The chatter phenomenon of machine tools (no. 2)

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The chatter phenomenon of machine tools (no. 2)

Citation for published version (APA):

Smits, G. W. M. (1963). The chatter phenomenon of machine tools (no. 2). (TH Eindhoven. Afd.

Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0085). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1963

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T ECHNICAL REPORT

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CONTENTS OF REPORT

NO. PAGES TEX'138

NO. CHARTS DRAVJING NOS. PHOTO NOS. APPENDIX PHOTOMICROGRAPHS COMPLETE COPIES

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Intc'oc 'ct"on to t.hc D na.r.1i Stabil1ty

of l'''C'l1ine-'J:ools

F 'ndt Jncnta 1 I!ltroduc ti on to thc ProbJ em

of i'~ Cfl11le tooI Stabillty

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TIle CuttlnG Proccss

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B. Bounda ry 0ta b11j ty Condition 20-26

C. D:::.::;cu~s10n of thc Theory of

Tlusty-Po1acek

27 -

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3

D. Concl us10ns from , v the Theorv ~

of Tlustr-Polacek 2 -30

TIlC Combincd Method Ol

Peters-Van Berck

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Ch8rac teri~ t:: c

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PropoCE 1 .. 0 Pract Cc 1 Test P:'of.;rr-m ·'or the ones & I«. mson 60° De Jree Node 1 30 Auto.

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_{no'd-\';n~1 " net' ,-.no~'!- -'0\-) ' .

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1[ 'Cl, ,'l :'!dC', :~Lpport nd keep tLe workp::'ccc :'rl plrc 8ucn

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'1' 1:":C: papel:' ::":.:, intcndec'i to ce :

r'(.'~"1.Jr,[é 0 :, 1:.'lJot 1,'S been do 0 :in 1,n~ïcl':::,t?nd:lnc t iC

cllt t:i:lL proce~::.:, ,:,ts efl'ect on tLc à, _l"";'IL_'_C ~tr.'b'l L:i.t:v

0 " ' 0 lilE cLJne-t0ül :::'ilCi tllc rpp1.1c.:"tJoll Oi' tL:i ~' l',:no1:.'lc(E):;

~!1 an O~i'02t t ULi1(] st<ll\le mé cninc too:ü,.

8. ) To (ü[' S - Fi Gen'] i ck b) rl'lm:ty - Polacck v

c ) Prof. Petcrn - vaD He~ck

c . f\ P 0 0::0::11 foI' t" e dyn0mic ~nvcrtiE?t:i on of n!l'C'hJne

-too1~j c.t JODe;"', ~: ,Lémson f-ipc!üne COlTlpo;n.~': Spr :i.ncfie1.(), Vermon,t) U. S.f\. .

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'l'j;c C'Llttjn;,~ f OI'CC C8ure3 dC'lO:::',é' tjon o,c' thC' l1l;:~d~:inc too] ;:'t.;rt)ct'l(rf~\·:rJcl:" rc:::::ults 1n a rel::'ti\'s ( c ,']ec[;jOii bet i':cen

t;J.c too 1 ë nd thc v.'orkpj.E' 'C:.

fI verJ t jon in thc cutUn. fo]'ce, C1",'8 1 rç r,:,Or:1C r:l:iStUr _b2rl_C_L Jn thc c;uttin[, pro('cE;s, pr'odv.cc.'s (0 V[;t":;~t:]on

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thcr'e12tl ~e c1ei'lectjon bC't\'It'cn t he tool cnei t:.c ·.'ork,p5c:'C' élnd th '2 0 ;' thC'

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p 21'138 be:i ng rcmovccJ .

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iH'ee ~ ç V3C'J, t.lon Q,l' thlG a· ... c l;i11 in tlTil Co '''':2e E' V<.:::,jr''c:iO)'

of t~e taal force.

/\ J:i ::=:tu:cbancc C'" tLuc C 1re a vi1n'8t:ion vJl",eh :::;h o'tJG i tsclI'

::: ~1 t liC farm of 2. \·.18VC on thc f.urfecc 0 thc \'JOY' p:icce.

11' Dm'.' tLC' amp1jtuc'lc of tr.e vibr2t10n é'nr; trws of t hc \'.'é:'Vir1c3S

on the ~3 l.Y',~'_ ce of tbc 'wrkpü~c(; increc ::0:::: lY'oill Cl~t to Cllt , \:c ::;r>y U:,r t ti C: Gystem js unstc::ble 8nc.t \'.'(; cr:}] t:',e reS1 Jt !1c hr ttcC!l.

If thc ampll t1.~de (yi' the I','C; ve dccrea 2CS t'I"on~ l'1.._'t to cut, ','c ('cf jnc the ::.:/stC'ITl a~ s têbl, .

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Ó:>t"(--l

Pé.lCe :)

Fo!' Cl L" Y'~;t :. pproY h E t j on, \'.'8 cai:) nNI mCikc t)-'Q rol TOv,'int;

cC;'E:r'::'th: :'cpre.c'(.l 'célt :Lon oL' tLc procc:o~: . (:1 C;. 2).

,

" t.nl:, (,C IY I, (\ L -- -- - - - -- .- ._--~ - ' cutt\\H.\ - pÇ;,:;ç ":C.:, ,c. ~:-'-.-t~~.e~

_

__

,

_

.

__

.

-1

~

, '

l_--b

t,

~,

~

~~ ~

--

-

----

l

_{_}

J

I1Ue

~

1'1 rd,\ '-'(.I

c\etl~d..

10\',

r

_________ , ____ .. _ ... _1

t:.

p

:

V Cl (.{ \ (l. h'or, 0

1

CA ( è ' ~', ~\

f

()

('

C

Loo];:lnC c. t eert in Cl.'t t ine; pr-ocer::: l'ri th ;J constant rot;-' V Oi:f' J

:c'y~cd SilO 8 constant .reed, i'ie r.1p.~,' nGS1.1ue t.h;- t the c ~tt':"nL: ~>orec

:'p p:coporLton::l to ttJe Vi '~ 8t:i or or th0 cros:J-:'cctlonpl ch:i.p

2 l'C2. Tél:ö,nc; '~~le lomjral depth of C1,lt 22

ye

u.~ r En :indlc;'te D va r1;' t :io (), thc nomj 11, 1 dcpth 0 / C'L:t r S b.

Y

.

From the !:1bove stftel1lerJt, 1,,0 C['11 C0 1C11 de th2t r'or

i: JdLî of cut b, ti~e var'1~..lt.:,on 0_' t 'jC c\!tt:i,ng ;'01'(;0

propo tlc)'léll to tl1e vDr:',.rtion'o;' t he nop1:ÏnD] d ptL

8 cOJ\st2nt

A'P

je-liel1ce 6.

p

=

R,6

Y

of Cl..t b.\(. I jn l'lh1ch:' Hl féCtO:".

'rLe '12 I'i a ,lon of the norün81 dcpth of c 'l1t 8 t [1 cert[ Jn t j me t

CDn 'Oe tlle sunm8tio~ 0:1.',;

1) Tl'Jc 21t crnéî tjoll of the penctré't:~on 0" ~~i';8 too]

po~nt at tlme=t or tlle relati~e ,jLTh t~on bctwce~

1 :i .. '

*

~"o!'"{piccc: 8i ei tooI \1!111.(;h Vle ':rUl i ,e lce te ~) \Jll

Y1

.

*

2) The ln' t'é~l vc.:.1p·t:;,on

Ya

of' the surfacc to be maclLined \,;'.:ich Cr'rl be Lle result ot tbc prc:cecUn_t:;

C":t 2t tir'e =(t-rr). (11' rr J8 the t i0C rOf' O!le

rot2 VLon of the \"!orkpiece ).

(14)

I

'

I \. , \ +(y) t_~

_

")

_~J

--,

J .,.

,

\

p

. ',-' -" ! 1 ( ) P:',:.:. ]0. 3 ):1 t;l1r:3 CRse 'cl1c v ... :::,':i.F,;:lon :'n cl :'1) t:".'C'l-C.lef::::: (C t t:lt'îc=t l~ccor.cf. :

6y

~

Y

o

-

y

,

.

