I‘. I . Q . ..
U.
_\ ' ‘ .
V‘fi. ‘ 1‘ ’. -. r’J‘”;'OI? .' ‘2’ ..' ‘ ’
. . . q .. . h . . , ' - ~Fr
.‘rbL H ".;'.v¢'- " £P,I 0.7 ' .‘I n - 0|. 0 - ’ —‘ , v v
. ‘r 3 . - I I o- l 9 ‘3. \0- u v-‘" ‘6 ’ - . ‘I r -MH‘v—w
a x - . ¢ F F C c-o f A o A ‘ ' ' ‘- ""
." ' _. ‘ I'. ,: . - _ _ . 5".’.“.'- 5". -¢_-- . 0,0-r0“‘: 0 ‘2 :9 V.“ s f, e. oq " _ . “l‘ ‘_7 .u "
(I. - -. | :Vl . I .‘ 3"; .3 .' ‘GCFQI"; ., L b 1.! 'P'.' aw“ r»; " 3" .U ‘ ‘ .O.‘ .-.‘ , .- o... p.‘ g ‘1‘...“ ~.
~ , ,. ~ .. , . - - 4 ~ ° . ~ . -‘ v .
30.1. p f . .r ' _“ . " :' ." ,' .‘.: .‘, '. . -_ . .- 49? . it" 4 ‘ ~. .1! . . . l-o. 6.. “ ‘ ‘U _ ‘J"... I :‘”'* "‘ Wfi . ..‘$~ . f.“‘»"‘. ._ . ..°V ‘:Q
. . . o , . ' - _«, Q a .- w tit; . 0 o Jg-a ‘ _ °O~ 0" _’.. H { ~0-fi'l- . r ' -4
‘fi' ‘ ' h I- ’ u‘ m u 4"“ ' ...' ‘V. ”w. an-" 1' d‘QO'.‘. -' ‘ w} ‘. 0 fi " vi.“ . ‘v‘~. ' I. t" ‘9'. to:
: _ n ' ‘ - . . r . o " u" " ‘ . --Q _ ‘ ".o-‘ ’ ’
I-.‘ g’ r" . o ,. 'v' ' 33-55- a. ‘va‘ I f?" '2' ' -_ ' ' ‘...0' . ,' L,» ..',o-‘o‘”o '3 ‘ . .J 'g' Ll |.. 0 Jr. .5 "F ' “ "‘. ' ". J
.. _. -, . ‘ *r 1.. r:' "*3." . .1- .- 342 -+ 4‘ f - - - , - L... «‘9 'r «3- ': ‘H- .' " 2' " -' ‘f‘ 5. '- V ‘ '
¢-'-. -.c . ~ ~ ' on. 3 . Is "1‘“4‘°"" W ' .‘°-- on o a- - ' - . H'- ‘ ' ‘ ",‘c u ...1 . 52 o
' .- o . o I o I. _ . n - . - .vl Y Q O ~ _ I 5 . , Q -
‘g.‘ 0 ¢ ‘4'. . '.‘- - ./_‘. ,1.- F‘ .‘ r ‘r. . 6 o. ' ."-',.:5 v - I, "W"... ‘. . ' ' 1.. t ‘H :0 .. ’ .l H b
, . . I, . - . .-.. .. ’ . . ~ . 0 -. ‘-'. '3” .' ' -.‘-.| _.. - d‘f’. -.:;‘.—’ ‘— n. o ’1'! J's-J. ._ .. ,- fi . . . ‘
L‘. ‘ ' .u ‘ .' ‘ r t 9 1|. no 3 a J a u- v ‘. 'fitvr O6 ‘, ' Z‘v‘l ' a? 04.. .3 ' . 4:: .Q ’- ‘ -‘P‘!‘; ‘0 1‘ \I . .’.'" ‘,a 1 ' ; a f ‘0? I -‘ ' ' .
.‘ _I Q - O I -o v“ ' ' ' y' .‘ h ‘ " u' ‘ O .« H . _‘.O ' ,3‘ fl. 1 . . n ' , A .
4'! ' .f '. . -¢~fltq-.f- - 4 0 [O ' ' . 'ff‘fgf‘u‘. “o" -." L— ~ .- . -~ “
‘ ' ' .~ ' 1 . u I A a” . y,‘ - _. J p". -- I o .- ..
10 M v . .. f . ,. - 'n';--' ‘0... ’ N :a.‘ 1' ' (tr ~c Pf.‘ 33.3.3.3‘, I “3, ‘- --' "4!? ' ‘J’.'0y‘“‘-‘-*— q ' .'¢_ 1' 0 5‘.) W - ’5'. I".
In- . ' Q ' Iod. . I" . . .. 0.4-." ° -' ' ~' ‘ -c. ' I 0'. ' 0| 0" . 9.9 O “ ° " 0 ’
‘ ~r r 9 ‘ J ‘ . . ‘ . ‘1. . .‘3J : . ‘.- g g {a ' _ ”,4 9 n‘ g...” -l V .QIJJ.‘ 0—... ‘ l ' ‘ ‘ '1 1 --
Q55 ‘ " 'J' 0 .- . ' ‘- 'r -.-. .- '- ' . "“ ”unw- ". ' .- - -"~"" .70211?" .7 ' -. "g ‘-"H' s.’ N
.l. - . . w' ' }.f' new.) "'.' u... ‘0 ° - ‘ .' .~'rr“ ’3 &‘ "" . " ' . v . .‘ 0 6" v
”.1... g . , . '7'“ 0‘ p. o —<v .0 1.7? f- l “‘°".' . . 0,0' ‘ .‘ ‘. r .‘ l ' s
“ .n . "fl . .‘4. ._. v- . o..\- . .H-.-";- Pa - ‘0 pt" \olo>-QJ-I~ \ C l,-‘ q‘ 0‘) if 3 6‘0 .‘ _ O.\ .‘n ‘.“-« .
.. ' o c O | 1 - -- — - ' , .. p . . 1 ..
1* .~.° .. ' .. ,1 “3.. .' .u can -" .:..- Tris .. rum-4» -. '; row'wu my? ,. . . . r'm .
l':..! 0 f o.- . '~ -' oo-tI' ‘.' . -a‘ on . ' ' 0 r o. ‘0 O.“"-' ' 'v n D... '. L ' . \a ~" I" . . -Q
.- " I I- . “a. 0,‘ f I § . '; .r . \ - . .. \ ' . 1..‘ . I h ' .4. .‘l‘( .'. la ‘1 ’
'1. y.‘ 1"! ‘ .. , o .‘. -.I ‘ .u. -' ‘4‘ noaqd'Ooo 0 OJ tog ,‘o’lo-‘d o- n' '- ,, - ‘A.
g o .. 0 ' r u . . 0- ‘ u' . . . J
( d’, ‘ ‘.' '. I, -l g... \ o .. .- “ . . ‘._ ' ' V r. o ‘ I. '. ‘J' ' ' Q. 0 ': ._.' Jul. . ' oY M" ‘ . ‘
or... ..' .' '.'. . ' 4' I . '- 0 ..h . '1 . I": . r “ . r ’r 0‘ ' ‘ f 9" .35!" - ‘.' "’.‘ O.
_- ' .‘- . . . ‘ o. ‘ I .u \ - c .r ‘ ' g o . .. - . . I a
‘.- . 1‘J‘ 'Io‘ ’° . , ' r. . ‘3 .,."‘.‘I' 1-“ f . - ¢ . f . ' 0-." 6 3..'o’- .,'. ‘_ '-. I) . . ’ ‘9‘?
ll “4 o'-’< . ' a .r. t I I, ’ ‘.“. . ' ' ‘ IJ ‘ - fi'. F" ”'0 W? :.. QI“‘ ('2‘. .. t}: . . “ ‘ 3 V '0‘; k.‘ . ’—".. -0
” .‘. I . ‘ '.. 'ol- o . - -..""£ 0 —- ’ av o
.I l,‘ o. q. . ' - | o. ' . ‘3 , . ’ ' ’ ‘ 5‘ . . J ‘ . .. v, g . -
°. 0 , -o J a .'~'~' $ ' . . . ‘ . - V 1 -v‘. - . ' ' ...-u . " .
o T. , ’ y.., .' z .. '0'. .. _ {:3}; ._.' ,_. brfl 'q..'.‘ ‘-F:.;- o - o-:, I J a". p .- o... ‘ _ s" > ‘ . t: . :‘ . 5-...)
},'-.’ g '0' -' '3. 0 . ' I 4 -.. . u- .‘ ,. . ‘,- -.; l . 3.. . 0...;o' .a..‘./.. .‘_o p - i; "a, o '. I -a
. . ' . n. " ‘¢ 0 ' 1. ~ ' o " ‘ . no '— . ‘ 6
| t a. l 1 WI 3 0‘ f ' 0 ' ‘ . _ on .. .
'1“... ... -.-.‘. .: y} .. 43"1. J. "l‘ ." -.4 0-" a . ' "-ovro’.“.( . w-t4‘-v _ ‘ ~ v ' - ~45
’.". '. o q I. (ft '... ‘J ".fi- .' .‘N' "3. rll-' ‘ é“"b A: . b v-10 ‘ -.‘ . n o. -‘4 'il. . ..
.. ‘. ‘ . .I u '. . . .' a d ,, V o h . . .. '_" o D
‘0. ,q . a.. .’J 'f"! c '. - 0‘ ‘4. . .0 ‘ '. f .4. '09 . . -.. . . - . .‘ . 'E‘ . I. ‘;_ _ a h _ ‘ .&V a . 6" . . (I: . O l ' .‘ .9.\ .'z 9
. o ‘ _ _ . . s v o _ . I . _ J -- .-_ o ,1 ..
I" ‘3‘ ”I ‘I :I' ' .~ ~ I u: H" '-, .‘ .i ‘4' 6o. . 4.: '3'.“ J. - .r-n- .t- ., v ‘ ‘3’ '- ! 3" . ' . ‘ t
‘ J 2 ‘. a". ‘J k. '. .g. ‘10 0-. .0 . a. I. .'r‘. ’1' V l ' o "'.1 I”. I l . ~ ‘ .. -. v.0 . I" .I r, . F. v . . . -. .Y ' O o. _ .p
O: ‘, 0' ‘ ‘ ‘ . - n . . . "‘ ‘ - 4 . . Dd ' ' o . I a.
‘5‘ ' H ' " ' ‘2»! \ ' ' ’v ‘ ’ ‘ n . I "' ‘ _' ‘ '. ‘ l ’ ‘- ‘ ' ' d, g 0 ° ° .0. p
a - - . ' . 0 ~ . . ~ . . . . - . - - a ' i ' o v - . :. o'- I
“4*- '-" ., "'”z""—13a ...\",,.' "‘ : '1.» ’~ wrvr Arm-543' -.-.' I. 43-— 1" v°-“"'.’d--‘ . =;:’.. " "W" ' ‘ -'
‘5‘ '- ' . o W 'rfi r.‘..t|. ‘ . “’- r I In ’ " . r.- > .’r - " . . ll. ’ ‘ .Q 0 ‘ - O ’o A. . - -
min "a. “1" ‘: " " . "'“" 7a,. 3 3." w- a .‘J' " ' v. ‘ 'r' "' ‘- .o ' ".‘r‘ ~ ~ "Y >._~.- . " 5‘.‘ . t. J
.' ---'-.o- . . ' - . in. . .. .-' - .. - ' .-. ..
1‘ V O ‘ o 4 v. ‘. ‘3 . l. r ' -_‘r'.?'o t ”"k ‘ ' .F‘l ' (3,.-.3 . v.30.- 1,. ‘ . -.l . a 1. n" ' “ ' "" ‘ 3" ’3 ’l h‘ - '1‘“. 1 i 1" f "
' on . ' . -.a.,. “J. ”Int- .._ 9 . I pr." . 0“ no. 1.- ‘ " ‘l ' " h '0 "‘
. . u ‘ i ‘ .. ,’ .u ‘ . a V g f‘ .. r ,‘ . -' b 3. . .§ ‘ Q‘. . .6 _ .JIJ I . .
‘9‘ ‘5 . k ‘ 6, . 0.. - .0" - ’ 0 73 n ‘fi‘37— . o o. 4"“ “A :I’ . ".1." 0 t “‘ :- . ‘ " o. .‘P‘ ‘ .' .u . k g .O ..' u: A'-,\- . o .
‘ .' ‘. — u v ‘ . . . .— .., v F". .. _. ' . . . y ‘ 4 - . . . u' . . .‘ -.1 I . . r1! . ‘ -
‘. 3 .' ' "' .3 -‘;. . h. ‘ V. .' ‘ 7". .‘O '.."IO.".'\?. 1- .- 0' '- I t-‘il : 'v‘ ' 5":r ‘, 'I." . f. -", ' ‘ " .".\J:J' " ' .‘ x . Q . a? ' . :‘H- ‘.":’.. ~50 ' ‘
- _. v V q. ' '.- .I 0‘ '0'}. . _< ‘ 'yI-‘p .f 0' f. 0‘ | 0 on?“ ' . O ‘ - W-q.’ - o'f.‘ ‘ ‘ I ‘ 'I ‘ 'Iy ‘ t a O
"zi. ' -_‘ .‘ 1“”. 4 .. ,fiq- - ' IIIPO 0t win-'a‘xp-d '.' ‘3.” m" "r. .‘. A“! {L'A'V‘I -. ' a: ~-dr~_.. ,( K - 3;“- v’_.""
t .' ~ .|. , '../.I . .;:' . “a '."'— 1‘“ r '.’"'_.—"1' -°" ,"r '0 ‘NJ c‘ru‘ ‘ In 2.!" C. I :OSY ‘K'Lu Q‘g'é'fi .‘7 ‘ ' _ “ .n'v ’ . ‘ - . I; ‘ '
I. ‘ J . . . «I '1‘95'0 ‘yr. ‘1.._ ;’ :.T:,afit Q. . ' "r ‘fo . o ' '... flit ‘ q "A "E.’ " I I,‘ ' 'I I '1””'|' .6. i ‘ :. '5‘:.:0!' .G"' \-. ' 'lcf C." " .
«a . ,.,~ 11-5? - r 5 ..\ .0 I’ .' .- .., v poo» . «4-,— Inr 4 or“ . . . ' \ _- ‘ . . v > . «a - ‘4: 2'
' ‘ “ ' o. r..- u" a .4. ' ' f-wffi.¢- 'M'~ n k " ' "" 7 g; '5' ‘ ‘ ' '.' ° ' " -I"r"~ .‘I '. I o ' 3- / ° ' '
' 'J . u'rnu‘ ‘r- n - °_- ' , --_ _. 0 :c .. ,0 . n. . .._. _ ; ‘ ~. .q . - . . _ -.. ‘ -. , _ .
' '3 - . x’. :1 '3, "‘.x;.".' ,-.: .'-. ., .." ;. 4422-1: ”3%. . ' _ .' Mr‘ :23“ (N “F - 5.1, gm...“- .,. _,. , p.
"4 ' .0 Way;- ‘buMI‘N-F- JJ’ ’ . ‘Ii- ”.3- - . ‘ “N 0- -. . ,‘ .-,; ..°.( ".1
" ’ ‘- :4. 2,!‘7'1'.‘.:"" ‘ .;:‘ «sum. Mr” ‘*‘ ..w-"‘* ' ‘ ' ' fr? “-
fi ’ 9‘ . ..‘:..' 'T 43"». ....'. I.) \ ‘z. or: I" “t J '
‘1; .Q‘0 . ' L' $ ’0‘. .f: ‘t‘ ' r.‘ (‘s' J.
. I . "-' . i..- ‘ L,“ 7,-3.._
.‘. v9 1 a ' g

3"...“ ‘ '0 - -
$3.??? ' 9-0 f N ' I
z: ' ,.

   
  
    
 
 
      
  

  

  
 

     
      
    
    
      
     
    
     
      

   

 

  

  

  
  
  

”W n ‘0...
.fl .0" V::._
‘0 'f .
; o..:‘
I.
‘ ,
.‘0'. I I I “ 5.. - I F“. _ ,.
'.’r-"-.. 1. ‘r Iéul‘xct r ‘.-'. ~ .‘i' I . ' — . :5 “ .‘O
. ..-_ M‘ ~' ' ‘f g M: . j ‘ ' , .‘. ‘A 0.. A... .‘t
. . ‘ . v _ u. a‘ _ - .
C‘II"'.’ ' gr.r%|rl*:“ . 3. ‘o- J oJ" u} - rt‘q J‘:.w{
- A . . . -_ .
' ‘...S“""a"‘"‘-' 3. “...-.“-: 4:- . 4H» in N' 1-
. ..3; ‘ .. J 04' y, “"0 .°. - ’2' __ '_ ‘
(."'I" ‘ ' ; -q.q. ' an . o J .\‘.ul ’.'”,Wd_‘r.." 9;. ‘
O - . 'f" I. o u .4.“ 'ifinflwQ'pl. o" ,quD ..,fi.“. -
"Ny‘ .' . 4...-e I I .1.- . - . g ,'l. _ u - I. ‘1'.’ U..-_.“" a o . Q
' - , - t L ~ A - o ‘ n p . '.
a...» '. "-:..r a...-:":‘o‘0:¢~.” “1.". ° I "7 ‘. ' . rr‘! -".: -V‘":'\“~1"r'."‘
a lo. ' .L‘ _ o'._'-'...:k.. - - ‘. - ‘. . _ .- -:_:.." .f...‘ . -, 4’..‘
tn.-. 1' 3,." wt: ‘... -‘ Jr ' - ‘0 J ‘ - . g” .1....?~$- g1... . ‘
. ~o fl . .. ".—-’rr‘,;# "0‘ 4.}. . . foaal 0 ~ .0. '.""."-r14; n40.09'.‘ 'w‘. ‘v'fl'i.~:q . ...h
Maw!“ for :..W ' . utifi a ti ' "' ' "ti: 1"..-. “ .
0:" . - ,M ':"'“‘.'f;'." 4 - -:o r-‘tdfioa—q‘.‘ {‘2 ~ana—oo-u4‘ \
, ' '-
ov - 7‘33!" - '9“ ""' '1'W“.":'_.' .. 00* "' ‘3‘ - u ‘ “$1 T‘“:J:’.“Z-
‘E’ ~ ‘-'.-7.-f‘:.—:;.o’-". .3” . ~ to. ' . | ‘5' it...» ‘ ,go-c‘.:‘
'W', O f. ,I- 4‘ n ‘_ 3' - 5 ‘, o r - 'I ' .F’ ('01: 1*. . (u‘ogu‘vr‘ .’g— ~o-i! ..';!o‘,!
.4 _.‘ . ,J ' o. I. 'f '0 ' “ I. u n s.
-I,. . '29-" .‘9 .. A 9 up; u-v .:::!:‘Jd~.‘o u 'ur' :2...’ a.,:'o- v r‘ ‘ . ' r " . fl . .‘. . 1.0-3, ::Q: ‘9‘. fOI_ :'-o
‘ a v - . - a .' , . . w “. - .
. “ll ‘ ’ 1’ I; ’ ".' (V';. v: 'V’f’" 9 'l': ' ‘4'"‘f l’::" '9. I"... h .' We ":".J~..‘_O "r 0". c 0 ~ 9’ r ' $19 'm1“?§"' I 0" I'
I.“-’ 'o ' - . .t ‘ . . . v a '..‘l . 4'. . '— v- '4’. .' f...- 1 r ‘ a K. g .. _ N . p..-’ ' ' . 'fi. .
. ' H." d .N .2 . _. "L." d 3 -0 O ‘ P' "1".“4'f-I’ a; I, ‘ ' "..’.l.r°." . .~'.-6'.: 4“ 4);..13“ ‘p t~‘I£J‘/';‘. :1'5‘ :f‘V' '2'. W" J . 0 190'”: '.~ ‘4 ‘ '.' :9 " ‘~‘; 6:? r“
4. 3- ' -,»' ~ I.‘ g I' ‘0 O ' ' - F ‘ (:0 ‘ 1‘ "019,.'0’- —. '5." 0.0;.9. I ‘ l‘.“ I‘V‘-Q:.'. " ~ ‘ 2'. 'V N .. 'fi ’1 4". v’.‘." "C“ ““-. " Vi"..' ..:.I§ 'r‘f’ . :1.
. .' ' h " ,‘ ‘ . .' ° no. ‘ o‘ - u- . u -- . .- ‘ . -. ' . .,, . . I . ’ '
""°.:--",-""-‘.o-u - r-n. nu _ .«-'.. .,5 - :~ '2‘: u: '-"':'..‘..2a 1»: ‘ .; 'Rf-‘p ”Pg-cc” w" .'~«~_:“-*-z. r' .t. ”31‘.“ “fir-ww-
o. . "n. -'- ..~ g,r:.'~ . _ o . . ”.«o I. w ,. . .r g , .1“‘ . an ' JI-nl' « «Pit. CIT-I [y 1“ (,‘IA ._. ‘3'" if. v. .t'.‘ : J 1 . . fl ‘0 , .. '-
‘ . . -. . .. on . . . -.. n. , .l g . - a ‘ n ... g . 'fl'. . . _ - I . l . . v. «a . . :r o Ionov-l
‘ 3., .‘r ¢: P‘ ‘- J y.- ‘.:‘H . _'° .9 :' )9.“ - .: r..‘§{ V l .. _| - ,Pfi. ‘ la ‘ ’-. ArJo - ’1‘ 03"". -:' a 0"2}~ -.l:"' , ’ -_ ;.‘.. . 1"..." 'fn. .:.‘, ‘-. ..‘..'._-.-‘. vst.” . ‘Jfir.
« -..;_ 3- .-‘.w- .~ '. ,:‘..“:..:.",".-;;;r-any: -. 11’3"» ' ‘ ~:-.-'~'o M" - 'wrt. 9,}; - , .- , '- .. .. W.
""""'.J u"! " "' u‘. ‘ . 3— -- I ' ‘. .' " ~ M” ,, " ° '” " :31 , '--"rtz.--r~- -.",. ’f"-'3"{j--52: "’,".-;:,.-f; “1"?"3153?
l“) I ‘ "‘.‘,ul‘~' ' ‘ . ‘0’ -~:‘ '5' . Id| ID. A 'o ‘ ‘1', .. .OJ’. .— )7! a...’ ' ‘ " u-‘Joflifi '0' ‘4‘ .5.» -
)' ‘-' ' ":’ “JO-‘v‘ ‘1' or. I vl -n . ..-:- -.. -' -_ g.::’ g. .1‘ - g. - 4 A J *n-‘f-fi ‘ _0 . .
:cf 0». °-'-"‘ r I "'."‘f’: [Ir-.3“: ~~-' ° at: "3" . ‘3'“ r . .. ‘ '. ‘
" \ -9 n. -' .u 4. I o . m.
‘fo ' c. v - . o .'
a“ .
o. ' "" '
u | I. I
7’. 0‘0‘ 0 J" ,1
Q t‘
a '.' Han; 0'1 .- .4
r0- ' y~ . C
. a. an I ‘ 4'00 4
' -—. 3 'O-c-o- ,‘h . o
—- ..\ . _ . ... w . ' 0 V 'b ud .l
‘1'. '9 . n . 9". . .4 o.. . .. .
. . ' o ‘; 4 o.‘-- .I. .1 a ”6 0,39 1.6) ”a; A
. u‘o. . ,c.— " a n; 4 .4‘ ' .r )l “
... _ . . . .
"H" .- .1. J {earl} a... cum.
I ‘v ' | r‘-I.‘ | I" .0 . l.-990'~'O:ng" o 3
:‘a. --- ‘ "‘2o-‘r’..’ .d
o-l‘u.‘ r I u ,-u 0" -9
J: 0. ._.0‘-ol." . ‘ - ‘ ’v f' '
6. v .. ~—' '
< . -
:t-pwr...‘ I.
.. 1..- 7...
1‘... .~ ': o#:- ' P
cu ‘

(:0... -' 0" l

or a «v I N

’ ~v- 1:!” full-'2: 0'

-0 you H v
[IQ 0"" 0" L' D.t-.'

  
 

  
     
    
    
 
   

f
clo"

-p
"N" .'
but oh—

 
   

'-'L:§. .t.‘
Q'o roz‘:
' -¢I

1' :‘o‘:COJ‘MII 1 .50, “5-4—0
" *1!“ :d‘::'::o‘..d an

. o

   

L"...

  

    
         
 
  
   
 
 
   

:0:
4'... .

       
 
 
   
 

1?." a .. . .
3r “3"" "my" . '. .. -' 1' ' . ’ '.’
3".0’3'1. '. . .ou .. o "'0 ' .I' 'o v no ’0 q... . g.‘c¢..

1" r. at ‘ ~ --:' m" .n r'c-r ~'

Jor" . .IH’K- 0 -‘ 0.. ‘04,,‘ --i
. . . . -‘ 'l‘|-"‘r.';’f
“ \ ”twins
0

\

J

1.3
'2'.
3

' ..
o
04r~.>—r.)1qonn ‘ ' I!“
.7- o. o - -¢- '5',
'dec‘o‘o pr“. — ' ..—-w r .g...
\I... l': van. or: .'
-v r - "to . oonpo‘ 6' “v“...
" v“ -1 fl ‘ o [‘1' l O. hoV '0,
.0. o. of-v‘.‘ v .- 5491.“ "A
0 a"..- NO ’u w. 'n- 0-" '0
'9 o.- .‘tou’g CPO" Ito. J .'.
’ '"j?.'::."‘ d'"v"-u" 'w‘:l 'z'. v
-' ‘0. O. 00 l " ' ° ° - ‘- up. I iv":
- L'u'.‘ 20 g I" — cl'v (00”: .‘lfll.l A:-
‘q' _'4. -' ‘-.. 'rvwv- P”! Q“ ...
Oo.- ‘shP‘... '-§'.’\‘~W¢'ovnuooa
on on u 6* «Or-r0 r1 5", at.
o «1‘ “"‘I0‘ I“"' 4",.1v ~09
_J ' '- o ' v'J', o l'v‘vo. p.-
. . n, . . . - '...«- .p‘wp .o.
.. c‘ ‘-.O 0'». ooofl -Q-...'-- .Oy‘lyr- .n:‘t0.
. " " '~or'Ooo-vh' In.
’ " ~ " ‘r UV‘ -' «r. p‘rd '
00‘ '0‘! " I ' O '- 5‘0- r- .-
J -']"' O I M 0-r ‘dfia'.-.’
_ o o Q a 9" 01". o,~K.:f—- .10
- 0‘ ‘ M'-‘.oJ ' O ' ‘§o '1'. O ‘o‘t‘1' ”H F"!
~ ‘0 ' ‘ u Va 'o a, vooo w! .. w:w«uuu.:
.00. .." g- 0- ‘v ."l'o‘O‘E-n
. “do | .“ 'o-tf- V”. 0-: 2.01:
a I. C I c O ' --' . 0 5' .'.H._ fi.:.th.;. .:.;..,'
a. I '."’ .- -. -. -O-- ' 0 q”. .. Inf
-... o. * .- r«0—- Jvt-u -—- r'ru ,'r u:—'o.’
‘ 1‘ O :i’, '0‘. . L: ‘E " ‘M ('n.
r-— .. ..
.. '3‘! ’. ') ... ;. ‘ it'OT a. 0....3A . 5 .1” iJrq'-
' l.‘.. " . 0 ~ '4 a
‘ "' H "1"" “"d‘u'l? :IQ :u:""::‘. m r‘, 017
A... ‘.‘l‘ a .‘vy'g" (‘c‘1..‘.,.'..:. ._ .9...
.-l-_.. “33"...-" ' -“vu tith'I’"

TAT IV RSI LIBRA

Illiiiiiiliil:Hiiiillllilllliiliiililllliilil

3 1293 00876 4759

 

 

This is to certify that the

dissertation entitled

NONLINEAR DYNAMICS OF SHAFTS ROTATING AT
NONCONSTANT SPIN RATES

presented by

Chih-Kao Ma

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mechanical Engineering

 

 

:UN'A-Cvilmzizima—

Mofessor
I 9 Hum

MS U i: an Affirmative Action/Equal Opportunity Institun'on 0- 12771

 

 

LIBRARY
MIchIgan State
Unlverslty

 

 

 

PLACE IN RETURN BOX to remove thie checkout from your record.

TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

|
[—71

MSU Ie An Affirmative Action/Equal Opportunity Institution
chmS-M

 

||

NONLINEAR DYNAMICS OF SHAFTS ROTATING
AT NONCONSTANT SPIN RATES

By

Chih-Kao Ma

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1991

ABSTRACT

NONLINEAR DYNAMICS OF SHAFTS ROTATING
AT NONCONSTANT SPIN RATES

By

Chih-Kao Ma

This study investigates both theoretically and experimentally the effects of
nonconstant spin rates upon the nonlinear dynamics of a shaft rotating about its longitudinal
axis. It is assumed that the spin rate can be written as the sum of a steady-state part and a
small periodic fluctuation. In particular, the spin rate is taken to be of the form

Q=O,+eQAsin(tot). The fluctuating component gives rise to time dependent

coefficients in the system's governing equations and thus a variety of resonant responses

result which depend on the relationship between (28, 89A, 0) and the system's natural

frequencies a)l and (02.

Both pro-buckled and post-buckled behavior of two main classifications, a
cantilevered beam and a simply supported shaft, are considered in this study. The
theoretical investigation involves obtaining approximate solutions to the governing
nonlinear differential equations by application of the method of multiple scales. A variety
of resonant conditions, including parametric resonance, main resonance, subharmonic
resonance and combination resonance, are investigated in detail. The influence which an

additional constraint of (02 ~ 3 to, has on the pre-buckled behavior of the simply supported

shaft is of predominant interest. Numerous numerical examples of these steady-state

solutions are presented which highlight a variety of phenomena such as non-existence of
steady-state motions, coexistence of steady-state motions and amplitude modulate motions,
and re-stabilization of trivial solutions. For the cantilevered beam case, Melnikov's method

is used to show that chaotic motions may exist.

Results from an experimental study on a spinning cantilevered beam is used to
confirm the presence of a variety of phenomena which were theoretically predicted to exist.
Finally, numerical simulations are employed to check a number of the approximate
solutions obtained by the method of multiple scales. They are also used to investigate non-

steady-state responses.

DEDICATION

To my parents,
and

To my wife.

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor Dr. Alan G. Haddow for
his advice, support, and guidance through this dissertation. I should like to state that it has
been a pleasure to work with him. I am very grateful to claim him as a friend and a mentor.

I would like to thank my committee members, Dr. Steven W. Shaw, Dr. Chang-Yi
Wang, and Dr. Joseph E. Whitesell, for their comments concerning this dissertation. Also
I would like to thank Michael Tamar, Leonard Eisele, and Bob Rose, who worked on the

complicated designs and equipment necessary for the experiments in this work.

I would like thank all of my friends at Michigan State, especially my officemate, Ole
Buesing for his friendship and effort in setting up the experimental equipment.

I would like to express my sincere appreciation to my wife, Yumei for her constant
support, encouragement and patience when I was in graduate school. Additionally I would

like to thank my parents, younger sister and younger brothers for their encouragement and

support.

Finally, I would like to thank my country, Taiwan, R. O. C., for giving me the
financial support while I studied in the United States. This work was partially supported
by NSF under grants MSS 8805985 and 8810390. This support is gratefully
acknowledged.

