Mechanical resonance theory and applications

Transcription

1 Mechncl resonnce theor nd lctons Introducton In nture, resonnce occurs n vrous stutons In hscs, resonnce s the tendenc of sstem to oscllte wth greter mltude t some frequences thn t others htt://enwkedorg/wk/resonnce Intercton of electromgnetc wves wth mtter, nucler mgnetc resonnce eg htt://enwkedorg/wk/nucler_mgnetc_resonnce, resonnce n electrc crcuts see eg htt://enwkedorg/wk/electrcl_resonnce or mechncl resonnce reresent tcl exmles of vrous tes of resonnce requenc, t whch the mltude of resonse hs locl mxmums, s clled the sstem's resonnce frequenc requences of self-osclltons corresond to resonnt frequences In mechncl sstems, self-osclltons occur onl n some secfc combntons of nertl, elstc nd vscose roertes Some mechncl sstems do not oscllte The sme holds for resonnce Sstem s behvor t resonnt frequenc or close to t drmtcll dffers from the sstem s behvor n other frequences Generll holds, resonse of sstem s outut on ts nut s dstnctvel hgher n the stte of resonnce Resonnce rncle s led n mn senstve mesurements NMR, EPR, RM nd others On resonnce rncle s lso bsed oerton of ntenns n electroncs nd functonng of mn muscl nstruments On the other hnd, resonnce n mechncs s often connected wth orgn of dngerous vbrtons whch s mortnt nmel n buldng engneerng nd ndustr Resonnce mesurements n mechncs RM, resonnce mechncl nlss, see lso Resonnce meters for vscoelstct mesurement-ttx re more senstve nd more recse lterntve to drect method DM, dnmc mechncl nlss, see lso htt://enwkedorg/wk/dnmc_mechncl_nlss of mesurement of vscoelstct Theor Mechncl resonnce of elstc sstems Smle mechncl osclltor g Sstem conssts of srng elstc element nd weght nertl element nlss of resonnce of ths sstem m be bsed on clsscl soluton of dfferentl equton of the sstem s movement Nevertheless, lcton of Llce trnsformton nd ourer trnsformton rovde more snotc methodologcl roch Oertonl clculus n vscoeltctdocx whch s lcble lso for more comlcted stutons Thus, the methodolog wll be descrbed ste b ste

2 srng weght závží self-osclltons energ nserted nto sstem g Smle mechncl osclltor orce equlbrum: t t t E I Energ s nserted nto sstem usng short mulse of force The force s the sum of the elstc force E nd of the nertl force I Consttutve equton: d t t H t M, d t where H s the srng constnt, s the deformton of srng, M s the mss of weght Self-osclltons lcton of Llce trnsform see Oertonl clculus n vscoeltctdocx leds to equton 3 H M, 3 where s new vrble, nsted of t or the mulse of the force holds eq 4 H M 4 or smlct, the vrble wll not be further wrtten, followng formuls Consequentl, for deformton t holds:

3 M H 5 H Roots of the denomntor of eq 5 re M or deformton n tme domn t holds: Where: t snt, 6 H 7 M Elstc rod s mechncl osclltor g nd 3 elstc rod mss n centrum of grvt externl mss b g Elstc rod wth externl nertl weght nd sngle elstc rod b n tensle lodng

4 b g 3 Elstc rod wth externl nertl weght, sngle elstc rod b n bendng lodng or resonnce of elstc rod, formule 7 holds But n relt, mss s ctng n the centrum of grvt of whole sstem Consequentl, the nertl force eq nd further must be dequtel converted see Oertonl clculus n vscoeltctdocx, ge 5 Resonnce curves of elstc sstems Resonnce curves m be derved from frequenc chrcterstcs of sstems ccordng defnton, mltudes of vbrtons re mxml t resonnce frequenc requenc chrcterstc m be obtned from b hel of ourer trnsformton see Oertonl clculus n vscoeltctdocx If ourer trnsformton s led on eq, t s obtned: H M 8 Consequentl t holds: 9 H M Equton 9 descrbes frequenc chrcterstc, rovded t nd t re hrmonc functons mltudes of vbrtons re mxml f ths holds: H M 0 0 Resonnce frequenc corresonds to eq 7 In resonnce, the mltude of vbrton s theoretcll nfnte In rel bodes, the mltude s lmted do to energ losses In rctcl stuton, entre resonnce curve m be mortnt Nmel, how shr s the resonnce curve Smulton s n resonnce elstckáxlsx Exmle s on g 4