(

V(~ctnr di~·fe,.'ençe f rnT:1 (" ," -+:;0 (2) ent : r.:l'l() U 18.CC'l'or c' tl..E'~ '\f;il':.!..é.t.' be ' 6.

P

= -

P,,( Y,-Yo ') I?Î~e ;)I,)O\TC ct, tr.;dcnt ~r:: t':''\.:2 J~Ol' ~'2c:~nc Cl'tf:' ,"l'ere t. c t ooI ~)o:i.nt é!h,lr':,!!3 L'olJ.o\,;f) ~,h0 [:211e De til ,

1l! ti c ce se 0 ;" ti r(;~ d Ci.' t ti!1S~

polnt (')1',;;:-;)'8 Cl·tG :i.n t, c cJ.e8:1

for :'I-:~ té.r:C(;" t 1:c ::001

Gl .. )' ... "<, cc ':LC ',:e CR,l ',1:-:': tc;- ' Yo=- \,>

,) I

rrhe-r'c "re ;;180 intern-;e .J,ctc c"ttJ.n.; ODCT'r' t JOllf! ro tJ,.c:, ,~

\:C cé.'l1 8C?:Je: . .'éltc ,ilO for ,l,~l<, s (2) é'n.~ (~-)) [J:; f:()~i~llg

é nel

A \j

=

U

y

o

-

'}

\

D,.

tJ

~

-

~,

(

YI - U

Yo

')

:'n ;(1.LC.l!: U i~;; rc.')et~ tion .~'é\ctor :'.1': t;he rë nr;c 0 ~ ~, i depcn0jng O~-! t.le Cl' tt:~rj , proceGf'. I

~

(15)

. _{c _}r 0'_~." ' _{_}t _{_...}~ n',", À I"". r -'i '1 J.. tJ' J. e (' I \ t c· I...:J : I r" 1'0 ,_ \.... .r·'~ \.... i. -·C·( r ( ' • t ... ~ 'I J_L-',• ~ C ~". "-' v Pl'" ~ .ot I.... ;'V l: '''C ..I. :<1 ,~"(;rö t re é lturlir·L.O:t 0 .' t:,e pCI e t :·['c ' Oli 0 ' ti.c tooI \)()~.'t Jc .

'/1

J

y1

8viilL 9:> ~ or-t:·.on::::· ;-;0 'c>o n'T'~2tjo:. 0';" t[ e.:!

C·:\CJ. J.Of'ce 'éJ,' L l' !] be c:;rpref:.:eC\ cr: 'ollO\'~~ :

( (~ )

~ I

~,(: (,;: ij iIC\': vi::U<.l:" ze '.;' l: p:'08ec: •. , t:' 2rJ. ope'l J.oop s· ~t 1:"l ·'.it:' ~'ccr '.: c:· ..

UV

t , . . ,.'2-.

L

. " .~. 1 Á

L

Á'I

:::

u

y

u

-

'i

\

Llp

=

R

\

Ä'/

"I,

-::

. r f;:.,. ~ :: RI \:("\.. U'fo t-

RIR

?.

In th:;-ory, t;r.e p o·olel.~ j E 'C[l1)~ fOJ Lplc r nd

2;' :·;terll cr~!l ',(2 ;.'0' net . Tl,ere ;:-PC, l',O~'i ' VEr, L.'o

dlH'inr: certpin cl;t t inC pCl'.,:o r';r; nco.

: t r·t):i.l.:i t~. 0 f' "L.r~e <.l .l ~' C' ~l. Cl) 11.;:~. e~: :

(16)

l n t:1C pa~~'l~ re';" yc~'rs m1,'.ch \';o:ck h08 beon done 'j r: t;;.,:, L f:jE:lr'

~':: 1\)1.,'8;; - Fü:l'.\,jicl: in EnGJ.énct é;nd b:. Tlt.:..:ty-PoldC:oÎ\ jn

C/;c('Lo~ 10 "/C k:lG .

To b :i.é1 2 -Pi [';;, \":j c:~ de vot

<:

t? ie 1::' a ttC'n t ion mB:i n 1:; 0 t:--' 0 C'A t t." ::!!~

n:.:Ücc:::::s in L ; ' I' ?~l;1 to C'zp18jn tr e p}:enO!'ilOilOl:l ~'c:;e!' th~t.

\:.0 .;'::nci (] ~)l'~ cl;j.r~cJ coh;t:~,Od .for JCI10 problcn.

m- t ~ ,~

1 J : S 'J -._ 0 1.[1 eek

t, (; _phenOmei1OI,

j.n t:·lC~.I' 3cz.::..'cl--: 1'0;:' ~ pr,rctJcal i:OJut:~o:; o~'

d('~vott.'c.l 'C;'e:] j,' ~ ttelîtion l;0 tbc: mrc}lj;-,c !.Ot 'let

Pro::'. Peter;:: <',;)) V,'Y1 IIGr'cl', frorr 3elClL',rn 'cested })olh tic tlJoorjo~:;

c' r:.'obiê8-Ficluiclc anc.) 'I:L.[.,t;,-Po]ée-.l( end tv':icè to bJ.CLU i.T~; tt-:o

ti:eo,.':·~es

In

tne

ne:;.:t d,<"IJtCl"i:' I 1;,'i1l t r:l to deGcr'~.bf, 3ilCl l~J2Cl.::::~_: t,-!c

tJ:rcc é'boï.re J:lcnVonec1 _s_· _s_ter.:_s _.

On(; chapt er of th~;: volt,n<.: \'J:~L le àe','otcc3 'co t !;C r.t·"l:.> OL'

niCcf,é)·,1:i.8S1 c:',,1:L~jti.·(;c c.::::,: ? 1:e pf'ul r'?':ci. in U:e (;E:~iL':l ~Lci

oe ':0 lopmc> ! t 0 [' w~ c:Li nc ~~oo 1 s .

In th:'~ JrO-:l~om \:E~ ·'.'~.J1 (jo~:cr:·.b pr~C't:lc?J. [':OPl'I):'(;!i of ·cc~~j.IC

t;!~" cte.,b:il:i.t:' 01' tLc ,'écl:~; ne-too1 .... (:;ruct"L·:cr.: enö i,o\'! l~ecl ~':j:;_c;:,l

(17)

'~3 n ,

IJ - !"'.-j

Page

la

CH! P1TER I.~ I

~)l)HIIjJ\R OF T"rlE T-JEOHY OF TOBIA3-FI~mHCK

Tobié1S JTJ21<es tbe ollmJ:lng aSE,umptïons :

I Thc machtne h:J.' suffici .ntly G ,. 11 damp:~rl'; so th;.~t t ca

be ià tr..at i-:-=- ',:br' t~.or. ch3r eter stics re comp ::>ed 0:"

t e S 'porpo:::··ti n of sepêr te motions of Ir.óny ~trl,ct\)ral

mode0.

2) The, t r1..<c turê ril des are so fa l' a pa rt i :~reql.~enc th? t one

can descr.be the

re

at

ve

mot~on between toel rd wo kp~cce

to c' tt:!.ng "orce vi. 'rat10ns 'oy an eq ü alect one-det;!'·>e of l'reedom s :stem .

•

ex + kx

=

dp

in "I]-.1e1-. : Je

=

d::1splacemer,t in tr:e pr'incjpr.-l dir ction of oEc111~tio~ between

too I < r.d vlO rx-pic ce

=

~'l m == eql. 1 \'8 len t ma s '"

e

=

tt

" r,princ co_ s t<.' 1 t

a rid k ca be óetel'r..ined f'rom

l'oson2 nce tos ts

bp= v2ri t:iOl1 0:' cl..ttlng force

TIle above formula c n thus be wrltten as :

m"/.,

+ cy, +-k'l\

=-

Ap

3) Chatter is a dynamic proces::> nd t us thc cutt_l18 fo:,ce

p j.s not only 2 funct10n of tr..e cr 1p thick eSf-' y ,1:t c l~o

o tr..c feed velocity Rf

nu

the rotaVon21 speed

wr

.

