Theoretical study on the vibrations of rudder and rudder post system of a high speed cargo liner

P1E 19B1

ARCHIEE

Lab. y. Scheepsbouwkurnie

Technische Hoges.chool

Deift

Reports of Research Institute for Applied _Mechanics

Vol. XVIII, No. 61, 1970

THEORETICAL STUDY ON THE

_{VIBRATIONS OF}

RUDDER AND RUDDER-POST

_{SYSTEM OF A}

HIGH SPEED CARGO LINER

By Jiro SUHARA and Nobuyoshj FUKtjcIII**

The possibility of arising the vibrations _{of the rudder and rudder-post}

system of a high speed cargo liner has been examined theoretically based on several assumptions.

Although it may not be predicted the vibration_{characteristics accurately}

owing to the complexities of the _{nature of the anticipated motions of the}

rudder and rudder-post system behind the propeller, the coincidence of the.

natural frequency of the fundamental mode _{of the system and the blade}

frequency of the propeller in service condition may be possible as a

thè-oretical result based on the assumptions which are thought to be reasonable. However, even though such coincidence arises unfortunately, it is considered

that the violent vibrations would not happen if the statically estimaed

maximum deflectjons of the rudder-post _{due to pressure around the propeller}

blades are less than the order of the clearances between the rudder pintles

and the gudgeons which _{cause the decrease of effective weight of rudder}

to the rudder-post i. e. the increase of the natural frequency _{of the system}

and check the resonance of the_system.

1. _Introduction

A cantilever type rudder-post attached with rudder of a high speed cargo liner was felt necessary to study _{on the possibility of occurrence of}

violent vibrations by the hydrodynamic disturbances by the propeller.

Theoretical estimations of the natural frequencies of the system have _been made on the bases of assumptions to simplify the theoretical _{treatments. The} fundamental data used in this theory are not yet sufficient to predict the _exact

value of the natural frequency of the _{system, because of the lack of}

example of measurement of an actual phenomena _{even for a similar case up to}

present However the theoretical study would, be _{useful for a rough prediction of the}

nature of the vibrations and the _{ranges of natural frequencies anticipated and}

for a design guidance of the scantlings of the system. * Professor, Fuculty of Engineering, Kyushu University.

**.Engineer Hiroshima Shipyard, _{Mitsubishi Heavy Industries. Co. Ltd.}

70 _{J. SUHARA and N. FUKUCHI}

2. Assumptions

Following assumptions are made as bases of the theory:

The rudder is connected to the _{rudder-post through pintles which} trans-mit the shearing forces only and_{sufficiently flexible not to transmit any moment} of force about horizontal _{or vertical axis from the rudder to the}

rudder-post. The horizontal inertia forces of the rudder including the effects of added weight of surrounding_{water apply continuously to the rudder-post through}

infinite number of pint les _{assumed to be distributed over the}

_{vertical axis at}

the aft end of the rudder-post.

Added weight o the rudder and rudder-post system for the motion of horizontal translation is estimated by the experimental formula obtained by Wereldsma of Netherlands Ship Model _{Basin [I] as follows:}

Added weight _{= C1 . r.B2d1}

where

rweight per unit volume of the surrounding _water.

Br= chord length of the rudder and the _rudder-post. dl= infinitesimal length of the system in vertical direction.

For the value of the coefficient C1, Wereldsma gave approximately 1/2 by

the model experiment of the Mariner rudder with nearly similar profile of _the system into consideration, however that

_{value is thought to be used for}

the whole system including the three _{dimensional effect. The value of C1 is to be}

variable along the length of the system when we estimate the local _added

weight per unit length at any section. _{In this report such effect, has been ignored}

and the C was taken as constant _{approximately.}

The estimated added weights were divided proportionally to the both _parts of the rudder and the rudder-posts by the ratio of respective chord _length.

