2DOFsystems&Vibration Absorbers
Transcript of 2DOFsystems&Vibration Absorbers
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(cushioning)Packaging
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vr4ACS
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, r! wesire sttitableniilalundifion,nu y*rrPv;brafefat' orteol +nunnafurql ref uet/dtu, Durittqveeuibrafion / one"l ln rwfuralfrnquena+fuTnplifudes"f +/'ufwodegree f fre"do,rr oordtntore re'lafed,t aspec;!,.c tannev and +he
con/,guratlor't s culled a "mormaluode,Pr'!ri2^1,&, w rtafqral?/tode""fv;brat,:onrt', "+-"deqreeo,l -frn"olont$1em hasfwonot^ql wodos+ t-,
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. As ;" qv;/ent frowt he fwa systers shoun bnlorel+hecovtguratioaP asys+en cauL,espec;{ieJb3A Set& i,tdependburtaordi,ta*eSuth aS, lenJth,MSV,w sowePftgsicalpavawe*er5ArJ s uchsdol coordivtole5s colleel t3eneralizedoorclinafu>'-
. Alfltougli+he equafiacol nnotio^ la twodE*sol gre"Jowsysteruare.SevterqlJJ*.!P.,l/" =o +hale.arh qwt:oi nvoluesallhe ra;irF6- ;t ;toloqgs' osebE to -FincJpartfa,rlaro.so0coor7d,inateisuch+hat eqih equ,atia4[ zuofton
coordirtales.
ra\ -) /
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wtians6knyotilt sa [otrct,t ;bro]iai'.
- ma s Egslern. a-baul Ee1'auL:*2trLi)
6 t+)
- | t z ,L f ' ' L V, L V- l ,Lr +-.ira---
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@Tt cartbe seen tha.t:
lhertu,Qaafrons O 2@ rapreson! sgsteu of t*oCoOl dcl lf'tonh'a e.q at ions.t
4J;c2 Uelsa.
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of undonndsysleus:
, , ,+,,ni(hittr) i; -: kz Xz=-o -@
*witta LAletl^ ed iiaincuu oscilafo baraonicalywa 44(!- hal-lljrh drlletat aiuy'litu
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1772b
trL ; t ;i i;4i iieel th:at
= O
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"1qle l, tlcn bv2
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@a {he ra t ios :Yr= ll / x,^'and;;: Y.d/x,h) (uttu'u' ttJ=ua )
s,4Y4* o2 ?1 l^otr (.U= Q/t- ard co"=a)r'-rol,an,z+Yr1tz)lzz -r112r.4,2+kr-+k3),,r=Y)')
,. '#= k z _yU, ifu.tks)u,' And u] cq.n e etptesssl esp"rtr-rlgQs:
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jr,l :-.tzt t _.^ + @, Th-otectors and X', a6,r1a slenob \
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(n"j .4 lht, tuo eqta+ia^s ^ t+xf)f l l l( -^ (#)"znl Xt - k X, =o
, t t l-r I't,Xt ILX.=- o .+> 4+= | -- rlA Z 'r (* *
^(any 4 thp fwo eguat,zns i.t t( i(
. lqnacl-fu An-p-atural
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I ' F llodntllz
-- lr\
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ht (2 )
*,ro-*if ii * ijJ-tiit: , i,", (r,*g, *o ,",'o, t+dz)@
@Ioutins eQua f i oas3
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l Q - + -
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lovs ional Jq-sWiS:U.Couside(a lorsiovtqlystenon ashak os shaanbelou:Mrr4llt'oreappled*orq leS f"g,I
n , t lconsisfil,sof fuo discs n74nfecl
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h A - k t , 0 1 a f u a * E d g z = A bFot thre rei fJibrationanalgsisQuat;ausA!@ rzdwfu:I
I t g, +((s,*L+) 9t - kt, 8z =of,. ii, - ktz 0ra (tltzTl/ts) 0z =o
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SySIen lteflllosioordiaales a dei q;he ifs Ciil;gurii;on.