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES
LIST OF SYMBOLS

CHAPTER 1 - INTRODUCTION
1.1 Motivation
1.2 literature Review
1.2.1 Rotating Beams
1.2.2 Rotating Shafts
1.3 Scope of the Thesis

CHAPTER 2 - MATHEMATICAL MODELLING OF THE SYSTEM
2.1 General
2.2 The Case of a Rotating Cantilever Beam
2.3 The Case of a Rotating, Simply Supported Shaft

2.4 Summary of the Chapter

CHAPTER 3 - ANALYSIS OF A ROTATING CANTILEVER BEAM
WITH 0. < 00

3.1 Inu-oduction

3.2 Equation of Motion

xi

xix

\JAUJN

10
10
13
19
23

24

24
25

3.3 Approximate Solution of the Governing Equation of Motion
3.4 Numerical Results
3.5 Summary of the Chapter

CHAPTER 4 - ANALYSIS OF A ROTATING CANTILEVER BEAM
WITH 0, > Q“

4.1 Introduction

4.2 Equation of Motion

4.3 Method of Solution

4.4 The Caseofto~2too
4.4.1 Steady-State Solutions
4.4.2 Stability of the Steady-State Solutions
4.4.3 Numerical Results

4.5 The case of to ~ (00

4.6 Simulation and Observations of Chaotic Motions
4.6.1 Melnikov's Method
4.6.2 The Primary Resonance: £0 ~ (Do

4.6.3 The Subharmonic Resonance: 0) z 2 (00

4.7 Summary of the Chapter

CHAPTER 5 - ANALYSIS OF A ROTATING, SIMPLY SUPPORTED
SHAFT WITH (2, < 9,
5.1 Introduction

5.2 Approximate Solutions of the Governin g Equations of Motion

5.3 Assuming (02 is Well Separated from 3o)1
5.3.1 The case of a) s 2 a),
5.3.2 The case of to ~ 2 (02

5.3.3 The case of to ~ (02 + (01

25
30
33

38

38
39
40
42
42
45
48

83

74
83
91

98
98

104
104
108
111

5.3.4 The case of a) as (92 — (01
5.4 Assuming (92 ~ 3 a),

5.4.1 The case of to ~ 2 (01

5.4.2 The case of to ~ 2 (02

5.4.3 The case of to ~ (02 + (01

5.5 Summary of the Chapter

CHAPTER 6 - ANALYSIS OF A ROTATING, SIMPLY SUPPORTED
SHAFT WITH (28 > 00

6.1 Introduction

6.2 Exactly Circular Cross Section
6.2.1 Equilibrium position
6.2.2 Coordinate transformation
6.2.3 A First-order Approximate Solution with pi = 0(1)
6.2.4 A First-order Approximate Solution with tti = o (g)
6.2.5 Numerical Results

6.3 Close to Circular Cross Section
6.3.1 Equilibrium position
6.3.2 Coordinate u'ansformation
6.3.3 Approximate Solutions

6.3.4 Numerical Results
6.3.4.1 The case of to s 2 a),

6.3.4.2 The case of a) . 031

6.4 Summary of the Chapter

115
119
119
136
138

142

144

144
145
145
148
149

151

153
154
161
163
164

166
167

172

181

CHAPTER 7 - EXPERIMENTAL WORK
7.1 Introduction

7.2 Experimental Setup
7.3 The Case of 0‘ < (20

7.4 The Case of O; > (20

7.5 Summary of the Chapter

CHAPTER 8 - SUMMARY AND CONCLUDING REMARKS
8.1 Summary of the Dissertation

8.2 Discussion and Future Work

APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
APPENDIX E
APPENDIX F
BIBLIOGRAPHY

182
182

182
186

191

201

202
202
204

207
209
211
212
213
215
216

Table
4.1
4.2
6.1
6.2

LIST OF TABLES

Summary of Bifurcated and Chaotic Motions for a) = 1.54.
Summary of Bifurcated and Chaotic Motions for 0) = 2.79.
Summary of Bifurcated and Chaotic Motions for a) = 0.947.

Summary of Bifurcated and Chaotic Motions for to = 0.4885.

page
82
83

172

178

LIST OF FIGURES

Figure Page
2.1 The Coordinate System. 11
3.1 Variation of the Steady-State Amplitude, a, with o. 31
3.2a Variation of the Steady-State Amplitude, a, with Os for a = -5.0. 31
3.2b Variation of the Steady-State Amplitude, a, with as for o = 0.0. 32
3.2c Variation of the Steady-State Amplitude, a, with (28 for o = 5.0. 32
3.3a Variation of the Steady-State Amplitude, a, with 89A for o = -20.0. 34
3.3b Variation of the Steady-State Amplitude, a, with EQA for o = 20.0. 34

3.4a Time History of the Envelope of Q (t) for o = -20.0; 0(0) = 0.2, 0(0) = 0.0. 35
3.4b Time History of the Envelope of Q (1:) for o = 0.0; Q(O) = 0.01, 0(0) = 0.0. 36
3.4e rune History of the Envelope of Q (t) for o = 0.0; 0(0) = 0.15, 0(0) = 0.0. 36
3.4a Time History of the Envelope of Q (t) for o = 15.0; 0(0) = 0.01, 0(0) = 0.0. 37
3.4e Time History of the Envelope of Q (t) for o = 15.0; 0(0) = 0.05, 0(0) = 0.0. 37

4.1 Variation of the Steady-State Amplitude, a, with o. 49
4.2a Variation ofthe Steady-State Amplitude, a, with 8291 for o = 0.0. 50
4.215 Variation of the Steady-State Amplitude, a, with 820A for o = -23. 50
4.3 Variation of the Steady-State Amplitude, a, with Q, for o = -28, 0, 10. 51
4.4a ea, 6, Q, - Plot for Stable Steady-State Solutions, (3-D representations). 51

4.4b ea, 0, 0' - Plot for Unstable Steady-State Solutions, (3-D representations). 53
4.5 ea, (0, Q, - Plot for Stable Steady-State Solutions, (3-D representations). 53
4.6 Bifurcation Diagram in the (D-EZQA Parameter Space for (28 = 3.70. 54

xi

4.7

4.8a

4.8b

4.9

4.10

4.11
4.12a

4.12b

4.13
4.14a

4.14b

4.15

4.163

4.16b

4.17
4.18

Effect of Q, on the Bifurcation Diagram in the (0'82 0A Parameter Space.

Time Trace of the Steady-State Solution Q for o = -28;
0(0) = 0.2, 0(0) = 0.0.

Time Trace of the Steady-State Solution Q for o = -28;

0(0) = 0.1, 0(0) = 0.0.

Phase Portraits of the Steady-State Solution Q for o = -28;
(i) 0(0) = 0. 2, 0(0) = 0.0, (ii) 0(0) = 0.1, 0(0) = 0.0.

Time Trace of the Steady-State Solution Q for o = -28;
0(0) = 0. 05. (2(0) = 0.0.

Phase Portraits of the Steady-State Solution Q for o = -30, -35, -40, -45, -50.

Trajectory of the Steady-State Solution Q in Non-rotatin g Coordinates
for o = -28; 0(0) = 0.1, 0(0) = 0.0.

Trajectory of the Steady-State Solution Q in Non-rotatin g Coordinates
for o = -23; 0(0) = 0.05, 0(0) = 0.0.

Variation of the Steady-State Amplitude, a, with 0.

Time Trace of the Steady—State Solution Q for o = -35;
0(0) = 0. 05, 0(0) = 0.0.

Time Trace of the Steady-State Solution Q for o = -35;
0(0) = 0.1, 0(0) = 0.0.

Phase Portraits of the Steady-State Solution Q for o = -35;
(0 0(0) = 0.05. 0(0) = 0.0. (11) 0(0) = 0.1. 0(0) = 0.0.
Time Trace of the Steady-State Solution Q for o = -35;
0(0) = 025. 0(0) = 0.0.

Phase Portrait of the Steady-State Solution Q for 0‘ = —35;
0(0) = 0.25. (2(0) = 0.0.

Variation of the Ratio 11 : 12(0)) with (t) for (2‘ = 3.70.

Variation of the Ratio 1112(0)) with 0) for Q: = 3.52, 3.53, 3.55,
3.60, 3.70, 4.0.

54

55

55

57

58
58

59

59

65

65

67

67
7O

70

4.19a

4.19b

4.19c

4.20a

4.20b

4.20c

4.21a
4.21b
4.21c
4.21d
4.21c
4.21f
4.21g
4.21h
4.21i
4.223
4.22b
4.22c
4.23
4.24a
4.24b
4.24c
4.24d
4.24c

Stable Manifold w8 and Unstable Manifold w“; 91194 = 0.74.

Stable Manifold w8 and Unstable Manifold w“; 9L?- = 1.036.

Stable Manifold w5 and Unstable Manifold w“, 9-1524 = 1.11.
Stable Manifold w5 and Unstable Manifold w“; 91794 = 1.48.
stable Manifold w8 and Unstable Manifold w“; 94% = 2.22.

Stable Manifold w8 and Unstable Manifold w“; flag—’4 = 2.96.

Phase Portrait of Period 1 Motion; c252,; 0.035.

Phase Portrait of Period 3 Motion; 2252,: 0.045.

Phase Portrait of Period 6 Motion; 2252,; 0.046.

Phase Portrait of Period 12 Motion; cznfi 0.04605.
Phase Portrait of Almost Periodic Motion; e252,; 0.04657.
Phase Portrait of Period 12 Motion; e205 0.0467.

Phase Portrait of Period 6 Motion; 820A: 0.0468.

Phase Portrait of Period 12 Motion; c252,; 0.04685.
Phase Portrait of Almost Periodic Motion; 80,: 0.0469.
Phase Portrait of Period 2 Motion EZQA= 0.047.

Phase Portrait of Period 6 Motion; c252,: 0.0915.

Phase Portrait of Period 3 Motion; 80,: 0.115.

Phase Portrait of Chaotic Motion; 60,: 0.12.

Poincare Map of Period 3 Motion; c252,: 0.045.

Poincare Map of Period 6 Motion; 8252,; 0.046.

Poincare Map of Period 12 Motion; 2.20,: 0.04605.
Poincare Map of Almost Periodic Motion; e’ofi 0.04657.
Poincare Map of Period 6 Motion; 2.20,: 0.0915.

71

71

72

72

73

73

75
75
76
76
77
77
78
78
79
79
80
80
81
84
84
85
85
86

4.24f
4.24g
4.25a
4.25b
4.25c
4.25d
4.25c
4.26

4.27a
4.27b
4.27c
4.27d
4.27c
4.27f
4.28

4.29

4.30
5.1

5.2

5.3
5.4

5.5
5.6a

5.6b
5.7

Poincare Map of Period 3 Motion; ezflA= 0.115.

Poincare Map ofChaotic Motion; e’ofi 0.12.

Phase Portrait of Period 2 Motion; czofi 0.07.

Phase Portrait of Period 4 Motion; e252,; 0.08.

Phase Portrait of Period 8 Motion; e252,: 0.0832.

Phase Portrait of Period 16 Motion; 220,: 0.0833.

Phase Portrait of Period 32 Motion; c295 0.0834.

Phase Portrait of Chaotic Motion; e’ofi 0.0835.

Poincare Map of Period 2 Motion; czoA= 0.07.

Poincare Map of Period 4 Motion; e295 0.08.

Poincare Map of Period 8 Motion; czoA= 0.0832.

Poincare Map of Period 16 Motion; 229,: 0.0833.

Poincare Map of Period 32 Motion; 229,: 0.0834.

Poincare Map of Chaotic Motion; 2.20,: 0.0835.

Time Trace of Displacement of Period 4 Motion; 829A= 0.08.
Time Trace of Displacement of Chaotic Motion; 2252,: 0.0835.

Comparison between Melnikov's Method and Numerical Simulations.

Variation of the Steady-State Amplitude, a1, with 913 (2, =7.0, 8 = 1.0.
Variation of the Steady-State Amplitude, a1, with 01; Os =7.0, 8 = 0.2.

Orbit in (UN) Projection; Os =7.0, 5 = 1.0.

Variation of the Steady-State Amplitudes, a2, with 01;
as =7.0, 8 = 1.0.

Variation of the Steady-State Amplitude, a1, with 61; Os =7.0, 8 = 1.0.

Trajectory obtained by Numerical Simulation in (U ,V) Projection;
0‘ =7.0, 8 = 1.0.

Trajectory obtained by MS in (U ,V) Projection; Os =7.0, 8 = 1.0.
Variation of the Steady-State Amplitudes, al and a2, with 0'1;
62 =0.0, 8 = 1.0.

xiv

86
87
88
88
89
89

90
92
92
93
93
94
94
95
95

96
107

107
109

112
116

117
117

122

5.8a
5.8b
5.9a

5.9b

5.10a

5.10b

5.11a

5.11b

5.11c

5.11d

5.11e

5.11f

5.11g

5.11h

5.123

5.12b

Orbit in (U ,V) Projection; 01 =l300, 02 =0.0.
Orbit in (U ,V) Projection; 61=1500, 02 =0.0.

Orbit in (U,V) Projection; 01:0.0, tr2 =0.0,
U(0) = 0.1,U(0) = 0.0, V(0) = 0.1,V(0) = 0.0.

Orbit in (U,V) Projection; 01:0.0, 02 =0.0,
U(0) = 2013(0): 0.0, V(0) = 10,V(0) = 0.0.

Variation of the Steady-State Amplitudes, a1 and a2, with 01;
9A = 2880, 0'2 = 11520.

Variation of the Steady-State Amplitudes, al and a2, with 61;
9A = 5760, oz= 11520.

Variation of the Steady-State Amplitudes, a1 and a2, with 61;
(IA = 5760, 02 = -5760.

Variation of the Steady-State Amplitudes, al and a2, with 01;
(2A = 5760, 02 = -2880.

Variation of the Steady-State Amplitudes, a1 and a2, with 61;
0A = 5760, 02: 0.

Variation of the Steady-State Amplitudes, a1 and a2, with 61;
(2A = 5760, 02 = 2880.

Variation of the Steady-State Amplitudes, al and a2, with 01;
52A = 5760, 02: 5760.

Variation of the Steady-State Amplitudes, a1 and a2, with 61;

Variation of the Steady-State Amplitudes, al and a2, with 01;
(2A = 5760, 0'2 = 17280.

Variation of the Steady-State Amplitudes, a1 and a2, with 61;
9A = 5760, 02 = 28800.

Variation of the Steady-State Amplitudes, al and a2, with 01;
0A = 5760, 02: 0, 8 = 0.1.

Variation of the Steady-State Amplitudes, al and 82, with 0'1;
DA = 5760, 02: 0, 8 = 0.5.

XV

124
124

125

125

127

127

128

128

129

129

130

130

131

131

132

132

5.12c

5.13
5.14a
5.14b
5.15
5.16

5.17a

5.17b
6.1a
6.1b
6.1c
6.2a

6.2b

6.20

6.2d

6.2e
6.2f

6.2g
6.3a

6.30
6.4a

Variation of the Steady-State Amplitudes, al and 112, with 0’1;
9A = 5760, 02= 0, 8 =1.0.

Orbit in (U ,V) Projection; 01:900. 62 =0.0.

Time History of the Steady-State Solution U; 0'1 =900, 02 =0.0.
Time History of the Steady-State Solution V; 0'1 =900, 02 =0.0.
Spectrum of the the Steady-State Solution U; 01 =900, 0'2 :00.

Variation of the Steady-State Amplitudes, a1 and a2, with 91;
02 = 0, 8 = 1.0.

Orbit obtained by Numerical Simulation in (U ,V) Projection;
02 = 0, 8 = 1.0.

Orbit obtained by MS in (U ,V) Projection; 02 = 0, 8 = 1.0.

Orbit in (U,V) Projection; pi =200 1:2, a, = a.

Orbit in (U,V) Projection; t1i =5001t2, a, = '65.

Orbit in (U,V) Projection; tri =10001t2, a, = a.

Orbit obtained by MS in (U,V) Projection; u; =10007r2, a, = m.
Orbit obtained by Numerical Simulation in( U,V) Projection;

ui =1Wfi2, a) = -;-5.

Orbit obtained by MS in (U,V) Projection; tri =10001t2, w = go.
Orbit obtained by Numerical Simulation in (U,V) Projection;

u; =10001t’, t0 = 2 a.

Orbit obtained by MS in (U,V) Projection; tti =1000n2, (0 = 2 a.
Orbit obtained by Numerical Simulation in (U,V) Projection;

ui =10007t2, (0 = 3 E.

Orbit obtained by MS in (U,V) Projection;in =10001t2, 01 = 3 to".
Orbit obtained by Numerical Simulation in (U,V) Projection;
pi=10001t2, (0:45 6.

Orbit obtained by MS in (U,V) Projection; u, = 1000n2, 0) = .6 a.

Trajectory of the Free Response in (U ,V) Projection; pi = 20 n2 , 8 = 0.5.

133
134
134
135
135

140

141
141
155
155
156
156

157
157

158
158

159
159

160
160
168

6.4b
6.5

6.6a
6.6b
6.6c
6.6d
6.7
6.8a
6.8b
6.9
6.10a
6.10b
6.10c
6.10d

6.11

6.12
6.13
6.14a
6.14b
6.15a
6.15b

7.1

7.2
7.3a

7.3b

Trajectory of the Free Response in (U,V) Projection; pi = 200 P, 5 = 0.5.

Orbit obtained by Numerical Simulation in (U ,V) Projection;
c0=2t01+e o, o from 576 to -1728.

Orbit of Period 2 Motion in (U,V) Projection; 9A = 864, 0) = 0.947.
Orbit of Period 4 Motion in (U,V) Projection; 9A =1296, to = 0.947.
Orbit of Period 8 Motion in (U ,V) Projection; 9A = 1340, (0 = 0.947.
Orbit of Period 16 Motion in (U ,V) Projection; Q, = 1350, (1) = 0.947.
Orbit of Chaotic Motion in (U ,V) Projection; QA = 1440, (0 = 0.947.
Map of Figure 6.7 in (U,U) Projection; 9A = 1440, (0 = 0.947.

Map of Figure 6.7 in (V,V) Projection; (2,, = 1440, or = 0.947.

Orbit of Period 1 Motion in (U ,V) Projection; DA = 288, t0= 0.4885.
Orbit of Period 2 Motion in (U,V) Projection; 9A = 576, to = 0.4885.
Orbit of Period 4 Motion in (U,V) Projection; DA = 1100, 0) = 0.4885.
Orbit of Period 8 Motion in (U ,V) Projection; Q, = 1110, (0 = 0.4885.
Orbit of Period 16 Motion in (U,V) Projection; 9A = 1116, a): 0.4885.

Orbit of Almost Periodic Motion in (U ,V) Projection;
9A = 1120, (0 =0.4885.

Orbit of Chaotic Motion in (U ,V) Projection; 9A = 1152, (0 = 0.4885.
Orbit of Chaotic Motion in (U ,V) Projection; 9A = 1440, 0) = 0.4885.
Map of Figure 6.12 in (U,U) Projection; a, = 1152 (0 = 0.4885.
Map of Figure 6.12 in (V,V) Projection; r2A = 1152, (0 = 0.4885.
Map of Figure 6.13 in (U,U) Projection;r2A = 1440, (D = 0.4885.
Map of Figure 6.13 in (V,V) Projection;t2A = 1440, (0 = 0.4885.
Schematic Diagram of the Experimental Apparatus.

Mechanical Construction of the Experimental System

Frequency Response Curve for Q, = 9.04 Hz.

Frequency Response Curve for Q. = 10.55 Hz.

168

169
169
170
170
171
171
173
173
174
174
175
175
176

176
177
177
179
179
180
180

184

185
187

188

7.30

7.4

7.5
7.6

7.7a
7.7b

7.8

7.9

7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17

Frequency Response Curve for (2' = 11.00 Hz.

Time Traces of the Fluctuation of Motor Speed and Beam Displacement.

Spectrum of the Fluctuation of Motor Speed and Beam Displacement.
Variation of the amplitude of beam displacement with 89A.

Frequency Response Curve for Q: = 13.58 Hz.
Frequency Response Curve for Q. = 13.70 Hz.

Time Trace and Spch'um of Beam Displacement for to = 11 Hz.
Time Trace and Spectrum of Beam Displacement for (0 = 8 Hz.
Time Trace and Spectrum of Beam Displacement for (0 = 7 .8 Hz.
Time Trace and Spectrum of Beam Displacement for (0 = 7.75 Hz.
Time Trace of Beam Displacement for a) = 7.1 Hz and 7.0 Hz.
Time Trace and Spectrum of Beam Displacement for (0 = 8 Hz.
Time Trace and Spectrum of Beam Displacement for (0 = 2.65 Hz.
Time Trace and Spectrum of Beam Displacement for to = 2.5 Hz.
Time Trace and Spectrum of Beam Displacement for to = 2 Hz.

Trme Trace and Spectrum of Beam Displacement for 0) = 2.64 Hz.

188

190

190
192

194
194

195
195
196
196
198
198
199
199
200
200

HI
e-J
K‘I

(2(1).1J(t)
V(t)

X, y, Z

0 (1)

‘28

LIST OF SYMBOLS

cross-sectional area of the shaft.

Young's modulus of elasticity.

moment of inertia of the cross section.

length of shaft.

the undeformed arc length of the neutral axis(z-axis).
the deformed arc length of the neutral axis.

the undeformed position of a particle on the shaft.
the deformed position of a particle on the shaft.

the unit vectors in the x, y, and z directions, respectively.
the tOtal kinetic energy of the shaft.

the kinetic energy per unit length of the shaft

the total potential energy of the shaft.

the potential energy per unit length of the shaft.

the displacement components of a particle on the neutral axis in the
x, y, and z directions, respectively.

the displacement components of a arbitrary particle in the cross-section in
the x, y, and z directions, respectively.

the nondimensional modal displacement in the x direction.
the nondimensional modal displacement in the y direction.
the rotating coordinate frame.

the spin rate of the shaft.

mean component of the rotational speed

xix

(Y

0

the amplitude of the sinusoidally oscillating part of the rotational speed

the flexm'al vibration frequency of small oscillations of the first mode of the
non-rotating shaft.

the frequency of the sinusoidally oscillating part of the rotational speed.
the spinning natural frequency of the pre-buckled cantilevered beam.

the natural frequency of the post-buckled cantilevered beam as it
vibrates about the buckled position.

the natural frequencies of the shaft.

the difference between In and I”, In = (1+ 5) I”.
the density of the shaft.

the square of slendemess ratio, which is very small.

the detuning parameter.
the external detunin g parameter.

the internal detuning parameter.
the internal and external damping constants, respectively.

the angle of rotation of an element of the shaft about the y axis.
the angle of rotation of an element of the shaft about the x axis.

_a_(_)_
32’

§_(_)_
at.

CHAPTER 1

INTRODUCTION

1.1 Motivation

In the design of turbomachinery one will always encounter rotating components.
The shaft is a common example among these components and often they spin at high
speeds. It is well known that high rotational speeds can induce Vibrations Via forces from
mass imbalances and from instabilities caused by destabilizing forces such as internal
damping, dry friction, hydrodynamic bearings, aerodynamic forces, magnetic, and
electrodynamic forces. These Vibrations will greatly limit the performance of the design
and can even lead to failure. Therefore, the study of the dynamic behavior of shafts and the
associated instabilities is an important consideration in turbomachine design.

A great deal of research has been undertaken in the general area of rotating shafts.
Attention has focused on such topics as support stiffness, shaft alignment and mass
unbalance, non-symmetric stiffness, internal and external damping, and bearing Stability.
The literature on these topics is immense (e. g., see the books : Dimentberg [1961], Bolotin
[1963], Tondl [1965], Vance [1987] and the references contained therein). Past Studies
have investigated the dynamics of the system during run-up and run-down conditions, i.e.,
when the speed of rotation changes, monotonically, from zero to its final rotation speed (or
Vice-verse) (see, Lewis [1932], Iwatsubo et a1. [1972], Nonami and Miyashita [1978,
1979], Victor and Ellyin [1981] and Ishida et al. [1987]). Other studies have been
concerned with the dynamics when the rotational speed is held constant. However, in

practice, the rotational speed of a shaft can only be kept approximately constant due to

1

2

external perturbations upon the system. Often, the fluctuating component of the rotational
speed is sufficiently small (as compared to the mean value of the rotational speed ) and the
rotating shafts will run at, or close to, a constant speed. However, even small fluctuations
can have large effects which may give rise to various types of resonances, such as two-
mode internal resonance phenomena, sub, super, and combination resonances, and

parametric instabilities.

The aim of this thesis is to report on the influence of an unsteady spin rate, fl(t),
on the nonlinear dynamic behavior of a flexible shaft rotating about its longitudinal axis. A
range of cross sections will be considered, varying from rectangular, in which the shaft is
assumed to be infinitely stiff in one direction, to circular. In particular, the spin rate is
taken to be of the form Q(t) - Q" + 8 9A sin((t)t), i.e., a constant spin rate, (2,, which
has a small sinusoidal fluctuation superimposed upon it. The fluctuating component gives
rise to time dependent coefficients in the system's governing equations (i.e., parametric
excitation) and thus a variety of resonant responses result which depend on the relationship

between (2., 89A, (0 and the system's natural frequencies. In this study, attention is

focused on the parametric resonances and the effect of the internal resonance (i.e.,

(02 ~ 3 0),, where (01 and (112 are the linear natural frequencies of the system).

1.2 Literature Review

It is well beyond the scope of this inuoduction to review the current literature
regarding rotating shafts in general and so only studies which are considered to have been
most relevant to this thesis shall be cited. _ For more general information, the reader is

refereed to the texts cited in Section 1.1.

1.2.1 Rotating Beams

The dynamics of rotating elastic beams have been the subject of many investigations
over the past years. Most describe the case in which the beam lies in the plane of rotation
(e.g., turbine blades and propellers). These are classed as radially rotating beams. Fewer
studies have been concerned with the dynamics of beams rotating about their longitudinal
axis. Amongst these are works by Shaw [1988] who showed, by using a version of
Melnikov's method, that chaotic motions exist for a slender beam rotating about its
longitudinal axis, acted on by pulsating torques. Wang [1982] investigated the bifurcation
branches of a fixed-free beam rotating at a constant rate. Odeh and Tadjbakhsh [1965] and
Atanackovic [1984, 1986] also considered axially rotating fixed-free beams whereas a
work by Bauer [1980] studied the linear dynamic behavior of a beam rotating with constant
spin about its longitudinal axis for a wide variety of end conditions. He also presented the
response to harmonically forced oscillations of the beam. Laurenson [1976] used finite
element techniques to determine the modal characteristic of rotating beams. Krousgrill and
Bajaj [1987] studied a single degree-of-freedom dynamic system rotating at a prescribed
rate about a vertical axis. The form of the prescribed rotation rate is given by a constant
spin rate which has a small sinusoidal fluctuation superimposed upon it. They found

chaotic motions resulting from period-doubling bifurcations.

In the early part of the last decade, spacecraft missions and satellites began to
require the use of long flexible appendages to accommodate spin stabilization and for other
reasons e.g., antennas and booms. These flexible appendages can be accurately modeled
as beams and/or shafts. Generally, the dimensions of these rotating appendages are very
large. The flexibility of these rotating appendages, therefore, could no longer be ignored.
The determination of the nonlinear dynamic behavior became and remains a subject of

prime importance. The effect of booms oriented along the axis of rotation has been neared

4

by Meirovitch and Nelson [1966]. Robe and Kane [1967] investigated the effects of elastic
deformation on the stability of a rotating satellite composed of two elastically connected
rigid bodies. They found the performance of the system can be highly sensitive to
dimension and spin rate changes. Kulla [1972] studied flexible satellite booms under

fixed-free boundary conditions in connection with the dynamic behavior of such a satellite.

1.2.2 Rotating Shafts

As long ago as Rankine [1869], problems were being studied regarding shaft
vibrations. Jeffcott [1919] proposed a linear model in order to analyze the response of high
speed rotating machines to rotor unbalance. Stodola [1924] made theoretical and
experimental studies of many fundamental phenomena (e. g., gyroscopic effect, secondary
resonance and stability, etc.) of rotating shafts. The critical speeds of shafts with
distributed mass were examined by Grammel [1929]. Kimball [1924] and Newkirk and
Taylor [1925] were the first to show the possibility of shaft instability in the post-critical
range due to nonconservative loads. Kimball found that the cause of instability was
internal hysteresis, whereas Newkirk and Taylor identified oil films in journal bearings as
another source of instability. Shaw [1989] used methods from dynamical systems and
bifurcation theories to investigate the instabilities and resonances of rotors caused by

internal hysteresis, fluid film bearing forces, and mass unbalance.

A book by Tondl [1965] presents an extensive collection of experimental results
and detailed theoretical analyses associated with the subject of rotor dynamics. Nonlinear
and internal resonance conditions are both treated. Yamamoto and his colleagues have been
very active over the years, dealing experimentally and theoretically with problems of

rotating shafts arising from nonlinear, parametric, combination, and internal resonances.

5

Lengthy reports have been presented, for example, Yamamoto [1960,1961], Yamamoto
and Hayashi [1963] and Ishida et al. [1986,1989,1990].

Parametric instabilities can be induced in rotating systems due to pulsating speeds
and torques. Eshleman [1967] developed the equation of motion and the corresponding
' boundary conditions of this problem, but he did not succeeded in solving the equations.
Unger and Brull [1981] presented an analytical and numerical investigation to determine the
stability regions of the shaft due to a pulsating torque applied at its ends. They investigated
cases of principal and combination parametric resonances and found that the most common
and dangerous parametric instabilities arise as a result of combination resonances.
Ariaratnam and Namachchivaya [1986] used the method of averaging to examine the
bifurcation behavior of parametrically perturbed rotating systems with nonlinear
characteristics. A paper by Kammer and Sehlack [1987] reported on the effects of
nonconstant spin rate on the linear dynamics of rotating shafts. They found that parametric
resonances exist for shafts whose cross sections have unequal principal area moments of
inertia. However, when the two moments of inertia are equal, the shaft cannot be

parametrically excited.

A book by Bolotin [1963] presents an extensive summary of the instabilities of a
rotating shaft due to the effects of nonconservative loads. Ehrich [1964] investigated the
instability in rotating systems induced by internal damping in the rotor and gave a stability
boundary defined in terms of the ratio of external damping of the system to the internal
damping in the shaft. Gunter [1967] and Gunter and Trumpler [1969] evaluated the
stability Of high speed rotors with internal friction on damped, anisotropic supports and
attempted to theoretically explain many of the experimental observations of Newkirk [1924]
concerning stability due to internal rotor friction. Genin and Maybee [1970] have used

energy methods and presented results in the form of boundedness and growth theorems for

6

the problem of whirl motions of a linear Viscoelastic continuous shaft. Crandall [1980]

gives detailed physical explanations of the destabilizing effects of damping in rotating parts.

The instability phenomena of asymmetric rotating shafts (caused either by the shaft
having a non-circular cross section or by slots on key ways) are significantly different from
those shafts which have symmetrical cross sections. The first extensive investigation of the
Vibration of asymmetric shafts was made by Smith [1933], who discussed the cases in
which the shaft, or the bearings, or both were asymmeuic. Taylor [1940] and Foote et al.
[1943] investigated the effect of an asymmetric cross section on the whirling characteristics
of high speed rotors. Brosens and Crandall [1961] investigated the motion of the
elastically supported rotors having unequal diametral moments of inertia. Crandall and
Brosens [1961] studied the stability of a rotating system whose rotor and shaft both are
asymmetric. Hull [1961] experimentally investigated the whirling for three shafts with
different cross sections in combination with uniform or asymmeuic stiffness bearing
supports. Ariaratnam [1965] studied the effect of both external and internal damping on the
transverse Vibration of unsymmetrical rotating shafts. Inagaki et a1. [1980] presented an
analytical method for the evaluation of the synchronous response of a general asymmetric
rotor-bearing system. Genta [1988] derived the equations of motion for a general (an
asymmetric rotor running on an asymmetric supporting structure), multi—degree of freedom
rotor, based on the finite element method. He then used a series solution to solve these
equations. Day [1987] used numerical simulations and the method of multiple scales to
investigate the nonlinear Jeffcott model which considered nonlinearities arising from
deadband, side forces and rubbing. Mazzilli [1989] used the method of multiple scales to
study the effect of a geometric imperfection on the large amplitude vibrations of a

horizontal, rotating shaft.

7

1.3 Scope and Organization of the Thesis

The main task of this study is to investigate by means of analyses, numerical
simulations, and experimentation, the dynamic response of a flexible shaft rotating at
nonconstant speeds. Particular attention will be focused on the nonlinear behavior of the

shaft.

There are two main classifications in this study:
(1) In which the shaft is assumed very stiff in one direction, i.e., we have a
beam with a rectangular cross section.

(2) In which the shaft is (a) exactly circular or (b) close to circular.

The general approach adopted to study both of these classifications is to first derive
the equations of motion. This is accomplished using Hamilton's principle, and the
resulting pair of coupled nonlinear, partial differential equations are reduced to two
ordinary differential equations by assuming that the first mode dynamics dominate the
shaft's response. The method of multiple scales (Nayfeh and Mook [1979] and Nayfeh
[1981]) is then employed to find approximate solutions to these equations. The accuracy of
these approximate solutions are then checked by direct numerical integration of the

equations of motion.

An important part of the overall study is the inclusion of a physical model. It is
used to obtain a realistic range of values for the parameters used in the mathematical model.
Moreover, the results obtained from the experimental part of the work will, it is hoped, add
credence to the simplifying assumptions used in the theoretical component of the study.
Experimental results may also uncover responses that the theory did not predict. Hence the

experimental work can be used to refine the theoretical approach. It is also believed that

8

working with the experimental set up will give better insight into the physics of the

problem

In particular, the arrangement of the thesis is as follows:

In Chapter 2, a mathematical model which characterizes the rotating shaft is
derived. Three partial differential equations are obtained using Hamilton's principal which
are then reduced to two nonlinear ordinary differential equations in terms of the first modal

amplitudes. These equations are the basis for all subsequent analyses.

The nonlinear dynamics of a cantilevered beam rotating at a nonconstant spin rate about
its longitudinal axis are investigated in Chapters 3 and 4. In Chapter 3 we restrict our
investigation to a mean spin rate Q. less than 90 (where 90 is the flexural vibration
frequency of small oscillations of the first mode of the non-rotating shaft). The equation of

motion is homogeneous with time dependent coefficients and cubic nonlinearities. The

principal parametric resonance is studied (i.e., (0 ~ 2 (00 where 0’0 is the natural frequency

of the beam as it rotates). If the mean rotating speed (28 is greater than 520, the beam will

buckle to a non-zero equilibrium position. This is the case discussed in Chapter 4. Using
a coordinate uansformation, the equations of motion governing the shaft's motions about
the buckled position are obtained and are found to be nonhomogeneous, containing
quadratic and cubic nonlinearities and time dependent coefficients. Attention is focused on
two resonances, to ~ 2 (1)0 and 00 ~ 0)., , and approximate solutions are obtained for
oscillations around the buckled position. Numerical simulation are employed to track the
beam's behavior as the motion changes from oscillating about one buckled position, to a
motion that encompasses both buckled positions. Bifurcated and chaotic motions are
observed in this instance and so use is made of Melnikov's method to determine the

parameter conditions for the possible existence of chaos.

