Vibrating Bars Subjected to Compressive Loads

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T im & r m o B iB s s u b jb o ^ s b f o

c o ]s fb b s 31? e lo â b s

Submitted, in partial Folfillment of the requirement» for the degree of

DOOTOH OP PHILOSOPHT at the POLïTKCHiriC INSTITUTE OF Bm OKLm

V

David Burgreen July 1949

ApproV k M

m

of Deog#tment

ProQuest Number: 27594655

All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is d e p e n d e n t upon the quality of the copy subm itted. In the unlikely e v e n t that the a u thor did not send a c o m p le te m anuscript and there are missing pages, these will be noted. Also, if m aterial had to be rem oved, a n o te will ind ica te the deletion.

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Approved by th e Guidance Oommltta»

M a jo r :

A p p lie d M echaalos

P ro fe s s o r o f A e ro n a u tic a l E n g in e e rin g

M in o r:

. M a th e m a tics

L C. L • d . H u tc h in s o n A s s o c ia te P ro fe s s o r o f M ath em a tics

M in o r ;

Aerodynam ics

A s s o c ia te P A e ro n a u tic

or of In e e r in g

A d d it io n a l Member:

VY L . é ^ er ^ ^ A s s o c ia te P ro fe s s o r o f A p p lle a M echanics

— -

TO U L L im

B io g r a p h ic a l S ketch

The a u th o r was b o rn i n New York. C ity on August 1 , 1917*

He re c e iv e d th e degree o f B a c h e lo r o f C i v i l Eng­

in e e r in g fro m th e C o lle g e o f th e C ity o f New York I n S ept­ ember 1943 » and f o r a s h o r t tim e t h e r e a f t e r w orked as a R esearch F e llo w in A e ro n a u tic a l E n g in e e rin g a t th e P o ly te c h ­ n ic I n s t i t u t e

o f B ro o k ly n *

I n November 1943 he was com m issioned an O f f ic e r i n th e U . S* Navy*

M ost o f th e y e a r o f 1944 was spent a t

th e N a val A ir E x p é rim e n ta l S t a t io n , P h ila d e lp h a P e n n s y lv a n ia , where he h e ld the p o s it io n o f P r o je c t Engineer i n th e Ex­ p e rim e n ta l S tr u c tu r e s D iv is io n *

I n 1945 and 1946 he was Eng­

in e e r ! % O f f ic e r # o f a N a val A i r T ra n s p o rt S quardron a t Alameda C a lifo r n ia .

A f t e r le a v in g th e Navy in F eb rua ry 1947 he was

a p p o in te d R esearch A s s is ta n t in A e ro n a u tic a l E n g in e e rin g a t th e P o ly te c h n ic I n s t i t u t e o f B ro o k ly n . He r e c e iv e d th e degree o f M a ste r o f A e ro n a u tic a l E n g in e e rin g in June 19 4 7 ,

I n September 1948 he was a p p o in t­

ed R esearch A s s o c ia te i n A e ro n a u tic a l E n g in e e rin g and in J u ly 1949 co m p le te d a l l t h e re q u ire m e n ts f o r Ph. D, in A p p lie d M e c h a a ic s .

AOKNOWLSDQMOTP

The a u th o r w ish e s to

express h is

s in c e re a p p r e c ia t io n to P ro fe s s o r H . J* H o ff f o r s u g g e s tin g t h i s

t h e s is t o p ic and f o r h is g u id a n ce

d u r in g th e c o u rs e o f i t s

in v e s t ig a t io n .

The a u th o r

w ish e s t o th a n k P ro fe s s o r V . L . S a le rn o and D r. F ra n ce s Bauer f o r t h e i r comment a and s u g g e s tio n s d u r in g th e d is c u s s io n o f t h i s p ro b le m .

Abstract

T h is d is s e r t a t io n d e a ls w i t h t h e dynamic b e h a v io r o f colum ns under p r e s c rib e d c o n d itio n s o f a x i a l end d is p la c e ­ m e n t.

The f i r s t

problem d is c u s s e d i s th e v ib r a t io n o f a

colum n w ith c o n s ta n t end d is p la c e m e n t.

This g iv e s r i s e t o a

n o n - lin e a r e q u a tio n i n th e tim e f u n c t io n w h ich i s

fo u n d to

have an e x a c t s o lu t io n i n te rra s -o f e l l i p t i c f u n c t io n s .

The

v i b r a t i o n p a tte r n s f o r a l l p o s s ib le c o m b in a tio n s o f lo a d and a m p litu d e o f o s c i l l a t i o n a re d is c u s s e d . blem i t

is

I n th e second p ro ­

p re s c rib e d t h a t th e end d is p la c e m e n t o f th e colum n

i s a l i n e a r f u n c t io n o f t im e , w i t h th e a m p litu d e o f o s c i l l a ­ t io n s assumed to be v e r y s m a ll,

A s o lu tio n is fo u n d i n term s

o f t h e B e s s e l fu n c tio n s and th e v ib r a t io n and b u c k lin g phenome­ na a re d is c u s s e d b e lo w and above th e E u le r lo a d .

The t h i r d

p roblem p re s c rib e s a u n ifo rm r a te o f end d is p la c e m e n t w it h la rg e a m p litu d e s o f o s c i l l a t i o n o r d e f le c t i o n .

The n o n - lin e a r

e q u a tio n t h a t r e s u lt s fro m th e s e s p e c if ic a t io n s i s

s o lv e d by

a n u m e ric a l p ro ce d u re in v o lv in g th e use o f power s e r ie s . t h is

la s t s e c t io n th e b u c k lin g process is more f u l l y

In

d is c u s s e d .

TABLE OF GONTEETS

Symbol».............. ........ ................. ......................................................................... , . . . , . . , . 1

General Part I

Intro du ctio n.

...........................................

....5

- Vibrating Oolmn With Conatant aid Dloplaoejaent In tro d u c tio n ................

Dlsouaslon.,

..,.,....5

......................................................

....8

Part I I ^ Vibrating Oolumn With Axial Load Applied As a Linear Ponotion o f Time Introdact Ion «

.......... .................... . . . . . . . . . . 5 1

D is c u s s io n .,...

.........................

Damping E ffects

...........

General Problem...............

Part I I I

....3 2

.52 .55

- Large Amplitude Golumn O s c illa tio n s W ith End Displacement . a Linear Function o f Time I n t r o d u c t io n ... . D1soussi on

R e f e r e n c e s . ... .... ...................

..................« . . . , . « .................