Headstock Live center Dead center
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Corp.)
,,--J-6
IzU)k2xr= ftr1ra 116,
k'(y - l\0) krb/.+ Ii0)
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nais ra and-massnomnl o4 iielld f,o abatlifs C.G2ksfirlJ on sprin1s af: .Jt:;lhnsies k,andl / z .
. Fat his twa degree f .fri"elon sgs.bn, ory o4 fh!filta,tt;1g.strt of coord;nafu ag be useel tp
t. Detlectiorrs ilt(t)ol rhe lothe AB.
3.4. DePle.ction(d * Poin!:e fo tla W 04 +14!C.G.
P locoful l a "ltsnnczond rofat;on |ft)
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2M=;ll
=
NiiH e- 1e J
ds coul{ be rea'trang'icland :,A S :
lfu,+k4 - (k,tr,-lr'!,)1Il(u,!'-u,A) k,l1*V,l)) t
The,wo n uatio,ii natuix .fo;rm
f: ttJ; [ o lt .It can be seea +hat eitc.l,t"h +lt equat;ons
Equat;ait up illotiaoUE;tr3 tt) abd \tt)i
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Zf = m'O-and
-b..ru- -0)a) [z-(!* l: g] - il,leAZM=p 6; k (e- :g) ,) u(y nl) AA-/ne*
- a
If E,l,'= k".!"" |fu tysttn atlt hare dynan;cet'( iieftiq) Zoupl;ng artlJ. , -' f n lhis caspj f Jfu latlz moaii np anif dbwn i't+hu g Qiiection, tfu inerf;a &rce nE; dhiali dals
Ni+a+heollaatry harccterisicsliheses.qsr'ils:
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a
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ecliatidns S rrtafiaa"f oru arto*lrl.
5.3 COORDINATECOUPLIN6
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The dillcrential equations f moiion IoI the 2-DOF s,vstem re n generalcoupled,nthat bolh coordinates ppear rl eachequation.ln he mostgenelalcasc,thewo equa-tions or the undamped ystem ave he fonnn;i + m;i+ &Lr,+ k,, i, = 0rytl + mtjt + kztrl+ 4)\: 0
fheseequationsanbeexpressedn matrix orm(seeAppendjx)asl n - l t , t . l , , * I , , l . i n lL , z , , r l l i , l l r / , l r r r l u lwhich mmediately eveals he type ot couplingpresenl.Massor d)ndmicalcouplingexistsf themassmatrix s nondidgon l.whereas t iffness rsl4l;c couprrs enists { thestiffness 1atrixs oondiagonal.It is also ossible to estabtishhe type ot' cotpling from the expressionsor thekinetic and potential energies.Crossproductsof coordinates n eilher expressio
denotecoupling,dvnamicor slatic,dependingon whether hey are found in I or UThe choiceofcoordinates stablisheshe t!?e ofcouplhg,and bothdynamicand staticcouplingmay bc present.
(5.3.1
(s.3.
IL s possiblc o ind a coordinatc ysten hat hasneither orm of coupling.ThewoequatioDs re thcn dccouplcdand eachequationcan be solved ndepeidcnlly of the
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orher. Such coordinales are c lladptincipal cootulinatz\ .^lsocalled nornal coordin!(s).Although il is alwayspossible o decouple hc equationsof motion {br theurdamped syslem- his is not ahvays he case br a damped systen The lbllowingmatrix equationsshow a syslcm hat has zero dynamicand static coupling.but thecoordinates re coupledb), hc dampingmatrlx.h ;.1{rr}=:} (53;.1{l}.::l{;l}.irIf in the foregoingequilion. cr2 crr = 0. then tl1edampjng ss idro beprcportional(to the stiffness rmassmatrix).and he system .luations ecome rcoupled.EXAMPLE,3.1Figure5.3. sbowsa riSidbar with its ccDkr ol mass ot coictudirgwilh ils geonetriccenter..c..lr + lz.md supported vtwo springs,kr ndk2.It represenis 2 DoFsyncni, becausewo coordinaicsarenecessaryo desdibe ts motion.Thcchoiceollhe coordiratcswill dcline he t]?e ofcoupLing lul can be inmediately determired tiom fte nlas ard slift rcss Dalices. Mass ordtndni.d! couplingexisrsf the mas lratrix is nondiagona- whereas liilicss or snni< ouptilgexists f thc sliltnessmairix is nondiagonal.Il s alsopossibleo haveboih tbrmso{ coitpLing.