= .....l

9

In Chapters 5 and 6 we deal with the dynamics of a shaft with circular and close to
circular cross section. The problem now involves two, coupled, nonlinear ordinary
differential equations, and hence there exists the possibility of an internal resonances
between the two rotating natural frequencies (01 and (02. Chapter 5 examines the pre-
critical behavior of the rotating shaft. In particular, four cases of external resonance are
studied: to a 2(01, (0 a... 2012, (0 = 011+ (02 and (r) 2012- (01, both in the presence of, and
the absence of, internal resonance. Chapter 6 is concerned with the post-critical behavior
of the shaft. When the shaft is buckled, low order internal resonances did not exist.
However, there is the added complication of the possibility of chaotic motions. This is

studied using numerical simulations.

Chapter 7 describes the experimental setup and presents the results obtained from
experiments completed on a cantilevered beam. These are compared, qualitatively, with the
theoretical results of Chapters 3 and 4. Finally, conclusions and recommendations for

future work are presented in Chapter 8.

CHAPTER 2

MATHEMATICAL MODELLING OF THE SYSTEM

2.1 General

This chapter is concerned with the mathematical modelling of the system which is
illustrated in Figure 2.1. The general configuration investigated in this study consists of a
long slender shaft of length I, rotating about its longitudinal axis at a nonconstant rate,
9(t), executing motions which can be described by deflections (u,v,w) measured relative
to a rotating coordinate frame (x,y,z) which is also rotating at Q(t). In the analysis the
following assumptions are made: (a) the effects of applied forces, torques and gravity are
negligible, (b) the thickness of the shaft is so small compared to the length, that the effects
of shearing deformation and rotatory inertia of the Shaft can be neglected, (c) the bearings
are rigid and axially symmetric, (d) the frequency of excitation is far below the first axial
and torsional resonances, and (e) that plane sections remain plane and so we may neglect
inertial torsional and axial effects. Hence, the shaft can be mathematically modeled using
Bernoulli-Euler beam theory. Having found the kinetic and potential energy of the shaft,
the equations of motion are derived using Hamilton's principle and the resulting coupled
nonlinear, partial differential equations are reduced to ordinary differential equations by
assuming that the first mode dynamics dominate the shaft's response. The equations of
motion contain nonlinearities up to order three. Two classifications of boundary condition

will be investigated in this study: (a) cantilever and (b) simply supported.

Employing the above beam theory assumptions, the displacement of an arbitrary
point in the cross section is given by

10

11

Y!v ‘¢

 

x,u

 

 

Figure 2.1 The coordinate system

fi(X. y. 2.t)=U(z.t) (2.1)
V (X. y. L!) = V(Zrt) (2.2)

WK. y. AI) = W(z,t)- X<P(z.t)- y¢(2.t) +862
=nr(2.t)+112(z.t)-xcp(z.t)-y¢(z.t)+eoz (2.3)

12

where the z-axis is the neutral axis of the shaft and the x, y-axes are axes of symmetry for
the cross-section (the principal centroidal axes of the shaft's cross-section); 11, V and w are
the displacement components of a point on the neutral axis in the x,y and 2 directions,
respectively; tp and 0 are the angles of rotation of an element of the shaft about the y and x
axes, respectively. Note that the axial displacement W(x, y, z,t) is composed of five
components: (a) displacement, 1] 1(2, t) , which is defined as being associated with uniform
elastic extension of the neutral axis; (b) displacement, 11 2(z,t), which is defined as being
associated with the "foreshortening effect", i.e., the axial displacement of the shaft due to
large transverse displacements; (c and d) displacements, xtp and y0, associated with fiber

strain of "plane sections" bending; (e) initial strain 802.

We assume 30 = 0 in this study. The displacement of an arbitrary particle in the

crosssectionoftheshaftcanthenbewrittenas

'r' = fix, y. z,t) i+V(x. y. z,t) i+W(X. y. z,t) E
= u(z,t) 'i'+v(z,t) j+[w(z,t)- x<P(Z.t)- y¢(z,t)] E (2.4)

where (1,], E ) are the usual unit vectors.

The velocity of this particle is given by

Vc=f+f2xf

=(u-vn)i+(y+no)'j+(w-x¢-yq’r)i€ (2.5)
where {2:9 k.

The kinetic energy per unit length of the shaft AT can be written as

13

(2.6)

In the modelling, we are not including the effects of rotary inertia and hence the total kinetic
energy of the shaft T is found to be

12%]; Asz

I , . -
=ipA l. [(u-v 9>2+<v+u m’W’] dz (2.7)

In the next two sections we will find the potential energy of the shaft and then use
Hamilton's principle to derive the equations of motion for two different systems, a

cantilevered beam and a simply supported shaft.
2.2 The Case of a Rotating Cantilever Beam

For this case the shaft is assumed to be infinitely stiff in one direction (i.e., a
beam). We also use the following assumptions: (a) If the beam is kept relatively short
(e.g., less than 20 beam width), the transverse vibration is purely planar (i.e., V = 0); (b)
The axial deformation of the neuual axis is negligible; (0) the beam has a moderately large

curvature.

Define extensional strain of the neutral axis as

1 dsz-dSZ

an: - ——
2 d5” (2.8)

14

where S is the undeformed arc length of the neutral axis and s is the deformed arc length of
the neutral axis. Using this definition, it can be shown (Ho, Scott and Eisley [1975]) that

the only non-zero strain component an in terms of displacement components is

eu=W’+-2‘-[u’2+w”]

= W""‘P'+iifi'z+wdi (2.9)

It is assumed that the longitudinal motions are small, i.e., w'2 a. 0- Hence we can obtain

_ I I l ’2

(2.10)
From assumption (c), it can be shown that (see Appendix A).
533 ~ 11” [1+1 u’z]
32 2 . (2.11)
The potential energy per unit length of the shaft AV can then be written as
Be 2
AV = IA —23h dA
= §[Aw’2+ Aw’u’2+ Iyy u”2 (l +lu’2) 2 +£u’4]
2 2 4 (2.12)

Note that the potential energy is composed of two components: (a) the elastic potential

energy due to bending; (b) the elastic potential energy due to stretching of the shaft.

The total potential energy of the shaft V can be found as

15
V = If, AV dz
E l ’2 I ’2 n2 1 12 2 A 4
=—j [Aw +Aw 11 +1 u (l+—u ) +—u’ dz
2 ° ” 2 4 (2.13)
Hence the Lagrangian function is given by L(t) = T(t)—V(t). According to Hamilton's
principle, we must determine the functions u(z,t) and w(z,t) which render stationary

:12 L(t) dt . A straightforward application of the methods of calculus of variations yields

the equations of motion:

pAw-EAi[w'+l u'2]=0
dz

2 (2.14)
pAii +EI”. ( u""+ u””u’2+4 u’u”u’” + u"3 ) -pAQ2 u
—EA3-[ '+l u’z] u’-EA [w'+l ed] u”=0
32 2 2 (2.15)

Equation (2.14) can be simplified by the following manipulations. Integrate it to obtain

EA [w’i-lu’z]: j; pAw(§,t) d§+C(t)
2 (2.16)

where C (t) is an arbitrary time dependent function. The normal force on any shaft cross-

section is

N(z,t) = [A 0’ dA
=jAE eu(z,t)dA

= EjA[w’-xtp’+-;-u’2] dA

= E A [W'+i“'2] (2.17)

Lye-l."

16

Now, since N(I,t) = 0, we may combine this with equations (2.16) and (2.17) to conclude
that

N(z,t)=—j: PAW(§.t)d§

(2. 18)
Substituting equations (2.17) and (2.18) into the equation (2. 15), we obtain
pAii + E1”, ( u””+ u””u’2+4 u’u”u”’ + u"3 ) — pAQ2 u + ueu —
pAw u'+u"]; pAw(§,t)d§ =0 (2.19)

where tie is a viscous damping term added to the equation to allow for some energy

dissipation.

The function w(z,t) can be eliminated from equation (2.19) by using the following

expression (see Appendix B)

~ _ _l_ :2
W(z,t)= 2 Eu (art) d: (2.20)
The governing integro-differential equation which determines u(z,t) is found by
substituting equation (2.20) into equation (2. 19) . The equation is:

pAii + El”. ( u"” + u””u’2+ 4U’ 11” u’” + u”3 ) — pAQzu + trail

2 a2
+%pA u’%;£ [u’2(§,t)] dt-é-pAu”£ 51—; {If [u’2(n,t)] dn }d§= 0
(2.21)

17

In order to transform equation (2.21) to dimensionless form, we use the following non-

dimensional variables:

Equation (2.21) then rescales to (dropping the overbars for convenience)

fi+(“m1+um1u12+4uruuum+un3)-92u+ucfi

1,32 ,2 1”182{§,2 }_

tin Wfiu (§,‘t)d§--2-u [W jo u (11.0411 dé—O

(2.22)
where

'__3_(_)

()-81

mi).

()-82

Equation (2.22) can be reduced to a nonlinear ordinary differential equation by using
Galerkin's method. We take a first mode approximation to the solution using the

eigenfunction , ¢(z), of the linearized fixed-free beam as the coordinate function.

Let
u(z,t) = Q0!) <9(2) (2.23)

18

where
Q(r) is the nondimensional modal displacement

¢(z) = cosh 912‘ cos 812- 0.7341 (sinh i312- sin [312)

51:1'8751'

Substituting equation (2.23) into equation (2.22) and applying Galerkin's method then
yields:

("2+4(Q’0+QQZ>+tt.0+bQ’+(noz-nz)Q=o (224)

where
a =1; [1; aide] 9'9 dz-li {L‘ [1582" ldfilw dz (2.25)

_l I”! ’2 l I II I» l ”3
b-joo o odz+joo<l> o Odz+joo odz (226)

93:9”.

Numerical values for the constant d and b are obtained by numerical integration of

equations (2.25) and (2.26), and they are

19

2.3 The Case of a Rotating, Simply Supported Shaft

For this case we assume: (a) the longitudinal inertia force is negligible, and (b) the
curvature is small (i.e., linear curvature). As will be shown, the source of the nonlinearity
is from mid-plane stretching which arises on account of the distance between the supports
being fixed.

The potential energy per unit length of the shaft AV can be written as

2
A 2

= 13-[Aw’2+ Aw’(u’2+ v’2)+ Iyyu”2 + luv”2 +£(u’2+ v’2)2]
2 4 (2.27)

The total potential energy of the shaft V can be found as

V=j{, Ade

= g. j!) |:Aw’2+Aw’(u’2+V’2)-t-Iyyu”2 +Iuv”2 +%-(u’2+v’2)2 dz
(2.28)

The equations of motion are then obtained by applying Hamilton's principle which result in

the following

pAw-EAi[w’+l(u’2+v’2)]= 0
dz 2

(2.29)
ee m, e 2 ° rr
pAu+EIwu -2pAQv-pAQ u-pAvQ+ue(u-VQ)
—EA-a—[ ’+-l-(u’2+v'2)] u'+u,u-EA [w'+l(u'2+v'2)] u”=0
dz 2 2

(2.30)

20
pAV+EIu v””+2pAQu—pAOZV+pAuO+uc(V+uQ)
—EA—a—[ ’+-1—(u’2+v’2)] v’+ uiv - EA [w’+l(u’2+v’2)]v” = 0
Oz 2 2
(2.31)

We have also simply added external and intenlal damping terms in the equations of motion
such that the external damping force is proportional to the absolute velocity, while the
internal damping force is proportional to the shaft's velocity relative to the rotating
coordinate system. The 2 component of these equations is simplified by assuming the
longitudinal inertial force pAw = 0 and by assuming that the shaft carries no initial axial
load; it is given by

1|: ’+-;—(u’2+v’2)]= 0

82 (2.32)

Integrating equation (2.32) with respect to z and using the boundary conditions w(O) = w(I)
= 0, yields

1
wr+_1_(ur2+vr2)=iJ‘0 (u’2+v’2) dz
2 21 (2.33)

The dependent function w(z,t) can be eliminated by substituting equations (2.32) and
(2.33) into equations (2.30) and (2.31) to yield

pAii+E1yy u”"-2pAQV-pA92u-pAVO-t-ucm-VQ)

. E1 9 II 1 I2 12
I ll 21 u I0 (11 ) (2.34)

pAV+EIn V”"+2pAQu-pA92v+pAuO+ue ( V+uQ)

+uiv -£-A-V”jé (u’2+v’2)dz=0

21 (2.35)

21

In order to transform equations (2.34) and (2.35) to dimensionless form, we use the

following non—dimensional variables :

I 45 ryyZ 45 ryyZ
EI _

r: —17;t=l‘t, (2:2
pAI I‘

and relationships:

1,m=(1+ii)lyy

Equations (2.34) and (2.35) can then be rescaled to (dropping all the overbars for

convenience)

2
ii + u””- 20v-n2u- vO—[rT’y] u” I; (u’ 2+V’2 )dz

2 2
+2[-r:’—] u,(u-VQ)+[—rfl] Uifi=0

I (2.36)

2
V + (1+ 8)v”” +2nv - 02v+ uQ-[flfl] v” [(1, (u’ 2+v’2 )dz
2

2
”[511] u,(v+uO)+[51] uiv=0
1 l (2.37)

22

Equations (2.36) and (2.37) can be reduced to nonlinear ordinary differential equations by
using Galerkin's method. To this end, a first mode approximation to the solution is

assumed using d>(z), the eigenfunction of the linearized simply supported shaft.

u = U(t) d>(z) (2.38)
V = V(t) ¢(z) (2.39)

where
U(t),V(t) are unknown functions of time t
¢(z) = «(2 sin 1t 2.

Substituting equations (2.38) and (2.39) into equations (2.36) and (2.37) and applying
Galerkin's method then yields

U+(noz-n2)U-2oV-VO+2eu,(U-Vn)
+ a uiU+e 002W U2+V2) = 0 (2.40)

V+((1+5)ao2-o2)V+2QU+UQ+2ett,(V+UO)
+ cu,V+et202V(U2+V2)=0 (2.41)

2

where 8 =(51'1)

23

It is important to note the definition and interpretation of a few of the terms in equations
(2.40) and (2.41):

no the flexural Vibration frequency of small oscillations of the first mode of the

non-rotatin g shaft.

8 the square of slendemess ratio, which is very small.

8 the difference between In and I”, In = (1+ 8) I”.

U(t) nondimensional modal displacement in the x direction.

V(t) nondimensional modal displacement in the y direction.

2.4 Summary of the Chapter

Equations of motion have been presented for a shaft rotating about its longitudinal
axis at a nonconstant rotational speed. The equations of motion were derived using
Hamilton's principle and the resulting coupled nonlinear, partial differential equations were
reduced to ordinary differential equations by assuming that the first mode dynamics
dominate the shaft's response. Two different systems were investigated in this chapter: (a)
a cantilevered beam and (b) a simply supported shaft. The equations of motion contain
nonlinearities up to order three.

CHAPTER 3

ANALYSIS OF A ROTATING CANTILEVER BEAM
WITH Os < 90

3.1 Introduction

In this chapter we investigate the nonlinear, planar motion of a uniform, initially
straight, elastic beam rotating about its longitudinal axis at a nonconstant spin rate. The
spin rate is expressed as the sum of a steady—state term (i.e., Os) and a relatively small
sinusoidal perturbation. The beam is considered to be fixed at one end and free at the
other. For a beam with these end conditions, nonlinearities can arise due to moderately

large curvatures and the longitudinal inertial forces (Atluri [1973]). We will resuict our

investigation to mean spin rates, $25, less than these required to buckle the beam, i.e.,
Q, < (20 where no is the flexural Vibration fiequency of small oscillations of mode one of

the non-rotating beam. In Chapter 4 we will analyze the case of 9, > no (i.e., post-

buckled), in which the beam stops oscillating about the static equilibrium position and

buckles to one side or the other.

The pair of coupled nonlinear, partial differential equations which were derived in
Chapter 2 are reduced to one ordinary differential equation by assuming that the motion can
be described by a single, in plane mode. Experimental work on a physical system indicate
that such an assumption is valid. Approximate solutions to the governing equation of
motion are sought using the method of multiple scales and the results are compared to these

obtained by direct numerical integration.

24

25

3.2 Equations of Motion

From Chapter 2 (see Section 2.2) we know the equation of motion will have the

following form:

0+4<020+QQZ)+u.Q+bQ3+tnr’-02)Q=o (3,1)
where Q(t) is the nondimensional modal displacement of the beam.

Note that equation (3.1) describes the first mode response of the rotating beam in the x-z
plane subject to assumptions outlined in Chapter 2. It is valid for any form of rotational
speed, 9. In the remaining sections of this chapter, attention will be focused on the

solution of equation (3.1) for rotational speeds below the buckling speed, i.e., Q < (20.
Moreover, the form Of Q will be taken as

Q=Q,+EQASinm (3.2)

where

e is an arbitrary small but finite parameter
Q, is mean component of the rotational speed

(2A is amplitude of the sinusoidally oscillating part of the rotational speed
to is frequency of the sinusoidally oscillating part of the rotational speed.

3.3 Approximate Solution of the Governing Equation of Motion

The method of multiple scales will be employed to obtain an approximate solution

to equation (3.1). To this end, we first reorder the equation by introducing

26

Q=e2 q (3.3)

Substituting equation (3.3) into equation (3.1), we obtain

éi'Hid(CIC'lz'l‘qzél)‘fZ’ELH'l'l'fib<l3+(902"§12)<1=0 (34)
where [.1.3 = 2 a 11.
We now seek a first-order uniform solution of the form
9(138)=90(T02T1)+591(T02T1) (3.5)
where Tn: 6%.

In terms of Tn, the time derivatives become

1=D0+8D1+82D2+ ooooooooo

dt (3.6)
iii-D 2+2er D +ez(2D D +D2)+m------

(11:2 0 0 l 0 2 1 (3.7)

 

Substituting equations (3.2), (3.6), (3.7) and (3.8) into equation (3.4) and equating
coefficients of like powers of 8, yields

27

D0290+(902'ar2)‘10=0 (33)

1302 (11+ ( Qo2 - 9:2 )91 = ‘2 DoDl‘lo ’ d [‘10 (Do‘lo)2 4' (1021302 ‘10]
.2 “Doqo-bqog‘f’zn'gASin (OT (10 (3.9)

The solution of equation (3.8) can be written as

where (00 = ‘t/ (202 — 0,2 and A is the complex conjugate of A.

Then, equation (3.9) becomes

D02 q, + (002 q, = 2 d (1102 A2 [A exp(3i(00To) + A exp(i(00To)]
— 2i 1.1010 A exp(it00To) - b A2 [A exp(3i0)oTo) + BK exp(i(00T0)]

(3.1 1)

where cc stands for the complex conjugate of the preceding terms and the prime stands for

the derivative with respect to T1. If 0) ~ 2030 we can see that additional secular producing
terms will arise in equation (3.11), i.e., a principal parametric response will occur. To
quantitatively describe the nearness of a) to the resonances, we introduce a detuning

parameter 0 defined according to

(0=2C00+80' (3.12)

Substituting equation (3.12) into equation (3.11) and setting the secular producing terms to

zero, yields

28

—2imo(A’+uA)+(2d0102—3 b) AZX-iXQA o,exp(ior,)=0 (3.13)
To solve equation (3. 13), it is convenient to write A in the polar form
2 (3.14)
where a and B are real.

Substituting equation (3.14) into equation (3.13) and separating real and imaginary parts
yields

, a
(00a =-umoa--§Q,QAcosy

 

(3.15)

3 2 3
amoy’=cmoa-3:a +dm‘2’ a +aQ,QAsiny (316)
where7=oT1—ZB. (3.17)

Of particular interest are the steady-state motions resulting from a’ = 'y’ = 0. Hence we

have a trivial steady-state solution, a = 0 or

 

 

. 1 32 2
sm'Y: [-O'U)o+—(3b—2d0)0 )]
9. 9A 4 (3.18)
cosy: 1 {-2 limo]
(2,0, (3.19)

Squaring and adding equations (3.18) and (3.19), we obtain

29

l
4 {a (00 :I:[(Q,QA)2- 4 112 (002] 2}
a2 =

 

(3b-2d0002) (3.20)

or, rearranging this equation, we have

a2

1
E(3b-2dm02)i[(Q,QA)2—4[12610212
0':

 

‘00 (3.21)

Using equations (3.3), (3.5),(3.10), (3.14) and (3.17) it is found that

I _ 2
Q = £2 a cos[912—Y] + 0(82)
(3.22)

Numerical examples of these steady-state responses will be given in Section 3.4.

Their stability can be ascertained by adding a small perturbation to the steady-state
value and checking if this perturbation grows or decays. For the trivial solution this results

in a unstable solution for ‘91 < O < 01 where

l

2 2
0'1 = [[ico‘lfi] _ 4 ”2]
0
(3.23)

The analysis to check the stability of the non-trivial steady-state value, a and 7, results in

the following inequality for a stable solution

2
-a—Q,QA sin'y [mod—32] < 0
2030 2030 (3.24)

30

The figures in Section 3.4 show the results of this stability analysis. A solid line denotes a
stable solution, a dashed line indicates an unstable solution. It can be proved that the upper

branch is always stable, whereas the lower branch is always unstable.

3.4 Numerical Results

In this section we present representative solutions of the equations derived in
Section 3.3. Values for various parameters are based on a physical system which was used

in laboratory tests. The parameter values used, unless otherwise stated, are:

I): = 2.8, 89A = 0.1, e = 0.01, 90 = 3.516, b = 40.44, d = 4.60, (and hence
010 = 2.1266), and 28D = 0.00422.

Figure 3.1 shows a typical frequency response curve for a non-zero mean spin rate
(S2I = 2.8). This shows the variation of the steady-state amplitude as a function of the
oscillating component of the spin. Clearly, the overall non-linearity of the system is of the
hardening type. To the order of approximation used in the present analysis, the stable and
unstable non-trivial solutions continue to a exist as O is increased. A higher order analysis
would rectify this deficiency which of course could not occur in practice. The results
obtained from directly numerically integrating equation (3.1) are also shown on this figure
and are discussed later.

A series of results presented in Figure 3.2 show the effect of the mean spin rate,

as, on the steady-state amplitude, a. Three plots are presented for different values of 0'.
In each, when 9, approaches 12,, = 3.516 the results become invalid since the analysis
used in Section 3.3 is restricted to O, < 00. It is interesting to note that for some values

of,o (Figures 3.2b and c), the steady-state amplitude decreases as (28 is increased. There

also exist regions in which the jump phenomenon can be observed.

31

 

 

 

 

 

0.5
0000 numerical simulation
—— stable
0.4 ~'"" m
0.3 +-
I
-In
to
0.2 -
0.1-
0-30

 

Figure 3.1 Variation of the Steady-State Amplitude, a, with O.

 

0.5
a=-5.0

SIJDIC

 

—- - - unstable

0.4 '-

0.3 -

 

 

Figure 3.2a Variation of the Steady-State Amplitude, a, with (28 for O = ~5.0.

 

 

 

 

32
0.5
080.0
"— stable
—-- unstable
0.4 —
0.3 -
a
u-IN -
to
0.2 -
o. ._L .l_..__l.'_. . . . .
0 0.5 1 1.5 2 2.5 3 3.5 4
9

Figure 3.2b Variation of the Steady-State Amplitude, a, with (2‘ for O’ = 0.0.

 

 

 

 

 

0.5
0:50
0000 numerical simulation
0'4 _ stable
—-- unstable
0.3 -
a
-IN
or
0.2 -
0.1 -
00

 

Figure 3.2c Variation of the Steady-State Amplitude, a, with Os for O = 5.0.

33

I Figure 3.3 presents results showing the variation of the steady-state amplitude as a
function of 89A, the amplitude of the oscillating component of the spin. Figure 3.3a is
plotted for a detuning value of -20 and shows that there is always only one stable steady-
state, whereas Figure 3.3b shows that for a detuning value of 20, multiple steady-states are

possible.

The results obtained using the method of multiple scales can be checked by directly
numerically integrating equation (3.1). This was undertaken for a number of different
cases and the results have been plotted as "numerical simulation" on Figures 3.1 and 3.2c.
The comparison is good in all the cases tested. In situations where two stable, steady-
states exist, the solution adopted depends on the choice of the initial conditions. A few
examples of these simulations corresponding to Figure 3.1 are presented as time traces in
Figure 3.4. It should be noted that only the envelope of the solution, Q (t), is plotted in
these figures.

3.5 Summary of the Chapter

We have investigated the pre-buckled behavior of a fixed-free beam rotating about
its longitudinal axis at a nonconstant spin rate. The spin rate was expressed as the sum of a
steady-State part and a relatively small sinusoidally varying component. The approximate
analytical solutions were obtained using the method of multiple scales and it was clearly
demonstrated that a principal parametric resonance can occur at mean spin rates well below
the fast critical speed of the beam. For this type of resonance, the nonlinearities were of a
hardening type. The perturbation solutions accurately predict the amplitude of the steady-

state motions and their stability as compared to the numerical simulation results.

 

 

 

 

 

34
0.5
oar-20.0
l—
nth
_.....-_ unstable
0.4 '-
0.3 ~—
I
r—Ifi 1—
O)
0.2 '—
0.1-
l l
00 0.1

 

 

 

Figure 3.3a Variation of the Steady-State Amplitude, a, with 89A for O = -20.0.

 

 

 

 

 

0.5
0:200
— suble
—--— unstable
0.4 —-
0.3 *-
a
-In '—
tr.)
0.2 -
4 \“
0.1 — \~.
\\
lt— \\
\\
0 r l J_ -_L-_ .1__-._l.-__-.l_.._l_--1_. __
0 0 1 0 2 0.3 0.4 0 5
E‘QA

Figure 3.3b Variation of the Steady-State Amplitude, a, with 89A for O = 20.0.

35

 

0.3
(rs-20.0

0.2

AAAAA
VVVV'VVVV

Displacement O
O
T
J
l
H

 

 

_03 lllllllllllllllllllllll
'O 500 1000 1500 2000 2500

 

Time

Figure 3.4a Time History of the Envelope of Q (t) for O = -20.0; Q(0) = 0. 2, Q(0) = 0.0 .

36

 

0.3
o=00

0.2-

Displacement 0
c:

L—

 

 

_0 3 l, L l 1-1 1 l 1 l l i l i i l l l l l l i i i
° an 1mm 13” 2CD ZED

 

Time

Figure 3.4b Time History of the Envelope of Q (t) for o = 0.0; 0(0) = 0.01, 0(0) = 0.0.

 

 

 

 

03
o = 0.0
0.2“-
[iii

OJ 1
c:
E .—
0.)
fr
0 0*—
.2
a- P
5

-0 1

WW‘
‘1!
-&2r-
_0 3 i i i l l l l l i L l .L l l l i l 1 1 l l l l l
' 0 500 1000 1500 2003 23K)
Time

Figure 3.4e Time History of the Envelope of Q (t) for o = 0.0; 0(0) = 0.15, 0(0) = 0.0.

37

 

0.3

0.2-—

 

Displacement 0
D
I
I
i
H

 

 

_0 5 l l i 1 l l l i l 1 1 i 1 l l 1 l i l l l i l
' 0 500 1000 1500 2000 2500

 

Time

Figure 3.4d Time History of the Envelope of Q (t) for O = 15.0; Q(0) = 0.01, Q(0) = 0.0.

 

0.3

'1”

 

 

Displacement 0
c3
1

 

 

 

 

_0 3 i i i i i l 1 1 1 l 1 i l i l l l l i I l l l l
' 0 500 1000 1500 2000 2500

 

Time

Figure 3.4e Time History of the Envelope of Q (t) for o = 15.0; 0(0) = 0. 05, 0(0) = 0.0.

CHAPTER 4

ANALYSIS OF A ROTATING CANTILEVER BEAM
WITH 0. > 90

4.1 Introduction

The nonlinear, planar motion of a uniform, initially straight, elastic beam rotating

about its longitudinal axis at a nonconstant spin rate is investigated in this chapter. In

Chapter 3 we restricted our investigation to spin rates (23 less than (20. However, in the
present chapter we will focus our investigation on (25 > no, i.e., under this condition, the
straight equilibrium position of the beam is unstable and the beam buckles to one side or the
other due to centrifugal effects.

When the mean rotational speed (2‘ is greater than {20, the linear stiffness term in
equation (2.24) will become negative with the consequence that small oscillations around
Q = 0 become unstable. To investigate the oscillations around the buckled position, we

transfer the coordinate of the equation of motion such that the new coordinate describes the
motion about the buckled position. The form of Q will be taken as Q = Q, + 82 9A sinan .

As in the previous chapter, we focus our attention on the principal parametric resonance
(0 ~ 2 (no where (no is the natural frequency of the beam as it vibrates about the buckled

position. However, as will be shown in the subsequent sections, the coordinate

transformation gives rise to a nonhomogeneous term and so we will also investigate the

case of a main resonance, (1) ~ (00. The accuracy of the approximate solutions will be
checked by direct numerical integration of the original equation of the motion. We also

investigate the possibility of chaotic motion which can arise since the system is essentially a

38

39

double-well potential problem. We use Melnikov's method to find the regions in the
parameter space where chaotic motion may exist. The results are then checked by
numerical simulations.

4.2 Equation of Motion

As in Chapter 3, we begin with the one mode equation of motion as it was obtained

inChapterZ.
Q+d(QZQ+QQZ)+2€uQ+bQ3+(Qoz-92)Q=0 (4.1)
where Q=Q,-I~£2§2A sinwt.

Before using the method of multiple scales to obtain an approximate solution about the

buckled position, we first transfer the coordinate of the equation of motion to the buckled

position. The buckled position, 5, is obtained by setting 9 = (2' and all time derivatives in
equation (4.1) equal to zero. Hence we have
(4.2)
Let the new coordinate be q, such that
Q = S '1' q (4.3)

Substituting equations (4.2) and (4.3) into equation (4.1), we obtain

4o

q+ar(2qq+qz)+az(q2q+qq2)+281151+°302q+71 c12'1’72 Q3

=ez(g,+g2q)sin(ut +154 (g3-1-g4 q)(sin (1)102 (4 4)

where (no, 7i, ori and gi are defined in Appendix C.

It is important to note that the equation (4.4) governs the beam's motion about the
buckled position. Comparing equation (4.1) with equation (4.4) we can observe that the
coordinate transformation has resulted in the addition of quadratic nonlinearities and a direct

forcing term.
4.3 Method of Solution

The method of multiple scales will be employed to obtain an approximate solution

to equation (4.4). We express the solution in the form

(1(T38) = 3 ‘11 (T0911, T2) '1' 82‘12 (T0,T1,T2) '1’ £3‘13 (T0,T1,T2)+° ' ° (45)

Substituting equations (4.5), (3.6) and (3.7) into equation (4.4) and equating coefficients
of like powers of 8 yields

Do2 ‘11 + (002% = 0 (4.6)

Do2 ‘12 '1' (002% = '2 D0 Dr ‘11“ 2 a! (111302 ‘11- “1(DOQr )2
'2 11l Do ‘11 - 71 Q12 ‘1' 8131“ (”To (4,7)

41

Do2 Q3 ‘1' ‘002Q3 = ‘2 D0 D1% ‘ 2 ‘11Q1Do2 Q2 ’ ‘12 Q1(DoQ1)2
- ( Dr2 ‘1' 2 130132 ) Q1 ‘ 4 arQrDoD1Q1" 0‘2 Q12 Do2 Q1

'2 ‘11 Q2 Donr’2 'Y1Q1Q2-3z 72 Q13
- 2 1»1 D1 Q1 '1' 82 Q1 sin ‘DTo (4.8)

The solution of equation (4.6) can be expressed in the complex form

ql = A( T1,T2 ) Cl (DOTO '1' K( 1‘sz ) c-l (”OTC (4.9)

where A is the complex conjugate of A.

Then equation (4.7) becomes

Dozqz-i-(uozqz =-2 i (no ( DlA+uA ) emoTO

+(3011moz-71 )r‘kzeiz‘°°T°—(onlcuozwy1 )AX+g—‘,e"°T°+cc
21 (4.10)

where cc stands for the complex conjugate of the preceding terms.

Depending on the inter-relationships between a) and (00, various conditions for

elimination of secular terms may be extracted from equations (4.8) and (4.10). In the next

sections we will consider two cases: (a) (0 == 2 (00 and (b) (0 == (00.

42

4.4 The Case of m— 2010

4.4.1 Steady-State Solutions

Eliminating secular producing terms from equation (4.10) yields

D1A=— 11A. (4.11)

Consequently, the solution of equation (4.7) becomes

 

2 2
01m - — 3am - ° T
q2=[ 1 02 YI]AA_[ 1302 YI]A26120)0 0
m0 030

__ igl HDTO
2(m02_m2)6 '1' CC
(4.12)

To investigate the resonance to ~ 2 (00, we introduce the detuning parameter 6 defined

according to

m=2mo+ezo (4.13)

Substituting equations (4.9), (4.12) and (4.13) into (4.8) and eliminating the terms that
lead to secular terms yields

_ 2
-2imoD2A-DfA—2uD,A+i(-6a,p,m02_§22+2y,ps)A Ce oTo

+[2a2w02-372-6a1F2C002+271( Fz‘F1)+2a1‘°ozF1]A2-K=O (4.14)

where

43

_ 2 (“13102-71 )
F1 (002

2
= 3 “1‘00 ‘71

F
2 3 (002

 

F=2§L
3 60),}.