.....6 0 61

...79

1#

LIST OF SYMBOLS

X

*

d is ta n c e a lo n g colum n c e n te r l i n e

y

= d is p la c e m e n t o f colum n c e n te r li n e

t

= tim e

P^

» lo a d on colum n when colum n i s i n s t r a i g h t p o s i t i o n , y “ 0

P

* lo a d on colum n

A

* c r o s s - s e c tio n a l a re a

L

= le n g th o f colum n

B

® Young’ s m odulus o f e l a s t i c i t y

a

» a m p litu d e o f o s c i l l a t i o n , o r i n i t i a l d e f le c t io n o f m id p o in t o f colum n

d

“ th ic k n e s s o f colum n

r

= r a d iu s o f g y r a t io n

I

= moment o f i n e r t i a

p

=» d e n s it y , mass per c u b ic in c h

p

n® EI l2

(O

_ TT^BAr* 1,2

” fu n d a m e n ta l fre q u e n c y , i n ra d ia n s p e r s e co n d , o f o f colum n under c o n s ta n t lo a d

F

“ tim e f u n c t io n a s s o c ia te d w it h fu n d a m e n ta l mode

T

® p e rio d o f o s c i l l a t i o n

v ib r a t io n

/9 '

« -

r

f

« frequency In cycles per second

f^

» fundamental frequency o f a

beam in cycles per second

ciJ^

» fundamental frequency o f a

beam in radians per second

c .p .s . * cycles per second R

» ra te o f loading in pounds per second

0^2

» a parameter p ro p o rtio n a l to the r a te

tg

* time required fo r the E uler load to be reached

^

* damping constant

TT o f loading » — — p jo r

General Introduction

I n th e l a s t fe w y e a rs a renewed in t e r e s t in th e problem o f colum n i n s t a b i l i t y has been m a n ife s te d .

Some o f th e papers th a t

a re re s p o n s ib le f o r p ro v o k in g th o u g h t on t h is

s u b je c t are th o se w r i t t m

b y Ton Karm an, R e fe re n ce s 1 , 5 ; T s ie n , R e feren ces 1 , 2; S h a n le y , R e f­ e rence s 3 , 4 ;

and ’^rager , R e fe re n ce 6 .

I n R e feren ces 5 and 6 Ton

Karman and Prager s ta te t h a t the mechanism o f b u c k lin g i n th e in e la s ­ t ic

ra rg e deperxis upon tiie h is t o r y o f lo a d in g o f th e co lu m n .

It

appears t h a t th e u s u a l methods o f a t t a c k are in a d e q u a te f o r s o lv in g th e most g e n e ra l b u c k lin g problem s as th e means f o r s tu d y in g th e colum n " h i s t o r y ” are n o t p r e s e n t.

The means, f o r s tu d y in g t h i s

" h is to r y "

may be p ro v id e d by th e in t r o d u c t io n o f a new v a r ia b le , n o t noYimally in ­ c lu d e d i n th e s tu d y o f b u c k lin g , n a m e ly , tim e . I t was suggested by P ro fe s s o r H . J . H o ff o f th e P o ly te c h ­ n ic

In s titu te

in t o th e

o f B ro o k ly n t h a t th e in t r o d u c t io n o f th e tim e v a r ia b le

study o f b u c k lin g in th e e l a s t i c range may g iv e a b e t t e r i n ­

s ig h t in t o b u c k lin g phenomena i n g e n e ra l and p ro v id e a fo u n d a tio n f o r th e 8 tu4y o f th e more c o m p lic a te d in e la s tio b u c k lin g p ro b le m .

The i n ­

t r o d u c t io n o f th e tim e v a r ia b le , i n a d d it io n t o p r o v id in g a param eter w h ich r e la t e s lo a d , d e f le c t i o n , e t c . , p e rm its th e fo rc e s o f i n e r t i a to be ta ke n in to a cco u n t in th e colum n e q u ilib r iu m

and changes th e

s t a t i c b u c k lin g problem in to a dynamic o n e . T h is pa per d e a ls w ith th e dy'^namics o f co lu m n s.

The

s o lu t io n o f th e d^^namic e q u a tio n o f colum n e q u ilib r iu m may e x h ib it

m o tio n o f the p e r io d ic o r a p e r io d ic t y p e , th e n a tu re o f th e m o tio n depending upon th e m agnitude o f th e a x i a l lo a d on the c o lu m n , and th e amount o f damping p r e s e n t.

The c o n d itio n s o f a x i a l lo a d in g o r

d is p la c e m e n t are p r e s c rib e d and th e r e s u lt in g m o tio n o f th e colum n is in v e s t ig a t e d . a m p litu d e ; i t

The colum n may o s c i l l a t e w it h c o n s ta n t p e r io d and

may o s c i l l â t e w it h c o n s ta n tly c h a n g in g p e rio d and

a m p litu d e ; i t may not o s c i l l a t e a t a l l b u t d e f le c t r a p id ly w ith th e t y p i c a l m o tio n a s s o c ia te d w it h b u c k lin g ; o r i t may b u c k le and o s c i l l ­ a te s im u lta n e o u s ly . The t e x t t h a t f o llo w s is d iv id e d in t o P a rt I a s tu d y i s made o f th e n o n - lin e a r v ib r a t io n s

th re e p a r t s .

In

problem t h a t r e ­

s u lt s by pre scribing th a t th e a x ia l end d is p la c e m e n t o f th e column re m a in c o n s ta n t •

I n P a rt I I an in v e s t ig a t io n o f th e b e h a v io r o f a

colum n s u b je c te d t o a u n ifo r m ly in c r e a s in g lo a d , i s c a r r ie d o u t .

In

P a rt I I I th e more g e n e ra l n o n - lin e a r p roblem o f a colum n s u b je c te d t o a u n ifo r m ly in c re a s in g lo a d is d is c u s s e d , t a k in g in t o

a c co u n t th e

e f f e c t o f th e l a t e r a l d is p la c e m e n ts o f th e colum n on th e lo a d .

P art 1

V ib r a t in g Column W ith C onsta nt End D isp la ce m e n t

In tr o d u c tio n ;

I n t h i s s e c t io n a s tu d y i s made o f a v ib r a t in g column th e e x tre m e tle s o f w h ic h are f ix e d t o

Immovable p o in t s .

T h is im poses th e

c o n d itio n o f c o n s ta n t end d is p la c e m e n t in s te a d o f th e u s u a l a ssu m p tio n t h a t th e lo a d on th e v ib r a t in g colum n rem ains c o n s ta n t.

The s o lu t io n

to th e p roblem o f th e v ib r a t in g colum n under c o n s ta n t lo a d may be found in R e fe re n ce 7 .

I n s p e c if y in g c o n s ta n t end d is p la c e m e n t, o r t h a t th e

d is ta n c e betw e en th e ends o f th e co lu m n rem a ins c o n s ta n t , i t

i s a p p a re n t

t h a t when th e co lu m n o s c i l l a t e s th é lo a d w i l l n o t rem ain c o n s ta n t but w i l l depend upon th e phase o f th e v i b r a t i o n , r e a c h in g a maximum when th e colum n i s

in i t s

s t r a ig h t £ X )s itio n and a minimum when i t

extrem e d e fle c te d p o s i t i o n .

is

in it s

The colum n v ib r a t e s th e n under a p e r io d ic ­

a l l y v a r y in g lo a d , th e v a r y in g lo a d b e in g in d u c e d b y th e v ib r a t io n o f th e c o lu m n . Under p r a c t ic a l c o n d it io n s i t i s

p ro b a b le t h a t a colum n

w i l l v ib r a t e i n accorda nce w ith th e c o n s ta n t end d is p la c e o ie n t r e q u ir e ­ ment r a t h e r th a n th e c o n s ta n t lo a d re q u ire m e n t.

