The effect of slamming impact on a ship hull structure

À r

_r..

fl-STEVENS INSTITUTE OF TECHNOLOGY

EXPERIMENTAL TOWING TANK

HOBOKEN. NEW JERSEY

THE EFFECT OF SAdflG BñPACT ON A SHIP HULL STRUCTURE

Kazuo Ochi

or 53 Panel SNÄME

E.T.TO Project

2067

Lab.

_{y. Scheepboijwkjrd}

Technische Hoschol

Deift

INTRODUCTION

i

EQUATION OF MOTION AND ITS SOLUTION

DISTRIBUTION OF EXTNAL FORCE AMONG VIBRATION MODES

_iS

1n'FXT OF EXTERNAL FORCE DURATION ₂₁

CPARISON OF DERIVED SOLUTION WITH FITE DIFFERENCE METHOD

(Polacheck' s Solution)

COMPARISON WITH EXPI1XENTAL RESULT OBTAINED ON A SHIP MODEL

₂₆

CONCLUDING RARKS ₃₀ AC1VLEDGE4ENT ₃₂ NIENCLATURE

33

REFERCES

35

APPENDIX (A) DETAILS OF THEORETICAL ANALYSIS ₃₇

DERIVATION OF EQUATION OF MOTION ₃₇

SOLUTION OF THE TRANSFORMED EQUATION OF MOTION

SOLUTION FOR RECTANGUlAR PULSE

_t7

(A-Li) SOLUTION FOR SUSODDAL DISTRIBUTED FORCE

₅₆

PREFACE

20 May 1959

This report was sponsored by the S-3 Panel, HuU

Structure Conriittee, of the Society of Naval Architecte and

Narine Engineers.

It

was prepared by Dr. Kazuo Ochi of the Transportation Technical Research

Institute, Tokyo, while

a visiting staff iiraber of

the Davidson Laboratory

(formerly

Experimental Towing Tank). (SNAME Purchase

Order No. 113,

D. T. Project _{Oo 2O67)}

The S-3 Panel has decided to issue the report in this

rough preliminary fox to a few interested individuals.

It

is hoped

that the

author will publish a paper on the subject in due course.

Meanwhile,

attention is

called to the method used for distributing

the external

force among

the

different modes of vibratIon (pp. 15-18). This procedure is somewhat

contre-versal

and is

not endorsed

by the S-3

Panel or the Davidson

TR0DUCT ION

The problem of hull stress in ships produced by a rapidly applied force, such as often occurs in a ship's slamming, is one of the most Ini-portant problems for a siip designer. Yet, few theoretical analyses of

a typical ship's structural response to an impact have been attempted be-cause of the difficulty of the transIent phenomena analysis.

The careful examinatIon of pressure and strain records measured on ships at sea or on model tests in towing tanks at the time cf slamming show that there exist two types of pressure and accompanying stress

dis-tributon0 In one type, the water pressure in the slamming area is

sud-denly applIed and soon removed. The duration of the load

iS

generally of the order ¿f 0.02 sec. or less for a full size ship, and a sudden change of hull stress occurs almost irrinediately and shows the maximum value at the first vibratory cycle after a slam. In the other

typE,

the pressure at slamming is gradually applied and may last for some

_3.5

sec. for a full size ship. A notable feature of this type

IS

that the critical 5treSs does not occur at the time of impact but some time later, say from the first to

the thIrd vibratory cycle after a slam. The typical examples of the former are seen in the records of the test made on the USCGC UNThAK at sea, (2) and model experiments at the towing tank while the typical example of the latter is seen in the record of the test on destroyer at sea

It is conceivable that the difference depends on ship form, ship speed and sea conditions etc., but this is not clear at present0 However, it is quite - certain that the vibratory stress, whIch seems to include several vibratIon

modes, appears a certain time after a slam in both cases0

In a slamming case, relationship between the applied load and the stress cannot be determined by the rules of statics, and it is necessary to inves-r tigate the elastic inves-response of a ship's hull stinves-ructuinves-re to a slamming impact0

In particular, it is important to evaluate the superpositIon of hull stresses resulting from various modes of vIbration and the resultant time history. A slaming impact excItes ship vibration of all modes and such a superpositIon of several modea defines the transient state0

The existing state of knowledge is briefly summarized in the _seakeeping monograph by Progessor Korvin-Kroukovsky.

(6)

The known methods of solving

-1-N - 97

the vibration problem, which were given by Timoshenko and others, treat mainly the steady state vibration reached under a continuously acting

per-iodic dstrbThg force, but do not obtain a satisfactory solution for a transient response. Taylor treated the vibration problem applicable to a transient phenomenon, but was primarily interested in the evaluation of the vibration frequencies0 He mentioned that a gradual shortening of the apparent period of vibration with tizne was oneof the characteristics

of the transient response. Frankland

(8)

treated the effect cf duration of force application and showed that the effect of duration becomes evi-dent and the 'namic load factor decreases when the duration of a rectang-ular pulse is short in comparison with the natural frequency of the struc-ture. McGoldrick obtained the solution for the response by the normal mode method and applied it to a steady state vIbration of a ship's hull, while Polacheck presented a finite difference solution for a partial

differential equation of vibration f ôr use with a high-speed automatic digital calculator. In the paper of Ormandroyd and others

(1,

the de-tailed mathematical analysis of the vibration process, including shear deflection and shear flow, was atteirn,ted, but. it is regretable that the application to ship slamming was not mentioned. The author attempted to treat approximately a transient phenomenon, such as ship slaniming, by the recourse to the Heaviside Operator calculation. In the latter of the author's papers, he discussed the basIc properties of slamming stresses, especially the hull stress which was represented as resulting from summatIon of all modes of vibration.

In spite of these many investigations, the available material is not sufficient to permit discusrn a! the magnitude and time history of slam-ming stresses. The greatest obstacle, which has not been treated before, is the problem of distribution of an applied exciting force among various vibration modes. Inasmuch as the slamming stress is expressed by the

superposItion of various vibration modes, the applIed force should be ra-tionally distributed among these modes. This problem is avoided in a steady state vibration by concentrating on the synchronous conditions and assuming that in all but an insignificant part of the impulse is applied to the mode under consIderation The distribution should be a

-3-function of the position of the applied force, damping coefficient (dif-ferent for each mode) and of the duration of exciting force. The thve

tigation of the damping factor at different frequencies is important in this connection. In the present paper an £ttempt is presented to obtain a theoretical solution of a transient phenomenon in case of ship slamming, taking the foregoing comments into account. The main purpose of the paper is to discuss the time history of slamming stress obtained by superposition of venous vibration modes and proper consideration of damping factors,

In particu.Lar, the exciting force distnibuticn among vibration modes is evaluated in this connection. This is accomplished by an appication of the

least work theem0 The theory and the numerical calculation are mainly

discussed for a rectangular pulse of about 0.02 sec. duration in a full size ships However, a theoretical solution for a slowly applied force (a sinusoidal distributed force) is also given in the appendix.

In solving the problem outlined by the foregoing discussion, certain simplifying assumptions are necessary partly because of insufficient level of the present 1awwledge and partly in oriler to simplify the troublesome calculation0 These assumptions are: (1) Virtual mass is taken to be

con-stant during slam0 (2) Shear and rotary inertia effects are not considered.

(3)

Section of ship is taken as uniform. These effects should be taken into account in future work. The effect of virtual mass is considered to be es-pecially important, as it nay be a function of vibration mode and time.