At this stage we have to employ the technique of reconstitution to continue with the

analysis. This involves recalling that

A=1AL=DOA+e D1A+82D2A

d ‘t (4.15)

 

and noting that DOA = 0 and an expression for DlA has already been found (see equation
(4.11)). From equations (4.15) and (4.11), we thus have

D2A=A+eztrA

8 (4.16)

Substituting equations (4.11) and (4.16) into equation (4.14) yields

. ._ 2
-2im0(A + euA) + 2211 (- 6 01,13, (002 - %1 + 2 71E, ) A e 8 0 TO + 112A

+ (2 a2 (1102- 3 72 -6 01le (0024-2 71(F2 -Fl )+2 (111(1102F)A2A}=(:4 17)

44

To solve equation (4.17), it is convenient to express A in the polar form given in equation
(3.14). Substituting equation (3.14) into equation (4.17) and separating real and imaginary
parts yields

é=-€ua-82[3a1F3moa+§12---YJ-§—a]cosy
41110 (no

 

 

(4.18)
2
a7=ez[u-+oa+2(3a1F3cooa+gJa-71F3a)siny]
010 41:30 (no
2 3 1 1 3Y2 NFL-F1) 3
-e ——orFo)+—aFco+—or0)- +
[(21202110220‘“Do 2000 a
(4.19)
where7=8201-2|3. (4.20)

Periodic motions of the rotating beam correspond to the constant solutions of equations
(4.18) and (4.19), which in turn corresponds to a = "y = 0. Hence we have a trivial steady-

state solution, a = O, or

c...) = -4.
e G (4.21)
2
siny=-1—[- o-E—+Haz]
2 G “’0 (4.22)
where
82 _ 71F:

 

G=3aFm+
‘3 ° 4a)0 coo

45

3 1 1 37 “HF-F)
H=-orF(o--aFm-—a(o+ 2+11 2
212 0 2110 2 0 4030 2‘00

2

Squaring and adding equations (4.21) and (4.22), we obtain
2 2
a2=i[o+"_12"02_117]
H (00 8

Substituting equations (4.9), (4.12) and (4.20) into equation (4.5), the steady-state

(4.23)

 

 

solutionhastheform
q=eacos(mt_7)
2
+82 a2 “1m02'71_3°‘1°’°2"71cosmic-7) +—§1fi-sinwt +O(e3)
20302 60102 (002—0)

(4.24)

Note that the last term in the above approximate steady-state solution comes from the direct

forcing terms 22 g1 sin an and does not depend on "a" .

4.4.2 Stability of the Steady-State Solutions

To investigate the stability of the trivial steady-state solutions, we will first convert
equation (4.17) from polar form into an autonomous Cartesian form. To this end we

introduce the complex coefficient, B, such that

iezot)

A = B cx
1" 2 (4.25)

 

where B=B,+iB,.

Substituting equation (4.25) into equation (4.17) and separating real and imaginary parts,
we obtain

2
B,=-euB,+r-:2[ —1’l—Bi+E-—G—B,-—PLBi Bf)
2100 2 (00 (00 (4.26)
2
2010 2 mo (00 (4.27)

To determine the stability of the steady-state solutions, we can add an infinitesimal
perturbation to the steady-state solution and check to see if this perturbation grows or
decays.
In vector form, this may be expressed as

9=$+y «am

where q) = ( 13,, Bi )7, e is the steady-state solution, and M << 1.

Substituting equation (4.28) into equations (4.26) and (4.27) and linearizing the resulting

equations, one obtains

:7 = M Y (4.29)

where

 

47

 

 

 

 

-e Ll _ 82 G 82 112
M = (no 2 (no
_ £2 112 —e 11 + 82 G
_ 2 (00 o, . (4.30)

The stability of the trivial steady-state solution is governed by the stability of y in equation
(4.29) and thus by the eigenvalues of M . The steady-state solution is stable, if and only if

the real parts of all eigenvalues are less than zero. For the trivial solution this result in an

unstable solution for a, < o < 02, where

2 I 2
“’0 E (4.31)
2 i 2
oz=—-u—+2 Gz-u—z
(no 8 . (4.32)

To investigate the stability of the non-trivial steady-state solutions we can follow the
same procedure. However, it is not necessary to first convert equation (4.17) into
Cartesian forms: the stability analysis may be completed directly on equations (4.18) and
(4.19).

InthiscasewefindtheJacobianmatrixMtobe

M- [4511- £26 cos? £265 sin'Y]

-282H 5 2226 cos? (4.33)

where the overbar denotes a steady-state value.

The results of this stability analysis are shown in subsequent figures of this chapter.

48

4.4.3 Numerical Results of Approximate Solutions

In this section we present representative solutions of the equations derived in
Section 4.4.1. Values for the various parameters are based on a physical system that has
been used in laboratory tests. Results from the experimental investigation will be reported
in Chapter 7. The parameter values used, unless otherwise stated, are:

as= 3.7, 1:212A =0.05, 2 e u = 0.00422, (20: 3.516 and e = 0.1.

Figure 4.1 shows a typical frequency response curve for a non-zero mean spin rate
of Q, = 3.7. Note that in this and subsequent figures, we plot ea which is the magnitude
of the first order term in the approximation of q (see equation (4.24)). Clearly, we can
observe that the overall nonlinearity of the system is of a softening type, whereas in pre-
buckling case (i.e., (2‘ < $20) the overall nonlinearity was of a hardening type. As before,
a stable solution is denoted by a solid line whereas a unstable solution is denoted by a
chained line. It can be shown that the upper branch is always stable, whereas the lower

branch is always unstable.

Figures 4.2a-b present results showing the variation of the steady-state amplitude as
a function of 820” Figure 4.2a shows a case for which there is only one stable steady-
state solution, i.e., o = 0.0, whereas Figure 4.2b shows a case where multiple steady-state
solutions are possible, i.e., o = -28.0. Figure 4.3 shows the effect of the mean spin rate,
(2., on the steady-state solutions, for three different values of a. For clarity, only the
stable steady-state solutions are shown. Because of the post-buckled condition, starting
points of all the curves in Figure 4.3 are slightly above the point co = 3.516 (since the
equation are not valid for Q, < (20). It is interesting to note that for o = 0.0 and 10, the

steady-state amplitude decreases as (2‘ increases. The overall effect on the response may

 

ea

49

 

 

0.11
o o o o numrieel simulation
0.11-
stable
0.09 t" -—--- unetoblc

 

 

 

 

Figure 4.1 Variation of the Steady-State Amplitude, a, with 0.

 

50

 

O-0.0

 

eteble

0.05 “‘ —""" unetable

0.06

ea

0.04

0.02

 

 

 

C'QA

 

  
    

 

stable

—"— unetable

  

 

 

 

:3 ‘\~
‘0 ‘\
.\‘
‘\
o 04 — x
\\
\\
.\\
\
0.02 - \
\\
o 1 1 1 \..._L-_._-._._-._.l
o o 02 0.04 o 06 0 08 o 1
I
5 QA

Figure 4.2b Variation of the Steady-State Amplitude, a, with 1:252A for o = -28.

51

 

0.08 ‘-

811

0.06 "

 

0.04

0.02

 

 

 

 

Figure 4.3 Variation of the Steady-State Amplitude, a, with (2‘ for a = ~28, 0, 10.

Anulitudes of the stable steady-state solutions

 

 

 

Figure 4.4a ea, 0, £25 - Plot for Stable Steady-State Solutions, (3-D representations).

52

be better visualized by considering the response surface in the (a, o, (2‘) space. A three-

dimensional representations of this for both stable and unstable solutions are shown in

Figure 4.4a and 4.4b, respectively, for fixed values of 11 and 829A.

In Figure 4.4, to move along a constant 0 line, we need to continually adjust to
(since (00 is a function of Q8 and 0) = 2mo+ 820’). However, in practice we would be
more likely to hold a) fixed and not try to constrain 0. Hence, Figure 4.5 has been plotted
to show contours of constant 0) values. Again, for clarity, only stable solutions have been
shown.

Figure 4.6 represents a bifurcation diagram which dictates which type of steady-
state solution exists in the (u- ezflA parameter space. This space is divided into three
regions by the curves: (3:112 [2(002im and 11 = e G. Note that the
boundaries of these regions are dependent on the coefficients of the quadratic nonlinear
terms and independent of the coefficients of the cubic nonlinear terms. In region I, only the
stable trivial solution exists. In region II, two solutions exist: the unstable trivial solution
and a stable non-trivial one. In region 111, there are three solutions: the stable trivial one

and two non-trivial solutions, one stable, the other unstable. Figure 4.7 shows the effect

of Q, on these regions. We observe that the region 11 decreases as Q, increases.

In order to check the results obtained by using the method of multiple scales,
equation (4.1) is numerically integrated. The results have been plotted as "numerical
simulation" on Figure 4.1 and represent one half of the peak to peak value of Q, obtained
by the numerical integration. The comparison is qualitatively good in all the cases tested.
Figures 4.8a-b present time traces of the steady-state soluu'on Q of the numerical integration
results for o = -28. Figure 4.8a results from one set of initial conditions from which the

solution is attracted to the trivial solution (i.e., a = 0). Note that even when a = 0, we

0.14

 

53

Mulituaee of the unstable steady-state solutions

 

 

 

   

- Mplituaes ol the stable steady-state solutions

 

 

Figure 4.5 ea, (0, {25 - Plot for Stable Steady-State Solutions, (3-D representations).

54

 

5' (7A

 

u-EG

 

#—
,-

 

£2 11,

 

 

 

Figure 4.7 Effect of Q, on the Bifurcation Diagram in the 03-82911 Parameter Space.

55

 

 

 

 

0.3
o.25~—
0.2-—
0 WW
'2
g 0.15 —
O
Q
0‘!
O
0.1-—
0.051—
0 1 1 1
11393 11398 11403 11408 11413
Time

Figure 4.8a Time Trace of the Steady-State Solution Q for 0' = -28;
Q(0) = 0. 2, Q(0) = 0.0.

 

 

 

 

0.3
0.25-—
0.21—
c:
E
5'5
Q 0.15
C
Q
0‘)
O
0.1-
0.05-—
0 A 1 1
88353 88358 88383 88388 88373

Figure 4.8b Time Trace of the Steady-State Solution Q for o = -28;
Q(0) = 0.1, Q(0) = 0.0 .'

56

would expect to see a small component of Q which arises from the direct forcing term
12’ gl sin (or (see equation (4.4)). The frequency of the response depicted in Figure 4.8a is
that of the excitation term, (0. Figure 4.8b shows the results for the same parameter values
as in Figure 4.8a, but using a different set of initial conditions. The solution is now

attracted toward the non-trivial solution set, corresponding to a at O . The main frequency

of thrs solutron rs -2- (1.e., subharmomc motion), which is cons1stentw1th the method of

multiple scales prediction of equation (4.24). The corresponding phase portraits of these

two time traces are shown in Figtu'e 4.9.

It is interesting to note that for yet another set of initial conditions, a third type of
steady-state solution can be found. In contrast to the preceding two, which were associated
with oscillation about the buckled position, we may also observe a large orbit which
encompasses both buckled positions (i.e., Q=:l:s). The period of this oscillation is twice
that of the forcing period (i.e., period 2 motion). The time trace of the steady-state solution
Q of this case is shown in Figure 4.10. If we decrease the a value further (e.g., down to -
30), numerical simulations can only find this type of motion, i.e., non-uivial oscillations
centered around a buckled position are no longer possible (but, of course, the small,
directly forced response can be found). Numerical simulations results for different 0
values are shown in Figure 4.11. From Figure 4.11 we observe that the snap-through
motion of the beam is of hardening type.

We also present the numerical simulations of equation (4.1) in terms of non-rotating
(i.e., laboratory) coordinates. Figure 4.12a shows a trajectory of the beam as it vibrates
about one of the buckled positions. The trace represents approximately 67 revolutions of
the beam. Figure 4.12b shows a trajectory associated with a motion that encompasses both
buckled positions.

57

 

(i) (ii)

Velocity Q
c:
1

 

 

I l l I l
0 0.05 0.1 0.15 0.2 0.25 0.3

 

Displacement 0

Figure 4.9 Phase Portraits of the Steady-State Solution Q for o = -28;
(i) Q(0) = 0.2, Q(0) = 0.0, (ii) Q(0) = o. 1, Q(0) = 0.0.

 

58

 

Displacanent 0

 

 

1 1 1
'0‘1‘1393 11398 11403 11408 11413

 

Tine

Figure 4.10 Time Trace of the Steady-State Solution Q for o = -28;
Q(0) = 0. 05, (2(0) = 0.0.

 

 

 

 

 

 

Velocity 0

 

 

 

 

 

 

 

 

 

Disploanent 0

Figure 4.11 Phase Portraits of the Steady-State Solution Q for o = -30, -35, -40, -45, -50.

 

 

       
       
  
 

           
     

 

 

59

0.3

0.2 —
S
'3 //
g 0.1 - W” t i
'5 // / / M
,.
c ) 1
.- o __ | I. ‘
E ,1 ~ \ \\\\
g \\ \‘_\\\ \ \ k
:6 —o.1 - \\\\\\§{\\\\\\§\\\\\\‘ 3 _ /
a \\\\\\ §§tszzééy

-0.3 A 1 l 1 1

-0 3 -0.2 .o_1 0 0 1 o 2 o 3

 

Displacanent in X direction

Figure 4.12a Trajectory of the Steady—State Solution Q in Non-rotating Coordinates
for o = -28; Q(0) = 0.1, Q(0) = 0.0.

 

 

 

 

 

 

 

 

0.4
0.3 h
g 0.2 -
‘5
0
.: o.1~
“O I
>-
C
e- o r—-
E
u -o.1-
O
Q
in
'o -o.2 *-
-0.3 —
_0 ‘ l A L l l l l
'-o.4 —o.3 -0.2 -0.1 o 0.1 0.2 0.3 0.4

 

Displocanent in X direction

Figure 4.12b Trajectory of the Steady-State Solution Q in Non-rotating Coordinates
for o = -28; Q(0) = 0. 05, (2(0) = 0.0.

 

4.5 The Case of (oi-st!)0

In this section the frequency to is taken to be close to (no and the proximity of co to

(00 is quantified by the external detuning parameter, 0, which will be defined by

m=mo+ezo (434)
The secular terms and small-divisor terms are now eliminated from equation (4.10) if

DlA = - 11 A - —1—g ei‘m'mm’
4‘00 (4.35)

Then, the solution of equation (4.7) becomes

2_ _ 2_ ,
q2=[alm92 71JAA—[3a1m0 2 7,]Azctzmo'ro + cc

“’0 3‘90 (4.36)

Substituting equations (4.9), (4.36) and (4.34) into equation (4.8) and eliminating secular

producing terms yields
-21(DODzA-Dle-ZtlDlA-i-[2112C002-3‘Yz—GGIF20302

+ 2 y,( Fz-F, )+ 2 alm02F1]A2 K = 0 (4.37)

As in Section 4.4.1, to proceed we use the technique of reconstitution. Substituting

equations (4.35) and (4.16) into equation (4.37) yields

61

. 2
-2i0130(A-1-1811A+t:F,,e15 0T0)

+ezl-u2A-uF4C€26T0+F5Ale=o (4.38)

where
4 (no

F5=(-2a2w02+372+6alF2m02-271(F2-F1)-2(110)02F1.

Substituting equation (3.14), the polar form of A, into equation (4.38) and separating the

results into real and imaginary parts, yields

2

 

é=-Eua-28F4COS‘Y+-e—u§siny
“’0 (4.39)
2 2 2 2 3
a’y=r~:zoa+2eF,,siny-1-a1’121+8 ”F4005? fl
”’0 ‘°° 80’0 (4.40)
where 7 = 82 o r-B. (4.41)

Again, it is possible to simplify these equations should only the steady-state responses be
sought. For such a case, equations (4.39) and (4.40) reduced

2
-28F4cosy+£—1‘-l-l—:‘-siny=eua
“’0 (4.42)

2 2 3 2 2
e F e F a e a
_ll ‘cosy=-ezoa+ 5 - 11

(Do 8 ‘00 2 ‘90 (4.43)

 

28F4Sin’Y-l-

62

Squaring and adding (4.42) and (4.43) gives

c3 a6+cz a4+cl a2+co=0 (4.44)
where

C3=F5

c2='8F5(112'*"°0‘5)

2 2
c,=—16(4tooot12+4030202+u‘+411820)")

 

4 12,2

2
(0
co=-64(—?—9-+112F42).

Given a set of values for the parameters, the steady-state amplitude "a" may be found
directly from equation (4.44). Since this equation is a 6th order polynomial in "a" with
only even powers present, it may have up to three non-negative real roots. Also, because

On it 0, no trivial solutions exist. Reconstructing the approximate solution to q, we find

q=eacos[tut-'y]

01012-7 301012—7
+132 [a2[ ‘zfooz 1' 151302 ‘60s12(m—1)]]]+0(23)

 

 

(4.45)

The stability of this case may be investigated by adopting the same procedure as
was used in Section 4.4.2. In this instance, the Jacobian matrix M obtained from

equations (4.39) and (4.40) is

63

 

 

' 2 —2'
-eu _825(6+_11__F5a)
M: 2 2 2 2030 8010
e 11 3F55'
— o+ - —e
_5( 2010 8000) ll _

 

 

where the overbar denotes the steady-state value.

The results of this investigation indicate that the steady-state motions are unstable when

 

2 2 -2 2-4
£2[(0+—E—)2-(O’+ ll )(FSa )+3F2 a2 ]+u2<0
2 000 2 (00 2 (00 64 000 (4.46)

and are otherwise stable.

A typical frequency response curve is shown in Figure 4.13. The parameter values
used are:

12,: 3.7.8212A =0.02, 2 e 11 = 0.1, oo= 3.516 and e = 0.1.

The nonlinear inertia terms bend the frequency response curve to the left (i.e., softening
type). Comparing the approximate solutions with the numerical simulations shows
qualitative agreement. Figure 4.13 shows, over certain regions, the response curves
become multi-valued. This gives rise to the well documented jump phenomenon. This
means the steady-state amplitudes can undergo spontaneous jumps due to an infinitesimal
change in 0'. Figures 4.14a-b show time traces of the steady-state solution of Q obtained
from numerical integration of equation (4.1) for a = ~35. Two different sets of initial
conditions are used. The corresponding phase portraits of these two time traces are shown
in Figure 4.15. These figures clearly demonstrate that the frequency response curve is

multi-valued. Once more, it is interesting to note that for yet another set of initial

 

a o a o nunericol sinulation

 

stable

1.6 1" \\ —"- unstable

 

 

 

0 11 1_341 .1 l L l l
—100 -80 ~60 -4O —20 O 20 4O

 

100

O)
O
00 N
0

Figure 4.13 Variation of the Steady-State Amplitude, a, with o.

 

 

 

 

0

—0. 1 L
(J
E
D
O
O.
m
o /

-0. 2 l—

-0. 1 1 1 3 1

2%880 26690 26900 26910 26920 269.30 26940
T ime

Figure 4.14a Time Trace of the Steady-State Solution Q for o = -35
Q(0) = 005. (2(0) = 0.0.

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0
-O 1 —
O
E
o
o
o.
'1
o
-o.2 ‘—
“0.3 i 1 l l 1
26880 26890 26900 26910 26920 26930 26940
Time

Figure 4.14b Time Trace of the Steady-State Solution Q for 0’ = -35;
Q(0) = 0.1, Q(0) = 0.0.

 

0.05 —

°‘ (ii) (i) .

Velocity 0

 

l l
'-0.3 -0.2 -o.1

 

 

Displacement 0

Figure 4.15 Phase Portraits of the Steady-State Solution Q for o = -35;
(i) Q(0) = 0. 05, (2(0) = 0.0, (ii) Q(0) = 0.1, Q(0) = 0.0.

66

conditions, a subharmonic motion which encompasses both buckled positions can be
found. The corresponding time trace and phase portrait of such a case are shown in Figure

4.16a and 4.16b, respectively.

4.6 Simulation and Observations of Chaotic Motions

In this section we will use an analytical method and numerical simulations to
explore the chaotic motions of a rotating buckled beam. A chaotic motion is a non-periodic
but bounded motion with a broadband spectrum and a high degree of sensitive to initial
condition. As a system parameter is varied, a periodic motion may undergo a series of
bifurcations. This series of bifurcations, if it continues, will lead to a chaotic motion. In
the simulations, we will concentrate on two regions: (a) primary resonance 0) ~ (00 and (b)
subharmonic resonance to ~ 2 (no. The parameter values used in the simulations, unless
otherwise stated, are: (23 = 3.7, 2 e p. = 0.1. Before using the digital computer to simulate
equation (4.1), we apply the Melnikov's method to provide a necessary condition of

chaotic motions.

4.6.1 Melnikov's method

Melnikov's method provides a measure of the separation between the stable and

unstable manifolds along the unperturbed homoclinic orbit (Guckenheimer and Holmes
[1987], Wiggins [1988]). It involves the computation of the Melnikov function Mao); if

Mao) has simple zeros the stable and unstable manifolds may intersect an infinite number

of times and chaotic motions may exist.

Following the procedure of this method, equation (4.1) is written in the form

67

 

 

 

 

0.3
0.2—
0.1—
o
E
u 0"
O
Q
I)
O
-0.1
-0.2
-0.3 ‘ .' ‘ ' '
26880 26890 26900 26910 25920 26930 26940

Time

Figure 4.16a Time Trace of the Steady-State Solution Q for o = -35;
Q(0) = 0. 25. (2(0) = 0.0.

 

0.05%-

Velocity Q
o
l

 

 

! =’ l i l
'-0.3 -0.2 -0.1 0 0.1 0.2 0.3

 

Displacement 0

Figure 4.16b Phase Portrait of the Steady-State Solution Q for o = -35;
Q(0) = 0.25, Q(0) = 0.0.

i=f(2)+eg(2.t)+82h(2.t). z=[9]=[x‘] 9- R2 (4.47)

where

 

f1(x,,x2) 2 $2 2 2
{(z)=[f ]= _dx,x2 +bx1+(flo —£2L):51
20‘1””) 1+ d x,2

O
g(z.t) = [8109. 19,10] = _ 2n x2 — 2 9.9;. sin(t01) xl
82(7‘1’7‘2") 1+ (1 x12

‘ 0
h(z,‘t) = [hl(xlax29t)] =[ (RA Sin m)2 x1]

h2(Xl, X2,1) 1 + d xlz

The Melnikov function is given by the formula (Guckenheimer and Holmes [1987], page
187)

Mao) = I;[f1(x10’ x20) 82 (x10, xat» T + To) " f2 (X10, x20) 81(x10’ x20, 7 + 10)] d‘
(4.48)

where (X10. X20) is the unperturbed homoclinic orbit.

For the system presently being studied, it is impossible to find an analytical solution
of the unperturbed homoclinic orbit. Therefore, we use the Runge-Kutta method to find an
approximation to the unperturbed homoclinic orbit by setting a = 0 in equation (4.47) with

an initial condition very near to (0, 0).

Equation (4.48) can be written as

Mao) = —2u 11+ 2 Q, 9,, [ 12(0)) cos mg + 13(0)) sin mo 1 (4,49)

69

where
2

“I" 1+ d x10 (4.50)

 

~ sinan x x
12(44):]; ( 1+d)x1::)2j£dT (451)

 

0050’: X X
1300):]; ( 1+c1)x1:,g Edi (4 52)

Then we numerically integrate the equations (4.50) and (4.51) by using Simpson's rule. It
is found that II > 0 and 13(0)) = 0 (since the integrand in equation (4.52) is an odd
function). Obviously, when 2 Q, 9A12(c0)> 2 [111, M(to) has simple zeros (i.e., the
stable and unstable manifolds intersect transversely). The ratio of I, and 12(0)) is plotted
as a function of 0) in Figure 4.17 for a value of Q, = 3.70. The regions above the curve in
Figure 4.17 are the parameter values where a homoclinic tangle exists. The influence of the
mean spin rates, (2,, on this curve can be seen in Figure 4.18. We observe that the

possible chaotic motion regions and the lower limit of the curves increase as (28 increases.

To add confidence to the results predicted by the Melnikov analysis, we numerical
obtain a number of the stable, W3, and unstable,W“, manifolds associated with the saddle
point of equation (4.47 ). To generate portion of W“, points on a small segment along the
unstable eigendirection centered at the saddle point are mapped forward in time in the
Poincare section. The W8 can be generated in a similar way, with time running backward
For the fixed parameter values 26}; = 0.1 and 0) = 1.54 ~ (00, Figure 4.19a-c show the
stable and the unstable manifolds of the saddle point (0,0) for three different values of
93A. As shown in Figure 4.1%, w8 just intersects w“ tangentially at 3th = 1.036, in

comparison with a theoretical value of 1.073. Figure 4.203-c shows the result for another

70

 

II/ I: (u)
I

4'—
05: 3.70

 

 

 

 

 

 

 

 

 

 

5 l
Qs= /
45% 3.52 355 35 3.0 30 40
‘l—
3.5
‘ Chaotic
/\ 3 2' notion
3
4' 25
\
N"
2..
'L ./
Periodic notion
0‘ l 1 l l I l
O 05 1 15 2 2.5 J 35 4
(0

Figure 4.18 Variation of the Ratio 11:12(c0) with (0 for as = 3.52, 3.53, 3.55,
3.60, 3.70, 4.0.

71

 

0.3 .

0.2 —' .

Velocity 0
O
l

 

 

 

 

6t»
(.1
l
o
N
s5
o
o
o
n
o
u

Disploca‘nent 0

Figure 4.19a Stable Manifold Ws and Unstable Manifold W“; 213-4 = 0.74.

 

0.3

 

Velocity Q
o
l

 

 

 

_0.J - A 1 1 l . I: ‘.
-O.3 -0.2 -0.1 0 0.1 0.2 0.3

 

Displacement 0

Figure 4.1% Stable Manifold ws and Unstable Manifold w“; —'—;’-&= 1.0.36

72

 

0.3

0'2 h. ‘- .._..-P‘”"...."0.I:.. . 0 I. . - . . . .'I 2...,

Velocilyé
O
l

 

 

 

 

-0—.3 ' ' -
-0.3 -0.2 ~O.l 0 0.1 0.2 0.5

Displacement 0

Figure 4.19c Stable Manifold ws and Unstable Manifold w“; 91% = 1.11.

 

Velocity 0

 

203 -02 -0.1 0 01 02 0.3

Displocmnt 0

 

 

 

Figure 4.20a Stable Manifold WS and Unstable Manifold Wu; 2'54 = 1.48.

73

 

 

   

    

ALI... ...
o In. . s .K.
- \|I rural...
. . .I. 1.. . .5...- ut.... . .
e I 0'0. U h nV- I .0
. . a . .. . . ..r... ..r:
e . a. n l o .- ...u I.“ so.
... . r .. ... .....u.
n a - up! I u are
..~\\ flasher}... .... 1. .... .
. ................... an... . .
. o .I. a .0.- ".6 red
... . .. vowa
u O. ..ne-

-0.

-0.2

 

 

 

o :_oo.o>

-0.2 —',.

-0..3

~03 "

Displocanent 0

2.22.

 

42,0,
ll

 

0.3

0.2

-0.

-O.2

 

 

Figure 4.20b Stable Manifold ws and Unstable Manifold wu

 

-0.3

-0.3 '

O >._uo_o>

Dieploeanent 0

$53 = 2.96.

fold wS and Unstable Manifold w“;

ani

Figure 4.20c Stable M

74

set of the fixed parameter values 2811 = 0.1 and 0) = 2.79 ... 2 (00. Note that the first

tangency (see Figure 4.20b) appears to occur about 3'34 = 2.22, compared with the

predicted value of 2.54. We conclude that the results obtained by Melnikov's method are
good. The slight disagreement between the analytical method and numerical Simulations
are possibly because the Melnikov's method is a first-order approximation.

In the next two sub-sections, we will vary the value of 8 9A to see what happens to
the solution Q (t) as one crosses the homoclinic bifurcation curve shown in Figure 4.17.
To this end equation (4.1) is simulated on a digital computer. After transient motions
decayed, the steady-State solution is recorded. The initial conditions used for the
simulation, unless otherwise stated, are Q (t) = 0.1 and Q(t) = 0.0. The time step size

equal to %, where T = 2;! is the forcing period. Note that, for small values of 8 9A , the

rotating, buckled beam will vibrate about one of the two buckled positions and the initial
conditions determine which buckled position the beam will vibrate.

4.6.2 The Primary Resonance: cos-000

Equation (4.1) was simulated on a digital computer for fixed values of (2s = 3.70,
2811 = 0.1 and 0) = 1.54. We present here only the effect of varying the amplitude,
22 9A ,of the small sinusoidal fluctuation of the rotational Speed. The results are shown in

Figures 4.21-23 and summarized in Table 4.1.

For values of 89A between 0.045 and 0.4605, we see a period doubling
sequence. This ends in an almost periodic motion ate 0A = 0.04657 . Between 0.0467

and 0.4685 there exists an island of periodic motion motions (12T and 6T) before one
again encounter an almost periodic motion at a (2A = 0.0469.

75

 

0.05 —

Velocity 0

l

-0.05

 

 

-0 1 l l l 1
'325 —0.2 -0.15 -0.1 -0.05 0

 

Dieplocanent 0

Figure 4.21a Phase Portrait of Period 1 Motion; e252,: 0.035.

 

0.15

0.05'-

Velocity 0

 

 

l l l l
-O.15 -0.1 -0.05 O

 

1.6
a

Displocunent 0

Figure 4.21b phase Portrait of Period 3 Motion; e295 0.045.

 

76

 

0.15

0.05 —

Velocity 0

-0.05 —

 

 

 

 

Displacement 0

Figure 4.21c Phase Portrait of Period 6 Motion; £29A= 0.046.

 

0.05 —

Velocity 6

-0.05 L

 

 

 

 

-0.15 1
0 .

Disploccnent 0

Figure 4.21d Phase Portrait of Period 12 Motion; ezofi 0.04605.

 

77

 

 

 

 

 

0.15

p
5
O
O

0 332.5

-0.05 —

0.25

0.2

spice-rent G

Di

 

 

 

F

0.25

0.2

0.05

 

 

Figure 4.21c Phase Portrait of Almost Periodic Motion; ezofi 0.04657.

_

5
O
O

o :_oo_o>

-0.05 '-

1

-0.

5

1

-O.

eplocmnt 0

Di

Figure 4.21f Phase Portrait of Period 12 Motion; 69,: 0.0467.

78

 

0.05 —

Velocity 0

 

 

 

 

-0.25 -0.2 -0.15 —0.1 -0.05 0

Displacement 0

Figure 4.21g Phase Portrait of Period 6 Motion; 6205 0.0468.

 

0.15

0.05’—

Velocity 0

-0.05r-

 

 

 

 

-0.15
0 0.05 0.1 0.15 0.2 0.25

Displacennnt 0

Figure 4.21b Phase Portrait of Period 12 Motion; ezofi 0.04685.

79

 

0.05"

Velocity 0

-0.05'-

 

 

l l l 4
0.05 0.1 0.15 0.2 0.25

 

 

Displacement 0

Figure 4.21i Phase Portrait of Almost Periodic Motion; 3294: 0.0469.

 

 

 

 

0.2
0.1“-
.0
>
._ 01—
U
0
o
>
-0.1 ‘
_o_2 l 1 1 l 1
-0 J -O 2 -0.1 O 0 1 0 2 0 3

Displacement 0

Figure 4.22a Phase Portrait of Period 2 Motion 820,; 0.047.

 

80

 

0.3

0.2 _

Velocity é

 

 

 

Displacement 0

Figure 4.22b Phase Portrait of Period 6 Motion; e251,: 0.0915.

 

0.3

0.2'-

('1

Velacuty
O
I

 

 

Displacement 0

Figure 4.22c Phase Portrait of Period 3 Motion; e252,: 0.115.

Velocity Q

81

 

 

 

 

 

 

 

Displacenent 0

Figure 4.23 Phase Portrait of Chaotic Motion; £20A= 0.12.

82

Table 4.1 Summary of Bifurcated and Chaotic Motions for to = 1.54.