FIGURE.3 .1 .Static coupling. Choosingcoordinatesr and 0, shown n Fig. 5.3.2,wherc r is thelinear displacement f the centerof mass,hc systemwillhavestatlccoupling. sshownby the malrix equation
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FIGURE .3.2. Coordinatescadnrg
k2l"+ 140)FIGURE .3.3. Coordin.le eading o dynamccouplinE.
' FIGURE ,3.4. Coordnratseldng lo sratic nddynamjc oupling.
Static and dynamic coupling.'If we choose = .rr at the end of the bar,asshown nFig.5.3.4,he equatioff of lnolion becomem n t ) \ ' . 1 1 ( t t . . r r _ i l r , I l n ln t , r l ; l I k . t k , t l . o ) l n l
and both staticand dynamiccouplingare nowpresent.
F r '--l
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a r-t" +-h, | \ Y il"JI
. fhis lguru shois the uar;alioi ol m aip];1o{, 4( ;;5:r;fioi"t +fu mach;ne U(fEtl wilh t!t! ilA.ttidA ipeea(a/ut).
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.Fd,andii,"1!
; k i X ' =
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ilor;ry+hdt:kz- = -LZ- . w, . lLlt - fA. /-nldz-l 'Tl = -ttz ' -n;' rt = -At \-aJrt /
i;" niti,atyg,rtdou"of"i:
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@ ThL variarion 4 -Q, ciln, and Sz.ldnL aslanct'iono* ffte ruAXS ah'a trlr/rt\ Are shoatqin tla ntyL lrque dl +hJ:!t- d;flvztni u"-Ait'o,4 ll* fuoquerrytvaflo th./tt/a,.It ca,n egzn tfut tfu a;ffu.rorr,ctpfiu@4
2.6
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.s'lt
c l5
2 42.2
0 2 0 4 0 6 0 . 8 t 0
-..-..t...-
{-,-- '-**u\n , " - ; ; - -^ - - - - - - - - - -f t t . . t r ' = r .o
ft r".ff=:o
EyanPleO: @A diesetong;no, egh;g 3000N) s supporleoln apedes+d
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'ynouytt,t has been bseruei+hat+he 4gtneaducesl;bratioin+ohesuruund;y3 o/e0*hrotgh ls pedestalourt af,anoperctivtyspeed"* 60ooyP71"ietetmirie heParan*uA al +neu;brationabsarbu+hatuJilved,uufu ulbrafian Nhwt mounled n +fuP;bdestalTh-amagi;lrdu o( 11'12Qxcihlj lorct is z5o N, and llat
awPlitqclaol uoti* '? nu aur;ltar3 ,noss ts fo be?i^;l"e! fo 2mn.Soluliio "
.\Iu lrequeng ol u;bra.tion of rl'uvv7a2la1'4e5LJ - - 60oo Pn poHt w *= tunfooo= 6283rad/s5 0
Srna +hzmol|an ,/ *t pnles+al s fo beEuel +o7ero2 ll'tn anPlifude o/ notio, el ilu awf torg mutsslauldbe equaland oPPnst:too.fW -l+/u o
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Gencrdtw
ahsotbt; : t :at ion
A ao'lsr generdtdt/gt r(shorrn in f;jute abou"), is desryala operutt irt +he speee) range o/ zooo la | 6oo frtt.H|w eucr, tht set is fo,tncl to uiol!'ltt/g uib rale ate speepl f Sooa rpa 'dut ta a s/ght uofu/onu iat44qnofw. ft is proposeslsoa,ilachacanlibtttrt ntoun/edlu,vpud n&9s absorbusysfrn lo aL;a;at7s Iaprobbm".ilnn a csili(uuh- crltJilg a triol massdl zkgl funello 30ooryil/.is a.tfacheclo t/>) +heresull;11 nalurolfr"4aeociei