élnd

c n th s be expressed as :

A \

~

Y ;- 11

~

t..

Rf

+

Fh

Á ~~ =: ~

p

In t':h j.ch :

ï

xpresses tt.e êr::'et::on of tr:e cutt::.r!i: ~'orcc due to v _ i2 t ion of tl1e chJp thiclmess.

'lzAR

f

expresse'" t he vDr:i::-t on o. the l/tt:ns i'orce duo to

vari t:ior of the pcnetrat~on o ,~ 'cLc tool po:::.~t::=: causing va iation jn the reed.

(>3)

(18)

Pé g~ IJ

f:h~\.~ cxprcsses t:,e va ~.ûtj_on of thc C1.,ttJr:g ;'OY'ce (H e to the v8rjê t~on::'1 thc rot. t .:on speed . Ca .Jsed .fo in::::t0ncIi:' DJ' ::'nd-'up 0 t~lC gcnr tr'cin, +;.I:'e

n.o'Jement 0':" tl)e .... pindlc, the nover.1c:t 0 the tool J .-.----etc .

The verlet or; or tr.e Y'otat om~J. speecl,.r\t'I" ca,n "be ne ]octecJ nà tnus the th: rd ter'l11 "'n/t1:e equf'tion

(g can be OM: tted . Equetion

(

9)

~us bccomes:

Let \1S loole c10scr to ormul,l (10). tooi \-:e ce.n ',;:;.':;' te ~"'or' ~y;

(10) .

nd Ol'

ld

<

f

;

(vector d~fference from cut -to-cut) (11 See Fi~.

3

Ché'pter I .)

A

~(=

-;ft

=

-

'I,

8nd Tobi s derives or the case of a single-erlgcd ~uttinb tooI ;

rt1

=

chip tL1cjr_ncs'1 _v_ar_j_at_i_on

- Tot81 t}irust vnr iati n fact or .

Rs is 2 st('t~c coef.'lc:Lei;t E'nd c;, be obtaL ed from ste2d~ strtc c'tt~nc testG .

(12

(19)

'lf1: ( , •

Ä

P

= ~I ("10-'11) - \..0'1( R ... - R,

J

Y,

Fr> can now combi e (3) and (15) and \'Je .. i nà,;

'm~:

+ c

y

1

+

k

Y

1

==

Q., ("/0 -

Y

1 ) -

~

( Q, '='

-or .

63 -

R-

l

P8.ge 12

(15)

which equat ion io a 11ncnr d"fferent ial eqlat lon of the second

order and 1'1 s consequcntly 2 solrtion of the type

(1

8 )

I t i s nOl' cIeo!' tha t foI'; " ') 0 the vibret::lo:-t

'I,

inc eDses j !'l

time ['nd t:-!ê.t th s thc c lt t ng cond::.tjon ['ppep-y's to be l..nlsté·ble

; }.. <:

0 tr.e vib '2t ion '/\ dccre8sec-. j:;

time énd thus ti·;e c'l;t tlnR: cond~t:lon ~'i:Îll be stebIc·.

~}Il be the bmnd.ry ~t~bilj ty

condit i on.

In ther "'Jords, thc: ~,t.çbil:lt~' c!':iteri[l!) ~-=o . .'c2~. lts in the

reql;irelî..Cn t tna t C ;-~

R

'!>

-

~ ~

1-

0 1 o'.:'öer to L<:: vc st" bIc

~j t-J<..tion. ~ Q \

It eho_'.J.lc.l be noL'cd tr:.a t the ir.f.luence r)f tree Fc3c1.jt:1on~ 1 dc:_'npi ng

~ (~':.-R..\) decrceses ·;~til incrca sin", Epee(l .

fl'he boundary s taDIe cond:~tion ~f.) .f'otr1d by c'ubst it;.,t:..ng \1'1 (17)

t_ e sa lut :'on '),

-=-

fî

Co~ Wt

\tIe can nO\'] fJnd those '/, JueG o~ Rl as funcL on o . ';'J{ 2t \'Jh::'c'~

(20)

6S

-

R-l

p!! ge IJ

'l'oc:'a ~)-F1sb d ck take the cÎ,:"'o thicl'nq§F5 r proportiona 1 to t E:

eLi p tlLckne s;s ~ v ri" t~ on J.~Ê e tor' ~._ . te thus erom f j nd f or e, cb

val')e of the rot tio al sp ec] ~,JR 0 for tlle nO!"l-dimensional

r tn t:l ona 1 speed

""

in.

,

thc mir,' mwn \T8!Lle of Rl

=

Rg or the

~lIinimul vnl~.e 01' b

=

bg for vJhich d Ia eter Cé<11 .:ust ce TIl..: ntaineà.

( ~

=

the nat1.a'al -frequency 0 the moóe Ull(jCl" eor.s:1.üerê lon) .

The r sult ca n Ge plot ted jon a ?;raph Cr lIed lip, ta t i l i ty eh rt "

n exaITlple of "::L1 eh c n

ce

seen j 11 Fig. 6. s:':1:' r:: r,ra ph .;h ie h

ollown _r_ro~_t:e_ee_le~lat~_on_s _c,n_be _{veri~:_ed} _by _expc_ri~ents_as

\-'0 CéJn plot a Er8 ph of bg A. S l'une tion of R (t:;e 0 ther C' tt:ï_n f,

cond.:J~,t40ns lLve to be left const;-·nt) (Pio • 7)

1il,o

v ,:..>

a

f

~ 1 0 "'0 __ - Jo(

l

- I \

I

"

./

/

~ ~ --;;. .J) -.j " - ' -,.<. o r; "" --5 .<:. .> '--' r:" ' 0

--

>< ...,I :t V 1f

-:5

_ç.. c:J o , I /

\

t .

?OL

_'

v!

___

_

30 - - ' -- -- - - - -- -'-o sc 100 F gure

G

*

St bi ity Ch rt af t er S. A. Tobja" and

:. Fi h~ick - En _ineering 205J 109, 239 (' 958)

\o

ef

"5°1

20j ... ____ ...--_ - - -... , _____

/'/,.-°

0

'So 10 0 \'30 200

,.

FJül.ire 7

---Sxper mentól J.j- de ::'ve S. . \. '0 b iE' S, P 1"'0 c. 1: 74 (J959 .

Stab l "ty Chart R~tel :

(21)

DJ "".CUf.S"on 'In COI" of' rrob:1as Cl d Pisti :' cl·':

.lere are t':iO great dit'.<.:icul'cie:s in tH? E:ppllcc:tion of

thc Theor' 01' Tobi.&s-.t'-,!:J- ·lick .

1) The Öeterrl:'.nat'l.or of bie chip thjckr!e .... s \T2-.>~êt10r.

factor Tt, ~ fl'o·, f r.,t'la (LJ ), ,

2) 'I'!:.e deterrr:inê t j on D ...Q... ,ne lîél t1.lI'c.l f'rcquency 01"

thc rlloc1e of v~br[:t:i.on Co 'or:üng the f. té·Jl1~t,: ol' tl:';

machir.e u:1dcr' o·o:..:crvêt.1on ..

Tl,':::; ü: of COL\ se no'..; cliffjcult ~[' ti, dLÏ'erent ;~ocJc;s ~~'

viiJretion oi.· tI e r.z:ch:nE: 2'ce ~·:e ... J SC:péL"é:.tcci. Dif:·~.ci.lt~es

8Y' .. <'0,Lm·J ver, :..<. tI~e :tni':Llle: t~.c 1 ;;.0<18" of.' \':;'l)rr~t;i::m "neiot ',J 11 sep:OI'2t u . In

t

:ût

(;83 , o-:e .. ="~~ t8 Cue:-,"! 'Ji;:i c:

t:le eq,;2

cj

01'1 (1'() f:'F~· to ... 8 sol 'led .for·.

p.., a '~n81 c()Ylcl~:G:;on J.t een DO c,"<c: t~1;

L-0;" L:e nCCCGSE'ry con:::tront js t00 ((.l:..ct' te;

·.·(.11~ i)10 :'0 .. t:~e ~~r'1C}-;:i.r;e-tr ol ::'rtd'lstr·:' .