EQUATION OF MOTION AND ITS SOLUTION

Assume a

ships

hull as a simple beam with

uniform

section shown in FIg. 1, the equation of the flexural vibration of the beam is given by the following formula0

ET

₍₁₎

where f

-Mass

of shIp per

unit

volume. A SectIonal area

E

Young's

modulous I -. Moment of inertia

C-fluid damping coefficient

C- internal damping coefficIent

F- external force per unit length, which

_{is the}

arbitrary function of x and t

The _{purpose of the present paper is to discuss the problem cf a}

tran-sient phenomenon when an external force due

_{to slamming Is applied to the}

ship hull, therefore it will be convenient

to introduce the Heaviside

Oper-ational transformation in

the above equation.

_{Taking into account the}

in-Itial condition as the displacemer.t

_{and velocity are zero when t}

_{O. The}

equation (1) iS transformed

as followa2

-dL4 T

- E1(li- /1)

(2)

where

-

ElCI

I

-P

p

J

where

out with respect to position. We get,

-t-0G

-'(o)t

ELCH-

(/)

-

'z.

(

) -

)

dt

q - Reaviside operator with respect

to position

tho,

)

(3)

the

trans-ut

(14)

into (3), the original equation of motion (1) is transformed

with respect to time and position0 That is,

%4

(o,)

(o,

₎

e

EI(!

Cy

₎

(S)

N

-

-S-p

- HeviSide operstor with respect to time

f(x)

- Position function of n external force.

øf(P)

- Transformed time function

of an external force.

In the next place, the i-jeaviside operational transformation is carried

5L(q)TransforTned position function of

an external force.

If a

single

force is applied to an arbitrary

point

Equation

(5) should be solved under the given

boundary conditions

and given

form of an external

force.

The

boundary condit±on of a beam

is considered to be both

ends free, that

is the bending moment and shear-ing force are zero at both

ends.

At

fIrst,

the following condition at x - O is taken,

namely

-= O

I

(Lt)(

= C

a3d

Equation (6) is transforned and put

into

equation () It

gives,

-4o)

dt

4_À4 EI)

By calculating

the inverse transformation with respect of q is carried

out

in equation (7),

+

t

(o,)

(8)

where

SL Ait-L) - J À(t-1)

C&4À)

(6)

=

(J2

/

)\))

N - LL97

The boundary condition at the other end of beam

iS

applied to the equation (8), and two un1own functIons _Ø

(o p),

(o p) are de-terrnined as follows: where (d,

) -

À-2)ì

Y)} g()

E1(i-a

_l(C

)__-

P()

z'r

(ÀQ)

4(cÀ-()

(4ÀL

1)

A(-.C)J 1(A)

_{p LAt-}}

pRZ)

From equation (8) and (9), the function (x p) Is evaluated.

(9) (Í-C)1 'Ck)- À(-:)}gcAL)

(i,

=

_À(i-)1

EL(ItC,)A3

J L C-X

1(ÀL)

A(-f(A)

-t-&(AQ.)

(lo)

The above formula is applicable for l x 10 But 11 the origin

x O Is taken at the other end of beam and substitute

_Cl

- l)i' or l

ail

-7-aboveforxiulas are also applicable for O x l ( the range of x is changed to i l x 1 in this case)o

Now a deflection of beam is obtained by carrying out the inverse transformation on equation (10) with respect to time. Prior to

proceed-ing with this transformation, the form of an external force çÇ (p) should be determined0 When a disturbance load is suddenly applied and removed

a short time t, as shown in Fig. 2(a), the function Ø, (p)

iS

expressed

f

--p

I (

F(-

)

EI(l1CyE)À3

tA()} A)

-13(AL)

fr-

Tt

The following calculation is for thIs rectangular impulse, the solution for slow2y applied sinusoidal distribution force is given later. The de-flection of beam for a rectangular puise is given by the following formula from ecuation (10) and (il).

r(À)-dA(-]

t)

-t-A)

(u)

(12)

In order to evaluate the deflection, it

iS

sufficiit for us to cal-culate the residues of the equation (12) at singular points. It gives,

-N - 1497

-8-F

-3)

IlL

(-L)9

(+3)(-!)

fA I 6

6o

2

(6k)f

(z-3)4

2- 3)

(E-)5

(3-4î)

(-L

420 l2P- 60 + 20

4k

where

-

ì473O,

7.853, 10.996,

iL,137 . . . .for each mode

4 k

-rEi

L()(S4

(j

-

S4)

-

Cl/A(J/L)4}

E -

[i4

L C

Distributed force in j-mode vibration at a specific point i

The ist group of the above f orrcula is a rigid body, rotatatory motion

of a beam which produces no

stress in the beam0 The 2nd group is a

sta-tical deflection, while the remaIning term

corresponds

to a transiit

a

s-(2ck).

60 _2CL3

[(rn)

_(w)

_k

-

()

₎ X -x I

F.

_{(Wi. \4'2Y}

(%

)2

[(

_T +

_{ç y) j j}

.rA%C'Y)}

t

Íc

)4_

,i

L 2YZ

(»_i J

t +

E

}

(13)

-9-deflectIon of the

beam.

Attention should be

called to the fact

that

the

equation (13) is a surrfnation of all

forces acting at arbitrary points

and also is a summation of all possible

vibration

modes which are caused by an

external force at the point.

So, the external force F1

at a specific point

(distance l from the origin of a beam) should be distrIbuted among

all the

possible vibration modes.

Let force F1 be the force producing the j-mode

vibration (j - 1, 2, 3

.

.

.

u).

In other words,

the distributed

enter-nal force

at a specific point produces the j-mode vibration, and

sunna-tlon of deflections caused by all these vibration modes

and

positions would be the final answer. The method of how to distribute the exciting force will be discussed in the next paragraph.

From the deflection f orxnula for a rectangular pulse shown in equation (13),

a transient bending moment is

evaluated as

follows:

H =

c-L3)x..2

(2í.)t3

(A. 16

-t-*iL

L X

»j

')

+

k-&)]

N -

¿497

-

lo

-C()4t

[ck)4 C24

L

}

fr

A steady state bending moment after the force is rcmoved is obtained

as follows by the same procedure as before.

M'F

where

{Cg

Lie

X z

.c4/ ()4;

!2fA _,2A CZ

(if

[4

t /2pA r _J

t

c

"

4

L

/2A '7? ,<

-()

[(.i)

r(»1) - (1j)

))

'Çì (/ ()4}t

i fA /2

[-L

tC

(J}2J

t WI J

(1g)

¿fA 4A

'î;

t

C4J

t7()4tJ

'c

-

il

-y2

As mentioned before, a solution for O x l is obtained y the

saine formulas as (114) and (15) when the origin x - O is taken at the other end of the beam and a substitt.e(l - l)f or l. In this case, the range of x becomes i

- l x 1.

It is of interest to mention here that a bending moment (stress) in higher vibration modes die out in turn. This is easily comprehended from equation (15), in which the damping term

i3

proporti nal to the fourth power of the constant So, the term e

- m

' will rap-idly decrease in the higher mode vibration and the bending moment due to the higher modes will die out

in

a very short time.

The bending moment for a slowly applied force, namely a sinusoidal distribution load as shown in Fig. 2(b),

iS

obtained r the same method f or a rectangular pulse

if

the function (p) is exr)ressed as the f ol-lowing formula instead of equation (11).