The 'masses to w h ic h

th e ends o f a colum n a re a tta c h e d a re u s u a lly la rg e » th e mass o f th e c o lu m n .

Because o f t h e i r l a r ^

i n com parison w it h

i n e r t i a , th ese end mass­

es w o u ld n o t respon d t o th e h ig h fre q u e n c y v ib r a t io n s

o f the colum n b u t

w ould behave s u b s t a n t ia lly as th o u g h th e y w ere s t a t i o n a r y . be t r u e

if

T h is w o u ld

th e co lu m n w e re one o f s e v e r a l i n a lo a d s u p p o r tin g s t r u c t u r e

or i f

i t w e re lo c a te d betw een th e head and base o f a t e s t i n g m a ch in e .

Even when th e co lu m n i s c a r r y in g a f r e e lo a d t h i s b e h a v io r c o u ld be e x p e c te d ; p r o v id e d , o f o a u rs e , t h a t th e lo a d on th e colum n was n o t v e ry s m a ll. W hether th e c o n s ta n t lo a d o r c o n s ta n t d is p la c e m e n t c o n d it io n i s e f f e c t i v e o f o s c illa tio n .

depends a ls o on th e m agnitu de o f th e a m p litu d e

Even when i t

th e colum n v ib r a t e s

is

known t h a t th e c o n d itio n s under w h ic h

are th o s e o f c o n s ta n t end d is p la c e m e n t , one may

s im p lif y the p ro b le m by assum ing t h a t the lo a d rem ains c o n s ta n t i f th e a m p litu d e i s v e r y s m a ll.

The e f f e c t o f th e

on th e lo a d i s shewn i n th e

a n p litu d e o f o s c i l l a t i o n

f o llo w in g ;

C o n sid e r a p in -e n d e d colum n a tta c h e d to two r i g i d w a lls . The arrangem ent o f th e colum n i s seen in F ig u re 1 . in g o f th e co lu m n

The t o t a l s h o r te n -

w h ic h i s c o n s ta n t i s g iv e n by th e sum o f th e

PL o f th e colum n — L ^ and t h a t due t o c u r v a tu re v h ic h i s l / 2 1 y * * d x . s h o r te n in g o f the c e n te r li n e

1/2 j

due t o th e lo a d

P

Then

y * d% AE

AB

2

or

Assume th e d e fle c te d shape t o be g iv e n b y th e h a l f s in e wave J

*

S e tt in g t h i s

e x p re s s io n in t o e q u a tio n (1 ) g iv e s

y = a s in ^ ^

,

7,

P = ?

-

(2 )

E q u a tio n { 2 ) Is s im p l if ie d b y d iv id in g th ro u g h b y th e E u le r Load

®BAr*

g lT in g P

a® (3 )

and s in c e

P

^0

3a2

I n d e f le c t in g th e m id p o in t o f a co lu m n by an amount " a " th e lo a d r e d w t i o n i s a p p ro x im a te ly g iv e n ly A

f

3 a^ P ^

,

T h is in d ic a te s th a t

d e f le c t io n s o f th e o r d e r o f m a g n itu d e o f th e th ic k n e s s o f th e colum n cause la rg e r e d u c tio n s in th e lo a d .

By d e f le c t i n g th e m id p o in t an amount

ecjxal t o th e th ic k n e s s th e lo a d i s re d u ce d t y 5P g, w h ic h , o f c o u rs e , p u ts th e

column in t o t e n s io n .

D e f le c t in g th e m id p o in t by an amount e q u a l

t o one q u a r te r o f th e th ic k n e s s re d u ce s th e lo a d b y a b o u t 1 9 ^ o f P , — E It

i s b o rn e o u t th e n t h a t w h ile we a re s t i l l i n th e range

o f th e l i n e a r colum n t h e o r y , r e q u ir in g t h a t th e d e f le c t io n s be s m a ll.

th e s m a ll d e f le c t io n s n e v e rth e le s s have a marked e f f e c t i n a l t e r i n g the lo a d o n th e colum n .

D is c u s s io n o f P roblem :

A vp in -e n d e d c y l i n d r i c a l colum n i s compressed betw een tw o p la te s as shown i n F ig u re 1 . The colum n i s compressed b y an amount P L AL = , c a u s in g a lo a d P to a c t on th e s t r a ig h t co lu m n . The AE ® end p la te s are now c o n s id e re d to be r i g i d l y f ix e d in t h i s p o s it io n and a l l v ib r a t io n s t h a t w i l l be c o n s id e re d ta k e p la c e w ith th e p la te s in t h is

p o s itio n .

A l l th e assu m p tio n s o f th e li n e a r colum n th e o r y

are assumed t o h o ld h e re — w it h th e e x c e p tio n t h a t th e lo a d may v a ry d u rin g th e v i b r a t i o n .

The e q u a tio n f o r th e v ib r a t io n o f a colum n is

d e ve lo p e d i n R e feren ce 7 and i s

The lo a d P depends upon th e d is p la c e m e n t o f th e colum n and i s g iv e n b y e q u a tio n ( 1 )

p a p ®

- iS 2L

dx

(1 )

J

o

To s t a r t th e o s c i l l a t i o n s th e colum n I s assumed t o in it ia lly

d e f le c te d in t o a h a l f s in e wave

y = a

s in

. L

be

For a

9.

pin ^e n d e d colum n t h i s I f th e

shape i s u s u a lly c lo s e ly a p p ro x im a te d .

s o lu t io n to e q u a tio n s ( 1 ) and ( 5 ) i s

y( x , t )

th e n th e bo u n d ry c o n d it io n s may be s ta te d as

y % o ;b h y (o

* 0

y (x ,o )

*

a s in * ^

’= 0

y (x ,o )

*

0

(6 )

The s o lu t io n to e q u a tio n s ( 1 ) and ( 6 ) i s ta k e n ta s

y » a s in ^ ^ L

P (t)

(7 )

P ( t ) b e in g some f u n c t io n o f tim e a s s o c ia te d w it h th e fu n d a m e n ta l mode a s in ^ —

. The bou n d ry c o n d itio n s ( 6 ) f u r t h e r r e q u ir e t h a t

F(

0

)

=

1

F (o )

=

0

(8)

S e tt in g e q u a tio n ( 7 ) I n to e q u a tio n ( 1 ) we o b t a in

P = P 0

- — r

coaf ^

2L

di L

J to

g iv in g

p = ?o -

iir

(91

10, or

4 p = p

- (p - ? ) ^ B O

0

(1 0 )

where = ^

(1 1 ) 4L

and 2

P u t t in g e q u a tio n s ( 7 ) and ( 9 ) in t o

.^ 1 ' ^

F + (p

jfi

L

- —

0

F^) (

e q u a tio n ( 5 )

s i n * ^ P }+ ^ A s in U i ÿ = 0

41^

^

L

/

L

S im p lif y in g b y u s in g e x p re s s io n s (1 1 ) we o b ta in

F

p

Where co

+ 0)^F +

= 0

is th e square o f th e fundam en tal

colum n under a lo a d

and

(1 2 )

flrequency o f v ib r a t io n o f a

is a f a c t o r p r o p o r t io n a l t o th e

squarè

o f th e a m p litu d e . E q u a tio n (1 2 ) is ob se rve d t o be a n o n - lin e a r th e second o r d e r .

