^ Technische Hogeschool
DeKt
g ^ l ^ C T ) Wave Induced Vibration % \ % ^ \ t
ffl ^ 3t
mm
m 4 0
sum
BH ?n 48 11
On t h e C a l c u l a t i o n M e t h o d s o f Wave I n d u c e d
V i b r a t i o n o f S h i p s
By F u k u z o T A S A I
I n t h i s p a p e r we h a v e d i s c u s s e d a b o u t t h e
c a l c u l a t i o n m e t h o d o f w a v e i n d u c e d v i b r a t i o n o f
a s h i p a d v a n c i n g i n w a v e s , a n d p r o p o s e d a
t h e o r e t i c a l a p p r o a c h to i n v e s t l g a t e t h e f o r c e s
-e x c i t i n g h u l l v i b r a t i o n .
233
Wave Induced Vibration f f ^ ^ f c
m t ^ - m ^m t m
*,15:i±. M^f7t£h\%X\,^?> wave induced vibration ©IfffffiCOTO^©'^•5 1 1 IC, ^ n f , C D H ^ ( D ^ a t t ! C ' 3 V > T s f t f t t l ^ f i 5 l i , # > ' i a t f Ü è ® - : . g ^ l C springing ItgL
I . ft a
2g5MKi*,2.v>i±3|ESCDMÜi=&a;Ts:ii:i5^i.5. rfLe,CDj|öö;MSii»±, ^ K ^ ^ S i ^ t i ® t j M Ü i 19 fc
Z<Di.'jfS:mmmcomm'mLXii, Saunders, H. E. ® i : # " S r f c | 3 i i 5 n - r f c l 9 , t ) ^ - M » t c i.-^xm^iZ^^'^Samm^i-i^lfL Ben and Taylor" JfC J ; T ^ ^ è f l . m # « a i ï t t ±
m^b" fc, ^^u-^ + i f K t l ï l i Ê I - ? . DW 76,500 ton CO f > -izmmU « 3 ^ 4 4 ^ 7 . ^ 7 0 ~ 8 ^ 1 2 H © F ^ , m^). « ©iKWtr. ± m fiïjfCióSöm, midship^üy--) ? 9 S ® ± T i P ; ^ S , «ömmidship^üy--)£cD2 (square station 9% t 9^midship^üy--) IC t >midship^üy--) O l t / i f f i i j ^ - f i ^ ï C i - o t ^ ' üi*ffi, /<-7-,f ^ ^ u ^ t - f P i c i ^ - t ^ S i t g | j S ® i t P J i t - ^ ? T ' i ^ © f ê ^ ® ï S : ^ J è g l i ~ S
8** izmt- mit2, H 3 f c 4 , 0 5 t 6 , mT tsfi<^t\^'n-!^cDnm^^x-^?>.
mi t2i,^i^hi-hi.Tiz, i&^mmi'/\^^\^i,zi,i^i^hh-nmm<D^mmi l o o g a i e ; ± ® J I T 2mmwn!>m±Lx\<^i,. th±Tzmi:nmm^(Dn^^7mmmiy'b^^^^iii^^i:x^^z. ttz±
0 3 ~ 8 iigmtSMfC J; S * 'J föiëS;^ 7 mo±S^itizmmLh'^(Dmm\X-kZ.
234 ^ 40 ^ lALTtal *' *r vertical accaltrallcn ,l F.R *f VtrllMI KCfnthK tt aSdMp *M xtrllnl accfrallto at AP. 1 m & s :f3, ±. T m
\
% CO E » if Horirontel aectlarallon at r.P.I
Bottom Prtssur» No.lAr Varfieel aeeataraffon at F.P, A* Vvlleal accalaraflon at mliMlp At Varlical tceahmtlon at A.K
m»<D Wave Induced Vibration t t ^ & l c H i - 5 - % ^ 235
y——^
Ar HoriloMal eectltratlon ot F.P. 70,^
m 4 m^Ê, m m f ^ t m m m o m u
Af Varlleat eecefaraflon ot F.P f ^ 1
At. WrtlcQt aceêttratSon at mfdsnip T el 8
As yêrtlcat aseéftratlon at A,fi fi
0 5 l ö l S f l E ^ I , ± T j t ! q a S t D | 2 g
± T J n a S ( i 2 g ^ | a i S L T V ^ 5 ) . C © l 2 g - C i ± ± T 2 i 5 ® ^ i ! i t a ] l S l ± 1 t ( c ^ t < li'ii^^is, Sffi
a±0i•5''d:S^si^^'^9^*iIll^^ci-p•CSleè^^5ffift^ii!lli springing t^J^ whipping, {5;f3ji springing stress ibSViti whipping stress fcDïlf l a t l ^ S ^ i s ^ f f ^ ^ ® K S V ; i - ' b f l J K 1 " • & tJ)? ®i7lc^l^5J'it-CV>5i'5-C-*)5.