Added weight moment of inertia about the shear center of the _rudder-post is estimated by Kumai's formula _[2]:

41= C2. r ir B where

B= chord length of rudder-post

C2=coefficient determined by the location of the center of rotation Any inertia couple of the rudder_{around its vertical axis passed through} the effective center of gravity of the rudder itself does _{not be transmitted to} the rudder post by the action of pintles' as it is assumed to be frictionless.

The relative angle between the rudder and the rudder-post _{may varies} in motion, consequently the effective position of the

_{center of gravity of the}

THEORETICAL STUDY ON THE VIBRATIONS OF RUDDER 71 rudder including the effect of the added weight of the surrounding water for the horizontal motion of the rudder-post is assumed to be m times of the distance between the center of the pintle and the actual position of the center of gra-vity of the rudder including the effect of the added weight of the surrounding water. The value of the factor m is supposed to be in the range of O and I, however it is thought to be difficult to determine this value exactly, we leave this value undetermined, and the natural frequencies are obtained in an range according to the variation of value m.

The effect of the clearances between the gudgeons and the pintles ¡s

negicted for the motion of the system in the theory at initial stage.

How-ever the discussions on this effect by the statical consideration are tried at the final stage of the study.

The rudder post is connected rigidly to the hull.

The stiffness of rudder stock is neglected, because it is supposed to be sufficiently flexible compared to the rudder post. However, its inertia forces including the added weight are taken into account by regarding this as a part of the rudder.

The effect of the shearing deflection is neglected ¡n the initial stage of the study, however it is esitmated approximately at the final stage.

The effect of rotary inertia of the system about horizontal axis is

neglected, because it is supposed to be neglesible order.

3. Fundamental Equations

The profile of the system is shown in Fig. 1. Any section of the

rudder-Fig. 1.

IO

Do.of Rudder

fW'

"'

and Rudder Post

{'t-ebx +m(axR-b)

EffeotivefCnfG(Y...

_{( R'X)} (e axp) Fig. 2. 4004' Cent, of Grv. of Rudder Center of Pint le ex b* UXR - Shear Center cxp Cent. of Grv. of Rudder Post

It is easily seen that the bending

vibration in longitudinal direction is

independent to the flexual vibrations in transverse direction and the torsional vibrations, owing to the symmetry of the sectional form and that the natural frequencies of the former ones are sufficiently high to meet the blade frequency

"

of the propeller.

The flexual vibrations in transverse direction and torsional vibration about

vertical axis passing through the shear center of the rudder-post may couple

because the shear center of the rudder-post and the effective center of gravity of the system are located separately and it is thought that these are the most important modes of the vibrations which we are going to study.

We take y-axiS in transverse direction passing through the shear center of the horizontal section at the lowest end of the rudder-post and taking z-axis vertically upward which has the same origin.

At initial stage, we neglect the effect of the shearing deflection to the

natural frequencies of the system, the warping

effect of the section of the

rudder-post, the rotary inertia of the system about horizontalaxis and the

devia-tions of the locadevia-tions of the shear centers

in the horizontal sections of the

72 J. SUHARA and N. FUKUCHI

post is symmetrical about x-axis in longitudinal direction and non-symmetrical about the axis y taken in transverse direction which passes through the shear center of the section of the rudder-post. (see Fig. 2)

\

THEORETICAL STUDY ON THE VIBRATIONS OF RUDDER 73

rudder-post from a same straight line (z-axis).

However for the effect of the shearing deflections, the approximate estima-tions are discussed at the end of this report.

