RELATION BETWEEN STATIC STIFFNESS AND FREQUENCY RESPONSE FUNCTION : APPLICATION TO THE HIGH SPEED MACHINE-TOOL

Transcription

1 RELATION BETWEEN STATIC STIFFNESS AND FREQUENCY RESPONSE FUNCTION : APPLICATION TO THE HIGH SPEED MACHINE-TOOL L. Hzé *, P. Ray *, G. Gog *, C. Barra **, O. Vida ** * Laboraoir d Rchrchs Appicaions n Mécani Avancé LaRAMA Insi Français d Mécani Avancé IFMA Univrsié Bais Pasca, Crmon II UBP Camps d CLERMONT-FERRAND / Ls Cézax BP Abièr, Franc. ** PCI SCEMM, Rchrch Dévoppmn R Coprnic Sain Einn, Franc. ABSTRACT. Th sdy of bhavior of a machin-oo nds o now h vas of dynamic and saic siffnsss. Th prpos of his aric is o b ab o sima hs vas wih h sam dvic, handy nogh. Diffrn mhods of drminaion of saic siffnss ar prsnd and rss ar compard bwn hm. Masr rss of h as mhod ar procssd by a sofwar dvopd wih Maab. KEYWORDS : dynamics, machin-oo, siffnss.. Inrodcion Rirmns raiv o h incras of prodciviy ad o h dvopmn of a nw concp of machin-oo. By nar, hos machins nd an incrasd siffnss in ordr o garan h bs machining aiy, howvr h machin is sd. Indd, sing h consrains imposd by high spd machining in rm of im, wor spd and accraion nd pda dsign nowdg on h dynamic bhavior of a machin-oo sing dvic Fig..

2 Figr : Machin oo sing dvic and is moding Machin-oo moding nds o now h vas of saic and dynamic siffnsss. Th aim is o sima hs vas wih a handing dvic. A dirc masrmn of h saic siffnss o h nos of h machin broach impis o carry o a rigid srcr arond h broach aowing h forc appicaion in varios posiions of h broach in h machin oo woring spac. Sch a srcr is vry cosy and som difficis xis o orin h forc in a dircions. Tha s why w propos o drmin h saic siffnss by sing h ransfr fncion of h frncy rspons FRF. Th FRF basd mhods has h advanag of providing icy h rigidiy of h sdid srcr if h frncis ar w nown. W prsn h advanags and h imis of ach mhod appid on a simp srcr and w aso prsn an appicaion on a high-spd machin oo. Ths mhods prmi s o sabish h maps of rigidiy in h hr dircions of h woring spac. 2. Probm Dring h anaysis of ransfr fncions [LAL 86][MCO 95], w sdy paricary h compianc HX/F X : dispacmn srcra rspons, F : xciaion oad. To g h dispacmn, signa from h accromrs nds a dob ingraion in h im domain. Th FFT anayzrs which w s can procd o his ransformaion of h signa. Th srcra rspons is spposd wih h sam frncy as h xciaion oad. Aso, h dob ingraion is don by a dob division by h psaion 2π.f in h frncy domain. This approximaion is corrc for high nogh frncis f > 20 Hz. I s no sab for vry ow frncis, whr h dob ingraion ads o infini

3 compianc. Aso, w hav o s anayic mhods sing h vaid par of h compianc crv, in ordr o caca is va a 0 for h saic siffnss. 3. Anaysis of h ransfr fncion From h dynamic compianc crv, h saic siffnss can b cacad sinc rsonanc mods ar no copd. Th mhod spposs ha h srcra bhavior for ow frncis is rd by h firs mod [LAU 98][MUS 98][LAU 99]. Aso, h moda paramrs of h mod ar dominaing for h saic characrisics of h srcr, in paricar h moda siffnss cos o h saic siffnss. This mhod was sd on a simp srcr whos propris ar nown : h bam wih a fixd nd. I was aso appid on a high-spd machin oo. 4. Moda sommaion mhod Using h sam ocaion for h accromr and h vibraor poin namd "Driving Poin", w can caca h saic rigidiy wiho having o mov h ransdcr. Th mhod assms ha any FRF mad p of µ mods for µ dgrs of frdom can b bron down ino µ FRFs ach of on dgr of frdom [COX 99]. For ach frncy pa of h rspons graph X/F moda paramrs can b an. For d poins of h srcr sd o appy forcs mar and o masr h rspons mar i, w sppos h xciaion as priodic and wih h foowing form : F d F From spaia coordinas x i, w considr xndd coordinas i dfind by : µ Ths, sing h Caghy condiion [PED 97][PIR] xi i i { ;...;d} [][ c M] [ ] [ ][ M] [] c, h aions of movmn can b ncopd and bcom : [ M ] x [] c x [ ] x F

4 [ ][ ][ ] [ ][][ ] [ ][ ][ ] [ ] F c M d F Q c m { } ;...;µ So, w obain : D Q m c Q 2 hn d µ i i F D x W now ha d i i F H x. Aso, for a cop of poins i rspons and xciaion, w can caca H i X i / F by idnificaion : µ i i i D H 5. Non-inar fiing From h Paria Fraciona Form sd and prsnd by Piranda [PIR][PIR 94][DEL 99], h FRF can b wrin as : µ s v H Givn h xprimna FRF crv y i for h accromr on h posiion i, w hav o minimiz h diffrnc :, i H y f

