10.6 Oscillatory Stress, Dynamic Loading and Vibrations

Transcription

1 Scn Oscllary Srss, Dynamc Lang an Vbrans Cr an rlaxan xrmns n rv cml nfrman cncrnng h mchancal bhavur f vsclasc marals. Ths xrmns usually rv s aa n h m-rang frm 0 scns 0 yars. I s fn f nrs knw h rsns f marals las f vry shr uran. Fr xaml, uran f h mac f a sl ball n a vsclasc blck may b f h rr f 0 5 sc. In rr b abl rmn h rsns fr such cnns, s ncssary knw h bhavur f a maral a hgh ras f lang r shr uran lang. Th chnus an aaraus fr nvsgang h rsns f a maral vry shr rm lang ar ffrn hs nvlv n lngr-rm sng. Fr vry shr m lang s mr cnvnn us scllary han sac lang, an n rr rc h bhavur f a vsclasc maral subjc an scllary la, n ns frmula h hry bas n scllary srsss an srans Oscllary Srss Cnsr a ynamc la f h frm cs 0.6. whr s h srss amlu an s h angular fruncy. Assum ha h rsulng sran s f h frm cs 0.6. s ha h sran s an scllan a h sam fruncy as h srss bu lags bhn by a has angl, Fg Ths angl s rfrr as h lss angl f h maral, fr rasns whch wll bcm clar lar. xanng h sran rgnmrc rms, cs cs sn sn 0.6. Th frs rm hr s cmlly n has wh h nu; h scn rm s cmlly u f has wh h nu. If h has angl s zr, hn h srss an sran ar n ynamc xrmns usually rv aa frm abu 0 8 sc. abu 0 sc. s hr s a smwha vrlang rgn whr aa can b ban frm bh ys f xrmn whn an scllary frc s frs al, ransn vbrans rsul a h naural fruncy f h maral hs sn u lavng h vbrans a h surc fruncy f n subsus 0.6. n h gnral cnsuv uan 0..7, n ss ha h sran an s rvavs cnan sn an csn rms, s ha h sran mus b f h gnral frm A cs B sn, whr A an B ar cnsans. Fr cnvnnc, hs can b wrn as C cs D whr C an D ar nw cnsans Sl Mchancs Par I 5 Klly

2 Scn 0.6 has as hans wh an al lasc maral, whras f /, h srss an sran ar cmlly u f has. cs Fgur 0.6.: Oscllary srss an sran cs 0 Th Cmlx Cmlanc Dfn cs, sn s ha cs sn Th uans an ar a masur f hw n, r u f, has h srss s wh h sran. Th frmr,, s call h srag cmlanc an h lar,, s call h lss cmlanc. Thy ar usually wrn as h cmnns f a cmlx cmlanc, : If n has a srss nu n h frm f a sn funcn, hn sn sn cs sn sn cs sn cs an agan h srag cmlanc s a masur f h amun "n has" an h lss cmlanc s a masur f h amun "u f has". Sl Mchancs Par I 6 Klly

3 Scn 0.6 Th Cmlx Mulus On can als rgar f h sran as h nu an h srss as h uu. In ha cas n can wr s agan h has angl by whch h sran lags bhn h srss cs cs cs cs sn sn Ths s n ffc h sam srss-sran rlansh as ha us abv, nly h srss/sran ar shf alng h -axs. Dfn nx h w nw uans cs, sn s ha cs sn Agan, hs uans ar a masur f hw much h rsns s n has wh h nu. Th frmr,, s call h srag mulus an h lar,, s call h lss mulus. As wh h cmlancs, hy ar usually wrn as h cmnns f a cmlx mulus 4, : 0.6. Agan, f n has a snusal sran as nu, n can wr sn sn cs sn sn cs sn cs 0.6. I s aarn frm h abv ha 0.6. whch s a much smlr rlansh han ha bwn h cr cmlanc funcn an h rlaxan mulus whch nvlv Lalac ransfrms, n ycal valus fr h srag an lss mul fr a lymr wul b arun 0 MPa, 0. MPa. Th ra f h amlus s call h ynamc mulus, /. Sl Mchancs Par I 7 Klly

4 Scn 0.6 Sl Mchancs Par I Klly 8 Cmlx Frmulan Th abv uans can b succncly wrn usng a cmlx frmulan, usng ulr's frmula cs sn Fr a srss nu, sn cs Th cr cmlanc funcn s h sran rsns a un la. In h sam way, frm 0.6.5, h cmlx cmlanc can b nrr as h sran amlu rsns a snusal srss nu f un magnu. Smlarly, fr a sran nu, n has sn cs an h rm n bracks s, by fnn, h cmlx mulus. Th rlansh bwn h cmlx cmlanc/mulus an h ffrnal cnsuv uan Pung an h rsulng sran n h gnral ffrnal rar frm f h cnsuv uan 0..9, n has Ths uan hus gvs h rlansh bwn h cmlx cmlanc an h cnsans,. A smlar rlansh can b fun fr h cmlx mulus: 0.6.8