Then the equations of the coupled bending and torsional vibration of the System are

-3-2-1E1

8z2 I. z 8z2 J - g 8:2

g 8:2

-[GI

8z t. a8zJ g 812 g 8:2

where following notation is used

ij=deflection at shear center of rudder-post in y-direction O=rotational angle of cross section of rudder-post E=modulus of elasticity

I=moment of inertia of section of rudder-post about x-axis ¡,,=sectorial moment of inertia of rudder-post {3J

GId=torsionai rigidity of rudder-post

p7==weight of rudder-post including virtual weight per unit length (used later)

PyR0. of rudder (do.)

mass polar moment of inertia of rudder-post including added mass around rudder-post only

g=gravitational accelerlation :=time

C= (1z)/1

1=effective length of rudder-post

e=distance between shear center of rudder-post and eflective center of gravity of system including added weight.

aXR= coordinates of the center of gravity of rudder-post and rudder including virtual weight respectively (used later)

b=distance between shear center of rudder-post and center of pintle. Putting =ifr,tq.t()sink, O=x0(C)sinIcr

kr=k lIT

b' -- e" = G!d012/EJW. EI,gf2

r'

' -

L.

ÈIgT'

- e"

'

_{- e"}

f,-

T",,014 e bZ+m(aXRbX)

- El0gT1

5=a0,,

=

a0,

(1)

74 _{J. SUHARA and N. FUKUCHI}

P,TP,R + /;p pa,2,ReZ +

r'!TmT'lP+,2,ke00 + p,pap

C:(C)_10(Z)/110 (for vertical member only)

Ci)m1(z)/I0 (

s, ) C,d()Id(z)/IdO ( " ) Cß,T(C)CR(C)+CP(C)

p,(C)

P,ro C0R(C) P,R(Z)/I.i,To,

C() p,p(Z)/ß,ro

C(C)

(z)/a0

Ca()caxp(C) CCR(C) . s + C(ç))

C1(C)_r'Ir(z)

- T'ID0 f p,,.0a20,0 C10(C)+ .-i;;-) C20p(C)CS2C00()+Cp(C))

r'4(z)

C10(C)

-The illustrations of above.mentioned quantities are shown in Fig. 3 to Fig. 10, where the suffix O indicates the average value of the quantity through the length of the rudder-post.

T=unit time (1 sec.)

m=factor to specify the effective position of center of gravity of rudder

including added weight in motion

Then we have fundamental equations in non-dimensional form as fol.

lows:

j Cjx(C) } = k2.b'(Cfl,T(C)ifr+ () C,(C)_{¡ --- z)} *0

{ c1(c)

= k2(j'Caa()*±r'C,p(ç)

z) The end conditions are as follows:

at the free end (ç=l):

=0, C,(ç)-!=O, C,4(x)-=O

and at the clamped end ¿ç=0)

*=0, x=O

I

(2)2 (2)3 (2)4

J

12

- 1.0

-0.4 0.2

W,ght of Rudder Post Including Its Vrtuol

Weight Per Unit Length (Non dmensionol)

CpP(C)Pyp ()/PyTo mean value

Ypo..76.o9 kg/cm

VituaI Wt.

Struct. Wt. I I t l I I I I I_...__

0

0.1 0.3 0.5 0.7

0.9

Fig. 3. Moment or Inert,a of Section willS respect to

e-OSis (Non dimenlonol

Cix() Ix(C)/Iao mean value Lao 3.786 alO cm1 X

JI.Jx

I 1.2 I.0 e-j, 0.8 0.2 - o o.i 3.0 o 2.0

o

Weight of Rudder Post Including Its

Virtual

Weight Per Unit Length (Nan dimensional)

CPRIC)=PYR ()/Yyro mean nalue

Ro.66.l8 kg/cm

Virtual W.

rupt.Nt.

09

Fig. 4.

Mass Polar Moment of Inertia of Rudder Post Including

Its

Virtual Weight with respect

to Gp (Non dimensional) Cipo () (1piÇ)/dIpo mean value '!po= 3.933 n lO kg cpi I I i e e t I I I L I i i $ 0.1

0.3

0.5 0.7

0.9

0

0.1 0.3

0.5

0.7 0.9 V Fig. 5. Fig. 6. - o' 3.0 2.0

o

¡I.0

o 2.0

4.0

Sectoral

Moment

of

Inertia of Rudder Post

(Non dirners;onai Cire (Ç)s 1w (C)/Iwo mean Value Iwo 1.832g tO em6 Fig. 8. 0.1 0.3 0.5 0.7 o

oi

d3

¿.5 ' ¿.7 '

-.-Distance

Between Center of Gravity of Rudder

Post

Gp (Including

Its

Virtual

Weight

and its Shear Cents, (Non dimensional)

Coxp (Cl-. asp (C)/axpo

mean value Oupo=-5.l7 cm 0.9 '-r en c X

-0

0.1 03 d.5

d.i

d.9 Fig. 7.