-lip /'r

(t

(1

(16)

N - 1497

12

-where

T2 - Duration of an external force for a sinusoidal d1strbution load0

i656)

't-f

where

\4'

I

4

_(4/

_¡rn

k)AtA(

_L()-{rA

_i2f4

I wI 4

t

2.

('Y

L

24.(Y)4fl

-The first term of (17) gives a forced bending moment during the force

is applied, while the second term gives a transient bending moment.

6 F

(j) i)sǻyj

a

ç-

-(17)

13

-p(A(!S(À'X) J

L

CV )

, 2 _\412 J ç

(XJ4A

C

) À)

ÂC)

ik

{{À()

(A)À(!-!:)

C

()4t

x 1-

Ï(;V42]

fr

The steady-state bending moment is giv

as follows,

H

fC/

-Ci/

[t

'24 y2

(W4]

(4 I

2e

(]]

IA /f4 C

C()4ÌJ

'1A /2

where

[ (»)

) () ]

['))

j

()

_C/

+

C;()4}t

r./

Ci

_Ci

()4J

j

+ ¿ L(

i)

24

¿rA' !2f

cy f

21

L

[(Y)-

'2f4 2f4

j

-I

+ %A

[(r_

%( r!J

i

N - b97

zj

(18),

The external force F applied at a specific point should be rationally distributed into each vIbratIon mode, because a deflectIon or a bending moment caused by an external force is expressed by the superposition of various vibration modes as mentioned before. Suppose these vibration modes are 1, 2, 3 . . ,u mode, an external force at a specific point F1

should be dIstributed in F11, F.2, F13. . .F1 for each mode. This is written as,

t).

(19)

A deflection at an arbitrary point

iS

expressed,

z

(20)

where is a deflection due to the j-mode, vibration caused by a distri-buted external force F. at a specific point i.

Bding manient at an arbItrary point is

M = EI Z1

(21)

Ci the other hand, a deflection is expressed by a product of func-tion of x and t as is easiiy seen in equation (13),

X() T

(t) (22)

X (y-),

where C is a constant, D is a function of j-mode andATj (t) are the func-tion of

x and

t respectively0

From (21) (22), a bending moment is expressed by

Ei C

Fa (23)

Strain enerr of a system is obtained as 2E1

=4'

Z

_1.1

11 (

d L £

t

-(it-i) (u-i) Lk

V

£

:

. t

=0

F1

=

-

2Y

---V

_o

F2

z --

-

o

'FL44)

Now the external force F is a summation of F1 as shown in (19) If we consider a force Ff1, F12 . . . _. are

independent

varIables and Fj is a

depend3t

variable. Then, by

applying

the least work theorum, we get,

(25)

16

97

Readers may question whether the least work theorem is applicable in this case. A brief explanation should be mentioned here. As is well known, the least work theorem means that the magnitude of moment or stress

is such as to make the strain energy of the system minimunì for a body in equilibrium under a certain force. This theorem would be applicable in the dynaini.c problem if we take the strain energy of a whole system and the whole time during which a force is applied. Also, in a strict

mean-ing,

the right hand side of equation (2g) would not always be zero but equal to a deflection at the point where a force is applied. For exinple,

the ist equation of (2g) would be expressed as,

j__

3I

, DF.,

where

Y11 Deflection of the ist. mode vibration at point i As is shcwn later (see equation 28), each term is wrItten as,

2KF,

o

K

C' D ( d? 2

K=

- C

dX

L 2 d. t .1 C (a) (c) - 17

-o

(b) -2 K F where

=

ci,.

,

Then, the equation (a) becomes

(2KL

C) F

-

=0

(d)

It is proved numerically that the value of Cil in equation (d) is negligibly small compared with 2K.1. For example, we suppose a single force F ₅₀ ton is applied at the point

_0.075

L aft of the forward perpendicular of a ship 500 ft. length, duration of load is 0.02 sec, without damping. _{This is an} example of numerical ciculation stated later in detail. For this case, the value of 2K is 2062 x

io2

as is shown in a later paragraph (see table 1). While the value of Cil is T1 (t), where

f T (t) K 1

As is clear in this example, C1 is negligibly small compared

with 2Kilo Then the equation (d) becomes

K_il F_il

-K

F

-O

iu iu

In other words, the above mentioned assumption _{is preferable and equation}

(25)

is permitted in our case.

Turning now to the proceding problem, V1, V2 . .

. .V1 in equation

(25)

are expressed by the following formulas _{respectively f rom equation (22)} and (2).). Et -

2'

_I 2 - 2

yCDF (T')

(t) dt -J L -r , - t ₂

;(

()dz

_1T(t)

= K

2 (26)

vfc

/ r.L

V --'

_¿

dX; dc o (e)

=

where the function _{K..(x, t) is designated as}

(e)

N - !97

18

-z

(

t) =

cL

91 ()d

J1 T.(t)

t

From (25) and (19), we get

;

_{= 2}

) V2

= o

F

o

VL

-

2 k

(27)

(28)

- 19)

Then, from (25) and (28),

K F

- K

F

= o

K K F

0

(29)

E

LL

::I

I

F:.

L'

Equation (31) gives the distrituted f ca'ce among each mode vibration.

(31)

N 197 - 20

Fill F e .Fj(u_l) are ritten from (29)

F11 - K1.JK11.

-

K/K

(30)

Fj(ul)

Ki(u_l)iu Fj

From(29)and (30) , F11, F12 . . . F are evaluated. They are,

N

21

-_____ EFFECT OF EXTERNAL FORCE DURATION

r It is of interest to discuss the effect of duration of loading on a

transient phenomenon0 The effect would be different according to a dif-ferent type of disturbance profile such as instantaneously applied rec-tangular type or sinusoidal one, so the discussion will be made on both cases0 _{Also, inasmuch as it is considered that the effect of damping} seems to be of

secondary

importance in discussing the effect of duration of loading, the discussion will be made omitting damping terms0

(a)

Rectangular Pulse

The factor N which is designated to show the

effect of duration of

loading is derived as follows from equation (ls).

N

[i

2.[(-

c:y(y;4

(32)

If dainpings are neglected,

N -

2

1-cos

(mj1)2

J

-

2

sin

(m1/kl)2 T1/2

}

(33

While the period of the j-mode vibration _{without damping is expressed.}

T1

2n/(_)

Coefficient C. is designated for convenience as

c

_j

_-

lj

(SS)

Then, equation

(33)

is simply rearranged

N - 2. sin

_(c'ii)

(36)

From the above formula, it is easily conprehended that an effect of

du-ration of loading will be

maximum when

_{a time duratin of rectangular}

pulse is one hal.f of a period of the j-mode vibration, But actually,

the value of C1 is about 0.1 _{or so in case of general ship}

I"

(b) Sinusoidal Distribution Load

The factor N ror a sinusoidal distribution load is from equatiofl (18)

If dampings are neglected

_(

C N

-

_p7)2

2(Ç;)

(::.

(Ì)/)4

or

where

tir' ' (2Jc/ _'\ S'

2(4)

- Circular frequency

4C C64 (c«rc J

(38)

(36!)

22

-I 2 A

iA

Lc

4Xrfl

N

-

[S- 'n42.

(

% +

(37)

The absolute value of N versus C is shown in

Fig0 3(a).