5 wih { }, v,,...,, s,..., µ sµ Evn if y i is a discr fncion d o h rsoion of h FFT anayzr, w considr f as coninos and diffrniab. In h nighborhood of h poin 0, w s a Tayor dvopmn of firs ordr : f 2µ 2 f 0,, f 0, d ε W ry o minimiz f by varying. Hnc, w hav o rsov a maricia probm by an iraiv mhod. Afr convrgnc, w obain which dfins niry h fiing fncion. 6. Appicaions 6. Appicaion on h fixd-fr bam Th fixd-fr bam sdid is mad of aminim Fig.2. Figr 2 : Fixd-fr bam Th hory prmi s o caca is saic rigidiy as 335 N.m -. By a random xciaion, w g h foowing ransfr fncion crv, rprsning h dynamic compianc, obaind hans o h sofwar Ps Fig.3:

6 [db/,00 m/s²/ns²] Frncy Rspons HSigna,Signa 5 - Inp Magnid Woring : Inp : Inp : FFT Anayzr [Hz] Figr 3 : Transfr fncion of a fixd-fr bam Each mhod drmins h va of saic rigidiy of h bam wih a prcision of arond 0%. 6.2 Appicaion on h machin oo 6.2. Prsnaion Th sam xprimna approach is appid o h srcr of a machin sing dvic. Th dvic was providd by h company PCI Fig. 4. Th broach carrir of h dvic is movd by wo inar moors p on h opposi sids of h boc. Th movmn is conrod by wo siding gidancs in h Y dircion.

7 command board inar moor broach carrir siding gidanc Figr 4 : Tsing dvic Mor Loads ar appid on h nos of h broach for diffrn posiions of h boc Masrs of dispacmn ar don in h sam ocaion and for diffrn dircions Appicaions Th masrs ar don in h hr spaia dircions, wih inrvas of 50 mm, for posiions of h broach varying bwn xrma posiions. Th figr 5 isras h xprimna mhod o drmin h FRF in h dircion X. y x z xciaion rspons Figr 5 : Masr in h dircion X

8 Th rss indica ha siffnss on h broach is nary consan for X and Z dircions, wha is ogica in rm of dvic dsign. D o h absnc of gidanc in h Y dircion o mainain srains, hr is an voion in his dircion. Dircion Z Saic rigidiy Dircion Y Dircion X Posiion mm Figr 6 : Saic rigidiy crvs dpnding on posiion of h broach 7. Concsions Masrs of h saic rigidiy of a machin oo is a diffic opraion, b anayic mhods i h moda sommaion mhod or h non-inar fiing ar siab for or appicaion. Indd, hir consrcion is sricy rad o h ransfr fncions hory. Thy ar ffciv on simp probms. Howvr, i s ncssary o no ha h drminaion of h moda paramrs on a rsonanc mod a ow frncis is vry snsiiv. In so criica siaions, informaions from accromrs ar no sr. Aso, h sdy has o b mor prcis for ow frncis o avoid probms.

9 Rfrncs [COX 99] COX, A., Mis n ovr d a msr ds rigidiés sais dynamis UGV, Rappor d Masr of Enginring IFMA Univrsié d Bah, Mai 999. [DEL 99] DELUCIS, N., Concpion d n srcr d ss dynamis sr n machin-oi UGV. Tchnis d xracion d rigidiés à parir d foncions d ransfr, Rappor d Pro d Fin d Eds IFMA, Jin 999. [LAU 98] LAUROZ, R., Comp-rnd ds msrs ffcés sr Mor 5, Rappor inrn IFMA, Ji 998. [LAU 99] LAUROZ, R., RAY, P., GOGU, G., "Raion nr rigidié sai rigidié dynami : appicaions ax machins UTGV", 4 Congrès Français d Mécani, Toos 999. [LAL 86] LALANNE, M., BERTHIER, P., DER HAGOPIAN, J., Mécani d vibraions inéairs. Ediions Masson, 2 édiion, 986. [MCO 95] McCONNELL, K.G., Vibraion sing : Thory and Pracic, Wiy Inrscinc, r édiion, 995. [MUS 98] MUSSET, H., Ed modéisaion dynami d n banc machinoi éipé d mors inéairs, Rappor d Fin d Eds IFMA, Jin 998. [PED 97] DEL PEDRO, M., PAHUD, P., Mécani vibraoir Sysèms discrs inéairs, Prsss Poychnis Univrsiairs Romands, 3 édiion, 997 [PIR] PIRANDA, J., Vibraions ds srcrs 3 pari : Anays Moda Expérimna, Cors d DEA Acoso-Opo-Ecroni Mécani ds Srcrs, Opion Mécani ENSMM, Laboraoir d Mécani Appié R. CHALEAT [PIR 94] PIRANDA, J., CHATELET, E., "Lissag d foncions d ransfr avc procédé d condnsaion", Rv Français d Mécani n 994-, p. 9-27, 994.