 

 

 

 

 

£9 . . Initial Conditions
A Figure Type of Motion (Q(0), (2(0))
0.035 Figure 4.21a 1T (0.10, 0.0)
0.045 Figure 4.21b 3T (0.10, 0.0)
0.046 Figure 4.21c 6T (0.10, 0.0)
0.04605 Figure 4.21d 12T (0.10, 0.0)
0.04657 Figure 4.21c almost periodic (0.10, 0.0)
0.0467 Figure 4.21f 12T (0.05, 0.0)
0.0468 Figure 4.21 g 61‘ (0.10, 0.0)
0.04685 Figure 4.21b 12T (0.15, 0.0)
0.0469 Figure 4.21i almost periodic (0.15, 0.0)
0.047 Figure 4.22s 21‘ (0.10, 0.0)
0.0915 Figure 4.22b 61‘ (0.10, 0.0)
0.115 Figure 4.22c 3T (0.10, 0.0)
0.12 Figure 4.23 chaotic (0.10, 0.0)

 

Increasing the value of 6 0A further, as one might anticipate, the beam will not
vibrate about only one of the buckled positions but will snap-through and adopt an orbit
enclosing the two buckled positions. A period 2T snap-through motion is easily seen by
the time c QA= 0.047 in Figure 4.22a. Once again, as we continuously increase 8 52A
beyond some critical value, a period demultiplying bifln'cation occurs with period 6T and
period 3T, as Shown in Figures 4.22b and 4.22c. In these two figures we observe that the
beam vibrates in a complex manner, first vibrates about one buckled position and then
snap-through between the two buckled positions. It should be stressed that this is a
periodic motion. Finally, a chaotic motion occurs with a 9,, = 0.12, as shown in Figure

4.23. The simulations indicate that as the value of 8 (IA is varied, period doubling

bifurcations, demultiplying bifurcations and chaotic motions occur.

 

83

Another way of showing the results just presented is to use a Poincare map. If the
response is periodic, then its Poincare map will only show a finite number of points.
However, when the motion becomes chaotic, its Poincare map will contain an infinite
number of points and reveal a fractal pattern (i.e., strange attractor). It is also known that
the Poincare map of an almost periodic motions is represented by line segments. Figure
4.24h-i depict the Poincare maps corresponding to a selection of the response shown in

Figures 4.21, 4.22 and 4.23.

4.6.3 The Subharmonic Resonance: (0"2010

In this sub-section we consider the case of subharmonic resonance (i.e., (0 = 2 (00) .

As in Sub-section 4.6.2, as we continuously change the parameter 8 9A , a period doubling

bifurcation occurs. For fixed values of (2s = 3.70, 261.1 = 0.1 and 0) = 2.79, the results

from the numerical Simulation are shown in Figures 425-26 and summarized in Table 4.2.

Table 4.2 Summary of Bifurcated and Chaotic Motions for 0) = 2.79.

 

      

, Initial Conditions

T of Motion .
”'3 (Q(0).Q(0))

   

 

 

 

 

 

0.07 Figure 4.25a 2r (0.10, 0.0)
0.08 Figure 4.25b 41‘ (0.10, 0.0)
0.0832 Figure 4.25c 81* (0.10, 0.0)
0.0833 Figure 4.25d 16T (0.10, 0.0)
0.0834 Figure 4.25c 321‘ (0.10, 0.0)
0.0835 Figure 4.26 chaotic (0.10, 0.0)

 

Figures 4.25a-e Show the phase portraits of the rotating, buckled beam with

different values of a 9A . These figures indicate that, as the value of 8 52A is varied, from

Velocity Q

84

 

 

 

 

 

 

 

 

 

 

0.1
-o
3‘
'5 0.05 -
3
5’
0 l_ l J l
—0.25 -0.2 —0.15 -0.1 -0.05 0
Displacement 0
Figure 4.24a Poincare Map of Period 3 Motion; EZRA: 0.045.
0
4105i-
-0 , l l l l
' 0 0 05 o 1 0.15 0 2 0 25

Displacement 0

Figure 4.24b Poincare Map of Period 6 Motion; 829,; 0.046.

85

 

4105r-

Velocity 0

 

 

-01 l l J l
’ 0 0.05 0.1 0.15 0.2 0.25

 

Displacement 0

Figure 4.24c Poincare Map of Period 12 Motion; e252,; 0.04605.

0.05

 

Velocity 0

 

 

l l l l
-0.1
0 0.05 0.1 0.15 0.2 0.25

 

Displacement 0

Figure 4.24d Poincare Map of Almost Periodic Motion; £20,; 0.04657.

86

 

VelacityO
o
l

 

 

 

-0.25
-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Displacenent 0

Figure 4.24c Poincare Map of Period 6 Motion; £29A= 0.0915.

 

0.3

Velocity O
O
l

 

 

 

-0..'5
-0..3 -0.2 -0.1 0 0.1 0.2 0.3

Displacement 0

Figure 4.24f Poincare Map of Period 3 Motion; #9,: 0.115.

VelocityO

87

 

0.3 *-

 

 

 

 

0.1 0.2 0.3 0.4

f:
8
i
i

Displacerient 0

Figure 4.24g Poincare Map of Chaotic Motion; £29A= 0.12.

Velocity 6

Velocity Q

 

 

88

 

0.2

 

 

 

 

 

-O.2 1 l
‘0”3 '0-2 -O.1 0

Displacement 0

Figure 4.25a Phase Portrait of Period 2 Motion; city: 0.07.

 

0.2

 

 

 

l l
—O.2
-0.3 -0.2 -0.1 0

Displacement 0

Figure 4.25b Phase Portrait of Period 4 Motion; e205 0.08.

Velocity 0

Velocity o

89

0.2

 

 

 

 

 

0 0.1 0.2 o_3

Displacenent 0

Figure 4.25c Phase Portrait of Period 8 Motion; 829A= 0.0832.

0.2

 

 

 

 

 

Displacement 0

Figure 4.25d Phase Portrait of Period 16 Motion; 820A= 0.0833.

Velocity 0

Velocity 0

 

0.2

0.1 —
01—-

 

 

 

 

Figure 4.25c Phase Portrait of Period 32 Motion; lazily= 0.0834.

0.3

Displacement

0

 

 

:’ n "I {I "..
t , {h

. [l
" \l‘ . '

i
H
r.

~ ’5 .:. y \\

\§ vsxétfixglz

fiesfit

‘—‘

,H
I

ll)

ll

1
t

D

:.--.\

\s . \\“
o l”€§§ x\}
I); it)»
s‘l:?_.//r.’i»
:55526

.
3'0
,.

Q ‘

/..—.--
0A5?" 1‘:
« -//;',"; \ ‘ \
//////),’ '..-\\ \
[Ll/[11¢ .

/ ' W,;0 3'.

’424Vfigfiflé§gb

011%JL h\

. . I‘ _. .
/ - ..' 30’" " ‘
w ,' . t’ - . H.

1 / I . . :_‘-_I ,,._‘s.- ‘

/ , _- _ _7 i \.
' II .\ ’

/ / '1' y I
. ,f

l

I).
t

{,‘l

TELQ¢V‘J\\
H. toe-’1, H,“
. .‘li‘llli iii

i

\,:l\ ‘3 l. 3’. ’r.." I I l
. ~\ '- \\.‘\ iii/fl 2, l,

~\\"\.‘l.~"“.‘.-)T7I’ ,',’, .4

A

\
_' \

 

. ,(/€,"o:l.i.‘.-
1-: 20.- ’/-"' ’."~
\ I ,l , - ‘ ‘ t
\ ' . l - t _ . -' Q . e. \
\ \ ‘x- {’xf' / :”v "‘ ‘-‘- ‘ \ it
\ ‘ ' I I ’/~
. ,. a ‘
\‘ ’/ I ’/,l
.\ //
1
l - 'I

\ .

l

 

 

-0.3
-0

Displacement

0

0.3

Figure 4.26 Phase Portrait of Chaotic Motion; 825%: 0.0835.

91

EQA= 0.07 to 8 52A: 0.0834, the beam undergoes a succession of period doubling
bifurcations (from period 2 motion to period 32 motion). Figure 4.26 shows a phase
portrait of a chaotic motion with 8 0A: 0.0835. The corresponding Poincare map of
Figures 4.25 and 4.26 are shown in Figure 4.27. Plots of the displacement versus time for
period 4 and chaotic motions are Shown in Figures 4.28 and 4.29.

Figure 4.30 shows a summary of the preceding numerical results. Clearly, chaos
was observed to occur well above the curve obtained by Melnikov's method. However,
this is consistent with the analysis since Melnikov's method gives a necessary, but not

sufficient, condition for steady-state chaotic motion to occur. It is also interesting to note

that for the resonance (1) = 1.54 an (00 we found almost periodic motions. No such motion
were found for the (0 = 2.79 as 2 too.case. Moreover, the sequence of period doubling

bifurcations is different in the two cases as is the strange attractor depicted in the Poincare

maps of Figure 4.24i and 4.27f.

4.7 Summary of the Chapter

The method of multiple scales was used to obtain a uniform second order expansion
for the response of a rotating buckled beam subjected to subharmonic and primary
resonances. For sufficiently small 8 Q, values, the beam was found to vibrate about one
of the buckled positions with a softening-type, nonlinear behavior. AS we increased the
a (2,, value beyond some critical value, the beam's orbit encompassed both buckled
positions. In the vicinity of the critical value of a 0A , a region was found where the beam

will displayed bifurcated and chaotic motions.

Melnikov‘s method was applied to predict the regions where chaotic motions might

exist. Numerical simulations were used to find bifurcated and chaotic motions in the region

Velocity 6

Velocity b

92

 

0.2

 

 

l l
'-0.3 -0.2 -0.1 0

 

Displacemnt 0

Figtn'e 4.27a Poincare Map of Period 2 Motion; 82 (IA: 0.07.

 

 

 

L l
'-0.3 -0.2 -O.1 0

 

Displacement 0

Figure 4.27b Poincare Map of Period 4 Motion; €29A= 0.08.

Velocity0

Valocity0

93

 

 

 

 

Displacement 0

Figure 4.27c Poincare Map of Period 8 Motion; e295 0.0832.

 

 

 

 

Displacement 0

Figure 4.27d Poincare Map of Period 16 Motion; £29A= 0.0833.

94

 

 

—

0.3

0.2

1

 

 

2 1 0

G
0.

o n._oo_o>

1

-0.2

Displacement O

ezofi 0.0834.

0
9

Figure 4.27c Poincare Map of Period 32 Motion

 

 

1
£02

0.3

0.2

1

1

 

 

0.3

 

a ».moo.e>

Displacement 0

amp 0.0835.

Figure 4.27f Poincare Map of Chaotic Motion

95

 

0.2"

Displacement Q

 

 

4.; L l l l L
22540 22550 22550 22570 22580 22590 22600

 

Time

Figure 4.28 Time Trace of Displacement of Period 4 Motion; 82%: 0.08.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.3
0.2'— n
0 1'-
0 U U U
E
u 0 " ’ ~
a
a
II
0 fl
-O.1 —
-0.2
4% l l l
4 100 45150 45200 45250 45300

 

Time

Figure 4.29 Time Trace of Displacement of Chaotic Motion; £29A= 0.0835.

[I /12(w)

96

 

   
 
    
 
 

 

  
  

  
 
 
   

 

 

 

10
9 — chaotic,
3T °
8 —
7 .—
6T.
8T.1GT.321' 5mm“:
6 _' 41'.
2T-
5 I—
4 .-
almst periodic
31,61,121!
3 .—
1T .
2 l—
1 _ Q,— 3.70
0 L l | l L I l
0 0 5 1 1 5 2 2 5 3 3 5 4
Ct)

Figure 4.30 Comparison between Melnikov's Method and Numerical Simulations.

97

of primary and subharmonic resonances. The values of 8 9A for which chaotic motions

occurred are much above the homoclinic bifurcation curve, indicating that Melnikov's

method gives a lower bound in the parameter space.

CHAPTER 5

ANALYSIS OF A ROTATING, SIMPLY SUPPORTED SHAFT
WITH 0, < a,

5.1 Introduction

In this chapter we investigate the nonlinear, non-planar motion of a uniform,
initially straight, elastic shaft rotating about its longitudinal axis at a nonconstant spin rate.
The Shaft is considered to be Simply supported and has immovable ends. For a shaft with
these and conditions, nonlinearities arise due to mid-line stretching. Two classifications of
cross section will be investigated: (a) exactly circular and (b) close to circular. We restrict

our investigation to mean spin rates, (2,, less than no (i.e., pre-critical). The post-critical

behaviors of the shaft will be studied in Chapter 6.

The derivation of approximate solutions to the system's nonlinear governing
equations of motion are presented in this chapter for the four cases of parametric
resonances, viz. 0) ~ 200,, 0) ~ 2012, 0) ~O)I+OJ2 and 0) arm-0),. Each case is analyzed
firstly in the absence of internal tuning (i.e., (02 is assumed to be well removed from 30)!)
and secondly in the presence of internal tuning (i.e., 032 is assumed to be closed to 3011).
We also presents a stability analysis of the steady-state solutions which were obtained by
the method of multiple scales for each case. The accm'acy of the approximate solutions will
be checked by direct numerical integration of the original equations of the motion. The

choice of parameters for numerical simulation is based upon an experimental model. The

case of (0 ~ 2 (01 with (02 ~ 3 (01 will be given a more detailed discussion.

98

5.2 Approximate Solutions

Approximate solutions to equations (2.40) and (2.41) are obtained by the method of

multiple scales. These approximate solutions will remain uniformly valid for all T when a

is small. Before beginning the analysis we note that for rotational rates below (20, the

internal and external damping terms have the same influence on the Shaft and so, with no

loss of generality, we will set it; = 0. (Note, however, when the rotational speed of the

Shaft is above (20, internal damping has a destabilizing effects (T ondl [1965]).

Consider equations (2.40) and (2.41) derived in Chapter 2, we seek a first-order
solution for small but finite amplitudes in the form

U(t; e) = uo(To,T,) + e u1(To.Tr) + 82 112(To.Tr) + (5,1)
V(t; e) = vo(To,T,) + e v,(To,T.)+ £2 V2(TorT1) + (5.2)
Where Tn = 8n 1.

Note that the small parameter, c, has a physical interpretation (see Section 2.3 in Chapter 2)
and is the square of the slenderness ratio. Unlike the beam case, it is therefore dependent

on the system's parameters.

Substituting equations (5.1), (5.2), (3.6) and (3.7) into equations (2.40) and (2.41) and

equating coefficients of like powers of e, we obtain

8°:

D02 “0 +( Q02 - (282)110 - 2 QsDovo = O (5.3)

100

D02v0+((1+5)Qoz-QSZ)VO+ZQsDouo=O
81:
D02 111+ ( 902 "" 9‘2 ) 111 ‘—' 2 Q.Dovl = -2D0Dluo — 902u03
+29Afl,uo sin (in + 9A0) v0 cos an +2£2Asin (m: Dovo

D02V1+((1+ 8 ) 902 — (2,2)V1-i- 2 9.130111 = .2D0D1Vo-Qoz V03

“902 Vouoz‘ 2 QrDIUO‘2 a! lle u0 ”zlleDoVo
+29Afl,vo sin (01 — 9A0) uo cos (m: —29Asin an D0110

The solution of equations (5.3) and (5.4) can be expressed in the form
1.10 = A1(T1) cxp( 1601 To) '1' A2(T1) cxp( 1012 To) '1' CC

V0 = A1A1(Tl) cxp( 10.11 To) + A2A2(Tl) CXp( iszo) + CC

where
1

c0 _[2 902+2QSZ+9028-003[16 032+89828+QOSZJE
I-

 

 

2

l
m _[2902+2932+9028+Qoi16952+89528+§20282J5
2-

 

 

2

_ .=__ 2“)le
“’1‘" ((l+ii)t2,,2-r2,2)-to,2

i

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)

(5.11)

101

. 20) as .
A = =_ 2
2 R2! ((l+5)flo’-Qs’)-w;’1 (5.12)
i = «/—1

cc denotes the complex conjugate of the preceding terms

Al and A2 are arbitrary complex functions of T1 which will be determined at the

next level of approximation.

In this section we focus on the shaft's behavior close to a main parametric response (i.e.,

(0 =- 2 (01) in the presence of the internal resonance, to; ~ 3 (.01. To this end we introduce

two detuning parameters.

OJ=2031+861 (5.13)
and

(02:30.11-1-80'2 (5.14)

where 01 is known as the external detuning parameter and 02 as the internal detuning
parameter. We note that (0 is an independent variable and that we can control 02 by

varying Q, (see equations (5.9) and (5.10)).

Next, in order to determine the solvability conditions of equations (5.5) and (5.6), we

express the form of their particular solution as

ul=Pucxp(i(DlTo)-I~Pu cxp(i(02To) (5.15)

102

v, = P21 exp( i co,T0 ) + Pm exp( i 662T0 ) (5.16)

and substitute equations (5.7), (5.8), (5.13), (5.14), (5.15) and (5.16) into equations (5.5)
and (5.6), equate the coefficients of exp( i (ulTo) and exp( i (112T0 ) to obtain

floz-QSZ-wlz -i293031 {Pll}_{Rll}
902 - 982 "' (1)22 — i 2 as (02 P12 _ R12
i2flsto2 (1+5)Qo2—Qsz-(l)22 P22 R22

where Rij are defined in Appendix D.

(5.17)

(5.18)

Since the determinant of the coefficient matrices of (5.17) and (5.18) are zero (this is how

(01 and (.02 were originally found), then for there to exist a non-trivial solution of the Pij we

 

 

musthave
Rn -129,0)‘
R (1+5) 2-92-(02 =0
21 Qo . 1 (5.19)
R12 -iZQ,(02
R 1+5 2-92- 2 =0
22 ( )00 s 032 (5,20)

 

 

At this stage it is convenient to consider two distinct cases, viz., 6 = 0 and 8 at O (i.e.,
symmetrical and unsymmetrical). In the former case the analysis can proceed by

considering equations (5.9) to (5.12) from which we find

Al=-i

103

A2: i
031:90“).
w2=Qo+Q.

l l
Qs=EQO+ZEOZ

Substituting (D1), (D2), (D3) and (D4) in Appendix D into (5.19) and (5.20), we obtain

the reduced equations
iA,'+ itteA,+2r20 A,2 Kr+ 4 floArAr 32= 0 (5.21)
iA2’+itch2+2 90 A22 K2‘1'490‘51‘3‘2;=0 (5.22)

where the prime denotes a partial derivative with respect to T1. The only steady-State
solution of equations (5.21) and (5.22) is Al: A2: 0. Hence for a perfectly circular shaft,
it is impossible to excite a main parametric resonance. This is consistent with Kammer and
Schlack [1987]. Moreover, the reduced equations are independent of the internal detuning
parameter and the above result also hold in the absence of internal resonance. For the other
cases of the parametric resonances (i.e., a) a 2 012, to ~ a)l + (02 and 0) ~ 012— 0),), it can

also be Shown that no non-trivial steady-state solutions exist.

We now return to equations (5.19) and (5.20) and consider the cross section of the
shaft to be close to circular (i.e., 8 at 0 and the shaft possesses unequal principal moments

of inertia). For clarity we will further subdivide the study by first investigating parametric

104

response in the absence of the internal detuning and then investigating the influence of the

internal resonance.

5.3 Assuming 012 Well Separated from 3011

In this sub-section we assume (1)2 is well removed from 3ml (i.e., no internal

tuning). The approximate solutions are obtained by the method of multiple scales for four

cases, viz. 0)~20),, (0~2(.02, cot-«001+c02 andm-scoz—(ol.
5.3.1 The case of cot-=20),

Following the same procedure as was used in Section 5.2., we obtain the reduced

equations

i A1, + i 613 CialTl K1 ‘1' i 615 “C A1+ GIG A12 3.1+ 617 Al A2 X2 = O (5.23)
iAz +i (323 “'6 A2 + 025 A22 X2 ‘1' 624 Al A2 X1: 0 (5.24)
where Gij are defined in Appendix E.

To solve equations (5.23) and (5.24), we write Al and A2 in the polar form

1 e
A1=—a,e“31

2 (5.25)

(5.26)

105

where 31,32,131 and 82 are real.

Substituting equations (5.25) and (5.26) into equations (5.23) and (5.24) and separating

real and imaginary parts yields
317’ = alO'I + 2 613 31 81117 -% 0‘6 813 - % GI? a] a22 (5.28)
93 = - Gas it. as (5.29)
a; 132: 7 G24 31 a2 + - G25 323 (5 30)
Where 7:01'1‘1-261' (5.31)

Of particular interest are the Steady-state solutions which correspond to
3;: a; = 7’: 0. This gives rise to two solution sets: a1= 0 and a2 = 0, age 0 and a2 = 0.

In the second set we can evaluate 81 by noting that

613 (5-32)

 

13 (5-33)

Squaring and adding equations (5.32) and (5.33), we obtain

1
2 [011 2 (6132- uezfi ]

312 =

 

Gm (5.34)

106

Reconstructing the first order approximate solutions of equations (2.40) and (2.41), using

equations (5.7), (5.8), (5.25), (5.26) and (5.31), we have

 

(tn-7

 

V0 = — kl a1 Sin(m2-Y)
(5.36)

The parameter values used for numerical results in this and subsequent sections,

unless otherwise stated, are:

oA = 5760,11e = 250, 8 =1.o.flo= 1:2 and e = (2+0)?

Figure 5.1 shows a typical frequency response curve for a mean spin rate Q, = 7.0. This
shows the variation of the steady-state amplitude al as a function of the frequency of the
oscillating component of the spin. Both stable and unstable solutions are shown. Dashed
lines denote unstable solutions and solid lines indicate stable solutions. The stability of the
steady-state solutions can be ascertained by adding small perturbation to the steady—state

solutions and checking if this perturbation grows or decays. For the trivial solution this

results in an unstable solution for 43,, < o < 0“ where

l
on = 2[ G132 " 11:2 F (5.37)

It can be proved that the upper branch of the non-trivial solution is always stable,
whereas the lower branch is always unstable. The general form of this curve is the same as

would be obtained by an analysis of Duffing‘s equation with a paramedic forcing term.

 

 

 

 

 

35
D
o o o 0 nautical limlotlon
30 l'" noble
—--- unstable
25 l—
20 _ ’./
OI. ’/
U /’
15 '— /
10 -- /
5 - /
i 6 ' ‘ °
0. -.l_--_L--__..L_-__L.--__L_ L A. a
-4000 —3000 -2000 —1000 O l000 2000 3000 4000 5000 6000

0:

Figure 5.1 Variation of the Steady-State Amplitude, a1, with 61; ()5 =7.0, 8 = 1.0.

 

 

 

 

 

 

.35
—— stable
30}..‘—‘"" unstable
25 '—
20 r—
d
15 ‘—
10 r—
5 ‘- /’,
l (5 1' 0 . 2
l
o l 1 1 _-L l 1 I A‘ ‘1 '
4000 -3000 -2000 -l000 0 1000 2000 .3000 4000 5000 6000

0'1

Figure 5.2 Variation of the Steady-State Amplitude, a1, with 01; (2, =7.0, 8 = 0.2.

108

Figure 5.2 shows another frequency response curve with 5 equal to 0.2. Comparing
Figure 5.1 with Figure 5.2, we observe that the width of the instability region of the trivial

solution and the amplitude al both increase as the 8 value increases.

The results obtained using the method of multiple scales can be checked by
numerically integrating the original equations (2.40) and (2.41). The results of such an
integration are also shown in Figure 5.1 and the comparison is very good in all the cases
tested. The response of the rotating shaft can be shown in various projections of the five

dimensional (U ,U,V,V,r) extended space. Hence, we plot the motion of the shaft in a

(U,V) projection to get a better visualization, as shown in Figure 5.3. The result is a
relatively simple elliptic orbit. Later, in Section 5.4, we will see how the structure of this

orbit becomes much more complex when the condition of internal resonances is added.
5.3.2 The case of 0) ~ 2 a),

In this sub-section the frequency a) is taken to be closed to 2o)2 and hence the

external detuning parameter, 02, is redefined as

Following the procedure used in Section 5.2, the solvability conditions of equations (5.5)

and (5.6) are now

1A, +i (3,, u. A, + Gm A12 X1 + On A1 A2 K2 = 0 (5.39)

iAz'H 0,, p, A,+G,, A12K2+i G22 eialT1X2+Gu AJ, A2=O
(5.40)

V Displacement

109

 

10

_4r-

_6r-

 

 

l 1
1c3-20 -15 —1o -5 o 5 10 15 20

U Displacement

Figure 5.3 Orbit in (U ,V) Projection; 9, =7.0, 8 = 1.0.

 

110

To solve equations (5.39) and (5.40) for Al and A2 , it is again convenient to inuoduce the

polar forms given in equations (5.25) and (5.26). Substituting equations (5.25) and (5.26)

into equations (5.39) and (5.40) and separating the result into real and imaginary parts, it

follows that
a; = _ (3,, u. a, (5.41)
, 1
81 Br: Gus 3:3 + Z 0:78: a22 (5 42)
a; = _ (323 pa a, — (3,, a2 cosy (5.43)
, . 1 1
a271 = 3101"" 2 622 a2 srn'y - 5 62432 312 — E 025 323 (5 44)
Where 7 = 61 Tl "' 2 61' (5.45)

Once more, it is possible to simplify these equations should only the steady-state responses
be sought. For such a case, equations (5.41) to (5.44) reduced to

(315 u. a1 = 0 (5.46)
aIBf = Gus 313 '1'; G17“: a22 (5.47)
623 he a2 + G22 32 cosy = 0 (5.48)
ale, + 2 On a2 sin ‘y - $— 62432 812 ":7 G25 a23 = 0 (5.49)

For non-zero damping there are two possible solution sets to these equations: either a1 = O

and a2 =0 or a1 = 0 and a2 at 0. In the second set we can evaluate 212 by noting that

022 (5.50)

111

sin'y: %[625 2122—20,]

 

22 (5.51)
Squaring and adding equations (5.50) and (5.51), we obtain
1
2 [0121: 2 ((3222- “62)? J
3.22 =
(325 (5.52)

Then the first approximations to the solutions of the original equations of motion have the

form

 

 

A typical frequency response curve is shown in Figure 5.4 with as: 7.0. Once again,

numerical simulations show very good agreement with the approximate solutions.

5.3.3 The case of m- (02+(ol

In this sub-section we investigate the combination resonance and hence the external

detuning parameter will be defined by

CD= (01+(02 +8 61 (5.53)

The solvability conditions are

112

 

 

 

 

 

 

35
a o o 0 nautical elrmlatlan
l)
30 ._ .3051.
—“- unetable /
25 —
20 *-
a
d
15 '—
10 '—
5 — /
', ILL.- 40
g L ll L 1 l i A
-%000 -1000 O 1000 2000 3000 $000 5000 6000
Or

Figure 5.4 Variation of the Steady-State Amplitudes, a2, with 01;
(2s =7.0, 5 = 1.0.

113

1A,' noun, A,+G,,,A,2 K,+iG,,c‘°lT1X,+G,7A, A2K2=o
(5.54)

1A,'+i oz, 11. A2 + (1,5 A,2 K2+ i on cml'li'A,+(3,4 A, 'A', A2 = o
(5.55)

Substituting the usual polar forms (5.25) and (5.26) of A1 and A2 into equations (5.54)
and (5.55) and following the normal procedure, yields

a; = ‘ 615 “e 31" 613 a2 00571 (5.56)
, . l 3 1 2
a = — G + — G + - G
151 13 a2 srn 'Yr 4 16 a, 4 1731 a2 (5, 57)
33 = ' (323 ll: a2 - G22 a160571 (5.53)
, . 1 1
a2 [32: — 622 a, srny, + -4- (32432 312 + 3 G25 2123 (5 59)
Where 7] = 01 T1 "‘ fll - B2. (5.60)
For the steady-state motions (i.e., a; = a; = y; = 0) we find
615 ”e 31"” G13 a2 00571: 0 (5-61)
623 11. a2 + 622 a, cosy, = 0 (5.62)
014432231 Sin'Yr‘lGu a12"'1'st a22+C'1a £2 Sinlr
a2 4 4 a1
_ 1 l 2 _
7; Gus ar 4 617 a2 - 0 (5.63)

By eliminating cosy, in equations (5.61) and (5.62), one obtains the following linear

relationship between a1 and 32

a2 = K al (5-64)

114

l

2
where K = [922.].
13

Substituting equation (5.64) into equations (5.61) and (5.63) yields
1
4 mic, 1-[KWT
022

C1

 

2-
a1—

 

where
c,=o,,+o,6+K2(G,5+o,,)
C2=Gan+G13 K.
Hence, the steady-state solution has the form
uo = a, cos[((r)l +8 EM -91]+ a2 cos[(co2 + 8 RM —02]

v0=—kla1sin[((01+8§,)1:-01]-k2 a2 sin[(o)2+e [Em-9,]

where

— a . 1 2 1 2
fi=’Gra-2‘31071+;G1631+zGr7az
31

(5.65)

(5.66)

(5.67)

115

- __ a . l 2 l 2
52— -0224811171'1’262431 +-4'Gfiaz
a,

7, = 0,-1- 02 , 01 and 92 are constants depending on the initial conditions.

A typical frequency response plot (equation (5.65)) is shown in Figure 5.5. This curve is

similar to the two cases already presented in Sections 5.3.1 and 5.3.2.

When (02 and to, are incommensurate, the resulting response is non-periodic. A clear way

to demonstrate this is to plot the trajectory of the steady-state response obtained by
numerically integrating equations (2.40) and (2.41) in a (U ,V) projection, this is shown in
Figure 5.6a and the corresponding result obtained by the method of multiple scales
(equations (5.66) and (5.67)) is shown in Figure 5.6b. A comparison between these two
figures shows remarkable agreement.

5.3.4 The case of 0)~ (oz—to,

In this subsection we consider to near to be the difference between to, and (112 and

we introduce the external detuning parameter as

The solvability conditions of this case are

iA,’ +1 6,511, A, +G,6 A12 K,+io,,e"“lT1A,+G,,A, A, K,=o
(5.69)

116

 

20

—— etable
re 1-

—“- unetable

 

#— " .n ‘0
2 / 1.1.

 

 

o . ..-—_ —__ ’- 1 4 L
-1000 ~500 0 500 1000 1500 2000 2500 .3000 3500 4000

01

Figure 5.5 Variation of the Steady-State Amplitude, a,, with 01; (2s =7.0, 5 = 1.0.

117

 

mricol eirrulation

‘E 2—

E

3 _

a. 0

n

c:

> ‘2 ‘
_4—

 

100 loreing perioae

 

 

_ 1 1 1 1 1 1 1 1 1
-10 -8 -6 -4 -2 O 2 4 6 8 10

 

U Dieploeanent

Figure 5.6a Trajectory obtained by Numerical Simulation in (U ,V) Projection;
()8 =7.0, 5 = 1.0.

 

V Displacement

-6—

 

100 lareing periods
-8 L l l l l l l l l
-10 -8 -6 —4 -2 0 2 4 6 8 10

 

 

 

U Dieplaeenent

Figure 5.6b Trajectory obtained by MS in (U ,V) Projection; £2s =7.0, 5 = 1.0.

118

iA,’+iG,,p,A,+G,,A,2K,+io,,ci°lTlA,+G,,A,X,A,=o

To solve equations (5.69) and (5.70) it is convenient to express the Al and A2 terms in

their polar form as defined in equations (5.25) and (5.26). Separating the resulting

equations into real and imaginary parts, yields

’-
31" ’ 615 “e 31‘ C'12 a2 90571

_ . l 3 1 2
3151- ' 612 a, 3197 'z 616 a, ’ z G173132

I —
a, - " Gza lie a2 -622 3190571

. l l
a, [33: ~G22 a, any, + 7 624a, a,2 +2» 6,, a,3

where 71: 5111" 131-52-

For the steady-state motions (i.e., a; = a; = 7; = 0) we find

Grs “a 31+ Grz a2 00571: 0
6,3 11, a2 + 02, a1 cosy, = 0

a . 1 1
61-1-an1- srn71+sz4 alz+szs a22
2

a . l 2 1 2_
+Gu-1srn‘y1-2-Gma, ‘anaz -0
a,

Following the same procedure as was employed in Sub-section 5.3.3, a linear relationship

between a1 and a2 is found in the form

a2=Kal

(5.70)

(5.71)

(5.72)
(5.73)

(5.74)

(5.75)

(5.76)
(5.77)

(5.78)

(5.79)

 

119

1

2
where K = [922 .
G12

It can be found that G22 always less than zero. Hence the trivial solution is the only steady-

state solution for this case of resonance.
5.4 Assuming (0, == 3 a),

The condition of internal tuning is now introduced. The nearness of (132 to 3(1)1 is
quantified by a, and this is controlled primarily by changing (I, (see equations (5.9) and
(5.10)). For example, if 5 equal to 1 and (25 equal to 5.46309, (132 will be very close to
3(1),. This section comprises three cases, two for the case of principal parametric resonance
and one for the case of combination resonance. Since many of the points discussed and

techniques utilized are common to all three cases, the first is presented in most detail.