Fig. 3(a) shows that a slight difference of duration of loading brings a marked effect on a bending moment (stress) in a transient phenomenon. In order to check this more clearly, n example of effect is numerically eval-uated for a destroyer and the factor N is shown in

Fig0 3(b).

_{In this}

cal-culation, the values of natural frequency for each mode _{are taken the} ex-perimental values on Destroyer C. R. WADE (length 383 ft.). On the other hand, the duration of loading is about

_3°

or

36

sec, for a destroyer

of nearly the saine length. _{By the way, inasmuch as the equation (38')} is indeterminate for 1/2, a special consideration has to be paid for this particular case0 But the value of _{is more than 3 even in the ist} mode in case of an ordinary ship slanudng, _{so the equation (38) is enough}

to evaluate the effect for ship slamming9

-COMPARISON OF DERIVED SOLUTION WITH FINITE DIFFERENCE METHOD (POLACHECK'S SOLUTION)

AS untioned in the introduction, Polachec1 presented a finite dif-ference solution for a partial differential equation of vibration for use with a high-speed automatic digital caicu1ator(:1. In order to check the propriety of the derived solution, comparisons were made between these two solutions under the sane conditions for two different dampings. The dim-ensions used in the calculation are as follows:

Length

Mass per unit length Sectional area

i -500ft.

fA

1.5

ton-sec2/ft2 2

A -5 ft.

N - ¿497

214

-Young's modulous _E - 2 io6 ton/inch2

Moment of inertia

I - 6

_{X ft.}

Damping Coefficient _{C/in- O and 1.}

External force

_{F -5oton}

(Rectangular Impact)

Position of applied force

-0.075 1 aft of the F. P0

(single force) Duration of Load

- 0.02 sec,

The eva1uaed distributed forces and bending moment _{at midship for} each mode vibration in case of no damping are shown _{in Table 1.}

The cal-culaticn formulas to evaluate _{are equations (31) (27) (114) and}

(15). The evaluated percentages of distribution force for each _{mode are plotted in} Fig. _{14. As is comprehended in the figure,}

the effect

of

_{damping is} rec-ognized: _{for instance, the distribution percentage with damping}

shifts to lower modes to a certain degree compared with _{no damping. It is clear} that a tolerable percent of

the external force is distributed in this _case, however it depends on the _{position where the external force}

is applied. In order to ascertain this, _{an example of the effect of an impact}

point

is

shown in Fig. _{in case of no damping0}

Fig. 5 is the one example, however

it

_{shows Important suggestions to the slamming problem.}

Because the con-siderable parts of an external force are applied the position _{between O.15L} and 0.25 L aft of the

forward perpendicular in case of slamming impact for ordinary merchant ships, however a tolerable force would be applied _{near O.1OL} aft of the forward perpendicular _{for full-form ships0}

Now, a superimposed calculated bending moment is shown in Fig. 6 and 7 comparing with Polacheck's solution. It is regretable that we have the claculating sheets by Polacheck's method recorded at every 0.02 sec., so we cannot plot exact curves

including

higher mode vibration; however, so much for the comparison between these curves, it may serve

to check the propriety of the present method. There is a considerable degree of correspondence between two curves. The reason for any des-crepancies may be considered in part due to the assumption that the present method does not include the shear and rotatory inertia terms0

!97

-COMP&RISON

wrrH

E)CPETh!ENTAL RESULTS OBTAfl'ED ON A SHTP MODEL

In order to apply the present method to a slamming case, the numerical claculation on the deck stress due to the slamming impact is mode for one of the model ships (V-form forward) on which the experiments were made by the author

The principal dimensions of the model are as follows:

(19.68 ft.) (27l ft.) (1.714 ft,) (1.63 inch2)

(1,575

lbs.)

(25143

inch14)

(15,358

x io3

psi)

(11.8 inch)

Model speed and draft under consideration were 2.141 rn/sec.

(F -

v/ /-jI

_O322)

and 20.0 cu

(787

inch) respectively, and the model.

suffered a heavy slamming under this condition. The measured duration of load was 0.005 sec. Virtual mass coefficient is assurued to be equal to each mode and is taken as 1.2,

In the numerical calculation, the applied impact force in ship bottom is obtained by multiplying the measured pressure

by

bottom pressure area, and this impact force is divided into five parts for the convenience of calculation, namely 00/lOL, 1/bL, 1.5/bOL, 2/1CL and

2.5/bL

aft of' the forward perpendicular0 The stress on deck is calculated by the super-position of the stress due to each distrIbuted force and up to 7th. mode0 The divided force into five parts are as follows:

N - 1497

-

26

-Length L - 600 cm Breadth B -

82.6 cru

Depth D - 53,0 cm Sectional area A -

_10.5

cine

Displacemit at light draft

W

7)5 kg

Moment of inertia

I - 10,550 cm Young's modulous (brass)

E - 10,7140

kg/xnim2

Distance between deck and neutral axis

These will

be

distributed among each mode vibration.

The derivation of damping coefficients used in this numerical calculation will be mentioned in aetail in Appendix (B). The main point of the derived method is that

McGoldrick's

formula (h1 is modified using Sezawas

paper(1,

as the model is made of brass. The damping coefficients

C1/2fA + cI/2?A (N/1)

By using these coefficients, the distribution percentage of the external force among

_{each mode}

_{at each point is obtained as shown in Fig0}

_8.

It is evident from this figure that the higher modes

_{should not}

be

neglected

in a

transmet phenomenon such as slamming _{case even though these are damped in}

a short time0

The

time history of the superposed _{stress amidship is shown in Fig.}

9

with the experimentally measured stress0 _{The superposed stress curve agrees} quite weLl with the experimental curve except on the ist cycle, and a great

deal can be learned from the comparison.

(1) _{The higher mode stress, say}

the 7th. mode stress, _{cannot be neglect} at the beginning stage of the calculation _{of vibratory stress due to slarrmiing} however it dies out quickly with large damping.

ed are as follows (see table 2)

Vibration mode

Cf + CI (M/i)

C1 +

CT

(M/l)L

2k

2rA.o 1 1.30

0020l

2

_3.033

_0.0171

3

.623

0.0168

9.677 0.0168 5 ]J4.1i57 0.0168 6 _20.196 0.0168 7

_26.877

_0.0168

27

-F,00 L aft of F.?. j,2 Kg

(70 Lbs0)

F,0.10 L

1008

(238)

F,00l L

28 2 (62.0) F,0020 L .0

(96 8)

F,002

L 30 .1.

(66.9)

Even though the

th mode stress is negligibly small from the

beginning, the 3rd mode stress should not be

neglected throughout the

time history, because its effect is still large near

the 14th cycle of

the vibratory stress and the maximum basic wave

sag$ng stress will be

superimposed near here.

Damping coefficients which are derived by the before mentioned

procedure are proved to be proper0

Now, the large discrepancy between the calculated and experimental

stress values is recognized in the ist cycle.

In the experiment, the

deck stresses were picked up by the resistance typo wire strain gauge and

megsured in the osoilograph. The natural frequency of the vibrators of

the oscillograph were about 900 cps or so, and were sufficient to record the

phenomena.

Also, the experimental stress value of the ist cyice are not

thought to be excessively high in this case, in comparison to the other case

in the series tests.

Therefore, the discrepancies of stress value at the

ist cycle may be considered dependent on some other reasonable causes, Lut

they can not be ascertained at the present stage.