"/tlu sgstn* aro lounrl tz be zsoi rpm
qild 3500 tPn"Desltr *he absorber a be ajlaolted by spnu/gi11tik mass qafl sf;!/ness so +htl l-h.a alurql fr"guenciu
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. It ;s rcquireclo ohfain ihi ualues *( -, 2 L.ihalut;ll resu-lt n: +haA
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3t ualut snatLozl(u equattz) 209-3rael/s
,J'(#)Q)rtrbzanes elnal
tlrrz/A)nt = Ilz ttrtt 6ndu)
/ \ 2lSLi | =\w,,f
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@/( &Lsf= r+ 1 +lA;u T7l : zrYrt+ TLu o!,uxol *,fotlowiE Qns'
r -, l l 'u4+l'-t -+ u1= tl ' tkfr 'Vl zrtt/h hLud hu n s",t,slj+ht
ehd @lu4#- ,F*7 >r(4tsr r//
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fritVl
2",/
3;;3tq.tb
---F iltlz to.32 kg --- kr=ilrr,{=60s",.Nou ]r/,{n'
FExtra nofeSon Lhe unalan nd u;brcofi'oa bsbrberSi'.
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Le*us &.0;ra*he' [email protected]{/= rllz /nt = nlags rafio
/ , f \p = dn"/U/n,= frncluenu ,aI; 6o Subs,liWirtq&r batln /r =nr/nt, ,e
y. (tz" ' nt, tu,") rA t =Fr
. // t l - / t t fa ^ t b - t r t t a ^- - n 6 / q v F t l
( L,, k. - il,t,*') (U;-ruzui') l,(;Lctt;ll
. Tlu absolc,h ualue"( tl* 4boueirr +14enlliwiyl !ry uT; il tfu u,P essc,n s Plffi/AIAL. l l= A,25
F'-@11[' (,h)],uP'
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p : 0.25B : 1O bE
IfNd
z
0 . 7 8 1 1 . 0 1 1 . 2 8 10.908 1.118
Frequency/ o \\ q /
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ol fr! sgstu4,!,o tn I*+t ;t ++udr; ,;ghrolrysh;Its"uch +kqfI XE f"l> t +Yteoru travrsnilledo+l,u rinaflsgsteu s oru?tif;d And+laObsotb.t'tg&n is no*c(t/t nProvernpntU(lL++aorilinal dAA n ,+ +l'1!-V}ina$ EJslert4.
, Tlu shado,l req,4 +h,agurn ,llusuatla+Lu tk!o?eralirgrartf; "+ th!-absorb!'Ldesgn :0.gog drlN
. l-lewe,{i:Ee driu;gguequenEttiJ+s ui+hiq his @
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stiad3- srnto uiEiaf;oi awP i+urde.e.rawirtiitj he. mags rafio A, Aid +ht frcquertg iafiirTJtes" uto diierrsioulessTuanlihesrdivec@ s7ec;{j bo;+lailassqnd St;ffrress +heabsotbtz glenr (n- P L").
F.
. {}fe twow h t .h
"*+ho fre4u"ngequahonlou,rd o be as lollows',( t7,Z JL?)
ttzTII
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"la@ arry, Hetra,;n;fs ML a+ba lwd orr++v wdvirnuulUala-t-"l X, bs #Lo Jes;jrrn.T\tt lollouliaq Lyawple ;lluszaf es aAabsorbta--Jd e s ; q A .