5 Scn 0.6 Agan n ss ha. Frm , h cmlx cmlanc an cmlx mulus ar funcns f h fruncy, an hus, frm h fnns 0.6.4, 0.6.6, 0.6.9, 0.6., s s h has angl. Thus s h rmary varabl nfluncng h vsclasc rrs whras m was us fr hs urs n h analyss f sac lang. Th rlansh bwn h cmlx cmlanc/mulus an h cr cmlanc/ rlaxan mulus I can b shwn 5 ha h cmlx cmlanc an h cmlx mulus ar rla h cr cmlanc an rlaxan mulus hrugh L L s s Hr, h Lalac ransfrm s frs akn an hn valua a A N n Fruncy, s 6. Fruncs blw 0. Hz ar assca wh ssmc wavs. Vbrans f srucurs an sl bjcs ccur frm abu 0. Hz 0 khz nng n h sz f h srucur. Srss wavs frm 0 Hz 0 khz ar rcv as sun - abv 0 khz s h ulrasnc rang. Fruncs abv 0 Hz crrsn mlcular vbran an rrsn an ur lm fr srss wavs n ral sls xaml: Th Maxwll Ml Th cnsuv uan fr h Maxwll ml s gvn by n. 0..6, Cnsr an scllary srss cs. W hus hav 7 cs sn sn cs cs sn usng Furr ransfrm hry fr xaml 6 an ar als rla ach hr as ar an by an vn mr cmlca rul knwn as h Kramrs-Krng rlan 7 h cnsan f ngran s zr assumng ha h nal sran s ha n h srng, /. Sl Mchancs Par I 9 Klly

6 Scn 0.6 Thus h cmlx cmlanc s 0.6. Ths rsul can b ban mr asly usng h rlansh bwn h cmlx cmlanc an h cnsuv uan: h cnsuv uan can b rwrn as Frm n , 0, whr,, 0 0, 0.6. / Als, h cmlx mulus s rla h cmlx cmlanc hrugh 0.6., /, s ha Fr vry lw fruncs, 0, sn / / /. ha fr a sac la,, an h rsns, as xc, rucs Fr vry hgh fruncs, / 0, an h rsns s / cs. Thus h sran s cmlly n-has wh h la, bu h ash- s n mvng has n m rsn a such hgh fruncs - h srng/ash- ml s racng lk an sla srng, ha s, lk a sl, wh n flu bhavur nrgy Dssan Bcaus h uans 0.6. sn, sn ar h aramrc uans fr an lls, ha s, hy rac u an lls fr valus f, h srss-sran curv fr an scllary srss s an llc hysrss l, Fg Th wrk n n srssng a maral r un vlum s gvn by Th nrgy ls Thus W W hrugh nrnal frcn an ha s gvn by h ara f h lls. Sl Mchancs Par I 0 Klly

7 Scn 0.6 W T T whr s sm sarng m an T s h r f scllan, n ns fr sran an srss hn gvs T /. Subsung W T sn cs T sn sn cs sn T , /, /, /, /, Takng 0 hn gvs8 Fgur0.6.: llc Srss-Sran Hysrss L W sn nrgy Lss Whn 0, h nrgy ssa s zr, as n an lasc maral. I can als b sn ha W an hnc h nams lss mulus an lss cmlanc. 8 h sam rsul s ban fr sn, sn r whn h srss an sran ar csn funcns Sl Mchancs Par I Klly

8 Scn 0.6 Damng nrgy Th nrgy sr afr n cml cycl s zr snc h maral has rurn s rgnal cnfguran. Th maxmum nrgy sr urng any n cycl can b cmu by ngrang h ncrmn f wrk frm zr u a maxmum srss, ha s vr n uarr h r T f n cycl. Thus, ngrang frm / whr 0 /, Fg cs W sn Th scn rm s sn / 4, whch s n uarr f h nrgy ssa r cycl, an s can b cnsr rrsn h ssa nrgy. Th rmanng, frs, rm rrsns h ara f h sha rangl n Fg an can b cnsr b h nrgy sr, W s cs / rucs h lasc slun W / whn 0. Th amng nrgy f a vsclasc maral s fn as W /WS, whr W S s h maxmum nrgy h sysm can sr n a gvn srss/sran amlu. Thus vng W by 4 s s cnssn wh h ngran vr a uarr-cycl ban h sr nrgy W W s an Damng nrgy 0.6. Thus h amng ably f a lnarly vsclasc maral s nly nn n h has/lss angl. 0 / / cs / / sn sn cs / Fgur0.6.: llc Srss-Sran Hysrss L 9 r n cul ngra frm zr maxmum sran, vr, / 0, gvng h sam rsul Sl Mchancs Par I Klly