An e x a c t s o lu t io n f o r i t

o f th e J a c o b i e l l i p t i c

may be o b ta in e d i n term s

f u n c t io n s , th e form o f

2 4 th e v a lu e s o f th e 'p a ra m e te rs W and k .

e q u a tio n o f

s o lu tio n dep a id in g upon

M u lt ip l y in g © q u a tio n (1 2 ) by

F and in t e g r a t in g th e r e ­

s u lt g iv e s

( F ) ^ + oo^F^ + k ^ F ^ = K

some c o n s ta n t

(1 3 )

S in ce e q u a tio n (1 3 ) must s a t is f y th e bou ndry c o n d it io n s F (o ) * 1 and P(

0

) = 0,

K

is fo u n d to be

and ( F ) ^ + w ^ F t k^F ^ =

f k^

g iv in g

y/ u )^+ k^-(w ^F ^ +k^F^ )

I n t e g r a t in g betw een th e l i m i t s

t = 0

and

t = t

g iv e s

F t =

1 f

dF

‘V

T h is e l l i p t i c

J,4

in t e g r a l has a r e a l s o lu t io n o n ly i f th e

q u a n t it y under th e r a d ic a l i s

p o s it iv e ; w h ic h means t h a t th e two f a c t o r s

under the r a d i a l must be e i t h e r b o th p o s it iv e o r b o th n e g a tiv e , o r

(1 5 )

k

w h ic h r e q u ir e s t h a t

“i f

■;> - 2

fo r

^

1

(1 6 )

and ^ < k"*

Except f o r

-2

fo r

2 -—. - - 2 k^

th e

d i c a t i r g t h a t some k in d o f m o tio n i s a m p litu d e and lo a d .

^

range o f

1

2 . -=^ i s u n lim it e d , i n k

p o s s ib le f o r a l l c o m b in a tio n s o f

The a c t u a l ty p e o f m o tio n t h a t w i l l be o b ta in e d

depends upon th e v a lu e s a s s ig n e d to th e q u a n tity the o n ly va lu e o f th e t i n e

f u n c t io n

e q u a tio n s (1 3 ) and (1 4 ) i s

P =» 1 ,

P,

,

When

-2

n o t im a g in a r y , t h a t w i l l

s a t is f y

2 S e ttin g th e l i m i t i n g v a lu e s

= - 2 and F = 1

k t io n (

to o b ta in the lo a d

p=p

or

P

0

-(p

E

- p ) - L ( i ) o _g

in t o e q u a -

The s ig n if ic a n c e o f th e above i s t h a t when th e column is d e fle c te d to th e p o s it io n a t w h ic h

P = Pg

and th e colum n w i l l re m a in s t a t io n a r y i n i t s I t is

of

d e fle c te d p o s it io n .

in t e r e s t t o in v e s t ig a t e th e c o n d itio n s th a t th e

in e q u a lit ie s (1 5 ) and f ir s t

no m o tio n w i l l e n s u e ,

(1 6 ) impose on th e lo a d

i n e q u a l i t y , th e f u n c t io n

P

P , C o n s id e rin g th e

w i l l ta k e on a l l v a lu e s i n th e

rnage

_1_ ^

^

w ith

being r e s t r i c t e d to

W ith t h i s

r e s tr ic t io n

of

P

2

= - 2 th e n P - P .

d e c re a s e s .

1

the la r g e s t va lu e th a t

P = 1 , a t w h ic h th e lo a d

When

^

?

w ill

As

P

w i l l a t t a i n w i l l be

have th e v a lu e

2

becomes g r e a te r th a n - 2 th e v a lu e

T h is in d ic a te s th a t when th e colum n i s i n

it s

e x­

trem e p o s i t i o n , (P = 1 ) th e lo a d must be le s s th a n th e E u le r lo a d f o r any v a lu e o f

^

2

1

in th e p r e s c r ib e d ra n g e .

k

I n th e second in e q u a lit y th e f u n c t io n v a lu e s i n th e

mngQ

P

t a le s on a l l

u-

p k

w ith

2 ÜA. r e s t r i c t e d t o

F

How the la r g e s t va lu e t h a t

F

w i l l have w i l l be

F = -1 -

and

k fo r th is

v a lu e o f

F

th e lo a d

" =

2 A g a in when W.k* comes le s s th e n

?

w i l l be

^ 1^0 - "e ' f î

has th e l i m i t i n g v a lu e o f -2

th e lo a d

P

'

ÿ '

-2 , P = P . ®

As

, ,2

be-

k*

decreases t o some value le s s th a n

th e E u le r lo a d , w h ic h d e m o n stra te s a g a in t h a t th e colum n i n i t s e x treme o u tw a rd p o s it io n w i l l a lw a ys be&mg a lo a d le s s th a n th e E u le r lo a d * S o lu tio n s t o th e in t e g r a l (1 4 ) were shown t o e x is t f o r

2 n e g a tiv e v a lu e s o f

,

in d ic a t in g t h a t o s c i l l a t i o n s

are p o s s ib le

k4 when th e lo a d on th e colum n in i t s E u le r lo a d .

s t r a ig h t p o s it io n i s above th e

The fo r e g o in g d is c u s s io n shows however t h a t th e s e o s c i l l a ­

t i o n s , under lo a d s g re a te r th a n th e E u le r lo a d , are p o s s ib le o n ly i f a t some tim e d u r in g th e o s c i l l a t i o n th e lo a d f a l l s below the E u le r lo a d * The p o s s i b i l i t y o f colum n o s c i l l a t i o n s under s t r a i g h t colum n lo a d s g r e a te r

than th e E u le r lo a d has been in d ic a te d i n a paper

1^,

by L u b k in and S to k e r , R e fe re n ce 8#

I n t h i s p a p e r, i n # i i c h th e b e -

h a v io r o f a column under a p e r io d ic a lly a p p lie d lo a d was d is c u s s e d , i t was p o in te d o u t t h a t such v ib r a t io n s ti.'ne d u r in g th e c y c le th e

are p o s s ib le o n ly i f a t some

lo a d f a l l s below.

T h is phenomenon may be e x p la in e d fro m a p h y s ic a l p o in t o f v ie w .

Suppose th e c o lu m n , i n i t s

s t r a i g h t p o s i t i o n , f in d s i t ­

s e l f c a r r y in g a lo a d g re a te r th a n th e E u le r lo a d . b u c k le .

B u t as i t

p ro ce ss t o

stop#

b u c k le s th e lo a d f a l l s The in e q u a lit ie s

g iv e lo a d s g r e a t e r th a n th e

I t s ta r ts to

o f f c a u s in g th e b u c k lin g

(1 5 ) p r o h i b it s o lu tio n s w h ich

E u le r lo a d f o r a l l phases o f th e c y c le

o f v i b r a t i o n , r e q u ir in g t h a t a t some phase o f th e c y c le th e lo a d f a i l below

Pg i n o rd e r to s to p th e b u c k lin g p ro c e s s ,

2

For c e r t a in v a lu e s o f th e p aram eter

th e in t e g r a l

k^ (1 4 ) g iv e s s o lu tio n s o f s p e c ia l i n t e r e s t .

