2dofsystems&vibration absorbers

7/29/2019 2DOFsystems&Vibration Absorbers

1/88

/^. t t lv? n ' J r , , , S , , - { a r n c 'I uo U4Sfees af lreet-ts,r

a

, Gnsrdiv +ha iltohx- pux,lps|slenl S below:owu

=_qj! ht Iol--stut-


2/88

atld ,t-ta ai iidppendiitrcoaidiio'fu-7-o spaCili

" Thia-+ht Sli]-eltr hal tuto ctgreel P *ee"loa- t rs

bod3 hau;y tati poi;ble lyp.esolatotion.(ft ltf|.A pwf|&,thini;a no-iw/ ta ipedfg rfu roktfiol"l *z mals aboul ;rs C.61..Sn0un be l0A:

(cushioning)Packaging

(rnass;)


3/88

@

alafiirr /rransb.hbt'tslong fu / anil y direct;onis')-., Thn eneral ule for fhe.of dugrresaf freedou

canpu+ation / +1,nu.betolcqa be sfstedastallows,'

r l ' n ' 4Nunqer r de$reesotA t,!reegls), o# +fteEls le4N4nberol o$eS n tfusJstenXrtuabolol Posr,&gfus .f rzofion [nofi

vr4ACS


4/88

, 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-,


5/88

. 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\ -) /


6/88

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---


7/88

@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.


8/88

of undonndsysleus:

, , ,+,,ni(hittr) i; -: kz Xz=-o -@

*witta LAletl^ ed iiaincuu oscilafo baraonicalywa 44(!- hal-lljrh drlletat aiuy'litu


9/88


10/88

1772b

trL ; t ;i i;4i iieel th:at

= O


11/88

"1qle l, tlcn bv2


12/88

@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:


13/88

jr,l :-.tzt t _.^ + @, Th-otectors and X', a6,r1a slenob \


14/88


15/88


16/88

(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


17/88

I ' F llodntllz

-- lr\


18/88


19/88

ht (2 )

*,ro-*if ii * ijJ-tiit: , i,", (r,*g, *o ,",'o, t+dz)@

@Ioutins eQua f i oas3


20/88


21/88


22/88


23/88


24/88

l Q - + -


25/88

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

i'l $iW.:W)ri.'re2 -bodu


26/88

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


27/88

SySIen lteflllosioordiaales a dei q;he ifs Ciil;gurii;on.

Headstock Live center Dead center


28/88

Corp.)

,,--J-6

IzU)k2xr= ftr1ra 116,

k'(y - l\0) krb/.+ Ii0)


29/88

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)


30/88

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


31/88

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:


32/88

a


33/88

ecliatidns S rrtafiaa"f oru arto*lrl.

5.3 COORDINATECOUPLIN6


34/88

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


35/88

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


36/88

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


37/88

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).


56/88


57/88

.Fd,andii,"1!

; k i X ' =


58/88

ilor;ry+hdt:kz- = -LZ- . w, . lLlt - fA. /-nldz-l 'Tl = -ttz ' -n;' rt = -At \-aJrt /

i;" niti,atyg,rtdou"of"i:


59/88

@ 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


60/88

.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


61/88

'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


62/88

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/&gt) +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


63/88

. It ;s rcquireclo ohfain ihi ualues *( -, 2 L.ihalut;ll resu-lt n: +haA


64/88

3t ualut snatLozl(u equattz) 209-3rael/s

,J'(#)Q)rtrbzanes elnal

tlrrz/A)nt = Ilz ttrtt 6ndu)

/ \ 2lSLi | =\w,,f


65/88

@/( &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//


66/88

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'.


67/88

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'


68/88

p : 0.25B : 1O bE

IfNd

z

0 . 7 8 1 1 . 0 1 1 . 2 8 10.908 1.118

Frequency/ o \\ q /

' . Nof.e,hd if a shoul drilt to eitfuwo.\glu, erl'? B 0U, +lrz coubined ysle,m uto,tld eEPerienQ- ,vesortancend fai[ sittce- lurua//! tl'u qdum/ frnquenrut


69/88

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 @


70/88

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

J_r\#)1ZI

-

DL!.L@l,rrA(g-

\t/az


71/88

F'.1

JZC


72/88

"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 .


73/88

gqatG


74/88

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'


75/88

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


76/88

----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


77/88

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)


78/88

0ontpe/ A3non;cU;botion A bsor ers. Thz"lymnic uibra*ion absorbe,r escr;bndn flu prevrousSeci;oucnztlesht or;qinol esondtlCeol{ irt the


79/88

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


80/88

(*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


81/88

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


82/88

thecrragn;tudes,, aridXz cqnbee{prett?d As, j" Qata{u)l"

2

z

n 2 ' tJ ]W

viv

=

r

6 26T

?a+t-+

TJ

1

?il" +'t+y'tTzJ

781:+' - l n l g

ii

T+

g

I

tt

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


83/88

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


84/88

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


85/88

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


86/88

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


87/88

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

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2dofsystems&vibration absorbers

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