9 Scn 0.6 Th uany an s knwn as h mchancal lss, r h lss angn. I can b cnsr b h funamnal masur f amng n a lnar maral hr masurs, fr xaml, an, c., ar fn us 0. Tycal valus fr a rang f marals a varus mraurs an fruncs ar shwn n Tabl Maral Tmraur Fruncy v Lss Tangn an Sahr 4. K 0 khz Sahr r 0 khz Slcn r 0 khz 8 0 Quarz r MHz 7 0 Alumnum r 0 khz 5 0 Cu-%Zn r 6 khz Sl r Hz Alumnum r Hz F-0.6%V C 0.95 Hz Basal r Hz Gran r Hz Glass r Hz W r Hz 0. 0 Bn 7 C -00 Hz 0. 0 La r -5 Hz PMMA r Hz 0. Tabl 0.6.: Lss Tangns f Cmmn Marals Imac Cnsr h mac f a vsclasc ball r frm a hgh h n a rg flr. Durng h mac, a rrn f h nal nal nrgy mgh, whch s nw knc nrgy mv, whr v s h vlcy a mac, s ls an nly sm s sr. Th sr nrgy s cnvr back knc nrgy whch rvs h ball u n h rbun, rachng a hgh h h, wh fnal nal nrgy mgh r. Th ra f h w hghs s r f h h r r s mgh mgh W s W W whr W s s h nrgy sr an W s h nrgy ssa urng h mac. 0 sm nvsgars rcmmn ha n uss h maxmum srabl nrgy whn 0, n whch cas h sr nrgy s / an h amng masur wul b W / W sn / frm Tabl 7. f Vsclasc Sls, by R. S. Laks, CRC Prss, 999 h cffcn f rsun s fn as h ra f h vlcs bfr an afr mac, v r / v, s f. S Sl Mchancs Par I Klly

10 Scn 0.6 Th mac vn can b arxma by a half-cycl f h scllary srss-sran curv, Fg Ingrang vr 0, / r /, /, n has W cs sn an s h hgh ls s gvn by W W f an W W W s s scllary half-cycl mac rcvry afr mac Fgur0.6.4: Imac arxma as a half-cycl f scllary srss an sran N sm hr arxmans ma: nrgy lsss u ar rssanc, frcn an raan f sun nrgy urng mac hav bn nglc n a ral mac, h srss an sran ar bh nally zr. In h currn analyss, whn n f hs uans s zr, h hr s fn, an hs wll nvably nruc sm rrr Damng f Vbrans Th nral frc n many alcans can b nglc. Hwvr, whn alng wh vbrans, h ruc f acclran ms mass can b arcabl whn cmar h hr frcs rsn. Vbranal amng can b xamn by lkng a a sml scllar wh n gr f frm, Fg A mass m s cnnc a wall by a vsclasc bar f lngh L an crss scnal ara A. Th mn f h sysm s scrb by h uans alhugh mgh b mr accura ngra vr 0, / 4 as mnn, hr s a ransn rm nvlv n h scllan whch has bn gnr, an whch s u vr m, lavng h sran lag bhn h srss a a cnsan has angl Sl Mchancs Par I 4 Klly

11 Scn 0.6 Dynamc uan: m x F 0 Knmac rlan: x / L Cnsuv rlan: ns n ml Assumng an scllary mn, x x, an usng h frs w f hs, mx A x L m L A L m A Th uany n bracks s h cmlx mulus As an xaml, fr h Maxwll ml s n s n an s A Lm, whch can b slv g A Lm L x vsclasc bar If m s small r s larg an ha an n has h am vbran Fgur0.6.5: Vbran / s n larg h r has a ral ar, v say, s / v Sl Mchancs Par I 5 Klly

12 Scn 0.6 x x x x c / / c c v c v Acs v Bsn v If, n h hr han, h mass s larg r h srng cmlan, n gs a ur magnary r, / v, s ha s ral an lss han zr an n has h arc amng x x c / v c / v Prblms. Us h ffrnal frm f h cnsuv uan fr a lnarly vsclasc maral rv h cmlx cmlanc, h cmlx mulus, an h lss angn fr a Klvn maral. u h frs w n h frm. Us yur xrssn fr h cmlx cmlanc rv h sran rsns a srss cs, n rms f,,,,, n h frm Acs Bsn Wha hans a vry lw fruncs? Sl Mchancs Par I 6 Klly