23Ö
0 6 mm, m m n t M ^ m t o m u
2) Whipping: ïS^-^SÖf*iiÜ0;*C§Vvhè V»Cr5-;5-be>-f, SSWKffi*'?' bow flare impact
'Jfmx\t, wave induced vibration (:Ht-5H-S;iÈt::':>V^-t®ïM^J:¥f5tè®Ü-r2.11 fc 1', S
2. Wave Induced Vibration
springing ^ w h i p p i n g ICJ: SjDnüS. ffiW^® < ftlf * - > > ^ fci 2>'iS^J'i t ' t l t ^ C L f c ^ S : CD£il'ifc®**>(fHl£, s a " ' " . Kaplan e t a l " , Goodman", Kumai and Tasai", Gunsteren »'
mii-CO Wave Induced Vibration ffS^Sfcl^-r5 237
g a * ' l±Wagner"> 0a^}cS-:^V^•r j | 0 J g » * ® t t E t ^ ® | | S t i £ t c o i ^ - t : ^ i ; T l > 5 . ^Oifê ^ . i E K i B ¥ O i E * ( i e ^ S S Ï ® « ^ ® f f i i I l - i ± ' < 5 - i : * ^ ! 2 f ê a ± f ' i a i - 5 i « , ® « S C J ; - , t S T 5 jK(*S«)t:iSb-Cti, M^f6»(rJ;5fc0:i5m'54r;*:§i/^0-C. whipping ®ISHfc b T l i S f t S S ^ t c
Saunders" ! i w h i p p i n g t t T . « J t S ? ® * } * » * t i a 3 B » ® ^ H ® 7 i < * « W i - 5 # i n H S t virtual mass ® 4 ' T , non-linear (mvir)„ t U, -?;®25^}-0±T>?I«lSSè > i f H l f non-linear tl reaction force F„ |±i?:3dCT-^A iöH2> t LTV-'S- EPt.
f'n=-^C('«wV)„>2. (1) iSöl"0±"F^ffi y TJS J-oSinOei (ü>e t ± ® t ® W é « S a ) , $=myacos<o,t -t>*>-=Tt,
H i s non-linear tt F„ immizm< h\-^^^7Ll5Xihi>. U.S. Navy © carrier ESSEX (--5V> T l t » - r 5 t , 2 « i j g l l ^ ^ U - S S : - r 5 g f i t [ > } ' ? 3 e > n 5 t ® - < 5 1 1 1 H , r ® : S K T ' flare impact ) & : f l c j ; 5 whipping è fc I t - ^ - f S Z t;5>-Tt 5 t l ^ U T V S .
g a " i±, ^©/J^èv^^®èg^^tf:liT)5^^Tr5^ö*^m^*Jli!l(4%^^^^:v^®H^jtv^ whipping
S n s ) H i - : > T S T 5 i I * ^ # i | g S ! t ' t f 3 . ^®4'®2|5MBi!3ai:raUM®:]KBE^J®:^I*«SSilfig:Öf^ ' i S t ^ U T l ^ S . t'S*?*,, g l 9 T » 7 K ® $ WoLo, WL t t S - S f f i f f l ± # i ^ S $
AF=AB+-^Umy) , (2) T - ^ ; i T V ^ S . AB=2pgby, Am^pizbyz t t ó l j i f
I
238
m
:t
H 40 ^ |0 9
i<
JV-f
M
<o
m m
AF=2pgby^^[^-y) (3) hti.y), 7 Hi-p-C b ^!3E^b-t5®-C, tzkpL y=rocx>sa>,t tL-Xl AF izit a>, £W©Sf2^®fig •
X'<jv-fizgi<tl Ft
F 4 = - f - { l + ©i cos cos <!),/+©, cos2a)«< + e,siii ZcDj+e^ cos 4a),n (4)
S60 2 a 5 S i i l S C ; i « ( 4 ) S 4 ' 0 f S i n * > i : - g t t - t * S t n t f whipping iih^. IP*, selective reso-nance |Cj;^-CSfil*>'S5. ^hizmt. ö f * g 10:5 h >®ffi^--^V^T K'=50 h > t L T $ l f ^ L , gegA=30m t 50m ® S K S Ï i r S 2 g S S S ) ® - h T J P a S * * » K S ! M H f i l > ® * ^ X V ^ 5 .