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gqatG
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di;ve,thornon;ctil\ y a rrtotor hal tulrrs +heblade ag ;l lu. i trratadelow.
vvtotorurts f canstuvftPaed ndproducesq 13-N orcoof t6o cSdu/uir t ecausesmulluqfulana llw rwtor'
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Thz ves,lnrrqonrn[ rt'btolia,,tlas la* deletil uutilolkt-+hoaw had baa *ra,rtularlurel'tht tvtartula,lrurerqrlis a JibrationabsavfuI'.fo dvi,le +Iu +abb.oscillaliaa Tergsinlp,lJby
des;gnrveno illinct qn absabtttsYtt +h!-baQ'
QDes;gur \/absorbpt- ssu,llig +hof l't '!,echl'tg- g*,Ifnnsspvov'rded y +Lufru{n gs is z6oo.N/w.Ln additioi-,u absotb.ut nus*;+ irrsi& +h!'+abg base- anclhenu-has a- m aY;dlua !bt;ott+ o'z crvl '
@ Atso2 alcu-lotot'u
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----E L(.-- F"//" = B N o.z ct^^=t3 { /o 'oozr. LSooU/t t ' t. Q;rta fu- abso,tW'ts aaoign"dsu&tthal:
t ? . t . f o S o o N / Ud = d , t , = N i z . ,
@ N,L +lfrwnu"f;,ltr /Fo= 1 e( /, lz, fo =-t
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s,,bat;+oljg/ bs+Lt lzt/rn;t z -t i,4 fu-' ' - l ' - ' - - ' - , t , 4 r ' Jth-roafs +fut spot;\ tkt ucW desgn ranyz:
(4 courg hs ,-h/ al_t-pga is t{e&l)
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0ontpe/ A3non;cU;botion A bsor ers. Thz"lymnic uibra*ion absorbe,r escr;bndn flu prevrousSeci;oucnztlesht or;qinol esondtlCeol{ irt the
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resovtctnactlrve ul\fu norhitte btf i,rlvocluul tadpealzs.. Thus*h.a nachineevper;entes l6rge an pl;+uclecas it Plsses hrsu3l'th-o irst peak 6/uv;ng tart-updnd stoPPitg.
T"la ntp!;fuAn *f flu mach;ne con be reolued byactdinj a danpd u;b,,atiou absotbt as shouh 'Nlout
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(*tc) \kn)
Dynanic vibration abso.be.
f lLLl - l
T
ht, i l r+ tr i i + kr(r, -%|+ cr( i ,- t t ,)=. F"s; iut
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nt,irt kz (r"-r) a c, (ia - x') = oBy assun;oj hz lollow;,t3 ohfion:
r / i q t/'(il= t\r e t / iw t/ z,tt)' l, Ethe loltouti,tj ?fuoelX- fafr- Solul;onsrk bnobtaird:
9y dnlningla lolloui't3:,4 " fllz /ftlt =**= Fo/kt= Sr{u. 6nllecltanl sgtbm
l rMass qt'6= lbsorborrnqssfiArn lnosS
g- -- foro,l f t.qaenS rafiaC." 2 lzasx=Crificaldoupiry Qilslattl" cz/c, dauitia3 atia
a
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thecrragn;tudes,, aridXz cqnbee{prett?d As, j" Qata{u)l"
2
z
n 2 ' tJ ]W
viv
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t)
nd- ! c ) "'-0 r'*2f
Tlu abo,t" quu.fiotu X,lS"t shows+hat+Lsanplilqcle ol vib,ot;"n .f rh"l/tait "nast is a*l',rt, f , ft and . The raphollx' /St+ltlu loral f ,equengtafio &= a/Un is Shattnand .u= //zo /tv aleu d flezo nttokt-otal7.