Distance Between Center of Gravity of Rudder (Including Virtual Weight

-and

Shear Center of Rudder Post

- 110

Cave(ç)= Oxrv(Ç)/axno (Non dimensional) mean Value

auRas 176.1 cm

r-X o I.0 E _a

Shear Center of Rudder

Post

in cm

J

0.8 A_..__

Distance

Between Center of Pintle

and

0.6.-\

90

--0.4-.

\

,.... "S-_r 0.2 i i t t i t t t I 80 0 0.1 0.3 0.5 0.7 0 4.0 3.0 k-n lorsonal

Rigdlty with respect

to Shear Center (Non dimensional Cid (C) = Gid (C)/Gido 2.0 mean vaIne Ido = 9.961 e 106 cro4 I.0 Fig. 9. Fig. IO.

THEORETICAL STUDY ON THE VIBRATIONS OF RUDDER 77 The equations (3) are simultaneous differential equations of the fourth and second order respectively with variable coefficients which are given numerically. These equations are hard to solve analytically for complexities of variations of the coefficients.

In such case the numerical method of solution is applicable directly by use of electronic digital computer.

At the first stage we assume the approximate solutions which satisfy the. end conditions (4) as follows:

(6ç2-4ç3+ç4) ₍₅₎

x(c)z(c)=2cc2

Substituting (5) into the right members of the equation (3) and integrating with the end conditions (4), and putting

=-=1

we have the second approx-imate values of (ç) and z (ç) except the value of k2.

Equating the maximum values (at ç=l) of the first and second approximate values of i' (ç) (the latter contains the unknown value of k2), we get the first approximate value of k2.

Using these approximate values of k2 and the latter, we get the complete values of the second approximate solutions (ç) and Xz (ç).

Repeating this process until we get the solutions with the accuray wanted. In this study, it has been taken following condition for the i th and ¡+1 th approximate values of k2. (k

k)/k12<lO3

By this process, the values of t. (ç) and z Cc) for any value of ç tend to

the respective limiting values with the convergence of k2.

Such process of the numerical calculations have been performed for various values of m which specify the effective locations of rudder-post.

4. Solution by Rayleigh-Ritz Method

To check the above results, the Rayleigh-Ritz Method are also applied to this problem. As the additional advantage of this method, we can estimate the effects of warping of the rudder-post.

For the system which are undergoing coupled vibrations, the strain energy U and kinetic energy T are

'EI /320 )2dz+ ¡ GI 30 2 -i.-- i-

r

(-)

dz =

1&

{3 (7)_Oe)

}2d r'p,,

[3(j-0a)J2a,+

'r

'30

J02g

Joj

3t J02g( 2

IIp,r(3\2d

l

327 30

_.dz+j

_{(--, dz}

J02g\3t)

j0g

1o2

_t-)dz+J

(6)

78 _{J. SUHARA and N. FUKUCHI}

Using the notations (2), the maximum values of the strain energy and kinetic energy are E10*0' r' C,,, dç+

E1,'

J'

maxU=__j C,,(c)

₂₁

--Gid, r' Cid max

T=k2[1302Í'

i ocilir (c) al ________ (i r

*oZoJ C0(ç)ifrzdç+

_{° Jo) (C1(ç)x)'dç}

gT2 Putting

where

Putting

max U=-4-N

and

_{max T=+Mkz}

N= (

E1,

) A*02+ (_)Bo2

r'4J\_.

