Analytical modelling of extruded plates

Transcription

1 paper ID: 56 /p. Aalcal modellg of erded plaes C. Pézera, J.-L. Gader Laboraore Vbraos Acosqe, INSA de Lo,5 bs a.j. Cappelle 696 VILLEURBANNE Cede Erded plaes are ofe sed o bld lgh srcres h hgh sffess. The are made p of o eeral plaes mechacall copled. The goal of hs sd s o compe broacosc qaes ho he se of FEM modellg, order o sae compao me. The proposed approach perms o calclae rsc parameers of he srcre (ege modes, modal des), qaes cocerg acosc radao for mechacal forces ad/or plae ae ecaos (mea sqare eloc, acosc poer, effcec), ad acosc rasmsso loss for plae ae or dffse feld ecaos. I addo, o he preseao of he prcple ad adaages of he approach, a es case s preseed ad compared h FEM modellg. Fall, mercal resls are sho for he rasmsso loss obaed h a dffse feld ecao of o almm erded paels sed ra srcres.. INTRODUCTION Erded plaes are made p of o prcpal plaes mechacall copled b ermedae plaes h arg agles. These pes of srcre prese he adaage o hae a hgh sac sffess, b her brao ad acosc characerscs are er dffere o hose of smple or doble plaes. The goal of hs ork s o elaborae a aalcal model of erded plaes, order o redce compao me compared o mercal models lke Fe Eleme Mehod. Ths paper descrbes ho he operaor of he srcre s esablshed h he assmpo ha he acosc coplg beee eeral plaes ca be egleced. Some resls are also preseed, shog pcal eamples oe ca hae for ras applcao. EXTRUDED PLATE MODEL.. Descrpo ad oaos As sho fgre, erded plaes are cosdered lke plae assembles h o parallel eeral prcpal plaes (plaes ad ) ad seeral eral ermedae plaes (plaes, ) aached o boh eeral plaes h a agle θ. α Plae O z z O θ d O z Plae β L Plae Fgre. Descrpo of a erded plae h oe ermedae plae.

2 paper ID: 56 /p. Iermedae plaes are aached o plae a coordae α (o as) ad o plae a coordae β (o as). Geomerc relaoshps beee α, β, θ ad legh L of a eral plae are: β = α L cosθ ; L = d / s θ () here d s he dsace beee prcpal plaes ad. Vbraor dsplacemes of all plaes are oed: (, ) : logdal dsplacemes of he plae he dreco (, ) : logdal dsplacemes of he plae he dreco (, ) : raserse dsplacemes of he plae he dreco z Bodar codos of plaes ad are cosdered smpl sppored... Applcao of he Hamlo s prcple The cos fco of oe plae (correspodg o he dfferece beee kec eerg ad eral poeal eerg) has he epresso (see []): ρ Ψ = S h ( ) ν ν 4 Eh ( ) ( ) d ds. Eh ν ν () here,, are braor dsplacemes, ρ he mass des, E he Yog modls, ν he Posso s rao ad S he srface of he cosdered plae. The global cos fco of a erded plae ca be re as follos: Ψ Ψ Ψ Ψ = = N () here N s he mber of ermedae plaes... Combao of dsplacemes Plae ad : These plaes are smpl sppored. Bodar codos are: = o bodares = ad =L ; = o bodares = e =L ; = o all bodares. Where =,.

3 paper ID: 56 /p. I order o sasf hese bodar codos, dsplacemes of plae ad are epaded o a se of fcos bass as follos: pπ qπ a pq cos s p q L L π π = p q b pq s cos (4) p q L L pπ qπ c pq s s p q L L here p ad q are dces correspodg respecel o he logdal ad raserse orders. Plae, : Bodar codos of eral plaes ms erf co of dsplacemes ad roaos h boh eeral plaes. These codos are: Beee plae ad : Beee plae ad : (,) = (, α) (5) (,L) = (, β) (9) (,) = (, α) cosθ (, α) sθ (6) (,L) = (, β) cosθ (, β) sθ () (,) = (, α) sθ (, α) cosθ (7) (,L) = (, β) sθ (, β) cosθ () (,) = (, α) ( ),L ( ) =, β () I order o erf hese relaoshps, he follog polomal decomposo of eral plae dsplacemes as sed: L () L (4) L L L L L L ( ) ( ) β, β sθ, β cosθ,. (5) L L L L (, ) = (, α ) (, β ) ( ) ( ( ) ( ) ), =, α cosθ, α sθ ( (, β ) cosθ (, β) sθ ) ( ) ( ( ) ( ) ) = α θ α θ ( α ),, s, cos,. ( ) ( ) Ths s he smples polomal decomposo permg he erfcao of dsplaceme coes beee eeral ad eral plaes. No addoal kos are rodced ad he dsplacemes of he erded plae s oall epressed h 6 kos per cople (p,q): a pq, a pq : geeralzed amplde for logdal dsplacemes b pq, b pq : geeralzed amplde for logdal dsplacemes c pq, c pq : geeralzed amplde for fleral dsplacemes