JIo?c\'er, t:.· method:is er:, veJ.ll'J.o ::-~:' j t .:.:~ves

,

.

S

8 , u~c1C'I'

~~é;1(Ün[; o_~ "t8b'::1:~ty as a;fcctei '.::":

of tl'e ::;:' f: tcn

The roté.'t~.o~,al ~p:inu.le s eeel

r;

---( ~"'>- RI ) t! e Ic:Le ... ·L·~tio, rate ::'é~ct r

Thc

(, • .I.

" l v

pilFG~n:.:;

t o Cl.. t .

(22)

0~-~-{-J Pr re 1::;

TI::!:: 'llJiEOJ\ r OF ~LU.:',TY -?CIACFK

',~ U',e t;;- or~' ,)" T.!.v~t~· -Polöcek pro'leü to te ver; 1- n('et'_c[:1

;'or

u'i

reet app' • CD t:' on 'n t ih...: r.lr C " 7 nc -tool j ndl... .:ot!':, ',.'e '1111

c;:plé:ir ':11e1... ":ork ll!ore extens::. .. el:' t:1f'D t!-:e t.1e l.'~' o~' nlOi..i2 3

-.f.' .-. sr·; 1

c

:

.

1,'.'e ml'st remembel' th8t tLè 8 :';n 0.:' r~1',~sty-P018ee:': ·.'['S to f'lnc} é prcC't::'c:::d i;<':':: to s01'.re 1fcj-rtter' n:--.en Der..:=: é'Là the;; ','ere tf,12re

-'ore uore concer!led \1:;.'..:;11 t;-.e st:'tJ('t;urc cr.ö ::t:" C1~TJC"I,1~ p r

-pertJes t1.2l! \ ~ til t!.~' ctJttlnc.; pr'ocess ~t:~eL .

A. Tne Cl t t:tl.g .Proce~:J

In t l-:eir E.ttC!:.pt to obt<J2.n a prêct';"ce.l :3(.h~t::.. I,~ T:!.v .. ·ty

-Polacek mêd(> tl'.c f'o~ 10\'rl.r~G ,::::;.ur.'f.:tions :

-(y)

1. 'l~lE c'\;t:; DG o!'ce p _ c c3jrectl:: pr', pO 1 't...0 l1F'1 to

t.Le ';ö.r::'at:l.or. 0;' tbe cross i'~cLlon(ll cl"jrJ [I·e2 ,

Thw,. S_~1CC cLrj:1G tj'~ c~·t th chjp hu: ë cm:

-::.:tant ,'.J:id.t.1J t:IC cuttj:.S; force p . 3 dJrc 'l~'

propor ion;' 1 to tI e VEll'.i ti on in cl:.' p thl c..lmef::: .

1'.le !81'lation of tl e C'utUng o:-,cc c,:~ tLus be

IH'itten os 8rrpl:l.t'\ .. àc ()_~ tLe \'~ I':~é t :i on 0 ' tr e I,~p tI~1c,mes". =- 'cctor ri j .. o'ercr .. ce '::'rOffi c 'ct

c

o

c:....t - .. _ . ~~"F-. / /

v,

BI

_

.

c!i:l _p _tI'_ic:1\.ne~s

\'ê riF L Ol ~[ctor

.j'-t

1.. ..J-. ) .

8 '

ForIT'.at:'on of the chip 1.: der r':-nE n ::.r' co::(J:~ tions a ~'ter

r.11u~t~'-Pol( CCtC

(23)

'\,

C3

-

H

-

l

P8gc 16

2. ~~ïe factor

j<.

L.> a re 1 o::it~ve rYln,1er end l~ r.~

r'.mct:io~ 0:' ~Le cl~tt~nc 8peecl~ tLe "eterj,_1 o· tLe

llorkpjccC', tr;e ,-,co,':r:)tr:' of thc: tooI. 2nd cfmeci2Jl~,' Fle lI:icJt'l of ,~1..~t ('0 . r.':'he I.'ddt·. of clit b . S tê~:e!!

proport ton~ J to the .:.'[ etor TIl.

J . The d:1recL on of t:le cutt':nl.:;; l'orcc 12 cOD"'tf'nt . 'TIjC:

c\.tt:"~C force p " e) only jn .. ··J.l.lenced t t!îe ')[' .~ect.io{

of -<;::0 l'012'.:1'\,'e \,jLr::--t:1on bct~.een t 01 ,'nó "f..'Jorlwlece

Ll tLe

C

yî

(::rection. TLe

c

y

)

directio!! beinc thc

àirecLl.o 1 perre,": 'lcvler t L.e i:12eh:;"ne(] üllri'óce .

1. ~:e :-1<' eh :'ne s t!'uc tu re [:8 f' ~ l.f :'Jc~; ent l.~/ Gma 11 da mpir.g

20 tj'c"..; it c~'n be s jd tl·.t its V:'bf'étion Ch~Y'éJcter'stj c

i;' cor:po~el 0'" tr,e S1..'P l'POs:i.tJon ci.' ceparcte moLton" 0"

rai .. J'l~ f.,;tr'uct·T~'l rlodes. T~ac' mode is c.~ef:lned by ó ~"'ixed

vj 1::r-8 t~ on i' rr:1 :~n \ir..i cL

::0

r:lpl~ t ,~àe oT the Io'.ot on o.

c ::' pü:...nt of tre r:1[>ci"".~ne .1.0110\':s t!.e 2r1plJtuöc<'reO'",ency

rel[Jt:'on 0 .... ~ singl deErec 0 .. ~'rcedor.1 of the r:ystem.

E8V~~!:G ngy·eed ·,lt;") tLe :';:,reI;01.nl; 8..JS1.r.lpt::1on2, -;':e CP!:' l":0\'.' l..i.se _ç

s~.nc;l(; desree 0,.' f'reedOT! 8:':'te:: te "y.pl'1~'.n t he relrt:i':o r:wtJuil

cct;',~en tooI c::.nd ' .. 'o1'l-:.p:i.ec:c (sec Fj (-. 9) ':.'hen tne s tructlJ.re

dorf 'ces in onc pa rtic :.Alar rr:o Ie,

\Yl

. ,

(24)

63-H-l.

P2e.;e 17

7:1 f.:~ste!':; Los;: \:".élSG H, a [OP inb cc lst?nt _( nd (3

dé'l!'DJnr-, cODftsnt c. ti H? ( x) cl i re C t 1 0 TI • XvJ't Corce

p

=

p

o

..Q (2 ) vle th2 n _18"\ e ti~ e follo\'JLl_ü eq'Jl" t ion. 11 I , v3t

m

x

-;-

C.

x

+ ,\:.; X

=

po

et

UOi': let: then (21)

?

o

/m

!Jjrn ~ ~ c

"1''IY1

-

\,~ '2- ,...-Cc '2

V

V1î

«..

:::

'2 VVi ..Q .f-Q=

d.Q

=

cS Cc 1::e ,et; -~ Q?. - ~ ~2-+ '2

t

6vJ -\JJ~ c,,-n t Ï:C rc 01:'0 ·:ri te : '1'11':'3 1'0 ~ ~.on ~:.8 .... l:O\·m :in Fjguro

9b

.

x

n.

d

~

p

vJ 6

=0-=

=

-=

--=

(21 ) COmpl0y. á 1P ~tude nr=: tur8 freq1.:e l .... y da mt>inc; rl" tio sprJ 15 con:::tar t corr,p] e: . .'orce

freO'l,en y 0 " exc j J.~ r jon der.rp:i n.~

re-

ct l'

(25)

C3

-

P

-

l

8'

c.:ce 1

\-Je élre f:OI'Jcver, ::nte:'ested :1 n the projection

y

in the

('I) direct5_on o;~ the arr.plitLic1e x ca:1.sed :\Y c fo:cce p

opcrc:'t:11g in t he ( ) c3irect4_0n ₍_.J_ee _F_-'g_. _'J_C).