When

= -2

it

was

k^ shown t h a t th e colum n re m a in s s t a t io n a r y u n d e r i t s

E u le r lo a d ,

g F o r y i l = -1 th e in t e g r a l (1 4 ) becomes

t = — \ k^ )

—= = = _ f / i - fS

-

sech

F

k

or

and

The load ? is

F = sech k ^ t

(1 7 )

y = a s in T L ^ sech k t

(1 8 )

16

^0 - (Po - Fgl ^ 00^

seoh^ k S

(19)

g For th e l i m i t i n g

case o f

= -1

we n o te t h a t th e

k4 m o tio n i s

a p e r io d ic

as shown in F ig u re 4 .

The colum n does n o t

o s c i l l a t e b u t approaches th e v e r t i c a l p o s it io n a s y m p t o tic a lly , 2

The i n i t i a l v a lu e o f th e lo a d

P+ t u_ n * f o r

== - i

is

h=o = %

-

or _ E ■

Po+Ft=0

,

(2 0 )

E q u a tio n (2 1 ) shows t h a t when th e E u le r lo a d i s a v e ra ^

th e

o f th e s t r a ig h t colum n lo a d and th e i n i t i a l lo a d o f th e

column in i t s molæ nturn to

d e f le c te d p o s itio n th e n th e column w i l l have s u f f i c i e n t

j u s t re a c h th e v e r t i c a l p o s it io n . When th e s t r a ig h t colum n i s under i t s

E u le r lo a d , t h a t

2 i s , when

= 0 , th e in t e g r a l (1 4 ) becomes t

1 ^

dP

J (l-î^)(l+p2) g iv in g

F = on ( V ^ k ^ t,'~ ) 2

( 21 )

17

and y “ a s i n — on

L

The p e r io d o f v i b r a t i o n

'

(2 2 )

*

is

T

(2 3 ) / X- i

V

a in V

2

and th e lo a d

P

p

-

^

cn^ ( / 2 k ^ t ,

4L^

The m o tio n o f th e colum n i s F ig u r e

Zm

The m odulus o f t h i s

^Œ,) 2

( 24 )

p e r io d ic o f the ty p e shown i n

Jacobi e llip t ic

f u n c t io n i s f a i r l y

s m a ll

so t h a t th e o s c i l l a t i o n p a t t e r n w i l l c lo s e ly resem ble th e c o s in e c u r v e . F o r th e g e n e ra l case vh e re th e a m p litu d e and lo a d re ­ la t io n s h ip . f a l l s

i n the raqge

> -2 w i t h k ty p e s o f s o lu tio n s , e x i s t f o r t h e in t e g r a l

F^< 1 ,

two d i s t i n c t

dF (14)

The f i r s t when

4 > - i

k4

and the

second when

—2
- 1 > o r when the s im p litu d e i s k^

I n o th e r w o rd s , i t

is

placem ent i n o rd e r to make i t

nece ssary to g iv e a la r g e i n i t i a l d i s ­ v ib r a t e back and f o r t h p a s t th e

y = 0

p o s itio n , When ment ia

r e la t iv e ly

2

-2 < —

< . - 1 , t h a t i s when the i n i t i a l d is p la c e -

s m a lle r th e

2 .„,4 1. k^

s o lu t io n to th e in t e g r a l y ie ld s

.

. ( k^t

: k®

20,

or

P = cLn (k^t ^

( 29)

- ) k^

and y = a s in 'îi

dn (k.^t

)

(30)

L

The period of o s c illa tio n s is IT ■n - 2 f ^

fe2l

/—

^ (3 1 )

% % 3 ------- —

and the load

p = I>0 - (-E ■ ^o’ ^

)

(32)

In Figure 5 i t is seen that the d e lta cosine e l l i p t i c fu n c tio n is always p o sitiv e , the o s c illa tio n s being confined to one side of the y = 0 p o s itio n of the column. r e la t iv e ly sm all amplitude betw een

The o s c illa tio n s are of

th e lim it s

y =

a

and y =

I t is noted th a t the period o f v ib ra tio n ia given ly

T

instead o f

^ f

^

1+ —

21

(P = k4

The d e lt a o o sin e e l l i p t i c

f u ix î t io n has tw ic e th e fre q u e n c y o f th e

c o rre s p o n d in g c o s in e e l l i p t i c th e colum n p a st th e

y = 0

f d n c t io n , s in c e one c y c le w h ich take©

p o s it io n c o rre s p o n d s t o two c y c le s Wien the

o s c i l l a t i o n are to one s id e o f the

y = 0

p o s itio n .

d isp la ce m e n t a re now ob se rve d to o s c i l l a t e w it h th e the d e lt a c o s in e e l l i p t i c b o th have th e

and th e square o f th e

same p e rio d o f v i b r a t i o n , 2 The l a s t case i s f o r < -2 , k

The lo a d and same p e r io d s in c e

d e lta c o sin e e l l i p t i c

> 1,

The s o lu t io n o f

the i n t e g r a l g iv e s

dn

(33

and y = a s in

tt

X

( t.i)

(3 4 )

The p e rio d o f v ib r a t io n I s

T

and th e lo a d

-k '

M .

(35)

22,

p = P „ 4 i? g - P „)

dn®

(3 6 )

er® , ,

P,

1- — I

(38)

8yCl,4r

a

2

L e t t in g / 3 = — and a) = ' r 0

^EAr^ 7“ U .4

» W

b e in g th e n a tu r a l c i r c u l a r 0

fre q u e n c y o f v ib r a t io n o f colum n under z e ro lo a d , i . e . o f a beam, we may w r i t e

/3 ®

8(1

/3®

2.)

R e w r itin g e q u a tio n (3 7 ) i n term s o f th e new ly d e fin e d p a ra m e te rs

/

. y3®

f =fL l2 L3H S L rr

(39)

4 J o

1 -

L.

s in ^ 0

2+

L e t t in g f ^ c .p .s .

=

Then

(4 )

, o r th e n a t u r a l fre q u e n c y o f v ib r a t io n o f a beam in

27.

f

(4 0 )

7"

2+ (3®

— is f o i n F ig u re 8 .

a g a in s t

— P-'

f o r v a rio u s va lu e s o f /3 * «S. / _

The in t e r s e c t io n o f th e cu rve s w ith th e

fo u n d by s e t t i n g fre q u e n c y i s

p lo t t e d

z e ro .