4 paper ID: 56 /p.4 The eremm of fcoal ψ ad ψ ges depede mar ssems for each dces cople (p, q) de o he fac ha he epaso (4) cocdes h modal dsplacemes for copled eeral plaes. Oe oba he mar relao: a pq {[ K ] [ M ]} b pq ω pq pq = (6) c pq here =, (de of prcpal plaes), M pq ad K pq are respecel mass ad sffess marces: ν π π ν π pπ q p q L L L L ν π L L E h π ν pπqπ q p L L L L 4 4 h pπ qπ pπ qπ L L L L [ ] K = ( pq 4ν ) ad [ ] ρ LL h pq (7) M = 4 The eremm calclao of eral plaes fcoal ψ ges smmerc mass ad sffess marces coplg ol orders he drecos M pql ad K pql (q ad l represe copled orders). For safe of bre, her epressos are o ge here. The ere comped sg a mahemacal smbolc ool sofare. Global ssem correspods o depede mar ssems for each logdal order p. Ege modes are he solos of hs global ssem here he rgh had sde s ll. I cases of forced braos de o acosc or mechacal ecaos, he compao cosss solg he global ssem, here he rgh had sde correspods o he geeralzed forces obaed b he projeco of her spaal dsrbo o he fco bass (4).. POST-PROCESSING The pos-processg s he compao of phscal qaes obaed from geeralzed amplde a pq, solos of he ssem descrbed he preos seco. I hs paper, ege modes (freqeces ad shapes) are preseed, hch ca be obaed from he ege ales ad ege ecors of he global ssem. I s also proposed o compe he acosc rasmsso loss obaed h a dffse feld ecao. The rasmsso loss db of he erded plae s obaed b: R = log (8) τ here τ s he rao beee he rasmed acosc poer h he cde poer. Whe he ecao s reerbera, s epresso ca be obaed h he epresso: τ= 6 π Rmm A rsm Rrsrs S c (9) m r s here R mpq s he real par of he modal acosc mpedace of a smpl sppored recaglar plae, A rsm deoes he erms of he erse mar of he global ssem (see [] ad [] for more deals). Wh hs approach, oe ca see ha he compao dffse feld does ecessae o sole seeral problems h plae ae ecao order o egrae hem. Ths s de o he fac ha geeralzed force epressos are drecl lked o he acosc mpedace ch are ell ko he case of smpl sppored recaglar plaes.

5 paper ID: 56 /p.5 4. NUMERICAL RESULTS 4. Ege modes I hs seco, s proposed o sho ege modes of a smple erded seel plae h ol oe ermedae plae (see fgre ). All plaes are mm hck ad he characerscs of he maeral are: E=,68 N/m, ν=,9, ρ=78 kg/m.,4 m, m m,5 m,5 m Fgre. Tes case sed for comparso beee FEM ad Aalcal models. FEM Freqec Aalc freqec order p Traserse shape 9.89 Hz 9.9 Hz.4 Hz.47 Hz.7 Hz.8 Hz Table : ege freqeces ad shapes of he hree frs modes. Ege modes (freqeces ad shapes) ere comped h logdal ad raserse orders. Table preses ege freqeces ad shapes of he hree frs modes. For comparso, he frs colm shos he ege freqeces of he same modes obaed b he FEM sofare Ideas (obaed shapes are he same). Resls are er close. 4. Trasmsso loss To erded plaes ere sded. The are almm paels m legh ad her profles are sho fgre. The hae 9 eral plaes placed ho smmer. Eeral plae s 5 mm hck ad ohers.5 mm hck. The dfferece beee boh cases, s ha eral plaes of case are orhogoal o he eeral plaes. The frs ege freqeces of cases ad are respecel 77 Hz ad Hz.

6 paper ID: 56 /p.6 5 cm 5 cm.44 m Case.44 m Case Fgre. Sded profles sed for rasmsso loss calclao.. Fgre 4 shos he rasmsso losses of boh cases obaed for dffse feld ecao. For freqeces less ha he frs ege freqec, all cres follos he sffess la. For pper freqeces, he rasmsso loss crease globall h 6 db per ocae (mass la), b fall do he he freqec correspods o a ege freqec. Effecs of resoaces are more sble for he case, becase s modal des s loer. I s clear ha arg agles, coplg logdal ad fleral braos (case ), perms o crease cosderabl he rasmsso loss of he erded plae. 9 8 Case Case freqec (Hz) Fgre 4. Trasmsso loss (db) of case (ble) ad case (gree) 6. CONCLUSION The proposed approach perms o cosrc a model for a pe of erded plaes. The prcple s based o he projeco of brao dsplacemes o he aalc ege shape bass of boh smpl sppored prcpal plaes. The eres of hs modellg s ha he compao s er fas, parclarl for he calclao of he rasmsso loss h a dffse feld ecao. For sace, mercal aldaos ere made ol o brao qaes h comparsos h FEM modellg. Of corse, epermeal ad ohers mercal aldaos are reqred for rasmsso loss resls. REFERENCES. J.-L. Gader, Vbraos des mle cos, Hermes Laoser, Pars,, 445p.. J.-L. Gader, C. Leser, Trasmsso of reerbera sod hrogh orhoropc scoelasc mllaered plaes, Joral of Sod ad Vbrao, Vol. 7(), pp.9- (98).. C. Leser, Raoeme acosqe des srcres, Edo Erolle, Pars, 988, 59p.