Kaplan et a l " l i , i ^ ^ I I J S f f ö i T S . fiï^®jlii)t±S'^®x K • / S l É T t t S L L T ver-tical shear force, bending moment J^^ft, W3UmWi\Z-o\,-^X\iÈM^b^M^o:>m^l3m^^ m^^X lateral shear force, lateral bending moment ^ ^ f c , ;H?>®jS^§S!K^fflV>-t^;SfllJ&
^(Oj&WMWUzmrf < bending moment Ti fc-®|§a®)tett05j¥%fi^Sft2':Si£$ i•fla3^L-CV^S.
ma^O Wave Induced Vibration fftgij^ftü-r 239
j',=vertical elastic deflection, c=damping coefficient, V',=shear force, /"(^r,/)= local wave exciting force.
tti r=deformation angle, KAG=vertical shear rigidity t t t l l i
dMtdx^V. + I , ^ . M=EI^, (6) è-C, P(.x,t)\t, # J l H S t t S H J ; 5 f c © § P^{x.t), ^ ^ 3 ® ^ ^ b H i ? , i ® * P^ix.t) fcL-C, P{x,
t)=p,ix,t)+p,{x,t)
t t - c v > s . z.h\mw hmm^sx^z-mzp,{x.t)
t t - r ( ± . e>:^P i ( x , / ) = ^ ( m „ t , . ) , ^ = - | — ( 7 )
^ i / f b «/,.=relative velocity=i-*Ö-l-Vö-Co. (8) S)*;, V=fiöi^, Z=±-Ffi§^{Sr, e=im^^, f o = g f f i ^ 0 orbital velocity ®±T^f|q]Sg^^,
=#iina<i#i;pSS-c, ffS®^^i0#^^SS^^f.^*ffi-c^^rv^sfê®#ilIlHgtMb§|v^^^t,®.
L T : ^ ^ S I M 4 ' ® « ö # : i 5 g : ® ^ ^ 5 1 M ^ è i t S b . bow flare slanunmg H i s whipp-ing bendwhipp-ing moment © ^ Ö i J ^ ' f b è fc^ftTV'S. carrier ESSEX .tZ-:3V>T3Slttt^èfTit£-p/;ifê
2 f ï ® ± T S Ü ( C i s bending moment (4^S:^# <^^,HS*<, 3 g f i ® ^ n t i ^ ^ - C ^ S f S i f .
1) mnm(Dm&\i. xmimnm^Mzm-^&s.t)tm\zm.m\z^L\^}iLtz. 2) )fö#:Si&;}iSgïl{fci(i:f^gi4^j^U7^;.
-/,=entrance length, m,=co^o= hhX^Mzè,^nmo'ti1smA, y=m^iS.(0±.^m&-® e . y=m^iS.(0±.^m&-® ^ x . : ^ f c g a " # i n i S Ö i l y=m^iS.(0±.^m&-® ^ { b H i S ^ ^ S « t « : A * % x , selective resonance tr i -3 r ± T M S i * « ^ S è n s t - t s t . ® - c * s .
( i « e 0 ^ f t i ; i : - 3 - c ï T ' j C 7 : 5 S H i - 3 T , g i i i i i i : ^ f c - i J A ' m i d s h i p Hfc^tssi^l^s:ü^&|f|?LTl/^ s.