M=

_{co2_2(!r)D*oxo}}

(

gT3 )°

gT2

j

Then we have 'r. = ivi 'in ₍₉₎

Equation (9) contains the amplitudes L0 and ,, for which the optimum values are given by 0

a0

i'

i '* '12_dç

[A(

Et,,,

=

1'

CCX\P

JoCid(ç) dç

where 2 = i means warping effect included, _{and 2=0 means warping'} effect neglected. And

Cß,

(Cß,r()*)'dç

Dm I'cßd(Ç)lfrXdÇ ,

K, m°

a'20 f D' \ Jo _T"po Q

_J0C,(ç)

p'

(C,(ç))2dçm

I C,,(ç)'dç Jo (7) (8) (10)

THEORETICAL STUD Y ON THE VIBRATIONS OT RUDDER ₇₉ (kb)2

=QÇ1( A'

P7,014 I)

(k \2

we get

x'}

i

- 2(1K1

(k)2=!ATB

_{r'iiz ()}

{ ((l+R2)±i/(1+Rz)z_4R2(1...K) ] k42(!I!2'0 (a0\ (D' (k12 (k0)2)

r'z) T)

₎ V'o k1(a,0\

(D\

-_7_) ')

(k42_(k,)2) ,

(i=l or 2)

Then we have two values _{of k which correspond to the circular frequencies} of the fundamental and second modes of_vibrations

By the way, the Lewis' formula[4J for estimating_{The natural frequency of}

the uncoupled flexual _{vibration of the cantilever beam is}

6O _Eg

11/2

27r12L(J1 (1_)2d)

(J ßy2 dç )

J

The result from this formula ¡s given in Table II _{for comparison.}

Results are shown in Table II _{in which the values for 2=1 show the}

re-suits taking accounted of the warping effect andÀ O means values neglected

it which correspond to the results obtained by

successive approximations from equations (3).

The functions _{t.(ç) and z(c) should be chosen so as to satisfy}

the end condition (4). _{In this case the final results}

obtained by successive approxima. tions have been used for _{them respectively.}

Thus obtained values of k are numerically coincided to the corresponding values of k obtained by the _{successive approximations}

These results verify that both processes of _{calculations are correct and}

practically usable. Flow _chart of this prosess is shown in _{Table I.}

5. The Effects of Shearing _Deflections.

The effects of shearing _{deflectjons are approximated}

by the formula [5J.

N,=N//

(14)

where N, is the corrected frequency and N is the _{calculated one from the} cou-I

(13'

f

80

j.

SUHARA and N. FUKUCHI No

(kk )/k

¡ 6k Yes Table I Flow Chart Start ¿ C2(6-4C'+C2)/3- *1(C) 2c(l Cz)_aXi(C) ¿ Calculatck0, k1, k1, k2. 0/* by energy

method using fr1(c) and x(C).

Stop

-,

axa c _(c)x-1(c) 1 82 (C1(C)0i*1a') i

i--)

MO k OC2 OC2 ¡ ¿

Calculate * (C) by Simpson's rule.

¿

*-(1)

_/

{-k_*1c1

} ---ak

(_-_ fr,(C)} X k --a

a -a k (J'C0(C>h(C) r'C1, (C)x1_1(C) } . [Ci(C)Ox(C)

Calculate x1(C) by Simpson's rule.

J,

¿

Calculatekb,kg,ki,kz by

energy method using exact solution *(C) and x(C).

Note: Suffix ¿ attached

to fr's, x's etc.

indicates the order of the approximation. x1(C)CXo/'fro) Xi (C)

k1-ak, (1= 1)

¿

Table

11.

Note : Blade frequecies:

Trial cond. 117X4=468

rpm and normal cond.

115X4=460 rpm.