~< w: b )~ (x), (n) :in one plene

The c ngle betv,1 en (;) ) and (x ) _. eX

TllC cnrJ.e bet',!een

C

'

)

end (p)

(

:)

\

.i!o~J the p~oiection y of x ~.s :

\ S:~?

""V :::. \~ C. 0 ':::> C. cJ.. --n\~) C S 0:.. ----rI ,, ( 2'': )

I --t< .{~~+~~60- ~<-

-CalJinr;; CoS(oI.·_~) Coc...,Q'- ,U(directiOné:l fector)~

vie CE n \':rttc : . Q'L

Y -.:::

?

4-

Q 2. -\-'2.

~

,

b

",-

cr

-

\.A.'j-

~

end there for'c (2~ ) A"" V lL Q ? r C \ Y)

=

--L-p -= -P:;::--_-_~ - - - ; -0.'" -r~

t

v.;t. -

w

'2

(25 )

\'Jf ~ cll e pression :J.8 celled the C:irf,cV'om'l

i' Hlct~ ri (a gr2phiea:1 representé'tion

ol' 'i:h:ic 1 shO\':n in F'je;. 9C)

he orcco~n[ now appl-.es to every mode of ~he Etructure .

For r systcm w:th n degrees OL f~eedom, ~e can thus,

2ccord~ng to the r~le of s~porposition, conclude that

on a.:.ternè!ting oree p In t.-.e (p) dlrecti on on a n

cJegrce o~ frcedorn system callses é' resvl ting • bration

of the l,'lholc syrtcr.1 in the (y) dj rect.J..on i'/;1Îch js tt1e

sl:mm:'1tio' of 811 thc pro ·'ectjon..: of x for the degrees

(26)

c: nd thus, a ccord:Lng for l'ormula (2h) end 1 it' (25)

n

y

=-

p

~

r

~

(.~)

:::::

pT

(V3)

\=

,

63-R-~ PE. ge .. 9 (26) cC)s~~-~) (oSo?~ (<-7)

(27)

PêGe 20

B. Boundary stabil "ty condi t ion

l\ v·~ br0 tion ca useà y so .1e rea son 0 e nother he t\'.'een

tool 2nd l,t;orl-::pj.ece HIJI ö~mt) Ol t :in - steble condj.t ioD. The vibre ti on ·,'1111 grOl".' to c11éltter in an unstûbIe con

-di t jon r nd :lt::: ampli -cl;de m ' rernB jn 1.mCflênn-ed jn é1

·oound":I') -ctr

'b18 conà:i tion.

Goin); back to 01X cutt Jng proces!::, '!C ',':il1 reP.1ember

tha t the v[)1'i<: tion of the Cl' t ting force

öp

ca u2e' by the

'Iv r -i e t:J on oi' the ' cross se cti onal cl-,ip (:; reê

6.y

eot.lel he cxpr'c ~ sed ~ y:

6. P

==- -

RI (

\j

1-'10

)

\',1e 1'0 nct i n (2S) that

y

-=

,?"f

("...J)

'llranslpi.;ed ~n the vartation of t he Cl.1tt:tng force 60 ene)

trc re18tJ.ve Plotion bett.-!een tool }o:lnt and l;JOy'](piocC \/ ₁

thc for~ 10 (2~ ) boeomes :

_ ,, _ n AI,'

yl

-

6. P

'"f( ,,,,-,,) - 6..

P

~

T

L =1 L

'.r:herefore by elimj.net"ng

6.p

res,' 1 ts :

the follow:lng equet i on

or the IIreprocJucti on l'a t io 11

CV

becomcG:

_

"I

, _

r-

(\,>v<+.)

CV

-

Yo

-

r(w) +

'/

R

I

The. rat io

Cv

VJh:lch expresse:=: the rptto of the

~ ~litude of the vibra tions from cut to cut in R

complex number since 11' ('IJ, ) ie complex (see ~:G-27 ).

[: \'JC saià before, t118 cut t jne ' cL .. on is stable 1f

tbe <~mplit1.:àe of the vi brêti otl f rom cut to cut rerk ins l.'ncha nged.

'l~1UC l\'e have a bounàary 3têb:Llitv cOLè:Ltion i C tr-!e

é'csolute l11.;mber O.c ~ js equal t~ unit:',: .

Thus 1f

\q\=

_I

(28)

nd thus

I~'

I

Dv

\

>

I

q.

I

<

I

Prom (2(,)

1=(\..\9) Cc.

tiJe cut t:'nr; eondition :is uï1.strt!le .

thc c tting eond~tion ;jr- 2t>b1c .

iC mm·] tl'I;' t

rew)

is COITlp1ex.

thus '. e l"'cpre3cnt ed by :

Ptt; 21

(32)

in I/hieh

=

reel part of d~rcction01

respor:se f'unction

r(

Q..)

TI

=

im~cinar~ part o. direction2_

I'ccponce '_~'~1ction r(\A9')

t

'"-= j me 1\5 n,_ ry '--mi t

~

V-l'

i'3 the j:;lé gi û3 ry P" r ts of t. e 1 U. rr- tor r!nd c:cnonin, tor

.

'1 re eo 1" '.:JP C2 n C orc 1ude th[, t ti' e y'eal pi' l'ts of C~3)

letermine tiie 3t2C~ Li t- or instc: bi lt:i ' 2ceorC:;:i.nzl~' c 2

\q\<

1

or

\q,

I

)

1

The ctç1oiL_ty J.s tlî1.. .. :::. eter ':J.ned b~.'

A

!:::J. t L.

'. I R,

In other \·:orc!.<:!: stability

bound r~' st hjlity

:1n .. t2'o- lity

(33)

(31~ )

The bounda - sté'bilit;:. condttion C8n thvs bc reprc"entcd

, , _.

.

\

q:

I

=/8

~

'/R,

1=

1

titus i

R

= - _ \

\

2 ~

( .., _:_.JC' _.₎)

(29)

63-1\-]

P::.lt.:"C 2r

S 1'10 j ndj en te the bound r'y :: V ti:i 1 ' t · condt tJ on \-,j t '1

the s 'b2e ipt ~ \';e th :;) cen 'Jri te :

TIle ~~stem j~ n 2 bound2ry strble con~~ l~on if

R -R

_{,- '}

_î

-

_-

- -

_{2 R}l _, In cener_l we find :

\

g'\

<

Î

\ ~'I

=1

I

qï

:>

1

or or or \

R

,

, -

28

\ ,~ -=

RC{-= -

'2 -I '0

11

\

R,

<.

-

28

Let u:::; look closer to the t)OUnd81'~; 8t21-;:' 1 t~ eoncJ:i t:i.on .

1. \'Jc é's::uned tr';- t Rl is a rCl], r'nd pO::-:itJv8 1':\.11:1 er.

Tl.is r1e2 J t:,[ t C3~ ) can only bc s,. t" sfied il' f

:13 rlefjéltivc or

'1 <0

2. He lc~ rned 1:'.a t the VI 1d t l of C',' t b :~s proport 10nD 1 to TI

J

,.,

.

