It appears that the three-node vibration stresses are negligible with respect to the twonode v i b r a -tion stresses. The frequency springing stress response function for the two-node vibration is the product of the resonance curve f o r unit excitation amplitude and the amplitude characteristic of the two-node component of the wave excitation, shown respectively in Figures l a and l b .

The resonance curve for unit excitation amplitude

shows only one peak, for co = co-j^^

The amplitude of excitation for the two-node vibration as a function of frequency shows several peaks at frequencies where the ship length is approximately a multiple of the wave lengths ( L / A = 1, 2 , . . . ) .

500. 000 t . tanker.

The numerical results of the computer-calcula-tions indeed show a pattern that can be explained by these two effects. The frequency response function has an absolute maximum f o r = co^ and local maxima at frequencies corresponding with L / A = 1, 2 , . . .

The wave bending moment stresses of the tanker are indicated with a dotted line. It can be concluded that in the frequency range where the wave bending stresses are relevant the springing stresses are of the same order of magnitude.

The effect of vertical ship motions on ^hese springing stresses is indicated with a dot-dash line.

For this tanker two frequency ranges are important.

- The region of resonance:

Vertical shipmotions are negligible in this area

- The region where L = A

effect on springing stresses in this area It seems therefore justified to neglect f o r this ship type the vertical ship motions in the analysis of springing.

Results of the calculations of springing in irregular long crested waves is shown in Figure 9. The characteristic peaks of the springing stress spectrum correspond with experimental results [4] for a corresponding shiptype. The low and the high frequency springing stresses, which correspond with the two peaks, can be treated as statistically independent events. Therefore, the probability that the springing stress amplitude exceeds a certain value can be calculated by separate treatment of the two narrow-band peaks of the springing stress spectrum.

7. Conclusions.

A theory of wave induced main hull vibrations in regular and irregular waves is developed, which has the following features:

- the vibrating hull is represented by a beam-element-model analysed with the theory, which is outlined in section 2 and is based on reference [1].

- the wave exciting for ces are determined with a strip theory for oblique waves, described in section 3.

- thetwo-node vertical vibrations are assumed to dominate.

Numerical results for a 500. 000 tons tanker have been obtained and discussed. For this ship type two additional assumptions are shown to be

valid.

- vertical ship motions can be neglected. - the sectional added mass can be approximated

by the asymptotic value for high frequencies. It may be concluded that the solution presented in this paper provides insight in the phenomenon of springing and is an efficient tool f o r its analysis.

There is a need for further experimental verification of the theory, especially of the wave excitation forces of small wave length ratios.

Acknowledgement.

The author wishes to express his appreciation to Prof. I r . J. Gerritsma, who initiated, super-vised and stimulated this work and to M r . J. de Vries, whoprovidedthe numerical results of the free vibration characteristics.

References.

1. Koch, J . J . , 'Enige toepassingen van de leer der eigen functies op vraagstuliken uit de toegepaste mechanica'. Doctor Thesis,. D e l f t , 1929.

2. Hartog, J . P . den, 'Mechanical v i b r a t i o n s ' . Mc G r a w - H l l l Book Co., New Y o r k , 1956.

3. T a y l o r , K . V . and B e l l , A. O. , 'Vi'ave-excited h u l l v i b r a t i o n stresses measurements on a 47. 000-tons deadweight tanker'. B . S . R . A . Report NS 115, 1966.

4. Mathews, S. T . , 'Main hull g i r d e r loads on a great lakes bulk c a r r i e r ' . Po^ceedlngs of the Society of Naval Architects and Marine Engineers, Spring Meeting, 1967.

5. Belgova, N . A. , Determination of overall bending moments caused by elastic vibrations of ships'. Transactions of the Leningrad Institute of Water Transport, Issue X W I I I , 1962.

6. Goodman, R . A . , 'Wave-excited main hull v i b r a t i o n i n large tanlters and bulk c a r r i e r s ' . Read on A p r i l 23, 1970, Royal Institution of Naval A r c h i t e c t s .

7. St. Denis, M . and P l e r s o n , W . J . , 'On the motions of ships In confused seas'. Transactions of the Society of Naval Architects and Marine Engineers, 1953.

8. Hylarldes, S. , 'Shlp v i b r a t i o n analysis by f i n i t e element technique'. Netherlands Ship Research Centre, Report No. 107 S, December 1967. 9. V r i e s , J . de, ' N u m e r i c a l calculation of v e r t i c a l hull

vibrations of ships by d i s c r e t i z i n g the v i b r a t i o n system'. Netherlands Ship Research Centre, Report No. 58 S, A p r i l 1964.

10. Mc. G o l d r l c k , R . T . , 'Buoyancy effect on natural frequency of v e r t i c a l modes of hull v i b r a t i o n ' . Journal of Ship Research, July 1957.

11. G e r r i t s m a , J. and Beukelman, W. , 'Analysis of the modified s t r i p theory f o r the calculation of ship motions and wave bending moments'. Netherlands Ship Research Centre, Report No. 96 S, June 1967.

12. Vugts, J . H . , 'The hydrodynamic forces and s h i p -motions i n waves'. Doctor Thesis, D e l f t , 1970. 13. Jong, B . de, 'Some aspects of shipmotions i n

I r r e g u l a r beam and longitudinal waves'. Doctor Thesis, D e l f t , 1970.

14. Gunsteren, F . F . v a n , ' H e t asymptotisch gedrag van de toegevoegde massa van dompende cylinders voor hoge frequenties'. D e l f t Shipbuilding Laboratory, Report No. 2 3 0 - M , M a r c h 1969.

y y L m c m m i N N ' P Q . 1 s t V X y z

\ \ \

X y z o o o ^w z z

Longitudinal radius of inertia of the ship

Length of the ship Variance

Sectional added mass Discretized mass

Number of transformation co-efficients. Sectional hydrodynamical damping coefficient Discretized loads Wave pressure Exciting forces Spectrum Time

Speed of the ship

Coordinates of the discretized system

Right hand coordinate system

Half width of designed water-line Heave displacement Heave amplitude Influence number Phase angle Wave amplitude List of symbols. a , b , P K K K c b , ip K p K p Coefficients of equation of forced vibration

Structural damping coefficient Fourier coefficients F ' , F ' , F ' , F ' Sectional hydromechanical 1 2 3 „ forces F , 9 K K H k Coefficients of springing in regular waves. Acceleration of gravity Frequency response function Wave number

Vibration amplitude of the discretized system

Pitch angle Pitch amplitude Wave length

Angle between ship course and direction of travel of the waves Density of water

Springing stress amplitude Volume of displacement Circular frequency

Circular frequency of encounter

Natural circular frequency of vibration.

Appe Asymptotic value of the sectional added mass.

The formula of the sectional added mass m ' (co oo) for multi-parameter forms reads as follows:

N 2

m V ) [1 + 2 a + 2 (a^ ^ (2n+1)]

2 1 n=o "^""^^ where:

p is the density of water a is the scale factor

and a are the transformation coefficients 2 n+1

corresponding with the following conformal transformation formula.

N

-(2 n+1),

- = ^ [ ^ \ ! , ^ 2 n + l ^ 1

The complex number w = y + i z represents a point in the physical plane or w - plane.

- j e

The complex number [ = j r e represents a point in the reference plane or [ - plane. The choice of the coordinate systems is illustrated i n Figure 10.

The derivation of the formula f o r the added mass m'(oo) is given in [14].

dix.

w - pi a n e

F i g u r e 10. D e f i n i t i o n of c o n f o r m a l t r a n s f o r m a t i o n coordinates.