5 mltud mltude 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0, 0,,4,6,8 frekvence frequenc Hz g 4 Exmle of resonnce curve of elstc sstem Prmeters: of: H= 00 N/m, M= kg Mechncl resonnce of vscoelstc sstems Resonnce of smle vscoelstc sstems Structure of smle vscoelstc sstem s on g 5 Sstem conssts of elstc element srng, nertl element weght nd dumng element dsh ot srng weght self osclltons dsh ot energ nserted g 5 Scheme of smle vscoelstc sstem

6 orce equlbrum t t t t E Energ s nserted nto sstem usng short mulse of force The force s the sum of the elstc force E, the nertl force I nd the dumng force D Thus t holds: d t d t t H t N M, dt d t where H s the srng constnt Hooke s coeffcent, s the deformton, N s the Newton s coeffcent, M s the mss I D Self-osclltons Llce trnsformton of leds to eq 3: H N M, 3 or the mulse of the force t holds: M N H 4 or deformton t holds: M 5 N H Resonnce occurs onl f ths holds: 4MH N 0 In ths cse, from theor of Llce trnsformton for deformton t holds: Where: e t sn t 6 nd 4M H N 7 M N 8 M Resonnce curves of smle vscoelstc sstems Resonnce curves m be derved from frequenc chrcterstcs of sstems ccordng defnton, mltudes of vbrtons re mxml t resonnce frequenc requenc chrcterstc m be obtned from b hel of ourer trnsformton see Oertonl clculus n vscoeltctdocx If the ourer trnsformton s led on eq, t s obtned: H N M, 9

7 0 H N M Equton 0 descrbes frequenc chrcterstc, rovded t nd t re hrmonc functons or rto of mltudes t holds: H N M where s the mltude of deformton nd s the mltude of force Consequentl: H M N or mxmum of vbrtons t holds: d H M dt N 0 3 Thus for resonnce frequenc t holds: Resonnce frequenc corresonds to eq 7 or the mltude of vbrtons t resonnce t holds: 4M H N f R 4 M R 5 N In rctcl stuton, entre resonnce curve m be mortnt Nmel, how shr s the resonnce curve Smulton s n vscoelstc resonncexlsx Exmle s on g 6 deformce mm deformton mm frekvence frequenc Hz Hz g 6 Exmle of resonnce curve of rel vscoelstc sstem Prmeters: of: H= 00 N/m, N= 0,0 Ns/m, wooden rod 00x4x4 mm

8 Resonnce curves of rel vscoelstc sstems Generl consttutve dfferentl equton for lner sstems s: m n x b where nd b re constnt coeffcents, nd re degree of dervtves, x s the nut quntt force, s the outut quntt deformton In Llce trnsformton t holds: m n b b or mulse of force n Llce trnsformton t holds: n 0 8 or deformton t holds? 0 9 Prevous equton s ossble to convert nto form: B B 30 Members wth coeffcent B re relevnt for resonnce: B B R 3 In tme domn t holds: t R t e B t sn 3 Resonnce occurs t locl mxms of the functon 3 In comlex mechncl sstem severl resonnce frequences m be found Resonnce of long rods nd cbles Introducton nlss of long rods nd cbles resonnce s mortnt nmel for buldng ndustr, nd engneerng Mn rctcl roblems re connected wth clculton of resonnce frequenc, dumng of vbrton,

9 elmnton of self-osclltons etc lcton of clsscl roch fls, s long structures re sstems wth dstrbuted rmeters Consequentl, models wth lumed rmeters eg clsscl rheologcl models cnnot be used Stsfctor theor of mechncl behvor of mechncl sstems wth dstrbuted rmeters currentl does not exst Nevertheless, rctcl solutons of some roblems on ths feld m be bsed on nlss of mechncl wve rogton Prncle of soluton for long rod s mechncl mednce of rod dffers from mechncl mednce of endngs of rod, mechncl wve conssts n nterference of drect wve nd wve reflectng from endngs Resultng re stndng wves Dstnce between nodes nd ntnode s one qurter of wvelength Mesurement of mechncl wve rogton g 7 m be bsed on ths fct L λ/4 node ntnode g 7 Prncle of mesurement of seed of mechncl wve rogton n rod Mesurement of seed of wve rogton Short mulse energ s nsertng nto rod The rod osclltes on resonnt frequenc f Resonnce frequenc f s mesured, 3 Seed v of wve rogton s clculted ccordng formul v 4 f L 33 Exmle of lcton clculton of mxm of vbrton longsde rod g 8 Postons of ntnodes corresond to loctons of mxms of deformton Postons of nodes corresond to loctons of mxms of stress

10 ntnodes node λ/4 λ/ λ/ λ/4 node g 8 Clculton of mxms of vbrtons nd stresses More detled nlss m lso led to clculton of dumng s well s to the w of vbrton elmnton