(

) shows frquency of 2nd mode

or torsional mode. m o ax (cm) 49.07 Natural frquencies (rpm) Uniform section Succ. Variable section App. Energy method Lewis App. Suce. N N Energy method N NT N Exact ¡Lewis N N8 N8 N N N N

7o

fr,x N8 489. 7 N N8 . X *o.Xo 1('.X 'e.. x 1=0 _404.4 ArzO _482. 2=1 1 541.7 (2361.2) 2=1 483.7 (2309. 2) 2=1 (1849.6) x X t;. 1=1 _{405.6 2138.8)} 1=0 2=1 1817.7 1=0 (1531.8)(1553.3) 408.4 404.4 (1807. 7 0.25 56. 33 481.0 539.4 _{2172. 1)} 481.4 (2081.5) 489. 7 1629. 3) (1413.8) 408. 4 0.50 63. 59 477. 0 536.7 (2007. 3) 478.6 (1889. 1) 489. 7 (1449.4) 401.7 403.0 1746.7 401.7 (1572. 3) 1430.5 (1283. 4)(1292. 2) 408. 4 0.75 70.85 473. 6 533. 5 (1864.0) 475. 5 (1734.8) 489. 7 1307. 5) (1186.8) 408. 4 1.00 78. 11 470. 1 530. 1 1739.4) 489. 7 472. 1 (1606. 3) (1532.7 470. 1 (1190.9) 398. 3 399. 7 _{1484.9) (1384.2)} 398. 3 1180.6 :1096. 5)(1095. 4) 408. 4 ax-0 489. 7 (3785. 549. 1 1) 489. 7 489. 7 478. 8 408. 4 408.4 (2725. 4) (2726. 7) 408. 4 402. 4

82 J. SUHARA and N. FUKUCHI

pled bending and torsional vibrations and y, is the ratio of the shear deflection to the coupled bending and torsional deflection. The value of , is given as

follows

(E\f

Ixo \ r'

'

_{1c d}

r= k

_{A,P )J0C4(ç) '}

ix (c)--j

c

where k is the coefficient which is known as, lying its value between 1 and 1.5. k = 1.O-.l.5, coefficient of shearing deflection

G 'Shear modulus, (E/G) =8/3

A effective sectional area of rudder-post for shearing deflection CA(ç) = A(z)/A0,

A0 = mean value of A through length of rudder-post

tfr(ç) approximately assumed solution of coupled bending and torsional vibration

Results are shown in Table III together with the effects of virtual mass factor ¿9 which means the ratio to be multiplied to the estimated added weight used above of translational motion for the reference, and used for introducing the effect of the clearance of the rudder pintle and gudgeon.

We can see that the both effects are fairly influential to the values of

natural frequencies for the fundamental mode of the system and the obtained

ones may meet the blade frequency of the propeller which is 460 rpm. Table III. Natural frequencies (rpm) (Effect of shear and

virtual weight factor)

(15)

B : Virtual weight coefficient.

k : Shear effect coefficient.

S.E.N. : Shear effect neglected.

ft k N m=l.0 j m=0.5 m=0.0 S.E.N. 470. 1 476. 8 ' 482. 1 1. 0 454.0 460. 5 465. 6 1.0 1.2 451.0 457.4 462.5 1. 5 446.6 452.9 458.0 S.E. N. 519.0 526. 3 532. 1 075 1.0 1.2 501.2 497.9 508.3 504.9 - 513.9 510.4 1. 5 492. 8 ' 500.0 505. 4 S.E.N. 587. 1 595.3 601.7 1.0 567.0 574.9 581.1 0.5 1.2 563.2 571.1 577.2 1.5 557.7 565.5 571.6

6. Estimations of Statical Deflections

The pressure raise on the surface of the shell at the nearly equal distance of the tips of the propeller blade may be order of 0.1 kg/cm2 as maximum in full load condition estimated from direct measurements for a actual and similar ship with a propeller of nealy equal diameter and blade frequency in service

condition, though it may vary with the power of the engine. Assuming the

magnitude of pressure applied to the system near the lower end of the rudder-post and its vertical distributions are taken variously as shown in Fig. 12. The statical bending deflection is given by

p,B14

I'g(c) 'd

/

\ Ç

1Q JO _'ix\.Ç)

where.