•= -1 S e t t in g

s in c e f o r t h i s 2

— ^2

a x is i s

va lu e o f th e r a t i o

^

th e

= -1

or

When th e a m p litu d e

i s o f th e m agnitu de d e fin e d by

e q u a tio n (4 1 ) th e fre q u e n c y is z e r o . Note th a t às ” a ” approaches zero P Pq — approaches 1 w h ic h i s th e v a lu e o f — f o r zero fre q u e n c y when % PE th e lo a d on th e column i s c o n s ta n t. When

i s e q u a l to

z e ro , t i i a t i s , when the colum n

o s c i ll a t e s w ith " z e r o ” a m p litu d e th e r a t i o

o f th e fre q u e n c y to th e

28,

n a t u r a l fre q u e n c y o f a beam i s

P

- / 1- —

f.

V

(42)

Pe

T h is e q u a tio n re p re s e n ts t h e curve i n F ig u re S la b le d

— = 0.

It

is

r th e t h e o r e t ic a l cu rv e f o r a v ib r a t in g

column in w h ic h th e v a r ia t io n in

lo a d d u rin g o s c i l l a t i o n i s n o t ta ke n in t o a c c o u n t* From th e c u rv e s i n F ig u re 8 i t

i s r e a d i ly

seen t h a t th e

fre q u e n c y in c re a s e s w it h d f e r e a s in g lo a d as w e ll as w i t h a m p litu d e . t io n

A t h ig h e r lo a d s an in c re a s e i n th e a m p litu d e o f o s c i l l a ­

has a more pronounced e f f e c t i n in c re a s in g th e fre q u e n c y .

e x p la in s th a n

in c r e a s in g

why i t i s

It

p o s s ib le t o o b ta in fre q u e n c ie s t h a t are g re a te r

th a t g iv e n by e q u a tio n (4 2 ) - e s p e c ia lly a t h ig h e r lo a d s .

R e feren ce 10 such a phenomenon has been e x p e rie n c e d .

In

E q u a tio n (4 2 )

was c o n s id e re d to g iv e th e t h e o r e t ic a l fre q u e n c y o f th e fu n d a m e n ta l mode re g a rd le s s o f th e a m p litu d e o f o s c i l l a t i o n .

E x p e rim e n t# gave

fre q u e n c ie s w h ic h were c o n s id e ra b ly i n excess o f th e t h e o r e t ic a l v a lu e s - e s p e c ia lly a t h ig h e r lo a d s . a t th e P o ly te c h n ic I n s t i t u t e

Some p r e lim in a r y e x p e rim e n ts

o f B ro o k ly n A e ro n a u tic a l L a b o ra to ry

a ls o in d ic a te d th a t r e l a t i v e l y

h ig h fre q u e n c ie s a re p o s s ib le i n t h e

v i c i n i t y o f th e E u le r lo a d . I n F ig u re 9 th e fre q u e n c ie s were p lo t t e d a g a in s t f o r v a r io u s v a lu e s o f

Pq — ,

For la rg e v a lu e s o f

—

r

and

P —2.

— ^ th e

-

fre q u e n c y appears to v a ry a lm o s t l i n e a r l y w it h th e .a m p litu d e , The p cu rve o f s p e c ia l in t e r e s t i s th e one la b ie d — = 1 , T h is c u rve is

29,

a p e r f e c t l y s t r a i g h t l i n e th ro u g h the o r i g i n , g iv in g an exact lin e a r r e la t io n s h ip between a m p litu d e and fre q u e n c y . it s

I f th e column i s u nd er

B u le r lo a d i n th e s t r a ig h t p o s it io n , th e n d o u b lin g th e a m p litu d e

w i l l d o u b le th e fre q u e n c y , t r i p l i n g fre q u e n c y , e t c .

th e a m p litu d e w i l l t r i p l e

th e

The e q u a tio n g o v e rn in g t h is c u rv e is o b ta in e d fro m

e q u a tio n (4 0 ) in w h ich

— rg

f

i;

is s e t e q u a l t o u n i t y .

T h is g iv e s

IT

'

^

? d0 T 1 , 2. 1 - — s in çb o/ Z

or f

=

_J3JÊ 4 x 1 .8 5 4

«

.4236 r

(4 5 )

W ith t h i s , the a n a ly s is o f th e v ib r a t io n o f a p in -e n d e d column w it h c o n s ta n t end d is p la c e m e n t i s c o n c lu d e d .

The i n i t i a l shape

o f th e colum n was p re s c rib e d as ib ilit y mode.

y ~ a s i n ^ ^ e lim in a t in g the p o s s L o f o b ta in in g modes o f v ib r a t io n o th e r th a n th e fu n d a ira n ta l

T h is a n a ly s is c o u ld e a s ily be extended to a colum n whose ends

a re c la m p e d , by assum ing t h a t th e i n i t i a l shape i s c lo s e ly a p p ro x im a t­ ed by

y =

j~ l- cos

2ttx y ^ = a cos —

F (t^ ,

A tra n s fo rm a t io n o f

y

a x is g iv e s

F ( t ) , and from t h i s p o in t on th e a n a ly s is c o u ld be

c a r r ie d fo w a rd i n th e

same manner as th e

fo re g o in g .

I f th e i n i t i a l

shape o f th e column were n o t p re s c rib e d as c o in c id in g w it h a p a r t ic u l a r

30.

mode, th e n an e x a c t s o lu t io n does n o t appear t o a p p ro xim ate s o lu t io n w o u ld have to o f n o n - lin e a r e q u a tio n s .

he p o s s ib le , and an

be o b ta in e d from an i n f i n i t e

set

F o r t u n a t e ly , th e f i r s t and second sym m etric

modes p ro v id e a good a p p ro x im a tio n o f th e i n i t i a l l y

d e fle c te d shape

o f a p in -e n d e d o r clamped end c o lu m n , p e r m it tin g a s o lu t io n to be ob­ ta in e d w ith o u t to o much la b o r .

o1 .

Part II

T ib r a t in g Colujiin W ith A x ia l Load A p p lie d As A L in e a r F u n c tio n o f Time

I n t r o d u c t io n ;

I n t h i s s e c tio n a s tu d y i s made o f th e m o tio n o f a co lu m n w h ic h i s b e in g com pressed by a lo a d t h a t in c re a s e s a t a c o n s ta n t r a t e . The m agnitude o f th e d e fle c tio n s are now assumed t o be very s m a ll so t h a t th e bow ing o u t o f th e colum n does n o t a p p re c ia b ly reduce th e lo a d on th e co lu m n . f l e c t i o n s , a re

When th e a m p litu d e s o f o s c i l l a t i o n , o r th e de­

sm a ll th e end d is p ls c e n e n t o f th e colum n w i l l be p ro ­

p o r t io n a l to th e lo a d , p e r m it tin g th e problem to be r e s ta te d as "A V ib r a tin g Column w ith End D isplacem ent a L in e a r F u n c tio n o f Time " , The c o n d itio n s o f th e problem a re f u l f i l l e d when a colunm r e s ts i n a t e s t i n g machine and th e head o f th e t e s t in g machine i s low ered a t a c o n s ta n t ra te o f speed. T h is is p ro b a b ly th e o n ly s it u a t io n in / w h ic h th e c o n d itio n s o f th e problem a re f u l f i l l e d , and in d e e d , i t is f o r th e purpose o f s tu d y in g th e b e h a v io r o f a colum n in a t e s t in g machine t h a t t h i s in v e s t ig a t io n i s b e in g c a r r ie d o u t . The purpose o f th e problem i s

to answer some fu n d a m e n ta l

q u e s tio n s r e g a rd in g th e b e h a v io r o f a v ib r a t in g colum n i n a t e s t in g m achin e.