240 ffl :t S 40 ^
mmmölzX^xm^Ltz 2^ ^fflV^-C_hTMSifS®co|fi:=£fTJiV\ i89fil!2fcffiO^-(bi:
spring-ing C!>|i^H-3V'>X|iD-Cl^5. È ^ ^ i ^ ^ H i S sprspring-ingspring-ing C07jc<i^^èï5^tV\ sprspring-ingspring-ing ] ^ selective resonance H i - 5 - C ^ S T S i t t ® ; 4 ^ » S 1 1 fc H, springing (D^ckWMMW^Oi^^
Ê l ± 0 | f - g L j Ê t * j S f i ï H M ' Ü f c 0 H , Goodman" t Gunsteren» ; n e . 2 A
MiSfc, P^4^i4fcv^•f*^fc-^i?©S^êl^^tL-C^!6i^ t t- 5 f c® T* , 5 .
CaCOs(*x-ha>.0. Ca=KM«i, *=e>V^=27rA ' l = S a <Ui=a)-FAK "Ci-S.
i='(:c,0=^Wsinfl>,i(ü>.=ttièraMii)|SO t*JSttf, njJi® mode ® S ï I i a » f M « l 0 ^ C ü 5 ^ { i e > :
ü-„Wsin(a).;-^„)]_ F(ï)f/„Wrfic
tztih ï/„(a:) = normal coordinate, A„=« S:®@^i
A = f ^ A ( ^ ) W W r f « = | ^ ( ' « + ' « ' ) f / n ' W < ^ ^ (13) I
C=|^(;V'-K-|^+c)f/„»(x)d;c. (14) I
; r - c w, m\ N' t6iiV= C' {±, S S S ^ i f l j ^ f i ï S è S t J ® ® * , » f f f i f f l ± T S i ö H * f T s # i n S a ,mzmmimii,
F = r ' ' ' F{x)Un{x)dxi-Wl
=2P8\y^U,,{x)e:,dx+\(N'-V^)u^Mt,dx+\m'U„{x)^,dx (15) -CJ^itf>n5. i^:/i-L., Ci=Coe-*<'icos(èx+£Ue<), Ci=-CaO>e-*<«isin(*A:-l-o,t). fi=-Coa>V-**JmW<D Wave Induced Vibration ft^iSicSS-r 241 Froude-Kriloff ©^j, mZT^tm^mi-^H^HmL^O) orbital S S ^ i J ; ZA'JqagHifcWITS di-ffraction force -CJi)5. la-g-fiM^CS^Js 03 * J i i / a 9 ^ t i Goodman" O - ^ x / i i C T ^ S .
^J;ttf «=2 t f c - i t i f , a i l M 4 ' ® 2 f 3 © g i I f i ( f g Ü i © S ^ f f i f S ^ * 5 V > i i , «ö<***ÏEH*3(tSffi
^h^comm^m-x, :^mmmmmmm ?,m 2 giiig!aiSffi^gK{Gij©iKifKi¥«**i^
Ê t ± © ' < / i wave induced vibration ©ffgCffitiS oH::;«cS!lèn5. fiPt>
Wagner"' ©^^ïf뮫7J<®Sg0a^!C^-?V^-C. x 5 ï > y m ^ ® ? ^ ; f t t ^ f ö » t ^ ® < ® » * E H
(2) ü s i s s i i
mi^K..^piimm&mm&3mt\zi.zm^Et3m:b(Dmtmt®#^ss$^«LT, *®4'®2
m^^^^Zm&WsCOfSMtlbti.. mm selective resonance (DMÖ^mi^^k LTffi{#;g|&êi+ ^ - f S (Saunders", g i a " , Kaplan et al«'. Kumai and Tasai". Kumai"', Kumai
(s) immmizi^mmi^
SJft4i®J8#iSSS®lt»Hffiv^f,n5x h y •.y:/SHJ:-3T,Sfêilt?i5êft{*tl(c radiation force i a i JA- Froude-Kriloff ®^j, jg© orbital ilS,JnagHifc|i1JTS:Ö=è#^L.fêV^fög®S^ïèHS^ • t - . 5 - i ^ H I S S a t L T i ^ § i a ^ H J : ^ T f f i f t g » - Ë H - » : - r . 5 f c ® T * . S (Goodman", Gunsteren").
3. Springing (Dtt»^IC•^4^r(7)#S
itfflitc®--<fc wave induced vibration © 3 a®©|fS!.i£©4'T, springing {r-:3V->Tt4(2)i: (3)®
i<M.^tzibtsi\i^(DX. *6^Tii2'3®ft^iSo^aS(cov^T^ft^èl)[^.^.. è ? > H * # ® B ^ è ® ' < s
3.1. a S K i S S S Ï
m 10 ® * i f ê T g ^ T i 9 i ^ ® § fc-o2ikTt&^m^Bm^M^xmmm yo X y=yosmcot 0 ±
' ^ + i ! : 0 = O , at >=0 (16) T - ^ ^ ?> H 5 - aej ^ T -fi:=cöV^.