R

_1

A~: IQ-=

2if

!;]C \tIj l l

in r n ~ncre Ging

R

_,_c

\',Iidt": 0" c" t b~

t

On the otl1er hAlld, (n

decY'ca:'::inc;

R

I ['ne;

ol' Cl l t b eo

.?l

find th t ;-' decrc2.~ing A re::uJ.t2 nd t hcreJ.'oI' _ 1n an :LncreasJ rJC

Jnere sjnc i'Ji l1 l"'cC1l1t i1 a

t hc rc' ore 11'1 :" d 8croc; ~ ' .. lg ~; ld th

A ::if\Y1r,l"L,:n neceU'J8 t'Jf.pljtudc 'lim vJ111 th 1S rest:,'t "

~ n.,nlnruffi R,~ ',:h"cn \.','8 ,,"'11 c .ll R, ~n; <: nd Jcher'c_o: e j n

(30)

Let 118 n01'! try t o f 'nè. è!l ey.prcssion '0 P-.

l".L'O;._ (~?f.») i'ie lmo\',' t.h~ t rOl' the Cl't tJ ng

r

roce r. s

J or (" G ""f (0)

=

t=\

+

6'

~ ...Ql'1. - -- - - - -- --- -~:'\. -\..3-'1..+ '2.. ~ 6~ 1.-<.9--.2L~ [( st,'l_ 1Jvct-'2) - '2. ~ 6~ \.0

J

(..0.

:

-

w'lf

+4 Ó~~?.

C

3 -

H

-

l

p[t

cc

~23

1'j-,i: !.'e::':'l~j_t ü : ,,-ery ::::r't:i_c:,,;,':i.nr, n:3 \.'111 be 0: L 1ncd

(31)

63-~·{-1 Pace "..1.

1. RLl\.Vl-) Je Jl;r:e rOe 1 pl.'r't o.~' t='-i.(v..9-; thc cUrect:'otlêl c"'pom:ic L' iw·;;J.o fo~ 2 do, Teo 0.,' ['recC10l:1

L

of tJ~c

~;tr1'C't1.:l'e. POl' cêch 'mportont cJei;;rce 0:' freecom 01'

r: C'tructu:t'o tbo 1': .ctJon rL(IA9') c~n oe ol.t<:incc1 _b" o:xc:itlnc; tLc .,tI'l,cture :1n t !(' (iJ!·(~ct~.o:Î 0:' t;le nodc

o~' ','j-IJ!'-t5,01llncle' ~in,Tef.'t..j:~é't" O:î ,,::i tl: ;, n ;::l~;cr'nGt.n~~ ~'OY'eo A

p

1. e rC~l OilLe !.lcr 2"l're j n ,tl· _ 3r.me

û:ir ct:or, 8: ( !:1l'ltlplJ.d ::it!' é.l octor ~ CJve3 [;::: -rl(~)

L'" i'!e mcpsurc t['e c.l_tern~ t h,C :L'(jpè't ','Ol eo ~?

'

tI e o"t'tp1.:t (ï:l.:::p1êC:O; ent 'F~c~) <'ne: thc plv'~:Öc rn;;1e bet\H;e:l t: c _{t '}.:oJ i'C Cc rl, i'::itL t ,e }\clp 0 -' r pO __ 2 r d:j.~',::r~'rn)

. l':1_D t Lc re;:-'1 X2t o. ""'F'i.,(L..9-) lJo:i

1e

9~( v9).

'l'j'c f\ .1ctjon Il;Cv.9-) ccn ,hr' '0 the :.:O:";~, ;:12 ::l'o~'Y' :~n F:tC. ] ',,- 0 r 2 , hm':i1 :L1 F~_[. 1 rcco!".li1C ;) ~ PO ... :1 c: \' , LO ,J"' t :ive d1 "ect:' on21 . 'r r. to:c

/.q

I ',2. I,C,

-

--:-

I.,J-

----n 2. "'F \ ("~ \ I -Il

1'h0 :!.'or".l 0 ;: tLc reel D(>f't;~ 0 :" tLe d:l.re~t:i orêl re::o:1c Lee CF"'VC r Ol' é.'

c-~'rtcn l.'t>C:i one (~(:f3;:t'CO of :-:"'ee(~o" <.'fte:e r~11uc:';~' cJnd ~)olE'coL .

A;

(IJ) bc C2il "'Cé"d S2.~ end l':e c< n ~plclJl[1te

1 r L'

Dne. c r.: \:~ll i)e cnl~· :il1ec lcJccr on.

3. r,'or en n. der: e " 0-:- ;I"~eC[~Onl ~~~'2 ~('rn, ~'?e \'Jj 11 [I.- 11d. 11 CL~r'\ es F] L (0)

fl.àded togct cr [;ré'ph:~c~' ,1;" , !',re Cc t ~'or the entirc structl1,re .

(32)

5 .

G

_..

·~ ) -'? , - ,

Pa ge ~

5

'l~1is Jo, ar; un C'XpmI)le, clone fol' Cl 2 c1cu rce of

,'eccàoITl 3~.,:~·tera in pJ g • 1~)

Tnc methocl of groph:;'cnl ca l cu10 U 0:1

oi' Gt2b11it<' _or rt t,·:o öegrce Ge

freC" Ol~. üy stel,l. J\ i'ter T11.16 t" D nd

Pol"" eek.

Prom (3{)) l:Je knot"l th"t the OOUl1d2}:'" 3to',.::ilitv oi' thc

s truc tliI'e ia de ter .linea h, Q,~-=-~ (nei [) s ~,,; .ir3 él I'e8,

ancJ. pos:lt:1..ve llITl)CY'

R'

1

i::: c1ete_!l~ncd by -che negp-ti\'c

ampJ:itl~ e oi' thc c~rve 8(\'-'). T'n;s ~\ x.':""\1.8 detcrr:1inc~c..i b:' t:le 1argest nC[';ê t:i \,e ;:; i :plJtllc1e oi,' tl.e curve 8 ( 0 )

bcine f1~V\.. 2t 'chc .:.~req1..iOncy W.t':'wv (See Fig. No. 12) 1.','c een rea d . rom 8c~)

en :i.mpor t 2nt in1'1 ucncc

JhJch degrees of :reedom h2ve

on the v::' br<.' t ins sy rtem .

(;. 'l .0 sr8I]- iCD l ly determineà Wt.'rvv fo!' \/:h1ch

R=

f=)Qw.., E. 01;,16

be thc Crccl"ucnc' pt Hl',c;' tbe d:;:-tter occurs <:1u:;:>11 g

tlie cutt:in.:..:, pf'ocess.

7. '.rJ.je rCD 1 :ir.:portance of tlle CTE'.pl:,

ti

(v...9) (rd the determ:i _l(1 -'clon of' R,t~Wv : ho "0 -er, 1::"e8 :Ln t i,o :~ _ ct tl:a t 'vIC Cf'Y:

oee l'rom tl1e [:rapbs ~','h()t in.1.111C;1CC ~lJe stever?l p;:~~l::eL:erS

of thc v:.lJI'ation syr.:te:-,1 hG' JO on tlJe~'in;:J.l ree:'\jlt3. :"e

can a180 Eee Lot! tllC st bJl:lt:' cl!;'l L::.s L' ~':e cLc.n~~,e one of tI C par20etera ₀ ₁ _the _s_ys_tc_m_.

Th:i.ro_, _n_Ol_/:, _'_;J'; _Cé:n _l1_.2}_{e _e_l _c2r_l;!it' _Ll_e _f_o_.1_o'l_J:i.nC _eX:?I.;o1e _.

Sl'PPOE;C \·je talce thc tvJü deCY'ee:: 0 :.' k reedom sy::t m f rom

Figure 12.

Ve eên 'ee '-'rom Fig. 12 t11êt Q,..k- js w~oinl;y dcterrünerl IJ;'

thc mode 0." v:ï.orat_o;1 3. (Curve f,' ) .

Let m; nov} ::'.1C1"C8 "e tLc s t :i.ffne2s

(33)

~:E1umin(· thc: t the m. ss clocs n t chlI'"tGc, tl'le

nat').!'é'l i'rcr;. 'one'- Q, [;08<:': up by " [',?ctor

V2-

0 nd llC ce t1 CD. 1 CUID to thc ne. ,q 1 (vJ-) .

63-1- .1.

Pé)

ce

2 ';

J\ssun l.nr urthernoro tr2t t ',0 second n:odc 0:'

vit· 8 t ion fl . re, in c> :nch nl";e<.1, i:C c Dr1 I lot

the re~1.~ltjl1G resonancc curve . (See Fl[;ure 13).

-- - - _ ..