5.4.1 The case of (Ir-2m,

Following the method of the multiple scales, the reduced equations for this case are
found to be

. A I . “02-0011 - i011] - G iO2T1 X 2 A
l l+lzeC A2+IGI3C Al+ 146 1 2

+ iG15 11¢ A14“ 616 A12 X1 + G1‘7 A1 A2 X2 = 0 (5.30)

I . .
i A, +1 (3,, (rm-“9T1 A, + (3,,5 6‘"le A,3 +1 (323 u. A,

+62515‘22 K2‘*'Gz4 A1A2K1=0 (5.81)

120

To solve equations (5.80) and (5.81), it is again convenient to introduce the polar forms
given in equations (5.25) and (5.26). Substituting equations (5.25) and (5.26) into
equations (5.80) and (5.81) and separating the result into real and imaginary parts yields

I l 2 .
31=‘Graa1°°S'Y1‘Grsuear’Grz3200572‘3'614 a, a, srn(y,+y,)

(5.82)
317;=alol+2613aISinYl-%616313+2 61232 Sin‘Yz
-';' GM a12 a2 00“ 71+72 ) ' % Gr? a1‘122 (5.33)
I 1 .
a2=—Gzza,cosh—623116a2+-4—G%a,351n(‘y,+y,) (584)
32(72‘%Yi)=32(°2’%°1)+C'22315in72+%62431232
1 1
+—G a3+-G a3cos +
4 25 2 4 261 (71 72) (5.85)
where
71=°rT1"ZBr (5.86)
72=(01-°2)T1+32‘Br (5.87)

The steady-state solutions of equations (2.40) and (2.41) correspond to the fixed points of
the reduced equations (5.84) to (5.87), which result from a; = a; = 7; = 73 = 0. Once
again we have two possible solution sets: a trivial set a1: 0 and a2 = 0 and a non-trivial set

a, at 0 and a, at 0. The nonlinear transcendental equations governing the non-trivial set are

1 .
613 51100371" 615 lie 31+ 612 a, 90572 + 2 GM 312 a, SIM 71+ 72 ) = 0
(5.88)

121
a,o, + 2 6,3 al sin 'y, -% G16 a,3 + 2 6,, a2 sin y,
-%Gr4 31232°°S(71+72)‘%Gr'131322=0 (5'89)
1 .
Gnalcos'y2+G23 ueaz-zGual3srn(y,+y,)=O (5.90)
a2 ( o,-% o, )+G,2 al sin'y,+%G,4a,2 a,

1 1
+-G a3+-G a3cos + =0
4 25 2 4 26 1 (71 72) (5.91)

Equations (5.88) to (5.91) can be solved numerically to determine the non-trivial steady

state solutions a,, a2, 7, and 7,. The first order approximate solutions to the equations

(2.40) and (2.41) can now be reconstructed using equations (5.25), (5.26), (5.86), (5.87),
(5.13),(5.14), (5.7) and (5.8) to yield

u0=aICOS( 2211-121)+32003(%0’1+72-'72—1) (5 92)

vo=—k,aisin(gT—lzl)‘kzazsm(%t+YZ-12L) (5 93)

Figure 5.7 shows the variation of the steady-state modal amplitudes, al and 212, as
functions of the external detuning parameter a, (i.e., frequency response curve).

Parameters were chosen to be representative of a physical system and a value of

Q, = 5.46309 was chosen such that we have perfect internal tuning ((32 = 0). The results

of a stability analysis are also shown on this figure. Stable solutions are denoted by a solid

line, unstable solutions by a dotted or chained line.

122

 

 

  
   
 
     
    
   
        

 
 

  
  

 

   

      
   

 

 

 

 

 

 

 

 

35
o o o a numerical sinulatlon
stable
30 -____
-—'- unstable
25 -— Hopf
bifurcation
_ L! 1 _ L .4_
420 420 O 320 620
" 20 -—
d trivial solution ‘
B
15 —
10 -
5 1-—
94000

Figure 5.7 Variation of the Steady-State Amplitudes, al and a2, with (3,;
(32 =0.0, (£2s =5.46309), 5 = 1.0.

 

123

In Figure 5.7, it is interesting to note that there is a region in the vicinity of 0, equal
to 2000 over which neither the trivial nor the non-trivial unstable solutions are stable. By
investigating the form of the eigenvalues we find that the non-trivial solutions have
undergone a Hopf bifurcation. Hence, the amplitudes and phases are not constant but vary
with time (i.e., quasi-periodic motions). To better demonstrate this point, we have
numerically integrated the equations of motion (2.40) and (2.41) for a value of ol= 1300
and compared this to the results obtained for the value of 61= 1500. Based on the
predictions of the approximate solution (see Figure 5.7) the former should yield a stable
steady state solution where the latter should give rise to a quasi-periodic motion. The
results are presented in Figures 5.8a and 5.8b respectively, in the form of an orbit plots of
U versus V. Both figures show the solutions after all transients have decayed and are
plotted for a length of time corresponding to 100 cycles of the forcing term. Clearly the
orbit in Figure 5.8a is periodic whereas the orbit depicted in Figure 5.8b shows a slow
amplitude variation. If this orbit where plotted for a longer period of time, more of the

(U ,V) space would become populated. This is typical of quasi-periodic motions.

With reference to Figure 5.7, it was initially thought that the trivial solutions were
unstable for values of [0,] < 2360. However, upon closer inspection we find that the
solutions re-stabilize over regions of 320< IOJ < 620. Moreover, at IO,I < 320 the trivial
solutions undergo a Hopf bifurcation and so we can predict the motion between IOJ < 320
will be quasi-periodic. Hence, there is a region in the vicinity of 01 equal to 0 over which
non-trivial steady-state motions and amplitude modulated motions are coexist. The motion
adopted depends on the choice of the initial conditions. Figures 5.9a and 5.9b show the

orbits corresponding two different initial conditions with 0, equal to 0.

Next, we present a collection of results which show how the steady state solutions

vary as a function of the system parameters. For clarity, we have not shown the stability of

V disploca'nent

V displacmnt

124

 

0'. 8m

10F Ut'°

 

 

 

—20 -15 -10 -5 O 5 10 15 20

U displacement

Figure 5.8a Orbit in (U ,V) Projection; 01:1300, oz =0.0.

 

15

10*-

 

100 [arcing periods

 

 

1 L 1 1 1 1 1
-20 -15 -1O -5 0 5 10 15 20

 

U displacement

Figure 5.8b Orbit in (U,V) Projection; 61=1500, 02 =0.0.

 

V disploemnt

V displocmnt

125

 

(.0. (0.1.0.0; 0.1.0.0)
0,-0
a.-°

 

100 lereing periods

 

 

_,5 1 1 1 1 1 1 1
-20 -15 -10 -5 0 5 10 15 20

 

U displacanent

Figure 5.9a Orbit in (U,V) Projection; 0, =0.0, (32 =0.0,
U(0) = 0.1,U(0) = 0.0; V(0) = 0.1,V(0) = 0.0.

 

15
ch (zLo;rom)

05-0
01-0

 

 

 

l 1 l L l l
-20 --15 -10 -5 0 5 10 15 20

 

U displocmnt

Figure 5.9b Orbit in (U ,V) Projection; 61=0.0, 62 =0.0,
U(0) = 20,0(0) = 0.0, V(O)=10,V(O)= 0.0.

126

the solutions. Figures 5.10a and 5.10b show the frequency response curves for two
different QA values. For a small (2,, value (Figure 5.10a), the four branches in the

frequency response curve are well separated from each other. In Figure 5.10b the value of
(2A has been increased slightly and a more complex combination of solutions exists, many
of which have been checked by direct numerical integration of equations (2.40) and (2.41).
Figures 5.11a-h show the trends association with changing the internal tuning. Clearly we
see that as the magnitude of the detuning, 02, becomes large the form of the response curve
reverts to that of Figure 5.1. This is to be expected as large [(3,] corresponds to a lack of

internal resonance, hence the case described in Section 5.3.1 is recovered. Figures 5.12a-c

show frequency response curves with different 5 values (at a small Ile= 4O ). Obviously,

both the amplitude, a1 and a2, increase as the 5 value increases. It should be note that no

distinction is made between stable and unstable solutions in Frgures 5.10 to 5.12.

Figure 5.13 shows an orbit with 01 equal to 900. Because of the presence of
internal resonance, the orbit is no longer an ellipse (c.f. the form of Figure 5.3). The
corresponding time histories of the steady-state solution in x and y directions obtained by
directly numerically integrating equations (2.40) and (2.41) are shown in Figures 5.14a
and 5.14b. Figure 5.15 shows the corresponding spectrum of the time history of the U
coordinate, which is obtained by using the Fast Foruier Transform (FFI’) algorithm. In all
the cases we checked, the numerical simulation results were indistinguishable from the

results obtained using the method of multiple scales.

127

 

65
so—QA-zsao

- 2
55__ a. 1150

50'—

40*-

a,

 

  

 

  

 

 

_, 1 1 l l l l l I 1 f
3000 -2500 o 2500 soon 7500 10000 12500 15000 17500 20000 22500 25000 27500 30000

 

01

Figure 5.10a Variation of the Steady-State Amplitudes, al and a2, with 01;
DA = 2880, (32 = 11520.

 

55
o a o a numerical sirmIation

50‘- QA- 5760
‘51-—
40 '—

35‘—

d 30"-

 

 

 

1 l. 1 A
4000 -’£000 0 2000 4000 6000 0000 10000 12000 14000 16000 18000 20000

 

0'1

Figure 51% Variation of the Steady-State Amplitudes, al and a2, with (31;
(2,, = 5760, oz = 11520.

128

 

 

 

 

 

 

 

 

:5
50— 03-4700
45-
‘0—
. 55-
3
U
25—
20—
15—
10—
s—
L——-—r 1 1 1
3000 a 4000 soon 12000 10000 20000
a,

Figure 5.11a Variation of the Steady-State Amplitudes, a1 and a2, with (3,;
0,, = 5760, (32 = -5760.

 

55

45—

as
..P

 

 

 

 

.fl a,
:kfms... "

at

 

Figure 5.11b Variation of the Steady-State Amplitudes, al and a2, with 0,;
QA = 5760, (I2 = -2880.

129

 

55

 

 

 

 

 

 

50” 02-0
45-
40—
c1. 55-
a,
. w—
d
25—-
20—
15—
10—
:1L 32
o 1 L 1 1
4000 4000 soon 12000 10000 20000
0'1

Figure 5.11c Variation of the Steady-State Amplitudes, a1 and a2, with 01;
(2,, = 5760, 0‘2 = 0.

 

w_ 0..“

45'—

35'-

as

25'—

 

 

 

 

 

or

Figure 5.11d Variation of the Steady-State Amplitudes, a1 and a2, with 0,;
0,, = 5760, (32 = 2880.

130

 

55

50— 0,-6700

40'—

35*-

as

30 —
a,
25 '—

20 — a,

 

 

 

Figure 5.11e Variation of the Steady-State Amplitudes, a, and a2, with (3,;
0,, = 5760, (32 = 5760.

 

50- q-rrsso

45—

35r-

as
__93

25' —

20""

10" a2

 

 

 

Figure 5.11f Variation of the Steady-State Amplitudes, a, and a2, with 0,;

131

 

 

 

 

 

at

Figure 5.1 lg Variation of the Steady-State Amplitudes, a, and a2, with 0,;
9,, = 5760, 02 = 17280.

 

as

(1,,

 

 

 

 

 

 

01

Figure 5.11h Variation of the Steady-State Amplitudes, a, and a2, with 0,;
9,, = 5760, 02 = 28800.

132

 

 

o e a 0 nautical sinulotien

7— 6-o.1

as

 

 

1
400-400

 

Figure 5.12a Variation of the Steady-State Amplitudes, a, and a2, with 0,;

 

 

 

 

“A = 5760’ 62: O, 8 = Oele
20
e a o a nunrleal sinulatlon
10 5" 6- 0.5
16 *-
14 '-
. a1 a,
12 — -
I
d
.: 1° r
d at
a 1—
s — l a.
‘ u—
a, a.
2 __ 2
l l L l I 4 7 l 1
30m -1500 -1000 —500 0 500 1000 1500 2000 2500 .3000
at

Figure 5.12b Variation of the Steady-State Amplitudes, a, and a2, with 0,;
Q, = 5760, 02: 0, 5 = 0.5.

133

 

 

 

35
o o o a numerical simulation
.10— 6- 1.0
at
25—
a» 20’-
at
d
151—
a,
a,
10—
a,
a2
5.—
as
a,
l. l l A l l l L 1 1
3M 4000 -2000 4000 0 1000 2000 3000 4000 5000 6000

 

01

Figure 5.12c Variation of the Steady-State Amplitudes, a, and a2, with 0,;
9,, = 5760, 02= O, 5 = 1.0.

134

 

10
mrieot simlation

8- op-un

(g-o

V Displacennnt

-4—

 

 

 

-10
-12 -10 -6 -6 -4 -2 0 2 4

U Displacanent

Figure 5.13 Orbit in (U ,V) Projection; 0, =900, 02 =0.0.

 

 

 

 

 

U displocanont
O
1

 

 

 

 

 

 

l l 1 l 1
-1
20444 26445 2 6446 26447 26446 26449 26450

 

 

Tine

Figure 5.14a Trme History of Steady-State Solution U; 0, =900, 02 =0.0.

135

 

 

 

 

 

 

 

V disploca'nent

-4-—

 

 

 

 

 

 

 

 

_ 1 1 1 1 1
20444 26445 26446 26447 26446 26449 26450

 

Time

Figure 5.14b Time History of Steady-State Solution V; 0, =900, 0, =0.0.

 

 

 

 

 

 

 

10
Frequency content of nunsricol sinulation data
8 _
a,
o 6 _
'0
a
’E
8‘
5 0’,- 90°
4 - as
nmrlcal
sinulation ws
Ch 040mm) same
2 '- 0., 3.70016 3.725
7, — 2.1002
0 1 L l l
0 2 4 6 8 10

Frequency (Hz)

Figure 5.15 Spectrum of the Steady-State Solution U; 0, =900, 02 =0.0.

136

5.4.2 The case of (ID-I20),

The solvability conditions of this case are

1A,, + 0,, 6°le X} A, + i 0,, u, A,
‘1' G16 A12 1:1 '1‘ G17 A1 A2 X2 = 0 (5.94)

I . _ _. .
1 A, + i (3,, 61°11”! A, + (3,6 e '°2"‘1A,3 + 1 (3,3 11, A,

‘1' C125 A22 76:2 + G24 A1A2 K1: 0 (5.95)

Substituting equations (5.25) and (5.26), the polar forms of A, and A2, into equation

(5.94) and (5.95) and separating the results into real and imaginary parts, yields

r_ l 2 .
a,--G,5 14:31 “'76“ a, a, srn 71

(5.96)

a,(2'y{+y,)=a,(2 0,-1-62)'-%Gr6313
_;G,, a,2 a, cos 11,-; (317 211a:2 (5.97)
a; = - C122 a, 00872 " 623 11, 32 + '1' G26 313 Sin Yl (5.98)

r_ - 1 2 l 3 __l 3
3.2 72—82 01-1-62232811172-3-6243, 3.2-362631 COS Y, 562532

(5.99)

where

71=02T1+Br3131 (5.100)

72=01T1-2132. (5.101)

137

These are the reduced equations for the case of 0) ~ 2 (0, with 00, ~ 3 (0,. Of particular
interest are the steady-state case which correspond to a; = a; = y; = 7; = 0. Equations

(5.96) to (5.99) then reduce to

a, (6,5 "e +11? G“ a, a, sin 71) = 0

(5.102)
311(2 01+ 0, )‘% Gus a12 "$014 a12 a, cos 71' % G” 322] = 0 (5.103)
(3,, a, cosy,+G,3 u, a,—-:-G,6 a,3 sin ‘y,=0 (5.104)
a, a, + 2 02, a, sin‘y2 - % 0,, a,2 a2 -% 625 a13 COS 71";st a23 = 0 (5.105)

This gives rise to three possible sets of solutions to these equations:
(1) a,=0anda,=0
(2) a,=0anda,¢0
(3) a,¢0anda,¢0

We first consider possibility (2) viz. a, = 0 and a, at O_ For this case equation (5.104) and

(5.105) reduce to
62311: a2+(322 3200871=0 (5.106)

. l 1
alol+2GnazsmYl-Ecflazalz-5625823=0 (5.107)

138

Equations (5.106) and (5.107) are exactly equal to equations (5.48) and (5.49) (i.e., the no

internal resonance case) and so are not discussed again here.

For possibility (3), we solved the roots (i.e., non-trivial steady-state solutions) of
equations (5.102) to (5.105) using numerical routine DNEQNF from the IMSL library.
This routine use a modified Powell hybrid method to find the roots. Although a great many
initial estimates of the solutions were used. The routine only found the trivial solutions.
The reason may be because the basin of attraction of the non-trivial steady-state solutions is
extremely small and/or irregular. This is consistent with the results obtained by directly
numerically integrating the original equations (2.40) and (2.41) in which all trajectories of

the cases tested eventually approach the trivial solution or possibility (2).

5.4.3 The case of m~ m,+(o,

The solvability conditions of this case are

I . _ - _
iA, +iG,3 1:”1T1 A,+ G,,e‘°2T1 A,2 A,

+ i 615 “e Al+Glé A12 K1+6” A1 A2 K2= 0 (5-103)

i A,’ + i (3,, 6"!“ K, + (3,, c"°2T1A,3 +1 (5,, 11, A,
+625 A22 Xz'l'c‘u A1A2K1=0 (5.109)

Substituting the usual polar forms of A, and A2 into equation (5.108) and (5.109) and
following the normal procedure, yields

a1: - G13 82 00871 " 015 “e 31"}6“ 8‘2 a: sin 72 6'1“»

139

31(71+Y§)=a1(01+°2)-616313+G1332 sin‘y,

-614 3'12 a2 00872- 61731322 (5.111)

I_ l 3 -
a,--G,, a, 00571'62311e 324':st a, sm'yz

(5.112)
3 , l , 3 l . 1
a,(Evy-44y,)=a,(10,-3-0,)+G,,a,srn‘y,—ZG,4a,2a,
__1_ 3-1 3
46,5 a, 4 Gwa, cosy, (5.113)
where

71:01T1‘Bz' Br (5.114)
72=02T1-3131+ Bz, (5.115)

Two possibilities of the steady-state solutions exists: either a, and a, are zero, or neither

one is zero. The steady-state solution has the form

 

 

u0 = a, cos|:(M - (11+ 72)]4- a, cosl:30M - (72 — 371)]

4 (5.116)

 

 

v0 = -k1a1 s,n[cor - (71+ 72)]_ ,2 32 si“[301 - (72- 371)]
4

4 (5.117)

Figure 5.16 shows a typical frequency response curve obtained by finding the roots
of the steady-state form of equations (5.110) to (5.113). The approximate solutions to the
full equations of motion can be checked by directly numerically integrating equations (2.40)
and (2.41). The comparison is good in all the cases tested. Figures 5.17a and 5.17b show
the orbits of the shaft obtained by numerical integration of equations (2.40) and (2.41) and
by the method of multiple scales (equations (5.116) and (5.117)), respectively. Note that,

140

 

 

 

 

 

 

6
5 _—
4 )—
I
d
a 3 )—
d
2 r-
1 o e o o nut-rice! olrrulctloo
Ital.
-—--- tractable
0 l I I l l
-200 ~150 -100 -50 0 50 100

0’,

Figure 5.16 Variation of the Steady-State Amplitudes, a1 and a,, with 0,;
o, = O, 8 = 1.0.

141

 

nmrieol limlolion

V Displocmnt

-4—

 

 

-8 1 1 1 1 1 1 L 1 1
-10 -8 -6 -4 -2 O 2 4 6 8 10

 

U Displocmnt

Figure 5.17a Orbit obtained by Numerical Simulation in (U ,V) Projection;
o, = 0, 6 = 1.0.

 

V Displocmnt

-5—

 

p—
I-.
I.-
p—
_
_
'-

 

 

L l
"510 -a -5 -4 -2 o 2 4 6 a 10

U Displocmnt

Figure 5.17b Orbit obtained by MS in (U,V) Projection; 0’, = 0, 8 = 1.0.

142

because of the condition of perfect internal resonance (i.e., (0, and a), are commensurate),

the orbits are fixed in the rotating frame.

The final resonance case (0 ~ 0), - (0,, with internal resonance, reduces to the case

of 0) ~ 2(1),, which has covered in Section 5.4.1.

5.5 Summary of the Chapter

This chapter has been concerned with a theoretical investigation of the nonlinear
dynamics of a perfectly balanced shaft rotating at a nonconstant speed. The nonconstant
speed gives rise to time-dependent coefficients in the equations of motion. Four cases of
parametric resonances, viz. 0) ~ 2 c0, , (0 ~ 2 (0,, (.0 ~ 0), + (0, and 0) ~ 0), - (0, were
considered. Each case was analyzed firstly in the absence of internal tuning (i.e., to, is
assumed to be well removed from 30),) and secondly in the presence of internal tuning
(i.e., to, is assumed to be closed to 300,). The investigation was restricted to the pre-

buckled case (i.e., Q, < 00) and approximate solutions were obtained using the method of

multiple scales.

No parametric resonances were possible in the case of a perfectly balanced
symmetrical shaft either with or without internal resonances. However, for a shaft that is
close to circular, it was found that parametric resonances occur. In the absence of internal
resonance we observed a single frequency response with a period twice that of the
excitation. This resulted in an elliptic orbit of the shaft in the rotating frame. When the
internal resonance is present, the response of the shaft contains two frequencies and the
frequency response curve is much more complex than that of the former case. In the later

case, we found the existence of such phenomena as non-existence of steady-state motions,

143

coexistence of steady-state motions and amplitude modulate motions and re-stabilization of

the trivial solutions.

CHAPTER 6

ANALYSIS OF A ROTATING, SIMPLY SUPPORTED SHAFT
WITH 0, > a,

6.1 Introduction

The source of the nonconservative forces which act on rotating shafts becomes
particularly important as one passes beyond the critical speed. These forces can be divided
into two categories. The first category contains forces defined as external frictional forces.
These are caused by contact of the rotating shaft with fixed components. The second
category contains forces defined as internal frictional forces. These have two fundamental
components: hysteresis forces and structural damping forces. The former arise as a result
of the rate of deformation in the elastic shaft, whereas the latter arise as a result of micro-
shifts between individual parts of the rotor structtu'e. As was pointed out by Tondl [1965],
when the rorational speed of the shaft is above 90, internal fiictional forces, under certain
circumstances, can cause instabilities, whereas external frictional forces always assist in

damping the vibrations.

The influence of the internal damping on the shaft's response is one of the effects
we shall study in this chapter. A number of the results are based on numerical integration
studies since it is extremely hard to obtain even a first order approximate solution to the
complex differential equations which govern the shaft's motion. However, some analytical
results have been obtained for the post buckled behavior of a perfectly circular shaft
subjected to a nonconstant spin rate. The analysis begins by converting the governing

equations from Cartesian coordinates to polar coordinates. This allows one to find the

144

145

buckled, or equilibrium position that the shaft will adopt when it is Spun at a constant rate.
It is motions about this buckled position which we then go on to investigate. A shift in the
coordinate system is undertaken and approximate solutions are sought to the resulting
coupled, differential equations, under the influence of the small speed fluctuations. In

particular, we look at the frequency, (1), of speed fluctuations being close to the whirl

speed, 6.

We also report on findings related to almost circular shafts. Unlike the pre-buckled

case, where as could be altered in order to obtain an internal resonance condition of

(0, =5 3(1),, this can not be accomplished for the post-buckled case. This results from the
fact that (02 >> 0),, for all Q, > (20. A limited amount of analysis is reported which shows

the type of response that might be expected and numerical simulations are completed to
investigate period doubling bifurcations and chaotic motions for the case of (1) ... (0, and

m~2mp
6.2 Exactly Circular Cross Section

In this section we investigate the post-critical behavior of shafts with exactly
circular cross section (i.e., 5 = 0). The influence of internal damping will be considered at
two different orders of magnitude.

6.2.1 Equilibrium Position

In order to find the buckled positions of the Shaft, it is convenient to introduce the

polar coordinates R and 0, defined by the relation

U+iV=Rei¢ (6.1)

146

where U and V are nondimensional modal displacements.

Substituting equation (6.1) and 8 = 0 into equations (2.40) and (2.41) and separating real
and imaginary parts yields '

114-2euek+eu,R+(Qoz-Qz)R-ZQRé-R‘iz'fenozR3=0 (6.2)

R$+RQ+2£u,(QR+R0)+euiR0+29R+2R0=0 (6.3)

Defining II- and 5 to be the equilibrium solution set of equations (6.2) and (6.3), they can
be found by noting that Q = Q, and by setting all time derivatives equal to zero. Hence we

have

(Qoz—Q,2)§+efloz'§3=0 (6.4)

2eueoj=0 (6.5)

The trivial solution '1? = 0 corresponds to the undeformed configuration of the shaft and is

always a solution. The non-trivial solution exists only when as > 00 and 11,, = 0

(i.e., no external damping) and is given by

1
i - (a 2 _ 8 2) 5
— 3 902 (6.6)

However, in practice there will always be damping and in such a case there is another

possible solution set in which It = R=0, R = fi and (i=0, 6:45, where a is defined

147

to represent the fact that the shaft whirls at a speed different from the rotational speed of the
shaft (i.e., nonsynchronous whirling).

Equations (6.2) and (6.3) now become
(Roz—{2,2)§+20,§6-§62+8002§'3=0 (6,7)

2eu,(EQ,-§as)-eu,'fifi=0 (6.8)

From equation (6.8), we have

ro'-—i—2“ 9' (ism)

2 lie 4’ iii (6.9)

Substituting equation (6.9) into equation (6.7), we have

1
R- : [(Q'-6)2-902] 3
8902

(6.10)
Noting that we want Ti to be real, we can solve equation (6.10) to find the value of Q, at

which a non-trivial value of i just appears. Calling this value of (2,, 0: , it is found to be

n:=oo+—&J2“ a
“i (6.11)

This is the first critical speed at which the trivial solution loses its stability and bifurcates
into a stable periodic solution which whirls at a speed different from the rotational speed of
the shaft (for stability details see, for example, Tondl [1965], page 20). Also, from

148

equation (6.11), we found that if u, = 0, then 9: -> °°. This means that, for a symmetrical
and perfectly balanced shaft, if we assume tr, = 0, whirling motions will never occur.

6.2.2 Coordinate Transformation

Before employing the method of multiple scales to obtain approximate solutions for
the motion about the buckled position, we first transfer the coordinates of equations (6.2)

and (6. 3) such that the new coordinates describe the motion about the buckled position.

Let the new coordinates be q and (i) , such that

R=R+q (6-12)

¢=—Et+q)+¢o (6.13)
where 410 is a constant depending on the initial conditions.

Substituting equations (6.12) and (6.13) into equations (6.2) and (6.3) and setting
(I: Q,+EQAsin(ct , we obtain

éi+2[(n.-'c3)’—flo’]q-2((2.—'c6)(q+i)(i>—(win)2
+3600qu§+eflozq3+e(2u,+u,)q=28(q+§)QAq')sinm

+28(Q,-(1)')QA(§+q)sin0n+ezn,,2(q+§)sin (1):2
(6.14)

149

(Q+§)¢+2<i¢+2(9.-5)€1+6(q+§)¢(Zusflls)
+28ueq(Q,—0)')-eu,qfi =-SQA(o(q+§)coscn
—2£QAQSin(0t—2£2u,QA(q+'§)sin0rt (6°15)

Note that these equations describes the response of a shaft about the buckled positions to

combined parametric and external excitations.

We will now seek a first order approximate solution to the two equations. Before
presenting this, we note that when the shaft operates above the first critical speed
(i.e., 9. > 9: ), internal damping has a destabilizing effect (i.e., it acts as a source of
energy to the system). We therefore might expect the internal damping to play a very

important role in the final form of the approximate solutions. This is indeed the case and it

is necessary to assume two different orderings for the size of (1,. In the analysis to follow

we will consider two cases: (a) the internal damping appears at the same order as
nonlinearity and (b) the internal damping appears in the lowest order of the perturbation

equations.

6.2.3 A First-order Approximate Solution with [1,: O (1)

In this sub-section we assume the internal damping will appear when the
nonlinearities appear. We seek a first-order uniform expansion by using the method of

multiple scales in the form

Q(T; 8) = 8 (110031) + 82 (120011) + ‘53 q3 (TO’T1)+' ' " ' " " (6.16)

(PCP; 8) = 3 ¢1(T0’Tl) "' 52 (9200.11) + 53 ‘P3(TmT1)+' """"" (6.17)

150

Substituting equations (6.16), (6.17), (3.6) and (3.7) into equations (6.14) and (6.15) and

equating coefficients of like powers of e, we obtain

Dozq,-2(Q,-'(§)§D02(p,+2[(9,-5)2’902] q:
:2 (Q.-'(i)')QA§sin0n (6.18)

fiD02¢1+2(Q,-E)Doql=-QA(DfiCOSGYC . (6.19)

The solutions of equations (6. 18) and (6.19) can be expressed in the form

q, =9“: +91.) (6.20)

§¢1=P1=P11+Prp (6.21)
where

q“, = A,(T,) + A,(T,) exp( i(1),To) + cc (6.22)

P111 = A1A1(Tl) + A2 A2(T1) ¢XP( i(”11.0) + CC (6.23)

qip=0 (6.24)

p,P = 24—1: (:0sz
a) (6.25)

151

m,=[6(n,-'oi)2—2Q,2]i

 

= (Q'-E)2-Qoz 1

A1 _.
(2.-0)

A, = 2(0, - 0)) i
0.),

cc stands for the complex conjugate of the preceding terms.

In equations (6.13) and (6.14), the excitation comes from two parts: (a) parametric and (b)

direct forcing. If the analysis is continued to the next order we can shown that the A, and
A2 decay to zero and thus the direct forcing term only influences p,p which in turn only
effects the whirling speed of the shaft, and not the radius of the whirl. Therefore, for a
perfectly circular shaft with a very small value of internal damping 11,, the orbit of the

whirling cannot be parametrically excited.
1
6.2.4 A First-order Approximate Solution with u, = O (E)

In this sub-section we investigate the effect of internal damping on the response of a
perfectly circular shaft rotating at nonconstant speed. We first order the internal damping

so that it will appear in the lowest-order perturbation equations. To this end we introduce
fi,, such that ft, = e 11,.

152

Substituting equations (6.16), (6.17), (3.6) and (3.7) into equations (6.14) and (6.15) and

equating coefficients of like powers of e, we obtain

Dozqr-2 (Qt-E)§D0¢l+fii D091+2[( as-E)2-QOZ] ‘11
=2 (9,-7.5)QAisin0fl (6.26)

R'Dozq),+2(Q,-E)Doq,+ft,§Doq),-fi,q,'(B=-QA(0§cosm

(6.27)

The solutions of equations (6.26) and (6.27) can be expressed in the form
(11 =9": “11;: (6.28)
‘11—‘91: Pr =plh +P19 (6.29)

By investigating the eigenvalues of equations (6.26) and (6.27) we find that the

homogeneous solutions will be damped out (i.e., q,,, and p,,I -)O). The particular

solutions can be found as follow

‘11,): Alp sin((m:+a,) (6.30)

Pro: A2,, cos(“‘09) (6.31)

where

153

A“, = ,lHfi + H,2, a, = tan’l-Iil

[_— _ H
A2p= H32+H42, (12';'tanli1
4

and

H,, H2, H3 and H,, are defined in Appendix F.
Hence, the steady state solutions have the form

_ A
U = Rcosq) = [ R + 8A,psin(0rr +a,)] cos[-Et + e -..—.2."- cos(0.)t — 0(2) + 4),,
R (6.32)

_ A
V = Rsind) = [ R + eA,Psin((01: +a,)] sin -'(Bt + (ht—.21 cos((or —a,) +00
R (6.33)

where 4),, depends on the initial conditions. Note that both the amplitude and frequency

parts in equations (6.32) and (6.33) have a small oscillation term with frequency (0 and the

amplitudes A,p and A2" which depend on (1,, (re, 05, {2A and (0.
6.2.5 Numerical Results

In this sub-section we present the numerical results using the set of parameter
values: Q, = 10.1, 11,, = 173 and e = (513?. The original differential equations (2.40) and
(2.41) are simulated on a digital computer. After transient motions decay, the steady-state
solutions are recorded and compared to those predicted by the method of multiple scales
(equations (6.32) and (6.33)).

154

We first demonstrate the effect of internal damping It, on the symmetrical shaft
response. Figures 6.1a-c show the orbits of the shaft for three different values of It, with
0) = 6)". From Figure 6.1a we observe that the shaft has a nonsynchronous whirling
motion with a circular orbit if the value of tr, is very small (note that U and V are rotating
coordinates). However, for larger values of hi, this circular orbit will change to a

"circular" orbit with a overhang. Note that the the size of the overhang and the radius of

the orbit increase as the value of 11, increases.

Figures 6.2a—i depict the results from both the numerical simulations and the

method of multiple scales for a selection of different (0 values, viz. (0 = n (0", n = l, in 2,

3. From these figures, we observe that the approximate solutions and the numerical
solutions are in close agreement. It should be noted that the phase angle constant 0,, in
equations (6.32) and (6.33) depends on the initial conditions, which in turn fixes the
orientation of the overhang. Also, from these figures we observe that if n is a integer, the
number of overhangs is equal to 11. If n is a irrational number, then the response of the
shaft is non-periodic. The corresponding results for such a case are shown in Figures
6.3a-b where 0) = 42 E. In this case the simulations represent approximately 10 cycles of

(r).
6.3 Close to Circular Cross Section
In this section we consider the shaft to have close to a circular section (i.e., 8 ,e 0) .