The distribution of the slamming stress on Ship deck at each cycle after

the impact force is removed is shown in Fig. 100

The measured one is also

written in the figure. From the consideration of this figure, it is of

in-terest to note that the maxinum stress is recognized somewhat forward of

ainidship, that is about 0.142 L aft of the forward perpendicular at the ist

and the 2nd cycle, and it will be amidship at or after the 14th cycle. The

experimental results show the same tendency as the theoretical results,

how-ever discrepancies are recognized beteen their absolute values at the ist

and 2nd cycle.

The property that the distribution of the slamming stress on deck shows

the maximum somewhat forward of amidship immediately after the siazmuing

im-pact was recognized in the results of the several model experiments.

(II)

This is now proved theoretically by the superposition method of all proper

vibration modes.

The next problem is to discuss at what time the highest stress occurs

after the slamming impact, or to see the instants of time various events

occur when the impact stress is superposed on the basic bending stress.

From

N - 1497

the exçerimental data, the basic hogging and sagging stresses ainidship are 0.12 Kg/mm2, (5 - -0.l Kg/mm2, period is 113 sec., and the

slam-ming occurs 032 sec, after the maximum hogging in this case. The behavior of the midship stress with time which is obtained by the superposition of the basic bending stress and the impact stress is shown in Fig. 11.

From the consideration of Fig. 11, the highest stress which is super-posed on the basic bending stress occurs in the interval between 0.11 and 0.23 sec. after the slamming impact except the ist cycle. This corresponds to bet-ween 0. and 1.1 sec. after the slamming impact in case of the full size ship. In other words, the highest stress occurs about the ¿th cycle, in the vacinity of which the maximum sagging appears.

The magnitude of the superposed high stress in the vacinity of the

xmax-imum sagging is about l.l. times the values of the basic sagging stress. This shows a little higher value than the value described in the _{experiments on the} full size ship (the value for the full size ship is said _{to be about 1.2 or}

1.30.). This discrepancy seems to depend on the fact that the experiment _was made in the light draft cOndition and very high speed, namely in the very

se-vere slamming condition.

-CCLUDThG RFLkRXS

0n the basis of the theoretical work on the elastic response of the slamming impact and from the comparison with the other solution and ex-perimental results, several conclusions can be drawn0

These are as follows:

jì

elastic response, namely a vibratory stress which is produced by the impact at the time of ship slamming seems to include several vibra-tion modes, The higher mode vIbration should not be neglected at the

be-ginning stage of slamming however it dies out. quickly with large damping0 The above mentioned high mode vibrations die out in turn. For example, the 7th mode vibration dies out after the 2nd cycle, th mode (though its amount is small at midship compared with the other mode) dies out after the 3rd cycle, and the

3rd

and the ist mode vibration remain

ip to the time the basic bending stress curve reaches the maximum sagging condition,

A distributIon of an external force due to slam shows that it seems to be a function of the position where the force Is applied, and also a function of duration of loading and damping coefficient.

(ti) The distribution of the slamiiing stress shows the maximum

some-what forward (00b2 L aft of the forward perpendicular) immediately after the impact force is removed, and the maximum shifts toward midship after several cycles0

() The stress superposed on the bas±c bending stress shows the high-est stress in the interval between O5 and l..1 sec, in the full size ship after the slain except the ist cycle0

The magnitude of the superposed high stress in the vicinIty of the maximum sagging is about lli. times the value of the basIc sagging stress by the numerical calculation0

External damping due to the generation of pressure wave seems to be not negligible in the ist mode vibration, it takes about 20 % of the

total damping in the ist mode. It is sufficient to consider only the in-ternal (structurai) damping in the other modes.

N - Lj.97

-N - L97

31

-(8)

The effect of a duration of loading for a sLwy applied force

is remarkable0 Â slight difference

of duration seems to bring a marked

ACOGET

The author wishes to acknowledge the kind advice and instruction of

Professor B. V. KorvinXroukovs1 and the encouragement and help of Professor E. V. Lewis, under whose guidances this work has been carried

out. Also, the author

would like

to extend hIs thanks to Dr. P. Kaplan who read through the mathatical part of the paper and kIndly gave some helpful suggestions0

N - t97

-N - L97

33

-N EJCLATURE

1 Length of beam

12 Length of beam between a specific point i and an original point

y Deflection of beam

f

Mass per unit volume A Sectional area

I

Momt of inertia

E Young's modulous

C Internal damping coefficient C. Fluid damping coefficient

F External force applied at a specific point i

Distributed force of F4 among the j-mode vibration V Strain enerr of the whole system

Strain iergr of the j-mode of vibration

p Heaviside operator with respect to time, namely d/dt q Heavisde operator with respect to posìtion, namely d/dx

Notation which means Heaviside operational transformation Notation which means inverse operational transformation (xp)Transfornied function with respect to tIme

(qp)Transformed functIon with respect to time and position

øf(xP)Transformed function of an external force with _{respect to time}

(f(qp)Transfor1ned function of an external force with respect to time and positior

T

Duration of an external force

' Phase lag

w

Phase lag

T _{Period of the j-mode vibration of beam}

C _{The ratio of duration of time to the period of j-mode vibration,namely}

/T

N _{Factor which shows an effect of duration of loading} mj _{Constants, suffix j means the j-mode vibration}

PAp 2+Cf P

Er + cup

k-"* o( (x)

-

(Cosh x + Cos x )

(xisj(Sinhx+Siflx)

' (x) -

(Cash x

- Cas x )

-

(Sixth x - Sin x )

(x) -

(Cash x Cos x 1)

(x)(CoshxCosx+1)

-

.

(Sixth x Gos X

- Cosh x Sin x)

2(x) -

.

(Sinh x Sin x )

97

31

-RTFEN CES

Jasper, N. H. and Birmingham, J. T. : Sea Tests of the USCGC

UNIMAK, Part 1. General Outline of Tests arid Test Results, DTMB Report No. 976

₍₁₉₅₆₎

Greenspon, J. E. : Sea Tests of the USCGC UNIMAK, Part 3,

Pressures, Strains, and Deflections of the Bottom PlatIng

mci-dent to Slamming, DTMB Report

_{No0 978 (1956)}

Akita, Y. and OchI, K. : Model Experiment on the Strength of Ships Moving in Waves. Trans. SNAME

₍₁₉₅₅₎

(Li) Ochi, K. : Model Experiments on Ship Strength and Slamming in

Regular Waves, Trans. SNAME

₍₁₉₅₈₎

(under printing).

Warnsick, W. H. and St. DenIs, M. : Destroyer Seakeeping Tra1s,

Proceedings of the Symposium on BehavIor of Ships in a Seaway(1957)

Korvin-Kroukovsky, B0 V0 : Seakeeping MOnograph

₍₁₉₅₇₎

Taylor, J. L. : Dynamic Longitudinal Strength of Ships, Trans0 INA

(19L6)

Franklarid, _J0 M. : Effects of Impact on Simple Elastic Structures, DTMB Report 1L81 (19b2)

McGoldrick, R. T. : Calculation of the Response of a Ship Hull to

a Transient Load by a DigItal Process, DTMB Report 1119

₍₁₉₅₇₎

(lo)

Polacheck, H0 : Calculation o± Transient Excitation of Ship Hulls by Finite Difference Methods, DTMB Report No. 1120

₍₁₉₅₇₎

Orrnandroyd, J., Hess, R. L., Hess, G. K., Wrench, J. W., Doiph, C. L., arid Schoenberg, C. :

Dynamics

of a Ship's Structure,University of

Michigan,

₍₁₉₅₁₎

Ochi, K. : On the Stress Distribution of Ships at the Slamming Speed,

Trans, Tech. Res. Inst. Report No.