F= - m'y -Ny- pgA^y (17)
-mJ-pgA^y -C^sS. o ^< O X'i> « TfciV^ti-a-. >»' t A^ti <Ü H i o T ^ i b T S i t i i i G
5 ^ ® # t n H f t t n - ^ b T ^ © ï | i ^ ^ è t n t f i j t t i # ; * « e È S M ± T /=o -c,m{ciiq:a*>-5VMi^èn2.:fê-a-li, « = o - r © g i ^ ® * f ^ f c 0 = ,
O httt), ü>->~ ©*ê-i:±<PI«5'i«it<*A*^«ië:, >«' © H - ^ i S t i a i u - r i . 5 .
(«.'>) = - » » ' y - : y - 8 7 = - " « ' i ^ - J » - ^ d^) t - r 5 ^ x > ? i i , K4nnin"' ^ W a g n e r " ' ('i-^TlÉ-tf ^ H ^ c i 71^:, ¥ J È © ^ # : ^ ^ ' * ® f ' S T Lx^mm^^&irzmwiw-t}'?m&iznts^j:}ii±^üm''hè <, motn-mM-r^tLxmi^
i l è * t ó i 7 fcT2.«)©-C*S. r<St/^Et^ra®i^^-Ci5S*><b. g È ^ S ^ f t i 0 = O - C * 5 f c # m.'fc ö » ^ » ffl^-ê-tiautófc'iS. y fctfcH!^<*:©;ïKSti*^iit'S^bT5^1iaSS©^2 ]S^i^sfeT#Tiv^ i © i 7 » ; a s i * f f l i É i i * 5 S , mmmw^-cwmmmLx\i-:hfmm<m#ts m^CO:^mmiT^ ^. g Ö ^ H± - r Ü 5^ S ® S i i! t t 5^ - t . ^ i t ' S i l t f f l L f i ^ t c ^ J i t S S i ? ^ ^ : ^ / © t t ^ H / S V ^ e j f l S a ü T i s S . ïiÉ-:>"C, Kaplan et a l " bow flare slamming H i 5 whippingmt^O Wave Induced Vibration immm-tZ-^m 243
li9^mW:X-^^, ± T 0 2ijfêK0@^||fflt±2g^föx-*>S. « } » S S i J 0 » r E 0 ^ b è 10m èr-rn
i f ? B = - ^ 1 " = = - ^ «Ïg§-Ci±0.25, 2a5®±Tfil&-Cii5.0 fcAjrS.
i ^ i - , RfêSlïJ: © ± T S t T 5 ^ i C S r S « t ^ : Ü i 4 , i t S S ^ ^ ^ t l C L - C lst-order,2nd-order
• • • © | ê l ! f e i ; & f ^ ^ i e ! © f f i f t A t L T ^ J 6 5 ' < # - C * , ^ 9 . J:J^©RMiC^S diffraction force b L-ct±iiS:a94'®±TS^fi:èS^fö-c*i^*>^fi.if, ïï<Kö5t;ii5È-cèi 9 .
3- 2- Goodman ö ) S g |
MS4MKi--^V>-CI± ^ï^^èV^Ö-rH^liaéi^-C iV'=0 bi^è, W i ± r ö ^ = o 0ffl$ffiV>S. Z ®IÉXTfflfll^©iiC©itH -^-9—f4'U> : * s ï ^ S t 5 ^ 2 : T j c . S . ZiXliT, h ') T ' S ! : i H i f S
^i±jT5i$sti-c*,5*>. ^Htxmm^tm.Lx\^zmmmmiznLx\i-mtimMx&?,. m' {--7V>-C{±^g#'»''"> ; è : | | i ; T V ^ 5 i 9 i-S&tt:®t3^!-*=-l^-tli3^^JTC^g^Ji^t < , dm'/dx fcjfö <*®SlI^-C#SLft'J^Sg-Cii^t::i£l-'OT', w' © S ^ ^ i c t ï k ^ a f ê - o T , -Vdm'/dx i'oV^-C^-» ® W ^ ^ 5 M i n s . Goodman ©S^fc»:©2J^{c-^v^-ï:Pfgg^5^^,Skl;'5• 1) 2giigSi©j; ^ i- ®p^èv>j@SSc®mT, X h y •fmi^mx^zot^') 2) m S 4 ^ t f ö < * g i ö H % n ^ - n ^ . ^ T S K ® ^ ^ S ï i > - ^ S i - ^ J i : S ® T i f f l ^ è J l f c ï i ' t t p : L i ^ 4- S
Ët± wave induced vibration ® l t S L f ë è 8 ! ^ L . ifêi:: springing ® f t ï J S t z M f s a t W è f ï ^ - o
springing ® 2-3®tt^aHi±^n^:nF4S*ï*>S. *Tfcg|!li,l;S^(± springing © f f g f f i t
L-cii^afiüTitcv> t ^ ^ t , n s .