-

.--

-Fj ,-,ure

j\ fter '.1: u;~ t:' <..llld Po la cc k.

H': iJtifi'en·.l1C th .. l:1ode Ol.' vi rr,-;.:jon 1, we n~ke

. PI sm lIer.

'1111s should ll?ve led to ' ::;'H1aller toté'll

A

K".

,

end t>llS a- Let teJ.' EtéJL·~l:lty . At thc ~~n:o time,

hO\:cver, tho l'C'sonê.ncc Cl~r re 1 I,oved l'P to the

l'ight in such e \:8" t llc.-t t 10 ::l~iIlrrlBtjon of' the 2

c rves hGlppencd to < dd lmf'Ol't 1,11( tely. UJth tLe reo..>l:lt t.hi't tlle "'t[lb' ~Lt·, d:i.d not 'iJecome botter.

(Cornpare Flc;ur'e 12 v!i tl: PiGu.re 1:';). r,[lhc on] y

'e::u 1 t I'IE S tllé) t tt:e cll~ t ter

.r req

V'ency moved to

(34)

(::::-H-l

Pé' j ? 27

C. DI;]CUSSImJ OF TEE _IJ~OHY OF TLU8TY -POI/I C:L::K.

1. The tr-:eory of ' l,'L~:::t~l'-Pol cclc includes 11 pr:inc:iple 011

the !1lodes 0 :' vjbr tlon or' t .e st ·~;,ctul"'e . 'J.rni<"1 j ,

controst to

we

(ppro,-,cll of Toui< ::-FJshvJic <: \'jr.:i.ch Vlas

limi ted to 01'11::' one decree 0 freedom.

é .• One Ool t Le m "n ci:Lf.~icl)lt:i.e!3 of tl~ 2pplic< t jon of the

theory o~' rEh C"'t~'-Pol: cel-: on Guch cOl.lpl icFted ~y~!tel7ls ['S

l'i1L ch::ne -tools ~::' thé:t one 11.98 to Getect t[,e jmport, ',t

dc[!;rees 0 ree om. It is cleDI' Ll-J.t: t tlüs ü: dl fficl11t

nei ted ~ OUf3 •

3 .

1\ si~ll m:i.st {C ~n tlle dE::t errdrk t:iOll 0 ' thc direction 0

,-11-.

vj ')T''''t:: Ol oL' ç cc" cc Ol .l'rccdom c(-;n cr'llse é. bie; r!ü~t('~(e

tn tbc l'inê 1 reSl lt bec,-,vse tltE:: 2.yr;tem 1[' very direct~,o~F 1

selsJt:ive.

Tl unty-Polo cel<- do not têke :i.nto é'cC01'nt the 1.n1'1\ cnce 0 f'

tLc ~'ot2t-':onêl ::peed of tLe rrrchine , s is c10ne b:' Tobies

-FJ,sht'i::'ck. i.:::;:> r esult oi' i'lhjch t lr .ictllod tellc]s to be

1c'<s prec::'se )l:t more p~."'8cticé~ 1.

rl'i1c ,.,t2bil'ï~~l c:18r'c o~' Th"st'l-Poleccl-:: ",8 is- shmm i ! Fit;. J~. y the cfore bccome~ / " / / / -' .~--

--,_

.

_-

-

-,-, FiGurc 14 St~biU,ty ClI rt '1ccorö' rE: to tne theor) of Tlusty-Polccec.

(35)

63-J1-1 Po ge 28

5 .

The C:l2 tter < ccorclinl3 to ~L'lust:j'-Po12ce~c ocel1"s ;:, l: t'le l1é l:m" 1 freqllCLcy or tLe mode of 'Ij br'~ tion

,'llich c;tves the larger-t nep::tive ~~rllpLi..tlJde oL' tl e Cllr ve

'1

('.A5) •

Excitation of resonEnce rcquires a f~xed phasc

bct~ een tl-'c excit·· ni'~ or'cc , no tllc r'cf,pon:::e 1;ihi.ch

:;n tl~:_s Cé'SE:' me ns betlieen thc h'o ::::urfDce T'; ves

' / , (I VI cl

,,/0

:

11' nm'] more tLan one Cl:tt cr t ootL j s Jn conté'ct

(mi111nc), the phoGC lJetvleen tllC t~;o 1;/[ ves <_ t tLe

Cl:tti nc ('(lee ... _ C~xed '0:\1 the t :i.ne elé'p:::il"!C -::s tbe teeth pé'E3 ê l'ixed po:i.nt on the r: rfD ce. 'l~li2·

1)h22jn~ js ~_n [ener01 not ('ol'respond:i;s to the

~.equjreö fjxecl phpsc .()~·:een tLe excitlnc; ['o.l.'(;e ;:'nel

the cxcite r [~litudc .

In oUcr i'Jords, rrlusty-Pol~cek 's theory bEn been

T:.'orkccl out 1'01' s':'n[:;lc pa th mE, eh :i.rün.:::;.

-In · pro ces ses 1.:;_ ere more thc:: n OilC tOotil 13 cut ttn[j,

tten the t;!CO ~~ : ;.' pess:i.wiE'tic 28 :in .oot c::'ses ',IC

ti,en

n:

_

ve an 0_-,--rc,-,onénce concij t:i on ciue to the

obove mcntl oned reason.

u. One shot;ld not nnclcrer:tlmptc ti e teci_j _o₁_s _tef3ts _anel

c21culations invo~ved jn the strDiQlt 2PPl:!CLt on

(36)

::~:~-l~-l " e

29

Th~::;t' -Polócek <'tate that ,_ Hflcilinc t ooI consi<'ts bêsic811y

0_ 0 n'."ltitude

o

t

l inec:r ::in_ó lc clce;ree oi' f' eec~om'sy:Jterns .

'llhere _'ore , t :e responses of eéch decree o~ r'reeclom 1n Cl

d_rectlon 02 lntereEt ror thc stcloil~ty ('('rl oe added togetr;er

nlcc bra i eB lly .

2" Frorll the 1'0 --'cgo:ing .p [CS IJC le2 rned thp' t wc <.lI'e r. a j.nly conccrned

bOl't thm,c mod S OJ. v1brc. t ion vllJ:Îcn ,_ppeer to c;ive consiöerEt1Jle

~.'e~:pon('es at tl':eü" vrriOlS r csonéJ.nce req1..encies ir: the dj rcc

-ti on o:C intereRt 1 () tho Gtc -oility.

Our main concern i3 ; OL' to d-'mjn1r:11 tlle rCi'pOlî~·es Ol." tbc

jr;:port"'nt modeG 0 vjbro'c--on jn the nei "hoorr:ooc1 of thejr !'eGonrnce frc(~l~.enc',' .

JI,ccord:1₁₁_C _to _f_o_rmu_le. ₍₂₄₎ _t_h_e _bE'_nic _rc"'p_onGc _of₍₎ s~.ngle _ë_e_g_re_e

oi' frecdorn jn 'ehe cJj -' cc tion of interest i 8 givcn r y:

/t,L û'). ).}v -D-""L

'j=

P

Jk

Q<-T2~öW---\A9'1-

==

?

-

~ sit+

~

6~

..Q.

w

_~'L

< nel this bccomes et t'_c reson"'nce .;:'rcc:\~cney;

DR

In order to hDve trc complex re~~ on'-'o ~ lî the 1<:5 p,;-,borhood

of 'cte rC:3on nce ;:reOl'enc:,' [\2 smf?-Il as po:J8i1)lc (8nc, rd.so

the real D'1rt 0 L tbe re sr. on c 1n ~','hic:r:. thc ::: t; 1,i 1i ty is

cxprcEscd) wi th EG

_

,ree

.

8 t ooI 108d

p

as

pO:Js5ble, wc crn:

a ) Decrcase the Circction 1 r-ctor ~~ .