19 (1956)

Och!, Ke : Some Consideration Concerning the Effect of Slamming

Impact on Hull Structure, ETT Note No. ).79 (1958)

(1L) (cGoldr1ck, R. T. : Comparison Between Theoretically and

Exper-ientally Determined Natural Frequencies and Modes of VIbration of Ships, DTB ReDort No0 936

_(195b)

Sezawa, K. : Damping Forces in Vibrations of a Ship, Trans0 of

Soc. Nay. Arch0 of Japan (1936)

Sezawa, K. : Die Wirkung des ddrucks auf die Biegungschuwingung

eines Stabes mit innere Dampfung, ZA1 (1932)

(1?) Suyeh±ro,'K. : c the Damped Transversal Vibration of Prismatic

Bors, 3u11. Earthquake Res0 Inst. Tokyo Univ. (1929)

Whittaker and Watson, : Modern Analysis (1915)

Taylor, J. L. : TJbration of Ships, Trans0 INA (1930)

Kumai, T. : DampIng Factors in the Higher Modes of' Ship Vibration,

Res. Inst. Applied Mechanics Report No. 21, Kyushu Univ. (1958)

N - 1197

-APPDIX (A) DETAILS OF THEORETICAL ANALYSIS

(A-l) DERIVATION OF EQUATI( OF MOTION

Suppose a ship's hull as a simple beam shown in

Fig0

1, the equation of a flexural vibration of the beam including damping terms is derIved by the following procedure. Tangential viscosity and Tensile viscosity are assumed being proportional to the velocity of shear and tensile strains

respectively, so they are written, Tangential viscosity

TensIle viscosity

where C! C" are coefficients, ' and a are shear and tensIle strains0

i i

A lateral motion of a bar per unit length including tangtial viscosity is given,

2 C'

__

2I5 ₊

(2)

at2 x G

where _f) Mass of ship per unit volume including virtual mass A - Sectional area

S - Shearing force

G - Modulous of rigidity

An angular motion of a bar per unit length including tangential and tensIle viscosIties Is given,

rT

-

-

+

where I - Moment of inertIa M - Bending moment

E - Young's modulous

By differentIating wIth x, it gives,

a3 I S _-' C M

r I

_--

_{t) =}

_-'-

_5t)

E

From (2) and (3'), the terms including S are eliminated.

II c ti4t c

2t

(1) C" i

(3)

(3')

37

-CA.

C:

M

-a

t

?L

L

-i-

-ì-WhIle a

bending

moment is given,

2

MaEI

Then,

equation

(b)

is rearranged,

pT

(\ Ef

-

cI

r

2 -

-

&

If a transverse dimension of a bar i

_{ll enough}

_{compared with its length,} the first term of the right hand side of equation

(6)

will be negligibly

small.

Then,

equatIon

(6)

is simply wrItten,

(7)

where a coefficient C. is substituted for C _for

_{simplicity, this corresponds}

to an internal

_{damping coeffIcIent Ó±}

_{bar0 Equation (7) was derived by}

ezawa(16)

and Suyehiro many years ago.

In case of

a forced vIbration due to an extsrnal force F (x, t) _{and If} a fluid

_{damping is considered, equation (7) would be}

_{as follows,}

-.44

A-

tEI----

F(t)

(8)

where Cf Fluid damping coefficIent.

The purpose of the present _{paper is to discuss the}

_{problem of}

a transient

phenomenon when an

_{external force due to slamming is applied to ship's hull,}

therefore it will be convenient to introduce the Heaviside operator in the above equatlon. First of all, the

_Heavlside

_{operational calculation is carried}

N - 2j97

-out with respect to time,

arid a tr.nsfonned term of y (x, t) ±s wrItten

as ' (x, p). That. is,

(9)

N

--

39

-where p - Heavislde operator with

respect to time,

d/dt

or it Is wrItten

symbolically

y (x,t)

' (x, p,)

(10Y

In proceeding

the transformation in

equation (8), the following transformation formulas

are used.

t)

.(z)

(o)

t)

p(2

o)-

(z,o)

(:L t)

d4

D

p7; ((zc)

An

tea1 force F (x, t)

in the right hand sIde of equation (8) is

transformed as

follows,

F (x,t) - f(x) g(t) £(x) (p) (12)

The initial conditions are taken as velocity and

dIsplacement are both

zero when

t

O, namely

at t - O, y (x,cJ) O

₍₁₃₎

(20)

N - li97 - 0

-Put (11) (12) into (8) and taking consideration of (13), the

trans-formed

equation of motion with respect to time is obtained.

_

(]J4)

where

I EL

(1g)

In the next place, the Heaviside operational

transformation is

carried out with respect to pos1ton x, and

the

transformed term of Ç' (x, p) ±s written as

(/i

(q, p)0

That is,

-(16)

where q is the Heavislde operator.with respect to

position,

d/dt or lt is written symbolically,

çi (x,p) _:

P (q,p)

(17)

In proceeding the ci1cu1ation, the

following

formula is used

4()_

(o) )

-(18)

The term of the external

force f (x)

Ç (p) In the right hand side of

equatic (iii.) is written by transformation,

f(x)

ciÇ (p)

_*f()

_f (P)

(19)

Put (18) (19)

into equatIon (Th), the transformed equation of motion

wIth respect to time and position is obtained as follows,

(

À) ()

+

+

-(ó,,b\

Here, we consider the

function of

(/

(q) for a single force at a

specific point x - 1,

as shom in Fig. 12.

If a force,

magnitude of which

is F,

iS

assumed to apply uniformly between x l and x - oo, it is

ext'ressed by the transformed form

as

F1

e1i

(21)

Similarly, a force applied uniformly between x -

l

+ L

and x

-00

is written,

+ 1 )

Fe

i i (22)

Therefore a force F1 between l + and i. is obtained from (21)(22).

:

Fe -

The.

;

- 2i.

(t-2. .2

1--

(L.)

-n

-(23)

For unit length Al. -1, the transformed function of Pf (q) is obtained,

Pf (q) - F1 q (2L)

Put

(2L) into

(20)

the original

equation

of motion (8) is transformed as follows,

(_

4)

(A-2) SOLUTION OF THE TRANSFORME EQUATION OF MOTION

We now solve the given transformed equation of motion (2g) wider the given boundary conditions and given form of an external force0 The boundary condition of beam is considered to be both ends free, that is the bending moment and shearing force are zero at both ends. At first, the condition at x - O is introduced. The condition is,

In the transformed form, (26) becomes

d2

l(O, p) O

(o, p) - o

3.