Employing the same procedure as was used in the Section 6.2, the post-critical behavior of
the shaft is investigated.

V displocanent

V displacement

155

 

60

40'—

20*-

 

 

 

U displacement

Figure 6.1a Orbit in (U,V) Projection; pi=2001t2, a) = ‘03.

 

60

20'-

 

 

l l l l 1,
-60 -40 -20 0 20 4

 

60

O

U displacencnt

Figure 6.1b Orbit in (U,V) Projection; u, =5001t’, co = (0'

156

 

60
nun-rice! aimlotion

40—

20"

V displocanent

-20 —

 

 

l l l l l
-60 -40 -20 O 20 40 60

 

U displacement

Figure 6.1c Orbit in (U,V) Projection; u; =10001t2, m = 6.

 

60

40'-

V Displacement

-20 '-

 

 

l l l l
.60 40 -20 0 2°

 

60

Sr

U Diaplocancnt

Figure 6.2a Orbit obtained by MS in (U,V) Projection; u,=10001t2, a, = a.

157

 

60
mrical aimlatian

40—

V displacement

40—

 

 

1 1 1 1 if
'50 -40 -20 0 20 4O 60

 

U displacement

Figme 6.2b Orbit Obtained by Numerical Simulation in( U,V) Projectio ;

2 -3—

 

V Displacement

 

 

 

 

'50-60 40 ~20 0 20 40 so

U Displacement

Figure 6.2e Orbit obtained by MMS in (U,V) Projection; u, =10001t2, m =

hfltd
cl

158

 

60
nmrlcal sinulation

40'—

20'-

V diaplacement

-40 )—

 

 

l 1 1
~60 40 ~20 O 20 4O 60

 

U displacement

Figure 6.2d Orbit obtained by Numerical Simulation in (U ,V) Projection;
11, =10001r2, (0 = 2 6.

 

60

20—

V Displacement

-20 '-

 

 

1 l L J
-so -40 -20 o 20

 

60

81-

U Displacement

Figure 6.2e Orbit obtained by was in (U,V) Projection; u,=10001t’, 0.) = 2 a.

159

 

60
nautical sinulatien

20—

V displacement

40'-

 

 

1 r 1 1 L
-60 -4o -20 a 20 4o 60

 

U displacement

Figm'e 6.2f Orbit obtained by Numerical Simulation in (U ,V) Projection;
u,=10001t2, to = 325.

 

60

20'—

V Displacement

 

 

! l L L
-60
~50 -40 -20 o 20 40 so

 

U Displacement

Figure 6.2g Orbit obtained by MMS in (U,V) Projection;j.t,=10001t2, 0) = 3 a.

 

 

 

 

 

Figure 6.3a Orbit obtained by Numerical Simulation in (U ,V) Projection;
115100019, (0:426.

 

 

 

 

 

Figure 6.3b Orbit obtained by MMS in (U,V) Projection; u, = 100018, to = a a.

161

6.3.1 Equilibrium Position

We first transform the equations (2.40) and (2.41) into polar form, by substituting
equation (6.1) into equations (2.40) and (2.41) and separating real and imaginary parts to
find

§+2£tteR+eu,R+(§02-Qz)R-ZQR0-R92
+139,2 113+ O,Rcos2¢=0 (6.34)

R$+RQ+22uc(QR+R0)+eu,Rq')+2QR

 

+2Ro—Odksin2¢=0 (6.35)

where

-- 2 2+8 2

.. O

90 2 0 (6.36)

and
-8
9d = 7 902. (6.37)

Setting (2 = Q, and all time derivatives in equation (6.34) and (6.35) equal to zero, we find
the equations govern the steady-states I? and 6 to be

(@2423)R‘+eO,2§3=-O,§cos2$ (6.38)

280,§Q,=Qd§sin 25 (6.39)

162

Three equilibrium positions can be found, either I? = 0 or

I

2_—2 2_ 2 2
"' [as (20 i‘JQd (zei‘leas) I, §1>§2

 

R,,,= c Q 2
° (6.40)

 

- _ -28},l Q
¢=ml[ 2 ‘2 c I2:2']

Note that in and 5 are independent of the value of 11,.

Again, we can find the values of (2,, say 9;, where equation (6.40) just begins to

generate real, positive values of K. They are

-1
n;_,=[rroz_2ezu,=r,/(mi-2e2tt.2)’—(fi.‘—a.’)]2 and 0:42;

(6.42)

 

For 9. < 9; only one stable trivial solution exists. For 9; < 0, < 9; , two solutions exist:
the unstable trivial solution and a single stable non-trivial solution. For 9; < 9., there are
three solutions: one stable non-trivial solution (i.e., E), one non-trivial solution (i.e., Hz)
and one trivial solution. The stabilities Of the latter two depend on u, and [1,, (see
Ariaratnam [1965] for details). For 0, sufficiently close to no, we only need to consider

one stable non-trivial solution (i.e., 13,).

163

6.3.2 Coordinate Transformation

Once more, we transfer the coordinates of equations (6.36) and (6.37) such that the
new coordinates describe the motion about the buckled position (i.e., 15 = i, ) .

Let the new coordinates be q and (p, such that

R = R + q (6-43)
and

¢ = 5 + (p (6.44)

Substituting equations (6.43) and (6.44) into equations (6.36) and (6.37), we obtain

(1+1:(211..+flo211;)c'1-2£2.(0+Ti)(ii-(<1+'Pli')<i>2
+2[ {2.2—502+J942—(28peflf ]q+e nozq2(q+3'fi)

 

2 3
+Qd(q+§)[cos2$ (-£-2—(2p—)-+-----)—sin2$ (2¢-Q§)_+ ------ ):|

= 2 eflAfl, (q +fi) Siam-+2 13 9A4) ( q+§ ) sinan-t- 82 9A2 (q-1--li)sin0)vt2
(6.45)

l
(q+§)<'t‘>+2n.ét+2¢q+c(2u.+u.)<q+i)¢-2[aaz-(2cu.fl.)2]5time

2 3
+Qd(q+§) [sinZE (—-(3(2P—)-+-----)+cos2$ (.S%fl+.....):|=

— e 9,, (o (q+§)cos0n-28 (2,6 smart-2 ezpenA(q+§)sinan
(6.46)

164

6.3.3 Approximate Solutions

Assume a first-order expansion in the form

‘1“; 8) = 5 ‘11(T0vT1)+ 82 ‘12 (T0111) ‘1' e3 ‘13 (T031) + ' (6.47)

‘P(Ti£)=€‘P1(TorT1)+€2‘Pz‘Tot'm‘l’ss‘PaaorTO'l'””°“" (4,48)

Substituting equations (6.47) and (6.48) into equations (6.45) and (6.46) and equating

coefficients of like powers of e, we obtain

 

I)02‘11' 2 as R- Do‘P1+2[ Qrz‘fioz'H/ 9112' ”3111:9021 ‘11

= 2 9,, Q, i sinan (6.49)

1
infant), Doq,+2 [Of-(2811,52,)2F iiq), =- QAEO) cosan

(6.50)

The solutions of equations (6.49) and (6.50) can be expressed in the form
‘11 =‘1111 +‘lrp (6.51)
i‘PI= P1 =P1b +P1p (6.52)

where

qta = A101) CXP( i‘orTo) + A,(T,) cxP( in); To) + cc

and

165

P11. = A1A1(T1) CXP( 1‘01 To) + A,A,(T,) exp( iszo) + cc
q,,, = A,p sin an

p,p = A2,, costar:

A _ 29,0)L
“- l
2[Q,,2-2(tt:p,,fzs)2 ]2—(1),2

 

 

298(1)2
2[ Qd2-2(81'leas )2 12 “(022
1 _
6952+4[Qd2—(28uens)2]5(1'8032)—2§oz-(6982-2£—202)2
(0,: 2
1 l
6Qsz+4lfld2—(2 eper:12(1+3QSZ)_2-Qoz+(6982_2§02)2 2
(02: ..

 

2

A,p and A2,, are depend on (2,, Q , (0, (Id, T20, '13 and 1.1, and cc stands for the

complex conjugate of the preceding terms.

Note that A, and A, are arbitrary complex functions of T, which can be determined

at the next level of approximation. However, due to the complexity of these equations it

166

was found impossible to proceed with analysis beyond this point. Instead, we employed

numerical Simulation techniques to explore the bifurcated and chaotic motions of this case.

6.3.4 Numerical Results

Equations (2.40) and (2.41) were simulated on a digital computer. The parameter

values used, unless otherwise stated, were:

as = 9.9, 5 = 1.0, lie = 256, u, = 200a2 and e = 1,1,7)?

Before investigating the shaft's response to a parametric excitation, we calculate the

buckled positions of the unperturbed case (i.e., 89,, = 0). From equation (6.1) we can

express these positions as

U- = R, cos¢ (6.53)

and

V = R, sin 4) _ (6.54)

Substituting the above parameter values into equations (6.40), (6.41), (6.53) and (6.54),
we find '17 = :1: 18.8489 and V = $0.017. Obviously, there is a big difference between '17
and V. Hence, we next try another value of 8 = 0.01 (i.e., the shaft has very slight
asymmetry) and we find U = :1: 18.6468 and V = :1: 1.69855. The value of U is still much
larger than the value of V. This suggests that, for an unsymmetrical, buckled shaft, its
post-buckled behavior may be similar to that of a rotating buckled beam. We will see that

this is the case in some of the numerical simulations to follow.

167

We now use numerical integration to find the trajectory of the free response of the
rotating, unsymmetrical Shaft by setting 80A = 0 in equations (2.40) and (2.41). Figures
6.4a-b show trajectories of the free response depicted in a (U ,V) projection for two

different 11, values with 8 = 0.5. In both cases the orbits approach the right hand buckled

position. A different choice of the initial conditions would cause the trajectory to go to the

other buckled position. Comparing Figure 6.4a with Figure 6.4b, we observe that the rate
at which the trajectory of the free response approaches the buckled position depends on u, .

We will next consider 89,, at 0 and show that the shaft will vibrate about one of

the buckled positions or encompass the two buckled positions. The following sub-sections

will consider the cases Ofa) z 20), and (I) ~ (1),.

6.3.4.1 The Case of arr-2(1),

We first use numerical Simulation to observe the trend of the frequency response

curve. Equations (2.40) and (2.41) are simulated on a digital computer for fixed I), = 9.9,
OA = 5760, 1te = 256, It, = 200, 5 = 1.0, e = (51512 (and hence to, = 0.4885) and to is
varied. The results are depicted in a (U ,V) projection in Figure 6.5. From this figure we

observe that the magnitudes of the displacement U and V increase as 0) decreases. That

means the nonlinearities are of a softening type.

Next, instead of varying a), we present the effect Of varying (2,, on the shaft's
response with a) = 0.947 18 2(0,. The results of the numerical simulation are shown in

Figures 6.6 and 6.7 and are summarized in Table 6.1.

 

enent

 

V displac

 

'5 _ t
"1'11'1’w 1111"

~60 ~40 ~2LO 0 20 4O 60

 

 

 

U Displacement

Figure 6.43 Trajectory of the Free Response in (U,V) Projection; (1, = 201:2, 8 = 0.5.

 

 

 

10
s 1.—
‘e'
g
a 0 ' C
n
.6
>
-5 —
-10 1 l 1 1 J
~50 ~40 ~20 0 20 40 60

 

U Displacement

Figure 6.4b Trajectory of the Free Response in (U ,V) Projection; 1,1, = 2001:2 , 8 = 0.5.

169

 

0.75

 

 

 

 

U
I 576
2 288
J 0
0.5 - 4 -un
5 ~57!
G ~0“
7 -1152
0 ~I6‘0
0.25).. 9 4720
E
u
3 _
a 0
D
:6
>
~0.25 _
~0.5 —
-o.75 ‘ 1
~24 ~20 ~16 ~12

U displacement

Figure 6.5 Orbit obtained by Numerical Simulation in (U ,V) Projection;
0) = 2 (o,+t-: o, o from 576 to -1728.

 

0.8 '-

V displacement

 

 

. . .1 . 1
~30 ~25 ~20 ~l 5 -1 0 ~5

 

U Displacenent

Figure 6.6a Orbit of Period 2 Motion in (U ,V) Projection; (2,, = 864, (1) =0.947.

170

 

0.8 T

V displacement

 

 

 

_,'2 1 41, 1 1
-30 -25 ~20 -1s -10 -s

U Displacenent

Figure 6.6b Orbit of Period 4 Motion in (U ,V) Projection; (2A =1296, (1) =0.947.

 

V displacement

 

 

 

 

'-30 -25 -20 -1s -10 -s

I
U Displacenent

Figure 6.6c Orbit of Period 8 Motion in (U,V) Projection; 0A = 1340, (1) =0.947.

171

 

 

 

 

1.2
0.8 —
0.4 -
E
0
O
a 0
O
'0
>
-o.4 -
-o.s —
-1.2 1 L 1 1
-so -zs -20 -15 -10 -s

 

U Oisplacmnt

Figure 6.6d Orbit of Period 16 Motion in (U ,V) Projection; (2,, = 1350, 0) =0.947.

 

 

V displacement

 

 

 

p-
p—
I-

 

l
2—30 ~20 ~10 0 1 O 20 30

U Displacement

Figure 6.7 Orbit of Chaotic Motion in (U ,V) Projection; 0,, = 1440, (0 =0.947.

172

Table 6.1 Summary of Bifurcated and Chaotic Motions for 0) = 0.947.

 

  

 

 

 

 

Type of Motion

 

Initial Conditions

(U(0).U(0); V(0).V(0)) ;

864 Figure 6.6a 21 (1.0, 0.0; 1.0, 0.0)
1296 Figure 6.6b 4T (1.0, 0.0; 1.0, 0.0)
1340 Figure 6.6c 8T (1.0, 0.0; 1.0, 0.0)
1350 Figure 6.6d 16T (1.0, 0.0; 1.0, 0.0)
1440 Figure 6.7 chaotic (1.0, 0.0; 1.0, 0.0)

 

 

Figures 6.6a-d Show a period doubling bifurcation depicted on a (U ,V) projection.

This series of bifurcations, if continued, would lead to chaos. The simulation indicates that

at Q, = 1440, a chaotic motion occurs as shown in Figure 6.7. In this case it is convenient
to describe the response of the shaft by projecting the trajectories on to the (U ,U) and
(V, V) planes and sample the values once per forcing period (as in a Poincare map).
Figures 6.8a-b depict the points in the (U,U) and (V,V) planes corresponding to the
motion shown in Figure 6.7. Note that Figure 6.8a is similar to Figure 4.24b in Chapter 4,

and the motion in the U direction dominates the shaft's response.

6.3.4.2 The Case of 01.0,),

In this sub-section, we consider the case of 0) = (1),: 0. 4885. A period doubling

bifurcation can occur as one varies the value of 911- The results of numerical simulation

are summarized in Table 6.2 and plotted in Figures 6.9-13.

For a small value of QA, the shaft vibrates about one of the buckled positions.
Figure 6.9 shows the limit-cycle attractor around the left buckled position with S2,, = 288.

173

 

10

U
n)
1

Velocity

_2-

' ”~'\' 18' - -

-4 —

~6 —

 

 

)—

 

-10 1 1 1 1 1
~30 ~25 ~20 ~15 ~10 -5 0 5

 

Displacsnent U

Figure 6.8a Map of Figure 6.7 in (U ,0) Projection; 9,, = 1440, 0) =0.947.

 

1}
.°
1

Velocity

9.9.99.9
I

 

 

 

 

Displacement V

Figure 6.8b Map of Figure 6.7 in min Projection; OA = 1440, 6) =0.947.

174

 

1.5—

V displacmnt

..1—

~1.5"

 

 

_2 1 1 1 1 1 1 1
~40 ~30 ~20 ~10 0 10 20 30 4O

 

U displacenent

Figure 6.9 Orbit of Period 1 Motion in (U ,V) Projection; Q, = 288, (0 =0.4885.

 

0.5 ‘-

V displacement

~0.5"'

~1.5 —

 

 

 

_2 1 1 1 1 L 1 L
~40 ~30 ~20 ~10 O 10 20 30 40

U displacement

Figure 6.10a Orbit of Period 2 Motion in (U ,V) Projection; (2,, = 576, a) =0.4885.

175

 

 

 

 

2
1.5 —
1.—
E 0.5—
U
0
a 0‘
1n
.1;
> 4.5—
~1.5 —
_2 l 1 l l J l 1
~40 ~30 ~20 ~10 0 10 20 30 40

 

U displacement

Figure 6.10b Orbit of Period 4 Motion in (U ,V) Projection; 0,, = 1100, 0) =0.4885.

 

 

V displacement

 

~1.5 —

 

 

_2 1 1 1 1 1 1 4
~40 ~30 ~20 ~10 0 10 20 30 40

 

U displacenent

Figure 6.100 Orbit of Period 8 Motion in (U,V) Projection; 9A: 1110, a) =0.4885.

176

 

1.5 —

V displacement

 

 

 

 

1 1 1 1 1 1 1
~40 ~30 ~20 ~10 O 10 20 30 40

 

U displacement

Figure 6.10d Orbit of Period 16 Motion in (U ,V) Projection; Q, = 1116, (0 =0.4885.

 

0.5 -

V displacement

 

-1—

 

 

 

1 1 1 1 1 41 1
~40 ~30 ~20 ~10 0 10 20 30 40

 

U displacement

Pigme 6.11 Orbit of Almost Periodic Motion in (U ,V) Projection;
(2,, = 1120, (0 =0.4885.

 

V displacement

 

 

 

 

~40 ~30 ~20 ~10 0 10 20 30 40

U displacement

Figure 6.12 Orbit of Chaotic Motion in (U ,V) Projection; (2A: 1152, (0:0.4885.

 

V displacement

 

 

 

 

U displacement

Figure 6.13 Orbit of Chaotic Motion in (U ,V) Projection; QA = 1440, (0:0.4885.

178

Table 6.2 Summary of Bifurcated and Chaotic Motions for a) = 0.4885.

 

 

Type of Motion Initial Conditions
(U (0).U(0); V(0).V(0))
288 Figure 6.9 1T (10.0, 0.0; 20.0, 0.0)
576 Figure 6.10a 2T (10.0, 0.0; 20.0, 0.0)
1100 Figure 6.10b 4T (10.0, 0.0; 20.0, 0.0)
1110 Figure 6.10c 8T (10.0, 0.0; 20.0, 0.0)
1116 Figure 6.10d 16T (10.0, 0.0; 20.0, 0.0)
1120 Figure 6.11 almost periodic (10.0, 0.0; 20.0, 0.0)
1152 Figure 6.12 chaotic (10.0, 0.0; 20.0, 0.0)
1440 Figure 6.13 chaotic (10.0, 0.0; 20.0, 0.0)

 

 

 

 

 

 

However, for larger values of DA, this attractor becomes unstable and gives rise to yet

another new, larger limit-cycle attractor that encircles both buckled positions, as shown in
Figure 6.10a. As we continue to increase the value of the 9A, this large outer trajectory
bifurcates, as shown in Figures 6.10b-d. This cascading of period-doubling bifurcations

will ultimately lead to a chaotic motion. However, before chaos is observed an almost

periodic motion is found for a 0,, = 1120, see Figure 6.11 which represents approximately

100 forcing periods. The trajectories of the chaotic motion associated with Q, = 1152 and

1440 are shown in Figures 6.12 and 6.13, respectively. Clearly the chaos in Figure 6.12
is not as well developed as in Figure 6.13, but it is certainly more than simple quasi-
periodic motion. Again, the trajectories in Figures 6.12 and 6.13 are projected on to the
(U ,U) and (V,V) planes and sampled once per forcing period. The results are shown in
Figures 6.14a-b and 6.15a-b, respectively. Note that the points in Figures 6.14a-b appear
to lie on a set of curves. However, the points in Figures 6.15a-b reveal a fractal-like
pattern (i.e., strange attractor) and Figure 6.15a is similar to Figure 4.27f of the rotating

beam case.

179

 

Velocity 0

_2-

-4—

 

 

 

-10 l l l 1 l l 1
~20 ~15 ~10 -5 0 5 10 15 20

Displacenent U

Figure 6.14a Map of Figure 6.12 in (U ,0) Projection; Q, = 1152 a) =0.4885.

 

 

~0.05 —

Velacrly V
c:
l
eenfi3==fia"

 

 

 

Displacement V

Figure 6.14b Map of Figure 6.12 in (V,V) Projection; (2,, = 1152, a) =0.4885.

180

 

 

 

10
a 1—
6 — r.
.4 ‘
”43-"
4 - (4.4-4"" . .
.4 " «V
a 2 ~ r- 4%
.1- 1.x .. v 8
3 II: . -‘ ‘
a 0 "' c . . s
a h ,1
; .3“. s " -‘-.-‘ .5
> - )—- 3 0‘ V . l r,
2 fl ?
Z... of
-4 — ° '
-6 --
_a .—
J L l l 1 l L l l 1 1

 

 

 

~10
~30 ~25 ~20 ~15 ~10 ~5 0 5 10 15 20 25 30

Displacemnt U

Figure 6.15a Map of Figure 6.13 in (U,U) Projection;OA = 1440, 0) =0.4885.

 

Velocity V

 

 

 

 

'-2 --1.5 -1 -o.s o 0.5 1 1.5 2

Displacement V

Figure 6.15b Map of Figure 6.13 in (v.17) Projection;nA = 1440, (0:0.4885.

181

6.4 Summary of the Chapter

In Chapter 6, we have investigated the post-critical behavior of a perfectly balanced
shaft by applying the method of the multiple scales and employing numerical simulations.
When the rotational speed of a symmetrical shaft is over the first critical speed, the shaft
will whirl with a circular orbit at a speed different from the rotational speed. The radius of
the orbit depends on (2', (1e and 1.1,. NO parametric resonances about the orbit of the
whirling were possible in the case of a symmetrical shaft with small internal damping.
However, the orbit of the whirling will be distorted (i.e., a circular orbit with a overhang)

in the case Of a larger value of internal damping.

When the rotational speed of an unsymmetrical shaft is greater than the
corresponding first critical speed, the straight equilibrium position is unstable and the shaft
buckles. Due to the complexity of the equations, no approximate solution were found for
this case. However, bifurcated and chaotic motions were found numerically to exist in
unsymmetrical, buckled shafts. The motions in the U direction dominated the shaft's
response. Therefore, for an unsymmetrical, buckled shaft, its post-buckled behavior
seems to be similar to that of a rotating, buckled beam.

CHAPTER 7

EXPERIMENTAL WORK

7.1 Introduction

The purpose of this chapter is to present an experimental investigation of a physical
model (i.e., a rotating beam) whose motion is governed by the differential equations which
have been analyzed in Chapters 3 and 4. The results of this experimental study confirm the

presence of the majority of the phenomena which were theoretically predicted to exist.

In Section 7.2, the details for developing the experimental apparatus are described.
Attention is focused on the measurement techniques of the motor's speed and beam's
response. The experimental procedures and corresponding results of both pre-buckled and
post-buckled cases are described in Section 7.3 and 7.4, respectively.

7.2 Experimental Setup

The laboratory test equipment consisted of:

1. AND AD-3525 FFI‘ Analyzer.

2. WAVETEK 2M Hz Variable Phase Synthesizer, Model 650.

3. PM6666 Philips 120Mhz Programmable Timer/Counter.

4. Stanford Dual Channel Low-Pass Filter, Model SR640.

5. PM 3365 Philips Oscilloscope.

6. Measurement Group Signal Conditioning Amplifier, Model 2210.

182

183

7. ALLEN -BRADLEY Bulletin 1326 AC Servo Motor.

8. Wendon Slip Ring, Part Number WSD-1750-102.

9. Analog Devices AD-650 Frequency-to-Votlage Converter.
10. Hewlett-Packard 6235A Triple Output Power Supply.
11. MASSCOMP 5550 Lab. Workbench 2.1.

The instrumentation schematic is shown in Figm'e 7.1.

The experimental model consisted of a spring steel beam with a length of 175 mm.
and 0.508 mm. x 12.7 mm. cross section rigidly clamped at one end which is driven by an
AC servo motor. The beam's first natural frequency was experimentally found to be 13.50
Hz by simply striking the beam and studying the frequency at which it responded. The
damping ratio of the beam was found to be approximately 0.01 from a transient decay test.
The experimental system was rigidly mounted on an aluminum plate which was vertically
hung on the wall in order to negate the effect of gravity. Details of the mechanical

construction of the experimental system is shown in Figure 7.2.

The experimental model was driven by an AC servo motor. The speed of the motor
was precisely controlled by a feedback system. The speed of the motor, (2 (t), was set by
the voltage output from a high accuracy signal generator. In all the experiments, the value
of the DC offset of the signal generator was set equal to 8 Vdc. An encoder attached to the
motor generated pulses at a rate of 1,000 per revolution. We then used two techniques to
monitor the rotational speed. Firstly, we used a counter to measure the frequency of the
pulses from the encoder and hence to measure the mean spin rate, (2,. A control box was
used to adjust the voltage of the DC signal to Obtain a desired mean spin rate. After
obtaining the desired mean spin rate, we superposed a small sinusoidal signal on the DC

signal to generate a periodic perturbation in the spin rate. Next, a Frequency-to-Votlage

184

£58.33. .35Etoeum— a... He Enema:— uzangom E. 9.35

 

 

 

 

 

_ .55..—

a

 

 

 

Stop—EU
533>.8.05=e9.h

 

 

>

2.5.5
coke.—

 

 

« , '— 3:35.5—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.3m
3 G n. J ‘
:34
crew '— 3525. I 5....— cmmm $689.:
«new 535
bee.— fiaE I became—.5. 3.2850
new.
35 ea . 1 + , _ .a
.832 no:
A 3:93.80 9
A .— ..Bm

 

 

 

 

 

 

 

 

 

 

 

.5685— 3.3.5
agen—

185

 

      

Motor Support /

 

 

Rigid Coupling

 

Ball Bearing

 

Slip Rings

Shaft

 

Inertia (optional)

Leveling

Clamping Device WaShch

Rotating Beam p

 

 

Q (t) Q?

Figure 7.2 Mechanical Construction of the Experimental System .

186

converter was designed and built to convert the frequency of the pulse from the encoder
into an analog voltage signal. This analog signal was monitored on an oscilloscope and a
PET analyzer.

The amplitude of the beam's response was measured by a strain gage which was
cemented near the clamped end of the beam and wired to form a half-bridge for the signal
conditioning amplifier. The signal of the strain gage was transmitted to the amplifier by
slip rings. Low-pass filters were used to filter out noise of the signals. The cut-off
frequency was set equal to 30 Hz for all the experiments. A two-channel FFI‘ analyzer was
used to monitor and record the response of the beam and the rotational speed of the motor

in the time and frequency domain .

We now present the experimental results of the two cases: (a) Q‘ < 00 and (b)

(2, > no in the following sections.

7.3 The Case of 0. < no

Based on the theoretical work described in Chapter 3, we would expect an

instability to occur close to to = 2(00. Using the result of Chapter 3, this will be in the

neighborhood of 2" 002 - (2,2 where (20 is experimentally found to be 13.50 Hz.

Figures 7.3a-c present the results obtained by completing an incremental frequency

sweep over the range 2000, while holding the mean rotational speed as constant at 9.04
Hz, 10.55 Hz, and 11.0 Hz, respectively. The amplitude, 89A, for all cases was set equal
to 0.0250‘. The "amplitude" term as it was plotted in Figures 7.3a.c corresponds to the

root mean square value of the signal from the strain gage. Representative examples of the

Amplitude (Va...)

187

 

 

5
00:13.50 Hz
Q,- 9.04 Hz
E
4 -— on
, l
O
o
_ o
3 Jo
0
° T
e
D o
o
o
2- . T
o
o
o
o
1—
° T
T I l
o
0‘ A B C ° H F G
19 19.5 20 20.5 21
0’ (Hz)

Figure 7.3a Frequency Response Curve for Q, = 9.04 Hz.

 

21. '

188

 

 

 

 

Figure 7.3b Frequency Response Curve for (2s = 10.55 Hz.

5
no- 13.50 H!
0
Q,- 10.55 H: °
4'- o o
o
o
o
,‘ e
g o
o
>- e
" 3 _. o
o e
o
g o
g, o
.- o
" o
E 2 ’ °
0
o
O
o
‘ —
0
° 0
o c ........ 1-- °°°¢ ......... ---
16 17 18 19 20
0 (Hz)

 

 

 

 

 

5
(2.-13.30m
c
Q,- 11.0091: o°
o
o
o
41— 0°
.0
e
o
g o
o
:’ o
v 3... 0
e
0' e
13 o
3 e
" o
2: o
o
E‘ " °
0
o
o
o
e
o
' _' e
e
o
0 3333333331 1 %333333£333333:33%
14 15 16 17 18
to (H2)

19

Figure 7.3c Frequency Response Curve for Q. = 11.00 Hz.

189

time traces obtained from the strain gage and from the signal representing the fluctuating
part of the motor speed, are shown in Figure 7.4 along with their associated Fourier

transforms in Figure 7.5, respectively.

In general, the form of the experimental response curves depicted in Figures 7.3a-b
agree well with the theoretical predictions, as presented in Figure 3.1. The trivial solution,
an amplitude of zero, becomes unstable over a region close to 2(00 and the resulting
response is of a nonlinear hardening type. There is also a range over which we can
observe multi-valued solutions. However, the type of behavior in the vicinity of the point
where the trivial solutions go unstable, is slightly different from that predicted by the
theory. The theory is based on a perfectly balanced, initially straight beam. In practice this
is hard to achieve and the introduction of small imperfections lead to imperfect bifurcations

(see, for example, Golubitsky and Schaeffer [1984], page 6).

To better explain the results depicted in Figure 7.3a, consider the points A, B, C,
D, E, F, G, H, I, and J as shown on this figure. First, we sweep the frequency, 0), up.
The experiment is started at a frequency corresponding to point A on Figure 7.3a. As a) is
increased, the trivial attractor is stable through point B until point C is reached. As to is
increased further, the beam jump from point C to point D (i.e., the beam moves to the non-
trivial attractor), after which the amplitude of the beam's motion is increased with
increasing to until point B is reached. As to is increased further, the beam jump from point
F. to point F (i.e., the beam returns to the trivial attractor again). Next, we sweep to down.
The experiment is started at point G, the beam stay at the trivial attractor through point P
until point H is reached. As to is decreased further, a small amplitude motion takes place
until point I is reached. As a) is decreased further, a jump of the beam takes place from
point I to point J, after which the amplitude is decreased with decreasing to until point B is
reached. As a) is decreased further, the beam returns once more to the trivial attractor. The

190

 

 

 

 

 

 

 

100m SpeedoftheMotor 32,3113;
(0:15.611:
TIME
V
~100m
0 Time(sec) 800.0m
5 . 000 ResponseoftheBeam
TIME
V
-5.000
0 Timelsec) 800.0m

Figure 7.4 Time Traces of the Fluctuation of Motor Speed and Beam Displacement.

 

100m SpeedoftheMotor n,-rrrrz
cos-15.6w

SPCTFIM

AMP
Vr‘
or

0.000 H

0 Frequz) 20.0
5 . 000 ResponseoftheBeun

 

 

 

SPCTFIM

NIB

AMP
VP

 

 

0.000

 

L
O Freq (Hz) 20.0

Figure 7.5 Spectrum of the Fluctuation of Motor Speed and Beam Displacement.

191

maximum amplitude corresponding to point B is attainable only when approached by

sweeping up. Figures 7.3b and 7.3c show the same trends, but the influence of the
imperfect bifurcation decreases as the spin rate, (28, increases.

Figure 7.6 presents the results from measuring the rms value of the strain gage
while slowly varying the amplitude 89A of the small fluctuation. The mean spin rate, 9‘,
for this experiment was set equal to 9.04 Hz and a) was fixed at 20.1 Hz. Once again, the
general trends as predicted by the theory are the same (cf. Figure 7.6 to Figure 3.3b). We
also note the influence of the imperfect bifurcation.

7.4 The Case of 0' > no

In this section we investigate the post-buckled behavior (i.e., (23 > (20) of the

experimental model. For this case the straight equilibrium position of the beam is unstable
and the beam buckles due to centrifugal effects.

We present the results of two experiments which were canied out on the physical

model. These experiments investigate the beam's response as functions of the frequency to
of the small sinusoidal oscillation. The mean rotational speed (28 is held constant at: (l)

13.58 Hz and (2) 13.70 Hz. It is hard to experimentally measure (00, the natural frequency
of the beam vibrating about the buckled position. However, we do have experimental
values for 00 and, of course 0‘, and therefore we can use the equations in Appendix C to
obtain a "theoretical" value of (00 based on experimental measurements. Hence, we find
(00 = 4.4 Hz when (2‘ = 13.58 Hz and (00 = 4.9 Hz when as = 13.70 Hz. We complete
experiments over the range of (0 = 2 Hz to 12 Hz and thus we will cover resonances at (00

and 2(00.

Mpl i tude (Va...)

192

 

 

 

 

00- 13.50 Hz
Q,- 9.04 Hz a
3 _c.) - 20.1 Hz 0 °
00
00°00
0°00
00

.5" 0°°

0°°

00
O
O
O

2 —
.5"-
1 —
.5 -'

O

00
o°°°
o, e A : £::o° I Li I I
0.1 0.2 0.3 0.4 0.5 0.6 0.7
:0, (Wm)

Figure 7.6 Variation of the amplitude of beam displacement with 62A.