^ ^ J ± , springing i±, S&fiSil^^TS^Sêftft^Jtcii; S-^i^l»i©^i•^C. ffitefjgltfc t o X
ummw-t\i=i5^^^iBiii©ffl#izi-pT^èHsfc®-c*>5t^it.s. tiÉT:>x-^mm%\wm<Dm^
(Dfzmz\i. mmz-z>\^XiimmmWtHzmti^\~>) v-fmo^^BO^M.
244 B ? ^iO^
im^=-^'ssLm^i^^mzm.^f>&tmmo)m^^nmt, ttz whipping © n - g : " H ^ L x i J i mmtLxmm^^xm<m^^mz^tii^, z(o/i^m'ki-<^-=i^iftts:-otz. v^sv>s«5iiM3K \,^tz^^iz^sT^-^mi^hmiiKmXÊbi>. ttz:^mzmnLtz wave induced vilmtion ©.OJJ.
# # S: itt
1) Saunders, H. E., Hydrodynamics in Ship Design, Vol. HI, The Society of Naval Architects and Marin Engineers, 1965, Chapter 16, p. 288-322.
2) Bell, A. O. and Taylor, K. V . , Wave-Excited Hull Vibration Measurements on a 47,000 DWT Tanker, Shipping World and Shipbuilder, Feb. 1968.
3) m^rmmn. a f l i « ö " B A " (Dmm^mom^, m^-^m^^n, m^zn, m^iè^^n. 4) a f a s s i . ) i & i s « « ! o K S t Whipping i c - p f x , mmmm^T^^, ^ 3 2 ^ . mfi^i^^n-5) Stmm^, Whipping ^ 4 © - ® H l t : - 3 0 T , B a ï B 4 3 ^ 2 H , a a E 5 i i § ± ü f f i * (ras]48^
12^) icmm.
6) Kaplan, P., Sargent, T. P. and Raff, A. 1-, An Investigation of the Utility of Computer Simulation to Predict Ship Structural Response in Waves, Second Techni cal Progress Report from Project SR-174, "Ship Computer Response" to the Ship Structure Committee, June 1969.
7) Goodman, R. A . , Wave-Excited Main Hull Vibration in Large Tankers and Bulki Carriefs, R. I . N . A . , April, 1970.
8) Kumai, T. and Tasai, F., On the Wave Exciting Force and Response of Whipping of Ships, European Shipbuilding, No. 4, 1970.
9) Gunsteren, F.F. van. Springing, Wave Induced Ship Vibrations, I . S. P., 1970. 10) Kumai, T . . Wave-Induced Force Exciting Hull Vibration and its Response, Hê?JSI6
^ 4 4 ^ , 1972.
11) Kumai, T . , Effect of Ship Side Wave upon the Resonance of Hull Spring, Rep. Res. Inst. Appl. Mech., Vol. XX, No. 66, March, 1973.
>th, S * ) g « ö ¥ è , raSJ44^7^.
13) Wagner, H . , über Stoss und GleitvorgSnge an der Oberftóchen von Fiüssigkeiten, Z. A. M . M . , 1932.
14) KirmSn, Th, von and Wattendorf, F. L . , The impact on sea plane floats during loading, NACA, T N 321, 1929.
16) Potash. R.L., Second-Order Theory of Oscillating Cylinders, Joural of Ship Resea-rch. Vol. 15. No. 4, Dec. 1971.