So, equatIon (25) becomes,

-(À)

(o)+

(28)

In the above equation, (O, p) and

₃

'(o,

p) are still unknown, but these are obtained by taking consideration of the boundary condition at the other end of beam. For thIs purpose, the inverse operational trans-formation is carried out with respect of q. Equation (28) is rearranged as

-C)

(28')

The ist term of equation (28') is inversely transformed by following procedure.

j1

4 (29) N - LL97

-

I2 -d3

dx3

t0

ci-Zoo

The integral path of

f.

means the path from C-ooalong the straight line parallel to the

imaginary axis of the q-complex plane.

hi1e, by

using the Jordan's inrna(18), the integral path

f

would be equa]. to the integral path

_f

,

that is the suui of thé integral along the straight line

parallel

to the imaginary axis from C-lao to C+iao and along the infinite closed circle from +iqo to C-ioo including the singular points

in this domain. Then, the solution of equation (29) is to evaluate the re-sIdues at singular points in the domain (see Fig. 13). Now, returning to

equatton (29), the residue evaluation on the 1st term is carried out as follows,

where - (t'(Q

h\

-____

t-

_'

-2A

r

-c

i-4--

1+-L

_J

c

1-

e1 )

4«o4)

L)(2

J

j.

¿Aa

N -

97

t3

-c

oÇ(A2..).

c)

(30)

ç,( (À)

_--

_Al

_{- C}

_AX)

₍₃₁₎

Similarly, the 2ns term of equation (28') is transformed,

q

(1

I

-1)

(32)

where,

The 3rd0 term is transformed,

()

F

A(-)}

45)

-À

ET('/)

A3

where

=

Ax -

A

_(3g)

By using transformed

formulas (30) (32) (3L), equation (28') is transformed as,

4(x

) -4- -

()

ê 1:)

(36)

The boundary condition at

the

other end of beaii is now applicable to

introduce in equation

(36) The boundary condition Is expressed as follows,

,t)(

o

Condition

(37)

is writtt in the transformed form,

o

N - L97

-(314)

(37)

(38)

From

₍₃₆₎

and

(38),

two unknown terms

(o, p),

.'(o, p) are obtained0

In the procedure of claculation, the differtials of function

(À.) (Ax),

(A))

are simply written as follows,

where

(hl)

-

(C4À-- CÀz)

The function

Ç' (O, p) and

'(O, p) are obtained,

4cc

)

=

c)

A((-)I

(()j

(A)

Ej((-t-C4

_t)À3

=

1' ¡)

À(!_L)} (A)

d.

_EI(HCr)Àl

where

)

(A)

c)

(CJL ('JA D

(39)

(L1o)

-

-From (!l) and (36), the _{function 9((x,} ) i evaluated as follows,

=

F E.L (i

_YE)'

H 1(

_{_À(-1) À)}

()43)

The above formula is _{applicable for l.. x}

l, namely for AB range of beam in Fig. 1. _{For BC range, all above formulas are also applIcable}

if the origin _{x - O Is taken at the other end of beam and substitute}

l_li

f or _{In this case, the range of x}

is l-li x l.

Now, a deflection of beam is

obtained by carrying out the inverse operational transformation _{on equation (L3) with respect}

to time, that is,

t)

-

q)(. p)

Prior to thx proceeding

integratIon, the form of the external force should be determined.

A

Y()

-

À(-f

O)

±

N - 1497

-(A-3) SOLUTI FOR RECTANGULAR PULSE

When a disturbance load

IS

suddenly applied and removed after a short tIme ?, as shown in Fig. 2(a), the applied force Is expressed in the form of the Heaviside operational transformat,ion as,

Then, a deflection of beam for a rectangular pulse is expressed by the fol-lowIng formula f rom (L3)

_{(144) (t5),}

-P1-Li

i .iiL) _EI((tC.,')A3

L

À(!-')} 1 i Àc)}

À)

It IS sufficient for us to evaluate the residues of the above formula at singular points.

(1. a) Residue at singular point p - O for O t T1

The 1st. term of equation

(6)

is expanded as a Series of p, considering p is small,

-. { (%

-EI(H-C1')

3 ErA3 2'V

(iY

3!

2

i_

2 p 1

(A)j-- O

J 147 -'LA )- 1\(k- ) TCÀ) -(146) 7 ZÀ -fr 7! (147)

Put (t7) into (L6), the residue of the ist. term of

(It6)

is obtained

3

---

(x-i)

E t

Similarly the residue of the 2nd term is obtained by

expanding

a series of

p. That

iS,

4F

(2f-3' i

(It9)

TA ,

Ei (.

J

The residue of the 3rd. term becomes,

/2 F

c-4.

s-r ₍

Á-!)

-4.;)2

-)z

(2-É)x.

_(so)

EI t /o 2.0

60

2O»

J

Therefore the deflection of a quasi static motion of beam is the si.munation

of

_(it8)

(149) (0),

that is F

4@D_3)

1pg)

-

-.----

-t- -

-S-O

PAL

-E.- Ç

(ì._Í)

+

3)(J-)

(2-3i)L4 (I2

42.0 /.2P

+

C3-4)Z

('-')

[2.

20

4L

20P

u

The

ist. and the 2nd. terms of the above formula are rigid body and rotary motion of beam respectively, while the remaining terms correspond to a statIcal deflection of beam0

N - 1497

148

-(148)

(I. b) Residue at singular point p - O for

The transfoniied function of tim (p) should be taken

(l_et1)

instead of i, as shown in equation (1). It is easily recognized that all residues are zero in this case.

(2 a) Residue at singular points which satisfy

(xi)

O for O t

Equation (146) has singular points which satisfy

L)=

(-Àt CÀ D o

(2)

If we write roots of the above equation as rnj (where j -2,

3, 14.

.

the value of mj are as follows,

=

_{r 1473O,}

_{± 7853}

_±:

_10.996

r ¿14.730,

ilO99ó

(3)

Each value corresponds to the ist, 2nd, 3rd . mode vibration respect-iveiy. Wow the residue is obtaned by carrying out the following calcu-lations,

-;-[3i(í4

_&

_{r:) -}

/À(-)

(J)

(Sb)

In equation (3), -_(Xl) is evaluated as dp where dA

L1t)

I(tt)t(Ej+çI)

4À3

(ic'i

CÀL

- C..uA2

_{LÀi )}

149

-(SS)

Then, the denominator of equation

(51) becomes,

-

-

EI

t

[An]

4-,

FIt(J

-

pA ')

-

4-while, p is evaluated from equation

(15)

where

4

f-/

f.A

'

Ei

Put

₍₅₇₎

into (SLL), the summation of

residues, nneiy

the deflection of beam is obtained. That is,

u

F.

LkAt

C:/()fJ4]2}

[

y.)

(i.) J(b)

[()

y1)_ () F')] ()

LL(' )

At

cV(y)4/t

{(-

%4t

C./()41

t E]

(59)

( /

f-

-

t

l

EI(Ct f-A')± n4(Er G1)

L

4À

_(EL i- (I)2

fr

Cs 2_fA

.i:1' 4

1

4\be

N - LL97

so

(6)

(58)

where

L

¿

f4-y

Attention should be called to the fact that the equation

(59)

is a

summation of all possible vibration modes which are caused by an external

force at all points.

So, the external force P1 at a specific point should

be distributed into the force which causes the j-mode (j - 1, 2, 3 .

.u)

vIbration, and thIs is written as F.

_lj

substituted for F in the

_i

equations.

In other words, an external force F1

at a specific

duces the j-mode vibration, and sirrnnation of deflections caused

these vibrations concerning vibratIon mode and position of external force

should be the final answer.

The method of how to derive the distributed

force for each mode of vibration was mentioned in the paper,

(2

b)

Residue at singular points which satisfy

(xl) - O for

t

The transformed function of time

should be taken (l-e'

)

instead of 1.