~Pjüc, flO\'JeVer, 1:1eans irr1..mecU;:'tely in mor.:t C8GCS [> ctrllcturr 1 rcdcsi[:n bCC8l.JSC " t Jmplj.c:: · re-orient j ng the too __ po.~nt to the \'.lorlepiecc.

h) Incrc8sc the ct mping conrtnnt C o~ thedecree of freedom

:i.n' ol ved.

In r.Ja ny cases J hm!ever, thc lmpro"emen t of the dampinr.;

constémt t'lil l be impractj.cal or even imposslble. Higrler

dampinc; vli11 be accornpli shco clthe b:/ u"'-"ne; other r:l' tcrü"?Js ~'I:lichJ hov/ever, oft'n meanG ' ess Gtren :tr_.; or by insuI'tnc: L:Lt,:ICT design stref):.:e3 . (Sec Page 207 J Hoyt, 1\1etalo...; nd

(37)

c) Dec'e Ge of m;:;::;8.

t ~_··.l_l

Page

:::0

Recluct_on of 111<:::::- iG l imitec1 b:' the c-trenctl, requirecl

\·:l.e~ thc ViOI'2t··on ie· eljm:'né:'ted ..

l·.'l pre vibrc tion stre"se3 11<: vc "'et tl'c~ 1:i_r,l_·_·_t _b_c::_f'_bre, J.o, d r: t"'e s "es 210ne ean :::et them 'i\I['en t rolir)l e20TYlC vi~:.>::'ction3 c' ).'(; èe~:isn(;ci o).:t . 1I(;é.'~. corwt:r 1. C't:ions

t~· c ~eun f8vore0 ~n t~e prst. -~ ~lovld however be

\'JC 1 notec: tltDtioef(edl.v.e rlOGSOS iil PIOSt C::'SC:::> ;:J01'( o 11y

detr:j:',ont~11~ to tbc :oroblem.

It jo o"ten 3eld t::dt Gn :il1cre?f.':c :;.i~ m[::::~ cor;t8inc

;'dd'i'G:'oik l dCl.lp:lnc. Th~.:J:i..: .0Jc tr'le-r.1êterj,>1 nl1~Gt

',.,'c:: :.:'trei:;~c(5 :;Y1 0 der to [~jvc c1,'1l11pJnr;. liE terlól \d~j ch

1:.: not ~·t..'ec~·ed cJoeG not 1"e1: the ;·t2uJl:it: bl,~t JE :i 1

cor:tré'st dctr::.L·,ent:-·l ':';0 ~t .

(1 ) JnCLCé.iCC o f tLc ct:J,fi'ne::::: .

JJeCJC1Ci: ir.,prov:~_nG tlie d;)IY12P.ÜC :;té:Î)::l:l'c~' 0 .;:' ('> st:r'W:.:t1.:rc~

'e.Je incrci"':'c 0:' tLe cti.f'fi!esc ['_.SCJ J:~:pro /cr: t}'c ctr.tj.c

:.;té'b~J:;":;:' . -'nprO\femént 0.: ,_·tir;'j1C:-'S ~~.3 tYe-.:' (orc of

~x :~·ér, ol:·lt :·mpo·~·t:,ncc ['Ol" mr'c';:i.n -tool:::.

c ) A c ~b:n tio')

0 ;

A, B, C anI D.

f, c1CC:('cDce o~' t "e rêtio 0 ' thc; lllé!2::: over i.;he :ct::. 'f'1e:::~, t f:C·L;_.C.~:' \ji.tl 1:c.J .. tnlrl dEr.1p~~ 1~t; rr10. C Jl()st ~'·rvoY~r'~..Jlc

C::.:.e::ct:lcn {';fcta:·.' 1:~~!.:_2 '::c t:,c gor' 1 {'ül' t".c (lc:::;~ c;!el' o.~' L12c:l::J.nc- J

-oolr:: . (~~co (3~)).

3. l,'-e S11011.1cJ ~ l~'Jé:~/[: rC'l:crl~'\CJ.-' th['t t~~lC! tLlpl)O'}er:ic:.1t 0;' :;: ~· .. e~l( mode

o ... ~ '::J.h"'r-J~:iolî of c :::'vr"'cture st:cUl(l I.'C donc ~n re12U on ~;o tbc

ot>cr 110(le2 of v:Jbr2J

c'ion ';-·hjc~:. pIL'"' é' 1'010 :111 ti:e :::;t.::'b:lLt~'

(38)

CE Pil ,H V

rrnr-:-

cor

mnn:m

rvmTIïOD OF' PEII'EnS-V N HEl CK

Iq . Tne JlI.-:Jchine cl~çrêctcriGtic

f:3-11-J

Pé ge 31

The tileOl:'y of rrlust~.'-Polacek :3PyE tl,,,t é'ccord:inc to the

rule UI 2upcrpositlon, ~n oltern2t lng force p in the (p) dil'ection on c n deg1'ee ot' 1'1'C dom G~'Dtem erl1~~es é! r€81l1t-·

ing v:ibrc.:tion oi' thc i': wIe syct(l i) the (~, ) eLi rect:i on

\"hich is tLe G_1.'r.1l11rt on 0_ (11 the pro,jecti ord: .;1 01' x:~. Por

tr.c dcgl'ces o ~' Fr'cedom: L (1'!herc l. :::, \ '2. '~ . ' '''. /'.\1 n)

and th E trlé t

l1Jl1s J nm':, re n.2 r'k Pe te T'S -Ilc n He rcl-c, l:=: "JC ["':. ~ n te re r- ; :.111['" • Bec w:c 1î. \';e ex itc <:' n::'eh:inc \':}.tL DiJ ['JtCl"l".t:ln3 f'ore:o

I

.r

)

p

".(1

tLe c1jrectjon of t J1c c'Ctt:ing lorcc, _LC 1;Jll1 _i'1cJ _'₂ (i:l~.pl< cement :in t"c cilrcc , on per'pend:'c' 1<,,1" to "Jje r.'rcll:inc6

Gur~'ace (bei: C the (, ) d~rc:ct:~on) .hich je thc 1'8S"'1. tant.

eo~~lex response y COl' th · total 8tructvre .

[1'2 tLi'" con be done for cach de3~l.recJ. i':,:,cquer:cy, Vle .: ' :._ bc 2ble t o l.':i .. Y("!) for the ent:':'rc moch·ine . i?: divt;;jnz:;

trüs rerult by thc C' c:~i.tc ·-ion ['ol'ce p, -;tie h["vc Lot T:'itho't..:t

én' dj;·.ficvltie:: the \'!bolc COL r;lcA d:i.:rectJoné.ll l'ecponse

fllnct:lon of t1;c S} stem, being;

-f

( 0 ) .

:C:'l mCc::3't..' inG tLc input fo ce p, t;~e OlJtpl't o·::pl .... cemclJl:

'-/( 0) and t :Jc Pf:::''''C érlf~l 'et~-eei1 t 'c t,·!O, ~!e eên

conr.t.rlJct 2 po18 . cl.'"cr21îl of' t!j(; entj-:'c ï:".2chlnc.

Prom tJl.C polar d:iec " Til i'.'C c,;n COnf'tI'l ct tI"e reEl p,rt 0['

b IG r.-:é'chinc r-er:po{wc, PO:lr.t LV pojnt, .:'or t:,(; ,.r1io1e ;'rer" /ency l'2"[~C o! ji1teroc:::t t:·;~s _be _'_rl;; ~1lL.)) ==?:.8~ (v..")

rl·,.I •••·•_t_.... J' !":: _{_} _t...1'1 're_. _'-Ql J t _. 111_•""lJ : • ' C rl'·.'11• "1... "'t,'" ~ _·D()~1_~., ··roc\__ k 0"t )1 ... .L ,J t; ~'ot ',.,,-~'.:-'J. t"'"Yt::.. _I. , -'V_ ;'1. e

tcdj01.lc p!'OCCd1L'C cl ~ccr '(e (; :in CLr?pter II . .

Vc l;.'ill àc,l J.') tll"~ d:i.,-,Cli:::;~-3jon V!::.th tl':e f.'L:.r~ler cOn::C(ZL~C::.ee;""