I t is d e s ire d t o d e te rm in e th e manner in w h ic h th e a m p litu d e

and p e rio d o f o s c i l l a t i o n are a ffe c te d as th e head o f th e t e s t i n g

32$

machine comes dow n, e s p e c ia lly in th e v i c i n i t y

o f th e E u le r lo a d .

T h is in v e s t ig a t io n f u r t h e r la y s th e fo u n d a tio n f o r th e

s o lu tio n

o f th e n o n - lin e a r problem o f P a rt I I I .

D is c u s s io n ;

The arrangem ent o f th e colum n is shown i n F ig u re 1 . colujnn i s

assumed t o

be a c y l i n d r i c a l b a r a tta c h e d to th e head and

base o f th e t e s t i n g m achine by means o f f r i c t i o n l e s s o f th e is

The

p in s .

The head

t e s t in g rm chine , a tta c h e d t o th e colum n a t th e p o in t

x = 0

assumed t o be moving downward a t a c o n s ta n t speed go t h a t the lo a d

on th e co lu m n is b e in g in c re a s e d a t the r a te o f A t the tim e

t = 0

th e lo a d is g iv e n by

R

pounds per second.

and a t any tim e

t

th e

lo a d is

P

N ote t h a t p re v io u s p ro b le m .

P

+ Rt

i s n o t reduced by

(1 )

AB

\

Ay 2 ( ^ ) dx

Since the a m p litu d e s o f o s c i l l a t i o n

v e r y s m a ll th e in te g ra n d t io n

-

dx

as i n th e

are ta k e n to be

is n e g lig ib le w h ic h makes t h i s

c o rre o -

in s ig n if ic a n t . As b e fo r e , the e q u ilib r iu m e q u a tio n o f a u n ifo rm c y l i n d r i c a l

colum n is

E

I ^ 0x4

+

P

^^2

+ pk /

^ ^^2

= 0

(2 )

33$

The i n i t i a l d e fle c te d shape o f th e p in ended colum n i s ta k e n as

y

t*=o

=

ITT a s in ~ L

(3)

p e r m it t in g a s o lu t io n f o r e q u a tio n ( 2 ) to he taken as th e p ro d u c t o f th e fu n d a m e n ta l mode and i t s a s s o c ia te d tim e f u n c t io n , o r

y * a s in — L

F (t)

(4 )

W ith th e f u r t h e r a s s u m p tio n t h a t th e v e lo c it y o f the column i s t

= 0,

th e s ix ho u n d ry c o n d itio n s g o v e rn in g

y (0 , t ) = y ” ( 0 ,t)

= 0

aaro a t

y ( x , t ) become

y ( x , 0 ) = a s in

T TX

L (5 y (L ,t)

= y ”( L ,t)

= 0

y & ,b ) = 0

w h ic h when put in t o e q u a tio n ( 4 ) r e q u ir e t h a t

F (0 ) = 1

and

S e tt in g e q u a tio n s (1 )

+ y o A a s in

L

F (0 ) = 0

and (4 ) in t o e q u a tio n ( 2 ) g iv e s

P

=

0

(6 )

344

or

-2 — (IL S L - P )

F . E & l

P + F

«

0

L e t t in g

0

and

lOPdr

g iv e s

F

E q u a tio n ( 7 ) is

+ u f F - CL^tP

=

0

{7 )

s im p l if ie d b y th e tr a n s fo r m a tio n o f th e independent v a r ia b le

y ie ld i n g

F

+ (X®t^F

When th e v a r ia b le t ^ (8 )

=

0

i s p o s it iv e

(8)

the

s o lu t io n o f e q u a tio n

is o b ta in e d in term s o f o r d in a r y B e ssel fu n c t io n s o f o rd e r

T

l/ s .

35$

These

n o tio n s are o f an o s c i l l a t o r y t y p e .

If

i s n e g a tiv e th e

s o lu t io n o f e q u a tio n ( 8 ) ia o b ta in e d i n te rm s o f th e m o d ifie d B e ssel

1

fu n c tio n s o f o rd e r

l/ 3

w h ic h resem ble th e e x p o n e n tia l f u n c t io n s .

The s i m i l a r i t y o f e q u a tio n { 8) to th e w e ll known ha rm o n ic e q u a tio n

••

F

i s r e a d i ly seen .

2

+ Ol> F

=

0

I n th e above e q u a tio n , when o f

is

p o s it iv e

the

s o lu t io n is o b ta in e d i n terras o f th e c i r c u l a r p e r io d ic f u n c t io n s . 2 lu

i s n e g a tiv e th e

When

s o lu t io n is o b ta in e d in te rm s o f th e h y p e r b o lic o r

e x p o n e n tia l f u n c t io n s . The tWQ d i f f e r e n t ty p e s o f s o lu t io n t h a t may be o b ta in e d f o r e q u a tio n ( 8 ) depend on w h e th e r terras o f

t

t^^

is

p o s itiv e o r n e g a tiv e , o r i n

w h e th e r

t

or

V® z t =

is

th e d i v id i n g p o in t between th e two s o lu tio n s th a t may be o b ta in e d ,

2 the q u a n tity

“

may be expresse d i n more fu n d a m e n ta l u n it s as

36.

S e ttin g t h i s is

lim itin g

m lu e o f

t

in t o e q u a tio n (1 ) i t

fbund t h a t

f

+ a

% -P o - r -

'

30 t h a t the t iir e a t w h ic h th e s o lu t io n t o

e q u a tio n { 7 j c h a n t s fo rm 2 P = Pp • i s th u s th e tim e f o r

co rre s p o n d s to th e tim e a t w h ic h

th e E u le r lo a d to be reached and may be d e s ig n a te d as

tg .

E

The s o lu t io n t o e q u a tio n ( 8 ) i s most e a s ily o b ta in e d by s e ttin g i t

in co rre sp o n d e n ce w ith th e s ta n d a rd fo m o f th e B e ssel

e q u a tio n {See R eference 11) w h ic h i s

1 —2 a

y =

(10

0

h a v in g as th e s o lu t io n

y =

J ^ {b x

)

The co rre sp o n d e n ce o f e q u a tio n (

F +OC^.P

8

)

(8)

t o e q u a tio n ( 1 0 ) r e q u ir e s t h a t

y = P ,x

=

1 2 t i , a = — , b = — oc , 1 ’ 2 3

0

3 = — 2

and

The c o m p le te s o lu t io n t o e q u a tio n ( 8 ) i s th e r e fo r e

F =

(111

U s in g th e fo rm u la e

i [ x

- “ j^ ( fc x ® )]= - k a x '

and n o tin g t h a t F ( t )

(tat®)

= - F ( t ^ ) , t h e , tim e d e r iv a t iv e o f ( 1 1 ) may he

w r it t e n as

i

= - A^oct, J_g/3 ( f ^ t ^ / ^ ) + B^OCt, Jgyg ( f a t / / ^

(1 2 )

The c o n s ta n ts Aj^ and c o n d itio n s ( 6 ) .