Then

ePt

in equation () is replaced by (l-e)e

proceeding the same calculation as before, a simunation of residues Is

ob-tained as follows,

\ /%i/ 4

'j%)

- C41.

-

C%4())4

I

'T/

(fi' 4

L' W

'12tA

(4

2J

)

former

point

gro-by all

N - 197

51

-X

[&()

?')

-. ¿

[çt

[

4t

₍₆₂₎

Jt

{%Át

4J(tt

_}

- ¿_

2'

4 Q_

For a simplicity of calculation we use the following temporaly notations.

lì

e

L f

Then the time depìdit term in

equation

(62) can

be rearranged as follows,

'a ('g.

j1 ir

(63)

N -

!L97 -

52

--' Ç

()Y

--

-

J

-

.) (_ ? ) J

_t

_r

-M.

e

C

L

C'e-nY

jw) t

-e

4:M) -

-) t

-t DM-t c c 7,

<L (

J' (4):

_t

_r

/_

I

-L

- 2C

e

t

g -t

/k;1

-2.

¡J

--w,

ml-; _{2,uT '} f

=

_{j -2C}

n-)

e. :

i

+

_(6h)

where

, , - -- t

_L

_¿4A

_{l-- W)}

(

tT _V

1-ne. tC(7i2.z;

u ç-7

ii

Therefore the equation (62) can be

written,

4 _c

4fl4

2e

J

(

4

ri

_{t T;}

'i L w

;i-()t {) r)

()J

(-)

uj)

t

-(6S)

-(66)

where

C%i

Y,:

[(-

+

(í)]

I ¡

r%Ø)4

c

[(r)4

+ A(

E

From deflection formulas for a rectangular

pulse as shown in equations

(S7) (9)

and (66), bending monients are evaluated as follows,

Vj

()

t

LI

_c'

-,+ 72tØ)

4 )(

£4

L)

ÍAt

%(r I

(67)

)

jj

(68)

N - tL97

-+

_{As mentioned before, a solution for O x l} is obtained by these f orrnuias when the origin x O is taken at the end of beam and substitute

l_li

for In this case, the range of x is i - l x 1, It is of interest to mention here that bending moments (stresses) due to higher mode of vibration die out in turn. This is proved from equatIon (69), in which the damping term is proportIonal to the furth power of constant m . So,

_ímt/T

i

the term '

1. '

would r'apidly decrease in the higher mode Ibration,

in other words, the bending moment due to higher rn e will soen die out at the beginning stage0

ft-s

Pv1--. L Li

C; / fi :/ ' r

- ¿2tA

/-

"

J s E-

₁

(

[1y.4./+

t ri. r_

[()

(-9

c)t

-)

11!

)'/

(?. _tA )

(r'!:'

s)]

ä(1--)

] t

-r-(69)

r y 5__5 -r

LA(4r

_t

_{try

--17

'(f

(A-Li) SOLIJTÏON FOR SINUSOIDAL DISTRI3UTIGN FORCE

Vhen an external force is applied i.ii a sinusoidal form with time and

removed after a short time as shown in Fig. 2(b), the applied force is

ex-pressed in the form of the following Heaviside operational transformation.

&v otT

1

1-î

-Jii-e.

) (TV

Li

EL(c(-A-Then, a deflection of beam for a sinusoidal distribution force applied

at a specIfic point is

cpressed as following formula from

(Li3) (1111)

and (70),

(M)

(71)

By the saine method as mentioned before, the integration in equation (73) is

evaluated by calculating the residues.

(1. a) Residue at singular points p

-

i (rr/-..)

forO tT

The residue in this case is obtained by carrying out the following

cal-cilatI on.

M - ¿4.97

-

-(jo)

1(A)

-LiJ

(A.)

_À-L)

_{À(/i ?(À)}

(Àx)

'Ç

-r

This is the calculation for a forced deflection which is caused when a sinusoidal distribution force is

applied.

Inasmuch as the effect of damping is considered to

be small in a forced deflection, damping

terms may be neglected

in

the procedure

of

calculation.

Therefore,

is simply written, neglecting

damping coefficients,

-

EICJ

El

in which

then,

where,

hile the term containing X in equation (72) shows always the same value

ir-respective of

var±ous

values of X.

For example,

(7g)

(73)

-

-L-1

4 t \I / 1T

-

¿

JÀ

_{À = -'}

-f

-(76) J

À(C-L)

J

Therefore, the term

ÀL-L)

d-i)

&A) frW)

o4(À)1-)\3

L

shows the constant value, incidentally the calculatIon Is sirnple By carrying out the calculation, equation (72) becomes,

-

t (

-

¿J

_(r

'v'id)

)})

À(Î-Nk-L) 1(u) Qf-J

¿Tt

-i/-tì

_iA)e

(4iT/q) N b97

-k -4 '() À(-I

r(À)

_A)

4(TÇ

) (77)

fri'

OtT2

(1. b) Residue at singular points p

-V

j(k.

)

=uj

In this case, the same calculatIon as shown in equatIon (72)is made

-p

multiplying (1 + e 2

₎

for

N - L97

-

-JÀ(-

(A)-A(-!:)}

)(()

A(-L)

(À)-3

(78)

ÀL) ==ItCTn1'T)

It is suffIcient to consider the time

dependent term

instead of

Sin in equation (77). While the time dependent term

becoines

St±/(t-'Tì= 2(t)Ca

=c

₍₇₉₎

Therefore the residue in

this case is

_{zero, In other words the beam has no} deflection

after a

time an ternal force is removed0

(2. a) _{Residue at singular points which satIsfy}

9(Xi)

-

_{O for o t}

The

equation 5'(i) - O has many roots in. as already

_mentioned0

_The

residues of

equation (71)

about

these

singular

points are evaluated by the

saine procedure as mentioned in the section (2. a for a rectangular pulse0 Tnat is, pt

A(--I)} (M)

JI

EI(j+ C p)À3

(1y7).

d A(- : )

T ()

AL)

+ (8o)

The equation

(5g) (56) (57)

are also apnllcable in this case, then a

sian-rnaticn of'residues is obtained as f o1lcws,

where

X 4 _I' +Ì

f)i

Cj// _C

C«K(}] [t

( )

+

_fA

[1Jïr-

7)_())J

₍₊₎

C:

(:

-fA ) LL(Th) 4

-C'

rA

izf4.

ç,

I "

ZJ/(7()4Ì

[

)-L

/2A

A(j

E. ).Z1

(2

_%A

_A(rJ

-

₍

N - ¿97

-60-ZJt }

(81)

(82)

(2. b) Residue at singular points which satsTy.(Xl) - O for

CalculatIon procedure mentioned iii (2. b) Thr a rectangular pulse is

also applicable in this case being replaced e in the equation (80) by

(I + e ¿it

The time depndt term in equation (81) is

14t

I, y2

L2.

L (- )

(t

)

n whichthe definitions of in, n are described in (63). The above formula can be rearranged às follows,

t

(

t

-

mt

wtj

Ç

t i- L (k (--) -rJ

i

I

-y

-j

y

I

-L

L

j

b

L-)

Jj where

u=

y ,

t

_{Ç }}

:

(ïCt-)

y

) t) - L 2'7j

C4

(_2)

1± e]1T (83) (BS)

- 497

mt

-

(

2

(

mt)

Tj ± L

f

- m) t -- 6, ± Ct }

(8h)

-.lt

Ç--

T44 +

,ss

J