p(z) = pressure at z applied to the system

p, = maximum pressure at the lowest end of rudder-post assumed to be 0.1 (kg/cm2)

B = chord length of rudder and rudder post approximately taken con-stant (490 cm)

g(ç) = P(Z)/po

The calculated values of maximum deflection are less than 1.6 mm (see

Table IV), which are same order or less than the clearances between the pintles and the gudeons. This means that the effectiveness of the inertia of the rudder and its surrounding water to the rudder-post should be decreased, i. e. virtual weight

factor ¡9 become lower than unity. It is supposed that complicate nonlinear

vibration may arise actually because the clearance effect is remarkable. in cm2

2.0-¡.0

0.5

THEORETICAL STUDY ON THE VIBRATIONS OF RUDDER 83

Effective Sectional Area of Rudder Post

for Calculation of Shearing Deflection

Mean Value=997 cm2

/Actual

0.1 0.2 0.3

04

0.5 0.6 0.7 0.8 0.9 (.0 Fig. 11. Assumed (16)

84

P kg/cm2

E

E 0.2

.E o.i 0.2

30.I

C)

ø 0.1

00

0.2 0.I Fig. 12.

J. SUHAR and N. FUKUCHI

Mea sured Point.

I.0

0 Table IV. g(C)=p(C)/po

max. def: maximum deflection in cm

Measured Points

Therefore the violent resonance phenomenon is supposed to be avoided in

actual condition because of the reduction

of vitual weight, the nature of

supposed nonlinear vibration and structural and hydrodynamical damping effect.

7. Comparison with Measured Trial Data and Discussion

Examples of measured data of the system are shown inFig. 13. The actual

amplitudes were far less than statically estimated ones obtained by above

mentioned way and the estimated fundamental frequency (460 cpm) was clearly recognized from them which are throught to prove correctness of the assump-tions in this theory within small amplitude of vibration. It is considered that the sufficient stiffness of the structure of rudder-post and the effect of friction between pintles and gudgeons made realized the vibration mode given by the

- ¡

300 'i 2

./

700 CPM 300 N'

y

' 3

70

CPM

is

(I 4

A

700 CPM 300 400 500 600 Fig. 700 13. CPM

p()=po

No. g(C) max. def.

0.I

i C 0.160

2 CZ 0. 128

3 C3 0. 107

THEORETICAL STUDY ON THE VIBRATIONS OF RUDDER 85 linear theory in this case and the effect of clearance mentioned above and miscellaneous damping effects prevented violent vibrations.

However it may not be guaranteed for the case of the rudder-post with

insufficient rigidity.

8. Conclusion

Methods of estimating the natural frequencies of rudder and rudder-post system of a ship of Mariner type have been developed.

If the possibility of arising resonance is anticipated by the linear theory, it is recommended to design the structure of rudder post so as to have sufficient rigidity of which order of magnitude should be determined by checking the rate of estimated statical deflection by induced pressure of propeller tips and the clearances of the gudgeons and pintles.

Acknowledgement

Authors wish to thank Miss J. Fukuda, Reserarch Assistant of Kyushu University who assisted us greatly in carrying Out this study.

References

[11 Ir. R. Wereldsma. Model tests for determining critical vibrations of the

rud-der-ppost of a "Mariner" Rudder. International Shipbuilding Progress. Vol. 6,

No. 57, May 1959.

[2.] T. Kumai. Added mass moment of inertia induced by tional vibration of

ships. European Shipbuilding. No. 6, 1958.

V. Z. Viasov. Thin-walled elastic beams. 2nd edition published for the National Science Foundation, Washington. D. C. and the Department of Commerce, U. S. A. by the Israel Program for Scientific Translations, Jerusalem 1961, pp.

263-272.

F. M. Lewis. A modification of Rayleigh's principle for calculating beam fre-quencies. Jour, of App. Mech. Sept. 1959, pp. 452.

F. H. Todd. Ship hull vibration. Edward Arnold (Publishers) Ltd. London 1961,

pp. 117-122.