!unxl*agcirtsldzr/= I
(=o \ jt 1 I i r :4/ +*:l612
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i ,{.4
L }\ !5l - i t( = 0 1 S{ " J1 8
0 9 1 _ 0+fi
1 , 3. 2_ l.7.6
, I! +heclaapirl!s zeyo Ccz-7=o),+hen ltesowsrruocruu af +lututo und,awpedesortant tEueilcies++lsgystreu, a resuk rhaf [ras alreadyadiida/ balore ,,lltsa
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lrt!. u,idsrped v\brationabsolbers terc disrtssld.. tt)hu *he &wpi,r3 becawteSnfir,,hn l= A)" +hewo t4asses
vtrdud lAz te , l i t+uql lg"lawrpedEethert 'and hesJsie/vl behaues essentiolljas a si49{-c,sgsten ,Nitha mass (qan1")=$s t i fJness l Vt . Tn this case dlsoJ
deg,teel fr""dav^ardksonavlc2cutao .9+S.ih Xt-+ oo of &= #!; = It -
*hese.imif1 lhe peanol X1
0 priwr .!yTunedV,$vafi6l Abgot er .It csnbeseu f,o* tlta !;gu,teol X, Stt lho! all +hecuriles interSect.f poi,rtsA and B regdrdlessFlhe, talue daupi,,g.These oilr+s -ube toco*edysubsu4,JirS+fu*Veffi
Thus+hepeo,kol Xros f-rt Cz=&.Soneulwa irt betweetui l l be Ai l in;( t tuhl .
is ir!fni+e loY Cz.= ASu, le l l
,Tht rwo ools "+ '4 kl ;'tdi alsfu- aluesffu lrequen3 afi6t tn ' ztta/u).andfls=corresponai-go+fuPornts t andB'^l'ute/u
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The ordirtafra 'l ,e anatB cqn b" lound bgcubsh'ht)ythe ualues"l 8r ondIt has been bserdedge tpsfeclite!1nlo qC"t'+hX*he?rusf---=
iWation absorbers otte or uhtC.h+haotd;ndu [*l,u poirrfs and B arc eqrl"al.This cottdifiouequitesthdt :
At absorbalsatis{girt1 he abottecurrectlq alled (c+he +uytedAt+hoEh tq
g iadicubrhouto fuile an obsorbetr,tdo"s o+ iria,r^h+fuopriaal alue t+h"danp;gra+ra I and the cortespondiyj a[ua"{ * 'Tfu pfinalualueel I canbe lound by making +hey'esponseurvt X,/Est as l/at' 4s Possib(z tpeaksI o'ndB. Thiscqn6p6tgl7;euec|oymak;'y +he currahori(orttalLt ei+h!/L w B 4s shoun below'
@- a covtd ti ortvibratian
cqnbefusorbet -
I tuld -J = =:1-- -'wn l+ (t4z/t't)
t210 t , = if=-L
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E
6,, '
2
curye horircotal ^ \ s
' 1 -O . h 0 7 0 . 8 u . q , u L l L l
For h;s 4 k) is subsfi{ul"dn & fil *e mak16,mat! the resulfinqequa-fionapPlicctble fvrh!. eauo! opfi/ilut/t/+ufingh., ii," uod;/,ed
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Q ro) is d;fknwliJeJ v;th rrespecl.+oftzh,td ffu slopt of th-e urut".t'X,/SylBqsefilnq lfu slope "yuol fo zetloat pointslanlll., w u"ob+oitt?zI . f - poialA
I - - 'dn4l '=
I ( t + t ) 3
A corruenienfverqqeal,rcel /z q;,,u. h-!+ht abop 7" ualies is (tseeJ, "otnsi3,So+L/at @^ z(IOPimum: 3,/ l 8 @
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g (t*t1z, The wrespondilSpfinalualwol be ufleS
@aNofri,
Aft canbe s*", h^ 4 rh tfuf t4eanpli*ucleoNtheabsorbtz-ass (Xr) is alwaclstuch grcufnthtu' lhnf rl f/a main nass (Xr).- Thus the desr3nshould be able ]o a-ccowrd/ahhr lorgeauphludes&f tlu absorbn rnoss.
@ Si,tu tfu anpllfuobs of rilz avee*pecle/ to belarge *he'absorhpz f i"J (k7) needsFobdesigned ho, alafgut pa;ntol uiatt.