I n te rm s o f

a re d e te rm in e d fro m th e houndry th e s e c o n d itio n s a r e

and

’< vS ' s e t t i n g th e s e i n i t i a l c o n d it io n s in t o

e q u a tio n s ( 1 1 ) and ( 1 2 )

and

a re d e te rm in e d as

4

•

iC (jjpt

'2/5 's 3 ' 4

-

3

------------------------------------------------

and

B -O J

The c o n s ta n ts

A^

and

may he e xp re sse d more co m pactly hy

s im p lif y in g th e d e n o m in a to r o f th e above e x p re s s io n s . g iv e s

The s i m p l i f i c a t i o n

39.

2 s in E z js 2 ir

01® ü?

(13)

H -

m aking 2 tt

60 “ S/S' Ot

V

b

:la £ * 3 o(.2

(1 4 )

and

3/g " CC^

-

2 /5

(£ ü £ ] 3 (x2

Thus th e tim e f u n c t io n f a c t o r o f th e fu n d a m e n ta l moc3s Is

p = 5 /3 ^

(15)

and i t s

d e r iv a t iv a

(16)

40,

The o s c i l l a t i o n o f th e colum n i n th e tim e range t < —5and under th e

c o n d itio n s s p e c if ie d f o r t h i s

pro b le m a re c o m p le te ly

re p re s e n te d by

Trx.(tj2

2tra CO

. . 1 /2

(1 7 )

The t y p i c a l o s c i l l a t i o n c u rv e t h a t i s re p re s e n te d by e q u a tio n ( 1 5 ) i s shown i n F ig u re 1 0 .

I t may be observed t h a t th e

p e rio d o f v ib r a t io n and a m p litu d e in c re a s e w i t h . t im e . 03^ 68

2

( ^

0^2

1 /2

t ^ 5 /2 same tim e t h e B e ssel f u n c t io n s J. / „ ( * ^ U & - t ) ) ' 3 oL^

At th e

/„(t ^ ( ^ —1 /3 o

p ro d u c t

t a p p ro a c h -

, t h a t i s , as th e E u le r lo a d i s approached th e f a c t o r ( ^ ^ - t ) *

approaches z e r o . and J

As

(^ *^ -t

cfE

r e s p e c t iv e ly approach z e ro and i n f i n i t y . / ( . £ o c ( i i^ - t 3 (X 2

The

becomes z e ro and th e p ro d u c t

1 /'^

. y ( .£ o ((u ^ - t ) ^ / ^ ) '-‘I / o 3

i n f i n i t y becomes f i n i t e

w h ic h i s th e p ro d u c t o f a zero and an

and may e a s i ly be e v a lu a te d as shown l a t e r ,

A v ib r a t in g colum n under a u n ifo r m ly in c r e a s in g lo a d w i l l t h e r e fo r e have a f i n i t e

d is p la c e m e n t when th e E u le r lo a d is re a c h e d , 2

Beyond th e E u le r lo a d o r f o r a tim s g r e a te r th e n t =

OL^

th e m o tio n o f th e colum n i s

d e fin e d by th e m o d ifie d B e sse l f u n c t io n .

A new in d e p e n d e n t v a r ia b le

t^ ^

is

d e fin e d as

t^ ^

t - y4- o r t , , = - t . 0^2

1

41,

in o rd e r t h a t

t^ . •‘•A

may be p o s it iv e f o r a l l tim e g r e a te r th a n

t

=

E q u a tio n ( 8 ) may now be w r it t e n as

P

- CX

( 8a

P = 0

w ith th e knowledge th a t th e minus s ig n i n th e e q u a tio n w i l l n o t change as lo n g as

t> (^

.

The s o lu tio n to é q u a tio n ( 8 a) i s o b ta in e d b y n o t in g i t s corre spond ence to th e s ta n d a rd fo rm o f th e m o d ifie d B e sse l e q u a tio n . R e fe re n ce 11 g iv e s th e s ta n d a rd fo rm as

y" + - i ^ y '

w i t h th e

(1 8

-

s o lu tio n

y =

i ~ ''

( ib x °

The co rre sp o n d e n ce o f e q u a tio n s ( 8 a ) and ( 18 ) r e q u ir e s t h a t

y = P , x = t.. ■ ‘•

g iv in g as a s o lu tio n to

,a ^

= ~ ,b 2

e q u a tio n ( 8 a)

= - ^ O C , o = - % - , n = l l 3 ^ 3

42.

2 1 /2

-1 /3

F "

3 /2

1

lo tit-

1 /2

2

1 /3

ÿ )

)

, 3 /2

2

(19

and th e tim e d e r iv a t iv e as

2 /3

,2

F = A ^ O t(t-^ )l

J _ 2 y /3 W t - g )

P +B_,OC(t11

The c o n s ta n ts th a t

F and F

c o in c id e a t

—2 / 3 J

1

(f^

Ai % and B 11

2 o ( rio c lt-^ ) 2/3 3 cx2

3/ 2

)

( 20)

a re e v a lu a te d hy s t i p u l a t i n g

o f th e r e g u la r and m o d ifie d B e sse l f u n c t io n s o lu t io n s

t

, 2 1 / ^I4i- - t )

9 3 /2

p

=

.

C o nsid er f i r s t th e f u n c t io n

P .2 3 /2 (-O ^ -^ -t) )'^ 0 ,

2

0,

J

oc^

F.

1 /5 3

«2

As

t ->■

p

o

1

9

5/2

^

) ~> CO

- 1 / 3 2 0C.2

and o ^

-1 /3

1/2

^

0

,

1

J

p 2 3 /2 t- w t)+0 , i /3 3 ^

The o n ly p o s s ib le f i n i t e

i/ 3 i

J

p p 3/2 ( ^ 0< i t - l ^ ) I-.0 -1 /3 3

p ro d u c ts are o f th e ze ro and th e

43$

The ooinGldence o f

infinity.

F

as

t

^

gives fro m equations

0(2 (1 5 ) and (1 9 )

0(2

,,£ -V 8 B, 11

1 /3 i

3 /2 J

0(2

( _ io t( t- < * t) _ l/3 3 «2

The p ro d u c t o f th e se ro and i n f i n i t y c o n s id e r in g th e s e r ie s e xp a n sio n o f

J

—n

(x )

and

)

(2 1 )

may be e v a lu a te d by i^ J

—n

( ix j.

These

a re r e s p e c t iv e ly

r(l-n)

S ( l- n ) ( 2 - n )

and 11 1 J

(tc)

(ÎF)

r

------( l- < i)

(1+ — ----- : ( l- n )

+

+ ..,) 2 ( l- n ) ( 2 - n )

As th e argument approaches zero

r(i-n) X -► 0

X -» 0

(2 2 )

44#

S u b s t it u t in g e x p re s s io n s ( 2 2 ) in t o

2 tt 5i£ ^

3/3

e q u a tio n (2 1 ) g iv e s

_____ £ I

j

-

2 /3

' 3 C(2''