193

The results of the experiments are presented as amplitude frequency plots in Figures

7.7a-b. In Figures 7.8-7.17 we present a selection of both the time trace and its
corresponding spectrum for different values of a) with (2‘ = 13.58 Hz.

The theory and the numerical simulations presented in Chapter 4 predict that for
forcing frequencies, (0, well above 2010 we will only see a small amplitude, directly forced
response, i.e., the beam will respond at to. Figure 7.8, which shows the time trace of the
beam response at a value of to = 11 Hz, is consistent with this. To theoretical make a
comparison with the results obtained using the method of multiple scales we have plotted
the response depicted in Figure 7.8 as a trivial response i.e., zero amplitude, on Figures
7.6. At f'u'st this may seem contradictory, however, it is the resonant response at 523 that we
are interested in. As is often the case in nonlinear studies, it is diffith to plot a standard
response curve (i.e. amplitude of response verse forcing frequency) since the response may
contain many frequencies in addition to the forcing frequency. As stated, we have chosen

to plot the amplitude of the 923 component on Figures 7.6a-b. This will then be consistent

with, for example Figure 4.1, as presented in the theoretical section.

As we decrease the forcing frequency, we reach a bifurcation point at 10 Hz (see
Figure 7.7a), below which a subharmonic response is found. Figure 7.9 depicts the time
trace and its spectrum for the case where to = 8 Hz. When we decrease the frequency
below this value we find a period four motion is introduced (see Figure 7.10). This was
not predicted by the method of multiple scales but it was by the numerical simulations of

the equation of motion, see, for example, Figure 4.28 in Chapter 4.

Further decreasing the value of to below 7.8 Hz we enter into a region of chaotic

response. Time traces and spectrums recorded in this region are presented in Figure 7.11.

M'pl i tude (Va...)

Nrpl i tude Wm.)

194

 

 

 

 

 

 

 

4
Q.- 13.50 a: o
o.- 13.5. H! o
3.5 — o
O
O
J - fl
0
O
0
2.5 " 0 Perl“ 2 Motion: (Snap-through)
O
2 — Perl“ 1 “that o
I . beetle Hellen-
0
I.5_ I. Perloddlhlieae
‘ -
0
MI“ I fill”. \ Furl“ 2 Unloa-
o.s — .1
O
0°s
0 I I L I I l I I I %g . I . I,
O 1 2 5 4 6 7 8 9 10 11 12

(.1 (Hz)

Figure 7.7a Frequency Response Curve for (2‘ = 13.58 Hz.

 

 

 

 

 

 

 

4
no. '3.” H! O
(7.- 13.70 H! °
3.5 - .
o
e
o
.3 " o
o tum a Innuen- (Inc-tum)
o

2.50-

2 _

For!“ C hllm
1.5 "
an". DUO-o
‘ _ 00. fluted : lieu-no
on...“ /
0....
o
0..
0.5 ’- o
.0
.0
O l l l 1 l l . ; ;
5 6 7 8 9 10 11 12

to (H2)

Figure 7.7b Frequency Response Curve for Q. = 13.70 Hz.

7.000

TIME

195

 

 

 

 

 

 

-7.000
0 Time (sec) 2.0
3.500 m-HHz
SPCTRM
AMP
Vr
O
O .000 A
O Freq (Hz) 20.0

Figure 7.8 Time Trace and Spectrum of Beam Displacement for to = 11 Hz.

 

7.000

TIME

 

v W

 

 

 

 

 

-7.000
0 Time (sec) 2.0
3.500 warn:
SPCTRM
AMP
VP 2
2
A .
0.000 fl_ -
0 Freq (Hz) 20 .0

Km 7.9 Time Trace and Spectrum of Beam Displacement for a) = 8 Hz.

 

 

196

 

7.000

TIME

-7 . 000

 

0 Time (sec) 4.0
3.500 0:1st

 

SPCTRM

AMP
Vr

{LDA
0 Freq(Hz) 20.0

 

 

p Ale
{1; N13

0.000

 

Figure 7.10 Time Trace and Spectrum of Beam Displacement for a) = 7.8 Hz.

 

7.000

TIME

-7 . 000

 

0 Time (sec) 8.0
3.500 (Ir-7.7511:

 

SPCTRM

AMP
Vr

(o
0 .000 AM JL,

0 Freq (HZT 20 .0

 

 

 

Figure 7.11 Time Trace and Spectrum of Beam Displacement for a) = 7.75 Hz.

197

To better demonstrate the chaotic motions, another two time traces also presented for 0) =
7.1 Hz and 7.0 Hz, as shown in Figures 7.12. Once again this is consistent with the
numerical simulation completed in Chapter 4.

For values of to below 7 Hz a period two motion remains but now the motion
encompasses both buckled positions. This motion was described as "snap-through" in
Chapter 4 and, as was found in the numerical simulations, is of a hardening type of
nonlinear response. This can be seen in Figure 7.7a which clearly shows the amplitude of
the snap-through motion decreases as a) is decreased, but increases if we increase (0.
Hence, for 10 Hz > a) > 7 Hz multiple steady-states exist. Time trace and its spectrum, are
presented for such a case with (o = 8.0 Hz in Figure 7.13. Comparing this with Figure
7.9, which was also obtained at (o = 8 Hz; clearly shows the existence of multiple

solutions.

The system having adopted the snap-through motion, can maintain this type of
response over a very wide range of to. Indeed the motion was recorded right down to
approximately a 0) of 2.65 Hz, at which time the motion changed to having a period of one.
Figures 7.14 and 7.15 show time traces and spectrums associated with the transition.
From these figures, we also observed that the third and fourth harmonics appeared The
period one motion was found to exist over a region of to = 2.5 Hz to 2.71 Hz, at which

point the period two motion was then once more adopted.

One other type of motion was found to exist for 0) = 2 Hz, as shown in Figures
7.16. This corresponds to a main resonance response, i.e., 0) ~ (00. It was extremely
difficult to experimentally obtain more data for this type of response as the snap-through
motion would intermittently occur. Indeed some chaotic motion was also observed in this

region. Figure 7.17 shows time trace and spectrum of a chaotic motion for to = 2.64 Hz.

7.000

TIME

-7.000

7.000

TIME

-7.000

7.000

TIME

-7.000

3.500

SPCTRM
AMP

Vr

0.000

198

 

 

 

 

 

 

 

00:7.11—12.
0 Time(sec) 8.0
to=7.0Hz
O Time(sec) 8.0

Figure 7.12 Time Trace of Beam Displacement for a) = 7.1 Hz and 7.0 Hz.

 

 

 

 

 

 

 

0.0 , Timelsec) 2.0000
2 00-81-12
2
0)
0 Freq(Hz) 20.0

Figure 7.13 Time Trace and Spectrum of Beam Displacement for (o = 8 Hz.

 

199

 

7.000

TIME

 

-7 . 000

 

0 Time (sec) 4.0
3.500 mazésnz

 

SPCTFlM

N|e

AMP
Vr

32
2
0)
0.000 1L Ii 3? - A
0 Freq (Hz) 20.0

 

 

 

Figure 7.14 Time Trace and Spectrum of Beam Displacement for to = 2.65 Hz.

 

7.000

TIME

 

 

-7 . 000

 

0 Time (sec) 4.0
3.500 m=2.5Hz

 

SPCTFlM

AMP
Vr

 

 

0.000 n A 3‘: A
0 Freq (Hz) 20.0

 

Figure 7.15 Time Trace and Spectrum of Beam Displacement for to = 2.5 Hz.

200

 

7.000

TIME

V mm

-7.000

 

0.0 Time(sec) 4.0000
3.500 (082112

 

SPCTFIM

AMP
Vr

 

 

2w
0 .000 {I 3‘,“ A
0 Freq(Hz) 20.0

 

Figure 7.16 Time Trace and Spectrum of Beam Displacement for a) = 2 Hz.

 

7.000

TIME

 

-7 .000

 

0 Time(sec) 20.0
3 .500 03:2.641-12

 

SPCTRM

AMP
Vr

 

 

0.000 .... z
0 Freq(Hz) 20.0

 

Figure 7.17 Time Trace and Spectrum of Beam Displacement for 0) = 2.64 Hz.

201

7.5 Summary of the Chapter

We have conducted a series of experiments on a rotating cantilever beam for two

cases: (a) (28 < (20 and (b) (2. > no. The results of the experiments of both cases show

qualitative agreement with the perturbation solutions and/or the numerical simulations. For
the pre-buckled case, we observed regions of imperfect bifurcations. For the post-buckled

case, we observed biftn'cated and chaotic motions.

CHAPTER 8

SUMMARY AND CONCLUDING REMARKS

8.1 Summary of the Dissertation

The effects of a nonconstant spin rate on the nonlinear dynamic response of shafts
has been investigated. The analytical study was based on equations of motion which were
derived in Chapter 2. These partial differential equations were quite general in nature, but
were reduced to ordinary differential equations for two particular cases, upon which the
remainder of the dissertation was based. The first case investigated the behavior of a
cantilevered beam rotating about its longitudinal axis, the motion in this case being modeled
by a single, second order, nonlinear differential equation with time dependent coefficients.
The second case was concerned with the dynamics of a circular shaft, simply supported at
either end. For this case, two coupled, nonlinear, second order differential equations were

used to model the behavior. Time dependent coefficients were once again present.

In each of the two cases the investigation focused firstly on the system's behavior
for spin rates below the critical speed and secondly, on their behavior above the critical
speed. The general approach was to seek approximate solutions to the governing equations
of motion using the method of multiple scales. The accuracy of these solutions was

checked using direct numerical integration.

In Chapter 3, the pre-buckled behavior of a fixed-free beam was investigated. It
was clearly demonstrated that a principal parametric resonance can occur at mean spin rates
well below the first critical speed of the beam. For this type of resonance, the

202

203

nonlinearities were of a hardening type. In Chapter 4 the post-buckled behavior of a
cantilevered beam was investigated. For sufficiently small a (2,, values, the beam was
found to vibrate about one of the buckled positions with a softening-type, nonlinear
behavior. As we increased the a 9A value beyond some critical value, the beam's orbit
encompassed both buckled positions. In the vicinity of the critical value of 8 (2A , a region
was found where the beam displayed bifurcated and chaotic motions. Melnikov's method
was applied to predict the regions where chaotic motions might exist. Numerical
simulations were used to find bifurcated and chaotic motions in the regions of primary and
subharmonic resonances. The values of a (2A for which chaotic motions occurred were
found to be much above the homoclinic bifurcation curve, indicating that in this case,

Melnikov's method gave a very conservative lower bound for the transition to chaos.

The case of the pre-buckled shaft was consider in Chapter 5. Four cases of
parametric resonances, viz. to ~ 2 0),, to ~ 2 0),, 0) ~ (01+ (02 and to ~ (02 - a), were
considered. Each case was analyzed firstly in the absence of internal tuning (i.e., (02 was
assumed to be well removed from 30),) and secondly in the presence of internal tuning

(i.e., (02 was assumed to be closed to 30),). It was shown that no parametric resonances

exist for the case of a perfectly balanced, symmetrical shaft. However, for a shaft that is
close to circular, it was found that parametric resonances can occur. In the presence of an
internal resonance, the shaft's behavior becarrre much more corrrplex. Such phenomena as
non-existence of steady-state motions, coexistence of steady-state motions, amplitude
modulate motions, and re-stabilization of the trivial solutions were all found to exist. They

were confirmed by direct numerical integration of the full equations of motion.

In Chapter 6 the case of the post-buckled shaft was studied. The analysis began by
finding the equilibrium position assuming a constant spin rate. The conditions for

nonsynchronous whirl were obtained and it was shown that even when a nonconstant spin

204

rate was imposed on the perfectly circular shaft, no resonances occurred if the internal
damping was small. However, the orbit of the whirling will become distorted (i.e., a
circular orbit with a overhang) for larger values of internal damping. In the case of the
almost circular shaft, no approximate solutions to the governing equations were found, on
account of the complexity of the problem. However, numerical studies showed the

existence of bifurcated and chaotic motions for the case of 0) ~ (01 and to ~ 2 to, .

The results from an experimental investigation of the cantilevered beam case were
presented in Chapter 7. The results of the experiments showed good qualitative agreement
with the perturbation solution for all the cases tested. When the beam buckled, bifurcated

and chaotic motions were observed.

8.2 Discussion and Future Work

(I) The coupled nonlinear, partial differential equations derived in Chapter 2 were
reduced to ordinary differential equations by assuming that the first mode motion
dominated the shaft's motion. At least for the case of a rotating beam, the experimental
results indicate that such an assumption is valid. Preliminary experimental results, not
reported in this thesis, show that a similar assumption would seem to be valid for the shaft

case.

(2) Although the approximate steady-state solutions obtained by the method of multiple
scales have been shown to be very accurate in a number of cases, the method does have its
limitations. For example, when a steady-state motion is found to loss its stability, the
method of multiple scales gives no indication as to the specific details of the resulting
motion, although the manner that the eigenvalues pass into the positive half plane do give

an indication of the type of response we might expect. Related to this is that no information

205

regarding domains of attraction can readily be obtained from the method. Also, close to
regions where degenerate bifurcations occur, the method of multiple scales can not give
details of the motion likely to be observed. A possible qualitative approach (e. g.,

bifurcation theories) might be an alternate way to gain insight into these problems.

Another practical weakness of this, and other perturbation methods, is that carrying
out the expansion to higher orders is very cumbersome, especially for multiple degree of
freedom systems. In practice one seldom goes beyond the third order unless the algebraic

manipulations are performed by a computer. Even then, there are limitations.

(3) In this thesis, the spin rate was expressed as the sum of a steady-state part and a
relatively small sinusoidally varying component. However, from an engineering viewpoint
a more complex periodic spin rate would be more physically realistic. If this were modeled
as a summation of sinusoidal terms (e. g., by a Fourier series), there would be the
possibility of multiple resonances occurring simultaneously. This would be an interesting

extension to the current work.

(4) In Section 4.6 we observed that Melnikov's method was very conservative in
predicting the transition to chaos. Moon [1987] used a different criterion based on the
heuristic idea that chaos may occur when periodic orbits become large enough to touch the
homoclinic orbit. It would be interesting to apply his ideas to the current problem.

(5) The mathematical models used in the present study did not incorporate the effect of
mass unbalance. Since this is a practical consideration, it would be useful to extend the
analysis to include this and thus study the interaction between nonconstant spin rates and
mass unbalance. Indeed, the experimental results presented in Chapter 7 show that

imperfect biftu'cation occurred. It is believed that this could be explained by incorporating

206

mass unbalance and other irrrperfections in general.
(6) The straight equilibrium position of a rotating beam is stable when (23 < 00 and

unstable when (28 > 00. However, when 9' is very close to no, the straight equilibrium
position is marginally stable. The stability of the straight equilibrium position will change
periodically if we superpose a relatively small sinusoidally varying component on as. This
corresponds to the problem of a system with a periodically disappearing separatrix. The
method of the multiple scales cannot capture the nonlinear dynamics of the beam under this
condition. This may be investigated by using averaging and elliptic functions (see

Coppola and Rand [1990]).

(7) Chapter 7 reported on the experimental investigations completed on a cantilevered

beam. This work should be extended to cover the shaft case.

APPENDICES

Appendix A

If (p is the rotation of an element of the beam about the y axis, then the slope of the

rotation can be expressed as

ul
tan =
(p 1+w’

 

a)

where (Y = — and S is the undeformed arc length.

BS

Differentiating equation (A.l) we obtain

,_ (1+w’) u"—u’w”
1+u’2+w’2

 

The elongation rate

I
e( S,t)=[(l+w’)’+u’2]5-1

(A.l)

(A.2)

(A.3)

is the measure of the axial deformation of the neutral axis. For a beam with a movable end

(e.g., cantilevered beam) the axial deformation of the neutral axis is negligible, i.e.,

e(S,t) = 0. Hence

207

(A.4)

208

Substituting equation (A.4) into equation (A.2) yields

II 1 I2 I
..u (1+2!) )3. ”(1+1 ’2)
-1+u’2+lu" u 2 u

4

I

 

(A.5)

Appendix B

Here we want to determine the function w(z,t) so that it can be eliminated from

equation (2.19). Note that

ds2= dx12+ d2,2 (13.1)

x1 (2, t) = u (z,t) (B.2)

x2(z,t) = z+w(z,t) (B.3)
therefore,

WET-301002] .2
z z (3.4)

2
as
An expression for (5;) is obtained from equation (2.8) by rearranging terms to get

a 2
{—5) = 1+2 eu(z,t)
82 (B.5)
Assuming eu(z,t) << %, we can get
[... )2
— El
31 (3.6)

209

210

The use of equation (B6) in equation (BA) and integrating with respect to z shows that

I
x2(z,t)5j; [1-u’2(t.t)]5dt+C1(t) (13.7)

The end displacement condition is that x2(0,t) = 0, hence

I
x2(z,t) s I; [1- u’2(t.t)]5 dz (B.8)

Finally, we combine equations (B.3) and (8.8) to obtain the formula for w(z,t):

l
w(z. t) s- j; [1 - u'2(§.t)] 5 dz - z (3.9)

In keeping with the first order approximation policy used herein, we expand the integrand
of equation (B9) in a binomial series, to obtain

I
I - 1 ’
[1- u 2w] 2 =1‘EI“ 290] ‘ """" (BJO)

This shows that

w(z,t)s —;1;[u'2<e,o]da . (13.11)

Appendix C

0, _._ds_
‘ 1+ds2 '

_ c
26(l+dsz) ’

 

u

_ 3bs
1 1+ds2

 

1= ZQIQES ’
1+ds2

4 _ QAzs
& ua?’

 

 

a _ d
2 1+ds2
(02- 2bs2
° 1+ds2
‘Y _ b
2 l+ds2
_200
2 1+ds2
_ 9A2
g4 l+dsi '

211

Appendix D

R,, = 2 (i (1),-A, 0, >11. A,+i em“) a, 0.7;, +(3 +A,2+2 A, X, ) 0,2 A,2 X,
“PW—“9T0 ( 2 A, (1),-i a, )QAA, +ei°°2T° ( 3+K, 2 + 2 X, A, ) 9,2 K,’ A,
+ ( 6+2 A, A,+2 A,X,+2 A, X,)o,2A, A, A,+2 ( i (1),—A, o, ) A,’
(D1)
12,, = 2 ( i A,(o,+ a, ) n, A,+i 2"”!To x,u, 0.x,
+(3 A,2 X,+ K,+2 A, )0, A,2 'A', + -e“‘°2'°l’T° (2 01),+i/\,Q,)£2,,A2
+ci°°2T° ( 3A 2A, +A, + 2 X, ) 0,2 A3 A,
+(6A1A2X2+2A1+2X2+2A2)002A1A2K2+2(iA,(D,+Q,)A,I
(D2)
R,,= 2 ( i co,—A, a, ) p, A, + (3 +A22+2 A, X, ) 0,2 A22 ‘A',
+e“<°1’°2”° ( ’—2—’:L‘-°l+ iQ,)12,,A,-1-e'“"2'r°(1+K,2)Q,,2 A,3
+ (6+2 A, A, +2 A,X, +2 A, A', ) (202A, A, K,+2 (i m,—A, o, ) A,’
(D3)
R,, = 2 (i A, m,+ a, ) 11. A, + (3A,2X,+X,+2 A, ) 52,2 A,2 A,
+c““’1"’2"b(——?=2 3‘” + i A,(Z, )QAA, +c"‘°2“° ( A,+A,3 ) (2,2 A,3

+ ( 6A,X,A,+ 2 A, + 2 'A', +2 A, ) 002A, A, K,+ 2(1A,m,+ a, ) A,’
(D4)

212

Appendix E
K,,=K,,=(2im,-2 A, 0,)(002-m,2)+2£2,2(it0,+1\, 0,)
+2 9028(iw1-A1 Q‘)

K12=2A2019A(002-(012)+19A Q‘(Q'2-Qoz-3 (012)

Hi),2 QA (2,5

+2ifloza),Q,(A,+2X,+3X,2A2)

K,,S = ( (20‘ + 90‘ 5— {202013—9020} )( 3+X,2+ 2 X, A, )
+2i9020),Q,(K,+2A,+3A,2X,)

K,,=K23=21m,(floz-m22)-2A,Q,(002-14022)
+2 9,2(ico,+A,Q,)+2S2028(ito,—A,Q,)

K22=—-:- A, a), a, (ml-0,2» go, o, ( 3002+ (0,2)

-QAQ,2(%A,0)2+iQ,)- 0,2 0A8(%A,w,—ifl,)

K24: 2 ( 904+Qo48-902m22-Q020.2 )(3+A1-A-1+A1A2+_A-1A2)
+4i002m,o,(A,+A,+X,+3A,A,X,)

K25: (904+90‘8-nozm22-0020‘2)( 3+A22+2X2 A2)
+2i0020),Q,(X,+2A,+3A22X2)

213

and

Appendix F

Let

11,=2[(Q,—'c1‘1)2-Q,,2]-c112
h,=t'it0

h3=2(Q,—?o')0)

h4=0)2

h5=fii5

h6=29A(Q,—E)R

H5 = (11,11,)2 + 1142(11,2 + 11,2) + 201,113)2 + 11,4 + 1134+ 211,11411,2

_ 11711301,2 + 11,2 + 11,114) + 11,01,11,2 + 11,211, + 11,113)
H5

 

H1

___ h7halh(h1‘ 1'4) ' 1161120122 + 1132 + 1142)
H5

 

H2

= 11,11,01,2 + 11,2 + 11,2) + 11,,11,11,(114 — 11,)

H
3 H5

 

 

_ -h7(hlh3z + 1112 114 + h22h4) + h6h3(h22 + [‘32 + h1h4)

H
4 H5

215

BIBLIOGRAPHY

BIBLIOGRAPHY

Ariaratnam, S. T., 1965, "The Vibration of Unsymmetrical Rotating Shafts," Journal of
Applied Mechanics, pp. 157-162.

Ariaratnam, S. T. and Sri Namachchivaya, N., 1986, "Periodically Perturbed Nonlinear
Gyroscopic Systems," Journal of Structure Mechanics, Vol. 14, pp 153-175.

Atanackovic, Teodor M., 1984, "Estimates of Maximum Deflection for a Rotating Rod, " Q.
J. Mech. Appl. Math, Vol. 37, pp. 515-523.

Atanackovic, Teodor M., 1986, "Buckling of Rotating Compressed Rods," Acta
Mechanica, Vol. 60, pp. 49-66.

Atluri, S., 1973, "Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia
Effects," Journal of Applied Mechanics, pp. 121-126.

Bauer, H. F., 1980, "Vibration of a Rotating Uniform Beam, Part I: Orientation in the Axis
of Rotation," Journal of Sound and Vibration, Vol. 72, No.2, pp. 177-189.

Bolotin, V. V., 1963, Nonconservative Problems of the Theory of Elastic Stability,
New York: The Macmillan Company.

Brosens, P. J. and Crandall, S. H., 1961, " Whirling of Unsymmetrical Rotors," Journal
of Applied Mechanics, Vol. 28, Trans. ASME, Vol. 83, Series E, pp 355-362.

Coppola, V. T. and Rand R. H., 1990, "Chaos in a System with a Periodically
Disappearing Separatrix," Pre-print.

Crandall, S. H. and Brosens, P. J., 1961, " On the Stability of Rotation of a Rotor with
Rotationally Unsymmetric Inertia and Stiffness Properties," Journal of Applied
Mechanics, Vol. 28, Trans. ASME, Vol. 83, Series E, pp 567-570.

216

217

Crandall, S. H., 1980, "Physical Explanations of the Destabilizing Effects of Damping in
Rotating Parts," Rotordynamic Instability Problems in High Performance Turbo
Machinery, NASA CP 2133, pp. 369-382.

Day, William B., 1987, "Asymptotic Expansions in Nonlinear Rotordynamics," Quarterly
of Applied Mathematics, Vol. XLIV, No. 4, pp 779-792.

Dimentberg, F. M., 1961, Flexural Vibrations of Rotating Shafts, London:
Butterworths.

Ehrich, F. F., 1964, "Shaft Whirl Induced by Rotor Internal Damping," Journal of
Applied Mechanics, pp. 279-282.

Foote, W. R., Poritsky, H. and Slade, J. J. Jr., 1943, "Critical Speeds of a Rotor with
Unequal Shaft Flexibilities, Mounted in Bearings of Unequal Flexibility-I," Journal of
Applied Mechanics, Vol. 65, pp A77-A84.

Genin, Joseph and Maybee, John S., 1970, "External and Material Damped Three
Dimensional Rotor System," Int. J. Non-Linear Mechanics, Vol. 5, pp. 287-297.

Genta, G., 1988, "Whirling of Unsymmetrical Rotors:A Finite Element Approach Based
0131, Complex Co-Ordinates," Journal of Sound and Vibration, Vol. 124, No. 1, pp. 27-
5 .

Golubitsky, Martin and Schaeffer, David G., 1985, Singularities and Groups in
Bifurcation Theory, New York: Springer-Verlag Inc.

Gragnmel, R., 1929, "Kritische Drehzahl und Kreiselwirkung," Z Ver. dtsch. Ing., N0.
2.

Guckenheimer, J. and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems
and Bifurcations of Vector Fields, New York: Springer-Verlag Inc.

Gunter, E. J. Jr., 1967, " The Influence of Internal Friction on the Stability of High Speed
Rotors," Journal of Engineering for Industry, pp. 683-688.

Gunter, E. J. Jr. and Trumpler, P. R., 1969, " The Influence of Internal Friction on the
Stability of High Speed Rotors with Anisotropic Supports," Journal of Engineering
for Industry, pp. 1105-1113.

218

Ho, C. H., Scott, R. A., and Eisley, J. G., 1975, "Non-planar, Non-linear Oscillations of

a Beam - I. Forced Motions," International Journal of Non-linear Mechanics, Vol.
10. PP. 113-127.

Hull, E. H., 1961, "Shaft Whirling as Influenced by Stiffness Asymmetry," Journal of
Engineering for Industry, pp. 219-226.

Inagaki, T., Kanki, H. and Shiraki, K., 1980, "Response Analysis of a General

Asymmetric Rotor-Bearing System," Journal of Mechanical Design, Vol. 102,
pp.l47-157.

Ishida, Y., Ikeda, T., Yamamoto, T. and Akita, T.,l986, "Internal Resonance in a
Rotating Shaft System," Bulletin of JSME, Vol. 29, No. 251, pp. 1564-1571.

Ishida, Y., Ikeda, T., and Yamamoto, T., 1987, "Transient Vibration of a Rotating Shaft
with Nonlinear Spring Characteristics during Acceleration through a Major Critical
Speed," JSME International Journal, Vol. 30, No. 261, pp 458-466.

Ishida, Y., Ikeda, T., Yamamoto, T., and Murakami, S., 1989, "Nonstationary Vibration
of a Rotating Shaft with Nonlinear Spring Characteristics During Acceleration Through
a Critical Speed," JSME International Journal, Vol. 32, No. 4, pp. 575-584.

Ishida, Y., Ikeda, T., and Yamamoto, T., 1990, "Nonlinear Forced Oscillations Caused by
Quartic Nonlinearity in a Rotating Shaft System," Journal of Vibration and Acoustics,
Vol. 112. PP. 288-297.

Iwatsubo, T., Kanki, H., and Kawai, R., 1972, "Vibration of Asymmetric Rotor Through
Critical Speed with Limited Power Supply," Journal Mechanical Engineering
Science, Vol. 14, No. 3, pp 184-194.

Jeffcott, H. H., 1919, "The Lateral Vibration of Loaded Shafts in the Neighborhood of a
Whirl Speed: The Effect of Want of Balance," Philosophical Magazine, Ser. 6, 37,
304.

Kammer, D. C. and Sehlack, A. L. Jr., 1987, "Effects of Nonconstant Spin Rate on the
Vibration of a Rotating Beam," Journal of Applied Mechanics, Vol. 54, pp. 305-310.

Kimball, A. L., 1924, "Internal Friction Theory of Shaft Whirling," General Electric
Review, Vol. 27, No. 4, pp. 244-251.

219

Krousgrill, C. M. and Bajaj, A. K., 1987, "Complex Nonlinear Dynamics of a Rotating

System." Developments in Mechanics, 20th Midwestern Mech. Conference, Vol.
14, pp. 832-838.

Kulla, P., 1972, "Dynamics of Spinning Bodies Containing Elastic Rods," Journal of
Spacecraft and Rockets, Vol. 9, pp. 246-253.

Laurenson, Robert M., 1976, "Modal Analysis of Rotating Flexible Structures," AIAA
Journal, Vol. 14, No. 16, pp. 1444-1450.

Lewis, F. M., 1932, "Vibration During Acceleration Through a Critical Speed." ASME
Journal of Applied Mechanics, Vol. 54, pp 235-261.

Mazzilli, C. E. N., 1989, "Subcritical Large-Amplitude Vibrations of Imperfect Rotating
Shafts," Appl. Mech. Rev., Vol. 42, No. 11, pp. 157-160.

Meirovitch, L. and Nelson, H. D., 1966, "On the High-Spin Motion of a Satellite
Containing Elastic Parts," Journal of Spacecraft and Rockets, Vol. 3, pp. 1597-1602.

Moon, F. C., 1987, Chaotic Vibrations, New York: John Wiley & Sons.

Nayfeh, Ali Hasan. and Mook, Dean T., 1979, Nonlinear 0scillations., New York: John
Wiley & Sons.

Nayfeh, Ali Hasan. 1981, Introduction to Perturbation Techniques, New York: John
Wiley & Sons.

Newkirk, B. L. and Taylor, N. D., 1925, "Shaft Whirling Due to Oil Action in Journal
Bearing," General Electric Review, Vol. 28, No. 8.

Nonami, K. and Miyashita, M., 197 8, "Problem of Rotor Passing Through Critical Speed
with Gyroscopic Effect," Bulletin of the JSME, Vol. 21, No. 151, pp 56-63.

Nonami, K. and Miyashita, M., 1979 "Problem of Rotor Passing Through Critical Speed
with Gyroscopic Effect," Bulletin of the JSME, Vol. 22, No. 169, pp 911-918.

Odeh, F. and Tadjbakhsh, I., 1965, "A Nonlinear Eigenvalue Problem for Rotating
Rods," Arch. Rational Mech. Analysis, Vol. 20, pp. 81-94.

220

Rankine, W. J. Mc.Q, 1869, "Centrifugal Whirling of Shafts," Engineer, 9 April, XXVI.

Robe, T. R. and Kane, T. R., "Dynarrrics of an Elastic Satellite, 1, II, III," International
Journal of Solids and Structures, Vol. 3, pp. 333-352, 691-703, 1031-1051.

Shaw, Jinsiang , 1989, "Non-linear Interactions in Rotordynamics," Ph.D. Dissertation,
Michigan State University.

Shaw, 8. W., 1988, "Chaotic Dynamics of a Slender Beam Rotating about Its Longitudinal
Axis," Journal of Sound and Vibration, Vol 124, No. 2, pp. 329-343.

Smith, D. M., 1933, "The Motion of a Rotor Carried by a Flexible Shaft in Flexible
Bearings," Proceedings of The Royal Society, Vol. 142, Series A, pp 92-118.

Stodola, A. A., 1924, Dampf und Gasturbinen, Berlin: Julius Springer.

Taylor, H. D., 1940, "Critical-Speed Behavior of Unsymmetrical Shafts," Journal of
Applied Mechanics, Trans. ASME, Vol. 62, pp A71-A79.

Tondl, A., 1965, Some Problems of Rotor Dynamics, London: Chapman and Hall.

Unger, A. and Brull, M. A.,198l, "Parametric Instability of a Rotating Shaft Due to
Pulsating Torque," Journal of Applied Mechanics, Vol. 48, pp. 948-958.

Vance, John M., 1987, Rotordynamics of Turbomachinery, New York: John Wiley &
Sons.

Victor, F. and Ellyin, F., 1981, "Acceleration of Unbalanced Rotor Through the

Resonance of Supporting Structure," Journal of Applied Mechanics, Vol. 48, pp.
41 -424.

Wang, C. Y. 1982, "On the Bifurcation Solutions of an Axially Rotating Rod," Q. J.
Mech. Appl. Math., Vol. 35, pp. 391-402.

221

Wiggins, S., 1988, Global Bifirrcations and Chaos: Analytical Methods, New York:
Springer-Verlag Inc.

Yamamoto, T., 1960, "Response Curves at the Critical Speeds of Sub-Harmonic and
Summed and Differential Harmonic Oscillations," Bulletin of JSME, Vol. 3, No. 12,
pp. 397-403.

Yamamoto, T.,1961, "On Sub-Harmonic and Summed and Differential Harmonic
Oscillations of Rotating Shaft," Bulletin of JSME, Vol. 4, No. 13, pp. 51-58.

Yamamoto, T. and Hayashi, S., 1963, "On the Response Curves and the Stability of

Summed and Differential Harmonic Oscillations," Bulletin of JSME, Vol. 6, N o. 23,
pp. 420-429.